INIIÌHN.VI IONAI II'I
(I
DIKMIIOIÌ,
I
I;
SKKIIÌS OI
Knowles,
M.
MONOCIIAIMIS
ON
PHYSICH
Hclimelling: Quantum cliromodynamics: Ini/li imi yi/
cj/icriiiniils and limil i/
III II DeWill: l'In• 1/fallili appnuich lo illuminiti fichi theory I li! I /.imi .Itisi ¡11: Quantum Jiclil III tori/ nini criticai phcnomc.na. l'ornili editimi III!, li M Munì: liroiriiiini molimi: fine I uni iotls, dynamics, inni application» III II Nisliimuri: Stati-itimi plti/sits of spili ylasscs inni informatimi pivcessinq: 1111 intrvduclioii llli
N. M. « u p n i n : / ' / n o i i / o / noiicqttìlibrìum
supcivovductivity
UHI 108. 107. I0(i I0V 101
A Aliaroni: Introducilo 11 lo lite I licori/ of ferromagnetism, Seco ti ti editimi li DoIjUs: / Idi 11111 Ih ree li WÌKliians: Calorimetri/ I Kiibler: 'ihcori/ of itincirint chetimi mni/nclism Y «inainolo. Y. Kitaoka: Dynamics of heavy clcctrmis I). ISaldili. (I. l'assarino: The standard model in the makiny
101
C ( ' . I il lineo. IJ. I.avoura. J . I'. Silva: CP
102. 101. 100. (IO. !I8. 1)7 •IO 'l'i II I 'Il 00 HO. H8. 87, 8
T. ('. C'hov: Effcctivc medium littori/ Il \raki: Mathematica! theory of quantum fichis I. M. l'ismen: Vortircs in noulinear Jichls I. Mifilrl Stellar muynclisin K. II. Hciincmaim: Nonlincar oplics in mctuls II Sal/iiiami: Alomic pliysics in Imi plasmas M Itrainhiila: Kinelic thnory of plasma waves M Wakataui: Stellamlor and licliotmn deviccs S ('hika/.iinii: l'Iiysics of ferromagnetism l( A Iteri Imanii: Anmnalics in t/uanluni fichi tlicory I' K (¡utili: Ioii Iraps I Siiiiiinck: Inhnmoyi 1 icous supcinmductors S. I.. Adler: Quatcmionic quantum mcclianics and i/unnlimi ficlds I'. S .loslii: ( 1 tolta! aspccls in gravitatimi and cosinolo!/}/ I-i li l'ike. S. Sarkar: 77» quantum theory of indiation
81
V. Z «resili, II. Morawitz, S. A. VVolf: Mcehunisms of conventminti
•il 80. 70 ,'M 7
and liiyli Tc
s u/ic cromi uct ivi 11/ ( ¡i-imo, .1. l'rost: 'l'In pliysics of liquid crystals Il II Unuutdeii, M. H. ('. McDowell: Charge exchangc and lln theory of ioti aloni cottiaion I leiir.en. A. li Markintosli: Kurt carili magnetism I( (iiihlinaiis I T \Vi 1 : 'l'Ile, ubiquitous photon l\ l.urliini. Il Molz: Ondulatori) and fnc-electron lascrs I' Weinliei>',ei : l\lectron scattering llieory Il Anlil, Il Kainillilll'a: The pliysics of tuli inclini/ clcclixius iti disunii reti systems I li I.UWMIII: l'In ¡iliysics of chanjcd ¡Hirliclc heams M Dui. S I' . Kdwnrds: l'In IInori/of polymer dynamics I' I. Wnll l'riiieiplcs of citi'troll tunneling spectmscopy Il « Menimi 1 Semiconductor contact* S l'liaiiilriiAcklllil 'l'hc mntlicmuticnl thcori/ of btack liolcs < ! Il Sul elite! lltnct nuclcfir ir.actions <' Molil i 'l'hc llicory of iilativity II I Sliililey liilioductimi lo plinsc. tinusitions and enfimi phcnomenn A Aluai'aiii l'rinciplcs of nuclcar magnetism I' A M Dirai-: l'cincijitcs of quantum inccliiinics II ' li l'elei I:, Quantum tìicory of solitls
Hit. I' (I dr
8.'
dolutimi
Quantum Chromodynamics High Energy Experiments and Theory
GUNTHER DISSERTORI Institute
for Partirle
Pliysics.
ET1I Zürich.
Switzerland
IAN G. KNOVVLES Foriiu iiii Università
of Edinburgh.
Scollami.
U.K.
MICHAEL SCHMELL1NG Mux-Planck-Institute
for Nuclear
Physics.
Heidelberg.
ò'/ Í
UNIVERSIDAD m,\dR«> BlßüOTHC* CIUMCIAS
CLARENDON PRESS • OXFORD
Germani/
Ills boo Has been printed digitally and produced in a standard specification in order to ensure ils continuing availability
F 1T
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78 0 1 8 0 72
A
Quantum Climmoduiiawics ( C ) was formulated as a nou-ahclian field theory in 1 )7. 5 and in the meantime lias evolved to the generally aeeepted theory describing strong interactions. T h i s book is intended to give a comprehensive overview over C studies in e e annihilation, lepton nucleoli scattering and hadron liadron scattering. T h e book conies at a time when the analysis of the L P data is being finalized and more t han ten years of most fruit ful research deserve to be summarized. n e of the goals of the book is to bridge the gap between theory and experiment, combining into a single volume theoretical description and tin1 experimental measurements. xperimental results are discussed with emphasis on their relevance with respect to strong interactions physics at high energies. Although not necessary for understanding the physics, it might occasionally be helpful to have some background knowledge with respect to the analysis of experimental data, such as the least-squares method or the interpretation of experimental errors and confidence level intervals. For an introduction to this subject., which is beyond t he scope of this book, we would like to refer the reader to any of the many excellent textbooks on the subject. ealing with particle physics we will express energy in units of e or mult iples of it. such as. for example. Me or c . In addition the so-called natural units will be used. that. is. a system of units where li = c = I. T h u s energy, momentum and mass are measured in units of energy, lengths in unit s of energy . nly iu cases where we want to be explicit about momentum or mass we will use e /r or e / r " . respectively. If desired, conversion to M SA-units is achieved by multiplying with appropriate powers of It and c. ere it is helpful to remember that, lie 0.1 7 e fin. c = .5 x 10 s m / s and r ss 1.(502 As. lectrical charges are measured in units of the positron charge and may carry an index specifying t he ident ity of a part icle, for example, c,, for the charge of a quark. An overview of the mathematical notation used in this book is given 011 page x. T h e presentation of the subject should be understandable for graduate students. and to some extent even undergraduates, but. also contains material useful for a research physicist. T h e basics of pert urbative C are presented in quite some detail. Anybody who has heard an introduction to field theory should be able to follow also t lie derivat ions and learn how t he building blocks of the C Lagraugian can be probed iu high energy physics experiments. T h e book focuses on basics. evertheless, some space in the presentation of the theory of C is also dedicated to advanced topics aiming to convey at least some impression of the general picture. For an in-dept h discussion of, for example, renormalization or I lie pat h int egral formalism the reader is referred to specialized textbooks.
vl
CHI I \< I
In order lo achieve a decpet understand ill}!,, some ol llir concepts discussed m tin- main text are illustrated hy means of problems. T h e solutions are given in I lie appendix. I )cpcndiug on I lie background of I he reader I lie apparent difficulty ol the problems may vary considerably. As a rough guideline we introduced a star-rating system for grading the complexity. A single star signals moderate difliculty. two stars are attached to what we consider really hard problems. Alter a short historical introduction, summarizing the steps which led to the formulation of QCI). the book focuses on the tests of QCD. First the t heory and phenomenology of pcrturbative Q C D for different kinds of reactions are explained at Horn level, t hen higher order correct ions are discussed. Considerable weight is also given to non-pert urbative effects such as the hadronization process. The currently available models are discussed in some detail both concerning i lie dilferent phcnomenological ideas and t heir implementation in Monte Carlo models. T h e experimental section starts with a short overview about accelerator and detector techniques and a detailed description of the general strategies of QCI) analyses. Then structure function measurements are discussed, which were vital in establishing (¿CD as a field theory and st ill are a t lu iving field of research. T h e focus then shifts to measurements done mainly in e + e ~ annihilation, which test I he detailed st ructure of stong interact ions as t hey can be probed by perturbat ive QCD. This covers measurements of the strong coupling constant as well as tests of thi- structure of Q C D . which finally led to the unambiguous proof for the existence of the gluon self-interaction. d o i n g from hard scattering to semi-soft, processes, the book describes how interference effects in higher order amplitudes alFeet. the properties of hadronic linal states. Finally studies of the hadronization process, the conversion from coloured parlous to colour neutral hadrons. are presented, as it can be probed by means of particle multiplicities or particle particle correlat ions. Clearly, within the limited space available it is only possible to cover a subset of all topics connected to QCD. T h e emphasis is put. 011 hard-scattering processes. Ot her subjects such as quark mass determinations, polarization phenomena. photon photon scattering, Q C D at thresholds, non-relativistic Q C D . nuclear collisions, lattice gauge theories, clnral perturbation theory, quark gluon plasma, or exclusive processes, if at all. are only briefly ment ioned. Again, for an in-depth discussion the reader is referred to external references specializing on the respective subject. Even after almost 30 years of research t here are st ill many open questions and plenty of opportunity for significant, cont ributions both iu t heory and experiment. Having worked through the book, we hope that the reader will have gained an overview of how Q C D developed in the twentieth century and where we stand with respect to a quantitative understanding after the turn of the millcnium. Many of the results collected for this book are likely to be superseded by improved measurements in the future, but the basic facts of (¿CD and the methods to extinct its defining parameters will remain valid.
r n t 1 ,\< 1
vil
lu collecting the material for this book we tried to do as complete as possible a survey of the relevant publications. Still, some important paper will have been missed, and we would like to apologize to all authors whose work is not yet fnllv appreciated in this book. Updates and amendments to the book, covering new developments as well as improvements to the exercises or error corrections can be found on the book's World Wide Web site. T h e link can be located via the catalogue of the Oxford University Press home page at: http://www.oup.co.uk/ November 2002
G.D.. I.G.K.. M.S.
Preface to the 2nd Edition For the second edition we would like to express sincere thanks to many readers, who gave feedback or pointed out errors in the first, version of the book. T h e mistakes have been corrected, and the? discussion of some aspects of (¿CD lias been re-worked. We hope that t his has led to an improved presentation of t he subject, which very recently has earned the highest recognition by the assignment, of the Nobel Prize in Physics 200
G.D.. I.G.K.. M.S.
ACK N( )WLEDG E.\i ENTS Our thanks arc due above all to Sonke Adlung of Oxford University Press for encouraging us to write this book, as well as to Anja Tschortner and Marsha Filion of Oxford University Press for their support during its making and completion. G.D. would like to thank the following people for their comments on the manuscript: C'ristiano Borean, Silvia Bravo. Barbara C'lerbaux. Maria Uorndl. Frederic Teubert. Andrea Valassi and Valeria Tano. 1.0. K. t hanks Sinead Farrington and the numerous other colleagues who have helped inform this work. M.S. would like to thank many colleagues and friends who directly or indirectly contributed to the results presented in this book, in particular Glen Cowan. Thomas Lohse, Ramon Miquel. Cristobal Padilla and Ron Settles, with very special thanks to Anke-Susanne Miiller for continued support and valuable input during the entire writing phase. For permissions to reproduce various figures and diagrams we are indebted to t lie ant hots cited in t he text and figure captions and to the following experimental collaborations and publishers: - AbEPII Collaboration, for Figures 5.7. (i.l and l()..r>. - American Institute» of Physics, publishers of AIP Conf. five. ~>.U Particles (mil Fit Ids: Seventh Mexican Workshop 1999. ©2000 American histitiili of Physics . for Figure 7.1. - American Physical Society, publishers of Pliys. lice. Lett. 78, 79, © 1997 hi/ the American Physical Society. Phys. Rev. Lett. SO. SI and Plry.s. Iter. D'nS'. © 190S by the American Physical Society, for Figures 7.13. 7.1 I. 7.1!)(a). 7.19(b). 7.20. 7.21. 7.25(b). 10.9(a) and 10.9(b). - Annual Reviews, publishers of Annual Review of Nuclear and Purtich ence. Volume /,!l ©I999 by Annual Reviews www.unnualiTviews.oty, Figures 6.9, (i.10. (i.ll and (i.l2.
Scifor
- Elsevier Science Ltd.. publishers of Nuel. Phys. B (Proe. Suppl.) (¡5 ©1998. Nut'.l. Phys. B/,70 ©1996 and Br>J,r, ©1999, Phys. Lett. B.','>(>. B4<)9 ©I999 and BJ,87 ©2000. Phys. Rept. 29/, ©1998 and Nuel. Instr. and Meth. A 860 ©1D95. for Figures 6.2. 6.8. 6.13, 7.9, 7.12(b), 8.6. 9.1. 9.3. 9.4, 10.1. 10.1. 10.7. 12.3, 13.1. 13.2 and 13.3. Ill Collaboration, for Figures 7.2 and 7.5. - Institute of Physics Publishing Ltd.. publishers of ./. Phys. G: Nuel. Phys. 22. for Figures 3.29. 3.33. 3.31. 3.35 and 3.36.
Part.
- Prof. .1. Friedman for the MIT-SLAC Collaboration, for Figures 2.2 and 2.3.
,w i\ INy )\\ 11 111 .r,,\u ,IN 1
Shaker Verlag, publishers of ISBN •'/-8¿(>!>-7/,• >'(>-2. lor Figures 5.1, 5.2, 5.3, 5.1. 5.5 and 5.6. Springer Verlag. publishers of Z. Phys. C57 ©199:}. C(>2 ©199/,. 67.7. C75 and C70 ©1997 and Eur. Phys. .1. Cl ©1998. C7. ClJ ©1999. CI2. Ct:l. ClJ, and Cl7 ©2000 and Cl9 ©2001. for Figures 6.3. 6.5. 7.3. 7.4. 7.6. 7.7, 7.10. 7.11. 7.12(a). 7.22. 7.23. 7.24. 7.25(a), 10.8, 11.6. 11.7. 12.1. 12.2. 12.4. 12.5. 12.6 and F.5. The Royal Swedish Academy of Sciences, publishers of Physica Scripta. Volume 51. for Figures 2.9. 9.2. I I.l. 11.2 and 11.3(left). World Scientific Publishing, publishers of the Procccdinys of tin 9th international Workshop DfS 2001. Boloyna. Italy, for Figure 7.8.
NOTATION Lorent.z four-vector indices {0.1.2.3} Euclidean three-vector indices {1,2,3}.«/' colour indices of the fundamental representation h. l>. <•....
colour indices in tli<' adjoint representation
n
Lorentz four-vector Euclidean three-vector
v (l
v
(i) , v)
l>
(E.P)
explicit components of a four-vector
or
four-momentum
'Iiip
'l"" = diafç(l- - 1 . —1. —1) metric tensor of special relativity associated c o v e d or to />'' /'/- = >hwp" scalar product, of two three-vectors V'l P\<1\ + P2I2 + P:\
"
unit matrix
A"
Gell-Mann matrices, a
T" = X" / 2 /"'" with
l
hc c
[T",T ']=if" T
{1
8}
generators of SU(3). n = {1
8}
structure constants of SU(3)
7'' with { 7 " . 7 " } = //""
Dirac matrices satisfying the Clifford algebra
/' = ')"p„ = 7 W>" i = j otherwise! f l( t.... With f( 0 )123... = + 1 ¿(a:) with /tb:/(3:)rf(x) = / ( 0 ) ,r < 0 6(./: otherwise
'slash' notation
fully antisymmetric Levi-Civita tensor
A/'
Hermit.ian conjugate matrix. M]- = M
H:
Kronccker-delta
Dirac delta-function 1 leaviside step-function
natural logarithm, hase 0 shorthand for <)/<).r''
"n 71;
.577 2ir> G 6 1 9 0 1 . . .
Euler Mascheroni constant 1 '
<(*)
Niemann zeta-function Yir
r(i')
Euler gamma-function
/
convolution /diy/;//(y)y(.i'/i/)
• .'/(-'•)
/"(»)
Mel lin transformat ion of /
Summat ion over repeated indices is implied by default .
CONTENTS
Notation
x
1
Introduction
1
2
The development of Q C D 2.1 Experimental evidence 2.1.1 T h e quark model 2.1.2 T h e quark parton model 2.1.3 Colour 2.1.4 Other puzzles 2.2 T h e Q C D Lagrangian
3
The theory of Q C D 3.1 Q C D as an SU(3) gauge t heory 3.2 T h e Q C D descript ion of basic reactions 3.2.1 Elect ron positron annihilation 3.2.2 Lepton liadron scattering 3.2.3 Hadron liadron scattering 3.3 Born level calculations of Q C D cross sections 3.3.1 e + e ~ annihilation to quarks at 0 ( n " ) 3.3.2 e + e ~ annihilat ion to quarks at 0 ( n ' ) 3.3.3 Gauge invariance of the Q C D Lagrangian 3.3.4 T h e evaluation of colour factors 3.1 Ultraviolet divergences and renonnalization 3.4.1 Self-energy and vertex corrections 3.4.2 Renonnalization 3.4.3 T h e renonnalization group equations 3.1.4 Calculating the KG E coellicient functions 3.1.5 T h e running coupling and quark masses 3.4.6 An explicit example 3.5 Infrared safety 3.5.1 Infrared cancellations 3.5.2 e + e ~ annihilation to hadrons at. NLO 3.5.3 Infrared safe observables 3.(5 T h e Q C D improved parton model 3.6.1 DIS at t he parton level 3.6.2 DIS at loading order 3.6.3 A heuristic treatment of factorization
<> 'i (i M 15 19 21 25 25 31 31 38 52 (12 ti'2 70 71 7!) 80 81 87 03 !)(> 08 103 I0(i 106 108 114 117 118 120 121
xll
CONTKNIN
3.6.4 3.(¡.5 3.(i.(i 3.6.7 3.(5.8 3.7 3.8
DIS at noxt-to-leading order T h e evolution of t he part,on density functions Leading logarithms T h e analysis of ladder diagrams T h e Droll Yau process
T h e treatment of soft gluons Hadronization models 3.8.1 Space time structure of multi-hadron events 3.8.2 Independent hadronization 3.8.3 String hadronization 3.8.1 Cluster hadronization 3.8.5 A comparison of the main hadronizat ion models
128 135 139 11 1 I I!) 153 157 158 161 162 168 171
4
M o n t e Carlo models •1.1 Fixed-order Monte Carlos •1.2 All-orders Monte Carlos 4.2.1 T h e partou evolution equations 4.2.2 Branching kinematics 4.2.3 Time-like Monte Carlo algorithm 4.2.4 Space-like Monte Carlo algorithm 4.2.5 Soft gluon logarithms 4.2.6 T h e colour dipolo model 4.2.7 T h e soft underlying event model 4.3 Multi-purpose event generators 4.3.1 Using event generators
17!) 17!) 181 182 184 185 187 190 196 200 202 202
5
Experimental set-up 5.1 Accelerators 5.1.1 Accelerator systems 5.1.2 Beam optics 5.2 Detectors at high energy colliders 5.2.1 Tracking detectors 5.2.2 Calorimeters 5.2.3 Passage of particles t hrough matter 5.2.4 Particle identification 5.2.5 ALEPH: an example of a LEP detector
206 206 208 211 213 215 216 220 221 223
(5
Q C D analyses 6.1 General concepts 6.1.1 Event selection 6.2 Observable« (5.3 Corrections (5.3.1 Detector corrections 6.3.2 Hadronization corrections 6.4 Systematic uncertainties
228 228 230 235 240 240 241 243
t
.
TI
T
Examples . .1 tructure function measurement at II . .2 . .
Inclusive et production at the T E V T R et rates at E
tructure functions and pnrton distributions .1 Charged lepton- nucl on scattering eutral current interact ions .1. .1.2 Charged current interactions .1. T h e Iow-.i and low-C 2 region .1.1 T h e glnon density in the proton .2 eutrino nucl on scattering .2.1 Experimental issues .2.2 easurements of and xF;t .2. T h e glnon distribution .2.1 T h e strange uar distribution . u m rules . .1 T h e dler sum rule . .2 T h e Gross lewellyn m i t h s u m rule . . T h e Gottfried sum rule . .4 T h e m o m e n t u m sum rule . . u m rules for polari ed struct ure functions .1 adron hadron scattering .4.1 T h e roll an process .4.2 T h e rapidity a s y m m e t r y .4. irect-photon production .4.4 Inclusive et production . Global C analyses The .1 .2 .
.4
.
.i .
strong coupling constant Theoretical predictions Comparison and combination of results Inclusive m e a s u r e m e n t s . .1 T h e ratios -, and Ri r . .2 easurement of o s from . . n s from s u m rules easurements of o s from heavy flavours .4.1 ecays of heavy uar onia .4.2 attice calculations caling violations . .1 caling violations in fragmentation functions . .2 caling violations in structure functions easurements at hadron colliders Global event s h a p e variables . .1 Theoretical predictions
il
2-1 211 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
4
1 2 2 4
1 12 1 14 1 1 1 2 2 21 21
I I
I IMS I
.7.2 vent shape variable s ,S. Analytical approaches to power law corrections . ) .lets in deep inelastic scattering .10 Summary of measurements
322 320 32 32
Tests of the structure of C .1 Part(in spins .1.1 T h e quark spin .1.2 T h e gllion spin .2 Flavour independence of strong interactions
331 33-1 331 33(i 340
10 T e s t s o f t h e g a u g e s t r u c t u r e o f C : colour factors 10.1 Three-jet. variables 10.2 Four-jet variables 10.3 Combination of three- and four-jet variables 10.4 Information from the running of o s 10.5 Information from jet fragmentation 10.(i Limits on new physics
344 340 351 35 35 362 363
11 L e a d i n g - l o g C 11.1 T h e structure of the parton shower 11.2 Momentum spectra 11.3 Particle multiplicities 11.4 Isolated hard photons 11.5 Sub jet multiplicities 11.6 Breit frame analyses
365 365 366 36 372 373 375
12
ifferences between quark and gluon jets 12.1 Theoretical expectations 12.2 xtracting quark and gluon jet properties 12.3 xperimental properties of quark and gluon jets 12.3.1 Topology dependence of jet properties 12.3.2 Multiplicities 12.3.3 .let profiles 12.3.4 Fragmentation functions 12.3.5 Particle content.
376 376 377 3 0 3 0 3 1 3 7 3 3
13 F r a g m e n t a t i o n 13.1 Identified particles 13.1.1 Multiplicities 13.1.2 Momentum spect ra 13.2 Inter-jet soft gluons and colour coherence 13.2.1 T h e string effect 13.2.2 Colour coherence in hadron hadron collisions 13.3 Two-particle correlat ions 13.3.1 Proton-ant iproton correlations
3 2 3 4 3 4 3 7 3 3 -100 402 402
C O N T K N
I S
13.3.2 Strangeness correlations 13.3.3 Bose instein correlations 13.1 Colour reconnnction
XV
101 -l li 110
14 S u m m a r y
113
Appendices
415
A
l e m e n t s of g r o u p theory A.l Basics of Lie groups A.2 The ( ) and S ( ) groups A.3 Colour factors
-ll-r 41 5 421 123
B
Building blocks of theoretical predictions B.I T h e Feynman rules of C B.2 Phase space and cross section formulae
425 425 127
C
imensional regularization C.l Integration in non-integer dimensions C.2 -dhnensional 7-matrix algebra C.3 -diinensional plia.se space C.4 seful mathematical formulae
420 420 433 433 435
-,, .I .2 .3 .4
F
i and T for a r b i t r a r y c o l o u r f a c t o r s T h e running coupling constant, and masses Theoretical predictions for T h e theoretical prediction for i T h e theoretical prediction for / r
430 430 43 43 440
S c a l i n g v i o l a t i o n s in f r a g m e n t a t i o n f u n c t i o n s .l efinitions .2 The flavour non-singlet case .3 The flavour-singlet case . l Fragmentation functions and liadron spectra .5 lect.roweak weight functions
442 442 443 14 1 445 440
Solutions
117
eferences Index
500 524
1 I T
CTI
arliclo physics is .he search for the fundamental constituents of matter and t in understanding of t heir interact ions. T h e concept of basic building bloc s for the entire physical universe dates bac to the Gree philosopher eino ritos. id .C.. who postulated that everything is built from indivisible entities, the so-called atoms , from the Gree word for indivisible or void, and that all conventional properties such as colour, taste, hardness etc. are a cons uence of interactions between different, arrangements of atoms. ith t he advancement of science different types of interact ions bet ween matter were identified, such as gravity or the electromagnetic forces, and continued st udy led to consistent theories providing a mathematical description of many phenomena. In addition it was reali ed that all nown substances could be synthesi ed from a finite number of chemical elements, which in themselves appeared immutable. T h e fact that chemical reactions always happened for fixed mass ratios was t he first experimental hint that the material world indeed was made of discrete entities, which again were called atoms. T h e final proof that, the chemical elements indeed are made of atoms then came around the beginning of the twentieth century together with the development of statistical mechanics and uantum physics. fter the discovery of radioact ivity it was. however, uic ly reali ed that the atoms which appeared immutable in chemical reactions are not the fundamental uantities of the Gree philosophers. Instead, they appeared to be complex ob ects where light electrons orbit around a heavy compact, nucleus, which was later found to be composed of protons and neutrons. closer loo at. t he atomic nucleus soon revealed that, in addition to the long nown and well established electromagnetic interactions new short range forces were also present a strong force which binds the nucl ons together, and a wea force which mediates the radioact ive beta-decay. Today we now that the strong force observed between nucl ons is only a van-der- aals type residual force of a more fundamental interaction between t heir constituents. T h e field t heory of st rong interactions is nown as i i i C . i e the other nown forces, it belongs to the class of so-called i lu a gauge theory, the fields are described bv representations of an abstract symmetry group, and the interaction between the fields, mediated by the is induced by the re uirement, t hat the agrangian is invariant, with respect, to arbitarv local transformations of t he fields. ow t his wor s technically is shown in the theory section of the boo .
Ii I l(
1 M II I ( IIS
T a b l e 1.1 Qiuilitutivi iiirlnrr nf tin fnuiltimriiliil .siitni/tIts v.mlmitcd fur Q I o Interaction strong
Approx. potential 12 r/23 n(
cm Q2
elect roinagnct ic weak gravity
-/ -)
Q2
Parameter values A « 0.2
e
.,
1/137
Mw « in i ni 2 Q
2
funis
0
e
the
ivhilivr
elative strength 1 1.4 x
e /c
0.7 x 1 0
with
P -
2.2 x 1 0 - "
3
1.2 x 10" 3 S
In t he language of gauge t heories electromagnetic interactions are characterized by an ( l ) gauge symmetry, weak interactions between left-handed fermions by an S ( 2) symmet ry and strong interactions by an S l ( 3 ) symmetry. All these symmetries are internal in the sense that they d o not act on space time coordinates. A gauge symmetry which is based on invariance with respect to arbitary local coordinate transformations finally leads to the decsription of gravity in the framework of general relativity. A quantitative comparison of the known interactions is not entirely trivial. Taking for instance two test part icles and asking for the relative strength of the different forces between them, t he answer depends both on the particle types and the distance of the two. T h e picture becomes most transparent when analyzing I In- situation in momentum space and looking at the Fourier transform of the potent ial which mediates t he interaction. T h e absolute value of t he t ransformed potential can then be taken as a measure for the strength of the interaction. In momentum space the distance is replaced bv the momentum transfer ( between the charges, which for this comparison is set to 1 e . T h e charges are unit charges of the respective interaction. For the strong, the electromagnetic and the weak force these are the electric charge, weak isospin and colour charge, respectively. In case of gravity, the relevant charge is the mass of a particle, which is a free parameter of the theory. In the comparison the proton mass is used. Table 1.1 gives a qualitative overview lor the known forces. At the scale of 1 e of the comparison one observes a marked hierarchy when going from strong interactions to gravity. ne also sees some characterist ics which neatly summarize the current picture of the Standard Model of particle physics. lectromagnetic and weak interactions are unified in the sense t hat at asymptotic energies both become similar, with a coupling strengt h evolving proport ional to o ( , l n . In the low energy limit the weak interactions are suppressed by the M term, which is due to the large mass of the gauge bosons of the interaction. In C there is no such low energy suppression. ne rather observes
IN I 11» ) I H "
I II >N
n growt h of t he interact ion st length wit h a divergence in t he ln(Q~'/A")-tenn. which implies t hat around and below a certain cut,-oil" energy A t his simple expression no longer describes t he physics of st rong interact ions. On t he other hand the coupling strength decreases with increasing Q 1 and eventually approaches the same value as that of the unified electroweak interaction. T h e exact point of this grand unification depends on the details of the theory. A hint that, it might exist can already be inferred from t he rough sketch given by Table 1.1. T h e fundamental fields known today are leptons and (¡narks, which are both spin-1 / 2 fermions. and spin-1 gauge boson fields such as the f/lnoil (g). the photon (",) and the \ \ : ± and the Z bosons, which mediate strong, electromagnetic and weak interactions, respectively. Isolated free quarks have never been observed experimentally. Bound states of three quarks form the so-called bart/ons. such as t he proton or the neutron, combinations of a quark and an antiquark yield a meson, such as. for example, the pion or tin» kaon. Mesons and baryons are collectively referred to as liadrons. heavy particles which are subject to strong interactions. In comparison to quarks, leptons. with the elect ron and its neut rino being the most prominent representatives, are rather light particles which do exist as free fields and are oblivious to strong interactions. (¿narks carry colour charge, electric charge and weak isospin and thus couple to gluoiis. photons and \ V ± and Z bosons. All leptons carry weak isospin and t hus are subject to the weak interaction, but only t he charged leptons have electric charge and thus also interact electromagnet ically. In the context of the Standard Model, all massive particles acquire their mass by coupling to the scalar Higgs field II. Finally, all energy couples to the spin-2 gravit.on field (G). which in a quant um theory of gravity is responsible for t he gravitational interaction. Table 1.2 .summarizes basic properties of the fields of t he Standard Model. T h e masses quoted in the table are taken from the 'Review of Particle Properties' ( P D G . 200(1). T h e considerable ranges given for the quark masses reflect t he difficult ies in dealing with masses of strongly interacting particles which are not observable as free fields. Tliey have to be understood as mass parameters of the theory rather than mass contributions to the bound states corresponding to observable liadrons. Table 1.2 shows that we currently have to deal with IS fundamental fields, and the natural question arises whether there might be a more fundamental level at which the picture becomes much simpler. Unification of the different interact ions including gravity addresses this quest ion, for example, in the context of the so-called sir hit/ theories. In such theories also the apparent, symmetry bet ween leptons and quarks and the repetition of weak-isospiu doublets iu both sectors may find a natural explanation. Promising alternative models to extend the minimal Standard Model exist, based on very good theoretical arguments. However, all experimental findings are perfectly described by t he Standard Model so far. T h e main focus of this book is Q C D in hard interactions. Structurally. (¿CD is a very straightforward theory, being a Yang Mills gauge theory based on an
I IN I l<< >1 M M
I II t[N
T a b l e 1.2 ¡'a ids of the iininiinitl Slandunl Moth l. The Jirst uroui> contains III< i/uarl.'s. tin second the Irjiluns and tile last one the bosoinc Jit Ids. All chaiycs arc i/inrii in anils of tin positron cliuiyc. Field
Mass/McV/r"
Spin
Charge/e
<1
3 - 9
1/2
-1/3
(1/2,-1/2)
3
u
1 - 5
1/2
-1-2/3
( 1 / 2 . + 1/2)
3
s
7.r) - 170
1/2
-1/3
(1/2.-1/2)
3
c
1150 - 1350
1/2
+2/3
( 1 / 2 . + 1/2)
3
b
• 1 0 0 0 - 1400
1/2
-1/3
(1/2.-1/2)
3
I.
174300
1/2
+2/3
(1/2.+1/2)
3
e~
0.511
1/2
-1
(1/2.-1/2)
0
< 0.000003
1/2
0
(1/2.+1/2)
0
105.06
1/2
-1
(1/2.-1/2)
0
< 0.19
1/2
(J
( 1 / 2 . + 1/2)
0
T~
1777
1/2
-1
(1/2.-1/2)
0
"T
< 18.2
1/2
0
(1/2, + 1 / 2 )
0
7 w-
0
1
(1
(0.0)
0
80419
1
±1
(l.+l)
0
Z
91188
1
0
(1.0)
0
O-»" <
0
1
0
(0.0)
8
H
> 114000
0
0
(1/2,-1/2)
0
G
0
2
0
(0,0)
0
I'~
(/•
Owi-ok
Colour states
mil>r
I NT II OJII NllKJ( 'I' NTH ' I II OIN N
basis, we briefly explain basic: ('OJ concepts experi",sis, Wi ' will wi ll bril'Oy l'xplli ll the tltl' l)/lsi \( ' ' p t~ of of high It i ~ll energy cllt' rg." physics physics ('x pl'ri ments. covering Hc('('lerntor accelerator n and detectorr technologies as well as dala data aanalysis, IIlo' IIl S. ('\1\'('J'ill); ile! (It·teei.. t.('d lllO lo;!;ies 1\::> ll a l~ ·s i s. and hen 1turn to ilt actual measurements off :;l.rll structure functions, the' st strong coupling /l lId tlliell ,111'11 Lo l.ll llJ 11I(,f1S lIl'('II IC II Ls o d m c fUll d iolls. cit rollg tO llpli ll:.( constant. he stru structure of Q (¿CD st udies of of l.hp the had hadroni/.ation process. (" lIls\ll IlI.. tests LeSt.s of tIIll' ct ure of '0 aand nd s\.lIdies l'OlIi zat.io ll pro(:css. Since it is pr'l('t practically impossible in O one book, we Sill(,(' ir'u lly ill l()(Issi hlp to review I'(' \'i('w all relevant n' I(,\o':1 l1 t results r('snlt,s ill il!' hn k. \\'(' focus studies performed high energy (:ollidl'l's. colliders. ro ' li S on ( II Q C'ID stlldies pf'\' for nwd at at. hi gh ('IIl'rg,v
2 T H E DEVELOPMENT OF QCD 2.1
Experimental evidence
I'lie liistorv <>f high energy physios in t.lie second half of I he twent i< * I h century was driven l>.v a sequence of increasingly more powerful particle accelerators, which allowed matter to !><• probed at ever smaller distances. In this chapter we will briefly recapitulate how t he experimental evidence accumulated bv experiments at those machines led to the theory of Quantum Cliroinodi/nainies (QCD). In the end. this chapter will bring us to the discussion of the (.¿CD Lagrangian and its physics implications. Schematically, tin» road from nuclear physics to Q C D can be separated into three phases. T h e lirst phase can be characterized as the era of hadron spectroscopy. which culminated in the formulation of the quark model. Then came a lirst series of deep inelastic scattering experiments which established the physical reality of quarks in the context of the quark parton model ( Q P M ) . Finally followed a set of improved measurements which probed the interactions between the quarks and allowed the first, quantitative tests of (¿CD. 2.1.1
The quark
model
T h e historical foundations of Q C D date back to t he early days of nuclear physics, when t he binding energy of the nucleus was realized to be due to a new kind of interaction between protons and neutrons. Scattering experiments soon showed t hat t he interaction is not only very strong, but also t hat it acts only over very short distances. It was Yukawa's insight t hat such a short range force could be understood by assuming that, the interaction is mediated by a heavy boson, a so-called meson. Scattering experiments also revealed a certain symmetry between protons and neutrons which was encapsulated in the isospin formalism introduced by Heisenberg and others. T h e meson theory of Yukawa did very well account for the phenomenology of those days by introducing the - - m e s o n s as force carriers for strong interactions. T h e picture was nicely confirmed when t hose pious were actually discovered as free part i d e s in cosmic ray studies and in accelerator experiments in t he liMOs. T h e mass was determined to be 11(1 M e V / c 2 and the lifetime around 2.6 x I t ) - 8 s. What came as a surprise was the discovery of many ot her new particles which all could be produced in interactions between nuclear matter. Interestingly, some of those particles were found to decay wit hin time spans as short as 10 -M s. while others had lifetimes many orders of magnitude larger. Since short lifetimes are related to a strong interaction mediating the decay, it followed that the short-lived particles decay via the strong force
I XI ' I I I I M K N T A I . K V I D K N C K
and i lie ot Iters through weak internet ions. T h e strange behaviour of those lunglived particles, which were produced in strong interactions and then decayed weakly, was explained formally by Pais and (¡ell-Mann who introduced a new quantum number, strangeness. which is conserved in strong interactions and can be violated by the weak force. The originally rather simple picture of the world of elementary particles had changed completely by t he mid 1960s, when so many different species were known that people were talking of a particle zoo. Fortunately, at that point, sufficient information had been gathered for some structure to emerge. While different schemes were being tried to quantify this, the most successful one was the ansatz by Gell-Mann and Ne'cman. who showed that the known hadrons could be classified in group-theoretical terms as multiple!.s of the special unitary Lie group SU(3). using isospin / and hyper charge V* = 13 + S. the sum of barvon number and st rangeness, as t he relevant quantum numbers. Thus, some kind of periodic table for elementary particles could be constructed. That this was more than just an abstract mathematical game became clear when vacant positions in the multiplets were filled by new particles. T h e most spectacular event was certainly the discovery of the 52 ", a barvon wit h strangeness S = —15. at Brookhaven in 1964. An introduction to the theory of Lie groups is given in Appendix A.I. Using group theory for the classification of the known hadrons was rather successful in explaining the observed regularities in the particle zoo. but there was the disturbing fact that all known particles were sitting in higher dimensional representations of SU(:i). while the fundamental one remained empty. In order to resolve this dilemma and against all experimental evidence Gell-Mann and Zweig finally made the step to postulate that a set of three particles corresponding lo the fundamental representation of SU(I5) should also exist. These new particles were called guarks by Gell-Mann. where, as he described in his book The Quark and I,lu: Jaguar (Gell-Mann. 1991). the sound was lirst and the spelling was adopted later from the line "Three quarks for Muster Mark" in .lames Joyce's book Finnegan's Wake.. Taken seriously, the quarks would constitute the basic building blocks of all hadronic matter. Unfortunately, according to -SU(.'5) they would have to carry electric charges il: 1 /-i and ±2/.'5 of the electron charge, somet liing that had never been observed. T h u s elementary particle physics at the t ime was faced with a situation, where the world of hadrons was beginning to be understood in terms of some hypot hetical part icles wit h absolutely no experimental ev idence in support of their existence!. T h e properties, that is. quantum numbers. of those quarks, commonly called 'u' (up), 'd' (down) and *s' (st range) and their ant ¡particles are given iu Table 2.1. Note that the ant .¡quarks of the various types or flavours. as the types are also referred to. have the signs of their additive quantum numbers reversed and. being s p i n - 1 / 2 particles, opposite parities. Baryons are constructed by combining t hree quarks; mesons are obtained as a combination of a quark and an antiquark. Wit h proper assumptions concerning the spin and orbital angular momentum, the actual hadrons and their excited
H
I III
I «1 VI- I U I ' M I ' N I ( ) |
1»
T a b l e 2 . 1 Quantum numbers of tin light quarks. IIn eleetrie charge r(, of tin quarks is given in units of tin positron charge. Quark ti <1 s ii (1 s
Spin 1/2 1/2 1/2 1/2 1/2 1/2
Parity +1 +1 +1 -1 -1 -1
e
'i +2/3 -1/3 -1/3 -2/3 + 1/3 + 1/3
I 1/2 1/2 0 1/2 1/2 0
+ 1/2 -1/2 0 -1/2 + 1/2 0
S 0 0 -1 (I 0 +1
B + 1/3 + 1/3 + 1/3 -1/3 -1/3 -1/3
stales can l>o constructed, like, for example, the particles in the s p i n - 3 / 2 harvon deeuplet. Pertaining to t he ground state, t he orbital angular moment tun vanishes, the parity is positive and the wavefiuictions are fully symmetric in both llavour and spin.
Baryon deeuplet A"
A"
A+
Quark content. A++
ddd
ddu dds
E-
>:+
dun dus
dss
mm mis
uss sss
sr Note that going horizontally from left to right the d-quarks arc substituted by u-quarks. one at a time, going down parallel to the left edge d-quarks are exchanged for s-quarks and parallel to t he right edge u-quarks are replaced by s-(|iiarks. In each direction one encounters an SU(2) sub-symmetry. horizontally the familiar isospin or 1-spin symmetry and in the other directions the so-called U-spin and V-spin symmet ries. Using the quark model it. is very easy to const ruct the known particles in a systematic way, which helped to keep the concept, alive until firm experimental evidence for the existence of quarks was found. It is also worth noting that not all possible representations of the llavour-SU(3) are realized in nature. Only t hose niultiplcts are allowed where the difference between the number of quarks and ant ¡quarks is a multiple of three, which ensures that all observed hadrons have integer electric charge. 2.1.2
The quark par ton
model
As higher energy accelerators became available the resolution at which matter could be probed also increased. When the momentum per particle passed
r.MT.KIMKNTAI. I'iVIDKNC'K
il
I ( i e V / f . according to do Mroglie's rclat ion
A . i .
(,,)
structures smaller than 1 fin. the size of a proton, could be resolved for t he first i ¡in«'. T h e point was reached where one actually could see the charge distribution inside the nucleus of hydrogen atoms and address the question whether there are pointlike constituents which serve as scattering centres. In other words, it became possible to do a Rutherford experiment for the nucleoli rather than the atom. T h e kinematics of such a deep inelastic lepton nucleoli scattering experiment is sketched in Fig. 2.1. A high energy electron with initial energy E and four-momentum /. via exchange of a virt ual space-like photon, is scattered oil a nucleoli with mass M and four-momentum />. which is at rest in the lab system. T h e scattering angle of the elect ron is 0. T h e final state is characterized by t he four-momenta /' of t he scat tered electron and />' of the hadronic system with an invariant mass IF.
FtC. 2 . 1 . Kinematics of deep inelastic electron nucleoli scattering
Starting from this comparatively simple diagram we will now show explicitly what kind of phenomenology one expects if the proton is a bound state of point like charged ob jects. As a first step it is convenient, to int roduce two now quantities, the energy transfer v from the electron to the hadronic system in the had roll's rest frame ,,=
E-E'=
JWi,
'J-f
(2.2)
and the squared momentum transfer Q~ carried bv the virtual photon.
V
- 2/» • />'
,r
,>'-
Ml
Mil. I M'.\ I I I MAI I MM I III' I ¿1 II
F.N l'I UIMUN'f \ l
Willi I lie quantities delincd in Fig. 2.1 o n e finds 2
Cf -- 2 A l h ( E + Mu - E') - A/,; - II" = 2A/,,i/ + M£ - W* < 2 M u « .
(2.4)
whore tlir equality for the hist term is obtained in the limiting case U'J. that is. for the case of clastic scattering. T h e deviation from clastic scattering thus can be described by the Bjorkeu-variable x n Q~ •>n = 7 r l T ~
with
»<*/*
2.5
I n proceed o n e needs to know the cross section for elastic scattering of an electron with a s p i n - 1 / 2 fertnion of mass M\, and charge See Ex. (2-1) for the elementary but rather lengthy calculation. O n e o b t a i n s da
I-";,,/',-;
E' f
dep = —QT-"~E V
2
Q2
ff
O S
2
+
2^
. S 1
"
2
o \ (
2/-
-(,)
l-;VII»KN( I-
11
does not happen with the nucleoli as a whole, but with exactly one of its constituents. Physically this picture makes sense when the energy of the projectile is sufficiently large to resolve the inner structure of the target. To describe this situation one has to partition the total four-momentum of the nucleoli between its constituents. Each constituent / thus carries the fraction .c, with a probability density f,(x,). the so-called par ton density function p.d.f.. meaning that the probability for .r, to fall into the infinitesimal range [r. .r-f-d.'] is given by /,(;e)d.r. From these a s s u m p t i o n s the structure functions 1F| and 11% can be calculated as superpositions of the elastic structure functions eqn (2.!)) with weights / , ( . r ) . To do this we n o t e that t he mass-shell constraint for the scattered constituent can be written (.t',/>)2 = (•'")/'+'/) 2 equivalent t o Q2 = 2x,M\,i/. We emphasise at this point that any constituent mass or transverse m o m e n t u m is negligible in a full calculation. T h i s suggest using the trick of replacing Mi, by x,M\, in eqn (2.!)) lo obtain the required structure functions, after s u m m i n g over all constituents and integrating out. the modilied ¿-functions. This yields
I'Viim this the double differential cross section with respect t o Q 2 and u can be derived. Start ing wit h the t rivial case of elastic scattering, where the mass-shell constraint on the scattered particle imposes the relation Q~ - 2M\,i'. one g e t s 2
d <7 w t u -
E' r - c r - z i ™
2
0
A
i
Q
2
à i f
.
2
0\
(
2 / n
Q
(2.7) m
ll'2(Q 2 .«/) = E /
T h e ¿-function guarantees that integration over /' yields e(|ii (2.6) with only one value of w contributing. In case of non-pointlike particles the double differential cross section has the s a m e structure as eqn (2.7) and. introducing two so-called and I I ' > { Q 2 . t ' ) . can he written as structure functions W\{Q2.i') ^ <1 Q*dv
|lE2(Q2.«/)cos2 Q
(2.1(1)
and
2
\ + 2W\ (Q2. v) sin 2 ^ | .
(2.8)
2.1.2.1 Elastic scattering From the derivation of eqn (2.8) the structure functions for elastic scattering of point.like particles wit h charge r<, are read olf immediately a s
1
' ( "
=
E
(
2
-
1
1
)
with .¡ u as defined in eqn (2.5). Il follows that in the part.on model the variable xit can be identified with the four-momentum fraction ./• carried by the struck parton. Note the subt le point that x n a s defined above is an experimental observable. while x is a parameter of the theoretical description of the nucleoli. T h a t the two are the same is a highly non-trivial finding. In the rest of this chapter and later on. we will usually use ./• and write x a only in cases where we want t o e m p h a s i z e an experimental measurement-. Finally, absorbing the total target mass Mi, and I lie energy transfer i> into a redefinition of the structure functions, one sees that d e e p inelastic scattering processes bet ween a charged unpolarizcd lepton and an unpolarized nucleoli can be described in terms of two functions F,(r) = M.IF, = \ £ « ? / . - ( * ) t
(2.!))
(2.12)
ami 2.1.2.2 The jHuion model Much more interesting is the case of t he part.on model, where inelastic electron nucleoli interactions are understood in terms of incoherent elastic scattering processes between the electron and pointlike constituents of the nucleoli. In other words, one a s s u m e s that a single interaction
E,(X) = IAV2^
•/,(.,•)
(2.13)
i which only depend on the sharing of the target nucleoli's four-momentum between its constituents. T h e structure function F\ measures the part.on density
I III: I »KVKI.OI'MKN I <>!• (J( I)
12
ns fiiiu t ion of x while F> describes the inoiiienl 11111 density, botli weighted with I he coupling strength to the photon probe. T h e fact that the observed cross sections depend only on a single diuiensionless variable x is also referred to as sealing behaviour. As shown above, it is a direct consequence of having point like diinensionsless scattering centres. Extended o b j e c t s would introduce a new energy scale into t in1 problem. 2.1.2..'5 Experimental findings T h e experimental observat ion of scaling was t lie first clear evidence for a partonic sub-structure in the nucleón, giving s u p p o r t to the concept of quarks as the building blocks of hadronic matter. S o m e early results (M1T-SI,AC C'ollab.. 1970) are shown in Fig. 2.2 and Fig. 2.3. T h a t the structure functions for deep inelastic electron nucleón scattering are mainly a function of j n and essentially independent of Q2 is illustrated by Fig. 2.2. An alternative way of showing scaling is to plot F> at. for example. Xn 0.25, as a function of Q~. T h i s is done in Fig. 2.3, where one sees that F> is indeed independent of Q2. 0.5 r
FIG. 2 . 2 . Scaling behaviour of //lF¿(u;) = F>(u>). uj = ranges. Figure from Ml I'-SLAC" Collab.(1970).
L/.r», for various Q2
,
(
i -IV,
2
4 2
Fit . 2 . . Value of v i r a IlC Collab. 1
2
GeV c
I
r
n
.2 . Figure from
.
nce it was possible to loo into the nucleoli, one could also try to determine the properties of those parlous. hile generic scaling behaviour is a universal feature of any parton model, t he details of course depend on the properties of the particles involved. The above derivation for electron nucleoli scattering, for example, explicitly assumed that the partons are s p i n - 1 2 particles. In this case a definite relation has to hold between F\ and F>. the so-called Callun Cross relation 2xFi(x)
=
F>(x)
.
2.14
T h e fact that the lepton nucleoli data are in very good agreement with this prediction shows that the struc partons indeed are s p i n - 1 2 fermions. dditional possibilities arise when different probes are used in deep inelastic scattering processes. sing, for example, neutrinos instead of electrons, the interaction is mediated by or bosons rather than photons. ow the coupling strengt h is given by the I bird component, of the wea isospin, which unli e the electric charge gives the s a m e coupling strength to all uar s. Comparing electron nucleoli and neutrino nucleoli cross sections thus allows to probe the electric charge of t he partons. Introducing uar densities for up and down uar s and their antiparticles in the proton. =
(*)
'I = dpi*)
'' =
V( ' )
d = dpix)
2.1
and assuming t he itar -charges as predicted bv the rpiar model, the st ruct ure funct ions for electron proton and electron neutron scattering are given by iT p .r
x l «
ri
i
rf
2.1
I I
K"(x)
II'.
I I I ' \
I ,1 A I L
\ I I M N
I
U L
1
I
I I
* { j | ( r / I ,!) + g ( « + « ) J .
(2.17)
Tin- transition from eqn (2.10) to eqn (2.17) is done by a simple isospin transformation: in the case of perfect isospin symmetry n-quarks in the proton are e(|iiivalcnt to d-quarks in the neutron and trice versa. It is also worth noting, that the ansatz for the structure functions takes antiquarks into account, the so-called sea-quarks, which are expected to contribute because of vacuum lluetuations. Contributions from heavier quarks are neglected at this stage, although they. too. are present in the nucleoli. Doing a scattering experiment with a targel material having equal numbers of protons and neutrons, like, for example, "'('a, the effective nucleoli structure function seen is the arithmetic average of the proton and the neutron contributions Ff
(.,-) = j-x 18
{ti + v + d + d} .
(2.18)
In the structure function describing neutrino nucleón scattering, where, for example. an incident union neutrino interacts via a charged W boson and is transformed into a because of charge conservation only negatively charged quarks cont ribute. One finds F.p\x)
=2x{a
fV'(x)
= 2x{
+ d)
(2.19)
«} .
(2.20)
where t he overall factor of 2 follows from the theory of weak interactions. Averaging the neutrino nucleón structure functions, one sees that there is a fixed ratio to the electron nucleoli structure function which is determined by the electric charges of the partons. Ff(-r) =
.
(2.21)
Confirmed by experimental measurements, this relation gave further support to the assumption that the partons found inside the nucleoli indeed are 11 it- quarks inferred from hadron spectroscopy. T h e physics potent ial of neutrino nucleoli scat tering experiments is. however, not yet exhausted. In high energy weak interactions only left-handed fermions and right-handed antifermions participate. Since the incident neutrinos are also left-handed, parity is maximally violated. From the angular distributions of the scattered leptons one can thus disentangle the contributions from quarks and autiquarks. which allows the extraction of an additional structure function /'t(.i'). £],(.»:) = ( « + < / ) - . ( « + J),
(2.22)
which is the difference of the quark and t lie ant ¡quark densit ¡es. Integrating over /•'i thus allows us to count the net number of quarks in the nucleoli. An early result for this Gross Llewellyn-Smith sum rule was
UNIVI USIIJAU AltTONOMA Dl MADRID BIIM.IOTl-.CA f Ml-NCl AS
KXPHlilMKNTAL KVIDIiNCIi
If,
, /
<\xF:t(x)
= 2.5 ± 0 . 5 ,
(2.23)
consistent with the expectation from the quark model. To summarize, up to this point the observation of scaling ill deep inelastic scat tering processes which allowed us to probe the interior structure of the nucleoli has shown that it contains pointlike constituents, the so-called partons. Further studies then revealed that those partons carry t he quantum numbers predicted bv the quark model, so that the two merged into the Q P M . 2.1.3
Colour
Despite its successes the Q P M still left, open questions, which indicated that it was not yet the complete story, lu the following pages we will go through the most important ones and show how the evidence accumulated that finally led to the field theory of Quantum Chroinodynamics. We will start with the findings that point towards a new internal quantum number, which later received the label colour. 2.1.3.1 The spin-statistics problem In the quark model the particles of the baryon decuplet have an s-wave spatial wavefunction and are fully symmetric in spin and flavour. This is most, easily seen for the particles occupying the corner positions in the = 1-3/2 state: |A + + ; + 3 / 2 ) = |u T) |u t) |u t) | A " ; + 3 / 2 ) = |d I) |«l t) |d I) \ii~ ; + 3 / 2 ) = |s I) |s I) | s | ) .
(2.24)
If this were the whole story, the complete wavefunction would be totally symmetric for identical fermions, which is a blatant violation of the Pauli-principle. A possible way out is to assume that the quarks carry an additional degree of freedom, colour, which can take on three distinct values. Then the Pauli-principle can lie restored by assuming that the wavefunction is completely antisymmetric in this new degree of freedom, which usually is labelled 'red', 'green' or blue". Denoting the colour state of a quark by an index to the flavour symbol, the wavefunction of. for example, t he A + + can be written as
|A
++
1 ; +3/2) = ^
:i
£ S i j k \ H i I)|u, T)|u* t) • ¡jA = l
(2.25)
where is the completely antisymmetric tensor with fi-vj = + 1 . A baryon thus is described by a totally antisymmetric superposition of all arrangements of the three basic colours bet ween the const it uent quarks. T h e name 'colour' is taken from the everyday experience that all ordinary colours can be composed from three basic colours. For ordinary colours, a superposition of equal amounts of the basic colours red. green and blue yields white, and somet hing similar also holds for the quark colours. If one assumes that t hose
i
l l
. I
I «
I
colours exhibit an I 1 synimet ry, I lien the colour part of I In bar von wavefu notion can be shown to transform as an singlet., that is, a baryon does not have any net. colour. It is white . This is a very important observation, since it implies that the new uantum number colour is effectively hidden inside the barvons and becomes visible only when it is probed at a momentum transfer which allows to resolve the individual parlous. ostulating that net colour is always confined inside hadrons immediately gives a heuristic explanation why free uar s are never observed. plausible dynamical explanation for colour conliiieineut appeared later in the context of C . t the current level of understanding. t he confinement hypothesis was. however, consistent in the sense t hat also the mesons could be understood as colour singlets. ade from a uar and an anti uar . its colour-wavcfunction is a superposition of colour antieolour slates of the form
t
l
I
l l 1
2-2
o far we have ust, post.ulat.ed that uar s come in t hree colours. elow we will now continue to discuss some of the evidence which supports this assumption. 2.1.. . 2 The Ailler Bell Jackiut anovialy Theories such as the tandard odel of electrowea interactions distinguish between left- and right-chirality lields. Technically I his means that the couplings depend on the irac-niatrix v - T h e treatment of in loop diagrams is very delicate and, as illustrated schematically below, can lead to unexpected results. Consider the reaction of 7 7 which in leading order proceeds via the triangle diagram given in Fig. 2.1. ote that in order to contribute to a physical process at least, one of the final state photons would have to be off mass-shell, since otherwise t he reaction would be forbidden by angular m o m e n t u m conservation. Here, however, we can ignore these details since we only want to show how such loop diagrams can give rise to problems. T h e coupling of the to the fcnnioii-loop is proportional to e I ft 7 . the stun of the vector and the axial vector coupling the coupling to the photons is proportional to the fermion charge . ccording to Furry s theorem the vector coupling of the does not contribute to the closed fermion loop, that is. only the axial contribution proportional to .-,. summed over all fennions. remains. T h e result is a term that would violate the gauge symmetry of the tandard odel since it is only associated with the -,-coupling. Thus, for the tandard odel to be consistent, the sum over all fennions must cancel. p-type and down-type fennions have i I 1 2 and n w - 1 2 . respectively. Ta ing only particles from t he first generation one obtains 2.2 where r is the number of colours for the uar s. T h e condit ion that the sum cancels immediately translates into V 1 . that is. the re uirement that the tan-
I
I I III
T
I,
I;VI
I
1
CI
F i e . 2 . 1 . Triangle diagram giving rise to the dlor oll that all charged fermions contribute in the loop.
ao iw anomaly.
ot.«
dard odel of electrowoa interactions is consistent, with fractionally charged uar s implies that there lias to be an additional internal degree of freedom for the uar s which can ta e on three different values. It is also interesting to note, that a cancellation of the d lor ell .lac iw anomaly generation by generation implies that the existence of the bottom and the top uar could he inferred already from the discovery of t he tan lepton. 2.1. .. The T decay rale triangle diagram of the type discussed above also describes the leading order decay amplitude of the neut ral pion Fig. 2.-r . Here, however, the diagram does not violate a gauge symmetry and gives rise to physical effects. ith the main contributions coming from u- and d- uar s in the loop, t he width of the TT is given by
I V
-
N-
(
-
,
2.2
where e and e,i are the electric charges of the it- and d- uar , respectively, expressed in units of the positron charge, and f- the pion decay constant. s pointed out by bbas 2 . these charges have to be nown from some external source in order to infer the number of colours 1 ,.. Ta ing them either from the stat ic uar model or measurements performed in deep inelastic scattering, one finds I V - .
-
F I ;
V
2.2
T h e experimental result. 77) . 4 .. r eV again gives strong evidence for the existence of a three-valued internal degree of freedom for the uar s.
IK
t'lllS I »KVKI.OI'MKN I OF QCI) 7
Fic;. 2.">. Diagram of tin*
( lecav
2.1.3.1 Elietroii ¡losihxm nnniliiliifion into liutlroiis Studying electron positron annihilation into hadronie final states, we are confronted with a situation where a very simple initial state of two light point like fennions is transformed into a complex multi-particle system of mostly pious, some kaons and a few harvous and antiharyons. T h e description of such a process by means of perturbation theory at first glance appears to be rather hopeless. T h e picture, however, becomes much simpler when ignoring the details of the multi-hadron final state. Assuming that '/. production is negligible, the dominant contribution to hadron product ion must, start with the creation of a quark ant ¡quark pair from a virt ual photon, which later 011 evolves into the complex system observed in the detector.
e~
//+
v~
antiqnark
Fie;. 2 . 0 . Leading order contributions to //-pair and multi-hadron production
Details of this process will be discussed at greater length later on. From Fig. 2.(i it is evident that the initial phase of multi-hadron production is very similar to t he creat ion of union pairs iu e + e ~ annihilation. T h e cross sect ions can be compared direct Iv. Taking the ratio, everything except t he coupling streugt lis cancels. T h e ratio /?-, thus directly measures the sum of the squares of the quark charges, that is. given the contributing flavours, it allows us to determine the number of quark colours. In the energy region above the Upsilon resonances, where five quark flavours contribute, we therefore expect
=
* ( e + e - _ hadrons) f
—>/:+//
)
=
^
|
=
'
1
t)
I!)
I \ I T 111 iMI'.N I A l . I A II i | ; N ( '!•'
I lie actual measurement yields A', ss 3.2. Again I here is experimental evidence li n three s t a t e s of I lie quarks, all hough it. seems that a value slightly above N r = 3 is preferred. Later this excess will liud a natural explanation in the context of higher order Q C D corrections. Here it is a first indication that colour may be more than just a quantum number which is needed to get some book keeping right. 2.1.1
Oilier
puzzles
Having dealt with the evidence for colour as a new quantum number carried by the quarks, we now turn to those findings which led to the realization that quarks are in fact strongly interacting fields. 2.1.1.1 The momentum sum rule As described earlier, deep inelast ic elect ron nucleoli scattering experiments allowed us t o measure the m o m e n t u m weighted probability density function. FV N (.r), of quarks and antiquarks in the nucleoli. Integration over ./• then yields the fraction of the nucleón m o m e n t u m carried by t he charged partons. T h e experimental finding was 18/' —
/
•'» ./a
Í' d x F < f { x )=
/
da: [«(./:)
h
-I- ñ(x)
-I- il(.r)}
«
0.5 .
(2.31)
./a
indicating that the charged partons which are probed by the scattering process carry only a b o u t one half of t he tot al momentum. Apparently, t here exist ot her components in the nucleón in addition to the quarks, which d o not carry electric charge and thus are invisible when using ;m electromagnetic probe. T h e y also are invisible in neutrino nucleoli scattering, that is. they d o not carry weak charges either. This means that those c o m p o n e n t s are either subject to an altogether new type of force, or. staying within the catalogue of known interactions, must lie specific to the strong interaction. 2.1.4.2 Scaling violations Scaling in deep inelastic scattering was derived from the assumption that inside the nucleoli there are non-interacting pointlike scattering centers. Although pheiiomenologically very successful, it is obvious that this simple picture can only hold approximately. Since the partons were found to be charged particles at least electromagnetic interactions between the constituents of t he nucleón have to be taken into account. With increasing Q2 the spatial and temporal resolution of the probe will also increase and become able to resolve vacuum fluctuations. This means that a quark which at lower Q2 is just seen as a point like particle will be resolved into more partons at higher m o m e n t u m transfers. A pictorial representation of t his scenario is given in Fig. 2.7. As a consequence, the total four-momentum of the nucleón is distributed over more constit uents, which implies a softening of the structure function. With increasing m o m e n t u m transfer the average fractional m o m e n t u m {:/:) per parton will decrease as sketched in Fig. 2.8. The amount of change in the structure functions will be proportional to the strength o of the
I I I I . I 71'. \ I I II \l I IN I III' 1(1 I ' low ( / '
lii K h Q*
P i c . 2 . 7 . Resolution of vacuum fluctuations at large Q~
interaction between the partons. that, is. we might expect a qualitative behaviour like
dQ2
to
|)ue to electromagnetic interactions we thus would expect scaling violations 1 d In F (2.:«) » *CIII KÌ7 d In Q~
F2 (X„)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
I
FlC. 2 . 8 . (Qualitative behaviour of the (/-'-evolution of st ructure functions
Scaling violations were indeed found experimentally. Taking for example measurements of Fj from deep inelastic electron or union nucleoli scattering, one
I III.
II F>(x . .Q' 2 which gives
111 G o V
m l i I, t ill
«
.1 and F2(x
d In F I In
. .Q-
1
GoV2
0.1(1.
-
.
.
2. 4
pparently, there is a strong force acting between the uar s, niticli stronger than electromagnetic interactions, which has to be explained theoretically. s an aside it may be worth noting, that, the almost perfect scaling observed in Fig. 2. results from the fact that x u .2 appears to be ust the fixed point, with respect, to t he softening of Fy. for values x u > .2 the probability density .2 it is growing. is shrin ing with increasing 2 . for values x u 2.2
The
C
agrangian
In a nutshell the evidence presented in the previous sections can be summari ed as follows hadrons are composed of fract ionally charged
uar s
uar s arc s p i n - 1 2 fermions t hey come iu three distinct colours there is evidence that colour exhibits an
symmetry
uar s are sub ect to a strong interact ion besides
uar s there are additional parlous in t he nucleoli
those parlous feel neither the electromagnetic nor t he wea
force
ote that the colour is distinct from and must not. be confused with the flavour discussed in ection 2.1.1. T h e pu les posed by these findings can be solved in a very elegant way by assuming that colour is a charge-li e uantum number, conceptually similar to the electric charge or the wea isospin. riginally proposed only as an index to got the boo eeping right, it is then understood as the source of a colour field. If the interaction mediated by this field is strong enough, then the large scaling violations observed iu the structure functions of the nucleoli find a natural explanation. T h e colour field apparently glues the uar s together to form the observed hadrons. motivating the name i/himis for the uanta of the colour field. If those glitons couple only to colour charge, then they are invisible in all deep inelastic scattering experiments using lepton probes and also the missing m o m e n t u m found from t he integral over F>(x) can be accounted for. Evidently there are good arguments iu favour of a field theory of strong interactions based on the colour charge of the uar s. T h e re uirement, that the theory bo renorinali ablo suggests a ang- ills gauge theory ang and ills. 1 4 . ssuming an unbro en g a u g e s y m m e t r y the general form of the agrangian is
CI
- IF;
F
1
i
- m
,, ,
2.
I |ii:\ l l,OI'MI NI
wln-re s u m s over repeated indices an- implied. T h e lield st.rength tensor F" the (ovarian! derivative D,, are given by the following expressions = O„ai
- D„AI -
{D,,),j
= ¿,jd„ + \aJTjK
(m,,),j
= niqSij .
I III'. (,»1 'l> I ,A< ¡H A N ( ¡ I A N
<)!• Q ( T >
and
!,rhrA';,K (2.30)
where A", are t he gluon fields.
A2 =
0
0
indexed by i. j = 1 . 2 . 3 . gluon colours by a.b.c.d.i 1 8. T h e three-gluon coupling between gluons of colour states a.b and <• is proportional to t he st ructure constant /"'"'. and the coupling between two quarks of colours / and j to a gluon of type
L
0
(2.37)
+ A(¡ =
0 0 0
0 0 i
i
i 0
a
1
(
i
0
H IV Ü0
0 0 i
(
j As = o )
a
!
1 0 0 \ 0 1 0 0 " 2 /
I[
¿al,
X.sf al ' C
flavours
The structure constants of the group. /"'" f„bc< defined through the commutation relations . (2.38) [T".T''} = if"hcTc are totally antisymmetric in their indices. For SU(3) the non-vanishing values arc Sva = 1 /i58 = /.Í7S = v / 3 / 2
b
«-0-Q-Q-ftfl.ftflf) -j—
QCD
A3 =
As =
In un t he field strength tensor consists of a free field term and t wo interaction terms where gluous couple to gluons. This coupling between gauge bosons is characteristic of a gauge theory based on a uon-ahclian group where the gauge bosons carry the charge of t he interaction, colour in case of QCD. and t hus arc able to couple directly to themselves. The fcrminnic part of the Lagrangiau is a sum over all quark flavours, again featuring a free lield term and a term for the quark gluon coupling. T h e triple-gluon and t he quark gluon coupling are proportional to the gauge coupling f/ s , the four-gluon coupling is proportional to i/;'. In addition the amplitudes associated with the individual couplings depend on t he detailed structure of the underlying symmetry group. Quark colours are
(2.39)
/117 = /tr.r, = f > it; = ¡a.io = f:m = f>r,7 = 1 / 2 . with permutations of the indices being understood. A more init.uitive respresentation of the Q C D Lagrangiau is given in Fig. 2.9. T h e gluonic part derived
d
9
Fl(J. 2.9. Pictorial respresentation of the Q C D Selunclling(1995tt).
t>TH a
•s
Lagrangiau.
Figure
from
T h e physics content of the (¿CD Lagrangiau is furt her discussed in the following chapter and in t he problems Ex. (2-2) and I';X. (2-3) given below. It. is shown explicitly, that there is a full symmetry in all colours with respect to physics, which is maybe not entirely obvious from the representation of the Gell-Mann matrices or the numerical values of the structure constants. One finds that the probability for gluon emission is the same for all quark colours, that the probability for gluon splitting into quark pairs is the same for all gluon states as is the probability of a gluon splitting into secondary gluons. Denoting the relative strengths of the splitting probabilities with C/.. C,\ and Ty for gluon radiation
'21
I III. I M A I I ( M'MI'.N I DI' (¿< II
oil a quark. gluon split l ing into two gluous and gluon splitting inlu two <|iiarks. respectively, Q C D predicts CV =
CA = 3
and
T,,=
1
-.
(2.40)
Sinec those numbers are proportional to the normalization of t he generators of the group, only ratios have a physical meaning. A convenient choice is
t - i
-
1-1«
l'Ile first of the ratios can he interpreted as the probability of gluoii emission olf a gluon relative to that of gli I on emission o f f a quark, that is. as the ratio of the colour charges of gluons and quarks. For an abclian gauge theory such as Q E D . where the gauge bosons do not carry any charge, this ratio would be zero. Q C D predicts it to be C'-t/CV = 2.2.r>, which means that the gluon has a colour charge more than twice as large as that of a quark. As will be discussed later, the two ratios introduced in eqn (2.41) are characteristic of the gauge group chosen in the bagrangian. Measuring them thus provides a way to probe experimentally the gauge structure of the fundamental interactions. Exercises for C h a p t e r 2 2 I Calculate the Born level prediction for t in; scattering cross section of an high energy electron and another s p i n - 1 / 2 fcrmion with charge c j and mass M at rest in the laboratory system. Neglect the mass of the electron. (Hint: Skip to Section 3.3 to learn about the technicalities of the evaluation of Feynman diagrams.) ' * * ' 2 2 Express the Cell-Mann matrices associated with the eight gluons as linear combinations of operators of the typo \C'){C\, which transform a quark of colour state |C) into |C'), and show that- the relative probabilities for a red. green and blue quark to emit a gluon are all the same. 2 3 From t he struct lire constants of SU(3) calculate explicit ly t he relative splitting probabilities for all gluon states into secondary gluons and secondary quark ant iquark pairs.
THE THEORY O F QCD 3.1
Q C D as a n S U ( 3 ) g a u g e t h e o r y
Quant um Cliromodyiiamics (QCD) is the gauge theory of coloured <|ttarks and gluous (Fritzsch ct til., 1973; Gross and Wilczek. 1973: Weinherg, 1973«), It is an example of a non-ahelian Yang Mills t heory. Its action is defined in terms of a Lagraugian density which for a single flavour of non-interacting quark is given by S = \Jd':»:£(*)
with
C(x.) = ijj(x)(i ¡¡> - vi)(h{x)
.
(3.1)
The index j on the Dirac fonr-spinors runs over the N r <|iiark colours. In practice we have N r - 3 but it is useful to leave it free and maintain generality. This expression is clearly invariant under a linear transformation, <jj > Uju'lk with l!il' 1. That is. when U is an element, of the fundamental representation of the unitary group U £ U{A'c). The unitary group is given by a direct product of groups U(A'V) = U( 1 )'X>SIJ(Ar,.). each of which can he treated separately. Here, we will concentrate on the SU(.'Vr) group. As shown in Appendix A. the group elements can be writ ten as a function of A"-' - 1 real parameters 0„
U(0) = exp(ifl„T")
(3.2)
where t he index ii runs over all generators T" of the group. When the parameter vector D is position independent we refer to U(0) as a global symmetry transformation. Similarly, if 0(x) is position dependent then we refer to U(x) U(f)(x)) as a local symmetry transformation. Due to the derivat ive term in eqn (3.1) it, is not invariant under local gauge transformations. To make the Lagraugian density invariant we begin by introducing .'V(: 1 real valued, gauge fields .4% and 1 replace ()'' by the eovariant. derivative D ', D" = 0" + \
with
A1' = AWT" .
(3.3)
The i»aramet.er f / s is called the gauge coupling. Local gauge invariance re(|uires tin* transformation property D"(A') equivalent to
D"(A')
=
U(x)ir(A)U(x)-1
= U(x)D"(A)q(x)
which is realized if the A1' field transforms as:
(3.4)
I III
A"
I III <MY < )!•' <¿t I»
1
U(x)A"Ulx)
+ — 10"U(x)]U(xy1. 9*
(3.5)
Interestingly. i lie presence of I lie second, hihouiogcncoiis term means that uon0. vanishing Range field configurations can he generated from the vacuum. A1' The above considerations lead to the locally gauge invariant Lagrangian density for the quark fields. £< |UI .rk =
»'U-w-M
=
- !,s4„Tjk]
(3.6)
Unfortunately, due to the lack of derivative terms, ()" A[\. in eqn (3.6) the gauge lields can be regarded only ¡us auxiliary fields associated with external sources. In order to make them dynamical we must find a new, gauge and Lorentz invariant term to add to eqn (3.6) which contains derivatives of A',\. In order to find a combination of fl"A'J. and possibly A", terms that has a simple behaviour under a gauge transformation we investigate the non-coimnutativity of successive covariaut derivatives: |D,,, D„\ = i <jaF,„, =>
F„„ = ()„A„ - d,,A„ + i g,[A„.
or, taking components
F°, = dflA", - i)„A"t - a»fabcA^A^
A„) .
(3.7)
This defines the gauge field strength (Lorentz) tensor. F,,„. The act ion of a gauge transformation on Ft,„ is easily derived by applying eqn (3.-1) to its definition as a commutator, see also Ex. (3-1). giving F„u ^
U(x)Fll„U(x)-1
.
(3.8)
Whilst F,,,, transforms non-trivially, as a tensor under S U ( N c ) . it is now easy to const ruct a suitable Lorentz and gauge invariant term to add to the Lagrangian density, =
{F„„F'"')
= ~ YF"-'F"""
•
The pre-factor. is purely conventional. The terms eqns (3.6) and (3.9) together define the classical Lagrangian density for (¿CD. Now eqn (3.9) is not the only invariant term which could be added to the Lagrangian density. Possible extra terms include r,(x) (D,tD")2,,(.,
) ,
[
(F ( I „F"")'
<•«'-
(-t-K')
If added to the Lagrangian density then all these examples would require coefficients carrying negative mass dimension, in order to ensure that the action remains a dimensionless number. However, the requirement of renonnalizabiliI V- discussed in Section 3.4. forbids all such terms. Thus gauge invariance and renormalr/ability prove to be highly restrictive with respect to the construction of a Lagrangian density. In part icular, gauge invariance also implies t hat gluons
(jCli AS AN Sll(:j) (! A11 (! I • I IIKOin
must bo massless, since n mass term for llie gauge fields, iir^A'^A"'1. would not lie gauge invariant due to the inhomogonoous term in ecjn (^t.r»). In addition to eqn (3.9) there is one other invariant term involving the gauge lields of mass dimension four or less which could he added to the standard l.agrangian density. In terms of the dual Held strength tensor. F*„ = - / „ „ " F a r
normalized such that
F = F .
(3,1 I)
I'/sUcKA"^
(3.12)
the so-called 0-terni is given by fjtn-
77" tr"W'
0 o»«T>2c""°T (^A';,<)aA'i + 1 (¡7T~ Us1'
The parameter 0 appearing above has nothing to do with the parameters ()„ in eqn (.'$.2). As the second form makes clear. Co can be expressed as the total divergence of a gauge dependent current. As such it contributes only a surface term to the action which naively may be neglected. Unfortunately, life is not so simple and the surface integral is related to a topological invariant, called the Pontryagin index. 1 T h e non-trivial topological structure of the vacuum in Q('I) is such that in practice the 0 - t o n n does give a non-perturbative contribution. This represents a serious problem since the 0 - t e n n violates both the discrete symmetries parity ( P ) and time reversal (T). which are known to be respected by QC'D to high accuracy, along with charge conjugation (C) invariance (Cheng. Ii)8S). Since T-violation is equivalent to CP-violation, one would expect a contribution to the CP-violating electric dipole moment of the neutron e.d.m. = 0 x l ( r ( l 5 - l 6 ) e - c m ,
(3.13)
where 0 is t he sum of 0 and the elect roweak. CP-violating phase in the quark mass matrix. Given the measured value, e.d.m. < l()""2rV-cni ( P D G . 2000). this requires 0 < 10 far smaller t han the CP-violation observed in weak interactions. This is the 'strong C P problem' for which several putative solutions are available in the literature, most prominent, of which is the axion. However, we adopt, a pragmatic approach and simply set 0 = 0: in any ease the 0 - t e n n does not give rise to any perturbative physics. T h e classical QC'D l.agrangian density £ , hiss £<,,,;,rk I described so far. is constructed to be invariant under local gauge transformations. However, this requirement, leads to difficulties in formulating the quant inn t heory. T h e crux of the problem is the large degeneracy between sets of gluon lield configurations which are all equivalent under gauge transformations. T h e treatment of t his 1
Loosely .speaking, I lie number of twists in the mapping of the .'i-spherc, at infinity, into i he St:(:{) gauge space.
w
mi
1111•:< >i<^ o r ' ( ¿ e n
problem requires I lie apparatus of gauge lixing and ghost lie-Ids. Here we provide a heuristic discussion of I lie solution using the Fevninan path integral method; more complete details can be found in any modern quantum lield theory text b o o k , such a s the one by Peskin and Schroeder (l!)!l.r>).
I'lG. 3.1. A schematic diagram of gauge field space showing t he 'fibres' of gauge equivalent field configurations. .-1". and the surface defined by t he gauge fixing condition. f(A°) = 0 In the Feyiunan path integral approach. quantities of interest are evaluated as averages over all configurations of the quark and gluon fields weighted by the exponential of the action for the fields. This is similar to the use of the partition function in statistical mechanics. It is the nai've functional integral over all the gluon fields, including the gauge equivalent copies, which causes a divergence. This divergence is completely unrelated to those discussed later in the context of renormali'/.atioii. The basic resolution, due to Faddeev and Popov (l!)(i7). is to split the functional integral into an integral over unique elements representing the sets of gauge equivalent lield configurations and a c o m m o n integral over the space of gauge transformations. The latter integral represents a constant (infinite) factor which can be safely dropped. T h e gauge degeneracy is broken by imposing a gauge fixing condition of the form f(A°) = 0. Here A" is the transform of the gauge field A under the action of U ( 0 ) . oqn (.'{.">). and f ( A ) is a function such that for a given A a solution exists for only one value of the gauge parameters 0. In a non-abelian theory this may be true only if we exclude topologically non-trivial gauge field configurations, which in any case give only very small contributions to the action and do not affect perturbation theory (Gribov. 1!)7<S). The situation is illustrated in Fig. .'5.1. By inserting the identity in a suitable form. c.f. I / d . < ; | d / / d j ' | i ( / ( . r ) ) , and not showing source terms, the fundamental partition function can be symbolically written as
Z = Jvij'Vii'VA
exp
p t f ddet et | d ' , x £ c | 0 „ ( t f v < M ) ) x y[vo
- J v ^ D c - D A V ^ V v exp ^
/
J d l x j£,.liiss -
^f(A)2
¿f(A")
S
Ml 60
(/{/!"))
<¿('11 A S A N K l l ( 3 ) C A H C K
I'll lit > m
2D
(3.1-1) Source terms are not. shown. In the second line the divergent. 0 integral lias lieen discarded as the remaining terms are act ually ^-independent. Formally, l his means that we have redefined the integration measure. T h e ¿-funct ion has been implemented in the action as the quadratic term. T h e parameter £ is arbitrary, contributing only to t he overall normalization, and as such it cannot, enter into any physical quantity, like S-matrix elements, though it may appear in intermediate expressions. As is made clear below, particular choices, such as £ = I. are often preferred due to the relative simplicity of the resulting gluon propagator. The determinant of the .lacobiau matrix is incorporated into the action as an integral over the octet of ghost fields. ;/". These are unphysical, complex valued. Lorentz scalars which obey Fermi Dirac statistics, that is. t hey are represented by Grassmann variables, and transform under the adjoint representation of the gauge group. Ghost fields only appear internally in loop diagrams, their physical role is discussed in a less abstract fashion ill Section 3.3.3.1. T h e result, of these manipulations is the addition of gauge fixing and ghost terms to the Lagrangian density. T h e gauge divergence in the path integral which is associated with t he gauge degeneracy of t he gluon lieltls manifests itself pert urbatively in the lack of a gluon propagator. T h e addit ion of a gauge fixing term allows this propagator to be defined. To see how t his works consider the popular choice of covariant gauge: <)i,A"'' 0. As indicated above, this requires two new terms to be added to the classical Lagrangian. ¿rtx+giiost = - ^ ( W n U X A " " ) + /)„,/"• (0"S" b + uJabcA*»)
(3.15)
Observe that the bracketed term in the ghost. Lagrangian is the appropriate generalization of the covariant derivative for the adjoint, representation: T"(A)i„. — i fnhc- It provides a kinetic term for the ghost fields and in this covariant gauge a ghost gluon coupling. Propagators are derived from the quadratic, free particle, terms in the action; for t he .4" field t hese are ^g.-uige - i | d ' . r I ( d „ A a v - d„Al){i)"A"v + i f d '.r/i;^./-)
,/•"<)- -
-\JdlpA»(p)
> n r -
-1J ( l -
- <)"A"1') + ^¡)llA"',i)„A""
•\,i,At(x)
+ C9(,l:!)|
(3.16)
In the second line, integration by parts is used whilst in the third a Fourier transform, 0,, —i}>,,. is used to go to momentum space. T h e gluon propagator. IK/;)"""'', is given by the inverse of the bracketed term.
NU
IÏIKOUY NR
q c p
» r y - ( i - 0 />"//' IK")
,7TT7
- , r +
(i
(3.17)
-
It is easy t o sec t hat t his inverse? would not exist in t he a b s e n c e of tin? g a u g e lixing term, that is. in the limit. £ —«> oc. S i n c e t h e n t he moment tun-vector />'' would lie an eigenvector of the inverse propagator with eigenvalue zero, this results in a matrix with at least o n e vanishing eigenvalue which cannot be inverted. T h e a term enforces causality. It can be traced to a d d i n g a term +ieA",A t t t ' to the action to ensure that the action integral is convergent. Another popular choice is the axial or physical g a u g e delined by 11 • .-1" 0 where n is a fixed Lorentz four-vector. S o m e t i m e s , the additional restriction I or i r = 0 is applied. T h e required g a u g e fixing term is ii 2 (3.18)
£ n , = - ¿ ( H • A-)(M - A " ) .
Since in this axial g a u g e the corresponding g h o s t term only c o n t a i n s t h e kin d ic piece and d o e s not. couple g h o s t s to any other lields. the g h o s t s m a y he t rivially integrated out and need not be considered further. T h e corresponding, m o m e n t u m space, gluon propagator is given by jtt'ub IK/')
i>2
Cllll
-V
+
n • ¡i
(" •
I>)
(3-If)
2
Now. in any g a u g e the gluon propagator can be d e c o m p o s e d into a weighted s u m of direct p r o d u c t s of polarization vectors ((/;)'' for the oil' mass-shell gluon: \ W "
= -T^-r- £ '' T.I..S
'f M
V
W
'
(:{"2,))
where, in general, the sum includes contributions from two transverse ('/'), o n e longitudinal ( L ) and o n e scalar c o m p o n e n t (S). Significantly, for an axial g a u g e in the on mass-shell limit, / r = t), only the physical, transverse polarizations propagate (C/. = C's <))• This proves t o be very useful in situations where physical a r g u m e n t s are to be used. T h e price t o be paid is t he relative c o m p l e x i t y of the propagator, in particular the presence of t he spurious singularities in (ii • /»)" 1 which require a careful treatment in terms of principal values (Leibbrandt. 1!)S7). In practice, it proves popular to use t he Feyninan g a u g e for higher order calculations and the axial g a u g e s to gain physical insight. Finally, we give the c o m p l e t e (JC'D Lagraugian density, in the covariant gauge: £ Q C I ) = >LJ(->:)[] V 1
\o„Al
"'M-'')
- d„Ai){cr
A"" -
ff'A"")
+
I III
11 I
-I-(< ) n
1
.
)(<>
+
ir I
II
nJ'-K'h (*)
- ~ ni,r n,i,
-i-
SUM»,,
Ah Ar A'lAl
il
l
.-ir .21
l liis lias b en separated into the uadratic parts and the remaining perturbations which are proportional to either or T h e first three terms give rise tu the uar , gluon and ghost propagators, the fourth and filth the uar gluon and host gluon vertices, whilst the sixth and seventh terms give rise to tripleand tiartic-gluon vertices. T h e corresponding Feynman rules are detailed in ppendix , together with those for the axial gauge. .2
The
C
description of basic reactions
In the following section, we a t t e m p t to give an overview of three basic aspects of hadron production in collider experiments. These are a summary of the actual properties of the events that are seen in lepton lepton. lepton hadron and hadron hadron collisions; an outline of these events formal description using C ; and an insight into t he physical pictures which guide people s t hin ing. In general, the use of C to describe a reaction means the use of perturbative C p C . This restriction is purely practical and merely reflects our present inability to calculate more than a few non-pert urbative properties within C . The applicability of perturbation theory relies on the strong coupling being small. very important, property of C is that the si e of the strong coupling varies wit h the si e of t he characteristic momentum transfer in a process. T h e coupling runs in such a way that it is small for large momentum transfers. Q , and large for small momentum transfers. To leading order one has s
,. -
T
.2
r
.22
ls an energy scale at which non-pert,urbative effects become imporHere tant. Experimentally, it is found to be 2 oV. that is. the mass scale of hadronic physics as given by t he pion mass or e uivalently the inverse of a typical hadron si e 11,,. T h e coefficient .in in e n .22 is defined in ection .l.-r . T h e appearance of the scale and the running of the coupling is a subtle aspect of renorinali able theories, such as C . which we shall discuss later. s a conse uence, the ma or part of this boo is dedicated, necessarily, to discussing hard processes that involve a large moment um t ransfer. This may arise naturally, as for example in the production of a heavy particle, or may be engineered. by. for example, only considering ets with large transverse energies. f course, we do need to discuss non-perturbat ive aspects of C . in particular liadroni at ion. Here, when detailed descriptions are needed, we must mainly rely on models rather than theoretically secure C predictions. Fortunately, the effects of liadroni at ion on p C predictions appear to be modest.
Restricting our attention to large momentum transfer processes, Qwith n - implies, by virtue of the uncertainty principle, that we see
I I I
( I
I
M
« ' I
lint lire i hi 11 si i mil. sul>-nuclear scale. At these scales had mi is appear I • t lie c< imposed of the (ant¡)<|iiarks and gluons which appear in the Q C D Lngrangiau. Furthermore. they are only weakly self-interacting t hanks to the running strong coupling. This allows t he individual, target hadrous to he characterized by part on density functions (p.d.f.s) describing the distribut ions of parlous as a function of the fraction of their parent hadron's momentum that they carry. In the parton model, cross sections for hard processes are calculated in terms of the tree-level scattering or annihilation of individual (anti)c|iiarks and gluons convoluted with the appropriate p.d.f.s. What gives this statement its power is the fact that the p.d.l.s are independent of the hard sit I »process. In essence what we have is that I he cross sect.ion can lie faetorized into a process dependent. short-distance, hard suhprocess. involving parlous, and a process independent., long-distance part , the p.d.f.s. describing t he hadrons involved. Now. many of the hard subprocesses of interest, are electroweak in nature so that Q C D really only enters via the higher order corrections. T w o important feat tires of this Q C D improved parton model are the dependence of the p.d.f.s on the hard scale of the interaction and the appearance of multipartoii final states. It is the QC'D improved parton model that provides the framework for most of what, follows. As we have just said, calculations in p Q C D are carried out in terms of the quark and gluoii degrees of freedom appearing in the Q C D Lagraugiau rather than the colourless hadrons observed in experiments. T h e confinement transition from the almost, free parlous to the bound state hadrons is still not well understood but must be addressed before making comparisons wit h experiment. Fortunately, given the necessary restriction to hard processes, it is believed that non-perl.urbative effects, which involve small momentum transfers. Q < AQC:». do not spoil parton level predictions. T h i s can be seen in two complementary ways. First , the disparity in momentum transfers argues that the periurbative features of an event can not be modified significantly by hadronization without introducing a new. pert urbative scale. Second, t he uncertainty principle can be used to relate the four-moment tun of a virtual particle. Q''. to the space time distance it travels. Q1' /Q2: see Ex. (-1-2). Thus, pert urbative physics takes place on short-distance scales, whilst non-pert urbative effects are long range in nature and can only have limited effect on the widely separated hard partous. T w o basic approaches are available to calculate hadrouic event properties within p Q C D . One approach is fixed-order perturbation theory, the other one is based oil a summation of leading logarithms. To describe a given type of event using fixed order perturbation t heory, its dominant, features arc identified, typically collimated sprays of hadrons known as jets, and these are associated with well separated primary parlous. In the absence of flavour tagging these may be either quarks or gluons. In this way the event is matched to a scattering amplitude containing the primary partous as external states. This amplitude is described by a sequence of ever more complex Feynuian diagrams which may be grouped into sets according to how
N I K q < ' l > D K K r m i ' T I O N 01'' I I A S I C U F A C T I O N S
3tt
many g a u g e couplings, g , ^/-ITTO,,, they contain. The simplest set of (tree) diagrams contribute to the cross section, which is proportional to the amplitude ,i|uared. at 0 ( o " ) where the power n is characteristic of the process. In gluon gluon scattering, for example, o n e has n = 2. whilst for three-jet production in 1. T h i s is the lauding oiiler (LO) approximation. T h e e 1 e annihilation it is n next simplest set of (one-loop) diagrams cont ribute at. O ( o " + I ) : this is the nextto-leading order ( N L O ) approximation, etc. Given a sufficiently small coupling, this perturbation series should converge t o the correct answer as more terms are added. In practice, the series is expected to be only asymptotically convergent, so that beyond a certain order the numerical evaluation of the series begins to diverge from the t rue answer. A complication arises in this approach because tree-level diagrams diverge whenever external partons become soft, or colli near and related divergences arise in virtual (loop) diagrams. T h i s is in addition to the ultraviolet divergences treated by renorinalizat.ion. Fortunately, in sufficiently inclusive measurements, such as the" total hadronic cross section, it. is guaranteed that, the two sets of divergences cancel. Unfort unately, in more exclusive quantities, which involve restricted regions of the external partons' available phase space, the cancellation is less c o m p l e t e and large logarithmic terms remain, generically of the form Since o S ( Q ' ) L is of order unity for Q2 » Q2. see eqn (3.22), L = \n{Q2/Ql). this can spoil the convergence of Unite order perturbation theory. In the second approach, the original perturbation series is rearranged in terms of powers of o s L . .la = £
rt„(asL)" n
+ as(Q2)
£ h„(a„¿)" n
+ •••
(3.23)
T h e first, infinite set of terms represent, the lending logarithm approximation (LI.A), then c o m e s the genuinely «„-suppressed next.-l.o-LLA ( N L L A ) and s o o n . Since t he enhanced regions of phase space involve near collinear or soft gluon emission, it is favourable for the primary partons t o dress themselves with a shower of near collinear or soft partons. T h e s e are the parton precursors of hadronic jets. An important feature of such multipartou matrix elements is that in the enhanced regions of phase space they factorize into products of relatively simple expressions allowing significant, simplifications in the treatment, of leading logarithms. In s o m e cases, it is actually possible to sum analytically the LLAand NLLA-series t o all orders in o s . T h e emerging picture of an event follows a sequence of decreasing scales. A genuinely hard subproeess produces a number of primary partons which then undergo semi-hard gluon radiation resulting in showers of soft partons which ult imately hadronize. T h e main features of an event are determined during its perturbative stages, thereby allowing tests of ( p ) Q C D . In the following subsections we describe the basic phenomenology of the three main types of particle collision and how QC'D applies to l lietn.
:n
nil
3.2.1
Electron
position
IIIKOHY <>l (VX'I)
itnn Unlnt ion
Electron p o s i t i o n annihilation to hadrons provides the simplest colliding beam processes that can be described using p Q C D . T h e simplicity follows from both I lie well-defined energies of t lie initial s t a t e part icles and the fact t hat t he Icptons interact via a weakly coupled, colour singlet, virtual photon. T h i s allows a clean separation of the initial and final s t a t e particles. T h e combined m o m e n t u m of t he incoming Icptons provides a large scale justifying the use of pQC'D. In t he parton model t he basic interaction is an electroweak process. e + e ~ — 7 * / Z —• (|(j: this has essentially the same cross section as the well established process e'e —> / / + / / ~ . It. is usually adequate t o consider single photon exchange due l o the small value of the electromagnetic coupling o ( , m = e~/(47r) s; 1/137. T h e structure of the hadromc final s t a t e d e p e n d s only on the centre-of-niomentinn (C.o.M.) energy, ^/s. of the collision and if polarized the polarizations of the incoming Icptons. T h e C.o.M. s y s t e m is often also referred to as "ccntro-of-mass" s y s t e m , since in the s y s t e m where the momenta balance, the centre-of-mass of the interacting particles is at rest. Dealing with relativist,ie particles, however, the name V-ontre-of-niomentuni' is more to the point. At low C.o.M. energies. 0 < y/s < 5 G e V . t h e most interesting quantity is the total hadromc cross section. This shows a lot of structure characterized by "steps' at quark thresholds together with strong resonances, associated with qq bound s t a t e s that possess the s a m e q u a n t u m numbers as the exchanged photon. vector meson: /i. uj. In essence the off mass-shell photon behaves as a ,I I>( ' = 1 (,'). .1 /(,'. T(L.b'). etc. T h e hadronic final s t a t e is characterised by low mult jplieitics an
/* = 3()GeV. by the emergence of three-jet features in a fraction. (9(1(1%). of t h e events. By identifying these jets with primary partons it is possible to test the nature of (¿CD's basic constituents and their couplings. For e x a m p l e , three-jet events are believed to be a manifestation of vector gluon emission in t he process e ' e~ —» qqg. An e x a m p l e of a three-jet event is shown in Fig. C.l. T h e rate of this three-jet, production gives a measure of t h e strong coupling. o s . whilst the angular distribution of the j e t s reflects the spin-1 nat ure of the gluon. At even higher energies, small fractions of well separated four, live and more jet events appear, allowing tests of the triple and quartic gluon couplings. Note that these j e t s are required to be well separated to avoid the collincar and soft enhancements that would invalidate lixed order perturbation theory, thereby complicating any comparisons to theory. A more precise definition of a jet is given in Section (i.2. On dimensional grounds the total cross section must take the form
(3.2-1)
nii; ci
i i--sc mr i I
it .sir H
CTI
tr.
where i lie represent the relevant masses, sneli as uar or hadron masses. The function f(x,) tends to a non- ero constant as x, . ince the uar and luulron masses are mostly small, their effect becomes negligible as .s increases as prescribed by t he photon propagator. This and I he cross section falls as .sremains true until around y s -l GeV when deviations begin to be seen this is I lie tail of the resonance which becomes dominant al y s 1 GeV. part from i he large enhancement in t he total cross section, the main effect, of exchange is to modify the flavour mix of produced uar s and to introduce asymmetries into the polar angle distributions of the primary uar s, compared to pure photon exchange. bove v s 1 GeV, photon and exchange remain of comparable importance, but the total Itadronic cross sect ion continues to fall and becomes of less relative importance as other product ion channels, such as e e V , open up. n example of a less inclusive uantity in e e annihilation is the cross seel ion for the production of a specific type of hadron in the final state. uppose this hadron. h. has momentum p'1. then the differential cross section can be written in the form of a convolution
da
- -h-v(/>.*)
=
( ,*)
Dll(z)
.
2
T h e first term. d r, is the hard cross section for the product ion of a parton a such that it carries momentum p1'/z. T h e second term is a fragmentation function. D\](z)ilz, which gives the probability that the parton ii produces the hadron h carrying a fraction c of the primary part.on s momentum. This fragmentation function is the final s t a t e analogue of the previously mentioned p.d.f.s, to be discussed more fully in ection .2.2. T h e product, of t hese two terms is summed over all the possible contributing partons and integrated over the momentum fractions. T h e factori ation is between a pcrturbatively calculable, short-distance cross section and a non-perturbative fragmentation function. It is important to reali e that a does not depend on the identity of the hadron h, which would be a long-distance effect, but only on the parton n and the colliding beams. Conversely. D\) does not depend on the short-distance, hard subprocesses; in this sense it is universal and can be applied to any subprocess that, produces the outgoing parton i. t the lowest order the relevant hard subprocess is e e , so that in e n .2 the sum is over uar s with 2m,t < \/s. This gives the parton model prediction for which, as indicated, the fragmentation function depends only on t lie momentum fraet ion T h e inclusion of C corrections complicates matters, though the basic faetori ed form remains the same. In particular, renormali ation re uires the introduction of an arbitrary renormali ation scale. ;. whilst, the factori ation procedure introduces a second, arbitrary factori ation scale, .-. This acts as a cut-off on the virl.uality of intermediate particles, e uivalent to a cut-off on the inverse distance it travels. T h e exact origin of the scales , and
.Ili
I III'.
iir,i u n i ir * ¿V II
///. will licciiiiic clear in Sod inns 3. I and H.li. T h e Q C D improved parton model predict,ion is drr^
C1C,<'
=
{^.s:,ll{.,lF)n):(z:,,,<.,„•).
(3.26)
where the parton sum now includes contributions from gluons. Here the coefficient function C is derived from the partonic cross section for the subprocess. e+e —» aX. Thanks to its short-distance nature C is calculable using p Q C D and is devoid of any divergences. On the other hand. L)\\ is not calculable with today's technology and therefore must be determined from experiment. That said, the dependence on the scale ///.-. which is introduced in order to separate short- and long-distance effects, is calculable. Recall that both / / a n d ///•• an* arbitrary as are aspects of the rcnormalizatiou and factorization schemes. However, the physical cross section, on the left-hand side of eqn (3.26). is independent of the particular scales and schemes used, provided that the same choices are used consistent Iv in both C and D)]. Whilst not. necessary, it is common practice to only consider the case t hat fin = /(/••(= V*)A simplification in the above description of electron positron annihilation is the assumption that the colliding leptons are mono-energetic. This is not true. In a process known as bremsstrahlung they decelerate into the collision by emitting photons which reduces the effective C.o.M. energy. T h i s initial .state photon radiation (ISR) may be treated using structure functions (perhaps more properly called electron density functions) (Kleiss rt id., 1!)89). T h e idea is that t h<' incident, electron is really surrounded by a cloud of photons and flirt her e + e pairs. What the structure function. fL./,.(.r, //.-). gives is the probability density for finding an electron in this cloud of particles carrying a fraction r of the parent electron's moment um when it is probed at a scale //. T h e electron positron collision is then between these constituents. Summing up all the contributions gives
/ fc/c(xi..s)fu/v(x2.s)da{s J I) Jo
= xix2.s).
(3.27)
On the assumption of massless electrons, the C.o.M. energy squared in the hard subprocess is given by s- = (-'"i />«. + .''j/',. • ) 2 = ci ./'_>.s. It is possible to calculate this structure function in Q E D perturbation theory. An approximate form is fc/c(x,
tr
) = ¡i{ 1 - x f -
1
with
/Hi'2)
=
(3.28)
In practice, one has 0 < 3 «C 1 so t hat f,./c(x. //") ac<|uires an integrable singularity for r —» 1. which favours soft photon emission. Whilst, t he singularity can be treated analytically, its treatment in a numerical implementation takes some care. (Computers don't handle singularities very well . . . )
I I I K (}( 'I ) | ) l S( Il I I ' I I O N I »1 M A S K • H I A t • I I O N S
'I'lir olfect of ¡nil,inl s t a l e photon ('mission on a total cross section is strongly influenced hv t he C.o.M. energy dependence ol' the hard snbproeess cross section. There is a trade-oil between the two terms in eqn (.'{.'27). Il' s is tuned t.o lie on a resonance then any ISR will reduce the effective C.o.M. energy, .s = xix-js, and (IÎT(.S') will be significantly reduced compared to d(7(.s). In this case, the effect, of ISR is modest, only distorting the resonance's line shape. However, if s is t uned to lie above a resonance then any ISR which reduces s to .s « mj { is favoured by the increase in cross section. For an illustration see Fig. 0.3. Such 'radiative returns' can have a major impact on the line shape and lead to individual events being thrown to the left fir right according t.o the energy imbalances in the post bremsstrahlung 1 optons actually entering the hard snbproeess.
FK;. 3 . 2 . Left, direct 7 7 interaction with Q F D photon quark couplings. Centre, singly resolved 7 7 interaction with the lower photon behaving as a collection of parlous characterized by its own p.d.f. Right, doubly resolved 7 7 interaction: in the upper photon the parlous can be traced to a point like component, originating from a perturbative 7 ' —* qq vertex, whilst in the lower photon the 'remaining' hadron-like partons have a non-perturbative origin. Not all the photons emitted as bremsstrahlung escape without interacting. A rich variety of processes result ing in resonance pairs, jet events etc. can occur due to photon photon interactions (Aurenche et ni. 1990). Indeed, the total cross section for e + e ~ —> e + e ~ 4- hadrons grows as In" (.s/nr.). A source of this complexity is that photons, as sketched in Fig. 3.2. are not as simple as might be na'ively thought. T h e reader is warned that the notations used to describe photons have become rather confused in the literature. Here we follow Chyla (2001). We are familiar wit h t he idea of a direct photon which has pointlike couplings and can lead to hadron production via the hard snbproeess 7*7* — qq. In addit ion, there are resolved photons that behave as dense clouds of (anti)quarks and glnons. Such a photon behaves like a hadron and is characterized by parton density functions. A point like component of these partons can be traced back to ail initial Q E D vertex 7 ' —> qq and the subsequent radiation of glnons which in turn may split into gluon-pairs or further qq-pairs (Witten. 1077).
1 1 1 1 .
I L L
I'.V
' I I )
I I I '
I ¿ I
I I
A hadron-likc < <>u11x>iifiit accounts for I lie remaining parlous which have their origins in non-perturhative physics, perhaps associated with tin- (negative virtua l l y ) photon lluctuating into a vector meson. T h i s opens up the possibility of effective lepton liadron. known as singly resolved, and hadron liadron. known as doubly resolved, collisions. These an- discussed in detail below. All these types of 77-evonts are characterized by low C'.o.M. energies and sizeable longitudinal momentum imbalances. Often these events constitute a hadronic background to the events of real interest in an experiment. We also ment ion one further complication. At very high energies it is necessary to use beams with very small transverse sizes in order to increase t he luminosity and compensate for the falling cross section. This gives very high charge density particle bunches whose intense electromagnetic fields can induce radiation in one another as they approach each other, the so-called beamstrahlung. The details of I his depend on t he specifics of t he beam profile, but can lie treated in a similar vein to breinsstrahlung (Palmer. 1!)!)0). if.2.2
Lepton
hadron
scattering
l.epton hadron scattering is a traditional method of probing the structure of hadrons. Since hadrons are now known to be composite particles with partonic. (nnti)quark and gluon. constituents, such collisions are more complex to describe than lepton lepton collisions. In essence, we view the observed scattering, / h —• I'X. as a manifestation of the hard subprocess fq —> Pq'. The advantage of lepton probes is that they undergo experimentally and theoretically clean, pointlike interact ions which are describable in terms of the exchange of a single, virtual, gauge boson. Multiple boson exchange, whilst possible, is suppressed by additional factors of the electroweak couplings, o 2 m or C y . A basic classification of t he event s is based on t he nature of the boson exchanged by t he initial lepton and quark. In neutral current events, characterized by / = ('. a photon or Z is exchanged. Whilst in charged current events, characterized by C. = e. //. r and / ' = v,.. v„, u r or vice-versa, a W ± is exchanged. For charged leptons the exchanged particle is predominantly a photon. Weak boson. Z or W ± , exchange is observed, however, at low Q~ <§: A/{y their contributions are many orders of magnitude lower. 0(Gy/a%,„). If a neutrino beam is used, only weak interactions can occur, making theiii a useful probe t o disentangle the contributions from quarks and antiqiiarks. Figure 2.1 illustrates the basic process underlying lepton hadron scattering as viewed from t he target liadron's rest frame. This frame coincides wit h the laboratory frame for fixed target experiments. To date, only the I IKK A machine at DKSY provides (asymmetric) colliding beams of electrons/positrons and protons. Once t he square of the C.o.M. energy. •>' = (/ + p)2. is fixed, the most important quantity for describing the scattering is the momentum transfer i/1' = P' P1' and the Lorentz invariant Q~ = —<¡~ > 0. The significance of Q - is that it characterizes the wavelengt h, or resolving power, of the probe. Also of interest is v. t he energy transferred to the target hadron. For a charged lepton neutral current
I N K q < 'I) I >KS< It IT I I O N (>1
MASK ' UKA<' I I O N S
event these can lie (leterinineil experimentally by measuring I lie energy and angular dellection 0 of the scattered lepton. Negleeting the relatively small lepton masses one finds Q* = - ( £ - ( ' ) 2
,, =
= + 1 E,Ef. sin~(0/'-)
(t-n-l>/Mu
= (E( ~ Ev)I,,,.
.
In addition, experiments may also measure details ol' the hadronie linal state. A". This can provide complementary, or indeed when C = // the only determination of (/* and v. The invariant mass. I f . of the hadronie final state is given by H/2 = (p + 'lf = A/,? + 2M\,u - Q2 . If
2
IK = A/,
2
=j.
(3.30)
i f = 2M\xu .
In the special case of elastic scattering, when the target hadron remains intact in the linal state, eqn (3.30) implies that Q2 and i' are not independent. The formal description of lepton hadron scattering is facilitated greatly by t he •factorized' nature of t he interaction. Details of t lie calculations can be found in tlie following sections. In terms of the Icplonic and hadronie currents the matrix element is given bv M(th
-
t'X)
= (C'\J„\P.)Otv
' 2gi,WX|./„lh) 2
(3.31)
where the elect roweak couplings have been factored out: (¡jx '•(•('Jy + "j\•)• see Appendix B. lu (high-Q") neutral current events t he matrix element should include both and 7. exchange contributions. Equation (3.31) suggests writing the inclusive lepton hadron scattering cross section in terms of two tensors /.,,,. and II'"' as 1
i 'ii
v
=
(i/n .'/liV)z
.
\
and
^\4\fh>((v,An
//"" = —V J • '177
•,,
(h\S"\x)(x\.r\h)(2*y,^(pX
- k -
p).
The hadronie tensor is suuuned over all t lie allowed final states and by convent iou includes a factor (
+ C'/,, - (Q2/2)ih,„
+ i CtV(,„rWT]
-I- 2 D(V >"},,„„ .
(3.33)
The last t wo terms are associated wit h parity violation. The coefficient C'tv 2n, \-1'(\ /(i'2\ + " n )
i in
111 r.v
il vir i ii II
leptou. For « photon it is C fi II, for a boson o n e lias , w -11 or f beam and C w 1 or I beam , respectively. T h e last, term, with i v -f i i f V , is suppressed by a relative factor coellicient y v v 2 and is almost always neglected. It is not straightforward to calculate the m hadrouic tensor. However, since it must be constructed from the only available T . it is possible four-vectors, and and the two isotropic tensors. and f to write down its general form as //
= -F\
1
\F2l> ,,
i FA f
+(F\-\
FrM-v
aT,r,,
T
+ {F,
+ Fr,<, <,
i (p
FWif
q)'1
.
. 4
I lere. t he liadrou specific structure functions. ,. arc dimensionless t han s to t he 1 . orent scalars. It is also c o m m o n t o see o n . 1 defined in factor of 1 [M2Jp-q)F2-(-,. terms of the e uivalent structure functions F\ and 11 o do not do this as it only adds unnecessarily to the uotatioual burden. If the spin of the colliding particles is specified, then extra terms, containing S and H, would be possible in o n . 1 . Terms involving further four-momenta would arise also if measurements are made on the hadronic final state. iiautum mechanics and s y m m e t r i e s impose important constraints on the Troininu ct at.. 1 2 . s defined in e n . I they are all real. T h e time reversal invarianeo of C implies that Fr, . s we shall learn in ection . .1. electromagnetic gauge invarianeo implies the following current conservation constraints q
=
and
q
=
.
.
similar, approximate constraint applies to the wea currents. y imposing e n . . see Fx. - . the form of the hadronic tensor is restricted further to
=
,( ,
+ ££)
)
+ £-(?
-
i>
A
)
i> q
. If we do not. impose o n . then we must add residual and F, structure functions, « la e ii . 1 . to e n . i but these would be suppressed as c.f. o n . . and will not subse uently trouble us . alfe and (t»tintl/Q')~. lewellyn- mith. 1 . Combining e n . with e n . . which satisfies also the e iiivalont of e n . . gives L
H' ' = F\2Q2+-—^I'll
4 p ()(P
I') - Mj*Q2]-Ctv
>
+ <')(]' .
.
P'l
where w e used cin rf' 'a,T. = -2 i this result in Tr - - q T' pplying e n . 2 gives tlie general expression for unpolari ed. inclusive lepton liadron scattering see also Ex. -4 d -V d E d cos
, ~
v ii I M2)'2
'2 Mh
I I IK Q C I ) H I M C l t l l ' l I O N O K I t A S K ' U K A O ' I I O N S
II
siii a (Ö/2) -I /•'• (I-VTn I
'IST
o n 'Ma (Q2 + M? )2
(|.i( I
x
( i - , - i s « ! ) f 2 - c v , , ( , - £ ) Ft I
+
dV1'
j/_ X
(3.38)
(l./d// '
Mere, we have defined d ¡ \ !ljy/{-It)T h e first form uses the energy and angle of the scattered lepton in the target rest frame. T h e second and third form use the horcntz invariant variables Q2, defined in cqn (3.29). and ./• and
./•
=
JJL. 2 Mhr E - E' <1P e-p E
(
Jl
Mr. •
•'II
In this framework all informal ion on the possible scatterings resides in the structure functions F\ _.(. These, in turn, may be only funct ions of diiuensionless ratios of Q2. p •
•2 Ml
Hp') [(p + p'yTi(Q2rM-)
-I i (//„ - 1 h„n""^,(Q2.-i\F)}
„(,,) .
(3.10) I Iere p\, is the magnetic moment, of the liadron measured in units of the nuclear magneton. < li/(2M]>). and T\ and J--, correspond directly to the electric and magnetic form factors of the hadrou. For the proton and the neutron //,, 1-2.793 and //„ = —2.913 respectively. These form factors can be related to the Fourier transform of the liadron ! s electric charge distribution. Using eqn (3.10) in e given in terms of T\ and T>. Empirically, the two form factors are both described well by the dipole formula which corres|)onds to a spherically symmetric, exponentially falling charge distribution. One has K-\r\/o
p(r)
=
S~f7
:1
1
f,(
(1
^M-y-
(3.11)
111 III I.t 111 III I
II
I ll« parameter o is minted to the hadron s iiicaii charge radiuss uared according tn i l, 1 'la1. Thai the structure functions vanish for 2 oc relleels the lac of high fre uency Fourier components in a smoot h charge distribut ion. t slightly higher C.o. . energies. uasi-elastic scatterings become important. Here, the target, hadron is excited and brea s up into a low multiplicity system of hadrons, for example. l ii r . gain this process can be described bv ec n . with form factors similar to those in e n .11 . 2 t larger C.o. . energies, high processes become inematically possible, a l l o w i n g t h e internal structure of the target hadron to be probed. In this regime, t he last falling uasi- elast ic cross sect ions vanish and the ma ority of collisions become inelastic t lie target hadron being bro en up. This is deep incla.stic sen lcvilli/ l I . ince the invariant mass of the hadronic final state. IF. is not determined. (J2 and u — p-q/Mi, are independent variables. gain e n . applies. but 111« form of the structure functions l-\ undergo a ualitative change. 2 Hat her t han vanish as oc they remain finite and become practically a function of the single variable x = Q2/(2Muu) e .1 ITC Collab., 11 2 . ri.
.p
.
i
.42
I ltis or en scaling or en. l i demonstrates that the exchanged vector boson now scatters off pointli e ob ects that have no mass scale associated wit h them. Furthermore, the effective constraint. Q~ x x 2 ,i 2xp q is reininiscent of elastic scattering, c.f. e n . . It is interpreted as being due to the lepton scattering elastically oil a charged, constituent anti uar which carries a fraction x of its parent hadron s momentum. In the parton model or en and aschos, 1 G Feyuman. 1 2 tin- hadron is viewed as a collection of independent, that is, essentially non-interacting or free, anti nar s and gluons each carrying a fract ion of the parent hadron s longitudinal momentum any transverse momentum is ta en to be small by comparison. T h e hadron is now described by giving t he probability density distributions for t he momentum fractions of its parton constituents x d.r
T(x'
- - d.r
q. q or g .
.4
T h e f ( x ) are nown as parton density /mictions p.d.f. or also, somewhat confusingly. as structure functions . Here, and in the following, we shall reserve the name structure function for physically observable uantities. These functions are similar to the fragmentation functions, which we met iu ect ion .2.1, but in a reverse sense. T h e hadron cross section is then formed as a sum of point li e anti uar cross sections weighted by t heir p.d.f.s, in direct, analogy to e n .2 . gain, in this faet.ori ed form the long-distance p.d.f.s are universal, that is, independent of the particular hard subprocess. s the exchanged vector bosons only couple to auti uar s. the presence of gluons in the hadron is felt only indirectly in I experiments.
m i
Q('ii ni s c n i i ' i iDN o r
U A S I C KI:A<"I I O N S
l.'t
Depending
+ D)+ l(U + U)
=
«
n.i xF?'
=2xJ2\D-U\ 1). u
i>.r
= 2 r £ [ i / - 0 ] n.r
o.r
(3.1-1)
Here. D represents any down-type quark ( d . s , b) and U represents any up-type quark ( u . c . t). Whilst t hese formal s u m s include t he heavy quarks (e. I>. t.) their practical contrihution is negligible if the probing boson is unable to resolve them. In the transverse plane, which is unaffected by Lorentz boosts along the beam axis, the size of the heavy quark is given by ~ l / . U q . whilst the exchanged boson sees scales > 1 /Q. Therefore, if A/Q > (}. the quark can be dropped front the summation. T h e coefficients in eqn (3.-I I) reflect the normalized electric and weak charges of the (anti)quarks. T h e constituent quark model (Close. l!)7!l) together with the conservation of flavour imposes a number of constraints on the p.d.f.s. For example, for a proton we have I <\x{s(x)
I d r [„(,•) - , , ( . , ; ) ] = 2
- s(x)]
=U
.A»
(3.15)
i tc. [ d.T [,/(*) - d(x)] = I Jo These equations state that the proton contains t wo units of up-ness. one unit of down-ness and no net strangeness. There is no such constraint on the gluons. IJ da: ;i(x). as the number of bosons is not conserved. It is usual to see the quark p.d.f.s separated into two components (Kuti and Weisskopf. 1971: Laudsholf and I'olkinghorne. I!)71): the valence quarks which carry all of the proton's quantum numbers and the sea quarks and ant ¡quarks which make up the remainder and carry no net charges. For example. u(x) = itv(.r) +
/ d.r li v (•'•) = 2
vs(x)
./(i
«(:x) = fl,(: r)
f
d.r[» s (:r) - r, s (,-)] -
(3.-IG) 0.
T h e sea quarks are commonly assumed to be produced in g —> qq splittings. This suggests the idea that the sea quarks are symmetric in the sense that IIS = i7s = f/ s = TL„ = .ss .ss = . . . . Whilst this makes many formulae simpler, it is known empirically not to be exactly true, though a theoretical understanding of how this comes about remains elusive.
I III'. I lll'iOIIY <>l
11
I)
Tlic pari,dm model interpretation o f ' d e e p inelastic lepton hadron scattering is only approximate and Q C D corrections should he taken into account. In the QC'D improved parton model the DIS cross section again can he written in a factori/ed form, but o n e which can now be proved formally to hold (Collins and Soper. I9.X7). v.-ith t h e structure functions given by
l f
U
W / ' ) =
Y .
/ ' f / h t p ^ ' / ' « ) ^
7
1
^ , - ^ - . ^ ) .
(3.47)
Here F,'1 " is a projection of t he cross section for the partonic scattering \ ' f —+ / ' appropriate to the /'tli structure function. Again, it has been necessary to introduce a factorization scale. ///.-. and scheme, plus a renonnalization scale. ////. and scheme. T h e (projected) parton cross sections. F,11 " . contain only shortdistance physics and are calculable in perturbation t heory. T h e y d o not depend on the hadron h. B y contrast, the p.d.f.s. f\,. know nothing of the hard subprocess and depend 011 the incoming hadron; they are not calculable using only perturbation theory. T h e proof of cqn (3.47) also justifies the assumption of incoherent scattering and provides a formal definition of the p.d.f.s. This definition shows that ¡11 a frame in which the target hadron has infinite m o m e n t u m the p.d.f.s reduce to the matrix elements, ( h | N / ( : r ) | h ) , where A r /(.r) is the number density operator for partons of type / with given m o m e n t u m fraction. Formally, etjn (3.47) only represents the first term in an operator product expansion for F , ° '"'(.r.Q 2 ) (Altarelli. 1982). This m e a n s that it is only exact for ( / ' —• oc. T h e expansion is organized in terms of the operators' twist ( mass dimension — spin). Thus, at finite values of Q~ there are higher t wist correct ions which are suppressed as
|iu((/-7^)]"'<"
Q"
'
when 1 11 = I for D1S. In general, t hese non-pertnrhat ive corrections are neglected, though there are situations where their effects should be taken into account. In eqn (3.47) the factorization and rcnormnlization scales are arbitrary. In = fij.-. This .simplifies practice, it is c o m m o n to set all scales equal. / r s Q2 = 1 f) the coefficient function, giving, for example. F, * ( i r . z : f i . / / ) oc rf(l - z) in the so-called DIS factorization scheme, which allows combinations of the / i , ( . r . / r Q2) to be determined directly in an experiment.. Whilst the fu involve longdistance physics, the scale /1 may still be sufficiently large that o s ( / r ) is small enough t o allow the dependence 011 the scale to be calculable a t least down to s o m e low scale //O^AQCD- This results in the p.d.f.s developing small. I1111 measurable, logarithmic d e p e n d e n c e s on //". Such scaling violations are described well by the coupled, integro-differential D G L A P equations (Altarelli and Parish
1 in-: q c i ) D K H c m r n o N 01 MASK: HKACTIONS
11)77; Cribov and Lipatov, 1972; Dokshitzcr. 1977).'-' At leading order these are I'.iven bv
=/ t S
[CI«»^*) +CM»(fv)]
<••>•*»
f — Q'O The kernel functions. I'„i,(z, o s ( / / 2 ) ) , are known as Altarelli Parisi splitting ftinctions and are associated wit h the branchings b —» a X . T h e y can be expanded as a powi-r series in o s . T h e leading order expressions are
r S ? u - c C <
s
>
r
( L ± f )
= M z * + (1 -
C W = 0 - >
+
= CF
.:)s|
+
(<7^7 | 1
+
( 1
."8'
3 1
•
-•>) +
- *) (3.10)
Away from z= I these are ordinary functions, but at z 1 the diagonal splitting functions. / ' , . must, be regarded as distribution functions. Details are elaborated in Section 3.(>.3. where also the meaning of the plus-prescription is explained. Since the virtualities involved in this initial s t a t e evolution are negative t hese are the space-like splitting functions. Equations very much like cqn (3.4!)) control the /
Ili
I III
I II l I
t I
densities Ilial in llii branching l> a. parton u will carry a fraction in the range c.c I ; of its parent, lis. nioineutuiu. and any other products. , a fraction 1 trictly, the branching probability densities are given by the distribution functions <)( 1 z)<\,I, I o s 2 ; T ) ' ' ^ ( ) which are regular functions away from z I. Thus, for example, the probability that a high virtnalily gluon. carrying moment inn fraction . . came from a low virtuality gluon. with a larger momentum fraction . is given bv
[**
/
' ^ C W i W <*- yz) = [
(
7
)
•
ll the terms in e n .4 have a similar interpretation. Figure schematically this interpretation.
.
shows
J)\n7f
Fit;. . . schematic interpretation of the space-li e G e uations whereby the scale dependence follows from the presence of parlous within other higher momentum parlous Given the above interpretation of e n .4 it is straightforward to anticiJ pate how the p.d.f.s will change with CJ . t low Q1. one might expect that there are few parlous in a had roil and that subse uently their p.d.f.s are s ewed to high momentum fractions. This picture is not too far from saying, for example, that a proton consists only of two u- uar s and one d- uar . each with momentum fractions smeared around the value x 1 . s the Q~ increases, the typical parton moment um fractions decrease as momentum is shared via parton branchings. Thus, we anticipate a growth in the small-.r component of t he p.d.f.s. Furthermore, we expect many of these small-x partons to be gluons. which have a high probability to undergo g gg branchings, and sea uar s such as u and s which arise in g branchings. s the p.d.f.s shift towards small x and t he sea grows we must respect the sum rules for the uar flavours, e n .4 . and the conservation of momentum.
=
i'drrLr) + V f(x) Jo if='i-
. 2
I III'
I) I >KS( 'HI I T U JIM < )!•' I I A S K ' | { i : A < ' H O N S
17
li is the application of tills sum rule which provides compelling evidence for the existence of ghions. currying over ,r)0% of the momentum, in a proton (LlewellynSmith. I!)72). Equation (3.49) allows us to calculate how the p.d.f.s change between the scales //(> and //. However, since t he /(.T, //o) ( / = q. <]. tj) only involve a nonpert urbative scale, we can not use p Q C D to calculate them. In principle, 11011porturbative techniques, such as lat tice calculations, may allow them to be calculated. However, at the present time only a few moments of their distributions have been obtained and we must rely on experiment. (Gapitani r.l at.. 2002). Their determination relies on an interplay between the use of cqn (.'$.49) and t he measurement of structure functions giving various combinations of p.d.f.s at different. Q2 scales. In essence, one tries to 'guess' a set of / ( . r . //o). evolve them to higher scales using cqn (3.49). and then optimize the fit to the measured combinations at the higher scales. Details of the procedure and results are discussed in Section 7..r). T h e overall consistency of this procedure gives evidence for the validity of the evolution equations and thereby pQCD. Many sets of p.d.f.s are available, for example the package I'DKMB (Plothow-Besch. 1993) contains a a compendium. As mentioned above, the formal proof of the parton model can be achieved using the apparatus of field theory. However, we can gain insight into its motivation by considering the space time structure of the collision. T h e hadron is pictured as a collection of partons sitting within clouds of further parlous that arc being emitted and absorbed constantly by one another . T h e virtualilies involved must be low, k 2 & M t 2 . if the hadron is to remain intact. Indeed, high momentum transfers are suppressed as [o s (Q")A/, 2 /A'"]". where ri = 2 for mesons and ii 3 for barvons. This, in turn, implies that the parlous have lifetimes ~ l / M i , . whilst the incoming exchanged boson interacts for a mere l/Q. Thus, to the incoming boson t he parlous appear almost, frozen having been formed well in advance of the near instantaneous collision. T h e struck parton has essentially no time to communicate with the other parlous and therefore behaves as if it were free. This also implies that the hard scattering knows nothing of the target hadron bevond t he probability that, it contains the struck parton. Returning to the struck parton, it is impulsively kicked out of l he hadron and leaves behind its cloud of partons. These remaining partons have been 'shaken free' and as they have nothing to be re-absorbed bv. they continue to By forwards on near collinear trajectories. T h i s ¡nil ial state radiation eont nines to shower and hadronize. resulting in a target, region jet. T h e struck parton behaves much like a quark produced in an e + e collision and fragments to produce a current region jet. Between the colour charge on the scattered quark and the ant ¡colour left behind on the hadron remnant is a colour field which converts into low energy hadrons lying between the two jets. Actually, since these intermediate hadrons are produced in a statistical Poisson-like process, it. is possible that no hadrons form bet ween t he two jets, alt hough t he probability for such a gap is expected to he exponentially suppressed as the distance between the jets increases. A typical
IK
mi
1111:0m 01
goi)
neutral current IMS event is shown in Fig. 7.2 and a typical charged current event is shown in Fig. 7.5. Jet-like structures start to become apparent in IMS for Q2 (1 G o V ) . As 2 the (J (G'.o.M. energy) increases multi-jet structures appear, just as in e + e ~ collisions. T h e LO hard subprocess is j — q'. which results in a far forward, target region, beam remnant and a more central, current region, jet. At ( 9 ( o s ) the NLO subproeosses are the Q C D C o m p t o n process. I'q —» gq'. for scattering off n(n anti)(|uark and boson gluon fusion. V'g —> qq'. for scattering off a gluon. Both of these processes can give rise t o two central jets in addition to the forward jet. T h e type of vector boson exchanged is strongly dependent on the event's Q 2 and type of lepton involved. For charged lcptous at low to intermediate Q'2 & (10 G e V ) - . tl 10 neutral current cross section, mediated bv a photon, is very much larger than t he charged current cross section, mediated by a \ V ± . Measurements are shown in Fig. 7.8. This difference essentially reflects the propagators of the exchanged bosons which lead to different ( } 2 behaviour: photons give a \ / Q x fall-off whilst W bosons give a nearly constant cross sect ion for Q 2 « M ' i -
W
a,Kl
d ^ s h , HrW
+ A W * ^ - -
(,,
-,3)
As the Q 2 increases further both cross sections begin to fall faster. This is because kinematics require higher Q2 events to have higher x values and t he p.d.f.s. /i,(.c ~ l.Q2). vanish as Q2 — oc. Also their difference diminishes until they become of equal magnitude for Q2 ^ (80 G e V ) 2 . An example of eleetroweak unification in action! Above (J 2 — (40 G e V ) 2 Z exchange starts to visibly contribute to neutral current events. This is manifested by the appearance of the parity violating F\\ structure function through 7 - Z interference effects, which start to reduce t n < ' ( / + 1 i ) compared to a x c ( t ~ h ) . In charged current events
-!j^(e+h)2 ax <1Q a(J' and
oc [ii + c + (1 - y)2(d
- l ^ ( e - h ) x [a + c + (1 - y)2(d 1 cto: (IQ~
+ s)] + 5)] .
(3.54)
For a proton, we expect qualitatively u(x) 2d(x) >
I III', g t
l> I ( I ' M It I I ' I l< IN ( II
IIA.SK ' U K A C T I U N S
I'l
Before continuing our discussion wo digress slightly in order to introduce* a natural variable for describing an outgoing particle. Rapidity, y. and pseudorapiditv, //. are defined with respect to an axis, typically the beam or a jet axis assumed to be pointing along the 2-direet.ion. by 1 (E l-/i. \ !/= ghi ( ¿ ^ f )
»1—0 *
'' = ln|cot(£»/2)] ,
(3.55)
where 0 is t he polar angle of the particle and m its mass: see also Ex. ((¡-2). Rapidity is small for central product ion, 0 ~ ~ / 2 . and large for far forward/backward production. 0 —• (I or TT. In a collision with C.o.M. energy Y/s the allowed rapidity is restricted kineinatioally to the range [— ln(^/if////), 111(^/5/7»)]. T h e usefulness of rapidity stems from its appropriateness in describing the Lorontz invariant phase space of t he filial state particle: (\:iv
= d/r' dr,xb/ .
(3.50)
The advantage of this form is the simplicity of the way in which each term I ransforms under a boost along t Ik* beam axis, lu particular for a boost of velocity 0 =
i'/c.
so that d.i/ is invariant, as are /r ± and Also, as we shall learn, soft particle production typically has a Mat. distribution in rapidity. In most DIS events the target hadron is 'blown apart.', resulting iu a trail of soft hadronic activity lying bet ween the colour connected remnant, target region, jet and one or more current region jets. However, at, III'.RA in a large fraction of those inelast ic events with small./'. and therefore large values of 11', t he total mass of t he outgoing hadronic system, the distribution of hadronic activity is markedly % Q2/x different (Ilebecker. 2000). T h e inverse relation II"2 = M?t + ( 1 r)Q-/x is easily derived from etpi (3.30). Whilst, central 'jet' activity occurs, it is isolated from the target hadron which is only slightly deflected and appears not to break tip. A rapidity gap, typically a region of size Ay ^ 3 in which no hadrons are found, lies between the 'jet' and the scattered hadron. What is seen in practice is no forward activity in the main detector and. in the absence of specialized, far-forward detectors, a target hadron which can be inferred to have disappeared down the beam pipe. Compared to a regular DIS event. 7M1 —» A*, this subset of events behave as 7M1 —* XY where Y is the scattered target hadron or possibly a low mass excitation of it . Empirically, both the square of the four-moment urn transferred to the forward hadron. t = (/) / / ) - < 0, and 1 he mass of the observed hadronic system, M\- = ( /'')"'• t hat, is excluding the scat tered hadron. are characteristically small, with a functional behaviour like
i mi
i i n « > m < >i i¿< i»
|H T h e dependence oil II (q I /<)-' and Q* (' — modest. In particular. I lie cross section for this type of event, stays constant, or even grows, as .s (( I />)""' or IF 2 increase, in marked contrast to the rapidly falling cross sections of IMS events. These are the so-called diffractive DIS events, characterized by t heir rapidity gaps and almost constant cross sections. T h e situation is illnst rated in Fig. .'J. I.
(IN Itadron current jet rapidity gap I .'/I.
FIG. 1. Schematic diagrams of the rapidity distribution for the number of hadrons produced in: top, a regular deep inelastic scattering: bottom, a diffractive deep inelastic scattering The requirement that the incoming liadron remains intact limits the momentum transfer to |/ < — f m a x & ( l / / ? o ) " . where /?
I NI': < J C I ) IN S C H U T I O N 0 1
M A S K * KKA<
I IONS
m
believed to dominate tlx- hadron hadron scattering cross section according to Regge phenomenology: tliis is discussed iu more detail in the next section. T h e formal description of these rapidity gap events follows similar lines to that of regular DIS events. T h e main difference is that the hadronie final state contains a hadron of known quantum numbers and momentum /»'. In addition to the usual variables. Q2, x and we introduce the quantities ' = (PI, - P'f
_
iW y
W
Q~ 2,i • (,, - ,,')
f ( f
2
+ Q-
(3.59)
Q*
*
,•„. ~ Q2 + M2
'
T h e approximations hold for low values of I < 0 and high values of II'-. such as are characteristic of high energy diffraction. A moment's reflection will convince you that, in analogy to the usual phenomenology of DIS, j:«« should be identified with the momentum fraction of the object . />{[> = ]>'' —I»'1' ~ Xip/^'. and fi wit h the fraction of the object's momentum carried by the struck constituent. Thus x. the constituents moinent.um fraction with respect to the incoming hadron. is given by x = .rip/l Both .rip and ¡1 lie in the range [0. 1). T h e four-fold differential, diffractive DIS cross section is given by
,
d.riP <\l d.rdQ-
=
xQ '
[1 + (1 1
-
'
v
f
\
•
v m
Here, for simplicity, we have neglected the small contribution from the longituIntegrating over dinal, diffractive structure function. /•'}" '' /•'!" — 2:r/;',')( /. which is often not observed, gives a t hree-fold differential dist ribution, now involving /•'! " 11 etc. As wit h ordinary DIS a factorization theorem has been proved {Collins, 1998), ,_> . - . U M )
r
-
E
i
J
.
.-J r p
g
z
, . (3.<>I)
Here ///.- is t he factorization scale (we have suppressed the renormalization scale is the usual DIS structure function describing a photon scat/in) and F-\'f\z) tering off a parton / carrying a fraction z of its parent hadron's momentum. T h e remaining terms are the new diffractive parton density functions (Berera and Sopor. 1994), also known as (extended) fracture functions (Trentaduo and Veneziano, 1994). T h e diffract ive p.d.f.s satisfy t he usual D G L A P evolut ion equations. At tempts have been made to go beyond eqn (3.til) using Regge factorization. This assumes that the Pomoron is a real object, whose coupling to the parent
I III'! T I I K O H Y O K
QCI)
liadron is described by a function of.in- and I mid whose parlon content is then deserilied by functions of Q and Q 2 . T h i s unproven assumption gives = /..,„(**,/.)F2D(2> ( f i = — . ( A . \ lip J
da: 0 >d/
(3.02)
where fw/\, is often referred to as the Pomcron flux factor. Measurements to date suggest t hat, t he Pomcron has a high gluon content and that there is a significant probability that, a gluon carries nearly all of its momentum. Such a picture has also been promoted in the context of hadron liadron collisions (Ingelmau and Schlcin. 198.r>). Unfortunately, it appears that the same Rcggc factorized structure fnnctious as measured in DIS will not be applicable, without at least, some modification, in the description of hadron hadron collisions (Collins et til.. 1!)!):$). Historically, dilfractive events have long been known in hadrou hadron collisions where a well developed phenomenology has arisen. Indeed, this was used to predict that sizeable dilfractive cross sections would occur at IIKHA (Dounachie and Landshoff, 1987). However, the discovery of such events at IIKHA (ZKUS Collab.. 1993: II1 Collab.. 1994) still came as surprise to many people and it has led to a resurgence of interest in the nature of diffraction. Deep inelastic scattering events, whether dilfractive in nature or not. are characterized by large values of Q2 i i (3 GeV)". There also exist events in which an incoming charged lepton emits via breinsstrahlung a quasi-real. Q~ ~ 0. photon which interacts with the incoming hadron: the so-called photo-product ion events. As mentioned earlier, such photons appear to have a rich structure and variety of behaviours. They may behave as a hadron. giving effectively a hadron hadron scattering. This in turn could be elastic, here meaning ->*h —> Vh with V a vector meson, dilfractive, soft, inelast ic or hard inelast ic. All these categories are elaborated below. T h e hard inelastic events are viewed as due to the scattering of (anti)quark or gluon constituents within both the hadron and photon. Thus we require p.d.f.s to describe even the photon. Of course, it is also possible that th<' photon remains intact and interacts direct ly with a quark or an ant ¡quark. 3.2.3
liadma
hadron
scattering
liadron liadron collisions exhibit a rich variety of reactions. These can loosely be divided into two classes. T h e first class involves soft interactions which have only small momentum transfers so that they are sensitive to long-distance effects and see a hadron as a coherent whole. These have typically large. (9(10 nib), cross sections which change slowly (logarithmically) with the C.o.M. energy. Examples include the total, elastic and single/double dilfractive cross sections discussed in more detail below. T h e second class involves hard interactions, defined by the presence of a large momentum transfer so that they probe the internal st ructure of a liadron. These have typically small to tiny cross sections and more pronouueed C.o.M. energy dependencies. Examples include high transverse energy jet. heavy quark and high mass lepton pair production. T h e non-pert urbativc
m i . g e n in s t ' m i i t k in n i
mask: r e a c t i o n s
nature of die physics involved in the first class of reactions means thai a more phcnomcnological approach is taken when describing tliein. Since p Q C D can be applied directly !<» the second class of reactions these shall be our main concern. The above classification of events is a little misleading. For the so-called soil events we intend that the characteristic m o m e n t u m transfers are small in comparison to the C.o.M. energy This leaves open the possibility that at high C.o.M. energies sufficiently large m o m e n t u m transfers may occur t o open up the possibility of applying p Q C D . For example, this is under active study for hard diffractive events. Whilst it is hard to apply Q C D to the bulk of hadron hadron reactions, it is nevertheless helpful to appreciate their basic properties. T h e general behaviour of the total cross sections is as follows. Initially, the cross section falls from 0 ( 1 0 0 nib) at very low C.o.M. energies to a broad minimum around y/s ~ 2 0 G e V before rising slowly. Below s / s 3 GeV resonance structure is apparent. Above i he resonance region simple quark count ing rules give an indicat ion of the relative cross sections. T h e rules posil that a total hadron hadron cross section is proportional to the number of (ant i)quarks in the projectile, as determined by t he constituent quark model, t imes the number of (anti)quarks iu the target . For s • oc one expects, for example. a u ,i ( f " p ) ~ p) and ) is also required by the Pomeraucliuk t heorem. Figure 3.5 shows these total cross sections as a function of the C.o.M. energy. T h e total cross section can be parameterized as > I
s
[1 +
F(s)]
" So where F(.s-). which vanishes as .s -> oc. describes the low energy behaviour. This parameterization automatically satisfies the requirement of uuitaritv as captured in the Froissart bound. <7tl)1 (.s) < ( : r / m 2 ) ln"(.s-/,s<)) for some unknown •Si (Froissart. 1901: Martin. 1903). We shall largely be concerned with high energy pp and pp collisions as this is where the search for new particles has focused the attention of experimentalists. In elastic scatterings the hadrons remain intact, without, excitation of any internal degrees of freedom. T h e y comprise a sizeable component, of the total cross section. <7, I(N) % 1/0 x <Ti„\(s). Elastic scatterings are specified by the space-like momentum transfer I = (j>\„ —/¿out) 2 < 0. which given s is equivalent to the C.o.M. scattering angle, 0 ' , via I = — l p ' 2 s i n 2 ( f l * / 2 ) ~ — s s i i r ( f l * / 2 ) . A number of /-ranges can be identified according to whether electromagnetic or strong forces dominate: the Coulomb region. |/| < 11.001 G e V 2 ; the interference region < 0 . 1 5 G e V 2 . Only 0.001 < |/| < 0.01 GeV' 2 : and the diffraction region 0.01 < the first region is well understood. T h e cross section is described by the /-channel exchange of a photon whose coupling to hadrons is described by two form factors. eqn (3.11). A typical behaviour for the differential cross section shows a strong peak below |/| = 0.01 G e V , then a steady fall until reaching a sharp minimum as |/| ~ 1.4 GeV" which is followed by a broad peak at |/| ~ 2 G e V . Above
I HK T H K O U Y OK (¿Ol)
n/7(GCV) PP
O
PP
•
rc~pa.„.
A
K'poM
KM)
K+p 80
PP o c i a s t i c PP °>.i»gl e difl'raciivc
60
PP ^double iliffr'.K'liVC
40
20
0 10
10 2
I0 3
V7 (GeV)
.'{.ij. Measured cross sections in hadron hadron collisions as a function of the C.o.M. energy. D a t a are taken from t h e Review of Particle Properties ( P D G . 2000) and from the D u r h a m reactions d a t a b a s e http://durpdg.dur. ac.uk/hepdata/roac.htinl.
I III-) < ¿ ( 1 » I »!•.•;« n i l ' I l< >N O l -
ItASK
HKA(' I IONS
I»' 10o % o o
10
10
10--'
10-1
A ISK. pp. \'s=30.4GeV 10-'
A ISK. pp. \'s = 52.8GeV •
5
to-
P
SI'S WA7. pp. p, i h = 30GcV/c
O SI'S WA7. pp. p,
I
a|)=50GcV/c
tf ti
O GDI-", pp. V s = l 8 0 0 G e V 10"
—I I » »1111 10-
10-7
10
10
1(1
l/l (Gl-V/C)
FIG. .'5.(i. T h e measured elilfcrential e-ross section in pp and pp collisions
|/[ 2 G e V - , the hard difi'ract.ive region, the differential cross section falls as I ". The so-called dimensional counting rules (Brodsky and Farrar. 1!)7">) suggest values n (i.8. 1(1 for meson meson, meson baryon and baryon barvon scat tering, respectively, though more explicit calculations based on gluon exchange between the constituent quarks modify these simple exponents (LandshoH", 1!)74). Figure 3.(i shows the /--dependence! of pp anel pp scattering. Approximate forms for the elilferential cross section in the elilfrae-tive region are given by
d<7cj d/
|/| < 0.4 Oe-V"' X
|/-"
|/| > 3.0 G e V 2
(3.0-1)
(
lie dip ni struct ure can be described by interference between wo exponentials. The -distribution can be related, via a Fourier cssol I ransloriuat ion. to the impact parameter space distribution of the scattering centres in the liadron. exp vu 4 exp b,.\flr), where h is the impact parameter. efining an effective slope by
( .( ,) the ineasureinents show that ,.ir increases for large s. Thus the liadron shrin s at higher energies. The very forward pea ed nature of the elastic scattering cross section indicates that low momentum transfers are dominant. This essentially straight through behaviour means that speciali ed low angle detectors, usually in conunction with low luminosity, are re uired to measure this large cross section. Interestingly the optical theorem provides a highly non-trivial connection between this forward f I differential cross section and the total cross section sec Ex. -11 .
F i e . . . ingle dilfractive dissociation, double dilfractive dissociation and central diffraction. Experimentally these events are characteri ed by a forward et separated by a rapidity gap from an intact , scattered, incoming liadron or two forward ets separated by a central rapidity gap or central activity separated by two rapidity gaps from t he scattered, incoming hadrons. The next important class of reactions involve dilfractive dissociation processes in which there is some brea -up of the scattering hadrons. possible way to view these events is as the -channel exchange of a colour singlet ob ect called a omeron. see Fig. . . nli e the case of elastic scattering, in a single double dilfractive dissociation event one both of t he hadrons is left in an excited state which then brea s up into a low multiplicity system of hadrons et , for example, p n;r . Typically the mass of t he excited hadronlc system is distributed as drr i . whilst, the -dependence of the cross section falls away exponentially with a coefficient which decreases as i n c r e a s e s . Experimentally the ey signat ure of these events is t he lac of any particle production in bet ween
I III <}< 1» I >1 St n i l ' t I O N O l ' H A S H • H I ' V
I IONS
I lie scattered/dissociated Imdrons. (¡(invent ionally. I his gap is quoted in units of rapidity. II the final state hadrnns have masses M t and A/j. then they have a rapidity gap of A;/ ln(.s/j\/| Since the size of the gap is usually there is a minimum C.o.M. energy , / s ^ 4 . 5 C l e V required for these events to occur. A related class of events, known as cent ral diffraction events, show two large rapidity gaps separating centrally produced jets from forward/backward going hadrons (UAH Collab.. 1988). These can be interpreted as the interaction of two Pomerons. as shown in Fig. 3.7. They are of particular interest because the jet activity indicates the presence of a hard scale and the possibility to apply pQCD to t heir description. At low energies the cross section for all these rapidity gap reactions equals approximately the elastic cross section, with the ratio of single to double dill'ractive events found t o be % 4 : 1. As the C.o.M. energy dependence of the cross cetioti for a fixed excited state is flat, the growth of the total dissociation cross cction with y/s can be attributed to new excitation channels opening up. Experimentally the double diffractive dissociation cross section grows faster than the single diffractive dissociation cross section. This is in accord with the naive expectation <7nn 555 <7.sd/"lot- The central (infraction cross section is a few per cent of the total dissociation cross section. At t he I.1IO. a ^/x — 1'lTeV. pp collider being built at C'ERN. predictions indicate that /•) = 350 MeV. and are uniformly distributed in rapidity, d.'V, i,/d// ~ 2. This implies that the multiplicity should grow logarithmically with the C.o.M. energy. (N) = .4 ln(.s/.s'o) + D/\/s. T h e pion. kaon. and baryon composition is observed to be roughly 85%. 5% and 10% respectively (UA5 Collab.. 1987). These soft particles also show short-range order characterized by positive correlations iu rapidity. This structure is often interpreted as being due to the production and subsequent decay into stable hadrons of 'universal clusters'. The properties of the soft particles show only a weak dependence on the C.o.M. energy of the colliding hadrons. The major components of total hadronic cross sections (elastic scattering, single/double diffractive dissociation and soft inelastic collisions) all feature 'small' transverse momentum transfers. This focuses our at tention on scattering in t he limit .s —• oc whilst I is held relatively small. Here, a successful phenomenology has been developed based upon Regge theory. This pre-QCD theory treats the angular momentum in a scattering amplitude as a complex variable and proceeds to derive consequences from analvtieity and crossing .symmetries (Collins, 1977). A typical /-channel exchange amplitude for a two-to-two process takes t he form
i in
O
1111•:<>in < >i (¿i i»
O
O
O
O
o
I
a dy
-
lu(\/s/2m)
+ lu(v/«/2m)
0
u
FLC:. 3 . 8 . A schemat ic diagram of a soft, inelastic collision and the associated rapidity distribution
M(s,t)
'( \ b - '
= (i + ,,)
with
f>(s.t.) =
Xm\M{s.t)\
« /»(»,()) .
The index i in the above equation denotes the flavour quantum numbers exchanged in the interaction. As the practically /-independent /> is (9(0.1). the amplitudes are mainly imaginary. For the exchange of a particle of spin ./ t he exponent, would lie o , ( / ) — ./. but in eqn (3.60) this has been "Reggei/ed' to include contribut ions from a whole family of particles lying on a linear Regge trajectory (3.07) o , ( / ) = o , ( 0 ) + o;./ + C>(/'2) . T h e parameters of the trajectories can b e found by fitting the spins and masses (.s-channcl poles) of real mesons and baryons using ./ = a , ( M ~ ) . T h e slope is almost universal with o ' « 1 GeV whilst the intercept depends on the flavour quantum numbers, i. being exchanged. One finds o , ( 0 ) % 0.5 for the leading (dominant) contribution from the non-strange vector mesons p,u!.n>. /•> The sub-leading pion trajectory has o , ( 0 ) % 0. T h e exponential /-dependence assumed in eqn (3.00) is empirical: it. implies !>,.^ = 2[/). More formally it is a measure of the coupling strength between the scattering and exchanged particles. T h e mass M, accounts for the dimensions and absorbs any numerical factors. Using eqn (3.00) one can derive compact expressions for. for example, total, elastic and singly diffractivo cross sections as
=
TT7» ( "772 )
M?
=
(1 + / r ) IGttA/;'
the
(3-68)
\M?)
( V M'f
\ J
-'<<>.<0-1) „21M i
11 IK Q C I » D E S O U I I ' I I O N O K B A S I C I t K A O ' l I O N S •-'(<.,(<)
i)
(3.70) A/;' A / f
\ M
For simplicity we have included only a single Reggeon exchange and omitted the electromagnct,ic contributions. Including the interference between the hndronic and. well known, electromagnetic amplitudes allows i> to be measured experimentally. These formulae provide a very good description of reactions which involve the exchange of flavour. These typically fall as ,s~'. To apply t hem to situations where no Ifavour is exchanged, where cross sections are constant, or grow as ,s- •> oc. a new dominant, contribution must be included. This is the I'omeron. It. has the quantum numbers of the vacuum and the Regge parameters au»(0) « 1.08
and
oj,. « 0.25 .
(3.71)
These values are derived from successful fits to a remarkably wide range of data (Donnachie and Landshoff 1902: 1991). This trajectory does not correspond to any presently known particles, though it lias been conjectured that it is related to t he predicted glueballs of Q C D . Actually, since o B »(0) > 1. t he Pomeron is 'supercritical' and. unless eqn (3.68) is modified, will lead to a violation of unitarity in the .s- —> oo limit . More apparent, is the absurdity T.-i/rr,<>( ^ (.s/A/j-)" n ' ( , , ) _ l > 1 for s sufficiently large. T h e inclusion of the necessary multiple Pomeron exchanges and unitarization corrections leads to a more complex theory (Khoze r.t. ul.. 2 0 0 0 ) .
It is important to remember that Regge theory has not. been derived from (¿CD. One should therefore be wary of regarding it as doing anything more than providing an accurate and economical, phcuomcnological framework for describing data in the Regge limit.. It also act s as a guide in framing the questions addressed in an experiment. T h a t said, p Q C D has been applied to the region .s » |/| > AQCI), leading to t he development of a hard Pomeron with an intercept significantly above one and a small slope. To distinguish it. the usual Pomeron is now often referred to as the soft Pomeron. This hard Pomeron is associated with the summation of leading logarithms of the form n s l n ( . s / / ) (Kuraev ct a I.. 1977: Balitskv and Lipatov. 1978). T h e simplest model for such an ob ject is the /-channel exchange of two glnons (Low, 197">: Nussinov. 197.r>) (or one 'Roggeized" glnon). which is suggestive of a glueball interpretation. T h e hard Pomeron also manifests itself in the small-.»' behaviour of structure functions where it sums leading [(vsln('l/:»:))" logarit hms. However, a word of caution should be sounded. As the hard Pomeron theory implies a rapid growth in t he number of partons then non-perturbative methods will be required ultimately. T h e search for the predicted hard Pomeron is an active topic of research. Finally, we turn to the rare, hard events which shall be our main focus of interest. By experimentally requiring an event to contain a large momentum scale we raise the possibility of applying p Q C D to its description. Furt hermore, t he short-dist ance scales suggest, working wit h t he quark and glnon constituent s rather than the colliding hadrons themselves. T h e situation is analogous to DIS and again a lact.orized formalism can be applied. This is illust rated in Fig. 3.9.
(¡(I
I III-: I I I K O H Y <)!•'
q c u
FlC. 3.?). A .schematic diagram for the production of final state particles <• and (I in a hard collision of hadrons IM and h->
T h e basic cross sect ion formula for the collision of hadrons ii j and h_> to produce particles c and il is given by d -
al) =
f dx,dx2 V
/„/,„ (XI. //"iOA/i,,(.'••>. /40d
(3.72) Here the /„/i,, and fi,/\,., are the same p.d.l'.s as arose in DIS. where the indices refer t.o partons a. I> <~ {<|, q, g } in the interacting hadrons hi and ho. Here there is a technical proviso that we are careful to use tin1 same factorization scheme iu the description of both processes. They are evaluated at the factorization scale /i/.-. which is typically 0(Q) a hard scale characteristic of tin" scat tering process. The use of i he same p.d.l'.s is possible because the presence of an incoming liadron does not cause the target liadron to modify its internal structure. This is the real significance of the factorization theorem and helps to make pQC'f) a predictive theory. Iu the matrix element, for the hard subproccss the part on momenta are given by />,'' .ri/'ii, and //{' = .i't/>{[,. In general, we do not expect ./'i - x-> so t hat the hard subproccss will be boosted with J — (./-| — x->)/(xi I- x>) with respect to the h|h-j laboratory frame, resulting iu the outgoing particles being thrown to one side or the other. T h e sum is over all partonic subprocesscs which contribute lo the production of c and il. For example, the production of a pair of heavy quarks receives contributions from qq —> QQ and gg — QQ. whilst prompt photon production receives contributions from qg —• q~, and qq • g7. These two-to-two scatterings give the leading. 0 ( < \ i ) and C?(o s o,. l n ). contributions lo the hard subproccss cross sin-tion. Beyond the leading order it is necessary to consider two-to-three, etc. processes, which gives rise to a perturbative expansion a — C i , o o " + C'M.()O" + I + CNNI,(>O" +2 + • • •• A complication arises with the higher order corrections as they contain singularities when two incoming or outgoing partons become collinear. It is the factorization of these singularities, order by order, into the p.d.f.s and fragmentation functions which gives them their calculable /ij. dependencies. This, logarithmically enhanced, near collinear
I'lIK < ¿<
> I »KSOItll'TION O f
i.l
H A S H ' IIKA( • I I O N S
raeliation is manifested as t lie appearance of init ial and linal s t a l e jets associated with eaeli of the incoming and outgoing partons. 'Flic mix of hard subproeesses which contribute* t o eepi (3.72) depends nont t'iviallv on I lie relative* sizes of hot h the cross sections and the p.el.f.s. The latter are iulluenceel by both the type and energy of the colliding be-ams and any rei|iure>ments placeel on the kinematics e>f the final state. For example. requiring the outgoing particles to be produced in a given rapidity range, perhaps e-ort'espoudillg t o the geometry of a detector element , directly affects the .r-ranges being sampled in the integral: see Ex. (3-13). To go further. we consider heavy quark production at the TEVATIION, a y/s = l.<STeV (now 2 T e V ) pp collider at. FKRMILAH. In the case of cent rally (y = 0) produced bottom qua rks one has x\ w x> ~ 2vi\Jy/s 2 x 5 / 1 8 0 0 = U.005G. whilst for top quarks it is n « r> ~ 2 x 175/1800 = 0.19. At small x gluoiis dominate the p.el.f.s, whilst at large x only valence (anti)quarks are present: this is particularly true at the higher scale appropriate for top production. Q ~ 2/«Q. Thus, bottom ejuark production is dominated bv g g —> bb scattering, whilst top quark production is dominated by the annihilation proc:css e|e| —» tt. Here, we* see- that in a high mass 'annihilation process' it pays to have au antihadron in the initial state. In this result the larger cross section for gg —» QQ is overwhelmeel by the p.d.f. contribution. As a seconel example we consider eli-jet production. In the» abseiie-e of any flavour eleterinillation the outgoing jets may be- seede»d by either a primary (auti)quark e>r gltion so that there» are» many cont ributing hard subproeesses: gg —» gg, ge| —» gq, eje|' —» <jej'. etc. Loosely speaking, the relative hard subprocess cross sections are in the ratio : C,\CF '• Cf.- ( 'tc.. reflecting the colour charges of the collieling partenis. This allows us to e-xpress the integrand in eqn (3.72) in terms of an effect,ive p.el.f. (Ce)inbrielge and Maxwell. 1984). see Ex. (3-14).
/.f
-
m)
with
/" , r (:r) -
<j(x)
Y , ^ J=
'
(:l7:i>
Here C'i- /C,\ = 4 / 9 ~ 1/2. 'Finis at moderate transven'se- jet e»nergies, ec|tiivale»nt to moderate x values, gg scattering will be eloniiuant. In addition to a hard suhproccss such hadronic scatterings alse> involve an underlying event arising from the» collision of the two beam remnants. In broad out line the underlying event is like a soft., inelast ic e-ollision bet ween two haehnns of reduced C.o.M. energy squared (1 X[X-i)s. Fortunately, the soft particles produe-cel have limited transverse» moment um and so elei not unduly obscure the high transverse energy partie les produced in the hard subprocess. Observationally there is an increased level of hadronic activity in hard events, even away from any jets, as compared to minimum bias events which are effectively equivalent to normal soft inelast ic collisions. T h i s is the so-calleel pedestal e-ffect.. Thus, more refined models builel in an interplay between the hard suhproccss and the underlying event (Sjostrand auel van Zijl. 19X7). OIK> possibility, which Iwcomcs more likely wit h increasing C.o.M. energy, is that a second hard scattering ex-curs
111 .(
between I In par tons in I lie Iii iiiii remnants. y t rent ing I In- two si at I n s as indei ii li iit the rate of double scattering can be estimated as y , nrif 2 i. i T h e assumption of independence is plausible provided all the momentum fractions remain small. In an experiment, it is necessary to supply a criterion to decide when to initiate the read-out of the detector. Typically, t his trigger condition is based upon nown supposed features of the events which are of interest. This introduces inevitably a bias towards ust such events. Therefore, it is also common to collect an unbiased data sample based upon a minimal trigger condition such as the occurrence of a bunch crossing or the presence of an energy deposit somewhere in the detector. Given the relative cross sections for the hadron hadron scatterings these miuimiun bias events coincide essentially with the soft, inelastic collision events. ince hadron hadron colliders are often viewed as discovery machines searching for very rare events, there is a need to use high luminosities. Given large hadron number densities in the colliding bunches it becomes li ely that more than one pair of hadrons from the colliding bunches may interact, most li ely in soft, inelastic collisions. Thus, even when a hard trigger is satisfied it is uite possible that the detector is seeing an event of interest together with several soft, inelastic events. For example, at nominal luminosity at the planned I.I IC at T . . each hard event is. on average, accompanied by 0( 1(1) simultaneous iinniinmn bias events. Fortunately, t hese extra pile-tip events produce mainly low transverse momentum particles, spread throughout longitudinal phase space, whilst the hard event must have high transverse momentum particles, typically restricted kincmatically to the central (y = 0) region. 3.3
B o r n level c a l c u l a t i o n s of Q C D c r o s s s e c t i o n s
lu this section, we shall review the calculational techniques required to evaluate basic tree-level processes. We shall concentrate on the process e + e ~ — qq. which is a paradigm for several important processes, together with its lowest. C?(o s ). tree-level. Q C D correction. e + e ~ —> qqg. which we will use in our discussion of the Q C D improved parton model. We will also look at the pure Q C D process qq —» gg which will give us an insight into the nature of gauge invariance. We do assume some previous familiarity wit h Dime spinors and working with Feyinnan diagrams. The interested reader can refresh t heir memory and find more details in any good text book, such as the one bv Ait.chison and llev (1989) or by IVskiu and Schroeder (l!)!).r>). 3.3.1
e + e ~ annihilation
to quarks at 0 ( o " )
T h e basic Feyuman diagram for e + e _ —» qq is given in Fig. .'{.10(a). Strictly speaking, this lowest C?(o") process is more an electroweak than a Q C D interaction. However, it remains of great importance in the description of e + e ~ annihilation to hadrons. and using crossing symmetry, also t o deep inelastic scattering. Fig. .'{. 10(b) and the Drell Van process. Fig. 3.10(c). Furthermore, by replacing t he leptou pair by a new quark pair (q q'). we can learn about di-jet production
IT< >KN MOVI' I C A L C H I
;+
\ I K >NS ( )!•' ( J< 'I ) ( 'IK )SS SI
q. j
<1
a) c + e ~ t o hadrons
<1
I K )NS
'I
l>) Neutral current DIS
e+
q'
c) Droll Yan process
q'
d) Di-jet. production
FlC. 3.1(1. E x a m p l e s of the basic processes contributing to hard lopton lepton. lcpton hadron and hadron liadron scattering. Our convention is such that tIH* F e y n m a n diagrams should be read front incoming s t a t e s on the left to o u t g o i n g s t a t e s on the right.
in hadron hadron collisions. Fig. 3.10(d). It should also be noted that since the elcotroweak couplings are relatively small. o,. l n ~ 10 2 . diagrams involving single photon e x c h a n g e should be sullieiont for an accurate description of processes (a), (b) and (c). On the other hand, since the strong coupling is relat ively large one might wonder about the size of the corrections to the single gluon e x c h a n g e diagram (d). A s y m p t o t i c freedom will h a v e s o m e t h i n g t o say here. T h e m a t r i x e l e m e n t for o + o rules in A p p e n d i x B.
M = (
+
)
• -i
= cv((+)~fflu(r)
7(
— qq is easily written down using the Feynnian
(I ) x 0 Q-'2)
—i »)"" 2
u(q) - i
x r.Whjri(q)rv(q)
v(q) .
(3.7-1)
Here, we use the particle n a m e s to also represent their four-momenta and introduce Q>' = ( f + |- £")'', t h e f o u r - m o m e n t u m transfer. For simplicity, only photon e x c h a n g e is included, which is appropriate for Q~ My. and we have chosen to work in the covariant, Feynnian g a u g e , ^ = 1. T h e <|tiark colours are specified by the colour indices and j which run from 1 to N c . N o t e that we have explicitly included a colour conserving Kronecker i5-function at the (juark photon vertex which ensures that the qq-pair forms a colour singlet. T h e order of the terms carrying spinor indices has been determined by working backwards along each fermion line. Next , we need to evaluate the mat rix element squared: | - ^Vt^Vi'. Now. whilst M is a c o m p l e x number it is formed from a product.
(I I
I III-: I III'.OKS O f
q<3>
ol nialrices, so that it is more convenient to use M * namely. \M\2 = MM1.
- ./V|* and rather evaluate
\M\' = c-- D(f+)7,I«(r)[f-(i+)7,«(r-)]t x(«g
3
x Q-x
• ' V V • «K^-y"wiv)[w(far)-»"wC7)]f
= c2 • H e + h „ * ( t - ) & ( t - h M i - )
x Q- >
X (re,,) 2 • Nc • &((
(3.75)
Here, care has been taken to keep the leptonic and hadronic terms separate and also to stun over the repeated indices. The llermitian conjugated terms in eqn (3.75) have been dealt with as a s|)eeial case of the following result. trf-vf [«r,r 2 ---r„<;i t = r t rj,--.r.U i"fo"
= '"'(Tori,70) • • • (7or^7o)(7orl 70)" with
7n{l.7S,7/.0>«T , 5- < T /«'} 7o = { + 1,-7, r >-+7/i-+7/x75.+<*;,/>}
Here F, represents any one of the five basic 1 x I matrices. At this point we pause to comment on the colour factor in eqn (3.75). Strictly speaking, the quark and ant ¡quark come with colour polarization vectors, so that M "x
= 4|
"(g)c«'(&)<, = <>c<. .
rol.pols
(3.77)
col, pots
appropriate for unpolarized quarks. For completeness we have included the equivalent result for a gluon. where now the indices {ft.c} = 1 . . . A r2 — 1. Thus. •j a(
= T
faifiijtijrflk
x
["* (q)A- ^ki «(q)/]
= / , fijjfiji = Av
(3.78)
Rather reassuringly, the reaction rate is found to be proportional to the number of quark colours. .V,.. If a quark or gluon appears in the initial state then the corresponding colours should be averaged, as described in Appendix B. In practice, it is standard not to write out the colour polarization vectors and instead simply keep the same indices on the external particles in both M and M1. A similar result, originally due to van der Wacrdcn, can be used to eliminate the spinor basis states still appearing in eqn (3.75). Denoting t he spinor indices by {/'../,/.-./} = 1 . . . I. the following relations hold "(l>)Ml>)j
= ^ [(/> + » 0 ( 1 + 75/%;
|M
« 7, [(/ + » 0 ( 1 T lr.)\,j
H O H N I.KVKI. C A M ' I U . A T I O N S O K 0< 'D I 'IIOSS SK< T l u N S
"(/')/•"(/')/
7, \(/>
"')(1 I" 7r./)|,, | 8 | p «
|(/> -I-»")(1 F 7r,)],,
(3-79)
The spin polarization state is specified by the space-like four-vector s which is orthogonal to />. s • /> = (I and which for a pure state is normalized such that I. The approximate form is appropriate to the high energy limit, HI <SC E. when the spin vector is parallel/ant ¡parallel to the particle's direction of travel: the so-called helicity basis. Often the incoming particles in a collision are unpolarizcd. that is. they are an equal admixture of all possible polarizations. It is t herefore conventional to include an average over the incoming particle spins: again see Appendix 13. Concentrating on the hadronic part o f e q n (."i.7-r>). making t he spinor indices explicit and assuming no spin sensitive measurements are made on the outgoing quarks this result allows us to write: E " ( ' / ) . '>:, " ( < / h ' ' ( ' / h y ' u « ( ' / ) / = spiiis
( X "('/)"'('/)• K spins
(
X •>(f~ihH
= (fi + '".,)/. • 7'j • (ff - '"
(3-80)
In reaching this point we have been careful to make explicit the individual steps involved. Consequently, t he derivation seems quite lengthy. However, with practice oik* can. in principle, go straight from A/f to the traces over propagators and vertices appearing in \ M \ 2 . One simply writes down a "»-matrix string from M followed by a second --matrix string from M 1 but with the order of the individual I'-terins reversed, including minus signs for any 75 and 7;,7r, terms present, see eqn (;i.7(>). and with spin-sums ( / i ± 111), as appropriate for the external spinors, inserted bet ween t hese st rings. To deal with such traces of 7-niatriccs we adopt the following strategy which is always guaranteed to work. First, expand out the brackets so that you have a sitin of terms of the form Tr {-¡I'lyi'i.. . y . . J Second, set all terms where 11 is odd equal to zero: here remember t hat. 7r, is the product of an even number (four) of -matrices. Third, for trace's of an even number of 7-matrices repeatedly use the following algorithm based on using the Clifford algebra. 7''"•/' 2i/'"' - ' to permute the first 7-matrix through the rest. We illustrate this for the case 11 = I. where we need to iterate three times, Tr { 7 " 7 " 7 " 7 r } = Tr {(2,,"" - 7 V ) 7 < V } = 2,,""Tr{7'T7r}
-Tr{7"7"7'T7r}
= 2//""Tr { 7 " 7 r } - 2//""Tr { 7 " 7 r } + 2i/" r Tr b ' V } - Tr{7"7"7r7"} =>
= //""Tr { 7 " - , r } - //""Tr { 7 V } + i/" r Tr {7"7"} .
(:i.81)
The last line follows because t he final trace equals the original one bv the cyelieity of t races. The algorit hm reduces t he number of --mat rices in a t race by t wo each
i n r . i HIM n i l
time it is applied. For n final result becomes
2 it gives 'IV
Tr {~,"Y~r"iT}
= -i \'rv"T
» H' i ¿1 i >
"}
//'""IV {I |
- >r>r
//'"' • I. Tims the
+ 'r>r\ •
Of course, in practical situations a number of tricks (short cuts) can often be used to speed up evaluations, for example. •" • 7/i • 1 • = + « " x • •' 7|i7i» • • • 7//^7'' • • • = - 2 x • • • (I
7,. # 7 " •••=
1« bx ••• 1 • • • =
, (3.83)
where tlx* dots are to remind us that the strings of consecutive --mat rices may lie embedded in a larger expression. Unfortunately, familiarity with these tricks only comes with practice. Given these trace results it is now straightforward to evaluate the liadron trace, eqn ( 3 . S O ) , t o yield Tr {(rf + / „ „ ) 7 " W - m „ ) 7 " } = Tr
- '""Tr { - „ ' , . }
= 4 [,,'•<-,''-<,<-,,,'•" = 4
[(¡"q" +,r
+
< / v - » v r ]
~(Q2/2)>r)
.
(3.84)
T h e leptonic trace, coming from eqn ( 3 . 7 5 ) , can be evaluated in the same way and gives essentially the same result, though with an extra factor 1/4 reflecting the spin average in case of unpolarized beams. If we collect our results so far. we liud
X > w i 2 = ¿ r e 2 • T i <(/ + - >»th,(r
+ »}
x ( c c q ) - . V r . Tr {(,? + « i „ b " ( j f - '"<1)7"} =
Ql
•(CC +
x(rrti)'2Nc =
-
{Q2n)'hA
• 4 [q"Q" 4- <]"<," -
1
— L,„,.//"". Q'
(Q2/2)q" (3.85)
Here is int roduced to denote a sum over final state and average over initial state spins and colours. Also, for future reference we have introduced the lcpt.on and liadron tensors /. ( ,„ and //'"'. T h e Lorent.z contractions are easily carried out to yield
=
(^c^fNc2
( 2 - H'2
+ p'2
C O S 2
(3.80)
In the second line the result is written in terms of the G'.o.M. variables 0 ' . the scattering angle between the incouiiug lepton and outgoing quark, and ¡1'. the
| U ) R N L K V K L C A I . C H I A H O N S (»!• <J<'I> C U O S S S I P
i,,
H O N S
velocity i»t' the final state quarks. In this C.o.M. frame, with massless loptons I ravelling along the ¿¡-direction and the scattering in the x-z plane, the fourlilomenta are given l>y
=
and
(->/'
'/
=
().(), ± 1 )
Jo2
(:{.<S7)
(1, ±fl' x sin 6>', 0,
cosfl*)
with
I ii* = J 1 -
.
Note that as a Lorentz invariant quantity |A4|- could only depend on the partii |es" four-momenta via their invariants, for example, their scalar products, ]>,-¡ij. In a two-to-two scattering. » + PI,)2 = (Pc T P,l)~
I = (/>« ~ Pr f = (pi, - P
II = (P„ - Pdf = (Pb - Pc)2 •
88x
Of those variables only two are independent since they are constrained to sat isfy n I I I ii iii2 -I- mf, 4- in2 I '»";• Since s > m a x { m 2 4- mft.mf. I infi} is always positive, one of I and ii. typically both, must he negative. Of course there is some freedom on which particle is label led c or d. T h e motivation for a specific choice is to try and ensure that the Mandelstam variables naturally arise in the propagators: .s in annihilation processes. for example. e + e ~ — qq. and I in scattering processes, for example. / q DIS. One refers to s-. I- or uchannol contributions. In terms of the Mandelstam variables, with I -= (
According to cqn (B. I), to obtain a (differential) cross section we need to include a Mux factor for the incoming particles and a (differential) phase space factor for the outgoing particles. In the uiassless-lepton limit the llttx factor, cqn (B.5), is given by I / ( 2 s ) . T h e evaluation of two-body. Lorentz invariant phase space, II 2 in cqn (B.(>). is relatively straightforward and gives
d-M,., = ± % ! d 8~ y/.S 1
(1/
cos 0 ' « * 2 77 (\
M5? | p f „ l \ / 5 2 -
(:j.9(l) '
T h e first expression for the two-body phase space element is appropriate for a description of the collision in the C.o.M. frame. In the second expression it is
li.s
I'lIK TIIEOKY <>l (¿CI)
given in ternis of the Mnudclstam variable I. After integration over I lie a/hnulhnl angle and introducing the frequently occuring line structure eiuistaiit n (3.91)
Iff
the final result becomes f>
d<7
rn/-'q~Ar
dcosfl*
2s
d/
.s
(2 - / <
2
cos2«')/*•
+
(3.92) .s-2
If both Icpton beams had a transverse polarization, then the matrix element cqn (3.8(i) would ac<|uire a non-trivial 0 dependence. Equation (3.92) is easily integrated to give the lowest, order expression for the total cross section for e + e ~ —• qq,
"o
2tto2 — "
-
(
f
\ J Vt *
Itto-',, ..
80.8nbGeV2 „ . * <»r
(3.93)
The approximation holds well above threshold. \ / s 2> 2///,,. equivalent to —• 1. It should be noted that the fact that the total cross section depends only on .s and //!,, is a result of the quarks (and leptons) having no sub-structure, that is. they are point like. Furthermore, t he polar angle dependence in cqn (3.86) is a direct consequence of the quarks (and leptons) having spin 1/2. Given (pseudo-) vector boson exchange the lepton and quark spins like to align at the two vertices: a positive (negative) helicity particle with a negative (positive) In-licit,y antiparticle. Thus, we have an initial sp'ui-1 state annihilating into a linal spin-1 state aligned at an angle 0 ' to the initial state. T h i s 0 ' dependence in cqn (3.80) is usefully rewritten as i C
.2
° — IX (1 + c o s 0 ' ) 2 + ( l - e o s f l * ) 2 + 8 ^ - 2 sill 2 0' . (I cos ft* .s
(3.94)
T h e first term corresponds to the contributions with a positive (negative) helicity Icpton going to a positive (negative) helicity quark; angular momentum conservation then favours 0' — » 0 over 0* —> TT. Likewise*, the second term corresponds to a positive (negative) helicity Icpton going to a negative (positive) helicity quark, which is favoured when the quark and lepton are antiparallel. T h e last term corresponds t.o a spin zero final state involving a spin-flip, which can only occur for massive quarks. That the first two terms contribute with equal weight is a consequence of Q E D (and Q C D ) being parity conserving. T h e photon couples with equal strength to the left- and right-handed ferinions. T h e weak interaction violates parity conservation and the Z couples differently to left- and right-handed ferinions. This changes the balance between the first, two terms in
I t O H N I I VI' l. C A M M i l . A H O N S < »1' ( ¿ C I ) ( ' l « ) S S SI-:*
I'lONS
(¡'I
i <|ii (3.9-1) mid leads in a term linear ¡11 c o s 0 ' which induces a forward backward iiHvnunetry. A similar o i l e d can be obtained bv polarizing o n e or more of the Ici'inions. I'o obtain similar results for the processes / <| -» C~q and qq —> . shown In Fig. 3.10. o n e approach is to calculate the matrix elonieni in exactly the inline manner as above. An elegant alternative is t o take the previous result for Al qq) and use crossing s y m m e t r y . T h e basic idea is to s w a p particles between the initial and linal s t a t e s . A n e x a m p l e is given by
M"h-Cd(p<,.ph.1>r.p,i) =>
\Mub~'c',{s,
^ M""~bC(pn.-p,l.
I. w)| 2 = \M'"'-'"
-p,„p<:)
(I. 'I. .s-)|2 .
(3.95)
When ferinions are involved the equality of the a m p l i t u d e s is m o d u l o an unobrtervable phase. Since the physical regions for the Mandelstain variables in the crossed and uncrossed process d o not overlap, the a r g u m e n t s of the second amplitude have to be analytically continued. T h a t this is possible places powerful const raints o n t he allowed form of the a m p l i t u d e . T h e only real, though minor, complication is to remember that the spin and colour averages for the initial state may need to be changed as appropriate. Using these results and neglecting masses we quickly obtain
J | > f ( r - q ^ r q ) |
and
2
=
( o % )
2
2
2
^
J>W(qq - r*+)|* = (,;
.
(:$.<J(i)
Observe that the result for qq • f.~f+ is essentially the s a m e as for f ~ f + • <|<|. except for the extra factor 1/JV 2 d u e to the average over the colours of the incoming quarks. In s i t u a t i o n s where a number of processes are related to o n e anot her be crossing s y m m e t r i e s it is c o m m o n practice to only q u o t e one matrix element (squared) and expect the reader to derive the o t h e r s using eqn (3.90). Finally, before finishing our discussion of these processes, we return to a very important property of the lepton and liadron tensors defined by eqn (3.85). S u p p o s e that we introduce a polarization vector for the exchanged (off massshell) photon and take f(Q)'1 oc then it. is easily verified that.
Q>'L,IV = 0 = LIIVQ"
and
Q"H„t. = 0 = H„„Q" .
(3.97)
which is t he einbodiinent of e l e c t r o m a g n e t i c g a u g e invariance. A s a conse(|iience of t his result , had w e chosen a g a u g e in which £ / 1, then t he e x t r a t e r m s in the numerator of the photon propagator, which are proportional to Q 1 ', would have given zero cont ribution. It w a s sullicient to use only —//,,„. T h i s is a trivial e x a m p l e of a m o r e general result which s t a t e s t hat the sum of a g a u g e invariant set of a m p l i t u d e s cannot d e p e n d o n the arbit rary g a u g e parameter £. To see why
7(1
llll
I IIKOUY O F
QOI)
a sum of amplit udes must vanish when a photon's polarization vector is replaced by it s four-ninmcntmu consider the Fourier transform of a Range transformation, A" >-. A" + c~x'0"0
=>
e(ky
f(k)" + e~xk"0 .
(3.98)
Since 0 is arbitrary, the scalar product (''M,, is only guaranteed to be gauge invariant, if we require k"M,, = 0. A similar, though more delicate argument holds in the uon-abeliau QCD. For more explanations see eqn (3.118) and the discussion in Section 3.3.3.1. Constraints such as this significantly limit the possible tensor structures of amplitudes and provide very useful checks on the intermediate stages of a calculation. 3.3.2
e + e _ annihilation
lo i/uarks at
O(al)
We now consider the leading, tree-level QCD correction lo the p r o c e s s e + e ~ — qq in which a glnon is radiated from either the quark or the antiquark. e + e ~ — qqg. Tin; two Feyninan diagrams are shown in Fig. 3.11.
e+
(
l
e+
q-./
FlC. 3 . 1 1 . The leading, tree level QCD corrections to c + e ~ —» qq. The moiuent inn shown at. the internal (|tiark flows in t he direction of t he fcrmion number, as indicated by the arrow. Concentrating on the hadrouic part of the amplitude, and again assuming only photon exchange, t he matrix element is
W + // + »'<.)
, A -(ti + fi) ; 7/1 + 7/1 , - , V,
M„ = i (•<•„<),Tl'j ft(q)
+ »'.,_
•) la »(
•
(3.99) The minus sign in the antiquark propagator arises because the moiueiitiun is Mowing in the opposite direction to the fcrmion number. Before proceeding to evaluate \ M \ 2 it is instructive to verify electromagnetic gauge invariance. Using Qi' = qi' ¿ji' -(- yi' and neglecting the constant overall factors we have l
1
M„Q"
OC
a(q)
id + it - IIIq
= »(
(ii + t + i ) + (4 + i + 4) ( - W + if) - » ' « , ) i
1
(fi +
4-»«.,)
W + 4 - »'<.) -('»..
+ it i 4)
(4 + Si + '»<.)
IU >UN I . H V I S I . < 'Al,< 'I l | . A I'M > N K < )!• (J< 'I M It« ) S S S i :« T l < > N S
"(
71
(3.100)
we wrote the propagators as inverses, using (/> - iii)(/> + m) = (¡r mill exploited I lie Dirae equation HI-IT
(li " »'.,)"(
nr) 1.
(f? + '»<,)«(
m n
I'liiis. we eonfirm that our expression for the amplitude is gauge invariant , provided we include the contributions from both diagrams. At this point, we set t he (¡nark masses to zero, as t hey serve only to complicate our calculations. This is a good approximation for the light quarks. Since we already know t he Icpton tensor we only need to evaluate the hadroii tensor. MtiaMlTt-(c,r({(j)T spins
=
M„„Ml°
.
(3.102)
spins
l i n e . we have expressed j\4„ from eqn (.'i.i)i)) as M/lof(ff)t{7
and used
pol
for the gluon's polarization tensor. We shall return to a consideration of this expression later. We start by considering the first diagram in Fig. 3.11. in which I lie glnon is emitted by the quark, which yields
(-'i •.'/)- X ]
M
<\M\
*
Tr
MT» if} + rfh»fo«(f} + Oh"}
spins
= +2Ti-m+iihMfi
i
-m
= +2Tr{#7 = + 2 • 2(
(3.104)
Mere, we used some tricks from oqn (3.83) to speed up the evaluation: 2(f. M — qg branching iu isolation. This can be achieved by expressing the numerator of the quark propagator. /». as a sum over bispinors. ! / ( / > . s ) v ( p . s). c.f. eqn (3.20). and pickinn <»ut t he term ii(rj)/'(jf)u(p). This expression has mass dimension one, and since t he only relevant quantity carrying mass dimensions is the virtuality of the
i in 1111:0m 01 gci)
72
propagator. wo have that ii must he proportional t o \J2
-
cos0 < l g ) .
(3.105)
it, is clear that this inverse propogator t e n d s to zero for vanishing gluon energy, 1. when the EK — 0. known as a soft singularity, and for a masslcss quark. opening angle vanishes, —» 0. known as a collinear or mass singularity. proceeds in t he s a m e manner. T h e result can T h e evaluation of $3 f i | l i l l g he obtained from oqn (3.101) by exchanging (/'' and <]''. T h e more cnmbersoine (•valuation of the cross-term yields (2<,-!,)(2f,
{ m . M , }
Spill*
— l(i [(
2('i,lQ„ +
i 2(q • Q)q„q„
-
2q,,q„) Q2q„q„ .
(3.106)
Combining these results, restoring the constant factor and evaluating the sum over the final s t a t e (¡nark colours, which is conveniently done using oqn ( A . 1 7 ) . Yv{T"T") = C¡. fl,, = Ci-Nc• gives the hadroiuo tensor _
Hcc^u'jCrN,. (
• <))(
• !/)
* { -Q"('/,-'/,
+ q,,
+ (q • Q) \q,iQ» + Q„q„] + (q • 'D [Q,'('/ - '
Q)2 + ('7 • Qf\>)„•' + (<1 • Q)[ (
(»•107)
+ Q„<1>\
+ ('/ •'/ + Q2/2)[q„
-
This can be contracted wit h the loptonic tensor, oqn (3.85). to yield t he following expression for the matrix element, squared:
I',.i'
= »(''
<%,) qsC /-'A, - —
Q-
—
('/".'/)('/ • U)
(3.108) Observe that only the first two terms in eqn (3.107) give non-zero contributions as the others are either proportional to ' and vanish by gauge invariance or are antisymmetric under // »-> (/. T o obtain the differential cross section eqn (13.-1) we need to include the lln.x factor, which is the s a m e as for e ' e —• qq, and the three-body phase space.
•
It
ii
l,l , v i i , C
11
II
o r
CI
R
ECTI
.1
. in - ii 1 . i . I lic evaluation of the phase space is best carried out in the .ii. . frame and yields I
1
, I
I
,
o
r
1 12 1 1
l r
2tt 1 12 ;r
-
. 1I 11i f t dii ;2 oii
I lere particles I and 2 can be any of the three filial state part icles; for the problem hand 1= . 2 and g is the natural choice. T h e second expression follows alter integrating over the decay plane s orientation and the third because, for example. ; 2 Q~ -I- i i 2 - 2\/Q- h o . Combining t hese results gives t he fully dilferent ial cross section tf
I
Q'2 x d
t
c
osC . s
d
, d E
i - - qf
+
q)2 + ( +
r
Q2
qf
+ (C+
f7)
2
]
{'luWi-o)
f
.
. .11
ften, in practice, we are not interested in the orientation of the decay plane. That the three particles lie in a plane, in the .o. . system, follows trivially from >1 I
( - , r +
r
(
.111
Here m may be safely ignored. This tensor manifestly satisfies e n . and may be thought of as t he spin-averaged polari ation tensor for the exchanged, off mass-shell, vector boson. Re-evaluating ,,,, t hen gives for the spin-averaged cross section for masslcss uar s .2
7 =
/l-'/r dr. dx,,
-
-
(q
g —— l)(q o)
.V? + n
2-
1 -. -,,
1
11 1
7
2
1 - . ;,,
.
.112
In the second form we have introduced the variables . ,. e ual to the energy fraction of t he t h particle in the C.o. . frame, defined bv
and such that .i , .r,, I .rR 2. ro ected onto the plane Fig. .12 . a alit plot, the allowed phase space lies in the upper-right triangle of the
71
IMI.
R I I K O U Y O K (.¿CI)
Fit!. 3 . 1 2 . A Dalit/, plot showing the allowed region of t he ./ ,, - r,, plane for a 7 —> ./•,, = 1 and 0 0 = > j:,, = 1. their intersection marks the position of t he soft singularity. .rK —< 0 = > x ( , = 1 = .»:,,. Also shown is the boundary line separating t he two- and three-jet regions based on the criterion mnxf.i:,,../ - ,,.jr R } < 7".
./•,, = ;(). 1] j , , = ¡0. l! s(|ttare. with lines of constant x R running parallel to the diagonal edge, where a:R = 1 . In eqn (3.112) we see very clearly the singularity structure for gluon radiation. If 0, m —» 0 then 2
(|(j —» gg and the niiut/c. invariant
QCD
Lai/mui/ian
We will now consider a pure Q C D process, qq —> gg. at leading order with, for convenience, niassless <|ilarks. Our approach will be to try to generalize the well understood theory o f a b e l i a n (¿ED to describe the non-abelian theory of (¿CD.
Fit:. 3 . 1 3 . T h e three leading order Feynman diagrams for qq — gg.
IN ) I ( N U ; \ I I C A I . C H I
\ HONS
or <
N
< moss
SI.»
I IONS
By direct analogy to 111 » - process c |< | • 77 we arc quickly led to the first, two fcynumn diagrams shown in Fig. 3.13. Their matrix elements arc given by (M,
-I• M„){c-(,„).(•
(,,2))
-Ì(JÌHQ)
"('I) • (3.111)
()liserve I hat we have included two non-commnt ing colour matrices at t lie «¡nark gluon vertices. This has significant consequences for gauge invariance. Replacing '(lli) hv = <1 + <] — n 1. c.f. eqn (3.100), we easily find (M,
+M„)[E'{(,X).<,2)
=
-I!,L(TBT"-T«T%MR(FHM
= -aifahcnmfUn)"('/) -
(3.115)
',ip I hat. unlike the ease of abelian C^FD. gauge invariance appears to be violated! Ilii" result is unaffected by including non-zero quark masses. Indeed this would lie the case if there were no other diagrams contributing. However, since gluons carry colour charge, we might anticipate t he existence of a t riple-glnon vertex giving t he third diagram in Fig. 3.13. In fact. looking at eqn (3.115). we see that the remainder has the form of a quark gluon vertex proportional to — i f/ s Tj] times a new factor proportional to +<7«/"'" . that is. proportional to the appropriate colour matrix for an adjoint representation gluon. T h a t the gluon should be placed in the adjoint representation makes sense from the group theoretical point 11I view, because when coupled to a particle in the representation / i only the adjoint representation, which for SU(3) is an octet. is guaranteed to be contained within the tensor product [{<%]{. Thus, only octet gluons can directly couple to particles from any other representation. The properties of this new triple-glnon vertex +{lsf",'r^l„.a(.!/i.i/2-!):i)with h„, \> and
(3.110)
as also given by the Fcynman rules iu Appendix B. Taking over t he form of the (Feyninan gauge) photon propagator for the gluon we have M ,(c'(f/i).c'
(f,2))
7ti
».(«/) • -\
IIIK
I IIKOltY OK «¿CD
• «(
x +gj"br
(.'/i +
X ['/„.,(//1 - iJ2)r + '/cr(.'Vl +
2), - i?„,,(2(/
1
(«• H7) -f (/,,)„! • e * (i/1)"('{)" .
and again replacing <'(
Ms
( < • ( / / ! )..'/*)
•Vt ' ' ( i l l ) 2gi • g>
I i / ; / " ' " 7];,-,(,;)
n(,,).
(3.1 IS)
(¡¡veil the choice of unit, numerical coefficient in cqn (3.1 l(i) this exactly restores gauge iuvariance. provided the first gluon is physical, that is. g\ • f(g\) (I.
scattering Having heen led t o introduce a t.riple-gluon coupling we ought to consider also the process g g —> gg, which now can proceed via the first three diagrams in Fig. 3.I I. If we test gauge iuvariance and replace f'(gi)'1 by //,'. the sum of the first, three diagrams in Fig. 3.11 gives the following contribution:
(M, + = -\<£[
Mu+Ms){e(ai).f(!l2).f'iO:i),H\)
+ /"'" rl'{u
• r:j
+ r"f<"»(fl
•cj, c a - ^ - f ,
+ f"'l, f'"r(e,
• 62 4 • <71 - ft • <5 ea • ill)]
• 94 - ft • HI f2 • <$) -c-if^-.j,) (3.119)
Here, we assume on mass-shell (g'f = (1). physical (g, • r(g,) = 0) external gluons and employ the .lacobi identity. c(|n (A.12). Once more, we find problems with gauge iuvariance for this subset of diagrams which can be remedied by int roducing a dimensionless. fully symmetric quart.ic-gluon vertex. T h e exact form can be read off directly from cqn (3.119) and can be found in Appendix B. T h i s vertex has exactly the same structure as that derived from the gauge-kinetic term in the Yang Mills ((¿CO) Lagrangian. What is more, these triple- and quartie-gluou vert ices are sufficient to guarantee the gauge iuvariance of all QOD processes to all orders no other gluon self-vertices are needed. 3.3.3.1 Physical states and ghosts In cqn (3.11 <S) we demonstrated effectively the gauge iuvariance of the qq —> gg amplitude provided that both gluons have
U O H N L E V E L ('A I CHI
VI I O N S O K Q C I ) C R O S S
SECTIONS
physical polarizat ions. Thin raises I lie issue of how to treat, the polarization tensor for the gluon: is the (¿I'll) form, eqn (3.1(1:5). really adequate for QCDY for a vector particle of moment um /.•'' its polarization vectors must satisfy f(A- s) • f'(/.:: s') =
A- • r(A-) = 0 .
and
(3.120)
I'he lirst equation imposes ortho-normality on the basis states whilst the second equation requires that, they are orthogonal to the particle's direction of motion. Now, because the glnon (or photon) is niassless it only has two physical polarization states whereas the second equation only imposes one constraint on the four components of a polarization vector. Thus, we require an addit ional constraint, which can be taken to be n-c = 0. where n is any four-vector subject, to n• A- 0. As the physical polarization sum. I c a n be constructed only from i]/n/. kf[ and II„. is required to satisfy k"T,„, = K"T,„, = II''Tl,„ = v"Tl„, — 0 and have trace '/"'' = —2, it. is straightforward to show that it must have the form
C
( A V (* )>' =
4
^
" („ . /,)-• '
(3.121)
pliys
see Ex. (3-15). This is precisely the form of the numerator of the gluon propagator in a physical gauge. There is quite some freedom in the choice of ?»'' which we can use to our advantage. In particular, having i r 0 simplifies things. Given the correct polarization tensor, eqn (3.121), you might, ask why this expression is not. used in Q E D calculations, since the arguments leading to its construction apply equally well to photons and gluons. T h e difference lies in the realizat ion of gauge invariance. In Q E D we have (3.122) which for a physical photon is independent of the polarization of all ot her photons. In Q C D all other gltions also had to be physical. Thus, the 'extra* terms in eqn (3.121) which subtract out the imphysical longitudinal and scalar polarizations and are proportional to A- give vanishing contributions and can be safely dropped. In Q E D . we can use i)lt„ for the polarization sum. Likewise we can use i/i,,, in Q C D . provided only a single external gluon is present. If two or more external gluons are present using - , will leave unwanted extra contributions. For example, using the full expression for the qq —> gg amplitude the additional contribution is given by:
M.rMl
T
.^\-n"0'} x (-#*)
-
- i f _
rr'
+
+
(<¡"«1 I "1
»-
n2
»
+ " W i ) _ ;;;.>
1I > U2
Ul'f' 2
11
(l>2 ' H2)2 \ J
I I I I : I I I K O K Y O K Q< I)
78
i.'/.:/"'" Tji
2Hi • !U
«"'('/)}>,«(
(3.123)
Here, svc see that any dependence 011 the arbitrary four-vectors n\ and n > \anishcs. whicli is in fact required l»y gauge iuvarianee. Tlie final result also appears lo he asymmetric. However, this is illusory: if we replace .then the bracketed term vanishes and we are left with the s a m e expression except thai now {¡., appears, and /"'"' becomes /'""'. that is. the result really is 1 2 symmetric. T h e obvious way in which to avoid the contribution due to unphysical glnon polarizations, such as in eqn (3.123). is to always use eqn (3.121) for external gluons. However, this is cumbersome and an alternative is available. We r e a m eqn (3.103) for tin- polarization sum and add yet another diagram to thos- in Kig. 3.13 whose contribution is designed to cancel exact ly t he unwanted contribution. Looking at eqn (3.123) the structure is t hat of a qqg vertex, followed by a glnon propagator and finally a term proportional t o ~
FlC. 3 . 1 5 . T w o of the diagrams contributing to the glnon self-energv At this point the rôle of the unphysical ghost fields appears to be only to facilitate a trick intended to simplify t he calculat ion of tree-level amplitudes. T h e situation is more subtle when loop diagrams occur, such as in Fig. 3.15. These diagrams can be viewed as either higher order cont ributions to the amplittice for qq scattering or. if the diagrams are 'cut' through the loop, as contribuì io is to the amplitude squared. M (left-hand side) x A4 1 (right-hand side), for qq — gg. This dual interpretation can be made more formal. In the first, interpret it ion there exists a region of the loop momentum integral (see Section 3.1 for an elaboration) in which both internal particles are real, that is. on their positive mass-shells, and generate an imaginary part in the complex amplitude. 11 t he
UOUN
IJ A I I . C A I C H I . \ l I O N S O l - ( ¿ C I ) C R O S S
SECTIONS
7!»
,c<(ni(| interpretation this same integral corresponds to the phase-space integral appearing in the total cross section for the production of a pair of real gltions. i I his non-linear relationship between S-niatrix elements is in essence the optical theorem.) T h e significance of this equivalence is that we must restrict the internal •luons in any loop to only physical polarization states or violate mutarity. This can be achieved in two ways. Eit her, use a physical gauge, so called because only physical polarizations propagate for an on mass-shell gluon. Or use a covariant gauge and whenever a gluon loop occurs add the contribution from a ghost loop, remembering to include the extra minus sign, thereby ensuring the removal of any non-physical contributions. 3.3. I
The evaluation
of colour
factors
I'he evaluation of the colour factors which occur in the expressions we will encounter for matrix elements can always be found using essentially two results, for any particular sub-amplitude the colour factor consists of an ordered product of T" and f"llr terms. T h e lat ter can be eliminated by applying the following Identity, derived in Appendix A. fabc
= ~—Ti{T"TbTc
- TcTbT"}
,
7>
(3.124)
to the fundamental representation. After applying eqn (3.121) we are left with a string of T" matrices. Now all internal indices are already summed over and once the amplitude is squared and colour averaged so are all external indices. Thus, t lie mat rices can be paired as T" T£t T*Tbx • • •. allowing the completeness relation to be substituted, (3.125) finally, by carefully contract ing the colour ¿-functions, a pure number, the colour factor, will remain. We illustrate this approach using the process qq • gg. Schematically, the amplitude may be written M = TaTbMt = T"Th(M,
-I- T'T"M„
+ i
+ A 4 , ) + ThT"(M„
f"'"TMs - At,) .
(3.120)
The first two terms are associated with the two orderings of the double breinsst rahhmg contribuì ion. T h e third term, which has been eliminated eit her directly using eqn (A.10) or less directly using eqn (3.124) and eqn (A.7). is associated with splitting of a radiated gluon via the triple-ghion vertex. T h e T"T'' and
ab tjkl
I III
ijkl
I I I K O H Y ()!•' ( ¿ C I )
"
b
=
-
= NrCfi
X TY
(SJRFKK -
-^-¿JK'HI^
.
(3.127)
After rather more work the cross-term can lie evaluated similarly to obtain the colour factor Tv{TT'T"T'1}
= NcCr
(c
r
" ^r) =
-
(3-128)
From these two results we can deduce that the colour factor associated with the |.M;j| 2 term is given by NECYC,\. Thus, we learn that \ M i s proportional to the number of qnark colours. JVC; the radiation of a gluon olf a(n anti)(|iiark is proportional to C¡-: whilst radiation olf a gluon is proportional to C . Also, we see a generic behaviour, the interference between two colour Hows, here T"T and ThT". is suppressed by a typical factor N~2 compared to the direct contributions. Again, in practice, a number of tricks may be applicable in special cases to speed up colour factor calculations. In the case of eqn (3.127) we could use eqn (A.17). ^2 aj T"jT" k . = C/.<>,;.. to immediately obtain the result. For eqn (3.128) one can substitute T"Tb = (ThTa -I- \ f',bcTr). The first term has just been evaluated. For the second term one can write i j-"'" Xr {J"''/''"/'"} = i /"'""Tr { r f< 'fhT" - TCT"T1'} = T,,f"'"f"hr/2
/2 (3.121))
= TFCA(N't - l)/2 , where we used the antisymmetry o f / " ' " , eqn (3.121), and eqn (A.17). These two terms when combined give the quoted result. However, the approach discussed first is guaranteed to work in all cases and is ideally suited to being done by computer algebra techniques, thereby reducing the risk of error and the tedium.
3.4
Ultraviolet divergences and renorinalization
When a Feynman diagram involves a closed loop, the momentum conserving 6functions at the vertices prove insufficient to specify fully the momentum of the part icles in t he loop an integral over a loop moment um remains. For example, in a self-energy diagram an integral of the following form may be encountered:
J
d
4
k
.T
A
i
k
(
).
+
J
-
1
m
A
A T A
f
w
I
,
A
(
0
h uncaring lie integral by mi ultraviolet eut-olf, . we sec that the integral is logarithmically divergent. Renormali atiou is the t reatment of such divergences which are associated wit h the high fre uency short distance components of the liclds. In essence. the procedure first involves using a regulator to artificially lender all integrals finite, so that they can lie safely manipulated. econd, new t ci nis are introduced into the theory in such a way that all the divergences cancel between the contributions from the bare agrangian and the new countertenns when the regulator is removed. ignificantly, these additional terms must have exactly t he same structure as the original agrangian. Finally, the finite matrix elements can be compared to experimentally measured numbers and. after iixing any free parameters, higher order predictions can be made. e will demonstrate this procedure at one-loop. ore details can be found in many of the standard held theory texts, such as Ilatnond. 1 or es in and chroeder, lil -ri . nfortunately, the crude cut-off used to ma e e n . .1 finite is not compatible with either orent or gauge invariance. Technically spea ing, the ard Identities are violated. However, it is easy to see that if the -integral involves fewer dimensions, then the integral again becomes finite but importantly now no longer violates the two symmetries. imensional regulari ation is the preferred choice in p C calculations. T h e method of dimensional regulari ation and the standard procedures for dealing with one loop integrals are described in ppendix C. .1.1
Sclf-v.nc.rfjH and vertex
cone.ctions
rmed with the nowledge of how to use dimensional regulari ation to calculate one-loop amplitudes, we now s etch the results for the uar and gliion self-energies and uar glnon and triplc-ghion vertices. T h e s e are the core corrections to the propagators and couplings in which all the external propagators are amputated . e shall wor iu a covariant gauge wit h arbit rary gauge parameter I. This will be followed by a discussion of how reuormali atiou handles I lie divergences which we isolate.
F i e . .1 i. T h e t hree tadpole-type diagrams and the uar glnon loop diagram which contribute to the uar self-energy at leading non-trivial order. os
2
I III . I II
F
CI
i a H folic diagrams, shown in Fig. . Hi. appear to contribute to the iiar self-energy. However, tin three tadpole diagrams give ero contributions. This is seen most easily by considering the diagram s colour factors. Tr and ,, . for the iiar and both the gluoii and ghost tadpoles. respectively. This leaves only the fourth diagram, which apart from a group theoretical factor, is the same as in the abelian theory F . The one-loop integral yields i\nk
+ (1-0<\nk = -i sfi')
'< (? +/>+
f -
CFSi
J
(2 (27,
-
m
pY -
}
.1 1
k k ' ]k
2
k-
This expression should be familiar from the discussion in coincides with e n C .l for 1. The full result, is -i E
i
CVrf,,
ppendix C and indeed
tf
-i> in
1
,o
2
2
2
1 -
e
r
,
d
l
-
,l o .1 2
.
o l o r . with (i\) = \in2 T h e calculation of the gluon self-energy proceeds iu a similar manner. t i s the relevant Feynman diagrams are shown in Fig. .1 . gain the four tadpole diagrams do not contribute. The first three diagrams are the same as for t he uar propagator and vanish for the same reasons, the fourth diagram is more interesting as its colour factor is non-vanishing. In dimensional regulari ation it is proportional to a momentum integral which possesses no intrinsic scale.
/
(2n)D
--
s a result there is no value of for which we can evaluate the integral. If D 2 then it contains an ultraviolet divergence, whilst if D 2 it contains an infrared divergence. In this circumstance we define such an integral to be identically ero Collins. l ) i . To help motivate this choice, if the integral were not ero then it would contribute a mass to the gluon. thereby violat ing gauge invariance. In fact, we could have also applied this argument to the loop integrals arising from the gluon and ghost tadpoles. The uar loop contribution, for a single uar flavour, is given by
- i n
= - (
h sll'fTv{T T
f
v,
Tr I
2
w2
l ,a
-
111 r n A V i m . 1
I I>IVKK<;I-;NM:S A N D
KKNOHMAUZATION
l ie. 3 . 1 7 . T h e four tadpole-type diagrams and I lie quark, gluon and (in a invariant gauge) its associated ghost loop diagrams, which contribute to the glitoti self-energy at leading non-trivial order. C?(n s )
(3. KM)
I'lic only subtlety in this contribution to the gluon self-energv is the overall 11 ti in is sign for the closed quark loop, otherwise the evaluation follows the earlier pattern and. if anything, is more st raiglit forward. Modulo the coupling and colour factor the result is the familiar one from (¿ED. In the uon-abeliau Q C D , the new coiitrihutions come from the gluon and ghost loops. T h e gluon loop is given by
¡11 ( / >'kk ){£ =
d"A( i l s l' ' -"""-""" f f . a h r o, ^ I I ro*r\t) X ['/"„(A- + 2 p ) a -
'/„/J('-/.• •»- !>)" -I- » / / ( A : -
/>)„]
(A"+ v T i k + v V 2 (A" + p)F
( ' r V
- o
(1
p)2
(A- +
x ['/"„(A" " p)i - 'M-r(2A" + ,,)" + ,h"(k y f
A:2 V''
— (I —
''
0
A-A,
,
I"»
+—p
+ 2!/>)*]
2
„,/
v'
6(1 - 0
(
-
1—3 4-
—p'P
A,
-
In
„
In ( J p )
- 2^
-
/,"/>")
I in: i illoitv OK <,>('!>
XI
i »(I
(3.135)
-o-o>->r
T h e overall factor of o n e half is to account for the diagram's s y m m e t r y factor. Despite its a p p e a r a n c e the calculation is tedious rather than dillicult a n d . given their relative simplicity, we have carried out the F e y m n a n parameter integrals. T h e g h o s t contribution is rather less involved and is given by
•n/ V"
- ' " K , '
/
= -('J*''
f
> SnrdSu.i
j
(A + 1 > ) n
f
~
{ 2 n ) l )
{ k +
"
]))H.-2
(3.13(5) Note that there is a n overall minus sign d u e to the ferinionic nature of t h e g h o s t fields. Referring back to the quark loop contribution to the gluon self-energy. given by e<|n (3.134), we see that it has a transverse tensor structure. /';,11,'J,',(/') = 0 = 1>>Ali,'.','(p). equivalent to 11,'^ oc ( p f y ' " ~ !>"}>")• T h i s structure is not s h o w n by either t he pure gluon. eqn (3.135). or ghost, eqn (3.130), contributions separately. However, t hey naturally form a set which w h e n added together gives - ¡ n ( « , „
1
-
= - i ( n s i + n!;;;,)
<3.137)
- *Y) •
h
A s we shall see. this is an important c o n s e q u e n c e of g a u g e invariance. A result all the m o r e surprising since our Lagrangian c o n t a i n s an explicit g a u g e breaking term and the "tree-lever propagator is not, in general, transverse.
'I-J
F i d . 3 . 1 8 . T h e two diagrams contributing to the quark gluon vertex at C ' ( o s ) T h e calculat ion of o n e - l o o p vertex corrections is more involved and results in rather c o m p l i c a t e d expressions which we c h o o s e not t o give in full: details can
VI
II
I.
I IHV
I
I
G
H
I.I
I
nr.
l
I i 2 11-
2, T .
n
f r
I li;. . .1 . T h e four basic diagrams contributing to the triple-gluon vertex at o . o t e that two further permutations of the first, diagram exist..
he found in the literature. Instead we only show, the relatively straightforward, divergent parts of the vertex corrections. Figure .f shows the two diagrams which contribute to the uar gluon vertex at C n . T h e first, is essentially the ime as in E . whilst the second only arises in a non-abelian theory such as CI . T h e result, for t he ultraviolet, divergent part of the vertex correction is
IT .
-wr
g
i
C
o -
1
f
1
CA
-l
,
u.v.
t
finite
.1 The first, term c o m e s from t h e E -li e graph, the second c o m e s from the non-ahelian graph, ure .1 s h o w s the four basic diagrams which vertex at o s . T h e result for the ultraviolet given by
, ( .,
T
=
( I
ITr I I it first, term is given by the lit E . this term would vanish which s t a t e s that any uar loop This is essentially because E s
) 1
+
T
l,
a
b
proportional to T r T , whilst proportional to Tl'f(li)CTc. Figcontribute to the triplc-gluon divergent part of the vertex is
(
2.
,
2)
(2 .V
linit
.
,
,,)
.1
fermions which contribute to the uar loop, according to Furry s theorem Furry. 1 . oined to an odd number of p h o t o n s vanishes. charge con ugation s y m m e t r y guarantees t hat
S(i
I III'. I I I F . O U Y O F
QOI)
the contributions from a <|uark and an antiquark going round the loop in the opposite direction exactly cancel. Furry's theorem does not ensure the vanishing of this diagram because in QC'D the order of the colour factors is important: Tr{T°TlTc} ^ Tr { T c T b T " } so that the <|iiark and antic|nark contributions do not cancel. T h e second term in eqn (.'{.13!)) arises from the pure gauge field (glnon and ghost) diagrams. As ¡1 result of these and other somewhat involved calculations we are able to write the one-loop corrections in terms of a divergent (in the t —> 0 limit) and a finite piece. Significantly, the coefficients of the divergent pieces follow the pattern established by the tree-level terms already present in the Lagrangian density no new divergent interactions occur. This will prove to be of crucial importance in our approach to renorinali/.ation. It is also a very non-trivial statement. For example, a term of the form I n ( p 2 / f i ' ) / f appearing in a correction would certainly invalidate this statement. Fortunately, a theorem due to S. Weinberg states that the divergences may only be proportional to a polynomial in the extern).I momenta and particle masses, of the order of the diagram's degree of divergence (Weinberg, 1900: 'tHooft., 1973: Weinberg. 19736). Before proceeding to consider the renorinali/ation of the theory it is useful to collect our results so far. Beginning with the quark propagator, the sum of t he tie:- and one-loop contributions may be written heuristicallv as SV + 6V(-i£)S„
+
O(n'i)
= 5V + 6 V ( - i E ) S > + S > ( - i S ) 6 > ( - i E ) 6 > + • • • = (Sp1
+ IE)"~' .
(3. MO)
lit •re B/.-1 = — i (/i — III) is t he inverse ol the quark propagator and X is the quark self-energy. This should be calculated front one-particle irreducible diagrams, that i;- diagrams which cannot be split in two by cutting a single propagator, so as to avoid double counting in the series. Now the interpretation of the first expression is somewhat confused since its second term contains a double pole. 1 /(l>~ ~ in')'• due to the two quark propagators. In order to obtain an expression containing only a simple pole we follow Dyson and sum the infinite set of terms given in the second line. T h e expression in the third line gives the sum of this geometric series. Even though .S'/.- is a 1 x I spinor matrix (and S n . below, is a two-index Lorent./. tensor), the usual trick of considering the difference of the geometric series and .S'/.-( i E ) times the series allows the sum to be calculated. T h e only proviso being that a little care is taken over the order of the noncommuting terms. Using the result eqn (3.132), the explicit expression for the summed (inverse) propagator is
,{,(,+
A. + «;£])-,„(. + l c , [ p OA. + Si;£]) + C,,;,}-'
(3.1-11) Similar considerations apply to the glnon propagator which can be summed in the same fashion to give
111.11< A V K >1.1'. I D I V K K O l î N O K S A N D
S„ I S„(
i ll).S'„ I Sn(
KUNOKMAI.IZATION
i ll).S'/((
i
I1)6'/J
87
I
= ((^'+¡11)-']"" - i [(1 -f I I 0 ) ( i r > r - i > ' , i > " ) + r , i > , , i > " } ~ ] 2
V
—(V (1 + Ho) V '
-I-
.P"P"
^F-) P J
(3.142)
Hero .s'„ is the inverse of the gluon propagator, as found in eqn (3.17). and II llii(/r//"" — I>''I>") is the gluon self-energy. We do not show the diagonal colour matrix. Observe that the 'gauge fixing", longitudinal, term is unaffected l»y tin- corrections which remain orthogonal to it to all orders. Using the results ni cqiis (3.131) and (3.137) we have the explicit expression I 1 IM/O = I +
ZTfu<
—
1 c
*
A
< + fhihe) +
) • (3.143)
Mere, we allow for n / flavours of quarks in the loop. The vertex corrections are iilinpler to treat. The sum of the tree- and one-loop level contributions to the (|liark gluon vertex is \
{l + ^
(
t c
F
+
£ ± * > c
A
(3. + fi'tihe) + O ( oÎ ) } - (3.1-14)
Whilst the similar stun for the triple-gluon vertex is i .'/,/<* fabc Wr(
- <JlT + >i"(u\ + 2!,-.)" - v"T(2
(3.145)
In the next section we discuss how to obtain physically meaningful results from these divergent expressions. 3.-I.2
Rc.normuUzatiov
The basic analysis that, led to eqn (3.132) can he applied to all the loop integrals encountered in pQCD. The loop integrals are first calculated in I) dimensions, using analytic continuation, and then expanded in < = (4 — D)/2. The ult raviolet divergences manifest, themselves as poles in 1/7 (Speer. l!)7-l: Breitenlohnerand Maisou. 1!)77). An AMoop amplitude has the Laurent expansion
(3.1 If.) m=0
I he eoeflieients {C„ } depend on combinat ions of the external moment a, typically arising via integrals over Keynman parameters, and an arbitrary mass //. Clearly, t he physical limit of an expression such as eqn (3.14(5) is not well defined. To
HN
llll
r i l U O H Y ()!•' Q C I )
make progress we must find a method of removing t.lie I¡, poles, tlius allowing the D — I (< —• 0) limit to be taken: this is renorinalizat on. T h e procedure we adopt is a pragmatic one. Using the original Lagraugian's Fcvnmnn rules, the divergent diagrams are identified and •waluated. T h e n knowing which interactions contain divergences, supplementary Feynman rules are added t o the theory, one for each divergent interaction. These have coefficients that are carefully chosen so that they completely cancel the divergent 1/f poles generated by the original terms. In essence, tin' Lagrangian is supplemented by a counterterm Lagrangian which, treated as a perturbation, generates the new interactions necessary to render the theory finite. Schematically, this can lie written as
C
» - I T nor IN
(1)
,2)
W
=£+£ — *- • " - r o n n l i - r +£ ' *-«!omili-r ~+C "-COMMIT
1
+•••
= £ + ¿counter •
(3.147)
The (9(o J term. £,(.,',I,IlU.r. is constructed to cancel the one-loop divergences generated by the original Lagrangian. C. T h e <3(n 2 ) term. i,1.",!,,,,,,,.. is constructed t.o cancel the 'two-loop' divergences generated by C + £,V,Jinl<.r etc. Referring back to eqn (3.132) we see that two divergences occur in the quark propagator. One is associated with the quark's kinetic term, -x f> — r i < f h \ and one with its mass term, -x in <—• met.". E(]uations (3.13-1) and (3.137) s h o w that another divergence is associated with the transverse part of the gluon's kinetic term, x (//-//"" - p''i>") — {i)"A"" - i)"A"',)i)llA';/. the longitudinal part of the gluon's kinetic energy term remains finite. Equation (3.138) contains a divergence of the same form as the quark gluon vertex, oc T"k")•'' 7 ' " , . / . ^ i . ' ; , . whilst eqn (3.13!)) contains a divergence of the s a m e structure a s the triple-gluon vertex. « /,./«-[»/"7('".4'". Proceeding in this way we find that the form for the C'?(o s ) counterterm Lagrangian. in a general covariant. gauge, is given by C
=
-
s z
Z c
-
+ s z < " (¿y'n^KO,,!,")
-lV6Z\1\ uslt' f„,A»„A';,)A'"'slc"
(3.148)
- <)Z\l! stir'
'/„,„•/„„,
A1'" A' "A]', A], .
Here, we have also added terms to cancel divergences which arise in the corrections to the ghost propagator, the quart.ic-gluon vertex and the ghost gluon vertex. We have chosen not to include a term proportional to ( i V A"t)-', as the longit udinal part of t he gluon propagator receives no corrections. As indicated earlier, in eqn (3.148) each term should be regarded as a perturbation to which new Fevmnan rules can be associated. The coefficients are chosen s o that the s u m of their individual contributions plus those of the corre-
III.TITA\- 101.1:I
DIVI'.iWìKNC.'ios A N I »
KKNOUMAI.IZATION
UH ICHIIIIS frinii olir enrlier calculations, S e d i o l i 3.'1.1, and reforring to the relai ioii111111) bot.wecn the (|iiantnin Lagrangian. cqn (3.21). and the Feynman rulos d e l l v d Ironi il. Appendix 15. we ean infer the values of six of the as K
=
KCVA,
F0\
+
4rr rf
-
A
^ 47T
~
ÓZ{!] , =
^ IT
sz%
=
( z*>«/
- (tCr
-
—-
CA 1 A, + Fa
)
+
A
F
'
Ai"f
^
(3.149)
•ITT
lite linil.e functions, Fv„ F,\. lrAl..,;.. FA> are arbitrary. Only the coellii iritis of the l / r poles arc prescribed. We shall return to the issue of how to 11 loose the /•", s later when we discuss renorinalixation schemes. In eqn (3.148) we have the beginnings of a remarkable result which saves our approach from being merely ml hoc. Comparing cqn (3.21) and eqn (3.1 IS) one is Immediately struck by their similarity: a fact which has been proven t o hold true lo all orders in perturbation theory. We can make this similarity more manifest by rcscaling, or if you will reuormali/ing, the fields, masses and couplings. To do t his, introduce Z = I + 6Z mid
where
SZ
„1/2 V'o - Z J
X = z\/2A>< «1/2
Vn = Z,,
»i
¿Z{
1
' + ¿Z(2)
+ òZw
"'(i = ZmH'Zv sZ
m
+ •••
(3.ir,o)
'"
m
(3.151)
io = ZAi
(Zi=ZA).
I his allows us to write the sum of cqns (3.148) and (3.21), with <js appropriate for D = 4 2e dimensions, as
• //,//' as
I
Ctcnorm = >l>o(ty ~
0)
-' ^ M d M A t i ' A ?
z
iiitr'
^ zi
-J„LJ.,,1, 4-
A[[„A;», .
In a renornializable theory all the ultraviolet divergences arising in loop diagrams can be cancelled by countortonns corresponding t o the linite nuiiiber of
1«!
i HI-
i i n <>IN O K Q O U
interact inns of mass (linionsioii lour or less. By contrast, in a nnn-rcnormaliKahlc theory oount.ortcims corresponding to new interactions must lie added at each new order in perturbation theory. Unfort unatelv, looking at eqn (3.152). we appear to have lost gauge invariance which, ¡is we learnt in Section 3.3.3 for tree-level calculations, requires very particular relationships to hold between the various terms in the Lagrangian. This is potentially a calamitous situation because the formal proof of renorinnlizability relies on the gauge symmetry to guarantee certain relationships amongst the theory's Green's functions. Now. this apparent loss of gauge invariauce already appeared at t he classical level because of the necessity to int roduce a gauge lixing (and ghost) term. Fortunately. Beeehi. Rouet and Stora ( B R S ) (1974) have found a rather unusual, but nonetheless exact, symmetry of the "broken Lagrangian'. Thus, a form of gauge invariauce can be restored with the proviso that all the gauge couplings are equal. This symmet ry t hen allows a number of relationships, known as Slavnov Taylor (Taylor. 1971; Slavuov, 1972) ident ities (or Ward identities in the abelian Q E D case), to be established between the Green's functions of the Yang Mills t heory. A consequence of t hose relat ionships is the requirement, for the following equations to hold:
= z
l
f z
v
z \ z , ,
= % z f
=
=
(3.153)
V
This allows us to introduce a single renorinalization factor Z,, and a unique gauge coupling (¡no in the quark gluon. triple-gluon. quartie-gluon and ghost gluon terms. Thus, the apparent proliferation of couplings in eqn (3.152) is illusory provided that we choose the finite parts of the counterterins in eqn (3.149) so as to respect eqn (3.153). T h e renormali/ation prescription results in a finite Lagrangian. eqn (3.152). whose form is exactly t he same as that of t he original Lagrangian. eqn (3.21). written in terms of rescaled fields Co. .4<» and //a and parameters iy.,n. ntn and £oThese are often referred to as the bare Lagrangian and bare fields and parameters, in essence what we have are a remarkable series of cancellations, for example. i!> = tl'o - (ZlJ
- l)ij>
in = m0 - [Z,„ - l ) m
etc.
(3.154)
Ilere both the bare (|uantities and the corresponding counter terms are divergent but their difference is finite. We now return to the question of how to choose the finite terms. /•",. in eqn (3.149). First, we should not he alarmed by their arbitrariness. This rellect.s nothing more than the need to experimentally measure the actual masses and couplings, something which would lie t rue of any t heory, irrespective of the need for renormali/ation. Actually this cannot lix the wavefunction renonnali/at ions. Z,.. Z..\ and Z,f. as they do not appear iu physical quantities, though this also
Ill.'l It A V I O l . l ' I I H V I .IM ¡ U N C I 'S A N I > K K N O H M A U / . A ' I
ION
means that we could live williout tlieni. T h e dilferent. clioiccs for the /•', fiincI Inn - constitute dill'eront renoriiiali/ation schemes. T h e most important point lo remember with choosing a reiiormali/ation scheme is to use it consistently throughout the calculation so as not t o spoil the critical relationships between (¡icon's functions. I'crimps the simplest s c h e m e is the minimal subtraction, or MS. s c h e m e ( t l l i i o f t , 1973) in which the couiiLcrtcrms only cancel the \./( polos. A variiiiil ol this is the modified minimal subtraction, or MS. scheme (Bardeen r.t. ill.. 11(78) in which the countcrterms cancel the full A , = l / ( + In(lff) — 7 u pieces. In eqn (.11 1!)) this a m o u n t s t o setting F, = 0 in all the expressions. T h e diiforence between the two schenies is a difference in the finite parts proportional to hit Iff) 7k = 1.954. equivalent to replacing the arbitrary //" by ft2 = •l-//-/e"" . I bus. in I he MS scheme a potentially large contribution to radiat ive correct ions i also removed, thereby aiding the convergence of the pert.ui'bative expansion. Despite iis s o m e w h a t abstract nature the MS scheme's simplicity makes it the must popular o n e for p Q C D calculations. A less manifest advantage of t he scheme r. I lie fact, that the dimensionless Z, do not. depend on the combination m / j i . As we shall see. this mass independence simplifies tin- discussion of the ronormall/ation group equations ( I l G E s ) . That t his holds t o all orders can bo seen by the following heuristic argument. T h e counterterins are constructed to have just, t he bare bones necessary to remove the divergences which occur at high m o m e n t u m . However, ill this limit,, we might e x p e c t any masses to bo negligible so that they iln not appear in the residues of the poles, and since /•', = (I. nor in the Z,. Ot her 'more physical* renorinalization schenies may also be used. For example. when focusing on heavy quark properties the on mass-shell scheme tnav be used. Here and F,„ are adjusted so that the (real part) of the pole in the quark propagator occurs at. the quark mass, / r - u r , and has unit residue. In a similar fashion F,.\ and F,\.< may be adjusted s o that, the triple-gluon vertex, eqn (3.145). plus oountorterm equals <js at a particular external m o m e n t u m configurat ion, typically chosen t o be miphysical s o as to avoid introducing extraneous singularities. In these schemes, it is c o m m o n for a mass ( / » / / ' ) dependence In be introduced via the finite parts of t he counterterins. l o finish this section we give the results of calculating the renoriuali/ation factors Z, to t wo-loop approximation in p Q C D . To be more specific, in a covariant gauge the MS proscription gives
Z„=
Iff f
, 3 Tin f + -Cy '1
(2)'K
r
o
1
TCfZA =
I- f
4 - / /.-/i, 3
(13-30
C
+
+ ^2)
(25 +
— —
C a
} c ,
C,t
!•: I I I F O H Y O K Q C I >
¡Ml:
2C, I V 2C
Vr A y , n
1 (3 I 2 0
y
, 4. ' "
1
(
+
7*
; <
-0
ir7
3 r
4
A
(r.!)f
(13-3Q,-. '
7
4.(a«Vf vTir/ \
~ 1
f5 r 7 [V2 1
n
C
1
f, - r/ ;. /i1i-
3
:i
9 2T,,UF - -C,, Z„ = 1
1iii Iff f
2 „,
II
3
(i
- / /. /(r
a
~
c,} i^Liillr ~ m ~ C A ( 3 5 - 3 ^ . CA
32 4r•( '/.- 12 r-( J1i
or'"f = i - ^-3CV + iff f
2(J),,. ( ,
m -
}c.
97
11 - - C
A
I }CR
C
(3.i-,r,)
Here, wo have done some work to dorivo tho expression for Z,,. which cannot ho obtained directly hilt is obtained indirectly using eqn (3.1.r)3). Typically. duo to its relative simplicity, the correction to the ghost gluon vertex is t<*
i n , i KAVIOI i i DIVI;UCKN(
•I I I
I'lii rciiornitiliziit ¡01> ¡/roup
i:s
\ N I > K I ; N O I < M A I , I / A i ION
equations
Ii hould not have gone unnoticed that there appears to be a disturbingly large (leedotn in Ihc application of the rcuorinalization procedure. First is the freedom lo select the linit.e parts of the counterterins. subject, to respecting the Hliivuov lavlor identities. We have already exploited this freedom to absorb a numerically large coefficient in the change from the MS to MS schemes. Second is the freedom in the choice of the unit mass. p. which sets a scale for t Ik problem. However, physical quantities can not depend on any of these arhlliary choices. All prescriptions are ultimately equivalent. For example, in two if hemes the quark masses are related as Z„,(Ii)mn = ino = Zm(Il')in n> so that \Z,„(Il')/Zm(n')]mn-, where the ratio is linit.e. because so is each !///,•. i veil though neither Z,„ is individually Unite. In this way the invariance can be encapsulated in the group structure of t he transformat ions connecting quantities (i/., in, if .etc.) in different, schemes. I lie seemingly simple invariance of physical quantities under changes in p leads to a very powerful differential equation collecting <js. in. r .etc.. defined at nlie scale p to t hose at a second scale. To see how this arises, suppose we calculate the amplitude, that is amputated Green's function, for an operator describing i lie 'scat tering' of n,;, (anti)quarks and 7/^ gluons (we need not consider external pilosis). This ainplit tide can be written in terms of either t he bare or renoruialized i|iiantit.ies. Here, we have assumed that the countert.erm for the interaction is proportional to it sell so that, the ronormalizatjon is m u l t i p l i c a t i v e . t h a t is.
F 0 (TTS0.M(,,6>.Q) =
.
(3.156)
I lore, we use t he single scale (} to characterize any external four-momenta present in the problem. For simplicity we only consider one quark mass. m. T h e left-hand Mile of cqn (3.156) is clearly independent of p. as must be the right-hand side. Differentiating with respect to p. using the chain rule, we obtain the following icnnruializat ion group equation
(I = r 0
V
^
-
o
{
i)
'
%
(3.157)
F(//,as,m,i,C?)
i) +
+
+
. o
) ~ * * *
"
, M 7
' 7
1
•
I'lie first term accounts for any explicit p dependence, whilst the remainder takes care of any implicit, dependences via //.,(//), m(p) and £(/')• Equation (3.157) serves to define the dimensionless coefficient functions 1
la g e n e r a l t lie c o u n t e r t e m i m a y involve o t h e r o p e r a t o r s of I lie s a m e m a s s d i m e n s i o n . A n e x a m p l e is p r o v i d e d by I lie pQC'D c o r r e c t i o n s t o a weak decay. In t h e case of s u c h o p e r a t o r inixiiiK it is neixissary l o c o n s i d e r linear c o m b i n a t i o n s of t h e o p e r a t o r s w h i c h a r e d i a g o n a l .
mi
!ll
/
hi
\
<>o,
u n i o n s 0 1 QOD /
in
c
\
1 /'
/
2
Sr s
/
m
\
V // (
i f ^
m
V
/'
cud
()
A* l'
,ii)Z
which an» all finite as r —> 0. You arc warned t hat variants of these definitions, dill'. •ring slightly in signs and normalizations, occur in the literature. Observe that ¡i a i d -,„,. like Zg and Zm. are both independent of A linear partial differential equation such as eqn (3.157) can be solved using the method of characteristics. To do this we introduce the functions J7(l). o s ( / ) , TFl(t) and i ( / ) which satisfy the differential equations
d/=?= li
/?(o s ,?/j///)
=g
m 7r..(«s.'"//')
=S
!
=-
£ (^(tt,,,777/71.0
(3.159)
and £(0) = and pass through the point /!(()) = /', <>s(()) = ° s - "*(0) : These functions connect t he parameters defined at the scale // to t hose defined at a second scale Ji — //e'. T h e 'bar-notation' serves to highlight that we are now thinking of o s , etc. as running parameters. Later on. except for 777. we will drop the special notation. The functions in eqn (3.159) define a characteristic parameterized by t. Since d£ x for ^ 0. then £ will remain identically zero if £(()) = II and we can ignore any ^-dependence in eqn (3.157): £ = 0 is the Landau gauge which we adopt. The solution of t hese equations is straightforward if ¡1. "fm (and i)c) do not depend on 7T7//7. In a minimal subtraction scheme, or more generally a mass independent scheme, all the functions in eqn (3.158) are independent of m / i i . Adopting this further restriction we have
m
-
C
"
^
-
—
Explicit solutions for o s and 777 require the actual expressions for ii and y,„. We shall derive these shortly but for t he moment we assume that the solutions have been found. This allows us to rewrite eqn (3.157) as
" »,.7.,•("*(/)) - ii,,7/t(a s (0)}
r(77(/),os(0,777(/),g) = 0 .
(3.101)
This ordinary differential equation is easily solved using an integrating factor:
r ( „ a s . , „ Q ) = exp
d x ^
- ,,
.
(3.102) The solution is a constant along our characteristic which we evaluate at / = 0. What it says is that the theory defined at ( / i . o s . m ) is equivalent to the
Ill.l
A
I
I.I
I I
i :i«:i .
i i:s A
MA
/A I
I
I li< Hi v ili'lined at //. provided I lull I In- coupling and mass are changed to lake I,he elicitivc values o s (/7) and m(Ji) and the fields present are scaled appropriately. It is useful to make explicit the dimensionality of the Green's function. T h e lliiihs dimension of I' is given by il.y = D — n,\
= s'1''T = s'ly exp (
^.os. /
da:
s/i lii I lie second form we have employed eqn (.'5.102). If we now choose JI I lien we can deterinine how the Green's function changes under a scaling of I he i eternal four-momenta.
I (/i,<.i s ,»;i..sQ) = s
1
exp
/
d.r —
fi(x) (3.1G4)
lu the second form we make the simplifying assumption that the 7, are conitant. Thus, the de|)endence on t he scaling factor .s can be taken into account by evaluating the Green's function at au effective coupling, o s (.s7i). and a scaled I'lleetive mass, m(s//.)/.s. togel her with an overall scaling of the Green's function. However, the overall scaling dimension differs from the canonical dimension, ily. by the presence of the 7, term. Reflecting this discrepancy with the naive expectation. the 7, are known as anomalous dimensions. As we shall see short ly, in practice one often uses // ~ Q. Note that even if in = (I. so that the classical theory were scale invariant, the anomalous dimensions and non-zero ^-function would still imply a breaking of this classical scale invariance in the quantum theory. This is possible because the renonnalization procedure necessarily introduces a m a s s / m o m e n t u m scale, here //. As the next section will show, in Q C D both fi and 7„, are negative. This means that the effective, or running, coupling, decreases logarithmically as the scale Q increases, eqn (.'{.22). This weakening of the strong force is the essence of asymptotic freedom (Gross and Wilczek. l!)7:5: I'olit.zer. 1974) and lies behind the success of perturbative QOD. T h e solution for Tn((J) shows that it is also logarithmically suppressed: this is in addition to the factor l / Q which appears in eqn (.'{.164). At this point, it might be tempting to neglect quark masses, in = 0 in eqn (3.164). at high energies. Q > Tn(Q). so that, all Q
I III'. I ' l l K O H Y O K
d e p e n d e n c e o c c u r s via
QOD
However, this can lead to problems with low-
energy or near colli near gluons so that for this approximation to make sense we must restrict ourselves to infrared safe q u a n t i t i e s a s discussed in Section 3.5. Before m o v i n g 011, w e mention t h a t a number of similar renormalization group e q u a t i o n s have been derived in the literature. Foremost, is the C'allan Svinanzik equation (Syuianzik, L971: Callan. 1972) which is obtained by studying the Green's function's d e p e n d e n c e on the physical m a s s in. T h e equation takes the form of eqn (3.157) with coefficient, functions t h a t only depend on //, ("/,„ = 1) and with the /i()/t)/i term replaced by an i n h o m o g e n e o u s term which may be neglected in tin- i n / Q — 0 limit. 3.-1.4
Calculating
the 'RGB coefficient
functions
T h e values of the coefficient functions ii. ->,„. ~,A. etc. defined in eqn (3.158) can be calculated as power series in o s using our previous results. Here, we illustrate the m e t h o d for the /¿-function. Consider the relationship, eqn (3.153). between the bare and renormalizcd couplings. gs0 = / / ' g s Z g . Since the bare coupling g„0 can know n o t h i n g of the arbitrary scale/», which was introduced only to facilitate renormalization. we must have , 'I/Aso = 0
() = //'
tgsZy
+ ,% i^ZtJ +
(3.165)
nJ-r^^j
<1//
From this we can obtain .1 = (<7.s/2TT)/?9. In a p p l y i n g the chain rule we have made the simplifying a s s u m p t i o n that we use a mass independent renormalization s c h e m e s u c h a s MS. T h u s Zg only d e p e n d s on the renormalizcd coupling !l„ and not o n 777(//)/// (nor o n 0 - E q u a t i o n (3.165) d e t e r m i n e s how much
+
(3.1«G)
»>1
On the other hand, we want b o t h //, and [ig t o be well defined in the limit < 0. that is to contain no poles in 1 / ( . T h u s , w e write [lg = .-1 - Be. it is easily confirmed that all higher powers o f f must, vanish. Substituting this into eqn (3.165) and collecting powers o f f gives [A«„_i 0 = (B+g,)e+A-\-g,Ba\
+----S
+(B
+ g.t)a„ 4- g„Aa'n_l
+
g,Ba'u]
+
•••,
(3.1(57)
where the prime indicates differentiation with respect t o gs. S e t t i n g the coefficients of c"\ !i,(g3)
in < I. t o zero gives = liin {f/g«! - ega}
and
«'„ = « , ( « „ _ i I-
-i) ,
n>
2
HI I K A V I O U
I DIVKItOKNCKS A N D IIKNOItMAI.I/A I ION
2 I
(3.168)
At lust sight, it may s e e m o d d that we can calculate the /¿-function from just t i c residue of the 1/r pole. However, the c o n d i t i o n s on the «„>•» ( ' t H o o f t . 1973). which ensure the absence of pole terms in ii. together with t h e boundary conditions ) 0 allow all the '/„>_> t o be calculated in terms of 0\. deferring to c
2/ — 1 477
4
r -Tm f
U - Cr- A
-
K-r
'fcA)r,nf-'fc:x
+
(3.109)
(4-r or-i
2C
/•
205 u '
2857
1415r, A
~ ~2T~
A
h f
)
~54
rite third t( Tin is given in (1 arasov i t ul.. 1980) for the MS s c h e m e and the 0{
_ 97 , - 3 C ' r - —C-a
2Tpnf
7/t = +
47,
I
3
-Cr
Cr
+ •
(25 I- 8£ + Q') Cr H
4
(13-30..
(3.170)
1 IT
+
CH) "s
( 4 C F + 5C/i )Tfii / -
(Izilr-
(5)
•r>T
(59- l l i - 2 i 2 )
8 (95 + 3 0
r
.
1
2
A
h • •
CA + •
Observe t hat, both ii a n d 7„, are independent, of the g a u g e parameter a fact which can be t raced t o the ( - i n d e p e n d e n c e of Zg and Z,„ in a m a s s independent renormalization scheme. This is not so for the wavefitnotion a n o m a l o u s dimensions 7^,. 7..1 and y , r T h i s raises t he issue of the s c h e m e (in)dopendence of our results. If we restrict ourselves to mass i n d e p e n d e n t s c h e m e s (and neglect possible 11011-perturbat.ive effects) then the first two terms in ii and the first t e r m s in 7m. " <;• a l l ( ' 1A are independent of t he specific choices of countertornis. B e y o n d these leading terms the results are s c h e m e dependent and. for e x a m p l e , the third and higher order terms in eqn (3.169) can be set equal to arbitrary values. In eqn (3.165) we a s s u m e d a m a s s independent renorinalizatiou s c h e m e had been used. If this is not the case, then the presence of ////// d e p e n d e n c e s in the Z,
11II' T H E O R Y O F
g e n
significantly complicates the RGEs. First, the ('valuation of H and the is made harder by, for example, the extra term, — l)OZg/<)('•—), in e(|ii (3.105). Second, the solution of t he eoupled. linear, differential equations for the effective coupling and mass, cqn (3.159), is made harder. Here, a possible a|>proach to avoiding these problems is to go to a regime where the mass(es) are negligible compared to p and all other scales (Q). 3.-1.5
The running
coupling anil (¡nark
musses
A key lesson from our study of the RGEs is the need to express our results in terms of the running, or effective, coupling and mass. If we use as argument. C f , the evolution equations in a mass independent scheme arc = -<»s(A> + th «„ + /fco 2 + • • •)
=
M&j ^^Tfp^ =
=
~" s ( 7 ° +
7,ft
*+
72
and
°*+ • ' •
(3.171)
(:u72)
e.f. c(|n (3.159). Here, we have in mind that #o-7n > 0 so that ii, y,,, < (I. Taking into account t hat Q2 d/dQ2 ( 1 / 2 ) Q d / d Q . the coefficients in the series expansions for the MS scheme can be read off from eqns (3.169) and (3.170). =
'
llC,. t - -17V,»/
0
=
(33 - 2»./)
12ff
I7C
j
2
12^
- (tiC,- -1- 1 0 C A ) T r n ;
=
(153 - 1 9 » ; )
24^2
2-lff2
3C r 7 0
=
1 =
17"
C,. (U7Ca + 9C/.- - 20T,. n/) 71
( 3 1 7 3 )
2
"
{JOff
*
(303 - 10»./) ~
72tt 2
Here /%, ii and 70 are c o m m o n to any mass independent renonnalizal ion scheme, whilst 71 is specilic to the MS scheme. T h e terms quoted above are given as function of the colour factors C/.-. C.\ and 7 > and thus are valid for any gauge theory with an unbroken gauge symmetry. Note that TV always appears ¡11 a product wit h » / . the number of active quark flavours. T h e coefficients on the light-hand side apply for the case of colour SU(3). that is. t hey are specific to QCD. Referring to cqn (3.160) and working t o next-to-leading order the solution for c*s(Q2) is given implicitly by r"AQ2) W o ) ~ ' - W ) &(* AQD "yj) O AQ*) djr
-
J
h ! 0 i •/(» "AQ?,)
M x2(/j„ + xfii)
00 _ & x2 x
ii j !% + xih
IH l it A V I O N . I I M V K I t C K N i ' K S A N I » I t K N O H M
1
, .
W J
L & !
=
+
*
(
M.IZATION
A
(3.174)
l ^ T T w ^ ) ) 9?,
I Ini . given tlic valili 1 of o s ( Q j i ) at o n e scale Qn. ¡1 is possible to solve for o S ( Q " ) III n Nrcniiil scale (J: the o n e proviso being that we remain in the perturbat.ivo 11>iiiiiiin where eqn (3.171) is valiti. An alternative and slightly simpler form of n|ii (3.174) can be obtained using the boundary condition O s ( A Q ( . | ) ) = oc. Q-
• in In
( n
^QCI)
m
s
•'a
'
">(Q~) +
)
(3.175)
^oAQ2))
I h i I• I lie parameter AQCIJ is equivalent to giving the coupling OS ((/"*) at a specific in nli • Q. At the (|iiani uni-level. Q C D is specilied by a dimension I'll I parameier. I In . is even t rue in the absence of any quark masses to set. a classical-level scale. I lie appearance of such a scale at. the quantum-level is known as dimensional I lansmiilatiou. In eqn (3.17-r>) o s ( ( , ) 2 ) is given implicitly. B y e x p a n d i n g in inverse powers of l n ( Q ' / A f j C I , ) an a p p r o x i m a t e explicit form can be derived. ft (Q~)
=
ln[ln(g-7A2c„)] (3.17«)
}
ihHQV^m)
t%
HQV^ru)
fij ' ^h.^Q'-VA^,,) In practice the last ( - 5 / 4 ) term iu this expression is often neglected. T h i s is equivalent to a redefinition of AQCI> by 0( I l(l'X) (Buras el
=
(3.177)
and the inverse of e<]n (3.175) is given e x a c t l y by eqn (3.22) with A Q ( U = Q~ expl l/,'V 0 o s (Qo)]. A s this expression for AQCD suggests, changing the value o f o s ( g - ' ) for a fixed Q d o e s not. really alter the theory but. gives the s a m e theory with its unit, of m o m e n t u m rescaled. In this s e n s e (inassless) Q C D is parameter tree ( C o l e m a n and Weinberg, 1973). Equation (3.17G) makes the a s y m p t o t i c behaviour of o s ( Q " ) manifest the Q C D coupling decreases as 1 / 1II(Q/AQ<:D) lor (J - > oc. It is important to realize that this decrease iu o s justifies the use of p Q C D . in particular the solution based on the first, few terms in eqn (3.171). As (} decreases the converse is e x p e c t e d and indeed a strong growth of is continued experimentally. However, the singularity at. Q = AQCD should not lie taken t o o seriously as large values o f o s invalidate eqn (3.171) and any solutions based upon it. In this low-Q regime Q C D is non-perturbative and no o n e knows yet how o s ( C / 2 ) behaves iu reality, nor can it be claimed that this is a proof of confinement in Q C D . though it d o e s make it more plausible. It is safer to regard
11)0
I I I I . T I I K O H Y (>!• Q C I >
Agci) ~ 20(1 MeV. roughly an inverse had roil size, as the scale at. which nonperturbative physics becomes important. Finally. returning to eqn (3.17G). if we substitute A q p u Q o c * P l — 1/A>°s(Qo)] w<> < - a "
with
(3.178) U)
\
Iff)
^ = 1 + ffo
IjJ )
(S)
which is accurate to next-to-leading order. T h e crucial fact for asymptotic freedom is that, ¡1 is negative, t h a t is ;i(t > 0. Referring to eqn (3.173) we see that quarks, and fermions in general, give a positive contribution, whilst non-abelian interactions amongst gluons. proportional to Cji, lead to an overall negative ti. provided n j < 17. In Q E D with abelian photons the /¿-function is positive and consequently electric charges grow as the scale of a measurement grows. How various particles contribute to the fifunction has been extensively studied and it is now known that only theories containing non-abelian gauge bosons give negative contributions (Coleman and Gross. 1973). Since many extensions to the Standard Model have been proposed, it. is interesting to see how a new particle would contribute to the .¿-function: see Ex. (3-22). At leading order only coloured particles can contribute to the Q C D ..'¿-function, t hough in higher orders all particles cont ribute. T h e contributions to fio consist of two components related to the particles' colour and Poincaré group representations. T h e general expression is
^ =
D r
E
' ><
(:U79>
-
colon m l parliclcs
Here I), equals —11 for a vector boson. + 1 for a Dirac fennion, + 2 for a Weyl fermion. -1-1 for a complex scalar and + 1 / 2 for a real scalar field, whilst Tr is a colour charge determined by the particle's SU(Av) representation. For example. 7/.- = 1/2 (by convention) for the fundamental (triplet) representation, Ta = 2.V,JV for the adjoint (octet) representation. (2A r r — 1 )'!).• for the sextet representation etc. For Q C D with a colour SU(3) octet of vector bosons and n / triplets of Dirac fermions this gives eqn (3.1(>!)). T h e next-to-leading order solution for Tn(Q-) follows similar lines to t h a t for o^(Q-). One finds
- Ja, <>JQ')
= 1/
>% Jo
tlx
+ (i\x)
x
fio + Pix
in i u A V I O N i i)ivi':i((!i:N('i:s A N I » H K N O I I M A I i / . A T I O N
Mil
«•ti-»! -
„
.,,. f W ) = f
. ("AQ-) M (
\ $ ( t j -
H i i
{Q*) \ n M ) ) t^oi
or
2
2
'i-'»)
2
m ( Q ) - m 0 («„(G ))
(:u80)
»+ §-"s(Q ) /»il
Ar.iin. Riveli Jit ono scalo Qo wo can calculate iii((}-) at. a second scalo (J. pinvided that, p Q C D , and in particular c<|ii (.'5.172), remains valid. T h i s oilers a 11 iix isc way of specifying a m i m i n g quark mass as t he mass when t he scale equals II mass: m m ( m 2 ) . In t he second form of the solut ion 777(| plays a similar ròle to A g r o . Specializing to the loading order result., = 0 = . 1 w o have
I
-
, , l W
'Hln(g/AQCD)j
•
( , U M )
I hits, we see t h a t t he q u a r k mass falls as an inverse power of a logarit hm as Q llHTonses. T h i s (|ii;int inn scaling violation, in addition to the classical iTi(Q~)/Q Mitppression. a d d s justilication to d r o p p i n g light quark masses from our calculations. Up to this point we have left unresolved the issue of how m a n y q u a r k s , n j . lo include in o u r calculations. T h e critical issue is the relative m a g n i t u d e of .1 q u a r k ' s mass to the overall scale Q. If t h e q u a r k has 7n(Q2) Q then it i >III only make its presence felt via internal loops and it is possible to remove these c o n t r i b u t i o n s by suitable choices of the count,orterms. T h i s decoupling theorem (Symanzik. 1!I7.'5: Appelquist and Carazzoue. I!)7.r>) moans that we can Ignore a q u a r k if v7(Q2) » Q. On the ot her hand, if the quark has 7Ff(Q~) -C (J. then we should include its contributions and infrared safe (plant it ios can be evaluated using t h e a p p r o x i m a t i o n ///,, = (I. T h e so-called light, q u a r k s , d, u and s, have ///,, < A Q O D S O that, in a p Q C D calculation, characterized by Q 3 > A Q O D . we always have iif > .'{. T h e issue is more delicate for the so-called heavy q u a r k s , i Ii and t.. in sit uations where TTiqi^"') ~ Q- Hero wo expect, significant., process dependent contributions from t h e q u a r k m a s s which we must therefore include In our calculat ions. F u r t h e r m o r e , we have to decide how to cope with t he change In the rf-fnnction above a n d below the quark mass threshold. Well above 777Q we can use t h e 'full' theory containing » / + 1 q u a r k s a n d o,' (Q 2 )- whilst well below m q we can use an 'effective' theory with i i j light quarks and o ~ {Q~). A t intern led iate scales. (J ~ Tn(j(Q J ), wo must m a t c h t h e two versions of the theory so as to ensure that they give consistent, results. T h i s matching has been carried out to next,-to-next-to-leading order for SU(H) in the MS scheme (Bernronther. 1983) and results in a relationship between t h e two running couplings given by "it«2)
<'."(Q 2 )
tiff
( ¿Tl~
I III; T I I K O H Y O F <¿(1)
wiM.
, =
.» 1 = ^ ) .
<M«>
If we evaluate this expression at the point at which t h e scale equals the r u n n i n g quark mass, IMQ — 1v(iiIq). that, is x = 0, then eqn (3.182) almost reduces to requiring o s to he continuous at t h e scale ihq (Marciano. 1984). "S"('"Q) =
"S~('"q) ~ y ^ i J ( « ¡ " ( ' » Q ) ) ' •
(3.183)
T h i s explains why it is c o m m o n to require o s t o be continuous a t (J = inQ r a t h e r t h a n at t he production threshold (J = 2?HQ. Of course, imposing continuity on o s implies a discontinuity in A q c d - which subsequently becomes dependent on the n u m b e r of active flavours, n j . For example, a t leading order it. is easy to verify that, t h e continuity of (v,(hiq) as 11 / ri/ + 1 requires
Aqcd
-
A
Q < ^ — j
)
- V D
•
(-*184)
Similar expressions can be derived at next-to-leading order given a specific equation for a s , equivalent to a definition for A q c d T h i s raises the issue of how t o q u o t e a measurement, of t h e running coupling. o s . T w o conventions in p o p u l a r usage a r e t o q u o t e a \ . [ M y ) or AQ ( .|J. In b o t h cases, this will typically involve having either to evolve o s or to match A q c d at flavour thresholds. I11 the case of A Q C D 11 ¡-s important, to be specilic as to which next-to-leading order equation is being used, for example, eqn (3.175) or (3.17(>) with or without the last t e r m . T h e value of A Q C D also depends 011 the rcnormalizatinn scheme, for example, A ^ | S = -1/Te -7 '- A J [ L S . Since there a r e more t r a p s involved in specifying A Q C D . t h e preferred option has become to q u o t e N S at the scale of the Z mass. An interesting aspect of converting a measurement at Q'2 t o an o N value at My is t h e effect on t h e m e a s u r e m e n t ' s error. By differentiating eqn (3.160) we find
So t h a t . if Q2 < My . t hen t h e error will shrink as we evolve from (J to My.. A second consequence of eqn (3.185) is that, a change in <\^(My ) only causes an O(njf) change in o s ( i / 2 ) . T h u s , an experimentally determine«! <|uantit.y. a :t: A
-
s
/LO^
/in
± ACo'i
»
±
— Q"
(3. ISO)
111, i It A\ K>ll I I HVKI« :i:n< r s
\NI> KKNOKMAU'/AI ION
I 'lint is. in addition to the leading 0 ( term we tu measure <\S(MY). In e(|u (Ti. 1.S6) we also included Hex! -to-leading order perturliative correction and a lion that is parameterized as a power law correction. i\„(My) can lie estimated as
KI:I
re<|iiire the t ) ( o ^ V | 1 ) term an estimate of the next-tonon-pertnrbative contribuT h e measurement error 011
Aas(Mg) _ nJiU}) 1
(3.187)
Looking at the first term, the "error telescoping effect' suggests using a small value of Q z together with an intrinsically higher order process, large N . However, the error associated with missing the second two terms favours nsiu^ larger values of Q:. wliere their contributions are smaller. It is also possible t h a t AC. AD \ N »11 that there is no advantage to larger ,Y. :i l.(i
An cxplic.il
example
lu the previous sections we learnt, that physical quantities are independent of I lie arbitrary renornializat ion scale // if they are made functions of the running coupling and mass. We now repeat this rather formal analysis in a particular case. We focus on QC'D corrections to the diinensionless I p a r a m e t e r defined ¡11 e ' e annihilation, eqn (2.30). Suppose we have calculated a pert tubal ive series for H. rt ( ^ « „ O ' 2 ) ) =
1
E
''m(Q7/'2K(/'2) •
(3-1SS)
We have removed an inessential factor Ncc'~ from e
.,
0
H 1 »1=0 ^
,
()
H
CM
'
'
The first few terms are given explicitly by •1 d»"t 0 = / / " +
( •• d»"2 \ •> I ~ i''tAi I <>s +
( •> d/'3 ~
-
1;)
~
1
\ -i J ^ —
'
(3. I'M) Since each coefficient must vanish individually we obtain a series of differential equations which are solved easily to give >;(') = c,
Illl
K< >HY <>!•' ( ¿( 'I>
r-,(t) = <••, + c, l%t /.((/) = c3 + (2c 2 A) + c\!ix)i rn(t)=cn
-I- c\;i~t~
+ — + c,(/fci)"~l •
(3.191)
lien- the //-dependence is via / In(i'~/Q~)illl<' {<"'} a n - i»>merical c o n s t a n t s . In g e n e r a l . / - „ ( 0 x t " ' . s o t l i a t the series contains t e r m s of the form ( o s i ) " . T h i s raises a potentially embarrassing problem, for when o s ( / / 2 ) I n ( f r / Q ' 2 ) > I. which is inevitable for sufficiently large (J. t h e series a p p e a r s not to be convergent. T h i s problem is easily finessed if we r e a r r a n g e eqn (3.191) t o take account of these so-called leading logarithmic terms. R(t,as)
= I + c,[L + fiQast
+ (tfoos/)2
+ •"
'K
+ [c:i + (2c-,,% + c, :i\ )t\n* + - • • 1
+ + ' ' 1 + , % o ?( /s /( -M ''' * ) l?n r( oQ -' // / / -M) s
li)2)
It is t hen apparent t h a t t he leading logarithms can be s u m m e d by t h e use of t h e one-loop running coupling « S ( Q " ) . Yon may wonder if this j u s t m e a n s that t h e convergence problem has been shifted to the next-to-leading logarithmic t e r m s oc ( n s / ) " n s . However, using t he two-loop r u n n i n g coupling s u m s hot h the leading and next-to-leading logarithms and shows that t h e NLL t e r m s are genuinely suppressed by o s . In fact we already know the result of carrying this p r o g r a m t o completion. Ii is given by eqn (3.162). R(l.as(Q2))
= I + c , « „ ( Q a ) + <'>oUQ 2 ) + 4 « s ( Q 2 ) •" • •
(31!'3)
a series whose convergence actually improves as (} —» oc. T h e coellicients in eqn (3.193) have been calculated to C (nj!) (Chetyrkin <1 al.. 1996«), T h e coefficient c\ is reuonnalization scheme independent, whereas r', for // > 2 d e p e n d s on t he scheme. We calculate the one-loop correction in Sect ion 3..r>. It is useful t o review t he above calculation from a d'liferent perspective which gives an insight into how t h e R G E s work. Earlier, we encountered Weinberg's theorem when discussing the form of t h e countert.ernis needed to remove ultraviolet divergences (Weinberg. I960). Once a d i a g r a m is rendered linite he went oil to investigate its a s y m p t o t i c behaviour as the scale of t h e external m o m e n t a becomes large. A typical behaviour is a dimensionful factor, Q'1. times a polynomial in l n ( ( / - / / r ) (Mueller. 19X1). T h u s , we expect a typical cross section to have the form rr = Q'1
(WQ2/l'2))
with
S„(:r) = a„<> + «„\x + ••• + <»,,„, *•"* . (3.194)
c.f. eqns (3.188) and (3.191). Now. because the t e r m s in t h e R G E eqn (3.157) a r e of different orders in o N . it interrelates $ „ of different ii and t heir coellicients (/„,„. Indeed these constraints allow the S„ t o be partially reconstructed. T h e
M
I R A V I O L I : I I>IVI;I«:KN< 'KS A N D K K N O U M A I I / A I ION
ii lilting s t r u c t u r e contains series ol leading ami next-to-leading logarithms etc., which correspond to expansions of the r u n n i n g coupling. Thus, the R G E enforces relationships between the coellicients such that, all the large logarithms can be iimnied by using an effect ive coupling with a scale a p p r o p r i a t e to the problem. Before leaving eqn (:i.l<S8) we comment <>n two features of eqn (3.191). First I', i he seemingly trivial observation that /• i (Q2/ii2) is a c o n s t a n t , c\. Since a n y // dependc nee in r u arises Irom the t r e a t m e n t of ult raviolet, divergences, this m e a n s I hat at one-loop t he Q C D correction to the 7 * q q vertex is finite; a result which must in fact hold at all orders if Q C D is not t o spoil electric charge conservat ion. Second is the effect of t r u n c a t i n g the series eqn (3.188). As we have noted t he coefficient, of the 0 « + l ) t e r m is related t o coellicients of the O(o-'J), 0 ( a " ~ ' ) . .., 0 ( < \ l ) t e r m s in such a way a s to remove the //-dependence t o ( 9 ( o " ) . For example, to two-loops one has "{2i
= 1+ <W/<2) +
<-•'-> - C i f o I n
t*2(,r),
(3.195)
which is //-independent, to 0(cv s ) b u t //-dependent at 0 ( < \ 2 ) . T h u s , a t r u n c a t e d neiies is //-dependent a n d in practice we m u s t decide what value(s) t o use for //. This is the scale setting problem. A conservative a p p r o a c h is to vary // in II range centred on the characteristic scale [Q/X.QX]. T h i s should cover more specific prescriptions whilst,, if A is kept, modest, avoid making the logarithm llirge and spoiling the validity of eqn (3.195). In this sense the m e a s u r e m e n t can he said to be of o s at t he scale Q~. A word of caution: it is sometimes claimed that by varying // one can e s t i m a t e t h e size of the next (uncalciilated) t e r m in a series. For example, t h e a 2 t e r m from the n s term in eqn (3.195), but since <••„. is a r b i t r a r y (until calculated) this procedure is not without risk. O t h e r more ambitious proposals for scale setting are available. T h e principle of minimum ensitivity ( P M S ) (P.M. Stevenson 1981) chooses // so as to make t h e t r u n c a t e d series locally independent o f / / . Applied to eqn (3.195) this yields •,d /?'"> d//~
, = <> = >
/''Ì.Ms =
/'f.Ms
tfi<wpf-Tr~ ~ È l ) • V A,c, Mi2)
(3.196)
The application is to eqn (3.195). Fastest, a p p a r e n t convergence (FAC) chooses // so t hat the first non-trivial term gives t h e s a m e result as t he s u m of the known terms. Applying this prescription to eqn (3.195) gives "u)(/4AC) = /?("Vfac)
=>
=
•
(3.197)
Again the application is to eqn (3.195). A third proposal by Brodskv, Lepage and Mackenzie (BLM) (1983) determines // from the requirement, that the /independence of the coellicients r, vanishes. In all cases, by going to higher orders in
mil
3.5
I in
n i l « »uv < )i ; g e i )
Infrared safety
Ultraviolet divergences are not the only complication which arises in Q C D . divergences also occur when real glnons are emitted with either very low energy or nearly col linear to the emitter. We already noticed this problem at tree level • qqg calculation in Section 3.3.2. Throughout this section we with the e + o ~ will illustrate t he basic methods and ideas used to deal with these infrared divergences using the important example of the 0 ( o s ) correction to electron positron annihilation to hadrons. T h e underlying process is e + e ~ —• ~'(Q) —» qq which, as we have seen, is also closely related to both deep inelastic scattering and the Droll Yan process. T h e 0 ( n s ) correction is given by gluon emission olT the final state quark or auti«|uark. though the following discussion applies with minimal modification for omission off a gluon. Recall the behaviour of t he quark propagator prior to emission, eqn (3.1(1")), i
=
i - .i„cos«
,f
h W
*
M =
1
/ i _ !!ia V n ' (3.198)
Mere wo see that there are basically two singular regions, which may overlap: 0 0 soft „.„-o (3.199) »II Ö,|(, — (I collinear. 2E 8 £ < 1 (1 - ß„) -1 (Propagator) T h e collinear singularity is also known as a mass singularity since the propagator is st rift ly only divergent for gluon omission ofT a massloss partou. C|iiark or gluon. In both limits the virtuality of the emitter tends to zero, so that it travels a large space time distance prior to the gluon emission. These divergences are therefore associated with the long distance, infrared behaviour of the theory. Similar infrared divergences occur also in virtual processes. This is because the integrals over loop momenta include phase space regions corresponding to the emission of both collinear and low-energy, real gluons for which t he propagators are singular. This leads to very long-lived virtual fluctuations. It must now be admitted that we glossed over this issue in our earlier discussions of ult raviolet divergc!iices. Since our calculations are pert urbativo. based on quarks and gluons. they will break down in these limits where non-pert urbativo contributions enter. In this section we shall consider how to cope with these divergences and under what oireunistaneos they cancel. 3.5. L
Infrared cancellations
and dimensional
rc.gularizat.ion
T h e key to treating infrared divergences lies in two observations. First, whilst real diagrams squared always give positive contributions to a cross section, interference involving virtual diagrams can give negative contributions. This opens up the possibility of arranging a cancellation between the singularities in t he two sets of diagrams at t he level of the amplitudes squared. Second, there is a striking similarity bet ween the two singular configurations, soft and near-oollinoar gluon
mv
INKHAUKDSAII-.n
emission, and I lit- sit uat ion where no emission at. all occurs. In practical situations a detector's energy resolution and granularity will not allow a sufficiently "ill m collinear emission to lie distinguished from no emission. If this is to be H'llected in the corresponding theoretical calculation then the two contributions in id in be added to give a useful result. In the case of electron positron annihilation In hadrons t he lowest order terms in the matrix element squared are given
by M<m
= M™
+ «MX
\M\2 = | > < f
+ ••••
+ ns ( | > C f
M,« = v/îv^C + '•• +
})+••••
(3-200)
There is no V/Fv7 cross-term in the squared result as there is no common final Mate: |qq) ^ |qqg). At next-to-leading order we should take into account both the real process e + e ~ —» qqg and the interference between the tree-level and oneloop. virtual corrections to the process e + e ~ —» qq. In eqn (3.200) we anticipate that the first and second terms at <9(o s ) will contain "+oc' and ' —oc' infrared divergences and that these will cancel when added to leave a finite result. As with the treatment of ultraviolet divergences, before we can manipulate any matrix elements we need to regulate any infrared divergences and make I hem finite. It is rat her pleasing t h a t dimensional regularizat ion again provides it suitable method. To see how this works we shall consider the tree-level process V • qq&- for inassless quarks. In the soft gluon limit the dominant contribution to the matrix element, squared comes from the cross-term, eqn (3.106).
\M[* X ,
M
~
1
'
.
(3.201)
which behaves as E"1. This expression, describing radiation off a colour anticolour dipole. also contains collinear singularities for (j - ' | | || 7j. where it behaves as 0~~. As we shall learn in Sections 3.7 and 3.6.7. this simplification of a matrix element squared in the soft and collinear limits is generic. T h u s the contribution to the cross section from a soft, gluon emitted nearly parallel to the quark is given by tin1 following /^-dimensional phase space integral
/
<»"•* Q ' + V ) , , w '(
" =
3
f l f dfl||dflxflx '
./ ( 2 * ) » J„ /' ...
(3.202)
r-n 1 r
2/^ ,
E„ sin'5-30(.„
- ,(/||] f ( î ï ) ic\D~:Hl KM)
11 IK IIIKOKY Ol q c i > HV>" r> ^ i> :
^ I E
5 K
> r>
o
j T i \ 0 m s i n ' • - ( 0 , J 2 ) a* ~*(0,j2)
j
In I lie first line we isolate t lie component of I lie gluon's m o m e n l u m parallel to the quark: /(Si) describes t h e (non-singular) angular dependence of the dimensionless combinat ion ( 4. as is the collinear, ()<w - (I. singularity. T h u s to t a m e infrared divergences we work in I) I - If dimensions, but now with ( < 0 so t h a t D > I. Readers may be aware t h a t a n o t h e r method for regulating infrared divergences is t o add a small mass to the gluon in intermediate calculations which can be removed iu the final result. However, this is a 'short-sighted' solution as a gluon mass term violates gauge invariance at. C')(o s ). As soon as two gluoiis are involved in a situation, a gluon mass cannot be used without destroying the basis for the theory. You may wonder if it is possible to extend t h e B R S symmetry even f u r t h e r t o a c c o m m o d a t e a gluon (or ghost.) mass term as well as the gauge fixing term. Unfortunately, this is only known to be possible for an abelian t heory such as (JED. 3.5.2
e+e"
annihilation
to liadmns
at
NLO
We can now start to calculate the cross section for electron positron annihilation to h a d r o n s including our infrared regulator. T h i s is given schematically as the product of a lepton tensor a n d a hadron tensor which is integrated over t he final s t a t e phase space, c.f. eqn (3.85). J_ 2Q2'
y
(3.203)
Observe that in the photon p r o p a g a t o r any t e r m s proportional to (1 £)Q''Q" do not contribute t hanks to electromagnetic gauge invariance which requires that both Q"L,IU = 0 = Q"L„„ and Q , , / / ' " ' = 0 = Q„H"". Now. t h e only available o b j e c t s that can carry the integrated hadron tensor's Loreutz indices a r e //'"' and Q,lQ". Add t o this the gauge invariance requirement, and we can restrict, t he integrated h a d r o n tensor t o t h e form fd<]HI'"'(Q) = with
H(Q')
7r +
= -Vlil,
jdH'"'(Q2)
H(Q-)
(3.204)
.
You may recognize t h e pre-factor as the a p p r o p r i a t e form of t h e spin averaged polarization sum for an olf mass-shell photon iu D dimensions, c.f. eqn (3.111). S u b s t i t u t i n g eqn (3.204) into eqn (3.203) and using the s t a n d a r d form of the (inassless) lepton tensor, eqn (3.85), then gives e 2 (D — 2)
infiiakkd safk i y
In
Ins way we have reduced t h e problem to the simpler one of o d e , . l a t h « the
m l ,
"fUi .' 1," i «with -rh4l, and integrating it over — l o confirm eqn (3 205) we apply it to the underlying lowJ o « t r piocess for a single, massless q u a r k flavour. r(0 =
2 ( 1 - 0 ^ (rr.y 4 c y (3 - 2e)
m
< / , 22
) A',. x
Q
,se X
d'I'jTr r(l-e)
(3 - 2e) 2
S
(i - 0
a
r(i-c)
'""" o •v' >(' 3 - 2 f ) f ( 2 - 2 f )
m
(3.20(i)
-
The evaluation of the hadron tensor essentially follows the earlier t r e a t m e n t . Section 3.3.1, however, we now work in D dimensions. This means a d d i n g a factor //' to t h e q u a r k ' s coupling and remembering t h a t = D = 4 — 2t when doing t h e -y-matrix algebra: see Ex. (3-19). We used eqn (C.22) for the phase space. T h e result coincides with eqn (3.93) in t he limit D • 4. At. C ) ( n J t h e most straight forward contri3.5.2.1 Tin- mil 0(<\s) contribution butions t o evaluate come from the real emission process 7 ' —+ q<|g: see Fig. 3.11 and the earlier discussion of Section 3.3.2. Working with massless q u a r k s in I) dimensions t he project ion of t h e qqg matrix element squared is ->hu.f!l't" (<'<^hii2')2C,,Nc 2(1 - f ) T r { l }
- e ) F t l \!)-
+
M ) f / q j
= 2(l-r)Tr{l} Hd - 2( I
-x.t)
(I
-x„)J +
(7 • <1)Q-
_
'
(n •'/)(// • 7) 2(1-*«) (»-•'.'.,)( 1
r ) T r {1}
-x„)
-2c
(3.207)
Men- we employ the usual energy fractions, ./•,. defined ¡11 eqn (3.113). In t h e first two lines it is easy t o identify t h e individual contributions from the d i a g r a m s corresponding to gluon emission off t he quark, emission oil' the antiquark and their interference. Iu this expression the collinear singularities a p p e a r a s single poles in t h e limits <j • <j — 0 (./:,, • 1) or <j • <j — 0 (./•,, — 1). T h e soft singularity 0 (.?:,, —> 1 and ./„ —» 1). Observe a p p e a r s a s poles in the limit r/i/ —» 0 a n d 7• tj that in this limit the interference term has a double pole. T h e a p p r o p r i a t e Ddimensional. three-body phase space integral is given by eqn (C.23), so that Hh(Q~) is given by Hu(Q2)-') =
<\W»(Q'2)
= o^^CVAWQ2
(
^
V 1(2 — 2f)
I iiKnitv o r - g e l »
I in
* I , b : < i / - , ; , l V i i ( i - *<,)(i f(1 _ f ) f l l ^ f l l ) + \ ( 1 — JV,)
x
j
v i ) ( J < I + - Di' +
(1 - •«'<,)/
(1 -
,:
a
-2f] .
(3.208)
This phase space integral looks very daunting, but can in fact be rendered quite simple by means of a change of variables. r ( | = 1 - r.r,,. and ,r(| r.
' ' " | i 2 ( l - x)o(l - »)]'
I
, i n -\(1~
t f )
r(2-or(i-e)r(-o r(:j - 3 f )
r3(i-0 L _ ( 1 _ 5 r(l-3e)(l-3f) W f
+
r(2 - o r a ( - f ) r ( 2 - :ie)
r:'(i-r)) T(3-3e)J
"4f)Nl (2 - 3e) /
The :r and r integrals are of the standard F.uler tf-fnnction form, eqn (C.27). and the resulting I'-functions have been manipulated using eqn (0.25). Mere we see a double, 1/f". pole which comes from the interference term and is associated with the soft gluon singularity. There is also a single, l / f . pole associated with the collinear/soft divergences. Combining eqns (3.20!)), (3.208) and (3.205) gives the real gluon emission cross section at C?(o s ), , < o - „ • » i k c y (*E!*X " " 2tt V Q2 ) 3.5.2.2
( - + - + - + 0(e)) . v r ( l - 3 e ) \c2 f 2 'J
(3.210)
The virtual C?(a s ) contribution We now turn our attention to the contribution coining from the interference of the virtual, one-loop cor-
Flt;. 3.20. The virtual, one-loop, Q C f ) corrections to the 7* -«• (|q vertex at 0(aH)
INI' I I , M i l 11 SAI I I N
Recalling our earlier discussion of rcnormalization, Section 3.1.1. we anticipa l.e that these virtual corrections contain both infrared and ultraviolet divergences. We begin by investigating the ultraviolet behaviour of the corrections. I'he divergence in the quark self-energy can be read olf from eqn (3.132). To obtain the divergent part of the vertex correction we note its similarity to the Hist. "QED-like" graph in Fig. 3.18 which is identical upon making the replace— C( ,l{T''Th)ij = ccHCi.u),r Thus, t he divergence can be read ment
. 1
+ y-('i) ^ y '
y ' i ^ E M + q- — lll~
£7"
+1
ViAQ.'i)"
A, + U.V. finite
i <'
(3.211)
Note t hat we only absorb half of I lie self-energy corrections int o 1 he renormali/.ed coupling appearing in t he third line. The other half goes into the wavefunction reuormahzatiou. We also use the Dirac equation acting on quark. "('/)• find antiquark. /'('/)• basis states to eliminate the two remaining divergent terms. I'he sum is then free of ultraviolet divergences. This result, for the "yqq vertex should be compared to that for the gqq vertex. Fx. (3-23). The absence of ultraviolet divergent QCI) correct ions to t he "/qq vertex is not an accident but an important requirement, for the acceptability of QCD. Its significancc lies iu the fact that QCD does not alfect the renormalization of electric charges or. if you wish, does not spoil electromagnetic gauge invariance. More formally, the electric charge operator commutes with the (¿CD Mamiltoniau. II it did not, then electric charges would not be conserved. For example;, consider an ant ¡neutrino interacting with an electron to give hadrons. u 0 c~ —< IT — elu. The initial state has charge —e and is unaffected by strong corrections, whereas the final state is potentially affected by them and so might have a rcnorinalizcd charge <' /- - < which is unacceptable. The -,qq. Zqe) and Wqq' vertices are free of strong interact ion divergences to all orders. This is the reason why we can calculate the charge' on a haelron from t he sum eif charges on its const.it uent quarks without regard to any complex, non-perturbative. stroiig interaction dynamics. Since we are focusing on the case of inassless, on mass-shell (anti)e)iiarks a number of the virtual eliagrams do not contribute', Recall e'i|n (3.131) for the quark self-energy, which in the case of ¡ r m~ = 0 reduces to I "A- 7 , M + />Y {•!-)» \k- + 2k • p]k2 (2
k"k" (I
0
I III', TIIKOUY ()!•' «¿('It
-(ya,,< S Cp6tJ/> I -
=
0 •
(3.212)
In I lie second line, we have made explicit the fact t hat the resulting loop integral is independent of any scale ( ¡ r = 0). This is the same situation that we encountered previously with the tadpole diagrams where the integral cannot he defined for any dimension D. In dimensional regularization these integrals are defined to be zero. Thus, the only one-loop diagram which can contribute is the vertex correct ion where the scale is set by (J2 — 2q • (]. T h e contraction of the had ron tensor describing the interference between the vertex correction and tree-level diagrams is given by the real part of t he following expression. r (|/>/. -*"""v' = x |TT
)-"lV { 7 * 7 * } + ihAt
~ Í)Vít"} -
i A,(A. +
WW
^
_ „)2
+
-1i)«V'}} • (3.213)
Evaluating the two traces is a straightforward if tedious exercise. + ¿h,M-áh01h"}
Tr
= + 8 ( 1 - e) [0a - 4(* •
fk2Q2)
Tr { 6 m + Ú h A t - i W ] = - 4 ( 1 - c)(fc +
(3.214)
T h e second result implies t hat t he gauge dependent contribution is of the same type as t he integral in eqn (3.212) and therefore vanishes. There is no £-dependence. In order to treat the remaining loop momentum integral we first combine the propagators using Feyinnan parameters, 1 k~(k +
_ =
/-1-0
/•'
[tt(k + q)2 + ;Hk-q)2 + (l - o ,1 ,1-0 J = / do / d/?— . Jo Jo [(A: + oq - Pq)2 + apQ*]* i
1
I
d/
;l)k2)* 3.215
This suggests the change of variables A " — k1' —ctq'' + 0q'' in the integral. Making this substitution in the integral's numerator the first trace, eqn (3.214). gives Tr {} = 8( 1 - e) { i / 1 - 4 [(A- • q)(k • q) - ap(q • q)2] - 2 [0q • q + nq • q] Q2 + ( [A'2 - 2o(1q • q] Q2 } = 8(1 - e ) Q 2 { [ l - a - 0+ (1 - e)«0]Q2
- (1 - e) 2 (2 - f)~lk2}
(3.216) .
In the first line we have discarded any terms that are linear in k and so give vanishing contributions in an isotropic integral. Whilst in the second line we
A
, . s
s
ss s s
)
, f A.
( .20. )
s
(( s s
s
)2(2
(1
) A
2
). T
s
( .27)
)
f\\n f\\r o ' o
( 1
)
(1
) )
r ( 2 - 2 c ) le 2
(
2 2 +
0 2 )2
s
.
s s
(i + (1 - f) x ) -
s s s s s
(
( .21 ) s 11
.
T
..
11
s s
( .21( ) s . T s ( .20 1)
{ 1
(
T
. s ( .21 )
s
A
)0
) + (1 0 ( )
(1
(1 0 1 r
2c +
( .218)
(1 20 s .)
s r
„i-i'V. / '
( .217) ( i'ir
\
s
s
l'(l +pr (l
-
s
.
( 2
( * * i ' 2 \ r(i + or-'(i - o ( 2 b - c p ) r(i 20
8
1
3 7
\ +
• ( .21 )
. . s . ).2. Tlic. combined ( s) ( .210), , ss , ( .2( ( ). ss s s ) ,
s contribution ( .21 )).
s
A
, s s.
2
(
.
e) \( 2
1)
s . 0(
s)
I III
111
lilHOItY <)!•' <}(!>
I'C I Q l ' C Hi - 2 0 [
+ °(0)
( [1 + P ( f ' ) ] (
= ^ o { l + -C,.-^
+ 0(c£)}
(3.220)
In the second line we made use of the expansion '&<{( I ) ' } = ' / ^ ' { e " " } = I 0(r') and eqn (C.26). As a result, it hecomes clear that, both the I (n¿/2)fI //-' and I /< poles cancel to leave a finite result. As a consequence we can safely take < — It and obtain the D = -I limit. Thus, for a suitably inclusive definition of the hadrouic cross section we avoid the potential infrared catastrophe associated with soft gluons. Before going on to identify the general characteristics of infrared safe observables we mention a succinct way of organizing t he above cancellation using cut-diagrams. T h e 0 ( i \ s ) cross section is represented by the diagrams shown in Fig. 3.21. T h e two basic diagrams have each been 'cut' in two ways. On the left-hand side of the cut we view it as the usual Feyninan diagram corresponding to M . On the right-hand side of the cut we view it as a "reversed" Feyninan diagram corresponding to M ' . Thus, the top-left diagram represents the interference between the two tree-level Feyninan diagrams, whilst the bottom-left diagram represents the interference between the tree-level and one-loop correction to the basic ~,qq vertex. You should notice that these are the same diagram cut in two different places. This one diagram encapsulates the contributions which must be taken into account, to ensure the cancellation of infrared divergences.
3.5.3
Infrared safe
observables
Our calculation of the cross section for electron positron annihilation to hadrons demonstrates that it is free of infrared divergences to at least 0 ( e i s ) . In fact, the KLN-t heorein (Kinoshita. 1SKÍ2; Lee and Nauenberg. 1964), and its generalizat ion to QOI) (Poggio and Quinn, 1976: St.crman. 1976). guarantees that such a fully inclusive observable is infrared finite to all orders. This theorem can be extended so as to apply to other observables (Sterman and Weinberg, 1977: Dokshitzer rl al.. 1980). T h e key requirement for t he cancellation is t hat a quark and a quark accompanied by any number of soft gluons a n d / o r colli near gluons and qq-pairs are treated the same. Likewise. |g) and |g + " i g s + "2g|| + ":j((l(~l)||) must give t he same contribution to an observable. Now. in practice cross sections are often weighted by functions corresponding to physical measurements on the parton final states. In general a measurement is described by an expression of the form
INI-'UAUKI) SAM' I Y
UNfVF.RSIDAD AUTONOMA DI MADRID RIBLIOTKCA nnNciAS
Fie. 3.21. T h e cut-diagrams which describe the comrihutions to the amplitude squared for 7 ' —» hadrons at 0 ( o s )
' = Ì
E Ì /'(I
{
(3.221)
/>„ — 1
I=i7
/>„ = <5(.V — X„(/>,))
I = -j"Y
(total cross section) (differential cross section) .
I'he sum includes the contributions from all n-parton final s t a t e s and. on the assumption t hat quarks, ant ¡quarks and gluons are not distinguished, we include I lie syuunetrization factor I/;/!. T h e weight functions of t he ii linai state partons. /»„(/'i /'„). define the measurement. For example, to obtain the total cross section. / = a . we simply use (>H(]>i) = 1- Often, we arc interested in the distribution of some variable A* which is given by t he function A',,(/*,•) for n partons. Typical observables discussed later include the event's T h r u s t , see eqn ((i.2). or a jet resolution parameter, see eqn ((5.3) or ((>.-1). To obtain / = drr/d.Y we have to use />„ - <S(.Y — A'i,(/>,))• However, in view of the cancellations required l'or infrared safety, the functions A'„+| and A'„ must become equal in the soft and collinear limits. 'V,1 + I(/>i X»+i(l'i
A/»„.(l - A); n ) ' J = A-„ (/'• I>n A))
!>,.)•
(3.222)
Whilst, this is a requirement imposed on theoretical grounds, you are reminded that it also has a basis in experimental reality. T h e results of a measurement
I III
1 HI
I IIKOUY OF <JCh
s111• 11111 be insensitive t<> changes in a detector's energy resolution or granularity. As a result of this requirement any observable which is not a linear function of the parton four-momenta will not lie infrared safe and should not be used in experiment or theory. If you do not heed this restriction then you will be sensitive to long-distauce physics. This means either introducing a non-perturbative model, upon which your results will depend, or accepting that, using pQOD will give infinity at NLO. Despite this established wisdom it is still too common to see variables such as Sphericity (not to be confused with spherocity) or (some) cone based jet finders which are infrared unsafe! A special situation arises for observables such as structure functions and fragmentation functions. By their n a t u r e the inclusive summation over initial or final states is restricted by the requirement to contain a specific particle. This results in residual collinear singularities which can be treated in a maimer which is reminiscent of renormali/.atiou: see Section 3.6. T h e first example of an observable specifically designed to satisfy the infrared liniteness requirement is the Sterman Weinberg jet definition (Sterinan and Weinberg. 1977). O t h e r examples of early variables are given in (Basham et i+l>,)2/Q2 < y for a given value of the resolution p a r a m e t e r ; / . By construction we liavey < 1/3. A two-jet event occurs for two basic phase space configurations. Either, the gluon is radiated sufficiently close in direction to the quark or with sufficiently low energy that ( ij we see that the three-jet region of phase space is confined to a triangular region. 2// < .r, < 1 (/. in the centre of the r ( l — plane away from the collinear and soft singularities, see Fig. 3.12 where y 1 - T. This means that calculating the 0 ( n s ) . three-jet. cross section is relatively straightforward as. unlike the two-jet cross section, it does not involve any infrared cancellations and therefore it does not. require a regulator. One can then use the known result a-> + <73 = rr,,[ 1 f (3C;.-a s /(4jr)], eqn (3.220). to infer the two-jet. cross section. Rather than do this in one step we first use eqn (3.221) with A"„=.i = min{ I — .r\. 1 — .r >. 1 — ,/*;t}. A' - y and the matrix element squared given by eqn (3.112),
0
*' J -t-1--
" (1 -x-)(l - i ) jr=l—y
+
[1-X)(l-X) t»t+(-I
mi
g e n iMi'i«>\ I ,D p a u t o n MOI>I:I.
.'/)• i '
c
PC
i •'•)
-••>•)
//(i
^ { i Z i + i Z i J „ ( • V - 3.V -I 2)
2-
?/(i - //)
- 2//^ _ V
?/
3
+ / / " ' - 2 ( 1 -.;•)(.»•-(/)] | ( 1 - •%)(! + y)
/
(3.223)
//
This expression shows n characteristic logarithmic divergence a s >/ —» 0. thai is. as tin' singular regions are approaclied. In passing, we mention t h a t this expression l1,Ives the Thrust distribution with T = I — y. Integrating this expression from II to the phase space limit 1/3, we can calculate rr:\ and hence the »-jet rates delined by f „ = n,JnM as A W = g c ,
mid
hiv)
{ 1 4 - %
= 1 - h(y)
-
+ (3 - % ) lu
•
(3.224)
Here, the dilogaritlnn or Spence function is delined by Li 2 (x) = -
f
(3.22 r ))
.
It arises frequently in Q C D calculations. E q u a t i o n (3.224) shows that in the limit '/ • f:i{u) diverges as ( n s C y / ~ ) In"//. T h i s can give t h e counter-intuitive rc(//) receives c o n t r i b u t i o n s from an interference t e r m . In calculating l.i(u) we restrict ourselves t o a subregion of t h e real phase space so that the full cancellation of divergences which occurred in eqn (3.220) is now only partially complete and residual logarithms remain. At higher orders we may a n t i c i p a t e terms of t he form ¡«• s C/p/(2/r) l n J / / ] " ' . Such large e n h a n c e m e n t s have obvious Implicat ions for t h e convergence of cross sections. Consequently a significant effort is dedicated to identifying and "rcsumming' such cont ributions. For reference we also give t h e 0 ( ( \ s ) . three-jet rate for t h e Sternum Weinberg j e t definition. M , 6 ) = g o v s {in ( i )
[in ( 1 - l ) - | ]
+
£
- 1
+
I n ^ In«,} .
(3.226) Here, it is
T h e Q C D improved parton
model
The naive (quark) p a r t o n model is independent of Q C D as such a n d indeed was invented before Q C D existed (Bjorken and Pasehos. 196!): Feynnian. 1972). It
I III
I l l l \ ( >KY <)l
QCI>
began as a (|uasi-classical model lor DIS. based upon the idea that a badron can be described as a collection of independent partons. with little transverse momentum, oil which a lepton can scatter via the exchange of a vector boson. I'lii- constituent quark model supplies q u a n t u m numbers to these partons and thereby suggests relationships between the various structure functions (Close, l!)7!l). At this tree level all that p Q C D supplies is support for treating the partons as independent, via asymptotic freedom, and candidates, the gluons. for the eleetrowoak neutral partons inferred from the apparent violation of the momentum sun rule (Llewellyn-Smit h, 1!)7'2). This parton model picture of DIS is easily generalized to hadron hadron collisions. The parton model comes to life when we add p Q C D corrections (Altarelli. 1982). This brings to the fore quant um effects and changes our picture of the partons wit hin a hadron. An essential feature of the parton model is the separation of a cross section into hadron independent «»efficient functions, which describe t he parton scatterings, and scattering independent p.d.f.s, which characterize the hadrons: see. for example. <'(111 (3.227). In order to maintain this separation ¡11 t he presence of QC'D corrections, we are obliged to make t he p.d.f.s scale dependent., that is. functions of both x and Q2. This introduces the idea that a parton contains within it further 'daughter' partons and that these are revealed when the Q~ of the probing vector boson is increased. This scale dependence is governed by the famous D G L A P equations and whilst it remains true that, in the absence of suitable iion-pert.urbat.ive techniques for QC'D. we cannot calculate the p.d.f.s from first, principles, we can deduce the p.d.f.s at one scale from a given set at. another scale. Introducing p Q C D also forces us to give a precise meaning to the idea of factorization, which has now been proved to hold in p Q C D (Collins and Soper. 1987). In doing so we give greater legitimacy to the QC'D improved parton model: so much so that, the QC'D improved parton model provides the conventional framework for carrying out. p Q C D calculations. Inevitably, when we add pQC'D corrections to the naive parton model, the necessary mathematics becomes more involved. However, we believe that the underlying ideas are not that complicated. Therefore, alter repeating the tree-level treatment of DIS we give a heuristic development of the NLO pQCD corrections to DIS and factorization. This is followed by the complete 0 ( ( \ s ) calculation. After this we switch at tention to the D G L A P evolution equations and their generalizations. This is followed by a discussion of how t hese equations take account of large logarithmic enhancements to a cross section. Finally, we show how the factorization formalism is applied to the Drell Yan process in hadron hadron collisions. 3.6. 1
DIS at tlic parton
level
T h e formal description of lepton parton scattering follows that for lepton hadron scattering. The partonic cross section. i \ n f f . is given by eqn (3.32) with two modifications: the hadron momentum p1' is replaced by the parton momentum yp1' and in the hadron tensor the state |h) is replaced by | / ) . / { q . q . g } . to
T I IK Q<'l> I M I ' K O V H I ) l ' A l t I O N M O D K L
give //,',), Again gauge hivnriance ensures that, this partonic tensor retains the form given in eqn (3.30) but with partonic structure functions. These partonic .11 net lire functions are functions of yp1' and which, thanks to Bjorken scaling (Bjorkcn. 1909), occur in the combination - =
£
f
dyMvW
{ t / )
(j)
.
(3-227)
I I1 is implies / C " V'A') =
,
£
f ( V P ' < l v ) Jo .'/
•
(3.228)
where the factor l / y can be traced to the1 scaling p1' —> yp'' used to obtain the lepton parton flux factor. Rather than work with the full hadronic tensor, it is helpful to project, out two combinations of struct ure functions, n{y],)
-v""tCh)
=
/ / j n " = p"p"fi^U) Q
>
(2 ,•)-'
,
^
I1 +
(3.229)
+
I his will simplify the expressions with which we have to work. Here, with a view to future use. we have chosen to work in D = 4 — 2< dimensions. Similar projections can be delined a t the parton level. In the case of //>• t his is straightforward but for Hi, we have t.o use the parton momentum, yp. in the equivalent of eqn (3.229). Referring t o eqn (3.228) we t hen have = E
« r
/' ^ A f o ) " ^ ' W ) =
= E
E
f
~ f u ( f ) W H z )
E
4
/ - / ' • ( - )
• (3.230)
1
2
Remember t h a t scaling implies that the i i \ ^ are functions of (¿ /(y2]> • <7-, ) = r/y. T h e advantage of the 'total' structure function, fly ^. is that, it is essentially the matrix element squared for the vector boson parton subproccss.
I I I I . I I I K O K Y <>!• ( ¿ C I )
I '.'I I
I'lic 'longitudinal' structure function. 11'^ 1 1 . i.s particularly nicc because many diagrams give vanishing contributions and those that do not vanish at 0 ( n „ ) are free of infrared singularities. In e<|n (.'5.22!)) we recognise the combination in square brackets as the longitudinal structure function: see Ex. (3-5). Once we have calculated / / v and ///. we can invert oqn (3.229) to give us the structure functions !•'•> and F \ . F,(-r)
„ ( V I . ) , (3 - 2 Q 4:r 2
1
(1-0
(1-0
• -l
Q 1
(yip
2
Illvf)(z)
<1-0 r/
^
I
(3 - 2r) - I " r ^ ' W (1-0 Q-
4j >
" M,l'h) 1
,
d: . / x \
U- i , i v f ) , .
(3.231)
'r
Here we have made the simplifying assumption that the 2xM\JQ terms are negligible. In what follows we shall also neglect all quark masses. This makes t he algebra simpler and will cast into sharper relief any eollinear singularit ies. 3.(¡.2 I)IS ill leading order We now calculate the leading order cont ributions to DIS in t he part.on model. We will focus on electromagnetic exchange in which a photon couples to the electrically charged part.ons: quarks and antiquarks. The charge conjugation symmetry of (JED and QC'D ensures that quarks and antiquarks give the same com ribntion. Thus a t C ( o " ) we need only consider the one tree-level subprocess -/'(| — q'. The treatment of Z and \ \ r ± exchange involves only minor modifications. To calculate fly ''1 we first require the matrix element squared. This is easily evaluated in D dimensions, Y , \ M ( l ' < l -
(3.232)
Here Q2 = — (//' — c/)2 = 2q • I). Next, we average over t he spin and colour polarizations of the incoming quark. 2Arc< and include the one-body phase space integral, oqn (C.19), to obtain I d < l » , ^ |,Vi(7'q - q')|- = 2 e 2 e 2 ( l - e)Q2 x 2*6(q'2)
.
(3.233)
Here we have used T r { l } 1 I. Since the struck (¡nark carries a fraction // of the parent hadron's momentum. ''. the ¿-function. which const rains the scattered quark to be on mass-shell, can be rewritten as q'2 = (,/p + (h)2 = g2p • q' - Q- = 2P • q'(g - .»•)
I III
g<'l) IMI'HOVKD I'AIM'ON
=>
MODI.I.
-<$(.'/-*). 2p • q-<
(3-234)
Here x is the usual Bjorken-.r. eqn (3.39). At this point we pause to observe t hat the origin of this 6(g - x) factor is purely kinematical and therefore we may anticipate that all the one-loop corrections to the 7*q —> q' vertex will also be proportional to i)(y x). Finally, following convention, we divide out a factor •Irrc~ to obtain H[?'l) =
d . l » , 2 |,W(7'c,
q')|- = <>2(l -
f
)
" -0
- c2(l - c)x6(y - x) .
(3.235)
The elfect of the ¿-function in e<|ii (3.235) is to select only those (anti)quarks with momentum fraction x. T h e presence of a ("¡-function also means that the partonic structure functions are formally distribution functions (in the mathematical sense) and so only have meaning when integrated with a suiliciently smooth ordinary function. The calculation of //} " l ) . oqn (3.229), is even easier as it vanishes. This follows because, assuming mnssloss quarks so t hat q) = 0. we have <1„M»(7*"
(3.23(5)
Given fl{y <|) . together with /"/J""1' = 0. we use oqn (3.231) to obtain the lowest oi'dor electromagnetic structure functions 2 x F ^ ' \ x ) = F?U\x)
= x £ c}fu(x) /-q.'l
.
(3.237)
I his confirms the Callau Gross relationship bet ween F\ and F> which holds at lowest, order for scattering off a spin-1/2 parton. As wo will see at 0(os) and beyond F>(x) ^ 2xF\ (x) and the two structure functions can no longer be regarded <us equivalent. In Ex. (3-24) the same result is obtained using the explicit hadron tensor. This also demonstrates, as one might expect of t he parity conserving QED. that F-]t~,U) = 0. This calculation has familiarized us with our notation and proved the Callan Gross relationship bet ween the structure functions appearing in the parton model at tree-level. We now wish to investigate how this picture changes with the inclusion of p Q C D corrections. 3.(1.3
A heuristic
treatment
of
factorization
A number of processes contribute to the structure functions at 0 ( n s ) . If the struck parton is a(n anti)quark we have the tree-level scattering 7*q —* q'g. the so-called Q C D Compton process, shown in Fig. 3.22. To this must, be added the interference between the' tree-level and the one-loop corrections to the basic scattering 7 ' q —> q'. The situation hero is similar to that encountered in the treatment of the pQCD corrections to the process 7* —• qq; see Section 3.5.
I III; I llKOUY ()!•' (.¿<'l>
'Piloro, both I lie ultraviolet and infrared singularities in the t wo sets <.l contribu1 ions cancelled to leave a finite result. Flectroweak vector bosons d o not. directly couple to gluons. In order for gluons t o c o n t r i b u t e to t h e s t r u c t u r e functions they m u s t first split into a charged qq-pair. T h u s their contribution is at least O ( o s ) . T h e lowest order contribution comes from the tree-level process 7 * g — <|(|. the so-called boson gluon fusion process, shown in 1'ig. 3.23.
F i g . 3.22. T h e two d i a g r a m s c o n t r i b u t i n g t o the Q C D C o m p t o n 7 « q _ ( ) 'g, a t leading 0 { o j a n d . reading right to left, gq' — 7*<ì
process.
FIG. 3 . 2 3 . T h e two d i a g r a m s c o n t r i b u t i n g t o the boson gluon fusion process, y j , _ q q . a t l leading 0 ( a j a n d . reading right to left. qq — 7 ' g
We shall c o n c e n t r a t e on the tree-level. C>(n s ). process 7*q — q'g an: is proportional to t h e m a t r i x element squared for the hard subprocess. T h i s can be obtained from that for the process 7* -
Y , |M(V
-
M's)!3 = »¿'Wrv
g-g' {T"T" )
!) • <1
+
<1 • ' / '
Q%q') (3.238)
T h e t h r e e t e r m s correspond t o emission of the gluon off the outgoing q u a r k ,
l lll-: <JCH I M I ' I I O V K I ) I ' A K T O N MOIH'.I.
oil tin' iucoiniug <|iiark a n d the interference between the two contributions. To obtain I I I " 0 we need to average over t he incoming quark spins and colours. 2.V,.. divide by the conventional factor Attc'2 a n d integrate over the two-body phase »pace of the final s t a t e particles. = =
/dW(7*q -
1
c,'g)| 2
/d, i f f I « + ./ 1.9 <7 fl 'l (
•
(3-230)
•'/').
Here we used Tr{Y'"7'"} = C/?Nc. Looking at the p r o p a g a t o r s , rx [/? K (1 — cosfl)]" ' . we see that, t his expn'ssion has a n u m b e r of singular regions; c.f. Section 3.3.2. T h e r e are col linear singularities when the gluon is emitted parallel to the incoming q u a r k . 0. T h e r e is also a oft singularity when the energy of t he gluon vanishes. T h e collinear singularit ies are associated with the vanishing of t h e q u a r k p r o p a g a t o r s just, prior to or just after the interaction with t h e p h o t o n . Such a low-virtuality i n t e r m e d i a t e q u a r k will travel a large distance. ./" = A " / k 2 . so t h a t the gluon is e m i t t e d eit her well before or well a f t e r t he h a r d subprocess. We may t herefore a n t i c i p a t e t h a t t h e Init ial s t a t e collinear singularity can be n a t u r a l l y associated with the incoming liadron a n d into which it might be a b s o r b e d . T h e final s t a t e collinear a n d soft gluon singularities both imply a zero mass particle in the final s t a t e . A' = (
•
= fa +
(
3
.
2
= Q2 ( 7 - 1) = Q
1 4
^
0
)
•
In the second form we let the incoming quark carry a m o m e n t u m fraction 1/. that is. 7'' = up'1 and we introduced z = x/y where x has its usual meaning, eqn (3.39). T h u s , s = 2 y • 7' —«• 0 is equivalent, to Q2(y/x - 1) —> 0 so t h a t these singularities involve the kinematics of the lowest order hard scattering to which they must consequently be associated. T h e y also raise the s p e c t r e of infrared singulai cross sections. Fort unately, for t he analogous process 7" —> qqg we learnt that, including t he contribution from t he interference between the t ree-level and ouc-loop corrections In I lie process 7 ' — qq ensured that all the singularities cancelled in the infrared safe total cross section, eqn (3.220): see Section 3..r>. Hence we might expect that t h e singularities present, in eqn (3.239) will cancel when we include the contribution coining from the interference I »'tween the tree-level a n d one-loop corrections to the process 7 ' q —» q'. Unfortunately, in DIS t h e probing photon can tell the difference between the charged quark in a collinear qg-pair a n d a quark <| with equal m o m e n t u m . T h u s , the equivalent cancellation for DIS is incomplete. In particular the final s t a t e collinear and soft, gluon singularities cancel but the col linear singularity associated with the incoming quark remains. This removes I lie danger of a singular DIS cross section, provided we have a means of dealing with t h e remaining init ial s t a t e singularity.
IJI
I III
In view of the above singularity present in eqn invariants. To evaluate y frame. Here t he momenta
I I I I O l t Y ()!• ( ¿ ( ' I I
discussion we will (3.239). E(|iiation •
analyse the initial state, collinear (II.210) gives us one of the Loronlz helpful to specialize to the C.o.M. q ' and g can he written as
<1"=PU,( 1 . 0 . 0 , 1 )
l±±p
A b =
2v/s
!l" = Pout(1. + sin 0'. 0. + cos0-)
,\mt =
(3.241)
.
These allow us to infer that 2it • <1 = 2 —
(1 - cost? )
1
>
~ O2 = ^-.(l-eosr)
^
— (1 + cos0 )
Q2" = £ ( 1 +cos0*).
(3.242) T h e two-body phase space integral is given by eqn (C.21) wit h, for the moment, < I). In terms of these C.o.M. variables eqn (3239) becomes //<;"'» = 4 c 2 « , ( 7 f — % / ' d cos 0' Mtt V/S J-1 [
(l-cos»•)
2 ( 1 — z)
[(1 — cos0*)
2(1 - c)
2 ; ( 1 ( COS*-)
(I - c)(l - c o s f * ) J
'
Referring to eqn (.'{.211) we see that the c o s 0 ' — 1 singularity in eqn (3.24:1) arises when the gtuon direction approaches that of the incoming quark. T h e soft gluon and final s t a t e collinear singularities manifest themselves as the c —< 1 singularity, see eqn (3.240). Now, rather than work with cos if*, we choose to use the transverse m o m e n t u m of the gluon measured with respect to the incoming quark direction. =
(3.244) Q
2
(l-z)
4
2
.... COS
_
'
2cos0' k
2 v
d cos»'
(1 + c o s 0 ' ) ( 1 - c o s 0 ' ) '
T h e limit cos0* —» 1 now becomes k 2 r —• 0. Making t his change of variables in eqn (3.243) gives =
(3.245) rQ
^dA-f 2 cos fl* J / f (1 I cos«*)
(1 - z)2 + (1 + cos f)')z (1 - z )
(
I - cos ft" 4(1 — z)
where c o s 0 ' is now implicitly given in terms of A-/-. Notice that we have introduced a cut-off. s2. on the transverse momentmn in order to regulate the
I III 0< 'I > IMl'UOV 1.1 ) I'AIM'ON MODKI.
loHincar singularity. At small opening angles, c o s 0 ' —• 1. the virtuality ol" the Intel m e d i a t e quark. 2 //•// in eqn (.'t.21'2). and the gluou's transverse momentum, 1 — z). Thus. is also a lower bound eqn (3.2'ly 2(f/ • ¡¡) = on the minimum virtuality of the intermediate quark, equivalent to an upper liound >in the distance it travels. This ml hoc prescription will he replaced by ilimensional regularization iu Section 3.().-l. Focussing on the col linear region we obtain Ili?'0
R"(z)
+ O ^ / C f )
. (3.2-lfi)
I'lie linal state, z —* 1. singularity is still present iu this result. Invoking the l ' ) ( o J corrections to the 7 ' q —+ q' vertex, which are proportional to ¿(1 - ; ) . this singularity is removed. A more proper treatment would show 1 is that the coellicient is actually a distribution, 1 4
(1 -
zh
CtmS( 1 - s) + "'(z)
I (3.2-17)
Here, we introduced 1/(1 prescription is defined by F(s)+
z)+ as a short hand for (1/(1
= F(z) - ¿(1 - z) I d y F ( , j ) Jo
z)).
where the plus-
(3.2-18)
Distributions only make sense when integrated with a suitable smooth function. We typically encounter the plus-prescription in a situation such as 9(<
f(z)
which is free of divergences provided u(z). and / ( ; ) . a r c non-singular. The full calculation would also have given us the form of the unspecified coellicients C',,,, and /»'(c). Later we shall use a physical argument to extract 3 / 2 . Given C, | ( | . then eqn (3.217) fully specifies the regularized, lowest order. Altarelli Parisi split t ing function /^"'(-i)- e and /•'|. Including t he lowest order contribution, <><jn (3.237). and. for simplicity omitting the sum over quark Havours, gives /•,(. r. (/-':/,)
I III
I2fi
I IIK< )RY OK QCD
(H.250)
« • / M i
In lliis expression we do not show explicitly any dependence on the renorinalization scale /in which anyway does not enter at C?(mJ. Now, having identified and isolated the initial state singularity in F>. we must decide how to deal with it. Equation (3.25(1) exhibits a large logarithm coming from the collinear singularity which we have identified with long-distance physics. What, we would like to do is to faetorize eqn (3.250) in such a way that, all long-distance physics is contained within the hadrou specific p.d.f.. whilst all short-distance physics is contained within a hard-subprocess specific coefficient function. In order to facilitate t he separation of the long- and short-distance contributions we int roduce a new factorization scale, ///.•, into eqn (3.250). Our aim is to move the logarithmic singularity into
(3.251)
+
¿ T
• © £ [ « < • > - ( £ )
i- n u z ) -
R!AZ)
This form suggests deliniug a factorization scale and scheme dependent p.d.f. which absorbs fully the collinear singularity < : k ) = ,<*) 4- f *
(i) g
[/>;:;>(,) In ( g )
4- < ( • : )
(3.252) The second term on the right-hand side of eqn (3.252) is logarithmically divergent as h~ —> 0. but we expect the p.d.f. on the left-hand side to be finite. In an argument that is very reminiscent of renonualization we claim that the 'bare' p.d.f.. 0 limit. / ( a : , < )
=
») g
[ 0 * > ( 4 )
4 <(*
(3.253) This 'physical' p.d.f. is now finite and so we may d r o p any reference t o the k regulator. In terms of eqn (3.253) we can rewrite eqn (3.250) to 0 ( o s ) as Fijx.Q1)
~2 <1
r,
= <1
a
r>/.'\
(r.lly.R^)
11 IK QOI> IMI'HOVKD I'AHTON MODKI.
127
llii- right-hand side of these equations appears to depend on ///.• and /?,''. However. all depeiideuee on ///.• and />',' cancels to the calculated order. O ( o s ) . and any dependence at 0(t\2) would also cancel if we included the neglected 0(n'~) terms in eqn (.'i.25'1). As the original expression, eqn (3.250), makes clear, the physical F>(:v,CJ~) is independent of both the arbitrary factorization scale and scheme. What we have gained, as (lie second form in eqn (3.251) makes clear, is that all the long-distance behaviour is contained within the finite p.d.f. whilst tin' process dependent coefficient, function only contains short-distance physics. The significance of ///.- is that it delimits t he boundary between short . Q > ///.-. and long. (J < ///.-. distance physics. In eqn (3.25-1) both the choice of///.- and ¡{¡' are arbitrary. For the scale, choosing ///.• = Q is clearly advantageous, as it yields the simple expression iM.i-.Q-) (3.255) The choice of which finite terms from /?,,,, in eqn (3.25(1) to absorb into /?,'" defines the factorization scheine. T w o schemes are in popular usage. In the (modified) minimal subtraction scheme only the singular term is absorbed into the parton density function, t h a t is. = 0. In the DIS scheme all of the finite term, together wit h the singular term, are absorbed into t he p.d.f.. that is. /?,. This scheme results in a particularly simple form for the structure function. (.'<•. Q-) = .
(3.256)
I lie above reasoning which leads to factorization applies equally well to l r \ ( r . ()ne might t herefore be tempted to define DIS p.d.f.s according to the e(]iiivalent of e(|ii (3.256) for F\. However, yon should be aware t h a t at C ( o s ) F> /- 2rF\ and the two schemes will not be equivalent.. Equation (3.256) is the conventional definition. It is significant that /•'/, F>/('2.r) — F\ is infrared finite and in particular contains no collinear. initial state singularities. This means that the same redefinition of the p.d.f.s used to render F> finite will also render F\ finite. Although we will not. demonstrate it. this is also true for /•'•(. As the discussion of factorization schemes makes clear, the p.d.f.s should not be regarded as physical quantities, since they depend on the scheme used to define them. However, when convoluted with the appropriate coefficient, function, eqn (3.251), they give rise to physical, measurable structure functions. The crucial point to remember with regard to factorization schemes is that the same scheme must be used for both the p.d.f.s and the coefficient functions. II this is not the case then the cancellation implicit in eqn (3.25-1) will not
nil
HIKOIIY OK QOI)
occur and ¡1 will hoi be rciiiivjilciit to <<<|ii (3.250). Since the p.d.f.s in the MS scheme carry no information that is specific to lepton hadron scattering, tliey are easier to use ii applications to hadron hadron scattering and thus often are the preferred choice. If DIS p.d.f.s were used to describe another process, then the new coefficient functions, describing that process's short-distance physics, would have t o include a compensating factor of take •n from the unrelate'd DIS process. 3.(5.4
DIS at next-to-leading
outer
The above discussion of factorization avoidexl technical eletails so as te> concentrate on the e-ore ideas. We now explicitly carry out this process fe>r the case of DIS. \Ye> will use dimensional regularization throughout to deal with all the singularities. 3.(5.4.1 The pmcess -j*<| —> q' at C?(u s ) There are two contribut ions to the process -y'e| — <1* which ne-e'el to be> considered at 0 ( « s ) . T h e real, tree-level scattering 7*q —• q'g. Fig. 3.22, and the virtual. e>ne-loe>p correlations te» 7*e| — q'. These» are both very similar to the- p Q C D corrections to the process 7* —» (|(j which we have already calculated: se'e Section 3.5. In fact , to obtain '/,„///'"' we can use; crossing, e-epi (.'i.i>r>). for the reepiire'd matrix elements without any further calculation. The D-ehmensienial amplitude! squared for the' process 7" —> ejeig is given by e'ejn (3.207). To obtain the> amplitude squared for the process 7 'ej —• e|'g we- ne-e-d to make the substitutions <j'' —» —
I
Here we have replaced Tr {T"T"} variables 2g • (/ = ^ - ( 1 _ z) , z
)2C,,Nr2
i - O
Tr {1}(1 - 0
—7 + [.'/ •'/ g-
(3.257)
+ 7—w—r: + (ft •
2 <
fJ
by Ci--Nc. As before we chose to use- the C.o.M.
2g q=^-v
anel
z
2q • q =
1 - v) .
(3.258)
which diller from e-epi (3.240) anel eepi (3.242) only in the replacement of c o s « ' by r (1 + c o s 6 * ) / 2 . In terms of these variables e-ejn (3.257) bee-omes Y.
)2CpNc2Tr v
{1 }(I - e) (1-01
(3.25!)) 2r
(1 - 0
n
\
To this expression we should aelel an average over the spin anel colour of the ineoining quark, 2Nr. divide by the conventional fae tor Inc 2 and include tlie* two-body phase space integral, e-epi (C.21). This gives
mi
g e n i m i ' u o v k d I ' a h t o n m o i mi.
Mv,)
">:.n
. 7 — f Z l^(7*q -
'
F(1 — 0e
4mVi \ M j
* J
1
(3.2(50)
q'g)r
- ,.)- | ( l
« (1-0
-0
+
2z
(1-0
(1 - 0)
)
(i
The r-integral is of t h e standard Euler 0-funeaiem type, eqn (C.27), and is évaluai eel to yield f •1 a
deiT'(l-t,)-{...}
-{a-
(1-0
2z r(-or(2-f) (1-0 r ( 2 - 2c) _r
8
(l-e) r r{l - 2 c ) \
F(1 - 2c)
r(3-2e) (
r2 2er2{1~(n r( 2 - 2 0 /
( j - e) (1-0
+-
1
+
(1-20(1
- 0 .
(1 — <
2(1-0(1-20
(1-20/
3 1^(1-0/ 1 i + z2 r(i - 20 t " n ^ ô ~
1
/ *
-
7
f
\
1
- 2(rrryJ ' t P W } • (3.2« 1)
The simplilicat ions in the second line have been achieved using e;epi (C 25) whilst ,llinl lino w o liave expanded out "' ing e-epi (3.2(il ) into eqn (3.2(50) gives
,,(>
2« s 1 1
/ X
{4 n/r
i
r
w
z
+ z2 ^
V
3 ~ m
xpression in curlv braces. Substitut-
F(1 - f)
1 —
/ )
+
-
;
(
7 G
\
- I F T I J
-, ^ *
•
(3.2(52) Identifying the- f
() limit in eqn (3.2(52) is a little tricky but using the identity IA/I
( i - o -
•
- ) ,) _ !4 _
2
/ 1 » ( 1 ~z)\ 1
I.12 ^
p
m
i;tn
r n i i TIIKOUY O F Q(T>
sec Fx. (ll-2(>). wo finally obtain /•/l->-l)
(
(1 - C)Hf
(3.264) T h e double pole. 1 / f 2 . is duo to tlio soft gluon singularity. A second eontribution to tlie total s t r u c t u r e function at 0{oj comes from the interference between t h e process "y'q —• q' at one-loop and at tree-level. T h e structures of the one-loop vertex and tree-level diagram are the same, so that we can combine t h e m into an effective vertex iF" = - i c c t f " -ic.,,7"
1
n «
// iïïiiY1N
Os
/ ^ W
'
r(i + pr2(i - o ( 2 1(1 —
(2
3
3 0
7T"
\1 \1 (3.265)
Here, the one-loop contribution has been inferred from eqn (3.21!)). t h e only difference being the absence of the ( — 1)' factor, reflecting the space-like n a t u r e of <7 in DIS. In t h e second line we used F(1 + r ) P ( l - f ) = 1 + ( f f 2 / 6 ) f 2 + 0(e' > ). Equation (3.265) has infrared singularities but is ultraviolet finite. T h e calculation of this additional contribution to ¡ l y ' <|) is straightforward, giving H™
= e*(l - f)rf(l - z)
(3.266)
Adding eqn (3.264) and (3.266) together we see that, the 1 / f ' terms, the soft gluon pole, cancel.
+ 3 - r -
Q
+
(3.267)
T h e remaining 1/c pole is associated wit h the colli near singularity for gluon emission off the incoming quark, its coefficient, is the regularized, one-loop. Altarelli I'arisi splitting function
I III ( ¿ r n IMI'HOVICI) I'A IM'ON M o I >101. I
~2
1 ^ ( 4 = Cr
131
vT + (1 Z T z)
1
P
^
(3.26S)
- • • m r ,
I'his calculat ion supplies us wit h the value of C m 3 / 2 in e<jn (3.247). We also need to calculate the longitudinal part of the hadronic tensor. T h i s is part ieularly easy to 0 ( « s ) since many of t he potential cont ributions vanish in the inassless quark limit. We have already seen, in eqn (3.236). t h a t t h e tree-level diagram, and by virt ue of eqn (3.265) its one-loop virtual correction, give no contribution. T h i s implies t h a t the longitudinal s t r u c t u r e function. /**/, oc ///.. is at least 0 ( n s ) . T u r n i n g to the 0 ( a s ) tree-level contribution and again a s s u m i n g w<> have inassless q u a r k s , so t hat ^/"('l) =
(¡,,M(7*"q
(ft' + J) , ,
- » q'g) oc u(
w + <,)lil
+
A i - i )
In
A)2
"(f//)o—"('i)f0(uY
u(
•
Thus, the d i a g r a m describing glnon radiation off the scattered quark gives no contribution, leaving only the d i a g r a m describing glnon radiation oil t lie incoming quark. Squaring this d i a g r a m and s u m m i n g over spins, where we can use if" for the lone gluon's polarization tensor, gives £
\
-
q'g)|2 = =
(.'/,/''
I
c)Tr { H u m }
=
- e)TV
=
c^(//,//')-C>A'c2(l-€)Tt{^}
= ' -';,(.'/,/'' ) ^ 7 , A v ( 1 - e)2q' • q Tr {1}
{¿M}
(3.270)
Here, we have used the t rick in Ex. (3-19) and repeatedly used jfjf = 2(j •
l'/,.X(7'"
= 1
2/
i 4 «2x
ts2r< Q2 n
\
1
I'""' f
" X
z 91 i4"''2 V r(2 - 0 * z V Q2 1 - z ) f ( 2 - 2 f )
1
f f di; v~ (1 - e)
I.TJ
I III
I IIKOKY Oh' (.¿( I)
(3.271)
k ^1 ^ + c x o . Iti Z •I
Given e q n s (3.2(i7) and (3.271) we can convolute tlicm with the p.d.f. to reconstruct / / v and ///. and hence obtain the O ( o s ) (anti)quark's contribution to F> and F\. e.f. eqn (3.231). as f ± „ m l xcl
2 n.L
/4£^Y £
: " i J i
(
' « ~> e T(1 - 2e) \ Q~ J
+ c. (I
^
/ln(l-;)\
-
1 - s
J
1+ +
In
(1-0
+ 3 -I- 2s 2(1-0+ • è / ' T ' ( i ) { - W W + cy
''
=
1
2x
2TT ./,
7"l-z)
C r Z
-
(3.272)
In the second expression for I]> we have expanded out the coefficient of the splitting function and introduced a more compact form for the remainder term. T h e l/f pole naturally arises in the combination A , . e q n (C.Hi). If we add in the leading ordier result . eqn (3.237). then eqn (3.272) takes the form of eqn (3.2-r)0) with i actiiug as regulator. If the factorization procedure removes just the \/< term we h a v e the minimal subtraction scheme, if it removes the addit ional terms. A , , we ha vie the modified minimal subtraction scheme. MS. In tin- DIS scheme bot h t he A , and finite terms are removed. 3.(¡.1.2 The 0(a„) process - * g — qq T h e calculation of the terms and qq follows the same lines as that for ( 7*
//}«» s
,H>>>' = 2c^Tr[z*
;ill!h,
11"1' =
+ ( l - z f ]
•.(1-0
I-
O ( c )
(3.273)
Using e<|ii (3.231) applied to the above results, which contain both the quark . and antiquiark terms, we obtain the 0 ( n s ) glnon's contribution to F% and
III!
<¿('11 I M I ' H O N I P I ' A I M O N
MOI>KI.
f) I'll - 0
riT^)
, , +
'"
\ /l 2 + T, \{z2 + (1 - z)21 In
" J+ M l ~ 0
T » ( f ) '=(.-2).
}
CUT-,,
:t.(i.1.3 The combined results for Ihr C?(o s ) DIS structure functions Tin* above results. eqns (.'5.272) and (3.271), can be eonibiued to give the NIX) formula for /'! 111 and F.[l'1'. In the modified minimal subtraction, MS, scheme we have ' I •>
-r^-(r.Q-)
=
T, X
/' f
E
(f.#*) k
- 0 1 £
(c<*>•» ?
• c ^ ( z ) )
(3.275) where we have also included the MS expression for F.J1 '". In eqn (3.275) i/\y gives the normalized strength of the exchanged gauge boson's coupling to the (anti)<|iiark, for example = < , r whilst t he coeflicient functions are given by .d'
cr ,
2
-
(
/'o
c f <" = d / '
0
I T O ' C - c , ( i i- z)
)
;)•
•*
r (3.27(i)
11 ,
Cf '
=
T,z
,2 . (
.
01
)
)
+
=
0
= 0 0
.. . s (. (
. )(,,
.
. . .s
.
.
.
=
s
, s
,
()
s
ss
.
. . .s
1
} (
)
s ( .277)
,
,
s
(
T
s,
.
. s
s. A
( s)
s
,
( .27 )
s. (0 = dv,,)(z) =
,.(
( .27 )
+
,,
s
.
. . .s ,s )1
2 (.
(
{
. ( ) .
,.,,. , .
ss
A
. .s s . .
( .280)
.
)
( .27(1) s
s
A
.
( .27 )
s s
( .277)
( .280) s s s. s . . .s s , . . . .s F> s s . . .s s
s s
s
s
ss.
s
s s
. s
s .
s s
s
s
s
ss s. s s
s
s s
s
. . .s
(. . s
),
s
. Ft . s s . . .s.
s
s
s
s
, s,
s . . .s f\,(x. s
s
Tlic evolution
of the jiurton densit
. . .
s s
s
if-).
ij.-)
s
( .2 2) s).
s 2
.
( .2 2),
'
> , ,
s .2.2.
s s ,
s
s . . .
s
s
s. . . . (
( ) li(cil)
s s s
) s
\
(
)
s
s
( ) s 0(
s. s. T s .
s
s s
ss s
ss
,
s
( ) s (
( .1 ) s ( ) s s. A
s , ( ) s s
( ) s
,->
) A
T
s s
( .2 1) , i '(x. ij. ) s s . . . . ..
.
( s s
. s
s
s ( .2 1), , s
) i(x. r)
),
ss s.
s s
, ( , s
.
s
ii
s s ( . 17). T s . . .. ( , s s ( .17) s s
functions
( .2 1) s
s
(
s
ss
s
T
q. g
X s
s
,( ..
. > A. T
s
.
s s .
ss s s
s)
( ). s (A ) s
)
2
2
0(
=
(
2
) =
(
2
)+'U
.'
(-.'i'2)
'h
) +
+
( ,,
2
(
)
)
)
=
, .
( .
)
(,
+
,)
,
.
( .282) T
s
,( . s b{z,
)
. s
s(
)) s
s
s
= .
(
s.
7
) +
s,
s , .
T )
s
(
)
,
.
0(
s
. T
s
.
=
1
(
.
.
s . T =
s
, .
s
s,
T
s,
s
T
s
s
s.
s s
s
s. s
s
T
s
s
s
s
( .282).
. 78).
( .282)
s
s
.
s
( .1 )).
,
,
s ss
s
).
s
ss
s
( ) s, s
s s
( .28 )
s
s (
(0 +
7
s
s
s
(. )
( .282),
s
.
ss
s ( s
s
s.
s .
1 )7
s
2
0( .
2
=
(
2
)+
.
1
(
)
.
(
11 A T
.
f=U l ( .28 ) s
s
( .1 )). T
s
s
ss
s
s
s
. s
s
s.
s
s
s
s s
s ( .)=
f\\xx . )
T
(
, 1
A s s s
{
=>
,
2
) =
s
A
s
r' i\nx-'f(n). r-i -x s
7(71
.
s
) ( .
s
s -. s s
) . s)
, 7( .
.
( .28 )
, s .
s s
s s
( ). T A
A .
),
, s
s
s
s
( (
s
f ( x ) = -^r
s
s
) 2).
, s
s
.
2
.
s
~lf x)
s
). T
s
s .
(
s
s ethod of moments A . T s
s
.) s
s
. 0( . . . .
(
s
. A . 1 82), s
s
s
s
, s
(
s
( .287) s
s
s
( .287) = s
(n.
s).
s
.
s ( ( T
s s
) ss
, s
s s s
A
s s
s
(
s. ,
s ( .22). As s . A s s ( .28( ),
s
s
s s
ss
s
s
s
s s s
( s
s
s, ( .288).
s . s
s
I NK THEORY OK QCI)
UK
sec examples Ex. (3-33) and (3-34). In the more general ease the Mellin transform of t he DGLAP equations leads to mat rix equations. These can he solved in essentially t he same way after they are first, diagonalizcd. We shall now use this moment space equation to determine the coefficient in eqn (.'5.217) whilst avoiding the need to evaluate any virtual corrections. Consider the leading order evolution equation for the difference of two quark p.d.f.s, such as (u —
.
(3.289)
All dependence on the gluon p.d.f. has cancelled. This is the evolution equation for t he non-singlet, in terms of its flavour SU(3) transformation propert ies, st ructure function. Exercise (3-8) investigates other useful combinations of p.d.f.s. Now. ./¡,d.r qg splitting function. That is.
rt
M
L
(, d; —
CV,<,<5(1 )+ ~ 2 ) ~~ 2
(3.290)
1- z
which supplies us with the value of and coinplct.es the expression for P,\"\z). The same result viewed from an alternative perspective is discussed in Ex. (3-30) and a similar approach based on using momentum conservation can be applied to
find Fx. (3-29). Of course these arguments rely on the physical interpretation of the p.d.f.s. In Section 3.6.-1 we proceeded by direct calculation to fully evaluate the splitting functions. Given the full expressions for the splitting functions, eqn (3.50), we can evaluate the anomalous dimensions appearing in eqn (3.287). At the lowest order we find: T i »
1 i -I(»-l}n (« + ! ) ( « + 2)
= 2C
1
«>(») = Cy '
n(n + I)
1 12
-w/,7>
(3.291)
^ m (3.292)
•>
^ — '
»1=1
2 4- ii -I- » '
¡a (3.293)
- r « ' ) = Tr //(// + 1 )(«• + 2) row \
[
2 + n + ir
"I
(3.294)
nil-: Q( Il IMI'ltOVKI) I'AH TON MODI I
0. thereby proving conservation of llavour. A number These show that 7<,?|)( I) of other conservation laws are also implied by «-(pis (3.291) (3.29-1). 0 = J'«*{&(*) 0 =
f <\zz{PVJ.(z)
!«•(,)
+ »/ [ f f n ( z ) - P^(z)}
}
+ 2vfP(U.(z)}
0 = j ^ \ z z { p ^ z ) -I- P™(z)
(3.295) (3.296)
+ P$?(z)
-I ,,f [P*,{z) + P,*,(;)] } (3.297)
See Tlx. (3-31) for further elaboration of how the momentum is shared within a hadron. 3.6.6
Leading
logarithms
Our derivation of the D G L A P equation focused on treating a region of phase «pace which has a logarithmically enhanced cross section. Recall that introducing / i j t o isolate the collinear singularity left behind a residual, large logarithm. There are two singular regions: the collinear region which gives logarithmic enhancements of the form o s In(Q"VQo) mid t he soft region which gives logarithmic enhancements of the form o s ln(l/.r). These regions can overlap and give double logarithmic enhancements of t he form o s \ii(Q 2 /Qo) h i ( l / x ) . Processes which invoke mult iple parton final states can have up to one h ^ Q 2 / ^ ) and one lu(l/.r) factor for each power of o s . The phase space regions which contribute these leading logarithmic enhancements are associated with configurations in which 'successive* partons have strongly ordered transverse. /.•/•. a n d / o r longitudinal. A/.( ;»:). momenta:
LLyA:
»
-»!•?,• » « 5
J " ' J DLLA: J " ! { Q* J- * I , . 1k a . x „ ( osLt « 1 LLj A:
( l \ L x Z !1 L ° s '-Q ^
«
J
"
xj
J « x„
' i «•>•<>
(3.298)
(3.299)
(3 300)
The solution of t he DGLAP equation sums over all orders in o v t he contributions from the leading, single, collinear logarit hms. [o s ln((/ 2 /Qii)]" and the leading, double logarithms [o s In((¿~/Qq) 1h( 1 /-'-)]"- This is t he region of st rongly ordered /. / and ordered x. It does not include the leading, single, soft singularities which me treated instead bv the B F K L equation (Kuraev it «/., I!I77: Balitsky and l.ipatov. 1978) which describes the ./•-evolution of p.d.f.s at. fixed Q1. Figure 3.24 .hows the ln(Q") ln(l/3:) plane and the regions which are described by the var....... ious hndinn Intiiii'ilhmir il.1,1 «¡mnmsit'.miiw 1i<>l-.ivim> >>II<> .>1" it«.
mi
riiKOHY OK <¿('i>
m 5 )
K ? )
F i e . :j.2-l. T h e In(Q 2 ) l n ( l / . r ) plane showing the regions in which the L L y , Lb,.. and DLL approximations hold. Also shown are the regions in which Reggc phenomenology applies and where s a t u r a t i o n / r e c o m b i n a t i o n effects hive t o be taken into account. constraints in eqn (.'5.298) or e<|n (.'{..'500) gives rise to a m xt-to-leading logarithmic 'NLL) enhancement to the cross section. These are suppressed by a factor '>., with respect to the LL-cnhanccmcnt. Including summed NLL-terms modifies t|ic D G L A P or B F K L equations whilst maintaining their general structure. We i|ow discuss the double leading logarithmic (DLL) approximation, the B F K L equation, the combined evolution equations which incorporate both D G L A P and B F K L evolution a n d t h e generalizations to include parton recombination. We shall make more explicit the relationship between these equations and the lead ng logarithms in the following Section 3.<>.7. 3.(>.(;. i The double hading logarithmic approximation At. small x and large Q~ we liiust sum the leading o s U i i Q ' / Q ^ ) l n ( l / . r ) terms. T h i s can be done directly froiii the D G L A P equations by keeping only the most singular 1 / ; t e r m s in the splitting functions. At. C9(o s ) only P ^ and P m have soft gluon singularities, but at O(njf) all splitting functions are singular as z — 0. In this limit t h e lowest ord(«r parton distributions a r e given by (Rujnla el id., 1971) f-2(x, Q2) ~ xg(x. Q2) ~ //(n 0 . Q5) cxp J ~ V .„ w,th
In
\a»\Q
)J
In
\
X
J
(¡5.301)
/C, h.KK^/o^/-'))
"
0 =
V
^RIAO
•
See Ex. (3-33), which also gives sub-leading terms. This solution shows a strong groivth in the sinall-.c partons and hence the structure functions, iu particular ,Q2) (Gliiek et id.. 199r>). T h e dependence on t he initial distribution is only
I IIKQCI) IMI'HOVKD I'AKTON MODKI.
via its i)|,-t.li moment. If this initial distribuì ion has too strong a small-.e growth, t hen the above solut ion will not hold: for example. xg(x, Q^) x A > 0 leads 2 2 to x;i(x,Q' ) -x ./• independent of Q . In what is known as "double asymptotic scaling", in the limit \/x. (J2 —* oc eqn (3.301) implies that for "soli" /(.'".Q,",) then I n Q 2 ) d e p e n d s linearly on ^ hi(o s (Q,"i)/o s (Q 2 )) x l n ( l / . r ) and is independent of t he complementary combination \ / h i ( a s ( Q 2 ) / n s ( Q - ) ) -:- l n ( l / . r ) (Ball and Forte, 1991). Sub-leading terms only slightly complicate this s t a t e m e n t . 5.(>.(>.2 The liFKlj equation At small x and moderate ( ) 2 > AQ C ; u . where Uluoiis are dominant, we must sum the leading o s l n ( l / x ) terms whilst keeping the full (^"-dependence. This means that we do not. have strongly ordered /.•/• hut instead integrate over the full range of A"/-. This leads us to work with the nnintegrated gluon p.d.f.. G(x.k2-). which is related to the usual p.d.f. via X!,(X.Q2)
= J
-jfQi.r.l-2).
(3.302)
In phenomenological applications it is conuuou to assume a narrow. Gaussian I;i distribution for the p a r t o n s in a h a d r o n . which is connnensurat.e with confinement. Predictions for s t r u c t u r e functions are then made using the so-called factorization (Catani et al. 1990«: 1991«). Ft(x.C?)
= j T f
/
.
(3.303)
Here Fj'"x is derived from the q u a r k box d i a g r a m s t h a t describe virtual-photon virtual-gluon scattering. 7*g* —> "qq" —» ";'g*. T h e nnintegrated gluon p.d.f. satisfies the B F K L equation (Kuraev et id.. 1977: Bali t sky and Lipatov. 1978): see also (Mueller. 1991) for au alternative derivation in terms of colour di poles. At leading order the B F K L equation is given by i)Q(x. A"2) _ C > " i) l,.(l/:r) " IT
r<WÌ''./,,-; 4
¡G(x.g2r)-Q(r.k2) \
Q(x, A"2) \ /IT/}" ! " / , [ ] '
(3.301) Given the nnintegrated gluon p.d.f. at one value of.ru. this equation allows you to calculate its value at. smaller values of x. that is. larger values of l n ( l / . r ) . If o„ is fixed, then the equation can be solved analytically. In the sinall-.r limit this basically gives a power law behaviour in x.
(3.30.r.) Pile solution follows by first, applying a Mellin transform and then using the saddle point method to evaluate the inverse. Here
i in
"
i in o i n o k g c i )
L A = 4 In 2
'
C',\<\s
+0.5
=
i/d^ \
and
2/
A" = 28C(3)
(.; Cios
\
'2 J (3.30G) (3.307)
<>.=(1.2 T h e numerical value of the Niemann zet a-function is (,'(3) % 1.202OGG9032. Due to e<|n (3.303) the behaviour Q oc x A feeds through to give F-> y. x~x. T h e /.••/ behaviour is typical of diffusion and reflects the lack of any Ay-ordering in I3FKL dynamics; in essence there is a random walk in A-/- as x decreases (Balitsky and Lipatov, 1978: Bartcls and Lotter. 1993). Actually, this observation highlights a problem. Given that the width of the Gaussian in \n(kj./k2-) is given by yj[A" ln(.ro/.r) .-1 j. then for sufficiently small x t here will be support for (/(./•. A'}' ) from the non-perturbative region in A- } . Thus, if we use a running coupling. i \ ^ ( k f ) . then it is necessary to introduce infrared cut-offs, for example k'ft > 0 in eqn (3.304). and other possible refinements such as including momentum conservation (Collins and Laudshoff. 1992: Bat tels cl. nl.. I99G). Whilst numerical evaluations show that similar [lower law behaviour in x and diffusion in ln(A-'f/A%y.) occurs (Askew cl
(3.308)
=0.2
Such a large, negative correction basically invalidates perturbation theory and. if taken seriously, leads to negative cross sections. T h e source of these large eorrections has been traced to large In{Q~/Qu) terms coming from phase space restrictions (Salam. 1998). There are a number of putative solutions to this situation, which include resnnunation (C'iafaloni cl .3 Combined evolution equations T h e D G L A P and B F K L e(|iiations describe evolution in two complementary regions. A number of a t t e m p t s have been made to give a combined description of bot h regions in a single equation. Amongst these are ail attempt to include l n ( l / x ) terms into the usual coilincar factorization bv adding summed corrections into the Pjy, kernel appearing in the D G L A P equations (Ellis c.t nl.. 1995; Ball and Forte. 1995). A second approach is given by the CC'FM equation which uses angular ordering to describe both the ./• and the Q 2 evolution and has the D G L A P and the B F K L equations as limiting cases (Ciafaloni. I9SX: C-atani et nl.. 1990ft). see also (Andersson ct id.. 1990«).
I III-: Q C I ) I M I ' I I ( ) \ 1:1» I ' A I I I ' O N
MOHKI.
((>.(». I Sim/lowing, i/Iiioii rccoiiibhiiilion tmil hot spots If left, unchecked, the rapid rise in the small-.r gluon p.d.f. predicted by both the D G L A P and B F K L equations would violate unitarit.y. It also leads to a breakdown in the parton model picture of.scat.tering oirindependent parlous. At sullicient.lv high densities it becomes possible for a second parton to overlap in space with the first, so-called 'ihndowing. T h e probability of this happening can be estimated as N(x.Q)
.
(:5.:509)
where the parlous, prcdominently gluons. are taken to have an effective area, N(x,Q)given bv a typical Q C D cross section, n ~ o*(Q' 2 )/Q~ i l l l ( ' number I he denominator is taken to be of order t he area of t he hadrou, wit h the radius It ~ /("(, = l/il/|,. In general, P s:vt is small but especially for small .r it may become large. When it becomes 0 ( 1 ) the hadrou is said to sat urate and the usual I X.JLAP equation may need to be modified to account for parton recombination. T [—(!Kl)2-
" J.r .'/
(3.310)
A similar modification can be applied to the B F K L equation. In this GI.R equation (Gribov ci ill.. 1983) the familiar first two terms lead to a growth ¡11 //(a:.//") due to emission whilst the third involves a suppression due to recombination, gg - • g. T h e competition between these two terms ensures t hat the gluon p.d.f. equilibrates below the unitarit.y bound. The validity of eqn (."i.."51(1) is not assured, but it. appears reasonable to use it to estimate the onset of shadowing (Askew ct ill.. 1993). It has been derived at DM. accuracy (Mueller and (Jin. 1986): however, this neglects l/A r , suppressed terms associated with pre-recombination interactions bet ween the gluons (Bartels. 199:5: Laenen and Levin. 1994). Its equivalent has also been derived for the I3FKL equation in the colour dipole approach (Kovchegov, 1999). More significantly. it must be admitted t h a t at saturation the high densities and lield strengths occuring. F'"' ~ \/<js, imply that the pert urbation theory is no longer valid. This has led to the development of a treatment in terms of a semi-classical, effective lield theory (McLerran and Venugopalan, 1999). which also leads to parIon recombination (lancu ct id.. '2000). T h e choice Ft = It 1, in eqn (:5..'51(l) corresponds to a uniform distribution of the Q C D fields across the hadrou. However, it has been conjectured that this may not be the case and that parlous inside the hadrou may concentrate in dense hot spots centred on the valence quarks (Mueller. 1991). In this case one should use an It < Iti,. Such a behaviour is predicted by the BFKL equation but not t he D G L A P equation. It predicts t he number of gluon jets per unit rapidity localized to a transverse region of size A.i j- ~ 1 //,-•}• as (hi
dln(l/.r)
CAas v/i-A'V^lnil/.r) '
(3.311)
I III
3.(i.7
The iiikiIi/sis of ladder
I I I K O H Y ()!• Q ( ' I )
diagrams
In our discussion of DIS. Sect ion (¡.3. wo encountered t he Altarolli Parisi splitting function / ' , , , ( ; ) . oqn (3.24(>). when investigating the limit of near collinear emission. A point which may not yet have been appreciated is the universality of i his result. That is. whenever we have a process which contains a q — qg vertex, then in the collinear limit its (a/iinuthally-avcraged) contribution to the cross section will l>e described by the same factor
Similar expressions describe the collinear limits of g —• qq and g —» gg vertices respectively. This factorization of the matrix with I',,,, replaced by P,ti. and element squared then leads to much simpler expressions for a cross section in i lie collinear limit. Furthermore, the collinear emission regions of phase space are very important because t liev are responsible for one of the dominant. leading logarithmic, contributions to the cross section. In the other dominant region of phase space, the limit of soft gluon emission, we also have that the cross section simplifies significant ly: see Section 3.7. In this way we can use simplified expressions to describe the bulk of a cross section. Of course, if our analysis focuses attention on a region of phase space which involves hard, non-colliuear einission(s). t hen t he approximate matrix elements may only be of limited use.
FlC. 3.25. The emission of a near collinear gluon oil an incoming quark in an // 4- 1 part on scattering To see how this simplification occurs, consider the situation sketched in Fig. 3.25 where a quark entering an //-particle scattering emits a real gluon. which we shortly will take to be near collinear with the quark. The matrix element for this process is given by -<'+1) = uJIMT^^-uMPV-{' .
(3.313)
Introducing a gauge vector //'' with, for convenience. //"' = 0 (and
iin-;»¿ I M N « ) V I - : I > I ' A H I O N M o n i - i .
(2„
I
( „ ' „ ) {
• • 0» - i ) [to • l M * ( » • M ] tf - 1 ) •• • )
(2,,'•,) ( n 2 . / ) ' " ' ( • • • « " ' < ' ' -
" 0 + (/''
+ ( n •/>)/*]•-•} • (3 31-1)
I'll.- clli|)sis iii these expressions represent the eont.rihut.inus from A^J."1 and . In line two we used nil identity based upon c o m m u t i n g 7 - m a t r i c e s t o obtain line three and then again c o m m u t e d 7 - m a t r i c e s in lines three and four, to obtain an exact result. Now we wish to plus using f)fj " <J2 = 0 = p 2 specialize to the near col I ¡near limit. To do this we use a Sudakov decomposition of the <|uark and gluon inoinentuin four-vectors (Sudakov. l!l.r)(i). q" = Zp" + fill" + F q"
.
(3.3 ir,)
Here 111' could have been any four-vector, subject, to n • p /- 0. but it proves most useful to make this the gauge vector whilst k'' is transverse to both />'' a n d p- k_ — 0 = v • /.*j_, and k- = — kj < (). T h e gluoirs on mass-shell constraint., z)i 1 • p). T h e (negative) virtualit.y of the
zMi I - ;
(3.3 Hi)
4
(3317)
Adopting these variables eqn (3.31-1) becomes u'A'i-àw
(1-2) 4
(1 h
(i
r {
+
+
2(71 •
p)
Now if we only wish to keep t he leading term ¡11 t he colliucnr limit , k-j we can d r o p t h e second two terms iu the square brackets to o b t a i n +1) 2
= ZS 2-
1 + uJ t C rr rfu'~ (\-z)
2
k
:
H i'
0. t hen
Tr { • • • / - • • } + 0 ( 1 ) (3.318)
Thus, in t h e collinear limit the matrix element factoriz.es into the product of the iinregularized. lowest, order Altarelli Parisi splitting function. /',,,,(;). and th<-
( (
s
ss ss ss s
s s
s +
ss
,
, s s
,
= (
-
A'.f,
^f^
tl
(z)
x
- ~
a .
( . 1 )
s s s
. (
).
T
A s . (
.
s s s s ).
.
s ss
.. s
, . s s
A
( . 18). ,,,, s .
s s s . , s
.
.2( .
s s
A
ss s s
s
s s
s
s s
ss s
s. T s (
s s
. 1 77). T
s s
I (¿CD IMI'HOVKI) I'A I M'ON MODEL
I IV
in identify those diagrams and the associated regions of phase space which give rise In the leading logarithms and then sum them. The treatment of the single collinear and double logarithms is a situation where the use of a physical gauge proves invaluable (Frenkel and Taylor. li)7(i: Dokshitzer r.l til.. 1980). It can then lie shown that the dominant contribut ions come from the so-called ladder graphs, such as Fig. 3.2(5. with strongly ordered branchings eqn (3.298) and (3.300). These ladders correspond to individual, tree-level Fcynman diagrams squared. Diagrams involving the quart ie gluon coupling give sub-leading contributions. Likewise. q u a n t u m interference between tree-level diagrams, which would give ladders with crossed rungs, only give sub-leading contributions. T h e strong ordering in A 2 . which effectively implies ordering of the virt.nalities eqn (3.31(i). allows eqn (3.319) to be ¡relatively applied. In the case of an »-rung gluon ladder the cross section. trn(j\ Q~), is given bv:
(3.320) Associated with each rung are a /.••/ and an .r integral, both of which may contribute a large logarithm in the collinear or soft limits, respectively. Equation (3.320) embodies an almost classical picture of a parton shower in terms of successive branchings. It is this which lies behind our interpretation of the Altarclli I'arisi equations and which will be further exploited in the development of all-orders Monte Carlo event generators. T h e usual collinear leading logarithmic approximation is characterized by A-f-r Qjj. strongly ordered transverse momenta. QA-flT 3> ••• 2> A'i r I'sing the one-loop expression for os(A••"}•) the nested transverse momentum integrals become: „
(r
m V • * > J Q, =
1
k*r
_L-/
\ fi' 2 2 2Jr,*,ln(fe r/A )
Q 3
i [_J_. »! [2tt,%
( l l
JQf>
„[h'fe/A2)l...
MftT * fcfr 2jt ¿in
fr
I hi(k'fr/A2)
rin(fc 2 r /A 2 )1
(HQ2/^)\}" \WQl/\*)J
I id In the second line, we have used a change of variables which makes the integrals simpler to evaluate. To do the moinentuiu fraction integrals it. is useful to work with the Mellin transform.
nil
I I I I o l t Y Ml
<>u>
(•{.322) l'In- result follows by repeatedly applying the fact that tin- Melliti transform of a convolili ion is the product of the Mellin transforms of the components. Combining e(|iis (3.321) and (3.322) gives
£
<7,,(/»,
f w ,
~
/"s(<St))
V"s
m )
gJYTO Q,ni I /,,•> V"s m(Q- ) ,2> ( <>„(/'»
=
.')('»•
(3.323)
which coincides with eqn (3.2SS). This demonstrates that the DGLAP ec|naiion sums t he leading o s \n(Q-/Q'f)) terms. In the double leading logarithmic approximation we approximate by 2('..\/z and impose st rong ordering on the longitudinal integrals. ./• ./„ ••• < j'2 .i'| 1. This is in addition to the strongly ordered transverse momentum integrals which we evaluated in eqn (3.321). T h e longitudinal integral becomes son(.r)
ot X
2( i J,
^
x„ r
f Jx ' •''"^
-7 | 2 C , log ( - 1 1
/ 2CA— Jr-i x2 -Jr., f
/ JXi
-.ItJ Ì l'i -
*t
2Cj\— g(.rt. Q{) -''J
^ (3.324)
G,(Qj{).
lu thesecond line we bave taken .n/(.i\ Qf t ) ('u(Qu). Again t he nested integrals are straightlbrward t.o evaluate and lead to a second I / n i factor. Couibining e<|ns (3.324) and (3.321) gives
f i r *
firSt
» ' • / i H i l H i ) -
**»
Here, we have recognised the sum ^]„(.'//2/i!)"" as the power series for the modified Hessel function /n(.'/) which li.us the asymptotic form e ! '/v/2~i/ (Arfken and Weber. 1!H)5). This result, coincides with eqn (3.3111). Strictly speaking, the ladder diagrams, such as Fig. 3.2(>. are only schematic. For example, at one-loop they should be understood t.o also represent diagrams
I III' (
It I M I ' I K >\ 11» I ' A U T O N M O h l l
.
s. T ( s. 20)
s
s
s
\n( A , s. A , ,
s
s
2
)
s
a
( s(
. )
.
2
)/ ii(
))
s s
s ss
s s
(. {.. {10) s
s
The Drell
Yan process
T
ss (
.
)7 )
s
s .
1
s . s
s.
s
s,
T s
,
s s s
s
(1 8 ) s, )(( ) 1 74 1 8 .
ss s
s
,
s s
s s
ss s
s
s ss s
,. s
,
s s
s
s
s
ss s
s
s. s s s T 1 78. ss
1
s
ss
2
.
s s s
0.
s s s
(
) s 7 (
)
ss s s . ss s + (
s
s
s
s. s
s
s
.
s
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ss
s
s ,
s ,
T
,
s
ss s
s
s
ss
T
ss
s
,
. (. { 7).
( .72)
s
— - j - ( l l | ll2 —• 7 * ( Q ) el Q C
efr, / tlx .( ,
s
.
. .8
.
et nl.. 1 8 ). s s
s
s
/
s (
s
/
( . 2( )
.
s
s s
s.
s
s
s
s.
2
s (
s
2
(
11
s
)
t
( . 27)
= x
['/hi ( i
ela' 0 1
(•''-') + '/in ( ' i )'/h-(•''->)] ~TTTr(
s
(
+
r){
,f)
till
f =
C
I
III:OHY
4r = — — S J*I J'j
()!• «¿CI»
and
r =
s
,
(3.328)
willi .s (j'l/M t-•>'•>i)>)' .I'I.r-j.s'. Note that in <*«|it (3.327) wo arc careful to include bot h cont ribut ions coining from t he quark (ant iquark) being in hadrou I (2) and nice versa. We have also temporarily suppressed the p.d.f.s' dependence mi t h e scale Q - . T h e differential cross section for t h e hard subproeoss is given by: « 2 1°) ie + n f ) 5 ( 3 3 2 9 ) = - f) • This has been obtained from f<jn (3.93) using crossing, allowing for the changed average over initial s t a t e colours and multiplying by unity in the form I = /df^*ii(i/' ! s). The factor r r , s e t s the scale for the cross section and contains its dimensions. Combining eqns (3.327) and (3.329) gives
w
- f
T
-
T
•
<«°»
If the p.d.f.s are scale independent, t h a t is, they do not depend on as in t h e iiiii've par ton model, then the differential cross section i\n/(\C)~ -x 1} 1 x a function of r . This result follows on dimensional grounds and is t lie s a m e scaling as we saw ill DIS. 3.(¡.S.l The 0(i\s) correct tons to the Div.ll Van pwe.ess T h e calculation of the 0 ( n „ ) corrections to the Droll Van process follows very much the same procedures as those used for DIS (Altarolli et til., 1979«: K u b a r - A n d r e and Paige. 1979). Hero we only outline t h e results using j' — f+l~ production for illustration: W and Z production follow t h e same linos whilst including t h e decay orientation of the lepton pair a d d s no new insights. A useful guide to t h e calculations is given by Willonbroek (1989). T h e r e are basically t wo new contributions a t ( 9 ( o s ) : charge conjugation symmetry relates quark and ant ¡quark initiated processes. T h e r e is the gluon broniss t r a h l u n g correction to t h e lowest order process, qq —> 7*. We expect, this t o show colli near singularities when the gluon becomes parallel to either t h e incoming q u a r k or ant ¡quark, but to be free of filial s t a t e singularities after we include the1 one-loop, virtual corrections to qq —> 7 ' . It is also free of ultraviolet singularities. T h e r e is also t h e gluon initiated process gq — 7*q'. We expect this t o contain a eollinear singularity when t h e scattered q u a r k lies antiparallel t o t h e incoming quark in the C'.o.M. frame. T h i s is equivalent to t h e gluon undergoing a near eollinear g —* qq branching in a fast moving frame. The a p p r o p r i a t e generalizat ion of cqn (3.327) is given bv
j( £ r O « i h 2 "
7'(Q) -
( + t ~ ) = / ' d r , [
THI'I Q C I ) I M I ' H O V K I ) P A K T O N
£
[.'/.I, (•'•! ) / , » ( * 2 ) + /.„(x,)
MODICI.
-
ir.i
7'q') J .
(3.331)
I. r e y , m a n d.agra.ns for t h e two new. hard subprocesses have been given in .gs 3.23 and 3.22 but now are read from right to left. Crossing also aÌlows n take over * T ™ d 0 1 , U " , t S W Ì t h " , Ì " Ì ' " i l 1 we« dire iv I k ove, t h e p h a s e space mtegrals as they involve different regions. T h e result of these calculations are
+ " q r,(f) (I<7
(,)
(
r
+ //K,(f)
(3.332) where for convenience we have introduced t h e coefficient functions =
C,, '1(1 +
' W O = 7>
n
I 1 -2
+ (1 - 2) 2 ] 111
III 2 +
i I 2z-
W - t ) ] 322
(3.333)
T h e s t r u c t u r e of eqns (3.331) and (3.332) is very similar to t h e corresponding expressions which arose in t h e N L O descript ion of DIS. e q n s (3.272) and (3.271). It is t h e power of factorization t h a t essentially t h e same separation of the cross section into factorization scale dependent long-distance p.d.f.s a n d short-distance coefficient functions will t r e a t t h e eollinear singularities in t h e NLO description of t h e Droll Yan process. Indeed, introducing the MS p.d.f.s. oqn (3.277). into cqn (3.331) gives d<7
-(h,ii2 - 7 * ( Q ) d<7-'
X (S(l - t) + g
f
e+(~)
a""
[2P<»>(f) In
/•'
/•' I.e.,
+ //„„(*)])
Mil
x
^
I IIKOKY OI1' (JC'I)
+
i* "'"'1)
} •
which is independent, of t he factorizat ion scale to (9(o s ). T h e MS coellicient functions in this expression depend on the factorization scheme, that is. which Imito terms are absorbed into the p.d.f.s and are therefore different in the IMS scheme; see Ex. (.'{-US). T h e 0 ( n ~ ) corrections to Droll Yan have also been calculated (Zijlstra and van Neerven. 1!)!)'2). .'{.(¡.8.2 Transverse momentum in Drell Yan processes T h e measurement of the W boson mass to high accuracy is very desirable as it facilitates tests of the Standard Model of electroweak interactions at the q u a n t u m (loop) level: see. for example, the report by Altarelli et al. (1989). Since the \V decays to a charged lepton and a neutrino its mass reconstruction at hadron hadron colliders must necessarily be indirect. T h e preferred method is based upon measuring the boostinvariant transverse momentum distribution of tl«' charged lepton. Assuming that the \V is produced with no transverse momentum and neglecting its width, one expects
^aúpr ^ n
1
i M V' - ^M\\W) V - " ^AIWA/
^
This strongly peaked distribution is very sensitive to M\y. but to be useful we must be confident that we understand the underlying transverse momentum distribution of the \V boson. At 0 ( o " ) the massive vector bosons produced in the Drell Yan process have zero transverse momentum. At G ( o ^ ) the f|ij — V'g and g<| —» Vi\' (gq • Vq') processes provide a good description of high-yi-/ vector boson prodiK-tion. Now. at order O ( o " ) the cross sect ion behaves as 1 dg a d pf
(f)
1 < / l „ . 2 „ - i hi 2 "- 1 ( pr,.
) -I
so that care must be exercised in the low-/>-/ region. M\ ~3> pr '>> A Q C I > . T h e need to sum these large logarithms was first recognized bv Dokshitzor et al. (1980) and an impact parameter space formalism for summing them developed (Collins and Sopor 1981: 1982: Collins et al. 1985). Impact parameters are the Fourier conjugate variables to pr- In computing t he effect of multiple gluon radiation it is important to impose momentum conservation. k„r = pr. on the (soft) gluon bremstrahlung (Parisi and Pctron/.io. 1979); this greatly reduces the possibility of obtaining p r = 0. A numerical implementation of this formalism (Ladinskv and Yuan. 1994) prove«I the necessity to include a Gaussian smearing of t he initial partons' impact parameter dist ribution in order to counter convergence and infrared problems. Analytic expressions for the coefficients in «•<|n (.'{.:{.'{(>) are available up to N ' L L accuracy (Kulesza and Stirling. 1999). In an alternat ive form of oqn (3.336) the right-hand side can be resuuuned by 'exponentiating' the leading logarithmic terms n " In"'4 ] (My//>'/')• which have been
11
T
AT
T
T
.,
A'
q + A ..
1
.
.27. T
ss
s
A s
s
s
.
ss ,
s til.. s
s
s. T
s
s
. 1
s s
s
. 2000)
T
s s
s
ss s ,
s
s (
,
s
s s s
s s ( .. 2001).
s ss s
. s
s
s
, s s
. (
s
s ,
s
s , ss ss . el til.. 1 8 ). As s s s s . s s s s s ss s s s s s , s 1 ( . ), ,. A = . A . .27. s
,
s s
s
s s
s
s s
s s
s s s
s . T
)
s, .
s
.. s
s s
s
.7
.
8). .T
ss
2000 . 2000
s
( s
s
) ) 7
s
t til.. 1 ) ) )). ss s
(
s
s ( s
2
s s s
, , ss
s s s s
s s
s s
,
s
s
s s
s. T ss
. T
s
s A ss s
, s s. T s
s, s
s s s s s { , ). T s s
s
s ,
s s s s
, s
s s s
s s
, s
s s
151
I 111 i )HY OF
I III
1
'I •
1
(7 I A)- - in-
- n W M M
1
-(rf W '
m ) 7 " -I- 2//" -I- 7 " f ] ' ' ' M > | ( / V > ( < / + A-) 2(i k
II • A
Here we have exploited, a f t e r a suitable r e - a r r a n g e m e n t . t h e Dirae equation satisfied by t h e basis spinor and neglected t e r m s proportional to A* compareproxiuiatiou. T h e s t r u c t u r e obtained is universal, that is, the same form would hold for emission olf an a n t i q u a r k . anot her gluon or even a coloured scalar: see Fx. (3-39). T h e reason is that the soft gluon has a correspondingly long wavelength and is unable to resolve t he spin s t r u c t u r e of t h e e m i t t i n g particle. Adding u p t he contributions from all the hard p a r t o n s we obtain
M ( A ' + , ) = fhf'(A:),, ( ¿ T T - ^ t ) V, • A\ ¿=1
>< M l N ) •
Here t he colour charge o p e r a t o r s . 7 " . g e n e r a t e t h e a p p r o p r i a t e colour factors for a gluon. colour n. e m i t t e d by p a r t o n /. T h e y act as follows:
•••,
^il-••;
• • •) = —T",\- • •: C/.j: • • •) t £ I"
(3.339)
<1- /»:•••) = - ' ' / « 6 c | - •••!)• <::•••)•
To obtain t h e mat rix element squared we need to use a n expression for t h e gluon polarization s u m . eqn (3.121), which allows us to write |jWlA'+,>|2 = - 7 . ^ - . / ' | A I ( J V > | 2
with
J " » ( k : p i ) = Y t " (-¡Tk " T T l ) •
(3.3-10) Here ./'' is known as the insertion current. For a colour neutral system, such as in an e ' e event, we have 7', = 0. so t h a t the second term, proportional to 1 the gauge vector ii '. can be safely d r o p p e d . It is also absent, in t h e Feynman gauge. Henceforth we d r o p it.. Including the phase space t h e n gives the following form for the soft gluon emission cross section (Marchesini and Webber. 1990). (h(N+l)
¿n
¿ttuu' P,
'¿J
• l b
(,,, • A-X/i, • A )
1
l>~,
2 (/>, • A-)2
1
p j. 2 ( P ) • A-)2
2JT
K IHlSA'rMKNT ()!•'«()!• T CîMIONS
,
"i«"'
n -
15!»
•
(3<341)
Notice that we have manipulated the diagonal terms with the aid of the ident ity /', ICiji, 0 ' s<>(> (3- K>). T h e last, expression is appropriate to t he case of inassless hard part.ons. In this limit we see that the only contributions to the cross section come from q u a n t u m mechanical interference: the soft glnon can equally he associated with the hard partons / or j. In Section 4.2.5 we will discuss an approximat ion which allows us to recover a classical picture of soft, glnon emission that is suitable for Monte Carlo simulation. T h e term in the sum is the classical expression for describing radiation olf a dipole. In essence, the hard part.ons form an 'antenna' and the soft gluons follow the associated radiation pattern. I he factor a.'"' ensures that it does not depend on the energy of the emitted soft glnon. only its direct ion. The energy spectrum is of the typical bremsstrahhmg form du;/u/. T h e cross section also contains collinear singularities whenever the direction of the glnon coincides with that of a hard external parton. k || />,. We illustrate the use of equ (3.341) with an historically important, example. Namely, the comparison of t he intra-jet energy How in 7 ' / Z —> qq-y and 7 * / Z —> qqg events which, modulo the couplings (c~o ( . m —> C/.-n s ), are described by the same matrix element. T h e focus of our attention is the radiation pattern, given bv ./-'. which is straightforward to calculate. For simplicity we assume t h a t the quarks are inassless and obtain J,
'Wm - %
P-
=> -
mm
^
=
for the characteristics of soft glnon radiation in a q q 7 event. Here, we employed colour conservation to substitute T,, — — T,t and that the (ptark's colour charge is given by T 2 = ("'/•'• T h e analogous calculation for glnon emission in a qqg event vields -
-.ï-(k)im
- C.
q"
, -
q"
'I •
(q-k)(u-k)
, 'J
q-q
'I • ¡1
(q • /, )(// • fc)
- (C'A -
2CF)
(q-k)(q-k) (3.343)
Again, we have avoided detailed evaluation of the colour factors by making use of a trick, 2f„
- 7;
= ( f „ 4 f s ) 2 - T* - i'i 2 2 - 7' 2 = (-T,,) - f = -CA
2T,| • 7;, = ( f „ + f , , ) 2 - f ; f - f = (-fK)2 - t ~ - T2 = CA - 2Cr
2
(3.344)
.
The two radiation pat terns show markedly different behaviour in their interjet regions. I11 qq7 events the energy flow is concentrated between the coloured
nil
I I I K O K Y
quark and ant ¡quark. In qqg events t lie energy llow is concent rated between I lie colour connected q-g and q-g pairs, whilst it is suppressed as (C \ 2Cy)/C,\ 1/A',. between the qq-pair. In both cases, the soft radiation is predominantly in the plane of the event. T h e strong collinear enhancements in tin- directions of the coloured partons give rise t o jets. Compared to the intra-jet energy Hows the inter-jet energy flows are only modest. T h e resulting radiation p a t t e r n s are illustrated in Fig. .'{.'28.
t o g e t h e r w i t h t h e u n d e r l y i n g c o l o u r Hows
Also shown in Fig. 3.28 are the associated colour flows based on representing the gluon as a pair of colour anticolour lines. This is the planar approximation, equivalent to replacing SU(A r r ) by U(N C ), which is correct to relative 0 ( \ / N 2 ) . It is so-called because in this approximation it is always possible t o draw any Feynman diagram such that no colour lines need cross. It suggests t hat a ghion should produce twice the amount of radiation as a quark by a factor of t lie ratio of t heir colour charges. CA/Cy = 2 + 0 ( 1 /.V 2 ). We also see t hat we might view t he qqg event as consisting of two colour dipoles in which t he soft gluon radiation off one part on is restricted bv the anticolour charge of its colour-connected partner. We shall investigate these ideas further in Section •1.2.5. Looking further ahead we will see t h a t t he distribution of soft gluons predicted by p Q C D coincides with the location of the string in the non-pertnrlmt.ive had ionization model of the same name (Section 3.8.3). Indeed, it was on the basis of this model that the 'string effect', the relative depletion of soft hadrons between the qq-pair in a qq-,- event compared to a qqg event was predicted. Today tlii- p Q C D explanation is preferred. T h e experimental situation is discussed iu Section 13.2. A more sophisticated introduction to the treatment of soft gluons a n d p Q C D in general can be found in the lectures by Dokshit.zer (!!)!)(>).
II \ l MlONIX/VI I O N M O D E L S
3.8
Ilndroiiizntion
models
I 'hi* proceeding sections have equipped us with much of the technology needed tn calculate properties of those inulti-hadrouic events that involve a hard scale. <> T h e usual description follows a sequence of decreasing momentum t ransfers, f i r s t . a set of primary partons is produced iu a hard subprocess t hat is described by a lixed-order matrix element. Second, a parton (or. oquivalontly, dipole) cascade evolves t hose primary partons from the hard scale down to a variable number of final state partons at a fixed cut-off scale Qu ~ 1 GeV. T h e leading logarithmic l.eli aviour of these cascades is governed by the same pt^CD evolution equations that were developed to describe the scale dependence of p.d.f.s and fragmentation functions. Third, the final state quarks and. predominantly, gluons arrange themselves into colour-neutral hadrons. Given the low scales inherent in this liadroni'/ation process, and the subsequently large strong coupling, this process is non-pert urbative and so. at present, beyond our means to calculate it. Fourth, pretaluilated decay tables ( P D G , 2(100) are used to sequentially decay any unstable primary particles into the stable hadrons. leptons and photons that are seen directly in the detector. As an illustration, the fragmentation function for finding a liadron h in a jet init in ted by an outgoing primary parton ti of maximum scale Q, such that it carriers a fraction x o f a ' s momentum, is given schematically by D]l(x.Q-)
= ( p Q C D evolution: Q~ -> Q2q) & (model: b — II)|
(3.315)
OC (tables: II — h. l l ' , . . . ) . I'he convolution runs over all partons b created in the p Q C D evolution, which then hadronize into all kinds of possible intermediate unstable hadrons II that iu turn decay into the final s t a t e particles seen in an experiment. T h e liadroni/ation process is believed to be essentially independent of Q. up to power corrections, but sensit ive to the arbitrary cut-olf In combination with t he p Q C D evolution the net result should, of course, be independent of the actual choice Qq. Now, it is important to remember that our present understanding of the hadronizatiou process, and hence its effects, is based largely on models and not (¿CD as such. Fortunately, it is expected to be both local in nature and not to involve any large momentum transfers. This helps to justify our belief that it is the first two calculable stages of an event, characterized by relatively large momentum transfers. Q 3> AQCD- that, are dominant iu determining its global features, such as energy dependences, event shapes, multiplicities etc. This makes possible meaningful tests of pQCD. However, this is not. to say t hat the effects of hadronizat ion are completely insignificant.. For example, in e ' e~ collisions, since pQCD predicts that soft gluon radiation in a three-jet event is preferentially in the plane defined by t he uul.nl quark ant ¡quark pair and t he first, hard gluon. it does have some influence on event shape variables which are sensitive to "out-ofplane' activity. It also determines the rates of production of identified particles and t heir correlations.
I III
s s
.
I I I K O I O < >!•' (v>< 'I»
s
s ,
. ss
s . T
s
s s s
s
s
Spaa'
s s .
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of inutti-hiidmn s . T
s s , , A s s s .
s s
time structure
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ss s
s. s
T8.1
s
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s
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(A. ) .
, , s .T s ss 1). T ss
. . . id. (1 8 . 1
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s A0(.
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t T
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. 1 .2
s. T s s
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ss s
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{ 1 =
ss
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s ()2 = ^ 'i
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ss ( {.
s s , . .
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,
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( . 4 ) s
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)
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, s /*
,,, s T
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.
=
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s
s n h -r
'rto =
s
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s =
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. =
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t,, .
q
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s, s s. T
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Kill
I 111 I III DlCN ()!•' ( M I)
Here the logarithmic A" dependence relleel.s the fact that the coupling is diincusionless. T h e logaritlmtic A' dependence follows from the ghion being nlassies«. Only the kernel function. P . which is of order unity, d e p e n d s on details of the parton gluon vertex. T h e distribution is thus governed bv longitudinal phase space. It predicts hadron production t h a t is uniformly distributed in rapidity and with typical transverse momenta ~ as measured from the directions of t lie primary partons. Now. not all q u a n t a are emitted at low l,-± and in a significant range, //,', ' A- < A- have /?oJO < Ii^/0^. the separation time of the hard gluon. and should therefore be associated with the parent, parton gluon system. This observation lies behind the strong angular ordering condition, it t). which defines the dominant phase space region ill a time-like parton shower. T h e coherence of soft emissions simply reflects colour charge conservation and is f u n d a m e n t a l to a gauge theory such as Q C D . We shall investigate f u r t h e r this colour coherence in Section 1.2.5: see also Ex. (-1-3). If we consider two gluers emitted at t he same angle, i). from the two hard partons. then when they hadroni/.e simultaneously their transverse separation is given by
»
•
(3.352)
T h i s large inter-gluer separation suggests that the h a d r o u i / m g gluers form a r a t h e r low density system in configuration space, thus limiting the influence one hadroni/ing p a r t o n system can have on a neighbouring parton system. Finally, the t i m e scales for the decay of the primary particles are set by the reciprocal of their mass-widths I". For t h e strong resonances typical values of I' ~ 1 100MoV give /
K.I
I I A D I I O N I / A T I O N MODI I S
In t in- resulting piel lire, at I lie end of I lie parlon shower we have a low density system of (¡narks and. predominantly, gluons, each dressed in a uniform sheath of strongly interacting gluers. What happens next is essentially a mystery hut as t he part.in virtualities d r o p below a critical value, Qo ~ 1 GeV, a phase transition occurs and t he relevant degrees of freedom cease to lie partons. q. q and g, and become mesons and barvons, that. is. c|<| a n d qq<| bound states. Now. due to the strength of the strong force at low m o m e n t u m transfers, it is energetically favourable for the vacuum to be populated with virtual qq-(Coopcr)-pairs of opposite flavour, moment um and helicity but non-zero net chirality. It is eoninion to all the hadrouization models considered here that the energy stored in the glucr field is used to promote these virtual qq and perhaps qq'-qq* pairs into real particles, which then arrange themselves into colour singlet hadrons. In practice, what, the models must supply are rules t h a t can be applied iternlively to determine which quark flavours are produced, which primary hadrons subsequently form and how much m o m e n t u m these hadrons carry away. T h e space time picture developed above suggests a simple ' t u b e ' model for describing hadronization (Feynmau, 1!)7'2). T h i s is suitable for semi-quantitative evaluations of t he effects of hadronization on event shape variables etc. However, for detailed studies of whole events it is necessary to specify every liadron produced in an event. This requires more sophisticated models as described below and in t he literature, for example, the review by Knowles and Lafferty (1!)!)7). 3.N.2
Independent
Imdrimizut ion
Independent hadronization is the oldest, and simplest realistic model for hadronization. Today, it is synonymous with the work of Field and Feynmau (1!)7<S). The idea is to iterate a sequence of universal branchings, qi —< q_> + h. based on the excitation of new quark, qaq-j. or diquark pairs. Gluons are treated by replacing them by a light, qq-pair. T h e basic '111111 cell' is illustrated iu Fig. 3.30. in
<1, : P + ,O.P_
Fit!. 3.30. T h e basic 'unit cell', showing the light-cone momenta, which is iterated in independent, hadronization models Unfort unately, the model has no strong theoretical underpinning so that it is rather arbitrary in its details. T h e flavours of the excited (di)quurk pair are selected in fixed rat ios. Empirically, it is found necessary to suppress both strange quarks and diquarks. d : 11 : s = 1 : 1 : /•,,. with r s ss 1/3. and q : qq' = 1 : /•,,,,.
I IIKOHY ()!• <¿<11
1(12
Willi /•,,,, % I/!). Furtlicr rules m e required to choose between the various low lying hadrons. h : (qiq_>), of a specified flavour. T h e hadronization process is described using light-cone coordinates. p. (E p.) a n d p . with the c-axis chosen to l»e the direction of the parent, p a r t o n . T h e mass-shell constraint is given in t e r m s of the transverse mass by p+p~ = nr - 111- 4 / r . T h e lightcone m o m e n t u m fraction of t h e liadron. r = p+/p'J.'• is specified by an arbit rary, longitudinal f r a g m e n t a t i o n function. A typical p a r a m e t r i c form is
f(x) = 1 - a + «(1 +1>)( l - .r)h.
(:{.:jr>:{)
though more sophisticated forms an* not uncommon when heavy q u a r k s (c,b) are present. The transverse m o m e n t u m is selected from a Gaussian distribution. exp( ¡r /2o2). As a result of this branching, tin 1 p r i m a r y q u a r k , q i . acquires a mass. T h u s the iteration may lie t e r m i n a t e d either at the point just before tile sum of the p a r t o n masses becomes greater than the available energy or just before the remnant q u a r k . q>. s t a r t s to travel backwards, p . < 0. Since each parton is treated in isolation, this requires an ad lio<- prescription to ensure overall m o m e n t u m a n d q u a n t u m n u m b e r conservation: only in the unit cell a r e these conserved automatically. Enforcing f o u r - m o m e n t u m conservation proves particularly troublesome, since in e ' e " collisions event s h a p e variables and hence n„ d e t e r m i n a t i o n s , see C h a p t e r K. are sensitive to t h e n a t u r e of the chosen solution (Bengtsson c.t til.. l!)S(i). More sophisticated variants of this basic scheme are still employed in the CO.IKTS (Odorico. 1 <)!>()) a n d ISAJKT (Paige and P r o t o p o p e s c u . 1!)8(>) event generators. However, independent hadronization coupled with their use of incoherent p a r t o n showers, typically with a r a t h e r large cut-off p a r a m e t e r . Qo ~ 3 G e V . does not give the best description of t o d a y ' s e x a c t i n g d a t a . T h i s is particularly t r u e iu e + e annihilat ion, where t hese p r o g r a m s are no longer widely used. i}.8..'$ String
hadronization
S t r i n g hadronization is often assumed to mean the ' L u n d ' model (Andersson i t at.. l!J83n). However, t h e r e a r e really several models, each based on a c o m m o n s t a r t i n g point. When an oppositely coloured q u a r k a n t i q u a r k pair moves a p a r t , as iu e + e — qq, it. is believed that the self-interacting colour field between t h e m collapses into a narrow 'llux tube" of uniform energy density per unit length, or st ring tension n. estimated to be n « I G e V / f m . T h i s model therefore corresponds to a linear confining potential. T h e transverse size of a string, (n ) = tt/(2k). is small compared t o its typical length. It is therefore plausible that its d y n a m i c s can be described by a niassless. one-dimensional, relativistic string possessing no transverse excitations. T h e classical e q u a t i o n s of motion follow from a covariant action equal to the space time area swept out by t h e string. In the solution lor a niassless qq-pair. seen from t h e string's rest frame, t h e end-point q u a r k s oscillate repeatedly o u t w a r d s and inwards at. the speed of light, passing through one a n o t h e r
M A I >L<< I N I / \ I II I N M ( >1 )L I . S
l' ic. 3 . 3 1 . Tlie space time evolution of a string as it breaks up into hadrons. Highlighted is a slowly right-moving string fragment , a liadron. exhibiting yo-yo modes. Observe t h a t t h e slowest, liadrons form at the earliest times near t he centre of t he string and t h e fast liadrons form latest near t h e ends of t h e string.
and t ransferring energy to and from t he string in what are termed yo-yo modes (Artru. 1983), seen as d i a m o n d s in Fig. 3.31. Solutions also exist that exhibit energy moment urn carrying 'kinks', which have been ident ified successfully wit h hard gluous (Andersson a n d Gust.afson. 1980). At, the end of t h e p e r t u r b a t i v e shower, colourless string segments form between neighbouring partons, each segment t e r m i n a t i n g on a quark and a n a n t i q u a r k . Figure 3.32 shows such a string network. T h e full three-dimensional evolution of such net works is complicated, hut robust treat m e a t s exist (Sjbst rand. 19K<1: Morris. 1987). Further complications arise for massive end-point q u a r k s when the classical equations become non-linear (Chodos and T h o r n , 197-1; Bardeen cl id.. 197(i). T h e motion of a st ring is described classically. Adding q u a n t u m mechanics introduces I he possibility of a string breaking up. ii hi the s n a p p i n g of a m a g n e t . via tin* creation of intermediate qq-pairs from the field energy in t h e string. T h e probability for this to occur is given by Wilson's exponential area decay law (Wilson. 1971) M
'
•
<
M
6
4
>
Here A is the space time area within the backward light-cone of the point, at which t h e qq-pair is created, see Fig. 3.31. and I], is a constant, reflecting t h e unifornmess of strings. It is important to note that the position of the b r e a k - u p point and the m o m e n t u m of t h e fragment a r e linearly related, E = i;Al and /> K A ; I - . T h u s , for example, t h e area decay law is equivalent to a mass-squared decay law, (VP/din2 = 6exp( bin2) with b = P()/'2K2. This implies that the average string fragment has ( r £ i m = t 2 - x 2 ) = I /P, t and (»'¡Tiring) = 2 k ' 2 / / n
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' n i l ? T I I K O K Y OI* g e l )
selection rules, the supposed dynamics of the process ( A I K I C I S S O I I <1 til.. 1980a). Since no transverse string excitations are allowed, the (¡narks have equal and opposite transverse momenta, ±p . and, according to their flavour, mass m ( | In order to supply the energy to produce a qq-pair of transverse mass 2m . it is necessary to 'eat* a length. ' I I I I ^ / H . of string. Assuming that the quarks in a qq-pair are created at a point, then they must, tunnel quantum-moehanieally to the classically required separation. This probability is estimated to be CXp
^ ( "' < " 1 +
0
= exp
K ? "«0 *
(Xp
'
(3-357)
This expression is only regarded as an approximation to the •true' formula and is essent ially used as a guide. Indeed, in t wo-dimensional QED. the exact expression for the product ion of a single qq-pair is known to be i r exp(-n-in-Jti), for which the lirst term is a poor approximation to the full sum (Caslier r.l. <¡1., 107!)). For example, a Gaussian transverse momentum distribution, which is independent of the flavour of the produced quark, is found to be compatible with d a t a but only if the width is increased from the 'predicted' value, (p ) - y/7/TT = 0.25 GeV to (p±) « 0.40GeV (ALEI'll Collab., 1998«: DELI'lll Collab.. 1990«; OPAL Collab.. 1990«: Doyle <1 til.. 1!)!)!)). Likewise, whilst eqn (3.357) implies that t he production of heavy c- and b-quarks in the string is negligible, ir is unable to quantify "v.*. t he degree to which strange quarks are suppressed relative to tin- light u- and d-quarks. In practice, several free parameters are needed to describe quark and diquark production. Finally, having selected the flavour and transverse momentum of the qq-pair and hence the flavour of the string fragment, it is necessary to decide which hadron it forms, in addition to the phase space factor 2.1 -f- 1. ./ being the h a d r o n s spin, consideration of a hadron's wavefuiiction suggests a suppression of heavy hadrons proportional to \ / n r (Andersson c.t til.. 1983a). In practice, the Lund scheme makes further free parameters available. The predictive power of the Lund scheme is somewhat undermined bv uncertainties in the properties of the (di)quarks. For example, what masses ///,,. /// s . '"mi. '"us and w s s are to be used in eqn (3.357). The UCLA scheme (Chun and Buchanan. 1993) a t t e m p t s to finesse this difficulty by only working in terms of t he known properties of hadrons. T h e essence of the difference is a reinterpretat ion of eqn (3.35(i) (Chun and Buchanan. 1987). In t he Lund scheme, eqn (3.350) is used to choose given a fixed transverse mass hadron. so that the normalization is given by AT"' = J
d-(1 ~
exp
= F(w'l) .
(3.358)
lu the UCLA scheme eqn (3.350) is used to choose both ; and i n s o that the normal izat ion becomes N~'
= £ ( C G ) 2 f c\piF(ml J ti
-I pi).
(3.359)
IIAI »IONIZATION MODKI.S
where ill«' sum is over all liudii>iis containing t h e end-point (juark anil CCJ represents ( lie corresponding c o n s t i t u e n t q u a r k model, SU(fi), Clebseh G o r d a n coellicients. Since i \ is now a c o m m o n constant , the transverse mass d e p e n d e n c e appearing in the exponential term in e<|ii (3.3")(>) immediately implies a suppresiini ol' heavy or large transverse moiiientmn hadrons. T h e actual UCLA implementation (Chun a n d B u c h a n a n , 1993) is slightly m o r e sophisticated t h a n described above, a s it tries to 'look a h e a d ' . For example, if the first, hadron leaves behind a u- or s - q u a r k . then the next h a d r o n is most likely to be a pion or a kaon. respectively, and the latter choice is (doubly) suppressed d u e to t h e larger masses of s t r a n g e h a d r o n s . T h i s is really an a t t e m p t to mimic tin 1 q u a n t u m mechanical projection of a whole s t r i n g on to a iiet of hadrons. Even in its more sophisticated implementation the scheme has remarkably few free p a r a m e t e r s . I lie relationship between t he various s t r i n g schemes discussed above is illusI rated bv t he family tree shown in Fig. 3.33.
FlG. 3 . 3 3 . A family tree for s t r i n g models. Figure from Knowles a n d I.affert,v( 1997). An interesting aspect of the s t r i n g models are the inferences that can be drawn from their associated space time pictures. We mention two here. At a siring break the <| a n d q are formed with equal and opposite transverse m o m e n t a . / .S',| x S\r must lie conserved a n d (L) ss 1/». the qq-pair must, typically form in a i s t a t e , particularly so at higher values of | p |. T h i s implies correlations between p and s))in for iieighbonring h a d r o n s (Andersson i t til., 1979/;). T h e second eonse(|iienc(i concerns identical particles (Andersson and Ilofinann. 19S(5: Artru a n d Bowler. 1!ISS). Provided that t h e y d o not have exactly the s a m e m o m e n t u m , t h e r e is an a m b i g u i t y in t h e o r d e r in which they may a p p e a r along the string. T h e s e two string configurations enclose different a r e a s with A / 1 <x (/'i PiY- Now, if the string d y n a m i c s and area decay law derive from t h e modulus squared of a n a m p l i t u d e . j\A exp[/.l(/.' I ¡Pq/2)]. c.f. eqn (3.35.I). then
1(18
T U B
T H E O R Y
O F
«.¿CL>
q u a n t u m interference will occur uixl tin- joint production probability becomes
which is clearly different from the naive sum of weights. This Bose Einstein correlation effect is largest for small A/L. t hat is. as (/J, p^) 2 • 0. This approach has close parallels to t he more geometric approach based on t he Fourier transform of the source distribution (Gvulassy et al.. 1979); see Section 13.3.3. M
B
B
B
M
B
FIG. 3.34. A schematic diagram of barvou production in the di
Cluster
liadronization
('luster-like structures have a long history in models of non-perturbative physics. They occur as intermediate states in string models ( A r t r u and Mennessier. 1974), in the statistical b o o t s t r a p model (Hagedorn. 1965: Frautschi. 1971). in multiperipheral models (Berger and Fox. 1973: Haiucr and IYicrls, 1973) and cluster
IIADIIONIZ M I O N MODI I
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I III
170
I I I K O U Y OK q ( ' l >
l'eri urlial ivi-
N o i l - p e r t urlmtive Model
primary part»»
labl.-N
shower
F i d . :{..'{(>. A schematic diagram for cluster hadronizatiou. Figure from Kuowles and Laffcrt.y(1997).
power of mass for a colour-coherent cascade (Bertolini and Marehesini. 1982). This is in marked contrast to the behaviour of the random, non-singlet mass spectra which broaden markedly with the energy of the primary quark. This behaviour holds also for down-type quark and gluon initiated jets. Similar observations apply to t he spatial sizes of clusters. T h e universal properties of these colour singlet, clusters, which arise naturally at the end of a coherent parton shower, has led to their use ;is the basis for a hadronization model (Gottsehalk. 1981: Webber. 1984). A typical mass of a few GeV suggests associating clusters with "super-resonances' that decay, independent, of one another, into the familiar liadron resonances. Here, simplicity is used as a guide to describing this process. Two-body decays are assumed for the majority of clusters, Clfqiffc) — h | -I- ho. T h e hadrons h | and ho are subject only to llavour conservation. If |/i|) = |qix). then we have |ho) = |xqo) with x = q ;t or qnqt: the field energy of the cluster being used to create either one or two qq-pairs. T h e probability of a given pair of hadrons being selected is made proportional to its available phase space. P(CI — h | + ho) rx (2.7|„ + 1)(2./|„ + 1 );>*(mci.»»»•!,,.m|12)0(-//!ei ~ '»h, - '»h,.) • (3.3GI) Here ¡>' is the C'.o.M. frame thrce-inoineiituni in the two-body decay. T h e 0 function ensures t h a t I he putative decay is allowed physically. T h e decay is taken to be isotropic in the cluster's rest frame. Referring to Fig. .'5..'in. two special cases suggest themselves. Some clusters will be too light to undergo any two-body decay. Such clusters undergo one-body decays with excess momenta being redistributed amongst neighbouring clusters. More significant are the limited number of rather heavy clusters found in t he tail of the distribution which occur when there is litt le pcrt.urbativc parton showering. Here, two-body, isotropic cluster decays would seem implausible, given the large
11/UNIONIZATION MODKI.S
171
momenta involved. I5y using I,lie (repeated) device of int roducing light (|(|-|)airs. I lies« clusters are split forcibly into lighter (laughter clusters whose directions of mot ion are aligned along the original <j• - qL> axis. This appearance of a preferred, colour Held, axis is reminiscent of the string model. The situation is summarized iu Fig. 3.30. As described above, cluster hadronization represents a well-founded a t t e m p t to go as far as possible with as little as possible. T h e original scheme is simple, compact and predict ive, though some special cases require ail hoc solutions. Since typical clusters are light. />' iu eqn (3.3<>1) is small and only limited transverse momentum can be generated during hadronization. Likewise. />' is reduced for heavier hadrons. leading to a suppression of barvons and strange hadrous. Furthermore. the spin ratios for iso-llavour hadrons are predicted partly from the (2.1 4 1) factor in eqn (3.361) and partly from the heavier mass of higher spin states. :t.S.r»
.1 comparison
oj the main hadronization
models
All the main hadronization models reduce to a set of rules t h a t are used iu recursively applied branchings. Their motivations range from the Q C D inspired, complex dynamics of strings, through the ininimalism of clusters, to t he simple expediency of independent hadronization. Table 3.1 compares their more important features. T a b l e 3.1 .-1 comparison
of tin main hadronization
model
ajijiroachcs
!ladrotiixation Model Siring l < al lire I'rinciple Loreiit•/. invariant flavour, charge ric., conservation Mass (lop. via Strangeness snpp. Itarvoii supp. ./'' ratios I / Ì I I I Ì I I H I //J_
Krng. fune. Ciit-otr (Qo) <M>. Stability 1.Miniai ions
Cluster Very simple Yes Automatic
Independent Simple No
Hadrons Predicted Predict«! Predicted Natural
Quarks l-'ree param. I'Yco param. l-Voe params. Knill in I'rec Very strong Collincar proli. Requires large (,)„
Significant Infrared prob. Massive dusters treat I'd like 'strings'
uii
hoc
l.tind Complex
UCLA l^-ss complex Yos Automatic
Quarks Hadrons Restricted params. Predicted Kestrictod params. l'rcdicted Restricted params. Predicted Built in Natural Restricted l>y 1. It syminotry Modest Stable Light strings treated as clusters
Kollceting its lack of a strong theoretical underpinning, the rather arbitrary nature of the independent hadronizat ion model is immediately apparent. A par-
()
172
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,
. 7 T
.0.(1.
s
, s
A
2
+
=
s s s s s
s ,
1 ( (
({, =
= (A ,. 0). ). s s ) 2
) (
s
s s
T
s
llll
171
I lll'.OKY <)!•• (¿CI)
3 8 As a precursor to solving the DC LAP equations it is helpful to tr\ to dccouple them as much as possible. Using e<|iis (3.282) and (3.28-1). derive the equations satisfied by the flavour non-s ,, s
3 9
3 It)
3 II
3 12 3 13
_ ,t and XbPi collide head-on to produce two outgoing particles of mass, rapidity and transverse momenta (m i. //i. p r ) and (m-j. n>. — />/•). measured with respect to the beam axis. Show that
•'•« = ^ - ( m / - , e + t " -I- m r 2 e + ^ )
x6 = - - K « - '
1
-I- w ^ c " » )
where in•/• \Jni- + pj is the transverse mass. 3 I I In a hadron liadron scattering there are many subprocesses which contribute to di-jet production. Below are the leading order. O(o~). matrix elements for the two-to-two QC'D scatterings of inassloss partons (Combridge r.t. til.. 1!)77). They are given as a function of the Mandolstam invariants .i = xix->s. / = ,s(l — eosl?")/2 and ii = - . s ( l -1- eosfl*)/2 = - ( . s + T), where 0' is the partons' C.o.M. frame scattering angle. T h e matrix elements are averaged over the initial state, and summed over the final s t a t e colours and spins, a factor //„' has been extracted. irtr —* tro • b»
9 / til lis si.\ — I A — 2 — —— — 2 I tU) 2 V A'
gg-»qq:
,-> .2, / I I {t'+v
3 1 \
IIADIIONIZATION M()l)i:i,S K'l • g
/-j . / ' I 3 1\ H (» I " ) -— I - V (> su 8 /2 J
•'I'll
8 J
8 .V2
-i
'I'l •'
9A '2 •1 .S2 + it2 -— !) f2 2 •1 .s- + ,> 9
S
)
ii 2 + .52 . >~'2 -1-12 \ 2 1 9 I f 2 ^ ) + it' •1 r9 .;2
'i'i — qq qq — <| q '
9 27 J7,
8H 2
Give the parton model expression, eqn (3.72). for di-jet production in terms of these two-to-two scatterings. Using the mat rix elements show t h a t their ratios, relative to that for the dominant gg —> gg scattering, are nearly constant, a s a function of c o s 0 ' and evaluate them in the dominant phase space region, cosfl" —» 1. Hence, show how to write the di-jet. cross section in terms of the effective p.d.f. defined in oqn (3.73). 3 15 Show that the polarization sum !)„,. eqn (3.121). for an on massshell gluon wit h four-momentum k1', follows unambiguously from the conditions k"T,t„ = k"T„„ = n"T„„ = II"T„„ = 0 and T' 1 ,, = - 2 . Here n is an auxiliary four-vector which sat isfies n • k ^ 0. 3 Hi Calculate the three-jet cross sect ion o + e — qqg for a scalar gluon. Assume inassloss (piarks and that the orientation of the event plane is averaged over. ' * * 1 3 17 By considering the Lagraugian oqn (3.21) in D dimensions determine the mass dimensions of the holds and show t h a t consistency requires the replacement
i:
d')xe.\p(-x2/2)
in both Cartesian and polar coordinates. 3 19 Derive the equivalents of the identities in oqn (3.83) sions.
for D dimen-
3 20 By considering m„ = ,nZm derive an expression for 7 , „ . defined '''Pi (3.158). ... terms of the Laurent expansion of Z,„. Assume
17(i
MI
IIIKOIIY OK Q C I )
a mass-independent rcnoi mali/at ion selieiue. so that Zm ^ni(n-)What constraints are placed on the coefficients in the expansions? Using <><111 (:i.ir»r>) evaluate your results. 1 ' :! 21 Using the n e x t - t o l e a d i n g order expression for AQCM> >>> terms of (Q~) a " d Q2 • e<|ii (3.174), show that for fixed i t j it is indepcn.;t. 3 2-r> Confirm the plus-prescription identity in e<|n (3.2(58). 3 2<> By considering the integral /n' d . r / ( j : ) / ( l — .r) l + < for arbitrary smooth f ( x ) show that I h n — i — = t—0 (1 — x}' +< In the "g —> qq. ( * * 1 3 28 Using eqn (3.275) and the identity / ¡ ' d . r / + ( . r ) = 0. calculate the 0 ( n s ) correction t o the Gross Llewellyn-Smith sum rule.
<|J . r | f l ( x . / / - ) 4- £
/(.r./r)|
(3.362)
By considering the //-' evolution <'(|uatiou for this quantity derive a constraint on the integral of the g —» gg splitting function and hence determine C K j.. the coefficient of the — ; ) term.
177
I I A I ) | ( < > N I ' / A I I O N M O I H I.S
I 30 Verify tin- result for 7,,,, . e(|ii (3.292). Next, working at one-loop precision, carry out the Melliu transform of the DCJLAI' evolution equation for a non-singlet quark distribution, eqn (F.2). and solve the result ing equation. W h a t is t he physical significance of the case 31 At one-loop level the Mellin transform of the coupled singlet-gluon equat ions, eqn (F. I), are given by
" rV V <'(»• I'2) ) ~
{ •
J I •*(»• I'") ) '
Solve these e(|uations for the case 11 = 2 and give the physical interpretation of the solution. ' * ' .'12 Show that in the DLLA the D G L A P equation for gluons can be written in the form <) ln(\n(Q-/\'2))<)
ln( 1 / x )
jrfo
* ''
and t hat eqn (3.301) satisfies this equation. II 33 In t he Mellin transform met hod of solut ion, the small-./- behaviour of the p.d.f. //(•/'.//*') is dominated by 7gg(" ~ 1). assuming a sufficiently non-singular f/(x. /if,). Consider a gluon-only theory and apply t he saddle-point method (Arfken and Weber, 1995) t,o evaluate xf/(x. //"). What assumptions have to be made a b o u t the position of the poles ill //(//./if,)? ' **' 3 3 1 At small x. when colour coherence is taken into account, the Mellin behaves as transform of the fragmentation function, D(n,Q2), ( l
I
2Ca
~nj(p)
3 3-r> 3 3(> 3 37
3 3N
(» - 1) .f.'WQ2)
(» - l ) 2
I 2tt " \CAn*(Q*)
\ J '
How does the mult iplioity of a jet behave as a function of Q and. using the sa
I III
178
()
)
)
( . .T
s
III (7 , + T,{)\q.r. 7 ., )
s .
= t
q.j) = 0. ( .
, 7
s
s ( .121). t,\ =
s.
11
( .
7)
. s 7).
. ( , , )2
. )
+t
s = C,\ 1. A s
)
,
,.j-.,,.l>) s
,.
s
f*
0 s
ss ( ) s
s
s
2
s ss
s
, s
(
(7
s. s
(
)
'
.
+
4 TE C R A
E
s s
s
ss s s s. T
s s ,
. T
s
s s s s, ss s s s
s s s
ss s s 1
,
s s.
s
s s ss
s
s s,
s
s
s (A s
. T
s ct ., 1 8 )). ct ul.. 1 ))
A(
s
s (1 80) (1 )8( ).
s s s s s
s
I
s
s .
s
.
s
. ,
s
s
.
. ,
s ct il.. .. 2000).
2 (A MIC ( s
s
s
c.t s
ss s .
s 1.1
s s s s
s. s ,
. s s s
s s. s s
s s
s s s
s ss
s,
s s
s . T s s
s s
s s
,
s
s s s. T s ss
s s s ,
s
s
s s
,
. s, s
, ss s
,
. T s
s, s
s
s
s
s ,
s
s
As ,
. 2.
s
s
s
s s 1 )) ) T
. ,
.
s
. . s s
,
,
, s
s s
s
s s
T
A 1.0
.
. s
s
s
s
s
s
s
s s
s s . T
s
s , s
s s. A ,
s, s
s s
s
s s
s s
s s
s
11
s
,
.
,
s
. T s s (A + )
s
A ,s
s
s
s s x 0. (. ) s x (. A s
s s
s
s
.. . . s s . As , s
s
. s
s
,
s.
(x)
s
s
s s
s s (0) ss
s s
s
(
( ) =
+ o
s
s .
, (11)
'
(
f
)
s
s
,
0
s
s
( 1.1) s
,
s
s . .
(4.1)
s
,
0.
(
)7
..
. 2001)
s s
.
s
s s. A
s
s
) )( + )
(
s
,
.
ss
s
ss
s ( ) s
s s
s
(. ) ./(l
s
..
1)) .
( ) .
. ( 2 ). . 1 ) )7 ). s s
s s
( 1.2)
•'
s
s
s . 10 ) )). s
s ss
(
( id..
(s).
s
(s).
0
) + 0(0)
+ 0(0'(0)«J) .
(-1.3)
AM, <)UI)|:itS MON'I I < 'AKI,OS
IKI
I'liis result is only exact in the limit A • 0. In practice, numerical values becoine independent of li when it falls well below any of the physical scales associated wit h the problem. A second, more significant, issue with one-loop calculations is that the cross section may be negative for p a r t i c u l a r phase; space configurations. I'liis means that only a weighted Monte C a r l o is possible without resorting to rat her artificial means. T h e principle a d v a n t a g e of a fixed-order Monte Carlo is t h a t it gives the exact result to a given order in p e r t u r b a t i o n theory. T h e y a r e suited ideally to describing well s e p a r a t e d , hard jets, which correspond to part.ou configurations away from t h e soft and collinear regions. In these regions we know t h a t large logarithmic e n h a n c e m e n t s imply the need to use resiiuuuerl calculations. T h i s is the domain of the all-orders Monte Carlos. T h e main disadvantage of a fixed-order Monte Carlo is t h a t it has partonic final s t a t e s r a t h e r t han realistic hadronic filial states. Again, this is the domain of all-orders Monte Carlos. Of course, it is in principle possible to generate a hard snbproeess using a fixed-order Monte Carlo and then pass the final s t a t e par tons to an all-orders M o n t e C a r l o for showering and hadroni/at ion. •1.2
All-orders M o n t e Carlos
All-orders Monte Carlos aim to simulate events in their entirety, t h a t is. to tinpoint of g e n e r a t i n g a fully specified set of final s t a t e hadrons. As such they are rather complicated programs, which must t a k e account of bot h port.urbat ive* and non-pert urbative QC'D. If they are to be useful, they must also conserve exactly f o u r - m o m e n t u m , flavour etc. T h i s m e a n s that t he p r o g r a m s m u s t deal with all of an event 's 'messy det ails', something which is often neglected at the level of accuracy of pQC'D calculations. T h e basic s t r u c t u r e of an event generator reflects t he factorized form of p Q C D cross sections. First, a hard snbproeess is generated according to the s t a n d a r d , fully differential, leading-order matrix elements a n d . if required, t h e p.d.f.s. Typically. t h e user specifies a set of sul¡processes of interest. T h e met hods used are essentially the s a m e as those for a fixed-order M o n t e Carlo, where t h e art lies in improving efficiency. Second, the •primary' p a r t o n s emerging from or entering into the hard snbproeess a r e showered to produce a set of 'final s t a t e ' partons. T h e generation of the incoming, space-like, and outgoing, time-like*, part.ou showers is baseel on the use of modified forms of the DC L A P equations for p.d.f.s and fragmentation functions. T h e s e stages are p e r t u r b a t i v e . T h i r d , a n o n - p e r t u r b a t i v e haelronizat ion model is use-d to convert the* final s t a t e partons into •primary hadrons". In a s c a t t e r i n g that involves initial state- haelron beain(s) there- is a need to treat the b e a m re-uinant(s) that are left behind when a part on is remove-el for the- hard snbproeess. T h e s e r e m n a n t s generate a soft underlying event, the- tnode-l for which is often elosely related to the haelronization model. Finally, many of the- p r i m a r y h a d r o n s are* actually highly unstable, st rong resonances or elcctromagnetically/weakly decaying particle's whose decays must, also be modelled. ()n t he whole. I liese- chains of elecaying particle's are gen-
M t t N I I C A I I L O M O D I I.N
crated using simple matrix elements and tables derived in pari from measured branching ratios ( P I ) G . 2000) a n d in part from p e r / i n - s p i r a t i o n . A c o m m o n exception are the lightest B h a d r o n s . whose decays a r e often treated as secondary hard subprocesses. followed by potential showering and hadronization. In some circumstances bot h these a p p r o a c h e s may prove i n a d e q u a t e to describe certain particles, for example, t a n s a n d Bs. In this case, t h e particle should be set stable in the Monte Carlo and passed on to a specialized decay package. 1.2.1
The. imlion
evolution
equations
The part-on showers convert highly virt ual, p r i m a r y p a r l o u s into low virtuality, final State p a r t o n s . T h e s e linal s t a t e p a r t o n s a r e either positive virtuality p a r tons just prior to hadronization. or negative virtuality p a r t o n s just emerging from any incoming beam hadrons. one per b e a m h a d r o n . We have learnt in Sections 3.2.2 and 3.0.5 that the Q-'-evolut ion of part.on-within-parton distribution functions are described by time-like a n d space-like D G L A P equations, eqns (3.285) and (3.49). T h e s e e q u a t i o n s s u m t he leading effects of repeated p a r t o n branchings. W h a t we would like to d o is t o generate explicitly these p a r t o n branchings a n d in the process c r e a t e a p a r t o n shower. To be a little more explicit, in Section 3.0.7 we learnt that in an axial gauge the D G L A P equations correspond to a sum of ladder d i a g r a m s (Frenkel and Taylor. li)70: Dokshitzer e.t til. 1980). with i n t e r f e r e n c e or crossed-rimg d i a g r a m s giving sub-leading contributions. T h i s ensures that the p a r t o n model language is a p p r o p r i a t e and facilitates a description of t h e leading contributions via a classical Markov process. Basically, each p a r t o n is assigned a set of probabilities for its possible branchings, or no branching at all. and is randomly evolved. T h e procedure is iterated for any daughters. T h i s jot calculus (Konishi et ui. 1979: Basset to et ui. 1983) led to the development, of Monte Carlo event generators (Fox and Wolfram. 1980: Field and Wolfram. 1983). Unfortunately, in t h e form given originally, the D G L A P e q u a t i o n s a r e not suitable for a numerical t r e a t m e n t , as t h e splitting kernels, cqn (3.50). a r e singular. However, we can reformulate them in an equivalent, finite form,
-/„/..(•'•• o £
/" Ji.
t) £
(-1.4) lt)
(4.5)
where h now represents a p a r e n t / d a u g h t e r p a r t o n . E q u a t i o n (4.4) p e r t a i n s to p.d.f.s. that is. a space-like evolution, while cqn (4.5) deals with the time-like
Alt
O U I ' H I S M O N I I < 'A It I.OS
evolution of f r a g m e n t a t i o n functions. We use the ( 9 ( n J expressions for the splitting kernels. /'(,„. which a r e associated with one-to-two branchings, a —* hi/, but omit the plus-prescriptions and ¿ - f u n c t i o n s in the diagonal n —• ng kernels. In Section 3.0 wo used a physical a r g u m e n t , based on the need to conserve quanlumbers a n d m o m e n t u m , to d e t e r m i n e t h e singular terms. However, in event generators these quantities are tracked explicitly and conserved trivially. By introducing suitable cut-offs on t he soft gluon m o m e n t a , the distribut ion function iem ulators can be replaced bv the equivalent subt ract ion t e r m s found in eqns (4.4) and (4.5). Their role is to provide correctly normalized branching probabilities. Formally, t h e s u b t r a c t i o n t e r m s follow from unitarit.y and the infrared fiiiiteness of inclusive observables and are given t o leading logarithmic accuracy by virtual diagrams of the s a m e order as t h e real emission d i a g r a m s . Strictly, b o t h the virtual a n d real emission d i a g r a m s a r c divergent with only their s u m being finite. In eqns (4.4) a n d (4.5) the divergent p a r t of the real emission is isolated below the cut-offs. Upon a d d i n g this •unresolved", real radiation to the virtual diagrams, the linit.e s u b t r a c t i o n term is o b t a i n e d .
a. t
a
-
t„
!>'. Iiy F t c . 1.1. A unit in a p a r t o n cascade, left., for a "forward evolving" time-like branching ti • h I //. a n d . right, a backward evolving' space-like branching n I- o ' -> I) As with the original D G L A P equations, both eqns (4.4) a n d (4..r>) possess simple classical interpretations, which prove very helpful in formulating the parton shower algorithms. We illustrate t h i s for t h e case of initial s t a t e radiation. First, recall that the p.d.f.. ft,/\,(.r. t)<\.r. is interpreted as the n u m b e r of p a r t o n s of type I) with m o m e n t u m fraction in t h e interval [.r,.r -|- d.r], within li at the scale t. Second, note t h a t t h e splitting kernels give the probability t h a t in evolving from the scale - I to - ( / | St) p a r t o n h will undergo a resolved branching into p a r t o n s a and n'. each carrying fractions c and (1 - ; ) . respectively, o f / / s moment um.
P(l> -
"(4
+ " ' ( 1 " z) when /
/ + St) = j ^ d l p ^ z ) .
(4.0)
T h i s basic unit is illustrated in Fig. 4.1. T h u s , in cqn (4.4) the first, (source) t e r m represents the possibility that a p a r t o n l> present at a slightly lower scale, t. with a m o m e n t u m fraction larger t h a n ./•. underwent a resolved branching, l> —> no'. yielding t h e specified p a r t o n n. T h e second (sink) term, which is s u m m e d over all the available decay channels n • re', represents the possibility t h a t the already
181
MON TI-: OA It I,O MODELS
present pallini a is n-ii loved from I lie specified momentum interval by a resolved Inanellili);. 1.2.2
Iinni ehm tj hint mat ics
As a preliminary to obtaining solutions to eqn (-1.4) and eqn (4.5) it is necessary to give the exact definitions of t lie evolution variable, t. and split t ing fraction. ;; see Fig. 4.1. 'These, in turn, determine the form of the z limits. (/,(/), appropriate tu each branching. In pract ice the definitions depend on the specific Monte Carlo implementation, as does t he choice of argument, for o s . Typical examples are |p|)„. which become equivalent in I i>~ or ¡ r and : = EI,/E„ or (E + I>:)I,/(E I he limit E„ —> oc. As a m a t t e r of convention, for initial s t a t e space-like radiation where virtuali!ics are negative, we take the evolution variable to be -t. Thus, for both initial ami final s t a t e evolution we have t ~ at the hard scattering, with decreasing values away from it until the respective cut-offs and io are reached. To see how the kinematic limits. ci,(t). depend on the precise meanings of t and c consider a time-like branching with t equal to the mother's virtuality and c the light-cone momentum fraction of b. Momentum (/>-) conservation implies p'ì = z( 1 - z)tn - (1 - ;)/,, - ztv
> I» •
(4.7)
Now. each partou species has a cut-off placed on its virtuality. below which pQC'D will no longer be valid and the shower evolution is terminated. For example. //, > t'f\ = (mi, + /»())*'. when- mi, is the parton's mass and m
]
~miii J
1 L , (tbo ~
2
t„
±
\{
.,('{; + 'o)
y
{
Cij-ff)2'
t„
/2
i - i
III Ml t
To reach the required level of accuracy (ill the Sudakov form factor eqn (4.11)) it is actually sufficient to use the simpler limits. ; <E [
fo(t„) = r
•
(4.9)
Finally, if a branching does occur, then eqn (4.7) implies the upper bounds lb < zta ,
t» < (1 - Z)l„
(4.10)
on the daughter virtualities. T h e space-like case is left as an exercise. Ex. (1-1).
A M OHI >I IIS M O N I I CAUI.OS
1.2.3
Timi-like
Miniti
Carlo
algorithm
I'roeoeding formally, a set of Sndakov form factors (Sndakov. I !).r)(>) is introdnccil. defined by
|"A„(m;;) = -
f~"{'\h<Miiììpca{s)
E n-.ee'JK
1
(4.11)
27T
'• •C')
= a—ce' 11 A a - c e ' O - Q ,
A„(t,t"0)
where t\\ is a cut-off associated with the time-like portoli a. T h i s allows the second term in eqn (4..r)) to be removed and the equations to be rewritten as
'I
(
M
)
=>
-
C
t) = A „ ( / . /;;)D,';(:,:. f||)
*
i
j
(.1.12)
r
t
^
-
^
a
-
^
)
•
111 t he integrated forili we used A„(/¡;./¡¡) = 1 and A „ ( / . t\\)/ A „ ( / \ I'(\) A,,(/./')• This set of (inhomogoneous. Volterà) integral equations is solved easily bv repeated back substitution to obtain the Neumann series solution. " ! ; ( ' . <)
* I
A
„ ( M S I D Í M )
t
f
r'iUi
y
r
^
f
' d ^ i
.Ir,; ' i Jx X A„(l. I, ) I
(
^
r'^u?
¿i .A,'; Z
l
}
.
^
h
^
M
l í,{t2)
r -
'
-
*
-
*
)
(\z.¿
.'x/;,
/ ^ . f c , )A),(/|. / , , ) ^ 1
^
.
[\h(z2)Ar(h. 1
3
)
This expansion forms the basis of the Monte Carlo implementation of linai st ate, time-like parton cascades. To give eqn (4.13) a physical interpretation we need to understand the significance of the Sndakov form factor A 0 ( M ¡ ¡ ) . It is the probability for parton a to evolve down from the scale I to /,'j. its cut-oil', without any resolvable radiation. This is most easily seen from t he fact t hat A„ satisfies the differential equation implied by
P(t -1- ¿t — f„ I no res. rad.) = . - E ii —re'
f1 ^ • '
a
m
m
4
*
m
*
>
1X6
MONTIS CAHLO MODELS
x'P(l —
| no res. rail.) .
(i.M)
which is essentially a radioactive decay law. In eqn ( 1.11) the siunination ovei all resolved branchings which remove the part on n — re' gives rise to the sul) probabilities A„_^ c< ' indicated in eqn (1.11). Armed with this interpretation, we see that the first, term in eqn (4.13) represents t h e possibility that the highly oil mass-shell p a r t o n n first evolves clown t o the low scale /¡¡, without any radiation being resolved. T h e n it undergoes a. by now non-pert urbative. transition producing h with a fraction .»• of its m o m e n t u m . Similarly, t h e second term represents t h e possibility that initially n evolves down to t h e i n t e r m e d i a t e scale. I \. at which point a resolved branching, a — lib', occurs. P a r t o n b. which receives a large fraction Z\ > .r of qg splitting function contains l n ( l c) terms, whilst in the case of t h e g —» gg splitting function there are also singular In c terms. By making t h e local transverse m o m e n t u m , /r' s= ; ( 1 c) a n d I n ; corrections a r e accounted for : ) ( } ' . the a r g u m e n t of o s these ln(l automatically (Amati cl til.. 1980); see Ex. (4-(>). T h i s choice does introduce spurious singularities into PK
A«,('.'o)|
lo.(l) . LJ:. I..A / A2
N i l " " '
I o It,A )
(4.15)
For «„(i) it rlecreases as an inverse power of I. whilst for o s | / r : = ; ( 1 — z)Q'\ it falls faster t h a n a n y inverse power. T h e difference is important for the preconfiuement propiMt.y of pQC'D (Bertolini and Marchesini. 1982). In both cases, eqn (4.1!>) shows that it becomes increasingly unlikely, as I increases, to evolve from I ~ Q- to /o without any resolvable radiation. T h e numerical implementat ion of eqn (4.13) is as a Markov process that proceeds in a series of iterated steps. Given an off mass-shell p a r t o n . ti. of m a x i m u m scale I > t[\. t he channel-by-channel Sudakov form factors. A „ _ C C ' ( / . ^ ). a r e used to select a possible intermediate scale at which a specific branching can occur. Of course, it is possible t h a t t h e p u t a t ive i n t e r m e d i a t e scale is below in which
Al.l. ( I l t l ' l 'll.s MON'I !•: CAIM.OS
case the p a r t o n is placed on 'mass-shell'. /¡¡. II a valid branching does occur, then t h e m o m e n t u m fraction is selected with distribution t\JJ.,z)Pc„(z). and the d a u g h t e r particles are given new m a x i m u m scales, /,. a n d derived from / a n d z. This procedure is repeated until all p a r l o u s are placed on their 'mass-shell'. It is worth remembering that the branching algorithms a r e designed p r i m a r ily to give good a p p r o x i m a t i o n s to t he full m a t r i x element in t he d o m i n a n t phase space regions. In non-soft, uon-colliuear regions, for example. e + e ~ —> qqg with the gluon recoiling against a near-aligned qq-pair. they may olfer a poor approximation. It is therefore c o m m o n to correct, the first branching t o t h e full, known m a t r i x element.. It may also be necessary t o apply corrections to p o p u l a t e regions of t h e full phase space which a r e not covered by the branching's ( / . : ) phase space. •1.2.1
Space-like
Monte
Carlo
algorithm
In view of the s t r u c t u r a l similarity between cqns (4.5) a n d (4.4) we can employ exactly t h e s a m e method to obtain t h e following formal series solution to eqn (4.4). ' /„/,,(.!',/) = A n (M'J )/„/., ( * ,
,/f" ' I
r'd/i Jr; 'l
JX
\Z \
¿1
rl-'>-'M(\zi Jx
O
r''i\i. J,';
h
r,~r:M-i)dz2 Jr/z,
x A „ ( ) . / | ) n s ( ^ : | > / l 4 : i ) A , , ( / „ k)°A[;_Z2) + •••.
J
*2
l\(z-2)
Ac(tj.
Q f
r / h
^ , t ^ (4.10)
Again, a simple physical interpretation is possible. T h e presence of p a r t o n n. inside h at t he scale t. for example virt uality. may be accounted for in au infinite number of ways. Reading from right to left. t e r m t wo. for example, represents t he possibility that at the cut-off /.¡J part.on h is found in h wit h a large m o m e n t u m fract ion x/z\ > x. Part.on l> then evolves without resolvable radiation to the scale 11 <E I'.'','] at which the branching l> • on' occurs, leaving p a r t o n o with energy fraction x. P a r t o n o then evolves from / | to the scale t. again without resolvable radiation. Similar i n t e r p r e t a t i o n s exist for all the other terms. Unfortunately, this forward evolution scheme, eqn (4.10). for t h e initial s t a t e is no longer generally regarded as suitable for Monte C a r l o i m p l e m e n t a t i o n . Its fatal Haw is t hat it is not possible to g u a r a n t e e t hat t lie generated parton cascade will produce a parton c o m p a t i b l e with t h a t required for the pro-selected hard scattering. Despite a t t e m p t s to improve this situation using protabulation techniques (Odorico. 1984). it remains a grossly inefficient, algorithm. Fortunately, t h e problem has been finessed by a reformulation as a backward evolution scheme.
ihvmi i i'i * i i i u A ' i\i w i . t ->'
I no
ill which t h e known p.el.l'.s. already used to choose tlie linrd suI »process, lire employed to guide t h e evolution of p a r t o n s from the h a r d - s c a t t e r i n g to the low-scale incoming hndron (Sjostranel, l!W.r>: C o t t s c h a l k . lH.Sli). By judiciously inserting 1 written as a ratio of p.d.f.s. eqn (4. Hi) can lie nianipnlated to o b t a i n an equivalent form, i = I L ('•'*;•'•)
f m ,
f
-
'./,.77./, ./,.. ' l
m
n
77
r'ijh 1
^
,
t
"»('t^i)fi
"
d ^
f
.fh i
u
' h
77" ^ ) i . (
T^tXi^
\
r\
-t/
f'-o<'v)
j
r
Z ,
+ ....
(1-17)
Here t he usual time-like Sudakov form factor, eqn (1.11). has been re|>laced by UJu:..r)
=
u u
)
f
f ^ .
(4.18)
T h e interpretation of [ ] „ ( ' • / " . x ) is as the probability that the partem n. present in h. evolves backwards from the scale I and m o m e n t u m fraction x to the scale i" < I and unchanged m o m e n t u m fraction. ;i\ without any resolved emissions. To see this, let / " „ ( / ' : / . . r ) d / ' be the fraction of p a r t o n s of t ype n present at I wit h m o m e n t u m fraction .r. which c a m e from a «•solved branching in t h e interval !/'. I' + d/']. T h e n one has r U M ! »
= 1
.
1
'
„,_£_ (
fu u{*.t) i, - A . C . O ^ r f -
or
u
v A,,(/'./)
) \ ) ( U 9 )
/i./id^'.')
In the third line, we have used the D G L A P equation, eqn (4.1). in a form equivalent to eqn (4.12) and when required used (t'.t) = A „ ' ( M ' ) . Recalling t h a t t h e D G L A P evolution suppresses high-;/- p a r t o n s . f \,(x.t) f \,(x.t ). we
A M OHDI'IHH MONTH OARI.OS
IN!l
ee I hat, the ellect of I he p.d.l rat io in eqn (-I.IS) compared to eqn (4.11) is to make II > A . T h e converse is t r u e at small x. T h i s rellects t h e observation that a high-j partem is less like-lv to have undergone a branching than a small r partou. A se'conel modilication in e-epi (4.17) a r e t he p.d.f. faetors aoeoinpanying the •plitting function. i,(z)fi, ll(x z. t) f \,(x. ). T h e presence; e»f fh il helps to guide: the' evolution to t h e correct partem content, whilst f n, emsurcs the- correct. inali/at.ion. Strict ly speaking, t hese p.el.f.s and t hose a p p e a r i n g in eqn (4.18). are the' solutions to eqn (4.4). However, in practice, t h e p.d.f.s used are those from the stanelard fits; see C h a p t e r 7. We are now able to interpret eqn (4.17) as a correctly normalized s u m of probabilities for all the chains of branchings that take a part.on a, present at the scale' I. back to an initial p a r e n t ' partem at the scale' I". R e a d i n g from left to right, the see:e»nel t e r m , for example, represents the probability t h a t theelesired part.on a . pre-sent at t. had e've»lve>el. without resolvable radiation, from the scale t , . At t\ it had be-on produced in t h e branching of a p a r t o n b of larger moment uui fraction x zj. which liael originally e-ome from the scale ^ without resolvable- emission. T h e bae kwarels evolution scheme' implicit in e-qn (1.17) is solved numerically using the following iterative a lg o r i t h m , which is similar to that for the time-dikeease. Give-n a space-like p a r t o n n of m a x i m u m scale t wit h m o m e n t u m fraction x. the modified Sudakov form faetor 1 1 „ ( / . . ; : ) is usee I tei.select a put a t i v e b r a n c h ing scale. If this new scale- is be-low the- space-like cut-off /',', or leaves insufficient phase space for any branching, then no resolved branchings a r e taken to have oe i nrreel and the partem is placed on 'mass-shell'. ;>2 = t . If a branching does occur the-ii the type, b tin'. and m o m e n t u m fraction. are selectee 1 according to o s ( / . z) h(z)fh \,(x z,t). New m a x i m u m virfualitie's are e-onstructcd for the daughter p a r t o n s anel the above p r o c e d u r e repeated for l>. whilst ti' is t r e a t e d using the til lie*-like algorithm. R e m e m b e r that only those p a r t o n s lying elirect.Iy em the chain linking the- hard subproeess to the incoming h a d r o n have ne'gaIive evolution variable-s (virtnalitie-s) anel only these p a r t o n s a r e governed by e-qn (4.4). All p a r t o n s on the siele- branches have' positive evolution variable's anel are elescribcel by the s a m e t iiiii'-like- e q u a t i o n s as above. A final issue is how to m a t c h t h e low-scale partem, at the- start of the initial state- shower, to the q u a r k content, of t h e incoming h ad r o n . lit the- e-a.se- of gluons and sea q u a r k s flavour conservation requires a minimum of up to two 'forced' branchings. For example, the simplest solution to the- worst, case of a s emerging from a proton requires a branching scqucncc such as p : (uu)el -* elg, g —> ss. Sine-e- we are now in the non-porturbativo regime, these branchings need not. follow the- s a m e prescription as in the cascade. though m o m e n t u m etc. conservation should st ill be- respected. In view of the pat tern's Fermi motion wit hin t he hndron it is ceimmou to give- the initiator partem, el in the above example, anel remnant mi. a Ga ussi a n transverse: m o m e n t u m .
MON I I < WKI.O MODELS
4.2.5
Soft i/htmi
logarithms
Logarithmic enhancements to a mulli-parton cross section occur not only in the coHinear limit hut also in the limit of soft glnon emission. It is therefore important t hat soft glnon elfects are included properly in Monte Carlo programs In Section 3.7 we learnt that the cross section for the emission of a soft glnon is dominated by interference between Feynman diagrams. T h u s in order to permit a Monte Carlo implementation we need to impose first a classical structure. In essence, we must find a way to associate the emission of a soft glnon with a particular hard parton. ensure positive emission probabilities and maintain the correct total cross section (Bassctto <1 n i . 1083). This will lead us to a simple modification of the D G L A P equations and the so-called modijtctl I<<1 • I'j (p, • k)(Vj • k)
ll'ij (k) = J2
=
_ 1 l>j 2 (p, • k)2
_ 1 1'j 2 (pj k)2
I [ j ^ L _ _ l
2 [Qk<ji.
Q.
.
(4.20)
(;,..
In the second line we have introduced the angular variables Cu = ,JlrpEli
ElJ
= l-fMjVosO.j
.
(4.21)
which range from (I — >i,;ij) w 0 to (1 I Haij) 555 2 as HtJ goes from 0 to jr. In the massless limit. —• 1. eqn (4.20) contains collinear singularities as either — 0 or Ojk * 0. This suggests the following decomposition, based upon adding and subtracting (I/OA- - 1 /<>*)•
=
Wj;\k)
+
Wjp(k) .
(4.22)
T h e first piece. H ' ^ ' ( k ) . has only a collinear singularity for k parallel to where H'^'^A) ~ (I
p,,
c o s 0 , j ) ~ K It is finite for k parallel to pjW Thus, it may
nat urally be associated with emission off parton i. In t he cross section formula, eqn (.'5.311). we think of the directions of the hard p a r l o u s as fixed a n d the direction of the soft glnon as variable. Therefore, let 11s consider W ^ ' ( k ) for lixed 0 t J and study it as a function of 0,/. a n d 0,. the a/iiiiut ha I angle w.r.t. pr In terms of these variables we have cos 0^ = c o s 0 , j cos | sin <),, sin On,. c o s ^ j and t hus
M4, O l t H I UN MON I l'i OA I (LOS
fi, - cosft1 ( 1 - li, c< is Ila
I
l!H
COS 0,k - liJ cos 0,J ¡ij |c( is 0,j cos 0 l k I sin 0,j sin Oik cos <j>, ]
(4.23) Figure 4.2 shows the angles and IV-'' as function of <j> for various (),, and f) i k .
F i e . 4.2. T h e angles between the soft glnon k. the hard parton i and its interference p a r t n e r j . T h e radiation function 26A- X U7^''(A1) as a function of (•>,. Left, jt/2 = Oij > 0,,. and 0ik = j r { 1 . 2 , 3 } / 8 for the dashed, solid a n d d o t t e d lines, respectively. Right, jt/4 = 0tJ < U,k = jt/2. In all cases ii, = 1 = f i j . A significant feature of eqn (4.23), as shown in Fig. 4.2. is t h a t it is everywhere positive-definite for 0 i k < () l} . In contrast, to this, for 0 i k > 0 t J the distribution goes negative, that, is, there is destructive interference. In fact, aziinuthal averaging gives
with H(i)(0
) = - ( " H C
=>
«(»„
~ vx^0,k
( os ,k
' "
~ !jJrn*"<J
A«-«») " y i c o s ^ - z v - ^ l
0, k )
for
B,.j -
1;
2
\
I ( l - ^ ) s h r t f j
(4.25)
see Ex. (4-4). This rather complex looking expression for //,')'(",A-) reduces to the lleaviside step-funct ion for massless. hard partons. This requires the opening angles to be nested. 0, k < 0 t J ( = Q k < (,,j). in order have any net soft radiation. This restriction is the basis of the a n g u l a r ordering prescription. T h e form of / / ' j '(fliA-) Ibr massive particles is shown in Fig. 4.3. It follows the same form as the step-function but the inclusion of particle masses softens its shape. For wide-auglo emission the first, term in eqn (4.25) is positive, whilst the second goes negative for cosOn,. < iij eosft,, and t h e funct ion falls away quickly. A new feat ure is the vanishing of radiation at. small opening angles, where the collinear singularity is shielded by the particle mass: t he inverse p r o p a g a t o r behaves as
MON I I C A I l l . o M O I H l,S
2(1 — H, eostf) ~ (111/E)2 I)2. Inspection of cqn ( l.2.r>) shows that the destructive interference begins when eosW,^ > i, ( t),i. < m,/E,). T h i s is I lie so-called "dead cone' for emission close to a heavy particle's direction of travel. In effect, soft radiation is restricted to tin- screened cone R\') defined via nr <
:
Hj cos 0, j < cos 0lk < fi,
<=>
< U £ Co •
(•' -26)
T h i s dominant region of p h a s e space is illustrated in Fig. 1.3.
F I G . 1.3. T h e screened cone for t he emission of t h e soft gluon h associated with p a r t o n /. whose interference p a r t n e r is j. Here we iissiuued A, ().!)!), r which iiuplies a dead cone up to « <S° and f, = 0.!). >, which softens the cut-off at 0,k = f>,j = TT/2.
Using these results we can rewrite eqn (3.3-11) in a form which is everywhere positive-definite. f|ffWr+i> =
^
. f , | B ( C 0 - d k ) W U > t o ) ! ! g < I G k 1- (/
.
(1.27) This version is for massless parlous. It is exact in t he angular ordered region and outside this region, where the soft ghion radiation is prohibited, correct on the azimuthal average. To include hard parton masses, replace (iCu by dCiA-/. 'i mid 0 ( 0 j ' Cik ) by so that the radiation is restricted to the screened cone. O n e might also include a multiplicative factor t o compensate for the neglected tails of the ¡¡Ij 1 distribution which lie outside the screened cone. T h e above reasoning applies equally well to ( J E D . where it is known as the C'hudakov effect (C'liudakov. l!).r>.r)) iu recognition of the observation of a lack of wide-angle soft photons emitted by t he e + e pairs found in cosmic ray showers. Henristieally. if the soft photon is emitted at too wide an angle to t h e hard
A M . O I I D I M I . I ¡\U1IN I I', I AlflAJM
lil.l
Icpt.ons, it will have uisullicicnt, transverse. resolving power to distinguish the .cparatcd charges and will only see their total charge, which is zero: see Ex. (4:i). In Q C D the argnnient is the same hut the total charge need not he zero. I'o understand the situation b e t t e r we will consider how to treat, successive soft gluon emission in a par ton shower. Consider a colour neutral system of hard p a r t o n s and let. us focus on p a r t o n s / and j . collecting t h e remainder together in Y l r T h e s q u a r e of the insertion current, is given by
- i J • •/' = t , • ( V , + £
- Y
t ^ j uf
f
if, + £
>) "
i
; <
J )
T< ' T i ' U '"' •
T, • t , I V,i - Y , t , • f j W j , - Y
i
f
(•' -28)
i/i'
lu the lirst line. we have decomposed IV tJ using eqn (4.22) and m a d e use of the identity Tj I- Tj I
'7/
(I. Re-arranging this expression gives
- i j - ./t = Ciwjp 4- CjWjp -Y
tr t, [(u //> - W$>) +
\lf
i
-Yf>
f
i
J
}
[ K
- O
+ < ]
-Yf>
w
t,'W„. .
(4.29)
where we have used T~ = (",, the q u a d r a t i c Casiinir. colour charge. In view of the third a n d fourth t e r m s we introduce a new function, IF,'.' = W ^ - W j p
such t h a t
(M>) x
=((ll' [()
; ,
),„ otherwise (4.:«))
where we took < (,',/• hi I F/(''(/>*•) t h e collinear singularity for pi.- parallel to ¡i, has cancelled and ill fact the function is only non-zero on average when the soft gluon is radiated at an angle greater than 0,J. but less than 0,/. Now. we make the assumption that, i and j are close in angle so that in the non-singular M'-functions. which are scale independent. /», and p j can be replaced by their sum p , pi + ] > j . In this a p p r o x i m a t i o n eqn ( 4 . 2 9 ) becomes
I./../t = c,
1
1 vjj 4-
Cj
11
' - £
7} - F S [ I F / ; ' + W ^ - Y ' F R T R W , , .
i
.
(4.:',1)
w
where Ts = 7', +Tj. T h e first two t e r m s represent (»mission oil" eit her / or j which is limited by their common opening angle and with weights C,. T h e n , if we think of s as the parent of /' and j the remaining terms represent, t h e s q u a r e of the insertion current for the .s- + ^ J / system prior to the '* • /. -f j branching' with the restriction that 'radiation from s ' must, be at an angle greater than 0 t J . Of
MM
MONTH CAKI.0 MOIMibH
course I lie parent .s is only n I heoret ¡oil construct which helps us to enenpsnlnte the effect of colour coherence in this small-angle limit, hut the concept proves very useful in developing a Monte Carlo implementation. By iterating the procedure which led t o e q n (-1.31) we see that we can capture the leading contribution from successive soft gllion emissions by generating a parton cascade iu which successive opening angles decrease. T h e cross section for generating successive* branchings is given by ( 1 „<,v + „ = 4
-7T ^—
I
\
(c,„ A i a - d ^ ' » . / <J>, ¿71 uj
(-U2)
where the interference partner. /'. is the parton produced in the same branching as i. This raises the obvious question: how does this expression relate t o e q n (4.0) which describes the colliuear singularities? Now. the diagonal splitting functions, /'„,,( 1 z) with .
(4.33)
so that, neglecting aximuthal correlations, the distribution of soft gluon radiation is correctly described by the usual A it arc-Hi Parisi kernel with an angtilar ordering constraint 0 ( C I N — C M ' ) imposed. A natural way to achieve this is to use tin* evolution variable l„ = Ef,( 1 - cos0i, c ) and the orderings //, < z~l„ and < (1 -z)~t„. Including the dist ribution of the aximuthal angle requires two sets of q u a n t u m mechanical correlations to be taken into account, those which are due to soft gluon interference, eqn (4.23), and those due to the spin polarization of hard gluons in a cascade (Knowles. 1990: Richardson. 201)1). T h e analysis of soft gluon coherence given above applies to the linal state. A similar approach holds for soft gluons in the initial state, though the analysis is more complicated due to the asymmetric kinematics of space-like branchings. T h a t said, the effect of colour coherence also can be included by placing a restriction on phase space iu terms of an angle-like variable which decreases away from t he hard scattering, towards the incoming hadron (C'atani el ill.. 1980). The precise variable is defined in terms of the incoming hadron momentum as I, = E, \ / C « E,ij,
with
Ci =
i U ~ 1 - cosp, . E,> hi,
(4.3-1)
where /' is the time-like daughter a t f s branching. Observe that in the ./• •• —» I limit (z, ~ I), ordering in /, becomes equivalent to ordering in Due to destructive interference this is more restrictive than the earlier ordering iu virtuality, though disordered transverse momenta are permitted by I, ordering. Iu the opposite .c —> 0 limit ( ; , ~ 0.5), ordering in I, is equivalent to ordering iu the branching's transverse momenta. Actually, in this small-.r region the DGLAP equation is superseded by t he BFKL equat ion: see Sect ion 3.0. This leads to st riot
AI I < HtlH.HS MONTK CAM.OK
Mulcting in y?, which is less restrictive than ordering in transverse momentum, anil the need for a new, non-local. non-Sudakov form factor, whose effect, is to dampen the l / c singularity in l'K)i(z) (Catani ct
T"T"(M, + Mi)
ThT"(M„
-
Mt)
I K : . I. I. T h e two colour Hows contributing to the amplitude for quark gluon scattering To see how this works in practice consider qg —> gq scattering. At leading order three Feyninan diagrams contribute to the amplitude: ¿--channel quark exchange wit h colour factor T"Th. »-channel quark exchange wit h colour factor r''T" and /-channel gluon exchange with colour factor ifabcT'' = T"T<> T"T". I'lie two colour Hows for this process are illustrated in Fig. I.-I. Referring back to Section T.'i.-l. the amplitude squared is given bv \T"Tb\Ms
rx C,,.\'r
+ Mi)
(»'-' + -s--)
+ TbT"(M„
-
M,)f (i.:jr,)
I he lirst and t hird terms correspond to t he squares of t he sub-amplitudes, which are used as weights when selecting a colour How. T h e second term corresponds
l'Ili
MON II ('A 1(1 o MODKI.S
i<> llir neglected interference t e r m . It is both colour suppressed. being of relative m a g n i t u d e C ( l / A : , " ) , a n d dynamically suppressed as it has neither a l s nor a I / singularity. 1.2.0
The colour dipole. model
T h e formulation <>f p Q C D evolution in t e r m s of initial a n d final s t a t e p a r t o n showers is not unique. A complementary, or dual, a p p r o a c h is possible in terms uf colour dipole cascades (Gustafson, l!)<S(i; Gustafson a n d Pettcrsson. 1988). T h i s is equivalent, to MLLA accuracy, to a p a r t o n shower with angular orderinn automatically incorporated (Aziinov ct id.. 19K.r>fi). T h e basic observation is that gluoii emission from, say, a qq-pair is described, in direct analogy with elect romagnet ism. by t h e s a m e formula as gluon emission by a colour dipole. To 0 ( \ / N 2 ) subsequent gluon emission may t h e n be described by radiation from two independent dipoles. T h e resulting picture is of a colour-ordered chain in which gluons join dipoles a n d dipoles join p a r t o n s . A close correspondence to the string model, in which gluons cause 'kinks' along a string, makes this formulation seem rather appealing. Colour dipole cascades are implemented in the program A R I A D N E (I.onnblad, 1992). Gluon «'mission from one of t h e three basic dipole types, - ' — qq — qig_>q;( with aligned quark spins, q — qg — c n g 2 g 3 . q — q g - » q t g j g . t or g -> gg - » gig_>g:t. is described by t he cross s<-<-tion formulae for inassless p a r t o n s
Here j'i.:t represent twice the energy fractions of the two e m i t t i n g p a r t o n s in the dipole's rest frame, so that, t he gluon has .r-j = 2 - .ri .r;t = 2/v K / v /s,ii, ( G [(I. 1). Here .s,ii,, is the s<|tiare of the di|K>le's invariant mass. T h e coefficient is given by Cjj 2/1$. Cxi = 3 / 1 C t:i for t h e t h r e e vertices, respectively. T h i s expression is designed t o be fully equivalent t o the conventional a p p r o a c h , see Ex. (1-7). T h e splitting g — qq does not arise n a tu r a l l y in t h e above formalism for gluon emission. Its treatment is inspired bv a p p e a l to the full 7* —> qq'q'q cross section (Audcrsson ct id.. 1990). In t h e limit of a low A - . low mass, intermediate gluon this factoriy.es. a f t e r aziiuuthal averaging, as .(:). which can b e s e p a r a t e d into two etpial p a r t s associated with the q-g and g-q dipoles. T h i s prescription for the dipole formalism enhances effectively g —. cjq splitting c o m p a r e d to the p a r t o n formalism (Seymour. 199.r>). T h e dipole formalism also uses a different, description of t h e phase space. In t he limit of soft gluon emission, oil' inassless p a r t o n s . t he following ' r a p i d i t y ' and "transverse moment urn' variables, bot h wit h respect t o t he mot her dipole's axis, suggest themselves.
with
A:'I = .s
(-1.37)
Al.l, (»Hill IIS MON I I CAIN.OS
197
I 'ic. -1.5. Tlic fractal phase space in the colour dipole model. T h e d o t t e d lines show t.lie allowed region neglecting the /,-L ordering constraint. T h e dashed lines indicate the additional constraint, when extended sources are present.
Here A is of the order of the Q C D scale A Q C D - P a r t o n mass effects can lie -Iincluded by replacing (1 — :C|)(1 - :'":i) in e q n s (4.30) a n d (4.37) by (1 — (inf ii|>)(l — 4— "if )/.s,ii,,). In light-cone coordinates the glnon has m o m e n t u m (A•+./. - ) = \ f k ' \ ( e + " . e _ ! / ) . A d o p t i n g these variables eqn (4.36) becomes
(I(T
= <¡02. \(! '¿nth l \
- e ^ - M " -f (1 v^di^/ \
y ^ J
~dv K
~ -dW K
.
(4.38) I'sing f o u r - m o m e n t u m conservation we find the following constraint, on // for a given A-x.
= 2A-J.cosh!)
=>
In
> l u ( 2 c o s h j / ) % |/y| .
(4.39)
This is an approximately triangular region iu the i/ — ln(A:x/A) phase space. When a glnon is e m i t t e d , t he p h a s e space for subsequent branchings is increased as indicated bv the projecting region in Fig. 4.5. Repeated gluon emission leads to a fractal-like s t r u c t u r e (Andersson cl til.. 1989«: C u s t a f s o n and Nilsson. 1991). When a g —» <|(| branching occurs, the phase space is effectively cut along the middle of t he projecting region. Given a branching a t (A'I_L, I / J ) . two new dipoles are formed a n d . a s shown in Fig. 4.5, the phase space forms two 'overlapping triangular regions' s e p a r a t e d by the glnon. Reflecting t h e gluon's d o u b l e colour charge, in the p l a n a r approximation. radiation is p e r m i t t e d on eit her side of the project ing region. T w o constraints a r e placed on the phase space for a second, subsequent branching.
IMS
M O N I I C A K I . O M O D I I,.S
<
kinematics dynamics.
A'II
(|
I'lie- kinematic condition follows from t h e masses of (.lie daughter dipoles. see Ex. (1-8), and can allow > A"i_. T h e d y n a m i c condition is required for the validity of the elipolc cascade approximation. Applying the stricter /.• ordering compromise's t he- dipoles" independence; hut guarantee's angular ordering. T h e cascade iinplcme'iitation of the dipeile- branching probabilities is similai to that of a conve-ntional partonie- cascade. Again, there is a divergence as A'x • II which is tamed by a Suelakov form factor. T h i s ine-lueles the- virtual corre'ctions and thereby provides a finite; probability distribution for t h e I: of glue>11 einissions. f o r example, the probability distribution for the- branching ejg — ejgg. first occurring at transve'rse m o m e n t u m A-x- is given by
X c x p { -
d p i
/
Y 1 /
|
•
Ih're / (A-~ ) is the branching probability e>f e-epi (4.38) integrated over the- rapidity inte-rval e'epi (1.39). This expression is applied te> each pe>ssible branching in is accepteel. turn and the e>ne> with the> largest A'x value above the> cut-e>ff. A-1 Given a particular type; e>f branching, a value o f / / can then lie- chosen according to the' a p p r o p r i a t e function / . ( s u b j e c t to t h e b o u n d a r y cemditions. e«|ii (1.39). Given values of A'x and ;/. it is possible t o reconstruct t h e opening angles between the partems. However. this leaves unspecified t h e two elegrees of freedom, polar angle' 0 and a / i m u t h a l angle . associated with t h e oricutatiem of the branching plane with respect to t h e original dipe>Ie>"s axis. These e let ermine the distribution of transve'rse' lnoine'utum recoils. T h e delist ribution is take'ii t o be; uuiforni. T h e 0-distrilmtion depends e>n the type* of branching. In t he case- of a e|-(| dipole;. spin e-onsielerations show t h a t , working in the C.o.M. frame', partem 1 or 3 should re;taiu its direction with a probability proportional to , r j and . r j . respectively (Kle-iss. 1986). In t he remaining cases, no such proscription exists anel it is postulateel t h a t t he recoils are chosen so as to minimize' the d i s t u r b a n c e to t h e neighbouring dipoles' colour Hows (Gust.afson and Pettersson. 1988). Specifically, the> gluon retains its process gg —» <j
A
( A
.( . () . As . T s s s ,
s
s
.
ss .
. ss
s
\. s . T s
s
, s
s ct. al.,
ss
s
s ) )
. s
.
s
s
s
s
s s
. T
s.
.
s s
s
s s
, £ ' = (1
r) . 1 s ( s ) s , s s, s (1 . ) s ss (1.12). s s . s , <
s
s
s
A
1= s .+ s
s
s.
s s
ss s
s
s
(A
s ss
s
s
s
ss s
, T
.
ln
s
s . A
,
s ss
s s
s
s s ( 1.12) (. ^V s
s
s
s s .
,
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,
, s s .
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*
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.
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..1).
s ,
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ill)
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TI'! C'Alll
M
S
r e m n a n t mid recoils is required ( A n d e r s s o n ct til., 19S9i ). T h e basic colour dipolc model has been e x t e n d e d lo i n c o r p o r a t e furthei physics. T h i s includes p h o t o n r a d i a t i o n from filial-slate e l e c t r o m a g n e t i c dipole,'. ( onnblad a n d niehl. 1092), t h e p r o d u c t i o n of heavy < uarkouia ( r n s t r o m a n d onnblad, l!)i)7: rnstriiin ct nl.. 1!)!)7) a n d a description of diliiact ive seal tcrinn. An a l t e r n a t i v e f o r m u l a t i o n of t h e dipole c a s c a d e for IS, known as t h e inked ipolc C h a i n model (Andersson ct til. 1996«. 19966) is also available as a n option ( harra .iha a n d o n n b l a d . 1998). T h i s is equivalent to using t h e C C F M e q u a t i o n : see Sect ion 3.6.6.3. T w o s i t u a t i o n s d o not lit easily into t h e dipole model. First, it d o e s not i n c o r p o r a t e boson glnon fusion in IS a n d this necessitates t h e use of a m a t r i x element m a t c h i n g s c h e m e for t h e first emission ( o n n b l a d . 1996«.). Second, in rell Van t h e dipole oins t h e two b e a m r e m n a n t s , so that without a special scheme to t r a n s f e r m o m e n t u m , t h e vector boson would a c q u i r e no t r a n s v e r s e m o m e n t u m , which is at o d d s with e x p e r i m e n t a l o b s e r v a t i o n s ( o n n b l a d . 1996b). 1.2.7
T e soft underlain /
event
model
In events t h a t involve o n e or two incoming h a d r o n s we must treat t h e b e a m r e m n a n t ( s ) , that is t h e particles which a r e left in t h e incoming h a d r o n ( s ) a f t e r e x t r a c t i n g t h e part on s h o w e r h a r d s c a t t e r i n g initiators. ssentially by definition, t h e relevant physics m a y only involve relatively small m o m e n t u m t r a n s f e r s a n d ilius necessarily p r o b e non-pert u r b a t i v e processes. T h e r e f o r e , o n c e a g a i n prog r a m s rely on models r a t h e r t h a n c a l c u l a t i o n s f r o m first principle. A n u m b e r of a p p r o a c h e s a r e available, including a p a r a m e t e r i z a t i o n of existing d a t a , a straight generalization of t h e hadroni at.ion model, a multiple s c a t t e r i n g mini- et model or oineron inspired models. At this point it is useful t o be a w a r e t h a t , whilst t h e basic physics of t h e soft u n d e r l y i n g e v e n t ( S ) is a s s u m e d t o b e r a t h e r similar t o t h a t of t h e soft h a d r o n i c collisions which d o m i n a t e m i n i m u m bias d a t a , significant, differences occur when a higli-Q ' h a r d s c a t t e r i n g is present,. In p a r t i c u l a r , t h e associated particle a n d energy flow in h i g h - Q events, for e x a m p l e on t h e s h o u l d e r s of high- >x ets, is significant ly larger t h a n in a m i n i m u m bias event with equivalent y s. T h i s pedestal effect' h a s been seen for et e v e n t s ( A 2 C o l l a b . . 1985«: A1 Collal).. 1988). W Z p r o d u c t i o n ( l ' A 2 Collab.. 1987) a n d rell Yan pairs ( l ' A 2 C'ollab.. 1985c). Typically, a n e n h a n c e m e n t in t h e a c t i v i t y of between 1.5 a n d I t i m e s t h a t in a m i n i m u m bias e v e n t is required. B o t h e x p e r i m e n t a l l y ( A2 Collab., 1985«) a n d theoretically (Ciaisser ct al., 1986; Marchesini a n d W e b b e r . 1988) a two-component p i c t u r e is favoured. Initial s t a t e QC' b r e i n s s t r a h l u n g provides a small c o m p o n e n t that grows with ~ a n d soft, physics provides a c o n t r i b u t i o n t h a t a p p e a r s t o s a t u r a t e for sufficiently large Q 2 . T h e l a t t e r is what we m e a n bv the S . A p p r o a c h e s based o n ul oc p a r a m e t e r i z a t ions of e x p e r i m e n t a l d a t a a r e exemplified by l l i : i m IC. T h i s t a k e s A 5 ' s G C M o n t e C a r l o ( A 5 Collab., 1987), fitted to t h e t h e n exist ing d a t a , a n d a d a p t s it. t o use its own cluster h a d r o n -
Alil
H
HH M
TH ('A I. S
'.>01
l ulinii s c h e m e . C l u s t e r s iin- g e n e r a t e d a s s u m i n g a llal. i n i t i a l r a p i d i t y p l a t e a u with G a u s s i a n tails a n d limited t r a n s v e r s e m o m e n t a . T h e r e s u l t i n g l i a d r o n s a r e I lien re< iiired t o c o n f o r m t o a n e g a t i v e b i n o m i a l m u l t i p l i c i t y d i s t r i b u t i o n , w h o s e p a r a m e t e r s a r e . s - d e p e n d e n t . A s p e c i a l p r e s c r i p t i o n e x i s t s lor t he l e a d i n g l i a d r o n s . o s u p p o r t i n g t h e o r y is a t t e m p t e d , so t h a t e n e r g y e x t r a p o l a t i o n s a r e o p e n t o i iiestion. It is i n t e r e s t i n g t o n o t e t h a t , in o n l e r t o r e p r o d u c e t h e p e d e s t a l seen in t h e h a r d s c a t t e r i n g d a t a for h a d r o u h a d r o n collisions, n o e n h a n c e m e n t , of t h e u n d e r l y i n g event w a s f o u n d necessary. e r t u r b a t ive s o f t g l u o n r a d i a t i o n p r o v e d t o he a d e q u a t e ( M a r c h e s i n i a n d W e b b e r . 1988). A c h a r a c t e r i s t i c a s y m m e t r y in t h e a v e r a g e mult iplicity in a et 's t w o s h o u l d e r s is ant i c i p a t e d . A s i m i l a r a p p r o a c h i i used by CO.IKTS ( d o r i c o , 1990), t h o u g h it w o r k s at t h e h a d r o n level a n d assumes a - s c a l i n g f o r m ( o b a c.t til., 1972) for t h e m u l t i p l i c i t y d i s t r i b u t i o n . Y T I I I A ( S o s t r a n d c.t til.. 2 0 0 1 ) p r o v i d e s two basic m o d e l s for t h e S . I lie first s i m p l y follows t h e c o l o u r flows t h r o u g h a n e v e n t , e n a b l i n g t h e c o l o u r e d b e a m r e n i n a n t ( s ) t o be c o n n e c t e d t o t h e rest of t h e event u s i n g a s t r i n g , which is t h e n h a d r o n i . c d a s u s u a l . A s i g n i f i c a n t l y m o r e a m b i t i o u s a p p r o a c h , b a s e d o n p e r t u r b a t i v e e s t i m a t e s for m u l t i p l e , s e m i - h a r d p a r t o n s c a t t e r i n g s , is a l s o available ( S o s t r a n d a n d van Zi l. 1987). T h e mini- et c r o s s s e c t i o n , a s a funct ion of a cut-off a n d n o r m a l i z e d t o t he f i t t e d , inelast ic, n o n - d i f f r a e t i v e c r o s s s e c t i o n , is used a s a m e a n s c a t t e r i n g p r o b a b i l i t y . M u l t i p l e s c a t t e r s a b o v e V1 a r e t h e n g e n e r a t e d f r o m a o i s s o n t y p e d i s t r i b u t i o n a n d t h e r e s u l t i n g event c o n n e c t e d using simplified s t r i n g d r a w i n g s . As a n o p t i o n , a d o u b l e G a u s s i a n s p a t i a l dist r i b u t i o n for t h e p a r t o n s w i t h i n a h a d r o n c a n be folded i n t o t h e p r o b a b i l i t i e s above. If a h a r d s c a t t e r i n g o c c u r s , t h e n a l a r g e h a d r o n i c o v e r l a p is m o r e likely .IIKI m u l t i p l e s c a t t e r i n g s c a n b e a n t i c i p a t e d , which n a t u r a l l y gives a n e n h a n c e d u n d e r l y i n g e v e n t . A p a r t i c u l a r feat u r e t o b e e x p e c t e d in such a m u l t i p l e s c a t t e r ing m o d e l a r e c o r r e l a t i o n s in a z i m u t h , a n d t o a lesser extent r a p i d i t y , for mini- et o b s e r v a b l e s ( W a n g , 199.'i). V a r i a n t s of t h e s e i n h i i - c t . s t r i n g m o d e l s c a n a l s o b e f o u n d in t h e I I T I ( i . 1992) a n d III. I C: (G.vulassy a n d W a n g . 1991) M o n t e C a r l o p r o g r a m s , which c o n c e n t r a t e o n unified descript ions of soft a n d h a r d Q C e v e n t s in h a d r o n h a d r o n a n d h e a v y ion collisions. A final class of m o d e l s for t h e S is i n s p i r e d by ' o i n e r o n p h y s i c s ' d e s c r i p t i o n s of soft a n d h a r d collisions. ISA. HT ( a i g e a n d r o t o p o p e s c u . 1986) uses t h e A G c u t t i n g r u l e s ( A b r a m o v s k i i cl < .. 1972). T h e b a s i c unit is a cut o i n e r o n . which gives rise t o a c h a i n of l i a d r o n s u n i f o r m l y d i s t r i b u t e d in r a p i d i t y omerons a n d w i t h a o i s s o n i a n m u l t i p l i c i t y d i s t r i b u t i o n . T h e n u m b e r of such a n d t h e m e a n h a d r o n p a r e a d u s t e d s e p a r a t e l y for soft a n d h a r d s c a t t e r i n g e v e n t s so a s t o r e p r o d u c e tin- d a t a . A f t e r s e p a r a t e l y a s s i g n i n g l e a d i n g b a r y o n s , t h e o m e r o n s a r e given rosea led l o n g i t u d i n a l m o m e n t u m f r a c t i o n s f r o m a unif o r m d i s t r i b u t i o n . Finally, e a c h o i n e r o n is f r a g m e n t e d in its o w n C . o . M . f r a m e , using a n i n d e p e n d e n t f r a g m e n t a t i o n f u n c t i o n m a d e e n e r g y d e p e n d e n t t o r e p r o d u c e t h e o b s e r v e d rise in < / i i w i t h s. T h e p r o g r a m I) I V J K T ( A u r e n c h e cl ill.. 1994) uses t h e m o r e e l a b o r a t e u a l T o p o l o g i c a l i i i t a r i z a t i o n m o d e l which a g a i n involves cut omerons.
I
202
4.
I I.
ulti-purpose event, goneintors
Whilst I here a r c a large n u m b e r of M o n t e C a r l o p r o g r a m s available t he m a o r i t y of t hese e i t h e r t reat, specialized event t y p e s o r s u p p l y p a r t o n i c final s t a l e s using fixed-order m a t r i x element- calculations. T h e r e a r e relatively few m u l t i - p u r p o s e event g e n e r a t o r s which offer c o m p l e t e a n d unified d e s c r i p t i o n s of m a n y t y p e s of events. Here we o u t l i n e briefly t h e m a i n f e a t u r e s of four of t h e m o r e widely used p r o g r a m s a n d refer t h e interested r eader t o t h e detailed descriptions in the p r o g r a m m a n u a l s a n d o u r earlier discussion. Y T I I I A ( S o s t r a n d et til.. 2001). which s u b s u m e s . TS T. is a general p u r p o s e event g e n e r a t o r that- has grown o u t of t h e well developed u n d s t r i n g model, which provides t he default h a d r o n i z a t i o n scheme. It c o n t a i n s a wide range of h a r d s u b p r o c e s s e s a n d relatively e l a b o r a t e models for soft physics. T h e s e typically c o m e with m a n y o p t i o n s a n d p a r a m e t e r s . T h e basic p a r t o n c a s c a d e uses virtuality o r d e r i n g with colour coherence imposed in t h e time-like c a s c a d e s via a veto on o p e n i n g angles. In a d d i t i o n , t h e nearest neighbour, intra e t . spin correlations a r e included. A IA ( o n n b l a d . 1992) only provides pQC' cascades, using t h e colour dipolc model; see Section 4.2.0. T h i s gives an a u t o m a t i c a l l y coherent a n d unified t r e a t m e n t of initial a n d final s t a t e c a s c a d e s which m e r g e n a t u r a l l y with string h a d r o n i z a t i o n . It must be interfaced t o a n o t h e r g e n e r a t o r , such as Y T I I I A , to h a n d l e t h e h a r d s u b p r o c e s s . h a d r o n i z a t i o n a n d particle decays. II HW' ; (Corcella et al.. 2001) is a general p u r p o s e event g e n e r a t o r which places its e m p h a s i s on t h e pert n r b a t i v e d e s c r i p t i o n of a n e v e n t . It uses c o m p a r a tively s o p h i s t i c a t e d p a r t o n showers that build in colour coherence a u t o m a t i c a l l y via o r d e r i n g of s u i t a b l e evolution variables. T h e first, b r a n c h i n g m a t c h e s t h e e x a c t result a n d in t h e semi-inclusive, x — 1. limit t h e a l g o r i t h m s a r e a c c u r a t e to . which allows Amo t o be related t o A^ g; see x. (4-6). Angular correlations a r e also fully included. H a d r o n i z a t i o n uses a cluster model. ISA. T ( a i g e a n d r o t o p o p e s c u . 1986) is a m o r e basic event g e n e r a t o r . It. employs relatively p r i m i t i v e incoherent, p a r t o n showers a n d independent h a d r o n ization. It allows fast st udies using its wide r a n g e of h a r d subprocesses. 4.3.1
Using event
generators
M o n t e C a r l o event, g e n e r a t o r s for t h e simulation of hadronic events a r e based on QC' . T h e r e f o r e t h e y should, in principle, have only seven free p a r a m e t e r s : t h e s t r e n g t h of t h e coupling a n d t h e six q u a r k masses. However, t h e need to use p c r t u r b a t i v e a p p r o x i m a t i o n s a n d n o n - p e r t u r b a t i v e models i n t r o d u c e s m a n y more p a r a m e t e r s . T h i s raises t h e q u e s t i o n s of how t o select these input values a n d what s y s t e m a t i c e r r o r s t o put on t h e subsequent predictions. Input p a r a m e t e r s a r e usually selected by e i t h e r a c c e p t i n g t h e default values. which m a y well be mere guestiiuat.es' or. b e t t e r , by t u n i n g t h e p r o g r a m to e x p e r i m e n t a l d a t a . T u n i n g a M o n t e C a r l o is b o t h s o m e t h i n g of a n art a n d comp u t a t i o n a l l y expensive, so it is not d o n e lightly. T h e basic m e t h o d is t o first select, a set, of d a t a t o lit a n d related p a r a m e t e r s t o t u n e . For e x a m p l e , t h e d a t a p o i n t s
Mill I I I'UUI'I )MK I A l )N I CKNI HA I OltS
may include I,lie f r a c t i o n of liadronic / d e c a y s w i t h 'I'hnist '/' (see Section 0.2) in I lie r a n g e T G (0.90.0.9r>|. or t he n u m b e r of I v + part icles wit h n i o i n e n t u i n f r a c t i o n in t h e r a n g e r ( , i [0.()1.().02] e t c . N e x t , by e s t i m a t i n g s t a r t i n g values MIMI r o u g h r a n g e s , o n e c a n c r e a t e a grid of v a l u e s in p a r a m e t e r s p a c e . S e c o n d , at each grid p o i n t t h e M o n t e C a r l o is r u n a n d a m e a s u r e for t h e d i f f e r e n c e b e t w e e n s i m u l a t i o n a n d e x p e r i m e n t a l d a t a , usually t h e is e v a l u a t e d for all t h e d a t a points; sufficient, e v e n t s s h o u l d b e g e n e r a t e d so t h a t t h e s t a t i s t i c a l e r r o r on t h e M o n t e C a r l o p r e d i c t i o n is negligible c o m p a r e d t o t h e u n c e r t a i n t i e s of t h e d a t a . I'br a single d a t a p o i n t , t h e is d e f i n e d a s I lie n o r m a l i z e d s q u a r e of t h e differences b e t w e e n t h e d a t a a n d p r e d i c t i o n . (D - T)~ jo1. It m e a s u r e s t h e q u a l i t y of t h e lit in u n i t s of t h e "error' s q u a r e d ( C o w a n . 1!)!)8). G i v e n t h e t o t a l v a l u e on tin* grid, a n i n t e r p o l a t i n g f u n c t i o n c a n b e c o n s t r u c t e d t o e n a b l e a M o n t e C a r l o p r e d i c t i o n t o be m a d e for a r b i t r a r y p a r a m e t e r values. T h i r d , t h e t o t a l is m i n imized a s a f u n c t i o n of t h e p a r a m e t e r s . T h i s s h o u l d result in a s e t of "best-lit' p a r a m e t e r values a n d their one-sigina e r r o r s . I .est t h e a b o v e p r o c e d u r e s e e m t o o s t r a i g h t f o r w a r d , you a r e r e m i n d e d of t h e large size of t h e d a t a s a m p l e s , t h e large n u m b e r of e x p e r i m e n t a l l y m e a s u r e d dis11 ¡buttons a n d t he size of t h e p a r a m e t e r s p a c e . T o t his s h o u l d b e a d d e d p o t e n t ial i'iimplications a r i s i n g f r o m c o r r e l a t i o n s b e t w e e n p a r a m e t e r s a n d possible i n s t a bilities in t h e m i n i m i z a t i o n . P a r t of t h e art is t o k n o w which subset of p a r a m e t e r s lo focus a t t e n t i o n on a n d w h i c h s u b s e t of t h e d a t a t o fit. T o d a t e , t h e m o s t c o m p l e t e p a r a m e t e r t u n i n g s h a v e been based o n LHP d a t a , w h e r e t.vpicallv a t w o - t i e r a p p r o a c h h a s been t a k e n in o r d e r t o m a k e t h e task t r a c t a b l e (ALHP1I C o l l a b . . 1908a: D E L P H I C o l l a b . . 1990«; O P A L C o l l a b . , 199(5«), A first t u n i n g c o n c e n t r a t e s on t h e " m a j o r ' p a r a m e t e r s , such a s AMC. t h e shower cut-ofl'(s) a n d t h e principle h a d r o n i z a t i o n m o d e l p a r a m e t e r s ,
201
MON II CAUI.O MODELS
p r o d u c e d in t h e s a m e or o p p o s i t e h e m i s p h e r e as the i n c o m i n g electron, li is directly related to t h e elect roweak mixing a n g l e s i n 2 f l w . A j e t c h a r g e is essentially a m o m e n t u m weighted s u m of t h e t r a c k c h a r g e s in a j e t . By using a M o n t e C a r l o m o d e l , t h e jet c h a r g e s c a n b e c o r r e l a t e d t o t h e d i r e c t i o n of travel of t h e p r i m a r y q q - p a i r coining f r o m t h e Z decay. T o q u a n t i f y t h i s c o r r e l a t i o n t h e following m o r e s o p h i s t i c a t e d a p p r o a c h was used. D i s t r i b u t i o n s were identified t h a t involved the s a m e physics a n d M o n t e C a r l o model a s s u m p t i o n s a s affected t h e jet charges: n ' . K , p. A, ft. i/. K* m o m e n t u m spectra, baryon-antibarvon correlations, etc. T h e M o n t e C a r l o p a r a m e t e r s were t h e n varied, s u b j e c t t o still describing t h e 'constraint d i s t r i b u t i o n s ' , a n d t h e allowed r a n g e of values of t h e correlation m e a s u r e d . By using a n u n d e r s t a n d i n g of t he m e a s u r e m e n t a n d t he M o n t e C a r l o ' s workings, a t t e n t i o n was focussed u p o n relevant p a r a m e t e r s , w h o s e values were then c o n s t r a i n e d by d e m a n d i n g a c c u r a t e d e s c r i p t i o n s of related m e a s u r e m e n t s . Of course, if t h e M o n t e C a r l o proves i n c a p a b l e of describing t h e d a t a a n d t h e c o n s t r a i n t dist r i b u t i o n s , it would be foolish t o use it. t o m a k e a n y inferences from t he d a t a . We leave t h e r e a d e r wit h a very i m p o r t a n t c a u t i o n .
Warning Monte C'urlo event generators are complicated programs that will almost incvitubln contain bugs, incorrici assumili ions and ¡licitoseli parameters. It is therefore vital that a user docs not take at least two compieteli/ ant/ results at face value. As <1 minimum indepi intent programs should be used in ting pln/sics studi/.
Exercises for C h a p t e r 4 I I For a space-like b r a n c h i n g , /<„ + p„> <— pi,, see Fig. 4.1. show t h a t II 0 ^< — ~ -
(1 " z) '
w h e r e / is t h e p a r t o u v i r t u a l i t y a n d ; is t h e light-cone m o m e n t u m fraction (E + />.-)„ = z(E 4- p:)h- If limits a r e placed on t h e d a u g h t e r v i r t u a l i t i e s . /„- > a n d //, > /'s'. a s s o c i a t e d with t h e n o n - p e r t u r b a t i v e physics of t h e p.d.f.s. w h a t is t h e allowed r a n g e z <= [¿mill-2||| IIX ]? If« 1 b r a n c h i n g o c c u r s , w h a t a r e t h e u p p e r limits o n t h e v i r t u a l i t i e s /,, a n d /„-? •I 2 An a l t e r n a t i v e form for t h e space-like S u d a k o v f o r m f a c t o r is given by ( S j o s t r a n d . 198!i):
II
expj
/ ' ^ / ' " ' " " T
7
^
f,./hi ''- ' ' )
MI M. l'I l'Ulti"« I.M I \ KN I CI-INKUATOKS
By c o m p a r i n g derivatives, show t h a t 11' is equivalent to e q n ( I.IN). W h a t is its physical i n t e r p r e t a t i o n ? I :i C o n s i d e r t w o coloiir-coniierted h a r d p a r t o n s . i a n d j . s e p a r a t e d by an a n g l e 0tJ a n d s u p p o s e partem i e m i t s a soft gluon k, a t an a n g l e On,.. By considering t h e wavelength of t h e gluon c o m p a r e d t o tin 1 d i s t a n c e between t he h a r d p a r t o n s a t t h e time of emission, s h o w t h a t t h e gluon will not resolve t h e individual charges in t h e ( r / ) - s y s t e i n unless 0,k < 0 t J . I I Derive t he expression for cosflj/,- a n d hence confirm e<|n (1.23) a n d t h e result for its a/.iinut hal average, eqn (4.2-r>). You may find it helpful to t r a n s f o r m an integral of t h e f o r m \t!<j>(A — Bcos)~1 into a c o n t o u r integral by m e a n s of t h e s u b s t i t u t i o n z = e ' * . ' * ' I "> He|>eat t h e derivation of eqn (4.31) for t h e colour n e u t r a l t h r e e - p a r t o n s y s t e m (ij)l. I (i In t h e limit £ —> I t h e two-l o o p s p l i t t i n g f u n c t i o n for q —> qg takes t he f o r m iiipi'-'), { 2?r
} l
(\lC 2-1
- z\
2n
A
-ir,.nf)
h.(l
-z)
By c h a n g i n g t h e a r g u m e n t of n „ ( ( J " ) a n d reseating A show how t h e leading o r d e r t e r m can also r e p r o d u c e t he n e x t - t o - l e a d i n g o r d e r t e r m in this expression. I 7 Show that in lln- limit .rj —» 1 a n d with .1:3 = z. eqn (4.3(5) yields t h e convent ional expression for a b r a n c h i n g in t e r m s of Alt arolli Parisi split ting kernels. I 8 C a l c u l a t e t h e masses of t h e two d a u g h t e r s in a dipolo b r a n c h i n g in t e r m s of /.¡± a n d 1/1 a n d hence t h e b o u n d on k->±. Show that those masses c o r r e s p o n d t o t h e «iprxcs ol t h e two triangles shown in Fig. 4.5. I !) In t h e colour dipolo model b r a n c h i n g s a r e ordered in /•-' . t h a t is vertically iu Fig. -1.5. How d o e s o r d e r i n g in virtualitv. Q-, or I = E2(, see t h e discussion below eqn (4.33). proceed in Fig. 4..r>?
EX PER I \ [ENTAL SET-UP In o r d e r to s t u d y QC'D or in general t h e i n t e r a c t i o n s between t h e fundaiueiital constit u e n t s of m a t t e r , o n e h a s t o go t o t h e highest possible energies. Front what h a s been discussed before, this m a y be e v i d e n t for t h e case of p e r t u r b a t i v e QC'D. where t he colour c h a r g e , a s for e x a m p l e in dee]) inelast ic s c a t t e r i n g , becomes only visible when p r o b i n g t h e nucleoli at a scale significantly below 1 fin. It equally holds for weak interactions, where t h e f u n d a m e n t a l scale a s set by t h e masses of t h e g a u g e bosons is in t h e 100 G e V range, a s well as for e l e c t r o m a g n e t i c interactions. In t h e l a t t e r case high energy processes allow sensitive t e s t s of t h e t h e o r y at t h e level of q u a n t u m corrections a n d t o p r o b e t h e s t r u c t u r e of t h e elect roweak unification, which is very dillicult at lower energies. In this c h a p t e r , we will discuss son«* a s p e c t s of tin 1 technological challenges involved in d o i n g e x p e r i m e n t s at high energies, b o t h f r o m t h e a c c e l e r a t o r a n d t h e d e t e c t o r point of view.
5.1
Accelerators
In collisions bet ween f u n d a m e n t a l particles t h e e n t i r e C'.o.M. energy is available to s t u d y t h e i n t e r a c t i o n s b e t w e e n t h e basic c o n s t i t u e n t s of m a t t e r . Ideally, o n e t h u s would like to a c c e l e r a t e particles which have n o internal s t r u c t u r e a n d m a k e I hem collide. T h i s may sound obvious, but it is certainly worthwhile t o think a bit m o r e a b o u t t h e c o n c e p t , since from an o p e r a t i o n a l point of view it does not m a t t e r w h e t h e r a particle is really f u n d a m e n t a l or built from m o r e basic c o n s t i t u e n t s , provided t he binding energy per c o n s t i t u e n t is significant Iv larger t hau t h e kinetic energy. To illustrate t h e point, let's s t a r t with t h e case of a p r o t o n , built out of t h r e e valence q u a r k s which c a r r y a b o u t half of t h e m o m e n t u m of t h e particle. Since t h e b i n d i n g e n e r g y of those q u a r k s is on t h e o r d e r of h u n d r e d s of MeV. a '20 MeV p r o t o n , which provides a convenient p r o b e in nuclear physics, can b e viewed a s a f u n d a m e n t a l particle. T h i s is no longer t h e case for .'515 G e V proIons stored a n d collided wit h a n t i p r o t o n s of t h e s a m e e n e r g y in t he CI'.HN' Super Proton Syiielirotwn (SI'S). Instead of 630 G e V t h e effective C'.o.M. e n e r g y in most collisions bet ween const i t u e n t s of t h e p r o t o n a n d t he ant ¡proton is much lower, on a v e r a g e even below 100 G e V . Still, it was sufficient for t h e discovery of t he W a n d Z b o s o n s a n d at t h e t i m e t h e only technologically feasible way to reach t h e required C'.o.M. energies in collisions between f u n d a m e n t a l particles.
\<
I I.I K A K >I<S
Tin- a p p a r e n t d r a w b a c k t h a t in l i a d r o n eolliilers I lie b e a m e n e r g y is s h a r e d beI ween t h e p a r t o n s , t h e r e b y l e a d i n g t o a r e d u c e d effective C . o . M . e n e r g y , c a n a l s o be t u r n e d i n t o a n a s s e t . High l u m i n o s i t y a n d a w i d e s p e c t r u m of effective collision e n e r g i e s m a k e liadron colliders ideal ' d i s c o v e r y m a c h i n e s ' . O n c e t h e e n e r g y scale for a p a r t i c u l a r p h e n o m e n o n is k n o w n , o n e would of c o u r s e prefer a well defined C . o . M . e n e r g y in o r d e r t o b e a b l e t o p e r f o r m precision m e a s u r e m e n t s . Here e + e colliders a r e u s u a l l y t h e m a c h i n e s of choice. A n o t h e r i n s t r u c t i v e e x a m p l e is t h e c a s e of a n o r d i n a r y golf ball. W i t h a typical m a s s a r o u n d 4 5 g a n d a d e n s i t y close t o t h a t of w a t e r , o n e c a n e s t i m a t e t h a t it. is a c o m p o u n d of s o m e t h i n g like 1 0 " ' i n d i v i d u a l molecules, held t o g e t h e r by hitornioleoular forces. Hit w i t h a n initial velocity of (50 i n / s t h e ball h a s a kinetic « XI .1 or 5 - 1 0 - ° e V . T h e kinetic e n e r g y p e r molecule t h u s e n e r g y of E = mv2/2 is only 0.5 m e V . which would b e available w h e n colliding t wo s u c h balls h e a d - o n . Since a t r o o m t e m p e r a t u r e t h e b i n d i n g e n e r g y p e r molecule must, lie larger t h a n t h e t h e r m a l e n e r g y of kT = 2 5 m e V . it follows t h a t u n d e r t h e c o n d i t i o n s of a g a m e of golf, t h e ball will indeed b e h a v e like a f u n d a m e n t a l o b j e c t . C o n t i n u i n g this t y p e of e x p e r i m e n t s a n d h i t t i n g t h e ball e v e r h a r d e r , o n e would e v e n t u a l l y realize t h a t it h a s m o r e f u n d a m e n t a l b u i l d i n g blocks. It is m a d e f r o m molecules, a n d w h a t a p p e a r e d t o b e a high e n e r g y i n t e r a c t i o n b e t w e e n t wo golf b a l l s was in fact a low e n e r g y i n t e r a c t i o n at t h e level of t h e molecules. R e p e a t i n g t h e s e s t e p s would in t u r n reveal t h a t t he molecules a r e m a d e of a t o m s , that, t h e a t o m s h a v e an e l e c t r o n cloud a n d a nucleus, a n d that, t h e n u c l e u s is a b o u n d s y s t e m of p r o t o n s a n d n e u t r o n s which in t u r n consist of q u a r k s . W i t h electrons a n d q u a r k s , a c c o r d i n g t o t o d a y ' s k n o w l e d g e , t h e f u n d a m e n t a l level is reached, t h a t is, even t h e h i g h e s t e n e r g y i n t e r a c t i o n s s h o w n o e v i d e n c e for a s u b s t r u c t u r e in t h o s e p a r t i c l e s . W h e t her o n e uses c o m p o s i t e or point like p a r t i c l e s d e p e n d s o n t he s c o p e of t h e e x p e r i m e n t . T a k i n g i n t o a c c o u n t t h a t a n ellicient. a c c e l e r a t i o n r e q u i r e s c h a r g e d s t a b l e p a r t i c l e s , a n d t h a t free q u a r k s d o not exist, t h e o b v i o u s c a n d i d a t e s a r e p r o t o n s , e l e c t r o n s a n d t h e i r ant ¡particles. Also h e a v y ions a r e of i n t e r e s t for s t u d i e s of m a t t e r u n d e r e x t r e m e c o n d i t i o n s , such a s for t h e c a s e of t h e q u a r k gluoii p l a s m a . At ext remely high e n e r g i e s a l s o u n i o n s live suflicicntly long t o lie of interest.. A union collider, however, is still b e y o n d t h e c a p a b i l i t i e s of t o d a y ' s technology, a l t h o u g h n u m e r o u s s t u d i e s a r e u n d e r way. T h e s t a n d a r d p r o c e d u r e for t h e s t u d y of f u n d a m e n t a l i n t e r a c t i o n s is t h u s to a c c e l e r a t e t h e most e l e m e n t a r y p a r t i c l e s w h i c h a r e a v a i l a b l e to t h e h i g h e s t possible energies. In a d d i t ion, t he i n t e r a c t i o n r a t e h a s t o m a t c h t he r e q u i r e m e n t s of a n e x p e r i m e n t . T h e r e l e v a n t q u a n t i t i e s in a collision of e l e m e n t a r y part icles a r e t h e C . o . M . energy, t hat is. t h a t part of t h e t o t a l e n e r g y of a collision which d o e s not go i n t o t h e kinetic e n e r g y of t h e e n t i r e s y s t e m a n d t h e l u m i n o s i t y C. defined a s t h e p r o p o r t i o n a l i t y f a c t o r b e t w e e n c r o s s section a n d e v e n t r a t e . 4 Ar = C a .
(5.1)
•JON
5. I. I
I XI'EHIMENTAL SKI III
Accclcmlor
systems
T h e most efficient way to m a x i m i z e t h e C . o . M . energy is t h e use of colliding b e a m s r a t h e r t h a n lixcd target, o p e r a t i o n s . In case ant ¡particles can he p r o d u c e d a b u n d a n t l y , an elegant technology t o achieve this is via particle a n t i p a r t i c l e collisions inside a s t o r a g e ring. B e c a u s e p a r t i c l e s a n d a n t i p a r t i c l e s have t h e s a m e m a s s a n d o p p o s i t e charge, t h e y can travel in o p p o s i t e direct ion t h r o u g h t he s a m e accelerator s t r u c t u r e . T h i s concept was realized for e x a m p l e in t h e I K V A T R O N at IT.KMII.AB. t h e C E R N S P S collider or t h e Latyc-Electron-Positron collider (LKP). T h e technical realization in all cases requires a r a t h e r int ricate s y s t e m of accelerators, which shall he illustrated here at t h e e x a m p l e of t h e C E R N s e t - u p . IP 5
Cf-RN MEYRIN Kl(l. 5 . 1 . Simplified view of t h e C E R N accelerator complex. T h e L E P ring has a c i r c u m f e r e n c e of a p p r o x i m a t e l y 27 kin. F i g u r e from Miiller(2ll()0).
A simplified view of this s y s t e m is sketched in Fig. 5.1. showing t h e Proton Snncliwt.mil ( P S ) , t h e S P S a n d t h e I.EP ring. T h e PS a s t h e oldest ring was originally built t o a c c e l e r a t e p r o t o n s to an energy of typically 25 CieV. In o r d e r to achieve higher energies t h e S P S was c o n s t r u c t e d to go u p t o 15(1 (¡cV b e a m energy, a n d is still in use lo s u p p o r t a lixed target physics p r o g r a m . Il can also be o p e r a t e d as a p r o t o n a n t i p r o t o n collider a nd served a s a pre-accelerator for the L E P . w h e r e four big e x p e r i m e n t s were located t o s t u d y t h e physics of e + e ~ a n n i h i l a t i o n s at C . o . M . energies f r o m t h e scale of t h e Z - r e s o n a n c e to ^/s >
A O I I.I KATOKS
200 G e V . In a d d i t i o n , liotli I lie I'S a n d I lie S I ' S a r c u s e d t o s u p p l y test b e a m s for research a n d d e v e l o p m e n t work on f u t u r e p r o j e c t s . T h e w h o l e s y s t e m is r a t h e r flexible, s u p p o r t i n g p h y s i c s w i t h a l a r g e v a r i e t y of b e a m s a n d a w i d e r a n g e of energies. O n e h a s t h e o p p o r t u n i t y t o work e i t h e r d i r e c t l y wit h t h e p a r t i c l e s f r o m an a c c e l e r a t o r o r . a l t e r n a t ively, w i t h s e c o n d a r y p a r t i c l e b e a m s w h i c h a r c c r e a t e d in lixed target, i n t e r a c t i o n s of t he p r i m a r y b e a m s . Ii is c l e a r l y b e y o n d t h e s c o p e ul I his b o o k t o d e s c r i b e all those possibilit ies in d e t a i l . Below we will f o c u s o n I he general s t e p s n e e d e d t o r e a c h t h e highest e n e r g i e s . S o m e key p a r a m e t e r s of LISP, as given in T a b l e 5.1. s h o u l d c o n v e y a n i m p r e s s i o n of w h a t is t e c h n o l o g i c a l l y feasible. Table storage
5 . 1 Characteristic ring
parameters
Parameter Circumference Magnetic radius Revolution frequency R F frequency Injection energy Achieved p e a k e n e r g y p e r b e a m Achieved p e a k l u m i n o s i t y N u m b e r of b u n c h e s Typical current/bunch
of
the
LISP
Value 26(558.88 m .{()!)(> m 11245.5 Hz 352 MHz approx. 20 G e V 104.5 G e V 4 pb~'/day 4, 8 o r 12 0.75 m A
At t h e b e g i n n i n g of a n y a c c e l e r a t o r c o m p l e x is a s o u r c e of p a r t i c l e s w h i c h a r e t o be a c c e l e r a t e d . T h i s is s i m p l e in t he c a s e of e l e c t r o n s o r p r o t o n s which c a n b e o b t a i n e d f r o m e i t h e r a h e a t e d f i l a m e n t or f r o m ionized h y d r o g e n , respectively. M o r e recent t e c h n o l o g i e s a r e b a s e d o n laser i o n i z a t i o n or mdio-frequency (RF) g u n s , w h i c h a l s o allow t o p r e p a r e p o l a r i z e d b e a m s . T h e f i r s t a c c e l e r a t i o n s t e p is usually e l e c t r o s t a t i c , like in a T V set w i t h a c a t h o d e r a v t u b e . T h e p a r t i c l e s p a s s t h r o u g h s o m e h i g h - v o l t a g e d i f f e r e n c e which b r i n g s t h e m u p t o a s p e e d w h e r e a n R F - s y s t c m c a n t a k e over by which t h e b e a m is a c c e l e r a t e d t h r o u g h t h e e l e c t r i c field i n s i d e a cavity. B o t h s t a n d i n g w a v e a n d t r a v e l l i n g w a v e d e v i c e s a r e u s e d . Since t h e e n e r g y t r a n s f e r c a n b e efficient o n l y if t he p h a s e v e l o c i t y of t h e w a v e is s i m i l a r t o t h e s p e e d of t h e p a r t i c l e , t h i s p r i n c i p l e w o r k s best for relativist.ic beams. So far only p a r t i c l e s a r e involved. For a p r o t o n a n t i p r o t o n o r a n e + e ~ collider, however, o n e a l s o n e e d s t h e c o r r e s p o n d i n g ant ¡part icles. T h e s e c a n b e c r e a t e d by d i r e c t i n g a p r i m a r y b e a m of sufficient e n e r g y on a fixed t a r g e t . A m o n g s t t he r e a c t i o n p r o d u c t s t h e r e will a l s o b e t h e r e s p e c t i v e ant ¡particles. A l t h o u g h by uo m e a n s trivial, it is t o d a y a well u n d e r s t o o d t e c h n o l o g y t o collect p o s i t r o n s or a u t i p r o t o n s f r o m such a r e a c t i o n a n d feed t h e m i n t o a n a c c u m u l a t o r ring which collects t h e a n t i p a r t i c l c s a n d r e d u c e s t h e i r l a r g e initial m o m e n t u m s p r e a d t o t h e
'¿Ill
I'APKItlMKNTAI. SKI HI'
Kl<:. 5 . 2 . S c h e m a t i c view of a q u a d r u p o l e m a g n e t . Also s h o w n a r e t he forces on positive p a r t i c l e s in t h e region b e t w e e n t h e pole f a c e s m o v i n g into t h e p l a n e of t h e d r a w i n g . F i g u r e f r o m Miiller(2000).
level w h e r e t h e y form a well collimatcd b e a m . W h e n e n o u g h current has been a c c u m u l a t e d , t h e b e a m s a r e injected i n t o f u r t h e r a c c e l e r a t o r s t a g e s a n d finally filled into t h e s a m e s t o r a g e ring a s t h e particles. L a r g e s t o r a g e rings such a s t h e LISP or t h e S I ' S o p e r a t e a s a s y n c h r o t r o n , which allows not only t o s t o r e p a r t i c l e s b u t also t o a c c e l e r a t e t h e m f u r t h e r a n d feed back e n e r g y losses. For t h e LISP this c o n t i n u o u s acceleration is vital. since e l e c t r o n s a n d p o s i t r o n s o n a circular orbit emit s y n c h r o t r o n radiation. In n a t u r a l u n i t s t h e p o w e r r a d i a t e d by a relativistic particle is given by d i v / d / = 0 , . , , , 2 / J w h e r e ¡i is t h e m o m e n t u m of t he particle, in its m a s s a n d /> t he m a g n e t i c r a d i u s of t h e ring. In m o r e familiar u n i t s t h e e n e r g y loss per t u r n of a n electron t h u s b e c o m e s A E = 0 . 0 8 8 5 ( £ / G e V ) ' / ( / ' / " O M e V . At t he highest LISP energies t h i s c o r r e s p o n d s t o a n energy loss of over .'i G c V per t u r n , which has t o be fed back by t h e R F - s y s t e m . Without, an a c c e l e r a t i n g s t r u c t u r e b e a m s of light p a r t i c l e s would t h u s be quickly lost. As a side effect, t h e H F accelerat ion in a s y n c h r o t r o n leads to a b u n c h e d b e a m , which needs to be taken into account w h e n designing a n e x p e r i m e n t . T h e o b v i o u s a d v a n t a g e is t h a t t h e t i m i n g of possible i n t e r a c t i o n s is precisely k n o w n , a d i s a d v a n t a g e m a y be that in case of high luminosity r u n n i n g o n e h a s t o cope w i t h s i m u l t a n e o u s m u l t i p l e i n t e r a c t i o n s . High luminosities require large b e a m c u r r e n t s a n d small t r a n s v e r s e d i m e n s i o n s in t h e i n t e r a c t i o n p o i n t s , which a r e achieved by t h e layout of t h e e l e m e n t s which g u i d e t h e b e a m t h r o u g h t he m a c h i n e . S o m e basics of t h e s e b e a m o p t i c s a r e discussed in t h e next section.
AU'KI.KU ATOMS
fociiviiij: quadaipolc
tlnlt SIXKC
tlcfocusinc , i|iiaun>po1c
211
ifcift space
iixii^inj: (|II.KIIII|>OIC
DM-IODO(VII
Fu;. .r»..i. L a y o u t Miillor(2000).
5.1.2
¡letmi
of
a
strong-focusing
FODO-strncture.
Figure
from
optics
I lie b e a m s a r e kept on their o r b i t by d i p o l e m a g n e t s . Since t h e b e n d i n g r a d i u s is p r o p o r t i o n a l to t h e m o m e n t u m , it follows that only particles with an energy close t o t h e nominal energy would have a s t a b l e orbit in a ring consisting entirely of dipole m a g n e t s . All o t h e r s would lie lost because they a r e deflected eit her t o o little or t o o much by t h e b e n d i n g field. Because of s y n c h r o t r o n r a d i a t i o n a n d o t h e r i m p e r f e c t i o n s t h e r e is always some energy spread in t h e b e a m . O n e t h u s has t o deal with particles which may have a s u b s t a n t i a l m o m e n t u m d e v i a t i o n , that is. a focusing m e c h a n i s m is needed t o bend t h e particle back into t h e accept a n c e of t h e a c c e l e r a t o r ring. T h i s is achieved by q u a d r u p o l e m a g n e t s (Fig. .r>.2). where t h e m a g n e t i c flux density li grows p r o p o r t i o n a l t o t h e d i s t a n c e f r o m t h e centre. Particles on the nominal o r b i t d o not see a field, b u t particles with an offset a r e deflected t o w a r d s t h e c e n t r e or a w a y from it. In such a s y s t e m particles wit h a m o m e n t u m offset start oscillat ing a r o u n d t h e o r b i t of a nominal part icle. Unlike a n optical lens, which usually is focusing or defoensing horizontally a n d vertically, a q u a d r u p o l e field which is focusing in t h e horizontal p l a n e is delocusing in t h e vertical one. a n d aire versa. Still, c o m b i n i n g a focusing a n d defocusing q u a d r u p o l e with a drift s p a c e in between, it is possible to achieve a net focusing. Intuitively this c a n be u n d e r s t o o d by t h e fact t h a t a p a r t i c l e which first has a large offset in t h e focusing q u a d r u p o l e sees a s t r o n g field which deflects it towards t h e centre. T h e r e f o r e t h e defocusing magnet is traversed closer t o t h e c e n t r e w h e r e t h e field is weaker. T h e d e f o c u s i n g is t h u s not as large a n d t h e net effect, is a focusing of t h e b e a m . A similar a r g u m e n t holds if t h e first q u a d r u p o l e acts as a defocusing lens. T h e b e a m then is directed o u t w a r d a n d e n t e r s t h e focusing q u a d r u p o l e at, a larger d i s t a n c e f r o m t h e c e n t r e . T h e focusing is again s t r o n g e r t h a n t h e defocusing a n d t h e whole s y s t e m perforins a focusing of t h e
. I.'
I\PI:UIMI:NI \LSI:I
HP
5
-5
23:00
3:00
7:(H)
1I:(H)
15:(K>
19:00
23:00
3:(K)
Daytime Fits. 5 . 4 . ffect, of t e r r e s t r i a l t i d e s o n t h e Miiller(2000).
IS
b e a m energy. Figure from
b e a m . F i g u r e 5 . 3 s h o w s t h e s c h e m a t i c layout of s u c h a F -struct.ure, where F s t a n d s for ' F o c u s i n g ' . for cfocusing' a n d () for t h e d r i f t s p a c e b e t w e e n two m a g n e t ic lenses. A c c e l e r a t o r s a n d s t o r a g e rings a r e c o m p l e x devices, which must be a b l e t o cont rol a p a r t i c l e b e a m wit h high precision over l a r g e d i s t a n c e s . For t h e LISP o n e is faced w i t h t h e s i t u a t i o n t h a t t h e b e a m e n e r g y is very s e n s i t i v e t o t h e effective r a d i u s of t h e ring. As a c o n s e q u e n c e it t u r n e d o u t . s o m e w h a t s u r p r i s i n g l y at. t h e t i m e , t h a t t h e LISP b e a m e n e r g y is a f f e c t e d by s u c h e f f e c t s a s t e r r e s t r i a l t i d e s , t h e level of t h e w a t e r in lake G e n e v a o r t h e t i m e - t a b l e of t h e F r e n c h Swiss r a i l r o a d n e t w o r k . T h e first t w o effects a c t t h r o u g h a slight d e f o r m a t i o n of t h e r i n g t h r o u g h t h e e l a s t i c f o r c e s o n t h e geological e n v i r o n m e n t of t h e region. B e c a u s e of t h e s e d e f o r m a t i o n s , t h e a v e r a g e LISP b e a m d e v i a t e s f r o m t h e n o m i n a l o r b i t a n d sees a d d i t i o n a l b e n d i n g fields f r o m t h e q u a d r u p o l c s . T h e so-called p asefocusing (see. for e x a m p l e . W i e d e m a n n 1993,1995) of t h e a c c e l e r a t o r s t r u c t u r e then automatically re-ad usts the beam energy to match the integrated bending field. A n e x a m p l e of h o w t i d a l forces c r e a t e d by t h e s u n a n d t h e m o o n a f f e c t t h e b e a m e n e r g y is s h o w n in Fig. 5.4. T h e m e a s u r e m e n t s a r e c o m p a r e d t o a n a b s o l u t e p r e d i c t i o n b a s e d o n t h e k n o w n geological p r o p e r t i e s of t h e g r o u n d . A n o t h e r e x a m p l e f o r e x t e r n a l i n f l u e n c e s on t h e LISP b e a m e n e r g y is s h o w n in Fig. 5.5. n t o p of a s m o o t h b e h a v i o u r t h e r e a r e m a n y e r r a t i c spikes which in t h e b e g i n n i n g w e r e n o t u n d e r s t o o d . Also t h e d r i f t of t he b e a m e n e r g y w a s kind of a m y s t e r y : but l e t ' s s t a r t wit h t he spikes. A first hint is o b t a i n e d f r o m t h e f a c t t h a t
l>r.T
l IIICII
IfCY <<> ,11>
20:00
00:00
S
4649X
46494
^
46490
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8
4 (>486
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X
46482
46478
46474 16:00
18:00
22:00
02:00
04:00
06:00
Time of day F i e . .r>..r). T y p i c a l evolution of t h e L F P b e a m e n e r g y d u r i n g t h e n i g h t . F ig u r e f r o m M iil'lor (2000).
the sit uat ion is very quiet, between midnight, a n d (i A M . t h a t is. it s o m e h o w s e e m s to be related t o h u m a n activities. T h e solution linally t u r n e d out to be related to t h e high speed t r a i n . T G V . f r o m G e n e v a t o Paris, which causes v a g a b o n d c u r r e n t s t h a t also reach t h e LISP t u n n e l a n d How a l o n g t h e b e a m pipe, t h e r e b y inducing a d d i t i o n a l m a g n e t i c fields which affect t h e b e a m energy. T h e individual spikes in Fig. 5.5 could in fact be correlated with t h e arrival a n d d e p a r t u r e times of t h e T G V a t some regional train s t a t i o n s . Residual m a g n e t i z a t i o n s caused by the c u r r e n t s associated t o t h e spikes linally explain a t least part of t h e drift in t he a v e r a g e b e a m energy.
.2
e t e c t o r s at. high energy colliders
A d e t e c t o r for high e n e r g y i n t e r a c t i o n s ideally should be able t o record all p a r t i cles including their p r o p e r t i e s which a r e c r e a t e d in t h e r e a c t i o n . M a n y excellent b o o k s ( B l u m a n d Rolandi. 1991; G r u p e n . 1996: Kleinknecht. 1999) exist which cover t h e s t a t e of t h e art of this s u bject in detail. Below, only a brief overview of the- most i m p o r t a n t techniques can be given.
I XI'I UIMI NTAI. SI'; i - l i | ' T a b l e 5 ,2 Particle /V l.± K.o.± A 7T°
r± 11 4e P± 1
'I)ipi
lifetimes of eli menUivii
¡mrtieles
Decay A p p r o x i m a t e lifetimes I. I x K ) " 2 1 s Strong 1.3 x I t r 2 3 s S t n >ng 5.5 x H P 2 1 s Strong E l e c t r o m a g n e t ic 8.4 x I t ) - ' 7 s Weak 2.6 x 1Q-* s Weak 2.6 x 1 0 - 1 0 s 1.2 x 1 0 - 8 s Weak 8.9 x I t ) - " s Weak Weak 5.2 x 10 s s 4.2 x 1 0 " ' 3 s Weak Weak 1.0 x I I ) - 1 2 s 1.6 x 1 0 " 1 2 s Weak 1.7 x 1 0 - 1 2 s Weak Weak 2.2 x l ( ) - ° s 2.9 x 1 0 - 1 3 s Weak 8.9 x 10 2 s Weak > 4.3 x 10 2 3 y e a r s Stable Stable > 1.6 x 10 2 5 y e a r s oc Stable
P a r t i c l e s c r e a t e d in high e n e r g y i n t e r a c t i o n s c a n be classified as short-lived resonances, which decay via s t r o n g interaction processes on a t i m e scale of I D - 2 3 s. particles which d e c a y t h r o u g h e l e c t r o m a g n e t i c forces with typical lifet i m e s of 1 0 " s a n d long-lived particles with weak decays. T a b l e 5.2 shows masses a n d lifetimes for a selection of particles, which i l l u s t r a t e t h e hierarchies of lifetimes observed in n a t u r e . E x p e r i m e n t a l l y s t r o n g a n d elect romagnet ic d e c a y s usually c a n n o t be resolved from t h e creation vertex of t h e m o t h e r particles. Weak d e c a y s of relat.ivist.ic part icles, on t h e o t h e r h a n d , o c c u r a f t e r macroscopic (light, dist ances. For heavy flavour particles with lifetimes a r o u n d 1 0 - 1 2 s these a r e typically in t h e r a n g e of a few h u n d r e d micron t o a few millimetres, a n d p a r t i c l e s wit h lifetime's of 10 ^ s or larger c a n b e considered a s absolutely s t a b l e since t h e y live long e n o u g h t o travel d i s t a n c e s of m a n y m e t r e s before decaying. An event, from a high energy i n t e r a c t i o n will u l t i m a t e l y consist of long-lived filial s t a t e particles which t r a v e r s e t h e e n t i r e d e t e c t o r . For a full r e c o n s t r u c t i o n o n e has t o m e a s u r e as precisely as possible these particles, b o t h charged and n e u t r a l ones. T h e p r i m a r y q u a n t i t i e s to be d e t e r m i n e d a r e c h a r g e , m o m e n t u m a n d m a s s of those particles. K n o w l e d g e a b o u t t h e precise direction close t o t h e p r i m a r y interaction vertex t h e n allows t o r e c o n s t r u c t heavy flavour d e c a y s or decay chains in general, which is i m p o r t a n t for detailed s t u d i e s of p r o d u c t i o n processes. T h e d e t e r m i n a t i o n of t h e electric c h a r g e is g r e a t l y simplified by t h e
i n i K( r o u s
\ i lit«:11 I-NI:K<^ < < >1.1.11 n :u.S
fuel I luil all c h a r g e d long-11 vet I part icles c a r r y eit her p l u s or m i n u s o n e p o s i t r o n c h a r g e . t h a t is. t h e d i r e c t i o n of c u r v a t u r e in a m a g n e t i c lield a l r e a d y allows t o Infer t h e c h a r g e of a part icle. In g e n e r a l , a d e t e c t o r c o n s i s t s of t h e so-called touching devices which d e t e r mine m o m e n t u m a n d point of origin of c h a r g e d part icles, calorimeters t o register also n e u t r a l p a r t i c l e s a n d m e a s u r e t h e i r energies, a n d p a r t i c l e i d e n t i f i c a t i o n devices. We will discuss t h e m in t u r n . 5.2.1
Trac ing
detectors
T r a c k i n g d e v i c e s exploit, t h e fact t h a t c h a r g e d p a r t i c l e s p a s s i n g t h r o u g h m a t t e r c r e a t e a n ion ization signal a l o n g t h e t r a j e c t o r y of t h e p a r t i c l e . Il is basically t h e s a m e principle for cloud c h a m b e r s , b u b b l e c h a m b e r s , e m u l s i o n s or all k i n d s of electronic t r a c k i n g d e t e c t o r s . T h e p r i m a r y p a r t i c l e passes t h r o u g h t h e m e d i u m a n d loses s o m e e n e r g y which g o e s i n t o t h e c r e a t i o n of e l e c t r o n ion pairs. In a cloud c h a m b e r c o n t a i n i n g s u p e r s a t u r a t e d v a p o u r , t h e free c h a r g e s t r i g g e r cond e n s a t i o n : in a b u b b l e c h a m b e r w i t h s u p e r h e a t e d fluid t h e y c r e a t e b u b b l e s f r o m local boiling of t h e fluid. In b o t h cases, t h e t r a j e c t o r y b e c o m e s visible a n d c a n b e a n a l y s e d . In p h o t o g r a p h i c e m u l s i o n s t h e c h a r g e s l i b e r a t e d bv t h e p a s s a g e of a high e n e r g y p a r t i c l e lead t o p h o t o - c h e m i c a l r e a c t i o n s t h a t a r e m a d e p e r m a n e n t by t h e d e v e l o p m e n t p r o c e s s . T h e ionization f r o m a c h a r g e d p a r t i c l e can also be r e c o r d e d d i r e c t l y by electronic d e t e c t o r s . A t y p i c a l e x a m p l e is a multi- ire proportional chamber (M W P C ) . w h e r e t h e p r i m a r y ionization in a g a s v o l u m e is a m p l i f i e d in t h e s t r o n g electric field a r o u n d a wire. T h e a m p l i f i c a t i o n s t a r t s w h e n e l e c t r o n s a p p r o a c h i n g t h e wire a r e a c c e l e r a t e d t o t h e point when» t h e y c r e a t e s e c o n d a r y i o ni z a t i o n . Since t h e s e c o n d a r y c h a r g e s a s well a r e a c c e l e r a t e d in t In* field a r o u n d t h e wire, an a v a l a n c h e d e v e l o p s which, for not. t o o high fields, is p r o p o r t i o n a l t o t h e prim a r y ionizat ion a n d large e n o u g h t o b e r e c o r d e d electronically. Since t h e p r i m a r y c h a r g e d r i f t s t o t h e closest wire, a signal on a wire yields a measurement, of o n e c o o r d i n a t e o n a p a r t i c l e t r a c k , w i t h a r e s o l u t i o n p r o p o r t i o n a l t o t he d i s t a n c e of t h e wires. An e n s e m b l e of such c h a m b e r s allows t o s a m p l e t h e e n t i r e t r a j e c t o r y a n d . placed inside a m a g n e t i c field, t o d e t e r m i n e .simultaneously t h e p a r t i c l e ' s c h a r g e a n d m o m e n t u m vector. A m o d i f i c a t i o n of t h e s c h e m e is a p p l i e d in t h e so-called time projection chamber (' C). Here, a n ent ire ionization t rack c r e a t e d inside a drift volume, typically a cylinder, d r i f t s in parallel electric a n d m a g n e t i c fields t o a n end p l a t e w h e r e it is r e c o r d e d by a MVV'I'C. A s b e f o r e , t h e signal is e n h a n c e d by g a s a m p l i f i c a t i o n at t h e wires. T o i m p r o v e t h e m e a s u r e m e n t , a d d i t i o n a l e l e c t r o d e s , so-called pads, a r e placed b e h i n d t he wires. If a p r i m a r y c h a r g e c r e a t e s a n a v a l a n c h e on a wire, a signal is also i n d u c e d o n t h e closest p a d s . R e a d i n g out t h e p a d s t h u s y i e l d s a p i c t u r e of t h e t w o - d i m e n s i o n a l p r o j e c t i o n of t h e t r a c k . R e c o r d i n g also t h e drift t i m e , t h a t is. t he t i m e b e t w e e n c r e a t i o n of t h e p r i m a r y i o n i z a t i o n by a p a r t i c l e track f r o m a n i n t e r a c t i o n a n d its arrival at t h e e n d plate, gives t h e t h i r d co o r d i n a t e . A T I T t h u s p e r f o r m s t h r e e - d i m e n s i o n a l i m a g i n g of c h a r g e d t r a c k s inside tin An
•21(1
EXPERIMENTAL SK'I'-UI»
a m a g n e t i c field. T h e m a g n e t i c lield act ually serves a d o u b l e p u r p o s e . Evidently, t h e b e n d i n g of t h e t r a c k s inside t h e field provides t h e basis for t h e m o m e n t u m m e a s u r e m e n t . In a d d i t i o n , a Z?-field parallel t o t h e electric drift lield reduces t r a n s v e r s e diffusion of t h e electrons, which in s t r o n g fields t e n d t o follow the B-field lines, thereby e n s u r i n g a high resolution image of t h e track. As a final point it is w o r t h m e n t i o n i n g that t h e electric lield is a d j u s t e d such that t h e drift velocity for t h e e l e c t r o n s is m a x i m a l . T h i s a g a i n h a s a d o u b l e benefit since it minimizes t he readout, t i m e a n d at t h e s a m e t i m e r e n d e r s t he d r i f t velocity insensitive against small local variations of t h e drift field. In o t h e r words, setting t lie drift lield to t he m a x i m u m of t h e drift velocity g u a r a n t e e s opt imal rcsolut ion in t h e longitudinal d i r e c t i o n . While MWPC's c a n be read out very fast , t h e event r a t e of a I P C is limited by t h e d r i f t velocity of charges. B o t h t y p e s of t r a c k i n g d e t e c t o r s have then preferred a p p l i c a t i o n s , a n d it is t h e e x p e r i m e n t a l e n v i r o n m e n t which drives t h e choice of technology. For e ' e ~ physics 'I P C s a r e excellent d e t e c t o r s , at high luminosity h a d r o u m a c h i n e s t he high event r a t e would r e n d e r t h e m useless. T h e y also cannot be used t o provide i n f o r m a t i o n for fast trigger decisions. For these kinds of a p p l i c a t i o n conventional wire c h a m b e r s or similar technologies have to be used. T h e t y p e s of t r a c k i n g d e t e c t o r s discussed so far a r e usually large devices which have t o follow a t r a c k over a long d i s t a n c e in o r d e r t o p e r f o r m precise m e a s u r e m e n t s of t h e deflection of high energy p a rt i c l e s inside m a g n e t i c fields. A n o t h e r task which is vital for t h e r e c o n s t r u c t i o n of s e c o n d a r y vertices, for e x a m ple from heavy flavour decays, is t h e precise d e t e r m i n a t i o n of t h e track direct ion a n d position close t o t h e p r i m a r y vertex. T h e high resolution needed for this task exceeds that of conventional wire c h a m b e r s by a factor of u p to 100. It is reached with silicon s t r i p or pixel d e t e c t o r s . Again, a charged p a rt i c l e passing t h r o u g h t h e d e t e c t o r ionizes t h e material. In case of silicon, electron-hole p a i r s a r e c r e a t e d , a n d d i o d e s t r u c t u r e s i m p l a n t e d on t h e wafer collect t h e p r o d u c e d c h a r g e s very m u c h like t h e r e a d o u t wires of a M W P C . Since t h e energy loss in solid s t a t e m a t e r i a l is much larger t h a n in gases, even in relatively thin detect o r s of typically •'{(!() //m thickness a m i n i m u m ionizing t r a c k g e n e r a t e s sufficient c h a r g e for direct readout by a charge-sensitive amplifier. No f u r t h e r amplificat ion is required. r
.2.2
Catorim
etcrs
N e u t r a l particles have t o be d e t e c t e d by a different technology called adorimcln/. T h e basic idea is to p u t a sullicient a m o u n t of a b s o r b e r m a t e r i a l for t he particles t o i n t e r a c t with t h e d e t e c t o r such t h a t t h e y deposit t he i r e n t i r e energy. T h e e n e r g y d e p o s i t t h e n is c o n v e r t e d i nt o a p r o p o r t i o n a l signal which can be read out electronically. W i t h p r o p e r calibration a c a l o r i m e t e r t h u s is able to m e a s u r e t h e energy of a n incident particle. A r r a n g e d in p r o j e c t i v e s e g m e n t s poi nt i ng t o w a r d s t he i n t e r a c t i o n point, it. also m e a s u r e s t h e direction of a n e u t r a l part icle a n d t h u s allows t o infer its m o m e n t u m vector.
DKI Ki TOIIN \ l IIIOII KNISHOY COM.IDKKS
•J 17
I wo t y p e s of c a l o r i m e t e r s a r e c o m m o n l y e m p l o y e d : e l e c t r o m a g n e t ic a n d Imdronic c a l o r i m e t e r s . A l t h o u g h b o t h follow t lie s a m e basic principle, t h e a c t u a l reali/at ions differ c o n s i d e r a b l y . A n e l e c t r o m a g n e t i c c a l o r i m e t e r is designed t o a b s o r b |i|lotons. At liigli energies t h e e n e r g y d e p o s i t i o n m a i n l y s t a r t s w i t h pair c r e a t i o n in t he s t r o n g electric field a r o u n d a h e a v y nucleus, w h e r e t h e e n e r g y of t h e incident p h o t o n is c o n v e r t e d i n t o a n e + e ' pair. T h e s e p a r t i c l e s iu t u r n r a d i a t e w h e n passing close t o a n o t h e r nucleus a n d t he- bremsst r a h l u n g p h o t o n s e m i t t e d in t he electromagnetic process c a n a g a i n b e c o n v e r t e d i n t o n e w e + c ~ p a i r s . A so-called shower d e v e l o p s , w h e r e t h e initial e n e r g y is successively d i s t r i b u t e d over a l a r g e n u m b e r of s e c o n d a r y p a r t i c l e s . T h e s h o w e r d e v e l o p m e n t s t o p s o n c e t h e e n e r g y pair-creat ion. T h e c h a r a c t e r i s t i c per p a r t i c l e falls below t h e t hreshold for e + e length d e s c r i b i n g t h e s h o w e r development, is t h e so-called radiation length. A'o. defined a s t h e m e a n d i s t a n c e over which a high e n e r g y e l e c t r o n loses all but a f r a c t i o n l / e s ; t)..'5(i<S of its e n e r g y bv b r e m s s t r a h l u n g . T h e r a d i a t i o n l e n g t h a s an a t o m i c q u a n t i t y is usually given in u n i t s of g / e n r ' . D i v i d i n g by t h e d e n s i t y of t h e act ual m e d i u m yields A'o in u n i t s of c m . A r o u g h est i m a t e , which is correct, to b e t t e r t h a n .ri% for all e l e m e n t s , is given by
y
_ 0
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.
~ Z(Z + l ) ( l f . 4 - In(Z)) c m ' 2
Here .1 is t h e a t o m i c n u m b e r a n d Z t h e c h a r g e of t h e nucleus. As a c o n s e q u e n c e of t h e elect r o m a g n e t ic n a t u r e t he d o m i n a n t q u a n t i t y for A'o is t h e c h a r g e Z . T h e most c o m m o n choice is lead, w i t h a r a d i a t i o n length A'o = (¡..'t7 g / e n r o r . a f t e r dividing by t h e d e n s i t y . A'o = ()..r>G c m . T h e t o t a l n u m b e r of p a r t i c l e s iu t h e s h o w e r is p r o p o r t i o n a l t o t h e e n e r g y of t h e incident part icle. Since t h e n u m b e r of p a r t i c l e s iu t h e s h o w e r g r o w s e x p o n e n t i a l l y with t h e l e n g t h , it follows t h a t t h e l e n g t h of a shower is p r o p o r t i o n a l t o t h e l o g a r i t h m of t he e n e r g y of t h e p r i m a r y p a r t i c l e . In a c a l o r i m e t e r t h e m e a s u r e m e n t of t h e e n e r g y of a n incident p a r t i c l e is t r a n s f o r m e d t o a m e a s u r e m e n t of t h e n u m b e r of s e c o n d a r y p a r t i c l e s c r e a t e d in t h e s h o w e r . T h i s c a n b e achieved in a variety of ways. O n e possibility is t o use wire c h a m b e r s a n d m e a s u r e t h e i o n i z a t i o n s ignal c r e a t e d by tin- p a r t i c l e s in t h e shower. A n o t h e r t e c h n i q u e is t o use s c i n t i l l a t o r m a t e r i a l s which emit light on t h e p a s s a g e of a c h a r g e d p a r t i c l e a n d r e c o r d t h e light signal by m e a n s of p h o t o m u l t i p l i e r s . In b o t h cases t h e c a l o r i m e t e r is built in a s a n d w i c h - l i k e s t r u c t u r e with a l t e r n a t i n g layers of a b s o r b e r a n d d e t e c t o r m a t e r i a l . Such a design a l w a y s is a c o m p r o m i s e b e t w e e n high density, which is needed for t h e shower evolution. a n d sensitivity, since p a r t i c l e s p r o d u c e d inside t h e s h o w e r s h o u l d not b e r e a b s o r b e d by t h e passive m a t e r i a l b e f o r e h a v i n g passed t h r o u g h s o m e s e n s i t i v e a r e a s . A m o r e f a v o u r a b l e s o l u t i o n would b e t o use h o m o g e n e o u s c a l o r i m e t e r s , such a s lead-glass blocks or high-/? s c i n t i l l a t o r c r y s t a l s . In b o t h e a s e s t h e entire v o l u m e a c t s a s shower m e d i u m . In lead-glass light is c r e a t e d in t h e f o r m of C h o r e n k o v radiat ion from t h e s e c o n d a r i e s , c r y s t a l s such a s BaK_>. B G O . N a l ( T l ) .
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of some
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/»/(g/cin3) 1.40 7.87 11.35 18.95
An/cm 14.0 1.70 0.5G 0.32
commonly A,/cm 75.7 10.8 17.1 10.5
C'sI(TI), P b W O , c u n t scintillation light which can he de t e c t e d with p h o t o m u l t i pliers. A n o t h e r kind of h o m o g e n e o u s c a l o r i m e t e r s uses liquid Argon or K r y p t o n as a h s o r h e r m a t e r i a l . T h e noble gases a r e p a r t i c u l a r l y well suited for calorimeters because t h e filled electron shells d o not a b s o r b electrons created in t h e shower E l e c t r o d e s i m m e r s e d in t h e liquid a r e used t o read o u t t h e ionization signal. W h i l e a lit {i lie I Argon c a l o r i m e t e r still requires lead p l a t e s t o get a sufficiently short radial ion length. Krypt on a l r e a d y h a s sufficient ly high Z t o act as an effective r a d i a t o r . It is t h u s possible t o s a m p l e a shower longitudinally, which makes it a very fast d e t e c t o r with excellent energy a n d position resolution. T h e typical e n e r g y resolution t h a t c a n b e achieved with large e l e c t r o m a g n e t i c c a l o r i m e t e r s is in t h e region a r o u n d v( )/ = U.U)/y/ / vV. T h e a b o v e discussion was mainly d e a l i n g with e l e c t r o m a g n e t i c c a l o r i m e t e r s , but i he basic principles of c o u r s e a r e equally valid for h a d r o n i c c a l o r i m e t e r s . T h e main difference is t h e t y p e of interaction which gives rise t o t h e shower develo p m e n t . E l e c t r o m a g n e t i c c a l o r i m e t e r s for t h e detection of p h o t o n s or electrons, which a f t e r t h e first s t e p develop an identical shower, a r e based on a cascade of b r e m s s t r a h l u n g a n d pair c r e a t i o n . B r e m s s t r a h h i n g is f a v o u r a b l y p r o d u c e d by light particles in s t r o n g electric fields, which is t h e reason why e l e c t r o m a g n e t i c c a l o r i m e t e r s a r e built preferably f r o m h i g h - Z r a d i a t o r m a t e r i a l . Heavier particles, such as muons. pious, kaons or p r o t o n s have a very small cross section for b r e m s s t r a h l u n g a n d t h u s h a r d l y shower in a c a l o r i m e t e r which readily a b s o r b s a high energy p h o t o n o r e l e c t r o n . H a d r o n s . however, can s t a r t a shower based on nuclear interactions. T h e length scale governing t h e evolution of a h a d r o n i c shower is t h e interaction length A/. which c o r r e s p o n d s t o t h e m e a n free p a t h of a h a d r o u b e f o r e an inelastic interaction with a nucleus. A rough a p p r o x i m a t i o n for A/ is given by
A/ = 35
cm-
.
(5.3)
As s t r o n g i n t e r a c t i o n s a r e insensitive t o t h e c h a r g e of t h e nucleus, t he interaction length only d e p e n d s on t h e n u m b e r of mtclcons for a given element a n d scales with t h e r a d i u s of t h e nucleus, t h a t is. p r o p o r t i o n a l t o t h e d i s t a n c e t ravelled inside nuclear m a t t e r . T h e a b o v e definition of A/ again defines an a t o m i c q u a n t i t y . Dividing by t h e a c t u a l d e n s i t y of t h e m e d i u m yields t h e interaction length in units of c m .
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shower a n d simply p e n e t r a t e s t h e d e t e c t o r JIS a m i n i m u m ionizing particle. Bec a u s e of th«' high energy of t h e incident n e u t r i n o , such ¡1 union can travel many m e t e r s even t h r o u g h a massive d e t e c t o r , w h e r e its m o m e n t u m is d e t e r m i n e d from t h e track c u r v a t u r e in t h e m a g n e t i c lield as m e a s u r e d by M W T C s . 5.2..'{
l'ossati
of particles
through
inalici•
M a n y of t h e m e t h o d s t o d e t e r m i n e t h e m o m e n t u m of particles created in high energy i n t e r a c t i o n s exploit t h a t charged particles ionize t h e m e d i u m t h e y a r e traversing. What we have neglected so far is that t h e p a s s a g e t h r o u g h m a t t e r also alfects t h e particle. Evidently, t h e p a r t i c l e m u s t lose energy d u e t o the creation of electron ion pairs. As a rule of t h u m b o n e can a s s u m e an excitation energy I as (10 ± 1) x Z e V .
(5.1)
where Z is the c h a r g e of t h e respective e l e m e n t . T h e average energy loss per unit length is described by t h e B et lie Bloch f o r m u l a |U 2 » M - W d;r
_
;/ >
_ 6
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2
M e V
c
'"" -
(5.5)
In t his expression d.r is m e a s u r e d in m a s s per unit area. T o convert t o c e n t i m e t r e s one has t o multiply with t he d e n s i t y of t h e m e d i u m . T h e q u a n t i t y c is I he c h a r g e of t h e incident particle in units of t h e p r o t o n charge. ¡1 = v/c. a n d 7 = 1 / y/1 — j f J a r e t h e usual relativist«- variables. E(|iiation (5.5) shows that for a given particle t h e energy loss is only a f u n d ion of t h e particle velocity. For small velocities it is d o m i n a t e d by t h e 1 ¡ ¡ I 1 t e r m , which takes into account that t h e energy loss is larger if more t i m e is available for an interaction between t h e particle a n d t h e m e d i u m . W i t h increasing energy t h e specific e n e r g y loss d r o p s until it- reaches a m i n i m u m before s t a r t i n g t o rise logarithmically p r o p o r t ional t o hi 7 $ . T h i s so-called relativist ic rise is caused by t h e t r a n s f o r m a t i o n of t h e electric field of t h e incident particle, which f l a t t e n s a n d e x t e n d s as t h e particle becomes relativistic. so that, t h e c o n t r i b u t i o n s f r o m distant collisions increasingly c o n t r i b u t e t o t h e energy loss. Since real media become polarized, t h e extension of t h e field is a t t e n u a t e d , which is described by <). t h e so-called density effect. At very high energies o n e h a s S 28.8« Vy/pjZ/A) - ~ In
I 1- In !iy — -
(5.0)
wit h t he density p of t he m e d i u m in u n i t s of g/cni"'. An i m p o r t a n t o b s e r v a t i o n is t h a t t h e e n e r g y loss for c h a r g e d particles ¡11 m a t t e r luts a m i n i m u m . D e p e n d i n g on t h e m a t e r i a l t h e value is typically in t h e r a n g e d / v / d . r m i „ ~ 2 MeV e n r / g . Since t h e rise is only logarithmic with t h e particle energy, this m i n i m u m c o n s t i t u t e s t h e typical energy loss for s e c o n d a r y particles from high energy interactions.
HI I K( I d l e . \ l 11K : II KNKKOY COU.IDKItS
A n o t h e r effect is elastic s c a t t e r i n g <>l c h a r g e d particles o i l ' a t o m s in maMcr. Mere I lie main offer I comes f r o m t h e electric lielrl of t h e nucleus, which reaches much furt her t han I lie s t r o n g nuclear force a n d t herefore d o m i n a t e s all mult iple .ca tiering. I.ike in t h e case of electromagnet ic showers t he a t o m i c q u a n t i t y t o parameterize multiple s c a t t e r i n g is t h e r a d i a t i o n length AO. Small a n g l e s c a t t e r i n g in a p l a n e is well described by a G a u s s i a n d i s t r i b u t i o n of w i d t h Ho. !:{.(> M e V Bo = 2
j
pep
r r 1 + 0.038 In J-rVA >
(s)
(r».7:
As before c is t h e c h a r g e of t h e particle in u n i t s of the p r o t o n charge, lie a n d /> are tin 1 velocity a n d m o m e n t u m of t h e particle a n d x/Xo t h e thickness of t h e s c a t t e r i n g m e d i u m in u n i t s of radiat ion lengths. S c a t t e r i n g angles g r e a t e r t h a n a few (-)o have t o be described by R u t h e r f o r d s c a t t e r i n g , which e n h a n c e s t h e tails of t h e d i s t r i b u t i o n c o m p a r e d t o a G a u s s i a n . •r».2.1
Particle
iilentijicjition
lu a d d i t i o n t o t h e d e t e r m i n a t i o n of t h e m o m e n t a of all final s t a t e p a rt i c l e s one also would like t o d e t e r m i n e t h e p a rticle type. For all practical p u r p o s e s this is equivalent, t o a m e a s u r e m e n t of t h e p a rticle mass. T h e direct m e t h o d s to achieve this a r e based on different, ways to m e a s u r e t h e velocity of a particle. I 'lie combined informat ion from velocity a n d m o m e n t u m t h e n allows to e x t r a c t t he particle mass. T h e m o s t s t r a i g h t f o r w a r d way t o d e t e r m i n e t h e velocity is via a simple time(if-Jlif/lil. ( T O F ) m e a s u r e m e n t . T h e m e t h o d is limited by t h e fact t hat t h e velocity of all relativistic particles a p p r o a c h e s t h e velocity of light, e. In typical applications the relevant observable t h u s is t h e difference to c r a t h e r t han t h e velocity, which p u t s high d e m a n d s on t h e t i m e resolution of a T O F - s y s t e m . A n o t h e r way t o d e t e r m i n e t h e velocity is f r o m t h e specific energy loss of a charged p a r t i c l e in m a t t e r . F r o m eqn (•r>.f)) o n e sees t h a t for a given c h a r g e d/i'/d.r is only a function of t h e velocity. Particle identification via d E / < \ x is au a t t r a c t i v e possibility since it also works for highly relativistic particles by exploiting t h e relativistic rise in t h e specific energy loss. It can be p e r f o r m e d simultaneously with t h e m o m e n t u m d e t e r m i n a t i o n in a t r a c k i n g s y s t e m by measuring t h e ionization c h a r g e t o g e t h e r with t h e position. A p o t e n t i a l p r o b l e m of t h e m e t h o d is related t o t h e fact t h a t ionization energy loss is subject t o large fluctuations. O n e t h e r e f o r e needs m a n y s a m p l e s along a t r a c k in o r d e r t o get a reliable e s t i m a t e for t h e a v e r a g e energy loss d E ' / d . r . P a r t i c l e identification via d / d . r then-fore is preferentially d o n e iu d e t e c t o r s with large drift c h a m b e r s a n d I PCs. T h e velocity of a p a rt i c l e can also be m e a s u r e d by m e a n s of C h e r e n k o v radiation. which is e m i t t e d when a charged particle travels faster t h a n t h e speed of light inside t h e m e d i u m . A charged p a r t i c l e in m a t t e r polarizes t h e m e d i u m . For a slowly m o v i n g p a r t i c l e t h e p o l a r i z a t i o n built u p in f r o n t of t h e p a r t i c l e c o m p e n s a t e s t h e relaxation behind it. However, if t h e velocity is larger t h a n the
.
IfiXPKRlMKNTAbSliT-UP
m
s p e e d of light in i lie inediiiiii, t h e n I h e r e is no p o l a r i z a t i o n in limit of t lie p a r t i c l e , s i m p l y b e c a u s e t h e p r e s e n c e of a c h a r g e c a n n o t 1»- c o m m u n i c a t e d a h e a d a n d t he r e l a x a t i o n of t h e p o l a r i z a t i o n c r e a t e d b y t h e p a s s a g e of t h e p a r t i c l e l e a d s t o t h e e m i s s i o n of r e a l p h o t o n s . T h e p h o t o n s a r e e m i t t e d a s s p h e r i c a l w a v e s a l o n g t h e t r a j e c t o r y a n d p r o p a g a t e w i t h t h e s p e e d of light in t h e m e d i u m . For a s t r a i g h t t r a c k t h e y t h u s a r e e m i t t e d a s a c o n e w i t h o p e n i n g a n g l e O,., c I 1 cos Be = - - = - — . n ii nti
(5.8)
w h e r e II is t h e i n d e x of r e f r a c t i o n in t h e m e d i u m . T h e n u m b e r of p h o t o n s emit ted per l e n g t h interval d.r a n d e n e r g y interval ( - ' l f . f l lie V d.rdE o r equivalent ly
2 ? d A iV
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of p h o t o n e n e r g y is given by
« 370s2 sin2 B,.(E) — — ,V J ' ' '
1
\
(5.9)
(5.10)
2
As in t he c a s e of t h e s p e c i f i c i o n i z a t i o n , a l s o t h e n u m b e r of C h e r e n k o v p h o t o n s is p r o p o r t i o n a l t o t h e s q u a r e of t h e p a r t i c l e c h a r g e . T h e t o t a l e n e r g y e m i t t e d i n t o C h e r e n k o v r a d i a t i o n c a n b e o b t a i n e d b y i n t e g r a t i n g e q n (•r>.9) o v e r t h e e n t i r e r a d i a t o r l e n g t h j• a n d t h e p h o t o n e n e r g y s p e c t r u m , w h i c h is finite b e c a u s e o n l y t h o s e r e g i o n s c o n t r i b u t e w h e r e n( ) > (i. T h e s i m p l e s t w a y t o exploit. C h e r e n k o v r a d i a t i o n for p a r t i c l e i d e n t i f i c a t i o n is realized by a t h r e s h o l d c o u n t e r . P a r t i c l e s w h i c h a r e f a s t e r t h a n r / n emit light, t h e o t h e r s d o n ' t . T h i s s i m p l e s c h e m e is o f t e n a l r e a d y sufficient, t o d i s c r i m i n a t e for e x a m p l e b e t w e e n p i o u s a n d h e a v i e r p a r t i c l e s . A m o d i f i c a t i o n of t his a p p r o a c h is a set of C h e r e n k o v c o u n t e r s w i t h different, indices of r e f r a c t i o n II. W i t h t w o indices II > o n e c a n t hen d i s t i n g u i s h bet ween p i o u s , k a o n s a n d p r o t o n s over c e r t a i n r a n g e s in m o m e n t u m . P i o n s give light, in b o t h r a d i a t o r s , k a o n s o n l y e m i t light in i i i . while p r o t o n s e m i t n o C h e r e n k o v r a d i a t i o n a t all. A m o r e elegant way t o d e t e r m i n e t he velocity of a n y p a r t i c l e which e m i t s C h e r e n k o v r a d i a t i o n is t o i n e a u r e t h e o p e n i n g a n g l e of t h e light c o n e . T h i s is d o n e w i t h t h e so-called R I C H . iny-Imagmg-CHere.n ov, counters. T h e basic p r i n c i p l e is t h a t a f o c u s i n g m i r r o r or lens i m a g e s t h e light f r o m t h e C h e r e n k o v cone o n a ring, w h o s e r a d i u s is p r o p o r t i o n a l t o t h e o p e n i n g a n g l e of t h e c o n e . T h e p o s i t i o n of t h e c e n t r e of t h e r i n g c o n t a i n s t h e i n f o r m a t i o n a b o u t t h e t r a c k d i r e c t i o n , w h i c h a l l o w s t o m a t c h a R I C H ring t o a p a r t i c l e r e g i s t e r e d by a t r a c k i n g system. N o t e t h a t t h e m e t h o d s for p a r t i c l e i d e n t i f i c a t i o n d i s c u s s e d s o f a r a r e u s u a l l y a p p l i e d u n d e r t h e a s s u m p t i o n t h a t t h e m a g n i t u d e of t h e c h a r g e of t h e p a r t i c l e is t h e f u n d a m e n t a l c h a r g e r . T h i s is n o p r o b l e m w h e n d e a l i n g w i t h t h e i d e n t i f i c a t i o n of k n o w n p a r t i c l e s , but m u s t b e t a k e n i n t o a c c o u n t w h e n looking, f o r e x a m p l e , for free q u a r k s w h i c h a r e e x p e c t e d t o c a r r y f r a c t ional c h a r g e s . .lust, a s a r e m i n d e r .
I »El H
l O R ' i A I I I I C I I KNI'lRCV COI.I.II >| |<s
a t r a c k i n g s y s t e m d e t e r m i n e s ;//>, a d f c ' / d r m e a s u r e m e n t y i e l d s ' f ( v ) a n d a 1 0 F o r a R I C H s y s t e m gives i>. The n u m b e r of p h o t o n s r e c o r d e d iu a R I C H also is p r o p o r t i o n a l t o 2 2 . T a k i n g i n f o r m a t i o n f r o m different s y s t e m s t h u s a l l o w s a s i m u l t a n e o u s d e t e r m i n a t i o n o f / / , in a n d . Iu a d d i t i o n t o t h e a c t i v e m e t h o d s of p a r t i c l e i d e n t i f i c a t i o n by m e a n s of dedicated m e a s u r e m e n t s , i d e n t i f i c a t i o n is a l s o possible by c o m b i n i n g i n f o r m a t i o n f r o m d i f f e r e n t d e v i c e s . E l e c t r o n s , for e x a m p l e , a r e u s u a l l y i d e n t i f i e d a s c h a r g e d p a r t i c l e s w h i c h d e p o s i t t h e i r e n t i r e e n e r g y in t he e l e c t r o m a g n e t i c c a l o r i m e t e r , c h a r g e d h a d r o u s a r e a b s o r b e d iu h a d r o n i c c a l o r i m e t e r s a n d m u o n s a r e recognized a s c h a r g e d p a r t i c l e s w i t h e x t r a o r d i n a r y p e n e t r a t i n g p o w e r . T h e y a r e t o o heavy t o c r e a t e e l e c t r o m a g n e t i c s h o w e r s a n d a s l e p t o n s d o not i n t e r a c t via t h e st r o n g i n t e r a c t ion, a n d t h u s even p a s s t h r o u g h c a l o r i m e t e r s j u s t leaving a n ionization t r a c k . 5.2.5
A L E P 1 I : an example
of a L E P
detector
A typical d e t e c t o r e x p l o i t i n g t h e t e c h n o l o g i e s d i s c u s s e d a b o v e is s h o w n in Fig. 5.0. It h e r m e t i c a l l y s u r r o u n d e d o n e i n t e r a c t i o n region at MSP. in o r d e r t o r e c o r d all p a r t i c l e s e m a n a t ing f r o m a n e + e ~ a n n i h i l a t i o n . O n l y p a r t i c l e s e m i t t e d a l o n g t h e beam pipe (I) a n d non-interacting particles such as neutrinos escape. T h e inner regions a r e inst r u m e n t e d wit h t r a c k i n g d e t e c t o r s , t h e n c o m e t h e e l e c t r o m a g n e t i c a n d t h e h a d r o n i c c a l o r i m e t e r a n d finally a g a i n t w o layers of t r a c k i n g c h a m b e r s t o identify high e n e r g y m u o n s . T h e i n n e r t r a c k i n g s y s t e m c o n s i s t s of a silicon v e r t e x d e t e c t o r (2) for h i g h precision m e a s u r e m e n t s of s e c o n d a r y vert ices f r o m h e a v y llavour d e c a y s . T h e next layer is a s m a l l c y l i n d r i c a l MYVPC (3) which p r o v i d e s fast, t r a c k i n g i n f o r m a t i o n for t r i g g e r i n g . In a d d i t i o n , it s e r v e s for e x t r a p o l a t i o n Iroin t h e m a i n t r a c k i n g c h a m b e r , a l a r g e I P C (5). t o t h e v e r t e x d e t e c t o r . T h e I P C w i t h a r a d i u s of 1.8 in is i m m e r s e d in a h o m o g e n e o u s s o l e n o i d a l /?-field of 1.5 T c r e a t e d b y a s u p e r c o n d u c t i n g coil (7) w h i c h allows p r e c i s i o n m e a s u r e m e n t s of t h e m o m e n t u m of c h a r g e d p a r t i c l e s . In a d d i t i o n it p e r f o r i n s part icle identification f r o m d / v / d . r by s a m p l i n g t h e specific i o n i z a t i o n u p t o 180 t i m e s a l o n g a t r a c k . T h e n e x t d e t e c t o r o u t s i d e t h e I P C is t h e e l e c t r o m a g n e t i c c a l o r i m e t e r (Ii). w h i c h is a l e a d - M W P C s a n d w i c h s t r u c t u r e . T h i s c a l o r i m e t e r is still i n s i d e t h e m a g n e t coil (7). which m i n i m i z e s t h e n u m b e r of r a d i a t ion l e n g t h s in front, of t h e c a l o r i m e t e r a n d t h u s t a k e s full a d v a n t a g e of t h e e x c e l l e n t p o s i t i o n r e s o l u t i o n of t.he M W P C s . A d d i t i o n a l s m a l l c a l o r i m e t e r s (1) close t o t h e b e a m p i p e a r e used t o m e a s u r e B h a b h a s c a t t e r i n g for precision m e a s u r e m e n t s of t h e l u m i n o s ity. O u t s i d e t he e l e c t r o m a g n e t i c c a l o r i m e t e r c o m e s t h e s u p e r c o n d u c t i n g coil a n d t h e n t h e h a d r o n c a l o r i m e t e r (8). T h e l i a d r o n c a l o r i m e t e r is a n iron s t r e a m e r - t u b e s a n d w i c h s t r u c t u r e . T h e s t r e a m e r t u b e s , like M W P C s , utilize g a s a m p l i f i c a t i o n t o d e t e c t i o n i z a t i o n signals, b u t iu c o n t r a s t t o t h e f o r m e r t h e y a r e not. o p e r a t e d in p r o p o r t i o n a l m o d e but a t h i g h e r v o l t a g e w h e r e t he s i g n a l s a t u r a t e s . T h i s res u l t s in l a r g e p u l s e s w h i c h c a n easily b e r e a d o u t via i n d u c t i o n p a d s . but. y i e l d s o n l y d i g i t a l i n f o r m a t i o n . T h e e n e r g y of a h a d r o n i c s h o w e r t h u s is p r o p o r t i o n a l t o t h e n u m b e r of p a d s a b o v e a noise t h r e s h o l d . T h e iron iu t h e h a d r o n c a l o r i m e t e r
I NM K I M K N T A I . S K I I I'
t o r V D U T . (3) i n n e r t r a c k i n g c h a m b e r I'l'C, (1) l u m i n o s i t y m o n i t o r I . C A b , (•r>) t i m e p r o j e c t i o n c h a m b e r I P C , (6a) e l e c t r o m a g n e t i c c a l o r i m e t e r E C A L ( b a r r e l [»art), ((ib) E C A I , ( e n d - c a p ) , (7) s u p e r c o n d u c t i n g coil, (8a) h a d r o n i c c a l o r i m e t e r I I C A I . ( b a r r e l p a r t ) , ( 8 b ) I I C A I , ( e n d - c a p ) . (!)) u n i o n c h a m b e r s . F i g u r e f r o m MiiHer(2000).
not o n l y s e r v e s a s s h o w e r m e d i u m , but a l s o a s inechaiiical s u p p o r t , of t h e e n t i r e d e t e c t o r a n d a s r e t u r n y o k e for t h e m a g n e t i c llux. It a b s o r b s all r e m a i n i n g particles f r o m t h e p r i m a r y i n t e r a c t i o n e x c e p t high e n e r g y u n i o n s , which t h u s a r e identified by t h e s i g n a l s t.hey leave in t h e t r a c k i n g c h a m b e r s (!J) m o u n t e d o n t h e o u t s i d e of t h e d e t e c t o r . T h e e x a m p l e of t h e A L E P I I d e t e c t o r i l l u s t r a t e s how different d e t e c t o r t e c h nologies c a n be c o m b i n e d t o cover t h e r e q u i r e m e n t s of a t y p i c a l high e n e r g y p h y s i c s e x p e r i m e n t . It h a s t o b e e m p h a s i z e d t h a t not o n l y a r e t h e t e c h n o l o g i e s i m p o r t a n t , but a l s o t h e g r a n u l a r i t y t h a t is n e e d e d in o r d e r t o r e c o r d reliably all final s t a t e p a r t i c l e s f r o m a p r i m a r y i n t e r a c t i o n . E a c h of t h e m a j o r d e v i c e s d e s c r i b e d a b o v e h a s b e t w e e n 10'1 a n d 10" r e a d o u t c h a n n e l s , a n d t h i s n u m b e r is g o i n g t o i n c r e a s e for f u t u r e d e t e c t o r s . T h e l a r g e n u m b e r of c h a n n e l s m a t c h e s t h e overall c o m p l e x i t y of t h e e v e n t s t h a t a r e t o b e r e c o r d e d , a n d it s h o u l d not c o m e as a s u r p r i s e t h a t t h i s n u m b e r h a s a t e n d e n c y t o a p p r o a c h t h e level of biological s e n s o r s , s u c h a s . for e x a m p l e , t h e h u m a n eye, w h i c h a l s o h a v e t o «leal wit h highly
D K T K O T O I T S A I I I I C I I KNKH.OY C O M . I I > K H S
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FlC. 5 . 7 . T h e working of t he A LISP 11 d e t e c t o r illustrat ed by s o m e typical e v e n t s as recorded by t h e e x p e r i m e n t -
complex p a t t e r n s . displays. T h e working of t he ALKPII d e t e c t o r is illustrated in Fig. .r).7 by event which p r e s e n t t h e i n f o r m a t i o n recorded by t h e d e t e c t o r in a g r a p h i c a l way. T h e view is a l o n g t h e b e a m direction. Going from the c e n t r e t o t h e outside, o n e sees the vertex d e t e c t o r a n d inner t ra c ke r, followed by t h e I PC. KCAI,. m a g n e t coil. 11 ( ' AI. a n d u m o n c h a m b e r s . Hits in a n y of t h e t racking d e t e c t o r s a r e m a r k e d and c o n n e c t e d by t h e r e c o n s t r u c t e d tracks. T h e amount, of e n e r g y seen in t h e c a l o r i m e t e r s is shown ill t h e f o r m of h i s t o g r a m s , w h e r e t h e h i s t o g r a m a r e a is p r o p o r t i o n a l t o t h e energy deposit,. T h e u p p e r left f r a m e of Fig. 5.7 shows a n e + o " a n n i h i l a t i o n i n t o a pair of t a n leptons. which b o t h decayed into a union a n d two n e u t r i n o s t h a t a r e not seen. Both c h a r g e d linal s t a t e particles pass a s m i n i m u m ionizing particles t h r o u g h t h e entire s y s t e m , leaving also hits in t h e union c h a m b e r s o u t s i d e of t h e magnet yoke. T h e event topology is t h e s a m e as for a n e + e ^ annihilat ion into a / / / / pair, except t h a t in t h e l a t t e r case b o t h unions would have t h e b e a m energy. For t he event shown, analysis of t he track c u r v a t u r e in t h e magnet ic field reveals t hat t h e unions a r e less energetic, which leads t o t h e conclusion that, o n e has a c t u a l l y observed t h e reaction c + e ~ —> r + r ~ . In t h e u p p e r right an c + c — T + T ~ event is shown, w h e r e t h e t + d e c a y s into a positron a n d t wo n e u t r i n o s a n d t h e T~ into a union a n d t wo n e u t r i n o s .
Ii
I M ' l .HI.MI IN I ,\l. M' I HI'
T h e n e u t r i n o s a r e not seen, t h e union again passes t h r o u g h t h e entire de t e c t o r . T h e p o s i t r o n e m i t s most of its energy into a h r e m s s t r a h l u n g p h o t o n a n d thus suffers a s t r o n g deflection in t h e m a g n e t i c field, t h e p h o t o n travels in a straight line until it is a b s o r b e d in t h e e l e c t r o m a g n e t i c c a l o r i m e t e r , w h e r e it d e p o s i t s a large a m o u n t of energy. T h e positron of co u r s e is also s t o p p e d in t h e F C A L . T h e lower loft shows a n o t h e r c + e ~ —» T+T~ event. Here t h e r + d e c a y s into a posi t ron a n d two n e u t r i n o s ; t h e r ~ d e c a y s h a d r o n i c a l l y i n t o t h r e e charged pious a n d a n e u t r i n o . T h e n e u t r i n o s d o n o t interact a n d e s c a p e unseen. T h e p o s i t r o n as a n elect roniagnetieally i n t e r a c t i n g p a r t i c l e is recognized a s a high m o m e n t u m track which is a b s o r b e d in t h e EOAL. T h e positive c h a r g e can be inferred f r o m t h e c u r v a t u r e in t h e m a g n e t i c field. A l t h o u g h barely visible in t h e ligure. it c a n be reliably m e a s u r e d in t h e I P C which h a s a m o m e n t u m resolution Ap/p l O _ 3 p / ( G e V / c ) . F r o m t h e t h r o e pious f r o m t h e r~ decay t h e two with t h e s a m e direction of c u r v a t u r e a r e 7r~s. t h e t h i r d o n e is a ti"+ . O n e of t h e 7 r - s is of so low e n e r g y t h a t a f t e r significant deflection in t h e I P C it is a b s o r b e d a l r e a d y iu t h e F O A L . T h e o t h e r two pass t h r o u g h t h e KCAL a n d a r e only s t o p p e d in t he liadron c a l o r i m e t e r , w h e r e t h e y finally d e p o s i t t h e i r energy. T h e lower right f r a m e of Fig. 5.7 finally illustrates t h e i m p o r t a n c e of a good vertex d e t e c t o r for t h e reconstruct ion of s e c o n d a r y vert ices a n d d e c a y s of b e a u t y a n d c h a r m - p a r t i c l e s . In c o n t r a s t t o t he previous e x a m p l e s it is a h a d r o u i c Z decay with a so-called t o-jet structure. Z o o m i n g iu t o t h e p r i m a r y vertex, o n e sees how it is possible t o reconstruct, t h e entire decay chain of a p r i m a r y B -m e so n, using t h e precise vertoxhig, a n d t he good m o m e n t u m m e a s u r e m e n t a n d particle identification capabilities of the d e t e c t o r .
Exercises for Chapter •r> I What, is t h e required e n e r g y for a p r o t o n b e a m on a hydrogen target in o r d e r t o reach t h e s a m e C.o.M. energy as a 7 T e V on 7 ToV p r o t o n p r o t o n collider? W h a t would b e t h e relativistic 7 - f a c t o r of t h e C . o . M . s y s t e m iu t h e lixed target, scenario? 5 2 T h e motion of a p a r t i c l e in an a c c e l e r a t o r s t r u c t u r e is p a r a m e t e r i z e d by its t r a n s v e r s e position (x. y) with respect t o I lie nominal orbit a n d t he direction (d.r/d.s. d y / d s ) . w h e r e .s is t h e longitudinal c o o r d i n a t e . C o n s i d e r i n g o n l y x. t he effect of an optical element can be described by a so-called transfer matrix M. which t r a n s f o r m s t h e s t a t e vectors of t h e b e a m , t h a t is. (x.x')r
C o n s t r u c t t h e t ransfer m a t r i x of a d r i f t volume of length L a n d t h a t of a q t t a d r u p o l e with focal length / . For t h e q u a d r u p o l e e m p l o y t h e thin-lens a p p r o x i m a t i o n , that is. ignore t h e length of t he device. Finally. by multiplying t h e t r a n s f e r m a t r i c e s of t h e individual c o m p o nents. calculate t h e combined effect of a focusing a n d a defocusing
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q t i a d r u p o l e si'|>aruled by a drill s p a c e a n d d e r i v e a e o n d i l i o n w h i c h h a s t o h e fnllilled il t h e e n t i r e a s s e m b l y is t o h a v e a net f o c u s i n g cllcct. < * * ' .'! S h o w t h a t in t h e a b s e n c e of m u l t i p l e s c a t t e r i n g t h e m o m e n t u m reso l u t i o n of a m a g n e t i c s p e c t r o m e t e r b e h a v e s like i\p/p ~ p. A s s u m e t hai t h e s p e c t r o m e t e r m e a s u r e s t h e r a d i u s of c u r v a t u r e for t h e p a r ticle t r a c k f r o m t h r e e e q u i d i s t a n t , p o i n t s o n a c i r c u l a r a r c . a n d t h a t the individual position m e a s u r e m e n t s have Gaussian errors. ' * * ' I A c h a r g e d p a r t i c l e t r a v e l l i n g a l o n g t h e ./--axis e n t e r s a m a g n e t i c field p o i n t i n g in t h e ¿ - d i r e c t i o n . T h e field i n t e g r a l seen by t h e p a r t i c l e is I IJ(\s — l.f) T i n . B e h i n d t h e magnet, t h e t r a c k d i r e c t i o n is m e a s u r e d by t w o M W P C s s p a c e d 1 111 a p a r t . Iu t h e a b s e n c e of m u l t i p l e s c a t t e r i n g , w h a t p o s i t i o n r e s o l u t i o n is n e e d e d in t he c h a m b e r s if t h i s spect r o m e t e r h a s t o b e a b l e t o d e t e r m i n e t h e c h a r g e of a pion w i t h l> 100 G e V / r w i t h a s i g n i f i c a n c e of t h r e e s t a n d a r d d e v i a t ions? H o w m u c h m a t e r i a l , in u n i t s of r a d i a t i o n l e n g t h s , c a n h e t o l e r a t e d in f r o n t of t he s e c o n d c h a m b e r if for a p a r t i c l e w i t h /> -r) G e V / c t h e e r r o r f r o m m u l t i p l e s c a t t e r i n g a n d c h a m b e r r e s o l u t i o n a r e allowed t o lie of e q u a l size? ' * ' ") S h o w t h a t t h e e n e r g y r e s o l u t i o n of a c a l o r i m e t e r follows d E / E ~ l/s/E. (> W h a t is t h e r e q u i r e d t i m e resolut ion of a T O F - s y s t e n i built f r o m t wo s c i n t i l l a t o r s w i t h a s e p a r a t i o n of 2 m t h a t h a s t o d i s t i n g u i s h bet ween a p = 1 G e V / c piou a n d kaon w i t h a significance of t h r e e s t a n d a r d deviations? 7 S h o w t hat t h e C h e r e n k o v light f r o m a p a r t icle m o v i n g a l o n g a st raiglit line is m a p p e d i n t o a ring at. a d i s t a n c e / b e h i n d a lens w i t h focal l e n g t h / . <*'
G QCD ANALYSES (!. 1
General c o n c e p t s
T h e p r o p e r t y of confinement implies t h a t we never observe q u a r k s or g l u o n s in our e x p e r i m e n t s , but only h a d r o n s a n d their decay p r o d u c t s , which c a n be of a hadroiiic or leptonic nat ure. T h i s m e a n s t h a t we a r e not able to observe, directly, t h e f u n d a m e n t a l particles of Q C D . t h e t h e o r y we w a n t t o t e s t , since t h e Q C D L a g r a u g i a n is built o u t of q u a r k a n d gluon fields, lu a d d i t i o n , until now n o b o d y has been successful in c o m p u t i n g t h e t r a n s i t i o n f r o m p a r t o n s t o h a d r o n s from lirst principles, a p r o b l e m which c a n n o t be solved by t he well known met h o d s ol p e r t u r b a t i o n theory. O n l y phenomenological models a r e available, a s h a s been described in C h a p t e r 3. T h i s obviously c o n s t i t u t e s a m a j o r c o m p l i c a t i o n for t h e e x p e r i m e n t a l i s t , w h o has t o find correlations between t h e p r o p e r t i e s of t h e observed h a d r o n i c final s t a t e a n d t h e p r o p e r t i e s of t h e p a r t o n i c s t a t e , t h a t is. t h e q u a r k s a n d gluons. T h i s is also in s h a r p c o n t r a s t to e x p e r i m e n t s which test theories with leptonic final s t a t e s , such a s tjiiuntam electrodynamics ( Q E D ) . T h e r e t h e a s y m p t o t i c s t a t e s of t h e t h e o r y a r e directly observable in t h e e x p e r i m e n t a l a p p a r a t u s . F u r t h e r m o r e c a l c u l a t i o n s have been carried out u p t o high o r d e r s in p e r t u r b a t i o n theory, which a l t o g e t h e r leads t o very precise m e a s u r e m e n t s , with m e a s u r e m e n t e r r o r s well below t he per mill«* level. Such a n a c c u r a c y is beyond reach for a n y test of t h e s t r o n g i n t e r a c t i o n s . T h e r e , t h e most precise m e a s u r e m e n t s r a t h e r have e r r o r s of a few per cent, in p a r t i c u l a r , t h e m e a s u r e m e n t s of t he s t r o n g coupling c o n s t a n t . A m a j o r theoretical b r e a k t h r o u g h in p e r t n r b a t i v e as well a s non-pert u r b a t i v e m e t h o d s would be a b s o l u t e l y necessary in o r d e r t o push t h e precision into a new regime. So. t h e challenge for t h e experimentalist is first t o m e a s u r e as precisely as possible t h e p r o p e r t i e s of t h e final s t a t e h a d r o n s . a n d second t o get i n f o r m a t i o n a b o u t t h e p a r t o n s . Of course, t h e l a t t e r task is not required if t h e e x p e r i m e n t only a i m s at s t u d i e s of t h e p r o p e r t i e s of h a d r o n s . In o r d e r to p e r f o r m t h e former task, we have t o build a n a p p a r a t u s which is a b l e t o m e a s u r e t h e m o m e n t u m a n d energy of charged a n d n e u t r a l particles, ¡us well as t h e decay vertices a n d decay p r o d u c t s of short-lived s t a t e s , a n d f u r t h e r more t o give e s t i m a t e s for t h e particle t y p e . A description of m o d e r n d e t e c t o r s is given in C h a p t e r -r>. Now consider, for e x a m p l e , e + e ~ a n n i h i l a t i o n s a t l.lil' at a C.o.M. energy of 91 G e V . w h e r e 20 c h a r g e d h a d r o n s a n d a similar n u m b e r of n e u t r a l particles a r e observed on average. Figure (i.l shows au event display of a h a d r o n i c event
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ill 1.101*1. m e a s u r e d w i t h t h e A I . E I ' l l d e t e c t o r . Most; of t h e c h a r g e d p a r t i c l e s a r e pious, w i t h in a d d i t i o n s o m e p r o t o n s , k a o n s a n d l e p t o n s ( e l e c t r o n s a n d m u o n s ) from d e c a y s d u e t o weak i n t e r a c t i o n s or f r o m p h o t o n c o n v e r s i o n s into lepton pairs. T h e s t a b l e n e u t r a l p a r t i c l e s a r e m o s t l y p h o t o n s , which s t e m f r o m t h e d e c a y s of n e u t r a l pions. S h o r t - l i v e d p ar t i c l e s include n e u t r a l k a o n s a n d t h e A b a r y o n . Finally, via r e c o n s t r u c t i n g t h e i n v a r i a n t m a s s e s f r o m t h e m o m e n t a of t h e s t a b l e d e c a y p r o d u c t s , even very s h o r t - l i v e d h a d r o n i c r e s o n a n c e s such a s t h e l> meson c a n b e identified. P a r t i c l e i d e n t i f i c a t i o n is o b t a i n e d f r o m m e a s u r e m e n t s of t h e specific ionization loss w h e n t h e p a r t i c l e s t r a v e r s e t h e d e t e c t o r m a t e r i a l , f r o m m e a s u r e m e n t s of t h e C h e r e n k o v r a d i a t i o n or f r o m t h e t ime of flight . F l a v o u r s e p a r a t i o n , t hat is. t h e d i s t i n c t i o n b e t w e e n h a d r o n s which c o n t a i n light, ( u p . d o w n , s t r a n g e ) , c h a r m or b e a u t y q u a r k s , is possible since t h e d e c a y s of h a d r o n s with h e a v y q u a r k s can be t a g g e d e i t h e r by m e a s u r i n g d i s p l a c e d vertices w i t h respect t o t h e m a i n event vertex, o r by l o o k i n g for high m o m e n t u m l e p t o n s f r o m t h e weak decay. T h e i n f o r m a t i o n o n t h e h a d r o n i c final s t a t e can finally be used t o recons t r u c t . for e x a m p l e , jets a n d t o s t u d y their p r o p e r t i e s . T h e precise m e a n i n g of t h e concept of j e t s will b e e x p l a i n e d l a t e r . In s u m m a r y , t h e d e t e c t o r r e q u i r e m e n t s a r e : excellent t r a c k i n g devices c o m bined w i t h g o o d p a r t i c l e i d e n t i f i c a t i o n c a p a b i l i t i e s , a n d g o o d e n e r g y m e a s u r e m e n t s f r o m c a l o r i m e t e r s w i t h high r e s o l u t i o n in e n e r g y a n d a ngl e s. For d e e p inelastic s c a t t e r i n g e x p e r i m e n t s , in p a r t i c u l a r a t t h e e l e c t r o n p r o t o n collider IIERA at D E S Y . t h e precise m e a s u r e m e n t of t h e f o u r - m o m e n t u m of t h e scat t e r e d e l e c t r o n is m o s t i m p o r t a n t , , s i n c e all t h e relevant k i n e m a t i c q u a n t i t i e s of t h e r e a c t i o n c a n b e d e d u c e d f r o m t h e e l e c t r o n ' s s c a t t e r i n g a n g l e a n d energy. T h i s a g a i n r e q u i r e s excellent c a l o r h n e t r y a s well a s precise t r a c k i n g . At H E R A a l s o t h e
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l i a d r o n i c final s l a t e is a n a l y s e d , since its e n e r g y a n d m o m e n t u m How allow for a n i n d e p e n d e n t d e t e r m i n a t i o n of t h e basic k i n e m a t i c v a r i a b l e s for e l e c t r o n p r o t o n s c a t t e r i n g . In a d d i t i o n , t h e l i a d r o n i c final s t a t e is a n a l y s e d in f u r t h e r d e t a i l , looking for j e t s a n d identified p a r t i c l e s . In c h a r g e d c u r r e n t i n t e r a c t ions t h i s i?. t h e only s o u r c e of i n f o r m a t i o n , since t h e p r o d u c e d n e u t r i n o e s c a p e s u n d e t e c t e d . Finally, at h a d r o n h a d r o n colliders t h e p r i m a r y t a s k of Q C D s t u d i e s is t h e a n a l y s i s of t h e p r o d u c t i o n of j e t s , i s o l a t e d p h o t o n s a n d d i - l e p t o n p a i r s . G o o d e a l o r i m e t r y is r e q u i r e d for a n a c c u r a t e d e t e r m i n a t i o n of t h e jet e ne rgi e s. In part i c u l a r . g o o d k n o w l e d g e of t h e a b s o l u t e e n e r g y s c a l e a s well a s t he e n e r g y resolution is m a n d a t o r y , a n d in o r d e r t o h a v e a reliable e n e r g y m e a s u r e m e n t u p t o t h e highest jet. e n e r g i e s ((9(100 G e V ) a n d m o r e ) , g o o d l i n e a r i t y of t h e r e s p o n s e of t h e c a l o r i m e t e r h a s t o b e a c h i e v e d . F u r t h e r m o r e , it s h o u l d b e p o s s i b l e t o s e p a r a t e well b e t w e e n l i a d r o n i c j e t s a n d e l e c t r o m a g n e t i c s h o w e r s i n d u c e d by e l e c t r o n s a n d p h o t o n s . G o o d h e r m e t i c i t y of t h e a p p a r a t u s will allow for a m e a s u r e m e n t of missing t r a n s v e r s e m o m e n t u m , w h i c h , for e x a m p l e , is i n d u c e d bv n e u t r i n o p r o d u c t i o n . Last but not le;ist. p re c i s e t r a c k i n g e n a b l e s f l a v o u r t a g g i n g a s in t he e a s e of e 1 e colliders. A f t e r t h i s brief o v e r v i e w we will now d e s c r i b e in s o m e m o r e d e t a i l t he individual s t e p s of a Q C D a n a l y s i s a n d give s o m e c o n c r e t e e x a m p l e s at t h e e n d of the chapter. (i. 1.1
vent
select ion
Event selection d e a l s w i t h t h e decision a b o u t which e v e n t s f r o m t h e r e a c t i o n bet ween t w o colliding b e a m s a r e of interest for a p a r t i c u l a r a n a l y s i s . It is clear I hat not all t h e e v e n t s o b s e r v e d in t h e d e t e c t o r a r e e x a c t l y of t he kind we w a n t t o m e a s u r e , t h e r e f o r e s o m e filtering h a s t o b e a p p l i e d . T h e filtering s h o u l d b e such t h a t t h e efficiency for t h e d e t e c t i o n of i n t e r e s t i n g e v e n t s is large, t h a t is. o n l y a few of t h e ' g o o d ' e v e n t s a r e lost. At t h e s a m e t i m e , a high p u r i t y of t h e selected e v e n t s a m p l e shall be a c h i e v e d , w h i c h m e a n s t h a t o n l y a few of t h e a c c e p t e d e v e n t s a r e a c t u a l l y f r o m p r o c e s s e s we did not w a n t t o select. In t h e following s e c t i o n , b o t h of t h e s e a s p e c t s will be d i s c u s s e d for t h e v a r i o u s collider t y p e s . (i.l.l.I lectron positmn anni ilation In Q C D s t u d i e s at L E P we a r e l o o k i n g for d e c a y s of t h e Z b o s o n i n t o q u a r k a i i t i q u a r k p a i r s , w h e r e t h e Z b o s o n s t e m s f r o m t h e a n n i h i l a t i o n of t h e i n c o m i n g e l e c t r o n a n d p o s i t r o n . Of c o u r s e , also highly v i r t u a l p h o t o n s c a n be e x c h a n g e d b e t w e e n t h e initial a n d final s t a t e . However, at t h e p e a k of t h e Z r e s o n a n c e t h i s p r o c e s s is s u p p r e s s e d w i t h respect to Z e x c h a n g e . In a n y c a s e , for t h e p u r e Q C D point of view it. d o e s n o t really m a t t e r if q u a r k s of a c e r t a i n flavour a r e p r o d u c e d f r o m a Z or a p h o t o n , as long a s t h e e v e n t o r i e n t a t i o n is not c o n s i d e r e d . T r i g g e r i n g o n l i a d r o n i c Z d e c a y s is r a t h e r easy. T y p i c a l l y , o n e looks for a n e n e r g y d e p o s i t in t h e c a l o r i m e t e r a b o v e s o m e t h r e s h o l d , of t h e o r d e r of several G e V . o r for a c o i n c i d e n c e of t r a c k s e g m e n t s w i t h h i t s in o u t e r c a l o r i m e t e r m o d u l e s , w h e r e t he t r a c k o r i g i n s h o u l d b e n e a r t h e i n t e r a c t i o n point . T h e s e very s i m p l e r e q u i r e m e n t s lead a l r e a d y t o a n efficiency g r e a t e r t h a n !)!).!•'/ for select ing
C I NI U A I , C O N C E P T S
>A I
h a d r o n i c event«. I lie next s t e p is I lie rejection of ¡ill possible b a c k g r o u n d s . T h e r e a r e backg r o u n d s which a r e not related t o t h e s c a t t e r i n g of t h e electron a n d t h e p o s i t r o n . Much as b e a m g a s interactions. T h e s e a r e a l r e a d y suppressed a t t h e level of t h e Inst trigger decision, mainly by requiring a certain energy d e p o s i t in t h e central calorimeter. More serious b a c k g r o u n d s would b e t h e leptomc d e c a y s of t h e Z boson, or p h o t o n p h o t o n s c a t t e r i n g , w h e r e e i t h e r p h o t o n is r a d i a t e d from o n e of I lie incoming b e a m s . The lcptonic Z decays distinguish themselves by their low multiplicity. T h e decay into electron or union p a i r s a t first o r d e r gives two back-to-back tracks, with t h e unions possibly identified by s o m e union detection s y s t e m . T h i s s i m p l e pict ure is a l t e r e d by brcmsstralilting a n d / o r s u b s e q u e n t conversion of t he bremss t r a h l u u g p h o t o n into lepton pairs. Tail p r o d u c t i o n can lead t o higher t r a c k multiplicities, since t h e tail lepton can d e c a y hadrouically. However, in t h e t a n decay also n e u t r i n o s a r e p r o d u c e d , which e s c a p e t h e a p p a r a t u s u n d e t e c t e d a n d thus lead to a reduced total m e a s u r e d energy. P h o t o n p h o t o n i n t e r a c t i o n s a r e c h a r a c t e r i z e d by a r e d u c e d effective C'.o.M. energy, a n d in contrast t o t h e genuine electron positron i n t e r a c t i o n s t h e momenta along t h e b e a m axis a r e not b a l a n c e d , which causes t h e final s t a t e t o b e boosted along t h e b e a m direction. T h e r e f o r e m a n y final s t a t e particles a r e lost undetected into t h e b e a m pipe. After the triggering s t a g e , t h e a n a l y s i s tries t o c o m b i n e all t h e i n f o r m a t i o n of t h e t r a c k i n g a n d c a l o r i m e t e r devices into a m e a s u r e m e n t of t h e energy (low. wit h a given t rack multiplicity a n d total energy o r invariant m a s s c o m p u t e d f r o m the r e c o n s t r u c t e d f o u r - m o m e n t a . By requiring, for e x a m p l e , at least live c h a r g e d tracks in t h e event, a n d a m i n i m u m a m o u n t of total energy o r invariant mass, all t h e b a c k g r o u n d s m e n t i o n e d a b o v e a r e rejected very efficiently. T h i s is illustrated in Pig. (t.2. where t h e m e a s u r e d d i s t r i b u t i o n s of t he t wo last mentioned q u a n t i t i e s are shown. Also t h e c o n t r i b u t i n g processes a n d t h e applied selection c u t s a r e indicated there. Already such a simple selection reduces t h e b a c k g r o u n d s t o t h e per inille level. T h e small r e m a i n d e r s t e i n s m a i n l y from h a d r o n i c t a n decays. It should be pointed out t h a t , when going t o C.o.M. energies well a b o v e t h e Z resonance, in p a r t i c u l a r a b o v e t he t h r e s h o l d for W pair p r o d u c t i o n a t 161 G e V or t h e threshold for Z pair p r o d u c t i o n at 182 G e V . a d d i t i o n a l b a c k g r o u n d s from I he h a d r o n i c d e c a y s of t h e \ \ 7 ± a n d t h e Z arise. T h e reject ion of this four-fcrmion buckfiround is s o m e w h a t m o r e difficult , a n d o n e e n d s u p with reduced efficiencies and purities. In a d d i t i o n , t h e p r o b l e m of t h e so-called nidiutivc return has t o be dealt with (c.f. Section .'5.2.1). Namely, t h e r e is a high probability t h a t o n e of t h e incoming electrons or p o s i t r o n s r a d i a t e s a very energetic p h o t o n such t h a t t h e effective C . o . M . e n e r g y of t h e e + e _ s c a t t e r i n g is reduced t o t h e m a s s of t h e Z. So. in o r d e r t o be a b l e t o s t u d y h a d r o n i c Z decays at. or very close t o t h e real C.o.M. energy. « 2 0 0 G e V at t h e final LI5P2 stage, it has t o b e checked that no very energetic p h o t o n is found in t h e d e t e c t o r , o r t h a t t h e h a d r o n i c s y s t e m h a s no s t r o n g boost along t h e b e a m line, which would be a signal for t h e r a d i a t i v e
0< Ii ANALYSES
F i e . 6 . 2 . D i s t r i b u t i o n of t h e invariant m a s s vs. t h e track multiplicity of events m e a s u r e d by t h e A L E P H d e t e c t o r . F i g u r e f r o m A L E P H Collab.(l!J!). r >«).
photon t o have escaped into tlie b e a m pipe. Figure (¡..'J shows the d i s t r i b u t i o n of which is t h e effective C.o.M. energy r e c o n s t r u c t e d in t h e d e t e c t o r , for a t o t a l L E P e n e r g y of l(if G e V . A clear two-peak s t r u c t u r e can be observed, where t h e peak close t o l(il G e V s t e m s f r o m e v e n t s w h e r e only a small a m o u n t of energy h a s been lost b e c a u s e of initial s t a t e p h o t o n r a d i a t i o n , a n d t h e peak a r o u n d t h e Z m a s s is d u e t o events with very h a r d p h o t o n r a d i a t i o n . If o n e is only interested in e v e n t s with t h e highest effective energy, o n e will usually c u t at large values of typically about !)()% of t h e total I.EP energy. It is clear from t he plot t hat in this case only a small fraction of t he collected e v e n t s r e m a i n s for f u r t h e r analysis, which is t h e main reason that m a n y of t h e Q C D a n a l y s e s at L E P 2 energies suffer f r o m limited s t a t i s t i c a l precision. T h e bulk of t he m o s t precise Q C D m e a s u r e m e n t s c o m e s from L E P 1 . with a b o u t four million h a d r o n i c Z decays registered by each of t h e four e x p e r i m e n t s A L E P I I . D E L P H I . L.'i a n d O P A L . T h e L E P I r e s u l t s at t he Z r e s o n a n c e a r e c o m p l e m e n t e d by i m p o r t a n t c o n t r i b u t i o n s f r o m SLD a t t h e SLAO linear collider SIX', based on a s m a l l e r d a t a s c t o b t a i n e d f r o m highly polarized b e a m s .
CKNIIIU 200 ISO > A
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OPAL (161 GeV) all radiative (Z/y)* four fermion background
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120 Vs* (OcV) I ' t c . 0 . 3 . Distribution of t h e m e a s u r e d effective C'.o.M. e n e r g y v/* 7 of t h e h a d r o n i c s y s t e m , for e + e ~ collisions at liil G e V collected by t h e O P A L detector. Figure f r o m O P A L Collab.(1997«).
(i. 1.1.2 Deep inelastic lepton micleon scattering In d e e p inelastic s c a t t e r i n g experiments, t h e basic process is t h e s c a t t e r i n g of a lepton (electron, p o s i t r o n , union or n e u t r i n o ) off a c o n s t i t u e n t ( q u a r k or gluon) of a h a d r o n , which for e x a m p l e is given by a p r o t o n at l l l i l l A . T h e s c a t t e r i n g should occur with large m o m e n t u m t r a n s f e r from t h e lepton to t h e h a d r o n . T h e reaction leads to a b r e a k u p of t h e h a d r o n into a m o r e c o m p l i c a t e d h a d r o n i c final s t a t e . O u t of this general s a m p l e of e v e n t s t h o s e with p a r t i c u l a r k i n e m a t i c c o n f i g u r a t i o n s a r e selected f u r t h e r , for e x a m p l e , e v e n t s with e x t r e m e l y large m o m e n t u m t r a n s f e r , o r events w h e r e t h e h a d r o n i c final s t a t e is such t h a t a large a n g u l a r region between the o u t g o i n g h a d r o n s is c o m p l e t e l y d e p l e t e d of a n y h a d r o n i c activity, so-called rapidit ap or diffraclivc events, c.f. Section .'5.2.2. T h e basic event selection criteria a t I IKK A. where electrons or p o s i t r o n s of about :5() G e V collide with p r o t o n s of r o u g h l y 820 G e V or 920 G e V energy, a r c briefly described now. T h e trigger looks for e n e r g y d e p o s i t s of a t least ~ 5 G e V in t h e barrel or t h e b a c k w a r d c a l o r i m e t e r , which should s t e m from t h e electron scat tered a t large m o m e n t u m t r a n s f e r . Here ' b a c k w a r d ' refers t o t h e p a r t of t h e d e t e c t o r being in t h e lliglit direction of t h e electron, t h a t is. b a c k w a r d s to t h e direction of t h e incoming p r o t o n a n d m o s t of t h e p r o d u c e d h a d r o n i c final s t a t e . In o r d e r t o s t r e n g t h e n t h e h y p o t h e s i s , s o m e isolation criteria in t e r m s of energy d e p o s i t s a r o u n d t h e electron c a n d i d a t e a r e i m p o s e d . T h e offline selection t h e n applies m a n y m o r e c u t s in o r d e r t o e n s u r e high purity. T h e m a i n criteria a r e t h a t the energy of t h e s c a t t e r e d electron exceeds a threshold of a r o u n d 1 0 G e V . a n d that t h e r e is g o o d m o m e n t u m b a l a n c e between t h e electron a n d t h e h a d r o n i c
2:» I
0<'l> ANALYSES
s y s t e m , in o r d e r t o remove e v e n t s with initial s t a t e r a d i a t i o n . B a c k g r o u n d s arise f r o m interact ions o u t s i d e t h e d e t e c t o r , for e x a m p l e , b e a m g a s s c a t t e r i n g , which can be pinned d o w n by looking at t i m i n g i n f o r m a t i o n , a n d m a i n l y f r o m e v e n t s w h e r e t h e electron h a s escaped u n d e t e c t e d into t h e beam pipe, but a p h o t o n or a h a d r o n a c t u a l l y fakes t h e presence of a n energetic lepton in t h e d e t e c t o r . T h e l a t t e r is s t u d i e d by m e a s u r i n g t he n u m b e r of e l e c t r o n s at very small angles with d e d i c a t e d d e t e c t o r s f a r d o w n s t r e a m with respect t o t h e main d e t e c t o r , or by a n a l y s i n g s i m u l a t e d events. W h e r e a s t h e trigger efficiency is very high, close t o 100%. t h e variety of t h e a d d i t i o n a l offline analysis c u t s reduces t h e overall efficiency q u i t e considerably. T h e r e m a i n i n g b a c k g r o u n d s a m o u n t t o a few p e r cent.
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F i e . (i. I. A s a m p l e of l e a d i n g o r d e r F e y n m a n d i a g r a m s of p o s s i b l e p r o c e s s e s t o o c c u r in h a d r o n l i a d r o n collisions ( V = Z o r \ \ T ± )
l a g m u l t i - j e t e v e n t s . P u r e l y e l e c t r o m a g n e t i c e n e r g y d e p o s i t s , w h i c h in a d d i t i o n a r e well i s o l a t e d , a r e a t rigger for p h o t o n s o r l e p t o n i c d e c a y s of t h e Z a n d t h e W boson, w h e r e t h e l e p t o n w a s a n e l e c t r o n o r a p o s i t r o n . Well identified u n i o n s w i t h high t r a n s v e r s e m o m e n t u m a r e a f u r t h e r i n d i c a t i o n for Z a n d / o r W * p r o d u c t i o n . In t h e c a s e of t h e W a l s o m i s s i n g t r a n s v e r s e e n e r g y s h o u l d b e looked for, c a u s e d by t h e e s c a p i n g n e u t r i n o . T h e s e t r i g g e r s a r e e x t r e m e l y efficient, a n d in t h e c a s e of m u l t i - j e t e v e n t s , w h e r e t h e event r a t e s a r e very h i g h , t he t rigger h a s e v e n t o be p r e - s c a l e d . if t h e e n e r g y t h r e s h o l d is low. P r e - s c a l i n g by a f a c t o r of n m e a n s t h a t o n l y e v e r y n t h t r i g g e r is a c t u a l l y r e c o r d e d . P o s s i b l e b a c k g r o u n d s a r e t he following: p h o t o n s a n d e l e c t r o n s c a n be f a k e d by h a d r o n i c j e t s : l e p t o n s w i t h large t r a n s v e r s e m o m e n t u m a r e also p r o d u c e d in s e i n i - l e p t o n i c d e c a y s of h e a v y q u a r k s ; i m p e r f e c t i o n s o r m e a s u r e m e n t e r r o r s in i lie c a l o r i m e t e r c a n lead t o m i s s i n g t r a n s v e r s e e n e r g y . A g e n e r a l p r o b l e m for t h e q u a n t i t a t i v e u n d e r s t a n d i n g of all p r o c e s s e s is t h e u n d e r l y i n g e v e n t . C a r e f u l st udies a r e n e e d e d in o r d e r t o u n d e r s t a n d t h e a m o u n t of e n e r g y a n d m o m e n t u m p r o d u c e d in t h e u n d e r l y i n g e v e n t a n d t h e e x t e n t t o w h i c h il is m i s - a s s i g n e d t o t h e h a r d - s c a t t e r i n g final s t a t e a t l a r g e t r a n s v e r s e m o m e n t u m . 6.2
Observables
O b s e r v a b l e s which s h o u l d b e c a l c u l a b l e w i t h i n t h e f r a m e w o r k of p e r t u r b a t h e Q C D h a v e t o fulfill a b a s i c r e q u i r e m e n t : T h e y h a v e t o b e infrared safe. T h i s m e a n s t h a t , given a n y final s t a t e c o m p o s e d of q u a r k s a n d g l u o u s , t h e value of t h e o b s e r v a b l e d o e s not. c h a n g e if a very soft p a r t o n is a d d e d t o t h i s final s t a t e , o r if t h e a d d e d p a r t o n h a s its m o m e n t u m p a r a l l e l t o o n e of t h e o t h e r p a r l o u s . If t his is t rue, t h e n t h e cancellat ion of s o f t a n d c o l l i n e a r s i n g u l a r i t i e s is e n s u r e d . T h e s e s i n g u l a r i t i e s a p p e a r w h e n c a l c u l a t i n g r a d i a t i v e c o r r e c t i o n s via real a n d v i r t u a l p a r t o n e m i s s i o n . A m o r e d e t a i l e d d i s c u s s i o n of t h i s very i m p o r t a n t aspect, h a s been given in S e c t i o n 3.5.
(¿(•|) A N A L Y S E S
Nnvv sonic e x a m p l e s for observables a r c given. A first r a t h e r o b v i o u s ease is the total liadronic cross section in c + o a n n i h i l a t i o n . Here, t h e n u m b e r ol a c t u a l l y observed liadronic events h a s to be c o u n t e d , a n d t h i s c o u n t i n g is not affected by t h e a p p e a r a n c e of very soft or collinear particles in t h e final s t a t e . T h e very soft part icles m i g h t simply not b e resolved a t all. a n d two collinear ones would be c o u n t e d as a single particle with t h e m o m e n t u m given bv t h e s u m ol their m o m e n t a . In e + e ~ a n n i h i l a t i o n , r a t h e r t h a n looking at t h e a b s o l u t e total liadronic cross section, it is preferred to m e a s u r e t h e r a t i o /?, =
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a ( e + e ~ —• leptons) T h e elect,roweak cross section for e + e ~ a n n i h i l a t i o n into a q u a r k
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.
(6.2)
E IP. I i=i T h e s u m r u n s over t h e t h r e e - m o m e n t a of all in final s t a t e particles. T h e maximization is p e r f o r m e d with respect t o a unit vector i i . t h a t is. one looks for t h e direction for which t h e s u m of longitudinal m o m e n t u m c o m p o n e n t s is m a x i m a l . T h e vector nr. for which this m a x i m u m is o b t a i n e d , is called t h e Thrust axis of t h e event. It is a r a t h e r simple exercise t o show that t h e T h r u s t variable is infrared safe (Ex. (6-1)). For t h e simplest case of a final s t a t e with only two back-to-back particles we have T = 1. For t h e o t h e r e x t r e m e case, n a m e l y a perfectly spherical »//-particle c o n f i g u r a t i o n with in - » oc. t h e T h r u s t is '/' - 1/2.
OBHI'iUVAHUiS
2.17
So, for every event , a T h r u s t value between t h e s e two e x t r e m e eases will he meaKiiretl. A nice aspect, of e v e n t s h a p e variables is t h e fact that t h e s h a p e of t he i r d i s t r i b u t i o n is r a t h e r sensitive t o r a d i a t i v e corrections, such as gluon r a d i a t i o n oil q u a r k s . T h e r e f o r e , t h e y offer a h a n d l e for m e a s u r i n g t he s t r o n g co u p l i n g cons t a n t . as will be described in C h a p t e r 8. T h e basic idea is that, t h e larger t h e coupling is. t h e m o r e gluou r a d i a t i o n will o c c u r from t h e initial q u a r k a n t i q n a r k s y s t e m . Part, of this r a d i a t i o n will o c c u r at, large angles a n d with high energies, leading t o m o r e spherical r a t h e r t h a n pencil-like event co n f i g u r a t i o n s , a n d t h u s e n h a n c e t he Thrust, d i s t r i b u t i o n a t low T h r u s t values. Anot her class of very useful o b s e r v a b l e s a r e q u a n t i t i e s re l a t e d t o jet. p r o d u c tion. which m e a n s that t h e final s t a t e particles are not d i s t r i b u t e d uniformly over phase space, but r a t h e r colliinated wit hin b u n d l e s of particles, t he so-called jets. On t h e p e r t u r b a t i v c level t h e reason can be found in t h e c o m b i n e d effect of t h e particular b r e m s s t r a h l n n g s p e c t r u m which e n h a n c e s logarithmically t h e p r o b a bility t o observe low-angle r a d i a t i o n , a n d t h e b e h a v i o u r of t h e r u n n i n g coupling, which increases logarithmically with decreasing energy scales, giving a n additional effective enhancement, of multiple low-angle a n d low-energetic r a d i a t i o n . T h e jet. p i c t u r e on t h e p a r t o n level is n o t a l t e r e d d r a s t ically by hadroni/.ation effects. provided t h e overall energy scale is sufficiently high, let's say a b o v e HOGeV or so. T h i s is b e c a u s e t h e t r a n s v e r s e m o m e n t a , which a r e p r o d u c e d d u r i n g hadronization a n d t r a n s f e r r e d t o t h e h a d r o n s . c a n not be m u c h larger t h a n t h e typical overall hadroni/.ation scale, which is of t h e o r d e r of A ~ 200 — 301) MeV. These j e t s a r e indeed observed a t colliders. F i g u r e (i.I shows an event, display where t h e j e t st r u c t u r e of the o u t g o i n g particles is evident. Now t h e goal is t o give a definition of a jet in t h e f o r m of an a l g o r i t h m i c prescription, which can be i m p l e m e n t e d in t h e theoretical c a l c u l a t i o n s a s well as in c o m p u t e r p r o g r a m s used t o a n a l y s e t h e e x p e r i m e n t a l final s t a t e s . T w o classes of algorit h m s exist. At e + e " colliders j e t a l g o r i t h m s of t h e ade t y p e a r e usually employed, w h e r e a s a t p r o t o n m a c h i n e s t h e cone bused jet. finders a r e preferred. T h e .JADI2 c o l l a b o r a t i o n (1IJ86, 1088«) proposed the following scheme: First, one defines a resolut ion criterion or m e t r i c for two particles based on their fourvectors,
vis
where E , . E j a r e t h e particles' energies, c o s i s the angle between t h e m , a n d /•-Vis is t h e energy s u m over all particles in t he final s t a t e . N o t e t h a t for masslcss particles i/j- c o r r e s p o n d s t o t h e invariant, mass squared of t h e two particles, normalized t o t he t o t a l e n e r g y s q u a r e d . T h e n we c o m p u t e t h e d i s t a n c e s yf for all particle pairs (i.j) in t he final s t a t e . T a k i n g t h e smallest one. we next c o m p a r e it lo some predefined cut-off p a r a m e t e r y r l l t . If i/<-.,ii- ' h e n t h e a l g o r i t h m stops, a n d all t h e particles a r e defined as jets. However, if //;' y < j/ c l l l . t h e n the t w o particles /,„;„ a n d j, , a r e combined or clusteied to form a new pseudoparticle. T h e r e exist several prescript ions t o p e r f o r m t h e c o m b i n a t i o n of t h e two
<}( L> \ N A I . V S I ; S
f o u r - m o m e n t a . I'ur e x a m p l e , in t h e so-called ' -selu un . t h e f o u r - m o m e n t a are simply a d d e d , w h e r e a s in t h e u-sclutuc lirst t h e f o u r - m o m e n t a a r e a d d e d , lull then t h e new three-inomeiituiii is rosealed in o r d e r t o have a niassless p s e u d o particle. Alter this clustering s t e p t h e n u m b e r of (pseudo)partieles in t h e event is reduced by one. a n d t h e a l g o r i t h m s t a r t s a g a i n by recoui|)iiting all u l j of lli. newly delined final s t a t e . T h e iteration c o n t i n u e s until t h e s t o p p i n g criterion •'Ami,.*.....
>
i s
1 <,a( ,,<,<,
'
•
•
At L K I \ a slightly modified version of this a l g o r i t h m h a s been widely used, t h e so-called Durhnui clustering a l g o r i t h m (Dokshitzer, 1990). T h e difference can lie found in t h e definition of t h e resolution criterion, which here is 2inii^Ej-'. Uij =
'j)(l -pi
- cosfl,j) (h ' )
vis
For small angles this c o r r e s p o n d s t o t h e t r a n s v e r s e m o m e n t u m of t h e less energetic p a r t i c l e with respect to t h e higher energetic one. T h e D u r h a m scheme is preferred f r o m t h e theoretical viewpoint, since using this prescription it is possible to resinn leading l o g a r i t h m i c t e r m s t o all o r d e r s of p e r t u r b a t i o n t heory, s o m e t h i n g which is not possible with t h e .lade scheme ( C a t a n i . 1991). T h e idea of t h e cone j e t finder arises r a t h e r n a t u r a l l y f r o m t h e characteristics of h a d r o n collider e x p e r i m e n t s , w h e r e line-grained ealorinietry plays a m a j o r role a n d where one has t h e problem of t h e u n d e r l y i n g event. R e m e m b e r t hat t he energy How of t h e underlying event, is d i s t r i b u t e d uniformly in r a p i d i t y a n d in a z i m u t h . Section T2..'S. T h e r e f o r e , looking for j e t s m e a n s looking for enhancem e n t s in t h e e n e r g y How a b o v e this b a c k g r o u n d in c e r t a i n intervals of r a p i d i t y a n d a z i m u t h . T h e c a l o r i m e t e r s a r e usually c o m p o s e d of p r o j e c t i v e towers segm e n t e d iu a z i m u t h . A«I>, a n d p s e u d o r a p i d i t y . A//. w h e r e // is related t o the polar angle by ij = l n ( c o t ( 0 / 2 ) ) . First, preclusters (seeds) a r e found by s u m m i n g t h e t r a n s v e r s e e n e r g y in c o n t i g u o u s towers, r e q u i r i n g a c e r t a i n m i n i m u m energy per tower, typically a b o u t I G e V at t h e T K V A T R O N e x p e r i m e n t s . T h e transverse energy £//• is defined by f = sin . w h e r e is t h e tower energy. For each precluster having a t o t a l t r a n s v e r s e energy larger t h a n a fixed t h r e s h o l d , for e x a m p l e . 2 G e V . a centroid is c o m p u t e d by t r a n s v e r s e - e n e r g y weighting. T h e n , a cone of r a d i u s / i — ^/(A«!')- + (A//) 2 is f o r m e d a r o u n d t h e centroid of each precluster. Here A// (A) refer t o t h e difference in p s e u d o r a p i d i t y ( a z i m u t h ) between a tower location a n d t h e c e n t r o i d . A typical values is /? = 0.7. All towers a b o v e a t h i r d t h r e s h o l d , for e x a m p l e . 0 . 2 G e V . in t r a n s v e r s e energy inside t h e cone form a cluster: t h e centroid is t h e n recalculated, a n d a new cone is formed. T h e process is r e p e a t e d until t h e list of towers inside t h e cone r e m a i n s u n c h a n g e d in successive iterations. Finally, if t w o clusters overlap, t h e y a r e merged if either cluster s h a r e s m o r e t h a n 50 l /i of its e n e r g y with t h e o t h e r : if n o t . t h e y r e m a i n s e p a r a t e a n d towers c o m m o n t o b o t h a r e assigned t o t h e nearest cluster. T h e jet t r a n s v e r s e energy is o b t a i n e d as t h e scalar s u m of t r a n s v e r s e energies of t h e towers belonging t o t h e jet.. T h e j o t ' s r a p i d i t y a n d a z i m u t h a r e c o m p u t e d as
I IHNKUVAMI.KS
2.1!)
weighted s u m s of lower rapidities a n d a z i m u t h s , with t h e t r a n s v e r s e energy of t he tower a s weight. In c o n t r a s t t o t h e .Inde-type a l g o r i t h m s , in t h e cone j e t finder not all p a r t i cles (towers) a r e necessarily clustered into a j e t . T h e energy t h r e s h o l d s help t o reduce t h e c o n t r i b u t i o n from t h e u n d e r l y i n g event. It is w o r t h n o t i n g that, t h e cone a l g o r i t h m in its basic definition a c t u a l l y is not infrared sa fe in all o r d e r s of pert u r b a t i o n t heory. A d d i t i o n a l a l g o r i t h m i c s t e p s have t o be i n t r o d u c e d in o r d e r to avoid this p r o b l e m . A d e t a i l e d discussion can b e found in t h e l i t e r a t u r e (Seymour, 1998). Having finally defined our jet s, we can c o n s t r u c t m a n y i n t e r e s t i n g obscrvables: at c + c colliders we look at. tile two-, three-, lour-, live- a n d m o r e j e t r a t e s a s function of t h e resolution p a r a m e t e r j/tui- T h e smaller it is chosen, t h e m o r e j e t s will be resolved, until we e n t e r t h e non-perturbat.ive regime where we startto resolve single h a d r o n s . F u r t h e r m o r e , j e t p r o p e r t ies such a s p a r t i c l e or s u b jet. production within t h e j e t a r e s t u d i e d , a n d o n e looks for j e t variables which d i f f e r e n t i a t e b e t w e e n a j e t c o m i n g f r o m a q u a r k , a n d a jet o r i g i n a t i n g f r o m a gluon. At h a d ton colliders o n e of t h e m o s t i m p o r t a n t m e a s u r e m e n t s is t h e d e t e r m i nation of t h e inclusive t r a n s v e r s e energv s p e c t r u m of j e t s . T h e cross sect ion h a s been m e a s u r e d over m a n y o r d e r s of m a g n i t u d e , a n d t h e tails of t h i s d i s t r i b u tion. where j e t s with very large t r a n s v e r s e e n e r g y a r e p r o d u c e d , act ually allow to pose rat her stringent limits on potent ial d e v i a t i o n s f r o m t he S t a n d a r d Model, such as q u a r k s u b s t r u c t u r e . O t h e r typical obscrvables a r e di-jet. r a t e s a n d t h e a n g u l a r distribut ion of t h e jets, a s well a s t he r a t i o of t h e cross sect ions for t h e p r o d u c t i o n of a \V~ t o g e t h e r with one jet,, a n d t h e cross section for a \Y : plus t wo jets, which is a n o t h e r m e a n s for m e a s u r i n g t h e s t r o n g coupling c o n s t a n t . S t u d i e s of event s h a p e d i s t r i b u t i o n s a n d jet p r o d u c t i o n a r e useful in o r d e r t o test p e r t u r b a t i v e Q C D , b u t of course allow also t o investigate t h e hadronizat ion phase. Here o n e m a i n l y c o n c e n t r a t e s on t h e t e s t s of t h e phenoinenological m o d e l s available, such as. for e x a m p l e , t h e L u n d s t r i n g model, e.f. Section 3.8. T y p i c a l obscrvables a r e inclusive charged particle d i s t r i b u t i o n s such as t h e multiplicity or t lie m o m e n t u m dist r i b u t i o n , or t h e p r o d u c t i o n r a t e s of identified h a d r o n s . for e x a m p l e n e u t r a l pious or vector mesons. It is w o r t h n o t i n g t h a t m a n y o b s c r v a b l e s s t u d i e d in e + e ~ e x p e r i m e n t s a r e also being investigated in d e e p inelastic s c a t t e r i n g at H E R A . If t h e filial s t a t e is analysed in t h e rc.it frame of reference, see Ex. (6-3). a n d if only t h e particles from t h e current, h e m i s p h e r e a r e considered, a s indicated in Fig. F.5. then t h e r e is a very close c o r r e s p o n d e n c e t o a single h e m i s p h e r e in e + e _ a n n i h i l a t i o n . T h e current hemisphere is t h e h e m i s p h e r e of t h e s t r u c k q u a r k or gluon. whereas the target h e m i s p h e r e refers t o t h e region where t h e p r o t o n r e m n a n t c a n be f o u n d . A Rreit f r a m e a n a l y s i s in d e e p inelastic s c a t t e r i n g has t h e a d v a n t a g e that Unavailable e n e r g y is variable, w h e r e a s in e 1 e a n n i h i l a t i o n it. is fixed. So within a single e x p e r i m e n t t h e energy d e p e n d e n c e of t h e obscrvables m e n t i o n e d a b o v e c a n be looked a t over a wide e n e r g y range.
TJCU VNAI/YSKS
(>..'{
Corrections
P e r f o r m i n g a m e a s u r e m e n t involves a r a t h e r i n t r i c a t e chain of d e t e c t o r s a n d analysis steps. As a consequence the result will b e a somewhat, d i s t o r t e d image of t he physical t rut h. a n d usually considerable effort goes into t h e u n d e r s t a n d i n g a n d u n d o i n g of these d i s t o r t i o n s . (¡..'5.1
Detector
collections
Every d e t e c t o r h a s linite resolution a n d a c c e p t a n c e . T h i s m e a n s t h a t we m e a s u r e t h e m o m e n t a a n d energies within s o m e u n c e r t a i n t y range, a n d we lose particles because t h e d e t e c t o r is n o t a b l e t o d e t e c t t h e m , a s is t h e c a s e for n e u t r i n o s , or because it is not perfect ly h e r m e t i c . At collider e x p e r i m e n t s , where t h e sensitive region only s t a r t s a b o v e s o m e m i n i m u m polar angle, t h e r e a r e unavoidable losses of tracks into the b e a m pipe, a n d t h e r e a r e losses because of regions in t h e d e t e c t o r which a r e not e q u i p p e d with sensitive devices, such a s t r a n s i t i o n regions between c e n t r a l a n d f o r w a r d d e t e c t o r c o m p o n e n t s . However, if we know our d e t e c t o r very well, t h a t is. if we know which regions of t h e d e t e c t o r cannot m e a s u r e a n y t h i n g , a n d if we know w h a t t h e resolution of o u r m e a s u r i n g device is, then we can correct for these d e t e c t o r effects. As a simple e x a m p l e , consider a n e x p e r i m e n t where we would like t o m e a s u r e t h e decay r a t e of s o m e r a d i o a c t i v e source. I m a g i n e we would like to m e a s u r e t h e n u m b e r of o decays per second. Now s u p p o s e t h a t we were a b l e t o build a perfectly spherical d e t e c t o r which is a b l e t o record every o particle passing t h r o u g h it. a n d we put t h e s o u r c e a t t h e c e n t r e of t h e d e t e c t o r . U n f o r t u n a t e l y , because of bad m a n u f a c t u r i n g , a q u a r t e r of t h e d e t e c t o r s t o p s f u n c t i o n i n g a t tinearly s t a g e of t h e e x p e r i m e n t . T h i s m e a n s t h a t from now on we have to correct for its linite a c c e p t a n c e , t h a t is, in o r d e r to get t h e t o t a l n u m b e r of d e c a y s A',,. the a c t u a l l y m e a s u r e d n u m b e r A/ m ,. as h a s t o be multiplied by a f a c t o r 1/3. since only 3/i- If this is t h e case, we will miss s o m e c o u n t s b e c a u s e t hey a r e t o o close t o o n e - a n o t h e r a n d we have t o correct for this when giving o u r final result. Of course, collider e x p e r i m e n t s a r e much m o r e c o m p l e x , but t h e basic ideas r e m a i n t h e s a m e . Very e l a b o r a t e c o m p u t e r m o d e l s of t h e d e t e c t o r a r e built, which then deliver (in simple words) t h e fraction of t he solid a n g l e which is e q u i p p e d with sensitive d e t e c t o r s , a n d t he resolution of t h e energy, m o m e n t a a n d t i m i n g m e a s u r e m e n t s . For e x a m p l e , in e + e _ a n n i h i l a t i o n s first h a d r o n i c final s t a t e s a r e g e n e r a t e d by s o m e M o n t e C a r l o model, a n d t h e d i s t r i b u t i o n Df " of some variable is c o m p u t e d f r o m t h i s final s t a t e . Technically, o n e usually a p p r o x i m a t e s t he d i s t r i b u t i o n by a h i s t o g r a m , a n d t h e index i i n t r o d u c e d a b o v e refers to t h e i t h bin of t h e h i s t o g r a m . T h e n all particles of these final s t a t e s a r e passed t h r o u g h t h e d e t e c t o r s i m u l a t i o n . At t h e e n d . o n e o b t a i n s a set of m e a s u r e m e n t s which should resemble a real set of m e a s u r e m e n t s , a n d from t h e s e new filial s t a t e s a d i s t r i b u t i o n D*"" is o b t a i n e d . For t h e sake of simplicity, a s s u m e t h a t t h e s a m e
I
binning a s before is used computed according to
I >1(111
1
I K
2\I
IINIL
Now. tin- so-called bin-hy-hin correction f a c t o r s a r e DT" c
r
=
~
•
(«••"»
Knowing t h e s e correction factors, the m e a s u r e d d i s t r i b u t i o n D"wns can he corrected for d e t e c t o r effects in o r d e r t o o b t a i n t h e true d i s t r i b u t i o n D ' " r r . D
corr
=
qlct
D
m«ns
((¡.(¡)
fliis final d i s t r i b u t i o n can bo c o m p a r e d t o theoretical c a l c u l a t i o n s or t o results from o t h e r e x p e r i m e n t s . Since those usually have different resolution a n d acceptance. it. would be meaningless to c o m p a r e directly m e a s u r e d d i s t r i b u t i o n s pnw» In this simple bin-bv-bin a p p r o a c h it is a s s u m e d that t h e r e is only a weak d e p e n d e n c e on t h e M o n t e C a r l o g e n e r a t o r used t o o b t a i n D*'". O n e must also take care t h a t t h e binning is in a c c o r d a n c e with t h e d e t e c t o r resolution, that is. d e t e c t o r effects should not i n d u c e t o o m u c h m i g r a t i o n f r o m bin i t o bin j , i •f- j. If m i g r a t i o n effects a r e i m p o r t a n t , t h e n a m o r e s o p h i s t i c a t e d solution is needed, based on t h e c o n s t r u c t i o n of a t r a n s i t i o n m a t r i x M f j ' . which relates Dj'" t o Dfm a c c o r d i n g t o Df'" = M'^f Df". T h e p r o b l e m is t h a t , in o r d e r t o get rr 1 ¿)' l " , a s i m p l e inversion of A/,''"' o f t e n leads t o u n s t a b l e results, so t h a t m o r e advanced m e t h o d s have to be applied in o r d e r to unfold t h e d e t e c t o r effects. We cannot go into t h e discussion of this topic here, but refer t h e r e a d e r t o t h e l i t e r a t u r e ( C o w a n . 1998; Sehmolling, 199-1). l>.:{.2 Hadronization
correct ions
If t h e d i s t r i b u t i o n Df"" i n t r o d u c e d a b o v e is o b t a i n e d with s o m e M o n t e C a r l o model which g e n e r a t e s h a d r o n i c final s t a t e s , o u r corrected d i s t r i b u t i o n D,('<"'1' c o r r e s p o n d s t o t h e m e a s u r e d d i s t r i b u t i o n of s o m e q u a n t i t y as o b t a i n e d f r o m a linal s t a t e built of h a d r o n s . Usually t h e e x p e r i m e n t s publish their results in this form, t h a t is. corrected t o t he hadron level, a s is t h e jargon. However, in o r d e r t o test predictions from p e r t u r b a t i v e ( ¿ C D . or t o d e t e r m i n e q u a n t i t i e s which e n t e r the p e r t u r b a t i v e predictions, first t h e s e predictions have t o lie corrected for t h e t r a n s i t i o n f r o m p a r t o n s t o h a d r o n s , called hadronization or fragmentation, and only then they can be c o m p a r e d t o D"'". T h e size of t h e h a d r o n i z a t i o n effects d e p e n d s very much on t h e observable a n d on I he overall energy scale Q. G e n e r a l l y speaking, t h e size of t h e correct ions decreases a c c o r d i n g t o s o m e inverse power of (}. Very well resolved a n d energetic jets, as well as t h e mult i-jet regions of event s h a p e variables, a r e less alfected t h a n for e x a m p l e close-by a n d low-energetic jets. S o m e q u a n t i t i e s can c h a n g e drast ically, such as total particle multiplicities. T h e solut ion of t h i s p r o b l e m is a g a i n o b t a i n e d bv r u n n i n g M o n t e C a r l o prog r a m s which lirst g e n e r a t e p a r t o n i c final s t a t e s a c c o r d i n g to exact or a p p r o x i mate perturbative Q C D , and then simulate the transition to hadrons according t o some phenoincnological models. T h e basic ideas of such p r o g r a m s have been
g c i ) \NAI.YNISH
'.'. i 2
4
A
L = In.r, FIG. 0 . 5 . H a d r o n i z a t i o n c o r r e c t i o n s for t h e event, siiape variable y-j, measured iu e + e ~ a n n i h i l a t i o n s a t two different C.o.M. energies. Figures f r o m ALKIMI Collab.(1997a).
described in C h a p t e r I. T h e n t h e a p p r o a c h is very similar t o t h e d e t e c t o r correction p r o c e d u r e described a b o v e . Analysing t h e p a r t o n ( h a d r o n ) level, we o b t a i n d i s t r i b u t i o n s Df" (£>J ,a
^ h . u l ¿JQCD
^
^QCD.corr
=
^«IFLQ™ .
£
(6
.7)
3 C a r e has t o be t a k e n if t h e c o r r e c t i o n s d e v i a t e t o o m u c h f r o m unity. It is clear t h a t a n o b s e r v a b l e which h a s h a d r o n i z a t i o n correct ions of. for e x a m p l e . 5 0 % will d e p e n d a lot on t h e a c t u a l h a d r o n i z a t i o n model applied, a n d it gets m o r e a n d more dillicult t o e x t r a c t reliably t h e p r o p e r t i e s of t h e p a r t o n i c s y s t e m . As a n e x a m p l e , at I.Kl' it is typically checked t h a t t h e h a d r o n i z a t i o n corrections d o not exceed t h e 10% level, if precise m e a s u r e m e n t s of basic q u a n t i t i e s such as n s a r e t o be o b t a i n e d . Figure 0.5 shows t h e c o r r e c t i o n f a c t o r CJ"" 1 for t h e event s h a p e variable 1/3. T h i s variable is found by a p p l y i n g t h e D u r h a m clustering a l g o r i t h m t o t h e final s t a t e ( p a r t o n s or h a d r o n s ) . until only t h r e e j e t s a r e left. T h e n 1/3 is given by //.( = mill//¡J. as described in Section 0.2. It d e t e r m i n e s t h e resolution value for which this final s t a t e would t r a n s i t f r o m a three- t o a twojet. c o n f i g u r a t i o n . T h e region of small values is p o p u l a t e d by two-jet like events, w h e r e a s t he region of large »/.•», o r equivalent ly small values of — In
S \ H I I M A I II • I'Nl 'KIM AIN I ICS
•Jl.'l
by. It can a l s o l>c o b s e r v e d t h a i I lie c o r r e c t i o n s slightly d e c r e a s e w h e n going t o a larger C . o . M . energy.
.1
y s t e m a t i c uncertainties
The t o p i c of s y s t e m a t i c u n c e r t a i n t i e s is a very dilficult o n e . a n d since t h e r e is s o m e lack of g e n e r a l a n d o b j e c t ive rules, it. is not a l w a y s free of cont roversy. T h e s o u r c e s of s y s t e m a t i c u n c e r t a i n t i e s d e p e n d very m u c h on t h e a n a l y s i s , a n d since it is p r a c t i c a l l y i m p o s s i b l e t o give a n e x h a u s t i v e discussion here, we will r a t h e r try t o o u t l i n e s o m e basic a s p e c t s t o b e c o n s i d e r e d . G e n e r a l l y , we c a n d i s t i n g u i s h b e t w e e n t wo large classes of u n c e r t a i n t i e s , t he e x p e r i m e n t a l s y s t e m a t i c u n c e r t a i n t i e s a n d e r r o r s r e l a t e d t o t h e t h eo r y . S y s t e m atic u n c e r t a i n t i e s b e c a u s e of t h e e x p e r i m e n t a l p r o c e d u r e arise b e c a u s e in f a c t we never u n d e r s t a n d perfect ly well o u r d e t e c t o r , a s we h a v e a s s u m e d in Section (i.'.i. T h i s m e a n s t h a t t h e M o n t e C a r l o s i m u l a t i o n of t h e d e t e c t o r , which we use t o e s t i m a t e a c c e p t a n c e c o r r e c t i o n s or t h e r e s o l u t i o n , is not. p e r f e c t . An easy check of t he s t a b i l i t y of a result is o b t a i n e d by a p p l y i n g d i f f e r e n t c u t s in o r d e r t o select e v e n t s a n d / o r p a r t i c l e s . If t h e d e t e c t o r s i m u l a t i o n would b e e x a c t , t h e n every set of c u t s s h o u l d lead t o t h e s a m e final result., since t h e c h a n g e in t h e event s a m p l e would b e t r a c k e d by a c h a n g e in t h e c o r r e c t i o n f a c t o r s . However, if t h e final result is not i n v a r i a n t , t h e n we h a v e a hint t h a t s o m e of t h e q u a n t i t i e s , w h e r e we a p p l y c u t s o n . a r e not. d e s c r i b e d c o r r e c t l y by t h e s i m u l a t i o n . As a trivial e x a m p l e , a s s u m e t h a t in t h e M o n t e C a r l o m o d e l t h e s e n s i t i v e region of t h e d e t e c t o r s t a r t s a t a p o l a r a n g l e 0MC\ w h e r e a s in t h e real d e t e c t o r it. s t a r t s only at > 0 M O If w e t h e n use s o m e overall e v e n t axis, such a s t h e T h r u s t axis, to d e f i n e t h e o r i e n t a t i o n of a n event., a n d a p p l y a n a c c e p t a n c e c u t o n t h e p o l a r a n g l e 0-ni of t h i s axis t o e n s u r e t h a t t h e event is well c o n t a i n e d w i t h i n t h e d e t e c t o r . o u r a c c e p t a n c e c a l c u l a t i o n will be correct a s long a s 0\-\\ > 0p|.;. However, if 0TH < 0|JB« o u r M o n t e C a r l o s i m u l a t i o n will predict a w r o n g a c c e p t a n c e , a n d t h e final result will c h a n g e . Anot her critical aspect of a n a n a l y s i s is t h e g o o d k n o w l e d g e of t h e d e t e c t o r resolut ion, such a s t h e jet, e n e r g y r e s o l u t i o n , a n d t h e overall e n e r g y scale. A g o o d underst a n d i n g of t h e l a t t e r c a n b e part icularly i m p o r t a n t for p r o c e s s e s w i t h cross sect ions t h a t fall s t e e p l y a s a f u n c t i o n of (t r a n s v e r s e ) jet. e n e r g y . O f t e n i n d e p e n dent p r o c e s s e s with respect t o t h e o n e a c t u a l l y u n d e r s t u d y c a n b e e m p l o y e d t o e s t i m a t e t h e e n e r g y scale a n d resolution, if t h e y offer s o m e physical c o n s t r a i n t such a s e n e r g y b a l a n c e or m a s s c o n s t r a i n t s . T h e final s y s t e m a t i c u n c e r t a i n t y on t h e original m e a s u r e m e n t m i g h t only b e limited by t h e stat ist ical u n c e r t a i n t y of t h e cross check analysis. T h e o r e t i c a l u n c e r t a i n t i e s s i m p l y a r i s e b e c a u s e of limited t h e o r e t i c a l knowledge. As an e x a m p l e , in Fig. (i.5 t h e hadroni/.at ion c o r r e c t i o n s for a n event s h a p e variable a r e p l o t t e d , a s p r e d i c t e d bv t h r e e different m o d e l s for t h e h a d r o n i z a t i o n process. Since we a r e still not a b l e t o c o m p u t e t h i s m e c h a n i s m f r o m lirst, principles. every m o d e l which is a b l e t o give a g o o d overall descript ion of t h e p r o p e r t ies of h a d r o n i c final s t a t e s h a s t o b e c o n s i d e r e d for t he c o m p u t a t i o n of h a d r o n i z a t -
<}CI> A N A L Y S E S
ion corrections. However, as we can see from Fig. (i.5, tliere a r e some differences in I lie p r e d i c t e d e o r r e e t i o n s . wliieli in ilie e n d can lead t o differences in some measured (|iiantilies such as o s . T h i s kind of u n c e r t a i n t y is called hadrouizatiim unccrtainti . A n o t h e r typical e r r o r source is a limitation of c a l c u l a t i o n s wit hin t he framework of p e r t u r b a t i v e Q C D . Every calculation h a s t o s t o p at s o m e o r d e r in tin' expansion p a r a m e t e r , which usually is given by o s . T h e p r o b l e m is t h a t this t r u n c a t e d p e r t u r b a t i o n series d e p e n d s on t h e unphysical r e n o r i u a l i / a t i o n scale, e.f. Section H.-l, a n d in principle every choice for this scale is equally well jusli lied. Different choices will lead t o different linn! answers, such a s different values for o s . As s y s t e m a t i c u n c e r t a i n t y o n e usually defines t h e variation of t h e liual results u n d e r v a r i a t i o n s of t h e r e i i o r m a l i / a t i o u scale over s o m e reasonable range. W h a t is "reasonable' is a question very m u c h u n d e r d e b a t e . As a filial warning it should be m e n t i o n e d t h a t usually t h e s y s t e m a t i c uncertainties a r e t r e a t e d a s if they h a d t h e s a m e m e a n i n g a s a statistical error. However, b e c a u s e of t h e a r b i t r a r i n e s s of s o m e of t h e e v a l u a t i o n p r o c e d u r e s , or simply because of t h e intrinsic non-probabilistic nat ure of certain s y s t e m a t i c uncertainties, t h e i n t e r p r e t a t i o n of specific results sh o u l d b e t a k e n with care. T h i s is p a r t i c u l a r l y i m p o r t a n t w h e n d e v i a t i o n s f r o m t h e e x p e c t a t i o n a r e observed, a n d if t h e significance of these d e v i a t i o n s is m e a s u r e d with r e s p e c t t o t h e q u a d r a t i c s u m of s t a t i s t i c a l a n d s y s t e m a t i c u n c e rt a i n t i e s .
1.
Examples
In the following sections we will give s o m e c o n c r e t e e x a m p l e s of Q C D measurements. We will d e s c r i b e in m o r e detail t h e main s t e p s t o b e t a k e n , t h e p r o b l e m s which a r c typically e n c o u n t e r e d a n d how t h e y a r e solved. It should b e clear to i In- reader that only a r o u g h sketch of an a n a l y s i s c a n be given, a n d t h a t a detailed description of all t h e a s p e c t s , such a s d e t e c t o r specific topics, a n a l y s e s of s y s t e m a t i c u n c e r t a i n t i e s or s u b t l e t i e s of t h e theore t i c a l f r a m e w o r k , would go far beyond t he scope of t his book. (j.5.1
Structure,
function
measurement
at IIEHA
In the first e x a m p l e , we will discuss t h e m e a s u r e m e n t of t h e s t r u c t u r e function F,(r. f r o m d e e p inelastic electron p r o t o n s c a t t e r i n g a t t h e HERA collider. An in-depth discussion of d e e p inelastic s c a t t e r i n g can b e found iu Section 15.2.2. In Fig. 2.1 we have d e p i c t e d t h e basic F e y n m a n d i a g r a m for t h e d e e p inelastic s c a t t e r i n g process. W h a t is shown is only t h e e x c h a n g e of a virt ual p h o t o n , since we will delimit our discussion t o m o m e n t u m t r a n s f e r s Q~ = —(I — I ' ) 2 {.r.(J-) and Fi.(x, 2).
d
^
=
[ 0 + (1 - -
2
)
- U2 F
2
)]
(0.8)
I X A M I ' I I:s
2 I.'.
where n,.,,, is t h e fine s t r u c t u r e c o n s t a n t . a n d /•'/. /'V 2o:f'\ is t h e longit udinal nlrnctiiro f u n c t i o n . As a rcinindcr. t h e k i n e m a t i c variables .1:. B j o r k e n ' s scaling variable, a n d y a r e defined a s Q2 2p •
i>'i (J
P •
„.„. I
(I < j r . y < I. W h e n neglecting particle masses we find s = (/» H- /)"' — 2/> • I for the C . o . M . energy s q u a r e d a n d Q2 = s.ry. Now let us c o n c e n t r a t e on t h e kineinatical s i t u a t i o n a t HICK A. T h e r e protons with a n energy of Ev = 820CJeV (later 9 2 0 G e V ) collide with electrons or positrons of E,. = 27.5 G e V . L e t ' s choose a f r a m e such t h a t /> (E,,.(), 0. Ep) a n d / ( E . . . 0 . 0 . -E,.). t h a t is, t h e p r o t o n s a r e moving in t h e positive ¿-direction. The C.o.M. energy is given by , / s = ^ / I E , , E r w 3 0 0 G e V . Since t h e m o m e n t a are highly u n b a l a n c e d , t h e C . o . M . s y s t e m is m o v i n g with r e s p e c t t o t h e laboratory s y s t e m in positive ¿-direction with a velocity (3 = v/c = |Ptotl/-^toi [Ev E,.)/(£,, I- Ec) « 0.935. T h i s h a s an i m p o r t a n t impact, on t h e d e t e c t o r design. In fact, t h e d e t e c t o r layout, is a s y m m e t r i c with respect t o z. In t h e forward (positive ¿) direction t h e d e t e c t o r is more densely i n s t r u m e n t e d t h a n in t h e backward direction, since m o s t of t h e t i m e a big fraction of t h e o u t g o i n g h a d r o n i c system will m o v e i n t o t h e f o r w a r d h e m i s p h e r e u n d e r r a t h e r small angles with respect to t h e b e a m pipe. In t h e b a c k w a r d direction a d e d i c a t e d c a l o r i m e t e r is installed in o r d e r t o m e a s u r e precisely t h e s c a t t e r e d electron or p o s i t r o n . T h i s most o f t e n is f o u n d at large angle, 0C > 150°, with respect t o t h e i n c o m i n g proton, t h a t is, low angle with r e s p e c t t o t h e b e a m line in n e g a t i v e ¿-direction. T h e kinemat ic variables can b e r e c o n s t r u c t e d in t e r m s of t h e m e a s u r e d energy and direction of t h e o u t g o i n g electron, w i t h o u t a n a l y s i n g t h e h a d r o n i c s y s t e m . T h i s follows from ( f = 21 • I' = 2E,.E;. (I -I- cos«,.) = Ql , V • I' y = 1- '—- = 1 -
K
(1 - cosfl t .) = yv .
(0.10) (0.11)
when neglecting particle masses. T h e index e indicates t h a t t h e variables a r e r e c o n s t r u c t e d with t h e so-called clr.c.tnm mc.l.liotl. W h e n expressing Ev in t e r m s of yt. we find Ql E'2 sin" I),. ~ /j?sin2flc 0.12 Qc = . ''o = — = — - T . r • !-?;«•
"Uu
sj/c(l-2/c)
From these expressions it is clear t h a t , before a p p l y i n g a n y e v e n t selection c u t s , t lie accessible r a n g e in Q2, x a n d y is limited by t he a n g u l a r a c c e p t a n c e of t he detector and the energy thresholds. In o r d e r to get a feeling for t h e n u m b e r s involved, let's t a k e a few e x a m p l e s . To start w i t h , a s s u m e that t h e electron is found in t h e b a c k w a r d c a l o r i m e t e r at 0,. = 160°, wit h E'c = 1 G e V . T h e n yc = 0,965, Q2 = 3-32 G e V 2 a n d xc = 3.8 x 10" r '.
I /I I' /\l\/\l,l .11',.-I
¿•ILL
I
10
I0:
IO4
It)'
io2
io'
((¡cV-)
(GcV-)
FIG. (¡.(i. K i n e m a t i c d o m a i n in t h e (Q2, x) p l a n e : (a) lines of constant E'.. 0r a n d //: (l>) lines of c o n s t a n t E,r (),, a n d //. w h e r e E,, a n d (),, a r e t h e e n e r g y a n d angle of t h e s c a t t e r e d q u a r k in t h e part,oil m o d e l . F i g u r e s f r o m Clerl>aux(1998).
W h e n going t o a higher energy, say E'c = l O G e V . we find //,. = 0.617, Q2 33.2 G c V * a n d x,. - 5.7 x I O - 1 . Finally, when E'e » Ee, t h e values yc = 0.03, Ql = 9 1 . 2 G c V 2 a n d xe = 0.034 a r e f o u n d . In Fig. G.G(a) t h e k i n e m a t i c d o m a i n in t h e ( Q 2 , x ) p l a n e is shown ( C l e r b a u x , l!)i)8). It is evident t h a t for i / £ 0 . 1 lines of c o n s t a n t Q 2 c o r r e s p o n d t o c o n s t a n t (),.. a n d ;/ is basically d e t e r m i n e d by t h e energy of t h e s c a t t e r e d electron for E'c < 2 7 G e V . An i m p o r t a n t f e a t u r e of d e e p inelastic s c a t t e r i n g within a h e r m e t i c collider e x p e r i m e n t is t h a t also t h e h a d r o n i c final s t a t e c a n l>e used for t h e d e t e r m i n a t i o n of t h e k i n e m a t i c variables. However, since a c o m p l e t e l~ a n g u l a r coverage is impossible to achieve, a n d since h a d r o n i c a c t i v i t y o f t e n o c c u r s at r a t h e r low angle, it is p r e f e r a b l e t o find variables which a r e r a t h e r insensitive t o losses into t h e b e a m p i p e a l o n g t h e p r o t o n direction. T h i s is achieved, for e x a m p l e , by t h e variable E =
- />;.„) = £ it
«i
E„ (1 - e o s 0 „ ) ,
(6.13)
where t h e s u m r u n s over all h a d r o n s u found in t h e final s t a t e . H a d r o n masses a r e neglec1<>d for t h e second equality. It is e v i d e n t t h a t this variable is r a t h e r insensitive t o particles lost a t very low angles with respect t o t he p r o t o n d i r e c t i o n , since these a n y w a y would not c o n t r i b u t e t o t h e s u m . It w a s first, p r o p o s e d by 2 •lacqnet a n d Blondel (1979) t o express (/ a n d in t e r m s of
y
=
2EV
=
yu.
Q- =
1 - i/i,
=
•
(fi
l 4
)
where P _ m = E „ s h \ 0 „ is t h e t r a n s v e r s e m o m e n t u m of t h e h a d r o n i c s y s t e m , a n d t h e s u b s c r i p t h i n d i c a t e s that t h e observables a r e c o n s t r u c t e d f r o m h a d r o n i c
EXAMPLES
'.! 17
variables. T h e derivation of these f o r m u l a s is discussed in fix. (I) I). There it is also shown that, the h a d r o n a n g l e defined as Pco
*
0h
= Pll
- E-
+
tf
(,Ur')
c o r r e s p o n d s to t h e angle of t h e struck q u a r k in t h e p a r t o n model. In Fig. (¡.(¡(Ii) again tin- k i n e m a t i c d o m a i n in Q ' a n d .r is shown, bill now for lines of constant f?l, a n d constant energy of t h e s t r u c k q u a r k a s derived in Ex. (6-4). A third m e t h o d for t h e k i n e m a t i c r e c o n s t r u c t i o n can b e employed in o r d e r to avoid t h e following p r o b l e m . F r o m equ (6.14) we see that I/I, d e p e n d s on E,.. Using t h e b e a m energy for /?,. is correct only as long as t h e incoming electron does not r a d i a t e p h o t o n s before u n d e r g o i n g I IK- a c t u a l h a r d i n t e r a c t i o n with t h e p r o t o n . If instead t h e r e is a large amount, of init ial s t a t e p h o t o n r a d i a t i o n (ISR), i lie relevant electron energy can lie rex I need s u b s t a n t i a l l y below t h e b e a m energy, and we would m a k e a large m i s t a k e in t h e d e t e r m i n a t i o n of ;/i,. T h i s effect, c a n be largely reduced by using t h e Siymn method (Bassler a n d B c r u a r d i . 15)95) for the r e c o n s t r u c t i o n , w h e r e E .'/>: =
, ri/. E + E{.(\
v
E'~ sin- 0,. g-r, - c.osOv)
1-2/s
6.16)
Again we refer t o Ex. (6-4) for a derivation of these f o r m u l a e . So far we have a s s u m e d that we c a n d e t e r m i n e the' k i n e m a t i c q u a n t i t i e s of t h e event with a r b i t r a r y precision. Of course this is not t h e case. In fact, for t h e choice of t h e final event selection c u t s t h e d e t e c t o r resolution plays an i m p o r t a n t role. T a k e for e x a m p l e t h e m e a s u r e m e n t of ;/. In t h e electron a n d t h e .facquet Blondcl m e t h o d s we find for t h e resolution ol'i/. for lixed 0 r i /ii/,. 1 — i/t. SE'. — = — ~FT ' .'/,• Ve E-.
Si/1, (iE — = v". Uu E
(6.17)
In t h e lirst. m e t h o d , t h e resolution on //,. is deterniined by t h e resolution of t h e energy m e a s u r e m e n t for t h e s c a t t e r e d elect ron a n d t h e a c t u a l value of //,.. At large ;/,. t he resolution is very g o o d , however, at low y t . it diverges. Th<; second m e t h o d docs not sillier f r o m t his p r o b l e m . Here, only t he resolut ion of t h e h a d r o n i c energy nieasurement. is of importance-. T h e s e f a c t s a r e illustrated in Fig. 6.7. w h e r e a M o n t e C a r l o s i m u l a t i o n of t h e III d e t e c t o r h a s been used (Gla/.ov. 1!)!)S) in o r d e r to show t h e t r a n s i t i o n f r o m a t r u e ( g e n e r a t e d ) t/ to a m e a s u r e d //,. (a) o r i/u (b) b e c a u s e of di-tcctor resolut ion effects. T h e d e g r a d a t i o n of t h e resolution on j/t. follow // is clearly visible, w h e r e a s it is r a t h e r c o n s t a n t over y w h e n t a k i n g yu- At large y t h e second m e t h o d is worse b e c a u s e of I he intrinsic lower resolut ion on t h e h a d r o n i c energy c o m p a r e d t o t h e elect ron energy. T h e s e p l o t s s u g g e s t t o c o m b i n e t h e two m e t h o d s for a best possible d e t e r m i n a t i o n of t h e k i n e m a t i c variables over a large a r e a in t h e ( Q 2 . . r ) plane. Namely, t a k e t h e electron m e t h o d for large //,
(}CH ANALYST'S
05 .ü S' 6
r
0.5 -I 10
10
10
F u ; . (¡.7. Resolution of >/,. (left) a n d
say a,- > ()••'{, t h e .lacquet Bloudel o r Sigma m e t h o d for t h e low ;/ region. a n d restrict t h e overall event selection t o regions with a c c e p t a b l e resolution. Finally t h e ( Q ~ . x ) p l a n e is divided into bins, t h e sizes of which a r e chosen such t h a t t h e bin-to-bin m i g r a t i o n from t r u e to r e c o n s t r u c t e d values because of t h e d e t e c t o r resolution is kept small. A f t e r a basic e v e n t selection such a s described in Section (i. 1.1.2, t h e d e t e r m i n a t i o n of t h e k i n e m a t i c variables, f u r t h e r event selection r e f i n e m e n t s a n d t h e choice of t h e r a n g e a n d b i n n i n g in Q~ a n d x, we can proceed t o t h e linal cross section m e a s u r e m e n t . For every bin ( L j ) in (Q , x) we have yilatii _
d'-V d.wlQ' 2
,yl>k«
•j
II. TC A , ' i s t h e n u m b e r ol m e a s u r e d e v e n t s in t h e relevant bin. a n d A , is t h e n u m b e r of e x p e c t e d b a c k g r o u n d events, usually d e t e r m i n e d f r o m M o n t e C a r l o simulations. Possible b a c k g r o u n d s have been described in Section (¡.1.1.2. T h e factor C,j s t a n d s for t h e a c c e p t a n c e correction a n d is also d e t e r m i n e d f r o m a M o n t e C a r l o s i m u l a t i o n . It is i m p o r t a n t here t h a t t h e simulation r e p r o d u c e s well t h e d e t e c t o r a c c e p t a n c e a n d resolution. F u r t h e r m o r e , it has t o be checked t h a t t h e correction f a c t o r s d o not d e p e n d on t h e choice of a priori s t r u c t u r e f u n c t i o n s needed for t h e M o n t e C a r l o generat ion of s i m u l a t e d events. T h e luminosity C is d e t e r m i n e d in t h e following m a n n e r . A process is taken for which t h e cross section is very well known theoretically, a n d t h e e q u a t i o n a
=
(Af < l u , n - iV l , k R )f C .
((¡.19)
is inverted w i t h respect t o £ . Again A f , l a , a ' l , k 8 ' is t h e n u m b e r of observed (expected b a c k g r o u n d ) events, a n d < t h e efficiency t o collect such events. T h e two l a t t e r n u m b e r s a r e t a k e n f r o m s i m u l a t i o n s a n d / o r a u x i l i a r y m e a s u r e m e n t s . At I IKK A a s u i t a b l e process is given by t h e B e t h e lleitler B r e m s s t r a h l u n g process
I X A M l'I.ICS
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5
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1.5
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t
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•
Q 2 = 20
.
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t
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eft
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S ,•
iituul. miJl.. until mud : Hill 0 unji .uni iJiadLxuiaiLiiiiJi I uiuJ . iiiLul amai : liiuJ liti-. liuti mull ii.uJ i litui i .uu 4 1 4 1 : 1 4 : 1 10 10 10 - 10 ' 10 10 10 10 10 10 10 10 10 4 10 10" 10 1 FIG. (¡.8. M e a s u r e m e n t , of t h e F> s t r u c t u r e f u n c t i o n b y III (1996) u s i n g t h e e l e c t r o n (closed circles) a n d t h e s i g m a m e t h o d ( o p e n s q u a r e s ) . T h e i n n e r e r r o r b a r is t h e s t a t i s t i c a l e r r o r , t h e full e r r o r b a r i n c l u d e s a l s o s y s t e m a t i c uncertainties.
e p —> e p 7 , w h e r e t h e o u t g o i n g e l e c t r o n a n d p h o t o n ( s ) a r e m e a s u r e d in coincid e n c e a t v e r y low a n g l e s , f a r d o w n s t r e a m of t h e d e t e c t o r . F i n a l l y . rfn<|<| r e p r e s e n t s all k i n d s of p o s s i b l e a d d i t i o n a l c o r r e c t i o n s , s u c h a s higher o r d e r r a d i a t ive correct ions t o t h e B o r n c r o s s sect ion, o r bin cent re c o r r e c tions which c a n b e of i m p o r t a n c e in r e g i o n s of a fast v a r y i n g c r o s s sect ion. H a v i n g d e t e r m i n e d t h e c r o s s s e c t i o n , c q n (6.8) c a n b e i n v e r t e d in o r d e r t o e x t r a c t /'l>(.r. Q 1 ) . T h i s is d o n e in t w o s t e p s . A t low ,i/ < 0.1 t h e c o n t r i b u t i o n f r o m F/, is negligible, t h e r e f o r e t h e e x t r a c t i o n of F> is s t r a i g h t f o r w a r d . At high // > ().(i t h e c o n t r i b u t i o n of F/. b e c o m e s m o r e i m p o r t a n t . H e r e t h e p r e v i o u s Fi m e a s u r e m e n t o r a n o t h e r i n d e p e n d e n t d e t e r m i n a t i o n of F> is evolved t o t h e relevant. k i n e m a t i c r a n g e u s i n g Q C D e v o l u t i o n e q u a t i o n s a n d i n s e r t e d in c q n (0.8). Then /•';. is e x t r a c t e d . Of c o u r s e t h i s m e t h o d is based on t h e a s s u m p t i o n t h a t /'•j a t large »/ is in a c c o r d a n c e w i t h t h e p r e d i c t i o n s of p e r t u r b a t i v e Q C D . F i g u r e I).8 s h o w s a m e a s u r e m e n t of t h e s t r u c t u r e f u n c t i o n F> in b i n s of r a n d Q~ (III C o l l a b . . 1996). T h e e r r o r b a r s in Fig. 6.8 i n c l u d e s t a t i s t i c a l a n d
(¿CI) A N A L Y S E S
s y s t e m a t i c u n c e r t a i n t i e s I'lic s t a t i s t i c a l c u i u s r a n g e liclwccn 1% a n d (>%.. T h e s y s t e m a t i c u n c e r t a i n t i e s a r e of t h e o r d e r of 5 % t o 10%, d e p e n d i n g on t h e region in i/. M a n y s o u r c e s of s y s t e m a t i c u n c e r t a i n t i e s c o n t r i b u t e , s u c h a s u n c e r t a i n t i e s on t h e l u m i n o s i t y m e a s u r e m e n t , , on t h e e l e c t r o n a n d h a d r o n e n e r g y s c a l e s in t h e r e s p e c t i v e c a l o r i m e t e r s , o n v a r i o u s r e c o n s t r u c t i o n efficiencies, a s well a s oil background expectations and radiative corrections. T h e m e a s u r e d s t r u c t u r e f u n c t i o n s c a n b e used a s i n p u t s for QC'D fits iu o r d e r to e x t r a c t p a r t o n d i s t r i b u t i o n f u n c t i o n s in t h e p r o t o n . A n a l y s e s of t h i s k i n d will be d i s c u s s e d in d e t a i l in C h a p t e r 7. (¡..r>.2
Inclusive
jet production
at tin
TEVATKON
The m e a s u r e m e n t s of tin? c r o s s s e c t i o n for inclusive jet p r o d u c t ion in p p collisions a r e a n i m p o r t a n t test for pert u r b a t i v e QC'D. a n d allow t o put const m i n t s o n t h e p a r t o n d i s t r i b u t i o n f u n c t i o n s a s will b e d e s c r i b e d in S e c t i o n 7.-1.4. F u r t h e r m o r e , d e v i a t i o n s of t h e d a t a f r o m t h e p r e d i c t e d c r o s s s e c t i o n s at tin- h i g h e s t jet e n e r g i e s could b e i n d i c a t i o n s for new p h y s i c s s u c h a s s u b s t r u c t u r e of q u a r k s , s i n c e w i t h jet t r a n s v e r s e e n e r g i e s of a b o u t 400 G e V d i s t a n c e s c a l e s a s s m a l l a s l ( ) - l ! l i n a r e p r o b e d , e.f. e q n (7.1). T h e p r o d u c t ion of j e t s is u n d e r s t o o d a s t he result of a h a r d s c a t t e r i n g of p a r l o u s f r o m t h e i n c o m i n g h a d r o n s . g i v i n g o u t g o i n g p a r l o u s w i t h large t r a n s v e r s e m o m e n t a w h i c h f r a g m e n t i n t o j e t s of p a r t i c l e s . T h e t h e o r e t i c a l i n g r e d i e n t s for t he d e s c r i p t i o n of t h e p r o c e s s h a v e b e e n o u t l i n e d in S e c t i o n .'5.2..'}. In t h i s s e c t i o n , we will r a t h e r c o n c e n t r a t e on t h e e x p e r i m e n t a l a s p e c t s of t h e c r o s s s e c t i o n m e a s u r e m e n t s . In p a r t i c u l a r w e will d e s c r i b e t h e m e a s u r e m e n t of t h e c r o s s s e c t i o n for p p —»jet I A', t h a t is, t h e p r o b a b i l i t y t o o b s e r v e a jet w i t h a c e r t a i n t r a n s v e r s e e n e r g y E-y a t a p s e u d o - r a p i d i t y //. without, c o n s i d e r i n g t h e r e m a i n i n g s y s t e m X . M e a s u r e m e n t s of t his t y p e h a v e b e e n p e r f o r m e d at t h e I S P p p collider at, a C . o . M . e n e r g y of J s = G e V (AI-'S C o l l a b . . 1983), t h e C E R N p p collider a t 540 a n d 630 G e V (IJA2 C o l l a b . 1985/>. 1991: t ' A l C o l l a b . 1986), a n d t h e F E R M I L A I J T E V A T K O N p p collider a t 1800 G o V ( C D F C o l l a b . 1989. 1992. 1996: DO C o l l a b . 1999a). A recent review c a n b e f o u n d in t h e a r t i c l e by B l a z e y a n d F l a u g h e r (1999). T h e selection of e v e n t s p r o c e e d s via s e v e r a l h a r d w a r e t r i g g e r s t a g e s a n d a final oflliuc s o f t w a r e selection. T h e first t rigger looks for a n i n e l a s t i c p p collision i n d i c a t e d by s i g n a l s f r o m d e t e c t o r s p l a c e d very closely t o t h e b e a m p i p e at b o t h sides of t h e i n t e r a c t i o n region. N e x t , t r a n s v e r s e e n e r g y a b o v e a t h r e s h o l d of a few G e V h a s t o b e m e a s u r e d in c a l o r i m e t e r cells d e d i c a t e d t o t h e t r i g g e r . A t t h e following s t a g e j e t s a r e r e c o n s t r u c t e d w i t h a c o n e a l g o r i t h m , d e s c r i b e d in S e c t i o n 6.2. As a n e x a m p l e , iu t h e DO e x p e r i m e n t at t h e T E V A T K O N t h e e v e n t s a r e r e c o r d e d a n d classified if at least o n e j e t w i t h e n e r g y a b o v e a t h r e s h o l d of 30. 50. 8 5 or 1 1 5 G e V is f o u n d . W i t h t h e s e r e q u i r e m e n t s i n t e g r a t e d l u m i n o s i t i e s of u p t o l O O p b " 1 h a v e b e e n c o l l e c t e d at. t h e T E V A T K O N R u n 1A a n d R u n 113. Finally, at. t h e offline a n a l y s i s level t h e c o n e j e t f i n d e r is a p p l i e d a g a i n , w i t h a t y p i c a l c o n e size of /? = 0.7. T h e t r a n s v e r s e e n e r g y of a jet. is c o m p u t ed a c c o r d i n g to
EXAMPLES H-X V1KW 3VHAM I'HIV 12 i^J | Kail Max KT- 145.4 GOV CAEH KT SUM- 968.0GOV VTX in Z.-S.4(cml
87288 Event
22«09 | 25-DKC- 199« 03:20
FlC. (>.9. Event display of a di-jet event in t h e 1)0 d e t e c t o r . F i g u r e from Bla/.ey a n d F l a n g h c r ( 1999).
=
= Y, I
i
E i sin
°< •
(C-20)
where t h e s u m goes over all c a l o r i m e t e r towers within t h e jet cone, with E, the m e a s u r e d tower energy a n d 0, t h e p o l a r a n g l e of t h e tower. N o t e t h a t a jet typically illuminates a r o u n d 20 towers. T h e j e t direction is d e t e r m i n e d by t h e energy-weighted average over a n g u l a r tower positions a n d tlx- location of t h e main event vertex. T h e l a t t e r is d e t e r m i n e d f r o m charged t r a c k s m e a s u r e d in the tracking system. F i g u r e (¡.9 shows a n event, display of a di-jet event, in t h e DO d e t e c t o r . An a l t e r n a t i v e way t o illustrate jet events is given in Fig. (i.10 for t h e case of a multi-jet event recorded by t h e C D F d e t e c t o r . Usually t h e cross section is i n t e g r a t e d over s o m e region in //. such a s |»/| < 0.5 (DO Collab.. 199!)«). T h i s central d e t e c t o r region h a s t h e best, energy resolution a n d uniformity. T h e bin-averaged d o u b l e differential cross section over a certain a n g u l a r r a n g e . (d-'«r/(d/v/-di/)). is d e t e r m i n e d bv s i m p l y c o u n t i n g t h e n u m b e r N of j e t s within a specific bin A E / - - normalized t o t h e bin w i d t h a n d t h e i n t e g r a t e d luminosity. /
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EXAMPLES
o 1)0
ata i
- -
liT AI)
250
300
< 0.5
500
;,«;cV)
Fit!, (i.l 1. T h e inclusive e t c r o s s s e c t i o n for ? < 0.5. m e a s u r e d by 1)0 a n d compared to a QC prediction. T h e error hand indicates the experi m e n t a l s y s t e m a t i c u n c e r t a i n t y : s t a t i s t i c a l e r r o r s a r e invisible. F i g u r e f r o m Blazey and Flaugher(l<)99).
b e a m m e a s u r e m e n t s . M o r e i m p o r t a n t l y , it is m o n i t o r e d in-situ d u r i n g t h e a c t u a l r u n n i n g of t h e e x p e r i m e n t by l o o k i n g for p h y s i c s p r o c e s s e s w i t h well d e f i n e d cons t r a i n t s o n t h e e n e r g y d e p o s i t s . For t h e e l e c t r o m a g n e t i c c a l o r i m e t e r t h e s e a r e Z d e c a y s i n t o e l e c t r o n posit ron p a i r s o r p h o t o n p a i r s f r o m TT d e c a y s . T h e h a d r o n i e r e s p o n s e is checked w i t h p h o t o n et, e v e n t s , w h e n - t h e precisely m e a s u r e d p h o t o n t r a n s v e r s e e n e r g y h a s t o b e b a l a n c e d by t h e e t . ther contributions to the a b s o l u t e et e n e r g y s c a l e c o m e f r o m i n s t r u m e n t a l effects, p i l e - u p f r o m p r e v i o u s b e a m b e a m c r o s s i n g s , a n d e n e r g y d e p o s i t s f r o m t h e u n d e r l y i n g event , w h i c h is t h e o r e t i c a l l y not so well u n d e r s t o o d . It is s t u d i e d in e v e n t s w i t h n o h a r d s c a t t e r i n g at all or f r o m t h e e n e r g y d e p o s i t s in r a n d o m c o n e s o r t h o g o n a l t o t h e ets. It. a m o u n t s t o about, 1 G c V for e t s w i t h cone-size l i = 0.7. T h e finite resolut ion of t he et e n e r g y m e a s u r e m e n t d i s t o r t s t he s t e e p l y falling e n e r g y s p e c t r u m . In t h i s r e s o l u t i o n is m e a s u r e d ( f r o m b a l a n c i n g r in et e v e n t s ) t o b e of G a u s s i a n s h a p e w i t h a w i d t h of a b o u t 7 G e V for r = l GeV.
(}< 'li A N A L Y S E S 35 Intuì error 30
energy scale (partially correlated) overall luminosity (lully correlatoti)
| £'
25
resolution ( liilly correlated)
20
relative luminosity (partially correlated) jet selection (lully correlata!)
/
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s (I 50
100
150
200
250
300
350
400
450
¿>«;cv) F i o . li. 12. C o n t r i b u t i o n s t o t lie systemat ir u n c e r t a i n t y of t h e cross sect ion meas u r e m e n t shown in Fig. (>.11. F i g u r e f r o m Blazey a n d F l a u g h e r ( 1 9 9 9 ) .
T h e d i s t o r t i o n is corrected by p a r a m e t e r i z i n g t h e s h a p e of a test s p e c t r u m , such as (a£-/-)(l 2 E f / • s m e a r i n g this s p e c t r u m with t h e known resolution f u n c t i o n , a n d d e t e r m i n i n g t h e p a r a m e t e r s f r o m a c o m p a r i s o n of t h e s m e a r e d s p e c t r u m t o t h e m e a s u r e d dist ribut ion. F r o m t h e r a t i o of t h e t h u s found test a n d smeared d i s t r i b u t i o n s bin-to-bin correction f a c t o r s a r e c o m p u t e d a n d applied t o t he d a t a . T h i s resolution correct ion reduces t he cross section observed by DO by (13 ± 3)% a t Er = (K)GeV a n d (8 ± 2 ) % a t -101) C e V . In C D F b o t h t he energy scale a n d resolution c o r r e c t i o n s a r e d e t e r m i n e d in a single s t e p by c o m p u t i n g response f u n c t i o n s with a M o n t e C a r l o s i m u l a t i o n . T h i s simulation is t u n e d t o describe well electron a n d h a d r o n t e s t b e a m results a s well a s jet f r a g m e n t a t i o n such a s charged track multiplicities a n d m o m e n t a . T h e y o b t a i n correction factors for t h e j e t e n e r g y b e t w e e n 1.0 a n d 1.2. T h e result of t h e inclusive jet cross section m e a s u r e m e n t by DO (1999a) is displayed in Fig. 0.11. T h e jet E r s p e c t r u m is m e a s u r e d for energies between 00 a n d 450 C e V . with a cross sect ion falling by seven o r d e r s of m a g n i t u d e . T h e good a g r e e m e n t within e r r o r b a n d s of a n e x t - t o - l e a d i n g o r d e r Q C D calculation over t h e full s p e c t r u m is r e m a r k a b l e a n d a success for t h e theory. T h e s e d a t a give no indication for new physics. A f u r t h e r discussion of t h e o r y - d a t a c o m p a r i s o n s for t h e inclusive jet cross section will follow in Section 7.4.4. Figure 0.12 shows t h e s y s t e m a t i c u n c e r t a i n t i e s of t h e DO m e a s u r e m e n t as a f u n c t i o n of E r - T h e t o t a l e r r o r varies f r o m 10'X at t h e lowest energies u p t o a b o u t 25% at t h e u p p e r limit of t h e s p e c t r u m . It, is d o m i n a t e d by t h e u n c e r t a i n t y on the energy scale.
EXAMPLES
(¡.r,.:{
.let rides at L E P
In o u r lasl e x a m p l e wo will d i s c u s s t h e m e a s u r e m e n t of jet. r a t e s in o ' o auniliilations at L E P . We will restrict o u r s e l v e s t o d a t a t a k e n at. t he Z r e s o n a n c e . T h e »-jet r a t e a s a funct ion of t h e resolut ion p a r a m e t e r «/<•„, is d e f i n e d a s t he n u m b e r of e v e n t s w h e r e exact ly » j e t s a r e f o u n d for a fixed value of //,-,,,. n o r m a l i z e d t o t he t o t a l n u m b e r of h a d r o n i c e v e n t s , Rn(y^) =
JVIiikI
•
(6.22)
.let f i n d i n g a l g o r i t h m s a n d t h e c o n c e p t of r e s o l u t i o n p a r a m e t e r h a v e b e e n int r o d u c e d in S e c t i o n 6.2. T h e m e a s u r e m e n t of j e t r a t e s is a basic t e s t for t h e d e s c r i p t i o n of g l n o n r a d i a t i o n oil' q u a r k s a n d allows for a d e t e r m i n a t i o n of t he st r o n g c o u p l i n g c o n s t a n t . T h e s t a n d a r d p r o c e d u r e of t r i g g e r i n g for h a d r o n i c e v e n t s h a s a l r e a d y been d e s c r i b e d in S e c t i o n 6.1.1.1. At t h e offline a n a l y s i s level f u r t h e r q u a l i t y c u t s a r e i m p o s e d . H e r e we will s u m m a r i z e t y p i c a l c u t s a s used in t h e A L E P I I e x p e r i m e n t . For e v e r y event, classified o n l i n e a s a h a d r o n i c event first a s a m p l e of ' g o o d ' c h a r g e d t r a c k s is s e l e c t e d . T h e s e t r a c k s must, h a v e a t least f o u r well m e a s u r e d c o o r d i n a t e s f r o m t he T i m e P r o j e c t i o n C h a m b e r , a p o l a r a n g l e of \ c o s f l | < 0.9. r i. a m o m e n t urn in excess of 201) M o V . a n d o r i g i n a t e f r o m w i t h i n a c y l i n d e r of r a d i u s 2 c m a n d lengt h 10 c m c e n t r e d a r o u n d t h e i n t e r a c t i o n p o i n t . T h e s e r e q u i r e m e n t s select t r a c k s w h i c h a r e well w i t h i n t h e a c c e p t a n c e of t h e d e t e c t o r a n d reject low m o m e n t u m particles which a t e possibly related to beam-induced background or n u c l e a r i n t e r a c t i o n s in t h e d e t e c t o r m a t e r i a l . The event is t h e n r e t a i n e d for f u r t h e r a n a l y s i s if a t least five s u c h g o o d t r a c k s a r e f o u n d , if t h e s u m of Unc h a r g e d e n e r g y e x c e e d s half of t h e C . o . M . e n e r g y , a n d if t h e p o l a r a n g l e of t h e T h r u s t axis, c o m p u t e d w i t h g o o d c h a r g e d t r a c k s only, s a t i s f i e s | cosflxiirust | < ().!). T h e last cut. e n s u r e s t h a t t h e e v e n t is well c o n t a i n e d w i t h i n t h e d e t e c t o r , w h e r e a s t h e p r e v i o u s c u t s reject b a c k g r o u n d f r o m Z —» r + r ~ —> h a d r o u s + A* a n d 7*7* —> h a d r o u s t o below a few p e r milk*. ( ) n c e a n event is s e l e c t e d , t h e j e t c l u s t e r i n g a l g o r i t h m of t h e D u r h a m - or .ladet y p e is a p p l i e d t o a list of r e c o n s t r u c t e d t r a c k s a n d n e u t r a l p a r t i c l e s . T h e l a t t e r a r e ident ified a s e n e r g y d e p o s i t s in t he c a l o r i m e t e r s a b o v e a t h r e s h o l d of several h u n d r e d M e V . T h e a l g o r i t h m ¡ s u p p l i e d for a fixed n u m b e r of //,.„, values, r a n g i n g for e x a m p l e f r o m 0.001 t o 0.1. T h e n u m b e r of j e t s f o u n d d e f i n e s t h e c l a s s i f i c a t i o n of t h e event for a specific ,,,. A f t e r h a v i n g a n a l y s e d all s e l e c t e d h a d r o n i c e v e n t s , which a m o u n t t o several million a t L E P . t h e »-jet. r a t e s a r e c o m p u t e d a c c o r d i n g t o e q n ( 6 . 2 2 ) . T h i s d e f i n e s t h e r a w d a t a dist r i b u t i o n Dnxw of j e t r a t e s . E x a c t l y t he s a m e a n a l y s i s is a p p l i e d t o a s a m p l e of s i m u l a t e d e v e n t s , w h e r e a M o n t e C a r l o m o d e l for q u a r k f r a g m e n t a t i o n a n d h a d r o n p r o d u c t i o n , such a s d e s c r i b e d in Section I. is c o m b i n e d w i t h a d e t a i l e d s i m u l a t i o n of t h e d e t e c t o r r e s p o n s e , g i v i n g / ? M < . F u r t h e r m o r e , t h e a n a l y s i s is a l s o a p p l i e d t o t h e set. of s i m u l a t e d h a d r o u s b e f o r e a n y d e t e c t o r s i m u l a t i o n , w h i c h d e f i n e s t h e ' t r u e ' dist r i b u t i o n D,n"'. T h e d i s t r i b u t i o n of j e t r a t e s a s a f u n c t i o n of ;/<•„,. [ ) " ' " . c o r r e c t e d
i ANALYSES
-lC I FIG. 6 . 1 3 . i s t r i b u t i o n of o t r a t e s m e a s u r e d by A I'LL as a f u n c t i o n of t h e r e s o l u t i o n p a r a m e t e r i c „t. F i g u r e f r o m A Iv II Collab.(19!)8a).
for d e t e c t o r a c c e p t a n c e a n d r e s o l u t i o n effects, is t h e n o b t a i n e d by c o m p u t i n g for every <-,,, value ntrue
=
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(6.23)
a s o u t l i n e d in Section 6.3.1. T o first a p p r o x i m a t i o n t h e c o r r e c t i o n f a c t o r d o e s not d e p e n d on t h e p a r t i c u l a r M o n t e C a r l o m o d e l used. A residual d e p e n d e n c e is checked for by c o m p u t i n g it with a d i f f e r e n t m o d e l . T h i s leads t o s y s t e m a t i c u n c e r t a i n t i e s of t h e o r d e r of 1 . F u r t h e r s y s t e m a t i c c h e c k s a r e p e r f o r m e d by rep e a l i n g t h e a b o v e p r o c e d u r e , b u t c h a n g i n g t h e selection c u t s on t r a c k a n d event quality. A g o o d test of t h e reliability of t h e M o n t e C a r l o s i m u l a t i o n is o b t a i n e d by r e p e a t i n g t h e a n a l y s i s u s i n g c h a r g e d t r a c k s only. In t h e c a s e of a p e r f e c t simulation every a n a l y s i s v a r i a t i o n s h o u l d lead t o t h e s a m e c o r r e c t e d d i s t r i b u t i o n . If t h e r e a r e d i f f e r e n c e s t h e y a r e t a k e n a s e s t i m a t e s of t h e s y s t e m a t i c u n c e r t a i n t y . B e c a u s e of t h e huge d a t a s a m p l e at t h e s t a t i s t i c a l e r r o r s a r e very s m a l l , a n d t h e t o t a l u n c e r t a i n t y is d o m i n a t e d by t h e s y s t e m a t i c u n c e r t a i n t i e s of a few per cent. T h e result of such an a n a l y s i s by A I I is s h o w n in Fig. 6.13 ( A I I Collab.. l!)!) ( ). We see t h a t t h e n u m b e r of e t s f o u n d i n c r e a s e s with d e c r e a s i n g ,„(, since a s m a l l e r resolution p a r a m e t e r resolves m o r e a n d m o r e g l u o n r a d i a t ion. T h e
EXAMPLES
(lain a r c c o m p a r e d In p r e d i c t i o n s o l ' s t a n d a r d M o n t e C a r l o models described Section I. T h e y give a q u i t e s a t i s f a c t o r y description of t h e d a t a , c o n s i d e r i n g tl fact that these »-jet r a t e s have not been used for t h e a d j u s t m e n t of t h e m o d parameters.
Exercises for Chapter (> (i I Show t h a t t h e Thrust, variable is indeed infrared safe, (i 2 In h a d r o n h a d r o n collisions t h e a c t u a l h a r d s c a t t e r i n g t a k e s place between two p a r t o n s coining f r o m t h e two h a d r o u s . a n d these two p a r l o u s c a r r y s o m e f r a c t i o n s r j a n d .;••_. (.Ti,-_> > 0) of t h e longitudinal m o m e n t u m of t h e h a d r o u s , leading to a longitudinal m o m e n t u m imbalance. 3••_>. T h e r e f o r e t h e final s t a t e will b e boosted along t h e direction of t h e b e a m axis, usually defined a s t h e ¿-axis. O n l y transverse to this axis is m o m e n t u m b a l a n c e o b t a i n e d , since t h e small ( a n d different.) t r a n s v e r s e m o m e n t a of t h e p a r t o n s can b e neglected in m a n y c i r c u m s t a n c e s . Show that, differences of t h e r a p i d i t y y of two jets. Ay = ;/j, - i/j 2 , a r e invariant u n d e r Lorentz b o o s t s a l o n g t h e z-axis. (i
T h e Breit f r a m e of reference is defined a s t h e Lorentz f r a m e in which t h e p h o t o n , which is e x c h a n g e d between t h e incoming lepton a n d a p a r t o n f r o m t h e p r o b e d h a d r o n . is purely spatial (E = 0) a n d collides head on with t h e h a d r o n . For a p a r t i c u l a r choice of t h e z-axis t h e p h o t o n ' s f o u r - m o m e n t u m takes on t h e form Q = ( 0 . 0 . 0 . Q). t h a t is. it h a s zero energy a n d only its ¿ - m o m e n t u m component, is nonzero. Show that a t r a n s f o r m a t i o n f r o m t h e l a b o r a t o r y f r a m e into t h e Breit f r a m e exists, a n d discuss t h e p h a s e s p a c e for t he incoming a n d outgoing scattered partons. ' * * ' (i 1 E x p r e s s t h e k i n e m a t i c variables Q - a n d y in t e r m s of energies a n d p r o d u c t i o n angles of t h e particles in t h e h a d r o n i c s y s t e m of a d e e p inelastic s c a t t e r i n g event at IIERA, a s s u m i n g t h e particles t o b e massless. Exploit t h e e n e r g y - m o m e n t i m i c o n s t r a i n t to derive t h e expressions of t he J a c q u c l Blondel a n d of t he Sigma met hod. A s s u m i n g t lie p a r t o n model, derive an expression for t h e angle of t h e s t r u c k p a r t o n in t e r m s of t h e previously derived q u a n t i t i e s .
I
TR CT RE F
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eep inelastic scattering (131S of l e p t o n s on h a d r o n s is t h e most f u n d a m e n t a l e x p e r i m e n t a l tool for t h e s t u d y of t h e s t r u c t u r e of h a d r o n s . In t h e e x p e r i m e n t s eit her c h a r g e d (electrons, unions) or n e u t r a l ( n e u t r i n o s ) l e p t o n s a n d a n t i l e p t o n s a r e used. T h e s c a t t e r i n g o c c u r s on single free p r o t o n s ( H A collider) or on h o u n d p r o t o n s a n d n e u t r o n s within a nucleus (fixed-target e x p e r i m e n t s ) . A t large m o m e n t u m t r a n s f e r t h e inelastic s c a t t e r i n g is described as t h e incoherent s u m of elastic s c a t t e r i n g off t he h a d r o n c o n s t i t u e n t s , t h e p a r t o n s . which a r e assumed to be pointlike. T h e s e c o n s t i t u e n t s a r e u a n d d q u a r k s in t h e simplest, q u a r k p a r t o n model, whereas Q C predicts t h e existence of f u r t h e r p a r t o n s within t h e h a d r o n s . such as heavier q u a r k s , a n t i q u a r k s a n d gluons. T h e leptou doesn't, feel s t r o n g i n t e r a c t i o n s , t h e r e f o r e in principle it can only p r o b e h a d r o n c o n s t i t u e n t s which c a r r y electric or weak c h a r g e s . G l u o n s c a r r y neit her of these charges, b u t their existence is p r o b e d by t h e sensitivity of t he m e a s u r e m e n t s t o r a d i a t i v e c o r r e c t i o n s a s p r e d i c t e d by Q C . T h e k i n e m a t i c s of t h e IS process h a s a l r e a d y been discussed in detail in previous c h a p t e r s . As a r e m i n d e r , t h e most, i m p o r t a n t q u a n t i t i e s a r e t h e mo= I — ', w h e r e 1(1') is t h e m o m e n t u m of t h e m e n t u m t r a n s f e r 1 = — 2 with incoming (outgoing) l e p t o n , a n d t h e B orken scaling variable x = ~/(2p ), where p is t h e h a d r o n ' s m o m e n t u m . In t h e p a r t o n m o d e l x c o r r e s p o n d s t o t h e fraction of t h e h a d r o n m o m e n t u m carried by t h e s t r u c k p a r t o n . Finally, t h e 2 /( '') c o r r e s p o n d s to t h e relative energy loss of q u a n t i t y y = (p j)/(p I) t h e lepton in a orenlz f r a m e where t h e h a d r o n is at. rest. T h e second equality is t r u e when h a d r o n a n d lepton masses a r e neglected, since t h e n t h e s q u a r e d C.o.M. energy is simply given by s = 2p-l. In Fig. 7.1. t h e accessible k i n e m a t i c d o m a i n in t h e (.r. Q2) p l a n e for different e x p e r i m e n t s is s h o w n . x p e r i m e n t s at 11 A (111. Z S) cover a very e x t e n d e d r a n g e in 2 as 0.1 - 10 1 GeV' 2 a s well a s in x 10 '' — 1 0 ' . In p a r t i c u l a r t h e very low . a n d very high 1 regions can not be covered by fixed-target, e x p e r i m e n t s , which i n s t e a d p r o b e t h e regions at i n t e r m e d i a t e ( 2 a n d larger .r in a nicely c o m p l e m e n t a r y way. It c a n also b e observed that t h e lower C.o.M. energy available in t h e fixed-target e x p e r i m e n t s c o m p a r e d t o II IIA effectively c o r r e s p o n d s t.o a rcscaliug t o lower values in y. It is worth noting t h a t the m o m e n t u m t r a n s f e r Q is a m e a s u r e of t h e ' m a g n i f y i n g power of t h e lepton p r o b e , since t h e p r o b e d d i s t a n c e scale il is given by t h e simple relation , l,c GeVfm f i w « 0 . 2 — — .
, (7.1)
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AND I'AIMON
/ l i t i s HIT 1995
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jjgj III 1 9 9 5 + I')'«)
¡¡¡J lll-.KA SVTX 1995 ; •
HliRA 1994
• Q HERA 1993 j P
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Fl<;. 7 . 1 . C o v e r a g e of t lie ( x . ( J 2 ) p l a n e bv v a r i o u s DIS e x p e r i m e n t s . F i g u r e f r o m Garcia Canal and Sassot(2000).
So we find t hat Q2 = -I GeV~ c o r r e s p o n d s t o <1 w 10 (/-' = -100(H) G e V 2 t o il « 10" 1 ( 5 c m .
11
cm. and correspondingly
T h e t h e o r e t i c a l i n g r e d i e n t s for t h e d e s c r i p t i o n of D I S e x p e r i m e n t s a r e a l s o d i s c u s s e d in p r e v i o u s c h a p t e r s . T h e r e it is e x p l a i n e d h o w t h e r e l e v a n t c r o s s sect i o n s a r e e x p r e s s e d in t e r m s of s t r u c t u r e f u n c t i o n s F ( : r . Q 2 ) . a n d t h e r e l a t i o n b e t w e e n t h e s e s t r u c t u r e f u n c t i o n s a n d t h e p a r t o n d e n s i t y f u n c t i o n s , p.d.f.s, (or p a r t o n d i s t r i b u t i o n s ) , which p a r a m e t e r i z e t h e s h a r i n g of t h e h a d r o n m o m e n t u m b e t w e e n lh<" p a r l o u s , is g i v e n . R e m e m b e r t h a t t h e s t r u c t u r e f u n c t i o n s a n d cross s e c t i o n s a r e p h y s i c a l ( m e a s u r a b l e ) q u a n t i t i e s in t h e s e n s e that, t h e y d o not d e p e n d o n t h e d e t a i l s of t h e c a l c u l a t i o n s , s u c h a s t h e n u m b e r of c o m p u t e d p e r t u r b a t i v e t e r m s o r t h e f a c t o r i z a t ion s c h e m e , w h e r e a s t h e p a r t o n d i s t r i b u t i o n s d o . T h e r e f o r e t h e p a r t o n d i s t r i b u t i o n s c a n b e viewed a s e f f e c t i v e q u a n t i t i e s s u c h a s a r u n n i n g c o u p l i n g . O n c e d e t e r m i n e d , tliev c a n b e u s e d t o p r e d i c t o t h e r s t r u c t u r e functions a n d / o r cross sections c o m p u t e d within the s a m e theoretical scheme a n d o r d e r of p e r t u r b a t i v e e x p a n s i o n . W h y a r c t h e m e a s u r e m e n t s of t he h a d r o n s t r u c t u r e so i m p o r t a n t ? F i r s t , t h e y
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III
im 12214
I vfitl ii'i.M i
( - = :Sllill(icV'. ,
ale I'MW I'WS
(!. >. .11 =211 ( i c V
I'lc:. 7 . 2 . v e n t display of a n e u t r a l current event, observed wit, li t h e III d e t e c t o r at t h e H A collider.
with P( ~) = tr ~/( ' + i t}). My is t h e Z m a s s , K 2 = l ( 4 s i n 2 0 w c o s 2 0 w ) is a function of t h e weak mixing angle 0 W . a n d vv = —1 2 1 2 sin 0 W a n d a,. = —1 2 a r e t h e vector a n d axial vector couplings of t h e positron t o t h e Z. Fn is t h e e l e c t r o m a g n e t i c s t r u c t u r e f u n c t i o n , o f t e n d e n o t e d a s F.f or simply a s Fj. which originates from p h o t o n e x c h a n g e only. T h e o t h e r s t r u c t u r e f u n c t i o n s d e s c r i b e t h e c o n t r i b u t i o n s from Z e x c h a n g e a n d f r o m Z 7 interference. T h e l a t t e r become only relevant for very large m o m e n t u m t r a n s f e r s , when 2 is of t h e s a m e order or larger t h a n M j . In p a r t i c u l a r , t h e c o n t r i b u t i o n from .rF. b e c o m e s visible, leading to a significant r e d u c t i o n in t h e < ' p cross section, only well a b o v e Q z — 5( l)0GeV . T h e c o n t r i b u t i o n f r o m tin- longitudinal s t r u c t u r e f u n c t i o n f . a m o u n t s t o a few p e r cent correction t o t h e cross section for y > 0.65 a n d (). I. In leading o r d e r Q C we have Q 2 < 1500 Go V 2 , a n d is negligible for . = I). At this o r d e r , a n d for small e n o u g h m o m e n t u m t r a n s f e r s where contrib u t i o n s f r o m Z e x c h a n g e can be ignored, t h e cross section is d e t e r m i n e d by t h e e l e c t r o m a g n e t i c s t r u c t u r e function F->{x. ~) = yt-tfrlf .. which is related t o t h e partem d i s t r i b u t ions a s F,(x. ~)
= x jj («( )
-1- c(x) + u(x) + c(x))
+
(il{x) + s(x) + ,l(x) +
s(x)) (7.5)
MVi
S I I t t i c i 11UIC FU NC I IONS AND PA I (TON DIS I HIMUTIONS
In lliis Q2 r e g i m e a n d at lal'gc x t h e c r o s s s e c t i o n is m a i n l y ( l e t e r i u i n e d by t h e n(x) tl(x) « u(x). valence ii-(|iiark d i s t r i b u t i o n Uv(x) For I lie d e s c r i p t i o n of t h e m e a s u r e m e n t s we will m a i n l y c o n c e n t r a t e 011 c x p o r i m e n i s a t H E R A , w h i c h cover t h e largest k i n e m a t i c r a n g e . T h i s is b e c a u s e of t h e large C . o . M . e n e r g y of ~ 3 0 0 G e V , a c h i e v e d by a head-011 collision of p r o t o n s w i t h Zi.',, 8 2 0 G e V , l a t e r e v e n 9 2 0 G e V , a n d p o s i t r o n s (or e l e c t r o n s ) close t o /•,',. 2 8 G e V . F u r t h e r m o r e , t h e collider e x p e r i m e n t s III a n d Z E U S a r e almost h e r m e t i c m u l t i - p u r p o s e d e t e c t o r s , w h i c h allow t o m e a s u r e t o high a c c u r a c y I he c o m p l e t e liual s t a t e of t h e i n t e r a c t i o n . T h e v a r i o u s a s p e c t s of a s t r u c t u r e function m e a s u r e m e n t at I IKK A a r e d i s c u s s e d in s o m e d e t a i l in S e c t i o n 0 . 5 . 1 . s u c h as t h e r e c o n s t r u c t i o n of t h e k i n e m a t i c v a r i a b l e s , t h e l u m i n o s i t y m e a s u r e m e n t a n d t h e m e a s u r e m e n t of t h e d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n a n d t h e s u b s e q u e n t e x t r a c t i o n of t h e s t r u c t u r e f u n c t i o n s . In Fig. 7.2 a n e v e n t d i s p l a y of a N C e v e n t in t h e III d e t e c t o r is s h o w n . T h e p r o t o n b e a m e n t e r s f r o m t h e right side. T h e p o s i t r o n a r r i v e s f r o m t h e left a n d is b a c k s c a t t e r e d i n t o t h e u p p e r left region of t h e d e t e c t o r , b a l a n c e d in t r a n s v e r s e m o m e n t u m by a h a d r o n i c j e t in t h e lower region. T h e p r o t o n r e m n a n t m a n i f e s t s il sell a s h a d r o n i c a c t i v i t y close t o t h e b e a m p i p e in t h e p r o t o n direct ion of flight. T h e event selection of N C e v e n t s is b a s e d o n t h e i d e n t i f i c a t i o n of a s c a t t e r e d p o s i t r o n w i t h l a r g e t r a n s v e r s e m o m e n t u m ¡'•/ •^ 1 0 G e V w i t h respect t o t h e b e a m line. as. for e x a m p l e , c l e a r l y visitile in Fig. 7.2. a n d a n i n e l a s t i c i t y ;/ s m a l l e r t h a n a b o u t 0 . 9 ¡11 o r d e r t o e n s u r e a precise k i n e m a t i c r e c o n s t r u c t i o n . S e v e r a l met h o d s a s d e s c r i b e d in S e c t i o n (i-r>. 1 a r e used for t his lat t e r p u r p o s e , s i n c e in N C e v e n t s t h e r e is r e d u n d a n t i n f o r m a t i o n f r o m t h e s i m u l t a n e o u s r e c o n s t r u c t i o n of t h e s c a t t e r e d p o s i t r o n a n d t h e h a d r o n i c final s t a t e . D e p e n d i n g o n t h e k i n e m a t i c region t h e m e t h o d w i t h t h e b e s t r e s o l u t i o n is a d o p t e d . M e a s u r e m e n t s h a v e been p e r f o r m e d u p t o t h e h i g h e s t values of Q~ » 4 0 0 0 0 G e V " by III (2000) a n d Z E U S ( 1 9 9 9 a ) . W i t h i n t e g r a t e d l u m i n o s i t i e s a r o u n d 4 0 pl>~ 1 f r o m t h e d a t a t a k i n g p e r i o d 199 1 97. t h e e x p e r i m e n t s h a v e collected u p t o 75000 N C e v e n t s in 1 5 0 - 4 0 0 0 0 G e V 2 . D e d i c a t e d m e a s u r e m e n t s of t h e l o w - Q 2 region t h e r a n g e Q2 will b e d i s c u s s e d in a l a t e r s u b s e c t i o n . T h e c r o s s s e c t i o n s a r e m e a s u r e d w i t h a few p e r cent s t a t i s t i c a l a c c u r a c y a n d s y s t e m a t i c e r r o r s of a b o u t -1%. d o m i n a t e d bv u n c e r t a i n t i e s in t h e e n e r g y scale a n d i d e n t i f i c a t i o n efficiency of t h e s c a t t e r e d posit ron a s well a s by u n c e r t a i n t i e s o n t he e n e r g y s c a l e of t h e h a d r o n i c final s t a t e . T h e l u m i n o s i t y a n d t h u s t h e overall n o r m a l i z a t i o n is k n o w n w i t h i n a precision of 1.5%. For Q ~ 5 > 5 0 0 0 G e V 2 t h e s y s t e m a t i c u n c e r t a i n t i e s d o m i n a t e , w h e r e a s at larger Q2 t h e s t a t i s t i c a l u n c e r t a i n t i e s i n c r e a s e s t r o n g l y b e c a u s e of t h e s t e e p l y falling cross s e c t i o n . T h e e l e c t r o m a g n e t i c s t r u c t u r e f u n c t i o n Fo(.i\ Q~) is e x t r a c t e d f r o m t h e m e a s u r e d d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n by c o r r e c t i n g for e l e c t r o w e a k r a d i a t i v e effects, t h e c o n t r i b u t i o n s f r o m Z e x c h a n g e a n d t h e c o n t r i b u t i o n s f r o m F t a n d /•';, as p r e d i c t e d by t h e N L O D G L A P e v o l u t i o n e q u a t i o n s . E x a m p l e s of m e a s u r e m e n t s as a f u n c t i o n of x a n d Q2 a r e s h o w n in Figs. 7.3 a n d 7.4. Nice c o n s i s t e n c y w i t h f i x e d - t a r g e t d a t a a t s m a l l e r Q 2 by N'MC (1995) is f o u n d . M e a s u r e m e n t s a t
( 'IIAKOKI1 LI I'I ( IN N I K ' L E O N SCA I II H INC
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F i g . 7 . 3 . M e a s u r e m e n t s of t h e e l e c t r o m a g n e t i c s t r u c t u r e f u n c t i o n F, a s a f u n c tion of a- for v a r i o u s Q2 values, by III a n d Z E U S , c o m p a r e d t o a N L O Q C D fit. F i g u r e f r o m 111 C o l l a b . ( 2 0 0 0 ) .
lower Q2 a r e s h o w n in Fig. (i.8. All m e a s u r e m e n t s a r e well d e s c r i b e d by t h e Q2 e v o l u t i o n of F, a s p r e d i c t e d by t h e N L O D G L A P e q u a t i o n s f r o m Q2 % 1 G e V 2 u p t o t h e highest m e a s u r e d Q2. For f u r t h e r d i s c u s s i o n see a l s o S e c t i o n s 3.2.2. 3.0 a n d 7.5. A p o s i t i v e s l o p e a s a f u n c t i o n of Q 2 is visible in Fig. 7.4 for t h e low-.)' d a t a p o i n t s a n d t h i s s l o p e d e c r e a s e s w i t h i n c r e a s i n g ./• a s e x p e c t e d f r o m QC'D. T h i s is b e c a u s e for low x a n d i n c r e a s i n g Q2 t h e e x c h a n g e d p h o t o n c a n resolve m o r e a n d m o r e p a r l o u s w i t h i n t h e p r o t o n , a r i s i n g f r o m partem c a s c a d e s like g l u o n r a d i a t i o n a n d g h i o n s p l i t t i n g i n t o q q - p a i r s . At l a r g e r ./• t h e p r o b a b i l i t y d e c r e a s e s t o find g l u o n s o r s e a q u a r k s . T h e r e t h e s t r u c t u r e f u n c t i o n is d o m i n a t e d by t h e v a l e n c e q u a r k d i s t r i b u t i o n , which varies m o r e slowly w i t h Q 2 . In Fig. 7.3
si itu< rum:
MINI
H O N S A N I » I ' A I M O N D I S I K I I U I I IONS
Q~
(GCVY
F l C . 7 . 1 . M e a s u r e m e n t s of t h e e l e c t r o m a g n e t i c s t r u c t u r e fi mei ion > as a function of Q . for various . values, l y III a n d M C , c o m p a r e d t o a I . QC lit,. F i g u r e from 111 Collali.(2000).
a s t r o n g rise of t h e s t r u c t u r e function at low . can be seen, which is i n t e r p r e t e d as t h e s t r o n g rise of t h e c o n t r i b u t i o n f r o m g l n o n s in t h i s k i n e m a t i c regime. In c o n t r a s t , a t t h e largest :r values t h e s t r u c t u r e f u n c t i o n decreases rapidly, d u e t o a rapili decrease of t h e valence q u a r k c o n t r i b u t i o n . 7.1.2
C arged current
interactions
If the exchanged particle in t h e d e e p inelastic s c a t t e r i n g is a V boson, wo talk a b o u t a c aitjed current ( 0 0 ) interaction. An e x a m p l e which is s t u d i e d at II A + is o p —> ,. A'. p t o small electroweak c o r r e c t i o n s t h e d o u b l e differential cross section for 0 0 e v e n t s is given liv
( T I A H O I I H I I ' T O N NUCI.I ON SCAT"I I If INC
•jdii
F i g . 7 . 0 . E v e n t d i s p l a y of a c h a r g e d c u r r e n t event., o b s e r v e d w i t h t h e 111 d e t e c t o r at t h e l l l i R A collider.
+ A ' r ) is t h e d e g r e e of p o l a r i z a t i o n of t h e l e p t o n b e a m , Here /',. = (A : i, — Nn)/(N\. : w h e r e A I.(H) is t h e n u m b e r of loll (right ) - h a n d e d p o s i t r o n s . We see that, t h e C C c r o s s s e c t i o n h a s a s t r u c t u r e very s i m i l a r t o t h e NC' c r o s s s e c t i o n , e q n (7.2). t h e o n l y d i f f e r e n c e b e i n g t h a t t h e fine s t r u c t u r e c o n s t a n t n,.1M is r e p l a c e d b y t h e Fermi c o u p l i n g c o n s t a n t G v a n d t h e p h o t o n p r o p a g a t o r t e r m 1 / Q 1 is r e p l a c e d by t he c o r r e s p o n d i n g YV p r o p a g a t o r . T h i s p r o p a g a t o r s t r u c t u r e tells u s i m m e d i a t e l y t h a t t h e C C c r o s s s e c t i o n is m u c h s m a l l e r t h a n t h e N C o n e . a n d relevant c o n t r i b u t i o n s c a n o n l y b e e x p e c t e d for Q 2 k , M f v . T h e t e r m \(•(•''• Q 2 ) However, let us w r i t e it d o w n i m m e d i a t e l y in t e r m s of p.d.f.s:
«¿CO = * [(«(•>••) -I- ¿(.T)) + (1 - I))2 (d(x)
+ ,s(.r))] .
(7.7)
T h i s e x p r e s s i o n is exact in L O QC'D. In o r d e r t o t a k e i n t o a c c o u n t q u a r k m i x i n g t h e i n d i v i d u a l t e r m s w o u l d h a v e t o b e w e i g h t e d by t h e relevant s q u a r e d m a t r i x e l e m e n t s of t h e C K M m a t r i x . Since for a n i n c o m i n g p o s i t r o n t h e e x h a n g e d W b o s o n h a s p o s i t i v e c h a r g e , t h e c r o s s sect ion is sensit ive t o d o w n - t y p e q u a r k s a n d u p - t y p e a n t i q u a r k s . In t h e c a s e of a n i n c o m i n g e l e c t r o n t h e e x p r e s s i o n would be c h a n g e d t o [(m + c) + (1 ! / ) 2 ( d + *)]• In a d d i t i o n , t h e p o l a r i z a t i o n f a c t o r is c h a n g e d t o (1 I Pv). T h u s we see t h a t C C i n t e r a c t i o n s c a n d i s t i n g u i s h be-
•ji;<;
s i u n c i HHI I ' U N C I IONS ANI» C A I M O N n i s i IIIIIU I IONS / I u s e e iw-i
F l C . 7.(i. M e a s u r e m e n t of t h e r e d u c e d c h a r g e d c u r r e n t cross section l>y Z F l ' S as a f u n c t i o n of r in different. Q 1 l)ins. Also i n d i c a t e d a r e NIX) QC'D fits a n d several p a r a m o t e r i z a t i o i i s of p.d.f.s. F i g u r e f r o m ZlvUS Collab.(20(K)«).
tween <|uarks a n d ant ¡quarks, which is not possible for p h o t o n e x c h a n g e in N C interactions. T h e e x p e r i m e n t s al IIKKA h a v e o b s e r v e d t h i s t y p e of inelastic s c a t t e r i n g . T h e event selection of O C e v e n t s is based on t h e i d e n t i f i c a t i o n of large missing t r a n s v e r s e m o m e n t u m / / j ! ' " " ^ l O G c V w i t h respect t o t h e b e a m line. An e x a m p l e of a C C event c a n d i d a t e o b s e r v e d w i t h t h e III d e t e c t o r is displayed in Fig. 7.-r). In t h e p l a n e o r t h o g o n a l t o t h e b e a m line, t h e .(//-plane, we see very clearly t h e m o m e n t u m i m b a l a n c e , s i n c e t h e ant ¡ n e u t r i n o lias e s c a p e d u n d e t e c t e d . There is h a d r o n i c a c t i v i t y with l a r g e t r a n s v e r s e m o m e n t u m in t h e lower region of t h e d e t e c t o r , a n d s o m e e n e r g y d e p o s i t s a r e o b s e r v e d very close t o t h e b e a m p i p e in t h e p r o t o n d i r e c t i o n s t e m m i n g f r o m t h e p r o t o n r e m n a n t . A f u r t h e r event
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solcet ion criterion is t o rest i iet t lie inelast icity y to a region, for e x a m p l e , . .'I 0.85. w h e r e a good kincuiat ic r e c o n s t r u c t i o n is possible. In this case t h e k i n e m a t i c reconstruction h a s to h e p e r f o r m e d using i n f o r m a t i o n from t h e h a d r o u i c final s t a t e only, a s described in Section 0.5.1. T h e b a c k g r o u n d s a r e more diflicult to re ect in this case. T r a n s v e r s e m o m e n t u m i m b a l a n c e c a n be caused by events where t h e s c a t t e r e d p o s i t r o n is not identified, by p h o t o - p r o d u c t i o n e v e n t s with Q2 0 w h e r e t h e positron escapes u n d e t e c t e d into t h e b e a m pipe, or by e v e n t s which a r e not induced by e + p collisions, such a s cosmic ray interactions. Because of t h e m u c h lower cross section a n d t h e t i g h t e r selection c u t s that a r e needed in order t o re ect b a c k g r o u n d s , t he n u m b e r of selected C C e v e n t s for a luminosity of about 4 0 p b 1 is of t h e o r d e r of 1000 only, to be c o m p a r e d to 7.r>000 C events. T h i s leads to larger stat ist ical uncertaint ies. Also t h e systemat ic uncertaint ies a r e a factor of two larger t h a n for a n a l y s e s of C i n t e r a c t i o n s . ecent m e a s u r e m e n t s are s u m m a r i z e d in p u b l i c a t i o n s by t h e H A e x p e r i m e n t s (III Collab., 2000: Z S Collab.. 2000«). Figure 7.0 shows t h e reduced C C cross section crcv. which u p to olectroweak corrections is equal t o c c , m e a s u r e d by Z S a s a funct ion of x in ( 2 bins from ~ = 280 G e V 2 to 2 = 17000 G e V 2 . G o o d a g r e e m e n t is seen w i t h a QC fit. Also shown a r e p a r a m e t e r i z a t i o n s of t h e p.d.f.s e x t r a c t e d by various groups, c.f. Section 7.5. As ant icipated, t h e cross section at large x is d o m i n a t e d by t h e s c a t t e r i n g off a d- uark. For lower x t h e cross section rises because of t h e increasing c o n t r i b u t i o n f r o m ft a n d c q u a r k s f r o m t h e sea. At t h e s a m e t i m e eland s - q n a r k s get suppressed b e c a u s e of t h e (1 - y)2 factor. e m e m b e r that low x c o r r e s p o n d s t o large y values. An interesting aspect t o st.udv in C a n d C C d e e p inelastic s c a t t e r i n g is t h e helicitv d e p e n d e n c e of t h e cross sections at large . . T h e e x p e r i m e n t s at. II IC A a r e sensitive t o t h e c o n t r i b u t i o n s from Z a n d V exchange, a n d t h u s to olectroweak effects. In p a r t i c u l a r , only left,(right ) - h a n d e d (ant.i)qnarks p a r t i c i p a t e in tin weak part of t h e interact ion. Since c e r t a i n spin c o n f i g u r a t i o n s a r e f o r b i d d e n by a n g u l a r m o m e n t u m conservation, a s shown in F x . (7-1). a n a s y m m e t r y in t h e p o s i t r o n s c a t t e r i n g angle 0'. defined in t h e p o s i t r o n - q u a r k C.o.M. s y s t e m a p p e a r s . T h e weighting f a c t o r (1 — y ) 2 in cc for d o w n - t y p e q u a r k s c a n be u n d e r s t o o d b e c a u s e of t h e relation eos (0 2) = I — y. In Fig. 7.7 t h e 111 m e a s u r e m e n t of t h e cross section t e r m < '( < is shown as a function of (1 y ) 2 . for various bins in t h e large x region. In leading o r d e r we expect a d e p e n d e n c e p r o p o r t i o n a l to (I - y ) 2 from positron q u a r k (d.s) s c a t t e r i n g , a n d an isotropic d i s t r i b u t i o n f r o m positron antiqiiark (u,c) s c a t t e r i n g . In fact, we observe a n almost linear d e p e n d e n c e of cc- with a finite offset, which decreases with increasing x. T h e r e f o r e t h e s e m e a s u r e m e n t s can help t o c o n s t r a i n s t r o n g l y t h e various q u a r k c o n t r i b u t i o n s in t h e large x region. Finally, t h e m e a s u r e m e n t s of t h e ( 2 d e p e n d e n c e of t he C a n d C C cross sections u p t o t h e highest values of 2 allow for a b e a u t i f u l visualization of t he unification of e l e c t r o m a g n e t i c a n d weak interactions. T h i s is illustrated in Fig. 7.8. At low Q 2 virtual p h o t o n e x c h a n g e d o m i n a t e s t h e C i n t e r a c t i o n s , a n d
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C C e v e n t s a r e s u p p r e s s e d by m a n y o r d e r s of m a g n i t u d e . However, wit h increasing Q 2 b o t h cross sections a p p r o a c h each o t h e r , showing that, t h e e l e c t r o m a g n e t i c 2 a n d weak c o n t r i b u t i o n s b e c o m e of similar size. o t e t h e fact t h a t for large t h e C'C cross section for e l e c t r o n s is higher t h a n for positrons. T h i s is b e c a u s e for electrons a W is e x c h a n g e d , which couples t o u p - t y p e q u a r k s . T h o s e a r e m o r e a b u n d a n t in a p r o t o n i hau d o w n - t y p e q u a r k s . In a d d i t i o n , for positron s c a t t e r i n g t h e helicity s t r u c t u r e of t h e interaction leads t o a n a d d i t i o n a l suppression in c e r t a i n p h a s e - s p a c e regions, as discussed a b o v e . In t h e highest, 2 region also t h e C cross section is larger for electrons t h a n for p o s i t r o n s . In this region t h e interference between p h o t o n a n d Z e x c h a n g e becomes relevant, which explains t h e observed a s y m m e t r y . Any deviat ions of t h e m e a s u r e m e n t s f r o m t h e S t a n d a r d Model prediction at, t h e largest 2 values would indicate t h e possible a p p e a r a n c e of new physics, such as t h e e x c h a n g e of a l e p t o - q u a r k instead of a p h o t o n . cpto-quarks arc
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h y p o t h e t i c a l p a r t i c l e s w h i c h c a r r y l e p t o n a n d b a r v o n q u a n t u m n u m b e r s . S o far n o i n d i c a t i o n s for such p a r t i c l e s h a v e b e e n f o u n d . 7.1.3
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So far we h a v e l e a r n e d a b o u t s t r u c t u r e f u n c t i o n m e a s u r e m e n t s in t h e m e d i u m a n d l a r g e Q 2 r e g i m e a t 11 H A. W e h a v e seen t h a t t h e r e is a s t r o n g rise in t h e e l e c t r o m a g n e t i c s t r u c t u r e f u n c t i o n I'-zi-' - Q 1 ) w h e n g o i n g t o s m a l l x , . T ^ . I . In t h i s r e g i m e t h e c o n t r i b u t i o n f r o m v a l e n c e q u a r k s is s m a l l , w h e r e a s g h i o n s a n d q u a r k s f r o m t h e s e a d o m i n a t e . In f a c t , t h e s t r o n g rise in F>(x. 2). which is of I he f o r m F> x x~x. A > 0. is a t t r i b u t e d t o a s t r o n g rise in t h e g l u o n a n d s e a q u a r k d i s t r i b u t i o n s . T h i s rise is a l s o g e n e r a t e d by t h e G A evolution equations, 2 if r e s t r i c t e d t o t h e p c r t u r b a t i v e r e g i m e Q ^ , 1 G e V ' . In t h e small-. ' r e g i o n t h e
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e v o l u t i o n is g o v e r n e d by (7.H) w h e r e P,IK is t h e Altarelli Parisi s p l i t t i n g f u n c t i o n ¡is d e f i n e d in S e c t i o n 3.2.2, a n d ij(.r) is t h e g l n o n d i s t r i b u t i o n . We u s e h e r e t h e s h o r t - h a n d n o t a t i o n W for t h e c o n v o l u t i o n i n t e g r a l a s e x p l a i n e d in e q n (3.49). In o r d e r t o solve tlie e(|iiatioii, lr>(.r. Q2) h a s t o be p a r a m e t e r i z e d a s a funct ion of x at s o m e s t a r t i n g s c a l e Qf)t s i n c e in c o n t r a s t t o t h e e v o l u t i o n t h e a b s o l u t e value of F ^ J : , Q f i ) c a n not b e p r e d i c t e d by p e r t u r b a t i v e Q C D . A t t h i s s t a g e several i n t e r e s t i n g ( j u e s t i o u s a r i s e . W h e n a p p r o a c h i n g t h e very low Q2 region, Q2 =s 1 G e V " a n d below, t h u s e n t e r i n g t h e n o n - p e r t u r b a t i v o r e g i m e , d o e s t h i s rise in F> for low x p e r s i s t ? O n t h e o t h e r h a n d , w h e n s t a y i n g in a r e g i m e w h e r e p e r t u r b a t i v e Q C D is a p p l i c a b l e , d o tin- D G L A P e v o l u t i o n s e q u a t i o n s give t h e correct p r e d i c t i o n for t h e e x t r e m e l y s m a l l l ( ) - - { region, w h e r e l n ( l / . r ) t e r m s might b e c o m e i m p o r t a n t ? Finally, w h a t is t h e a s y m p t o t i c b e h a v i o u r of F> for x —< (.)? S o m e t h e o r e t i c a l i n t r o d u c t i o n t o t h e s e t o p i c s is alr e a d y given in Sect ion 3.6. H e r e w e look at t he s u b j e c t a l s o f r o m t h e e x p e r i m e n t a l point of view. In o r d e r t o t a c k l e t h e first q u e s t i o n , w e d e f i n i t e l y h a v e to a d v o c a t e n o n p e r t u r b a t i v o m o d e l s . D e e p inelastic l e p t o n p r o t o n s c a t t e r i n g a t Q 2 £> I G e V " is r e l a t e d t o t he c r o s s s e c t i o n for t h e s c a t t e r i n g of a virt ual p h o t o n 7* a n d a p r o t o n . a'' which is pro|>ortional t o F>- A p o s s i b l e m o d e l for t h e d e s c r i p t i o n of 7 " p s c a t t e r i n g is b a s e d o n t h e so-called Generalized Vector Meson Dominance ( G V M D ) m o d e l ( S a k u r a i a n d Schildknecht., 1972) c o m b i n e d w i t h i d e a s f r o m Rogge t h e o r y i n t r o d u c e d in S e c t i o n 3.2.3. At very low (} 2 t h e v i r t u a l p h o t o n f l u c t u a t e s i n t o v e c t o r m e s o n s t a t e s , for e x a m p l e , /».u;,(f>. w h i c h t h e n interact with t h e p r o t o n . Rogge t h e o r y p r e d i c t s t h e e n e r g y d e p e n d e n c e of t h e t o t a l c r o s s s e c t i o n for l i a d r o n h a d r o n s c a t t e r i n g . ÎT 1 "-'" oc / I s " " - '
-I- B s a v ~
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w h e r e s is t he e n e r g y s q u a r e d of t h e h a d r o n h a d r o n i n t e r a c t i o n , o m : O.fi is t h e [{e.fiycoii c o n t r i b u t i o n , d e s c r i b i n g t h e e x c h a n g e of h a d r o n i c s t a t e s w i t h d e f i n i t e ( | u a n t u m n u m b e r s s u c h a s v e c t o r m e s o n s , a n d ou> « 1.1 is dm- t o Pomcron e x c h a n g e , a n a p p a r e n t l y colour-less m u l t i - g l u o n s t a t e w i t h t h e q u a n t u m n u m b e r s of t h e v a c u u m . T h e r e l a t i o n t o d e e p i n e l a s t i c s c a t t e r i n g at low x is given by t h e f a c t t h a t t h e e n e r g y s q u a r e d I F " for t h e -7*p (or v e c t o r - m e s o n p r o t o n ) scat t e r i n g c a n b e w r i t t e n a s I F 2 « Q2/x. e q n (2.4). P u t t i n g all t hese i n g r e d i e n t s t o g e t h e r , a b e h a v i o u r for F-z such as (7.10)
is e x p e c t e d , w h e r e t h e first t e r m in b r a c k e t s is m o t i v a t e d by G V M D , a n d t h e s e c o n d bv Rogge t h e o r y . W e see t h a t F>(x — 0 ) oc ". wit h a rat h e r s m a l l
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A,.ir ~ •). 1. So a very soft, a n d ( / ^ - i n d e p e n d e n t .r-behnviour is p r e d i c t e d , without a n y s t r o n g rise. In o r d e r t o test these predictions experimentally, the e x p e r i m e n t s at II ICR A have u p g r a d e d t heir d e t e c t o r s a n d used d e d i c a t e d 11 ERA runs. Very small Q- values have to b e m e a s u r e d , which is equivalent to detect ing t h e o u t g o i n g positron or electron a t very small angles. Special d e t e c t o r s very close t o t h e b e a m pipe have
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been installed, such a s t h e Beam Pipe Calorimeter (13PC) a n d t h e Beam Pijx Teacher ( B P T ) iu t h e case of Z E U S . T h e s e d e t e c t o r s a r e positioned at a d i s t a n c e of a b o u t 3 in f r o m t h e m a i n i n t e r a c t i o n vertex, iu t h e p o s i t r o n - b e a m d i r e c t i o n . A n o t h e r w a y of accessing s m a l l e r s c a t t e r i n g a n g l e s is t o shift t he i n t e r a c t i o n ver m tex by a b o u t 70 cm in t he p r o t o n - b e a m direction, as was d o n e for a s h o r t period ii mil 1995. P r e v i o u s l y u n e x p l o r e d regions such a s 0.045 G e V 2 < Q2 < 0.05 G e V 2 a n
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li 10 7 < .J- < 10 have I litis been s t u d i e d ( Z E U S C'ollab. l!l!l!)/>, 2000/;; I I I Collab. I0!)7i/). T h e main b a c k g r o u n d for t h e s e m e a s u r e m e n t s is p h o t o - p r o d u c t i o n , where t h e positron escapes u n d e t e c t e d into t h e b e a m pipe, a n d a signal is lakcd in the low angle d e t e c t o r s by e l e c t r o m a g n e t i c showers induced mainly by TT {I S. T h e e x t r a c t i o n of t h e s t r u c t u r e f u n c t i o n a f t e r t h e measurement, of t h e d o u b l e differential cross section follows mainly t h e lines described in Section 0.5.1. In Figs. 7.!) a n d 7.10 t h e m e a s u r e m e n t s of F>(x,Q2) a r e displayed, f r o m 2 2 2 2 Q 17 G e V down to Q = 0.015 G e V . a s o b t a i n e d by t h e 11 ICR A e x p e r i m e n t s a n d by t h e fixed-target experiment' E 0 0 5 (1!)!)(>) at F E R M I L A B . A d e c r e a s e of t h e rise for low x is very nicely visible when going to smaller a n d smaller Q2 values. Iu t h e p e r t u r b a t i v e region Q2k, 1 G e V 2 a N L O Q C D lit, d e s c r i b e s t h e d a t a well, whereas t h e Regge model predicts a t o o soft. .r-dependencc. However, t h e l a t t e r m o d e l , eqn (7.10). h a s been lilted t o t h e d a t a ( Z E E S C'ollab.. 20006) 147.8 ± l.tiy/b. at Q2 < 1 G e V * . a n d indeed a good lit is o b t a i n e d , with A OIK = 0.5. B 02.0 ± 2.3/ib. o r = 1.102 ± 0.007 a n d = 0.52 ± 0.04 G e V 2 . T h e q u o t e d u n c e r t a i n t i e s c o m b i n e s t a t i s t i c a l a n d s y s t e m a t i c errors. T h e t r a n s i t i o n f r o m t h e p e r t u r b a t i v e t o t h e Regge b e h a v i o u r for l ' \ has been s t u d i e d by m e a s u r e m e n t s of A,.|r = d hi F>/d l n ( l / ; r ) , o b t a i n e d by lilting F-> = o.r A '" t o Z E U S a n d E 0 0 5 d a t a with x < 0.01 ( Z E U S C'ollab., l!)!)9/j). For Q2 < 1 G e V " t h e d a t a indicate a ( ^ - i n d e p e n d e n t p a r a m e t e r A,.|f 0.1. as suggested 2 by Regge theory. At larger Q values A,.ir increases with increasing Q2 t o values a r o u n d A,.ir ~ 0.3. a n d t h e d a t a a r e nicely described by a N L O Q C D lit based on t h e D G L A P evolution e q u a t i o n s . It is w o r t h n o t i n g t h a t , for very small x. t e r m s of t h e f o r m ( o s l n ( l / . r ) ) " b e c o m e relevant for every o r d e r in t h e p e r t u r b a t i v e series. T h e D G L A P evolution e q u a t i o n s only resuin logarithmic t e r m s ( o s I n ( ? " ) " . A r e s n n i m a t i o n of t h e l o g a r i t h m s (ln(l/.'•'))" is achieved by t h e H F K L e q u a t i o n s , see Section 3.0, which iu leading o r d e r predict A,.ir s i 0.4 0.5. However, s t u d i e s of N L O c o n t r i b u t i o n s t o this prediction indicate r a t h e r large corrections, so that, f u r t h e r development in this a r e a is t o be awaited before final conclusions can be d r a w n . At this stage t h e third question about t h e a s y m p t o t i c b e h a v i o u r is t o be faced. At. large Q2 t h e rise in t h e gluon d i s t r i b u t i o n is s t r o n g , a n d at lower Q2 t h e sea increases steadily. However, t h e rise cannot persist for x • 0. as we hit basic physical b o u n d a r i e s . If t h e n u m b e r of part.ons within t h e p r o t o n b e c o m e s very large, t h e y c a n n o t b e regarded a s free particles a n y more, as a s s u m e d in t h e p a r ton model. Dynamical m e c h a n i s m s such a s gluon r e c o m b i n a t i o n should lead t o p a r t o n s a t u r a t i o n , c.f. Section 3.0.0.4. T h i s is a developing Held of Q C D . a n d f u t u r e precise IIERA d a t a should help t o c o n s t r a i n f u r t h e r tin- various models which a r e on t h e market.. 7.1.4
The gluon density
in the
¡noton
A s t a n d a r d m e t h o d t o infer t h e gluon d i s t r i b u t i o n in t h e p r o t o n f r o m DIS d a t a is t o p e r f o r m a global a n a l y s i s of scaling violations in t h e s t r u c t u r e f u n c t i o n Fi(x. (¿2). T h i s will be described in detail in Section 7.5. However, this m e t h o d
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in t h e p r o t o n by 111. based product ion (b). A c o m p a r scaling violations. F i g u r e s (b).
t wo or m o r e j e t s wit hin t h e a n g u l a r a c c e p t a n c e have been m e a s u r e d as a funct ion of variabilis such as t he j e t t r a n s v e r s e energy. T h e f r a c t i o n of di-jet. e v e n t s in N C DIS varies s t r o n g l y wit h Q2. between 1% at. Q2 = . r )GeV 2 t o ~ 20% ¡it Q2 = JiOOOGeV". W h e n rei|iiiring ¡in a p p r o x i m a t e relative t r a n s v e r s e m o m e n t u m of k r > l O G e V between j e t s , using a D u r h a m t y p e jet. linder. Ill finds a b o u t 11000 (2000) di-jct e v e n t s in t h e low (high) Q 2 s a m p l e . T h e r a t e s a r e m e a s u r e d within s y s t e m a t ic u n c e r t a i n t i e s of a b o u t 10%. m a i n l y d o m i n a t e d by t h e u n c e r t a i n t i e s on t he c a l o r i m e t e r energy scales. T h e m e a s u r e d r a t e s a r e well described by M o n t e C a r l o s i m u l a t i o n s based on pert.urbat.ive Q C D calculations. T h i s gives confidence for t h e e x t r a c t i o n of t h e p.d.f.s i] l '(.r) in t h e p r o t o n a c c o r d i n g t o ]). T h e coefficients r K a n d cv a r e «•¡deniable in pert.urbative Q( 'I) T h e only remaining u n k n o w n s a r e o s a n d ;/(x). T a k i n g t h e world average Value of o s , ¡i direct d e t e r m i n a t i o n of y(x) is possible. T h e 111 d a t a allow for such ¡1 d e t e r m i n a t i o n within t h e range 0.01 < x < 0.1. T h e result o b t a i n e d for a factorization scale ¡I 2 , = 2 0 0 G e V ~ is shown in Fig. 7.12 (a), w h e r e t h e s h a d e d b a n d reflects ex-
S T U I I C T U U I : I H N C I IONS A N D I - A K T O N DIS I K I I U I I I O N S
p o n n i o n t a l a n d t h e o r e t i c a l u n c e r t a i n t i e s . G o o d a g r e e m e n t is o b s e r v e d with tin« r e s u l t s o b t a i n e d f r o m global a n a l y s e s of sealing violations. By i n t e g r a t i n g iy(x) 2 it is found that, at a scale of 200 G e V g l n o n s c a r r y a b o u t 2 3 % of t h e p r o t o n m o m e n t u m over t h e m e a s u r e d .r-interval. A n o t h e r direct t e s t of t h e gluon d i s t r i b u t i o n is o b t a i n e d f r o m t h e measurement of t h e p r o d u c t i o n r a t e of heavy q u a r k s . Heavy q u a r k p r o d u c t i o n proceeds almost exclusively via boson gluon fusion, a s s h o w n in Fig. 7.11 (e). where the t w o q u a r k lines now represent a e- or b - q u a r k . A d d i t i o n a l theoretical input, is needed in o r d e r t o d e s c r i b e t h e t r a n s i t i o n of a heavy q u a r k t o a heavy meson, such a s c -> D*. which is d e t e c t e d in t h e e x p e r i m e n t . For t h i s p u r p o s e plienoinenological f r a g m e n t a t i o n f u n c t i o n s a r e e m p l o y e d , which model t h e fraction of t h e q u a r k ' s e n e r g y t r a n s f e r r e d t o t h e h e a v y m e s o n . An e x a m p l e is given by t h e P e t e r s o n <1
K
(1!)!)!)«) h a s p e r f o r m e d such a m e a s u r e m e n t for N C e v e n t s . In a d a t a s a m p l e of C = 9.7 pi) ' t h e y find e s t i m a t e d efficiency for r e c o n s t r u c t i n g t h e s c a t t e r e d of a r o u n d 4 2 % T h e gluon d i s t r i b u t i o n is e x t r a c t e d
[ , , < * „ . / / 2 ) a ( : r g . , i * ) A [ : r f \ ; « : K . / / 2 ) ] -I- <7„(.T*>bs) .
(7.12)
w h e r e s) is t he m e a s u r e d D* cross sect ion a s a f u n c t i o n of t h e r e c o n s t r u c t e d gluon m o m e n t u m f r a c t i o n . .r R is t h e t r u e m o m e n t u m f r a c t i o n . i y ( j ' g , / / 2 ) is t h e gluon d i s t r i b u t i o n at a given scale p2. a{xK.p2) is t h e p a r t o n i c cross section for c h a r m q u a r k p r o d u c t i o n f r o m boson gluon fusion, a n d s w h e n t h e a c t u a l t r u e value was xK. T h i s m a t r i x is o b t a i n e d f r o m M o n t e C a r l o s i m u l a t i o n s . T h e a v e r a g e scale / / J relevant for t he III m e a s u r e m e n t is d e t e r m i n e d t o be 25 G e V 2 . T h e r e s u l t s of t h i s a n a l y s i s
NH
I K I N ' I NIK I.I O N SCA'I I C K I N O
277
a r e displayed in Fig. 7.12 (b) for DIS a n d p h o t o p r o d u c t i o n ( 7 p ) . W i t h i n t h e statistical a n d s y s t e m a t i c u n c e r t a i n t i e s good agreement, is f o u n d with t h e indirect d e t e r m i n a t i o n s f r o m scaling violation a n a l y s e s . 7.2
Neutrino
nucleoli s c a t t e r i n g
After t h e discussion of c h a r g e d l e p t o n nucleoli scat tering, we now focus on neut r i n o nucleoli o r equivalently neut ral lepton nucleoli scat tering. T h e r e a c t i o n of interest is i^(Pf)N — C(f.)X. w i t h a n i n c o m i n g ( a n t i ) n c u t r i n o s c a t t e r i n g on a nucleoli N. which m a y a c t u a l l y bo inside a nucleus such as iron, a n d t h e corr e s p o n d i n g lepton as well a s a h a d r o n i c s y s t e m A' in t h e final s t a t e . T h u s , t h e s a m e sit u a t i o n a s d e p i c t e d in Fig. 2.1 is given, except, t h a t t h e i n c o m i n g c h a r g e d lepton is replaced by a n e u t r i n o , a n d t h e e x c h a n g e d v i r t u a l p h o t o n by a W boson. Note also t h a t by i n t e r c h a n g i n g t h e o r d e r of t h e n e u t r i n o a n d t h e l e p t o n . the c o n f i g u r a t i o n for c h a r g e d c u r r e n t DIS is r e s t o r e d . T h e p a r t i c u l a r interest in n e u t r i n o nucleoli s c a t t e r i n g is u n d e r s t o o d f r o m t h e following o b s e r v a t i o n . T h e n e u t r i n o couples t o o t h e r particles only t h r o u g h weak i n t e r a c t i o n s , via t h e e x c h a n g e of a W boson, which can p r o b e different, q u a n t u m n u m b e r s of t h e p r o t o n ' s c o n s t i t u e n t s c o m p a r e d t o a v i r t u a l p h o t o n . In p a r t i c u l a r , il is possible t o distinguish between particles a n d a n t i p a r t i c l e s . In I his sect ion we will c o n c e n t r a t e on c h a r g e d c u r r e n t interact ions. However, neut ral current, processes w h e r e a Z is e x c h a n g e d w i t h consequent ly a n e u t r i n o instead of a lepton in t h e final s t a t e , can o c c u r a n d a r e of interest- for precision t e s t s of t h e S t a n d a r d Model of elcctroweak i n t e r a c t i o n s . As a n e x a m p l e , t h e weak m i x i n g a n g l e s i n 2 f l w is m e a s u r e d in such e x p e r i m e n t s . A very c o m p r e h e n s i v e review on m e a s u r e m e n t s w i t h high e n e r g y n e u t r i n o b e a m s can be f o u n d in t h e l i t e r a t u r e ( C o n r a d c.t til.. 1998). which s u m m a r i z e s e x p e r i m e n t a l t e s t s of Q C D a s well a s elcctroweak theory. Let us lirst .summarize briellv t h e k i n e m a t i c s a s well a s t h e relevant cross sections a n d s t r u c t u r e f u n c t i o n s before going t o a d e s c r i p t i o n of t h e e x p e r i m e n t a l results. T h e relevant Lorentz-invariant. q u a n t i t i e s a r e t h e s a m e as for c h a r g e d lepton DIS, n a m e l y t h e m o m e n t u m t r a n s f e r Q2 = -(¡2. t h e Bjorkon scaling variable :r = Q*f{2p-q), t h e inelasticity y = (p•'¡)/(¡>• I) a n d t h e e n e r g y t r a n s f e r " (/'" w h e r e M is t h e p r o t o n m a s s . All n e u t r i n o e x p e r i m e n t s a r e fixedt a r g e t e x p e r i m e n t s , w h e r e t h e nucleoli is a t rest, t o a very g o o d a p p r o x i m a t i o n . T h e r e f o r e , in t h e l a b o r a t o r y f r a m e t h e k i n e m a t i c variables c a n be e x p r e s s e d in t e r m s of t h e m e a s u r e d e n e r g y E,, a n d a n g l e <),, of t h e o u t g o i n g lepton (typically a union) a s well a s of t he e n e r g y E\\ of t h e h a d r o n i c s y s t e m :
" -
E
< <
• » -
w
h
.
• * -
-
w
• -
-
•
™
w h e r e we have used a s m a l l - a n g l e a p p r o x i m a t i o n a n d E„ E,, f E\\. neglect ing t h e p r o t o n rest e n e r g y in t h e initial s t a t e . T h e C . o . M . e n e r g y s q u a r e d is s (2E„M + A / 2 ) ss 2 E „ M .
SI KMC r u m
!• U N C T I O N S A N D I ' A K T O N I M S I I t l l t l
I IONS
T h e cross section for n e u t r i n o nucleoli s c a t t e r i n g is again expressed in t e r m s of s t r u c t u r e f u n c t i o n s . dV"">N
G?. C I ( 2 zx W
Mi' V + Ml-J
. l - y - ( ^ )
2
2 + 2/?' / ' (l>>N (.r. Q~)
)
,
i x f j ' ^ ^ I f O - l )
(7.M)
w h e r e t h e ± is + ( ) for /'(/>) s c a t t e r i n g . Instead of F\ (x. Q-) here we have used t h e s t r u c t u r e f u n c t i o n /?/.. which can he i n t e r p r e t e d a s t h e r a t i o of t h e longit udinal a n d t r a n s v e r s e virtual boson a b s o r p t i o n cross sect ion.
err
2xFi (x,
Q2)
In of is as
leading o r d e r Q C D a n d neglecting t h e p r o t o n m a s s we have /?/. = t h e Callan G r o s s relation F? = 2xF\. In Ex. (7-2) t h e cross section c o m p a r e d t o t h e cross section for charged current lepton nucleoli discussed in Section 7.1.2. In t h e naive q u a r k - p a r t o n model t h e n e u t r i n o s t r u c t u r e f u n c t i o n s f u n c t i o n s o f . r . a n d c a n be expressed iu t e r m s of p.d.f.s. F';l'(x) xF^'(x)
0 because eqn (7.14) scattering a r e simple
= 2x I d ( x ) + s(x) + u(x) + r(.r)] = 2x [
(7.10) .
(7.17)
T h e s e a r e valid for n e u t r i n o p r o t o n s c a t t e r i n g . We o b s e r v e t h a t only negatively charged p a r t o n s c o n t r i b u t e t o t h e interaction. This is b e c a u s e in n e u t r i n o s c a t t e r i n g a W ' + is e x c h a n g e d . By isospin s y m m e t r y t h e s t r u c t u r e f u n c t i o n s for n e u t r i n o n e u t r o n s c a t t e r i n g a r e simply o b t a i n e d bv replacing in the* a b o v e expressions t h e d- (u-) q u a r k s by t h e u- (d-) q u a r k s . T h e sea q u a r k d i s t r i b u t i o n s , s(x).e(x). a r e a s s u m e d t o be t h e s a m e in p r o t o n s a n d n e u t r o n s . T h e s t r u c t u r e f u n c t i o n s for an isosealur target a r e o b t a i n e d by t a k i n g t h e average of t h e p r o t o n a n d n e u t r o n s t r u c t u r e f u n c t i o n s , since a n isoscalar target h a s a n equal n u m b e r of p r o t o n s a n d n e u t r o n s . T h e result is F.f(.r) xF^(x)
= x [u(x) + d(x) + 2s(x)
+ u(x) + d(x) + 2c(.r)j
= x [»(as) + d(x) + 2.s(x) - v(x)
- d(x) - 2c(x)] .
(7. IS) (7.1»)
Ant ¡neutrino scat t e r i n g o c c u r s via t h e e x c h a n g e of a \Y . t herefore only positively charged p a r t o n s c a n c o n t r i b u t e t o t h e process. If we a s s u m e .s(.r) = .s(r) = xFg" - l.r ¡.s(.r) - c(x)]. a n d c(x) = r(.r). we find f T N = F%N a n d xF^ Looking at t h e a b o v e expressions we can s u m m a r i / e : FX
a n d / ' T n m e a s u r e t h e s u m of q u a r k a n d a n t i q i i a r k d i s t r i b u t i o n s .
NEUTHINO
• xFn uv(x)
NUDLISON SCATREIUNC!
MI
[/2(xF'i'fi I rF\'N) m e a s u r e s t h e s u m of valence q u a r k d i s t r i b u t i o n s = u(x) fi(x) a n d dv(x) d(x) tl(x).
• T h e difference A(xFn) x/•"'('N c h a r m c o n t e n t of t h e nucleoli.
xF'{*
is sensitive t o t h e s t r a n g e a n d
• Comparing F a n d c o n s t i t u t e s a test of t h e fractional c h a r g e assignment to q u a r k s a n d gives sensitivity t o t h e s t r a n g e q u a r k d i s t r i b u t i o n (see Ex. 7-.'i). Iu a d d i t i o n , t h e search for two unions in t h e final s t a t e gives a direct h a n d l e t o t e s t t h e s t r a n g e q u a r k c o n t r i b u t i o n . T h i s m e a s u r e m e n t is described iu m o r e detail below. O f course, t h e s t r u c t u r e f u n c t i o n s a b o v e a r c modified b e c a u s e of s t r o n g interactions. which g e n e r a t e scaling violations, t h a t is. t h e p a r t o n d i s t r i b u t i o n s a r e functions o f . r a n d Q~ a n d sensitivity t o t h e gluon d i s t r i b u t i o n is o b t a i n e d f r o m their Q1 d e p e n d e n c e . An o f t e n used n o t a t i o n is F2(x.(f) = x xF-,(x.Q-)
= x
J2
'/.(•'•• Q - )
Y .
I Ux.Cf)
- Q i M
2
)
(7.20)
,
(7.21)
where f/,(.r. Q1) s t a n d s for a (piark distribut ion. Note that, in t h e D I S factorizat ion s c h e m e (Altarclli rt id., li)7X). Section .'i.ti. this expression r e m a i n s valid also beyond leading o r d e r in n s . 7.2.1
Expcrhne.nUil
issues
N e u t r i n o b e a m s a r e g e n e r a t e d by firing a high i n t e n s i t y p r o t o n b e a m i n t o a t a r g e t , such a s beryllium. In t h e i n t e r a c t i o n s w i t h t h e target pious a n d kaoiis a r e p r o d u c e d , a large f r a c t i o n of which d e c a y scinilept.onically into u n i o n s a n d uiuon-ncut.rinos, u,,. As a n e x a m p l e , t h e n e u t r i n o b e a m for t h e f ' C E R e x p e r i m e n t at. E E R M I L A B consisted in a b o u t 86.4% //„. lf.:5% u „ . a n d 2.:?% v, a n d u c . D e p e n d i n g on whet her t h e r e a r c sign- a n d m o m e n t u m selecting m a g n e t s behind t h e t a r g e t , t h e n e u t r i n o b e a m can cover a n a r r o w or wide energy range, s u c h as from .'50 to 3 0 0 G e V for t h e C ( T K e x p e r i m e n t . T h e x a n d Q~ r a n g e s typically covered a r e 0 . 0 1 < x < 0 . 8 a n d 0.1 G e V 2 < Q2 < 100 G e V 2 , as shown in Fig. 7.1. T h e accessible r a n g e is set by t h e b e a m energies a n d by t h e e x p e r i m e n t a l acceptance. T h e effect of t h e small i n t e r a c t i o n cross section. er"]'r/E„ « 0.7 x 1 0 _ - i s c n i 2 / G e V . is o v e r c o m e by m o d e r n e x p e r i m e n t s t h r o u g h t h e use of high-intensity b e a m s coupled to massive d e t e c t o r s , resulting in luminosities of a b o u t 10'"' c m " " s - 1 a n d c o r r e s p o n d i n g l y in collected d a t a s a m p l e s of u p t o a million e v e n t s . A short description of a typical n e u t r i n o fixed-target d e t e c t o r has a l r e a d y been given in Section ¡5. An important, aspect of t h e d e t e c t o r is t h e a b s o r p t i o n m a t e r i a l , o f t e n also employed as a b s o r b e r for t h e c a l o r i m e t e r . J u s t t o m e n t i o n a few. iron is u s e d b y C ( T R
( n o w N U T K V ) at E E H M I L A B a n d C D I I S W a t C E R N ,
marble
S I LUKTUHL
I 11N< ' I ' L O N S A N D P A I ( T O N I ) I S T I ( I U W T I O N S
(CaCO.-j) by C H A K M . glass l>y C I I A K M I I a n d n e o n o r d e u t e r i u m by HliMC. nil l o c a t e d a t (lie C K U N SI'S n e u t r i n o b e a m . W h e r e a s m a r b l e a n d d e u t e r i u m a r e isoscalar t a r g e t s , iron is n o t . T h e r e f o r e t h e s t r u c t u r e f u n c t i o n s o b t a i n e d f r o m iron lirst h a v e t o In- c o r r e c t e d b e f o r e b e i n g c o m p a r e d t o paramct.criznt.ions s u c h a s in e q n s ( 7 . I S ) a n d (7.19). F u r t h e r m o r e , b e c a u s e of tin' s c a t t e r i n g 011 h e a v y t a r g e t s , n u c l e a r e f f e c t s h a v e t o b e t a k e n i n t o a c c o u n t , in p a r t i c u l a r w h e n e x t r a c t i n g s t r u c t u r e f u n c t i o n s . An e x a m p l e is gliton r e c o m b i n a t i o n , which c a n o c c u r in a l a r g e n u c l e u s b e t w e e n p a r t o n s of n e i g h b o u r i n g n u c l e o n s . l e a d i n g t o a n A ( = a t o i n i c n u m b e r ) d e p e n dent d e p l e t i o n of low-.r p a r t o n s (Nikolaov a n d Z a k h a r o v . 1975: Mueller a n d Q i u . 1980). O t h e r e f f e c t s a r e F e r m i m o t i o n of n u c l e o n s in t h e n u c l e u s , o r t h e E M C e f f e c t , w h i c h is a s u p p r e s s i o n of s t r u c t u r e f u n c t i o n s f r o m high-.4 t a r g e t s c o m p a r e d l o d e u t e r i u m for 0.2 < x < 0.7. A d i s c u s s i o n of t h e s e e f f e c t s a n d o t h e r s s u c h a s t a r g e t a n d "higher-twist,' e f f e c t s c a n b e f o u n d in t h e review by C o n r a d a n d c o l l a b o r a t o r s (1998) a n d r e f e r e n c e s t h e r e i n . An i m p o r t a n t e x p e r i m e n t a l issue is t h e precise d e t e r m i n a t i o n of t he n e u t r i n o llnxes. in o r d e r t o get t h e n o r m a l i z a t i o n for t h e c r o s s s e c t i o n m e a s u r e m e n t s r i g h t . Of c o u r s e , a d i r e c t m o n i t o r i n g of n e u t r i n o b e a m s is difficult , t h e r e f o r e v a r i o u s indirect met h o d s a r e e m p l o y e d . O n e met h o d c o n s i s t s in m o n i t o r i n g t h e p r o d u c e d s e c o n d a r y pion a n d kaon b e a m s , w h e r e a s a n o t h e r m e t h o d is b a s e d on t h e m e a s u r e m e n t of t h e 1/ d e p e n d e n c e of t h e c r o s s s e c t i o n for —* 0 . o r equivalent ly / i n — 0 . a s d i s c u s s e d ¡11 E x . (7-4). Finally, a M o n t e C a r l o s i m u l a t i o n of t h e n e u t r i n o b e a m a n d its i n t e r a c t i o n s in t h e target c a n b e i t e r a t e d by v a r y i n g t h e b e a m c o n d i t i o n s unt il g o o d a g r e e m e n t is o b t a i n e d b e t w e e n d a t a a n d calculal ions. T h e (lux m e a s u r e m e n t s cont r i b u t e a l a r g e tract ion t o t h e filial overall s y s t e m a t i c u n c e r t a i n t i e s 011 t h e cross sect ions. 7.2.2
Mca.simnitnts
of Fi mid xF%
Several e x p e r i m e n t a l t e c h n i q u e s exist in o r d e r t o e x t r a c t t h e s t r u c t u r e f u n c t i o n s f r o m c r o s s s e c t i o n m e a s u r e m e n t s . For e x a m p l e , t h e C'C'FR C o l l a b o r a t i o n (19!J7) used t h e following m e t h o d , which in t h e e n d led t o r a t h e r s m a l l s y s t e m a t i c u n c e r t a i n t i e s of t h e o r d e r of 2Vc. a l l o w i n g for precision t e s t s of N L O Q C D . A f t e r h a v i n g d e t e r m i n e d t h e n e u t r i n o flux <1>(/T,/). t h e n u m b e r of o b s e r v e d e v e n t s ,V"' 1S is c o m p a r e d t o t h e p r e d i c t e d c r o s s s e c t i o n for e v e r y x a n d bin a c c o r d i n g t o N"h*
=
PLNA
[ J r Inn
<\Q2 [
d.r I 1
. I Q bin
./nil energies
AE.ME,,)-^—
.
(7.22)
(13
w h e r e ¡i is t h e target, d e n s i t y . L is t h e t a r g e t l e n g t h a n d is t h e A v o g a d r o n u m b e r . T h e s t r u c t u r e f u n c t i o n s a r e e x t r a c t e d bv a n i t e r a t i v e p r o c e d u r e , w h e r e input s t r u c t u r e funct ions for t h e M o n t e C a r l o predict ion of t h e c r o s s s e c t i o n artvaried until g o o d a g r e e m e n t in eqn (7.22) is f o u n d . T h e r e s u l t s a r e s h o w n in Fig. 7 . 1 3 f o r F> a n d Fig. 7.14 for x.F;\. Very g o o d a g r e e m e n t w i t h a N L O Q C ' D fit based on t h e I ) C L A P e v o l u t i o n e q u a t i o n s is f o u n d , t h a i is. s c a l i n g v i o l a t i o n s a r e c l e a r l y o b s e r v e d . It s h o u l d b e n o t e d that
Ni l 1 I KINO NIK'I.KON SC/VI I KltlNO
FlC. 7 . 1 3 . M e a s u r e m e n t s of t h e n e u t r i n o s t r u c t u r e f u n c t i o n F? by C'CFH. T h e e r r o r s a r e statistical only. Figure f r o m ( ' ( ' I ' l ? Collab.(1997).
for fixed Q~ /*•> increases with decreasing x. whereas xF-.i lirst increases a n d then decreases a g a i n . T h i s is seen m o r e clearly in Fig. 7.15, where t h e results of several e x p e r i m e n t s a r e shown as a function of ./•. 13ccau.se of t he a v e r a g i n g over Q'2, t he r e s u l t s a r e n o t e x p e c t e d t o agree perfect ly, since t h e e x p e r i m e n t s m e a s u r e d different Q2 ranges. T h e low-./- b e h a v i o u r of F> is u n d e r s t o o d by t h e increasing i m p o r t a n c e of t h e gluon a n d sea q u a r k c o n t r i b u t i o n s . For ,/7'':( we h a v e seen t hat it is sensitive t o t h e valence q u a r k d i s t r i b u t i o n s . 1F3 tx ./'¡'y(.z') <j(x)\ — xqv(x). In t h e simplest, q u a r k p a r t o n model without s t r o n g i n t e r a c t i o n s we would expect this d i s t r i b u t i o n t o have a narrow peak a r o u n d 1 / 3 with s o m e s m e a r i n g d u e to Fermi m o t i o n , since t h e t h r e e valence q u a r k s should each c a r r y a b o u t o n e - t h i r d of t h e p r o t o n m o m e n t u m . Because of Q C D effects t he d i s t r i b u t i o n is s h i f t e d t o lower values a n d s m e a r e d out much f u r t h e r , a s seen in Fig. 7.15. An i m p o r t a n t a s p e c t of t he m e a s u r e m e n t s of F> a n d xF: 1 a r e t h e a s s u m p t i o n s
S I l U K T t l K K I U N C I I O N S A N D I ' A H I O N I »IS I II11II' I IOINJS Mix
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Fit:. 7.1-1. M e a s u r e m e n t s of t h e n e u t r i n o s t r u c t u r e f u n c t i o n xF.t by CCI-' . T h e e r r o r s a r e s t a t i s t i c a l only. F i g u r e from C Fit Collal .(1997).
m a d e about >' .. which e n t e r s in t h e cross section prediction cqn (7.1-1). f t e n it is simply sel to zero, or its Q C prediction is used. However, it can be m e a s u r e d by fitting t h e f u n c t i o n
(7.23) which c a n b e derived from eqns (7.1-1) a n d (7.15) for . „ 2> M. ~ M2V a n d xF'{ = .rF'(. Because of t h e last a s s u m p t i o n a correction for A(. - : ) has (1 - i ) 2 is t h e polarization of t h e to be applied. T h e t e r m f = 2(1 - y ) [ 1 virtual V boson. T h e r a t i o y is e x t r a c t e d f r o m linear fits in c to F in fixed . a n d Q 1 bins. esults a r e shown in Fig. 7.Hi. t o g e t h e r with m e a s u r e m e n t s f r o m charged lepton I)IS. W i t h i n t h e still limited precision of t h e s e m e a s u r e m e n t s it is seen t h a t ? . II for increasing Q . crturbat.ive Q C p r e d i c t s ;. -x o s ( ( , ) ) .
I III
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FIG. 7 . 1 5 . eut rino st ruct lire funct ions !'> a n d . )( a s a funct ion of r . averaged over ( . o b t a i n e d by several e x p e r i m e n t s . Figure f r o m C o n r a d i t .(1998).
so t hai t h e b e h a v i o u r of Hi, can bo u n d e r s t o o d f r o m t h e d e c r e a s e o ( < S (Q 2 ) with increasing Q 2 . Iu o r d e r t o s t u d y u s c a t t e r i n g on p r o t o n s a n d n e u t r o n s separately, experi m e n t s at. C have used tli<- Big u r o p e a n Bubble C h a m b e r B B C (l!)<S. ). 1994) (illed with d e u t e r i u m . A u or u i n t e r a c t i o n was identified a s c o m i n g f r o m a n e u t r o n if it had either a n even n u m b e r of charged tracks, a s s u m i n g conservation of electric c h a r g e which is zero iu t h e initial s t a t e , or an o d d n u m b e r of charged t r a c k s t o g e t h e r with a low m o m e n t u m p r o t o n , i n t e r p r e t e d a s s p e c t a t o r in t h e reaction -neutron within t h e d e u t e r i u m . All o t h e r e v e n t s with a n o d d n u m b e r of charged t r a c k s a n d t h u s a net. t o t a l c h a r g e were classified as -proton interactions. C o r r e c t i o n s were applied for mis-identifications by s t u d y i n g M o n t e C a r l o s i m u l a t i o n s . From t h e m e a s u r e d s t r u c t u r e f u n c t i o n s a n d F ' it, is observed that. F',' w 2F'P' over almost, t h e full k i n e m a t i c r a n g e , a n d in p a r t i c u l a r at larger . ' w h e r e valence q u a r k s d o m i n a t e . T h i s can be u n d e r s t o o d since t h e v i r t u a l W + f r o m a u s c a t t e r i n g m u s t interact with a negatively charged q u a r k , which at large . is with high p r o b a b i l i t y a d - q u a r k . In a n e u t r o n t h e r e a r e twice a s m u c h d valence q u a r k s as u - q u a r k s , which e x p l a i n s t h e larger n e u t r o n st ruct u r e function. So these d a t a clearly i n d i c a t e t h e flavour sensitivity of n e u t r i n o
S I l i n c i l'in
H' < H
S A I) l ' A I
MISI HIII 'I I
S
S 04 . «
0.2
o.l 0.0
0.5 04 0.3 0.2
0.1 .
05 04 *
0..1 0,2
0.1 00 05 04 0.J EC 0.2 0.1 00
0.5 0.4 O.J 0.2
0.1 0.0
05 0.4 O.J a: (1.2 0I
0.0
-O.l f
-
F i e . 7. Hi. M e a s u r e m e n t s of Ry l>v n e u t r i n o a n d charged lepton DIS experim e n t s . Figure f r o m C o n r a d et «/.(1!)9S).
scattering. An interesting test of t h e universality of p a r t o n d i s t r i b u t i o n s is o b t a i n e d l»v c o m p a r i n g t h e results of F-t measurements f r o m c h a r g e d lepton to n e u t r i n o d e e p inelastic s c a t t e r i n g . In E x . (7-3) t h e r a t i o of s t r u c t u r e f u n c t i o n s is derived as n Ts
i
+ a) - «(* r • i((/ + <))
(7.2-1)
where (/ + <) r e p r e s e n t s t h e s u m over all q u a r k llavours. T h i s r a t i o is d e t e r m i n e d by t h e electric c h a r g e a s s i g n m e n t t o q u a r k s , a n d t h e expression a b o v e s t a y s valid at all o r d e r s of p e r t u r b a t i o n t h e o r y if defined in t h e DIS factorization scheme. W h e n going t o large x, w h e r e t h e sea c o n t r i b u t i o n is small, we expect
NKUTKINO NIK 'MOON S< 'AT I I .KINC
285
/•'J' N /F-2 N • 5/18 0.277s. Iinli'cd, c o m p a r i n g n e u t r i n o d a t a from ('OI K a n d ODIISVV (198!)) to union d a t a f r o m I K ' D M S (1987) a t high x yields a r a t i o of 0.278 I 0.010. in exeellent agreement with t h e e x p e c t a t i o n . 7.2.3
The gluon
distribution
A siiiiiillaneous analysis of scaling violations in t he n e u t r i n o s t r u c t u r e f u n c t i o n s / , a n d ,iF;í allows for a d e t e r m i n a t i o n of t h e gluon d i s t r i b u t i o n in t h e m e d i u m a n d large x d o m a i n , hi p a r t i c u l a r , t h e b e h a v i o u r of xF;i <x x(g <¡) is ex|)loited. We know from Ex. (3-8) t hat, t h e evolution e q u a t i o n for non-singlet q u a r k dist r i b u t i o n s r/1NS = i¡, - i/, only d e p e n d s on o s a n d t h e non-singlet, d i s t r i b u t i o n itself, w h e r e a s t h e evolution of flavour singlet d i s t r i b u t i o n s also d e p e n d s on t h e gluon d i s t r i b u t i o n
IS = ns P w
®
NS
.
= o , ( P m ® r/ s + />,)R 0
^
(7.25)
This allows for a precise d e t e r m i n a t i o n of n s without a detailed knowledge of t he gluon d i s t r i b u t i o n , o r equivalently t h e gluon d i s t r i b u t i o n can be d e t e r m i n e d largely n n c o r r e l a t e d with o s . which is in c o n t r a s t t o t h e s i t u a t i o n in c h a r g e d lepton IMS described previously. An analysis by OOI'K (1997) yields x
G 5 ± 0 C8
(7.2(i)
iu t h e region 0.04 < x < 0.7. T h i s result is then evolved to Qz = 3 2 G e V " a n d c o m p a r e d t o 1IEKA results (111 Collab.. 1995«), as shown iu Fig. 7.17. G o o d agreement between t h e various m e a s u r e m e n t s a n d t h e lits from global a n a l y s e s of part.011 d i s t r i b u t i o n s is f o u n d . It, is w o r t h n o t i n g t h a t t h e 1IKKA results discussed iu Section 7.1.4 a r e m o r e recent t h a n t hose of Fig. 7.17. 7.2.4
The strange
quark
distribution
( - h a r m p r o d u c t i o n via n e u t r i n o s c a t t e r i n g off s t r a n g e q u a r k s can be used t o isolate t h e s t r a n g e q u a r k d i s t r i b u t i o n s(.r) in t h e nucleón. T h e distinct experimental s i g n a t u r e for c h a r m p r o d u c t i o n a r e two oppositely signed unions iu t h e linal s t a t e , which is u n d e r s t o o d t o o c c u r via i'„ + N — / / " T c + X I--> s -I- //+ +
.
Indeed, c h a r m p r o d u c t i o n is a s u b s t a n t i a l f r a c t i o n of t h e t o t a l cross section, a s large as 10% at. high energies, E „ « 300 G e V . T h e cross section for di-miion prod u c t i o n is t h e p r o d u c t of t h e cross section for c h a r m p r o d u c t i o n , a f r a g m e n t a t i o n function l)(z), with z = l>i)/l>\"nx. which describes the t r a n s i t i o n of a c - q u a r k t o a c h a r m e d D meson, a n d t h e b r a n c h i n g rat io B, for the subsequent semileptouic decav of t h e D meson. d : 'ff(f/„N
/ / - / / + A ' ) _ d 2
d£d//ds
~~
c it~X)
d£ dy
D(z)Bc(
c - , ,
+
A').
(7.27)
m u m
I u n r < I' u r n
i iwnn
/mm ' i /mi i win
inn i iiiiii'i
luiin
d ±1 n gluon error iiomt'CTK daia O ±\n gluon error from liM>5 + Cilnons from IN jcls — --
Fit:
CIT.Q4M GRV94IIO MRS K .
7 . 1 7 . T h e gluon d i s t r i b u t i o n m e a s u r e d by O C T K . F i g u r e f r o m C o n r a d at
Here instead of t h e B j o r k e n sealing variable x t h e variable £ a p p e a r s , which is a p p r o x i m a t e l y given by £ = j:(l + in2/Q2)( 1 - x2M2/Q2). T h i s so-called slow reseating ( H a r n e t t . 197(i: Gcorgi a n d Politzer, 197(i) t a k e s effects of t h e c h a r m q u a r k m a s s ///,• a n d of t h e t a r g e t m a s s A/ into account . T h e leading o r d e r cross section for c h a r m p r o d u c t i o n is rr'-'V/.N -
c.Y) oc [(*«(*,/< 2 ) + t d f o l ? ) ) IK,i| 2 + 2£ S (£, / / 2 )|V' C ! t | 2 ] .
(7.28)
Note t h a t this expression holds for a n isoscalar t a r g e t , w h e r e also t h e u-dist r i b u t i o n aj>pcars b e c a u s e of isospiu invariance between p r o t o n s a n d n e u t r o n s . u(x) = u]'(x) = d"(x). We see t h a t t h e cross section d e p e n d s on t h e C K M m a t r i x elements. T h e c o n t r i b u t i o n f r o m s c a t t e r i n g off d- or d - q u a r k s is C a b b i b o s u p pressed. |Ks|- T h e sensitivity t o .s(.r) b e c o m e s o b v i o u s f r o m e q n (7.28). In t h e case of // s c a t t e r i n g a b o u t 50% of t h e c h a r m p r o d u c t i o n is d u e t o s c a t t e r ing olf s - q u a r k s . w h e r e a s for /> s c a t t e r i n g 9 0 % c o m e s f r o m s. since only d f r o m t h e sea can c o n t r i b u t e in t h i s case. T h e largest d a t a s a m p l e s a r e available f r o m t h e C'CFK (1995) a n d ( ' I ) I I S (1982) C o l l a b o r a t i o n s . An i m p o r t a n t e x p e r i m e n t a l issue is t h e control of t h e b a c k g r o u n d s , such as unions f r o m pion a n d kaon decays. Fortunately, t h e dense c a l o r i m e t e r s with short i n t e r a c t i o n lengths minimize t h e s e c o n t r i b u t i o n s . Furt h e r m o r e . a control of t h e c o n t r i b u t i o n s from u versus u is of i m p o r t a n c e . S y s t e m a t i c u n c e r t a i n t i e s a r i s e f r o m t h e limited knowledge of t h e c h a r m f r a g m e n t a t i o n function D(z). T h e size a n d s h a p e of t h e s t r a n g e part on d i s t r i b u t i o n a r e d e t e r m i n e d in t h e following way: T h e m e a s u r e m e n t s of F> a n d ./•/';$ a r e used to c o n s t r a i n t h e s u m of a n t i q u a r k d i s t r i b u t i o n s .
Ni l i I I I I N i i N U C I I O N
SCAITKHINC
OcV)
•2H7
->
Fit:. 7 . 1 8 . (a) D i s t r i b u t i o n of t h e visible e n e r g y for t h e C'CFK //- a n d //-induced di-muoii events. Indicated a r e also t h e c o n t r i b u t i o n s f r o m t h e s t r a n g e sea (b) M e a s u r e d q u a r k q u a r k s , s. a n d t h e valence a n d sea d - q u a r k s . dv a n d a n d s t r a n g e s e a d i s t r i b u t i o n s at, //* IGCV". T h e grey b a n d a r o u n d t h e s t r a n g e q u a r k d i s t r i b u t i o n i n d i c a t e s t h e ± I r r uncertainty. F i g u r e s f r o m C o n rad et «/.(1998).
F , w ( f - i / N S = 2q = 2(?7 -I- <1 + .s) = 2(/"/ + d) -|- (.s + 5) .
- /••» x
(7.29)
T h u s , a s s u m i n g s(x) = s(x). t h e s t r a n g e d i s t r i b u t i o n can b e related t o t h e nons t r a n g e d i s t r i b u t i o n s . A possible p a r a m e t e r i z a t i o n is
* =
{'i
< [
;
r l M
;
r ) +
J0 d.r [./•!?(:/:) +
' ' " f ll
xd(x)}
•
*
M
-
-
+
•
® M
where t h e p a r a m e t e r s /,• a n d scat tering. T a k i n g t h e world a v e r a g e values for t h e C K M m a t r i x elements, t h e p a r a m e t e r s t u r n o u t to be n = 0.477 ± 0 . 0 5 1 a n d o = - 0 . 0 2 ± . 0 . 0 3 ( O C F U C'ollab.. 1995), which a r e o b t a i n e d t o g e t h e r with m e a s u r e m e n t s of H, a n d ///,.. T h i s m e a n s that wit hin t h e current, e x p e r i m e n t a l precision t h e s h a p e of t h e si r a n g e q u a r k <list ribut ion cannot, be distinguished from t he n o n - s t r a n g e component.. However, k < I d e m o n s t r a t e s a violation of S U ( 3 ) flavour s y m m e t r y in t h e q u a r k sea. Intuitively, we could explain this by t h e larger s - q u a r k m a s s with respect t o u a n d d. which would s u p p r e s s t h e gluon s p l i t t i n g i n t o s t r a n g e q u a r k pairs. However, non-pert.urbative
p h e n o m e n a could h e I m p o r t a n t us well. T h e e x t r a c t e d p a r t o n d i s t r i b u t i o n s a r e s h o w n in Fig. 7 . 1 8 ( b ) . An a n a l y s i s by C O I ' l l in o r d e r t o d i s e n t a n g l e .s(:r) f r o m s(x) w a s not yet conclusive. A n o t h e r h a n d l e on t h e s t r a n g e q u a r k d i s t r i b u t i o n (iF{ ,N could he o b t a i n e d f r o m t h e a p p r o x i m a t e relation .r.s + xs i)/3F.J' N ( E x . 7-3).
.
u m rules
S u m rules a r e integrals over s t r u c t u r e f u n c t i o n s or p a r t o n d i s t r i b u t i o n s , expressing usually t h e c o n s e r v a t i o n law for s o m e q u a n t u m n u m b e r of t h e p r o t o n . Basic s u m rules h a v e a l r e a d y been m e n t i o n e d in Section 3.2.2, a n d a f u r t h e r discussion is f o u n d in E x . (7-5). Q C D p r e d i c t i o n s exist for a wide variety of s u m rides. T h e s e p r e d i c t i o n s a r e t est e d e x p e r i m e n t a l l y by i n t e g r a t i n g t h e m e a s u r e d struct lire f u n c t i o n s /-VI(.I\ ~) over t h e full x r a n g e at a given 2. A basic problem arises f r o m t he fact t h a t only a s u b r a n g e [ X „ , i „ . .R„LAX] C |<>. 1] is accessible e x p e r i m e n t a l l y . A possible solution is t o c o m b i n e m e a s u r e m e n t s f r o m different e x p e r i m e n t s which cover different k i n e m a t i c r a n g e s . O t h e r w i s e various s t r u c t u r e f u n c t i o n m e a s u r e m e n t s a t different scales c a n be evolved t o a single scale using QC'D evolution e q u a t i o n s or a p p r o x i m a t i o n s such a s s i m p l e power laws. T h e result s h o u l d cover a wide e n o u g h r a n g e iu x for t h e integral to be c o m p u t e d reliably. Nevertheless, usually som e a s s u m p t i o n s a b o u t t h e b e h a v i o u r at x —• t) and x 1 h a v e still t o be m a d e in o r d e r t o c o m p l e t e t h e integrat ion r a n g e . T h e s e a s s u m p t i o n s lead t o s y s t e m a t i c u n c e r t a i n t i e s o n t h e s u m rule m e a s u r e m e n t s . A l t h o u g h it is q u i t e c o m m o n t o call all t hese integrals over s t r u c t u r e f u n c t i o n s simply sum rules, o n e s h o u l d be a w a r e t h a t a c t u a l l y t here exist different classes of s u m rules. We might call a s u m r u l e a n exact CD sum rule if its result found within t h e c o n t e x t of t h e p a r t o n model is not a l t e r e d by a n y r a d i a t i v e or nonpert u r b a t i v e c o r r e c t i o n . An e x a m p l e is given by t he A d l e r s u m rule. Next t h e r e a r e s u m rules w h e r e t h e p a r t o n m o d e l result is modified by r a d i a t i v e c o r r e c t i o n s a n d m a y b e by p o w e r - s u p p r e s s e d t e r m s , such as t h e G r o s s Llewellyn-Smith or t he Bjorken s u m rules. Finally, we have s u m rules which a r e st rongly affected by nonpert u r b a t i v e physics, t h u s deviat ions of t h e m e a s u r e m e n t s f r o m t h e p r e d i c t i o n s of p e r t n r b a t i v e QC'D might not be a s u r p r i s e . E x a m p l e s a r e t h e G o t t f r i e d a n d t h e Ellis . l a l f e s u m rules. A last p a r t i c u l a r case is t h e m o m e n t u m s u m rule. W h e n going beyond leading o r d e r Q C D . it sh o u l d be seen a s a c o n v e n i e n t c o n s t r a i n t on t he definition of t h e p.d.f.s r a t h e r t h a n a basic Q C D s u m rule. 7.3.1
The
dler
sum
rule
T h e Adler s u m rule (Adler. 19(53) is defined a s
/A = / ' — ( F 7 M .o x
2
)
- f-T(-r. -))
= 2 f d.r ( « v ( . r ) - dv(x)) o
= 2 .
(7.31) T h i s result is o b t a i n e d by i n s e r t i n g t h e p a r t o n model e x p r e s s i o n s for t h e s t r u c t u r e functions, F = 2 x ( d ( x ) + s(x) + u(x) + c(x)) a n d F = 2 x ( u ( x ) 4- s(x) +
'¿Mil
SIIM HUM S
i/(.c) I '•(.' )). Tlic s u m ruli' expresses I lie dilfercnee v pertiirbntivc or non-pert urbat.ive (.¿CD c o r r e c t i o n s b e c a u s e of l lie conservation of t h e c h a r g e d weak c u r r e n t . T h e s a m e result c a n be o b t a i n e d from - F^(x.Q2))
/A = / ' - (Fp(x.(?) x Jo
= 2 .
(7.32)
using t h e const raint. J 0 ' d:r(c - c) = J 0 d.r(s - .5) = 0 (ef. E x . .'i-(i). An e x p e r i m e n t a l p r o b l e m arises f r o m t h e fact t h a t usually isoscalar t a r g e t s a r e employed. For such t a r g e t s we have F i ' ^ = . and the contributions from p r o t o n or n e u t r o n s c a t t e r i n g a r e difficult t o disentangle. T h e only measurement, in tin- l i t e r a t u r e is f r o m t h e I5EBC C o l l a b o r a t i o n (1984, 1985). We have described previously how t h e y used n e u t r i n o a n d a n t m c u t r h i o s c a t t e r i n g on a d e u t e r i u m t a r g e t iu o r d e r t o d i s e n t a n g l e /"j' 1 ' f r o m /•?/". T h e i r result for t h e Adler s u m rule is = 2.02 ± 0 . 4 0 . in agreeineni with t h e theoretical e x p e c t a t i o n I = 2. A re-evaluation of t h e HEHC d a t a ( C o n r a d el ul.. 1998) using t w o different a p p r o x i m a t i o n s iu order t o cover t h e u n m e a s u r e d low-.'»: region gave = 1.87 ± 0.15 a n d = 2.05 ± 0.15. d e p e n d i n g on t he e x t r a p o l a t i o n m e t h o d . Again b o t h results a g r e e w i t h t h e e x p e c t a t i o n . 7.3.2
The Cross
Llewellyn
Smith
sum
rule.
The G r o s s Llewellyn-Smith ( G L S ) s u m rule (Gross ami Llewellyn S m i t h . 1909) is defined a s
/ c a s = / ' —xF:i(x. ./o ®
Q 2) = f d x (FZ(x, Jo -
Q2) + F\'(x. Q2))
.
(7.33)
Iu the simple part.on model t h i s s u m rule is t h e integral over t h e valence q u a r k d i s t r i b u t i o n s , with an e x p e c t a t i o n of 3 for p r o t o n s or n e u t r o n s , t h a t is.
/GLS -
[
d.r (uv(x)
I- dv{x))
= 3 .
(7.34)
T h i s leading o r d e r e x p e c t a t i o n is modified by radiative corrections from Q C D discussed in Ex. (3-28) a n d Section 8.3.3. T h e r e it is shown that t h e m e a s u r e m e n t of t h e G L S s u m rule can be used in o r d e r t o o b t a i n a m e a s u r e m e n t of o s . M e a s u r e m e n t s of x.Fn by several n e u t r i n o e x p e r i m e n t s have been c o m b i n e d ( C o n r a d el ui. 1998), a n d t h e G L S s u m rule is d e t e r m i n e d to b e /c:i,s = 2.04 ± 0 . 0 0 . In a recent analysis of d a t a f r o m C'C'I'H (1998) a n d o t h e r n e u t r i n o experiment«; t h e G L S integral has been e v a l u a t e d at various a v e r a g e Q 2 values, giving for e x a m p l e /<;i,S
=
2.49
± 0.08stnt
±
(I. l O s y s i
/ ( ! , . s = 2.78 ± 0.06,1., ± 0.19 s y s i
a t (Q2) =
2
GeV2
at ( Q 2 ) = 5 G e V "
NI HIU I HUI I U N I ' l l l >NN /\N|1 l'AK | U|N DIN I HIIU 1 I l< 1|\N
Ir.I.S = 2 . 8 U ± (1.13.1,,1 I 0.l8 H y i l ,
al. (Q'¿)
12.0CeV' .
It. is clearly seen t h a t t h e results satisfy /<;i.s < 3. a s e x p e c t e d f r o m eqn (8.10), since I he Q C D c o r r e c t i o n s a r e negative. Flirt her more, we c a n see that, t h e integral a p p r o a c h e s t h e naive e x p e c t a t i o n of /c:i_s = 3 w h e n going t o larger Q'. since at larger scales t h e s t r o n g coupling; c o n s t a n t a n d t h u s t h e r a d i a t i v e c o r r e c t i o n s decrease. 7.3.3
The Gottfried
sum
rule
The A d l e r s u m rule is m e a s u r e d in d e e p inelastic n e u t r i n o s c a t t e r i n g e x p e r i m e n t s . A c o m p a r a b l e s u m rule for charged lepton s c a t t e r i n g is given by t h e G o t t f r i e d s u m rule ( G o t t f r i e d , 1907),
Ir.
f
Jo
-
X
(FP(x,Q*)
- J T ' M
2
) ) = \ + t .5
•>
f
JO
d * («(*)
- d(x))
.
(7.35) which can be o b t a i n e d by inserting eqn (7.5) a n d using isospin s y m m e t r y . that is. u(x) u]'(x) = d"(x), d(x) — iP*(x) = «"(.r). S t a r t i n g f r o m t h e a s s u m p t i o n that t h e sea q u a r k d i s t r i b u t i o n s ü(x) a n d d(x) in t h e nucleoli a r e t h e s a m e , we e x p e c t /<; = 1 / 3 . However, t h e NM( • e x p e r i m e n t m e a s u r e d I a (./' = 0.004 - 0 . 8 ) = 0.227 -1- 0.0G7si¡,i ± 0.014 s y s , a t Q - = I G e V 2 , a n d /<¡ = 0.240 i 0.010 w h e n ext r a p o l a t e d t o t h e full x r a n g e (N'.MC Collab.. 1991). T h i s result is significantIv different f r o m 1 / 3 . A l t h o u g h m a n y possible e x p e r i m e n t a l biases have been considered. no e x p l a n a t i o n could be found except a difference in t h e sea q u a r k d i s t r i b u t i o n s of p r o t o n s a n d n e u t r o n s , or equivalentlv t h a t t h e r e is a light q u a r k flavour a s y m m e t r y in t he sea. (\x(Ti(x) - d(x)) = 0.140 ± 0.024 / 0. Lately this observation h a s been confirmed by t h e K860 e x p e r i m e n t , as will be discussed later. N o n - p e r t u r b a t i v e processes a r e thought, t o be at t h e origin of t h i s a s y m metry. For a m o r e i n - d e p t h discussion we refer t o t h e l i t e r a t u r e ( K u i n a n o a n d L o n d e r g a n . 1991: FSOO/NI'SEA Collab.. 1998/;). 7.3.4
The momentum
sum
rule
As will be described in Section 7.5. a global Q C D analysis of deep inelastic s c a t t e r i n g e x p e r i m e n t s results in a set. of p.d.f.s for q u a r k s a n d gluons. It is shown ( B o t j e . 2000) t h a t t h e integral over t h e s e d i s t r i b u t i o n s gives t h e following values when e v a l u a t e d at Q~ — 1 G e V . /,, =
f
d.i: x V
[
= 0.594 ± 0.018 .
¡K = [ < \ x x u ( x ) = 0.394 ± 0.018 . .lit T h e e r r o r s c o m b i n e s t a t i s t i c a l a n d s y s t e m a t i c u n c e r t a i n t i e s iu q u a d r a t u r e . We see that t h e q u a r k s c a r r y only a b o u t (>()% of t h e nucleoli m o m e n t u m . T h i s was
•J! 11
SUM HULKS
ii big s u r p r i s e w h e n observed for t h e lirsi t i m e , a n d s u g g e s t e d t h e p r e s e n c e of f u r t h e r p r o t o n const it ueiils, c.f. Sect ion 2.1. Indeed, we now know t h a t t h e g l u o n s c a r r y t h e very large missing fraction of t h e p r o t o n m o m e n t u m . 7.3..r>
Sum
rulen for polarized
structure,
functions
So far we have only discussed s t r u c t u r e f u n c t i o n m e a s u r e m e n t s which a r e ext r a c t e d f r o m s p i n - a v e r a g e d cross sections. However, if t h e p r o b i n g b e a m anil t h e t a r g e t a r e p o l a r i z e d , t h e n similar e x p e r i m e n t s can be p e r f o r m e d in o r d e r to m e a s u r e polarized s t r u c t u r e f u n c t i o n s . We will only give a very s h o r t overview here. T h e interested r e a d e r is referred t o t h e l i t e r a t u r e for a m o r e detailed discussion (Ellis el id.. 19ÍJÍ56), or a general review ( W i n d n i o l d e r s . 1999) on recent results iu spin physics. T h e basic o b s e r v a b l e in polarized s c a t t e r i n g is t h e s p i n a s y m m e t r y , defined as t he a s y m m e t r y of t h e c r o s s s e c t i o n s d a " (d
'4(J
-Q
>
=
dat 1(*.Q*)
=
+ da"(x,Q>)
PnPrf
"
(
'
I'lie second e q u a l i t y s h o w s t h e relation bet ween t h e o b s e r v a b l e a n d t h e m e a s u r e d a s y m m e t r y .4„„. a s . T h e f a c t o r s P¡HT) K'v<" Hie d e g r e e of p o l a r i z a t i o n of t h e b e a m ( t a r g e t ) , which for m o d e r n e x p e r i m e n t s such a s H E R M E S at 11 ICR A r a n g e f r o m 10% u p t o 9 0 % . T h e d i l u t i o n f a c t o r / a c c o u n t s for t h e f r a c t i o n of p o l a r i z a b l c iiuclcous in t h e t a r g e t . T h e u n c e r t a i n t i e s o n t h e s e t h r e e f a c t o r s a r e i m p o r t a n t sources of s y s t e m a t i c u n c e r t a i n t i e s on t h e spin a s y m m e t r y A. For t h e k i n e m a t i c r a n g e relevant, for m o s t of t h e lixcd-target. e x p e r i m e n t s (where only p h o t o n e x c h a n g e n e e d s t o b e c o n s i d e r e d ) t h e s p i n a s y m m e t r y is (•''• Q1)(F\ (•''. Q*). w h e r e r/i is t h e polarized equivalent of given bv A(x,Q~) fli t h e s t r u c t u r e f u n c t i o n F\ f r o m u n p o l a r i z c d s c a t t e r i n g . T h u s a m e a s u r e m e n t of •l a n d /•"i will d e t e r m i n e <j\. In t h e p a r l ó n m o d e l a p p r o x i m a t i o n <j\ is delined a s 01 ( * ) = i E ' M I •I
:
+
±
(7.37)
with c 2 t h e electric c h a r g e s of t h e q u a r k s . T h e part.on d i s t r i b u t i o n s relevant for s p i n - a v e r a g e d s c a t t e r i n g a r e q =
Í/A 11
.'/V
0.209 ± 0 . 0 0 1 .
(7.38)
11 e r e / / A / f l v is t h e r a t i o of a x i a l - v e c t o r over vector couplings m e a s u r e d f r o m t h e n e u t r o n f-dccay. T h e s u m rule is modilicd by r a d i a t i v e c o r r e c t i o n s , e x p r e s s e d a s
S T Kl i l l
Kl
M I N C I IONS A N I > I ' A K T O N DIS I l< 11 It ' I I O N S
a power scries in t h e s t r o n g coupling c o n s t a n t , a n d hence was also employed to m e a s u r e o s , a s discussed in Section 8.3.3. Similarly, when considering only t h e p r o t o n , t h e Ellis J a f f e s u m rule is obtained (Ellis a n d Jaf f o . 1974),
£ -i
/,, -
cM' =
±
i/A
(1 + C) = 0.185 ± 0.003 ,
(7.39)
w h e r e C' is a correction n o t c o m p u t a b l e within pert.urbative Q C D . It can be o b t a i n e d , for e x a m pl e , f r o m an a n a l y s i s of n —< p a n d —» n decays (Ellis til.. 19! )(>/>: Close a n d R o b e r t s . 1993). A c o m p a r i s o n with e x p e r i m e n t a l results, such a s /,-:., = 0.142 ± ().()()8.sli„ ± 0 . 0 1 U y s , at. ~ = K l G e V " ( S M C Collab., 1994). reveals a large discrepancy, which c a n n o t be c o m p e n s a t e d by r a d i a t i v e corrections. A basic a s s u m p t i o n which e n t e r s t h e theoretical prediction eqn (7.39) is t h a t t h e st range q u a r k sea is u n p o l a r i z e d . T h e r e f o r e t h e discrepancy could be an indication for polarization of t h e s t r a n g e sea q u a r k s . An a d d i t i o n a l i m p o r t a n t c o n t r i b u t i o n should c o m e f r o m t h e giuons. T h i s c o n t r i b u t i o n cancels out in t h e Bjorken s u m rule. T h e m e a s u r e m e n t s of these s u m rules a s well as of t h e s h a p e of t h e polarized p a r t o n d i s t r i b u t i o n s a r e a field of a c t i v e research at existing a n d f u t u r e e x p e r i m e n t s , such a s COMPASS at (T.K.N. 7.4
Hadron-badron
scattering
I n f o r m a t i o n on t h e p.d.f.s can be o b t a i n e d not only from s t r u c t u r e f u n c t i o n s m e a s u r e d in DIS e x p e r i m e n t s , but also f r o m analyses of liadron h a d r o n s c a t t e r ing. In o r d e r to r e d u c e t h e c o m p l e x i t y of these r e a c t i o n s , a n d t o allow for cross sections t o be calculable, t h e final s t a t e is required to have p a r t i c u l a r properties. For e x a m p l e , one looks for l e p t o n -p a i r or p h o t o n p r o d u c t i o n , b o t h final s t a t e s being of non-hadroiiie nat uro. Ilowever, even h a d r o n i c final s t a t e s a r e s t u d i e d , since j e t s p r o d u c e d in h a d r o n h a d r o n collisions with very large t r a n s v e r s e m o m e n t u m can be u n d e r s t o o d iu t e r m s of a simple u n d e r l y i n g h a r d p a r t o n p a r t o n s c a t t e r ing. T h e sensitivity t o t h e p a r t o n d i s t r i b u t i o n s b e c o m e s clear when looking at t h e general expression for a cross section, a s given in eqn (3.72). Section 3.2.3. T h e r e we also have discussed various possible p a r t o n p a r t o n processes relevant for dilferent. final s t a t e s observed in t h e experiment.. Iu t h e following we will briefly review several e x p e r i m e n t s which help t o c o n s t r a i n p a r t i c u l a r p.d.f.s. 7.4.1
77« Dri ll Yan
process
A basic discussion of t h e h a r d s u b p r o c e s s , relevant for Droll Yan p r o d u c t i o n of lept.on pairs iu nucleoli nucleoli s c a t t e r i n g , has boon given ill Section 3.0.8. We look for t h e e x p e r i m e n t a l l y very clean s i g n a t u r e of two oppositely charged leptons. typically unions. T h e lept.on pair is p r o d u c e d by t h e a n n i h i l a t i o n of a qqpai r. t he q u a r k s being const it uents of e i t h e r of t h e two h a d r o n s . They a n n i h i l a t e i n t o a p h o t o n o r a Z boson, which s u b s e q u e n t l y d e c a y s into t h e two leptons (c.f. Fig. 6.4, lower rightmost F e y n m a n d i a g r a m ) . T h e invariant m a s s of t h e
IIADKON
I I A D H O N SUA'I I'EH I N C
21KT
.VI
I K;. 7 . 1 9 . (a) T h e r a t i o fr , " l /'- of Droll Van cross s e c t i o n s m e a s u r e d by E8G6 a n d c o m p a r e d l.o various part on d i s t r i b u t i o n s , (h) T h e r a t i o of d v a s a funct ion of x in t h e p r o t o n , d e t e r m i n e d by E8G6. Also s h o w n is t h e result bv N A 5 1 a n d a c o m p a r i s o n t o a p a r t o n d i s t r i b u t i o n o b t a i n e d f r o m a global Q C D a n a l y s i s . F i g u r e s f r o m IVSGG/NUSKA C o l l a b . ( 1998A).
l e p t o u |>air is given by M 2 = sx.\x2, w h e r e s is t h e overall C'.o.M. e n e r g y a n d .1*1 CJ) is t h e h a d r o n ' s m o m e n t u m f r a c t i o n c a r r i e d by t h e first ( s e c o n d ) q u a r k . T e s t i n g t h e r e f o r e d i f f e r e n t regions of t h e invariant m a s s d i s t r i b u t i o n allows t o p r o b e different, x regions of t h e p a r t o n d i s t r i b u t i o n s . Since for t h e a n n i h i l a t i o n a q q - p a i r is r e q u i r e d , t h e Droll Yan p r o c e s s is p a r t i c u l a r l y useful t o test t h e sea ( a n t i ) q u a r k d i s t r i b u t i o n s . E x p e r i m e n t a l l y , it h a s been s t u d i e d a t f i x e d - t a r g e t e x p e r i m e n t s a n d colliders, such a s t h e T E V A T R O N . H e r e we will c o n c e n t r a t e on a recent F E H M I I . A B f i x e d - t a r g e t e x p e r i m e n t . E8GG. which h a s c o n f i r m e d t h e flavour a s y m m e t r y of t h e light q u a r k sea in t h e p r o t o n , a s i n d i c a t e d in Section 7.:{..'{. T h i s e x p e r i m e n t m e a s u r e s t h e m u o n - p a i r yield f r o m t h e Droll Yan p r o c e s s in a 8 0 0 G e V p r o t o n b o m b a r d e m e n t of liquid d e u t e r i u m a n d h y d r o g e n t a r g e t s . Using different, t a r g e t s t h e y a r e a b l e t o test t h e sea q u a r k d i s t r i b u t i o n s in p r o t o n s and neutrons, and ultimately determine the d u and d ii d i s t r i b u t i o n s in t h e p r o t o n over t h e r a n g e 0.020 < x < 0.345. T h e o b s e r v a b l e is t h e r a t i o of Droll Y a n l ' 2a ' ', which t h e y h a v e m e a s u r e d cross sect ions for d e u t e r i u m a n d h y d r o g e n . f r o m 140000 Droll Yan p a i r e v e n t s w i t h a t o t a l s y s t e m a t i c u n c e r t a i n t y of less t h a n 'X (ESGG/NUSEA C o l l a b . , 1998a). For a p a r t i c u l a r k i n e m a t i c region t h i s r a t i o c a n bo a p p r o x i m a t e d a s pit 2rr>'i'
1 +
«2 J
.
(7.40)
w h e n - th(> indices 1(2) refer t o t h e p a r t o n s b e l o n g i n g t o t h e i n c o m i n g ( t a r g e t )
S I I I I ' C I ll|{|
MINI
I'lONS A N D P A I M ' O N DIS I II11II
I IONS
l i a d r o u . T h e s e n s i t i v i t y t o d/ii is nicely seen in t h i s r a t i o , w h i c h wonhl he u n i t y if t h e light q u a r k l l a v o u r s e a w e r e s y m m e t r i c . However, t h e d a t a s h o w a d e a l d e v i a t i o n f r o m u n i t y (Fig. 7.151(a)), which i n d i c a t e s a n e x c e s s of d o v e r // sea q u a r k s . T h i s is f u r t h e r i l l u s t r a t e d in Fig. 7 . 1 9 ( h ) . w h e r e t h e m e a s u r e d cross s e c t i o n r a t i o h a s b e e n t r a n s l a t e d i n t o a d i s t r i b u t i o n of ( / / / i i ) ( . r ) . Several p a r t o n d i s t r i b u t i o n s , w h i c h a r e o b t a i n e d f r o m g l o b a l QC'D a n a l y s e s , a s s u m i n g a nons y m m e t r i c light f l a v o u r s e a . a r e a b l e t o r e p r o d u c e t h e low-./- rise of (d/u)(x), hut not t h e fall-off t o w a r d s u n i t y a b o v e ./• « 0.2. N o t e t h a t a very r e c e n t g l o b a l Q C D a n a l y s i s by t h e C I ICQ C o l l a b o r a t i o n l e a d s t o a set of p . d . f . s . called O ITCQ5M, w h i c h a r e a b l e t o d e s c r i b e t h e r a t i o d / v over t he e n t i r e ./' r a n g e (CTICQ C o l l a b . . 2 0 0 0 ) . A s a l r e a d y m e n t i o n e d previously, t h i s a s y m m e t r y in t h e sea q u a r k s must h a v e its origin in n o n - p e r t u r b a t i v e p h e n o m e n a of st r o n g i n t e r a c t ions. 7. 1.2
T/zc W rapidity
asymmetry
S i m i l a r t o t h e Droll Yan p r o c e s s . \ \ ; ± b o s o n s a r e p r o d u c e d in p p collisions m a i n l y by t h e a n n i h i l a t i o n of u- (d-) q u a r k s in t h e p r o t o n w i t h a d - (ii-) q u a r k f r o m t h e a n t i p r o t o u . T h i s is p a r t i c u l a r l y t r u e at TICVATUOX e n e r g i e s of a b o u t l . S T e V . As global Q C D a n a l y s e s s h o w (Section 7.5. Fig. 7.2 I), u-<|uarks c a r r y o n a v e r a g e m o r e m o m e n t u m t h a n d - q u a r k s . T h e r e f o r e t h e \ V + b o s o n s t e n d t o follow t h e d i r e c t i o n of t h e i n c o m i n g p r o t o n a n d t h e \ V ~ b o s o n s t h a t of t h e a n t i p r o t o u . w h i c h l e a d s t o a c h a r g e a s y m m e t r y in t h e a n g u l a r ( r a p i d i t y ) d i s t r i b u t i o n of t h e M2y t h i s \V r a p i d i t y a s y m m e t r y is s e n s i t i v e t o p r o d u c e d \Y b o s o n s . For Q2 t h e n- a n d d - q u a r k d i s t r i b u t i o n s in t h e p r o t o n , in p a r t i c u l a r t o d/a. a s c a n b e seen f r o m t h e following a p p r o x i m a t e r e l a t i o n s h i p .
W{J>
_ d(./•, )il(:!'•>) - d(.ei )//(•/:•_,) _ /?(./•,)~ //(.r, )) +
I)
H(rI)
wit h Ii(x) d(x)/u(x). .'T(2) = •''() <'X])(±;Y) . xn - Mw/^/s. y being the rapidity a n d \ / s t h e C . o . M . e n e r g y . T h i s r e l a t i o n is d e r i v e d u n d e r t h e a s s u m p t i o n of //''(./•) = Tt^ix) a n d (P'(x) =
Il \ l UK IN II \ | ) H O N S('A I H I I I N C
•.»m
0.25 ( DI- IW2-')5(ll(iph
(1.2
1
e + /i)
-0.1
-0.1 S -0.2
0
0.5
1.5
Kit:. 7 . 2 0 . T h e l e p t o n c h a r g e a s y m m e t r y a s a f u n c t i o n of r a p i d i t y , m e a s u r e d by C D F , w i t h s t a t i s t i c a l a n d s y s t e m a t i c u n c e r t a i n t i e s a d d e d in q u a d r a t u r e . T h e d a t a a r e compared to N L O Q C D predictions based on various p a r t o n d i s t r i b u t i o n s e t s . F i g u r e f r o m C D F C o l l a b . ( 1998).
a s y m m e t r y e f f e c t , t h e o r i g i n a l W r a p i d i t y a s y m m e t r y is d i l u t e d . N e v e r t h e l e s s , t h e T F V A T R O N d a t a still h a v e e n o u g h s e n s i t i v i t y t o c o n s t r a i n t h e p a r t o n dist r i b u t i o n s f r o m t h e m e a s u r e m e n t of t h e l e p t o n r a p i d i t y a s y m m e t r y . T h e C D F C o l l a b o r a t i o n h a s a n a l y s e d t h e d a t a t a k e n d u r i n g 1992 95, c o r r e s p o n d i n g t o a n i n t e g r a t e d l u m i n o s i t y of 110 p b 1 o r a b o u t 9 0 0 0 0 e v e n t s a f t e r all selection c u t s ( C D F Collai)., 1998). T h e y h a v e looked for m u o n s a n d e l e c t r o n s over a r a p i d i t y r a n g e 0 < |i//| < 2.5, w h i c h a l l o w s t o c o n s t r a i n t h e q u a r k mom e n t u m d i s t r i b u t i o n s in t h e p r o t o n for 0.OCX» < x < 0.34 at Q2 « M2X. T h e result is s h o w n in Fig. 7.20. w h e r e t h e d a t a p o i n t s a r e c o m p a r e d t o different N L O Q C D predictions, based on various p a r t o n distribution sets. T h e d a t a are c o m b i n e d for p o s i t i v e a n d n e g a t i v e l e p t o n r a p i d i t y . R e c e n t g l o b a l ( ¿ C D a n a l y s e s ( C T I Î Q C o l l a b . , 2000: M a r t i n et «/.. 1998) w h i c h i n c l u d e t h e s e C D F d a t a give p a r t o n d i s t r i b u t i o n s , in p a r t i c n l a r (
Direct-photon
production.
D i r e c t - p h o t o n p r o d u c t ion in h a d r o n h a d r o n collisions is not. o n l y a useful tool for t e s t i n g p e r t u r b â t i v e Q C D . b u t in p a r t i c u l a r t o c o n s t r a i n t h e g l u o n d i s t r i b u t i o n in t h e p r o t o n at m e d i u m a n d l a r g e x values, x « 0.2 — 0.0. T h e r e D I S e x p e r i ment s h a v e less s e n s i t i v i t y t o t h e g l u o n c o n t r i b u t i o n . D i r e c t - p h o t o n p r o d u c t i o n o c c u r s m a i n l y v i a ( ¿ C D c o m p t o n s c a t t e r i n g gq —» 7 q a n d a n n i h i l a t i o n p r o c e s s e s q q —» 7 g . t h a t is. at O(o,.,„«•„). T h e r e l a t i v e i m p o r t a n c e of t h e t w o c o n t r i b u t i o n s d e p e n d s o n t h e t y p e of h a d r o n s c o l l i d i n g a n d t h e x ss ./ /• = 1p\jy/s r a n g e . For
STItlKTI'ltl
I'lIN*
I IONS A N D I ' A I M ' O N DIS I HIUI H O N S
/> r (Gc Vfc)
Fili. 7.21. T h e inclusive TT" a n d d i r e c t - p h o t o n cross sections a s f u n c t i o n s of t h e transverse m o m e n t u m m e a s u r e d by t h e E706 e x p e r i m e n t . T h e d a t a a r e c o m p a r e d to N L O QC'D predictions with different p a r t o n d i s t r i b u t i o n s , in one case s u p p l e m e n t e d wit h int rinsic t r a n s v e r s e m o m e n t u m for t h e incident p a r t o n s . F i g u r e f r o m K70G Collab.(l!)!)8).
e x a m p l e , t h e Q C D C o m p t o n process d o m i n a t e s in p p collisions a n d a t large .) •/•, where t h e anti(|iiark sea is suppressed. T h e r e f o r e fixed-target e x p e r i m e n t s a r e p a r t i c u l a r l y suited for t e s t i n g t h e gluon d i s t r i b u t i o n . In p p collisions t h e annihilation c o n t r i b u t i o n can be significant., b e c a u s e t h e u a n d <1 d i s t r i b u t i o n s in t h e ant iproton a r e t h e s a m e a s u a n d '/'. for e x a m p l e . 3..r> G c V < p-p < 12 G e V for t h e fixed-target e x p e r i m e n t K70G at I KUMILAM, balanced by a recoiling jet on t h e o p p o s i t e side of t h e exp e r i m e n t . T h e e x p e r i m e n t a l a d v a n t a g e of d e t e c t i n g p h o t o n s is t hat t he i r energy a n d m o m e n t u m can b e m e a s u r e d m o r e precisely t h a n for jets. P h o t o n s d e p o s i t all their energy within a few fells of t h e e l e c t r o m a g n e t i c c a l o r i m e t e r , w h e r e a s j e t s a r e m o r e s p r e a d o u t a n d d e p o s i t a large fraction of their energy also in the h a d r o n i c c a l o r i m e t e r . Overall this leads t o a b e t t e r e n e r g y resolution for p h o t o n s . In a d d i t i o n , j e t s have t o be defined by s o m e a l g o r i t h m , a n d a n a m b i g u i t y arises from t h e assignment of particles belonging t o t h e u n d e r l y i n g e v e n t or t o t he jet. T h e e x p e r i m e n t a l d i s a d v a n t a g e is t h e relatively low r a t e of single p h o t o n s c o m p a r e d t o jet p r o d u c t i o n , n a m e l y C)(o ( , l n n s )/C?(o"') = O ( o , . m / o s ) R: 10 .
II \ I ill! IN II \ l i l t O N S( ' A T I K K I N C
2! 17
I n! t h c n u o r c , direct p h o t o n producti<m suffers f r o m a very large b a c k g r o u n d of ,i" decays into I w o p h o t o n s which o f t e n form o v e r l a p p i n g clusters in t h e calorimeter. T h e y can lie idenlilied by s t u d y i n g t h e prolile of t h e e n e r g y d e p o s i t s in t h e calorimeter, which is different for single or two n e a r b y p h o t o n s . Also isolation criteria a r e a p p l i e d , since 7r°s a r e usually a c c o m p a n i e d by o t h e r high energy Itadrons. w h e r e a s direct p h o t o n s a r e not. An i n t e r e s t i n g result h a s been o b t a i n e d ill a recent m e a s u r e m e n t by t h e K7<)<> experiment. (1998). T h e r e t h e inclusive TT" a n d d i r e c t - p h o t o n cross sections a r e measured for 530 a n d 800 G e V proton b e a m s a n d a -r> 15 GoV n " b e a m incident on beryllium t a r g e t s . T h i s fixed-target, e x p e r i m e n t f e a t u r e s a large lead a n d liquid argon e l e c t r o m a g n e t i c c a l o r i m e t e r a n d a c h a r g e d p a r t i c l e s p e c t r o m e t e r . W h e n c o m p a r i n g t h e 10706 d a t a t o NI,() c a l c u l a t i o n s r a t h e r large d i s c r e p a n c i e s of factors of two a n d m o r e a r e found over almost, t h e entire />•/- r a n g e . Fig. 7.21. T h e s e discrepancies c a n not b e reduced by using different p a r t o n d i s t r i b u t i o n sets or c h a n g i n g t h e l e n o n i i a l i ' / a t i o n a n d / o r f a c t o r i z a t i o n scales. A possible solution is to a s s u m e that, t h e incident p a r t o n s have G a u s s i a n t r a n s v e r s e m o m e n t u m dist r i b u t i o n s with o n a v e r a g e (/.'•/•) 1 - 1 .5 G e V . which could be d u e to m u l t i p l e soft gluoii r a d i a t i o n . However, it is w o r t h not ing t h a t results f r o m o t h e r m e a s u r e m e n t s , for e x a m p l e by ISU e x p e r i m e n t s , o r by t h e WA70 a n d UA(> C o l l a b o r a t i o n s , do n o t lead to t h e s a m e conclusions. T h e r e t h e d a t a a r e in b e t t e r a g r e e m e n t with t h e NLC) Q C D predictions. Since there a p p e a r t o be even inconsistencies between t h e various e x p e r i m e n t a l dat a, m a n y issues c o n c e r n i n g t h e p h e n o m e n o l o g y of directp h o t o n p r o d u c t i o n r e m a i n to be clarified. For f u r t h e r r e a d i n g we refer t o a recent, discussion of t h i s topic by Auroncho et tit. (1999. 2000). 7.1.1
nclusive
jet
production
The m e a s u r e m e n t s of t h e inclusive cross section for jet p r o d u c t i o n as a f u n c t i o n of t h e jet t r a n s v e r s e e n e r g y at t h e IK VAT 1U)N d u r i n g recent, y e a r s have triggered some excitement, a n d s t i m u l a t e d i n t e r e s t s for s t u d y i n g p a r t o n d i s t r i b u t i o n s , in p a r t i c u l a r t he g l u o n d i s t r i b u t i o n , at large ./'. A basic description of t h e process a s well as a detailed discussion of I he e x p e r i m e n t a l issues can be found in previous c h a p t e r s . T h e origin of t h e excitement is u n d e r s t o o d by looking at Fig. 7.22. T h e r e a recent m e a s u r e m e n t by C D F is c o m p a r e d to a N L O Q C D prediction convoluted with a s t a n d a r d p a r t o n d i s t r i b u t i o n set, C T K Q 5 M (C'TKQ Collab., 2000). From t h e c o m p a r i s o n of t h e cross sections a n d even m o r e clearly f r o m the r a t i o d a t a / t h e o r y a n excess of t h e d a t a over t h e prediction is found for energies a b o v e 2 5 0 G e V . Such a n excess at t h e largest energy scales could be d u e to new physics, such as s u b s t r u c t u r e of q u a r k s . T h e d i s c r e p a n c y can not be reduced bv c h a n g i n g t h e r e n o r m a l i z a t i o n as well as f a c t o r i z a t i o n scales w i t h i n reasonable ranges, since this changes t h e predictions a b o v e ~ 1 0 0 G e V only by 2 - 9 % (Blazey a n d F l a n g h e r . 1999). However, it is found t h a t t h e t h e o r e t i c a l u n c e r t a i n t y f r o m t h e p a r t o n d i s t r i b u t i o n s can give rise to variations u p t o 30%. It is f u r t h e r p o i n t e d out t h a t t h e m e a s u r e m e n t s f r o m C D F a n d DO a g r e e well
di m n
I u n . r , r WM% I i wi-».-»
* I m i « v/ii
. ,-•
Kali»: l'rei. (lulu / Nl.O Ql'l) (CI I 0 5 M ICTI-Q5IIJ )
CHI-
1.4 CTEQ5M : noun, factor: CTIIQ5IIJ:
1.00 1.04
1.2 I 0.8
I nel. J ci : /)/
1
Data / CTEQ5M CTEQ5IIJ / CTEQSM CDF Data (I'rcl.) CTEQ5UJ CTEQ5M
dn ip,
( IO"Mnb GeVf>) 0.6
0.4
0.2
0 50
100
150
200 250 /MGeV)
300
350
400
F i e . 7 . 2 2 . C o m p a r i s o n of tlx* inclusive j e t cross section f r o m C D F w i t h t w o different p n r t o n d i s t r i b u t i o n sets by t h e C T E Q g r o u p . T h e u p p e r plot shows t h e r a t i o d a t a / t h e o r y . F i g u r e f r o m O T K Q Collab.(200()).
within t h e e x p e r i m e n t a l s y s t e m a t i c u n c e rt a i n t i e s . Since t h e inclusive j e t cross section at t h e largest j e t energies is very sensitive t o t h e large-.r regime, theoretical effort h a s been invested in o r d e r t o scrutinize this region f u r t h e r . T h e precise d a t a which a r e d o m i n a t e d by s y s t e m a t i c u n c e r t a i n t i e s with k n o w n correlations, t o g e t h e r with d a t a f r o m D1S. help t o c o n s t r a i n t h e par t o n d i s t r i b u t i o n s a n d in p a r t i c u l a r t h e gluon c o n t r i b u t i o n in t h e r a n g e 0 . 0 5 < .r < 0.25. T h e jet d a t a p r o b e t h e gluon a t a much larger e n e r g y scale t h a n for e x a m p l e t h e d i r e c t - p h o t o n m e a s u r e m e n t s , a n d in a d d i t i o n t h e y a r e not plagued by m u l t i p l e soft-gluou r a d i a t i o n effects. T h e C T E Q g r o u p h a s p e r f o r m e d a d e d i c a t e d global Q C D a n a l y s i s ( C T E Q C o l l a b . . 2000) where they have a d j u s t e d t h e p a r a m e t e r i z a t i o n of t h e gluon d i s t r i b u t i o n in o r d e r t o g e n e r a t e a significant e n h a n c e m e n t a t large .r. T h i s p a r a m e t e r i z a t i o n , called C T E Q 5 H . I , leads t o a good d e s c r i p t i o n of t h e full set of DIS a n d Droll Yan d a t a , with only a marginally worse lit result c o m p a r e d to t h e s t a n d a r d p a r t o n d i s t r i b u t i o n sot. However, in a d d i t i o n t h e CTEQ5I1.J d i s t r i b u t i o n s a r e a b l e t o r e s t o r e good agreem e n t b e t w e e n t h e o r y a n d high-EV jet; d a t a , as seen in Fig. 7.22. Similar s t u d i e s have b e e n p e r f o r m e d based on m e a s u r e m e n t s of t h e
Wl,mi/M q u i ) ANAI.YSKS
211!)
differential cross section as a liuiciion of ilio «li-jet m a s s is sensitive t o t h e p a r t o n • listrilnitions. T h e stime conclusions as a b o v e can be d r a w n f r o m t h e s e s t u d i e s (Bla/.ev a n d F l a u g h e r . 1999). Using s t a n d a r d p a r t o n d i s t r i b u t i o n sots, again a n excess of e v e n t s a t t h e largest di-jet masses is found. However, t h e t h e o r y c a n a c c o m o d a t e also this excess by i n t r o d u c i n g an e n h a n c e d gliion c o n t r i b u t i o n at largo .r.
7..r>
Global Q C D analyses: parton distribution fits
We have learned previously t hat. Icpton h a d r o n or liadron h a d r o n cross s e c t i o n s can bo expressed as a convolution of pal toni«- cross sect ions a n d p a r t o n m o m e n t u m dist ributions. T h e f o r m e r a r e calculable within pcrturbat.ive Q C D , w h e r e a s the l a t t e r a r e n o t , since t hey describe n o n - p o r t i i r b a t i v e l o n g - distance p h e n o m e n a of st r o n g i n t e r a c t i o n s . However, t hey a r e universal, t hat is. process-independent.. So i hey can be d e t e r m i n e d f r o m e x p e r i m e n t a n d s u b s e q u e n t l y be used t o predict cross sections for different process«« a n d / o r o t h e r energy scales. Similarly, when g o i n g b e y o n d leading o r d e r , a l s o t h e s t r u c t u r e f u n c t i o n s a r e expressed as convolutions of p e r t u r b a i i v o l v calculable f u n c t i o n s a n d p a r t o n d i s t r i b u t i o n s . M e a s u r e m e n t s of s t r u c t u r e f u n c t i o n s a r e t h e r e f o r e p a r t i c u l a r l y useful t o d e t e r mine p a r t o n d i s t r i b u t i o n s . A precise d e t e r m i n a t i o n of p a r t o n d i s t r i b u t i o n s e t s is of p a r a m o u n t i m p o r t a n c e not. only for t e s t s «>f ( ¿ C D . but also for precision m e a s u r e m e n t s of S t a n d a r d Model p a r a m e t e r s such as t h e W mass, as well a s t he d e t e r m i n a t i o n of signals a n d b a c k g r o u n d s for new physics searches at h a d r o n colliders. Iu o r d e r t o o b t a i n such a d e t e r m i n a t i o n o n e h a s to use a large set. of d a t a which t o g e t h e r cover a large r a n g e in r a n d Q~ a n d put s t r i n g e n t c o n s t r a i n t s on th<' various p a r t o n t y p e s within t h e p r o t o n . T h r o u g h o u t , this c h a p t e r we have s u m m a r i z e d m a n y m e a s u r e m e n t s which a r c used for global Q C D a n a l y ses. We have seen t h a t t h e s t r u c t u r e f u n c t i o n m e a s u r e m e n t s in charged Icpton and n e u t r i n o D I S e x p e r i m e n t s c o n s t r a i n t h e q u a r k c o n t r i b u t i o n s , a n d indirectly, via scaling violations, also t h e gliion d i s t r i b u t i o n . Sea q u a r k d i s t r i b u t i o n s a r e c o n s t r a i n e d by n e u t r i n o d a t a ( s t r a n g e q u a r k s ) , by Droll Van d a t a («/, «) a n d heavy q u a r k p r o d u c t i o n ( c h a r m ) . T h e l a t t e r also gives a h a n d l e on t h e gluon d i s t r i b u t i o n , t o g e t h e r with d i r e c t - p h o t o n a n d jet p r o d u c t i o n iu h a d r o n h a d r o n collisions. T h e s e global Q C D analyses have been p e r f o r m e d by several g r o u p s (CTKQ (1997. 2000). MR S T ( M a r t i n ci al., 1998). G R V (Gliick el al.. 1998)). A n o t h e r recent, d e t e r m i n a t i o n was p e r f o r m e d by B o t j e ( 2 0 0 0 ) . We will now s u m m a r i z e t h e s t e p s to b e taken in such a n analysis, avoiding however a discussion of d e t a i l s which can bo found in t h e references m e n t i o n e d above. T h e first s t e p consists in a choice of e x p e r i m e n t a l d a t a , with possible restrictions on t h e ./• a n d Q r a n g e iu o r d e r t o avoid e x p e r i m e n t a l l y a n d / o r t h e oretically critical regions. T h e n an a n s a t z for t h e functional form of the p a r t o n d i s t r i b u t i o n s a t a fixed scale Qft is chosen, for e x a m p l e . Qfi = 1 GeV~ (Bot.je. 2000) or I G e V " ( C T K Q Colla!».. 2000). As a n e x a m p l e ( B o t j e . 2000) t h e following p a r a m e t e r i z a t i o n s for t h e gluon d i s t r i b u t i o n (a:«/), t h e sea q u a r k d i s t r i b u t i o n
— • • O • A A • * 0
g e l ) fii
C ; (CKV J )
ZI-.US L<W.« III 1994 K665 NMC BCDMS SI.AC .
3000 (+12.0) 2000(+I I-J) 1500 (+10.7)
1200 (+10 I > 800 (+9.5) 650 (+9.0) 450 (+8.5) 350 (+8.0) 250 (+7.5) 200 (+7.0) 150 (+6.5)
120 (+6.0) 90 (+5.6) 70 (+5.2) 60 (+4.8) 45 (+4.4) 35 (+4.0) 27 (+3.6) 22 (+3.2) IX (+2.8) 15 (+2.4) 12 (+2.0)
10 (+1.6)
8.5 (+1.2) 6.5 (+0.8) 4.5 (+0.4) 3.5 (+0)
FIG 7 . 2 3 . T h e p r o t o n s t r u c t u r e f u n c t i o n F, v e r s u s * a t h x c d ya ues of Q f r o m d a t a included in t h e Q C D a n a l y s i s of Ref. ( B o t j e . 2000). I he full h u e i n d i c a t e s t h e r e s u l t of t h e Q C D lit. T h e c o n s t a n t s in b r a c k e t s a r e a d d e d t o
xS, t h e d i f f e r e n c e of d o w n a n d u p a n t i q u a r k s ( * A ) a n d t h e valence c,uark distributions
(xii
v
,xd
xS(x
v
)
are used:
Q l )
=
=
2*(S
+ d +
-s) =
~ -'•)"•(!
.
+ 7
s
x) •
(7-43)
(7.44)
<||.W|I/U. q l II A IN A 1.1 NI'iN T a b l e 7.1 milli
ters
are
Parameter A 'l
7
Tin
unities
defined
oj in
tlir
tin
/miiiim
lert.
The
Ins < rrors
obtained are
front
u ¡/lobnl
statistical
only.
xS a: A X<) 0.81 ± (1.03 0.31 ±0 . 1 1 ) 1.32 0.15 ± 0.01 0.57 ± 0.09 - 0 . 2 6 ± 0.03 3.90 ± 0.19 7.47 ± 1 . 0 0 5.19 ± 1.53 - 0 . 5 2 ± 1.42 - 1 . 3 2 ± 0 . 0 6
2.72 0.62 ± 0.02 3.89 ± 0 . 0 2 2.79 ± 0 . 3 1
The
pa
xdv 1.98 0.05 ± 0.02 3.07 ± 0 . 1 8 -0.82 ± 0 . 1 2
(7,15)
x(d - d) = A , .r'"(L - , r ) " " ( l + 7,|.'-') .
(7.40)
= x(u
x d A x . Q l )
=
I IIIIS t h e p a r a m e t e r s
v
Jit.
(1 - , r ) " " ( l + 7 „ : r ) .
riiv(x.Q'ft)
-
a)
= .4,,
xu
QCD
(?/.) govern t h e large (small) .r b e h a v i o u r . T h e s t r a n g e
q u a r k d i s t r i b u t i o n x(s I .s)
'2.rs is fixed t o b e a f r a c t i o n i; of t h e n o n - s t r a n g e sea
as f o u n d by n e u t r i n o e x p e r i m e n t s (Section 7.2.1). T h e n o r m a l i z a t ion p a r a m e t e r s .1. a r e fixed by t h e m o m e n t u m s u m r u l e a n d valence q u a r k c o u n t i n g rules / n ' da:
/,,'(•''.'/ T
itv(x)
=
xS
2 and
4-
xav
j0' d.c
+ d
v
xdv)th: (x)
=
-
I
I. T h e
r e m a i n i n g free p a r a m e t e r s have t o b e d e t e r m i n e d f r o m t he fits t o t h e d a t a . O n c e t h e p a r a m e t e r i z a t i o n a s well a s t h e t h e o r e t i c a l s c h e m e , t h a t is. r e n o r m a l i z a t i o u a n d f a c t o r i z a t ion s c h e m e , such a s MS. a r e fixed, t h e DC!LAP e v o l u t i o n e q u a t i o n s at N L O p e r t u r b a t i v e Q C D a r e e m p l o y e d t o c o m p u t e t h e p a r t o n dist r i b u t i o n s a n d s u b s e q u e n t l y c r o s s s e c t i o n s a n d s t r u c t u r e f u n c t i o n s a t t h e scales relevant for t h e various e x p e r i m e n t s . T h e p r e d i c t i o n s a r e t h e n fitted t o t h e d a t a by a l e a s t - s q u a r e s m e t h o d . In B o t j e ( 2 0 0 0 ) 1578 d a t a p o i n t s a r e f i tted, resulting in a n excellent lit w i t h a \ - / p o i n t of 0.97. T h e lit, t o F> s t r u c t u r e f u n c t i o n m e a s u r e m e n t s is s h o w n in Fig. 7.23, a n d t h e r e s u l t i n g lit p a r a m e t e r s a r e given in T a b l e 7.1. Similarly, t h e O T K Q C o l l a b o r a t i o n uses a b o u t 1000 d a t a p o i n t s f r o m DIS. L10 of Droll Yan a n d 57 of j e t e x p e r i m e n t s , w i t h e q u a l l y g o o d lit results. P l o t s of t h e fitted p a r t o n d i s t r i b u t i o n s a r e given in Fig. 7.2-1 for (a) Q- = K l G e V " ( B o t j e . 2000) a n d (b) Q 2 = 2 5 G e V 2 (CTI2Q C o l l a b . . 2 0 0 0 ) . It is nicely seen t h a t t h e valence q u a r k d i s t r i b u t i o n s p e a k a r o u n d x - 1(1 1 a n d vanish a t s m a l l x. w h e r e a s t h e gluon a n d sea q u a r k d i s t r i b u t i o n s i n c r e a s e with d e c r e a s i n g x. It. s h o u l d b e n o t e d t h a t t h e g l u o n d i s t r i b u t i o n h a s t o b e scaled d o w n in o r d e r t o bo s h o w n on t h e p l o t , w h i c h d e m o n s t r a t e s t h e d o m i n a n c e of g l u o n s in t h e p r o t o n at s m a l l x values. T h e s e p a r a m e t e r s can b e used t o c o m p u t e o t h e r p r e d i c t i o n s w i t h i n t he s a m e o r d e r of p e r t u r b a t i o n t h e o r y a n d t h e s a m e t h e o r e t i c a l s c h e m e . T h e y a r e implem e n t e d in c o m p u t e r p r o g r a m s to he used t o g e t h e r with a n a l y t i c a l c a l c u l a t i o n s or M o n t e C a r l o s i m u l a t i o n s . B e c a u s e of pract ical r e a s o n s t he e n e r g y d e p e n d e n c e of t h e p a r a m e t e r s is o f t e n p a r a m e t e r i z e d by s i m p l e power laws, which a v o i d s t h e need for c o m p u t i n g t h e full D G L A P evolut ion every t i m e a n e w p r e d i c t i o n is needed. T h e r e a r e m a n y s u b j e c t i v e choices w h i c h have t o b e m a d e w i t h i n a global a n a l y s i s . T h i s u n a v o i d a b l y l e a d s t o d i f f e r e n c e s in t he r e s u l t s f r o m d i f f e r e n t g r o u p s
K i c . 7 . 2 1 . Tlu* p a r t o n m o m e n t u m d i s t r i b u t i o n s in t h e p r o t o n as e x t r a c t e d by Hot jo (•_>()()()) (a) a n d tlie C T K Q g r o u p (200(1) (b). T h e s h a d e d a r e a s indicate t h e b a n d s of u n c e r t a i n t y .
a n d consequently in intrinsic s y s t e m a t i c u n c e r t a i n t i e s . T h e s e choices cover • t h e selection of d a t a sets a n d k i n e m a t i c b o u n d a r i e s . As an e x a m p l e , t h e use of d i r e c t - p h o t o n d a t a is p a r t i c u l a r l y d e b a t e d b e c a u s e of t h e discrepancies between d a t a a n d NLC) predictions (c.f. Section 7.•!..'{). • t h e theoretical s c h e m e used t o c o m p u t e t h e p e r t u r b a t i v e predictions. In p a r t i c u l a r t h e t r e a t m e n t of heavy q u a r k s (threshold a n d m a s s effects) is not free of a m b i g u i t i e s . • t h e choice of the st rong coupling constant. o s . which is st rongly correlated with t h e gluon d i s t r i b u t i o n . Recently t h e preferred a p p r o a c h h a s b e c o m e t o fix t h e coupling to n s ( A I $ ) = 0.118 a n d t o s t u d y t h e effect on t h e p a r t o n d i s t r i b u t i o n s when varying it wit hin a c e r t a i n r a n g e , for e x a m p l e , by ± 0 . 0 0 5 . • t h e correction for n o n - p e r t u r b a t i v e , so-called higher-twist c o n t r i b u t i o n s . O n e ansat.z is F j 1 T = K 2 Q C D ( 1 + (x) 2). w h e r e (x) is a polynomial func t i on. Its p a r a m e t e r s a r e d e t e r m i n e d f r o m t h e global lit. • corrections for t a r g e t m a s s a n d nuclear effects of d a t a from fixed t a r g e t e x p e r i m e n t s . O n l y pheuomcnological models exist for such c o r r e c t i o n s , a n d t he g r o u p s differ in their choice of d a t a t o b e co r r e c t e d o r not. • t h e p r o p a g a t i o n of e x p e r i m e n t a l a n d theore t i c a l u n c e r t a i n t i e s i n t o uncertainties of t h e p a r t o n dist r i b u t i o n s a n d u l t i m a t e l y t h e uncert aint ies of cross section predictions. C u r r e n t l y this is a field of a c t i v e s t u d i e s a n d discussions. T h e user of a p a r t i c u l a r set of p a r t o n d i s t r i b u t i o n s should be a w a r e of these p r o b l e m s a n d carefully e v a l u a t e their impact on t h e resulting predictions.
1(1.1
til.« > Il/Y I, <)( I) ANALYSES (ni
| 1.5 .
gluon VEitiiiiions tli.it .ur consistent with DIS+Dioll Yanitiliiscli / +1*«. y y
III
(I.S
"
-15%
0.5
0.6
C? = 5 GcV
Mn 0.4
1.5 + 10% 1.0
0.2
-10% 0.5
Q= 100 GcV
0 10*10 X
1
10 -
0.1 I'ailon x
0.2 0 3 O.-l 0.5
Fit:. 7 . 2 5 . (a) Tlie r a t i o (d+d)/(u + R) v e r s u s x a t Q2 = l O G e V 2 f r o m t h e ( ¿ C D lit s of ( B o t j e . 2000) a n d (C'I EQ Collai).. lil!)7). (h) S t u d y of t h e u n c e r t a i n t y on t h e gluon d i s t r i b u t i o n ( H u s t o n ci a I.. 1!)!)8). Shown is t h e r a t i o t o t h e d i s t r i b u t i o n o b t a i n e d in ( C T E Q Collab.. 1997) at two different. (J values.
A f t e r m a n y y e a r s of global ( ¿ C D a n a l y s e s our knowledge of t h e p r o t o n s t r u c t ure is q u i t e rich a n d precise. T h e valence q u a r k d i s t r i b u t i o n s a r e r a t h e r precisely known. T h e gluon d i s t r i b u t i o n is well c o n s t r a i n e d over t h e r a n g e 0.05&i::£0.25. however, t h e very low-:/: b e h a v i o u r is much less well k n o w n , a n d also at large .1:^0.25 it suffers f r o m large u n c e r t a i n t i e s (Fig. 7.25(1))). In t h e large-;;: region also the d / u r a t i o is very much u n c o n s t r a i n e d , a s can be seen f r o m Fig. 7.25(a). T h e large-./ - d o m a i n could definitely gain f r o m a b e t t e r theoretical control of directp h o t o n p r o d u c t i o n in h a d r o n liadron collisions. C o n c e r n i n g sea q u a r k s , t h e fact t h a t d ^ n is well established bv now. F u t u r e n e u t r i n o e x p e r i m e n t s should help to u n d e r s t a n d if s = .s. T h e c h a r m q u a r k s e a still suffers from theoretical a m biguities in h a n d l i n g heavy q u a r k effects. Finally, t h e q u e s t i o n of u n c e r t a i n t i e s on t h e p a r t o n d i s t r i b u t i o n s a n d their p r o p a g a t i o n into u n c e r t a i n t i e s on t h e cross sections has t o be addressed f u r t h e r in t h e near f u t u r e .
Exercises for C h a p t e r 7 7 I Show that in a d e e p inelastic lept.on proton interaction the Icpton s c a t t e r i n g a n g l e ()'. is given by cos 2 (
S IKIKTUNK
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I IONS A N D I'AIM'ON DISI NIHU I IONS
7 2 Write down t he expressions for charged c u r r e n t p and e ( p scattering in t e r m s of t h e target h a d r o n s p.d.f.s. C o m m e n t on t h e helicity s t r u c t u r e of t h e c o n t r i b u t i n g hard suhproeesses. 7 '.i Derive a n expression for t h e r a t i o of t h e F¿ s t r u c t u r e f u n c t i o n s in lepton nucleoli a n d n e u t r i n o nucleón s c a t t e r i n g , in t e r m s of t h e nucleoli's p.d.f.s. 7 I Show t h a t t h e i/ d e p e n d e n c e iu t h e limit i) — 0 of t h e n e u t r i n o nucleón s c a t t e r i n g cross section can he used t o d e t e r m i n e t h e relative fluxes of n e u t r i n o s a n d n n t i n e u t r i n o s . 7 5 T h e p r o t o n h a s electric c h a r g e 1 (in units of c). b a r y o n n i u n b e r 1 a n d zero st r a n g e n e s s a n d c h a r m . Use these c o n s t r a i n t s iu o r d e r t o derive s u m rules for t h e valence q u a r k dist ribut ions, such a s L d.r u v ( . r ) = 2.
THE
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II o n e a c c e p t s that. QC.'D is described by an u n b r o k e n SU(3) g a u g e s y m m e t r y , then, apart, f r o m q u a r k masses, t h e t h e o r y of s t r o n g i n t e r a c t i o n s c o n t a i n s only one free p a r a m e t e r , t h e s t r o n g coupling constant. a s . Of course o n e has t o verify e x p e r i m e n t a l l y t h a t SU(3) really is t h e u n d e r l y i n g g a u g e s y m m e t r y of t h e t heory, a topic which is a d d r e s s e d in a later c h a p t e r . Here we will t a k e for g r a n t e d t h e a r g u m e n t s which m o t i v a t e d t o build Q C D on t h e g a u g e g r o u p S IJ (.'5) a n d d e s c r i b e t h e various a p p r o a c h e s t o m e a s u r e t h e co u p l i n g s t r e n g t h . T h e r a t i o n a l e behind using as m a n y m e t h o d s as possible t o d e t e r m i n e o s is to d e m o n s t r a t e thai. Q C D really is t h e c o r r e c t t h e o r y of s t r o n g i n t e r a c t i o n s by showing t h a t o n e universal coupling constant, describes all s t r o n g i n t e r a c t i o n s p h e n o m e n a . T h e processes discussed below cover a large r a n g e of b o t h space-like a n d time-like m o m e n t u m t r a n s f e r s . R e a c t i o n s include n e u t r i n o a n d charged lepton nucleoli s c a t t e r i n g , proton-(anti)prot.on collisions. e + e ~ a n n i h i l a t i o n a n d d e c a y s of b o u n d s t a t e s of h e a v y q u a r k s . O b s e r v a b l e s a r e . for e x a m p l e , cross section m e a s u r e m e n t s , scaling violations, b r a n c h i n g ratios, global event s h a p e variables a n d t h e p r o d u c t i o n r a t e s of h a d r o n j e t s . To leading o r d e r all t hese processes can be described as i n t e r a c t i o n s between s p i n - 1 / 2 f e n n i o n s a n d spiti-1 g a u g e bosons. W h i l e t h e involved spins will det e r m i n e t he general s t r u c t u r e of a n g u l a r d i s t r i b u t i o n s a n d correlations between initial a n d final s t a t e particles, t h e r a t e s can a l w a y s be a d j u s t e d by i n t r o d u c i n g effective c o u p l i n g c o n s t a n t s . T h e real test, of Q C D comes with controlling higher o r d e r effects t o t h e level w h e r e precise q u a n t i t a t i v e predictions can be m a d e a s function of t h e f u n d a m e n t a l coupling of t h e theory. If a universal coupling exists. then all d a t a within t h e respective c o m b i n e d e x p e r i m e n t a l a n d theoretical u n c e r t a i n t i e s must b e consistent with t h o s e calculations. O v e r t h e p a s t y e a r s significant, progress h a s been m a d e b o t h e x p e r i m e n t a l l y a n d in t heoretical calculations. As explained in detail in previous c h a p t e r s , nexlto-lead-inf) order ( N L O ) p r e d i c t i o n s a r e generally available. For s o m e inclusive q u a n t i t i e s also t h e ncxt-lo-next-to-leading order ( N N L O ) corrections have been calculated a n d e s t i m a t e s of t h e next higher t e r m s exist. In a d d i t i o n , all-order r e s u m m a t i o n s of leading-logarithmic (LL) a n d ncxt-to-lcu ng-logurithmic (NLL) corrections have been p e r f o r m e d in s o m e cases. Power law correct ions a r e t r e a t e d either by heuristic m e t h o d s which i n t r o d u c e new free p a r a m e t e r s , or m o r e rigorously in t h e f r a m e w o r k of t h e operator product expansion ( O P E ) o r t h e r e s u m m a t i o n of r e n o r m a l o n chains.
I III SI 1« >N<; < ' I I U I T I N i : < (IINS I \ N I
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Tin- Q C D p r e d i c t i o n for a c r o s s section at a n e n e r g y s c a l e (J c a n lie exp r e s s e d as s u m of p e r t u r b a t i v e t e r m s <5|>-p v a r y i n g l o g a r i t h m i c a l l y w i t h energy, a n d n o n - p e r t urliat ive p o w e r law c o r r e c t i o n s . In n e x t - t . o - l e a d i n g o r d e r t he p e r t u r b a t i v e p a r t is of tlie f o r m V r = o s ( / t 2 M + a 2 ( / / 2 ) ( t f + A,% In
.
(S. 1)
Here /i 11a is tli«» a r b i t r a r y r e n o r i n a l i z a t i o n s c a l e of t h e c a l c u l a t i o n a n d .fn t h e l e a d i n g o r d e r coefficient of t h e Q C D /^-function. S e t t i n g //"' = Q~ tile a b o v e e x p r e s s i o n c q n (8.1) simplifies t o a p o w e r series in < i s ( Q 2 ) w i t h e n e r g y i n d e p e n dent coefficient f u n c t i o n s for t h e L O a n d NLC) c o n t r i b u t i o n s , respectively. T h e f u n c t i o n s /I a n d 13 a r e d e t e r m i n e d by t h e L o r e n t z s t r u c t u r e of t h e c o n t r i b u t i n g F e y n i n a n d i a g r a m s a n d t h u s essentially by t h e s p i n s of t h e i n t e r a c t i n g lields. Q C D only e n t e r s t o t h e e x t e n t t h a t it d e f i n e s t h e r e l a t i v e w e i g h t s for t h e different k i n d s of c o u p l i n g s . For /1 1 = Q~ t h e e n e r g y d e p e n d e n c e of c q n ( S . l ) conies o n l y f r o m t h e r u n n i n g of t h e c o u p l i n g <*AQ 2 )W h i l e t h e choice //-' - ( J 2 m a y a p p e a r t o b e t h e n a t u r a l o n e . t h e r e is a priori no reason t o single o u t a specific r c n o r n i a l i z a t i o n scale s i n c e t h e full t h e o r y is i n v a r i a n t w i t h respect t o //. For t h e p e r t u r b a t i v e e x p a n s i o n t h i n g s a r e not so o b v i o u s . If t h e r c n o r n i a l i z a t i o n scale a n d t h e scale of t h e p h y s i c s p r o c e s s a r e very different . | l n ( / / 2 / Q 2 ) | S> 1. t h e n t h e p e r t u r b a t i v e e x p a n s i o n is n o longer in t h e s t r o n g c o u p l i n g b u t in a n effective e x p a n s i o n p a r a m e t e r o s ( / i 2 ) l u ( / r / Q 2 ) which c a n h e larger t h a n u n i t y or e v e n negative;. W h i l e a l w a y s b e i n g f o r m a l l y a valid e x p a n s i o n of t h e p r e d i c t i o n for a p h y s i c a l o b s e r v a b l e , t h e c o n s e q u e n c e s for a n u m e r i c a l e v a l u a t i o n of t h e series c a n b e d i s a s t r o u s . A c o n v e r g e n t s e r i e s in o , ( C / - ) c a n b e c o m e d i v e r g e n t in o s ( / f 2 ) , t h a t is. while t h e full t h e o r y d o e s not. d e p e n d on t he choice of ¡i. t l»' t r u n c a t e d p e r t u r b a t i v e e x p a n s i o n c e r t a i n l y d o e s . O n e a l s o sees t h a t t h e r e n o r m a l i z a t i o n s c a l e s h o u l d b e c h o s e n not t o o different f r o m t he n a t u r a l scale, w h i c h a t least g u a r a n t e e s c o n v e r g e n c e of t h e pert u r h a t i v e series for a s y m p t o t i c e n e r g i e s w h e r e o s g o e s t o z e r o . T h e a b o v e d i s c u s s i o n s h o w s t h a t v a r y i n g t h e r e n o r m a l i z a t i o n scale is a w a y t o s h i f t a r o u n d h i g h e r o r d e r c o n t r i b u t i o n s . S c a l e v a r i a t i o n s t h u s a r e a convenient way to p r o b e t h e s e n s i t i v i t y of t h e p r e d i c t i o n t o u n c a l c u l a t e d h i g h e r o r d e r s a n d thereby to assess theoretical uncertainties. O n e must, however, emphasize that a s m a l l scale d e p e n d e n c e at. t h e c a l c u l a t e d o r d e r d o e s not m e a n t h a t h i g h e r o r d e r t e r m s a r e negligible. In g e n e r a l , a s o u n d a s s e s s m e n t of t l u ' o r e t i c a l u n c e r t a i n t i e s will h a v e t o t a k e o t h e r w a y s of e s t i m a t i n g t h e o r e t i c a l e r r o r s i n t o a c c o u n t a s well, s u c h a s d r o p p i n g t h e highest- o r d e r t e r m w h i c h is k n o w n , v a r i a t i o n of t h e socalled matching schcmc w h i c h is used t o m e r g e a fixed o r d e r c a l c u l a t i o n a n d a resumnied prediction, or using P a d e approximations to rewrite the p e r t u r b a t i o n series. W i t h o u t g o i n g i n t o d e t a i l s it is fair t o s a y t h a t all t h e s e m e t h o d s c a n be i n t e r p r e t e d a s a t t e m p t s t o g u e s s a n d resiim t h e u n c a l c u l a t e d h i g h e r o r d e r s . U n f o r t u n a t e l y , t o d a t e n o g e n e r a l l y e s t a b l i s h e d way t o a s s e s s t h e o r e t i c a l e r r o r s
COMPARISON
.1
ANI» C O M M I N A I I O N O K K K S U L T S
exists. T h e m e t h o d s m e n t i o n e d a b o v e a r c used in various c o m b i n a t i o n s , which makes it very dillicnll t o c o m p a r e theoretical uncertainties. Nevertheless, a n d even if il is not. possible t o assign confidence levels in a strict, m a t h e m a t i c a l sense t o theoretical errors, they a r e b e s t e s t i m a t e s of t h e a c t u a l u n c e r t a i n t i e s c o n s t r u c t e d such t h a t it is reasonable t o interpret, t h e m like conventional OS1/ confidence level intervals.
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P i c . S. 1. E s t i m a t e of theoretical un c e r t a i n t i e s for a m e a s u r e m e n t of t h e s t r o n g coupling from global event s h a p e variables. A detailed discussion is given in the text.
T h i s is illustrated in Fig. <8.1 by m e a n s of s o m e m e a s u r e m e n t of t h e s t r o n g coupling c o n s t a n t p e r f o r m e d on global event s h a p e variables by t h e AI.KI'II collaboration. T h e variables will be described later. T h e left plot shows error b a n d s in m e a s u r e m e n t s of n s ( M $ ) based on t h e LO, N L O a n d N L O + N L L A predictions for the two-jet r a t e /?•> as function of ln(//'~/Q 2 ). T h e w i d t h s of t h e b a u d s indicate what h a p p e n s when switching from t h e p e r t u r b a t i v e prediction of to that of In R i . T h e t heoretical e r r o r was taken t o be t h e r a n g e of values covered by t h e p r o j e c t i o n of t h e respective b a n d s over 1 < In l i 2 / Q " < 1 on t h e abscissa. T h e right figure shows how t h e central values a n d e r r o r s o b t a i n e d this way for t hree different s h a p e variables converge with i m p r o v e m e n t s in t h e I heorv. That, this p r o c e d u r e yields r e a s o n a b l e e r r o r e s t i m a t e s is d e m o n s t r a t e d by t he fact that for a fixed level of theoretical precision t h e e r r o r s cover t h e s c a t t e r bet ween t he different variables, a n d t h a t t h e y also m a t c h t he convergence observed w h e n using b e t t e r predictions.
.2
Comparison and combination of results
To c o m p a r e m e a s u r e m e n t s of t h e s t r o n g coupling which were p e r f o r m e d at different scales, o n e h a s t o t a k e into account that o^ is energy d e p e n d e n t . Me a su re -
•ION
I III', N T H O N O O O U I ' I . I N O ( O N S T A N ' I
i n c u t s of NS c a n In- c o m p a r e d , eit her by e v o l v i n g b a c k w a r d s t o I he point A<J<-D. t h a t is. A j ^ w h e n w o r k i n g in t h e M S - s c h c i n c . w h e r e o s diverges, or by evolving t o a c o m m o n r e f e r e n c e scale, which in recent, y e a r s h a s b e c o m e t h e '/. m a s s . Details a b o u t how t o p e r f o r m t h e e v o l u t i o n i n c l u d i n g t h e p r o p e r t r e a t m e n t of llavour t h r e s h o l d s c a n b e f o u n d in A p p e n d i x D. 1. In a d d i t i o n it is i n t e r e s t i n g t o plot a m e a s u r e m e n t of t h e s t r o n g c o u p l i n g for t h e scale w h e r e it h a s been meas u r e d in o r d e r to see t h e r u n n i n g . In c a s e s w h e r e a r e a c t i o n b e t w e e n point like p a r t i c l e s h a p p e n s at a fixed C . o . M . energy, this e n e r g y is q u o t e d a s t h e scale (J of t h e m e a s u r e m e n t . If t he d a t a cover a r a n g e | Q , „ i „ . ,,:,*]. t h e n eit her t h e c e n t r a l scale (¡noted by t h e a u t h o r s is given o r . since o s varies wit h t h e l o g a r i t h m of t h e energy, t he scale (} = y / Q m \ n Q „ m x which satisfies I11Q = ( l n Q m i l l + l n Q I l i n x ) / 2 . For a n overview over v a r i o u s a d m e a s u r e m e n t s o n e is not only i n t e r e s t e d in t h e final r e s u l t , but a l s o in a b r e a k d o w n of t h e u n c e r t a i n t i e s i n t o e x p e r i m e n t a l a n d t h e o r e t i c a l ones. W h i l e t h e f o r m e r a r e fixed for a given m e a s u r e m e n t , t h e l a t t e r in principle could b e r e d u c e d by f u r t h e r p r o g r e s s on t h e t h e o r e t i c a l side. O n e t h u s a l s o s h o u l d k e e p t r a c k of t h e c u r r e n t level of c a l c u l a t i o n s used in t h e analysis. All m e a s u r e m e n t s p r e s e n t e d below a t least a r e based on N L O predict ions, m a n y a l r e a d y on N N L O or N L O p l u s a l l - o r d e r resununat.ioii of leading a n d n e x t - t o - l e a d i n g l o g a r i t h m s . I n d e p e n d e n t of which kind of t h e o r e t i c a l p r e d i c t i o n h a s been used a s a basis for a part icular a n a l y s i s , all r e s u l t s a r e c o m p a r a b l e since t h e t h e o r e t i c a l e r r o r s s h o u l d a c c o u n t for t h e effect of missing higher o r d e r t e r m s . W h e n c o m b i n i n g r e s u l t s o n e h a s t o keep in mind t h a t t h e precision of m a n y m e a s u r e m e n t s of t h e s t r o n g c o u p l i n g c o n s t a n t is limited by t h e o r e t i c a l u n c e r t a i n ties. T h i s leads t o a s i t u a t i o n w h e r e m e a s u r e m e n t s f r o m different, e x p e r i m e n t s , while being s t a t i s t i c a l l y i n d e p e n d e n t , a r e c o r r e l a t e d t h r o u g h t h e u n c e r t a i n t i e s in t h e t h e o r e t i c a l p r e d i c t i o n . A global a v e r a g e h a s t o t a k e t h i s i n t o account.. U n f o r t u n a t e l y it is p r a c t i c a l l y i m p o s s i b l e t o q u a n t i f y t h e s e c o r r e l a t i o n s o n a e a s e by c a s e basis since t h e m e t h o d s t h a t h a v e been e m p l o y e d in t h e d e t e r m i n a t i o n of t h e t h e o r e t i c a l u n c e r t a i n t i e s v a r y c o n s i d e r a b l y b e t w e e n different m e a s u r e m e n t s . T h e s e c o r r e l a t i o n s usually d o not affect t h e v a l u e of a w e i g h t e d a v e r a g e , but. t h e y have t o b e t a k e n i n t o a c c o u n t in o r d e r t o derive a realistic e r r o r e s t i m a t e . In situ a t i o n s w h e r e t h e v c a n n o t b e q u a n t i f i e d reliably, t h e best o n e c a n d o is t o treat h i d d e n c o r r e l a t i o n s in a n effective way. M e t h o d s t o d o t h i s iu a s y s t e m a t i c way h a v e been p r o p o s e d by o n e of t h e a u t h o r s (Sclunelling 1 !)!)!">/;. 2000). w h e r e a n e s t i m a t e for t h e size of t h e c o r r e l a t i o n s b e t w e e n m e a s u r e m e n t s is derived f r o m t h e t y p i c a l s c a t t e r of t h e r e s u l t s a r o u n d a c o m m o n a v e r a g e c o m p a r e d t o t h e q u o t e d e r r o r s of t h e m e a s u r e m e n t s . T h e basic idea is t h a t if t h e a v e r a g e s c a t t e r is s m a l l e r t h a n t h e a v e r a g e e r r o r s , t h e n t h e e r r o r s a r c c o r r e l a t e d . In t h e following s e c t i o n s we will d e s c r i b e t h e most i m p o r t a n t m e t h o d s t h a t a r e used t o m e a s u r e o s a n d t h e s t a t u s of o u r k n o w l e d g e reached at t h e t u r n of t h e m i l l c n h u n . T h e field is still r a p i d l y evolving, a n d t h e r e a d e r is a l s o r e f e r r e d t o t h e l i t e r a t u r e for f u r t h e r i n f o r m a t i o n . G o o d reviews a r e found for e x a m p l e in ( B e t h k e . 2000: Hinchliffc a n d M a n o l l a r . 2000). R e s u l t s a r e collected in Table iS.:{ a n d . unless m e n t i o n e d o t h e r w i s e in t h e t e x t , q u o t e d f r o m ( B e t h k e . 2000).
IN< I UMIVI
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Inclusive m e a s u r e m e n t s s
s
s
.
)
s
ss, s
s 0(
),
(
c.t il.. . T
s { . .3.
s
. T
s s
7 ).
s
s A
The ratios
T
s
f -t and
.
i
s
s s
s s s s
s
.
s
(
)
. T
s
s .
.2. T s s s
s
s s
s
s
s
s s,
s s s
s
.
s s
s
s
,
s
,
ss
Ms.
s
s
. s
s
s
=
ss s
s)
<7i,oni(e+e- — leptons) s
s +
s
ss ss
.
,
s s
ss s
s s
s s, s,
s s
ss s s
s. ss
s
, s
s
s
s s.
s
,
;tio
11115 s i i i o c ; COUPLING CONSTANI
R = I{K\V (I + <>qcd + <>',,. + <
) ,
(8-3)
w h e r e t h e overall f a c t o r RKW d e p e n d s on t h e clect.rowoak c o u p l i n g s of I he q u a r k « . T h e c o r r e c t i o n s a r e d o m i n a t e d by t h e pert u r b a t i v e Q C c o r r e c t i o n rfgci). Tli ot her t e r m s >m a n d t a k e into account 111<' finite q u a r k masses a n d t h e noli perl u r b a t i v e c o r r e c t i o n s . Since t h e leading n o n - p e r t u r h a t i v e effects a r e { l ') + a n n i h i l a t i o n processes. M a s s clfeets, t h e y a r e negligible for high e n e r g y e e in p a r t i c u l a r f r o m t h e large t o p - b o t t o m m a s s s p l i t t i n g , h a v e been c a l c u l a t e d ( C h e t y r k i n v.t ol.. 1996i>) t o C ( o f ) . Ignoring m a s s c o r r e c t i o n s a n d non-pert urbat ive effects, t h e t heoretical prediction for becomes
^
w h e r e the s t r o n g c o u p l i n g h a s t o b e e v a l u a t e d at t h e scale of t h e C . o . M . e n e r g y ^ s of t h e e x p e r i m e n t . Various d e t e r m i n a t i o n s of t h e s t r o n g coupling « „ ( Q 2 ) based on eqn (8.4) h a v e been p e r f o r m e d , s p a n n i n g t h e e n e r g y r a n g e f r o m us! a b o v e t h e lib-threshold ( C Collab., 1998), = 10.52 G e V , until .'M G e V ( B e t h k e . 2000) a n d = 42.4 G e V ( H a i d t . 1995). T h e results a r e given in Table 8.3. Also t h e theoretical prediction for ? . w h e r e a n e l e c t r o n a n d a p o s i t r o n a n n i hilate into a Z b o s o n , is known t o third o r d e r in «... B e c a u s e of t h e s i m u l t a n e o u s c o n t r i b u t i o n of vector a n d axial c u r r e n t s in t h e Z fern l ion c o u p l i n g s a n d tindifferent n a t u r e of m a s s c o r r e c t i o n s in t h e presence of axial c u r r e n t s , t h e coeffiAn c i e n t s in t h e t h e o r e t i c a l p r e d i c t i o n a r e slightly different f r o m t h o s e for v explicit d e s c r i p t i o n t o c a l c u l a t e t h e t h e o r e t i c a l p r e d i c t i o n including t h e leading o r d e r c o r r e c t i o n s is given in A p p e n d i x .3. A m o r e c o m p l e t e c a l c u l a t i o n leads to the following p a r a m e t e r i z a t i o n (Tournelier. 1998) a s f u n c t i o n of < s (My ).
, = 19.934
-I- 1 . 0 4 5 ^ + 0 . 9 4 ^ -
(8.5)
sing I lie c o m b i n e d result f r o m all four I' e x p e r i m e n t s ( A b b a n e o i t « .. 2(1011), , = 2 0 . 7 6 8 0.024, o n e finds o s ( M ) = 0.124 0.004. w h e n - t h e e r r o r is p u r e l y s t a t i s t i c a l . T h e full result i n c l u d i n g t h e o r e t i c a l u n c e r t a i n t i e s f r o m scale v a r i a t i o n s a n d t h e u n c e r t a i n t y of t he Higgs m a s s is given in T a b l e 8.3. 8.3.2
easurement
of o s from
RT
An o s - m e a s u r e m e n t can also be o b t a i n e d f r o m RT — Auulr < ' h e r a t i o of t h e h a d r o n i c t o t h e electronic b r a n c h i n g r a t i o of t h e tail l e p t o n . Here Q C radiative c o r r e c t i o n s a f f e c t t h e h a d r o n i c final s t a t e f r o m a W decay, t h e leading o r d e r d i a g r a m of which is s h o w n in Fig. 8.3.
IN< I I SIM MKASUUKMKN IS
I' l c . 8 . 3 . L e a d i n g o r d e r d i a g r a m for t he decay of a T l e p t o n i n t o liadrons. Q C D c o r r e c t i o n s m a n i f e s t t h e m s e l v e s a s glnon r a d i a t i o n off t h e q u a r k lines.
In c o n t r a s t t o « „ - d e t e r m i n a t i o n s f r o m R -, or Ri. t h e m a s s of t he h a d r o n i e s y s t e m is n o t fixed in r d e c a y s , t h a t is. energy scales f r o m I he m a s s of t h e pion t o I IK- mass of t h e r lepton cont r i b u t e . For t h e I heoretical p r e d i c t i o n of R T o n e I litis essentially h a s t o i n t e g r a t e t h e Q C D c o r r e c t i o n t o R-, over t h e c o n t r i b u t i n g m a s s range, weighted with t h e m a s s s p e c t r u m of t h e h a d r o n i e s y s t e m . C o m p a r e d t o lì-, a n d Hi t h i s m a k e s t h e q u a n t i t y I t r d o u b l y inclusive, t h a t is. i n t e g r a t e d over all h a d r o n i e final s t a t e s at, a given m a s s a n d i n t e g r a t e d over all masses b e t w e e n \ l - a n d M T . E x p a n s i o n ill powers of n s ( : V / - ) yields t h e following expression: LT T = 3.0. R »82 ( V T - ^
+ 5 . 2 0 2 3 ^ J 4- 2 3 . 3 6 6 ^
H----Ì
(8.6)
Despite t h e low e n e r g y scale t h e p e r t u r b a t i v e e x p a n s i o n still s e e m s t o be well b e h a v e d . I n d e e d , t h e r e is n o reason t o r e - e x p a n d tin- i n t e g r a t i o n over t h e m a s s s p e c t r u m which resuins c e r t a i n t e r m s of t h e p e r t u r b a t i v e p r e d i c t i o n lo all o r d e r s and t h u s results in a n even m o r e convergent expression. T h e e v a l u a t i o n of this r c s u m m e d p e r t u r b a t i v e c o r r e c t i o n t o R r is d e s c r i b e d in A p p e n d i x D.-l. However, w o r k i n g at a low m a s s scale o n e has t o worry a b o u t t h e n o n p e r t u r b a t i v e c o r r e c t i o n s <1"M>. T h e S h i f m a u V a i n s t a i n Z h a k a r o v (SVZ) a n s a t z ( S h i f m a n rt al.. l!)7!l) c o n t r o l s n o n - p e r t u r b a t i v e c o r r e c t i o n s in t h e f r a m e w o r k of t he O P E . «»XI
,„
.
(8.7)
with coefli< - ients C„ t hat, can be d e t e r m i n e d in p e r t u r b a t i o n t heory, a n d universal v a c u u m e x p e c t a t i o n values of o p e r a t o r s (),,. so-called c o n d e n s a t e s . Values for t hose c o n d e n s a t e s o b t a i n e d f r o m p h e n o m e n o l o g i c a l a n a l y s e s can b e found in t he l i t e r a t u r e , for e x a m p l e ( B r a n t e n el al.. l!)i)2). A l t e r n a t i v e l y , t h e y c a n b e e x t r a c t e d l.oget her w i t h t h e st rong c o u p l i n g f r o m higher m o m e n t s of t h e m a s s s p e c t r u m of h a d r o n i e r d e c a y s (Lo D i b e r d e r a n d Pieli. 1992«). Assuining t h a t t h e SVZ a n s a t z holds, it t u r n s out t h a t t he n o n - p e r t u r b a t i v o correct ions t o R j a r e s u r p r i s i n g l y small, below- 1%. T h i s , t o g e t h e r with t h e finding (Girono a n d Neither!.. 199(5) t h a t p e r t u r b a t i v e Q C D m a y be a p p l i c a b l e d o w n t o m a s s scales below I G e V . implies t h a t a n o s - i n e a s u r e n i e n t based on R r is p o t e n t i a l l y very a c c u r a t e .
I III STH
:tl2
C
I'I.I C C
SI A I
x p e r i m e n t a l l y t r <': )) b e d e t e r m i n e d in a variety of ways, a n d it is quire i n t e r e s t i n g t o see how it is inferred from at first glance completely unrelated i n p u t s . T h e obvious way to proceed is of course to m e a s u r e t h e h a d r o n i c a n d t h e electronic b r a n c h i n g f r a c t i o n s , ?i, a , r a n d ,.. a n d t a k e t h e ratio. A s s u m i n g lepton universality a n d t h e c o m p l e t e n e s s relation A?i,a
g ,, lr
-
e ~
f
W i t h i n t h e S t a n d a r d Model it is also possible t o infer ,. a n d t h u s j comp a r i n g mass a n d lifetime of t h e tail lepton t o t hat of t h e m u o n . Since t h e p a r t i a l w i d t h of weak d e c a y s is p r o p o r t i o n a l to t h e fifth power of t h e mass, a n d a s s u m i n g again lepton universality, o n e h a s
ro(/0
r t o l (/t)
I'c(T)
(8.9)
,Tu,t(r)
T a b l e 8.1 s h o w s t h e a v e r a g e s for s o m e basic p a r a m e t e r s of t h e tan l e p t o n as t h e y a r e given by ieli (2000). T h e value for is t h e weighted average of t h e t h r e e i n d e p e n d e n t m e a s u r e m e n t s t h a t c a n be e x t r a c t e d from those numbers. sing t h e formalism described in A p p e n d i x .-l. this value t r a n s l a t e s into a m e a s u r e m e n t of t h e s t r o n g coupling o s ( A 3 ) = 0.317 0.012 or (\s( ) 0.121 0 . 0 0 2 . ifferent a p p r o a c h e s in t h e theoretical prediction ( B r a a t o n ct til., 1992: c i b e r d e r a n d ieli, 1992«; Ball ct al.. 1995: G i r o n c a n d e u b e r t . 1990) i n d i c a t e t h a t t h e t h e o r e t i c a l u n c e r t a i n t i e s a r e s o m e w h a t larger, which is t a k e n into account for t he values a n d e r r o r s given in T a b l e 8.3. T a b l e 8.1 tan lepton
verages
for some
Quantity Mr TT t
Rr
8.3.3
o s from sum
parameters
of the
Value (1777.0r. (290.77
: (t ) M e V c 0.99) fs
(17.791
0.054)
(17.333
0.054)
3.043
0.010
,
rules
S u m rules a r e discussed in detail iu Section 7.3. e t e r m i n a t i o n s of t h e s t r o n g coupling c o n s t a n t based on s u m rules a r e fully inclusive m e a s u r e m e n t s at very
A
.
A
.A
.
s . ),
.
. . s (
s
s .
s
)
et ni..
7
) ).
.
I
.
(. ) =
1
0
,
s
8
s s(
s )( ) s
)
s
s
,
s
s s s ( A). T . s
.
s A
s (
s
ss s .
,
)
s s
s
s
s
s,
s. s
s s s, tl.. 1 ) )( ) s s
s
s
s et ni.. ) = 0.010 s s s s s s T s s . ss , (
s
As s
s ,
.
s s
A
s s s .
,
s
s s
M e a s u r e m e n t s of o s from heavy flavours s s s
s
s A A
s
s et
.4
s s
.
s s. T
( s
).
. , 1 87). s . . 1 ) )8). N S s s
s
s
s ss
. (8.11)
. s. T
s
s
= s
.
s ( s el til. ( ) )
(8.10)
2
s
s (
2
.
.
20.8
s ITE
s
s
18. 7
.,
s
s
s
.. 8
- il" (•!')) =
s
s (
) ) ). T ,
s
L
s. A s 0( ).
) s
s
s
, s
s
A
(
)
s
.1
.
s
s s
( s
ss
s
s
s s
.
s
s s
s, s s
,
I III S I HONG COUPLING CONS I AN I
:u i «1
<1 r(qc,
leptons)
F(qq —> h a d r o n s )
Fit:. 8 . 1 . Horn level a m p l i t u d e s for t h e d e c a y s of a spin-1 heavy flavour system into leptonic a n d ha dronic final s t a t e s
example, hadronic t o leptonic final s t a t e s , or from t h e level splittings between different o r b i t a l e x c i t a t i o n s of the b o u n d s y s t e m , which p r o b e t h e Q( ' p o t e n t i a l between q u a r k a n d a n t i q u a r k . Since t h e latter a p p r o a c h requires a good q u a n t i tative u n d e r s t a n d i n g of t he non-pert u r b a t i v e i n t e r - q u a r k potent ial, it is p u r s u e d in t h e c o n t e x t of lattice g a u g e theories. 8.1.1
Deca s of lira a
i uar onia
A precise « „ - m e a s u r e m e n t is o b t a i n e d f r o m t h e r a t i o of t h e h a d r o n i c t o t h e leptonic w i d t h of a spin-1 b o u n d s t a t e of a heavy q u a r k a n t i q u a r k pair. T h e d o m i n a n t Born level amplit udes for t h e decay into leptons a n d h a d r o n s a r e shown in Fig. 8.4. While t h e leptonic channel can proceed via a n i n t e r m e d i a t e p h o t o n , colour conservation p r e v e n t s t h e analog process t h r o u g h a single gluon since t h e two q u a r k s form a colour singlet s t a t e , while t h e gluon is a colour o c t e t . As far a s colour is concerned, a decay into two gluons would be allowed, b u t t hat again is forbidden by a n g u l a r m o m e n t u m conservation because t h e gluons a r c massless a n d t h u s cannot form a spin-1 system. Finally, since t h e s t r o n g coupling is m u c h larger t h a n t h e electromagnetic coupling, t h e annihilation of t h e heavy q u a r k pair into light q u a r k s is only a small correction to t h e t h r e e gluon decay. T h e leading o r d e r d i a g r a m for h a d r o n i c decays t h u s becomes t h e decay into three gluons which subsequently hadronize. It follows t h a t t o leading order t h e r a t i o of t h e h a d r o n i c t o leptonic width is proportional to ti^ o 2 ,,,. T h e p e r t u r b a t i v e prediction is known t o . Considering the ratio h a s t h e a d d i t i o n a l a d v a n t a g e t h a t t h e b o u n d s t a t e wave function at t h e origin cancels, m a k i n g a m e a s u r e m e n t of t h e s t r o n g coupling independent of precise knowledge of t h e Q C potential between t h e q u a r k s . n e has. however. t o take into a c c o u n t relativist'«- corrections which a r e p r o p o r t i o n a l t o t h e average (e <- 2 ) of t h e q u a r k s . Here we will present a result ( obel. 1092) based on t h e following p a r a m e t e r i z a t i o n F(qq — hadrons) r(qq-e+e )
Ml ASHIIF.MI N IS (H 01« FltOM IIKAV \
FLAVOURS
lor I lie N L ( ) dcscript ion ol I lie decay of a colour singlet, (((¡-stale. Tlie eoellicicnts 1 a n d Ii a r e k n o w n f r o m ¡ieri.ni lialion I lieory. I ) is a free ¡larainctcr of t h e model. A s s u m i n g D t o lie a universal c o n s t a n t , it w a s e x t r a c t e d ( K o h e l , 1992) t o g e t h e r wit 11 o s in a c o m b i n e d a n a l y s i s of T a n d t h e .l/'I' decays. T h c o r e t ieal uucert aiiit ies in t h e p c r t u r b a t i v c sector were s t u d i e d by i n t r o d u c i n g ad hoc N N L O t e r m s , renoriiiafixation scale variation a n d by Pade-like rewriting (1 / i o s ) a s 1/(1 -) / ¿ o j . T h e result a t a scale Q = 11) G e V given in T a b l e 8 . 3 is d o m i n a t e d by thcoret ieal u n c e r t a i n t i e s . T h e C I . l i t ) C o l l a b o r a t i o n (1996) h a s p e r f o r m e d a similar a n a l y s i s using t h e r a t i o F ( T —» gg7)/I"(Y —> ggg). which t o leading o r d e r is p r o p o r t i o n a l to o,. M I /°s- T h e result, d e t e r m i n e d with t h e h e l p of a M o n t e C a r l o model for t h e h a r d p h o t o n s p e c t r u m at a scale Q 9.7 G e V . is also given in T a b l e 8.3. Here, t he s y s t e m a t i c e r r o r s of t h e m e a s u r e m e n t , reflecting t he diflicult.v of d i s c r i m i n a t ing against t h e p h o t o n b a c k g r o u n d in h a d r o n i c Upsilon decays, a n d theoretical uncertainties contribute equally t o the total error. N.1.2
L
calculations
Precise d e t e r m i n a t i o n s of o s were also p e r f o r m e d in t h e a n a l y s i s of level s p l i t t i n g s between t h e S- a n d t h e P-stat.es in t h e Y - s y s t e n i (Davies ct ai. 1996: E l - K h a d r a . 1990) by m e a n s of l a t t i c e c a l c u l a t i o n s . T h e physical scale of t h e s e d e t e r m i n a t i o n s of t h e s t r o n g c o u p l i n g is Q = 10 G e V . T h e l a t t i c e a p p r o a c h tries t o r e p r o d u c e t h e observed energy levels in b o u n d s t a t e s y s t e m s by a b - i n i t i o c a l c u l a t i o n s b a s e d on t h e full Q C D L a g r a n g i a n . Technically this is d o n e by letting QC'l) lields evolve on a discret i'/.ed s p a c e t i m e l a t t i c e using a M o n t e C a r l o m e t h o d . W h i l e t h e q u a n t i z a t i o n of t h e s p a c e t i m e c o n t i n u u m m a k e s t h e p r o b l e m a m e n a b l e t o numerical s i m u l a t i o n s it also i n t r o d u c e s so-called lattice a r t e f a c t s which vanish only iu t h e c o n t i n u u m limit. A good u n d e r s t a n d i n g of how t o e x t r a p o l a t e t h e n u m e r i c a l r e s u l t s t o t h e c o n t i n u u m limit, is essent ial for a precision m e a s u r e m e n t of t h e s t r o n g coupling. In t h e past years significant p r o g r e s s h a s been m a d e iu r e d u c i n g l a t t i c e s p a c i n g e r r o r s , in t h e conversion f r o m t h e l a t t i c e t o t h e c o n t i n u u m MS c o u p l i n g c o n s t a n t a n d iu t he t r e a t m e n t of f c r m i o n s in loop c o r r e c t i o n s . D u e t o t h e a n t i c o n n i i i i t i n g n a t u r e of t h e fcrmion fields a n u m e r i c a l t r e a t m e n t on t h e l a t t i c e is highly non-trivial. C a l c u l a t i o n s exist with n j = 0 a n d n j = 2 d y n a m i c f c r m i o n s . which give only m a r g i n a l l y different r e s u l t s a n d t h u s allow a safe e x t r a p o l a t i o n to t h e physical c a s e of / i / = 3 light flavours. T w o early results ( F l y n n . 1990) a r e a S ( M % ) = 0.1 IS ± 0.00.5 ( N R Q C D C o l l a b o r a t i o n ) a n d os(<\/|) 0.116 ± 0.003 ( F N A I. C o l l a b o r a t i o n ) . A m o r e recent c a l c u l a t i o n using a different numerical t e c h n i q u e (Spitz ct
UNlVBiSIDAD AHTONOMA d l
I III STUONO COUI'L.INC CONSTAN I
.no
8.5
Scaling violations
S c a l i n g v i o l a t i o n s a r c a n i m m e d i a t e c o n s e q u e n c e of tin* fact t h a t (¡narks a r e s t r o n g l y i n t e r a c t ing p a r t i c l e s a n d t h u s p r o v i d e a r a t h e r direct, a n d theoretically clean way t o m e a s u r e t h e s t r o n g c o u p l i n g c o n s t a n t . In IMS p r o c e s s e s wit h spacelike m o m e n t u m t r a n s f e r s c a l i n g v i o l a t i o n s a r e o b s e r v e d a s a s o f t e n i n g in t h e nucleoli s t r u c t u r e f u n c t i o n s , t h a t is, t h e a v e r a g e m o m e n t u m f r a c t i o n p e r part o n b e c o m e s s m a l l e r w i t h i n c r e a s i n g m o m e n t u m t r a n s f e r . In e + e ~ a n n i h i l a t i o n i n t o h a d r o u s , which p r o c e e d s v i a t i m e - l i k e m o m e n t u m t r a n s f e r , t h e y manifest t h e m s e l v e s a s a s o f t e n i n g of t h e f r a g m e n t a t i o n f u n c t i o n s , w h i c h m e a n s t h a t t h e a v e r a g e m o m e n t u m f r a c t i o n p e r final s t a t e p a r t i c l e d e c r e a s e s w i t h i n c r e a s i n g Q~. N e i t h e r s t r u c t u r e f u n c t i o n s n o r f r a g m e n t a t i o n f u n c t i o n s c a n b e c a l c u l a t e d wit hin p o r t u r b a t i v e Q C D . b u t given a t a c e r t a i n scale. Q C D p r e d i c t s t h e e n e r g y evolut ion as a s o f t e n i n g w i t h i n c r e a s i n g Q 2 , d hi F / d In Q 2 -x o s ( Q - ' ) , w h i c h is des c r i b e d by t he D G L A P evolut ion e q u a t i o n s ( G r i b o v a n d L i p a t o v . 1972: Altarelli a n d Parisi. l!)77: D o k s h i t z e r , 1!)77). c.f. S e c t i o n 3.0. For s t r u c t u r e f u n c t i o n s t h e s o f t e n i n g c o m e s about, b e c a u s e h i g h e r m o m e n t u m t r a n s f e r s resolve m o r e p a r t o n s f r o m v a c u u m f l u c t u a t i o n s iu t h e nucleoli, for f r a g m e n t a t i o n f u n c t i o n s t h e g r o w i n g p h a s e s p a c e p e r m i t s a d d i t i o n a l g l u o n r a d i a t i o n a n d t h u s p a r t i c l e p r o d u c t i o n in j e t s . T h e t h e o r y ( C u r e i el «/.. 1980: F u n n a n s k i a n d P e t r o n z i o , 1980: Altarelli c.t. al., 10796; N a s o u a n d W e b b e r . 1994) is k n o w n t o n e x t - t o - l e a d i n g o r d e r a c c u r a c y . In a d e t e r m i n a t i o n of a s t h e n o n port u r b a u v e e f f e c t s c a n b e d i s e n t a n g l e d by t heir e n e r g y d e p e n d e n c e . H e r e D I S processes a r e favoured because n o n - p e r t u r b a t i v e effects decrease rapidly with 1 /(/"'. In e + e ~ a n n i h i l a t i o n t.hev d e c a y o n l y p r o p o r t i o n a l t o i/Q. which r e n d e r s t h e m e a s u r e m e n t s of o s f r o m f r a g m e n t a t i o n f u n c t i o n s loss p r e c i s e t h a n t h e o n e s f r o m DIS. even if t h e y a r e p e r f o r m e d at m u c h l a r g e r m o m e n t u m t r a n s f e r s . T h e a n a l y s i s of s c a l i n g v i o l a t i o n s in DIS is covered in S e c t i o n s 3.6 a n d 7.5. Here we will now c o m p l e m e n t , t h a t d i s c u s s i o n by d e s c r i b i n g h o w a n o^ m e a s u rement. is p e r f o r m e d by a n a l y s i n g s c a l i n g v i o l a t i o n s in f r a g m e n t a t i o n f u n c t i o n s . T h e t h e o r e t i c a l f r a m e w o r k a s well a s t h e p h c n o m c n o l o g i c a l i n p u t s a r e d i s c u s s e d iu d e t a i l in o r d e r t o i l l u s t r a t e t h e l i m i t a t i o n s of s u c h a n a n a l y s i s . E x p l i c i t exp r e s s i o n s for t h e f u n c t i o n s e n t e r i n g t h e a n a l y s i s a r e collected in A p p e n d i x E . 8.5.1
Sailing
violations
in fragmentation
functions
T h e single inclusive part icle s p e c t r u m p r o d u c e d in t he p r o c e s s e + e ~ —• h a d r o u s c a n b e w r i t t e n a s a s u m of a ' t r a n s v e r s e ' (T) a n d a ' l o n g i t u d i n a l ' (L) c r o s s sect 1011 * 1
,h7(.s)
a,OT(»)
d.r
1
d
ou>[ (s)
da:
=
where the transverse component
|
CTU„
1
i\0'-(s) (S)
tlx
has an angular distribution proportional to
2
I I c o s 0 and the longitudinal one a distribution proportional to sin~0. Here ' t r a n s v e r s e * a n d ' l o n g i t u d i n a l ' refer t o t h e p o l a r i z a t i o n s t a t e s of t h e i n t e r m e d i a t e p h o t o n or Z b o s o n . T h e a n g l e 0 is m e a s u r e d w i t h r e s p e c t t o t h e i n c o m i n g b e a m s . T h e L o r e n t z i n v a r i a n t v a r i a b l e ./• is d e f i n e d t h r o u g h .r = 2(1: • Q)/(Q'
Q)
where
SI AI.INO VIOLATIONS
, s
( . .
.
s s
+
s , T
s
s
s
s s
+
s
.
1 2
(s).
s.
ss
ss
,
s
s
,
s s
s
s
. .
s s
s s s
s
(. )
ss s
s
s
s
s. T s
a
s
=
s s
,
s s Dv(.r.s).
s
s
),,
s .
s . T
s s
ss s s s
s
s. T . ss
s
s)
crUtl (. s (.s)
. s
). ,
s
s
=
^
s
. T ss s
s
s . .
s .
. T
s 2
s
( s
,
.
s
A
s
s
s
s
s
ss
T s.
s
*
s
' )
,,
l
, . s
s s
non-sini lct . ( ..s) ( . .s) =
sinijlil
D
.'t s
s
s
Y = . ..,
s s
s
s A
s
s s II\.(S) — 1. T
s s
s
(-s.M)
s
s. T s ss ss s s s . T ,
,
s
z
.s
.
S(.r. s) =
.
s
s
s s
s .
s s
s
(
. T
s
s
s .
. .
s
s ) ( . .s)
'
*) =
D
> (* 's')
s A ,( .s) s s). S(x.
s) .
(8.1 .)
H I E STKONC COIU'I.INC ('ONSTAN'I
1111>111-
Measurement
D(.r.,rr(>))
-
<
Q C D O m « )
J
#
at
-
y/>T,
^
1 Q C D ( f i r , I'it) Measurement. at. ^ f s ] D f r M f ) )
-
QCD(^-(/)..>v)
-
^
-
^
F i c . 8 . 5 . S c h e m a t i c of a n a n a l y s i s of scaling violations. T h e variable r is t h e mo m e n t u i n fraction in t h e p e r t n r b a t i v e s e c t o r , x,. t h e e x p e r i m e n t a l l y observed q u a n t i t y which includes non-pert u r b a t i v e c o n t r i b u t i o n s .
O n e then o b t a i n s for t h e non-singlet p a r t s
s±Ni(x, s) = f t\zPs*(z, s, ,IR)N, (Is
,/ x
\Z
(-. '
s)
.
(8.1(>)
T h e evolution of t h e singlet c o m p o n e n t s is described by a coupled s y s t e m :
s— d.s
s(x. s)
= J
<\Z | / g f ; ( s .
l'it)G
( f . * ) + PQQ{Z,S.IIU)S
( ^ . s ) j s . 18)
with /> = Q or G are calculable in p e r t u r b a t i o n Again t h e kernels / , N S a n d theory. T h e ronorinalization scale / / / ( r e l a t e s t h e s p l i t t i n g kernels at t h e C . o . M . energy >/s t o t he s t r o n g c o u p l i n g c o n s t a n t at, t h e energy scale / / / ( . N o n - p e r t n r b a t i v e c o r r e c t i o n s which a r e e x p e c t e d t o show u p a s c o r r e c t i o n s p r o p o r t i o n a l t o 1 / s/s t o t he logarithmic scaling violations f r o m pert u r b a t i v e Q C D a r e discussed in d e p t h by N a s o n a n d W e b b e r (1994). A simple way of p a r a m e t e r i z i n g n o n - p e r t n r b a t i v e effects is a c h a n g e of variables, relating t h e p e r t n r b a t i v e variable x t o t h e m e a s u r e d q u a n t i t y x, t h r o u g h x = <j(xe,$) = .;:,.(1 -\ h / s / s ) . T h e c o n n e c t i o n b e t w e e n t h e p e r t n r b a t i v e prediction drx/d.r a n d t he e x p e r i m e n t a l l y observe;«! cross section drr/d./:,. c a n t h e n be derived from t he contra int. of energy conservation before a n d a f t e r t h e t r a n s f o r m a t i o n J d . r . / d a / d . r = I'd./:, .('«. d f f / d x , . . which gives
drr =
xc
(s J{))
d.c
A t this point t h e set, of ingredients needed for an o s - m e a s u r e m e n t f r o m scaling violations in f r a g m e n t a t i o n f u n c t i o n s is c o m p l e t e . T h e explicit expressions of
SI'AI.INO V l n l \ l IONS
:1I9
i lie liinel ions needed lo p e r f o r m t h e a n a l y s i s a r e given in A p p e n d i x E. Sclien mill nllv a Q C D test, based on m e a s u r e m e n t s of inclusive cross sections at different 0 o. M energies is sketched in Kig. S..r>. T h e input is a set of f r a g m e n t a t i o n I'uncIIons l ) ( . i . i i j . ( i ) ) at an initial f a c t o r i z a t i o n scale ///••(>'). P e r t n r b a t i v e Q C D then i d u l o s t h o s e f r a g m e n t a t i o n f u n c t i o n s t o a n o b s e r v a b l e cross section which, a f t e r IIH In .ion of t he non-pert urbat ive corrections, can be c o m p a r e d with e x p e r i m e n tal d a t a at. an initial C.o.M. energy (horizonta l a r r o w s ) . T h e energy evolution • •I the f r a g m e n t a t i o n f u n c t i o n s is described by p e r t n r b a t i v e Q C D (vertical a r rows), where t h e energy variation a t t h e scale // is expressed as f u n c t i o n o f / / a n d 1 he I ('normalization point ///,•. T h e evolution (filiations yield t h e f r a g m e n t a t i o n Inactions at a new f a c t o r i z a t i o n scale ///.• where they again can be r e l a t e d t o observable cross sections at. a new C.o.M. energy ^ s j . T h e test, of Q C D conU'.is in showing t h a t all available d a t a a r e described consistently with o n e set of liagnientation f u n c t i o n s , a universal c o u p l i n g constant, a n d a global p a r a m e t e r li lo account for non-perl urbat ive effects. In a n a c t u a l analysis t h e f r a g m e n t a t i o n llllictlous n s a n d h a r e a d j u s t e d in a global lit t o t h e available d a t a . For t h e p a r a m e t e r i z a t i o n of t h e f r a g m e n t a t i o n f u n c t i o n s s o m e g u i d a n c e is provided by p e r t n r b a t i v e Q C D in t h e f r a m e w o r k of t h e modified leading-lot] approximation ( M L L A ) , which predicts t h a t t h e m o m e n t u m s p e c t r u m of final s t a t e particles should exhibit an a p p r o x i m a t e l y G a u s s i a n peak in In./: (Pong a n d Webber, 1 99 1). F r o m t h i s o n e would infer a functional form for t h e f r a g n i e n t a t i o n function
«TT^T7 ~
<>X1>
~
1,1 x
')2)
~ e x p (-<•• I n 2 . r ) x2al-1
.
(8.20)
C o m b i n e d with t h e e x p e c t a t i o n t hat t h e m o m e n t u m s p e c t r u m falls off wit h s o m e power of (1 - ./ ) for ./• — I. then finally yields t h e a n s a t z D(x)
= N e x p ( - c h r .r):i:''(l - ./;)".
(8.21)
where A' is a n o r m a l i z a t i o n c o n s t a n t a n d o. I> a n d e a r e free p a r a m e t e r s which have t o be d e t e r m i n e d f r o m t h e d a t a . T h e p a r a m e t e r s o a n d h d e p e n d on t h e q u a r k m a s s , t h a t is. t.liey a r e different for light q u a r k s , c - q u a r k s . b - q u a r k s a n d glnons. w h e r e a s c should be flavour i n d e p e n d e n t . A priori t h e scales ///.• a n d pn a r e u n c o n s t r a i n e d . W h e n c a l c u l a t i n g to all orders, a n y d e p e n d e n c e on t h e choice of t h e scales vanishes. In finite o r d e r pert u r b a t i o n t h e o r y a residual scale d e p e n d e n c e is related to iiiicalculated higher o r d e r t e r m s . In o r d e r t o avoid large logarit h m s in t h e theoret ical predict ions t h e P r ( f ) / s j = iqjpj= 1. Varying t h e scalesn a t u r a l choice of scales is pj.-U)/-= is one possible way t o e s t i m a t e t h e theoretical u n c e r t a i n t i e s of t h e prediction. A c o m m o n choice to p a r a m e t e r i z e t h e scale d e p e n d e n c e is t h r o u g h a c o m m o n factor / ]i'2r(i)/s, = ¡r,,(f )/sj t o p r o b e t h e f a c t o r i z a t i o n scale d e p e n d e n c e a n d for t h e lenorinalizalioii scale d e p e n d e n c e . a n o t h e r f a c t o r fr ~ I'jt/pj:
TILLÍ S T K O N G C O U P L I N G
CONSTAN''I
T h e full N L O theoretical f r a m e w o r k lias b e e n used in two d e t e r m i n a t i o n s of t h e s t r o n g coupling c o n s t a n t f r o m scaling violations in f r a g m e n t a t i o n functions ( A L E P I I Col lab., lü!)r>fo; D E L P H I Collab., 1997«), B o t h a r c based on inelusive d i s t r i b u t i o n s a n d flavour enriched d a t a s a m p l e s f r o m h a d r o n i c Z decays, I hat is, moment um spool ra for light q u a r k s , c - q u a r k s , b - q u a r k s a n d gluons. combined with i n e a s u r e m e n t s f r o m lower C.o.M. energies d o w n t o y/s = Í4 C e V . The s t r o n g c o u p l i n g c o n s t a n t was d e t o r m i n e d t o g e t h e r wit h p a r a n i e t e r i z a t i o n s for tin- f r a g m e n t a t i o n f m i c t i o n s of t h e different, p a r t o n t y p e s a n d a power law correction describing n o n - p e r t n r b a t i v e effects. T h e l a t t e r was found t o b e small for t h e energy r a n g e u n d e r c o n s i d e r a t i o n . T h e c o m b i n e d result is given in T a b l e 8.3.
8.5.2
Scaling
violations
in structure
June!ions
T h e s t u d y of scaling violations in s t r u c t u r e f u n c t i o n s is o n e of t h e classical a p proaches for t h e d e t e r m i n a t i o n of t h e s t r o n g coupling c o n s t a n t . M e a s u r e m e n t s were p e r f o r m e d in DIS with n e u t r i n o b e a m s a n d charged loptons on t a r g e t s of heavy a n d light nuclei. T h e q u a n t i t y e x t r a c t e d f r o m t h e s e m e a s u r e m e n t s was t h e Q C D scale for four active flavours. Since t h o s e e x p e r i m e n t s covered ¿¿•-ranges between 0.5 G e V 2 a n d 260 G e V 2 . with a typical c e n t r a l scale a r o u n d /( = 7.1 G e V , t h e c o n t r i b u t i o n s from heavier q u a r k s a r e negligible. T h e result from a combined a n a l y s i s ( V i r e h a u x a n d M i l s z t a j n , 1902) of S L A C - B C D M S meas u r e m e n t s of t h e s t r u c t u r e f u n c t i o n !•"•>(.>:). based on a N L O theoretical prediction is A ^ 263 ± 4 2 MoV. T h e e r r o r is t h e c o m b i n e d e x p e r i m e n t a l a n d theoretical uncertainty. In t his analysis also t he gluon d i s t r i b u t i o n a n d n o n - p e r t n r b a t i v e cont r i b u t i o n s have been fitted. Ill a similar analysis ( K a t a e v et ai. 2000) of C'CFR d a t a (1997), using N N L O theoretical predictions, t h e s t r o n g coupling c o n s t a n t was e x t r a c t e d from t h e evolution of t h e flavour singlet d i s t r i b u t i o n l'\{:v) as measured in n e u t r i n o nucleón s c a t t e r i n g . Finally we should also mention a N N L O analysis ( S a n t i a g o a n d Y n d u r a i n , 1999) based on t h e evolution of m o m e n t s of !•'•>, where iu a d d i t i o n to fixed target m e a s u r e m e n t s f r o m electron a n d union scattering also II Eli A d a t a in t h e (/-'-interval f r o m 2.5 G e V 2 t o 230 G e V 2 were used. I 'hi* respective results a r e s u m m a r i z e d in T a b l e 8.3.
8.0
M e a s u r e m e n t s at hadron colliders
As we have learned, high e n e r g y h a d r o n h a d r o n collisions with large m o m e n t u m transfer arise from t h e s c a t t e r i n g of st rongly i n t e r a c t i n g p a r t o n s . Ii follows t h a t t h e cross section is sensitive t o o s . In general, however, d e t e r m i n a t i o n s of t h e s t r o n g coupling in h a d r o n i c i n t e r a c t i o n s a r e less precise, on o n e h a n d because of u n c e r t a i n t i e s iu t h e p a r t o n density functions, a n d on t h e o t h e r d u e t o t h e underlying event, a n d o t h e r soft c o n t r i b u t i o n s f r o m t h e s p e c t a t o r p a r t o n s which d o not p a r t i c i p a t e in t h e h a r d s c a t t e r i n g . In a d d i t i o n , we have to keep in mind that the value of o H also e n t e r s in t h e d e t e r m i n a t i o n of t h e p a r t o n de nsity functions. T h i s i n t r o d u c e s a n a d d i t i o n a l , a l t h o u g h higher o r d e r , u n c e r t a i n t y , in a n y '
: 1
t-xmvledi'o of t h e liar ton
CI.OHAI l A E N I SI IA I'l' VAHIAMI.KS
density f u n c t i o n s as i n p u t . Nevertheless, c o m p e t i t i v e m e a s u r e m e n t s were performed from d i r e c t - p h o t o n p r o d u c t i o n processes, m e a s u r e m e n t s of t he 1)1» cross sect ions a n d t he inclusive jet cross section in high energy nucleoli (anti)nuclcon collisions. D i r e c t - p h o t o n p r o d u c t i o n in h a r d p a r t o n p a r t o n s c a t t e r i n g is a C o n i p t o n - l i k e process of ( 9 ( o s n , . n l ) . T h e N L O Q C D c o r r e c t i o n s a r e k n o w n , so t h a t a reliable m e a s u r e m e n t of t h e s t r o n g coupling is possible. In t h e cross section difference ">A") t h e sea q u a r k a n d gluou p a r t o n density f u n c t i o n s of t h e nucleoli cancel, that is. only t h e well known valence q u a r k d i s t r i b u t i o n s a r e needed a s e x t e r n a l i n p u t for a n « „ - d e t e r m i n a t i o n . Still it is a difficult measurement t o d i s e n t a n g l e p r o m p t p h o t o n s f r o m b a c k g r o u n d c a u s e d by TT" a n d 1/ d e c a y s into pairs of p h o t o n s . T h e result f r o m t h e HAG C o l l a b o r a t i o n (1!)!)!)). o b t a i n e d at t h e C E R N SI'S collider is given in T a b l e 8.3. Despite a p p C'.o.M. energy of 030 GeV t h e typical scale of t he hard p a r t o n p a r t o n scat tering for this measurement was only Q = 24.3 G e V . T h e theoretical error of t h i s m e a s u r e m e n t covers uncertainties from the choice of t h e r e n o r n i a l i / a t i o n scale a n d t h e variations of t h e p a r t o n density functions. Heavy q u a r k s which a r e not. present in t he initial s t a t e a r e p r o d u c e d by q u a r k ant ¡quark a n n i h i l a t i o n or g l u o u gluou fusion processes which a r e of < 9 ( « ; ) . T h e theory is developed to N L O . E x p e r i m e n t a l l y those reactions can b e tagged by t h e decay c h a r a c t e r i s t i c s of t h e heavy h a d r o n s . Table S.3 lists o n e such measurement (IIA1 Collab.. l!)!)(i) from a n analysis of b b I jets p r o d u c t i o n ¡11 p p collisions. A measurement, of t he st rong coupling (Giele el. ai. 1!)!M>) is based on t he jet cross sections d a / d / v y , with E r t h e t r a n s v e r s e energy of a j e t . measured bv t h e C'DE C o l l a b o r a t i o n (1902) at a C'.o.M. energy y/s = 1.8 T e V . T h e a n a l y s i s was p e r f o r m e d in bins of Er using a M o n t e C a r l o model based on t h e N L O Q C D matrix e l e m e n t s to give o s { / v ; ) in t h e r a n g e 30 G e V < Er < 400 G e V . 8.7
Global event shape variables
Many d e t e r m i n a t i o n s of t h e s t r o n g coupling c o n s t a n t t h a t have been p e r f o r m e d iu t he past, years a r c based 011 global event s h a p e variables, t h a t is. observablcs designed to d e s c r i b e t h e s t r u c t u r e of h a d r o n i c filial s t a t e s in e + e ~ a n n i h i l a t i o n by o n e c h a r a c t e r i s t i c n u m b e r or d i s t r i b u t i o n which in p e r t u r b a t i o n t h e o r y can be related t o t h e s t r o n g c o u p l i n g c o n s t a n t . A large n u m b e r of t h e s e variables has been invented in t h e past years, a n d it is beyond t h e scope of this book to deal with all of t h e m . I n s t e a d , we will focus on t h e most, i m p o r t a n t ones, discussing various a p p r o a c h e s t o describe t he p r o p e r t i e s of i n u l t i h a d r o n e v e n t s in a q u a n t i t a t i v e way. s.7.1
Theoretical
predictions
I lie theoretical prediction for all global event s h a p e variables is known t o N L O O(aj), based 011 a numerical integration (Kunszt rt at., 198!l) of t h e E R T matrix e l e m e n t s (Ellis el al., 1!)81). For s o m e variables also leading-logarithmic a n d
.
.1 1
s, ,
s
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s s s s 1
s s,
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(
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s e.t ill.. s s s s.
.
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. T
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ss 100
A
s
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). A.
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s
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.
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s
)
s
s
s
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2
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s
s
s
s 0.1
s s
s
s s
s
s
. T
a = o n +
s
s
s
ss
s. T
s s
s ss. A
s
s
s
s
s
1 s
s
s s , s
s
s s
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shape.
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s
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.
s
s s
s ss s.
s
variables
s
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. vent
T
.
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s
, s s. T
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8.7.2
ss
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+
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, .
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.
(ILOIIAI
EVENT SHAPE VAUIAHLKS
:M
is a m e a s u r e of t h e s t r o n g coupling c o n s t a n t . However, in o r d e r t o p e r f o r m a measurement of o s o n e need • to know t h e p r o p o r t i o n a l i t y c o n s t a n t between t h e ratio llw/R'i a n d o s which d e p e n d s on such details as t h e j o t finding a l g o r i t h m a n d tin" selected k i n e m a t i c s . It is also affected by higher o r d e r c o r r e c t i o n s , which because of t he large value of t h e s t r o n g coupling constant usually a r e not negligible. T u r n i n g t he s i m p l e concept, of global event s h a p e variables into a precision m e a s u r e m e n t of o s t h u s is a r a t h e r c o m p l e x business. In o r d e r t o be useful for a m e a s u r e m e n t of t h e s t r o n g coupling, a n event liape variable must be calculable in p e r t u r b a t i o n theory, that is. tin- soft a n d the collinear singularities m u s t cancel. T h e variable must b e ' i n f r a r e d ' o r 'soft and collinear safe', c.f. Sections 3.-r> a n d (i.f. T o i l l u s t r a t e t h e point , we will first discuss s o m e variables which a t first glance m a y a p p e a r a s possible q u a n t i t i e s to extract o s . but which d o not s a t i s f y t hose criteria. A f t e r that, we will t u r n t o describe 'safe' obsei'vables. An e x a m p l e for a variable which is n e i t h e r soft nor collinear safe is t he final s t a t e multiplicity, which is increased bv o n e unit if an a d d i t i o n a l low e n e r g y p a r ticle is emit ted o r if a given m o m e n t u m is split into two collinear ones. Alt hough the n u m b e r of particles is expected t o grow with o s . a pert u r b a t i v c c a l c u l a t i o n a n d t h u s a m e a s u r e m e n t of o s is not possible. A n o t h e r event s h a p e variable which is widely used for event classification is t h e Sphericity S. defined a s
3 . E/>/4 S = - miu ' ,,
(8.22)
¿^p V~ where t h e s u m r u n s over t h e t h r e e - m o m e n t a of all particles in t h e event. T h e d e t e r m i n a t i o n of S involves finding t h e direction, with respect 10 which t h e s u m of t h e s q u a r e s of t h e t r a n s v e r s e m o m e n t a /
Y
as
IPI
XM
RILL'! s n n i N c C O U H . I N C CONSTANI
C = 3 (BnB22 +
+ BxiBn
»^«la
023©«»
«31«:») •
(8.24)
In (IK- definition o l ' H i j t he s u m is a g a i n over all final s t a t e m o m e n t a />. />, d e n o t e s t h e /tli c o m p o n e n t . /' 1 . 2 . 3 , of t h e m o m e n t u m vector />. For ¡111 ideal two-jet event of t w o b a c k - t o - b a c k s y s t e m s of p a r t i c l e s with v a n i s h i n g t r a n s v e r s e m o m e n t a one h a s C (I. An isotropic flow yields C -- 1 / 3 . E q u a t i o n (8.2-1) is equivalent to a definition of t he O-pnrainot.er based on t h e eigenvalues {At.A2.A3} of which is usually f o u n d in t h e l i t e r a t u r e : C = 3(A|A-.> + A2A3 + A3A1). A n o t h e r set of event s h a p e variables is based 011 t h e division of t h e event into t w o h e m i s p h e r e s . o a n d l>. by a p l a n e p e r p e n d i c u l a r t o t h e T h r u s t axis n-/ D e s p i t e t h e n u m e r i c a l p r o b l e m s ¡11 t h e d e t e r m i n a t i o n of t h e T h r u s t axis in a mill t i h a d r o n event, t h e h e m i s p h e r e definition based 011 T h r u s t is ¡1 very n a t u r a l one when looking at t h e p Q C D p r e d i c t i o n for d i s t r i b u t i o n s of event s h a p e variables. T h e leading o r d e r Q C D correction t o t h e two-jet cross section is t h e emission of one gluon f r o m e i t h e r of t h e p r i m a r y q u a r k s . B e c a u s e of m o m e n t u m c o n s e r v a t i o n t h e t h r e e - p a r t i c l e final s t a t e is a p l a n a r c o n f i g u r a t i o n , w i t h t h e T h r u s t a x i s coinciding with t h e highest, e n e r g y part.on. T h e t w o h e m i s p h e r e s a a n d l> t h u s have o n e p a r t i c l e a l o n g t h e event axis recoiling against a s y s t e m of two particles. T h e p r o b a b i l i t y of a l a r g e a n g l e gluon emission g r o w s wit h increasing n s . Since large angle emissions also m e a n large invariant m a s s e s of t h e t w o - p a r t i c l e s y s t e m , o n e is led t o a definit ion of event s h a p e variables based 011 h e m i s p h e r e masses. W i t h \ / „ a n d Mi, t h e i n v a r i a n t m a s s e s of t h e two h e m i s p h e r e s , t h e n o r m a l i z e d h e a v y jet m a s s />// is defined JUS ,,/I
=
- L „ M X(\%.AI2), vis
(8.25)
where t h e q u a n t i t y Evif, is t he t o t a l visible e n e r g y in t h e event. T h e s u m goes over all p a r t i c l e s which c o n t r i b u t e ¡11 t h e e v a l u a t i o n of t h e event s h a p e variable. T h e n o r m a l i z a t i o n to Ef.is serves two p u r p o s e s . F i r s t , it normalizes t h e r a n g e of values for t he s h a p e variable, m a k i n g d i s t r i b u t ions r o u g h l y i n d e p e n d e n t of t h e C . o . M . energy of t h e reaction, second it reduces t h e size of s y s t e m a t i c u n c e r t a i n t i e s in a n o s - d e t e r n > i n a t i o n since part of t h e efficiency a n d a c c e p t a n c e c o r r e c t i o n s of t h e d e t e c t o r cancel. T h e event s h a p e v a r i a b l e s discussed so far c h a r a c t e r i z e t h e s t r u c t u r e of a uiultiliadron event by j u s t o n e n u m b e r . A l t e r n a t i v e l y o n e can also m e a s u r e t h e s t r u c t u r e of t h e m o m e n t u m How by m e a n s of a c o r r e l a t i o n f u n c t i o n which yields a d i s t r i b u t i o n for each e v e n t . T h e classic e x a m p l e for t h i s kind of event s h a p e m e a s u r e s is t h e c.nergu t neryy-corrc.Uition function ( E E C ) ( B a s h a i n ct nl. 1978«. 1978/;), defined a s t h e d i s t r i b u t i o n of t h e weighted o p e n i n g a n g l e s between all p a i r s i.j of final s t a t e particles with energies E, a n d E},
Here .V is t h e total n u m b e r of e v e n t s t o be a n a l y s e d . T h e E E C is defined a s t h e c o n t i n u o u s a v e r a g e of t h e d i s c r e t e d i s t r i b u t i o n s m e a s u r e d from t h e individual
( .
s.
11A
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(8.2( )
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s s s
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, s.
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s
A
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(A
2 A . (
C
rg -( iicrg -cornd it.ion-asi nnni tr
). A
( ) =
(180
)
( )
s
(8.27)
s
s
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s
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I III
STItONG C O U P L I N G OONSTAN'I
that is. both q u a n t i t i e s c o n t a i n essentially t h e s a m e i n f o r m a t i o n , f r o m t h e exp e r i m e n t a l point of view t h e .' -dist r ibution is t o b e preferred since here relative frequencies at different values of 3 a r e statistically i n d e p e n d e n t . A l t h o u g h at. first glance dealing with t h e s a m e i n f o r m a t i o n , dilferent e t alg o r i t h m s or generally event, s h a p e variables have different unealeulated higher o r d e r corrections in t h e theoretical prediction a n d dilferent sensitivities t o hadr o n i a t i o n a n d d e t e c t o r effects. S t u d y i n g a large set of ohservahles t h u s gives a feeling for t h e s y s t e m a t i c l i m i t a t i o n s of a n « „ - m e a s u r e m e n t . M a n y m e a s u r e m e n t s of t h e st rong coupling based on global event s h a p e variabilis have been p e r f o r m e d in e + e a n n i h i l a t i o n r e a c t i o n s in t h e energy r a n g e between 22 GeV u p t o 2 energies. Historically, t h e first set of precise meas u r e m e n t s was based on theoretical predictions. a t e r improved c a l c u l a t i o n s b e c a m e available, w h e r e leading- a n d n e x t - t o - l e a d i n g - l o g a r i t h n i i c c o n t r i b u t i o n s could be resumined to all o r d e r s . As m e a s u r e m e n t s based on these improved predictions t u r n e d out t o b e m o r e precise, a n d since also older d a t a have been re-analysed in view of t h o s e theoretical i m p r o v e m e n t s , we now a r e in t h e lucky s i t u a t i o n to have m e a s u r e m e n t s over t h e r a n g e f r o m 22 G e V t o a r o u n d 200 GeV based on a h o m o g e n e o u s theoretical f r a m e w o r k . M e a s u r e m e n t s from m a n y aut h o r s ( A H Collab. 1998. 1999: T A Z Collab. 1993; A I I Collab. 1998«, 1999; H I Collab. 1 a; 3 Collab. 1997«; A Collab. 1993«. 1990 . 1997«. 2000«: S Collab. 1995) c o n t r i b u t e t o t h e values listed in T a b l e 8.3. 8.8 A n a l y t i c a l a p p r o a c h e s t o p o w e r law c o r r e c t i o n s A l t h o u g h it is c u r r e n t l y not possible t o c a l c u l a t e non-pert,urhativc effects for global event s h a p e variables f r o m first principles, t h e r e exists at least a phcnomenological a n s a t z . which allows to relate n o n - p e r t u r b a t i v c corrections for different s h a p e variables t o a few m e a s u r a b l e p a r a m e t e r s . T h e basic a r g u m e n t ( okshitzer a n d W e b b e r . 1995: W e b b e r , 1995) goes as follows. C o n s i d e r a dimensionless q u a n t i t y F( ). for e x a m p l e , t h e average T h r u s t m e a s u r e d at a C . o . M . energy . T h i s a v e r a g e will have c o n t r i b u t i o n s from gluon r a d i a t i o n s p a n n i n g a r a n g e from very small m o m e n t a , which a r e essentially 11011perturbat.ive. u p t o t h e h a r d scale . Formally o n e can write with
f(
) = « F « s ( A - 2 ) — — - T 0 ( « 2 ) for A- -
0
(8.29) w h e r e t h e leading o r d e r t e r m s of ( ) can be d e t e r m i n e d in p e r t u r b a t i o n theory. ote t h a t t h e r u n n i n g coupling « s { ~ ) effectively r c s u m s a p a r t of t h e higher order perturbat.ive c o n t r i b u t i o n s . T h e purely p e r t u r b a t i v e prediction for F( ) is of t h e form F( )
= FyoJui)
+ (' -> + Po 1 V lri
>2M ) + 0(a a)
(« « )
t h a t is. a power series in t h e s t r o n g coupling c o n s t a n t evaluated at a r e n o r m a l ization scale q u a t i o n (8.29) can now be used t o e x a m i n e t h e low energy
ANAI.Y I 11 .'AI AI'I'UOACIIKS I'O I'OWKIT I.AW < 'OLTLTKI "HONS
i mil i ¡lull ions In F(Q), 111> l<> ii m a t c h i n g scale /// wliieli s h o u l d lie s o m e w h e r e ill I lie 11 ansit ion region liel ween I lie perl u r b a t ive a n d l lie n o n - p e r t u r b a t i v e r e g i m e . 2 GeV. One then can introduce A M ., •• /'/ <SC Q. A typical value would lie /// p a r a m e t e r s o,,(/
//,,+
l
(S.:H)
M — n ^ q ) . I' T I
which a r e m o m e n t s of t h e low-energy b e h a v i o u r of t h e s t r o n g c o u p l i n g c o n s t a n t . These m o m e n t s a r c I he key element of this a p p r o a c h . In t h e a b s e n c e of d e t a i l e d knowledge of t h e low e n e r g y b e h a v i o u r of n s (A - 2 ). o n e a s s u m e s t h a t t h e t r u e physical funct ion is suflicient ly r e g u l a r for I he m o m e n t s t o e x i s t . As s u c h I hey a r e an effective way t o p a r a m e t e r i z e low e n e r g y Q C D . i n c l u d i n g b o t h p e r t u r b a l i v e and n o n - p e r t u r b a t i v e c o n t r i b u t i o n s . A l t h o u g h t h e m o m e n t s c u r r e n t l y c a n n o t h e c a l c u l a t e d , tliev a r e universal p a r a m e t e r s of Q C D , which in principle c a n be m e a s u r e d . T h e y a l s o offer a convenient way t o t r e a t n o n - p e r t u r b a t ive e f f e c t s in « „ - m e a s u r e m e n t s based on event s h a p e variables. Willi eqn (<S.."11) t h e t o t a l c o n t r i b u t i o n t.o F ( Q ) u p t o t h e m a t c h i n g scale /// is given by
[ "
-
^
' M / < 7 ) = A T + /HP.
(S.32)
T h e p e r t . u r b a t i v e p a r t in t h i s e x p r e s s i o n can b e d e t e r m i n e d by s u b s t i t u t i n g t h e pert urbat ive r u n n i n g of t he st r o n g c o u p l i n g . o s (A:-) = o s ( / / / , ) - I W M )
~T + I'It
•
to yield
Subt ract ing t h e pert urbat ive part which is a l r e a d y c o n t a i n e d iu oqn (S..'{()) f r o m c<|ii (S..T2) t h u s allows t o e x t r a c t t h e n o n - p e r t u r b a t ive c o n t r i b u t i o n . It d e p e n d s on t h e coefficient «/• which c a n b e d e t e r m i n e d a n a l y t i c a l l y for a given event s h a p e v a r i a b l e /•'. a n d t h e m o m e n t d o ( / ' / ) . which is a universal p a r a m e t e r of Q C D . T o give a lew e x a m p l e s , for t h e m e a n value of I - T o n e h a s «•/• = ICF/TT. for t h e m e a n v a l u e of /'// it is titl ~ 'IC'/.-f- a n d for t h e m e a n value of t h e C'p a r a m e t e r t he result is ac = (i("/.'• A p p a r e n t ly, n o n - p e r t u r b a t ive effects a r e m o r e p r o n o u n c e d for t h e C - p a r a m e t e r t h a n for T h r u s t or h e a v y jet m a s s . T h e f o r m a l i s m h a s been a p p l i e d t o t h e m e a n values of I T a n d ¡>n by t h e DKI.ITII C o l l a b o r a t i o n (1!)!)!)«). c o v e r i n g t h e e n e r g y r a n g e b e t w e e n I I G e V a n d 1«S:< G e V . Tin- result is s h o w n iu Fig. <S.ti. O n e clearly sees t h e i m p o r t a n c e of n o n - p e r l u r b a t i v e c o r r e c t i o n s u p t o t h e highest. LKP energies, but. o n e a l s o observes an impressive agreement between the theoretical prediction and the
I III'. S T I M > (! C
I
G
C
STA
'I
\'s (GeV)
it;. 8.(i. M e a s u r e m e n t of o s based oil m e a n values of global event s h a p e variables m e a s u r e d b e t w e e n /s = I I G e V a n d /s = 1<S.' G e V . T h e e x p e r i m e n tal d a t a are c o m p a r e d t o the purely perturbat.ive Q C prediction a n d a n e x t e n d e d theory including power law corrections. Figure from H I Collab,(1999a).
t.wo-parameter Iii t o t h e e x p e r i m e n t a l d a t a d e t e r m i n i n g s i m u l t a n e o u s l y < s (My ) a n d t h e elfective non- )erturl)ativc coupling oo( l GeV '). T h e lit results a r e given in Table 8.2. A l t h o u g h t h e overall agreement between d a t a a n d t h e o r y is very satisfactory, t h e quality of t h e lit as expressed by t h e ndf-valui's points to s o m e internal p r o b l e m s with t h e d a t a . T o a c c o u n t for this t h e value q u o t e d in T a b l e 8.II was taken to b e t h e result f r o m 1 — T. with t h e e r r o r s scaled bv v FT^Finally, it lias t o lie m e n t i o n e d t h a t p o w e r law c o r r e c t i o n s c a n not only be calculated for m e a n values o r o t h e r scalar observables. but also for d i s t r i b u tions of event s h a p e variables. Here t h e correction t o t h e purely pert u r b a t i v e
II I!. IN 1)1 I I I N M A S T I O SC'A'I l l . l t I N O Table 8 . 2 Results from a simullunions Jit <>J m n (A/.¿) ami uon-pertuitxitive effective eoujihnij oo('l G o V " ) to tin eneri/H evolution i/lob/il i vent slmiie variables
tin of
Observable
o0(4GcV2)
cvs(j\/|)
\-2/"df
(1 - T)
0.193 -J: 0.009 :fc 0.004
0.1191 ± 0.0015 ± 0.0051
50.3/2G
0.550 ±- 0.024 ± 0 . 0 1 3 0.1192 ± 0.0022 ± 0.0037
2.65/15
{MfJE%, i s )
prediction can be p a r a m e t e r i z e d a s a shift in t h e a r g u m e n t of t he d i s t r i b u t i o n f u n c t i o n ( D o k s h i t z o r . 1099). for e x a m p l e .
a dC
8.i)
= — ^ v r ci|t*|* d c
(C - Dr/Q)
.
(8.35)
J e t s in d e e p inelast ic scattering
Also in d e e p inelastic lept.011 nilcleon s c a t t e r i n g it is possible t o a n a l y s e d i s t r i b u tions of event s h a p e v a r i a b l e s for t h e f i n a l - s t a t e h a d r o u i c s y s t e m a n d ext ract a m e a s u r e m e n t of t h e s t r o n g c o u p l i n g c o n s t a n t . So far t h e most precise m e a s u r e m e n t s a r e f r o m t h e s t u d y of j e t p r o d u c t i o n r a t e s . O n e c o n t r i b u t i o n t o multi-jet p r o d u c t i o n in op-collisions is, for e x a m p l e , gluou r a d i a t i o n oil" a q u a r k s c a t t e r e d by a large Q 2 v i r t u a l p h o t o n . Q u a r k a n d g l u o u e m e r g e a s t w o j e t s in a d d i t i o n to t h e jet f r o m t h e p r o t o n r e m n a n t . T h e p r o d u c t i o n r a t e //_>+i of t h o s e ( 2 + 1 ) jet final s t a t e s is k n o w n t o n o x t - t o - l o a d i n g o r d e r 0(o~) ( K o r n o r et at.. 1989: B r o d k o r b et «/.. 1989). A very a p p e a l i n g f e a t u r e of this kind of a n a l y s e s is t h a i k i n e m a t i c selection of t h e s c a t t e r e d electron allows to c o n t r o l t h e Q2 of t h e process. t h e r e b y m a k i n g it possible to e s t a b l i s h t h e r u n n i n g of t h e s t r o n g c o u p l i n g c o n s t a n t w i t h i n a single e x p e r i m e n t . M e a s u r e m e n t s f r o m t h e '/K1'S C o l l a b o r a tion (1995) a n d t h e III C o l l a b o r a t i o n (1995/;. 1998. 19996) covering a n e n e r g y r a n g e f r o m 0 G e V t o 100 GoV were c o m b i n e d in (Betlike. 201)0) a n d q u o t e d in T a b l e 8.3. 8.10
S u m m a r y of o s
measurements
A s u m m a r y of t he various t y p e s of m e a s u r e m e n t s discussed a b o v e to d e t e r m i n e t h e value of t h e s t r o n g c o u p l i n g c o n s t a n t is given in T a b l e 8.3. T h e s a m e inform a t i o n is displayed g r a p h i c a l l y in Fig. 8.7 a n d Fig. 8.8. T h e r e s u l t s a r e s o r t e d a c c o r d i n g t o t h e e n e r g y scale that is p r o b e d by t h e different m e t h o d s . T h e t a b l e c o n t a i n s t h e value of r»s at t h e scale of t h e m e a s u r e m e n t a n d . for a global comparison. t h e value a t t h e scale of t he Z mass. In b o t h cases, t h e t o t a l u n c e r t a i n t y of t h e m e a s u r e m e n t is split into a n e x p e r i m e n t a l a n d a t heoret ical e r r o r . T h e last c o l u m n i n d i c a t e s t h e t h e o r e t i c a l f r a m e w o r k e m p l o y e d in t h e m e a s u r e m e n t . T h e precision is at least N L O . If in a d d i t i o n leading a n d n e x t - t o - l e a d i n g l o g a r i t h m i c c o n t r i b u t i o n s have been r e s u m i n e d this is specified by Tesuin". T h e t e r m ' L O T ' refers to c a l c u l a t i o n s based on lattice g a u g e theories.
I NK S I HONG GOUI'LING OONS I AN'I
T a b l e 8 . 3 Sum mar;/ of mea.suiv.ments Observable
g/GeV
of the strong
a,{Q)
coupling
constant Tlieory
«.(A/|) o ¡jo7
±0.002
NNLO
^
±0.005
NNLO
0 . 1 18 ± 0 . 0 0 1
±0.003
NNLO
0 1 ) 1 1 1
0 . 1 18 ± 0 . 0 0 5
±0.003
NNLO
0.208 :t 0.007 ±
0.007
0.117 ± 0.002
±0.002
NNLO
7.1
0.177 ± 0.008 Ì
}{•{}],',
0.113 ± 0.003 ± 0.00 1
NLO
9.7
0.103 ± 0.009
±0.010
0.1 I I ± 0 . 0 0 4
NLO
10.0
0.107 ± 0.002
ÌJJDJ^
0.113
10.0
0.171 ± 0.000 ±
0.007
li,
10.5
0.200 ± 0.060 Ì
¡{{¡¡£
,.,„ + 0.021 + 0 . 0 0 3 1-1 - 0 . 0 2 9 - 0.002
|>p —1)1)+jets
20.0
1 , 1
+0.012 +0.013 - 0.010 - 0.016
« , , » + 0.007 + 0 . 0 0 8 " " - 0.006 -0.009
NLO
e ' e~ J e t s / S h a p e s
22.0
0.161 ± 0.009 Ì
0.121 ± 0.005 Ì o'oolj
resina
21.3
0 . 1 3 5 ± 0 . 0 0 6 + {¡
0.1 IO ± 0 . 0 0 4
NLO
op l)IS/Jets
2-1.5
0.1 18 ± 0 . 0 1 3 ±0 . 0 1 1
0.118 ± 0.008 ±
IT,
31.0
0.160±
0.133 ± 0.013
e 1 c~.lots/Shapes
35.0
0.145 ± 0.002 Ì
e + c ~D{x)
30.0
lij-Suinrule
1.73
1 1 : 4 2 0
( ¡LS-Suniriilo
1.73
0.280 ± 0.061
UT
1.78
0.323 ± 0.005 ±
0.030
5.0
0.214 Ì q
016 ±
f
5.0
/•M-') T Decays
k
.//>!', T Decays Q(J
IXIIIIKI s t a t e s
PP. P P
- * 7 + A'
12.1
e ' c~ J e t s / S h a p e s
<1-1.0
,, ,
| r
Ì oor.j
±0.01« f
¡{ ¡ j : ^
(J'OOG ^
0.019ÌQ003
0 . 1 18 Ì 0.112
±
0.005
NLO
±0.001
0.115 ± 0.000
±0.(K)3
ijl'oo^
Ì
0.IMJ7 o
o('Ì)
LGT NNLO
NLO NNLO
0.123 ± 0.002 Ì O'OOJ
resimi
0 1,17 + o.ooo a. oU Mo rSi
0.125 Ì
NLO
0.144 ±
0 . 1 2 6 ± 0 . 0 2 2 Ì I O'(K> )
NNLO
0.123 ±
resimi
o!oof
" -0.010 *
'
0.029 Ì o
003
o'oo7 ±
0.009
0.139 ± 0.004 Ì
o oo7
• )t).(i
0.131 ± 0.003 ±
0.008
0.119 ± 0.002
±0.007
NLO
e ' o~ J e t s / S h a p e s
58.0
0.132 ± 0.003 ±
0.(M)8
0.123 ± 0.003
±0.007
resimi
<•' e~.lets/Shapes
9I.2
0.121 ± 0 . 0 0 1 ±
0.006
0.121 ± 0 . 0 0 1
±0.000
resimi
Hi
ili.2
0.124 ± 0.004 Ì
o oo'j
(7(pp — jets {ET))
111).
0.118 ± 0.008 +
0.005
e ' e" Jets/Shapes
133.
0.113 ±
e ' e" Jets/Shapes
+
e e"
(1
-T>
r
0.00.3 Ì
o'oo5
0.124 ± 0 . 0 0 1 Ì o
oo2
NNLO
0 . 1 2 1 ± 0 . 0 0 8 ± O.IH).r>
NLO
±0.000
0.120 ± 0.003 ±
0.006
rosimi
101.
0.109 ± 0.004 ± 0.005
0.118 ± 0.005 ±
0.006
resimi
e ' c~ Jcts/Sliapes
172.
0.101 ±0 . 0 0 4 ± 0.005 0.1 I I ± 0 . 0 0 5 ± 0 . 0 0 6
resimi
e 1 e " .lets/Shapes
183.
0.109 ± 0.002 ±0 . 0 0 4
rosimi
e ' e " Jets/Sliapes
189.
0.110 ± 0.001 ± 0.001 0.123 ±0 . 0 0 1 ± 0 . 0 0 5 rosimi
0.003
0.121 ± 0.002 ± 0.005
SOMMAMI Ol n , MKASUHKMKNTS
Uj-SR GLS-SR
Q= (?=
V
«i
V it! V
moincots(Fi) f'2(x) V dccays J/i/z. Ì' dccays V-systcm LGT li
Q= Q= Q= V
Q= Q= (?= (?= (?=
jcts/shapcs cT(pp V -> —> jcts) + c e" -> jcts/shapcs c V —> jcts/shapcs c+e~ —> jcts/shapcs c V —> jcts/shapcs c *c~ —> jcts/shapcs
0.04
V
-
c V —> jcts/shapcs
i
V
pp —> hh+jcis c*c~ —> jcts/shapcs pp. pp -> ; +X c p —> jcts li e ' c —> jcts/shapcs c V IHx) 1< c*c~ —) jcts/shapcs <1 -T>
i
V
I
V V V V V V
V
(?= (?= Q= Q=
V V
C>=
V
{?= Q= Q=
V
V V
I 0,0.8
0.12
0.16
0.2
FlC. 8 . 7 . M e a s u r e m e n t s of t lie s t r o n g coupling constant evolved to t he scale of t h e 7. mass. Also q u o t e d is t h e e n e r g y scale of t h e a c t u a l m e a s u r e m e n t . The d a t a a r e c o m p a r e d t o t h e global average given by t h e P D G (2000).
All n u m b e r s given in T a b l e 8.3 a r e r o u n d e d to t h r e e decimal digits. T h i s is still perfectly a d e q u a t e in m o s t cases, a l t h o u g h in s o m e places t h e intrinsic precision has reached a point where four digits would b e preferable. At low energies, s u m rules allow a m e a s u r e m e n t at t h e scale close t o t h e
I III
S I KONG C O U P L I N G C'ONSTAN'I
0.5
0.-1
0.3 \<02) 0.2
0.1
0 Q (GEV)
FlC. 8 . 8 . M e a s u r e m e n t s of t h e s t r o n g coupling constant plotted at t h e scale of l.lie respective m e a s u r e m e n t s . T h e r u n n i n g is clearly visible. T h e e x p e r i m e n t a l d a t a a r e c o m p a r e d t o t h e Q C D e x p e c t a t i o n for t h e world average value of o s a s q u o t e d by t h e P D G (2000).
nucleoli mass. Also t h e h a d r o n i c b r a n c h i n g rat io of t h e t a n lepton is very sensit ive t o t h e s t r o n g coupling. A n o t h e r g r o u p of m e a s u r e m e n t s with a still fairly low scale comes f r o m t h e m e a s u r e m e n t of scaling violations iu s t r u c t u r e f u n c t i o n s a s seen in d e e p inelastic lepton nucleoli interactions. Here t w o m e a s u r e m e n t s a r e based on t h e evolution of t h e s t r u c t u r e f u n c t i o n s F>(.r) a n d F : t(.r). respectively, a third result, c o m e s f r o m t h e evolution of t h e m o m e n t s of F>. Also d o n e at c o m p a r a t i v e l y low scales a r e m e a s u r e m e n t s using d e c a y s or level s p l i t t i n g s iu heavy q u a r k o n i a . G o i n g to higher energies most m e a s u r e m e n t s a r e derived from t h e s t u d y of final-state j e t topologies o r in general f r o m event s h a p e variables. T h i s covers j e t s in d e e p inelastic lepton nucleoli s c a t t e r i n g , in p r o t o n ant ¡proton collisions o r in c + c a n n i h i l a t i o n . In a d d i t i o n t h e r e a r e m e a s u r e m e n t s based on t h e total
SI I MM,AMY I »1 ... MKASUKKMKNTS
li.idronie cross sections, It. and Hi, heavy llavonr or direct-photon production in proton a n t i p r o t o n interact ions or scaling violations in fragmentat ion funct ions. I he overall agreement of tin* results is r a t h e r impressive, giving convincing evidence that Q C D really is the universally correct theory of s t r o n g interactions. The m e a s u r e m e n t s cover space-like and time-like m o m e n t u m transfers, energy scales between the mass of t h e nucleoli and the highest Mil* energies and are based on lepton lopton. lepton h a d r o n a n d hadron liadron interactions. T h e ol>servables are sum rules, branching ratios, level splittings in hound s t a t e systems, sealing violations, cross section ratios o r event s h a p e variables. Within their respective experimental a n d theoretical uncertainties all results a r e consistent. I low to merge all available m e a s u r e m e n t s into a single combined result is not entirely obvious (Sehmelling l!)(l(>. •2()()(l). Applying t h e m e t h o d proposed in the latter reference to all d a t a quoted ill Table S.:5 yields a global average of o«(.l!'•}) = (1.1180 ± 0.00-13. One has to keep in mind t h a t the error obtained by lliis m e t h o d (Sehmelling, 2000) t e n d s to be overestimated if some of t h e input d a t a are quoted with overly conservative errors. T h i s kind of bias can be reduced by a careful selection of the d a t a which are used in the average. Examples can be found in t h e literat ure (Bethke. 2000: P D G . 2000). Here, Bel like arrives at a global average t \ s ( M ' j ) = 0.1181 ± 0.0031, while the Partiee D a t a G r o u p quotes a s ( M % ) 0.1181 ± 0.0020. T h i s average is compared lo the individual measurements in Pigs. 8.7 and 8.8. Note that t h e e s t i m a t e s for t h e central value of tin* global average essentially agree. T h e e r r o r e s t i m a t e s differ by as much as a factor of two. rolled¡ng different a s s u m p t i o n s about intrinsic precision a n d correlations between individual results. Over t h e past years t h e global average has stayed remarkably stable. T h e value quoted iu 2001 by the Particle Data G r o u p ( P D G . 200-1) is o s ( A / £ ) = 0.1187 ± 0.002. t h a t is. within the inherent uncertainties from the d e t e r m i n a t i o n of the average, the new result is consistent wit h the one from 2000. Exorcises for C h a p t e r 8 8 I Show how the Sphericity of an event can be calculated from t h e eigenvalues of the Sphericity-tensor UptWj , J
-
Epi>•-' "
E p l> P'
EpP2
where p, is t he /'tli component., i = 1.2.3. of the m o m e n t u m vector p. (Ilint: It may be convenient t.o work with the form of II' where the tensor p r o d u c t s a r e expressed in matrix notation, using column vectors for the individual m o m e n t a . ) ' * ' S 2 Show t hat Sphericity is not collinear safe. 8 3 Show that the C - p a r a i n c t e r is collinear safe. 8 I Show t hat t he C - p a r a n i e t e r has a value between C = 0 for ideal t wojet. events and C !/•< for events with isotropic m o m e n t u m flow.
9
TE T
F THE
TR CT RE
F
C
W h e n d i s c u s s i n g t e s t s of Q C - so f a r flic m a i n e m p h a s i s h a s been o n t h e d e t e r m i n a t i o n of t h e s t r o n g c o u p l i n g c o n s t a n t , w h i c h , a p a r t f r o m t h e q u a r k m a s s e s , is t h e o n l y free p a r a m e t e r of t h e t h e o r y . If t h e t h e o r y is c o r r e c t , t h e n t h e v a l u e for n „ ( A g ) m u s t h e t h e s a m e for all t y p e s of s t r o n g i n t e r a c t i o n p r o c e s s e s . H o w e v e r , iu o r d e r t o g o f u r t h e r a n d check t h e e n t i r e t h e o r y of Q C . o n e a l s o h a s t o verily t h a t t h e q u a n t u m n u m b e r s of t h e i n t e r a c t i n g fields, s u c h a s spin a n d colour c h a r g e , a r e w h a t t h e y a r e a s s u m e d t o b e in t h e Q C a g r a n g i a n . Here we will s t u d y t h e spin of t h e q u a r k s a n d t h e g l u o n a n d t e s t t h e f l a v o u r i n d e p e n d e n c e of o „ , t hat is. t h e a s s u m p t i o n t h a t all q u a r k s c a r r y t h e s a m e c o l o u r c h a r g e . T h e r a t i o of th<- c o l o u r c h a r g e b e t w e e n q u a r k s a n d g l u o n s will b e e x a m i n e d in t h e following c h a p t e r . 9,1
arton
spins
n e f i n d i n g f r o m t h e p h e n o m e n o l o g y of IS p r o c e s s e s w a s t h a t q u a r k s s h o u l d b e s p i n - 1 2 part icles, c.f. Sect ion 2. sing a Y a n g Mills g a u g e t h e o r y t o d e s c r i b e s t r o n g i n t e r a c t i o n s b e t w e e n q u a r k s by glnoii e x c h a n g e l e a d s t o t h e p r e d i c t i o n t hat, g l u o n s a r e spin-1 fields. B o t h a s s u m p t i o n s c a n be t e s t e d e x p e r i m e n t a l l y in
9.1.1
T e
uart
spin
T h e m o s t s t r a i g h t f o r w a r d way t o p r o b e t h e q u a r k s p i n in e + e a n n i h i l a t i o n is by l o o k i n g a t t h e a n g u l a r d i s t r i b u t i o n of t h e event axis, w h i c h c o i n c i d e s w i t h t h e d i r e c t i o n of t h e p r i m a r y q q - p a i r . For s p i n - 1 / 2 f e r i n i o n s a n n i h i l a t i n g i n t o a vector b o s o n , c o n s e r v a t i o n of a n g u l a r m o m e n t u m p r e d i c t s a d i s t r i b u t i o n
dI c o s « I
~ 1 4- cos" B
(9.1)
if t he linai s t a t e p a r t i c l e s h a v e s p i n - 1 / 2 a n d da d|cos0|
1 - cos" B
(9.2)
for s c a l a r p a r t i c l e s . H e r e B is t h e p o l a r a n g l e w i t h r e s p e c t t o t h e d i r e c t i o n of t h e i n c o m i n g p a r t i c l e s . If p a r i t y is c o n s e r v e d , t h a t is. w h e n t h e a n g u l a r distribution does not have t e r m s proportional to c o s B . t h e absolute values can b e o m i t t e d . For e + e ~ a n n i h i l a t i o n i n t o h a d r o n i c final s t a t e s t h e s e s i m p l e pred i c t i o n s a r e m o d i f i e d by h i g h e r o r d e r c o r r e c t i o n s a n d n o n - p e r l urbat ivo effects.
l'Ali I ON S P I N S
l'Ile first m e a s u r e m e n t s ( S c h w i t t e r s < / ul.. l97- r .) of t h i s t y p e w e r e p e r f o r m e d at l a t h e r low e n e r g i e s . B e c a u s e of t h e r u n n i n g of t h e s t r o n g c o u p l i n g , h i g h e r o r d e r c o r r e c t i o n s a r e s m a l l e r at L E P e n e r g i e s. In a d d i t i o n p o w e r law c o r r e c t i o n s a n d n o n - p e r t u r b a t i ve effects in g e n e r a l a r e s m a l l e r a n d t h e T h r u s t a x i s t o a very good a p p r o x i m a t i o n a l i g n s with t h e d i r e c t i o n of t h e p r i m a r y q u a r k s . T h e r e s u l t ing a n g u l a r d i s t r i b u t i o n f o u n d in ( A L E P I I C'oliali.. 1998a) is s h o w n in Fig. 9.1. Since n o a t t e m p t w a s m a d e t o d i s t i n g u i s h q a n d q a n d in o r d e r t o a v e r a g e o v e r t h e f o r w a r d - b a c k w a r d a s y m m e t r y in t h e c o u p l i n g of t h e Z t o t h e p r i m a r y qqpair. t h e a n g l e B b e t w e e n t h e i n c o m i n g b e a m a n d t h e direct ion of t h e final s t a t e q u a r k s is a l w a y s t a k e n in t h e r a n g e 0 < 0 < 9 0 ° .
COS
<6|!HUM>
FIG. 9 . 1 . A n g u l a r d i s t r i b u t i o n of t h e T h r u s t a x i s in h a d r o n i c Z d e c a y s . F i g u r e f r o m A L E P I I C o l l a b . ( 1998a).
T h e e x p e r i m e n t a l d a t a a r e c o m p a r e d t o a M o n t e C a r l o c a l c u l a t i o n which i n c l u d e s t h e full d e t e c t o r s i m u l a t i o n . T h e d a t a a r e in perfect a g r e e m e n t w i t h tins p i n - 1 / 2 a s s i g n m e n t for t h e q u a r k s . T h e sensit ivity t o t he q u a r k spin is i l l u s t r a t e d by c o m p a r i n g t h e m e a s u r e m e n t s t o t h e e x p e c t a t i o n f r o m a s p i n - 0 a s s i g n m e n t , which is clearly e x c l u d e d . For large p o l a r a n g l e s t h e s h a p e is p e r f e c t l y d e s c r i b e d by t h e s i m p l e e x p e c t a t i o n 1 + cos" B . T h e s h a r p d r o p in t h e d i s t r i b u t i o n a r o u n d cosB (I.S is d u e t o t h e event selection c u t s a n d finite d e t e c t o r a c c e p t a n c e a n d is a l s o well r e p r o d u c e d by t h e M o n t e C a r l o s i m u l a t i o n .
:<:<(>
9.1.2
I ESTH Of
Tin: gluon
I'llE S T H U C T I I I t K O P (¿OI>
spiii
W h i l e t h e q u a r k s p i n c a n h e d e t e r m i n e d a l r e a d y hy l o o k i n g at two-jet e v e n t s , t he s t u d y of t h e s p i n of t h e g l u o n r e q u i r e s at least t h r e e jets, t h a t is, e v e n t s w i t h at least, o n e h a r d g l u o n in t h e final s t a t e . T a k i n g for g r a n t e d t h a t q u a r k s a r e s p i n - 1 / 2 p a r t i c l e s , i n f o r m a t i o n a b o u t t h e s p i n of t h e g l u o n c a n h e o b t a i n e d f r o m i n t e r n a l c o r r e l a t i o n s in three-jot e v e n t s , o r t h e a n g u l a r d i s t r i b u t i o n of ( he event p l a n e a s f u n c t i o n of t h e T h r u s t . M e a s u r e m e n t s h a v e b e e n p u b l i s h e d l»v m a n y e x p e r i m e n t s ( T A S S O C'ollab.. 1980: P L U T O C'ollab.. l!)8l)« : C E L L O Col lab.. 1982; 1.3 C'ollab., 1991«; O P A I . C o l l a b . . 1991«: D E L P H I C o l l a b . . 1992«; AI.KI'II C o l l a b . . 1998«), T h e s e n s i t i v i t y of t h e d i f f e r e n t i a l q q g c r o s s s e c t i o n t o I he spin of t h e gluon a r i s e s f r o m t h e f a c t t h a t t h e c o u p l i n g of t h e v e c t o r g l u o n t o t he q u a r k s is helicitv c o n s e r v i n g , while a s c a l a r Higgs-like g l u o n would i n d u c e a spin Hip. C o m p a r i n g m e a s u r e m e n t s t o t h e p r e d i c t i o n s f o r b o t h h y p o t h e s e s t h u s a l l o w s t o g a u g e t he sensit ivity of a n o b s e r v a b l e t o t h e s p i n of t h e g l u o n a n d check I he s p i n - 1 assignment.. The k i n e m a t i c s of a t h r e e - j e t event is t h a t of a t h r e e - p a r t i c l e Z d e c a y i n t o a q u a r k q . a u t i q u a r k q a n d g l u o n g. In t o t a l t h e r e a r e n i n e d e g r e e s of f r e e d o m . A f t e r i m p o s i n g t h e const r a i n t of e n e r g y a n d m o m e n t u m c o n s e r v a t i o n , o n l y five a r e i n d e p e n d e n t , t h r e e of which d e s c r i b e t h e overall o r i e n t a t i o n of t h e e v e n t , r i m s , t h e r e r e m a i n o n l y t w o i n d e p e n d e n t v a r i a b l e s t o s t u d y t h e t o p o l o g y of t h r e e - j e t e v e n t s b e y o n d its a n g u l a r o r i e n t a t i o n . Let x, d e n o t e t h e jet e n e r g i e s normalized to t h e b e a m energy, x, = ~
i = q. q. g ,
(9.3)
w i t h .(•<, + .r,, + ,rK = 2 by e n e r g y c o n s e r v a t i o n . T h e l e a d i n g o r d e r c r o s s s e c t i o n s for t he v e c t o r a n d t h e s c a l a r g l u o n h y p o t h e s i s a n d m a s s l e s s p a r t o n s . n o r m a l i z e d to t h e B o r n level c r o s s s e c t i o n
IL
=
2TT
£
26.
R- + « - V ^
1
.
+
^
(9
.4)
>
w h e r e tin* s u m s r u n o v e r all c o n t r i b u t i n g q u a r k f l a v o u r s , w i t h (••<, a n d
, n (1 - J-,,)(l - a v , )
«««I
(».5)
T h u s , t h e n o r m a l i z e d d i f f e r e n t i a l t h r e e - j e t c r o s s s e c t i o n is t h e s a m e oil a n d oil' t h e Z r e s o n a n c e . In c a s e of a s c a l a r g l u o n t h e s i t u a t i o n is d i f f e r e n t . H e r e t h e kuicmat.ical f u n c t i o n s a r e
l ' A l i I O N SI'INS
x
\
cos /.
F l o . !).2. E n e r g y d i s t r i b u t i o n of t h e lowest energy jet, a n d t h e Ellis K a r l i n e r angle in e ' e anniliilat ion at t h e Z resonance. Figure from Scliinclliug( 1995a).
= (1 - .i: f ,)0 - .7:,,)
i
"' < l
S
" =
-
2( 1 +
*•> '
In t h e a b s e n c e of ( | i i a r k / g l u o n identification t h e t h r e e jets a r e energy ordered iis > .!••> > x ; ! . T h e leading o r d e r cross section for a given c o n f i g u r a t i o n is o b t a i n e d by s u m m i n g t h e llavoiir-ideiitified m a t r i x elements, c. .r^} to {.;:<,. x,,. x„ }. T h e s u m over all six p c r m u t a t ions yields 1
d'-'a 1
<7<| d x u l x j
a,Cr
x\{ + x% + xg
(9.7)
TT (1 - X j ) ( l — X-,)(l - X 3 )
and I
d-V dxidx-2
a„C,.- f xr(l -
^
- x , ) + 4 ( 1 - x , ) -f- .,5(1 - ./•:,) (1 - x , ) ( l - x 2 ) ( l - X : , )
1 0 £ 4 E
+ ^ J
(9.8) for a spin-1 a n d a scalar gliion. respectively. On t h e Z resonance t h e constant olfset in t he scalar gliion ease takes on t h e value 10 ]C " q / 1 C rtq + ('<Ì ~ 7.15. In both cases, t h e differential cross section is singular for .r :l » 0. p r o p o r t i o n a l to l / x 2 for t h e vector gliion a n d pro]>ortional to I/.X3 for tlx' scalar gliion. T h e cross section for t h e l a t t e r is less singular, because the spin flip induced by t h e emission of a soft scalar gliion results in a final s t a t e with a u t i p a r a l l c l s p i n s follile two lerinions. which has less o v e r l a p with t h e vector s t a t e of t h e initial Z bosou. In a c t u a l studies, c o n f i g u r a t i o n s close to these singularities a r e avoided by restricting t h e p h a s e s p a c e to a singularity-free s u b s p a c e . for e x a m p l e bv imposing t h e D u r h a m algorit hm with ?/,.,, 1 = 0.008 to define three-jet linai s t a t e s .
;t:iK
11 ;s i s o l
i S T H U c r i i m - : <>i- q c n
T w o i n d e p e n d e n t variables which a r c sensitive t o t h e difference in t h e sin g u l a r i t y s t r u c t u r e of t h e leading o r d e r m a t r i x e l e m e n t s a r e . lor e x a m p l e . I lie /•(-distribution or t h e Ellis K a r l i n e r a n g l e A|.; (Ellis a n d K a r l i n e r . 1!)79). t h e a n g l e between t h e highest e n e r g y j e t a n d t h e t w o lower e n e r g y ones in t h e restf r a m e of t h e t w o lower e n e r g y j e t s . It is r e l a t e d t o t h e scaled jet energies via cosAi;k (-'"a ./':i)/.i'i. F i g u r e 9.2 s h o w s m e a s u r e m e n t s bv t h e L.'t C'ollabor a t i o u c o m p a r e d to t h e e x p e c t a t i o n for a vector a n d a s c a l a r gluon. T h e d a t a a r e well described by t h e m o r e s i n g u l a r b e h a v i o u r e x p e c t e d for t h e vector gluon h y p o t h e s i s , a n d a r e in clear d i s a g r e e m e n t with t h e s c a l a r g l u o n h y p o t h e s i s . An a l t e r n a t ive choice for a pair of t e s t v a r i a b l e s is 9
x, e
I
1
and
1 1 Z - —7= (;/>> - x ; i ) € 0. V3 v3
(9.9)
How these variables c o r r e l a t e with different- event topologies is illustrated in Fig. 9.3.
Mercedes siar configuration
F l C . 9 . 3 . P h a s e s p a c e a s f u n c t i o n of./'i a n d Z for e n e r g y - o n l e r e d jet configurations. ./'i > ./'•> > ./':(. T h e a r r o w length is p r o p o r t i o n a l t o t h e energy. F i g u r e f r o m A I. F P U C o l l a b . ( 1 9 9 8 a ) .
I.ike t h e Ellis Karliner a ngl e , t h e Z - v a r i a b l e is also sensitive t o t h e spin of t h e gluon. Figure !).-l shows t h e e x p e r i m e n t a l Z - d i s t r i b u t ion c o m p a r e d to various t h e o r e t i c a l models. T h r e e - j e t e v e n t s were selected using t h e D u r h a m inetric. p r o j e c t e d o n t o t h e event p l a n e a n d t he jot e n e r g i e s r e c o n s t r u c t e d f r o m t he jet directions, using t h e formula , -
• , • SIU >1'i-j + Sill V23 + Sill U';t|
(9.10)
with {i. j.l a n y p e r m u t a t i o n of { 1 . 2 , 3 } . a n d <1'jk t h e a n g l e between jets / a n d /.-. T h i s f o r m u l a st.rietly o n l y holds for massless j o t s . Since t h e e m p h a s i s lies on a
l ' A l i I O N SI'INS
Z Kit:. 0 . 4 . ¿ - D i s t r i b u t i o n : P l o t t e d a r e t h e corrected d a t a with full e r r o r s a n d two a l t e r n a t i v e gluon spin models. For b o t h m o d e l s t h e leading o r d e r ( L O ) a n a lytical formula a n d a M o n t e C a r l o s i m u l a t i o n including higher o r d e r «'fleets is s h o w n . F i g u r e f r o m AI.KPII Collah.( 1908«),
c o m p a r i s o n between d a t a a n d theoretical m o d e l s for inasslcss p a r t o n s . eqn (0.1(1) was taken a s t h e definition of t h e e x p e r i m e n t a l observables. Both for t h e vector a n d t he scalar gluon hvpot hesis. t h e r e s u l t s a r e compare«I to t h e predictions from t h e leading o r d e r m a t r i x elements. Already w i t h o u t t a k i n g into account, higher o r d e r s a n d h a d r o n i / a t i o n effects, t h e q u a l i t a t i v e a g r e e m e n t between d a t a a n d t h e o r y is q u i t e s a t i s f a c t o r y for t h e veetor-glnon model. Higher o r d e r effects only induce minor corrections t o the theoretieal prediction, which f u r t h e r i m p r o v e t h e description of t h e d a t a by t h e veetor-glnon model. The s c a l a r gluon is clearly excluded.
I r..~> i n n i
.1 I H
I ill'. .-» I IIIIV M ill I HI' I
"I I
T l i r sensitivity of t.lie o r i e n t a t i o n of t h e event p l a n e t o t h e spin of t h e gluon is a p a r t i c u l a r feat ure of t he different, vector a n d axial-vector couplings of I lie Z boson t o t h e p r i m a r y q u a r k s ( K o r n e r vt al., 1987: Schiller a n d Korner. 1989) T h e a n g u l a r d i s t r i b u t i o n of t h e n o r m a l vector of t h e e v e n t p l a n e with respect t o t h e direction of t h e incoming b e a m s can be described by
d cos0„
1+
(T) cos" <)„ ,
(9.11)
wit h an asyininet ry p a r a m e t e r (T), which in general is a f u n d ion of t h e T h r u s t /' of t h e event. For vector gliions t h e prediction is (T) = —1/3. for t h e scalar gluon (T) increases w i t h d e c r e a s i n g values of T. E x p e r i m e n t a l results from t h e D E L P H I C o l l a b o r a t i o n (1992«) a r e found t o b e in good agreement with the vcctor-gluoii hypothesis while t h e sealar-gluon m o d e l can be ruled o u t . 9.2
F l a v o u r i n d e p e n d e n c e of s t r o n g i n t e r a c t i o n s
A n o t h e r important, test of Q C D is t h e llavour i n d e p e n d e n c e of t h e s t r o n g coupling c o n s t a n t . Effectively, this m e a n s that all q u a r k s c a r r y t h e s a m e colour charge. Flavour-universality of t h e coupling constant is a f e a t u r e of a n y nonabelian gauge theory. It is not required in a b e l i a n t heories such a s ( J E D . where different electric c h a r g e s for different flavours a r e allowed. T h e difference c o m e s about because of t h e self-interaction between t h e g a u g e fields in uon-abelian theories. T o i l l u s t r a t e t h e p o i n t , consider q u a r k gluon C o i n p t o n s c a t t e r i n g , lu Q C D t h e t h r e e d i a g r a m s s h o w n in Fig. 9..ri c o n t r i b u t e , a n d they all have t o a d d u p with t h e correct weights t o yield a g a u g e invariant s c a t t e r i n g a m p l i t u d e . It follows that t h e gluon gluon coupling is linked t o t h e q u a r k gluon coupling with a lixed r a t i o d e t e r m i n e d by t h e s t r u c t u r e of t h e g a u g e g r o u p , a n d t h u s that tlust l o n g coupling m u s t be flavour i n d e p e n d e n t . In abelian theories, on t h e o t h e r h a n d , t h e r e is no self-interaction between t h e g a u g e fields, a n d already t h e s u m of t h e first, two d i a g r a m s iu Fig. 9..ri is gauge invariant. Since there is no link t o gauge-field self-interaction, a r b i t a r v c h a r g e s a r e allowed for different ferniions wit hout violating t h e g a u g e s y m m e t r y .
<1
k
q
q
g
<1
(
l
K g ^ ^ q a ^ ^ ^ ^ g Fit;. 9 . 5 . D i a g r a m s c o n t r i b u t i n g t o q u a r k gluon C o i n p t o n s c a t t e r i n g
Indirect evidence for t h e universality of t h e s t r o n g c o u p l i n g constant is alr e a d y o b t a i n e d f r o m t h e f a d t h a t t h e m e a s u r e m e n t s of from a inultil ude of reactions a n d energy scales a p p e a r to be consistent with a single c o m m o n
I LAVOUIt INI >1 l'I NI'I NCI (>!•• S I ItONC IN I KIlAC I IONS
valile. M o r e s e n s i t i v e t e s t s « ¡ili Ite pei Ini 11it'd wit li <|c1 M r o n g c o u p l i n g constiinl bused 011 d a t a s a m p l e s w h e r e t h e c o m p o s i t i o n ol t he ne t ivo q u a r k f l a v o u r s c a n b e c o n t r o l l e d e x p e r i 11 lent ally. A n y o b s e r v a t i o n of a flavour d e p e n d e n c e would ho a st r o n g i n d i c a t i o n for physics b e y o n d t he S t a n d a r d M o d e l , e i t h e r bv e s t a b l i s h i n g a d e p e n d e n c e b e t w e e n q u a r k f l a v o u r a n d c o l o u r c h a r g e 01 by p o i n t i n g t o w a r d s now p a r t i c l e s t h a t c o u p l e t o t h e s t r o n g i n t e r a c t i o n s s e c t o r . T a b l e 9 . 1 Mcasur/inc.iits of ratios of the stronfi coupling constunt for different flavour combinat ions. The errors arc the combined statistical, siisteinatie and theoretical uncertainties of the results. Observable
Result
Reference
os(b)/os(ndsc)
1.(1(12 :l: 0.02:1
A L F P I I C o l l a b . ( 1995c)
Os(uds)/os(be)
(l.!)71 ± 0.02:5
AI.KPII C o l l a b . ( 1995c)
as(b)/os(udsc)
1.0(1 ± 0 . 0 5
DELPHI Collab.(1993a)
os(b)/os(udsc)
1.00 ±0.08
I..Ì Collab.(1991/;)
os(b)/os(udsc)
1.017 ± 0.0:i0
O P A L C o l l a b . ( 199:5/;)
os(c)/os(ndsb)
0.918 ± 0 . 1 1 5
O P A L C o l l a b . ( 199:5/;)
ns(s)/as(udcb)
1.158 ± 0 . 1 0 I
OPAL Collab.(19936)
os(uds)/os(cb)
1.0:58 ± 0.221
O P A L C o l l a b . ( 11193/;)
ns(b)/os(udsc)
0.992 ± 0.010
OPAL Collab.(1995«)
i>s(uds)/os(ildscb)
0 . 9 8 7 ± (l.0:55
SI.I) C o l l a b . ( 1 9 9 0 )
os(e)/os(udseb)
1.012 ± 0 . 1 7 1
Sl.fi Collab.(1990)
os(b)/ns(u
1.020 ± 0 . 0 0 5
S L D C o l l a b . ( 1990)
'»S(c)/«s(uds)
1 ,o:56 ± 0.001
S l . l ) C'ollab.( 1999«)
os(b)/«s(nds)
1.001 ± 0.011
S L I ) Collab.(.1999«)
A l l o w i n g for a f l a v o u r d e p e n d e n c e of t h e s t r o n g c o u p l i n g c o n s t a n t , t he s e c o n d o r d e r Q C D p r e d i c t i o n for t h e d i f f e r e n t i a l c r o s s s e c t i o n of a g l o b a l event s h a p e v a r i a b l e f r o m h a d r o n i e Z d e c a y s i n t o a q u a r k a n f i q u a r k p a i r of flavour / c a n be written as = A ( s . f ) n M ) + B(x, / ) « „ ( / ) . T h e f u n c t i o n s A(.r.f) a n d It(.r.f) a r e k i n e m a t i c f u n c t i o n s w h e r e t h e flavour d e p e n d e n c e e n t e r s t h r o u g h t he d i f f e r e n t q u a r k m a s s e s . For b - q u a r k s t hose p u r e l y k i n e m a t i c e f f e c t s amount- t o a c o r r e c t i o n of t y p i c a l l y 5% in t lie t h r o e - j e t cross section, which c a n n o t b e ignored in a n y precision test of t h e llavour i n d e p e n d e n c e of n... For t h e o t h e r q u a r k f l a v o u r s m a s s e f f e c t s a r e negligible. T h e q u a r k m a s s d e p e n d e n c e of A(.r. f ) h a s long been k n o w n (loffe, 1978; I . n e r m a n n a n d Z e r w a s . 1980), t h e m a s s e f f e c t s in tlx' noxt.-to-leading o r d e r c o r r e c t i o n s were d e t e r m i n e d o n l y r e c e n t l y ( B a I lest r e m et al. 1992, 199 I: B e r n r e u t h e r et at. 1997. B r a n d e n b u r g a m i I h v e r 1998, R o d r i g o et al. 1997).
iiìsts«)i-
iili-:STIUN• iURF<>i
T a b l e !).2 Mens un tin ills of liti striniti coujiliiitj Jlavours. The errors are the total uncertainties. Observable
Result.
g<'D constant
for
all
quark
C o r r e l a t i o n Coefficients
os(u)/os(incl.)
0.951 ±0 . 2 0 ! )
L~~
os(d)/«s(incl.)
0.933 ± 0 . 1 9 5
-0.531
1.
1.141 ± 0 . 1 4 8
-0.348
-0.386
1.
os(c)/os(incl.)
0.912±0.091
-0.180
-0.345
-0.051
1.
"hOOA'SO"'1)
1.021 ± 0 . 0 2 0
-0.010
-0.002
-0.036
-0.207
1.
Experimentally, various t a g g i n g techniques c a n be applied to select d a t a samples with different flavour c o m p o s i t i o n s . R e q u i r i n g for i n s t a n c e a lept.on with large t r a n s v e r s e m o m e n t u m relative to t h e T h r u s t axis or a displaced s e c o n d a r y vertex in a n event yields a b - q u a r k enriched s a m p l e . A m i - t a g g i n g on lifetime or •.imply requiring a leading particle in t h e event with a m o m e n t u m of m o r e t h a n 70% of t h e b e a m m o m e n t u m enriches light flavours u. d a n d s. P r i m a r y c - q u a r k s can be enriched by selecting j e t s with a h i g h - m o m e n t u m c h a r m e d meson, like a I)*, or by looking for s e c o n d a r y vertices f r o m d e c a y s with typical lifetimes of e-mesons. Selecting j e t s w h e r e t h e m o s t energetic p a r t i c l e is a Kij e n h a n c e s t h e fraction of p r i m a r y s - q u a r k s . A possibility t o e n h a n c e u-t.ype q u a r k s would b e t o select events with a h a r d p r o m p t p h o t o n in t h e final s t a t e . Most of t h e met h o d s listed a b o v e h a v e been a p p l i e d in o n e or m o r e m e a s u r e m e n t s t o select specific flavour-enriched event samples, w h e r e t he st rong c o u p l i n g constant, t h e n was d e t e r m i n e d by o n e of t h e s t a n d a r d m e t h o d s based on jet r a t e s o r o t h e r global event, s h a p e variables. T a k i n g t h e r a t i o of t h e flavour-selected o s a n d the st rong coupling found in t h e inclusive s a m p l e or a c o m p l e m e n t a r y d a t a set., m o s t of t h e o t h e r w i s e d o m i n a n t theoretical u n c e r t a i n t i e s cancel. T h e r e remains. however, t h e u n c e r t a i n t y on t h e precise flavour c o m p o s i t i o n of i he selected events, which is usually e s t i m a t e d by M o n t e C a r l o s i m u l a t i o n s a n d is needed t o correct a m e a s u r e d r a t i o of c o u p l i n g s t r e n g t h s t o a r a t i o of p u r e flavours. S o m e results f r o m L F P a n d SI.I) a r e collected in T a b l e 9.1. All r a t i o s a r e c o m p a t i b l e with unity, showing that within t h e c u r r e n t e x p e r i m e n t a l precision of a few per cent t h e s t r o n g coupling c o n s t a n t indeed is flavour i n d e p e n d e n t . Ultimately, o n e would like to express all results c o n c e r n i n g t h e flavour ind e p e n d e n c e of o s t h r o u g h t h e r a t i o s o „ ( / ) / o s ( i n c l . ) with ¿=u.d,s,c.h. T h i s inf o r m a t i o n can be o b t a i n e d from a n y set of five d a t a s a m p l e s with sufficiently different flavour c o m p o s i t i o n . A first analysis of this kind was p e r f o r m e d bv t h e O P A L C o l l a b o r a t i o n (19936). In a d d i t i o n to an u n t a g g e d s a m p l e , d a t a sets with different flavour c o m p o s i t i o n were selected using high-/;-;- leptons, D " . K" a n d simple m o m e n t u m - b a s e d leading particle tags. From these t h e «„-values for t h e individual q u a r k flavours were unfolded. All ratios a r e c o m p a t i b l e with unity. T h e result t o g e t h e r with t h e full correlation m a t r i x (Biebel a n d M a t t i g , 1994) is given in T a b l e 9.2.
I I.AVnUll INI<1 l'I NIlKNC'K OK S T K O N G IN I KKAC I l o N S
E x e r c i s e s f o r C'liapl<>r !» 9 I A s s u m i ' negligible f c r m i o n unisses nini p u r i t y c o n s e r v a t i o n t o s h o w that angular m o m e n t u m conservation predicts an angular distribut i o n ( 1+COS- H ) for e 1 e a n n i h i l a t i o n i n t o a p a i r of s p i n - 1 / 2 ]>art icles. a n d (1 cos'-'B) if t h e filial s t a t e p a r t i c l e s h a v e s p i n 0. ' * ' + ' 2 lu e e ~ a n n i h i l a t ion I lie s p i n of t h e gluoii c a n a l s o b e i n f e r m i f r o m t h e d i f f e r e n t i a l c r o s s s e c t i o n of e v e n t s h a p e v a r i a b l e s . Use t h e l e a d i n g o d e r m a t r i x e l e m e n t s t o c a l c u l a t e t h e Thrust d i s t r i b u t i o n for a s p i n - 1 and a spin-0 gluon. ' * * ' '( .'5 lu a n ex|)erinicnt.. t h e d i r e c t i o n of a jet is o f t e n m o r e f a i t h f u l l y recons t r u c t e d t h a n its d i r e c t l y m e a s u r e d e n e r g y . A s s u m i n g m a s s l c s s p a r ticles a n d w o r k i n g in t h e C . o . M . f r a m e , d e r i v e e x p r e s s i o n e q n (!).10) for t he jet e n e r g i e s in t e r m s of t h e inter-jet. o p e n i n g a n g l e s , for a n e + e a n n i h i l a t i o n event i n t o t h r e e j e t s .
10
TE T
F THE G
GE
TR CT RE F CT R
F
C
C
R
In Sect ion 2.2 we have a l r e a d y learned that, t h e probabilities for a q u a r k to r a d i a t e a gluon, or a gluon t o split into a gluon o r a q u a r k pair, a r e related t o t h e s t r o n g c o u p l i n g c o n s t a n t a n d s o m e f a c t o r s which a r e d e t e r m i n e d by t h e underlying g a u g e g r o u p . Iu this c h a p t e r , we will present e x p e r i m e n t a l tests which have shown that these f a c t o r s a r e indeed consistent with t h e e x p e c t a t i o n s , namely, that SU(3) is t h e g a u g e g r o u p for t h e t h e o r y of s t r o n g interactions. Before we s t a r t to describe t h e various m e a s u r e m e n t s , let us s u m m a r i z e once m o r e what a r e t lie ingredients. For a general g a u g e t h e o r y based 011 a s i m p l e Lie g r o u p t h e couplings of t h e ferinion fields t o t h e g a u g e fields a n d t h e gauge-field self-interactions in t h e 11 11abeliau case a r e d e t e r m i n e d by t h e coupling constant, a n d t he Casiinir o p e r a t o r s of t h e gauge g r o u p . M e a s u r i n g t h e eigenvalues of t h e s e o p e r a t o r s , called colour factors, p r o b e s t h e u n d e r l y i n g s t r u c t u r e of tin* t h e o r y in a g a u g e invariant way. Considering t h e case where A r /' a n d ,\ a r e t h e d i m e n s i o n s of t h e f u n d a m e n t a l a n d adjoint r e p r e s e n t a t i o n s of I he g a u g e g r o u p with s t r u c t u r e c o n s t a n t s f bc a n d generators t h e following r e l a t i o n s hold: NA u,
(T T )u
= \jCr.
A',, ]T T ' r -= »,jp=l
iV,v nb
s Tr .
'
' ' '=
'c .
a.h— 1
(10.1) where«./; (i.j ) r e p r e s e n t . g a u g e (ferinion) field indices a n d C/.-. C,\ a n d Tp a r e t h e colour factors. T h e o p e r a t o r s C/.'rf (J a n d C,\ ) ' a r e t h e Casiinir o p e r a t o r s of t h e f u n d a m e n t a l a n d adjoint r e p r e s e n t a t i o n of t h e g r o u p , respectively. As a s t a n d a r d normalization condition 7V = 1/2 is chosen. T h e n for SU(A'V) one finds Ca = N
c
,C
f
= ? 1 ^ .
(10.2)
In t he case of QC1). we have t h r e e f u n d a m e n t a l colours, t h e r e f o r e c = 3. a n d f r o m that we derive C.\ = 3 a n d C' = -l '-l. In E x . (2-2) a n d (2-3) it is shown that t h e probability for a q u a r k t o r a d i a t e a gluon is p r o p o r t i o n a l to ('/. 4/3, independent of its colour s t a t e . Similarly, t h e probability for a gluon t o split into two gliions or into two q u a r k s is shown t o be p r o p o r t i o n a l t o C,\ = 3 a n d T/•• 1/2. respectively, again not d e p e n d i n g on t h e colour s t a t e of t h e decaying gluon. T h e s e relations are illustrated in Fig. 10.1. In t h e case of a gluon s p l i t t i n g
I ESTS Of
I III (iAIICK SIIMH T i m i . ( )F (
'It
2
/
N.
^
CA
^
y, / /
1 2
Fit:. 10.1. Relations between basic vertices a n d colour factors. Figure f r o m Disscrtori( 1!)!)8).
into a q u a r k pair, a n a d d i t i o n a l f a c t o r n j has to b e taken into a c c o u n t , if t h e glnon can split into u j different, q u a r k flavours, a n d if t h e final s t a t e is not differentiated with respect t o its flavour c o n t e n t . For e x a m p l e , at. L E P we have nf r>. since t h e energies a r e large e n o u g h to p r o d u c e a n y of t h e six q u a r k s a p a r t f r o m t h e t o p q u a r k , which is t o o heavy. T h e physical q u a n t i t i e s to !>e tested a r e
IA =
and
C /•
J
r
= -JF , C /••
(10.3)
as well as o s C'/.-. N o t e t h a t f a n d /•;• a r e i n d e p e n d e n t of t he n o r m a l i z a t i o n chosen for t h e g r o u p r e p r e s e n t a t i o n , a n d t h a t a n y new choice of n o r m a l i z a t i o n affecting ("'/.• can be a b s o r b e d into a redefinition of o s . R e m e m b e r that t h e s t a n d a r d definition is o s = f/-'/(-l~). T h e r a t i o s f . a n d /•/• can be i n t e r p r e t e d in t h e following m a n n e r : J \ measures t h e relat ive s t r e n g t h of t h e glnon glnon coupling wit h respect t o t h e q u a r k glnon coupling, w h e r e a s /•;• d e t e r m i n e s t h e r a t i o of t h e n u m b e r of possible colour degrees of f r e e d o m carried by q u a r k s over t ho n u m b e r of gluons. T h i s follows f r o m t h e relation T
r
A
=
C r.
r
(10.1)
I KSTH i )!• I III- O AI "CK SI It IK I IMtK OK <J< l>
derived in A p p e n d i x A . l . w h e r e t h e d i m e n s i o n of t h e adjoint r e p r e s e n t a t i o n of t h e g a u g e g r o u p , A'..\. c o r r e s p o n d s t o t he n u m b e r of g l u o n s . a n d I IK- dimension of t h e f u n d a m e n t a l r e p r e s e n t a t i o n , iVp. e<|iials t he n u m b e r of colour degrees of freedom of t h e q u a r k s . Now. what is t h e real interest in a m e a s u r e m e n t of t h e s e colour factors'.' A very stringent test of Q C D would b e given by a s i m u l t a n e o u s m e a s u r e m e n t of t h e s t r o n g c o u p l i n g constant. o s a n d t h e colour factor ratios, as. a p a r t f r o m the q u a r k masses, t h e former is t h e only free p a r a m e t e r of t h e theory, and t h e l a t t e r show w h e t h e r t h e d y n a m i c s is indeed described by a n u n b r o k e n SU(3) s y m m e t r y . Although it is known t h a t q u a r k s c o m e in t h r e e 'colours', t h e relation bet ween these internal degrees of f r e e d o m a n d t h e d y n a m i c s of s t r o n g i n t e r a c t i o n s is not lixed a priori. A s s u m i n g t h a t all t h r e e c o l o u r s a r e c h a r g e s of s t r o n g i n t e r a c t i o n s suggests a s i m p l e Lie g r o u p such a s SU(:i), SO(.'J) or a n a b e l i a n U(l):<. Only t h e a d d i t i o n a l i n p u t t h a t t h r e e q u a r k s or a q u a r k a n t i q u a r k pair can exist in a colour n e u t r a l s t a t e singles out SU(.'i). A c c e p t i n g that q u a r k s t r a n s f o r m as SU(.'l) triplets, it is still conceivable t h a t not. all internal degrees of f r e e d o m c o n t r i b u t e t o t h e d y n a m i c s of QC'D. In o t h e r words, only a s u b g r o u p of a global SU(3) colour s y m m e t r y is also a local g a u g e s y m m e t r y . In this case, s u b g r o u p s of SU(:i) such a s SU(2). S()(2) or U ( l ) b e c o m e possible c a n d i d a t e s for t h e g a u g e s y m m e t r y . However, o n e would have to i n t r o d u c e a d d i t i o n a l m e c h a n i s m s which force t h r e e q u a r k s o r q q - p a i r s into colour n e u t r a l s t a t e s . Going o n e s t e p f u r t h e r , o n e can also i m a g i n e s t r o n g i n t e r a c t i o n s t o b e described by a s p o n t a n e o u s l y broken SU(:i) s y m m e t r y . T h e resulting massive g a u g e b o s o n s would lead t o a d y n a m i c a l s t r u c t u r e which would d e v i a t e from t h e SU(3 ) e x p e c t a t i o n . Finally, d e v i a t i o n s can also be caused by I he existence of new physics which couples to t he st rong i n t e ractions sector. An e x a m p l e for t h e l a t t e r is t h e case of a light gluiuo, t he s u p o r s y i n m o t r i e p a r t n e r of t h e gluou, which at effectively c o n t r i b u t e s t h r e e additional fermionic degrees of f r e e d o m in e + e ~ a n n i h i l a t i o n processes. All t h e s e different h y p o t h e s e s a r e c h a r a c t e r i z e d bv a specific set of values for /A- f r a n d nTherefore m e a s u r i n g t h e s e n u m b e r s should allow t o rule out s o m e of t h e h y p o t h e s e s . P r a c t i c a l l y all of t h e m e a s u r e m e n t s have been p e r f o r m e d at LEI', b e c a u s e t h e s t a t i s t i c s is very large a n d t h e e x p e r i m e n t a l conditions a r e very clean. In general, a cross section for o + e ~ a n n i h i l a t i o n into h a d r o n i c final s t a t e s has t h e s t r u c t u r e „ = f(asC,--.~.nf^r). C L y
(10.5)
In o r d e r to p r o c e e d , we have t o find specific processes a n d observables with a cross section t h a t has large sensitivity t o t h e colour f a c t o r s . 10.1
Three-jet variables
We d e n o t e a s I line-jet varia les those q u a n t i t i e s for which t h e pert nrbat ive prediction s t a r t s at C'7(ns ). E x a m p l e s a r e event s h a p e d i s t r i b u t i o n s such a s T h r u s t .
I l l l l l I II I VAHIAHI.ES
¡el m a s s e s , jet. b r o a d e n i n g * or t h e d i f f e r e n t i a l two-jet r a t e , which h a v e been discussed a l r e a d y in d e t a i l in p r e v i o u s c h a p t e r s . For a g e n e r a l event, s h a p e d i s t r i b u tion I/. which v a n i s h e s in t h e limit of perfect two-jet t o p o l o g i e s , t h e d i f f e r e n t i a l cross s e c t i o n c a n b e w r i t t e n a s :
A 2 f|.n«l I d?/ = ° « < / ' M ( I / ) + " " ( > ' ) (\ « ( • ' / ) + M
—
M
—
S) ) +
•
(»«•«)
Here
C f W —^ 1-
_ " s ( » ) { ,1 ''I " „ ( * ) .1 1 1 \ = "s(/' ) = - 7 "' w \ bfj w )
Using t h e a b o v e d e f i n i t i o n s of f A . f r a n d u/. of t h e QC'D ^ - f u n c t i o n a r e
'<> -
- \"llr
a n d fc, =
« ' = 1 -t-i>0«s(••>') 111
$
.
(10.7) t h e first, t w o coefficients /i() a n d l>\
^ J A + 1) " f f r •
(HI.S)
Not«' t h a t w h e n t a l k i n g about, "colour f a c t o r s ' in t h e following, we will a c t u a l l y m e a n t h e c o l o u r f a c t o r r a t i o s f A a n d /•/•. T h e coefficient f u n c t i o n s A(y) a n d B(y) a r e o b t a i n e d by i n t e g r a t i n g t h e fully d i f f e r e n t i a l Ellis Ross T e r r a i n ) ( E R T ) m a t r i x e l e m e n t s (Ellis c.t
= B,,(y)
+ fABA(y)
f nff
T
B
T
(y) .
(10.9)
T h e funct ion B/.-(y) g e t s c o n t r i b u t i o n s f r o m d o u b l e g l u o n breinsst r a h l u i i g . which o c c u r s at a r a t e p r o p o r t i o n a l t o Of.-, t h e f u n c t i o n BA(y) a c c o u n t s m a i n l y for processes w i t h t h e t r i p l e - g l u o n c o u p l i n g a n d Br('.l) for p r o c e s s e s w i t h g l u o n splitting into quark pairs. F i g u r e 10.2 s h o w s t y p i c a l F c y i n n a n g r a p h s t o be t a k e n i n t o a c c o u n t for t h e c a l c u l a t i o n of t h e c r o s s sect ion of a three-jet. v a r i a b l e . T h e u p p e r left g r a p h is o n e of I he t w o d i a g r a m s ( t h e o t h e r o n e is s i m p l y o b t a i n e d by at t a c l l i u g t h e g l u o n t o t h e o t h e r q u a r k line) w h i c h d e t e r m i n e A(y). a n d it is r a t h e r e a s y t o see t h a t t h e o n l y c o l o u r f a c t o r d e p e n d e n c e a t t h i s l e a d i n g o r d e r h a s t o be of t h e f o r m o s O/.'.4(//). Tlii' o t h e r t wo d i a g r a m s in t he u p p e r r o w a r e e x a m p l e s for r a d i a t i v e correct ions t o t h e l e a d i n g o r d e r t e r m . Very s i m p l y s p e a k i n g , t h e s e c o r r e c t i o n s a r e t a k e n i n t o a c c o u n t bv r e p l a c i n g t h e b a r e c o u p l i n g w i t h a r e n o r m a l i z e d r u n n i n g c o u p l i n g , g i v i n g as(/i,i)A(y). However, in a full u e x t - t o - l e a d i n g o r d e r c a l c u l a t i o n a l s o g r a p h s of t he t y p e s h o w n ill t he s e c o n d row a r e c o n s i d e r e d . F r o m t h e last e q u a t i o n s it b e c o m e s c l e a r t h a t i n f o r m a t i o n o n t h e c o l o u r factor r a t i o s e n t e r s o n l y in n e x t - t o - l e a d i n g o r d e r via t h e coefficient f u n c t i o n B(y).
11 s i s m
i ni (¡Ain ri-: s'i u n e i n u i ' . 01 o< *i »
«I
<1 «I
which m e a n s that t h e sensitivity to these ratios will not be very large. However, additional d e p e n d e n c e at O(ii^) a n d higher orders enters t h r o u g h t h e r u n n i n g coupling, mainly via l>{>. if t h e rcnoruialization scale / r is chosen to be different from the hard scale .s (see Ex. 10-1). For event s h a p e variables it has been found (AI.KI'II Collab., 19!)Ire OPAL Collah.. 1992«) t h a t a r a t h e r small scale //'-' « : .s has to be used in order to achieve a good description of the d a t a . In this case missing higher orders, which a p p e a r to be i m p o r t a n t , a r e mimicked by t e r m s generated by the expansion of the r u n n i n g coupling. For several event s h a p e variables it is possible to resuin t he leading and nextto-lcading logarithms In y to all orders in o s . restoring t h e choice //-' « .s as the n a t u r a l one. In those cases a function «ns(/<J) In //) is added to the expression in e<|ii (10.0). T h u s h,, enters again in connection with the leading terms, which introduces ¡i large correlation between the e s t i m a t e s of f , \ a n d /;•. For example, in t h e case of the differential two-jet rate the leading t e r m s in third order are of tin* form (10.10) with L = — In 1/3. Summarizing, it. can be s t a t e d that three-jet variables arc suited for measuring a s ( M y ) a n d a function of f , \ a n d /•;•. namely bo. This is illustrated in Fig. 10.3, which shows t he kind of information that can be expected from a colour factor m e a s u r e m e n t based on three-jet variables compared to analyses based on four-jet variables. While the latter will give a compact confidence region such a s tin 1 ellipse in the ( J A . / r ) - p l a n e . the former will limit I lie possible combinat ions of colour factors only to t h e band corresponding to a narrow r a n g e of values a r o u n d bo. It is worth noting that b,, also contains information on i i f . As a reminder, il is the n u m b e r of fermions which give c o n t r i b u t i o n s to loop corrections and to
I 111(11 II . I VAHIAItl.KS
Itili
Fit;. 1 ().:{. C o n f i d e n c e level c o n t o u r s in t h e ( f . . . n / f i ) plane a s o b t a i n e d in m e a s u r e m e n t s b a s e d 011 t h r e e - a n d f o u r - j e t variables
gluon s p l i t t i n g processes. At M i l ' n / = "> is e x p e c t e d , hut if a d d i t ional fcrinionic degrees of freedom with colour c h a r g e exist, t hen a sizeable elfect on l>o should be observed. As a l r e a d y m e n t i o n e d , a possible c a n d i d a t e for such a d d i t i o n a l fermions would be a very light gluino in t h e m a s s r a n g e G e V , which is predicted bv p a r t i c u l a r siipersyinnictrio extensions of t h e S t a n d a r d Model (Fayet. l!)7(i: F a r r a r . I !)!)">). For e x a m p l e , such a gluino would c o n t r i b u t e a d d i t i o n a l fermion loops a s d e p i c t e d in Fig. 10.2. A first colour f a c t o r a n a l y s i s of three-jet variables has been p e r f o r m e d by t h e A LEI-MI c o l l a b o r a t i o n (1908«), based on a b o u t 111)000 three-jet e v e n t s . A theoretical prediction of t h e form e q n (1().(>) with corrections for h a d r o n i z a t ion effects h a s been lifted t o t h e m e a s u r e d two-dimensional d i s t r i b u t i o n of two linearly independent- c o m b i n a t i o n s of jot. energies. In o r d e r t o got a d d i t i o n a l orthogonal in f o r m a t i o n on t h e colour f a c t o r s a n d t h e s t r o n g c o u p l i n g c o n s t a n t , they also m e a s u r e d t h e two-jot r a t e , which in second o r d e r p e r t u r b a t i o n t h e o r y is t h e c o m p l e m e n t t o t h e t hree-jet a n d four-jet r a t e :
Tlia.l
1 -
(10.11)
Here, t h e sensitivity t o t h e g a u g e s t r u c t u r e c o m e s from tHo 0 ( o j f ) c o n t r i b u t i o n s to t h e three- a n d four-jet r a t e s .
IKSTSOI'
ir.o
r i l K C A H C K S T U U C I UHKOI'' Q C D
1111 [ 111111 • • ' 1 • ' • ' • i • • i ' ..•!•••• 1.4
;
•
•
Q C D = .SU (3)
/
—
/ /
Q massless gluino
/
AI.EPH 2
/
-
jet events
•
i t i / /
'
; •
/
0.8
*
»
k o
<3
-
ix
-- "
0.6 i
i
0.4
OPAL event shapes
-
"
0.2 0 iiiii » i » ' -0.2
pir 1 %
m
I
1
1.25
1.5
1 1
•••
•j - measurement j., ; . , i . . . . I . . . . I "
••'"••'
1.75 eye,
C o l o u r f a c t o r m e a s u r e m e n t s b a s e d o n t h r e e - j e t variables T I . e b a n d ^ x p e c t e d sensii ivity f r o m m e a s u r e m e n t s b a s e d on t h e r u n n i n g
of a • F i g u r e f r o m D i s s e r t o n ( 11)08).
T h e d o m i n a n t e r r o r s o u r c e were h a d r o n i / . a t i o n n n e e r i a i n t i e s a s well a s u n c e r t a i n t i e s b e c a u s e of t h e v a r i a t i o n of t h e rcnoruuilizat ion scale. T h e e x p e r i m e n t a l u n c e r t a i n t i e s w e r e f o u n d t o b e r a t h e r s m a l l . T h e final r e s u l t ( w i t h s t a t i s t i c a l a n d s y s t e m a t i c e r r o r s ) was o , ( M $ ) C r = 0 . 2 1 0 ± 0.016»,,,! ± 0 . 0 4 8 s y s , CA/C,.-
= 4 . 4 9 ± 0.75stn, ± 1.12sysl
Tr/CF
= 2.01 ± 0 . 4 9 s t „ , ± 0 . 8 6 s y s t .
T h i s m e a s u r e m e n t had st ill r a t h e r l a r g e e r r o r s . However, t h e c o n f i d e n c e level for t h e S U ( 3 ) e x p e c t a t i o n w i t h II/ — 5 w a s 5 2 % . w h e r e a s a b e l i a n g a u g e g r o u p s with C'.\ = 0 a n d //•• > 0 were f o u n d t o b e inconsistent w i t h t h e m e a s u r e m e n t , h a v i n g a c o n f i d e n c e level below 2 x 10 ''. It is w o r t h n o t i n g t h a t t h e r e s u l t s w e r e highly c o r r e l a t e d , for e x a m p l e , t h e c o r r e l a t i o n coefficient b e t w e e n t h e fitted values for C \ C'i a n d T . C'i is /> = 0.90. T h e o r i g i n of I his l a r g e eorrolat ion c a n b e t r a c e d back t o t h e fact t h a t t h e m a i n s e n s i t i v i t y is t o l>tl. which is a l i n e a r c o m b i n a t i o n of JA a n d /-/•. T h e result is s h o w n in Fig. 10.1.
I OIIU II I VAKIAIIIKS
.'{51
lu a n a n a l y s i s by t h e O P A L c o l l a b o r a t i o n (19956) t h e event s h a p e d i s t r i b u tions for Thrust., h e a v y jet m a s s a n d t h e t o t a l a n d wide jet h r o a d c u i n g s have been s t u d i e d . For t h e s e v a r i a b l e s n o t only t h e full n e x t - t o - l e a d i n g o r d e r predictions a r e k n o w n , but a l s o t h e r e s u n n n a t i o n of all l e a d i n g a n d u e x t - t o - l e a d i n g l o g a r i t h m s . Very similarly t o t h e t e c h n i q u e s of t h e « „ - m e a s u r e m e n t s f r o m event liape variables, a s d e s c r i b e d in C h a p t e r 8. t h e y e m p l o y e d t h e s e variables for a Miiinltaneous lit of I he s t r o n g c o u p l i n g c o n s t a n t a n d o n e of t he colour f a c t o r s at .1 time, fixing t h e o t h e r s t o t h e Q C D e x p e c t a t i o n . A fit w i t h all c o l o u r f a c t o r s lis free p a r a m e t e r s did n o t c o n v e r g e . A g a i n , t h i s c a n b e u n d e r s t o o d a s a conseq u e n c e of t h e fact t h a t s e n s i t i v i t y is m a i n l y t o />„. T h e results, with s t a t i s t i c a l and s y s t e m a t i c e r r o r s , were = 3.2 ± O.lstat ± 0 . 5 s y s ,
With
O s ( A / | ) = 0.112 ± 0 . 0 1 2
C,- = 1.2 ± 0 . 0 s t „ t ± ().:$SVs.
with
o s ( A / 2 ) = 0.122 ± 0.019
n f = 4.3 ± 0 . 3 s , n , ± 3 . 0 s y s t
wit h
a s ( M ' i ) = 0.1 13 ± 0 . 0 1 2 .
c,\
I'lie result for IIf h a d been o b t a i n e d b y fitting (iijT ), a n d t h e n fixing T = 1 / 2 . Of c o u r s e t he result c a n a l s o b e i n t e r p r e t e d a s a fit. for 7V by fixing II J - 5. T h e major contributions to the systematic error a r e hadronization and renoriualIzation scale u n c e r t a i n t i e s . T h e r e s u l t s a r e displayed in Fig. 10.4. T h e c o l o u r factors f o u n d in t h i s a n a l y s i s a r e nicely consistent, w i t h t h e Q C D e x p e c t a t i o n s . However, c o n c e r n i n g t h e m e a s u r e m e n t of / / / . t h e e r r o r s were still t o o large t o strongly c o n f i r m or rule o u t t h e light g h u n o s c e n a r i o . 10.2
Four-jet
variables
()no of t h e o b v i o u s m o t i v a t i o n s for m e a s u r i n g t he colour f a c t o r s is t o test t h e nonabelian n a t u r e of t h e g a u g e g r o u p e m p l o y e d t o c o n s t r u c t t h e t h e o r y of s t r o n g i n t e r a c t i o n s. A c o n s e q u e n c e of t h i s n o n - a b e l i a n n a t u r e is t h e d i r e c t c o u p l i n g between g l u o n s , iu c o n t r a s t t o Q F D . w h e r e p h o t o n s d o not c o u p l e d i r e c t l y to each o t h e r . So t h e goal s h o u l d be t o look for e v i d e n c e of s u c h a g l u o n g l u o n coupling. W h e n looking at t h e F e y n m a i i d i a g r a m s of Fig. 10.2 we see t h a t iu e + e a n n i h i l a t i o n s such a g l u o n g l u o n i n t e r a c t i o n will o n l y c o n t r i b u t e t o final s t a t e s w i t h a t least, four p a r t o n s . Of c o u r s e , a s d i scu ssed in t h e p r e v i o u s c h a p t e r , there a r e less d i r e c t effects of t h e triplc-gluon vertex on t h e r u n n i n g of o^ a n d on t h e two- a n d three-jet cross sections, if t h e f o u r p a r t o n s of t h e final s t a t e a r e clustered i n t o a t h r e e - p a r t i c l e final s t a t e by s o m e j e t - c l u s t e r i n g a l g o r i t h m . But. it is clear t h a t w h e n looking at four-jet e v e n t s , we s h o u l d b e a b l e t o resolve m o r e directly t h e gluon gluon i n t e r a c t i o n . So t h e s t e p s t o p r o c e e d a r e t h e following: we h a v e t o c l u s t e r t h e event t o f o u r jets, d e f i n e d by s o m e resolution p a r a m e t e r i/, 1 ^, c o n s t r u c t a n o b s e r v a b l e X o u t of t h e f o u r j e t - m o m e n t a , c o m p u t e t he pert.urbat ive c r o s s section for t his o b s e r v a b l e as a f u n c t i o n of t he colour f a c t o r rat ios /.., a n d /•/•. correct for h a d r o n i z a t i o n a n d d e t e c t o r effects, a n d finally lit this c o r r e c t e d p r e d i c t i o n t o t h e d a t a by v a r y i n g tlie colour f a c t o r r a t i o s .
i i s i s < >i
nn
CAUCI-: s i K i K
i inn
01
IJCH
T h e pert u r h a t i v e predict ion lor t he dilferent in I cross seel ion in I his observable A . which we will generally call four-jet mriubic. is given by
— ^ Tin,.i
= < ( D < < X ) + C7 T - D A ( X ) + h j J f D r W ) F CrV
+
(10.12)
I he f u n c t i o n s D,,(X). t = F. .T a r e o b t a i n e d by i n t e g r a t i n g t h e fully «lillerent ial E R T m a t r i x e l e m e n t s c o m p u t e d front d i a g r a m s such as depicted in the second row of Fig. 10.2. T h e f u n c t i o n D .-(X) gets its c o n t r i b u t i o n s f r o m double g l u o u - h r c m s s t r a h h m g . D,\(X) f r o m g r a p h s c o n t a i n i n g t h e t-riple-glnon vei lex a n d Dr(X) f r o m g r a p h s with a gluon s p l i t t i n g i n t o q u a r k s . At t h e t i m e when t h e i n e a s u r e m e n t s of four-jet variables wore p e r f o r m e d , only t h e leading o r d e r expression oqn (10.12) was known. Only recently t he n e x t - t o - l e a d i n g ordei O(II^) c o n t r i b u t i o n s have been fully calculated (Dixon a n d Signer 1997: Nagy a n d Tri'x'sanvi 1997. 1999. a n d references t h e r e i n ) , a n d re-analyses of t h e I.KIM d a t a sol using t hose improved calculat ions have j u s t been c o m p l e t e d ( B r a v o 2001; OI'AI. Collab. 2001). T h e lack of knowledge of t h e higher o r d e r c o r r e c t i o n s c a u s e s t he total four-jet cross section t o d e p e n d very strongly on t h e renorinalizat ion scale //-. which inI induces a large theoretical uncertainty. (>t herwise. t his observable would be well suited for testing t h e t t iple-gluon vortex, since in an abelian t heory wit h C..\ 0 wo expect fewer four-jet events t h a n in Q C D . a s follows from e
!•'<)IH( Il f VAHIAHI.KS
(IN -
• AI.ITII ilala — QCDSUtf) - - alvlian model
0.07
yj* y1
O.IK,
7/
/ /
0.05
*
Ç
0.01 O.O.I 0.02
• ALIiPH data — QCI>SU<3) - - ahclian imxlcl
0.01 o o.i 0.2 0.3 o.4 o.5 o.(> 0.7 0.8 o.v
I -O.S -0.6-0.4 -0.2 0
0.2 0.4 0.6 O.S
COS 3TW
I COS/,,/1
F l o . 10..r>. P r e d i c t e d a n d m e a s u r e d dist ribul ions of I lie B e n g t s s o n Z e r w a s a n g l e (a) a n d t h e a n g l e b e t w e e n t h e t w o lowest e n e r g e t i c j e t s o : i | (h) f r o m four-jet e v e n t s . T h e d a t a a r e c o r r e c t e d for hadronizat.ion a n d d e t e c t o r effects, t h e e r r o r s a r e s t a t i s t i c a l onlv.
variable called Bengtsson 1989),
Zerwas
angle ( B e n g t s s o n a n d Z e r w a s , 1988; B c u g t s s o n .
Vnz = ¿[(P. x p,)•(/>, X P4)l = | ! 7 J | X P j ! M } P : ^ P ' i | .
(10.13)
| ( P i x P 2 ) l MPs x p . , ) |
w h e r e ; > , . / = 1, -J a r e t h e e n e r g y - o r d e r e d (£", > E> > E-,\ > E.\) m o m e n t a of the four p a r t o n s (jets). In Fig. 10.5(a) we see t h e p r e d i c t e d d i s t r i b u t i o n s for t h e B e n g t s s o n Z e r w a s a n g l e for a n a b e l i a n m o d e l a n d for Q C D . The a b e l i a n m o d e l s h o w s s o m e enh a n c e m e n t a r o u n d 90° w i t h respect t o t h e n o n - a h c l i a n p r e d i c t i o n . T h e d a t a were o b t a i n e d w i t h t h e AI.KI'II d e t e c t o r a n d c o r r e c t e d for h a d r o n i z a t . i o n a n d d e t e c t o r effects. T h e y c l e a r l y f a v o u r t h e Q C D p r e d i c t i o n o v e r t h e a b e l i a n m o d e l . Next we d e s c r i b e a s e c o n d test, v a r i a b l e , t h e ( g e n e r a l i z e d ) Naclitmanu Reiter angle. ( N a c l i t m a n u a n d R e i t e r . 1982). w h i c h is b a s e d oil t h e following c o n s i d e r a t i o n s . A g a i n we n a m e t h e m o m e n t a of t h e p r i m a r y p a r t o n s a s pt a n d p.,. a n d let. p.j a n d p , be t h e m o m e n t a of t h e s e c o n d a r y p a r t o n s , o r i g i n a t i n g f r o m t h e d e c a y of a n i n t e r m e d i a t e g l u o n . N o t e t h a t w e d o not c o n s i d e r d o u b l e gliion radia t i o n for t h e m o m e n t . T h e n we c h o o s e t h e p a r t i c u l a r p h a s e - s p a c e c o n f i g u r a t ion l>\ I P> P:i + 1>i = 0 w i t h Ei = E> 3> Ex - E.|. T h u s , t h e t wo h i g h e r - e n e r g e t i c jets a r c b a c k - t o - b a c k , a n d so a r c t h e t wo l o w e r - e n e r g e t i c ones. In t h i s c o n f i g u r a t i o n t h e i n t e r m e d i a t e g l u o n is a t rest a n d h a s zero hclieity w . r . t . t h e d i r e c t i o n ol t h e p r i m a r y p a r t o n s , a n d h e n c e hclieity c o m p o n e n t s i l in a n y o r t h o g o n a l d i r e c t i o n . S t r i c t l y s p e a k i n g t h i s is o n l y e x a c t l y t r u e if t h e t w o p r i m a r y p a r t o n s a r e r a d i a t e d a l o n g t h e d i r e c t i o n of t h e i n c o m i n g e + e~ p a i r . H o w e v e r , it is slill t r u e t o l e a d i n g o r d e r iu a n e x p a n s i o n in E-.\/E\ ( N a c h t m a i m a n d R e i t e r . 1982).
I KS I s O K M I I OAUCJI; s I u I U 111 U IV o r CI« "I»
:IM
^ 8
A
t g o KD o P o o
q
'
—*
—*
r> <* q q
q
O O to g t g
A
for lidden
all« »wed
^
q
Kit:. 10.(i. Four-jet c o n f i g u r a t i o n s for a p a r t i c u l a r p h a s e - s p a c e point as explained in the t e x t . T h e a r r o w s i n d i c a t e t h e direction of t h e m o m e n t a a n d spins.
T h e consequences of such a configuration a r e illustrated in Fig. 10.6. T h e primary (secondary) p a r t o n s a r e d r a w n horizontally (vertically). If t h e pair of secondary partons is r a d i a t e d at 5>110 a s shown in t h e figure, ii h a s t o c a r r y lidicity ± I. On the l e f t - h a n d side t h e i n t e r m e d i a t e gluon d e c a y s into two f u r t h e r villous. Since these a r e massless bosons, they c a r r y helicity ± 1 each, a n d their total helicity has to be 0 or ± 2 . T h e r e f o r e , t h e process as d r a w n in Fig. 10.0 is forbidden, whereas il would be allowed if t h e s e c o n d a r y p a r t o n s were r a d i a t e d along t h e direction of t h e p r i m a r y pair. On t h e r i g h t - h a n d side, t h e gluon splits into a qq-pair. and t h e total helicity of this s y s t e m of two s p i n - 1 / 2 particles can be :1 1. so this c o n f i g u r a t i o n is allowed. On t h e o t h e r h a n d , t h e decay of the secondary quarks parallel t o t h e p r i m a r y o n e s would be s u p p r e s s e d . Since iu Q C D we have more gluon d e c a y s i n t o t w o gluons r a t h e r t h a n into a qq-pair, we expect a smaller relative r a t e of events of t h e c o n f i g u r a t i o n s h o w n iu Fig. II).(i than in a theory without t h e triple-gluon vertex. Now, we can define t h e N a c h t i n a n n Reiter a n g l e l>su a s >S
=
i
P:il
•
»12 ~ <>m « 1S0° ,
(10.14)
which is sensitive t o t h e relative c o n t r i b u t i o n s from q q g g a n d q q q q e v e n t s to I he discussed four-jet c onfigurations. However, il is clear t h a t t h e requirement I)|.j « 0:\\ % 180° restri c t s t h e p h a s e s p a c e t o o m u c h , a n d e x p e r i m e n t a l l y we would end up with an ext remely small n u m b e r of event s. T h e r e f o r e a t cncnilizcd Nacht.niann Reiter a n g l e h a s been p r o p o s e d ( Betlike ct «/.. l!)i)l). <>h\
A ( P i - Pj)AP:i ~ Pi))
0° < 0' S H < !)0°
(10.15)
K U ' I i II I VA II IA III,KS
wliicli c o n t a i n s W^u as a special case. So. we h a v e t o m e a s u r e t h e a n g l e b e t w e e n t h e «inference of m o m e n t a «»I t h e t w o h i g h e r e n e r g e t i c j e t s a n d t h e t w o s o f t e r ones. At t h i s p o i n t we h a v e t o pul a w a r n i n g . In t h e d i s c u s s i o n of t h e B e n g t s s o n '/.et was a n d t he Nacht i n a n n R e i t e r a n g l e w e d i d n o t c o n s i d e r t he c o n t r i b u t i o n s Irom d o u b l e g l u o n b r e i n s s t r a h l u n g , which lead t o t h e s a m e linal s t a t e a s e v e n t s with a g —> gg s p l i t t i n g . It could h a p p e n that, o t h e r h y p o t h e t i c a l g a u g e g r o u p s lead t o d i s t r i b u t i o n s v e r y s i m i l a r t o t h e QC'D p r e d i c t i o n . S o we c o n c l u d e t h a t t hese t wo a n g l e s a r e r a t h e r sensit ive t o t h e r e l a t i v e c o u t r i h u t ion f r o m q q q q e v e n t s , b u t in o r d e r t o o b t a i n c l e a r e v i d e n c e for t h e t r i p l e - g l u o n v e r t e x we h a v e t o lind additional discriminating variables. As a n e x a m p l e for s u c h a n a l t e r n a t ive v a r i a b l e we p r e s e n t t he Körner Schierholz Willrodt nnnle ' I ' K S W ( K ö r n e r et
A(l>\
* PI).(P2
X
P.I)1
.
<
'I'KSW <
180° .
(10.10)
In t h e o r i e s without, t h e t r i p l e - g l u o n v e r t e x t h e p l a n e s o r t h o g o n a l t o t h e v e c t o r s p , x p i a n d p., x ])A a r e u n c o r r e l a t e d . a n d b e c a u s e of p h a s e - s p a c e r e s t r i c t i o n s t h e a n g l e 'I'KSW b e t w e e n t h e s e t w o p l a n e s is f o u n d p r e f e r e n t i a l l y a r o u n d 9 0 ° . However. if t h e r e is a t r i p l e - g l u o n v e r t e x , t h e n t h e p o l e s t r u c t u r e of t he p r o p a g a t o r for t h e i n t e r m e d i a t e gluon leads t o a p r e f e r e n c e for s m a l l a n g l e s b e t w e e n t h e t w o s e c o n d a r y g l u o n s . a n d a c o r r e l a t i o n b e t w e e n t h e p l a n e s is i n d u c e d . B e c a u s e of t he e n e r g y o r d e r i n g , t h e p l a n e s t u r n o u t t o b e a n t i p a r a l l e l m o s t of t h e t i m e . A d e t a i l e d d e r i v a t i o n c a n b e f o u n d iu t h e p a p e r by K ö r n e r et
1/2
{ ¿ [ ( p . x p , ) , ( p . , x p.,)] + / [ ( / > , x pA).(p->
x p.,)]} .
(10.17)
f i n a l l y , a simplified version of t h e K ö r n e r S c h i e r h o l / W i l l r o d t a n g l e is o b t a i n e d by l o o k i n g at t h e a n g l e b e t w e e n t h e t w o lowest e n e r g e t i c j e t s . So we a r r i v e at t h e d e f i n i t i o n of a f o u r t h a n g u l a r v a r i a b l e ( D E L P H I C'ollab.. 199.U;), O:u = 4 P . V P , ]
.
0 ° < O:,., < 180° .
(10.18)
In a n a l o g y t o «I'KSW- t h e a n g l e o;¡.| d i s t i n g u i s h e s b e t w e e n t h e r e l a t i v e c o n t r i b u lions f r o m d o u b l e g l u o n r a d i a t i o n p r o c e s s e s a n d g l u o n s p l i t t i n g i n t o g l u o n p a i r s . G l u o n r a d i a t i o n f r o m t h e t w o p r i m a r y q u a r k s o c c u r s m o r e o r less i n d e p e n d e n t l y , a n d b e c a u s e of t h e e o l l i n e a r c h a r a c t e r of b r e i n s s t r a h l u n g a n d t h e e n e r g y o r d e r ing of t h e four j e t s we e x p e c t r a t h e r large a n g l e s b e t w e e n t h e s e c o n d a r y p a r t o n s . G l u o n s p l i t t i n g i n t o s e c o n d a r y p a r t o n s on t h e o t h e r h a n d will lead t o r a t h e r small o p e n i n g a n g l e s . T h e d a t a a s s h o w n in Fig. I()..r>(I>) a r e a g a i n in v e r y g o o d a g r e e m e n t w i t h t h e CJCD p r e d i c t i o n a n d d i s f a v o u r a n a b e l i a n m o d e l .
I KS'IS (il
I III <¡AUOT. S I HUC I H U E (il Q< I •
For I li<'so last t wo a n g u l a r (list ribut ions wo d i d not c o n s i d e r t h e c o n t r i b u l ions f r o m q<|<|<| e v e n t s , wliicli might lead t o a r e d u c e d d i s t i n c t i o n b e t w e e n Q C D a n d o t h e r t h e o r i e s . W e c o n c l u d e that, a n e x p e r i m e n t s h o u l d m e a s u r e not o n l y o n e ol t h e a n g u l a r v a r i a b l e s d e s c r i b e d a b o v e , b u t r a t h e r a s e n s i b l e c o m b i n a t i o n of s o m e of tliein. in o r d e r t o o b t a i n g o o d s e n s i t i v i t y t o all k i n d s of possible processes. I n d e e d , t he v a r i o u s e x p e r i m e n t s h a v e e m p l o y e d d i f f e r e n t s e t s of v a r i a b l e s a n d c o m b i n a t i o n t e c h n i q u e s . T h e D E L P H I c o l l a b o r a t i o n (19936) h a s p e r f o r m e d a b i n n e d l e a s t - s q u a r e s lit o f e q n (111.12) t o t h e t w o - d i m e n s i o n a l d i s t r i b u t i o n in t h e v a r i a b l e s 0' S H a n d n.j.| in o r d e r t o find est i m a t e s of t h e c o l o u r f a c t o r rat ios. T h e s e t wo v a r i a b l e s a r e s e n s i t i v e t o d i f f e r e n t t y p e s of g r a p h s a s we h a v e l e a r n e d a b o v e , a n d lilt iug d i r e c t l y t h e t w o - d i m e n s i o n a l dist ribut ion t a k e s i n t o a c c o u n t t he c o r r e lation b e t w e e n t h e m . A s i m i l a r t e c h n i q u e w a s a p p l i e d by t h e O P A L c o l l a b o r a t i o n (199/ir). However, t h e r e a t h r e e - d i m e n s i o n a l d i s t r i b u t i o n w a s m e a s u r e d by using also t h e a n g l e \ i r / . L a t e r it h a s b e e n s h o w n by t h e D E L P H I c o l l a b o r a t i o n (19976) t h a t t h e s e n s i t i v i t y of t h e s e a n g u l a r d i s t r i b u t i o n s t o t h e c o l o u r f a c t o r s c a n be f u r t h e r i m p r o v e d bv t a g g i n g t w o of t h e four j e t s a s o r i g i n a t i n g f r o m h or e q u a r k s . T h e i r m e t h o d gives a n efficiency of a b o u t 12% t o t a g b o t h p r i m a r y j e t s c o r r e c t l y a n d a p u r i t y of 7 0 % . w h e r e a s w i t h e n e r g y o r d e r i n g only iu 12% of all e v e n t s d o t h e t w o most e n e r g e t i c j e t s o r i g i n a t e f r o m t h e p r i m a r y q u a r k s . Finally. t h e A L E I ' l l c o l l a b o r a t i o n (19976) h a s m e a s u r e d all f o u r a n g u l a r v a r i a b l e s a n d f i t t e d t h e m s i m u l t a n e o u s l y t o t h e t h e o r e t i c a l p r e d i c t i o n e q n (10.12). T h i s m e a s u r e m e n t will b e d e s c r i b e d in m o r e d e t a i l in t h e n e x t s e c t i o n . G e n e r a l l y , in t h e s e m e a s u r e m e n t s t h e p r e d i c t i o n at p a r t o n level w a s c o r r e c t e d for h n d r o n i z a t i o n e f f e c t s by m e a n s of b i n - b y - b i n c o r r e c t i o n f a c t o r s . Similarly, t h e d a t a w e r e c o r r e c t e d t o c o r r e s p o n d t o a h a d t o n level d i s t r i b u t i o n w i t h o u t a n y d e t e c t o r - s p e c i f i c d i s t o r t i o n s , a g a i n by e m p l o y i n g b i n - b y - b i n c o r r e c t i o n s . B o t h t y p e s of c o r r e c t i o n s w e r e o b t a i n e d f r o m M o n t e C a r l o s i m u l a t i o n s . T h e s t a t istical u n c e r t a i n t i e s o n t h e m e a s u r e d c o l o u r f a c t o r s a r e sizeable, d e s p i t e t h e l a r g e L E P I d a t a s a m p l e s . T h i s is d u e t o t h e r a t h e r s t r o n g p h a s e - s p a c e r e s t r i c t i o n w h e n a s k i n g for f o u r j e t s w i t h a c e r t a i n i/ c „t r e s o l u t i o n p a r a m e t e r , a s well a s d u e t o limited s e n s i t i v i t y t o t h e colour f a c t o r s . T h e l a t t e r is p a r t i c u l a r l y t r u e for T / . / C / ' . Applying a heavy-quark tagging algorithm reduces the sample further. T h e s t a t i s t i c s a v a i l a b l e r a n g e d f r o m a b o u t 10000 four-jet e v e n t s w i t h b - t a g g i n g t o a b o u t 170000 f o u r - j e t e v e n t s in t h e A L E I ' l l a n a l y s i s ( 1 9 9 7 6 ) . T h e d o m i n a n t s y s t e m a t i c e r r o r s a r i s e f r o m u n c e r t a i n t i e s in t h e h a d r o n i z a t i o n c o r r e c t i o n s a n d f r o m e s t i m a t e s of t h e u n k n o w n h i g h e r o r d e r c o n t r i b u t i o n s . In t h e e a r l y d a y s of L E P , t h e A L E I ' l l c o l l a b o r a t i o n ( 1 9 9 2 c ) followed a different a p p r o a c h based on a m a x i m u m likelihood lit of s e l e c t e d four-jet e v e n t s t o t h e t h e o r e t i c a l p r e d i c t i o n for t h e five-fold d i f f e r e n t i a l f o u r - j e t c r o s s s e c t i o n . T h e idea is t h e following. If t h e overall event o r i e n t a t i o n is not m e a s u r e d , t h e n a n event w i t h f o u r niassless p a r t i c l e s iu t h e final s t a t e is c h a r a c t e r i z e d by live i n d e p e n d e n t v a r i a b l e s (see E x . 10-2). for e x a m p l e t h e live scaled invariant m a s s e s y,j in~Js of p a i r s of p a r t o n s i a n d j w i t h i = 1.2: j 2.3,-1: i < ./'. If w e e v a l u a t e t h e m a t r i x e l e m e n t s a s a f u n c t i o n of t h e s e live v a r i a b l e s , we exploit all a v a i l a b l e in-
Ft H 'It II I VAIMA HLKS
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cA/c, F k ; . 10.7. C o l o u r f a c t o r m e a s u r e m e n t s b a s e d o n four-jet v a r i a b l e s . T h e s h a d e d a r e a i n d i c a t e s t h e a v e r a g e of all m e a s u r e m e n t s . F i g u r e f r o m Dissert ori( l!)i>S).
f o r m a t i o n c o n t a i n e d in o u r event, s a m p l e , w h i c h is i m p o r t a n t in c a s e of limited s t a t i s t i c s . In c o n t r a s t , w o r k i n g w i t h t h e a n g u l a r d i s t r i b u t i o n s d i s c u s s e d a b o v e we d o lose s o m e i n f o r m a t i o n , s i n c e t h e y c a n n o t r e p r e s e n t a c o m p l e t e b a s i s in t he l i v e - d i m e n s i o n a l s p a c e of k i n e m a t i c f o u r - j e t v a r i a b l e s . Part, of t h e i n f o r m a t i o n is lost, by i n t e g r a t i n g out s o m e regions of t h e p h a s e s p a c e . In p r a c t i c e , t h e c o l o u r f a c t o r s were d e t e r m i n e d f r o m tlx 1 d a t a by m a x i m i z i n g t h e so-called li elihood function ...
v-*.
n, ( C
Cj••. 7'/.- ',.-)
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with r e s p e c t t o C .\ Cr a n d TI /C'I--. T h e s u m r u n s o v e r all se l e c t e d four-jet e v e n t s . For e a c h e v e n t n , d e n o t e s t h e four-jet c r o s s s e c t i o n , which a g a i n is of t h e f o r m eqn ( 1 0 . 1 2 ) . e v a l u a t e d for t h e p a r t i c u l a r m o m e n t u m c o n f i g u r a t i o n of t his e v e n t . S i n c e n o p a r t o n - t y p e i d e n t i f i c a t i o n w a s p e r f o r m e d , all p e r m u t a t i o n s of p a r t o n - t y p e a s s i g n m e n t s t o t h e f o u r j e t s h a d t o b e c o n s i d e r e d . H e r e <7|,,( is t h e t o t a l c r a s s s e c t i o n , t h e r a t i o
I I SIS ol
I III o M ' ( : i : s i h i k i • u n i
<>i•
' l ' a l i l o IO. I Tallir of restili* of colour factor tunisini minis using fonr-jt i ncuts, ii f = 5 is assumed ever here. 'The first error is statistical, tin second s stematic Measurement A UOPI 1 92 DELPHI 93 DELPHI 97.b-tag O P A L 95 Combined
CA C, 2.24 ± 0 . 3 2 ± 0 . 2 5 2.12 ± 0 . 2 9 ± 0 . 2 0 2.51 ± 0.25 ± 0.1 3 2.11 ± 0.10 ± 0 . 2 8 2.27 ± 0 . 2 7
Correlation Tr C, 0.58 ± 0 . 1 7 ± 0 . 2 1 +0.04 0.40 ± 0 . 1 3 ± 0 . 1 3 -0.30 0 . 3 8 ± 0 . 0 9 ± 0.05 0.0 -0.45 0.40 ± 0 . 1 1 ± 0 . 1 4 -0.10 0.40 ± 0 . 1 2
a n d h a d r o n i z a t i o u elfocts. T h e m e a s u r e m e n t s u l f e r e d f r o m sm a l l s t a t i s t i c s of only a b o u t 1000 e v e n t s . In l a t e r a n a l y s e s a l e a s t - s q u a r e s a p p r o a c h w a s t a k e n i n s t e a d of t h e m a x i m u m likelihood m e t h o d , since for largo d a t a s e t s it is m u c h less t i m e c o n s u m i n g . F u r t h e r m o r e , t h e c o r r e c t i o n p r o c e d u r e in a m a x i m u m likelihood lit is not a s s t r a i g h t f o r w a r d a s in a l e a s t - s q u a r e s a p p r o a c h , w h e r e first t h e d i s t r i b u t i o n s a r e c o r r e c t e d , a n d o n l y t h e n t h e lit is p e r f o r m e d . T a b l e ltl.l s u m m a r i z e s t h e r e s u l t s of t h e v a r i o u s m e a s u r e m e n t s , which a r e m e n t i o n e d a b o v e a n d w h i c h w e r e b a s e d on four-jet v a r i a b l e s only. A g r a p h i c a l r e p r e s e n t a t i o n is given in Fig. 10.7. T h e a v e r a g e h a s been o b t a i n e d u s i n g t h e m e t h o d p r o p o s e d by S c h m o l l i n g (19956. 2000). Excellent a g r e e m e n t w i t h t h e e x p e c t a t i o n f r o m Q C D is f o u n d , o r in o t h e r w o r d s : t h e e x i s t e n c e of t h e triplegliion v e r t e x is c o n f i r m e d e x p e r i m e n t a l l y . O t h e r h y p o t h e s e s such a s a u a b e l i a n m o d e l a r e definitely ruled o u t . T h e m o r e precise m e a s u r e m e n t s a l s o d i s f a v o u r a light g l u i n o h y p o t h e s i s . T h e results f o u n d for t h e c o l o u r f a c t o r s c a n b e t r a n s l a t e d i n t o a m e a s u r e m e n t for t he n u m b e r of g l u o n s Ar..\. o r e q u i v a l e n t ly, t he d i m e n s i o n of t h e a d j o i n t r e p r e s e n t a t i o n of t h e g a u g e g r o u p , by e m p l o y i n g e q n ( 1 0 . 4 ) . = C'r A V / 7 V = 7 . 5 ± 2 . 3 . Here, we have a s s u m e d t h e d i m e n s i o n of t h e f u n d a m e n t a l r e p r e s e n t a t i o n .'V/.' t o b e N r = 3. t h a t is. we a s s u m e t h a t q u a r k s c a r r y t h r e e c o l o u r d e g r e e s of f r e e d o m , which is a well e s t a b l i s h e d a s s u m p t i o n a s e x p l a i n e d in t h e i n t r o d u c t o r y c h a p t e r s . T h e e x p e r i m e n t a l r e s u l t is c o n s i s t e n t with t h e e x p e c t e d n u m b e r of eight c o l o u r d e g r e e s of f r e e d o m for t h e g l u o n s . 10.3
C o m b i n a t i o n of t h r e e - a n d f o u r - j e t v a r i a b l e s
In t h e p r e v i o u s t w o s e c t i o n s we h a v e learned how t h e different, t y p e s of olwerva b l c s a r c d i f f e r e n t l y s e n s i t i v e t o t h e c o l o u r f a c t o r s . In Fig. 10.3 t h e e x p e c t e d s h a p e s of the c o n f i d e n c e level c o n t o u r s h a v e b e e n a n t i c i p a t e d , a n d t h e m e a s u r e m e n t s d e s c r i b e d l a t e r e o u l i r i n e d t he e x p e c t a t i o n s . T h e i n t e r e s t i n g o b s e r v a t i o n is t h a t t h e o v e r l a p of t h e c o n t o u r s for t h r e e - a n d four-jet v a r i a b l e s is r a t h e r s m a l l , t h e r e f o r e a c o m b i n e d a n a l y s i s s h o u l d help t o c o n s t r a i n f u r t h e r t h e allowed region in t h e colo ir f a c t o r p l a n e .
INFORMATION I KOM l lll
RUNNING OK h k
The A1.1-11 * 11 e x p e r i m e n t (19976) lias p e r f o r m e d sncli a c o m b i n e d m e a s u r e In //;* i n c u t . As three-jet v a r i a b l e i lie d i s t r i b u t i o n of I lie event s h a p e v a r i a b l e lias been <'iiiployed, w h e r e i/;i is t h e niiniiniun d i s t a n c e scale // t J , c o m p u t e d no c o r d i n g t o t h e D u r h a m p r e s c r i p t i o n , a f t e r c l u s t e r i n g a n event t o t h r e e jets. I'liis variable is a l s o called d i f f e r e n t i a l two-jet r a t e , a n d t he r e s m n m a t i o n of leading a n d n e x t - t o - l e a d i n g l o g a r i t h m s is a v a i l a b l e for it . a s h a s a l r e a d y been discussed in p r e v i o u s c h a p t e r s . As lbiir-jct variable's all four a n g u l a r d i s t r i b u t i o n s deliued in Section 10.2 h a v e been used, wit h j e t s o r d e r e d in e n e r g y a n d no h e a v y llavoni t a g g i n g a p p l i e d . In t o t a l . 2.7 million h a d r o n i c e v e n t s h a v e been a n a l y s e d , giving a large s a m p l e of a b o u t 170000 four-jet. e v e n t s . A l e a s t - s q u a r e s lit h a s been perf o r m e d s i m u l t a n e o u s l y t o all five o n e - d i m e n s i o n a l d i s t r i b u t i o n s by t a k i n g into a c c o u n t t he c o r r e l a t i o n s . T h e result, is a s ( A / . j ) = 0.02-14 ± 0.0()()3 slill ± 0 . 0 0 0 9 s y s t Ca CF
= 2.20 ± 0.09 s l i l , ± 0 . 1 3 s y s t
T,, Cr
= 0-20 ± 0 . 0 5 s l i l l ± 0 . 0 0 s y s l
wit h a correlat ion of p(C..\ Ci--. T C ) 0.-17. The s y s t e m a t i c u n c e r t a i n t i e s include e o n t r i b n t ions f r o m e s t i m a t e s of u n k n o w n higher o r d e r t e r m s , f r o m liadronization effects a n d biases i n t r o d u c e d by t h e d e t e c t o r s i m u l a t i o n , a n d finally f r o m t h e e s t i m a t i o n of m a s s effects. T h e g r a p h i c a l represent at ion of t h e result is given iu Fig. 10.iS. N o t e t h a t a l s o for this a n a l y s i s m e t h o d new m e a s u r e m e n t s , based on N L O c a l c u l a t i o n s , h a v e been p u b l i s h e d very recently ( B r a v o 2001: O I ' A L C o l l a b . 2001). C o m b i n i n g t h e r e s u l t s a b o v e with t h e m e a s u r e m e n t s f r o m four-jet variables, we o b t a i n C C,.-
= 2.27 ± 0 . 1 8
y y / c y = 0.35 ± 0.09 with a c o r r e l a t i o n of A/CF-.TY C'i--) - 0.25. T h i s also i m p r o v e s t h e precision of t he m e a s u r e m e n t of t he n u m b e r of gliious, N
a
=
r y
N
F
/ T
F
=
8.5 ± 2.2 .
T h e fact that a precise s i m u l t a n e o u s m e a s u r e m e n t of b o t h c o l o u r f a c t o r r a t i o s , b a s e d on jet physics, d o c s not. g o b e y o n d a relative precision of a b o u t 8 % for CA/CF a n d a b o u t 2 0 % for T . C .- is i n d i c a t i v e for t h e difficulties which h a v e t o be faced in t h e s e t y p e s of m e a s u r e m e n t s . T h e y arise f r o m t h e u n c e r t a i n t i e s of t h e t h e o r e t i c a l d e s c r i p t i o n of h a d r o n i c jet p r o d u c t i o n , a s well a s t h e limited s t a t i s t i c a l sensitivity. 10.4
I n f o r m a t i o n f r o m t h e r u n n i n g of o s
T h e Q C D i l - f u n c t i o n . S e c t i o n 3.4. which g o v e r n s t h e r u n n i n g of o s . is k n o w n u p t o t h i r d o r d e r iu o s . It is a l s o a f u n c t i o n of t h e colour f a c t o r s , a n d t h e explicit
I KSTS OK I Hi; OAHOK S I I M K T U R K ()K Q< •!)
I
I
68'* C L contour
95% CL contour SO(3).E8
0.8 0.6
QCI) = Sl)(3)
O
masslcss gluino
SU(2),SI > (2) o
•SOI 4)
U.
*
G2 [
SO(5).K4 .
0.4 F.7
E6
0.2
-•0.2
I
1.25 1.5
1.75
2.25
2.5
2.75
3.25 3.5
CAICF Fie.
1 0 . 8 . R e s u l t s of t h e c o m b i n e d a n a l y s i s of t h r e e - a n d f o u r - j e t v a r i a b l e s
by A L K P I I in t e r m s of t h e c o l o u r f a c t o r s C A / ( ' , . • a n d T F / C F - F i g u r e f r o m A L K P I I C o l l a b . ( 19976).
INFORMA I ION M t O M I II!' Rt ' N N I N O O F o „ 0«) gauge Ilici ii y gauge theory willi gluino 68.3% CI. contours QCI) value 1.8
2
2.2
2.4
2.6
— 68.3% CL — 95.4% CL . - - 99.7% CL . QCI) <>h C./C.
F i g . 1 0 . 9 . C o n f i d e n c e c o n t o u r s for t h e c o l o u r f a c t o r r a t i o s f r o m a s t u d y of t h e r u n n i n g of a s f r o m R, a n d Rr ( l e f t ) . O n t h e right, s i d e a l s o t h e r e s u l t s f r o m f o u r - j e t s t u d i e s a r e i n c l u d e d . Ellipses w i t h t h r e e c o n t o u r s s h o w t h e r e s u l t s when including gluinos. Figures from Csikor and Fodor(1997).
have been c a l c u l a t e d t o t h i r d o r d e r in n s , a n d t h e explicit d e p e n d e n c e o n t h e colour f a c t o r s is k n o w n : c.f. A p p e n d i x D. R
F
T h e idea is t h e following. For a given set of c o l o u r f a c t o r r a t i o s ( C A / C F . / C F ) t h e s t r o n g c o u p l i n g c o n s t a n t « s (eqn 10.7) is d e t e r m i n e d by m i n i m i z i n g =
*
( / y u i a s _ /flhroy1
OHRT)
2L,
(10.20)
T h e t h e o r e t i c a l p r e d i c t i o n s a r e f u n c t i o n s of t h e st r o n g c o u p l i n g c o n s t a n t a n d t h e colour f a c t o r s , = ltf™0{U&,CA/CF.TF/CF). T h e s u m g o e s o v e r measurem e n t s R " " '^ a t v a r i o u s e n e r g y scales, w h e r e of c o u r s e
.Ili*
I r.s i is « u
ini
a i u . i ' ^ i mi i i u n i ' i u i;i i )
Information from jet fragmentation
lu previous chapters, we have learned that h a d r o n production can he described by a so-called p a r t o u shower, which is a chain of successive b r e m s s t r a h l u n g processes. followed by h a d r o n f o r m a t i o n which cannot be described pert.urbat ively. Since hreinsstrahluug is directly proportional t o the coupling of the r adiated gluon to the r a d i a t o r , which can be a q u a r k or a n o t h e r gliion. we expect the ratio of gluon multiplicities from a gluon a n d a quark source to be equal to the ratio of the colour factors, C . \ ¡ C y = 9 / 4 , in the limit of very large energies where phase-space a n d non-pert urbat ive effects become negligible. Of course the radiated gluoiis give rise to t h e production of h a d r o n s . so t h e increased radiation from gluoiis should be reflected in a higher h a d r o n multiplicity iu gluon induced jets and also in a stronger scaling violation of the gluon fragmentation function. T h e shortcomings of a measurement based on t h e above idea are the following: At finite energies there might be pert urbat ive a s well as n o n - p e r t u r b a t i v e effects which cause the ratio of hadron multiplicities to be different from C,\/C[--. Another important aspect is the scale at which quark and gluon j e t s a r e compared to each o t h e r . As it t u r n s o u t . simply taking the jet energy is not a good choice. Considerations based on calculations of colour coherence rather suggest to use a transverse-momentuin-likc scale, such as (10.21)
Q = /?,.., s i n ( 0 / 2 ) .
where /ij,., is t he jet energy a n d 0 the angle to the closest jet. Using such a scale, the hadron multiplicities of q u a r k and gluon j e t s can be writ ten as (Kt)(Q)
= K
(1\HQ)
= N" + Nvr(Q)
+ AMG)
(10.22)
Here N " ^ are n o n - p e r t u r b a t i v e t e r m s introduced to account for the differences in the transition to h a d r o n s for q u a r k s a n d gluoiis. When measuring the ratio of the derivatives of the above expression with respect, to the scale, these t e r m s d o not contribute. T h e p e r t u r b a t i v e prediction AY i for hadron multiplicities in quark j e t s has been calculated within the framework of the modified leading logarithmic approximation (see. for example, Ellis <1
Hvr(Q)
= K (os(Q))b
exp ( \Vn*(Q)J
) [l + O i ^ ) ]
(10.23)
I IMIT.H ON N E W P H Y S I C S
:«i.'<
I Iii• r a t i o r { Q ) lias liri'n c a l c u l a t e d by C n l f n o y a n d M u e l l e r (1985), a n d it is d i r e c t l y p r o p o r t i o n a l t o C,.\/C'rThe e x p e r i m e n t a l dillieulty lies in tlic need t o (list higuish b e t w e e n q u a r k a n d g l u o n j e t s . T l i c simplest nict.liod relies on t h e fact that, in three-jet. e v e n t s t h e I'.luon jet is p r e d o m i n a n t ly t he lowest e n e r g y jet., which is s i m p l y a c o n s e q u e n c e of t h e \ / E b e h a v i o u r of t h e b r e m s s t r a h l u n g s p e c t r u m . T h e r e f o r e , we h a v e t o select t h r e e - j e t e v e n t s , o r d e r t h e t h r e e j e t s in energy, classify t he lowest e n e r g e t i c jet a s t h e g l u o n j e t . a n d finally correct, t h e m e a s u r e m e n t , for t h e i m p u r i t i e s , which o b v i o u s l y a r i s e f r o m e v e n t s w h e r e t h e t h i r d jet in e n e r g y is not i n d u c e d by a g l u o n . A n i m p r o v e m e n t of t h i s s i m p l e m e t h o d is o b t a i n e d by l o o k i n g for b - h a d r o n s in t h e j e t s . If t w o out of t h e t h r e e jets a r e identified a s o r i g i n a t i n g f r o m a b - q u a r k , t h e t h i r d o n e h a s t o b e a g l u o n jet by d e f i n i t i o n . However, this m e t h o d s u l f e r s f r o m a s t r o n g r e d u c t i o n in s t a t i s t i c s . T h e D E L P H I e x p e r i m e n t h a s c a r r i e d o u t a m e a s u r e m e n t , of C,\/Cp (19996), based 011 t h e ideas o u t l i n e d a b o v e . T h e y i n d e e d o b s e r v e a c l e a r e x c e s s of t h e h a d r o u m u l t i p l i c i t y in g l u o n jets w i t h r e s p e c t t o q u a r k j e t s , f r o m w h i c h tliev extract CAJCF
= 2 . 2 4 6 ± 0 . 0 0 2 s t a t ± <>.080 svs , ± 0.095,1,
which is of t h e s a m e precision a s t h e most a c c u r a t e m e a s u r e m e n t s b a s e d o n three- and four-jet variables. 10.(»
L i m i t s 011 n e w
physics
In t h e p r e v i o u s s e c t i o n s we h a v e a l r e a d y s t a t e d several t i m e s that, t h e v a r i o u s t e s t s of t h e c o l o u r f a c t o r s all g a v e r e s u l t s c o n s i s t e n t wit h t h e e x p e c t a t i o n s f r o m S U ( 3 ) , a n d t h a t o t h e r h y p o t h e s e s such a s a n a b c l i a n m o d e l without, a g l u o n s e l f - i n t e r a c t i o n a r e ruled o u t . A n o t h e r e x t e n s i o n b e y o n d s t a n d a r d Q C D . which has c a u s e d m u c h m o r e t h e o r e t i c a l interest t h a n p o s s i b l e d e v i a t i o n s f r o m S U ( 3 ) , is th<- s t r o n g i n t e r a c t i o n s e c t o r of s u p e r s y m m c t r y . A s u b c l a s s of s u p e r s y m m e t r i c m o d e l s p r e d i c t s a g l u i n o g. t h e s u p c r s y i n i n e t . r i c ferrnioiiic p a r t n e r of t h e g l u o n . with m a s s e s a r o u n d a few CieV. T h e e x i s t e n c e of s u c h a g l u i n o s h o u l d m a n i f e s t itself in t h e m e a s u r e m e n t s d e s c r i b e d ¡11 t h i s c h a p t e r , since it would c o n t r i b u t e t o loop c o r r e c t i o n s a n d t h u s a l t e r t h e r u n n i n g of t h e s t r o n g c o u p l i n g c o n s t a n t , a n d it. c o u l d lead t o final s t a t e s of t he t y p e q q g g via t h e s p l i t t i n g of a n i n t e r m e d i a t e g l u o n i n t o a g l u i n o p a i r . P o s s i b l e now b o u n d s t a t e s c o n t a i n i n g g l u i n o s h a v e not boon s e a r c h e d for in t h e t h r e e - a n d four-jet. m e a s u r e m e n t s at L K P . In S e c t i o n s 10.1 a n d 10.2 we h a d a l r e a d y a n t i c i p a t e d t h a t t h e g l u i n o h y p o t h esis is d i s f a v o u r e d by t h e m e a s u r e m e n t s of t h e c o l o u r f a c t o r s . In loading o r d e r t h e c o n t r i b u t i o n of a m a s s l e s s g l u i n o c a n b o p a r a m e t r i z e d b y c h a n g i n g t h e exp e c t a t i o n for 7 > / C > f r o m 0 . 3 7 5 t o 0.0. l e a v i n g C..\/C'r u n c h a n g e d . T h i s point in t h e c o l o u r f a c t o r p l a n e lies o u t s i d e t h e c o n f i d e n c e c o n t o u r s of t h e m o s t p r e c i s e measurements. T h e A L E P I I c o l l a b o r a t i o n h a s used its c o m b i n e d a n a l y s i s of t h r e e - a n d f o u r jot v a r i a b l e s for a m o r e d e t a i l » ) s t u d y of t h e g l u i n o h y p o t hesis. Inst e a d of a s s u m ing live ferinionic d e g r e e s of f r e e d o m a n d f i t t i n g for t h e c o l o u r f a c t o r s a n d t h e
.'Mil
IKSTSOl'
Mil (1AIICK STI
s t r u n g c o u p l i n g c o n s t a n t , tlic a r g u m e n t was t u r n e d a r o u n d by a s s u m i n g Sl)(U) t o b e t h e correct g a u g e g r o u p , which lixes t h e values of t h e c o l o u r f a c t o r s , a n d l i l t i n g for o s a n d n f . T h e r e s u l t s w e r e i v s ( M | ) = 0 . 1 1 0 2 ± 0 . 0 0 1 2 s l a , ±().l)(IIO H y - l and nf 1.21 ± 0 . 2 y s , a t ± l.t- r > s y s l . T h e ineasurenient. of Ot»(M$) is in a g r e e m e n t w i t h t h e world a v e r a g e , a n d t h e result for n / is c o n s i s t e n t w i t h t h e e x p e c t a t i o n of live. At l e a d i n g o r d e r a m a s s l e s s g l u i n o w o u l d lead t o a n e x c e s s a b o v e live ol t h r e e u n i t s . I lowcver. m a s s e f f e c t s c a n lower t his excess. F r o m t his m e a s u r e m e n t . A LEI 'I I c o m p u t e d a n u p p e r limit on t he e x c e s s of AN/ < 1.0 at !)-r>% c o n f i d e n c e level, f r o m which t h e y d e d u c e d a lower limit o n t h e g l u i n o m a s s of in,-, > 6 . 3 G c V A c a v e a t in t h e ALKIMI result w a s t h a t t h e g l u i n o e x c l u s i o n limit w a s b a s e d oil a r g u m e n t s valid in l e a d i n g o r d e r only. T h i s c o u l d b e o v e r c o m e by t he a n a l y s i s of Csikor a n d F o d o r (1!)!)7). b e c a u s e by t h e n t he t h r e e - l o o p calculat ions for t h e .'¿-function a s well as for /i; a n d / ? r . fully i n c l u d i n g g l u i n o c o n t r i b u t i o n s , h a d b e e n c o m p l e t e d . So t h e y c o u l d r e p e a t t h e i r a n a l y s i s of t h e c o l o u r f a c t o r s by a s s u m i n g t h e e x i s t e n c e of t h e g l u i n o . which l e a d s t o dilferent c o n f i d e n c e c o n t o u r s iu t h e c o l o u r f a c t o r p l a n e , a s s h o w n in Fig. 10.!). Still a g r e e m e n t w i t h S U ( 3 ) would b e e x p e c t e d , but t h e r e s u l t s s h o w t h a t t h e S U ( 3 ) p o i n t is not covered by t h e c o n f i d e n c e region a n y m o r e , r u l i n g out t h e SU(.'i) I g l u i n o h y p o t h e s i s . A more quantitative analysis showed t h a t , when combining their analysis with t h e f i n d i n g s f r o m four-jet s t u d i e s , t h e y c a n rule out g l u i n o s w i t h m a s s e s below in; t = .r> G o V at m o r e t h a n !)!)% c o n f i d e n c e level. Exercises for C h a p t e r
10
10 I S h o w t h a t by e x p a n d i n g t h e e x p r e s s i o n for t h e r u n n i n g c o u p l i n g cons t a n t in p o w e r s of o s ( / / 2 ) a n d i n s e r t i n g it i n t o eqn (10.(i). a d e p e n d e n c e o n />d at. t h i r d a n d h i g h e r o r d e r s a p p e a r s . 10 2 S h o w t h a t a final s t a t e of four p a r t i c l e s is u n i q u e l y c h a r a c t e r i z e d by a set of five k i u e m a t ie v a r i a b l e s , if t h e event o r i e n t a t i o n is i n t e g r a t e d over. F i n d such a set of v a r i a b l e s which a r e Lorcnt.z-invariaiit.
11
LEADING-LOG QCD I'll«- classical a p p r o a c h t o a r r i v e at t h e o r e t i c a l p r e d i c t i o n s for e x p e r i m e n t a l ol>servahles is t o go o r d e r hy o r d e r in | ) e r t u r h a l ion t heory. An a l t e r n a t i v e is provided by so-called leading-log Q C D . where s o m e cont ribut ions a r e rcsumined t o all orders. T h e modified leading-log a p p r o x i m a t i o n ( M l , L A ) for e x a m p l e is essentially equivalent t o t h e coherent p a r t o n shower picture, where interference effects between subse(|tient emissions a r e taken into a c c o u n t t h r o u g h a n g u l a r o r d e r i n g , that is. d e c r e a s i n g emission angles, a s t h e p a r t o n shower evolves. T h e o r e t i c a l predictions in t h e f r a m e w o r k of MLLA (Azimov ct al.. 1986«: Fong a n d Webber. 1991) can be calculated analytically for various obscrvables. N o t e that t h e MLLA predictions still refer t o t h e p a r t o n level. T h e connection t o t h e a c t u a l l y observed h a d r o n level is established by t h e hypothesis of local p a r t o n h a d r o n duality ( L P H D ) (A'/.imov ct id.. 19S.ri6). which a s s u m e s t h a t t h e cross section at hadron level is p r o p o r t i o n a l to t h e o n e at p a r t o n level. lu t h e following sections we will lirst describe s o m e m e a s u r e m e n t s which s u p p o r t t he generic p i c t u r e of a Q C D cascade and then t u r n t o specific results t h a t p r o b e details of t h e MLLA prediction.
11.1
The structure oft.be parton shower
lu previous sections we have described I he evolution of an ¡nit ial q u a r k ant ¡quark pair into a mult i h a d r o n final s t a t e using t h e pict ure of a p a r t o n shower, w h e r e au initial f o u r - m o m e n t u m is d i s t r i b u t e d between a large n u m b e r of final s t a t e part icles in a self-similar c a s c a d i n g process. Of c o u r s e t h e i n t e r e s t i n g q u e s t i o n is w h e t h e r e x p e r i m e n t a l evidence for such a self-similar st ruct ure can be found. lu fact, c o r r o b o r a t i n g evidence is o b t a i n e d f r o m t h e s h a p e of t h e charged particle multiplicity d i s t r i b u t i o n . Based on rat her general a s s u m p t i o n s o n e e x p e c t s from a c a s c a d i n g model that t h e multiplicity d i s t r i b u t i o n s h o u l d be described by a log-normal d i s t r i b u t i o n ( C a r i u s a n d Iiigclman. 1990: Szwed a n d W'rochna. 1990), which is in good a g r e e m e n t with t h e e x p e r i m e n t a l findings ( A L E P H Collab.. 1!)!)1 /;: O P A L Collab.. 19926). A n o t h e r piece of evidence comes f r o m t h e s t u d y of intermit,tencv. t h a t is. non-Poissonian f l u c t u a t i o n s of particle multiplicities in restricted p h a s e - s p a c e intervals. Here t h e a s s u m p t i o n of a self-similar c a s c a d i n g m e c h a n i s m leads t o t h e prediction, t h a t t h e size of t h e f l u c t u a t i o n s as m e a s u r e d by t h e so-called factorial moments should grow p r o p o r t i o n a l l y t o an inverse power of t h e size of t h e p h a s e - s p a c e interval (Bialas a n d Peschanski. l9.S(i). E x p e r i m e n t a l results
LF.ADINO-I.OO <¿< I)
:M¡(¡
. OI'Al tint» |» ' — liilci|H,tili„ii Imicdim
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ItttTlI'llll"!! tlUÉvlll«, I
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Imlíi,,' s OPAL dala > | ¡> liiicipiibiiim funciiiHi
7
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F i e . 11.1. Distill u n i o n s 1 / n d i r / d £ . £ = l n ( l / . r , , ) . o í c h a r g e d p a r t i c l e s f r o m h a d r o n i c 7. d e c a y s . F i g u r e f r o m Sclmiellingí l!)!).riu).
(Al.F.I'll Col lab.. l!)!J2í/: D K I . f ' l l l C o l l a h . . 1ÍJÍH); O P A L Collal».. H>!)l/>) c o n f i r m this behaviour. H a v i n g e s t a b l i s h e d t h e validity of t h e g e n e r i c p a r l ó n s h o w e r pict ure, t h e next s t e p is t o check predict ions which a r e specific t o t he d y n a m i c s of Q C D . such as c o h e r e n c e e f f e c t s in s u b s e q u e n t s t o p s in t h e p a r t o u c a s c a d e .
11.2
Momentum
spectra
E x p e r i m e n t a l e v i d e n c e s h o w i n g t h e e x i s t e n c e of c o h e r e n c e e f f e c t s c a n be o b t a i n e d f r o m t h e s t u d y of inclusive m o m e n t u m s p e c t r a . Here t h e c o n s e q u e n c e of a n g u lar o r d e r i n g is u n d e r s t a n d a b l e f r o m a s i m p l e m o d e l w h e r e all e m i s s i o n p r o c e s s e s h a p p e n w i t h t h e s a m e t r a n s v e r s e m o m e n t u m r e l a t i v e t o t h e m o t h e r p a r t o u . Having t h e t r a n s v e r s e m o m e n t u m lixefl. d e c r e a s i n g e m i s s i o n a n g l e s i m p l y i n c r e a s i n g t o t a l m o m e n t a in s u b s e q u e n t e m i s s i o n p r o c e s s e s . This forced i n c r e a s e d o e s not exisi in t h e a b s e n c e of c o h e r e n c e . In o t h e r w o r d s , c o h e r e n c e effects lead t o a c h a r a c t e r i s t i c s u p p r e s s i o n of low m o m e n t u m p a r t i c l e s ( A z h n o v i t ul.. I'lSli«). T h i s effect is o b s e r v a b l e in t h e v a r i a b l e £ = hi( 1/J:,,), w h e r e ./:,, 2/i/Y/s is t h e sealed m o m e n t u m of a final s t a t e p a r t i c l e . C o h e r e n c e e f f e c t s lead t o a
M< iMEN II IM S | ' F ( "I UA
.«¡V
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(a)
• DATA
— yen A,.„ = I 17 Ml-V s
A,„= 14X5 MeV
ni
r i
4-TK 0.2
^
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(I
£ = FlG. 1 1.2. Dist r i b u t i o n s l/ou do,,/d£. Z d e c a y s . F i g u r e f r o m Schniclling( 1995«).
CD
1 I
l
i 3
i I
S
of n e u t r a l p a r t i c l e s f r o m h a d r o n i c
(I»
yen CMI.LAi
I0 :
10 VS~(GCV I
\ s (GcV)
FIG. 1 1 . 3 . T h e p e a k p o s i t i o n for different, p a r t i c l e t y p e s a s f u n c t i o n of t h e C . o . M . e n e r g y . F i g u r e o n t h e left f r o m Schmellhig(H)i)r»«).
Table
I I . 1 1'itlk punition,s
Particle
all all all all
7T° Tt* charged charged charged charged KKi! K Kg Kg '/
p.p AA AA AA AÂ — ~
Mass/GeV 0.135 o . i :tr> 0.1 1(1 0.22 0.22 0.22 0.22 0.1!) 1 0.498 0.498 0.498 0.498 0.547 0.547 0,!):5.s
as /miction fs
4.11 3.96 3.81 3.018 3.(i7 3.71 3.603 2.63 2.63
J: ± ± ± ± ± ± ± ±
2.62 2.89 2.91 2.(i0 2.52 3.00
± ± ± ± ± ±
of tin particlt
mass
Uelerenee 1,3 C o l l a b . (19!)le) 0.18 1.3 C o l l a b . (1994) 0.13 0.02 O I ' A L C o l l a b . (1994«) 0.028 ALEPII Collab. (1992«) 0.10 D E L P H I C o l l a b . (1!)92b) 0.05 1.3 C o l l a b . (1991c) 0 . 0 4 2 O P A I . C o l l a b . (1990) 0.04 O P A L C o l l a b . (1994«) 0.04 A L E P I I C'ollab. ( 1 9 9 4 « ) D E L P H I C o l l a b . (1992/;) 0.11 1.3 C o l l a b . (1991) 0.05 O P A L C o l l a b . (1991c) 0.04 0.15 1.3 C o l l a b . ( 1 9 9 2 « ) 1.3 C o l l a b . (1994) 0.10 0.09 O P A L C o l l a b . (1994«)
1.110 1.116 l . t 16
2.67 ± 0 . 1 4 2.82 ± 0 . 2 5 2.83 ± 0 . 1 3
A L E P I I C o l l a b . (1994«)
1.116 1.321
2.77 ± 0 . 0 5 2.57 ± 0 . 1 1
O P A L C o l l a b . (1992c) O P A L C o l l a b . (1992c)
D E L P H I C o l l a b . (19926) 1.3 C o l l a b . (1994)
s u p p r e s s i o n ¡it l a r g e v a l u e s of £ sucli that d
The v a r i a t i o n of t h e p e a k p o s i t i o n is p r e d i c t e d w i t h o n l y o n e f r e e p a r a m e t e r A 0 jf w h i c h d e t e r m i n e s t h e c u t - o l f of t h e p e r t u r b a t i v e p h a s e a s d e s c r i b e d by a p a r t o n s h o w e r . A s t h i s cut-olf is e x p e c t e d t o g r o w w i t h t h e m a s s of t h e linal s t a t e p a r ticles. t h e p e a k p o s i t i o n s h o u l d shift t o s m a l l e r v a l u e s ( l a r g e r m o m e n t a ) for h e a v i e r p a r t i c l e s . T h e l e a d i n g t e r m in e<|ii (11.1) p r e d i c t s a n e n e r g y e v o l u t i o n w i t h t he logarit h m of t he s q u a r e - r o o t of t h e C'.o.M. e n e r g y . T h i s e n e r g y d e p e n d e n c e is c h a r a c t e r i s t i c for a c o h e r e n t p a r t o n s h o w e r w i t h a n g u l a r o r d e r i n g . A n incoherent, p a r t o n s h o w e r w o u l d lead t o a n e n e r g y e v o l u t i o n of t h e p o s i t i o n of t h e m a x i m u m w i t h t h e l o g a r i t h m of t h e C'.o.M. e n e r g y .
(
.
A L3 A 1.
10
(
. 11. 1. T s
s
(,
s
s
(
s s
s ,
s s A,, s
s
s
. 1 . 11.
(
, )
s
s
). T ). A s s
s
s
s s
s
s
T
11.2
s
s A
ss
s s s s s s
,
s ss. T . s ss. T s s s
s
1.
s. s
s.
s s T
ss
.
s s
s
. 11. s
s
A .
(1
s 11.1. T s s . s
s
.
s (
. s s s s ss,
4 ).
s
A
s
s
s. A .
s
A s s
.
s s
(
s
s.
A s
s
s 1.2
c,
s s
)
1 .
, ,,, 1.2 . 11.2. T s s s ss. s
. 11.1 s
s
. 1
s
1.2
ss
s
(
s
.
s
)
s
A
I i \|»IN<: I.OG QCI> TABLE I 1.2 I'riilc positions y/s/GW
mill mean
chun/cil
(»,•!,)
puiticli
multipliai
Reforonce
12.0
8 . 1 0 1 OTTÎÏ
J A D E C o l l a i » . ( 1Î1XIÎ)
12.3
8.70 ± 0 . 6 0
P L l ' T O Collab. (19806)
9.30 ± 0.41
l'ASSO Collal».( 1990.
11.0
2 . 3 5 3 ± 0 . 0 13
17.0 22.0 22.0
2.051 ± 0.041
27.0
1989)
9.40 ± 0.70 11.20 ± 1.00
P L I J T O Collai». (19806) P L l ' T O C o l l a b . (19806)
11.30 ± 0 . 4 7
l'ASSO Collai).( 1990.
1989)
12.00 ± 0 . 8 0
P L U T O Collai». (19806)
12.80 ± 0 . 6 0
T P C / 2 7 Co)lah.(1988.
29.0
12.87 ± 0.30
11RS C o l l a b . ( 1 9 8 6 )
30.0
13.10 ± 0.70
J A D E Collab.
30.6
12.30 ± 0 . 8 0
P L U T O C o l l a b . (19806)
13.59 ± 0 . 4 6
l ' A S S O Collal».( 1990.
29.0
35.0
2.866 ± 0 . 0 6 0
3.063 ± 0.024
35.0 44.0
13.60 ± 0 . 7 0 3.120 ± 0.054
15.08±0.47
52.0
15.99 ± 0 . 2 3
55.0
16.85 ± 0 . 2 7
55.0
3-147 ± 0.093
57.0
us
1987)
(1983)
J A D E Collab.
1989)
(1983)
l'ASSO Collab.(1990.
1989)
T O P A Z Collab. (1988) l'OPAZ Collab.
(1988)
A M Y Collab.
(1990«)
17.19 ± 0 . 4 9
A M Y Collab.
(19906)
91.2
3.618 ± 0.028
20.85 ± 0.2 I
A L E P I I C o l l a b . ( 1992A.
19916)
91.2
3.670 ± 0.100
20.71 ± 0.77
D E L P H I Col lab. (19926.
1991)
91.2
3.710 ± 0.050
20.79 ± 0.52
L3 Collal>.(l991c.
91.2
3.603 ± 0.042
21.40 ± 0 . 4 3
O P A I . Collab.( 1990,
23.84 ± 0 . 7 3
D E 1 , P W Collab.
130.0 133.0
3.944 ± 0 . 0 5 9
24.04 ± 0 . 4 4
ALEPII Collab.
133.0
3.940 ± 0 . 1 2 1
23.40 ± 0.65
OPAI. Collab.
161.0
4.098 ± 0.093
26.75 ± 0 . 7 8
ALEPII Collab.
25.46 ± 0.58
D E L P H I Collai».
161.0
19926) 19926)
(19966) (19986)
(19966) (19986) (1998«)
161.0
4.000 ± 0.050
24.46 ± 0.40
OPAI. Collab.
172.0
4.040 ± 0.089
26.45 ± 0 . 7 8
A L E P I I Collai).
26.52 ± 0.76
D E L P H I Collab. (1998a)
172.0
4.031 ± 0 . 0 5 3
25.77 ± 1 . 0 5
O P A L Collab.
183.0
4.110 ± 0 . 0 7 3
26.44 ± 0.60
ALEPII Collab.
27.05 ± 0 . 4 2
D E L P H I Collab.
172.0
183.0
(1997«) (19986) (2000«) (19986) (2000«)
183.0
4.075 ± 0 . 0 4 4
27.04 ± 0 . 4 9
L3 Collai). ( 1998)
183.0
4.087±0.033
26.85 ± 0 . 5 9
O P A L Collab.
189.0
1.081 ± 0 . 0 2 8
27.42 ± 0.34
ALEPII Collab.
27.47 ± 0.45
DELPHI
189.0
(2000«) (2000«)
Collab.
(2000«)
189.0
4.121 ± 0 . 0 3 8
20.95 ± 0.53
O P A L Collab.
(2000«)
196.0
4.145 ± 0 . 0 3 1
27.42 ± 0.49
ALEPII Collab.
(2000«)
200.0
4.127 ± 0 . 0 3 4
27.83 ± 0.52
ALEPII Collab.
(2000«)
200.0
27.58 ± 0.49
D E L P H I Collab. (2000a)
206.0
27.98 ± 0 . 2 3
ALEPII Collab.
(2001)
371
QCD (MLLA+LPHD)
• *
_l
50
1
1
1
100
ALEPII DELPHI
* L3 • OPAL O JADE • PLUTO A TASSO * TPC/2 Q I IRS • TOPAZ 9 , AMY 1 1 150 200
V7(GcV) Fie
1 1 . 5 . M e a u chargée! [»article iiiultiplieity f r o m e + e annihilation into h a d r o m e final stat.es for C . o . M . é n e r g i e s b e t w o e n £,.,„ = 12 C e V t o E,.„ = 206 C;
LP11D (Kiinszt- rt 11L. 1989) a s a f u n c t i o n of t h e s t r o n g c o u p l i n g c o n s t a n t . T h e e n e r g y d e p e n d e n c e of t h e m e a n m u l t i p l i c i t y is t h u s u n d e r s t o o d a s a c o n s e q u e n c e of t h e r u n n i n g of o s :
( " e l , ( £ „ „ ) > = A'I.MID • o ' ! ( E r i l l ) • e x p ( — = J L = = j . \ v/Os(Km) J
(11.2)
T h e n u m e r i c a l v a l u e s of t h e c o e f f i c i e n t s 11 a n d 6 in e q n (11.2) a r e p r e d i c t e d by Q C D as n \/(>-I2/23 and 6 4 0 7 / 8 2 8 . F i x i n g n s by e x t e r n a l m e a s u r e m e n t s , t h e o n l y f r e e p a r a m e t e r in e q n (11.2) is /v'l.rni). E x p e r i m e n t a l d a t a for t h e m e a n c h a r g e d p a r t i c l e m u l t i p l i c i t y {11,1,) f r o m + e e " a n n i h i l a t i o n i n t o h a d r o n s a r e c o l l e c t e d in T a b l e 11.2. T h e n u m b e r s refer lo t h e c o n v e n t i o n w h e r e all p a r t i c l e s w i t h a m e a n l i f e t i m e below I n s a r e f o r c e d
.172
I . I : , M ) I N < : !,<><: ( ¿ E N
t o decay while t h e o t h e r s a r e a s s u m e d to he a b s o l u t e l y stable, lit some eases t h e published d a t a wore corrected for differences in t h e a n a l y s i s (Schmolling. 1995«), For t he entire r a n g e of C'.o.M. energies b e t w e e n 12 CJoV a n d 2(J(i GoV. that is, for all available d a t a a b o v e t h e p r o d u c t i o n t hreshold for b - q u a r k s . o n e finds excel lent agreement. between experiinontal d a t a a n d t he QC'D e x p e c t a t i o n . T h e energy d e p e n d e n c e of t h e m e a n c h a r g e d p a r t i c l c multiplicity is perfectly described by using t h e loading o r d e r expression for t h e r u n n i n g c o u p l i n g constant o s ( / r ) , fixing its value at t h e scale of t h e Z m a s s t o t h e global a v e r a g e <\*(My ) = 0.1 IN. a n d a d j u s t i n g only t h e pheuoineuological p a r a m e t e r A'l.pim- T h e results a r e shown in Fig. 1 1 . N o t e t h a t a priori such a good fit could not be e x p e c t e d , since t h e s t r o n g c o u p l i n g c o n s t a n t a p p e a r i n g in eqn (11.2) should be u n d e r s t o o d a s an effective coupling. T h e good fit thus indicates that higher o r d e r corrections may be small. I I .'1
Isolated hard
photons
So far we have established t h a t t h e partem shower is a self-similar c a s c a d i n g process w i t h c o h e r e n c e p r o p e r t i e s in agreement w i t h t h e predictions of t h e modified leading l o g a r i t h m i c a p p r o x i m a t i o n . F u r t h e r i n f o r m a t i o n a b o u t its s t r u c t u r e beyond self-similarity a n d a n g u l a r o r d e r i n g c a n be o b t a i n e d from m o r e detailed studies, such as t h e s t u d y of isolated h a r d p h o t o n s o r s u b j e t multiplicities. Isolated h a r d p h o t o n s p r o b e t h e early p a r t o u s h o w e r i n g s t a g e . Being e m i t t e d with a large m o m e n t u m t r a n s f e r , t h o s e p h o t o n s test t h e short d i s t a n c e p r o p e r t i e s of the multi-part on s y s t e m which f o r m s t h e shower. For a given f o r m u l a t i o n of the p a r t o u shower in t e r m s of gluon radiation from q u a r k s a n d gluons. also t h e p r o p e r t i e s of a p h o t o n e m i t t e d in t h e cascade a r e fixed. As both a r e inassloss vector particles, t h e only difference between a p h o t o n a n d a gluon is t h a t t h e f o r m e r couples t o t h e electric c h a r g e of t h e q u a r k s a n d t h e l a t t e r t o t h e colour charge. T h e r e a r e no new free p a r a m e t e r s . As t h e p h o t o n is blind t o colour charges, once e m i t t e d , it p e n e t r a t e s without f u r t h e r i n t e r a c t i o n t h e c o m p l i c a t e d colour fields of t h e partem shower, t h e r e b y p r o d u c i n g a kind of X-ray p i c t u r e of t he cascade. E x p e r i m e n t a l results were presented by t h e I.El' C o l l a b o r a t i o n s (OI'AL Collab.. 19926: A L E P I I C'ollab., 1993; \M C'oilah., 1992c). F i g u r e 11.6 shows how a measurement of the isolated p h o t o n r a t e a s a f u n c t i o n of a resolution p a r a m e ter i/ r hi c o m p a r e s t o various p a r t o u shower models. Here t he p a r a m e t e r i/cui is t h e d i s t a n c e t o t h e closest j o t s in t h e J A D E metric. O n e finds t h a t for large isolat ion p a r a m e t e r s //,.„| > 0.1 t h e p r o d u c t i o n r a t e is r e a s o n a b l y well described by t h e J E T S E T a n d IIEKWIC! models a n d slightly o v e r e s t i m a t e d by AH IAD NIC. For small resolution p a r a m e t e r s t h e p i cture changes. T h e r e b o t h A R I A D N E a n d IIEKWIO provide a good description of t h e d a t a while J E T S E T u n d e r e s t i m a t e s t h e p h o t o n r a t e . As t h e various p a r t o u shower m o d e l s only (filler in t h e way in which next-to-leading logarithmic effects a r e i m p l e m e n t e d , these findings d e m o n s t r a t e t h a t t h e detailed s t r u c t u r e of t h e p a r t o u c a s c a d e can be s t u d i e d with isolated h a r d p h o t o n s .
:»7.l
y«n F l c . 1 1.0. Measured r a t e of isolated h a r d p h o t o n s a s f u n c t i o n of t h e s e p a r a t i o n from t h e closest, j o t . c o m p a r e d to M o n t e C a r l o model calculations. F i g u r e from A L E P I I Collal>.(1993).
As a side r e m a r k , it m a y be w o r t h m e n t i o n i n g t h a t a discrepancy between t h e d a t a a n d a p a r t i c u l a r model d o e s not necessarily imply t h a t t h e model is c o n c e p t u a l l y inferior t o its c o m p e t i t o r s . O n e always has to keep in mind t h a t usually t h e model p a r a m e t e r s a r e a d j u s t e d such that, t he overall event p r o p e r t i e s as measured by t h e final s t a t e h a d r o n s a r e well r e p r o d u c e d by t h e c o m b i n e d p a r t o u showering a n d h a d r o n i z a t i o n stage. P e r t u r b a t i v e a n d n o n - p e r t u r b a t i v c effects a r e t h u s e n t a n g l e d . M e a s u r e m e n t s with isolated p h o t o n s , which see t h r o u g h t h e h a d r o n i z a t i o n phase would allow to t u n e t h e description of the p e r t u r b a t i v e p h a s e independently of t h e h a d r o n i z a t i o n s t o p a n d t hus offer a way to d e c o u p l e t h e p a r a m e t e r s describing t he two regimes.
11.5
Subjet. multiplicities
A n o t h e r a p p r o a c h t o test t h e s t r u c t u r e of t h e p a r t o u shower is by m e a n s of s u b j e t multiplicities, defined in t h e following way: in a first s t e p a h a d r o n i c final
L E A D I N d I.OC!
:j7i
s t a t e is c l u s t e r e d wit li a jet r e s o l u t i o n p a r a m e t e r ;/i i n t o a well dcliiicd muul>er of initial j e t s . T h e n t he s a m e event is looked at w i t h a d i f f e r e n t r e s o l u t i o n p a r a m c t c i .'/(i < y/i a n d t h e n u m b e r of c l u s t e r s s t u d i e d a s a f u n c t i o n of ;/u. T h a t way t h e s e new j e t s c a n b e u n i q u e l y a s s o c i a t e d w i t h o n e of t h e original j e t s . In t h e limit i/o —« 0 t h e s u bj e t m u l t i p l i c i t i e s b e c o m e e q u a l t o tin- n u m b e r of p a r t i c l e s in t h e j e t s . T h e interest a r i s e s b e c a u s e v a r y i n g i/o is e q u i v a l e n t t o v a r y i n g t h e A y , in t h e D u r h a m a l g o r i t h m , a t w h i c h t h e event is p r o b e d : A " " " T h u s using a largo //,, p r o b e s p o r t u r b a t i v e p h y s i c s whilst d e c r e a s i n g ;/u a l l o w s us t o i n v e s t i g a t e w h e n a n d t o w h a t e x t e n t h a d r o n i / a t i o n b e c o m e s i m p o r t a n t . T h e i n c r e a s e in t h e n u m b e r of jets w h e n lowering t h e j e t r e s o l u t i o n p a r a m e t e r is p r e d i c t e d in p o r t u r b a t i v e Q C ' D w i t h a rosu initiation of l o a d i n g a n d n e x t - t o - l e a d i n g logarit h u m ( C a i a n i et
(a)
v, =0.010 Jl
1.4 V
1.4
12 l F.
i 5
OPAL data COJHTS623 ARIADNR3I COJETS6I2
• OPAL daia o.x
V
--
()<«.) QCD.A=0.35GcV
JETSET73
OS
HHRWIG 55
— NLLAQCt). A=0.35GcV mil ........1 , . , I It) 4 10 ' 10 s v«i
10 2
It)''
10 4
10 '
10 2
Vii
FIG. 1 1 . 7 . R a t i o of s u b j e t m u l t i p l i c i t i e s in two- a n d t liree-jet e v e n t s c o m p a r e d t o Q C D a n d m o d e l p r e d i c t i o n s . F i g u r e f r o m O R A L C'ollab.( 1994b).
Let M-.i a n d M2 b e t h e j e t m u l t i p l i c i t i e s at a r e s o l u t i o n //„ w h e n t h e initial m u l t i p l i c i t i e s at 1/1 a r e 3 a n d 2 j e t s , r e s p e c t i v e l y . F i g u r e 1 f . 7 s h o w s e x p e r i m e n t a l r e s u l t s for t h e r a t i o of t h e a d d i t i o n a l j e t s (.\/ : t - .'{)/(M> 2 ) a s a f u n c t i o n of ( O R A L C o l l a b . . 1994/;). A f t e r a s h a r p rise w h e n g o i n g t o s m a l l e r v a l u e s of 1/0, t h e r a t i o s t a r t s t o level oil" a r o u n d 3/0 = 4 • I d - ' . T h e rise i.s well d e s c r i b e d by t h e p o r t u r b a t ive Q C ' D p r e d i c t i o n i n c l u d i n g a n a l l - o r d e r s r c s u i n u i a t i o n of l o a d i n g a n d
URI I I I R A M I
ANALYSES
n e x t - t o l e a d i n g l o g a r i t h m s . T h e i m p o r t a n c e of r o s u m m i i i g t h o s e l o g a r i t h m s is i l l u s t r a t e d bv c o m p a r i n g t h e r e s t i i n m c d p r e d i c t i o n w i t h a lixed o r d e r c a l c u l a t i o n which fails t o d e s c r i b e t h e d a t a . However, in n e i t h e r c a s e t h e levelling oil" of t h e e x p e r i m e n t a l d a t a at v e r y s m a l l v a l u e s 1/0 is r e p r o d u c e d . It follows that, t h e d y n a m i c s at. v e r y low scales is d e t e r m i n e d by p o r t u r b a tive h i g h e r o r d e r s b e y o n d t h e m o d i f i e d l e a d i n g - l o g a p p r o x i m a t i o n a n d by imn p e r t u r b a t i v e effects. T h e l a t t e r a r e a n a t u r a l c a n d i d a t e for p r o d u c i n g a u n i v e r s a l b e h a v i o u r , i n d e p e n d e n t of t he n u m b e r of j e t s a t t h e h a r d s c a l e 1/1. It is. t h e r e f o r e , interesting t o confront t h e d a t a with model calculations which take higher order Q C D a n d n o n - p e r t u r b a t i v e e f f e c t s i n t o a c c o u n t . Using m o d e l s t h a t h a v e been t u n e d t o r e p r o d u c e t h e d a t a , o n e f i n d s t h a t t h e coherent, p a r t o n s h o w e r m o d e l s with c l u s t e r o r s t r i n g h a d r o n i / a t i o n s u c c e s s f u l l y r e p r o d u c e t he e x p e r i m e n t a l res u l t s over t h e full 1/0 r a n g e . T h e m o d e l s b a s e d o n a u i n c o h e r e n t p a r t o n s h o w e r a n d i n d e p e n d e n t f r a g m e n t a t i o n o n l y r e p r o d u c e t h e q u a l i t a t i v e t r e n d , but h a v e diflicultics in d e s c r i b i n g t h e d a t a q u a n t i t a t i v e l y . 11.6
Droit, f r a m e a n a l y s e s
I11 t h e p r e v i o u s sect ion t h e d i s c u s s i o n d i d f o c u s o n o b s e r v a b l e s d e f i n e d iu e + c ~ a n n i h i l a t i o n r e a c t i o n s . S i m i l a r a n a l y s e s c a n in fact a l s o b e p e r f o r m e d in e l e c t r o n p r o t o n collisions w h e n c h o o s i n g a n a p p r o p r i a t e r e f e r e n c e f r a m e . T h a t f r a m e is given by t h e so-called Brcit f r a m e , a s d i s c u s s e d in Sect ion (¡.2. Viewed f r o m a p a r t o n in t h e p r o t o n , w h i c h s c a t t e r s off a v i r t u a l p h o t o n f r o m t h e e l e c t r o n , it. is t h e L o r o n t z f r a m e w h e r e t h e p a r t o n ' s longit u d i n a l m o m e n t u m b e f o r e t h e s c a t t e r i n g p r o c e s s is +Q/2, a n d a f t e r s c a t t e r i n g —Q/2. In o t h e r w o r d s , it is like s c a t t e r ing o n a u i n f i n i t e l y m a s s i v e brick-wall. T h e s c a t t e r e d p a r t o n is t h e n e s s e n t i a l l y equivalent t o o n e of t h e initial q u a r k s in 0 1 0 a n n i h i l a t ion i n t o ha
tjo
12
DIFFERENCES BETWEEN QUARK AND GLUON JETS .lets ¡nil iat.cd b v p r i m a r y q u a r k s a n d g l u o n s . w h i c h d i f f e r in b o t h i heir s p i n s a n d c o l o u r c h a r g e s , a r e p r e d i c t e d t o h a v e different p r o p e r t ies. T h i s h a s raised siguilicant t h e o r e t ical a n d e x p e r i m e n t a l interest, e v e r s i n c e t he d i s c o v e r y of g l u o n j e t s , f o l l o w i n g o n f r o m e a r l y q u a l i t a t i v e s t u d i e s . 1,EI' h a s h e r a l d e d a new q u a n t i t a tive e r a ( G a r y , 1994: K n o w l e s el ul.. 1996). In t h i s c h a p t e r we begin by r e v i e w i n g t lie I lieoret ical e x p e c t a t ions a n d e x p e r i m e n t a l p r a c t i c a l i t i e s b e f o r e d i s c u s s i n g t lie established results. 12.1
Theoretical
expectations
An e a r l y p r e d i c t i o n w a s t h a t o n a v e r a g e t h e m u l t i p l i c i t y of a n y t y p e of p a r t i c l e in a g l u o n j e t c o m p a r e d t o a q u a r k j e t s h o u l d b e a s y m p t o t i c a l l y e q u a l t o f . i / C/.2.2.r> ( B r o d s k v a n d G t m i o n . 1970). At N L O t h i s r a t i o b e c o m e s a f u n c t i o n of t h e s t r o n g c o u p l i n g c o n s t a n t o s a n d t h u s e n e r g y d e p e n d e n t . If we use (A : ,,) a n d (.'VK) t o d e n o t e t h e s e m e a n m u l t i p l i c i t i e s t h e n at NNLC) t h e i r r a t i o is given by ( G a i f n e y a n d M u e l l e r . 19S.r>)
1+ 2
^
>sCt 18tt
-
f
^
i
)
(12.1)
/25
ai^TV
„»/7KVN
U
2 CA
-
CI
)
nsc-/t IS-
T h i s p r e d i c t i o n h a s since b e e n refined t o a c c o u n t for e n e r g y - m o m e n t mil c o n s e r v a t i o n in t h e s h o w e r ( D r e n i i n a n d N e c h i t a i l o , 1991: E d e n . 1998) a n d t a k e n t o N : ! L O ( C a p e l l a el ul.. 2 0 0 0 ) . At t h e s c a l e of t h e Z m a s s t h e r a t i o is p r e d i c t e d t o b e (.V K )/(.V < .) s i 1.7. Also t h e w i d t h s of t h e m u l t i p l i c i t y d i s t r i b u t i o n s . a K a n d f r , r a r e p r e d i c t e d t o differ. At l e a d i n g o r d e r o n e e x p e c t s (12.2) Corrections to this and higher m o m e n t s are also available (Malaza and Webber. 1986: Dremiii a n d H w a . 199-1). T h i s implies t h a t g l u o n j e t s h a v e l a r g e r lluctua l i o n s in iiiultiplicity by a n a s y m p t o t i c f a c t o r \/C\/C¡.- = 3/2. The relat ive f a c t o r C'.\/C/.- is t h e r a t i o of t h e c o l o u r c h a r g e s of a g l u o n a n d a q u a r k . Il a r i s e s b e c a u s e t h e e v o l u t i o n of a q u a r k - i n i t i a t e d jet is d o m i n a t e d by I he e m i s s i o n of t h e first v i r t u a l g l u o n w h i c h t h e n effectively gives a g l u o n j e t .
XT ACTI G <)
lll
A
(!I H
.I I
TI S
T h e h i g h e r a v e r a g e m u l t i p l i c i t y In t i n n i i e t s m e a n s 1,1ml tliev must luive n sillier f r a g m e n t a t i o n fillietiou t h a n q u a r k e t s . At. s m a l l x o n l y t h e p r i m a r y part,oil's c o l o u r c h a r g e is i m p o r t a n t . , s i n c e l o n g w a v e l e n g t h g l u o n s d o not s e e t h e d i l f e r c n t s iins. a n d so b o t h f r a g m e n t a t i o n f u n c t i o n s h a v e t h e s a m e s h a p e lo I . A . a G a u s s i a n in l n ( l . r ) . b u t t h e r e l a t i v e h e i g h t s of t h e p e a k s dilicr bv a f a c t o r CA/CF ( o k s h i t z e r et. ni. 1991: F o n g a n d W e b b e r , 1991). In o r d e r t o conserve m o m e n t u m t he g r e a t e r n u m b e r of soft p a r t i c l e s in a g l u o n et t hen l e a d s lo a r e l a t i v e s u p p r e s s i o n iu t h e n u m b e r of h a r d p a r t i c l e s . T h e a n g u l a r s i z e s of q u a r k a n d g l u o n e t s a l s o d i f f e r ( i n h o r n a n d W e e k s . ITS). T h i s c a n b e s e e n u s i n g t h e S t e r n u m W e i n b e r g d e f i n i t i o n e q n (3.226) of I lie t w o - e t f r a c t i o n , > = 1 — ; , in e + e a i u i i h i l a t i o n . If t h e t w o e t s a r e r e q u i r e d lo c o n t a i n a f r a c t i o n (1 - r ) of t h e t o t a l e n e r g y inside b a c k - t o - b a c k c o n e s of hall angle S. t h e n t h e p r e d i c t i o n for s m a l l is given by (1 <
exp
,)
IC,-os(.s) hi(I.A)
and
\
= =
\aC,/CA
(12.3)
The g l u o n et result, follows by s u b s t i t u t i n g C'A for CP in e< ii (3.220). T h u s at I lie s a m e e n e r g y , g l u o n e t s a r e w i d e r t h a n q u a r k e t s . A c t u a l l y t h i s p r e d i c t i o n for gluon e t s h i g h l i g h t s a p r o b l e m a s it a p p l i e s t o a c o l o u r singlet gg s t a t e such a s might b e p r o d u c e d in t he d e c a y of s o m e heavy, -I 1 '' = 0 + r e s o n a n c e . A' — > gg. Here t h e p r o p e r t i e s of t h e g l u o n c a n b e i n f e r r e d , p e r h a p s wit h t he aid of a n a x i s such a s t h e T h r u s t , a x i s , f r o m t h o s e of t h e h a d r o n i c s y s t e m a s a whole. T h a t is. t here is no need of a et f i n d i n g a l g o r i t h m . T h i s sit u a t i o n is not t y p i c a l of most e x p e r i m e n t s . At t he d e c a y Z —> g g is f o r b i d d e n by t h e a n d a u Van t h e o r e m so that, g l u o n e t s must, b e identified as. for e x a m p l e , t h e ' t h i r d ' et in a s a m p l e of e + e — qqg events. 12.2
xtracting quark and gluon
et
properties
To d a t e h a s d o m i n a t e d I lie q u a n t i t a t i v e s t u d y of q u a r k a n d g l u o n et p r o p erties. T h e r e f o r e we focus on t h e issues w h i c h a r i s e in t h e a n a l y s i s o f e + c ann i h i l a t i o n e v e n t s . It is s t r a i g h t f o r w a r d t o i d e n t i f y a n d m e a s u r e t h e p r o p e r t i e s of q u a r k e t s in e i t h e r t h e inclusive o r t w o - e t s u b - s a m p l e of h a d r o n i c e v e n t s . F u r t h e r m o r e , t h e e n e r g y of t h e e t is well d e f i n e d a s y s 2 a n d t h e a n g l e t o its c o l o u r c o n n e c t e d p a r t n e r fixed at 180 . T h i s m e a n s t h a t t he q u a r k e t p r o p e r t i e s a r e o n l y m e a s u r e d at. o n e scale, d e f i n e d by t h e b e a m e n e r g y , w h i c h is l a r g e r t h a n t h a t of g l u o n e t s in t h e s a m e e x p e r i m e n t . Iu o r d e r t o o b t a i n r e s u l t s at lower scales o n e c a n c o n s i d e r e + o —> q q 7 e v e n t s ( H I C o l l a h . . 1990c) o r e x t r a c t t h e m a s a b y - p r o d u c t ' of t he g l u o n et. a n a l y s i s . T h e a n a l y s i s of g l u o n e t s is m o r e s u b t l e . T h e s t a r t i n g p o i n t is o b t a i n i n g a s a m p l e of t h r e e - e t e v e n t s f r o m which o n e c a n i d e n t i f y a n d e x t r a c t t h e g l u o n e t ' s p r o p e r t i e s . T h e s e e v e n t s a r e a s s u m e d t o c o n t a i n o n e g l u o n et a n d t wo q u a r k ets, w h e r e t h e q u a r k a n d a n t i q u a r k i n i t i a t e d ets a r e not d i s t i n g u i s h e d . T h i s is t y p i c a l l y d o n e u s i n g , for e x a m p l e , t h e u r h a m et clustering algorithm with a lixed cut-oil v. in = 0-008 t o s e l e c t e v e n t s w i t h t h r e e e l s . This e x c l u d e s e v e n t s
:I7
II-T H
C S III IAV
( IIAH
A
(II,H
I IS
willi four 111111011 a t I. I'. 00110 based ot tinders have also I I'M used so as t o facilitate c o m p a r i s o n s with results f r o m liadrou liadion colliders ( I'AI. Collab.. 199-1 c). .lots a r e I lion classified using energy ordering . 1 > .' > > .':( a n d o t h e r tags. Allowing for a difference in t h e d i s t r i b u t i o n s ol s o m e observable A', a r i s i n g f r o m light (< =uds) a n d heavy ( Q = c b ) cpiark e t s t h e m e a s u r e d distribut ion for o n e such s a m p l e of ots c a n ho d e c o m p o s e d a s X j = P(j
= g)X
4 V ( j = t,)X„ 4 V ( j = Q ) A ' q .
(12.4)
Here t h e probabilities, or purities, satisfy V( j = g) 4 V ( j = t ) I P(j = Q ) I = and Z ^ T h u s given a n u m b e r of such m e a s u r e m e n t s with differing q u a r k a n d gluon c o n t e n t s t h e intrinsic d i s t r i b u t i o n s can b e unfolded ( I'AI. Collab.. 1993c). Hero, a first difficulty lies in b o t h linding s a m p l e s c o n t a i n i n g differ out q u a r k gluon a d m i x t u r e s a n d in defining t h e respective probabilities. At I. IM energies it is i m p o r t a n t t o distinguish between light q u a r k ots a n d I - ets. T h e p r o p e r t i e s of t h e l a t t e r a r e d o m i n a t e d by t h e decay p r o d u c t s ol t h e h - h a d r o n which c o n t r i b u t e s a b o u t half of t h e e t ' s multiplicity. C o m p a r i s o n s of b - q u a r k a n d gluon e t s a t a scale of '2-1 G e V show t h a t t h e i r multiplicities, a n g u l a r w i d t h s a n d f r a g m e n t a t i o n f u n c t i o n s a r e very similar t o one a n o t h e r , whilst significant dilforences a r e found in c o m p a r i s o n t o light (uds) q u a r k e t s ( I'AI. Collab.. 1990«), To illustrate t h e basic m e t h o d we consider the s y m m e t r i c . Y - s h a p e d , e v e n t s defined by > > .t 22 G e V . equivalent to t wo inter- et angles being in t h e r a n g e 1 5 0 4 10 . T h o s e wore popularized in the early I. I' studies. ecalling t h e c h a r a c t e r i s t i c , s o f t , b r e i n s s t r a h l u n g s p e c t r u m of g l u o n s . t h e energy ordering allows us t o say that ot 1 is almost c e r t a i n l y not a gluon. By using s y m m e t r y we then have P(2 = g) — P('.i = g) i . T w o enriched s u b - s a m p l e s of e t s c a n t h e n b e o b t a i n e d by t a g g i n g t h e presence of a q u a r k in o n e of t h e low energy e t s a n d t h e r e b y t a g g i n g t h e o t h e r low energy et as a gluon. T h e t a g m a y e i t h e r bo for heavy flavours, asking for a displaced vertex, impact p a r a m e t e r or p r o m p t lopton. o r light flavours, by t h e presence of a high m o m e n t u m track w i t h , for e x a m p l e . . .; > (1.15. T h i s allows t h e p r o p e r t i e s of light q u a r k , heavy q u a r k a n d gluon e t s t o b e measured a n d s e p a r a t e d . f c o u r s e q u a r k tagging inevitably i n t r o d u c e s s o m e biases into t h e e t s a m p l e s , but these can be assessed a n d corrected for using M o n t e C a r l o st udies. A n o t h e r p o p u l a r set of events a r c t h e threefold s y m m e t r i c e v e n t s defined by 1 > s: 30GeV . equivalent t o inter- et angles being in t h e range 1 2 0 10 : t h e s e a r e t h e so-called 'Mercedes' events, hi t h e absence of any tagging, s y m m e t r y allows us t o infer that for each e t P(j = q) 4 T(.i = Q ) = 3 = 2 V ( j = g). G i v e n t h e l a r g e final d a t a sots available f r o m I. IM it has also boon possible to s t u d y t h r e e - e t e v e n t s in which two e t s a r e tagged a s c o n t a i n i n g heavy flavours leaving t h e t h i r d as an almost p u r e gluon e t . In t h e a b o v e discussion it r e m a i n s for us to fully q u a n t i f y t h e probabilities being used. T h e most c o m m o n a p p r o a c h is based u p o n t h e use of r e c o n s t r u c t e d
IXTHACTI
C ( IIAH
A
CI.
I
I'
I'IIIM'I S
37! i
M o n t e C a r l o ( vents. 'I'liiH a l l o w s a t.liree- et event at d e t e c t o r level t,o a l s o h e rec o n s t r u c t e d a s a t h r e e - e t event at t h e l i a d r o n a n d p a r t o n levels. S c h e m e s b a s e d u p o n m m h u i z u i g a n g u l a r s e p a r a t i o n s b e t w e e n e t s c a n t h e n b e used t o m a t c h t h e d e t e c t o r level e t s t o i n d i v i d u a l l i a d r o n level e t s a n d t h e n t o o n e of t h e t h r e e p a r t o n level e t s . At t h e p a r t o n level t h e e t s a r e classified a s ( a n t i ) q u a r k or g l u o n a c c o r d i n g t o t he p r i m a r y , q q g . p a r t o n s t h e y c o n t a i n . T h u s , tin- p r o b a bility for a p a r t i c u l a r reconst r u c t e d e t t o b e a q u a r k or g l u o n c a n be specified. nfort u n a t e l y t h i s is not possible in all cases. v e n t s which a r e r e c o n s t r u c t e d a s t h r e e e t s a t t h e d e t e c t o r level m a y a p p e a r a s t w o e t s a t t h e p a r t o n level. T h e a n g u l a r m a t c h i n g s c h e m e m a y a l s o p r o v e a m b i g u o u s in s o m e cases. r it m a y not be p o s s i b l e t o a s s i g n p r o b a b i l i t i e s a t t h e p a r t o n level if. for e x a m p l e , t h e original q q - p a i r g o i n t o t h e s a m e et. A d e g r e e of effort, h a s g o n e i n t o d e v e l o p i n g m a t c h i n g s c h e m e s which m i n i m i z e t h e e f f e c t s of s u c h a m b i g u i t i e s , a n d t h u s t h e s y s t e m a t i c e r r o r on t h e r e s u l t s ( A I . I I C o l l a b . . 19906). As a n a l t e r n a t i v e t o t h e e l a b o r a t e m a t c h i n g s c h e m e s a n elegant, s o l u t i o n is p r o v i d e d by t h e u s e of I I C o l l a b . . 1997c: t h e t h r e e - e t m a t r i x e l e m e n t e q n (3.112) ( F o d o r . 1991; A A C o l l a b . . 20006). T h i s c a n be used t o d e f i n e p r o b a b i l i t i e s . :i ( 3 - g) j= which c a n b e a p p l i e d at t he d e t e c t o r , l i a d r o n or p a r t o n level, t h e r e b y a v o i d i n g t h e a m b i g u i t i e s of t h e m a t c h i n g s c h e m e s . n l y t h e usual d e t e c t o r a n d h a d r o n i z a t ion c o r r e c t i o n s need t o b e a p p l i e d . If h i g h e r o r d e r m a t r i x e l e m e n t s a r e c o n s i d e r e d t hen new d e f i n i t i o n s b e c o m e possible. n c e t h e q u a r k a n d g h i o n e t p r o p e r t i e s h a v e b e e n m e a s u r e d a s e c o n d dillic u l t y a r i s e s . A s n o t e d e a r l i e r t h e t h e o r e t i c a l c a l c u l a t i o n s a r e for idealized, b a c k t o - b a c k q q a n d g g e v e n t s . T h e s e o t s a r e u n i q u e l y c h a r a c t e r i z e d by t h e scale = ,, ( = f s / 2 . T o s o m e e x t e n t t h i s is a l s o t r u e w h e n a n a n a l y s i s o n l y selects e t s f r o m specific p h a s e - s p a c e c o n f i g u r a t i o n s , s u c h a s t h e Y - s h a p e d e v e n t s . However, in g e n e r a l a o t is s e n s i t i v e t o its e v e n t ' s overall t o p o l o g y a n d t h e e t ' s e n e r g y ( a n d t y p e ) is insufficient t o c h a r a c t e r i z e its p r o p e r t i e s . T h e t o p o l o g y d e p e n d e n c e of e t p r o p e r t i e s c a n b e u n d e r s t o o d by r e c a l l i n g o u r d i s c u s s i o n s of s o f t g h l o n s . w h i c h d o m i n a t e a e t ' s m u l t i p l i c i t y a n d small-.)' f r a g m e n t a t i o n funct ion; see S e c t i o n 3.7 a n d S e c t i o n 1.2.0. These long wavelengt h q u a n t a d o not s e e i n d i v i d u a l p a r t o n s but. c o l o u r a n t i c o l o u r d i p o l o s . F i g u r e 3.28 i l l u s t r a t e s t h e colour flow in a n o + o —> q q g e v e n t . F o c u s s i n g o n t h e q u a r k we see t hat a s t h e inter- ot. a n g l e t o t h e g l u o n . 0 q , g e t s s m a l l e r t h e q g - d i p o l e s h r i n k s a n d t h e c o l o u r c h a r g e of t h e q u a r k is m o r e effectively s h i e l d e d by t h e a n t i c o l o u r c h a r g e on t h e g l u o n . T h i s s u p p r e s s i o n of t h e r a d i a t i o n s u g g e s t s t h e u s e of a transverse-momontum-like energy scale to characterize the quark ot: i =
isill(0 iB/2)lo.iB=jr i.
QT = EJOI s i n ( f l m i „ 2 ) .
= (12.(5)
,ii r. i n i ., in n i M r i i i ¿11; \ ii |\ /»IN II I11,1 u
.11. I
I lie a p p r o x i m a t i o n follows b e c a u s e t i n ' g l u o n jet uiosl o f t e n is t h e nearest jet t o n (|unrk. T h i s m a k e s ¡t r e a s o n a b l e t o use Q \ which is m o r e convenient in p r a c t i c e In t h e case of t h e gluon several topological e n e r g y scales have been s u g g e s t e d , (A I.KI'I I Collab.. 1997«), Qt ( R B I - P H I Collab. 1999ty such a s Q„ = S/Qk„Qr,, 2(100/;: O P A L C o l l a b . 20006) a n d Q'r = v/('/ •.
12.3
Experimental properties of quark and gluon jets
In t h e following we will discuss a large s p e c t r u m of e x p e r i m e n t a l results which all show that g l u o n - h u t i a t c d j e t s indeed a r e different f r o m q u a r k jets, a n d in p a r t i c u l a r , t h a t t h e o b s e r v e d differences can be u n d e r s t o o d in t h e f r a m e w o r k of QCD. 12.3.1
Topology
dcpcndcnrc
of jet
pnipcrl.ies
T h e first s t u d y t o d e m o n s t r a t e t h e need to use topology d e p e n d e n t scales looked at t h e a v e r a g e multiplicity of q u a r k a n d gluon j e t s a s a f u n c t i o n of b o t h their e n e r g y a n d t he a n g u l a r s e p a r a t i o n . <),„¡„. t o t h e closest of t h e o t h e r t w o j e t s ( A L E I ' l l Collab.. 1997c). For a fixed E)r{ t h e jet s purity, a s used in t h e unfolding, is essentially a c o n s t a n t independent, of 0 u n n which is allowed t o s p a n t h e whole of t h e available p h a s e space. F i g u r e 12.1 s h o w s t h e r e s u l t s for p u r e light q u a r k j e t s p l o t t e d a s a f u n c t i o n of t h e j e t - e n e r g y scale QK - E\,.\ a n d t h e topological scale Q i\ cqn (1'2.0). Every point i n t e g r a t e s over a n e n e r g y r a n g e of = "> GeV a n d an a n g u l a r interval A
X
ItlM
'l
I. I'lt
I
i ll S
F gilA
A I
CI.H
III I S
:ISI
( llillk clv
ft--«: oV)
0 r (< icV) F i e . 12.1. T h e a v e r a g e m u l t i p l i c i t y in ( iiark e t s a s a f u n c t i o n of t h e e t - e n c r g v s c a l e I = fc' ,., ( t o p ) a n d t h e t o p o l o g i c a l s c a l e Q,„i„ « - ( b o t t o m ) . For clarity, in t h e t o p g r a p h et e n e r g i e s a r e s h o w n offset w i t h i n e a c h A JV, bin such t h a t e t s in lower W,„ ri b i n s a r e s h o w n w i t h slightly lower e n e r g i e s . F i g u r e from A II Collab.(f' 97c).
g r o w t h in t h e i r p o p u l a r i t y . S t u d i e s of inclusive a n d idcntilied p a r t i c l e inultiplieities. s e a l i n g v i o l a t i o n s in f r a g m e n t a t i o n f u n c t i o n s e t c . h a v e s i n c e a d o p t e d t h e scale y = £ j e t s i n ( ( 9 , „ i „ 2 ) for b o t h q u a r k a n d g l u o n e t s ( D E L P H I C o l l a b . I!)!)!t' . 20006; O P A L C o l l a b . 20006). T h e c o n s i s t e n c y of t h e d a t a s e t s in t h e s e a n a l y s e s , b o t h i n t e r n a l l y a n d . in t h e c a s e of q u a r k e t s , a l s o w i t h e x p e r i m e n t a l r e s u l t s o b t a i n e d at lower C . o . M . e n e r g i e s , a s well a s t h e a g r e e m e n t w i t h t h e o r e t i cal e x p e c t a t ions h a s given c o n v i n c i n g e v i d e n c e for t he utility of t h e s e t o p o l o g i c a l energy scales. 12.3.2
Multiplicities
A m a i n f o c u s of e x p e r i m e n t a l a t t e n t i o n h a s b e e n t h e m e a s u r e m e n t of q u a r k a n d g l u o n e t s ' ( a v e r a g e ) c h a r g e d p a r t i c l e m u l t i p l i c i t i e s a n d t h e t e s t i n g of c q n ( 1 2 . f ) ( G a r y . I!)!)-l). W h i l s t e a r l y r e s u l t s s u g g e s t e d t h a t g l u o n e t s h a v e a h i g h e r mill-
D I I T K H F N C K S 111 IVVKKN (¿UAItK A N D O N I O N J!•:IS
liplicity ilian cpiark j e t s , t h o u g h by a s i g n i f i c a n t l y lower r a t i o t h a n C \ / ( ' r . ¡1 was not until I . E l ' t h a t r e l i a b l e , q u a n t i t a t i v e m e a s u r e m e n t s b e c a m e a v a i l a b l e T h e lirst I.KP r e s u l t s , b a s e d oil t h e st u d y of Y - s h a p e d e v e n t s a n d t h e use of hq u a r k t a g g i n g , y i e l d e d v a l u e s s u c h a s ( . V g ) / ( A ' ( | ) = 1.27 ± 0 . 0 7 . for j e t s of e n e r g y 24 CieV ( O P A L C o l l a b . . 199Hc). which c o n f i r i n e d e a r l i e r q u a l i t a t ive st tidies. I lowever. t h e s e r e s u l t s used t h e j e t ' s e n e r g y a s its scale a n d m o r e s i g n i f i c a n t l y were sensitive t o t h e choice of jet a l g o r i t h m used in t h e a n a l y s i s ( D E L P H I C o l l a b . . I996c; O I ' A I . C o l l a b . . 1995i/). la)
I I1I I I I I I I I I I I I I I iI •I I• I1 1I 1I I i • • •I I• I iI1I1 I 11
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F t c . 1 2 . 2 . C h a r g e d p a r t i c l e m u l t i p l i c i t y d i s t r i b u t i o n s for inclusively d e f i n e d g l u o n j e t s of e n e r g y 41.8 CieV ( t o p ) a n d light ( u d s ( c ) ) q u a r k j e t s of e n e r g y 45.6 G e V ( b o t t o m ) , c o m p a r e d t o predict ions f r o m c o h e r e n t M o n t e C a r l o event generators. T h e error bars are the statistical and systematic errors combined in q u a d r a t u r e : t h e s m a l l e r b a r s g i v e t h e s t a t i s t i c a l e r r o r a l o u e . F i g u r e f r o m OPAL Collab.(1998«).
As n o t e d a b o v e , o n e w a y l o avoid t h e s i d e - e f f e c t s of a jet a l g o r i t h m is t o u s e t h e inclusive d e f i n i t i o n of a g l u o n a s t h e w h o l e of t h e h e m i s p h e r e o p p o s i t e l o o n e c o n t a i n i n g t w o t a g g e d h e a v y q u a r k j e t s . A c o r r e s p o n d i n g d e f i n i t i o n for a
EXPERIMENTAL I ' l i o i ' l III II ' ¡ o r QUARK AND CI HON .IKTS
1)1 I I'lll < iVxai»«» / «>IAII hi I I'lll iMjten-n»x CI lOV(IS) < H'AI. availing vliion
( > 7 8 9 10 y i (GcV)
20 25 C'I(CK-V)
30
35
l i t ; . 12.3. Average charged particle multiplicities (left) a n d multiplicity r a t i o s (right) for light q u a r k a n d gluon j e t s as a function of t h e topological scale QT- T h e s h a d e d h a n d indicates t h e r a t i o of t h e derivatives. Figure f r o m DKLPIII Collab.( 19996).
<|uark is as o n e half of a h a d r o n i c event. F i g u r e 12.2 s h o w s t h e m e a s u r e d c h a r g e d particle multiplicity d i s t r i b u t i o n s of light (uds) q u a r k j e t s of energy 45.6 G o V a n d gluon jet« of e n e r g y 41.8 GoV ( O P A L Collab.. 1998a). O n l y light q u a r k s a r e included so as to b e t t e r c o r r e s p o n d t o t h e a n a l y t i c c a l c u l a t i o n s which employ inassless kinematics. T h e gluous have a larger m e a n multiplicity, s h o r t e r tail a n d more Gaussian-like d i s t r i b u t i o n t h a n q u a r k s . A f t e r using a M o n t e C a r l o model to correct for t h e slightly higher q u a r k jet energy, t h e following ratios for the lirsl m o m e n t s of t h e dist r i b u t i o n s a r e o b t a i n e d at. 11.8 G c V . Mean: Width: Skexvness: Kurtosis:
// o 7 n
= = = =
(N) y / ( ( N - /0'-') ((N - ¡tfS/o* ((N - it)4)/a* - 3
/ ; K / / i ( | = 1.171 :fc().()2 l ± 0 . 0 4 3 7g/7<1 kk/K(1
= i.or>r> ± 0 . 0 1 6 ± 0 . 0 5 5 = 0.47 ± 0 . 1 6 ±0 . 2 1 = 0.19 ± 0 . 3 7 ± 0 . 3 3
The N ' L O . a n a l y t i c prediction for t he r a t i o of m e a n multiplicities is in modest agreement with t h e d a t a , b e i n g a b o u t Lr>% higher. B e t t e r agreement is found with t h e predict ions of t h e coherent M o n t e C a r l o event g e n e r a t o r s , also for t h e higher m o m e n t s . T h i s suggests that it is i m p o r t a n t t o fully t r e a t energy m o m e n t u m conservation in t h e calculations. F u r t h e r results a r e available for t h e normalized factorial a n d cmnulant. factorial m o m e n t s of these d i s t r i b u t i o n s t o g e t h e r with their ratios ( O P A L C'ollab.. 1998«), Again fair agreement is found with coherent Monte C a r l o event g e n e r a t o r s a n d with a n a l y t i c p r e d i c t i o n s at N N L O . T h e value of t he r a t i o of m e a n multiplicities is predicted t o vary with scale beyond LO. In ( D E L P H I Collab.. 1996c) such a d e p e n d e n c e 011 t h e energy of t h e p a r l o u s identified within three-jet e v e n t s was e s t a b l i s h e d . T o d o this for
DIKKKHHNOKS III I'W I'.KN QUAKK AND (¡LUON .IK I S
inclusively d e f i n e d j o t s , w h o s e sealo is \ / 3 / 2 . i n e v i t a b l y involves t h e c o m p a r i s o n of r e s u l t s f r o m different e x p e r i m e n t s . A n a l t e r n a t i v e is t o s t u d y t h e v a r i a t i o n of t h e q u a r k a n d g l u o n jot p r o p e r t i e s over t h r e e - j e t p h a s e s p a c e a s a f u n c t i o n of a t o p o l o g i c a l e n e r g y s c a l e ( D E L P H I C o l l a b . , 19996: O P A L C o l l a b . . 20006). F i g u r e 12..'5 (left) s h o w s t h e m e a n m u l t i p l i c i t i e s of q u a r k a n d g l u o n jets a s f u n c t i o n o f Q i . G l u o n j e t s s h o w a s t r o n g e r growt h t h a n q u a r k j o t s w i t h i n c r e a s i n g scale, a n effect which is s t r o n g e r a t larger scales. T h i s effect is c o n f i r m e d b\ t h e m u l t i p l i c i t y r a t i o a s a f u n c t i o n of t h e scale. Fig. 12.3 ( r i g h t ) . T h e highei o r d e r , a n a l y t i c p r e d i c t i o n for t h e s h a p e of t h e m u l t i p l i c i t y r a t i o , oqu (12.1). is in q u a l i t a t i v e a g r e e m e n t w i t h t h e d a t a hut r e m a i n s t o o h i g h , p a r t i c u l a r l y so at lower scales. T h e s h a d e d b a n d in Fig. 12.3 ( r i g h t ) is t h e r a t i o of d e r i v a t i v e s of t h e a v e r a g e m u l t i p l i c i t i e s , which h a s a n a l m o s t c o n s t a n t v a l u e 1.07 ± 0 . 1 0 ( D E L P H I C o l l a b . , 19096). A s i m i l a r m e a s u r e m e n t by t h e O P A L C o l l a b o r a t i o n (20006) g a v e 2.27 ± 0.20. T h e r a t i o of t h e d e r i v a t i v e s is e x p e c t e d to b e less s e n s i t i v e t o h a d r o n i / . a l i o n c o r r e c t i o n s t h a n t h e r a t i o of t h e m u l t i p l i c i t i e s a n d t:. p r e d i c t e d t o h a v e t h e a s y m p t o t i c value C.\/C'I•• a n d a value a r o u n d 1.92 at present scales (C'apolla el til.. 2000). An indirect a p p r o a c h t o q u a r k a n d g l u o n jet mult ¡plieities which a v o i d s s o m e of t h e need t o a s s i g n p a r t i c l e s t o j e t s is p r o v i d e d by a st u d v of t h e t o t a l mult iplieitv in a t h r e e - j e t event ( D E L P H I C o l l a b . . 19996). B a s e d u p o n M I . L A c a l c u l a t i o n s t h i s is p r e d i c t e d t o h e ( D o k s h i t z e r <1 ill.. 1988)
(12.8) = .V,..,. ( v ^ v 7 / ) + IHQ'R)
[A"'"
AN,
In t h e s e c o n d f o r m u l a we h a v e e x p r e s s e d t h e qiuirk j e t m u l t i p l i c i t y in t e r m s of t h e t o t a l e v e n t m u l t i p l i c i t y a n d t h e g l u o n j e t m u l t i p l i c i t y in t e r m s of t h e q u a r k jet multiplicity. T h e f a c t o r H{Q'Y) is given by e q n (12.1) a n d AA r o is a n e m p i r i c a l olfset d e s i g n e d t o a c c o u n t for a l a r g e r l e a d i n g p a r t i c l e m u l t i p l i c i t y in q u a r k j e t s . A g o o d s i m u l t a n e o u s lit of e q n (12.8) t o t h e t o t a l a n d three-jet m u l t i p l i c i t i e s , a s a f u n c t i o n of t h e inter-jet a n g l e s , is p o s s i b l e w i t h C.\/CJ2.240 ± 0 . 1 3 9 ( D E L P H I C o l l a b . , 19996). As a spin-off of t h i s a p p r o a c h o n e c a n r e a r r a n g e e q n (12.8) t o give a d e f i n i t i o n of t h e g l u o n m u l t i p l i c i t y a t t h e scale Q'j.. A d o p t i n g t h i s d e f i n i t i o n gives s i m i l a r r e s u l t s t o t h o s e q u o t e d a b o v e . A useful v a r i a n t of t he s t a n d a r d i n v e s t i g a t i o n s i n t o t h e m u l t i p l i c i t y d i s t r i b u t i o n s in e v e n t s a r e p r o v i d e d by t h e siibjet m u l t i p l i c i t i e s ( A L E I ' l l C o l l a b . . 20006: D E L P H I C o l l a b . , 19986) i n t r o d u c e d iu S e c t i o n 11.5. U s i n g a jet c l u s t e r i n g algor i t h m . a u e v e n t t h a t is f o u n d t o c o n t a i n t h r e e j e t s a t a r e s o l u t i o n p a r a m e t e r I/I c a n be re-clust.ercd a t a s e c o n d s m a l l e r c u t - o f f . //o <
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l<;. 12.-1. M o a n a n d w i d t h of tlio s u h o t m u l t i p l i c i t i e s in q u a r k a n d gluoii o t s a s a f u n c t i o n of t ho s u h e t r e s o l u t i o n p a r a i n e t o r (>. Also s h o w n a r e t h e g l n o n q u a r k r a t i o s of t h e s e q u a n t i t i e s . T h e d a t a a r e c o m p a r e d t o predictions from various coherent M o n t e Carlo event generators. Figure from A Il Collah.(20006).
I>11-1KHKN< I S III I WKKN (il lAUK AND (¡1.1 ION .11. I S
F i g u r e 12.-I s h o w s t h e m e a n s a n d w i d t h s of I lie s u b j e t m u l t i p l i c i t y d i s t r i b u t i o n s for identified q u a r k a n d g l u o n j e t s t o g e t h e r w i t h t h e i r r a t i o s ( A L K I ' I I Collal>.. 201106). T h e d a t a were o b t a i n e d f r o m t h r e e - j e t e v e n t s , defined using t h e D u r h a m (A-/ ) a l g o r i t h m w i t h a p r i m a r y r e s o l u t i o n p a r a m e t e r in = 0.1 (/.•/ 2 8 . 8 C e V ) . T h e r e s u l t s a r e for j e t s i n t e g r a t e d o v e r t h e w h o l e a v a i l a b l e p h a s e s p a c e : c o n s e q u e n t l y t h e q u a r k j e t s h a v e a h i g h e r a v e r a g e e n e r g y (31.0 G e V ) t h a n t h e g l u o n j e l s (28 G e V ) . T h e a v e r a g e m u l t i p l i c i t i e s m i n u s o n e a r e s h o w n in o r d e r t o f o c u s a t t e n t i o n o n t h e a d d i t i o n a l , r a d i a t e d p a r t o n s in t h e j e t . As r e q u i r e d bv t heir d e f i n i t i o n (A',,. K (t/o) — I) b o t h rise s t e a d i l y a s ;/<> d e c r e a s e s , f r o m 0 at IJo = i/I t o close t o ( o n e less t h a n ) t h e p a r t i c l e m u l t i p l i c i t i e s at i / o ^ l l l (/, /• = 0.1 G e V ) . A s i m i l a r g r o w t h is seen in t h e w i d t h s ir<| K(v/o) a s i/o d e c r e a s e s f r o m t/i- Looking at t h e r a t i o R„(yo) = {Af g (i/o) l)/{iV<,(i/o) - 1) a s a f u n c t i o n of j/(l we find t h a t it c l i m b s t o a p e a k value « 1.87 for i/o ~ 0 . 0 0 3 (k-r — 5 G e V ) b e f o r e falling back t o « 1.21 in t h e i n d i v i d u a l p a r t i c l e limit. T h e LLA p r e d i c t i o n is C,\/C/• 2.2-r>. S i m i l a r l y Ra(y») = iT g(i/»)/ . T h e LLA p r e d i c t i o n is s / c j c ^ = 1-r,. E v e n t h o u g h t h e n u m e r i c a l values of t h e e x p e r i m e n t a l r e s u l t s a r e s e n s i t i v e t o t h e d e t a i l s of t h e event a n d jet selection (DELIMII C o l l a b . . 19986), t h e m e a s u r e m e n t s c a n n e v e r t h e l e s s be c o m p a r e d t o a n u m b e r of p r e d i c t i o n s . At t h e h a d r o n level coherent M o n t e C a r l o event, g e n e r a t o r s , s h o w n in F i g . 12.-1. p r o v i d e fair d e s c r i p t i o n s of all tin- d i s t r i b u t i o n s w h i c h s p a n u p t o t h r e e o r d e r s of m a g n i t u d e . However, if c o m p a r i s o n s a r e m a d e t o t h e final s t a t e p a r t o n level p r e d i c t i o n s t h e n d e v i a t i o n s a r e seen for / / o ^ l O (A /- = 2.9 G e V ) . i n d i c a t i n g t h a t below I his scale h a d r o n i / a t i o n correct ions start, t o b e c o m e ini])ortant.. For t h e m e a n s u b j e t m u l t i plicities t w o a n a l y t i c c a l c u l a t i o n s a r e a v a i l a b l e . T h e first is b a s e d o n t h e l e a d i n g o r d e r ( L O ) 0 ( < \ 2 ) m a t r i x e l e m e n t s , t h e s e c o n d o n e is i m p r o v e d by t h e inclusion of s u m m e d l e a d i n g a n d n e x t - t o - l e a d i n g h i ( t / i / j / o ) l o g a r i t h m s (LO-l N L L A ) (Seym o u r . 1990). T h i s L Q + N L L A c a l c u l a t i o n c o n t a i n s h i g h i T o r d e r t e r m s t h a n f o u n d in t h e p a r t o n s h o w e r i m p l e m e n t a t i o n s but d o e s not r i g o r o u s l y i m p o s e e n e r g y m o m e n t u m c o n s e r v a t i o n . T h e s u m m e d r e s u l t s m a y be e x p e c t e d t o offer a b e t t e r d e s c r i p t i o n of t h e s m a l l i/o r e g i o n , w h e r e t h e 11i(//1/(/(>) t e r m s b e c o m e i m p o r t a n t , but. t h i s must, be c o u n t e r b a l a n c e d by t h e e x p e c t e d g r o w i n g i m p o r t a n c e of t h e m i s s i n g h a d r o n i z a t i o n c o r r e c t i o n s . T h e L O p r e d i c t i o n o n l y d e s c r i b e s t he a v e r a g e s u b j e t m u l t i p l i c i t i e s for t h e highest ;/f a g r e e m e n t is not a s g o o d a n d b o t h t h e LO a n d L O I N L L A p r e d i c t i o n s a r e o n l y a d e q u a t e for high i/u values. For t h e w i d t h s of t h e s u b j e t m u l t i p l i c i t y d i s t r i b u t i o n s . c ( | . K (//o), a n a l y t i c p r e d i c t i o n s a r e o n l y a v a i l a b l e in LLA whilst for t h e i r r a t i o . R„. a i/o i n d e p e n d e n t . N L L A c a l c u l a t i o n is a v a i l a b l e ( M a l a z a a n d W e b b e r . 1984). b u t see a l s o D o k s h i t z e r (1993). However. t h e a g r e e m e n t b e t w e e n t h e s e c a l c u l a t i o n s a n d t h e e x p e r i m e n t a l r e s u l t s is
K X I ' K I H M K N T A L I ' L L T H T I I T I I S ()!•' (¿11 A U K
\ N I > C I . H O N .11 I S
g e n e r a l l y not us g o o d us lor tlx* m e a n values. I2.:t.;i
Jet
profiles
As not,ed e a r l i e r , difference's a r e e x p e c t e d in t he t r a n s v e r s e sizes of q u a r k a n d gliion j e t s . T h i s h a s heen i n v e s t i g a t e d u s i n g a n u m b e r of v a r i a b l e s , i n c l u d i n g i lie r a p i d i t y a n d t r a n s v e r s e m o m e n t u m d i s t r i b u t i o n s of p a r t i c l e s w i t h i n a jet ( O P A L C o l l a b . . 1999u) a n d t h e j e t b r o a d e n i n g ( A L K P I I C o l l a b . , 2000/;; DKL1*111 C o l l a b . . 19986). T h e l a t t e r is d e f i n e d , in a n a l o g y w i t h t h e event, s h a p e varia b l e s w i d e a n d n a r r o w jet b r o a d e n i n g ( C a t a n i et til., 1992/;). t o be n
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w h e r e Tij,., is t h e jet d i r e c t i o n . Il is e s s e n t i a l l y a u e n e r g y w e i g h t e d a v e r a g e over t h e a n g l e s of t h e p a r t i c l e s in a jet w i t h respect t o t h e jet axis. F i g u r e 12.!i s h o w s t lie m e a s u r e d jet b r o a d e n i n g d i s t r i b u t i o n s for q u a r k a n d gliion j o t s ( A L K P I I Collab.. 2000/;). T h e d a t a is b a s e d on three-jet e v e n t s selected u s i n g t h e D u r h a m a l g o r i t h m w i t h j/,.,„ = 0 . 1 .
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FIG. 12..r>. .let b r o a d e n i n g a s m e a s u r e d in q u a r k (left) a n d gliion j e t s ( r i g h t ) , a s measured over three-jet phase space, compared to predictions from coherent M o n t e C a r l o event g e n e r a t o r s . F i g u r e f r o m A L K P I I Collab.(2000/;).
T h e jet b r o a d e n i n g for q u a r k s c l e a r l y is m o r e tight ly dist r i b u t e d a b o u t lower values t h a n is t h e c a s e for gliions, i n d i c a t i n g t h a t q u a r k j e t s a r e indeed n a r r o w e r t h a n gliion j e t s . H o w e v e r , we s h o u l d c a u t ion t h a t t h e s e q u a r k j e t s h a v e o n a v e r a g e I 1% m o r e e n e r g y t h a n t he gliion jets a n d that t h e effect of a b o o s t , t h o u g h s m a l l in t h i s i n s t a n c e , is t o collimat.e a j e t . At large B¡,.,. p ( J C D e f f e c t s a r e e x p e c t e d t o d o m i n a t e a n d i n d e e d t h e d a t a a r e d e s c r i b e d by coherent M o n t e C a r l o event g e n e r a t o r s at t h e p a r t e m level. At s m a l l e r B j r , . h a d r o m z u t i o n a f f e c t s t h e p a r t o u i c j e t s t o g i v e r b r o a d e r l t a d r o n level j e t s w h i c h a r e in b e t t e r overall agreement, w i t h
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F i e . 1 2 . 0 . T h e q u a r k a n d g l u o n f r a g m e n t a t i o n f u n c t i o n s at. t h r e e v a l u e s of t h e t o p o l o g i c a l s c a l e Q-y. T h e c u r v e s a r e t h e result of a s i m u l t a n e o u s lit s a t i s f y i n g the time-like D G L A P equations. F i g u r e from D E L P H I Collab.(20006).
t h e d a t a : t h e s e a r e s h o w n in Fig. 12."). T h e s c a l e d e p e n d e n c e of t h e a v e r a g e jet b r o a d e n i n g h a s a l s o b e e n i n v e s t i g a t e d ( D E L P H I C o l l a b . , 19986). It is f o u n d t h a t (/?j,.t)(Q'l') is a p p r o x i m a t e l y c o n s t a n t , w i t h {Z? J^.,)/(/?[4,.t) w 1.5. r e f l e c t i n g t h e fact t h a t for p a r t i c l e s in a jet ¡>\. a n d p r a p p r o x i m a t e l y s c a l e w i t h ( ¿ y . In c o n t r a s t )(/Tj < -,) falls wit h E j , i . i n d i c a t i n g t h a t h i g h e r e n e r g y j e t s a r e n a r r o w e r . 12.3.1
Finf/mcntation
functions
E a r l y L E P m e a s u r e m e n t s u s i n g light q u a r k a n d g l u o n j e t s a m p l e s e x t r a c t e d f r o m lixed t o p o l o g y e v e n t s h a v e c o n f i r m e d t h e t h e o r e t i c a l e x p e c t a t i o n s for t h e f r a g m e n t a t i o n f u n c t i o n s ( A L E P H C o l l a b . . 1990/;: D E L P H I C o l l a b . . 19986: O P A L Collab., 1995c). L a t e r L E P s t u d i e s h a v e e x p l o i t e d t he full t hrce-jet p h a s e s p a c e u s i n g t h e t o p o l o g i c a l scales, s u c h a s Q-y, t o s t u d y t h e e v o l u t i o n of t h e f r a g m e n t a t i o n f u n c t i o n s ( D E L P H I C o l l a b . , 20006). T h e c o r r e c t c h o i c e of s c a l e is i m p o r t a n t as it specifies b o t h t h e s t r e n g t h of t h e c o u p l i n g a n d t h e size of t h e a v a i l a b l e p h a s e s p a c e , which we e x p e c t t o b e r e d u c e d by a n g u l a r o r d e r i n g effects. We c a n h a v e f a i t h in t h i s a p p r o a c h s i n c e t h e q u a r k f r a g m e n t a t i o n f u n c t i o n , e x t r a c t e d f r o m t h r e e - j e t e v e n t s a s a f u n c t i o n of Q-y. is in g o o d a g r e e m e n t w i t h t h o s e o b t a i n e d f r o m e + e e x p e r i m e n t s r u n at lower C'.o.M. energies. As Fig. 12.0 s h o w s , t h e g l u o n d i s t r i b u t i o n is s i g n i f i c a n t l y s o f t e r t h a n t h e q u a r k d i s t r i b u t i o n w i t h a higher p e a k at. s m a l l v a l u e s of xy; = E\m,{,,m/Ej,., and a faster, approximately e x p o n e n t i a l , fall off in t h e r e g i o n x-n —> 1. F u r t h e r m o r e , t h e f r a g m e n t a t i o n f u n c t i o n s get. s o f t e r w i t h i n c r e a s i n g Q-y. T h a t is. a b o v e xy; ~ 0.1 t h e f r a g m e n t a t i o n f u n c t i o n s d e c r e a s e a s Q-y i n c r e a s e s whilst below xy; ~ 0.1 t h e y increase. T h i s s o f t e n i n g is m o s t p r o n o u n c e d for g l u o n j e t s . T h e r e s u l t s in t h e s i n a l l - x region, which is d o m i n a t e d by wide a n g l e soft g l u o n s , a r e s o m e w h a t s e n s i t i v e t o t h e choices of jet. a l g o r i t h m a n d t o p o l o g i c a l scale v a r i a b l e . T h u s , a s vet, o n l y q u a l i t a t i v e s t u d i e s a r e j u s t i f i e d . T h e p e a k re-
KXPEHIMENTAI
l'IIOI'l III I I S »)!•' Q U A K E AND CI.HON .11 I S
:tsi>
glotis of t h e q u a r k anil g l u o n f r a g m e n t a t i o n f u n c t i o n s a r c b o t h well d e s c r i b e d l>v G a u s s i a n (list ribnt.inns in ln( 1 /:>-'); o.f. S e c t i o n 11.2. F n r t l i e n n o r e , t h e titled p e a k position ln( L/.r,,,.;,(,) is c o i n p a t i h l e wit h a 0 . 5 I U ( G t / Q o ) h e h a v i o n r . that. is. the s a m e s c a l e d e p e n d e n c e t h a t w a s seen in t h e overall f r a g m e n t a t i o n f u n c t i o n lor e + e e v e n t s at. d i f f e r e n t C'.o.M. e n e r g i e s . In t h e large-.? - region. x ¡ ¡ > 0.15. t h e f r a g m e n t a t i o n f u n c t i o n s a r e s t a b l e against t h e d e t a i l s of t h e j e t a l g o r i t h m . H o w e v e r , it. is still i m p o r t a n t t o use Q i a s t h e scale w h e n t e s t i n g t h e s c a l i n g v i o l a t i o n s p r e d i c t e d by t h e t i m e - l i k e D G L A P e<|uatiojis. e q n (3.285). ( N a s o n a n d W e b b e r . 15)94). T h e result, is a n excellent s i m u l t a n e o u s lit. o v e r wide ./:/.; a n d Q-¡ r a n g e s , of t h e q u a r k a n d g l u o n f r a g m e n t a t i o n f u n c t i o n s a s m e a s u r e d in a single e x p e r i m e n t . F u r t h e r m o r e , t h e r a t i o of t h e r e l a t i v e r a t e s of c h a n g e of t h e g l u o n a n d q u a r k f r a g m e n t a t i o n f u n c 2.20 ± 0.10 ( D E L P H I C o l l a b . . 20006): tions allows t h e e x t r a c t i o n of CA/CF see Ex. (12-2). 12..'1.5
Partirle
content
In p e r t u r b a t i v e Q C D t h e e v o l u t i o n w i t h e n e r g y s c a l e of a jet is d e s c r i b e d by a p a r t o n (or d i p o l e ) s h o w e r . T h e h a d r o n i / a l i o n . which c o n v e r t s liual stall- p a r t o n s at a low. lixed v i r t n a l i t y i n t o h a d r o n s . d o e s not d i r e c t l y d e p e n d o n t h e initial scale of t he j e t . I n d e e d , t h e s t a n d a r d h a d r o n i / a l ion m o d e l s m a k e n o dist i n c t i o n bet ween t h e p a r t o n s p r o d u c e d a t t h e e n d of a q u a r k o r a g l u o n i n i t i a t e d s h o w e r : see S e c t i o n 3.8. C o n s e q u e n t l y , p e r t u r b a t i v e p r e d i c t i o n s for t h e d i f f e r e n c e s between q u a r k a n d g l u o n j e t s a r e u n i v e r s a l a n d a p p l y e q u a l l y well t o a n y identified p a r t i c l e s jlist a s t o t h e filial s t a t e p a r t o n s . T h u s , for e x a m p l e , e q n (12.1) is exp e c t e d t o hold for all t y p e s of identified p a r t i c l e s . F u r t h e r m o r e , t he relat ive r a t e s of p a r t i c l e s w i t h i n a q u a r k o r g l u o n jet a r e not p r e d i c t e d t o vary w i t h t h e j e t ' s scale. T h a t , s a i d , s m a l l d i f f e r e n c e s a r e a n t i c i p a t e d d u e t o n e g l e c t e d e f f e c t s w i t h i n the p Q C D calculations. T h e s e include effects coming from unequal phase spaces d u e t o different p a r t i c l e m a s s e s a n d p e r h a p s m o r e significant ly t h e ' l e a d i n g p a r ticle e f f e c t ' c o m i n g f r o m t h e p r e s e n c e of f l a v o u r iu a q u a r k i n i t i a t e d j e t . fu t h e most ext reine e x a m p l e , s i n c e h - q n a r k s a r e o n l y r a r e l y p r o d u c e d in a s h o w e r (Seym o u r . 1995: A b b a n e o et at.. 2 0 0 1 a ) . t h e n u m b e r of B - h a d r o n s o c c u r i n g in g l u o n j e t s c o m p a r e d t o q u a r k j e t s will b e m u c h lower t h a n p r e d i c t e d by e q n (12.1). T h e a b o v e e x p e c t a t i o n of u n i v e r s a l i t y is g e n e r a l l y believed t o be r o b u s t . T h e r e are. h o w e v e r , s o m e u n o r t h o d o x h a d r o n i z n t i o n s c h e m e s w h i c h suggest t h e p r o d u e tion of g l u e b a l l s a n d o t h e r e x o t i c s t h a t f a v o u r e n h a n c e d isoscalar p r o d u c t i o n , for e x a m p l e , of // o r //' m e s o n s in g l u o n j e t s ; for s o m e d i s c u s s i o n s e e ( A L E P I I C o l l a b . . 2000c) o r ( K n o w l e s a n d L a f f e r t y , 1997). S t u d i e s of identified p a r t i c l e p r o d u c t i o n in q u a r k a n d g l u o n j e t s h a v e foP\[(Q). cused on m u l t i p l i c i t y r a t i o s , s u c h a s R*(Q) = (A 7 *(Q))/{.*>%((})) a n d fíh(Q) = {N*(Q))/(N¡l(Q)) a n d n\, = R f j n \ \ = /.',,//?,',,. w h e r e t h e i n d e x "If d e n o t e s a n identified l i a d r o n a n d ' c h ' all liual s t a l e c h a r g e d p a r t i c l e s . Pert u r b a tive Q C D p r e d i c t s that. Pft(Q) a n d P[\(Q) a r e i n d e p e n d e n t of t h e s c a l e Q a n d also H\t(Q) ~ /?,.|,(Q) so t h a t t h e d o u b l e r a t i o P'u((¿) % 1. T h e a d v a n t a g e t o
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KXI'KKIMKN IAl I ' l t o l ' l I d IKS u l
Q U A R K AND CI i j o N .11 I S
(li.-jet energy /vj,., (Ol'AI. ('oliali.. 199X6). I'lie iiKiiueiitiini s p e c t r a in q u a r k a n d g h i o n j e t s a s a f u n c t i o n of ,r ;) Pu/l't< i liave been m e a s u r e d for identified pious, k a o n s a n d ( a n t i ) p r o t o n s ( D E L P H I C'oliali.. 2000c). T h e s e coiilirm t h a t t h e g h i o n jet 's m o m e n t u m spect ra a r e all s o f t e r t h a n in a n e q u i v a l e n t q u a r k j e t . F u r t h e r m o r e , t h e e x c e s s of p r o t o n s h a s b e e n t r a c e d t o t h e region of m o d e r a t e l y large m o m e n t a 0.10 < ./•,, < 0.50. A G a u s siau lit t o t h e ( = l n ( l / . e , , ) d i s t r i b u t i o n for identified p a r t i c l e s s h o w s t h a t t h e peak position g r o w s w i t h t h e j e t s c a l e Q y iu a way t h a t is consistent, w i t h llu- l o g a r i t h m i c g r o w t h seen for a s m e a s u r e d w i t h all c h a r g e d p a r t i c l e s ; see Section 11. Iu q u a r k j e t s it h a s b e e n f o u n d , a t a given Q-y, t h a t > ~ whilst in g h i o n ¡(its > > Also c o m p a r i n g t h e i n d i v i d u a l p e a k p o s i t i o n s for ?r. K a n d p m e a s u r e d in q u a r k a n d g h i o n j e t s of e q u a l Q-y, o n e f i n d s > (.,]• iu <|iialitative a g r e e m e n t w i t h t h e p Q C D p r e d i c t i o n ( F o n g a n d W e b b e r . 1989).
(12.10) A s i m i l a r s h i f t in (,' h a s b e e n seen in t h e c h a r g e d p a r t i c l e mull iplicitv d i s t r i b u t i o n ( D E L P H I C o l l a b . . 20006). As n o t e d e a r l i e r , t h e pC^CT!> c a l c u l a t i o n s of identified h a d r o n s p e c t r a a r e exp e c t e d t o receive c o r r e c t i o n s d u e t o p a r t i c l e m a s s e s a n d o t h e r n o n - p e r t u r b a t i v e effects. O n e w a y t o a c c o u n t for t h e s e c o r r e c t i o n s is t o u s e M o n t e C a r l o event g e n e r a t o r s wit h t h e i r exact k i n e m a t i c s a n d d e t a i l e d h a d r o n i / a t ion m o d e l s . T h e s i m u l a t i o n s a l s o a l l o w t h e specifics of t h e jet d e f i n i t i o n t o b e t a k e n i n t o a c c o u n t . It is t h e r e f o r e c o m m o n t o c o m p a r e t h e d a t a on r a t e s a n d m o m e n t u m s p e c t r a t o M o n t e C a r l o p r e d i c t i o n s . T h i s h a s p r o v i d e d m i x e d r e s u l t s , h o w e v e r , it a p p e a r s I hat t h e 11ERWK! c l u s t e r s c h e m e is less s u c c e s s f u l at d e s c r i b i n g t h e d a t a t h a n for e x a m p l e t h e J E T S E T m o d e l ( A L E P H C o l l a b . . 2000c; D E L P H I C o l l a b . . 2 0 0 0 c : O P A I . C o l l a b . , 19996). Exercises for C h a p t e r
12
12 1 A set. of o n e f o l d s y m m e t r i c e v e n t s is d e f i n e d w i t h t w o inter-jet. a n g l e s e q u a l t o 150° a n d a set of t h r e e f o l d s y m m e t r i c e v e n t s w i t h all interj e t a n g l e s e q u a l t o 120°. At t h e Z. what a r e t h e e n e r g i e s a n d Q-Y scales of t h e j e t s in t h e t w o event t y p e s ? 12 2 B y c o n s i d e r i n g t h e time-like D C L A P e q u a t i o n s , s h o w t h a t t h e r a t i o of t h e r e l a t i v e sizes of t h e s c a l i n g v i o l a t i o n s in t h e f r a g m e n t a t i o n f u n c t i o n s of q u a r k s a n d gliions a p p r o a c h a constant.. ¿HnD^rVyain/t2 3
0IuDn(x,ii )/d111//
ill the limit .r— 1. »*>
CA 2
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13
FR G
E T TI
F r a g m e n t a t i o n is tilt process by which t h e p r i m a r y p a r l o u s p r o d u c e d in a h a r d snliproeess convert into t h e s p r a y s of h a d r o n s seen by e x p e r i m e n t s . We have learnt that this process is usefully s e p a r a t e d i n t o t w o s t a g e s , a c c o r d i n g t o the m o m e n t u m t r a n s f e r s involved. e r t u r b a t i v e Q is a p p l i c a b l e t o t h e t r a n s i t i o n f r o m t h e highly v i r t u a l p r i m a r y p a r t o n s t o a set of low-virtuality final s t a t e p a r t o n s . T h i s is p i c t u r e d as a c a s c a d i n g process t h a t is d o m i n a t e d by t h e eollinear a n d soft emissions of gliions a n d mainly, t h o u g h not exclusively, light q u a r k a u t i q u a r k pairs. By c o n t r a s t , m o d e l s a r e used t o d e s c r i b e t h e n o n - p e r t u r b a t i v e t r a n s i t i o n f r o m these final s t a t e p a r t o n s to h a d r o n s which then may be decayed according to tables or f u r t h e r models. F o r t u n a t e l y t h e calculable, p e r t u r b a t i v e s t a g e is responsible for much of an e v e n t ' s s t r u c t u r e , even including t o some extent t h e m o m e n t u m s p e c t r u m of t h e p r o d u c e d h a d r o n s . T h a t said, i m p o r t a n t details d e p e n d on t h e specific t y p e of h a d r o n p r o d u c e d a n d a t present this can only be modelled. T h u s , for e x a m p l e , whilst we m a y be a b l e t o predict s o m e t h i n g of t h e m o m e n t u m s p e c t r u m of kaons. we find it more difficult to say how m a n y will b e p r o d u c e d in a given s i t u a t i o n . A p a r t i a l exception is t h e total r a t e of h a d r o n s c o n t a i n i n g heavy q u a r k s , e a n d b. which a r e n o t believed t o be p r o d u c e d during hadroni ation. Whilst m a n y t y p e s of particles a r e p r o d u c e d in a high energy hadronic event, only a few live long e n o u g h t o be d e t e c t e d directly. In a typical experiment these are e , . TT . p. p a n d which a r e e i t h e r absolutely s t a b l e o r effectively so. having lifetimes longer t h a n 1 0 - h s . which p e r m i t s t h e m to t r a v e r s e t h e d e t e c t o r b e f o r e they decay. h o t o n s can b e d e t e c t e d w i t h a n e l e c t r o m a g n e t i c calorimeter, n e u t r o n s a n d mesons can in s o m e cases still be recorded with a h a d r o n calorimeter, while n e u t r i n o s e s c a p e u n d e t e c t e d , e.f. Section 5.2. T h e presence of all o t h e r particles can only be inferred f r o m t h e i r long-lived decay products. x t r a p o l a t i n g p a r t i c l e t r a c k s t o w a r d s t h e p r i m a r y p r o d u c t i o n vertex, experi m e n t s with high-resolution t r a c k i n g d e t e c t o r s , ideally silicon s t r i p d e t e c t o r s , may still be c a p a b l e of seeing t h e c h a r a c t e r i s t i c displaced vertices associated with weakly d e c a y i n g p a r t i c l e s of modest lifetimes, such a s r f T leptons, t mesons, h y p e r o n s . that is. liaryons c o n t a i n i n g o n e or m o r e s t r a n g e q u a r k s , a n d t h e lightest c- a n d b - h a d r o n s . T h i s is still relatively easy in t h e case of t h e or t h e h y p e r o n s . since these particles may travel tens of c e n t i m e t r e s before decaying. n t he o t h e r h a n d , r e c o n s t r u c t i n g decay vertices of c- o r b - h a d r o n s . which travel at most a few millimetres, can be q u i t e d e m a n d i n g . Apart f r o m finding
I'MAflM
TATI
:t!i:t
n s e c o n d a r y vertex, in more generally, since I lie p r i m a r y decay may l>e into one 01 more short-lived d a u g h t e r particles, a sequence of s e c o n d a r y vertices, o n e also needs t o identify t h e final s t a t e particles in order to r e c o n s t r u c t t h e exact decay chain. T h u s , it is also i m p o r t a n t t o have a d e t e c t o r with g o o d particle Identification capabilities. articles with s t r o n g a n d e l e c t r o m a g n e t i c decays travel no discernible distance from their p o i n t of p r o d u c t i o n , a n d t hus h a v e t o be r e c o n s t r u c t e d by ot her means. T h e basic t e c h n i q u e for finding c a n d i d a t e s is based u p o n i d e n t i f y i n g s e t s of p u t a t i v e d e c a y p r o d u c t s whose invariant masses a r e calculated to lie within a given interval a b o u t t h e e x p e c t e d r e s o n a n c e m a s s . For e x a m p l e , t o find (982) c a n d i d a t e s via their decay to + o n e would consider all t h e possible p a i r i n g s of opposite-sign charged t r a c k s in a n event. sing t h e m e a s u r e d t h r e e - m o m e n t a a n d assigning kaon a n d pion m a s s e s to t he t r a c k s o n e can c a l c u l a t e the resulting invariant, mass. T h e d i s t r i b u t i o n of t h e s e m a s s e s M should c o n t a i n a roit W i g n e r peak.
d
r (M -
2
MR)
+
, T2 4 '
(13 1) '
coming from t h e correct pairings of genuine decay p r o d u c t s on t o p of a combinatorial b a c k g r o u n d , which may b e p a r a m e t e r i z e d by a s i m p l e p o l y n o m i a l or a n o t h e r s m o o t h f u n c t i o n . T h e p a r a m e t e r s F a n d M in oqn (13.1) a r e t h e w i d t h and t h e peak position of t h e resonance. For very n a r r o w resonances, such a s or .I 1 !', t h e finite d e t e c t o r resolution o f t e n leads t o s i m p l e G a u s s i a n peaks. T h e removal a n d control of t h e c o m b i n a t o r i a l b a c k g r o u n d poses t h e greatest e x p e r i m e n t a l challenge t o finding short-lived heavy h a d r o n s . M e t h o d s for t h e reduction of b a c k g r o u n d , with preservation of signal, include t h e use of particle identification. In t h e e x a m p l e above, a good s e p a r a t i o n of pion a n d kaon t r a c k s would allow m a n y w r o n g c o m b i n a t i o n s t o b e d i s c a r d e d , in a d d i t i o n k i n e m a t i c cuts may b e applied. For e x a m p l e , decay p r o d u c t s from heavy resonances m a y be e m i t t e d with large t r a n s v e r s e m o m e n t a a n d in general have a h a r d e r m o m e n t u m s p e c t r u m . Seen from t h e heavy h a d r o n ' s rest f r a m e , t h e g e n u i n e d e c a y p r o d u c t s a r e also e m i t t e d with larger angles t h a n t h e bulk of t h e ' b a c k g r o u n d h a d r o n s ' . T h e select ion a n d o p t i m i z a t i o n of such e x p e r i m e n t a l c u t s is clearly d e p e n d e n t on t h e s o u g h t - f o r h a d r o n . but. in general t h e n a r r o w e r a n d m o r e distinctive a decay channel t he b e t t e r t he c h a n c e s of finding a significant, signal. M a n y kinds of s t u d i e s a r e possible with identified particles observed in t h e f r a g m e n t a t i o n of high e n e r g y q u a r k s o r gluons. In t h e following, we will focus on results o b t a i n e d in e + e a n n i h i l a t i o n at. tile scale of t h e Z a n d above. Apart from mult iplicities a n d m o m e n t u m spect ra of identified particles, m e a s u r e m e n t s a r e also available for two-particle c o r r e l a t i o n s . Here, o n e can st udy kinemat.ical correlat ions in rapidity, azimut hal a n g l e a n d polar angle, but also Bose instein correlations for identical particles. A n o t h e r interest ing sub ect is q u a n t u m n u m ber c o r r e l a t i o n s such a s for st rangeness or b a r y o n n u m b e r . T o g e t h e r , these r e s u l t s place i m p o r t a n t c o n s t r a i n t s ou I lie hadronizat ion models a n d give us insights into
t h e u n d e r l y i n g physics. Tests of perl u r b a t ivc Q C ' D a r e a l s o possible, especially lor heavy (|nark c o n t a i n i n g h a i l r o n s .
13.1
Identified particles
We now review briefly t w o of t h e most i m p o r t a n t p r o p e r t i e s of identified p a r t icles n a m e l y t h e i r m u l t i p l i c i t i e s a n d t h e i r m o m e n t u m s p e c t r a . Most of t h e r e s u l t s q u o t e d h e r e a r e b a s e d on t h e high s t a l ¡ s t i e s d a t a collected at LHP a n d S L O f r o m ha/Ironic Z d e c a y s . 13.1.1
Multiplicities
'The b r e a d t h a n d q u a l i t y of t h e a v a i l a b l e d a t a is i n d i c a t e d in T a b l e s 13.1 a n d 13.'.! A c o n v e n t i o n which is a p p l i e d c o m m o n l y in t h e d a t a is t o t r e a t all p a r t i c l e s w i t h lifetimes g r e a t e r t h a n , for e x a m p l e . l ( F ' s s a s a b s o l u t e l y s t a b l e , while all o t h e r s a r e a s s u m e d l o h a v e d e c a y e d . For a d e t e r m i n a t i o n of t h e pion m u l t i p l i c i t i e s t h i s would i m p l y t h a t t h e y 'include t h e d a u g h t e r p a r t i c l e s f r o m K[! d e c a y s . N o t e t h a i t h i s t r e a t m e n t m a y differ b e t w e e n e x p e r i m e n t s so t h a t c o r r e c t i o n f a c t o r s must be a p p l i e d b e f o r e m a k i n g a n v direct, c o m p a r i s o n s . A r e l a t e d point c o n c e r n s t h e o r i g i n s of t h e lower lying h a d r o u s . For e x a m p l e , d e c a y c h a i n s s u c h a s B " I f - — DT>TI — m a k e it. clear t h a t t h i n e a r e m a n y s o u r c e s of a p a r t i c u l a r final s t a t e p a r t i c l e o t h e r t h a n direct p r o d u c t i o n in t h e h a d r o n i / a t i o n . a n d c a r e h a s t o b e exercised t o c o r r e c t l y a c c o u n t for d o u b l e c o u n t i n g w h e n i n t e r p r e t i n g t lie d a t a . T h e d a t a in T a b l e s 13.1 a n d 13.2 i l l u s t r a t e a g e n e r i c b e h a v i o u r of h a d r o i u c e v e n t s , n a m e l y t h e d o m i n a n c e of pion p r o d u c t ion wit h a p p r o x i m a t e l y e q u a l n u m b e r s of c h a r g e d p i o u s a n d p h o t o n s , c o m i n g f r o m t he d< '('clVS 7% " —• - p . in t h e final s t a t e . K a o n s a n d b a r y o n s c o n t r i b u t e a f u r t h e r ~ 1(1% each t o t h e t o t a l multiplicity. T h e r a t e s of heavy, e a n d b. q u a r k c o n t a i n i n g h a d r o u s a r e e s s e n t i a l l y fully a c c o u n t e d for bv pert u r b a t i v e p r o d u c t i o n , e i t h e r in t he initial Z d e c a y o r subsequent gliion s p l i t t i n g s . The m e a s u r e d r a t e for h e a v y q u a r k p r o d u c t i o n iu t h e p a r t o n s h o w e r is c o m p a t i b l e w i t h t h e p Q C D p r e d i c t i o n ( S e y m o u r . 1995). As a case in point c o n s i d e r T a b l e 13.3. F r o m t h e k n o w n b r a n c h i n g r a t i o s of t h e Z i n t o h e a v y q u a r k s , t h e p r o d u c t i o n r a t e s of h e a v y q u a r k s f r o m g l u o n s p l i t t i n g in t h e p e r t u r b a t i v e p h a s e a n d t h e n u m b e r of c / c - q u a r k s p r o d u c e d iu t h e d e c a y c h a i n of b o t t o m q u a r k s o n e c a n est i m a t e t h e e x p e c t e d n u m b e r of c h a r m q u a r k s , a n d t h u s c h a r m h a d r o u s . p e r h a d r o n i c Z d e c a y . 'The direct m e a s u r e m e n t is in g o o d agreement with the expectation. L o o k i n g in a little m o r e d e t a i l we see t h a t w h e r e m e a s u r e m e n t s h a v e p r o v e d possible t h e r e is e v i d e n c e for q u i t e s u b s t a n t i a l p r o d u c t i o n of o r b i t a l l v e x c i t e d m e s o n s a n d b a r y o n s . For e x a m p l e , t h e k n o w n t e n s o r ( 2 + + ) t o v e c t o r ( 1 ~ ~ ) r a t e s , f\>/f>". }•>/ a n d K j / K * . a r e all % 2 0 % . P r e s u m a b l y m a n y o t h e r higher s t a t e s a r e p r o d u c e d , but just, a s t h e i r l a r g e w i d t h s m a k e s t h e m elusive t o find, it a l s o m a k e s t h e r e l e v a n c e of t h e i r fleeting e x i s t e n c e q u e s t i o n a b l e . A s u p p r e s s i o n of s t r a n g e n e s s c o n t a i n i n g s t a t e s is evident for b o t h light a n d h e a v y h a d r o u s .
IKD PAHTICLKS
:jfl[j
T a b l e 1 3 . 1 l'In aueratp multiplicities of mesons ¡mulini ti tu the liatlmnic titan/ of ii Z. The sum oj particle ami unliparl.ic.lc. rules is implicit: for example, 77 1 M presents also n~ . .-1 duyijcr ( f ) indicates internal conflict within the experimentu! mensurt incuts. Iiefc.re.nces art quoted with A , D , L , 0 . M , S used as shorthand for sec Al FIMI. D E L P H I , 1-3, O P A L , M A R K I I and S I . D . For the definition of xE Section 13.1.2. Particle
Average
charged
21.05
± 0.12
References/Remarks A D L M O ( f 9 9 5 i / . 1999c, 1991c. 1990. 1995c)
7
20.97
± 1.15
0(1998c)
7T+
10.99
± 0.20
A D O S ( 1998c. 1998c. 1994«, 19996)
9.82
± 0.23
A D L ( ) ( 1 9 9 7 i / . 1996//. 1996. 20006)
0.95
± 0.07
L O ( 1 9 9 6 , 1998c)
0.282
± 0.022
A(1998a)
0.17
± 0.02
L O ( 19976, 1 9 9 8 c ) '
0.004
± 0.014
A(1998a)
2.40
± 0.43
0(1998/:)
71
>l' l>
+
0.1 < 0.1 <
x,.:
P0
1.242
± 0.093
A D ( 1 9 9 6 c , 1999//)
0.098
± 0.003
A D O S ( 1996c. 1996c, 1998//, 19996) t
tjJ
1.083
± 0.088
A L O ( 1 9 9 6 c . 19976. 1998c)
«1.(980)
0.27
±0.11
0(l998c)
/o(980)
0.147
± 0.011
D O (1999//, 1998//)
/2(1270)
0.109
± 0.021
D O ( 1 9 9 9 / / , 1998//)
/ a i 1525)
0.012
± 0.006
D(1999//)
K+
2.25
± 0.05
A D O S ( 1998/:. 1998/:, 1994«. 19996)
K°
2.020
± 0.024
A D L O S ( 1 9 9 4 r t , 1995a, 1997c, 20006, 19996)
K' (892)
0.714
± 0.044
A D O ( 1 9 9 8 a , 1995a, 1993d)
K*°(892)
0.739
± 0.022
A D O S ( 1 9 9 8 « , 1996c. 19976. 19996)
K J 0 (1430) K'n(1430)
0.073
± 0.023
0(1999//)'
0.19
± 0.07
0(1995/)'
D*
0.187
± 0.014
A D O (1994 6. 1993c, 1996//)
D"
0.462
± 0.026
A D O (1994 6. 1993c, 1996//)
0.1833
± 0.0081
A D O ( 1 9 9 4 6 . 1993/.-. 1998c)
0.131
± 0.021
0(1996//)
2.9
± 0.7
0 ( 1997c)
0.28
± 0.3
D( 19956)
+
D'+
.1/«'A'ct «,''{3085) T( 1.2.35)
0 . 0 0 3 8 6 ± 0.00024
A D L O ( 19926. 1994. 1993. 1996e)
0.0075
± 0.0030
D L ( 1 9 9 4 , 1993)
0.0036
± 0.000(1
D O (1994, 1996c)
0 . 0 0 0 1 0 ± 0 . 0 0 0 0 6 0 ( 19967)
xE
< 0.3
0 . 6 < 3:n
rrii
I
II
\L .¡\LL',|N
I I \ I I I MN
T a b l e 1 3 . 2 Tin urciage multiplicities of burnous produced in tin liaihiinii rules is implied: for examdeeag of II Z. Tin• sum of /mi li<:lc anil untiparliclc ple. also represents X) . .-1 dagger ( f ) indicates internal conflict within the experimental measurements. References an quoted with A.D.I..O.S used us shorthand for A I . E P I I , D E L P H I . L3, O P A L and S L D .
P A++ A A(1520)
References/Remarks
Average
Particle 1.04
±0.04
A D ( ) S ( 1998c. 1998c. 1994«. 19996)
0.088
±0.014
D()( 1995c. 19951/)*
0.374
± 0.003
A D L ( ) S ( 1 9 9 4 n . 1993«. 1997c. l!)97i/. 19!)9/<)
0.0225 ± 0.0028 D()(200ll(/. 1997«/) 0.107
± 0.010
DL()(1995rf. 2000. 1997c)
0.082
± 0.007
DDO(1995
0.078
± 0.008
A D L O ( 1 9 9 8 « . 1990/, 2000. 1997c)
+
£ •( 1:185) 0.0237 ± 0.0014 A D O ( 1998«, 1995r/. 1997#/) £ " ( 1 3 8 5 ) 0.0237 ± 0.0010 A f ) ( ) ( 1998«. 1995
—
E"(1530)
0.0204 ± 0.0008 A D O ( 1 9 9 8 « , 1995
A D O (1998«, 1995«/, 1997«/) f
0.0010 ± 0 . 0 0 0 3 A D O (1998«, 1990/, 1997«/) K
0.078
± t).017
0(1990«/)
T h i s s u g g e s t s that only about, an eighth of t h e light q u a r k s p r o d u c e d d u r i n g h a d r o n i / a t i o n a r c s t r a n g e q u a r k s . M u l t i p l y - s t r a n g e s t a t e s a p p e a r t o he even f u r t h e r s u p p r e s s e d , as is t h e p r o d u c t i o n of b a r y o n s c o m p a r e d t o mesons. A n o t h e r a r e a w h e r e a t t e n t i o n h a s focused is t h e d e p e n d e n c e of t h e p r o d u c t i o n r a t e s on a particle spin. All o t h e r t h i n g s being e q u a l t h e r a t e might be expected t o be p r o p o r t i o n a l t o t h e n u m b e r of s t a t e s for a given spin .J. '2.1 + 1. W h i l s t t h e r e is s o m e e v i d e n c e for t h i s behaviour, the s i t u a t i o n is not clear c u t , since it is h a r d to d e t e r m i n e t h e c o n t r i b u t i o n s m a d e bv directly p r o d u c e d h a d r o n s a n d t hose coining from higher r e s o n a n c e decays. Specific m e n t i o n should b e m a d e of t h e c o n t r i b u t i o n from b - h a d r o n s . especially since t h e b r a n c h i n g r a t i o of t h e Z into b - q u a r k s is larger t h a n 20%. T h e average charged particle multiplicity of a I v h a d r o n d e c a y is (i«,i,(b)) 4.955 ± 0.002 ( A h b a n e o el ai. 1998). Since t h e t o t a l charged multiplicity in Z — lib events is not drastically different, f r o m light-quark events, this implies that in a Z —• lib event about 50% of t h e particles c o m e f r o m t h e two b - h a d r o n s , whilst t h e f r a c t i o n of t r a c k s f r o m p r i m a r y b - q u a r k s is 10% for t h e n a t u r a l mix of Z — qq events. T h u s it is i m p e r a t i v e t o have a good u n d e r s t a n d i n g of b - h a d r o n decays in order t o be able t o d r a w l i n n conclusions c o n c e r n i n g d e t a i l s of t h e h a d r o n i / a t i o n process from t h e d a t a . U n f o r t u n a t e l y , t h e present decay t a b l e s for b - h a d r o n s a r e still s o m e w h a t s p a r s e , which forces us t o model their decays. We e x p e c t , however, t h a t with u p c o m i n g results f r o m t h e B - f a c t o r i e s which a r e a l r e a d y r u n n i n g o r u n d e r c o n s t r u c t i o n , this s i t u a t i o n will improve .significantly.
II'I T a b l e 13.3 me.nsuiv.mcnts expected Br(Z
xpected peld of c arm purticlcs for adronic dermis of llie Z
num er
of c arm
> cc)
I pQC
(g -
I [(Br(Z +pQC
cc) bb)
( g — l>i>))
xrate(l) — c c X ) ] x2 measurtxl
+
13.1.2
.
I ll'll 1 I'AIM ICI. S
num er
.
Momentum
of c arm
. i >, A , t . . . .
(anti) uar s
per
:i!l7 compared
lo
cxpcrinienl.td
event
0.1702
0.0031
( A b b a n e o ct ai.
0.0299
0.0039
( A b b a n e o ct al.. 2 0 0 1 « )
[(() .2165
0.0007
( A b b a n e o et ai,
+0.0025
0.0005)
( A b b a n e o ct al.. 2 0 0 1 « )
x 1.220
0.000]
( A b b a n e o c t « ., 2 0 0 1 6 ) ,
0.1686
0.021!)
0.9372
0.0438
particles
> 0 . 862
per
2001«) 2001«)
event
0.010
T a b l e 13.1. T a b i c 13.2
spectra
l lie m o m e n t u m s p e c t r u m . o f t e n a l s o r e f e r r e d t o a s t e fragmentation function, is 1 usually q u o t e d a s t h e d i f f e r e n t i a l l a t e <7 d
I'lN
i n . \ t ; M I : N I \ I I< >IN
e l u d e d . T o see t h i s . a s s u m e I lint I lie f r a g m e n t at ion funel ion e a n lie writ t e n a s t lie convolut ion of ii pert ni l»ntive ( P T ) a n d a n o n - p o r t u r b a t ive ( N P ) piece. D(x) i/p'i < i ( x ) gives a s h a r p l y f o r w a r d - p e a k e d distri0.8 for Ey.x -- MyJ'2. C o m but ion w h i c h v a n i s h e s at x - 1. It p r e d i c t s ( x ) ) > T p a r i n g t h i s w i t h t h e v a l u e m e a s u r e d for b - q u a r k s a i I.I'll', (x) 0.702:1:0.008 (Abba n e o c/ ill... 1!)!)S). o n e not o n l y sees t h a t n o n - p e r t u r b a t i v e e f f e c t s a r e n o t negligible, but a l s o t h a t t h e n o n - p e r t u r b a t ive f r a g m e n t a t i o n f u n c t i o n h a s t o b e r a t h e r h a r d ' , t h a t is. p e a k e d t o w a r d s large values of x . A s i m p l e a r g u m e n t ( S u z u k i , 1!ITT: B j o r k o n . li)7S) s h o w s t h a t d u e t o t h e h e a v y ( ¡ n a r k ' s i n e r t i a t h i s is a c t u a l l y e x p e c t e d . T h e a r g u m e n t g o e s a s follows. S u p p o s e t h e h e a v y (¡nark Q a c q u i r e s a light (¡nark t o f o r m a s y s t e m of rest m a s s i n q + ///,,. T h e h e a v y q u a r k ' s e n e r g y f r a c t i o n is IIIQ/(IIIQ + //»»,) ~ 1 HI,JIHQ w h i c h is u n a f f e c t e d by t h e b o o s t i n t o t h e overall C . o . M . s y s t e m . A s t h e m a s s of t h e light (¡nark, which is inversely p r o p o r t i o n a l t o its d e Broglie w a v e l e n g t h , d e t e r m i n e s t h e size of t h e h a d r o n , Hi, ~ 1 / » / , , . o n e t h u s e x p e c t s {x)NV ~ 1 - /i„ ' / " ' Q - w ' ' h t-110 t y p i c a l size of a h a d r o n . I Ieavier h a d r o n s a r e e x p e c t e d t o h a v e h a r d e r f r a g m e n t a t i o n f u n c t i o n s . In t h e l i t e r a t u r e several f o r m s a r e p o s i t e d for t h e n o n - p c r t . u r h a t i v e f r a g m e n t a t i o n f u n c t i o n . T h e o n e most, c o m m o n l y used is t h a t d u e t o P e t e r s o n cl ul. ( 1 9 8 3 ) .
f /N,.(x)
= -
(l - - -
X \
X
1
—
X )
"2
with
(13.2) 7T
w h e r e t h e p a r a m e t e r (Q c a n b e i n t e r p r e t e d a s /=Q = J{^2/IIIQ.
For e<|ii (1:1.2) o n e
f i n d s (x) N p = 1 - Jc^ s : 1 R^/VIQ. A l a r g e v o l u m e of d a t a is a v a i l a b l e for e - h a d r o n s . m a i n l y c o i n i n g f r o m exp e r i m e n t s a t t h e Y(-I.S') r e s o n a n c e ( P D G . 2000). T h i s a l l o w s t h e f r a g m e n t a t i o n f u n c t i o n s of i n d i v i d u a l h a d r o n s t o b e m e a s u r e d . T h e d a t a c a n b e d e s c r i b e d by e q n (13.2) p r o v i d e d t h a t r g is t u n e d s e p a r a t e l y for e a c h h a d r o n . In g e n e r a l , t h e h e a v i e r t h e m e s o n o r b a r y o n t h e h a r d e r is their m o m e n t u m s p e c t r u m , w i t h orbitally excited mesons being particularly hard. T h e fragmentation function for b - h a d r o n s h a s b e e n m e a s u r e d at L E P 1 . Here t h e a v a i l a b l e s t a t i s t i c s o n l y p e r m i t a m e a s u r e m e n t - a v e r a g e d over t h e n a t u r a l l y o c u r r i n g mix of b - h a d r o n s . A g a i n eqn (13.2) c a n d e s c r i b e t he d a t a but o n e s h o u l d be a w a r e of d i f f e r e n c e s in t h e d e f i n i t i o n ol'.r a n d of d i s t i n c t i o n s m a d e b e t w e e n p r i m a r y a n d s e c o n d a r y b - h a d r o n s c a s c a d i n g d o w n f r o m e x c i t e d s t a t e s . At t h e Z, t h e r e l a t i v e r a t e s of prim a r y b - h a d r o n s a r e r o u g h l y given by 13 : 13" : 1 3 " a: 1 : 3 : 2. T h e e x p e r i m e n t a l result for f , - / f | , « 10 is c o n s i s t e n t w i t h t h e m a s s r a t i o (III\,/IIIC)2 • 13.2
Inter-jet soft gluons and colour
coherence
Iu t h i s s e c t i o n we s h a l l i n v e s t i g a t e p r e d i c t i o n s for i n t e r - j e t c o h e r e n c e effects: t h a t is. t h e d i s t r i b u t i o n of p a r t i c l e s lying b e t w e e n h a r d jets. T h e o r e t i c a l l y , t h e a p p r o a c h is t o r e g a r d t h e h a r d p a r t o n s a s a c t i n g a s a n a n t e n n a a n d predict t h e d i s t r i b u t i o n of r a d i a t e d , soft g l u o n s u s i n g t h e t e c h n i q u e s of S e c t i o n 3.7. T h e s e s t u d i e s c o m p l e m e n t t h o s e of i n t r a - j e t c o h e r e n c e e l e c t s in S e c t i o n 11.2.
INI Kit . I I I S O I I CI.IK INS AND OOI.OI Hi < •< »IIKKKNOI'
13.2.1
The sir nil)
effect
I lie classical lest- of inter-jet c o h e r e n c e is t h e "string effect.' seen in e + o ~ a n n i h ilation t o t h r e e j e t s . T h i s p r e d i c t s a d e p l e t i o n in t h e How of p a r t i c l e s , o r e n e r g y , lying ill t h e e v e n t p l a n e b e t w e e n t h e q a n d q r e l a t i v e t o t h a t b e t w e e n t h e q a n d g o r q anil g. A s t h e historical n a m e s u g g e s t s , it w a s first, p r e d i c t e d on t h e basis ol t h e L u n d s t r i n g h a d r o n i z a t i o n m o d e l : h e r e t h e g l u o u d r a w s t h e s t r i n g in t h e d i r e c t i o n of i t s m o t i o n a n d a w a y f r o m t h e q q - a x i s . c a u s i n g t h e d e p l e t i o n ( A n ilcrssoii el ill.. 19806). L a t e r a p Q C ' D e x p l a n a t i o n w a s a d v a n c e d (Aziniov el til.. 1985/t): see e q n (3.3-13) a n d Fig. 3.28. T h e earliest m e a s u r e m e n t s , c a r r i e d o u t a t ^ s = < 9 ( 3 0 G e V ) , used e n e r g y ord e r i n g t o i d e n t i f y t he g l u o n jet a n d m a d e c o m p a r i s o n t o M o n t e C a r l o e v e n t generators based upon string a n d independent fragmentation models (JADK Collab., 1985: T A S S O C o l l a b . . 1985; T P O / 2 7 C o l l a b . . 1985). W h i l s t t h e s t r i n g m o d e l w a s f a v o u r e d , t h e t e s t s were o p e n t o criticism of k i n e m a t i c a l biases a s s o c i a t e d w i t h the difference between narrow quark jets a n d wide gluon jets. A second, more conclusive, series of t e s t s h a s since b e e n c a r r i e d out. at. L K I ' l ( A I . K P I I C'ollab.. 1998//: 1.3 C o l l a b . . 1995; O P A L C o l l a b . . 1991//). T h e b a s i c m e t h o d is t o select, t h r e e - j e t e v e n t s that- a r e well c o n t a i n e d w i t h i n t h e d e t e c t o r , using, for e x a m p l e , a t y p i c a l r e s o l u t i o n p a r a m e t e r //(-,lt = 0 . 0 0 8 for t h e D u r h a m jot. finder. E v e n t s w i t h f o u r o r m o r e j e t s a r e e x c l u d e d . If t h e j e t s a r e e n e r g y o r d e r e d , Ey > E} > E.\. t h e n jet 1 is a l m o s t c e r t a i n l y a q u a r k or an ant.iquark j e t , a n d iu 0(70%) of all c a s e s jet, 3 o r i g i n a t e s f r o m a g l u o n . In t h e a b s e n c e of f u r t h e r c o n s t r a i n t s o n t h e j e t s ' e n e r g i e s it. is useful t o s t u d y t h e p a r t i c l e or e n e r g y How ¡11 t h e event, ¡»lane a s a f u n c t i o n of s c a l e d a n g u l a r v a r i a b l e s , such a s 0 = (0 - \)/(2 - 0\)w h e r e 0\;>.:<, a r e t h e a n g l e s of t h e j e t s in t h e event p l a n e a n d //» t h e a n g l e of t h e p a r t i c l e or e n e r g y d e p o s i t . C o n c e n t r a t i n g on t h e c e n t r a l r e g i o n s , 0 . 3 < n < 0.7. t h e r a t i o of t h e i n t e g r a t e d p a r t i c l e o r e n e r g y How b e t w e e n j o t s 1 a n d 2, in most, c a s e s t h e q u a r k j e t s , a n d j e t s 1 a n d 3. d o m i n a n t ly o n e (¡nark a n d o n e g l u o n j e t , h a s been m e a s u r e d t o b e s i g n i f i c a n t l y below unity. In a r e f i n e m e n t of t h i s a n a l y s i s prompt, l e p t o n l a g s m a y b e used to i d e n t i f y h e a v y flavour j e t s a n d i n c r e a s e t h e g l u o n p u r i t y of t h e t h i r d j e t t o 0 ( 9 0 % ) , t h e r e b y m a k i n g c l e a r e r t h e effect of c o l o u r c o h e r e n c e . A s k i n g for a l e p t o n i n s i d e a jot e n h a n c e s t h e f r a c t i o n of h e a v y f l a v o u r s a n d t h u s s u p p r e s s e s t h e g l u o n c o n t a m i n a t i o n . S u c h f l a v o u r t a g g i n g is e s s e n t i a l in t h e c a s e of s y m m e t r i c e v e n t s w h e r e t h e j e t e n e r g i e s a r e a l m o s t e q u a l so t h a t e n e r g y o r d e r i n g n o l o n g e r distinguishes between quarks and gluons. T h i s d a t a h a s boon c o m p a r e d t o a v a r i e t y of M o n t e C a r l o m o d e l s a n d a n a l y t ical predict ions w i t h t h e c o n c l u s i o n t h a t c o l o u r c o h e r e n c e is r e q u i r e d t o d e s c r i b e t h e d a t a . A s i g n i f i c a n t l i n d i u g w a s t h a t u s i n g . I P ' T S H T ( P Y T I I I A ) it is necess a r y t o i n c l u d e b o t h a n g u l a r o r d e r i n g in t h e s h o w e r a n d t h e s t r i n g h a d r o n i z a t i o n m o d e l in o r d e r t o d e s c r i b e t h e d a t a . T h e i n t e r p l a y b e t w e e n t h e s e c o n t r i b u t i o n s h a s boon e v a l u a t e d in Kltoze a n d L o i m h l a d (1990). Iu p r a c t i c e , t h e inter-jet- p a r ticles. chiefly p i o n s . h a v e e n e r g i e s of o n l y a few h u n d r e d M o V . What, t h e d a t a suggest t h e n is t h a t p Q C D e n a b l e s us t o c a l c u l a t e t h e s t r e n g t h of t h e u n d o r l y -
I I I A< ¡MEN'I A I ION
II III
iug c o l o u r field which p r o d u c e s t h e h a d t o n s . K u r t h e r i n o r e . t h e h u d r o n i z u t i o n ii< a s o i l , n o t a h a r d , p r o c e s s w h i c h d o e s n o t d i s r u p t o u r p r e d i c t i o n : t h a t is. local p n r t o n h a d r o n d u a l i t y w o r k s ( K h o z c a n d O c h s , 1!)!)7). A s e c o n d t e s t , p r o p o s e d in (Aziinov ct til.. 198(16). is t h e r e l a t i v e d e p l e t i o n of p a r t i c l e s b e t w e e n t h e <|<| s y s t e m in qq-;, c o m p a r e d t o t h e p a r l i c l e How in q q g e v e n t s : see E x . (13-3). S i n c e t h e p r o d u c t i o n c r o s s s e c t i o n for qq- t e v e n t s is sii|i p r e s s e d by t h e r a t i o of e l e c t r o m a g n e t i c a n d s t r o n g c o u p l i n g , r ~ o ( m / ( 0 ' ' > s ) , t h i s s t u d y rec|iiires l a r g e i n t e g r a t e d h u n i n o s i t i e s . G i v e n t h a t , a n d by c a r e f u l l y m a t c h ing t he e v e n t s e l e c t i o n c u t s , it is p o s s i b l e t o c o m p a r e i n t e r - q u a r k r e g i o n s t hat a r e k i u e i n a t i c a l l y alike. D i f f e r e n c e s a r i s i n g f r o m t h e a l t e r e d p r i m a r y q u a r k flavoill mix a n d dilferent c o n t e n t s of t h e o p p o s i t e h e m i s p h e r e s h a v e b e e n s h o w n t o be m i n i m a l . A g a i n , e a r l y s t u d i e s ( J A D E C o l l a b . , l!),SHb: M A R K I 1 C o l l a b . , 198(1; T P C / 2 - , C o l l a b . , 198(5) h a v e b e e n followed by s t u d i e s a t EE P I ( D E L P H I Collab., 199(ic; 1.3 C'ollab.. 1995; O I ' A L C o l l a b . . 1995/j). O n c e m o r e t h e c o n c l u s i o n is t h a t only fully c o h e r e n t M o n t e C a r l o s i m u l a t i o n s a r e c a p a b l e of d e s c r i b i n g t he m e a s u r e d r a t i o of p a r t i c l e flows, p(qq~')//»(qqg) ~ 0.K5. C o l o u r c o h e r e n c e is o f t e n s u m m a r i z e d by t h e p h r a s e ' a n g u l a r o r d e r i n g of successive p a r t o n b r a n c h i n g s ' . However, it g o e s f u r t h e r t h a n t his a n d a l s o p r e s c r i b e s t h e a z i i n u t h a l d i s t r i b u t i o n of soft g l u o n s a b o u t t h e e m i t t e r , which p r e f e r a b l y lie in t h e p l a n e b e t w e e n t h e e m i t t e r a n d its c o l o u r p a r t n e r , s e e e q n (4.23) a n d Fig. 4.2. E x p e r i m e n t a l l y o n e e x p e c t s t o find a n a s y m m e t r y in t h e m o m e n t u m How for a q u a r k o r a n t i q u a r k jet wit hin a t h r e e - j e t e v e n t s u c h t h a t t h e low m o m e n t u m p a r t i c l e s l e n d t o lie on t h e s i d e t o w a r d s t h e g h i o n . E v i d e n c e for t h i s elfect h a s a c t u a l l y been seen in t h e L E I ' l d a t a . 1.3.2.2
Colour
cohr.iv.iicc
in hadmn
hadron
collisions
T h e c o m p l i c a t e d n a t u r e of h a d r o n h a d r o n collisions h a s r e s u l t e d in fewer t e s t s of soft g l u o n c o h e r e n c e . O n e except ion is a s t u d y of t h e d i s t r i b u t ion of t h e lowe s t t r a n s v e r s e - e n e r g y j e t , a s s u m e d t o b e a g l u o n j e t in t h r e e - j e t e v e n t s at t h e I E V A T R O N ( C D F C o l l a b . . 199-1: DO C o l l a b . , 19996). T h e basic idea is t o consider an underlying Q C D two-to-two hard scattering and derive the distribution of r a d i a t e d soft g l u o n s u s i n g t h e s a m e i n s e r t i o n c u r r e n t t e c h n i q u e a s d e v e l o p e d in S e c t i o n 3.7. It is a l s o p o s s i b l e t o u s e t h e e x a c t t h r e e - j e t m a t r i x e l e m e n t s ( M a u g a n o a n d P a r k e . 1991) which give t h e s a m e r e s u l t s in t h e soft g l u o n limit. T h e s e p r e d i c t i o n s must b e w e i g h t e d by t h e a p p r o p r i a t e p a r t o n d e n s i t y funct ions, s u m m e d over all c o n t r i b u t i n g s u b p r o c e s s e s , i n t e g r a t e d o v e r t h e o b s e r v e d p h a s e s p a c e a n d finally L P H D invoked: a t a s k n a t u r a l l y d o n e using a n e v e n t g e n e r a t o r . T h e m a i n d i f f e r e n c e w i t h t h e s i t u a t i o n e n c o u n t e r e d in e + e ~ a n n i h i l a t i o n is t h a t t h e c o l o u r llows connect, i n c o m i n g a n d o u t g o i n g p a r t o n s . see. for e x a m p l e , Fig. 4.4. so t h a t c o h e r e n c e m u s t b e t a k e n i n t o a c c o u n t iu b o t h t h e initial a n d liual s t a t e r a d i a t i o n (Ellis ct E2 > E-. f. a n d t h e a n g u l a r d i s t r i b u t i o n of j e t 3 a b o u t j e t 2 is m e a s u r e d . S t u d i e s s h o w t h a t j e t 3 is most likely t o be i n i t i a t e d by a g l u o n . whilst, j e t 2 is most likely
INI I K II I .'¡1)1 I I.I HONS \ N I i c o l , O H M < •< »III 1(1 .N< I
mi
in lie t lir c o l o u r purl HIT of jet ,'t. W o r k i n g in t h e ;/ p l a n e , useful v a r i a b l e s in a d d i t i o n t o t h e p s e u d o r a p i d i t y of jet 3. //.(. a r e R a n d (i divined by
R = v/('/•'» - Va) 3 + i
(i = tan
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sign('/-.»)(0:i ~ 2) ('/:» - »ft)
Here ¡1 is t h e a n g l e of jet 3 a b o u t jet 2 w i t h z e r o d e f i n e d t o b e in t h e direct-ion of t h e b e a m n e a r e s t t o j e t 2. For ¡1 = I), n j e t 3 lies wit hin t h e e v e n t p l a n e , s p a n n e d by t h e colliding b e a m s a n d jet. 2. a n d for = ± 7 t / 2 it p o i n t s o u t of t h e event p l a n e . C o l o u r c o h e r e n c e p r e d i c t s e n h a n c e m e n t s for (1 = TT a n d t o a lesser e x t e n t a l s o for fi = 0. T h i s is seen ¡11 t h e d a t a for b o t h c e n t r a l l y prod u c e d j e t s ( C D F C o l l a h . . 1994: DO C o l l a b . . 1999/;) a n d m o r e c l e a r l y for f o r w a r d j e t s (DO C o l l a b . , 199!)/;). A11 o b s e r v e d b r o a d e n i n g of t h e ;/;( d i s t r i b u t i o n is a l s o predicted. C o m p a r i s o n to M o n t e C a r l o simulations show that t h e incoherent ISA.JliT o r P Y T I I I A w i t h a n g u l a r o r d e r i n g s w i t c h e d off a r c i n c a p a b l e of d e s c r i b ing t h e d a t a , whilst, t h e c o h e r e n t part.011 s h o w e r m o d e l s l l l \ U W I C o r P Y T I I I A reproduce the data. A r e l a t e d s t u d y h a s i n v e s t i g a t e d t h e d i s t r i b u t i o n of h a d r o n s in W I jet. e v e n t s (DO C o l l a b . , 1999/;). T h e o r e t i c a l l y t h i s is a s i m p l e r s i t u a t i o n t o a n a l y s e s i n c e a t lowest, o r d e r only t w o p r o c e s s e s c o n t r i b u t e : C|Cj' — W g a n d q g —> q ' W (qg > q ' W ) ( K h o z e a n d S t i r l i n g , 1997). T h e s e l e c t e d e v e n t s e a c h c o n t a i n e d a c e n t r a l l y p r o d u c e d W a n d a t least o n e j e t wit h |;/j,.| |. |.vw| < 0.7. w i t h m o d e s t t r a n s v e r s e energies, / ? y ( j e t ) . / Y i W J ' f c l O G e V . In c a s e s w i t h m o r e t h a n o n e jet t h e m o s t e n e r g e t i c jet in t h e a z i n i u t h a l h e m i s p h e r e o p p o s i t e t h e W w a s s e l e c t e d . T w o a n n u l a r r e g i o n s w e r e t h e n d e f i n e d in t h e <:> p l a n e such t h a t 0 . 7 < R < 1.!» wit h R = Y/(A;/)-' -I- (Ac';)-, a n d p s e u d o r a p i d i t y a n d a z i n i u t h a l a n g l e m e a s u r e d w i t h respect t o t h o s e of e i t h e r t h e W o r t h e jet.. T h e caloriiiietric a c t i v i t y w a s t h e n m e a s u r e d a s a f u n c t i o n of t h e a n g l e (). d e f i n e d a s in c q n ( 1 3 . 3 ) . in t h e a n n i i l u s . B y c o m p a r i n g t h e r e s u l t s for t h e colourless W . w h i c h d o e s n o t c o n t r i b u t e a n y soft gliions, a n d t h e jet it. a l l o w s m i n i m i z a t i o n of d e t e c t o r e f f e c t s a n d c o n t a m i n a t i o n s f r o m t he u n d e r l y i n g e v e n t . In t h e event, p l a n e t h e d a t a s h o w m o r e a c t i v i t y a r o u n d t h e jet t h a n a r o u n d t h e W . ¡11 a c c o r d w i t h e x p e c t a t i o n s f r o m c o l o u r c o h e r e n c e a n d g o o d a g r e e m e n t , w i t h t ile c o h e r e n t M o n t e C a r l o g e n e r a t o r s . In c o n c l u s i o n , several s t u d i e s clearly e s t a b l i s h t he need for c o l o u r c o h e r e n c e in t h e d e s c r i p t i o n of e + e " a n d h a d r o n l i a d r o n collisions. At I.F.I' t h e l a r g e b o d y of d a t a h a v i n g a b e a r i n g o n i n t e r - a n d inl r a - j e t c o h e r e n c e e f f e c t s st r o n g l y d i s f a v o u r s M o n t e C a r l o m o d e l s w h i c h e m p l o y e i t h e r i n c o h e r e n t s h o w e r s or i n d e p e n d e n t h a d r o n i z a t i o u . A s a c o n s e q u e n c e t h e CO.IF. I S M o n t e C a r l o g e n e r a t o r is used only r a r e l y n o w , o t h e r t h a n a s a s t r a w m a n . a n d its d e v e l o p m e n t , h a s c e a s e d . At I lie T K V A T H O N t h e i n a d e q u a c y of M o n t e C a r l o p r o g r a m s w h i c h use i n d e p e n d e n t f r a g m e n t a t i o n m o d e l s is less e s t a b l i s h e d a n d so ISA.JKT is still e m p l o y e d , t h a n k s in part t o t h e large n u m b e r of sit I »processes t h a t it s u p p o r t s .
nr.!
I.'{..'J
•HAOMKN I VI ION
Two-particle
correlations
W h e n s t u d y i n g d i s t r i b u t i o n s for g r o u p s of t w o o r m o r e p a r t i c l e s in l i a d r o n i c s y s t e m s , it is i n t e r e s t i n g t o look for d e v i a t i o n s of t h o s e d i s t r i b u t i o n s f r o m I he e x p e c t a t i o n s b a s e d o n s i n g l e - p a r t i c l e p r o p e r t i e s . Very g e n e r a l l y , we c a n m e a s u r e a q u a n t ilv s u c h a s c
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w h e r e P\(,P\ • Pz) | s t h e joint p r o b a b i l i t y t o o b s e r v e s o m e p r o p e r t y /»i of p a r t i c l e I t o g e t h e r w i t h a p r o p e r t y p> of p a r t i c l e 2. n o r m a l i z e d t o t h e single p r o b a b i l i t i e s /' s . For u n c o r r e l a t e d p a r t i c l e s we w o u l d h a v e C = 1. W e shall call d e v i a t i o n s f r o m t h i s e x p e c t a t ion multi-particle correlations. A l m o s t all of t h e e x p e r i m e n t a l i n v e s t i g a t i o n s h a v e been p e r f o r m e d for t w o - p a r t i e l e s y s t e m s . W h a t c a n be at t h e origin of s u c h c o r r e l a t i o n s ? Q C D p a r t o n s h o w e r a n d h a d r o n d y n a m i c s , s u c h a s soft g l u o n c o h e r e n c e , r e s o n a n c e s a n d final s t a t e i n t e r a c t i o n s , c a n i n d u c e c o r r e l a t i o n s . Local q u a i l i urn n u m b e r c o n s e r v a t i o n c a n lead t o c o r r e l a t i o n s of p a r t i c l e s w h i c h a r e close iu p h a s e s p a c e . Finally, q u a n t u m m e c h a n i c a l e f f e c t s s u c h a s B o s e E i n s t e i n a n d Fermi D i r a e c o r r e l a t i o n s s h o u l d b e c o n s i d e r e d . C o r r e l a t i o n s h a v e b e e n s t u d i e d in p a r t i c u l a r w i t h i n t h e c o n t e x t of b a r y o n p r o d u c t i o n , w h i c h in g e n e r a l is less well u n d e r s t o o d . Single p a r t i c l e s p e c t r a c a n b e r e p r o d u c e d m o r e o r less well by t h e most i m p o r t a n t phcnoiuciiological m o d e l s , n a m e l y s t r i n g a n d c l u s t e r h a d i o n i z a t i o n . H o w e v e r , c o r r e l a t i o n s t u d i e s allow for a d e e p e r look i n t o t h e prod u c t i o n d y n a m i c s of b a r y o n s , which a r e q u i t e d i f f e r e n t l y m o d e l l e d , a s d e s c r i b e d iu S e c t i o n 3.S. B a r y o n s . a n d in p a r t i c u l a r s t r a n g e b a r y o n s , h a v e t h e a d d i t i o n a l a d v a n t a g e t o p r o b e d i r e c t l y t h e p r o d u c t i o n m e c h a n i s m . Iu g e n e r a l , t h e h e a v i e r t h e p a r t i c l e t h e m o r e o f t e n it s t e m s direct ly f r o m t h e h a d r o n i z a t ion p h a s e a n d not f r o m d e c a y s of o t h e r h a d r o n s . F r o m b a r y o n n u m b e r c o n s e r v a t i o n w e e x p e c t t h a t a b a r y o n is a l w a y s a c c o m p a n i e d by a n n n t i h a r y o u . so. p o s s i b l e p a i r i n g s a r e for e x a m p l e p p . A A o r Ap. However, f l a v o u r c o n s e r v a t i o n in s t r o n g i n t e r a c t i o n s s u g g e s t s t h a t t h e b a r y o n p a i r s m a y b e p r e f e r e n t i a l l y p a r t i c l e ant ¡particle p a i r s . If w e finally a s s u m e t h a t t h e c o n s e r v a t i o n of q u a n t u m n u m b e r s o c c u r s r a t h e r locally, a n d t h a t t h e concept of local p a r t o n h a d r o n d u a l i t y holds, t h e n we e x p e c t t h a t t h e b a r y o n s a r e p r o d u c e d close by in p h a s e s p a c e . A s u m m a r y of recent m e a s u r e m e n t s w i t h a c o m p r e h e n s i v e list of r e f e r e n c e s c a n b e f o u n d in t h e r e v i e w by K n o w l e s a n d L a f f c r t y (1!M)7). 13.3.1
Proton
antiproton
correlations
We s t a r t w i t h t h e a n a l y s i s of p r o t o n a n t i p r o t o n c o r r e l a t i o n s , which h a s been p e r f o r m e d a t L E E a n d a l s o at lower energies. T h e m a i n e x p e r i m e n t a l issues a r e p a r t i c l e i d e n t i f i c a t i o n , t h e s u b t r a c t i o n of b a c k g r o u n d s f r o m d e f i n i t e l y nnc.orrelat.ed p a i r s , a n d t h e d e f i n i t i o n of s u i t a b l e r e f e r e n c e s y s t e m s a n d o b s e r v a b l e s . P a r t i c l e i d e n t i f i c a t i o n is t y p i c a l l y p e r f o r m e d by m e a s u r i n g t h e specific ionization e n e r g y loss, which for p a r t i c u l a r r e g i o n s of m o m e n t a a l l o w s t h e s e p a r a t i o n
I WO l'AIM H I i COHUKI.AI IONS
ni p r o t o n s IVoin pious, kaons anil elect runs. B a c k g r o u n d s in a selected s a m p l e ol proton a n t i p r o t o n pairs can arise f r o m p a r t i c l e misideutilication or f r o m t h e g r o u p i n g of p a r t i c l e s a c t u a l l y belonging to dilfercnt b a r y o n a n t i b a r v o n pairs. It i found t hat t he distribut ion of so called like-sign pairs, t hat is. p p or p p . reproduces well t he dist ribut ion of t hose b a c k g r o u n d s , a n d t h u s is used for b a c k g r o u n d suht met ion. Most of t h e e v e n t s in e + e ~ annihilat ion a r e two-jet events. In this case, we assume t h a t h a d r o u s a r e p r o d u c e d out of a c o l o u r field of t h e f o r m of a flux t u b e stretched between t h e p r i m a r y q u a r k a n d a n t i q u a r k , which fly a p a r t in o p p o s i t e directions. T h e r e f o r e , a relevant reference s y s t e m should contain a n axis which coincides with t he direct ion of t h e flux t u b e . An e s t i m a t e for this axis is given for e x a m p l e by t h e T h r u s t or t h e Sphericity axis, a s described in previous c h a p t e r s . I'ypical observables which have been m e a s u r e d a r e rapidity, aziniuthal a n d polar angle with respect t o this axis. F i g u r e 1:5.1 (left) (A I.KIM I Collab.. 1908«) shows the corrected like-sign subtracted r a p i d i t y d i s t r i b u t i o n for ant .¡protons, given that a p r o t o n has been found in a c e r t a i n r a p i d i t y region, as indicated by t h e s h a d e d horizontal bar. T h i s way of p r e s e n t i n g t h e correlation had been p r o p o s e d by t h e O P A I . c o l l a b o r a t i o n (I!)!):{<•) in a s t u d y of s t r a n g e b a r y o n p r o d u c t i o n . We observe a clear local compensation of b a r y o n n u m b e r . Generally, it h a s b e e n d e t e r m i n e d that for a given rapidity //,, of t h e p r o t o n t h e r e is a b o u t a 7(1% probability that t h e a n t i p r o t o n is found w i t h i n ¡//,> — //,>J < 1. A marginally weaker s h o r t - r a n g e Ap c o r r e l a t i o n h a s been m e a s u r e d , too. In c o n t r a s t , when p p p a i r s a r e found iu a n event, a l m o s t no correlation between t h e r a p i d i t y of t h e two b a r v o n s is o b s e r v e d . Q u a l i t a t i v e l y these f e a t u r e s a r e r e p r o d u c e d by t h e cluster a n d s t r i n g hadronization models, as i m p l e m e n t e d in t h e IIEKWIO and J E T S K T M o n t e C a r l o p r o g r a m s , respectively. However, q u a n t i t a t i v e l y t he st r e n g t h of t he correlation is overestimated by IIKKWIC a n d by .IKTSKT without t h e ' p o p c o r n m e c h a n i s m ' , c.f. Section :5.8.:i. T h e p o p c o r n model is a convenient e x p l a n a t i o n for reduced correlat ions between a g e n e r a t e d b a r v o n ant ¡baryon pair, since hi this model a d d i t i o n a l m e s o n s c a n be c r e a t e d between t h e t wo b a r v o n s . A very i n t e r e s t i n g test of h a d r o n i z a t i o n m o d e l s can be p e r f o r m e d by m e a suring t h e angle 0' between t h e axis of t h e baryon a n t i b a r v o n pair in its rest f r a m e a n d t h e event axis. As m e n t i o n e d a b o v e , in s t r i n g models t h e event, axis is aligned with t h e colour field, a n d since iu t h e s e models most of t h e m o m e n t u m t r a n s f e r r e d f r o m t h e s t r i n g t o t h e particles is longitudinal r a t h e r t h a n transverse, t h e b a r v o n s should r e m e m b e r t h e direction of t h e colour field. T h u s , their mom e n t u m difference, or equivalent lv t h e axis formed by t he pair in its rest f r a m e , is expected to form a small angle with t h e event axis. T h e angle 0'. m e a s u r e d in t h e d a t a , p e a k s at small values a s we observe in Fig. K5.1 (right), a n d this is r e p r o d u c e d by .IKTSKT. which i n c o r p o r a t e s t h e L u n d s t r i n g m o d e l . O n t h e ot her h a n d , t h e IIKRW'IO model fails t o describe t h e d a t a , b e c a u s e in t h e cluster h a d r o n i z a t i o n model t h e clusters decay ¡sottopically. In s u m m a r y , we have seen that, t h e e x p e c t e d local c o m p e n s a t i o n of b a r y o n
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v(A)-.v(AI l i t : . 1 3 . 2 . T w o - p a r t i c l e e o r r e l a t ions for s t r a n g e b a r y o n s a s a f u n c t i o n of r a p i d i t y difference. F i g u r e f r o m A C I I C o l l a b . ( l!)98fl).
a large u u i n b e r of e v e n t s w i t h h a d r o u s of t y p e h,. T h e m e a s u r e m e n t s give (A,A) = -li) 0 . Whilst ( A , A ) - 13 1 . ( A , g ) = 17 2 a n d '( i. i !) 21) 1 . S o o u r e x p e c t a t i o n is c o n f i r m e d by t h e d a t a . In o r d e r t o u n d e r s t a n d w h y P(A. A ) is not closer t o u n i t y , it s h o u l d b e r e m e m b e r e d t h a t s t r a n g e h a d r o u s a r e p r o d u c e d not o n l y in s t r o n g i n t e r a c t i o n s , but. a l s o d u r i n g weak d e c a y s of h a d r o u s c o n t a i n i n g h e a v y q u a r k s ( b . e). T h e fact that. ' ( . ) lies s o m e w h e r e b e t w e e n P(A. A) a n d P(A. A) c a n b e u n d e r s t o o d s i n c e n e u t r a l - m e s o n s d e c a y o n l y w i t h a p r o b a b i l i t y of 5(1 a s -st.ates. T h e t w o m a i n s o u r c e s for t h e p r o d u c t i o n of s t r a n g e a n t i s t r a n g e h a d r o n p a i r s are: l e a d i n g h a d r o u s a s s o c i a t e d wit h t h e s s p a i r g e n e r a t e d in t h e h a r d i n t e r a c t i o n s (for e x a m p l e . Z —> ss), a n d p a i r s p r o d u c e d locally out of t h e e v e n t ' s c o l o u r field. In o r d e r t o d i s t i n g u i s h t h e m , we s h o u l d look for p h a s e s p a c e c o r r e l a t i o n s . T o first a p p r o x i m a t i o n we expect, a l o n g - r a n g e c o r r e l a t i o n b e c a u s e of t h e f o r m e r p r o d u c t i o n m e c h a n i s m , a n d a s t r o n g s h o r t - r a n g e c o r r e l a t i o n iu r a p i d i t y w i t h respect, t o t h e T h r u s t axis, a s d e f i n e d a b o v e for p r o t o n a n t i p r o t o n c o r r e l a t i o n s , b e c a u s e of t h e l a t t e r . In Fig. 13.2 a m e a s u r e m e n t of t h e c o r r e l a t i o n of AA p a i r s as a f u n c t i o n of t h e i r r a p i d i t y d i f f e r e n c e is s h o w n . T h e c o r r e l a t i o n f u n c t i o n is defined a s n . ' i - . ' ) = Arha
(l
/>)
(.' I ) (i 2) '
(13.0)
w h e r e n- ) is t h e r a p i d i t y of t h e first ( s e c o n d ) s t r a n g e h a d r o n , A : m ,i t h e t o t a l n u m b e r of h a d r o n i c e v e n t s , a n d refers t o t h e n u m b e r of e v e n t s w i t h o n e o r two h a d r o u s f o u n d in a specific r a p i d i t y r a n g e . For u n c o r r c l a t e d p a i r s w e h a v e
11)1.
I It A( ¡MENTA I H)N
('(.'/1 •.'/-') : 1- 11«' normalization is m e a s u r e d l«y I n k i n g l i a d r o n s f r o m dilfcrcut events which a r c definitely iineorrclatod. T h e m e a s u r e m e n t s h o w s t h a t a s t r o n g positive short,-range correlation is present for A A pairs, as was also seen in t he lirst analysis of this kind by I lie O I ' A L c o l l a b o r a t i o n (1993c). Similar m e a s u r e m e n t s for K"K<| a n d A K " pairn show a weaker positive correlation at small a n d very large r a p i d i t y differences, a n d for AA pairs a s h o r t - r a n g e autocorrelation is o b s e r v e d . T h e cluster hndroniz at ion model o v e r e s t i m a t e s t h e s h o r t - r a n g e c o r r e l a t i o n , w h e r e a s t h e L u n d s t r i n g model with a fraction of Tit)'/, of all b a r v o n s p r o d u c e d by t he p o p c o r n mechanism describes t h e d a t a well. Increasing t h i s f r a c t i o n , which reduces t h e correlation, or switching t h e p o p c o r n m e c h a n i s m off leads t o a s o m e w h a t worse agreement with d a t a , but t h e r a t h e r large e r r o r s prevent us f r o m m a k i n g m o r e q u a n t i t a t i v e s t a t e m e n t s . M e a s u r e m e n t s of t h e overall p r o d u c t i o n r a t e s of s t r a n g e liadrons confirm t h e preference for a 5 0 % p o p c o r n probability. In s u m m a r y , we can s t a t e that t h e e x p e c t e d s h o r t - r a n g e c o r r e l a t i o n s I'm st r a n g e liadrons have been observed in t he experiment's, a n d that s o m e t h i n g like t h e p o p c o r n m e c h a n i s m , which in general decreases t h e correlations, is needed in o r d e r t o o b t a i n a good description of t h e d a t a . 13.3.3
Bose
Einstein
corn hit ions
F r o m q u a n t u m mechanics we known t h a t a s y s t e m of identical p a r t i c l e s with integer spin (bosons) has to obey Bose Einstein s t a t i s t i c s , that is t o say. t h e wave f u n c t i o n of t h e s y s t e m h a s t o be s y m m e t r i c u n d e r t he e x c h a n g e of a n y t wo of t h e particles. H a d r o n i c final s t a t e s a r e m a d e out. of m a n y bosons (mesons) and fern lions ( b a r v o n s ) . In p a r t i c u l a r , pious a n d kaons a r e p r o d u c e d very copiously, a n d usually their m o m e n t a a r e m e a s u r e d r a t h e r precisely. So. we m i g h t wonder if it is possible to observe effects predicted by s t a t i s t i c a l q u a n t u m m e c h a n i c s in high energy collisions. Indeed, when looking at t h e two-particle differential cross section for likesign pion p r o d u c t i o n (identical bosons), a n e n h a n c e m e n t with respect t o t h e unlike-sign cross section is observed when t h e two pious have similar m o m e n t a , a s predicted by Bose Einstein s t a t i s t i c s . T h i s was lirst r e p o r t e d e x p e r i m e n t a l l y when st inlying pairs of charged pions p r o d u c e d in p p collisions ( G o l d h a b e r et
(13.7)
T h e correlat ion funct ion is usually defined as t he r a t i o of two-particle cross sections for like-sign a n d unlike-sign p a r t i c l e pairs. T h e f u n c t i o n ft is p r o p o r t i o n a l t o t h e Fourier t r a n s f o r m of t h e spatial density d i s t r i b u t i o n />(x) of t h e pion source, a n d 0 < A < 1 is a m e a s u r e of t h e d e g r e e of c o h e r e n c e in t h e source, with A (I
I W< I'All I ( I I I '< nul l.A I I
S
IIIV
lor a completely c
(13.8)
if we a s s u m e a spherically s y m m e t r i c source with a G a u s s i a n f o r m ,
where r 2 = x
4- y 2
z 2 . Generalizing this result t o ( 7 ( 1 , 2 ) = 1 -I- A e - y V ' 1 .
(13.10)
the i n t e r p r e t a t i o n of a a s t h e s p a t i a l extension of t h e source is less obvious. T h e ansntz ( '(1.2) = f( 2) is valid only for a very rest ricted class of sources, such as those for which >(. ) = />(t~ x~). For m o r e realistic sources we also expect more complicated space tim e d i s t r i b u t i o n s . For e x a m p l e , in t h e case of a n e x p a n d i n g source t h e specific pion pairs will m a i n l y p r o b e t h e region of t h e s o u r c e which, for a c e r t a i n t i m e p e r i o d , e x p a n d e d parallel lo t h e direction of their m o m e n t a , and t h e e x t r a c t e d size p a r a m e t e r c a n n o t give i n f o r m a t i o n about t h e overall size and s h a p e of t h e source. Finally, if t h e model for the source d i s t r i b u t i o n is not a p p r o p r i a t e , t h e i n t e r p r e t a t i o n of t h e e x p e r i m e n t a l l y m e a s u r e d p a r a m e t e r A as an indicator for coherence might also be wrong. A very t h o r o u g h discussion of these p r o b l e m s c a n be found iu t h e article by Bowler (1985). So m u c h a b o u t t h e l i m i t a t i o n s of t h e physics i n t e r p r e t a t i o n of t h e nieasureinents of Bose instein effects. Since practically all e x p e r i m e n t a l t e s t s a r e based on t he model of cqu (13.10), we a r e going to st ick t o it for f u r t h e r discussions. ater, we will see that these m e a s u r e m e n t s have p h c n o m c n o l o g i c a ! implications, so they a r e nevertheless of i m p o r t a n c e . xperimentally, a q u a n t i t y
-
fei
1
-
can be m e a s u r e d , w h e r e ++( ) (A r + ((,))) is t h e n u m b e r of like-sign (uiilikesign) charged particle pairs as a f u n c t i o n of t h e f o u r - m o m e n t u m dilference. T h i s q u a n t i t y should give a good a p p r o x i m a t i o n of t h e correlation f u n c t i o n C ( l , 2 ) .
I IIACM
I H
IAI I
M e a s u r e m e n t s liave boon p e r f o r m e d lor . I ' and ! p a i r s , a s well + a s for TT-Tr-TT a n d - ; r r t r i p l e t s , b y e x t e n d i n g t h e d e f i n i t i o n of r Q ) accordingly. A review m a y be f o u n d in n o w l e s a n d a I forty (l!l!)7) a n d reference.-, therein. T h e m a o r e x p e r i m e n t a l p r o b l e m is t o d e f i n e a r e f e r e n c e s a m p l e w h i c h is c o m p l e t e l y f r e e of B o s e instein correlations. s i n g uulike-sign c h a r g e d pion p a i r s m i g h t give a g o o d a p p r o x i m a t i o n , b u t t h e p r o d u c t i o n of p i o u s d u r i n g I li
miA )
w h e r e .\!LUIX(Q) is t h e n u m b e r of e v e n t - m i x e d p a i r s a s a f u n c t i o n of Q. C a r e h a s t o b e t a k e n in o r d e r t o m e a s u r e t h e m o m e n t a of t h e p i o u s f r o m dllforont e v e n t s w i t h r e s p e c t t o a well d e f i n e d c o m m o n r e f e r e n c e f r a m e , s u c h a s t h e o n e given by t h e e i g e n v e c t o r s of t h e S p h e r i c i t y t e n s o r . At t h i s s t a g e t h e r e might still b e s o m e c o r r e l a t i o n s p r e s e n t , which a r i s e f r o m e n e r g y m o m e n t u m c o n s e r v a t i o n or r e s o n a n c e d e c a y s . If a g o o d M o n t e C a r l o s i m u l a t i o n of t he overall p r o p e r t i e s of h a d r o n i c final s t a t e s is a v a i l a b l e , t h e n a double ratio Mln Q)
-
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c a n b e m e a s u r e d . H e r e r M ( is t h e r a t i o m e a s u r e d w i t h t h e s i m u l a t e d e v e n t s . If t h i s s i m u l a t i o n c o r r e c t l y a c c o u n t s for all p h y s i c s e l f e e t s a p a r t f r o m Bose instein c o r r e l a t i o n s , t h e n b y c o n s t r u c t i o n t h e a b o v e d o u b l e r a t i o s s h o u l d be close t o 1. a p a r t f r o m a p o s s i b l e e n h a n c e m e n t , a t low Q d u e t o B o s e instein correlations. In F i g . 13.3 a m e a s u r e m e n t , of t h e s e d o u b l e r a t i o s by t h e A I . I ' I I collabor a t i o n (19!)8m) is s h o w n . Here, t h e c o r r e l a t i o n f u n c t i o n s a r e lirst m e a s u r e d for c h a r g e d p a r t i c l e p a i r s , a n d t h e n c o r r e c t e d for t h e pion p u r i t y a n d for r e s i d u a l C o l o u m b r e p u l s i o n a t t r a c t i o n e f f e c t s . A c l e a r e n h a n c e m e n t is o b s e r v e d below C . i i G o V . w h i c h is i n t e r p r e t e d a s e v i d e n c e for B o s e i n s t e i n c o r r e l a t i o n s . In t he d i s t r i b u t i o n for ?., ((GcY) c a n b e seen. F u r t h e r m o r e . ?,„ix(Q) is not flat at l a r g e ( as e x p e c t e d , a n d t h e m a x i m a l e n h a n c e m e n t is not t h e s a m e a s in ? + ( Q ) . T h e s e p r o b l e m s a r e p r o b a b l y d u e t o a n i n a d e q u a t e M o n t e C a r l o s i m u l a t i o n . It d o e s not s i m u l a t e final s t a t e s t r o n g i n t e r a c t i o n s , a n d not all p a r t i c l e p r o d u c t i o n r a t e s a n d r e s o n a n c e s h a p e s a r e well m o d e l l e d . T h e i m p e r f e c t i o n s a r e a c c o u n t e d for by t h e s y s t e m a t i c u n c e r t a i n t i e s o n t he final m e a s u r e m e n t r e s u l t s . Finally, t h e m e a s u r e d t w o - p a r t i c l e c o r r e l a t i o n is f i t t e d w i t h a f u n c t i o n a s given by c q n (13.10). T h e lit r e s u l t s for o a r e c o n v e r t e d t o a d i m e n s i o n of lengt h,
I W<) I'AI '
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0.25 0.5 0.75
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l i e . 1.' .: . C o r r e c t « ! (loul)le r a t i o s ? , (C^) (left) a n d ulix( ) (right), as meas u r e d by t he A I.HI'I I e x p e r i m e n t . ( A I I C o l l a b . . 1998«), T h e c u r v e s represent fits of a m o d e l b a s e d o n a G a u s s i a n s p h e r i c a l l y s y m m e t r i c s o u r c e .
using t h e c o n v e r s i o n f a c t o r e = 0 . 1 9 7 G e V f m . T h e r e s u l t s a r e t y p i c a l l y w i t h i n t h e r a n g e 0 . 5 - 0 . 7 fin, w h i c h is of t h e o r d e r of t h e p r o t o n size, <9(1 fin). T h i s is a l s o t h e relevant s c a l e for n o n - p e r t , u r b a t l v e s t r o n g i n t e r a c t i o n s . T h e r e s u l t s for t h e c h a o t i e i t v p a r a m e t e r A v a r y m u c h m o r e s t r o n g l y , a n d for s o m e m e a s u r e m e n t s t h e y a r e e v e n l a r g e r t h a n 1. T h i s i n d i c a t e s s h o r t c o m i n g s of t h e f i t t e d model a n d o r of t h e e x p e r i m e n t a l t e c h n i q u e t o o b t a i n a c o r r e l a t i o n - f r e e r e f e r e n c e sample. A s a l r e a d y o u t l i n e d a b o v e , t h e i n t e r p r e t a t ion of t h e fit ted p a r a m e t e r s is not c o m p l e t e l y u n a m b i g u o u s , but t h e o b s e r v a t i o n of t he B o s e instein correlations has p h e n o i u e n o l o g i c a l i m p l i c a t i o n s . For e x a m p l e , in h a d r o u i c Z d e c a y s t h e s e c o r r e l a t i o n s , w h i c h in p r i n c i p l e a r e relevant o n l y for identical p a r t i c l e s , lead t o a g e n e r a l c o l l i m a t i o u of t h e e t s . All p a r t i c l e s a r e b r o u g h t closer t o g e t h e r in m o m e n t u m s p a c e , w h i c h t h e n l e a d s t o d i s t o r t i o n s in t h e mass spectra, e s p e c i a l l y at low m o m e n t u m w h e r e t h e m u l t i p l i c i t y is largest,. If t h i s effect, is not t a k e n i n t o account., t h e n t h e s i m u l a t i o n of r e s o n a n c e s s u c h a s t h e >(770) a n d t h e d(9.S2) will not be a b l e t o d e s c r i b e t h e m e a s u r e d s h a p e s . a t e l y , a n o t h e r i m p l i c a t i o n of Bose i n s t e i n correlat ions h a s g a i n e d t h e o r e t ical a n d e x p e r i m e n t a l i n t e r e s t . In t h e p r o c e s s e + e — W+ V —> h a d r o n s at I. I'2 c o r r e l a t i o n s m i g h t not o n l y a r i s e b e t w e e n p i o u s s t e m m i n g f r o m t h e s a m e W . but a l s o b e t w e e n p i o n s c o i n i n g f r o m different W s . S u c h a n e f f e c t c o u l d result in a shift of t he reconst r u c t e d W m a s s in m u l t i h a d r o n i c W d e c a y s , b e c a u s e t h e r e c o n s t r u c t e d e t m o m e n t a a r e d i s t o r t e d ; see, for e x a m p l e , onublad and S o s t r a n d ( 1 9 9 8 ) a n d r e f e r e n c e s t h e r e i n . Since t h e r e is n o u n a m b i g u o u s e x p e r i m e n t a l p r o o f yet for t h e e x i s t e n c e of c o r r e l a t i o n s bet ween p i o n s f r o m different W b o s o n s , t h e m e a s u r e d W m a s s is a s s i g n e d a s y s t e m a t i c u n c e r t a i n t y of t h e o r d e r of 2 5 M o V . Y e t a n o t her e f f e c t , r e l a t e d t o s t r o n g i n t e r a c t i o n s in h a d r o n i c final s t a t e s , w h i c h c o u l d possibly i n f l u e n c e t h e m e a s u r e m e n t of t h e W m a s s , is
Ill)
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d i s c u s s e d in t h e following s e c t i o n . I.'f.-l
C o l o u r reconnect, ion
T h r o u g h o u t most, of I lie h o o k we h a v e d i s c u s s e d I he p r o d u c t ion of h a d r o i i s l>y I he s e p a r a t i o n of c o l o u r c h a r g e s w i t h i n a c o l o u r singlet s y s t e m , such a s Z — ( ( hat Irons. I l o w e v e r . a n interest ing s i t u a t i o n a r i s e s w h e n t w o c o l o u r singlet s y s t e m s d e c a y close-bv in s p a c e t i m e . T h e r e t h e q u e s t i o n a r i s e s of how t h e r a d i a t i o n f r o m t h e t w o c o l o u r s i n g l e t s is a f f e c t e d bv t h e o v e r l a p of t h e t w o s y s t e m s , a n d at w h a t level, pert u r b a t i v e a n d o r noil-pert u r b a t i v e , t h e d i s t r i b u t ion of t h e liual s t a l e h a d r o n s is d e t e r m i n e d . S u c h s i t u a t i o n s c a n a r i s e in weak d e c a y s such as B .! >!' I A . in Z — q q e v e n t s p r o v i d e d t h a t t h e r e a r e t w o perl u r b a t i v e g qq vert ices in t h e p a r t o u s h o w e r b e f o r e s t r i n g f r a g m e u t a t i o u o r at least t lirce c l u s t e r s in t h e c a s e of c l u s t e r f r a g i n e n t a t i o n . ill W - p a i r p r o d u c t i o n in e ' e annihilation, w h e r e b o t h W s d e c a y h a d r o n i r a l l y i n t o q u a r k p a i r s , a s well a s in c + c —< ZZ, 4 , etc. e'e 2 H , p p p p - - VV-1I. II W + W - , it - b V V b T h e c a s e w h i c h r e c e n t l y h a s b e e n s t u d i e d in m o s t d e t a i l is t h e o n e of fully — <[I s h o u l d be s u p p r e s s e d by 1 A r = 1 9 , f r o m s i m p l e c o u n t i n g of p o s s i b l e c o l o u r s t a t e s . e v e r t h e l e s s , if it h a p p e n s , w e typically expect a n e n h a n c e m e n t of p a r t i c l e p r o d u c t i o n bet ween t he e t s f r o m different W's. St rictly s p e a k i n g , a s p o n t a n e o u s r e a r r a n g e m e n t of c o l o u r d i p o l e s is not. what is m e a n t by c o l o u r r e a r r a n g e m e n t in t h e f r a m e w o r k of p e r t u r b a t i v e Q C . T h e r e , i n t e r f e r e n c e e f f e c t s b e t w e e n t h e t w o r a d i a t i n g s y s t e m s h a v e t o b e c a l c u l a t e d . In S ostrand and h o z e (1994) it is s h o w n t h a t s u c h i n t e r f e r e n c e e f f e c t s c a n o n l y o c c u r at ( 9 ( n f ) . t h a t is. w h e n a g l u o n is r a d i a t e d f r o m b o t h c o l o u r d i p o l e s f o r m e d by t h e original q u a r k p a i r s . T h e d e t a i l e d c a l c u l a t i o n s h o w s t h a t such e f f e c t s a r e s u p p r e s s e d by n 2 (.'V, I). In a d d i t i o n , b e c a u s e of t he ( m i t e s e p a r a t i o n of t h e I wo V b o s o n s , o n l y c o n t r i b u t i o n s f r o m soft g l u o n s with ^ V w are to be expected, which m e a n s t h a t o n l y a few low-energy p a r t i c l e s s h o u l d be a l l c c t e d . c h a n g i n g t h e m u l t i p l i c i t y by A . v c o ' u r w ' m ^ ( r i ) . T h e effect on t h e t o t a l c r o s s
i '< tl ( I'll III i i >
< I'lo
iTI.inn lor W paii proihict ion is est i m a t e d lo lie even s m a l l e r t h a n nit. S o il is r o n c l n d c d t h a t c o l o u r reconnect ion at t h e pert.urhat ive level s h o u l d lie negligible. The sit n a t i o n is different w h e n g o i n g t o t he h a d r o n i z a t ion p h a s e , which o c c u r s ai d i s t a n c e s c a l e s of t h e o r d e r of I fin. m u c h l a r g e r t h a n t h e typical s e p a r a t i o n ol t h e d e c a y i n g VVs. T h e r e f o r e , l a r g e r e f f e c t s c o u l d h e e x p e c t e d . T h e p r o b l e m is t h a t for t h i s level o n l y p h e n o m e n o l o g i c a l m o d e l s exist, r a t h e r t h a n c a l c u l a tions f r o m first, p r i n c i p l e s . For e x a m p l e , w i t h i n t h e p i c t u r e of t h e u n d s t r i n g m o d e l , we k n o w t h a t at t h e e n d of t h e pert u r h a t i v e p h a s e , w h i c h g o v e r n s t h e s h o w e r i n g of p a r t o n s , s t r i n g s a r e s p a n n e d wit hin t h e original c o l o u r dipolos. wit h possible k i n k s d u o t o r a d i a t e d g l u o n s . o w we could i m a g i n e that, t h e original st ring c o n f i g u r a t i o n is c h a n g e d if t h e r e is a s u b s t a n t i a l o v e r l a p of t h e st r i n g s (or colour fields) f r o m t h e t w o d i p o l e s . I l a d r o u p r o d u c t i o n t h e n o c c u r s f r o m t h e t w o r e a r r a n g e d s t r i n g s , l e a d i n g t o a c h a n g e in t h e final s t a t e a s d i s c u s s e d a b o v e . S o s t r a n d a n d h o z e (li)!M) h a v e s t u d i e d t h e p h e n o i n e n o l o g i c a l c o n s e q u e n c e s of different m o d e l s , d e p e n d i n g on t h e d e t a i l e d s p a c e t i m e d e s c r i p t i o n of t h e s t r i n g . T h e p r o b a b i l i t y for s t r i n g r e a r r a n g e m e n t m a y b e c o m p l e t e l y fixed by t he model, o r m a y be a f r e e p a r a m e t e r t o b e d e t e r m i n e d f r o m t h e d a t a . I l a d r o u d i s t r i b u t i o n s s u c h a s r a p i d i t y o r m u l t i p l i c i t i e s a r e c h a n g e d m o r e o r less s t r o n g l y by different, m o d e l s . In a n y c a s e , it t u r n s o u t t h a t t h e finite d e t e c t o r r e s o l u t i o n a n d a c c e p t a n c e t e n d t o w a s h out. t h e effects. In a d d i t i o n , s t r i n g o v e r l a p m a y I ' s u b s t a n t i a l o n l y for a p a r t i c u l a r s u b s e t of all possible W W d e c a y t o p o l o g i e s , which r e s u l t s in s t a t i s t i c a l l i m i t a t i o n s . A s a n e x a m p l e , t h e f o u r I.HI e x p e r i m e n t s h a v e m e a s u r e d t h e c h a r g e d p a r t i c l e m u l t i p l i c i t y in fully h a d r o n i e W d e c a y s a n d s u b t r a c t e d t w i c e t h e m u l t i p l i c i t y f r o m s e m i - l e p t o n i c W e v e n t s . For t h e l a t t e r n o c o l o u r reconnect ion c a n o c c u r . W i t h i n t h e i r u n c e r t a i n t i e s t h e d i f f e r e n c e s a r e f o u n d t o lie c o n s i s t e n t w i t h zero, that is, f r o m I he d a t a t h e r e is 110 i n d i c a t i o n for s i z e a b l e c o l o u r r e c o n n o c t i o n effects. However, t h e e r r o r s a r e t o o l a r g e t o really d i s c r i m i n a t e b e t w e e n m o d e l s w i t h o u t reconnoct ion a n d m o d e l s w i t h r e c o n n o c t i o n , w h i c h predict, o n l y a s m a l l effect for s o f t p a r t i c l e s . In a d d i t i o n , t h i s s o f t region is t h e o n e m o s t a f f e c t e d by d e t e c t o r e l f e c t s . A m o r e p r o m i s i n g a p p r o a c h could be t h e s t u d y of p a r t i c l e How bet ween e t s a s s i g n e d t o t h e s a m e W a n d b e t w e e n e t s f r o m different W s . T h e r e , o b s e r v a b l e s c a n b e c o n s t r u c t e d which a p p e a r t o be m o r e d i s c r i m i n a t i n g t h a n t he c h a r g e d p a r t i c l e m u l t i p l i c i t y . T h e s e s t u d i e s a r e o n g o i n g . For a n o v e r v i e w of t h e present s i t u a t i o n we refer t h e i n t e r e s t e d r e a d e r t o t h e p a p e r by o n g (20(11). T h e most, i m p o r t a n t effect, is seen 11 t h e m e a s u r e m e n t of t h e W m a s s . F r o m m e a s u r e m e n t s b a s e d on M o n t e C a r l o m o d e l s w i t h or w i t h o u t c o l o u r r e e o n n e e lion a s y s t e m a t i c u n c e r t a i n l y of AM\\- -= 10 MoV is a s s i g n e d , which d o m i n a t e s t h e c u r r e n t u n c e r t a i n t y of t h e I . I d ' m e a s u r e m e n t f r o m fully h a d r o n i e d e c a y s of A : V v ( t o t ) = 62 M o V a n d A A W ( s t a t ) = .HI MoV ( B a r b e r i o r.t « ., 2 0 0 1 ) . A r e d u c t i o n o r at. least a solid u n d e r s t a n d i n g of t h i s u n c e r t a i n t y could lie o b t a i n e d by f i n d i n g o t h e r o b s e r v a b l e s which clearly d i s c r i m i n a t e bet ween d i f f e r e n t r e c o n noction s c e n a r i o s , o r by r e s t r i c t i n g t h e m a s s m e a s u r e m e n t s t o t o p o l o g i e s a n d o r r e c o n s t r u c t i o n m e t h o d s which a r e t h e least s e n s i t i v e t o p o s s i b l e effects. An in-
I'KAOMENTA'I ION
11-2
d i l a t i o n t h a t roconnoction olfcets might not he largi\ a f t e r all, is given by i h e fact that t h e (inference of t h e W masses m e a s u r e d f r o m fully h a d r o n i e a m i semileptoiiio decays is consistent with zero. A j \ / \ v ( q q q q <1 <\(T/f) 18 :h l(> M e \ ( B a r b e r i o ci ni. 2(1(11).
Exercises for Chapter 13 1» 1 C o n f i r m that, if D(x) = flvr NI' t h e n (.R) = |{-'')NI>13 2 T h e p o p u l a r P e t e r s o n ci ul. f r a g m e n t a t i o n f u n c t i o n , eqn (13.2). c a n lie derived from t h e i n d e p e n d e n t f r a g m e n t a t i o n model by a p p l y i n g old fashioned p e r t u r b a t i o n t h e o r y t o t h e process Q — ( Q q ) + q . w h e r e t h e m o m e n t u m /> of t h e heavy q u a r k Q is s h a r e d between a heavy meson Q q a n d a leftover cpiark q. W i t h AE t h e e n e r g y used for t h e f o r m a t i o n of t h e meson a n d t h e leftover q u a r k t h i s yields , , \ i t J C ' Q , , -l q | H | Q ) | "N!>(•'') oc ( p h a s e space) (AE)'' Given that t h e ( l o n g i t u d i n a l ) p h a s e s p a c e is p r o p o r t i o n a l t o f / . c a n d a s s u m i n g that t h e m a t r i x e l e m e n t is c o n s t a n t , d e r i v e eqn (13.2) a n d show t h a t e q = R ~ ~ ¡ m f y 13 3 Use oqns (3.312) a n d (3.343) to c a l c u l a t e for s y m m e t r i c 7* —> qijg a n d 7* —• q q - e v e n t s t h e relative p r o b a b i l i t i e s for a soft glnon to be p r o d u c e d d i r e c t l y b e t w e e n t h e q a n d t h e q. d i r e c t l y b e t w e e n t h e q a n d t h e g or 7 a n d p e r p e n d i c u l a r t o t h e event p l a n e . 13 4 Show t h a t t h e t w o - p a r t i c l e c o r r e l a t i o n f u n c t i o n for t w o ident ical bos(Ax) o n s (for e x a m p l e pious) b e h a v e s like 1 + c o s ( A f c - A r c ) . w h e r e A k is t h e difference of t h e i r m o m e n t a (space c o o r d i n a t e s ) . Use t h e plane wave a p p r o x i m a t i o n for t h e particle wave f u n c t i o n s . T h e n i n t r o d u c e a s p a t i a l d e n s i t y d i s t r i b u t i o n />(x) for t h e s o u r c e of pion p r o d u c t i o n , a n d show that in this case t h e Bose E i n s t e i n e n h a n c e m e n t effect in t h e correlation f u n c t i o n is p r o p o r t i o n a l to t h e s q u a r e of t h e Fourier t r a n s f o r m p(Ak) of t h i s s o u r c e d i s t r i b u t i o n . SOUK* h i n t s t o w a r d s t h e solution m a y be found in t h e article by Bowler (198.r>). ( * ' 13 -r> Discuss t h e i m p l i c a t i o n for B o s e E i n s t e i n c o r r e l a t i o n s of c o h e r e n c e between pion sources, a n d show t h a t t h e c o r r e l a t i o n f u n c t i o n is enh a n c e d for r a n d o m l y f l u c t u a t i n g sources. (For s o m e hints see. a g a i n . Bowler. l!)S.r>.) 13 (» Derive t h e t w o - p a r t i e l e c o r r e l a t i o n f u n c t i o n for a spherically s y m m e t ric G a u s s i a n s o u r c e (in t h r e e s p a c e d i m e n s i o n s ) , using t h e results of E x . (13-4).
SUMMARY
In t h e c h a p t e r s of this book we have tried t o give an overview how Q C D evolved I rom t h e early b e g i n n i n g s in harlron s p e c t r o s c o p y to t h e precision m e a s u r e m e n t s 111 h a r d i n t e r a c t i o n s t hat b e c a m e available with high e n e r g y collider e x p e r i m e n t s . T o d a y Q C D is a well e s t a b l i s h e d part of t h e S t a n d a r d Model of e l e m e n t a r y particle physics, e x p e r i m e n t a l l y t e s t e d with a precision a t t h e p e r cent level for m a n y results. A m a j o r p r o b l e m a n d challenge in t h e s t u d y of Q C D is t h e confinement property. that is. c o l o u r c h a r g e s a r e not o b s e r v e d iu a s y m p t o t i c free particle s t a t e s . Nevertheless, it has been possible to p r o b e t h e s t r u c t u r e of t h e Q C D L a g r a n g i a n with g r e a t a c c u r a c y , s h o w i n g t h a t s t r o n g i n t e r a c t i o n s a r e indeed described by a 11011-abeliaii g a u g e t h e o r y based on a n S U ( 3 ) s y m m e t r y . T h e c o n s e q u e n c e s for liadron p r o d u c t i o n processes in high e n e r g y i n t e r a c t i o n s a s s t u d i e d at all t y p e s of high e n e r g y colliders a r e well u n d e r s t o o d . T o d a y m u l t i - h a d r o n p r o d u c t i o n is u n d e r s t o o d to o r i g i n a t e f r o m a c o h e r e n t p a r t o n s h o w e r i n g process. Still, m a n y o p e n q u e s t i o n s r e m a i n , in p a r t i c u l a r ¡11 t h e low e n e r g y regime where t h e s t r o n g coupling b e c o m e s so large t h a t p e r t u r b a t i o n t h e o r y b r e a k s down. T h i s field could be t r e a t e d only very superficially in this book, but it h a s to bo e m p h a s i z e d t h a t also here significant p r o g r e s s h a s been m a d e in t h e past years. T h e field is still r a p i d l y evolving, closely coupled to progress iu a n a l y t i c a l as well as numerical m e t h o d s a n d high p e r f o r m a n c e c o m p u t i n g . So. d e s p i t e t h e laet t hat t h i s m o n o g r a p h h a s b e c o m e a r a t h e r largo volume, t hen* a r e si ill m a n y s u b j e c t s t h a t could not. be covered. We nevertheless h o p e that we have m a n a g e d to give t h e reader s o m e insight into how Q C D evolved t o its current s t a t u s a n d how it can be tested in high energy r e a c t i o n s bet ween e l e m e n t a r y particles. Historically we a r e c u r r e n t l y in a t r a n s i t i o n p h a s e . T h e I.EP p r o g r a m m e has finished a n d now i n f o r m a t i o n f r o m e 4 e a n n i h i l a t i o n s h a s to a w a i t t h e s t a r t u p of a p r o p o s e d linear collider o r allot her. similar, facility. For t h e n e a r fut ure wo can e x p e c t anot her set of precision m e a s u r e m e n t s of t h e nucleón s t r u c t u r e f u n c t i o n f r o m lepton nucleoli s c a t t e r i n g e x p e r i m e n t s at DESY a n d CEHN. T h e s e e x p e r i m e n t s will p e r f o r m h i g h - a c c u r a c y m e a s u r e m e n t s of t he nucleón s t r u c t u r e funct ions, including t h e spin s t r u c t u r e , which will allow a reliable e x t r a p o l a t i o n to e n e r g i e s iu t h e niiilti-TeV range. T h u s t h e y will p r o v i d e i m p o r t a n t i n p u t for a q u a n t i t a t i v e u n d e r s t a n d i n g of t h e physics at t h e Large l i a d r o n Collider. I.IIC. a t C E H N . Q C D is a n d will r e m a i n a very a c t i v e field of research, a n d we a r e looking forward t o now m e a s u r e m e n t s a n d t h e o r e t i c a l d e v e l o p m e n t s , which will shod
III
S MMA Y
li Miller light on I he s t r u c t u r e of hnryoiiie m a t t e r a n d t h e i n t e r a c t i o n s between its c o n s t i t u e n t s . T h e s p e c t r u m ranges from st udies of h o u n d s t a t e s at low energy scales, over few-body interact ions at high energies, t o m a n y - b o d y s y s t e m s at the e x t r e m e c o n d i t i o n s of a q u a r k gluoii p l a s m a , as most likely existed in the early universe. Today, t h e f o u n d a t i o n s for u n d e r s t a n d i n g s t r o n g i n t e r a c t i o n s a r e firmly established, but t hat is c e r t a i n l y not t h e end of t h e story. Much exciting physics still lies a h e a d .
IVI-R. I A U T O N O M A NIMADRID
I I TEC CIE CI
A E E
E
IX
T
F GR
A THE
R
G r o u p I heory r a n be viewed a s t h e i n a t h e m a t i e a l description of syininetries. Since syininetries o f t e n play a key role in physics, g r o u p t h e o r y e n t e r s into t h e discussion in m a n y places. Here we will give a short, s u m m a r y of t he most i m p o r tant a s p e c t s of g r o u p theory, wit h p a r t i c u l a r e m p h a s i s on ie g r o u p s . For furt her studies t h e reader is referred t o d e d i c a t e d t e x t b o o k s ( o n e s . 1!)!H): T u n g . l!)S.r>: C a l m . l!)S'l) a n d references given t herein. A.l
B a s i c s of
ie g r o u p s
A g r o u p . . is a set of e l e m e n t s . A B.C.... . t o g e t h e r with a b i n a r y o p e r a t i o n , t h e g r o u p multiplication o. which satisfy t he following r e q u i r e m e n t s : t h e product of any t wo e l e m e n t s is again a m e m b e r of t h e g r o u p , a p r o p e r t y known as closure. AoB
= C
,
(A.L)
the multiplication is associative, Ao(BoC)
= (AoB)oC,
(A.2)
there exists a u n i q u e g r o u p e l e m e n t , t h e identity m e n t s invariant, u n d e r t h e g r o u p m u l t i p l i c a t i o n
. which leaves all g r o u p ele-
Ao
=
oA=A,
(A. )
a n d for each g r o u p element .1 there- exists a unique inverse .4 ' . such that A o A - 1 = A~'
o A =
.
(A. l)
ote that in general t h e multiplication is not c o m m u t a t i v e , that. is. Ac>B BoA. n l y for t h e special case of a cliuii g r o u p s o n e h a s A o B — B o A. T w o g r o u p s a r e isomorphic to o n e a n o t h e r if t h e r e exists a one-to-one m a p ping between t h e m , which preserves t h e g r o u p ' s product s t r u c t u r e : f ( A ) o f(B) f(A o B). An i s o m o r p h i s m which m a p s t h e g r o u p (dements o n t o s q u a r e matrices a n d t h e g r o u p multiplication o n t o o r d i n a r y m a t r i x multiplication is called a r e p r e s e n t a t i o n of t h e g r o u p . From t h e general p r o p e r t i e s of t h e matrix mult iplication it is evident t h a t t h e delining p r o p e r t i e s of a g r o u p can be satisfied. In fact, it can be shown t h a t infinitely m a n y r e p r e s e n t a t i o n s exist for every g r o u p , which differ in t h e dimensionality of t h e matrices. T h e lowest dimensional non-trivial r e p r e s e n t a t i o n is also referred t o as t h e f u n d a m e n t a l or delining r e p r e s e n t a t i o n of a g r o u p a n d d e n o t e d by t he index .
Ill,
I.
T
C
I
HI
I n t e r p r e t i n g mat rices a s t r a n s f o r m a t i o n s of a vecior. for e x a m p l e , r o t a t i o n s nr reflections, i l l u s t r a t e s t h e conneetiou between g r o u p t h e o r y a n d s y m m e t r y . Il also shows how different r e p r e s e n t a t i o n s of a g r o u p realize t h e s a m e symmetry in diirereut-dimensional spaces. A c o m m o n e x a m p l e a r e t r a n s f o r m a t i o n s which preserve t h e o r t h o u o r m a l i t y of a set of basis vectors. Working on complex numbers, these are x matrices U. which satisfy UjU = 1. T h e s e mat rices form t he f u n d a m e n t a l r e p r e s e n t a t i o n of t h e u n i t a r y g r o u p of dimension :Y. ( ). sing t h e u u i t a r i t y constraint it is easy t o see t h a t det<(( ) = 1: if in addition t h e m a t r i c e s have ( l e t ( U ) = + 1 . then we have t h e special u n i t a r y g r o u p . S ( ) , a s u b - g r o u p of ( ). eal x matrices satisfying ' = I form the o r t h o g o n a l g r o u p of d i m e n s i o n A'. 0 ( ) . a n d if in a d d i t i o n det.( ) 11 t h e special ort hogonal g r o u p . S ( ) . T h e special u n i t a r y a n d o r t h o g o n a l g r o u p s a r e e x a m p l e s of so-called Lit groups. T h e s e a r e defined as c o n t i n u o u s groups, in t h e sense that it is possible to p a r a m e t e r i z e each g r o u p e l e m e n t in t e r m s of a set of real p a r a m e t e r s ()„.(! 1 . 2 . . . . . in such a way t h a t t h e product of t wo g r o u p elements is p a r a m eterized by a n a n a l y t i c f u n c t i o n of their p a r a m e t e r s . T h e n u m b e r of p a r a m e t e r s is specific t o a given g r o u p . A familiar e x a m p l e is t he u l e r - a n g l e p a r a m e t e r i z a t i o n of t h e s p a t i a l r o t a t i o n s g r o u p S ( 3 ) . which has t h r e e p a r a m e t e r s . If we a r r a n g e that 0„ = 0 p a r a m e t e r i z e s t h e identity a n d i n t r o d u c e t r a n s f o r m a t i o n s T . i = 1 . 2 . . . . . associated with t h e individual p a r a m e t e r s , t h e so-called generators of t h e g r o u p , then t h e continuity p r o p e r t y allows g r o u p elem e n t s in t h e n e i g h b o u r h o o d of t h e identity t o be w r i t t e n as U(S0)
= 1 + iSf)„T
+ 0(60-)
.
(A.5)
o t e that, here a n d in t h e following, even if not explicit ly w r i t t e n , s u m m a t i o n over identical indices is implied. T h e f a c t o r i = — l h a s been i n t r o d u c e d for later convenience. By a p p l y i n g a sequence of infinitesimal s t e p s we can write a macroscopic' g r o u p element as U(0) =
hn
+ i
= eitf
7
.
(A.6)
T h u s , within a c o n n e c t e d g r o u p , each e l e m e n t can be reached f r o m t h e identity a n d w r i t t e n as a n e x p o n e n t i a l of a linear c o m b i n a t i o n of t he g e n e r a t o r s . All inform a t i o n a b o u t the st ruct ure of t h e g r o u p a n d its s y m m e t r y c a n be e x t r a c t e d from t h e g e n e r a t o r s . T h i s is a n e n o r m o u s simplification for t h e s t u d y of t h e p r o p e r t i e s of t h e g r o u p , since one only has to deal with a finite n u m b e r of g e n e r a t o r s r a t h e r t h a n wit h a c o n t i n u u m of g r o u p elements. An o p e n question at this point is t h e a c t u a l choice a n d n o r m a l i z a t i o n of t h e g e n e r a t o r s . vidently, given o n e set of g e n e r a t o r s t h e g r o u p can equally be c o n s t r u c t e d from any linear c o m b i n a t i o n of t h o s e g e n e r a t o r s with s u i t a b l y t r a n s f o r m e d p a r a m e t e r s . While t h e g r o u p a s such is invariant with respect to t h e act ual choice for t h e g e n e r a t o r s , t h e st udy of t he g r o u p p r o p e r t i e s is g r e a t l y
MASH
• <>I I II
117
(¡KOIII'S
i.implilicd when using it sol of g e n e r a t o r s wliiiii most, cloarly reflects t h e internal '.vuiinct ries of tlie g r o u p . As a heuristic argument., consider t h e c h a n g e of a n . in general complex valued. vector r u n d e r an infinitesimal t r a n s f o r m a t i o n with a n y of t h e g e n e r a t o r s /'", given by d r " 6T"v, with 6 a n infinitesimal global scale p a r a m e t e r . A m e a sure for t h e t o t a l c h a n g e of r is t h e s c a l a r p r o d u c t d r " L l r " , which for a good choice of t h e g e n e r a t o r s should on average, t h a t is. when i n t e g r a t i n g over all vectors r . he t h e s a m e for all g e n e r a t o r s . Likewise, the. scalar p r o d u c t of c h a n g e s lioin i n d e p e n d e n t g e n e r a t o r s . dv\dv'\ should vanish on average. Formally t hese condit ions t r a n s l a t e to {{Tuv)\Tbv))
~ Id'%
T r {vv^T^T1'}
- T r {T"*Th}
= T,{6"h
.
(A.7)
T h e first p r o p o r t i o n a l i t y is simply t h e definition of t h e average, t h e second o n e holds b e c a u s e t h e integral over t h e direct product of two vectors is p r o p o r t i o n a l to t he unit m a t r i x w h e n integrat ing over t h e ent ire space. By construct ion M" 1 ' Tr { T"irrh} is a posit ive-demidefinite hermitiaii m a t r i x . H e n n i t i c i t y follows from (M1'")*
= T r {ThiT"Y
= T r {Tb*Ta}*
= T r { T " t ' T b } = M"1'.
(A.8)
where t h e s t a r d e n o t e s c o m p l e x c o n j u g a t i o n , a n d positive-semidefiniteness f r o m the fact that t h e q u a d r a t i c form E >'},M"h>'i> = X ] T r { ( v l T " ' ) ) } = T r { 5 * 5 } nh
with
5 = ^ n
T"..„
is non-negative for every non-zero vector r . T h u s , it is always possible t o s a t i s f y e<|ii (A.7) by fiiuliug a n a p p r o p r i a t e t r a n s f o r m a t i o n of a given set of g e n e r a t o r s . T h e p r o p o r t i o n a l i t y constant /"/?. t h e so-called D y n k i n - i n d e x . is a n o r m a l i z a t i o n p a r a m e t e r t h a t can be chosen freely for o n e r e p r e s e n t a t i o n of t h e g r o u p . Ii is then fixed for all o t h e r r e s p r e s e n t a t i o n s . In t e r m s of t h e e x p o n e n t i a t e d form, t h e product of two g r o u p e l e m e n t s is given by t h e Baker C a m p b e l l I lausdorff f o r m u l a . As is easily d e m o n s t r a t e d by e x p a n d i n g t h e e x p o n e n t i a l s t o second o r d e r a n d not a s s u m i n g coiniiiutativity of the factors o n e o b t a i n s
exp(i0)exp(i4>) = exp j i (0-f ) - ^\(),
(A.!))
Here 0 a n d a r e linear c o m h i n h a t i o n s of t h e g e n e r a t o r s with real coefficients. T h e c o m m u t a t o r . \().<}>\ = ()
11«
ELEMENTS O F « H O U P TIIEOKY
t o lie ¡1 linear c o m b i n a t i o n of tlie g r o u p ' s g e n e r a t o r s with real-valued coefficients rtibc [T",T6] = i J " , K T .
(A.Ill)
The eoellicients f"')r a r e called structure constants a n d serve to define t h e Lie algebra of t h e g r o u p ' s g e n e r a t o r s . If t h e /"'"" vanish, t h e n t h e g e n e r a t o r s a n d hence g r o u p e l e m e n t s c o m m u t e a n d t h e g r o u p is a b e l i a n . E x p a n d i n g out the commutators [{'T.Tb\.Tc]
+ [\Th.TC\,T")
+ [[T\T"}.Th]
= 0 .
(A.ll)
one sees that t h e s t r u c t u r e c o n s t a n t s satisfy t h e .laeobi identity jnlHl filet
jhnlJ'I'lf
JC/ldjdht
_ Q
^
J.,)
By c o n s t r u c t i o n , t h e s t r u c t u r e c o n s t a n t s a r e manifestly a n t i s y m m e t r i c in t h e first t w o indices. If t h e g r o u p is semi-simple a n d c o m p a c t , t h e n o n e can also find a basis for t h e g e n e r a t o r s w h e r e t h e s t r u c t u r e c o n s t a n t s a r e completely a n t i s y m m e t r i c in all indices. For an i n - d e p t h discussion of these c o n c e p t s t h e reader is referred t o t h e l i t e r a t u r e . Here, we will only show how this p r o p e r t y comes a b o u t for s o m e g r o u p s that f r e q u e n t l y a p p e a r in physics, such a s SU(N) or S O ( N ) . T h e g e n e r a t o r s of these g r o u p s a r e also known a s q u a n t u m mechanical ope r a t o r s a c t i n g on s t a t e vectors a n d t h u s a r e a s s o c i a t e d w i t h physical observables. T h e r e f o r e , t h e y a r e represented by h e r m i t i a n matrices, t h e eigenvalues of which c o r r e s p o n d t o observable q u a n t u m n u m b e r s . For h e r m i t i a n m a t r i c e s , = T,<6"''. one o b t a i n s which s a t i s f y T " f = T", eqn (A.7) b e c o m e s T r {T"Tb} from eqn (A. 10) T r { \ T " . T>'\TC} = T r {i f«MT''Tc
} = i f"hdTnV,c
(A. 13)
a n d thus, e x p a n d i n g t h e c o m m u t a t o r in eqn (A.13) /•»<"• = _
' Tl. r T i T b T c _ I ii
T
bT,Tc}
(A.14)
We see t h a t with t h e choice of eqn (A.7) for t h e g e n e r a t o r s t h e s t r u c t u r e c o n s t a n t s form a completely a n t i s y m m e t r i c t e n s o r . It is clear t hat t h e r e is a very close relat ionship between t h e g e n e r a t o r s of a g r o u p a n d their Lie a l g e b r a , a n d t h e s t r u c t u r e of t h e g r o u p itself. In p a r t i c u l a r , if we can u n d e r s t a n d t h e a c t i o n of t h e g e n e r a t o r s on t h e g r o u p ' s basis s t a t e s , we can u n d e r s t a n d t h e a c t i o n of t h e g r o u p itself. T h e r e f o r e , we c o n c e n t r a t e on investigating t h e m a t r i x r e p r e s e n t a t i o n s of t h e Lie a l g e b r a . In m a n y a p p l i c a t i o n s t o high energy physics, wave f u n c t i o n s of physical s t a t e s a r e represented by vectors a n d t h e g e n e r a t o r s a c t as o p e r a t o r s t r a n s f o r m i n g those s t a t e s . T h e a l g e b r a defined by e q n (A.10) does n o t have a u n i q u e realization. S u p p o s e we have a n jV-dimensional r e p r e s e n t a t i o n of t h e g r o u p g e n e r a t e d by a set of A r x N
II \'.K
< »1 I II O H O l t p S
matrices '/'*'(/?) which satisfy eqn (A.10). II t h e g e n e r a t o r s a r e represented by hermitian matrices, then one easily sees t h a t t h e g e n e r a t o r s T"(R) = —T"'(R) r also form a r e p r e s e n t a t i o n of t h e g r o u p . W i t h T'" = ( T " ) ' a n d t h u s T"(J{) (T")1 (R) one finds \T"(R).Th(R)}
= [Th(R).T"(R)]r
= i f'""'(Tr)r(R)
= i f"'"V(R)
.
(A.Hi)
Note t h a t t h e m i n u s sign in t h e definition of t h e g e n e r a t o r s T"(R) results in s t a t e s t r a n s f o r m i n g a c c o r d i n g to t h e c o m p l e x c o n j u g a t e r e p r e s e n t a t i o n h a v i n g I he o p p o s i t e sign eigenvalues a s t h e s t a t e s t r a n s f o r m i n g u n d e r T"(R). T h u s , if l{ is used for p a r t i c l e lields. then R is t h e nat ural r e p r e s e n t a t i o n for t h e a n t i p a r t ieles. If t h e two r e p r e s e n t a t i o n s l"(R) a n d ]"(R) a r e equivalent, that is. if a non1 singular m a t r i x S exists such that ST"(R)S~ T"(R) for all ii. t h e n it is called a real representat ion. For example, all r e p r e s e n t a t i o n s of SU(2) a r e real: a fact which relies on t h e eigenvalues of a n y of t h e g e n e r a t o r s o c c u r r i n g in o p p o s i t e sign p a i r s ±A. On t h e o t h e r h a n d , t h e f u n d a m e n t a l r e p r e s e n t a t i o n of SU(3) a n d ils complex c o n j u g a t e a r e not equivalent. In flavour SU(3) t h e q u a r k s t r a n s f o r m under t h e former, t h e ant ¡quarks u n d e r t h e lat ter representat ion. An i m p o r t a n t realization of t h e algebra is t h e adjoint r e p r e s e n t a t i o n .1. defined I hrough t he m a t r i c e s P W V
= i/"'"-.
(A.l(i)
T h a t these m a t r i c e s s a t i s f y e q n (A.10) follows s t r a i g h t f o r w a r d l y from t h e . l a e o b i identity eqn (A.12). T h e dimension of t h e adjoin t r e p r e s e n t a t i o n is t h e s a m e as t h e n u m b e r of real p a r a m e t e r s required to specify an element of t h e g r o u p . As such, it also becomes the r e p r e s e n t a t i o n u n d e r which t h e g a u g e fields t r a n s f o r m in a g a u g e theory. An i m p o r t a n t q u a n t i t y t o c h a r a c t e r i z e a r e p r e s e n t a t i o n R of a g r o u p is t h e eigenvalue C'a of tin* q u a d r a t i c C a s i m i r o p e r a t o r . T~(R)
= YjT"{R)T"\R)
.
(A. 17)
»
For t h e special case of h e r m i t i a n g e n e r a t o r s t h e value of C'/t can be d e t e r m i n e d by s i m p l y e v a l u a t i n g t h e s u m in eqn (A.17). As shown below, since that s u m c o m m u t e s with all g e n e r a t o r s , a n d t h u s also with all g r o u p elements, it is by Sclnir's first l e m m a p r o p o r t i o n a l t o t h e identity m a t r i x . For t h e c o m m u t a t o r \T2.TU\ of t h e C a s i m i r o p e r a t o r with a n a r b i t r a r y g e n e r a t o r one finds Y^\T"T".Th\ <>
- ^^ T"\T".y''] n
+ \T".Tl'}T"
=J2'f"hc
+ Tcr").
(A.IS)
Since t h e / " ' " a r e completely a n t i s y m m e t r i c in all indices, whilst t h e t e r m in parenthesis is s y m m e t r i c u n d e r e x c h a n g e of n a n d c. t h e s u m vanishes a n d 7' 2 has t o be p r o p o r t i o n a l to t h e unit m a t r i x .
I I i :\IKNTS <)!• CHOI II' I IIKOHN
120
In I In* context. ill Q C D l lif e i g e n v a l u e s C'F a n d C,\ a s s o c i a t e d wit ii t h e Inn daini'iital a n d t h e a d j o i n t r e p r e s e n t a t i o n . respectively. a r e a l s o k n o w n a s coloiu f a c t o r s . An i m p o r t a n t r e l a t i o n is o b t a i n e d in c o n n e c t i o n w i t h e q n ( A . 7 ) . C o m p a r i n g t h e d e f i n i t i o n of t h e D y n k i n index Tp a n d t h e e i g e n v a l u e Cp, b o t h lot t h e f u n d a m e n t a l r e p r e s e n t a t i o n of t h e g r o u p , o n e h a s YT-JTJ> 0
=
T
'-S"b
a,Kl
Y T ' i "•}
r
' H =
c
(
A
1
°
)
S u m m i n g over all indices, b o t h e x p r e s s i o n s yield t he s a m e r e s u l t . a n d o n e o b t a i n » 7 > • Na
= CF • NP •
(A.20)
w h e r e NA a n d NF a r e t h e d i m e n s i o n s of t h e a d j o i n t a n d t h e f u n d a m e n t a l r e p r e s e n t a t ions. respectively. In p r a e l i c e we o f t e n e n c o u n t e r sit u a l i o n s w h e r e t wo p a r t i c l e s , w h i c h I r a n s f o r m a c c o r d i n g t o r e p r e s e n t a t i o n s U\ a n d ll> of t h e s a m e g r o u p , f o r m a c o m b i n e d s y s t e m which a l s o e x h i b i t s t h e s y m m e t r y of t h e g r o u p . T h e g e n e r a t o r s a c t i n g on t h e c o m b i n e d s y s t e m a r e t h e n given by t h e t e n s o r p r o d u c t 7 f t , « f t * = Tn• ®
+
1
®
T
n> •
(A.21)
In t h e p r o d u c t - t e r m s t h e first f a c t o r a l w a y s a c t s on t h e p a r t i c l e t r a n s f o r m i n g a c c o r d i n g t o /?, a n d t h e s e c o n d o n e o n t h e o t h e r p a r t i c l e . It is i m m e d i a t e l y seen t h a t t h e q u a n t u m n u m b e r s of s u c h a s y s t e m , t h a t is. t he e i g e n v a l u e s of t h e g e n e r a t o r s t h a t a r e s i m u l t a n e o u s l y d i a g o n a l i z a b l o . a r e t he s u m s of t he q u a n t u m n u m b e r s of t h e c o n s t i t u e n t s . T h i s m a k e s it s i m p l e t o c o n s t r u c t t h e so-called root diuijitirn of a mult i - p a r t icle s y s t e m , which d i s p l a y s t h e e l e m e n t s of a r e p r e s e n t a t i o n in a p i c t u r e u s i n g t h e eigenvalues of t he s i m u l t a n e o u s l y d i a g o n a l i / a b l e g e n e r a t o r s , t h a t is. t h e q u a n t urn n u m b e r s of t he s t a t e s , a s ort h o g o n a l a x e s . An e x a m p l e is given in Fig. A. 1 lor t h e c o m b i n a t i o n of a n SU(.'i) triplet a n d a n t i l l i p l e t . t h a t is. q u a r k a n d ant ¡ q u a r k . T h e p o s s i b l e q u a r k s t a t e s a r e r e p r e s e n t e d by t h e c o r n e r s of t h e d o w n w a r d p o i n t i n g t r i a n g l e , t h e a n t ¡ q u a r k s t a t e s a r e o b t a i n e d by its m i r r o r i m a g e u n d e r collection a t t h e h o r i z o n t a l axis. S h i f t i n g t h e c e n t r e of t h e q u a r k t r i a n g l e t o t h e c o r n e r s of t h e a i i t i q u a r k t r i a n g l e t h e n yields a figure w h o s e c o r n e r s d e f i n e all possible quark aiitiquark states. A d e e p e r result w h i c h will not b e d e r i v e d h e r e , but w h o s o e s s e n c e s h o u l d be f a m i l i a r f r o m t h e C l e h s e h G o r d a u series of a n g u l a r m o m e n t u m , is t h a t t h e t e n s o r p r o d u c t s c a n bo r e d u c e d i n t o d o s e d s u b s e t s of e l e m e n t s w h i c h t r a n s f o r m i n t o o n e anot her u n d e r I he act ion of t ho g r o u p , but which d o not involve' t h e rem a i n i n g e l e m e n t s . T h i s d e c o m p o s i t ion i n t o so-called irreducible rciircseiilid.iov.s is d e p i c t e d on t h e r i g h t - h a n d side of Fig. A . I . T h e d a r k d o t s s h o w t h e q u a n t u m n u m b e r s of t h e o c t e t , t h e o p e n dot in t h e c e n t r e is a singlet s t a t e which is invariant u n d e r S U ( 3 ) t r a n s f o r m a t i o n s . N o t e t h a t a t t h e c e n t r e of t h e root diag r a m t h e r e a r e t h r o e s t a t e s built f r o m t h e s a m e q u a r k f l a v o u r s , only t w o of which
I III
M(N) \NI> S l l ( N ) O I I O H I ' S
I'I
Fit:. A . I. T h e root d i a g r a m for t h e a d d i t i o n of t h e triplet a n d a i i t i l r i p l e t of Sl'(:5) mult ¡plots a n d its d e c o m p o s i t i o n i n t o i r r e d u c i b l e r e p r e s e n t a t i o n s
• dsn h a v e t h e s a m e q u a n t u m n u m b e r s . I r r e d u c i b l e r e s p r e s e n t a t i o n s a r e a s s o c i a t e d with p h y s i c a l s y s t e m s . In t h e a b o v e e x a m p l e , t h e black d o t s a r e a s s o c i a t e d w i t h t h e meson octet.. As a f u r t h e r e x a m p l e , t h e b a r v o n d e e u p l e t c a n b e c o n s t r u c t e d in a s i m i l a r way a s t w o of t h e i r r e d u c i b l e r e p r e s e n t a t i o n s of t h e t e n s o r p r o d u c t of t h r e e q u a r k s .
A.2
T h e U ( N ) and S U ( N ) groups
Since u n i t a r y g r o u p s p r o v i d e t h e basis for t h e g a u g e t h e o r i e s of t h e S t a n d a r d Model we c o n s i d e r t h e m in a little m o r e d e t a i l . As a l r e a d y m e n t i o n e d , t h e d e f i n i n g r e p r e s e n t a t i o n for t h e u n i t a r y g r o u p U ( N ) c o n s i s t s of N x N c o m p l e x m a t r i c e s satisfying = 1
->
|
(A.22)
A general c o m p l e x N x N m a t r i x h a s 2 N 2 real p a r a m e t e r s . Since t h e i i n i t a r i t v c o n d i t i o n i m p o s e s N2 c o n s t r a i n t s , t h e n u m b e r of r e a l - v a l u e d p a r a m e t e r s for U is o n l y N2. B y a p p l y i n g a c o m m o n p h a s e f a c t o r o x p { i 0 o } we c a n a r r a n g e t h a t d o t ( / * ) = + 1 . I m p o s i n g t h i s e x t r a c o n d i t i o n gives t h e S p e c i a l U n i t a r y g r o u p . S U ( N ) . T h u s we c a n d e c o m p o s e t h e u n i t a r y g r o u p i n t o a direct p r o d u c t of I wo g r o u p s , U ( N ) = U ( 1 } < & S U ( N ) . In t e r m s of g e n e r a t o r s we m a y w r i t e U
=
exp(i
0„
+
I O „ T " )
,
(A.23)
w h e r e t h e i n d e x U r u n s over t h e N2 1 g e n e r a t o r s of S U ( N ) . e q u a l t o t h e n u m b e r of real p a r a m e t e r s for a m e m b e r of S U ( N ) . I he U( 1) a n d S U ( N ) c o m p o n e n t s c a n be t r e a t e d s e p a r a t e l y . S p e c i a l i z i n g t o a n inlinitesinial 0. t h e S U ( N ) g e n e r a t o r s a r e easily seen t o s a t i s f y : t/t/1 and
=
1 +\S0N(T"
-
7'"') + • • • = !
—
T"
=
r
r t
(A.21)
122
.1
( ) = s
(
{
)
s s
T
a'
s (A. ). s ( ). T
s
s
T
(
s s
T
r {T
1
ss
s
s
s. 1 s
s . (.
1)
s ( ). T
HI = 1
ss
ss . .
s,
s . ).
s
s
s III
s
s
,
s. T ss
—III
s.
s
s s
s
,
s s
1
s
( . 1. 2 ) . s ( ) s
s
s
s
s
1.0)
s
s
( )
( .
s
s
s
. ()). 1
N — 1.
.
s,
1s
s { 1. 1.
.
s s
s
.
s
.
s.
, .
( ) s
s
s
s
s. s, s
s s
.
s s
s s
s
s
( ). T ( ) (
s
s
s, s
s
s
.. 1
s.
, 8).
s
s
. T
s
T
s
s s
s s
.
( ). s
s \T .Th
A
.
s . T
= 0 . (A.2 )
s. s
s
s
{
s
s
(2)
1
.
+
. . l ' ' = 0.
=
T
(A.2 ) .
= ^
s s
f ' T
s s
{7 .T s
=
T Tb
= \
T.
= 1
T {Tn
= 0 ,
{
=
l
l,cr
r
s
, .
(A.27) s
.
s
+ ('/"'"" +
T
r
.
-
)
(A. 10) .
( . 1)
A
l\'{
T'-r)
T
(.
, 2.(
,
s
).
(A.32)
s r ' - r jabr(,, al
+
r' + j c l . c + jUUr(,
.
= =
(A s (
)
=
{.
F . Fb] = s
),
=
)
. Db] =
s
s
,,,,
.,
(
f'
)' .
(A.
)
s
4 ) + (
,
,
,
) ,
= ().
(A. 7) (A. 8) (A.
)
(A. 10)
=
(A. T). {
T {F FbF \
= i^r
T {
=
Tr {D D''F'
= '
T
=
,
)
(A.42)
=
= A ,. s
A.
))
= Tv{F>F'' = ,U.
=
F F
(A s
TT{D = . f 'f''r
)
s
r' -F'-
= T {
2
(A
.
(A.4 )
.
(A.44) .
(A.4 ) f ''
(A. 6) .
(A. 17)
s s s
s
s
.
s
7 C C,,. s
s
s C
Cr ,
s
.
,
s s
,
s
s
s.
•
Group
, I > I K M
• •>
W R
V M W I ' I
I
TF/C'I••
U I ' . U I N
C'A/C,.
SU(N)
N / ( N ' - \ )
2N-/(N-
- 1)
SO(N)
2/(/V - I)
( 2 N - 4 ) / ( A ' - 1)
Sp(2N)
2 / ( 2 N + 1)
(4;V
U( 1 ),v
N
<»
G2
1/2
2
F4
3/2
2
E(>
18/13
9/2«
E7
24/19
8/19
E8
1
1
4)/(2JV + 1)
I h<. g r o u p s S U ( N ) have been discussed ill detail before. S O ( N ) a r e t h e rotalion g r o u p s in -V dimensions, and Sp(2N) t h e so-called syinpleotic g r o u p s . Since t h e g e n e r a t o r s of t h e l a t t e r a r e not r e p r e s e n t e d by h e r m i t i a n matrices, they are of lesser i m p o r t a n c e in t h e context of q u a n t u m field theory. U ( l ) . v a r e t h e g r o u p s of p h a s e t r a n s f o r m a t i o n s . All these g r o u p s come with a n index JV that defines t h e dimensionality of t h e f u n d a m e n t a l r e p r e s e n t a t i o n . In addit ion to those, t h e r e arc only five so-called exceptional g r o u p s G 2 . F 4 . Eli. E7. E8. which a r e also listed in t h e a b o v e table.
APPENDIX
B
BUILDING BLOCKS OF THEORETICAL PREDICTIONS 13.1
T h e Feynman rules of Q C D
Fovnman d i a g r a m s provide a very useful pictorial device in which each comp o n e n t of a d i a g r a m r e p r e s e n t s a p a r t of t h e algebraic expressions for t h e corr e s p o n d i n g S - m a t r i x a m p l i t u d e . W h e n using t h e Feynmaii rules, to go from a d i a g r a m to a n algebraic expression, you a r e advised to pay careful a t t e n t i o n t o the d i r e c t i o n s of t h e m o m e n t a a n d t h e o r d e r of any indices. T h e s e d e t a i l s a r e i m p o r t a n t because they impact on t h e relative signs of t h e various t e r m s which in t u r n help t o e n s u r e g a u g e invariauce. E x t e r n a l q u a r k s a n d gluons c o r r e s p o n d to basis s p i n o r s a n d p o l a r i z a t i o n vectors as shown below. G h o s t s a r e s c a l a r s a n d t h e r e f o r e have trivial unit basis states. Basis s t a t e s p, — pf -> «(/'.)
¡•(P,)
Propagators p—>
— u ( p
—
•
f
)
i — > » —
V(PF)
„
N S M S S F R
j
i
,,(/M'.;").6„ p- — in- -f if
H
p- + 1 1 <(l>,)''
OML*
KQQSLi f * ( / ' / ) "
"
*
b
i
2
' .f6,m
lut e r n a l part icles corresptaid to p r o p a g a t o r s , which a r e colour diagonal, a s s h o w n . T h e sign of t h e infinitesimal, i m a g i n a r y part i r is chosen so as to e n s u r e causality. Ill t h e case of t h e gltion t h e Lorentz tensor ) d e p e n d s on the choice of t h e g a u g e fixing t e r m a n d t h e g a u g e p a r a m e t e r T w o c o m m o n choices are:
+ (I
—
0——
COVariant g a u g e .
) = 1
(
(» • ]>)'*
physical gauge.
(B.l) In t h e physical g a u g e s g h o s t s do not a p p e a r in t h e F e y n m a n d i a g r a m s , but in t h e covariant g a u g e s they a r e required in o r d e r to preserve iinitarit.y: see Section 3.3.3.
IIIJII DINC III.OCKS ()!• I Ill-OKI-. IK 'A I I 'III I >1' I IONS
P a r t i c l e i n t e r a c t i o n s a r e r e p r e s e n t e d by vertices. W e h a v e t h e gluoii i|iiark, g l u o n g h o s t , I riplc-gluon a n d qiinrt.ic-glnon v e r t i c e s c o r r e s p o n d i n g t o t h e following a l g e b r a i c f a c t o r s :
li.d
IHsT"!
!ls f,il„ I'll
ij >!• \
0-2.fi. I>
,/•,. A. I,
.7.1 • ' ) • < •
f/i .(\.
r
f/i.n.«
+ i f * / « t e [ + i),,.i()-, + VihiOi
~ Oa)a
+ >l-.,Aoa - .'/1>.-*]
' • ' / / ; [ + fni» /«-«*. ('/«-.'/.*>
¿- <1
iioA'i.ti)
+ farefflti* ('/,„>'Mi ~ '/.. T fmlI flm-Ofaii'hi
I'M)
- '/
In all t h e s e g r a p h s t h e c o n v e n t i o n is t h a t all m o m e n t a a r e o u t g o i n g a n d so s u m t o zero. In t h e gluoii g h o s t v e r t e x t h e m o m e n t u m is t h a t of t h e o u t g o i n g g h o s t . b i t lie gluoii s e l f - c o u p l i n g s , o b s e r v e t hat hot h v e r t i c e s a r e s y m m e t r i c u p o n i n t e r c h a n g e of all t h e labels o n a n y p a i r of legs. Implicit in e a c h of t h e s e v e r t i c e s is a f o u r - m o m e n t u m c o n s e r v i n g ¿ - f u n c t i o n . E a c h i n t e r n a l line is a c c o m p a n i e d by a n i n t e g r a l over its f o u r - m o m e n t u m . T h i s r e s u l t s in a n overall f o u r - m o m e n t u m c o n s e r v i n g ¿ - f u n c t i o n w h i c h is a b s o r b e d i n t o t h e p h a s e s p a c e definit ion. Since q u a r k s a n d g h o s t s a r e ferniionic. for e v e r y closed l o o p involving t h e m in a d i a g r a m , a n a d d i t i o n a l f a c t o r 1 must b e i n c l u d e d . F u r t h e r m o r e , w h e n a p a i r of identical f e r m i o n s is present in t h e e x t e r n a l s t a t e of t w o d i a g r a m s , t h e n t he ' c r o s s e d ' d i a g r a m a c q u i r e s a m i n u s sign r e l a t i v e t o t h e ' u n c r o s s e d ' d i a g r a m , a g a i n t o a c c o u n t for t h e a n t i c o m m u t a t i v i t y of f e r m i o n s . Finally, when n identical p a r t i c l e s a r e present in t h e liual s t a t e of a set of d i a g r a m s , t h e n a s y m m e t r y f a c t o r 1 / n \ must be i n c l u d e d in t h e a m p l i t u d e . N o t e that, we h a v e s u p p r e s s e d t h e s p i u o r indices on t h e q u a r k p r o p a g a t o r s a n d vertices. T h e correct o r d e r i n g of t h e s e t e r m s is given by w o r k i n g b a c k w a r d s a l o n g t h e i n d i v i d u a l f e r m i o n lines. T h i s p r e s c r i p t i o n will a l s o give t h e c o r r e c t
I'll AS
M A< I A
« Id )SS S i t I I
M
A
127
o r d e r i n g of a n y eolotn mat rices. For c o m p l e t e n e s s we a l s o i n c l u d e s o m e r e l e v a n t F e y n i n a n r u l e s f r o m t h e s t a n d a r d elect rowcak t h e o r y . Here t he p r o p a g a t o r s for t he v e c t o r b o s o n s . V = 7. W HI ,. a r e given by
,
l>- - My
71
+ Ie
(-'
'
1
0 >-., -
My
-+ . I f )J
(B.2)
whilst t h e i r c o u p l i n g s t o f e r m i o u s t a k e t h e f o r m - i e«7 i(v
+
r,)
(B.:i)
The i n d i v i d u a l c o e f f i c i e n t s a p p e a r i n g in t h i s e x p r e s s i o n a r e collected in t h e following t a b l e : Boson
Vf
J 0
7 1 ( 2 s i n 0W cos 0,,.)
. W
l{ - 2 c sin 0W
Vjr/(2y/2slu0 )
1
-1
H e r e for t h e l e r m i o u . 1 is its e l e c t r i c c h a r g e m e a s u r e d in u n i t s of t h e p o s i t r o n c h a r g e c > (): is its t hird c o m p o n e n t , of w e a k isospin. = T ^ for u p t y p e q u a r k s o r n e u t r i n o s , a n d .( = for d o w n - t y p e q u a r k s o r c h a r g e d l e p t o n s . For t h e c o r r e s p o n d i n g a n t i p a r t icles t he s i g n s a r e r e v e r s e d . For c h a r g e d c u r r e n t i n t e r a c t i o n s involving q u a r k s , t h e c o e f f i c i e n t s V f a r e t h e r e s p e c t i v e e l e m e n t s of the C a b b i b o o b a y a s h i M a s k a w a m a t r i x , t h e d o m i n a n t e l e m e n t s of which a r e r —Vct a n d V', , s ; I. For l e p t o n s o n e effectively I „,1 s ; (I.!t7. i x: l7,. . F u s s : 0 . 2 2 2 0.22.5. h a s V„,c — i ) f f . T h e p a r a m e t e r 0 W is t h e w e a k m i x i n g a n g l e w i t h sin 0 W
B.2
h a s e space and cross section
formulae
n c e t h e a m p l i t u d e s q u a r e d for a p r o c e s s h a s b e e n e v a l u a t e d , it is n e c e s s a r y l o i n c l u d e t h e Mux f a c t o r a n d t h e ( d i f f e r e n t i a l ) p h a s e s p a c e in o r d e r t o o b t a i n t h e ( d i f f e r e n t i a l ) c r o s s s e c t i o n . W e c o n s i d e r t h e g e n e r a l p r o c e s s p„ -{-pi, — p for which t h e c r o s s s e c t i o n is given s c h e m a t i c a l l y by (lfT =
finx
X l V,
'
'
X (l<1>
'
(
-0
I lere it s h o u l d b e u n d e r s t o o d t h a t t he c r o s s sect ion a n d p h a s e s p a c e a r e t y p i c a l l y m u l t i - d i f f e r e n t i a l q u a n t i t i e s . For h e a d - o n collisions t h e Mux f a c t o r is given Influx = I /(p, 1 I ,)'2 ~
{tiiuiiii,)-
= 4 p;, vA = ilplr'lm,,
s = (p„ + P )2
(B.5)
2s . In t h e s e c o n d line t h e flux is given in t e r m s of t h e C . o . M . m o m e n t a , p >, . 1, 1 and the laboratory variables p and p ' ( ,. 0 ) . T h e t h i r d line is a p p r o p r i a t e
m m . D I N O BLOCKS OI
I IIKOKI- I U AI I - H I D U I IONS
in Mie l'unit (il negligiblo p a l l i d o masses. In t h e case of a p a l l i d o deeay t h e llux faetor is given hy twieo t h e docaying parlicle's mass, llux = 2 M . A différential element of t h e L o r c n t z invariant u - h o d y p h a s e s p a c e for ilio o u l g o i n g parl icles is given hy d„(/>„ + l'I, • Pl,
l'n) f
= (2TT) ' ¿ ' " L + , > „
¿/', ) n i / 1=1
d V -
(2?r) 3 À ,
(2ir) 3 2£ l ,
In t h e second version t h e on mass-shell ¿ - f u n c t i o n has boon explicitly integrated out a n d t h e positive e n e r g y solution E, = + \ / p j + nr selected. Using oqn (B.ti) a n d c(|u (B.-l) it is easy to verify t h a t t h e d i m e n s i o n a l i t y of t h e p h a s e space is given by 3 » . whilst t h e m a s s dimension of \M\~ must be -l - 2n. In m a n y practical s i t u a t i o n s t h e i n c o m i n g p a r t i c l e s a r e u n p o l a r i z e d a n d the spins of t h e final s t a t e particles are not m e a s u r e d . T h e s a m e applies for their colours. T o t a k e this into account o n e h a s to s u m t h e a m p l i t u d e s q u a r e d over t he spins a n d colours of t h e o u t g o i n g particles a n d a v e r a g e over t h e s p i n s a n d colours of t h e incoming particles. T h u s , in oqn (B.-l) we use \M\2
I>>
2
-
n
à;t
K <>.!>
x
s ^ spin.colour
•
(B.7)
where t h e colour d e g e n e r a c y is Nu X, for a (|itark or an a n t i ( | u a r k a n d N2 - I for a gluon a n d w h e r e we allow two spin p o l a r i z a t i o n s for t h e e x t e r n a l fonnions a n d inassless e x t e r n a l g l u o n s or p h o t o n s .
APPENDIX
C
DI lM E XSIO N A L R EG U L A RIZ AT 10 N ('.I
I n t e g r a t i o n in 11011-integer d i m e n s i o n s
Dimensional regularization is the preferred m e t h o d in QC'D for r e n d e r i n g ultraviolet divergent loop integrals finite. Tito basic idea is t o work in D I 2f space t i m e dimensions. T h e n , given s u i t a b l e definitions, we e v a l u a t e t h e loop m o m e n t u m integrals with a n y divergences a p p e a r i n g as poles in l / e . T h i s renders t h e t h e o r y finite, for D < l. so that we can c a r r y o u t t h e r o n o n n a l i z a t i o n procedure a n d a f t e r w a r d s lake the limit f —> 0. In D d i m e n s i o n s t h e si m e t tire of t he Q C D Lagrangian is unaltered: it c o n t a i n s tin- s a m e kinetic a n d interaction t e r m s a n d . therefore, h a s t h e s a m e F e y m n a n where // is an rules. T h e r e is only o n e c h a n g e , t h e r e p l a c e m e n t yH -> a r b i t r a r y unit m a s s ( ' t f l o o f t . I!l?:i). T h i s is needed to e n s u r e that each t e r m in the L a g r a n g i a n d e n s i t y h a s t h e correct, m a s s dimension: see Ex. (3-17). Before e x p l a i n i n g t h e m e t h o d , it is useful to i n t r o d u c e a few s t a n d a r d manipulations which makes t h e final integrals easier to carry o u t . We illustrate this a p p r o a c h using t h e following typical integral which arises in t h e calculation of I lie fcrmion self-cnergv. ,
* A-o
fI
dfc
7"(M-/' + m)7„ l(h- + p)2 + if] [A:-' I- if]
(C.I)
l or t h e m o m e n t , we have not set D -I but. left it. free. We have also i n t r o d u c e d an a r b i t r a r y m a s s // which serves to preserve t h e canonical dimension of t h e integral for D ^ I. T h i s integral h a s a superficial degree of divergence D 3. o b t a i n e d by c o u n t i n g t h e n u m b e r of powers of t h e loop m o m e n t u m in t h e integrand, suggesting a p o t e n t i a l linear divergence in D = 4 dimensions. At t h e expense of i n t r o d u c i n g e x t r a integrals, oqn ( C . l ) is simplified by c o m b i n i n g t h e t wo t e r m s in t h e d e n o m i n a t o r using t h e i d e n t i t y 1 /I'/M."2
• • • A\
[
r(»,)i (»a)---r(nfc) ./„ k) .A,
(oiAi -i
(C.2)
Here t h e e x p o n e n t s j / i , } need not be integer. T h e { o , } a r e known as F e y m n a n p a r a m e t e r s . A p p l y i n g this result, to oqn ( C . l ) . a n d at t h e s a m e t i m e i n t e g r a t i n g out t he ¿ - f u n c t i o n , gives
DIMKNSIONAI
Ifl'CUI.AKI/.AriON
(«¡(A- + j>)~ - ///"-' + U\ + (I - a ) [A-2 + it])* =
q
y - n
f ^ L ./ ( W
/'do ^ + 2 3 + o.(l - a)p2 ./<» ((A- + o p ) - am
- . + if)'
(CM)
In t h e s e c o n d line w e h a v e ' c o m p l e t e d t h e s q u a r e ' , w h i c h a f t e r s h i f t i n g t h e mom e n t u m v a r i a b l e . A-'' — » A'' — o p ' 1 , y i e l d s
./ (27T)
(A-2 - A + i f ) 2
y (1
(A,
= rf,I d« y |d - ^ + ,„]7, J ^
,„t ,,k), • (c-D
H e r e , we i n t r o d u c e d /l = o n e ' — o ( l — o ) p 2 . T h i s c h a n g e of v a r i a b l e a n d t h e r e - o r d e r i n g of t h e i n t e g r a l s is l e g i t i m a t e b e c a u s e we will c h o o s e D t o m a k e t h e integral c o n v e r g e n t . T h e A-'1 t e r m v a n i s h e d b e c a u s e t h e i n t e g r a n d is i s o t r o p i c a n d n o longer h a s a p r e f e r r e d d i r e c t i o n . T h e p1' d e p e n d e n c e is now via p~ in A. T h i s m e a n s t h a t t h e a p p a r e n t linear d i v e r g e n c e of e q n (C'.l) is in r e a l i t y only a logarithmic divergence. At t h i s point y o u a r e r e m i n d e d t hat in M i n k o w s k i s p a c e A--' = E~ — k". s o that t h e t e m p o r a l a n d s p a t i a l c o m p o n e n t s a r e not o n a n e q u a l f o o t i n g . T o r e m e d y t h i s sit uat ion we t r a n s f o r m t o E u c l i d e a n s p a c e , E •-> iA'o, so t h a t k~ — - k f - = \ k~. For t h e c a s e at h a n d t h i s gives
r
i - 0 0 2jr _.
<1/J~ ' A-
J
1 (E2
r + 0 Q dAp
- k
2
- A + U)2
r d"-'A:
1 2
./ ( 2 * ) » - ' (-A- - Ar - /I + i f ) 2
A s u b t l e t y in t h i s m a n i p u l a t i o n is t h e role played by t h e i n l i n i t e s i m a ] if in t h e d e n o m i n a t o r . Essentially, we h a v e used a closed c o n t o u r iu t h e c o m p l e x E - p l a n e t h a t goes a l o n g t h e real a x i s , d o w n t h e c o m p l e x a x i s a n d closes iu t h e first a n d t h i r d q u a d r a n t s . Now t h e i n t e g r a n d h a s p o l e s at A-n = ± ( \ f k 2 — /I — i f ) which, t h a n k s t o t h e if t e r m (f > 0 ) , lie j u s t o u t s i d e t h e c o n t o u r a n d t h e r e b y e n s u r e t h e e q u a l i t y of t h e t w o i n t e g r a l s in e q n (C.f>). Following t h e s e m a n i p u l a t i o n s t h e e x a m p l e i n t e g r a l e q n (C'.l) b e c o m e s
/ =
rfS-»/'da
v
[(1 - a ) >
+
H T > / ¡ B ^ ( A ^ r W
'
(C
'6)
I
II
ltATI
I
-I
I l i II li
IM
SI
S
131
Vc n o w e x p l a i n i h e m e t h o d of d i m e n s i o n a l r e g u l a r i z a t o n a s a p p l i e d t o e( M (C'.(i). F i r s t , we i n t r o d u c e p o l a r c o o r d i n a t e s whilst, still k e e p i n g free, which yields /
m
-
7
"
id - < 0 /
+
,„]* / g j »
.
C. Since by design t he rc(|nired i n t e g r a l e<jn (C'.(i) is isotropic, t h e a n g u l a r i n t e g r a l s c a n b e t r e a t e d s e p a r a t e l y . T h e / ^ - d i m e n s i o n a l e x p r e s s i o n for t he a n g u l a r i n t e g r a l s is given in t e r m s of t h e E l d e r r - f u n c t . i o n . e<|ii (C.2-1),
/
*
-
C
= m m -
T h i s result c o i n c i d e s wit h t he st a n d a r d e x p r e s s i o n s . 2TT, -ITT. 2 ~ " \ . . . for p o s i t i v e i n t e g e r s II 2.3,-1 However, t h a n k s t o t h e use of t h e F - f u i i c t i o n . t h e result is a n a l y t i c in D so t h a t we c a n use a n a l y t i c c o n t i n u a t i o n t o d e f i n e t h e r e s u l t for n o n - i n t e g e r a n d even c o m p l e x v a l u e s of D. T h e d e r i v a t i o n of t h i s result c a n b e f o u n d in E x . (3-18). Finally, t h e r e is t h e /.' i n t e g r a l which we t r e a t a s a r e g u l a r integral. It is of t h e s t a n d a r d Filler .i-lunctioii f o r m : f,u.
-
n a
2 ) l > - Dm
C o m b i n i n g e i | n s ( C . 8 ) a n d (C.f)). w i t h II [ 2
< l
(C
,n
2. allows e q n (C.(>) t o be w r i t t e n a s
° V Id-«)/'-I'»] 7,/l(o)"/2-2
r1
/ w \ x 0/2-2
= i ^ F ( 2 - D / 2 ) l dn V |(1
R.L.
.n 2-n
.(C.10)
+
For f u t u r e r e f e r e n c e t h e basic D - d i m e n s i o n a l integral is given by (Bollini I I I . 1973) d"/,/
(A-2)»
'(2jt)" (/.'-'- /!)'"
=
r ( n + Df2)F(m
M M
(<1TT)'j/2
- n -
D 2)
F(D/2)F(m)
(C.ll) T h e p r o c e d u r e l e a d i n g t o t h i s r e s u l t w a s i l l u s t r a t e d for t h e c a s e of a s c a l a r i n t e g r a n d . If t h e i n t e g r a n d d e p e n d s o n o n e of t h e c o m p o n e n t s of A-/.;, s a y A'j. t h e n we w r i t e t h e i n t e g r a l a s (/"/,/(/,-i./.-2)
= jd/;~
'/.-(IA-1/(A'I.A 2 ) = jt\,i- u\ssa-ii\ if{ x. ^
4- *-') .
(CM 2) In t h i s way. t h e i n t e g r a l o v e r \ is t r e a t e d a s a n o r m a l i n t e g r a l a n d t h e Dd i m e n s i o n a l t r e a t m e n t is reserved for t h e r e m a i n i n g ' i s o t r o p i c ' c o m p o n e n t s of .
HIMENSIt )NAI, H E O U L A I U Z A T I O N
B e f o r e invest ¡ g a t i n g t h e D —> I limit of e q n ( ( M O ) we m u s t hist. d e a l with t h e 7 - i n a t r i c o s . T h i s is d i s c u s s e d ¡11 Sect ion C . 2 . U s i n g t h e D - d u u c n s i o n a l a l g e b r a ot -mat,rices it is easy t o s h o w that, 7 " I d " « )/' + "'1 7,i = ~(D
- 2)(1 - a)/> -I Dm .
(C.13)
S u b s t i t u t i n g t h i s result in e q n ( C . 1 0 ) a n d a t t h e s a m e t i m e w r i t i n g D
I
2»
)
.
gives
7 =
1
(M^
= i
r ( f )
f
< l n
H 2
"
2 0 ( 1
ji'a« |_2(1 _
"
o ) /
n ) /
'
+
"
2 f ) m l
'
+ 4 m + ,(2(1 - a)f, - 2m)]
(
«
(C.14) I11 t h e s e c o n d line we used e(|ii ( C . 2 5 ) t o m a k e explicit t h e pole a s s o c i a t e d w i t h t h e D —> 4. ( -< 0 limit. U s i n g e q n ( C . 2 6 ) t o g e t h e r w i t h ./•' = e' '"• r . we c a n now i n v e s t i g a t e t h e ( —> 0 limit of e q n ( C . 1 4 ) . w h i c h b e c o m e s / = ig
ji
rfrt|(-2(l
- a ) j + 4 m ] [ J - 71c + M * * ) ~ I"
+ 2(1 - o ) / - 2 m |
= 1^
{ ( - / + 4m)
( ^ r )
+0(f)
- 7I-: + l n ( 4 i r ) ) + f> [ l + 2 jf rfn(l - a) h.
( ^ r )
In t h i s e x p r e s s i o n it m a y b e n o t e d t h a t t h e inclusion of t h e m a s s // t a k e s c a r e ol t h e d i m e n s i o n s in t h e l o g a r i t h m . W h a t we find is t h a t t h e ultraviolet d i v e r g e n c e in eqn ( C . I ) is now isolated a s a s i m p l e \/< pole. E x p e r i e n c e will c o n f i r m t h a t 1 / r a l w a y s o c c u r s in t h e c o m b i n a t i o n A , = - + ln(47r) - 7K .
(€.16)
(
T h e r e r e m a i n s a finite p a r t given in t e r m s of t e d i o u s but c a l c u l a b l e o - i n t e g r a l s . It m u s t be a d m i t t e d t h a t o u r a p p r o a c h t o d i m e n s i o n a l r e g n l a r i z a t i o n h a s been a little cavalier. T h a t o u r r e s u l t s hold is t h a n k s t o t h e work of o t h e r s ("t.Hoofl a n d V e l t m a n . 1!)72). I11 essence, what we h a v e d o n e is t o first i d e n t i f y t h o s e d i m e n s i o n s . D < I in t he e x a m p l e a b o v e , for which t h e d e s i r e d i n t e g r a l is finite; t h i s m e a n s free of b o t h u l t r a v i o l e t , k — oc. a n d . if niassless p a r t i c l e s a p p e a r ¡11 t h e loop, i n f r a r e d , k. k • // — 0. d i v e r g e n c e s . T h e i n t e g r a l is c o m p u t e d a n d t h e n e x p r e s s e d a s a n a n a l y t i c f u n c t i o n of D which c a n b e used t o c o n t i n u e t h e integral i n t o t h e vicinity of D = 4.
/> DIMENSIONAL
('.2
/J-diincnsioiml
,-matrix
• MAI ItIX AI.OEHHA
algebra
In a c c o r d w i t h t h e d i s c u s s i o n of S e c t i o n 3.3. I. also in t h e g e n e r a l c a s e of D n s i o n s we r e q u i r e 7,', = + 7 » a n d 7 = - 7 , for i = 1 . 2 , 3 : tlie s a m e dill'ord algebra { 7 ' ' . 7 " ) 2 ; / ' " ' l ; a n d t h e l i n e a r i t y a n d cyclicity of t r a c e s . As these rules e s s e n t i a l l y c o i n c i d e w i t h t h o s e a s s u m e d e a r l i e r , we c a n use t h e s a m e m a n i p u l a t i o n s on t h e t r a c e s of 7 - m a t n c c s in D a s in I d i m e n s i o n s . O n e w o r d of w a r n i n g is t o r e m e m b e r t h a t if1' = D. w h i c h l e a d s t o e x t r a t e r m s p r o p o r t i o n a l in D I a p p e a r i n g in s o m e results; c o m p a r e e q n (3.83) a n d Ex. (3-19). Finally, r e m e m b e r i n g t h a t each t r a c e is p r o p o r t i o n a l t o t h e t r a c e of t h e D - d i m e n s i o n a l unit m a t r i x , we d e f i n e T r { l } f ( D ) w h e r e / is a n y well b e h a v e d f u n c t i o n of D subject to the boundary condition / ( 4 ) 4. T h e s i m p l e s t c h o i c e is f ( D ) 4. Since ill p r a c t i c a l a p p l i c a t i o n s we will a l w a y s t a k e t h e limit r —> 0. a n y d i f f e r e n c e f(D) - I O ( f ) c a n o n l y c o n t r i b u t e t o divergent, g r a p h s , oc \/e". a n d . a s we shall learn, t h e a d d i t i o n a l t e r m s a r e e q u i v a l e n t t o a c h a n g e iu t h e finite p a r t of I he count e r t e r i n s a n d so i i n o b s e r v a b l e bv r o i i o r i n a l i z a t i o n g r o u p i n v a r i a n c e . Whilst it d o e s not a r i s e in p u r e u n p o l a r i z e d Q C D c a l c u l a t i o n s , for t h e s a k e ol r a d i a t i v e c o r r e c t i o n s t o c h i r a i w e a k p r o c e s s e s , we m e n t i o n t h e t r e a t m e n t of 7 v T h e u s u a l p r o p e r t i e s of 75 a r e : 75 = fr>. (7.1)" = 1 a n d {7r,,7 ; ,} = 0. Unf o r t u n a t e l y . if we r e q u i r e r e s u l t s t h a t a r e a n a l y t i c f u n c t i o n s of D t h e n t h e a n t i c o i i m i u t a t i v i t y p r o p e r t y o b l i g e s TV{757,,, •••-),,„} = 0 for a n y /». However, in i) 1 d i m e n s i o n s we c a n realize 75 a s 75 i 7 o 7 i 7 2 7 3 a n d o b t a i n t h e result IV {*:T.7/I7«/7«T7T i i <•/««"•• ' f h i s conflict h i g h l i g h t s t h e fact t h a t 75 is i n t r i n s i c t o four d i m e n s i o n s . O n e r e s o l u t i o n is t o use t h e D I d e f i n i t i o n of 7-, a n d m o d i f y t he ant ¡ c o m m u t a t o r s ( t lloolt a n d V e l t m a n . 1972) t o o b e y
7* = ¡ 7 0 7 , 7 * 7 *
= >
|
7 5 7
"
=
T
7
"
7
'
( 757/I = + 7 / ( 7 5
' ' R "
1
-
2
-
otherwise.
1
( C . 17)
The price of t h i s s o l u t i o n is t h e loss of L o r e n t z i n v a r i a n c e . T h u s , w h e n 7r, is p r e s e n t we must treat, s e p a r a t e l y t he sots of c o m p o n e n t s // < I a n d // > 1. C.3
/7-di i n c i s i o n a l p h a s e s p a c e
T h e g e n e r a l i z a t i o n of t h e / / - b o d y p h a s e s p a c e t o D d i m e n s i o n s is s t r a i g h t f o r w a r d .
'=l
( (2Sr)(»-')2E, *
A g a i n t h e rf( + ) (./:) , = 2 ^
, +
)(p'f -
m|)| I/'1 /'..+/».
(C.19)
I:t I
IUMKNSIONAI. KKOUI.AHIZATION
We will illustrate ilio uso <>!' «-<111 (C.18) for Ilio caso n o c c u r in ¡1 t.wo-to-t.wo s c a t t e r i n g , wliicli givcs I'J-1.,
,II>--\.
R
I (^k(-">'
- / - • < - /
2. suoli a s migli!
v m < < J
~ -
»>
llcro. wo have E, = \/pf + w'f a n d , a f t e r i n t e g r a t i n g out t h e s p a t i a l m o m e n t u m c o m p o n e n t s of t h e second filial s t a t e particle, p-> Q Pi - We shall now special0 ) . which is generally t h e most convenient, ize t o t h e C.o.M. f r a m e . Q 1 ' a n d a s s u m e t h a t t h e r e is a n explicit, d e p e n d e n c e on t h e longitudinal c o m p o n e n t , pi. p e o s f * . of t h e o u t g o i n g m o m e n t u m in t h e i n t e g r a n d . T h e p h a s e - s p a c e integral b e c o m e s
/
^
-
S
W
=
«
4
/
^
J
*
-
»
1
J
Hi-«)
~ V'
~ "< )
^ 7 1
r + l> |
4 ( 2 j t ) 0 - 3 r ( i D - m2 ) / 2 )L (\TTY
«
' {"r
HhEl 9_(/J 2 )/2
1 p
-
/
7f
( /
'
<1 cos 0 sin-2< i i
- ¿ ^ ( p ) ' n i b ) j f ^ ' - r -
111 t h e final line we have changed variables t o t> = (1 + cos0}./2. which proves useful for some a p p l i c a t i o n s . If t h e i n t e g r a n d is a scalar, t h a t is. does not d e p e n d explicitly on /J/., then we can reduce eqn (C.21) t o
/
( i,i,.,
-
= - L A (JLV 4* Vi K P J
Z i i n i l r ( 2 - 2e)
(c o2) - J
1
We will also need eqn (C.18) for t h e case n — li. which can be w r i t t e n as 1
* ! ' 2 [ ( l - a * ) ( l -afc)(l (C.23) At this point we also m e n t i o n t h e n u m b e r of s p i n - p o l a r i z a t i o n s t a t e s which should be nsixl when a v e r a g i n g t h e m a t r i x clement s q u a r e d in 1) dimensions. Given I he s t a n d a r d choice Tr ( 1 } = 1 . t h e ( a n t i ) q u a r k s as usual should be taken t o have two s p i n - p o l a r i z a t i o n s t a t e s . O n t h e o t h e r h a n d , it is conventional to give inassless gliions D 2 2(1 f ) spin-polarization states.
l ' I I HI MAI 11 KM A I If 'A I !•'( MlMULAI.
C. l
Useful mall
formulae
Here we collect some useful m a t h e m a t i c a l results. F u r t h e r discussion c a n be lound in t h e s t a n d a r d m a t h e m a t i c a l physics texts, such as t h e book by Arfkou and Weber (l!)!).r>). We m a k e f r e q u e n t use of t h e Eider F-funct ion, which c a n lie defined by t h e convergent integral r(z)=
/ ./ii
,1/t*-1 o '
~R.('.{z} > 0 .
(C.2'l)
Integration by p a r t s will confirm the important, identity r ( i + s) = z r ( z ) .
(c.25)
T h u s , for positive, integer values of z. F ( c ) = (c I)!, which explains t h e alternative n a m e 'factorial f u n c t i o n ' . E q u a t i o n (C.25) can also be used to shift t h e a r g u m e n t a n d define t h e F-funct,ion when 7vr{c} < I). T h i s also shows t h a t t h e r e a r e s i m p l e polos at ; 0. - 1. — 2 . . . . We also need t h e e x p a n s i o n
1(1 + f) = 1
~7K-+ ( S
+
' 2 + °{(i)
h*)
•
(c 2,i)
-
where 7|.; 0 . 5 7 7 2 1 .r)(ìfì l filli • • • is t h e Filler Mascheroni Constant.. We will also often use ilio relaled Euler J-liinoiion integrai.
L
n
. > - • •
M
APPENDIX
I)
/>',. /?, A N D l i T F O R A R B I T R A R Y C O L O U R
FACTORS
T h i s c h a p t e r c o n t a i n s a c o m p i l a t i o n of t h e i n g r e d i e n t s t h a t go into t h e theoretical prediction for Iifj a n d R y • A " expressions a r e given for a r b i t r a r y colour factors, which allows to e v a l u a t e not only t h e QC'D-SU(3) predictions, b u t also t he predictions for a l t e r n a t ive theories wit h an u n b r o k e n g a u g e s y m m e t r y based on a simple Lie g r o u p . T h i s is needed, for e x a m p l e , by a n y a n a l y s i s which a i m s at a m e a s u r e m e n t of t he colour factors from R, a n d R r • Keeping t h e colour factors. it is convenient, to redefine t h e coupling c o n s t a n t such t h a t t h e a m p l i t u d e for gluon emission f r o m a q u a r k is independent, of t he g a u g e g r o u p of t h e t heory. A b s o r b i n g a f a c t o r 2n as well yields t he redefined coupling
<4 T h e predictions of t h e t h e o r y for /»/ q u a r k degrees of f r e e d o m t h e n can be expressed as f u n c t i o n of t h e free p a r a m e t e r
= ~
(./.-
,
fr = ~ w--
and
f„ =
C/.
.
(D.2)
All ex]>ressions a p p l y for t h e MS renoriiialization s c h e m e a n d cover at least the d o m i n a n t c o n t r i b u t i o n s . In s o m e cases, t h e higher o r d e r expressions a r e known but a r e not q u o t e d here, since t h e main o b j e c t i v e of this section is t o provide simple expressions that allow a fast evaluation of the respective effects.
D.l
T h e running coupling constant and masses
The variation of the strong coupling constant
-n = d In ¡i
-
/ > i < - & 2 « ( D . ; i )
- .VI "s • • • •
i ri iv (L).4)
T h e p a r a m e t e r s b, a n d
\j„
(D.5)
I IIK UUNNINO C o U I ' l . l N O C O N S T A N T AND MASSES
/'i
TR/'I
IT/.L / I I
(D.r.)
/..
() .1 r 28. ,7 :t 1415 , 3 ^ - 2 1 6 ^ +
7! I 1
0
8
205 ^
11...,
+
+
(D.7)
anil (D.8)
f/n = 3
3
97
5 .
u\ = 7 + " p j / . i ~
(D.9)
•
Equation (D.3) determines how the strong coupling constant evolves lor a lixed number of act ive flavours, whereas in practical applications one often has to relate a value of o„ from a scale m with iif = I active quark flavours to the measurement at a scale /¿r, with " / = 5 flavours. The treatment of flavour thresholds is described in Bernreuther and Wetzel (1982). Bernreuther (1983). Marciano (1981). and Rodrigo and Santamaria (1993), e.f. Section 3.-1.5. With ".. " s ("/•/')• the coupling constant a s ( ± ) = " s ( ' i / ± I. /') for a different, number of flavours, but at the same energy scale //. can he expressed as a power series in the original coupling. To O(a^) the expansion is given by n K ( ± ) = « s =F " i ^ f r L
«2
\fT'1)
T
(f//l/''
+ 2 / r
)
T
{ P
A h
~
V2h)
(D.10)
where L = ln(777(77i)//j) is the logarit hm of the ratio between the fixed point of the MS running mass of the extra quark flavour 777(777) and the matching scale //. Note that the matching condition eqn (D.IO) implies that two measurements at the same energy scale with different, numbers of active flavours, in general, will see a different, coupling strength. Only for a point // close to 7n(7fi) is the coupling continuous, as one would naively expect.. The numerical value of the point of continuity depends on the order of the pert urhativc expansion. Up to NLO. it. coincides with 777(777). In the context, of arbitrary colour factors it, would be preferable to express eqn (D.10) as a function of the pole masses M of the quarks rather than the MS running masses 777. since the latter already absorb part, of the radiative corrections of t he specific theory. To leading order the pole mass M is related to the running mass according to m { M ) =
(D
'
U )
From the leading order term, eqn (I).-l). one obtains r n \ -"sfln
m(p)=m(M)(i-)
'
,
(D.12)
Il .11, ANI» HT l'Oit AKMN KAKY COI.OUK I'AlTOIt.S
a n d t h e lixed-poiiit condition ///(/<)
// immediately yields (D.13)
77i(m) = A / ( l + 2 a s ) 1 / ( l + « s i / » ) . 'Io lending order in the s t r o n g coupling one thus has
L = In
777(777)
.| „ — M 4 2rt + O(n'i) . s
(D.ll)
which is sufficient to rewrite the third order matching condition, «-(in (D.10). as a function of I. = hi M / i t . «s(±)
=asJ-t,;-frL
+
. (D ir,)
W i t h these ingredients, t h e evolution of t h e s t r o n g coupling constant from t h e (9(1 G e V ) scale upwards can he realized by using it/ = 3 up to /1 = 2 M c . then ttj = 4 u p to it = 2M\, and it/ = 5 until the t o p mass threshold // = 2Mt. At each llavotir threshold t h e m a t c h i n g condition eqn (D.15) has to he applied. T h e theoretical error of the p r o c e d u r e may he e s t i m a t e d by varying the m a t c h i n g scale between M a n d 2 M . D.2
T h e o r e t i c a l p r e d i c t i o n s for R -
T h e Q C D corrections b o t h for R 7 a n d R\ a r e related t o t h e Q ( ' l ) correction <5° of /?-,. =
,(o+e-
-vhadrons)
^
+
(D
,(!)
which is known to order «jj (Gorishny t l til.. l!)!l 1: Surgnladze and Samuel. l!)!ll).
,
W
I
l nj.
(D.l 7)
,, r
V
' ')
For the s t r o n g coupling constant taken at t h e C.o.M. energy of t h e hadronic s y s t e m tin* coellicients are /v'i = r2 .
(D.18)
A' a = - 1 + / , , (»9
- 1 K;«) - /., ( y /127
143
. \
- ICi) • ,.,/90445
(H. 19) 2737.
55 . \
I III /.,„ / « i f
I III I nil I li Al. rill'.l >l("l ION l'Oit U, \ m •
/004
U
< /"'"V/"
,,r
/ 3880 890 . 20 \ w » \ - 2 7 — r ^ f ^ )
j
152 . \
- i H
7T2 / l l ,
Ts =
m
4
(D.20)
; (D-2»
V"'
/ U _ c. \
"CT" U>
(D.22)
7 '
The numerical values of the Hiemanu £ functions are C:i 555 1.2020509 and ss 1.0309278. T h e coellicients
T h e t h e o r e t i c a l p r e d i c t i o n f o r /?,
T h e t heoret ical prediction for Hi is o b t a i n e d from that for //., by t a k i n g into account q u a r k mass effects a n d the fact t h a t , in t h e coupling of t h e primary q u a r k s to the Z. vector and axial-vector c u r r e n t s c o n t r i b u t e differently (Hebbeker, 1991). T h e prediction can be written as F(Z —» hadrons) = ~FP) 1 (Z —> i—: leptons)T =
,
. 'V. + *» + <>.., + A.) •
„ (D.23)
Here. R) is the purely electroweak prediction without Q C D corrections, ¿a is t he Q C D correction for the case of massless q u a r k s which is c o m m o n t o t h e vector and t he axial current , while <1,, is an additional t e r m which only c o n t r i b u t e s to t h e vector current. T h e two remaining terms a r e mass correct ions, r5m. a cont ribution to t he leading; o r d e r coefficients which mainly comes from t h e axial couplings of the q u a r k s to the Z. and . a second order correction in the axial current, due to t h e large mass splitting bet ween t o p and b o t t o m quark masses. Using t h e effective parameterization of both the t o p and the Iliggs mass dependence from tin' TOPA/.O p r o g r a m ( M o n t a g n a tl til. 1993». 19936) given by Hebbeker r.t til. (1994). o n e o b t a i n s R}:w
= 19.999 ( \ - 2 . 2 - 1 0 " ' h i ( ^ T O
1 i1 ~'L7"
• (D-24)
For M1 150 G e V and ;Wn 300 G e V this expression reproduces the value > ,u /i '' 19.903 given bv Passarino (1993) based 011 TOPAZO for the s a m e mass parameters. Not«' that the measured value of the top mass is M, ss 175 G e V and t h a t the mass of the Iliggs particle is expected to be M\\ < 200 GeV. T h e dominant part of the Q C D correction is t h e s a m e as for R v with the exception of t h e contribution proportional to 7':( where only the vector current contributes. So = Kin,
I- Kju'i -I- (A'.i + /?:,)«;' •
(D.25)
/»•
It,
A N I > It i I ' O I I A K I t i I ' l t A K Y ( X ) I . O U K
l'A('TOIt.N
I n t r o d u c i n g (/,, a n d
(|
sin" Ww
and
n,, = H* .
(D.2G)
t h e c o n t r i b u t i o n f r o m '/it can he w r i t t e n as :t> , r r 6« = ( 2 3 « , ) •
''."j -j. (Xl,, "
•
m o7\ (D.271
'I'lie second f r a c t i o n is t h e relat ive conl ribut ion of t he vector current to t h e total cross section, which here is expressed s i m p l y a s a f u n c t i o n of t h e electroweak couplings. Due t o threshold effects which go p r o p o r t i o n a l t o (3 — •i~)i'i/2 for t h e vector current a n d for t h e axial coupling (Djouadi ct al., !!)!)()). t h e r e is also a slight d e p e n d e n c e on t h e q u a r k masses. On t h e Z resonance these effects a r e small, a n d within t h e precision of these calculations, t h e y can b e ignored. T h e leading o r d e r m a s s correction <),,,. expressed as a f u n c t i o n of t h e pole m a s s of t h e q u a r k s , is given by
(D.28)
An improved mass correction can he o b t a i n e d by a b s o r b i n g large l o g a r i t h m s into r u n n i n g masses 7/7,, ( C h e t y r k i n a n d Kiilin. 1990: C h e t y r k i n ct at.. 1992). For a d e t e r m i n a t i o n of t h e s t r o n g coupling from R/. however, t h e difference t o Unloading o r d e r t e r m is negligible. Details a b o u t t h e t o p m a s s correction c a n b e found in t h e l i t e r a t u r e (Kmelil a n d Kiiliii 1989. 1990). T h e leading o r d e r t e r m c o m e s from an i n c o m p l e t e cancellation between two t r i a n g l e d i a g r a m s Z •• gg. whore via t h e axial current t h e Z couples to a It-quark or a t - q u a r k loop. T h e colour s t r u c t u r e of this cont r i b u t i o n is of t h e t y p e {T«T',j)(T!kTkl) = TyN-»• h e c a u s e of t h e identity A'..\ - S yCy/Ty. equal t o t h e p r o d u c t TyCyNy. S e t t i n g t h e n u m b e r of q u a r k degrees of freedom t o Ny t h e colour f a c t o r d e p e n d e n t correction b e c o m e s
- < f r
"A. y¿-«I r-•! 4 '- <1'
N Mt
(Mr
' 27 V Mt
(0.29)
Note that only t h e fraction of t h e cross section with b - q u a r k p r o d u c t i o n cont r i b u t e s to t),. T h e n e x t - t o - l e a d i n g o r d e r correction t o <>,. p r o p o r t i o n a l t o ir*. is known a n d a m o u n t s t o 15'X of this leading o r d e r correction ( C h e t y r k i n a n d T a r a s o v . 1991). D.4
T h e t h e o r e t i c a l p r e d i c t i o n for
Rj
T h e theoretical prediction for I f - is also related t o R.t. Detailed discussions can be found in t h e l i t e r a t u r e (Braat.cn ct ul. 1992: l.e D i h e r d e r a n d I'ich 1992«.
I III
I III < till 11» \l I'lll I'D 'I ION l o l l U ,
III
1992 b; I'icli I!)!I2). Here, we will present only n short siuiuiiary. Similar t o t h e ruses of R , a n d li'i t h e s t a r t i n g point is
r =
r(
r ; I (r- —
r
isTi cc-)
=
(
+
+ Jo + <>-„„) •
(D..10)
3.0582 d e n o t e s t h e purely elect.roweak e x p e c t a t i o n , which is modiHere /t'!/ u lied by a residual correction o. which can be c a l c u l a t e d in p e r t u r b a t i v e Q C D . T h e a d d i t i o n a l t e r m <)"„,, covers t h e nonpert urbat.ive corrections. T h e main difference f r o m t h e case of Ri is t he fact t h a t t h e liadronic s y s t e m |iroduced in T d e c a y s is not. at a fixed mass, but rat her e x h i b i t s a m a s s s p e c t r u m ranging f r o m ,U T t o M r . As a consequence, t h e Q C D correction t o t h e liadronic width is o b t a i n e d by integrat ing t h e correct ion t o over t h e m a s s s p e c t r u m . Expressing t h e r u n n i n g coupling c o n s t a n t t h r o u g h its value at t h e scale M r a n d t urning t h e integral over t h e m a s s s p e c t r u m i n t o a c o n t o u r integral, o n e o b t a i n s (Le Diborder a n d Pich 1992«. 19926; P i r h 1992) <S0 = A*iAt + A ' j / l j + A':t.4:j | • • • .
(D.31)
with
= - K c [ <\
(D.32)
where « s ( s) a n d i^( r') a r e related via eqn (D.3). From t h e e x p e r i m e n t a l d a t a , t here a r e i n d i c a t i o n s ( B r a a t e n r.l at.. 1992: A L E P I I Collab.. 1998rA O P A L C o l l a b . , 1999c) t h a t t h e n o n - p e r t i i r b a t i v e c o r r e c t i o n s a r e slightly negative a n d below 1%. <•>„., = - 0 . 0 0 5 ± 0.005.
APPENDIX C
I G VI
TI
I
E
FR G
E T TI
F
CTI
I lore. I lie explicit expressions of t h e various funct ions a r e given, which a r e needed for a N L O analysis of scaling violations in f r a g m e n t a t i o n functions. As before, we m a i n t a i n generality a n d give t h e expressions for a r b i t r a r y colour factors. E.l
Definitions
The leading o r d e r s p l i t t i n g kernels, which also a p p e a r in t h e N L O expressions, are
. . « . M
E.l ^
.
= m(x)=x*
(E.2) (E.3)
+
+ ll - x )
2
.
(E.-l)
Also, t h e following q u a n t i t i e s a r e used frequently. S,(.r) = - L i 2 ( l - x ) = [
S
l n ( l — c) ,
Jo
S-,(x)
= -2U,
( j - ^
) +
(E.5)
* 2
*-
2
C + ' )+ Y
(E.G)
All funct ions given below a p p l y for a n v t h e o r y wit h s p i n - 1 / 2 ferinions i n t e r a c t i n g via lnassless spin-1 g a u g e bosons. T h e choice of a specilic t heory is m a d e by s u b s t i t u t i n g t h e a p p r o p r i a t e colour fa c t o r s C . C\.\ a n d T . For Q C D (SU(3)) one h a s t h e values C = 4 / 3 , C,\ = 3 a n d T = 1/2. Below, t h e r a t i o s
°S =
'
= §7
au
/« = " / j i r ;
(E.7)
will be used. T h e N L O kernels P { x ) at a scale y/s a r e a power series to s<'cond o r d e r in t h e s t r o n g coupling. In t h e following sections they a r e always given for t he renonnali/.ation scale //// = y/s. t h a t is. r(x)
= « s / l ( * ) +nj (x)
with
(E.8)
I III II W O l ' H NON-SlNOI.Ki C 'ASK
T h e t r a n s l a t i o n t o a r b i t r a r y r c n o r m a l i z a t i o n scales is d o n e in t h e usual way by >( . ) =
) {x)
-I-
( {x)
+
(x)l>{\ In
(E.9)
with l><>
-Jn . I)
(E.10)
.1
Some of t he splitt ing kernels q u o t e d below a r e singular a t x I a n d t h u s require legitimization when s u b s t i t u t e d into t h e evolution e q u a t i o n s . Here, t h e so-called ( I )-schenie has t o be used, indicated by a subscript ( T ) at t h e respective function a n d defined t h r o u g h
E.2
T h e flavour non-singlet case
Collecting t h e f o r m u l a e for t h e evolution of t h e non-singlet p a r t s of t h e f r a g m e n tation f u n c t i o n s f r o m t h e l i t e r a t u r e (C'urci r.l «/.. 1980) o n e linds t h e following building blocks for t h e N L O part of t h e split ting kernels. ,,(x)
= 2
m(x)
[hi x In(1 - ./•) - In 2 ar] - 5 + &r
4 c( r)
= '
_ 5 -
(x)
—
m(x)
(
m(
= S,(x)
In 2 x
+(1 + ')l" x
(E. 13)
+ = Inx\
( E . 14)
x) + 2
2;.: 1 ( 1 + x) l u x .
(E.l!>)
C o m b i n i n g these f u n c t i o n s with t h e a p p r o p r i a t e colour fa c t o r s yields = P'<3:) + f M - > ' ) + h P „ ( x ) . (x)
(E.12)
20x
+ Y = -
+ ^
-^hix+ih.2*)
(. ) 20
(x)
In x + Q
= (2-f
)
(x)
(E.Hi) (E.17)
and >+ (x) = a \ (x)
+ "s (/*»<*> + O
j :
> )
•
(E. 18)
from which the N L O splitt ing function governing t h e evolution of non-singlet f r a g m e n t a t i o n f u n c t i o n s is finally o b t a i n e d as (x)
=
'
)]
(+)
I
*26(
- x) f
\z
(z)
.
(E.19)
SCALING VIOLATIONS IN !• MAC MENTATION F U N C T I O N S
III
E.3
r
riu; flavour-singlet case
T h e s p l i t t i n g kernels for t h e flavour-singlet p a r t s of t h e f r a g m e n t a t i o n funct i o n s ( F i i r m a n s k i a n d P e t r o n z i o . 1980) a r e slightly m o r e involved. T h e N L O c o n t r i b u t i o n s for (lie f e n n i o n f e r m i o n s p l i t t i n g a r e :
/?.,<,(x) = 2S->(x)P,M(-x) +Pm(x)
-
'»•'•
4
+
i )
I n x — 2 I n 2 x + 2 l n a l n ( l - a:)J
^
11 c , , ¿W*) = y -
- 1+ x -
(E.20)
„ , v / f > 7 - :j~ 2 11 1 2 \ ^ + — I n x + - In x j + Pw(x) 18
-SAx)Pm(-x) . 7
, „
""(x)
40
(E.21)
52
=
28.r +
I12.r
l(u;2\ ,
(
+
~
)lnj'
+ ( 2 + 2x) In 2 x - Pw(x)
^
+ H ln.r)
(E.22)
For t h e f e r m i o n gltton s p l i t t i n g o n e has:
R
= - 5 + y
- ( 8 - | ) l n * + 2.x-ln(l - a ) + ( 1 -
-P<m(x) $«(*)
^
= Si(x)PfK(-x)
- l n 2 ( l - x) - 4 In .r l n ( l - x) + 8 5 , (a-))
+
^
-
^
+ ^ + ( 2 +
- 2 . r )n(l - x) - (4 + x) h i 2 x + -Max
In 2 a-
12:r +
(E.23)
In*
Pm(x)
- 1 h i 2 x - In 2 a ( l - x) + 8 . S ' , ( . r ) j
(E.24)
T h e g l u o n fermion s p l i t t i n g is d e s c r i b e d by: /? K< i(:r) = - 2 + 3:r - (7 - 8.r) In x - 4 ln( 1 - x.) + (1 - 2x) In 2 a- 2 P m ( x ) ( 5 - 7T2 - In a: + In(1 - x) + In 2 x( 1 - x) - 8 5 , (.1:)) ( E . 2 5 ) Sm(x)
, „ , , = 2S3(x)Pm(-x)
+( 2
40 + 152a: — lOO.r 2 —
, , + 8a-) In 2 x + Pm(x)
+ I^|,i(l
/ 1 7 8 — 21 rr~ f
-I + 7(i.r , , — — In.r + 4 l n ( l , , 4 1 0 6 , (a:) - - In r
— x) — In 2 x + 8 In x ln( 1 - x) + 2 ln 2 ( 1 - x) j
x)
I'ltACMEN'l A I'lON I IINI l'l( INS ANH IIADItON S|'E<' I It \
%sM
= - § -
/;,,» ' ) ( ^
111»-'-11H1
1 ir,
(E.27)
4 }
Finally, for t h e gluon gluon s p l i t t i n g o n e finds: Kil.1 2 « ' 0
= ^!).r - 4 + 12a: I
!)
I(i 10a' 2 — + 1(1 + 14a: + — — In a: 1 (2 + 2a:) In 2 x 3a3
t. , \ , ,,, , S ^ x ) = 2 S2(x)P^(-x)-
,
134 27 27a— + — — IS./2 2 / 44 11 2 5a: \ (i^ + y - — ( 4 + 4 a : ) l n
, . / (¡7 + P« ( • * ) (
22 + — hi a-
2
(E.28)
134a; 2 18 , . ,r
\ 3 In" a- I- 4 In x I n ( l - x) J
(E.29)
29a- 2
29
(E.30) From t h e s e t h e c o m p l e t e N L O uiircgularizcd .splitting kernels for t h e e v o l u t i o n of t h e llavour-singlet f r a g m e n t a t i o n f u n c t i o n a r e c o n s t r u c t e d a s follows: ¿QQ(*) = « S -P«|.iM + «* ( ^ K i ( ' ) + / a 5 I | C | ( X ) + f„Tm(A:)) Pqc.M = a M ' )
+ " s ( * . « < * ) + /A5, i b (.»-))
/ , oq(.') = « s 2 / „ P m ( x ) + < (f„Rm(x) Pac.(')
(E.31) (E.32)
+ f„fASm(x)
= « * 2 f A P m ( x ) + « 2 (f„Rm(x)
+ f,J„Tm(x))
+ fAfASm(x)
(E.33)
4 f„fAT^(x))
(E.34)
Here, t h e d i a g o n a l p a r t s of t h e s p l i t t i n g f u n c t i o n a r e s i n g u l a r at x regularized s p l i t t i n g kernels for t h e e v o l u t i o n e q u a t i o n s a r e givt-n by
ami
Pii(:(.r)
= -r [ . r ^ G ( . x - ) ]
PQQ(X)
=
- \xP*Q(x)} X
E.4
+)
L
- 6 ( 1 - x) J* dzzP?;Q(z)
-M + )
S(
1 - x) f
dz zP*G(z)
1. T h e
(E.35)
.
(E.30)
./(I
Fragmentation functions and hadron spectra
T h e relation b e t w e e n f r a g m e n t a t i o n f u n c t i o n s , given at a f a c t o r i z a t i o n scale //,.•. a n d t h e liual h a d r o n s p e c t r a at a C . o . M . e n e r g y ^/s. is given by a c o n v o l u t i o n of t h e f r a g m e n t a t i o n f u n c t i o n s with coefficient f u n c t i o n s for t h e l o n g i t u d i n a l a n d
III.
S C A U N O VIOI.A'I IONS IN !• It A< ¡MENTATION I UNCI IONS
t r a n s v e r s e part o f t h e inclusive cross sections. In N L O those f u n c t i o n s (Altarelli it ill.. I!)7!>6) t a k e into account t h e effect of t h e first, h a r d gluon emission on the inclusive p a r t i c l e s p e c t r u m . T h e y a r e given by: 1 - s
(E.:i7) (E.:w)
PcAz.s-l'i-
= "s (/'/••) {
1+
(!-;)-
_
PZ(s,s,fiy)
1 I
=¿(1
2
[ln(l — z) + "2 In 2]
l z i _ l ± l L z i ) !
<>S(/'F)
"
2
/ i }
(E.30)
JJ)
i ± i ! „ ,
1 - 2
l n
•-ite)
Note that t o leading o r d e r all t e r m s p r o p o r t i o n a l t o fi s can be neglected a n d t h e h a d r o n s p e c t r a a r e directly given by t h e f r a g m e n t a t i o n f u n c t i o n s . Also, only t h e t r a n s v e r s e part c o n t r i b u t e s for q u a r k s .
E.5
Electroweak weight functions
T h e electroweak weight f u n c t i o n s «',(.s) specify t h e relative f r a c t i o n s of t h e prod u c e d q u a r k flavours a s a function of t h e C.o.M. energy. «><(*) =
, ''ft
, ,
With
/ = { u . d . s . c . 1,} .
(Ell)
T h e relat ive cross sections r,(.s) a r e given by t h e electroweak t heory:
' '
, (
s
)
=
c
'
+
(M.j -
s)
-'
+
M.jy i
Mr, - .« 4slcw
H>4,i
Here s w — sinW w a n d c w = c o s 0 w a r e t h e sine a n d cosine of t h e weak mixing a n g l e 0 W . u, l \ a n d r, - /'j - 2r,.s' 2 . t h e axial a n d t h e vector coupling t o the Z. c, t h e f e n n i o n c h a r g e a n d /•![ t h e t h i r d component, of t h e weak isospin. T h e s u b s c r i p t e d e n o t e s t h e electron.
APPENDIX
F
SOLUTIONS
I i'lir differential cross section lor a 2 —• 2 process of two incoming particles with loi ir-momenta p\ and po. which are scattered into outgoing part .ides pn and ¡i |. I Is given l»y the expression
(F.l) Here, the lirst term is the scattering matrix clement., the second one with the •¡(|iiare-root, is the flux of incoming particles, then comes a four-dimensional <>function for energy momentum conservation and finally the phase-space factors for the outgoing particles. The total cross section is then obtained by integrating over all possible kinematic configurations. The various powers of 2~ come from the normalization of the wavefunctions as given in momentum space. Note that the phase-space factors for tin- final state particles are essentially just t he volume elements of the four-momenta subject to the constraint that the particles be on mass-shell, with E > 0. (1
''P ,1 0 / 2 ,2\ _ = ,,,) (i, - m ) .
I'lie matrix element can be calculated in perturbation theory using the Fevnman rules of Section B.l wit h the elect ric charge of the respective fcrmions instead of I lie gauge coupling (¡s. The amplit ude j\A for /-channel scattering of an electron with another spin-l/'2 particle carrying a charge r. • <• f as shown in Fig. F.L modulo irrelevant phase factors, thus, is given by M = cfr2[r/(R!)7""(/'t)] Note
['"'(/'t)-Y„"(P2)] •
that the spinors are normalized such that, t he polarizat ion sums are ui'i spins
in t- i>
aixl
^ ^ vti — in — />. spins
lii the case of unpolarized [»articles the [woper matrix element is obtained by summing over final state and averaging over initial state polarizations, that is.
SOLUTIONS
I IN
FlC. F . 1. Born level d i a g r a m for e l e c t r o m a g n e t i c scat tering of a n electron a n d a different. s p i n - 1 / 2 fermion with c h a r g e c • c.j
\M\2
M M
= i
'
spins e2e4
5 1 [' 7 (/'-i)""''(/'l ) " ( / ' ! )7"»(/':t)] {"(p\)~/,,2)"{l>3)l„"(m)]
=
•
spins
A p p l y i n g t h e usual t r a c e rules, t h e a p p r o x i m a t i o n m i = 0 which is perfectly a d e q u a t e for t h e case of oleetron-nueloon s c a t t e r i n g , a n d sett ing m-> = M . o n e obtains IX|- = ^ T r
{ i x f W k i h l n h l » +
•
Using t h e ident ¡ties T r { / V / " / , 7 " } = 4 (P'M
+ p f f l ~ '/""(/'. " PA))
«>».1
Tr
fr'V'"}
= 4,,"" ,
t h e result simplifies t o \M(2
=
'^T
[(/'l
Pi)(P3-Pl)
+ (P\
Pl)(P2 •/':») - M~(p\
•/'.))] •
F u r t h e r calculations now a r e conveniently d o n e in t h e l a b o r a t o r y s y s t e m . W i t h t h e explicit f o r m of t h e f o u r - m o m e n t a . p\ = (E. p , ) , p-j = ( A / . 0 ) , />;i = (E',p:{). a n d t h e s c a t t e r i n g angle 0 between p , a n d p , . t h e various scalar P\ (E\,p.{) products appearing above become PI
•
=
EM
p-> • pi = E'M P
I P 3
,
.
= EE'(
p3 • p, =
I-case). +
1 - cos 0) .
SOLUTIONS
p , •/>., = E'M
- EE'(I
II!)
- cos0) .
which leads t o | M\2 = c j J ^ E E ' M
2
| ( 1 + cos0) +
^ ( 1 - cas») J .
For elastic s c a t t e r i n g processes one can s u b s t i t u t e E - E'
//
if
M
M
2 A/2
and with t h e usual t r i g o n o m e t r i c relations . , i 0 1 + cost/ = 2 c o s -
and
. 0 I — cos0 = 2sin" —
one filially o b t a i n s for t h e m a t r i x element \M\
=cjc
—
A/
|cos
2 + 2ÂP s»'
5} •
T h e next s t e p in t h e calculation of t h e cross section is t h e d e t e r m i n a t i o n of t h e phase-space factors in e<|ii ( F . l ) . E v a l u a t i o n of t h e flux f a c t o r a n d i n t e g r a t i o n over p.,. using t h e constraint of energy conservation, yields
=
+ A /
M ' û n à m
- * -E
i )
•
where E\ now is a funct ion of t h e r e m a i n i n g variables. E.1 = \/M2
+ ( p , - p . ) 2 = v ^ / 2 I- /•:-' -I- Eri + <72 - 2 E £ ' .
Here, tin" relation between s c a t t e r i n g a n g l e 0 a n d m o m e n t u m t r a n s f e r Q 2 h a s been used. Q2 = 2EE'(\ c o s 0 ) . For t h e i n t e g r a t i o n over ]>:i t h e available p h a s e space c a n b e p a r a m e t e r i z e d in a variety of ways. T h e most c o m m o n o n e is t h e use of spherical c o o r d i n a t e s , which, using t hat t h e s c a t t e r e d p a r t i c l e is a s s u m e d to be inassless, leads t o d'V:j = E'2i\EW cosfldf/j. Instead of i n t e g r a t i n g over c o s 0 one c a n also d o it over W i t h t h e .lacobian 2 E E ' t h i s leads t.o
d'/'i =
^ \ E \ \ Q
2
.
Now. c a r r y i n g out t h e integrat ion over ( \ E ' in o r d e r t o get. rid of t h e ¿ - f u n c t i o n , a n d a s s u m i n g a z i m u t h a l s y m m e t r y , which gives a n o t h e r f a c t o r of 2
lyW|2 1
HOI H I IONS
a n d filially s u b s t i t u t i n g t h e m a t r i x clement calculated earlier, together with the definition of the line s t r u c t u r e constant o - c - / ( l~). t hen gives t h e dillerciiUnl cross section for elastic s c a t t e r i n g between two s p i n - l / 2 ferinioiis: -ITTn~m< -f £? ' (
«Iff W -
ê V
Q'
~
20 OS
.
Q2
2ÄI
2 Sl
2
"
0 \ 2/
'
Q u a r k s can come in a s u p e r p o s i t i o n of t h r e e basic colour s t a t e s ]/?). |G) a n d |//) which a r e represented by o r t h o g o n a l unit vectors, t h a t is.
I'/) = r\R) + (j\G) + b\D) = r U j
+ <j ^ l j + b ( „ )
.
Gliions describe t r a n s i t i o n s between colour s t a t e s . In t h e basis introduced by the C e l l - M a n n m a t r i c e s t he eight gluoii s t a t e s a r e given by A, =
\G)(R\
A2 =
\\G)(R\-
A:, =
\R)(R\
-
A, =
\B){R\
+
A* =
i|/i){/?|
-
1 B)(G\
A« = A- = As =
^\
+
\R)(G\ \\R)(G\ |G>(C?| \R)(B\ \\R)(B\ \G)(B\
+
\\B)(G\ R )(R\ +
\\G)(B\ ± \ G m
Intuitively these expressions can be read as superposit ions of colour-anticoloiir states, with t h e ket-vector representing t h e colour part a n d t h e lira-vector the antieolour eoniponcnt. From the a b o v e expressions t h e probability t h a t a red q u a r k emits a gluon is o b t a i n e d by incoherently a d d i n g t h e c o n t r i b u t i o n s of the t r a n s i t i o n s from red to red. red to green a n d red t o blue. Interference t e r m s d o not a p p e a r since t h e filial s t a t e s a r e distinguishable t h r o u g h t h e colour s t a t e ol t h e e m i t t e d gluon. R e m e m b e r i n g t h a t t lie Q C D couplings a r e given by T" = A„/2 one then finds: A(R
-
qg) = A(R
-
/?) + A(R
-> G) + A(R
-
B)
= | ( / ? l ^ : , | / ? ) | 2 + |(/?,^Vs|/?)i 2 + | < G | ± A , | / l ) | 2 + | < G | 5 A 2 | f f > | 2 + Ktfl5A.1l/?)! 2 + | ( C | I A 5 | / / ) | 2
sot 11 r i o N s I,. I 1 .( I 1 - 1 1 1 1 1 1 + l ) = 1 .{ = A(G -
A(G
-> c|g)
A(G
- > qg) = A(B
R) I A{G R) + A(B
G) + A(G -
G) + A(B
B) = ~
— B) = Î
1 I he gluon splitting a m p l i t u d e s a r e given directly by t h e s t r u c t u r e c o n s t a n t s of SU(3) M|ii (2.38). Because t h e fuhe a r e totally a n t i s y m m e t r i c in their indices onlv splittings into m u t u a l l y different colour s t a t e s a r e allowed. Since t h e final state's are distinguishable t h e individual a m p l i t u d e s have t o be s u m m e d incoherent Iv. One finds t h e following cont ributions:
A(l
— Kg) = 2 ( / f * . + / , 2 i r + / 2 C ) >1(2 —• 1(3 —> rrtr\ = 2 ( / | , 2 + Ilr, + Ihr,) .4(1 —1 gg) = 2(/?58 + II17 + Ihr, + A(r> — gg) = -(/¿is + Ii10 + Iki + >1(0 — Kg) = 2 (Ihs + Iiis + + -4(7 — g s ) = 2(/2cs + /?,., + f i r , + Od \ = 2 ( / | , + / | >1(8 » f»i->/ 5 r>7)
= 3 = 3 = 3 fir,) = 3 fh~) = 3 fi37) = 3 fir,) = ."i = 3
T h e f a c t o r of two in t h e a b o v e e q u a t i o n s takes into a c c o u n t that for each a m plitude t h e o n e with t h e second a n d t h e t h i r d index exchanged also c o n t r i b u t e s . One finds a perfect s y m m e t r y for all gluon s p l i t t i n g processes into s e c o n d a r y gluous. T h e splitting of a gluon into secondary q u a r k p a i r s is a g a i n governed bv t h e quark gluon couplings a s defined t h r o u g h the C e l l - M a n n matrices. Each m a t r i x element is directly p r o p o r t i o n a l t o t h e a m p l i t u d e of the gluon described by that particular m a t r i x splitting into a defined quark- a n t i q n a r k s t a t e . S q u a r i n g t h e a m p l i t u d e s in o r d e r t o gel t h e relative probabilities shows perfect s y m m e t r y of physics wit h respect t o colour.
SOI.II I IONS
I Wi
A|,
— (l«l)
A(l
2
-1-
Ai, 2
+
AL 2
+
A|2
2
2 Ai, o
A( 7 — cm) -
A?,
-4(8 —
2
2
1 2
—
1 ~ ö
2
•>
A§ 3
1
2
2
[UA„U~l
+
+
2
i I P r o c e e d i n g directly: !•]„> =
i),,Au
+ \gaA,lA„
- (/t <-» i/)
-I- ig. \UA„U~X - (/< -
+ igJl(()„U)U~l]
\g:\dvU)U~l]
v) U~x - U [.A„(OJJ-')
= U [(t)„A„) + \()SA,,A„\ -i.'/;1
- (d„U)(<)„U-1)
- (// -
+
A„(()„U-1)]
- (d„U)(dU~]
)]
v)
= UF„„U~l In t h e t h i r d line wo m a d e nso of t h o identity supplied and r e g r o u p e d t h e t e r m s so a s t o m a k e m a n i f e s t t h e s y u u n e t ry u n d e r // <-» i> of t ho second a n d third g r o u p of t o n u s . T h e s e l a t t e r t e r m s cancel t h e r e b y explicit ly c o n f i r m i n g eqn (3.8).
5 2
Let t h e I b u r - m o m o n t u m of t h e virtual particle in t h e l a b o r a t o r y f r a m e be Q1'. In the particle's rest f r a m e it t h e n has t h e f o r m Q'1' - (\/(J2. 0). According t o the 1 / \ f ( p and e n e r g y t i m e u n c e r t a i n t y relation this particle lives for a t i m e I ' t h u s travels the f o u r - d i s t a n c e . r ' " = ( l . O ) / ^ / ^ . T h o rest f r a m e a n d l a b o r a t o r y f r a m e a r e related by a boost in tho direction Q with -i - \ Q \ / Q ° a n d -, that QU/\/Qi-
<
r
^
)
1 _
=
„ ( Q 0
Q
1 ) _
or
()>'
= £
.
|
Here, we a s s u m e d (}'' is time-like. t he s a m e result holds if Q1' is space-like. T h i s result can lie applied itéraiivoly t o a chain of virtual particles by s u m m i n g their d i s p l a c e m e n t s to build u p a s p a c e t i m e p i c t u r e of a s c a t t e r i n g event.
5 3 T h e h a d r o n i c tensor / / ' " ' is a rank-2 Lorentz tensor. Ignoring spin degrees of freedom, the free indices can only c o m e f r o m t h e f o u r - m o m e n t a />'' a n d
SOLUTIONS
ir.:i
I lie c o n s t a n t t,elisors //'"' a n d < ' " ' ' " . In a d d i t ion, a n y r a n k - 2 (elisor const,met,<><1 f r o m i liese i n g r e d i e n t s c a n lie mull ¡plied by ;i s c a l a r f u n c t i o n of Lorenl/. invariiuil q u a n t i t i e s . U s i n g t h e not.-it,ion f r o m S e c t i o n 3.2. t h e most, g e n e r a l e x p r e s s i o n is
= -m< ^ . /' •'/
r M » -
f P • <1
/' •'/
' P'l
p-'l
wM •
I he const r a i n t s s u p p l y t wo v e c t o r e q u a t i o n s w h i c h m u s t hold for a r b i t r a r y />'' a n d
= -
+ I'll'" "
W"
. qu = -Fxq" - n ~ u
+ <> + F'xq" -I- (,,-/,,
/O = - F , + F i + (
+ /-V -
• ,,)Fty
+ (
q
(q2/p-q)Fr,q"
1 F'tq'' +
jO = -Fl + Ff> + (q2/p-q)FG \Q = +F2 + (q2/pq)F'i
q" p"
T h e c o n t r a c t i o n s w i t h t h e c, t „„ T v a n i s h b e c a u s e of its a n t i s y m m e t r y .
I For t h e a z i m u t h a l l y i n t e g r a t e d , invariant p h a s e s p a c e e l e m e n t of t h e final s t a t e lept.on o n e h a s E r d cos0 s: E c d E f d cos0 . Now. it is most c o n v e n i e n t t o work in t h e t a r g e t h a d r o n ' s rest f r a m e , w i t h s A/,-' 2 M u E ( = Q2 /(.''.'/) a n d t he d i f f e r e n t k i n e i n a t i c a l v a r i a b l e s given by: „2 o r r Q- = 2EfEf'(
n
m
\ - cos0) , x =
EtErl 1 - cosfl) — MU(E,-Er)
,, r . is = Er - Er
. y =
E, Et
From this t h e J a c o b i a u s for t h e c h a n g e of v a r i a b l e s a r e e v a l u a t e d a s
<)(Er.co*0) 0{Q\x)
cos 0)
= 2 EfEf
Q2 Er xy.M i,
ij. M i.
£?£?.(!-ocwfl)
2x
" ~ A/|,(E, - £i<)2
Q2
V• 2x
Er -
cos 0)
Mh(Er
-
£,.)
and the t r a n s f o r m e d cross sections are obtained using the rule i\Er d cos0 =
d(a. b) ¿)( Er.
co* <>)
(\,n\b .
w h e r e (ti.li) d e n o t e s a n y of t h e a l t e r n a t i v e p a i r s of v a r i a b l e s .
Er
•
SOI,II I IONS
•154
\ -ri Referring id eqn (3.3(>) a n d noting t h a t < \ • • fx)(p
(• wo have
• t'x)— + W A ' A W — p•q p q
Noting that f-. • c^ = —1 and f ± • p = (I whilst r() • M\> \/i'2 + Q2/Q we derive
= +1 and i,\ • P
ll± = Ft T F:i \ l + % ~ F\ T F* > 0 V2 //F2 ( Q\ „ /••> „ 2.7- \
/>- /
2.r
T h e a p p r o x i m a t i o n s follow since Q2/i'2 = (2xM\,)2/Q2 -C 1. 1 polarization e'J tc
Since t h e scalar limit Q2/i/2 I T h i s inability to 2:rJ'\.
i (i Recalling t hat t h e weak current, couples u to ( c o s f l c d - t - s i n 0 r a ) . a n d ignoring any c o n t r i b u t i o n s from heavier q u a r k s , t h e s t r u c t u r e f u n c t i o n s a r e given by t~
1
Fih
= cos 2 0,d 4- u + sin 2 0cs
and
x~1 F.?h = cos 2 0r,l + « + s i n 2 0c.s .
T h e n using t h e c o n s t r a i n t s for t h e third c o m p o n e n t of t h e weak isospin a n d t h e electric c h a r g e of t h e liadron h. 'A(h) = j f dr
[«/(*) - '/(:'•)] + ^ [»(•'•) - '"'(•'•)] }
Q(I.) = f \ \ r | - 1 [,/(.,•) - d{*)] + | [«(*) - f i ( j ) ] - i [.s(.r) - 5(.r)] |
.
one immediately obtains i
f
— [ * £ ' • ( * ) - f T " ( . r ) ] = 2T : l (h) + s i n 2 « , . [ 3 Q ( h ) - 47' 3 (h)] .
- Jo
T h i s is t h e original Adlcr s u m rule (Adler, 19ti.'5). If c h a r m p r o d u c t i o n is allowed kinematically. then t h e t e r m p r o p o r t i o n a l t o s i n - ( ) r is a b s e n t .
5 7 C r o s s i n g s y m m e t r y gives tlin-c relationships: g • gg gives V ^ z ) l / ( . K ( 1 — c). q
sol.iinoNs
ir.s
I H W r i t i n g t h e c o n v o l u t i o n s s c h e m a t i c a l l y a s p r o d u c t s (M) of s p l i t t i n g kernel a n d £ a n d <j i n t o e q n (il.282) p.d.l.. t h e n suhst.il,liting tlie e x p r e s s i o n s for ry ( Ns . violi Is
1 / 2
I I
, ±_
-
èI = S
/ e \
7
-
(F.2)
_ t v ( [r™
+
+ "fU*
I /§,)]
,
,
F
n
T h e e q u a t i o n s for t he flavour-singlet, a n d t h e g h i o n p.d.l'.s d o not d e c o u p l e .
I ') W i t h t h e d e f i n i t i o n F2>,p2) = . r ^ o
2
[,,(,:. p 2 ) -I- , / ( , , , , * ) ] .
I he e v o l u t i o n e
r ' 0 ( . , p
2
)
=
f d z \ p f f ( z ) F 3 (f.„*) s
+
L
'i
In t h e limit x — II. t h e t e r m p r o p o r t i o n a l t o F.J o n t h e r i g h t h a n d s i d e c a n b e neglected. W r i t i n g out e x p l i c i t l y P,\"](z). t d f q .
2,
„,.
y-
e q n (3.50). t h e e ( | u a t i o u b e c o m e s
,2, "nss (( /p' '^2))
f
,
r
,,
n i - z f ] c
±)
.
W i t h t h e a n s a i / G(x) ~ ;r"(l - x)'" t h i s h a s b e e n used t o e x t r a c t t h e g l u o n d e n s i t y f r o m t h e e v o l u t i o n of F.J (Ellis c.t til.. 100-1).
I III In t h e limit of small values for |/| a n d A t h e re(|tiired r e s u l t s follow i m m e d i a t e l y from 2'/ • (P - />') = (
l>')2 - '12 - (P ~ P?
= Ml
+ Q2 ~ I « M'i- + Q2
2t, • p = (t, + p ) 2 - t f - p 2 = I E 2 I Q2 - A / 2 « I I ' 2 + Q2 .
The smallest value of —t. o c c u r s w h e n t h e i n c o m i n g h a d r o n c a r r i e s oil u n d c l l e c t e d . T h e a c t u a l c a l c u l a t i o n s a r c most c o n v e n i e n t l y p e r f o r m e d in t h e C . o . M . s y s t e m , w h e r e o n e finds -I
~(p - P'iL,
= -2A/.T -I- 2 (E'E"
- ^(E-i
- AI*)(E"*
-
M2]\
ir,(i
SOLUTIONS
*-2Alh\-Mh
Mh
E> E n
( I F 2 + A/ 2 + i ? 2 ) / 2 H ' ami
Substituting E-
= ( I F 2 + A/,2
A / 2 ) / 2 l F HlOJl
gives the result ~
? 11 Using (>(s,t)
(ii
+ A/ 2 + q 2 ) ( i V 2 + Ar 2 - A/;()
1h\M{s,t)\llm\M{»
11 1
"
« p(.s.O). oqn (:U><>). one has
In practice, a cross section is related to the n u m b e r of events seen using the beam luminosity, a = A ' / C . T h e non-linear relationship is useful because it allows 11». t o avoid the need to know t h e luminosity. If N c \ and A', m .| a r e t h e numbers <>l elastic a n d inelastic events then (dA r t .|/dQ < = ( )
lfa """ " ( 1 + 7 )
+ Arin,t) '
Here we assume t h a t t h e detection efficiencies for t h e two event t y p e s a r e the same.
< 12 With t h e s u b s t i t u t i o n s />.-
p c o s 0 a n d E = s//>2 + in2 ~ p. one immediately
obtains
5 13 Using t h e identity (E + p:)(E
- pz) = E2 - p2: = in2 + p2, =
'"f
one can express the rapidity, eqn (:i..r>.r>). of a particle as { E + Ih)2 = In ^ „ —- I h, = - In 2 E - p: 2 (E + P i ) ( E - p s ) wr
= In
E -
and thus E ± p: = mTe±v
.
From energy m o m e n t u m conservation one has E\ + E2 = xap„ + J i.pi,
and
which a r e easily solved for x„ and .11,.
/>,, + pZ2 = XaPu - *kPb •
• Pz
SOI IITIONS
I t I here a r e m a n y subproccssos which c o n t r i b u t e t for /i,, (xi )}\,.{.r >), the full expression is rl
r ^ d:i:|d:<:2 |(/ii/2 dA(gg - » gg) +
111
d
<1
+
T
[mh
I- /i.<7 2 ]dà(gq -> gq) +
[ f l [ '< 2
1
Y^
dfffqq — <|q) + ^
[>li
['/!'/•> T '7t'/2]d
d
- »
h ' / 2 +
f
7H72]d(T(qq — qq)L >
T h e gg — gg m a t r i x element is very strongly forward backward peaked where the i f f 2 a n d 1 / f / 2 poles d o m i n a t e . T h i s gives the classical l / s i n ' ' ( 0 ' / 2 ) R u t h e r ford s c a t t e r i n g formula, characteristic of spin-1 particle exchange. By c o n t r a s t , non-diagonal scatterings, qq — gg. gg -» <j<j a n d qq —» q ' q ' have lesser l / l or l/.i 2 poles and make little contribution. Figure F.2 shows the ratio of t he m a t r i x element squared to t h a t for gg • gg. Since we do not distinguish between outgoing piirtons the m a t r i x elements were symmetrized under i — 11. T h e s e ratios are reasonably c o n s t a n t as a function of cost?*. In the limit c o s 0 ' — 1. that is I • 0.11 —» .s, it. is easy to verify, by identifying t h e coefficients of t h e leading \ / i poles, that the ratios become l/i). (-I/9) 2 or 0. 0.5 0.4 0.3 .2 § <= 0.2
0.1
I 111• I 0.2 0.1
0.3
11 1 I1 1 1 1 I 1 0.4 0.5 cos (If)
0.6
0.7
0.8
0.9
F t c . F . 2 . T h e ratio of partonic s c a t t e r i n g m a t r i x elements squared to that for gg —> gg a s a function of the cosine of the p a r l o u s ' C'.o.M. f r a m e s c a t t e r i n g angle; from top to b o t t o m : gq • gq. qq —> qq, qq' qq' = q q ' — qq', qq — qq, qq gg. gg q q and qq — q ' q ' .
This a p p r o x i m a t i o n id lows a significant simplification of t h e di-jet cross section.
SOM
ir,«
( =
cb iibri j i.^ l
+
]+
1
1
(
7 2+ q.q'tq 1 . .( 2
,
T \ h\
+
)
( ^
) '
'
i
)>+
(
(T1)
(. ,,)
(
)
) 4(
E
+ 0 + 0
' -1
+
4( n+
,
1
f \h = .
+
Y
HONS
T(
, ~ E
,
h
(
)
)
7 ) s
.. .
A1
s ' ,
=
ifi , +
,,
+ Cn,,nv
s s
s
=
n''T
+ )+
(ii
=
n
+
(n
) =
= 0 2
s
)n
0 = .4 + 0 = C( II
0
=
n,,
.
. T
(n )
(n
s
j
)
) NV
+ D{n
+ Cn2 + D{n (n ) + u2 ss s II". T
= 4" '
s
+ 0 +
+ Cirn
0.
1 T in
~( ' ,
+ n,, )
)n
-f
ir
A)
. s
s +
s
r
'^ ',,
s 2 = s
A c2(-ii
+
s
A T
T
+ D ,,n
s
,,, = 4
=
1 ,
( .121).
s 2
+
''
) i,
( .111).
s s
,
,
' ' s
IMI
SOLUTIONS
g a u g e ' i n v a r i a b l e, c a l l lie s i m p l i l i e d t o
< ~ i ) ' " ' C f / 3 . T h e hadl'Ollic m a t r i x e l e m e n l
is g i v e n liy
M„
=
cc,t
. ( f i + è)
_
it+
(>! + !,J2'"
^(O
ii'lii
W
2// • q
2(1 • <)
4)
,
+ '/r
w h e r e w<" used t h e D i r a e e q u a t i o n t o e l i m i n a t e t h e fj a n d ¡J t e r m s . It satisfies = 0. S u m m i n g over t h e e x t e r n a l p a r t i c l e spins, t h e first t e r m s q u a r e d gives
(2 g - q \ ,,
C i A ' , . ..
'( <<<,!l*)~ Z ' ' y . ' ï i u •
MrCttfsfCrK— • aq •• qn
The t h i r d t e r m s q u a r e d is o b t a i n e d f r o m t h i s result bv e x c h a n g i n g q1' a n d tj1'. T h e i n t e r f e r e n c e t e r m is g i v e n by )
2
{
T
' T KrTv
- ( A •
{fiii'Ml"
Cl N
) = 2(1 r ^ f
' '
•
*(>'
.¡'ISFCL.-NR
.
C o m b i n i n g t h e s e r e s u l t s yields for t h e spin a n d o r i e n t a t i o n a v e r a g e d m a t r i x element.
3
I L I H • '/
Q
»(g%gfV.Q I
,v
P
- •' ( < , * r ' i ! ! * ) ' r
ÌÈ ' • '/)(// •'/)
' i
!L1 + 2 .'/ " <1
+
I
Q-
J
Y •
i
' ( 1 - .r.,)(l - * „ ) "
N o t e t h e f a c t o r r-(/-/'A f r o m I he lopton t e n s o r . I n c l u d i n g t h e flux f a c t o r . I / ( 2 Q ~ ) . a n d t he t h r e e - b o d y p h a s e s p a c e element,, t h e differential cross s e c t i o n is giv<'n by
,,
_
1
N
& 3Q*
d.i.jdXi,
1
= <7<)—C/. 4JT (I
4ff
'j
Q-
-
-
.!•,,)( 1 - . / ' , , )
where o s = /(47r) is t he st r o n g c o u p l i n g for a s c a l a r g l u o n . As a n a s i d e , if t h e vector boson c o u p l i n g t o t h e p r i m a r y q q - p a i r h a s a n a x i a l - v e c t o r c o u p l i n g , a s is relevant for '/ e x c h a n g e , t h e n only t h e sign of t h e i n t e r f e r e n c e t e r m c h a n g e s .
soi.iiri
Kill
s
t 17 In n a t u r a l units t h e action is dimensionless, which requires [
=
= 2 [ /> + 1
d„A d>lAl
=
; = 2 [A + 2
=
= 2 [ ;] + 2
[il> =( -
f) 2 .
[A = (
- 2) 2 .
[t,
- 2)/2
= (
.
ext consider a n y interaction t e r m , for e x a m p l e : [fla'W] = ga =2-
=
[r;, + 2[il> + [A = [i/s
/2 = (4 -
)/2
+ (
- 1) + ( /2
- 1)
= e .
All interaction t e r m s give t h e s a m e result. T h u s , a consistent, dimensionless a c t i o n with a dimensionless gauge coupling < s can he achieved using t h e replacement ( s —' < ., '. where t h e a r b i t r a r y p a r a m e t e r p has m a s s d i m e n s i o n one.
:i 18 W i t h t h e
-diniensional volume element in C a r t e s i a n a n d p o l a r c o o r d i n a t e s d' r = d' -
r >-'dr
t h e integral is given by
J
/
,i>-1
=
Jo
2 it '2
r(D/2)
n t h e left-hand side we have re-ex pressed t h e e x p o n e n t i a l of a s u m as a product of exponent iaIs a n d used the s t a n d a r d result for t he G a u s s i a n integral. n t he r i g h t - h a n d side we have t r a n s f o r m e d t o spherical p o l a r c o o r d i n a t e s a n d . a f t e r t h e c h a n g e of variables. : = r 2 2 . recognized t h e integral definition of t h e - f u n c t i o n . T h e final result n a t u r a l l y generalizes t o non-integer d i m e n s i o n s .
si H.11 r i o N S
M.I
l'i I'lir « I n i vat ion follows t h e usual p r o c e d u r e based upon a p p l i c a t i o n of the Clilfonl algebra, 2//"" I bill now r e m e m b e r i n g t hat, ?/ '' = D so l.liat. 7,,-)" = D. O n e finds:
H = =
= o ^ J - i Y l " + l";")
=
ova,,v""l
W i .
i j l " = 7 , . ( 2 « " 1 - 7 ' V ) - H . - U,/ = -(/J) - 2 V . M h "
= 7 ^ ( 2 ( > " 1 - 7"!>) = 2/lff - 7 ( ^ 7 ' 7 ' = 2 ( 2 « • /;1 - # ) + ( D -
2)#
= +4«-/>l + (D - • ! ) # . U f ' h " = 7V,#(2C"1 - 7 " / ) - 2 / # - % , # 7 " / ' = 2/-(2« - M W ) + (-1« • hi + ( / } - l ) ( / / i ) / - 2 / W - (£> -
!)#/• •
Observe t h a t t h e s e r e d u c e to e q n (3.83) w h e n D -
I.
Vii S t a r t i n g f r o m t he d e f i n i n g e q u a t i o n ///,, = Z,ltni a n d t a k i n g i n t o account i hat t h e scale d e p e n d e n c e of Z,n conies f r o m t h e scale d e p e n d e n c e of t he g a u g e co upling ;/.,. using t h e chain rule o n e finds d'/io <>'» , àZ,„ ()!h Now use t h e e x p a n s i o n s Z,„ = I I ]Cn>i b„/(", li, (¡f\
a
"d
b'n = //,/>„_J + //s«Î'»h-i.
Referring t o eepi (3.1 r>.ri). a n d t a k i n g into account that o., /»-coefficients a r e given by 5 t •A1'"'
-V' 4
« > 2 . I/~/-ITT.
-)7r ~ 12
t h e leading
C,, .
Irom which eqn (3.17(1) for 7,,, can b e i m m e d i a t e l y c o n f i r m e d . For 11 r e c u r r e n c e relation c a n only be tested to yielding
2 the
SOM 1 HONS
I <>2
-4
2Tmf
9^ - -CF
-
11 ^ —CA CF
= —GCF
where a c o m m o n factor (iy,/4jr) 3 h a s been removed. Yon a r e invited t o repeal t h e exercise for 7..\ where one must also allow for £ d e p e n d e n c e .
3 21 Differentiating eqn (3.174) with respect t o Q'2 and using eqn (:i. 171 ) gives: , Q2 ^°A|CK
• "
W?
1 m
= ,% + <»;(/*<> + / i t « s )
.01 ( I h \ # k
Pi Y •h+fhcyj
-¡h .«ÎM +
fto«).
= 0 . You a r e invited t o confirm a similar result for 7ïî,t in eqn (3.180).
3 22 T h e leading o r d e r //-function is given by eqn (3.171) with a modified ;i 0 - Using (>qn (3.179) we have gluons + gluinos: <|iiarks + sqnarks:
127r/?o = + 11 - 3 — 2 - 3 12*-//,) = ^ - 4 • i - 2 x 1 • ^ J n j
C o m b i n e d : 127T0o — 2 7
3«/,
where n j is t h e n u m b e r of complete families t h a t cont ribute.
3 23 At C ( o J t h e corrections t o t h e leading o r d e r gqq vertex - i < l s T ' , j Y a r e given by self-energy corrections to t h e gluon. given in Fig. 3.17. e<|n (3.134) and eqn (3.137). t h e self-energy corrections t o t h e o u t g o i n g q u a r k s , given in Fig. 3.20 and eqn (3.132). a n d t h e vertex corrections, given in Fig. 3.18 and eqn (3.138). R e m e m b e r i n g again t h e conventional factor ^ d u e t o t h e wavefunction renormalization. a n d collecting t h e respective t e r m s , t h e s u m of t h e divergent p a r t s is p r o p o r t i o n a l t o
- M
)
( « V * - s ' Y ^ ( - » . + a • - ( f - f ) f
- C r [iff - (3 + i ) m ] a ± i | | ) y . - ^ k t i g c r l - «
~ 0> +
< H
SOI.U I IONS
i C
\
,
+
«¡:i
S i i l e ,
- -r,..,,.') V . where we have e x t r a c t e d a c o m m o n factor i q ^ T " : ( o s / 4 f f ) A , . In t h e second line we used t h e Dirac e q u a t i o n to e l i m i n a t e t h e (tf | in). (<} - in) and j) (fi I " ' ) I (# — »*) terms. T h e resultant coelliciont of t h e divergence is t he leading term of t he Q C D //-function. e<|n (3.109). The divergence is cancelled by a d d i n g the c o n t r i b u t i o n s from t h e coiinlert.erins. eqn (3.149).
I Hat her t h a n calculate directly it is simpler to o b t a i n the t e n s o r describing 7*q q' from t h a t describing 7* -• qq using crossing, eqn (3.!).r>). T h a t is. s u b s t i t u t e (/'' •
M
( V I -
(
l ' ) X ' ( 7 , / l "> q') = e2eClNc
[,]„,,'„ + q'^q,, - (q • q')q„„) 'lY {1} .
I his has the couplings a n d colour factors restored. Next, we average over t h e incoming q u a r k ' s spins a n d colours, 2NC. include t h e o n e - b o d y p h a s e space factor d'I'i = 2itS{q r i ) using eqn (3.234) a n d divide by t h e conventional factor 1 ~ c 2 . finally, we e l i m i n a t e in favour of q„ + q^ a n d in favour of ]>,,/y t o o b t a i n ¿r>
f 'I'ImE-V(-;;,
'/;.»/ T
C(7,;q
q')
%
-q~ • to o b t a i n t h e expected tensor s t r u c t u r e . ConvoHere we used 2q • tf> lili ing this expression with t h e p.d.l'.s. e<|ii (3.228). the resultant e x p u l s i o n can he c o m p a r e d with eqn (3.3(i) t o give 2.eF,
,2fu(.r
(.r) = F ™ ( x ) = ,- £
and
F.\"'h)[x)
= 0
f=l-'i
l'i T h e identity
can easily be proven by using t h e definition of I he plus-prescription eqn (3.248) 011 t h e left-hand side. O n e o b t a i n s p1 -_ r - "'(I - *) / ' ? ./i>
1~
~ - //
" t ~ 4 t + 0 + ; 2 M 0 - *)
/ ' d ^ J L ./„ I - !/
St >1,1 l'I IONS
M.I
=
- z ) f \ i y l-^— .Ai V i -.'/
- V1 ^ i -.'/ /
=
- *) / dw(l Ai
- 2) = 2<5(1 -
Here, we h a v e used (1 +
+ w) = ¿(1 -
*)f
z).
2(> Using t h e hint o n e finds (1 - ••'-)
./.,
(1 - • ' • ) '
Jo -
-
j
y
-
^
^
o
-
^
-
'
-
B
+ O(c) .
Ill t h e second line, t h e coefficient of < is finite, a s can h e c o n f i r m e d b y Taylor e x p a n d i n g f ( x ) a b o u t x - 1. In t h e t h i r d line we recognise t h e d i s t r i b u t i o n f u n c t i o n f / ( l - x ) + a n d place t h e / ( l ) / e t e r m back u n d e r t h e integral u s i n g a ¿ - f u n c t i o n . Since f ( x ) is a r b i t r a r y t h e i d e n t i t y is e s t a b l i s h e d a n d we c a n c h a n g e - f >-> (. In a s i m i l a r way. a g a i n s t a r t i n g with n e g a t i v e r . o n e finds
( 111.I'• + • •• JII
-« [
d a : / ( . r ) - ^ [ ì + f l n ( l - : » : ) + •••] 1
./H
+ /-'
I"/
1
\
( l n ( I - a:) \
<5(1-a:)
Using t h e a r b i t r a r i n e s s of / ( . r ) a n d s u b s t i t u t i n g - e :? 27 T o o b t a i n t h e m a t r i x e l e m e n t s q u a r e d a p p r o p r i a t e l o t h e c r o s s i n g
IMï'g -
0(f2) .
Ina:
r t h e n gives eqn (:5.2(>:i). ' / ^ / / ' " ' ( " f g -> q q ) apply
= T r ( l } c 2 c 2 ( i / s p ' ) 2 T l - { r " T " } 2 ( l - c) (1
T h e n , i n t r o d u c e t h e variables
QH<1 • '!) V/y • '/
(
y ' iJ
-
(.'/ •
2f
S o l , H I IONS
*
(2po„i f
2
2
(I -
.
2
Kir.
and
217 •
w h e r e w = (I + c o s t ì * ) / 2 a n d ; = x/y. T h e v a r i a b l e a: h a s its usual m e a n i n g , eqn (3.3JJ). a n d y is t h e m o m e n t u m f r a c t i o n of t h e s t r u c k p a r t o n w i t h y" - yi>''. I h e n . use T r {1 } I. i n c l u d e a n a v e r a g e o v e r t h e I) - 2 = 2(1 - r ) t r a n s v e r s e - I ), gliion p o l a r i z a t ions, a n d t he N'~ - I colours, w r i t e T r [ T " T " } = Tr{N2 divide by t h e c o n v e n t i o n a l f a c t o r -ITTI - a n d i n t e g r a t e over t h e t w o - b o d y p h a s e space, eqn ( C . 2 1 ) , t o o b t a i n / ' < I < I , J S 1^(7*6 -
=
qq)l"
r v , 1 Potll ( \ ' 1 ( j j J r(T^T)
2 =
x /"doc-'o _ , . Jo 2as/r.
/ I»//" \ '
r
{(1 - 4 U + — I I 1 - '' V
-
V{ 1 - />)
4 J
1 ro^T)
; c
r
2
(l-0
f
F ( 1 — 2e) 1
2(1-t)2
4z(l-z)
c(l - 2t)
e
2f
\
(1 - 2r) J
The integral was e v a l u a t e d using eqn (C.27) a n d t h e result m a n i p u l a t e d using eqn (C.2. r i). T h e m a t r i x e l e m e n t a p p r o p r i a t e t o t h e c a l c u l a t i o n of
j t"(.'/) J I A q q ) = - i c W < T " , , ( , , ) { ó j ^ J L y , 2 + 7<7"(<7)f { (n-'i)('/-il) J I • '/
.'/ • '/ J
'(
As a p r e l i m i n a r y t o s q u a r i n g t h e a m p l i t u d e , in t h e s e c o n d line we h a v e used (i(i fi2 - <1 a n d s w a p p e d t h e o r d e r of „ a n d y„fj so a s t o b e a b l e t o use !?'•('/) = " a n d »('/)(? = 0, respectively. U p o n s q u a r i n g a n d s u m m i n g over s p i n s , using - i f " for t h e g l u o n , t h e d i a g o n a l t e r m s vanish since
I!h.M(Y"a
-
q q ) | 2 = 2e 2 e 2 (/;, v p' ) 2 T r { T " T " } — ; ' l ' r II- 'HI • <1 2 2 2 = 2C C (, A „P' ) T r { T " T " } ( 2 , , • ij)Tr {1} .
T h i s c o n t r i b u t i o n t o t h e l o n g i t u d i n a l c o m p o n e n t of t h e h a d r o n t e n s o r is free of singularities. N e x t , we i n c l u d e t h e a v e r a g e over t h e g l u o n ' s s p i n a n d c o l o u r .
S
TI
S
«livide by ili«' conventional lactor lT< a n d i n t e g r a t e over t h e p h a s e s p a c e using e« ii (C.22). since t h e i n t e g r a n d is a c o n s t a n t . T h e linai result is ti
= ~ >
f d'l' ^ IH,M(
,<
f,,
=
-
T
r
2-
'
G
F
I
H
«i« ) -
1 f i7r,r
zy
il
^
1
)
- Q r
V
- 1W
1(1
(l
- o
1 -e r 2-
+ m
ote that in any s u m m a t i o n over q u a r k flavours it should be r e m e m b e r e d thill Iliese expressions for li{ ) a n d p ' c o n t a i n t h e c o n t r i b u t i o n s from one flavour of q u a r k a n d a n t i q u a r k .
5 2S
sing e« ii (-'i.27-r>). t h e
cont ribut ion of a single q u a r k to - it is given by
I
=
I dzC>(l + 2)x f .a . ii ' iCy r> os — X dyy(y). I 1 Ju
i yit(u)
To decouple t h e two integrals, we lirst s w a p p e d their o r d e r , ,,'dc
d . l ' c . a n d then changed variables f r o m r t o ; = .r/ .
t h e fact that
V,
,,'d.r
/
—
In line two we use«I
is a p u r e plus-liinction so that its integral vanishes.
ikewise
1
in t h e MS s c h e m e most of t h e integral of C ' - ^ ' ' . e« ii (.TiTti). vanishes, leaving a n expression which e q u a l s C . ^ i n correction is p r o p o r t i o n a l t o t h e
the
IS scheme, «'qn (3.279). T h e
expression. S u m m i n g this result over the
a p p r o p r i a t e c o m b i n a t i o n of p.d.f.s then gives t h e one-loop correction t o t h e ( s u m rule (Altarelli el ui.
1! 7S: B a r d e e n el
S
il.. li)7 ). T h e final result including
III«' leading o r d e r t e r m is /«L*-
T h i s result is independent of the factorization scheme.
.
S
i
TI
ircct a p p l i c a t i o n of t h e lending o r d e r
x
c w
2
^ { / ' « I « +
C
G AI
2
4
C s
> ]
e q u a t i o n s , oqn (.'i. If)), gives
C
C
x [« . . . .
C H
zz[rW(z)+
l
S
ir)
? (?y.p2)
x I'd
f-«2
n
.
In t h e last e q u a t i o n we h a v e e m p l o y e d t h e p r o p e r t i e s of t h e Mellin t r a n s f o r m to d e c o u p l e t h e c o n v o l u t i o n , c.f. x . (3-28). Since t h e result s h o u l d hold for a r b i t r a r y g l u o u a n d s u m m e d ( a n t i ) q u a r k p.d.f.s. t h e i n d i v i d u a l coefficients must vanish. For t h e g l u o n , direct, c a l c u l a t i o n gives
= [ d
)+ 2
[ p ^
fA* 11CA -
p ^
)
+ ( I .(1
V2njTF 11 - JCA
l l f
) + ra(l -
) + Cm S([
)
-
)
+ (1 - ; ) 2 ]
2
+ C ^ T - T , VJ ATpnf
The coclficient of t h e ( a n t i ) q u a r k p.d.f.s is identically zero with t h e previously d e t e r m i n e d value of G,, ( = 3 2 .
,111 T h e only s u b t l e t y in c a l c u l a t i n g t h e a n o m a l o u s dimension is t o r e m e m b e r to treat c o r r e c t l y t h e 1 ( 1 — t e r m . T h i s gives
id-,)'
= CF
- J
[
+
~ l+
2( ~-
+
+
)}
K< )I,U I IONS
1 t h e r e s u l t , is 7, < ", ) (1) =
For n =
l\¡i|s<,M
P™
and P ^
= 0.
0 . I n <'<|ii ( F . 2 ) , ¡it o n e - l o o p l e v e l , w e
have
I n c a r r y i n g o u t t h e M e l l i n t r a n s f o r m it i s u s e f u l t o
introduce a ¿-function which leads to
/'cb-.r"-y^(x.„ J,i (>P, 1 - A
2
) = ^
[\\.rx"-lV(.r.,c)=
f\\xx"~l ./„
f \ \ z f\\,,H.r J,, ./(I
[\lzz*-lP$(z)
<>/'" 7II
!fz)P™(z)V(„)
f d y y " 'l'(//)
./„
>''$("•>'*)
-
./(i
=
I
Mere, we have V =
l>.p
7
<«
{U)
h ,
t
2n
) = K(,.p„)exp|
7
( / Q
["''
tln((/A2)
2«fio
—
h
i
^
<1/
^
^
j
j
Since 7q(<'1,(l) = 0. t h e q u a n t i t y V ' ( l . / i 2 ) = J j ' d x [
51 At one-loop order, using c q ns (.1.25)1) (.1.294) t o o b t a i n t h e a n o m a l o u s dimensions. t h e II 2 e q u a t i o n s a r e given by ,JJ_ / £(2,p2)\
_ » s f - t C r
'' <>H~ y
. + jf n f T , \
2n
( £(2.p2)\
, - § « , ! > J {
T h e s e e q u a t i o n s a r e diagonalized by t h e linear c o m b i n a t i o n s E(2 , p 2 ) + E(2.p2)-
eigenvalue eigenvalue
(I -(jjC>+§»/!>)<•» •
T h e diagonalized e q u a t i o n s can t h e n be i n t e g r a t e d as in Ex. (3-30) to yield E ( 2 . p 2 ) + <7(2.p 2 ) - ¿ ( 2 . / i f i ) +
and
SOLUTIONS
409
( •"
i • l >'• ' ,.•> t
2( /.• I he first, e q u a t i o n implies m o m e n t u m c o n s e r v a t i o n , eqn (3.3(52). T h e solution t o t h e second e q u a t i o n gives t h e r a t i o of m o m e n t u m carried by ( a n t i ) q u a r k s a n d gliions al a s y m p t o t ic scales. J„'d.r•*•£(•;•• p 2 )
„fT,, _
/¡'d.rx<7(.?-,p2)
"
2CF
;iUf
~ TtT
~ ' '
Above t h e i n q n a r k t h r e s h o l d g l n o n s c a r r y a p p r o x i m a t e l y half of t h e p a r e n t hadron's momentum.
12 In DEI.A. g l n o n s drive t h e evolution of t h e p.d.l'.s a n d s p l i t t i n g f u n c t i o n s can he a p p r o x i m a t e d by their s m a l l - : limits. T h u s , using t h e one-loop expression for in eqn (.'{.49) t h e glnon p i l l s evolution is described by •Uhtjx.Q2)
j
2
OQ
i)ln(
r\\y
77.% •
'
J
Defining LR = l n ( l / . r ) . L A = h i ( l n ( Q 2 / A 2 ) ) an<\ c = the solution, eqn (3.301), b e c o m e
DL„
a
2 * ^ 1 , 1 « ^ ) . / ,
I/.,•)<•) l n ( l n K / - 7 A - ' ) )
0LX
2Ca
0LR
I V
V
' • CA/TT.%.
this e q u a t i o n a n d
J
L „
2\fcLrL„)
l.l Near n = 1. eqn (3.291) gives
=
" —I
+ « + 0 ( n - I)
with
,; = - ^ - n 1, ^ (>
.
I n t r o d u c i n g t h e n o t a t i o n L s = l n ( l / . r ) a n d L„ = l n ( < i s ( p ~ ) / o s ( p 2 ) ) a n d u s i n g e
IVII
SOLUTIONS
1
P
«I» -,
gv
-V + X. rC-f-i
i =
27TÎ
2/i in
i
« - L r d n (/(/)„.//'-')cxpi 2 f f i Jr.. iV " /'o + ••• +
+
^
'
'
}
« S
1
•
M
(// - 71 o ) '
C,\L,,
ox|)<J . / — — L „ L, j
^•''«'Kïéfef)
2C '.i + /.• -I// - 1
« } -
»'0
J
lii line t h r e e wo e x p a n d a b o u t t h e sa / ( » *()/',)• Inspocl ion shows t h a t I he c o n t o u r of steepest descent has c o n s t a n t real part., II N{, \ i/. a n d evaluating the resulting G a u s s i a n integral gives t h e solution. T h i s solution a s s u m e s t h a t t h e r e a r e no poles in ()(n. //,",) t o t h e right of i/ (l . For e x a m p l e , if:riy(j".//¡"¡) ~ . r - ' . then «/(//./'o) ~ («• — f — a n d this behaviour would domin a t e r/(.r.//-'). T h e a b o v e solution applies also t o t h e full gluon a n d singlet q u a r k evolution but with a modified value o f / . ' = C . j / t i - f //y27V(C'..\ 2 C'i.-)/('.lC,\).
Ill T h e multiplicity is given by t h e II = 1 m o m e n t , so t h a t V) = 0 ( 1 , Q 2 ) = (N)
f
~ ex,, f - j - , / 2 C * y h y *aH(QJ)
j J
Higher order corrections t o this formula a r e available (Mueller, I'lNli). In anticip a t i o n of t h e result for t h e s h a p e w r i t e i
i r
=
•1%'S(Q2) '
2 IT
and
CA<>m-)
C = "" /n \
•2Ca nnj(p)
As in F x . (:{-:{:{) we find .rD(x.Q2)
i
n
^ <>X,,
d/l e x p
(
n 2(7-'
L
l
2a2
J
'
h —
(»»-!) I
( ¿ x - D
:;< n i i'i'ioNS
r, i
" t e r m into (lie e x p o n e n t i a l a n d c o m p l e t e d In line ( wo we have moved I In- ;r I lie square. T h e q u a d r a t ic (;/ I) a n d C t e r m s just c o n t r i b u t e t o tin- fragment,atioii f u n c t i o n ' s overall n o r m a l i z a t i o n . Higher o r d e r c o r r e c t i o n s distort t he s h a p e of this G a u s s i a n in ln(l/:i:) which b e c o m e s d o w n w a r d skewed a n d p l a t v k u r t i c (f'bng a n d W e b b e r . 1991).
11 T h e p h a s e s p a c e e l e m e n t is Lorentz invariant so we can work in a n y convenient f r a m e . C h o o s i n g />" = £ ( 1 . 0 , 0 , 1 ) a n d n" = ( 1 . 0 . 0 , - 1 ) w i t h k" (0. k± i . k_2> 0) <-(in (3.315) becomes
I
* = ((1
•
which h a s t h e on mass-shell const r a i n t .
8z
(1-2)'
A s s u m i n g a z i m u t h a l s y m m e t r y t h e gluon p h a s e s p a c e element t hus b e c o m e s d'Vy
,l(i T h e m a n i p u l a t i o n s for g iu Section 3.0.7:
„>r„T„Tl. /
1
d
1
dc
,
> qq a r e a direct a n a l o g u e of those for q —> q g given
u - f n
(
„.,,,
\ m + m \
(ii-
2 1 ( . ' V 2 - 1) . = Is F \} «H-fc x = A .." • a 2d • 11 1
x T r ( • • • [ » • (a - <j)(fj -]}) + (n •
2
Hs'F
xTr
¡y
h-k>—p—
• • ([z* + (1 - z f ] Ü + (2z - l)Jf_, +
= (A f 2 - 1 )2(,iTr
+ (1 - c ) 2 ] ^
^
T
r
••-J {• • - {, •. •}
\
Si il,H I IONS
•172
= ( A ' 2 - D2
M
"
•
In t h e final expression t h e c o l o u r f a c t o r lias b e e n a r r a n g e d suoli that t h e f a c t o r s of ( N 2 - 1) a n d 1 / N r c o m p e n s a t e for t h e a v e r a g i n g over c o l o u r p o l a r i z a t i o n s w h e n t h e i n c o m i n g gluoii is replaced by a n i n c o m i n g q u a r k .
i 37 lu t h e h a d r o n ' s C . o . M . f r a m e t h e colliding q u a r k a n d a n t i q u a r k m o m e n t a are - l ) / 2 . so t h a t Q" given by />'/ = x , ^ ( 1 , 0 . 0 . l ) / 2 a n d pi,' = x2^(1.0.0. />',' +1>!', ' i - r•,)/'!. T h e r e f o r e ( a s s u m i n g Q2 = S) Q2 = J'i x2S, = £ l n ( x i / a > j ) , which gives or r = . r , . r 2 . a n d / / " = ± l n | ( Q ° + Q:)/(Q° +!/ >_!/ = T a n d .r 2 = \ / ™ ( / ' i)- hi oqn (:i.."i:i0) c h a n g e variables from .eI = v / r e (.r ).i-| to (/. d.r/.r d//. to o b t a i n t h e result = <,<»>
[«..(v^c**)«.^-»): + ^ ( ^ » w ^ - » ) ]
c2.
5 3S T h e exercise is t e d i o u s 1 lilt not difficult. We f o c u s on t h e '/!,,%_, t e r m hi oqn (3.33-1) a n d a s s u m e a n a r b i t r a r y f a c t o r i z a t i o n s c h e m e . W o r k i n g t o C ? ( o J we find J d x ^ ^ S ( 1 - f ) + £
x (,,(:,)- g
f
, ( 1
+ <,(r)])
X > < f ) (C(*> h
»
)
(/ffUO
f
x ( « . m - g / f
- è./r
[->/'<;;»(r)ln g
E
- »« ( 4 ) ]
-
( c < * > [A. ( $ ) ] - *?<*>) / | ' I ( y ) (C<*> ( f )] - « f w ) ) } •
T h e triple integral in t h e second line is simplified using the following m a n i p u l a tion, f d x ^ x - A ô (l -
«/.„(.n)/„, ( y )
St )|.l I'liONS
= yd.ir,
.
f i r s t . we e l i n . i n a t e d x2 using - t / t , x 2 ) = (r/.i:, )ó(a- 2 - r / . r , ), t h e n we eliminated ; in favour of x 2 = T/(nz). T h i s t h e n gives t h e first t e r m in oqn (3.334). / d , , d . r 2 c 2 I !/,„(./•, )
2/'<;;»(f)h. ( J )
+
)
A, - In
+7hI,,(.'' 2 ) X : f=i/.<j
K
- '» ( 4 ) 1
^
-
'<Ï(Ï)))}
\ / ' / j
/ / j
Picking o u t t h e f/i,,
I
A t — In ( —
+ "£,(*) I < ( f ) +
< ( f ) ) l ,
which coincides with t h e original expression provided that = //.,.,(c)
n'f;(z).
- <(.) -
In t h e D I S s c h e m e flic finite t e r m s a r c /Y,1,"8 =
= /?, l , l s a n d / ? l ) l s = C ^ 8 '
Thus, t h e form of oqn (3.334) is t h e s a m e with f ^ ( x . / r r ) t h e new coefficient f u n c t i o n s
f{?ls(x./i'j.)
and
•l'I Slart.ing froin,
(7 + A')-2 - 7/?"1 follow e x a c t l y t.lie saine Nlep>. m. loi tlie q u a r k case. T h e équivalent, e x p r e s s i o n for r a d i a t i o n off a h a r d ( e r ) gluiiu in glven bv
M«*»(!l.k)
-- i g J ^ M i g I A-)\. » H »
'
f
ria *!! • K
+ *)*
SOI H i IONS
171
X |(A-
+ (2<J + h)„V,'\
~ (.'/ + 2k) l ,ìix l t \f'r(!i) h ''('(/>•)">' ,
whioh redi Kos tr:r I O(UJ) a n d n e g l c c t i n g 0 ( u ; ) t o r i n s in t h e i i i m i e r a t o r .
15 III T h e r e s u l t s follow by a p p l y i n g t h e d e f i n i t i o n s in oqn (3.339) t o a n arbit r a r y s t a l e ; for e x a m p l e , t2\-
f;; ( 7 * | . • - : q.j;
•••))=
T?jTJk\-
k: • • •)
U s i n g |i/. • I'/-': '/•./• fl< b) 'x Tj'j a n d |'/i. t>'. (/•>.(•:
<7-./) 'x
c a t i o n of t h e d e f i n i t i o n s gives ( f q + f„)|i7,/: < i J ) - x T ; - - T ; ) (t„ + r„ + fK)|r/.i; (7' Kl + '/"„.j -f
7. j: ;,.!>) -x
- 7£7?t - i
/ „ ^
//2.e: r/:t.
all of which a r e zero, e i t h e r m a n i f e s t l y o r d u e t o t h e c l o s u r e p r o p e r t y e q n ( A . 1 0 ) or t h e .lacobi i d e n t i t y oqn (A. 12).
3
II T o o b t a i n t h e leading b e h a v i o u r of a m a t r i x e l e m e n t ( s q u a r e d ) in I ho soft-glitoil limit, r e t a i n t h o s e t e r m s w i t h t h e m o s t f a c t o r s of f/'' in t h e d e n o m i n a t o r a n d least in t h e n u m e r a t o r . In t h i s i n s t a n c e , oqn (3.107). s u b s t i t u t e q • Q zs q • Q fa q • q K Q2/2 a n d Q" = q'' +
//„„(qqg) = =
r x H,iv(qq) (q •
/2)Vll„]
.
-I I T h e result follows f r o m f o u r - m o m e n t u m c o n s e r v a t i o n , in p a r t i c u l a r p _ c o n s e r v a tion w i t h (E - i>z) = ( p j 4- vr)/(E
+ p~),
In a chain of b r a n c h i n g s , since x =
Zi z-j • • • z„. we r e q u i r e c > .r a n d
thus
-min = •''• T o o b t a i n a n u p p e r b o u n d we work in t h e s t r o n g l y o r d e r e d limit ti,
/„, s o t h a t
•SOLUTIONS
t) < — — z 1 - z
175
; < 1
k — . t„ + ta.
I'lie u p p e r limit is largest w h e n !„• is smallest., s o t h a t we h a v e
[ * « » , 2...axl = [x, 1 - < < ( / „ ) ]
With
<,(/„) =
< €„-(/„) . I'" + '„ )
Naively, t h e requirement. t h a t q2L > 0 gives I), < t.Jz a n d /„< < / „ ( I z)/z. However. t h e first, limit d o e s n o t g u a r a n t e e s t r o n g o r d e r i n g . //, < ' „ . which t h e r e f o r e has t o lie i m p o s e d d y n a m i c a l l y .
T h e / - d e r i v a t i v e of In f l ^ . r . / . t s ) c a n h e r e a d off f r o m its d e f i n i t i o n . P r o c e e d i n g l o ll„(.r. /./_,). we o b t a i n 1
()
— ' T i r ' « / " . (•<••') +
<)f
i7r
ti—cc' '."CI 11» fu/!.(•'••')
f^ ' " d i i d t , ! )
=
J
,
.Is
w h e r e we h a v e used o(|ii (-1.1) t o e l i m i n a t e t ( ) f a / h f ( U . T h u s b o t h d e r i v a t i v e s a r e given by
t M-
J
H
=
<>t
¡>i
=
' - ^ " ' ( i c o ^ M I) .)_ I iih(~)—-rI /ii/ldJ".')
I'lirtherinore. H „ ( . r . / „ . / , ) |"l,,(x. t„. t.a) 1 so that ll„(.r././,,) = for all / by i n t e g r a t i o n of t h e a b o v e e q u a t i o n .
n;,(x.M,)
C o n c e r n i n g t h e i n t e r p r e t a t i o n , t h e f o r m li;,(.r./. f s ) satisfies t h e following e ( | u a t i o n for t h e p r o b a b i l i t y of p a r t o n
+ 61
/ , I n o res. r a d . ) = P(l — / v | no res. r a d , ) I
r>-'"'{n(\znJt,z)
4t
/r
\
e.f. e(]ii (4.14). T h e t e r m in s q u a r e b r a c k e t s gives t h e f r a c t i o n of p a r t o n s of t y p e ii a t scale / -(- St which did not c o m e f r o m a b r a n c h i n g h an' in t h e interval / t o t I- St. T h a t is. it is t h e f r a c t i o n of p a r l o u s of t y p e n which evolved f r o m / + St t o / w i t h o u t a n y resolved r a d i a t i o n .
R e f e r r i n g t o Fig. F . 3 a n d using t h e u n c e r t a i n t y principle, E x . (:t-2). t h e dist a n c e travelled by t h e i n t e r m e d i a t e p a r t o n p r i o r t o emission is given by
SOLUTIONS
'170
(/'. 1 /«•)/(/», I A) 2 « l>,/(2p, • A-) l / ( 2 u ; ( l - «osfljjt)). Likewise, t h e s e p a r a t i o n between p a r t o n s i a n d j at t h e t i m e of emission is given by
I
sin 0,k
t a n 0,
<
2 tan
2 ( 1 - c o s 0,k)
0ik
< t a n 0,J
0ik < Ojk
F t C . F . 3 . Tlie ralliât ion of a soft gluon from t wo colour-connected partons
I I Using t h e angles defined in Fig. 4.2 we p a r a m e t e r i z e t h e directions of t h e hard p a r l o u s a n d soft, gluon using n , = (0.D.1). n } - (sinW^.O.cosi/,,) a n d n * (sin 0,k cos O. sin 0,k sin (p. cos Oik), which implies cos Ojk = rij • 7i k = sin 0,j sin 0,k c o s 0 + cos 0,, cos 0,k • T h e c o r r e s p o n d i n g f o u r - m o m e n t a a r e given bv />, a n d A = u / ( l . n j t ) so t h a t eqn (4.22) b e c o m e s (1 - ¡1, cos 0ik) - (1 - J f ) 1 - fij COsOik
(
E,(\.
if,ii,).
Pj = Ej(i, f i j i i j )
(1 ~ fk<<>*<>,j) - ( I - / / , cos Oik) I - f i j cos Ojk
3i
J, — cos Oik
cos Oik — fij cos Oij
20 k
1 - (ii cos 0^
1 - 0j <'os Ojk
which is eqn (4.23). T h e a z i i n n t h a l a v e r a g e of t h e first t e r m is trivial, as it does not d e p e n d on . T h e second t e r m d e p e n d s on <j> via cos Ojk- T o t reat this second t e r m convert t h e integral from 0 to 2/7 into a c o n t o u r integral a r o u n d t h e unit (c + z~l)/2. and thus circle using ; = e i c i , giving d = — i c " ' d c a n d c o s 0 /•-'" dç> ./o
I
_
f dz_
2 ^ (A - B c o s 0 ) ~ J
2
27\ (2Az
B(z+
_i_ 2
- B - Bz )
~ ~B f
- z - ) " v/,4-' - B~ '
f
d: (z - z_ )(c -
;+)
S( >1.11'l'IONS
where we have i n t r o d u c e d . I (I , ij ci is 0,j c< >s 0,k) > 0 a n d B !ij sin (),, sin 0 i k . I he poles o c c u r at (.4 ± - B-]/B of which only is within ( lie unit < irele ( z + ----- 1 / ; _ ) . it h a s residue 2tti/(Z- z+). Finally. A* - B- = ( I - ¡ij cos 0 o cosOm)' 2 - $ sin- 0 U sin" 0 i k 2 f i j cos 0, , cos 0,k + 02 [cos- 0,j (1 - sin- Oik) - Sin 2 0tJ s i n 2 0,k}
1
= I - cos 2 Oik + (cosOn- - / ? j C o s f l , j ) 2 - .'ijsin 2 Ojk = (1 - fi2) s i n 2 0,k + | cos Oik - fij cos 0,j j2 which leads t o t h e q u o t e d result, eqn (4.25). for t h e aziinnt hal average.
We proceed in e x a c t l y t h e s a m e way a s for t h e general case: -./ •./» = - f i • tjWij
- t , • t,W„
- tj •
t,Wji
= t , - ( f i + f , ) \ \ f + (Tj + T<) • TjWlp f~IF'0 T
2
"
T
- f , • f,IF,,
- tj -
fiWji
- • f<
W f f - Tj • f , [(IF«/) - IF, 1 /') + II î/(<)
= CiWW-ti'Ti
|IF),
II
JI
I Cj wjp' — f j
f t W)ï
Next, for t he t e r m s in s q u a r e brackets, we replace p , a n d ¡>, by ps obtain \ j • ./t = C,IF,';' + c X f - f t i 4 T j ) • t ,
where in t h e last line we have i n t r o d u c e d f s
, H f 1 p, ) pr
and
+ H-<')]
- t , I 7) = -7',.
Within t h e braces we recognize t h e coefficient of ln(l - ; ) as eqn (3.173). Now using t h e lowest o r d e r expression for t h e r u n n i n g c o u p l i n g we find ft,[(l
- z)Q2}
= ns(Q2)
~ 0o hi( 1 - z)«2(Q2)
I- O ( o ; ' ) .
so t h a t m a k i n g t h e a r g u m e n t (1 - z)Q2 in t h e L O expression for n s giv<-s t h e ln( 1 z) contribut ion of t he N L O t e r m . Likewise, using fC'A(iu A m c
=
c x p
{
- 3tt 2 ) - 20T,.nf\
3(11 C
A
- 4 T m
f
)
, )
gives t h e c o n s t a n t p a r t of t h e N L O t e r m . A similar result, holds for t h e g — gg s p l i t t i n g f u n c t i o n . T o g e t h e r t h e v m o t i v a t e t h e use of t r a n s v e r s e m o m e n t u m
soi.M
17«
IONS
a s ilio a r g u m e n t of o s . In a carefully f o r m u l a t e d coherent shower, t h e a b o v e relation allows : \ M S t o be m e a s u r e d f r o m AMC by using seini-inelusive. high-.r d i s t r i b u t i o n s ( C a t a n i
I 7
First we c h a n g e variables f r o m (x1,-1:3) t o (2 = .1:3,/."' = (1 — X | ) ( l - x;i)-s',ii|>) in ei|ii (-1.3(5) to o b t a i n
'.-i
>
1 — Z
Z7Z
I — Z
A"7
with .i | - 1 - A , 2 / ( l — ;).s,iip. For emission off a q u a r k " — - a n d t h e /'¡¡¡'(c) s p l i t t i n g function is i m m e d i a t e l y a p p a r e n t . For emission oli a gluon n 3 but we must be careful lo include also t h e emission olf t h e n e i g h b o u r i n g , colour c o n n e c t e d d i p o l e w i t h ? —• 1 — 2. which t h e n yields 1+
+
1 1 ( 1 - 2) : t
=
9
1
+
+ 2(1-2)
1 - 2
I S T h e i n v a r i a n t m a s s s q u a r e d of t h e first d a u g h t e r d i p o l e is given bv .s-1•> (p| + 2 2 2 2 f>>) = (Q P3) =
l'I
=
.'/ = [.'/i + l n ( v ^ / f c i x ) ] / 2
for t he I 2 dipole a n d l..(A-1/A)-2/=
lu(Js^/A)
hi(A"x/A) + y/ = lu(A'i_/A) + i/i
A-2 = e + . ' " A - I i v / S ^ II = [//. -
ln(y*äip/A'ix)]/2
for t h e 2 3 dipole.
I !) We will a s s u m e t h a t it is p a r t o n I which b r a n c h e d so t h a t Q 2 = s ( 2 a n d I if.'; + E>)2(1 - cos612) = (E\ -I- E2)2.sl2/(2EiE2). R e f e r r i n g t o F x . (4-8). Q2 =
S< U N I IONS
A /S-,;,,, HO s
(
7A )
( (A
. = x3 S^ 2 + c~v) 2. T 1
2
R /;—
,
(A
A)
= ,
(
,s, 1 2 = . , =
—SII
\2
(
A
+. A
)( +
+
)A s,. ,,
2
.s
)(
+
,,.
s )
s
s
s . T
s
. .
p,
s
s s
if
t)'-
= { \> +
.
.. .
)2
(ph
s s
plt =
s ss
, . A
s
= ( , + , . T
s
s
.
)
s
s
s.
2
2(1
s
A
T
-
2E-"A_V^I-
(
s
s (£,
s
2
2(1 +
s s
2. s
s
. T
s \
=
t
,
7 A + A) .
.. . (4. 7)
, + 2 -i) = ( +
I7!l
s
,
0
pt, m.
,
, =
s, s s = 14 T
t
in - 0 .
.
2
8
, .s = ( u + in)2 - p2 = ( l
2
) + 2
h
m 4
2
= 2(
u
m + in2)
.
s , = s
104 178 T
. T 7 =
As
- in
2m
toi
s
u
=
7
s
s
+ , \s -= = — - » 7463 . S 2m
.
s , ,
s
s
,
s
0.1
.
12
T
s
, ( )
s s
s
s s
s
.r s
x' ,
m=(;;-) • T
,,( )
s
s s
,
.11
low
.
its displacement. In the thin-lens approximation there is no displacement at the ua«lrupole. and one finds
*
*
C
4
f
f
~
]
A d e f o c u s i n g <|iiadiupole is d e s c r i b e d by t h e s a m e m a t r i x w i t h a n e g a t i v e value for t h e focal lengt h. T h e c o m b i n e d s y s t e m of a f o c u s i n g a n d a d e f o c u s i n g (|itadril pole s e p a r a t e d bv a drift s p a c e is t h e n o b t a i n e d by t he p r o d u c t of t h e r e s p e c t i v e matrices, giving
I
U
=
-
m i O = (
[
/
- \ ~
H
f
l
h
,
, )
\
T h e e n t i r e s y s t e m h a s a net. f o c u s i n g effect if t h e lower left field of t h e m a t r i x is n e g a t i v e , t h a t is. for
/> - h
-
T h e t r a n s f e r m a t r i x for a s e q u e n c e w h i c h s t a r t s w i t h a d e f o c u s i n g q u a d r u p o l c a n d e n d s w i t h a f o c u s i n g gives t h e f o c u s i n g c o n d i t i o n /•_> - }\ < /.. T h u s , s e t t i n g /1 h yields a n a c c e l e r a t o r st ruct lire w h i c h f o c u s e s t h e b e a m in b o t h t r a n s v e r s e dimensions.
{ In a h o m o g e n e o u s m a g n e t i c field t h e m o m e n t u m /> = |p{ a n d t h e r a d i u s of c u r v a t u r e r a r e r e l a t e d by t h e c o n d i t i o n t h a t t h e L o r e n t z a n d c e n t r i p e t a l force are equal. •j c\v x / i | = cv = . t h a t is. p = ( r . H e r e v is thi' velocity of t h e p a r t i c l e . T h e m o m e n t u m , o r m o r e precisely t h e r a t i o p i t h u s c a n b e m e a s u r e d f r o m t h e r a d i u s of c u r v a t u r e of t h e t r a j e c t o r y in t h e m a g n e t i c field. T h e m i n i m u m i n f o r m a t i o n n e e d e d t o d e t e r m i n e r a r e t h r e e p o i n t s on a s e g m e n t of t h e circle. For s i m p l i c i t y let's a s s u m e that t h e p o i n t s a r e e v e n l y s p a c e d , s u b t e n d i n g a n a n g l e 0. a n d t h a t t h e secant f r o m t h e first t o t h e t h i r d m e a s u r e m e n t h a s t h e l e n g t h L. It follows . sni - = —
2
mid
*= r
(
1 — cos -
2r
2
w h e r e s is t h e s a g i t t a of t h e t r a j e c t o r y . W i t h r 3> /- o n e linds
H
'
-
^
H
-
t hat is. a n e x p r e s s i o n w h e r e t h e r a d i u s of c u r v a t u r e is g i v e n a s a f u n c t i o n of t he observables . and T h e length is fixed t h r o u g h t h e p o s i t i o n i n g of tlx* d e t e c t o r
SOLUTIONS
IS I
elements, s is I lie resnll of « position measurement which can he assumed to have a Gaussian error, l'ho uncertainty of \/\> is thus proportional to the standard deviations of s. and standard error propagation yields const. i
ï
)
<
)
dp — ~ p . I>
—»
=
A particle with charge i¡ and momeutuiu p entering a magnetic field with flux density B feels a Lorentz force which dolleets it perpendicularly to its velocity and the direction of the field. Denoting the unit vector in the direction of the particle by n. the direction of li by b and p |p| one has /i • (v x B) = — =
or
d.s-
(¡13 (n x b) = / i ^ .
d.s
Hero v is the velocity of the particle and .s the longitudinal coordinate along its trajectory. In case t he magnetic field is ort hogonal to v the cross product in the above expression is again a unit vector and the deflection angle q> is obtained by (ln t A = -l — l- = —d.s
g i•v i•n g
n, <, / > =<1- / [ Bus
.
W i t h <j = zc. w h e r e t is t he p r o t o n c h a r g e , p in G e V a n d | / i d s in T m = V s / i n . o n e a r r i v e s at t h e c o n v e n i e n t e x p r e s s i o n
(Jg,/.s)/(T.n) =
0
-
U
p/GeV
•
T h u s , f o r a field i n t e g r a l / Btls l.f> T i n t h e d e f l e c t i o n a n g l e of a 100 G e V r pion is o n l y -l.. > m r a d . In o r d e r t o d e t e r m i n e t h e c h a r g e of t h e pion w i t h a s i g n i f i c a n c e of t h r e e s t a n d a r d d e v i a t i o n s o n e h a s t o m e a s u r e t h i s a n g l e w i t h a precision of l . f j m r a d . G i v e n t wo p o i n t s a n d a lever a r m of L 1 in. t h e a n g l e of 0 - 4 . 5 m r a d l e a d s t o a t r a n s v e r s e d i s p l a c e m e n t of I) = I, • = 4 . 5 111111 b e t w e e n t h e t w o c h a m b e r s , w h i c h h a s to b e m e a s u r e d wit h a n e r r o r of n/> = 1.5 111111. T h u s t h e r e q u i r e d p o s i t i o n r e s o l u t i o n ¡11 each of t h e t w o c h a m b e r s is a = 1.5 111111/ \ / 2 = 1 n u n . I g n o r i n g t h e s m a l l l o g a r i t h m i c c o r r e c t i o n , t h e p r e s e n c e of m u l t i p l e s c a t t e r i n g leads t o a G a u s s i a n s m e a r i n g of t he deflect ion a n g l e , ^(«¡>)MS =
1,U>
pc
— V X
.
w h e r e .V is t h e t h i c k n e s s of t h e sea Merer in u n i t s of r a d i a t i o n l e n g t h s , f o r ¡i = 5 GeV one has )i> = 0 . 0 0 1 5 f r o m t h e c h a m b e r r e s o l u t i o n for A' « (1.3. G i v e n t h i s a m o u n t of m a t e r i a l in f r o n t of t h e s e c o n d
St >1.11 I l< >NS
chamber. the momentum resolution is limited I v multiple scattcring lor p r IcV and by the chamber resolution for larger momenta. I lic above example illustrates how t lio combined clfect of multiple scattering and chamber resolution on a momentum measurement can be parameteri ed dding both contributions to the uncertainty in the deflection angle in uadraHire yields
*
2
( ? )
=
m j
=
t o r * *
i
^
with p in units of CeV. ids in Tin and general structure is of the form (7-2 ( -M
V/v
\
a
l
+
( ^ r )
x
)
'
in units of radiation lengths. The
l>2 . ~ II 2 -fI —
i>-
where t lie constant term depends on t he spatial rcsolut ion of t he t rac ing system and the p-dcpendcnt term on the amount of material built into it.
Inside a calorimeter the energy of a primary particle is distributed over secondaries. The readout system then yields a signal which is proportional to the number of those secondary particles. ince the number of particles created in a shower is sub ect to oissouian fluctuations one has \
d.
v V
\~
or
s
.
The relation between particle mass in. momentum p and velocity c is given by i
/'
^ p - + in-c-
'
The dilference in the arrival times of a I CleV c pion and a aon travelling over a distance of I. = 2 m is thus
separation at the level of three standard deviations thus re uires a time resolution around rr, 2. ps.
I.I
I
V I he transfer matrix formalism iulinduced earlier to describe the beam transport system of a particle accelerator can e ually l e applied to conventional optics. Here, we are dealing with Cliercn ov light emitted from a relativistic particle, which falls onto a focusing lens with focal length and is viewed at a distance behind the lens. t the position of the lens it can be described by state vectors (.r..r')' and ( . ') which specify the transverse position (x. ) and the directions x' d.r d and di dr of the photons. The resulting image can he determined by multiplying the state vectors wit h the t ransfer matrices of the lens and that of a drift space of length . For t he . -direction one gets
an analogous result is obtained for . The image of the photons is then given by the coordinates X / . I / I ) = ( f x ' . f . For a particle travelling along the optical axis of the imaging system, the photons are emitted on the cone (x'.i ) sin 0 cos ; . sin 0 sin . and the image is a ring around a centre which corresponds to the direction of the particle. The radius of t he ring is proportional to the product s i n f l . that is. given the radius and the index of refraction of the radiator medium one can extract the velocity of the particle.
I
T(M\
m 1
e have to show that r
i/i-fl
I P. •
£ ,
max n
with ?,/
r/i
l
E l vr .
it 1 E I ,-I i=i
T
m
n
l
max , '=•
if one of the (iii 4- I particles energy vanishes, or if t wo of the (m 1 particles are collinear. et us consider the first, case. For niassless particles we have to study the situation - pt I . For simplicity we choose -- in + 1. ow il is easy to see that the numerator as well as the denominator show the expected behaviour, namely
£ iPii = £ ip-i + i ^ i »=1 i=i m-fi
E
if-1
m
£ IP. • n ' l ^ L \ >> i=i i=i
n
'\ + I
»'I - £ i=i
t ,
n'\
for p . I. ince now both denominator and numerator are the same as for 1 . it is clear that I lie maximum under variation of n luus to be found for
I.SI
SOI.H I IONS
tin- sunn direct ion max as hi lie case l particles. This optimal direction in H 11 generally referred to its the Thrust axis « . o in this limit we have i
),
xt we have to test the collinoar safety. so we have to study the situation ecause of lh - properties of parallel vectors, ) .. et us call p ) -I p).\PJ • N'\
+ |PK
n'\ = | p •
ri|
,
\ j\ + \p -\ = IPl
•
it follows again that the denominator as well as the numerator reduce to the , u 1 -particle case, therefore .
The definition of rapidity is 1
.
n hi
+ - p.
ow let us apply a orcnt . transformation to the particles momentum, in particular a boost, along the c-axis with velocity . If the four-momentum in the original then the transformed momentum ( '.p'r.p'.p' ) frame is p = ( .pr.pv.pz). is £' = -,(
with v
-
ip.)
1
. p' = pv
. p' ='i(p--
ii )
.
From the above e uations it follows that
£' + />; =7(i-
Thus the rapidity
. p'T=px
+
• F'-P\
=->(1
+ i)( -p ).
in the boosted frame is
that is. the boosted rapidity is the same its the original one up to a function of i. only. The difference in rapidities of two particles is invariant under orcnt . boosts, since f ii cancels out iu t he difference. Ident ifying t he et directions with the directions of primary scattered partous. it follows that, one actually does not need to now the boost of the two-parlon system along the c-axis in order to infer the C.o. . scattering angle.
e start our discussion by defining the relevant momenta in lie laboratory frame, where the c-axis coincides with the proton s direction of flight. Thus the proton has four-iiiomeiitiiin ,,, . I.p. . and the exchanged virtual photon al , ,-nl . I , with e ll f I. s a first has the four-momentum step we apply a boost to these momenta in order to transform them into the rest frame of the proton. uch a transformation exists, since the proton has a finite
MOM J I'lONS
mass />-' ¡\l~ > (I. I'll«' necessary boost, velocity ii\ is found by the requirement. // 0. where />'. is t lir '.-component of the proton's moment iiid in its rest frame. o = p\ = 71 (pz - 01 £p) => ih = Pz/Ep € 1-1.1] , wit 11 Vf ( 1 — j } 2 1 . Applying this t.ransformat ion to hot It part icles, we find new lour-momenta p'z = (A/.0.0.0) and q = (R/O. qr.q,r IJ:) = ( !h = q'Jq'z 6 | - 1 , 1] .
The last statement is true because of in')'2 = (
< o => \q'0\ < Iv'J .
Let us briefly discuss t he sign of the photon momentum and t he last boost . If we start out with the proton moving in the positive and the electron in the negative ¿-direction, then after the boost into the proton's rest frame the photon will have a negative ¿-component of its momentum. < 0. The new proton momentum is p" = 0,0, —~f>ii>M). This can be rewritten using q2 , _ , , __, ('/;,)- = 12 m 2 (
,
,
"
I'/'I
In t he last. e(|iiation we have introduced the delinition u = E"x — E"">. which is the energy loss of the scattered electron in the proton's rest frame. So. linally. we liud the proton four-momentum as
where t he delinit ion of the variable x = Q*/(2Mv) from Section 2.2 has been substituted. For the following considerations we restrict ourselves to the parton model. In this model the proton is built out of partons. each parton / carrying a fraction x, of the proton's longitudinal momentum. The actual scattering process between the electron and the proton is described as incoherent scattering between the virtual photon and one of the partons. As was shown in Section .'{.2.2. to leading order one has ./:,• = x.
s< >I,I H O N S
(l-.v)(?/(2i) target
vV
+Q/2
< * v
-Q/2
\ current
F i e . F . 4 . C o n f i g u r a t i o n of longitudinal m o m e n t a in t he Breit. f r a m e of references
Now we see that in t h e Breit f r a m e of reference, t o lirsl a p p r o x i m a t i o n , t h e s c a t t e r i n g process o c c u r s between a p h o t o n with m o m e n t u m q" = ( 0 . 0 , 0 . 7 " ) = ( 0 , 0 . 0 , - Q ) moving in the negative z-direotion. a n d a (inassless) p a r t o n with m o m e n t u m
moving in the posit ive ¿-direction. T h e z e r o - c o m p o n e n t is o b t a i n e d front (/>JInr.j„) = 0. T h e s c a t t e r e d (outgoing) p a r t o n has to have m o m e n t u m
PU
= />;:,„,„+
•
T h i s c o n f i g u r a t i o n of m o m e n t a is represented graphically in Fig. F.-l. T h e h a d r o n s p r o d u c e d from t h e o u t g o i n g p a r t o n can have a m a x i m u m longitudinal moment u m with respect t o t h e p r o t o n of r,Q. a n d all particles c a r r y i n g negative longitudinal m o m e n t u m have to s t e m f r o m t h i s p a r t o n a n d lie in t h e so-called cariv.nl hemisphere, a s depicted ¡11 Fig. F.5. T h e o t h e r p a r l o u s in t h e p r o t o n , which d o not p a r t i c i p a t e in t h e s c a t t e r i n g , obviously have to c a r r y longit udinal moment inn (1 - x)Q/(2.r). T h e r e f o r e , all t h e h a d r o n s p r o d u c e d out of this remaining system will have m o m e n t a within the so-called target hemisphere, that is. positive longitudinal m o m e n t a with respect to t h e p r o t o n ' s m o m e n t u m , with t h e limit set by p¡""* = (1 - x)Q/(2x). a s shown in Fig. F A
SOI 11 I I O N S
IS7
I'lC. F..r>. T h e c u r r e n t a n d t h e target h e m i s p h e r e for t h e Breit f r a m e of reference. T h e variables a r e t h e m o m e n t u m t r a n s f e r Q, t h e Bjorken scaling variable .r a n d t h e longit udinal ( t r a n s v e r s e ) m o m e n t u m p, (pi,). F i g u r e from ZKl.'S Collab.(1999c).
For inassless p a r t i c l e s energy m o m e n t u m conservation gives t h e following constraints: E
h\, + E,, = E'c +
<> •
E
v ~
<•
= E'e cos 0q + E a
E„ cos 0„ .
(F.5)
where t h e s u m r u n s over all h a d r o n s a iu t h e linai s t a t e . F r o m t h e r e we derive 2Ee = £
E„ (I - cos«,,) I- E'„ (I - cos0„) .
(F.ti)
a Now. we deline £ = E„ (1 - c o s 0 „ ) . a n d insert. T. = 2E(. - E'.{ 1 t h e expression for y o b t a i n e d from t h e electron m e t h o d , e q n (0.11).
e o s f l c ) in
The t r a n s v e r s e m o m e n t u m iu t h e initial s t a t e is zero, therefore, the linai s t a t e h a s to sat.isfv h ;'. sin (),. + Px J, = 0 .
I \ J, =
E„ sin 0„ .
( F,S)
Si a r t i n g again from the expressiou in t he elect ron mctliod. e<|n (0.12), it is now easy to express Q~ in t e r m s of t h e h a d r o u observables:
1 _
.'/
1
- Uh
I he expressions for 1 he sigma inethod a i e siniply o b t a i n e d by replacing t h e f a e t o r 2E,, in oqn (F.7) by eqn (F.6), a n d replacing by i/v in t h e expressiou for Q - .
IMS
M M U | IONS
>;
m = v£ +. rE', v, ,(1
,
TTT • Qi: -
- cos0„)
ErJ sin2 o„ L • I - '/>:
{F.lffl
Filially, w h e n g o i n g t o t h e p a r t o n m o d e l . I lie st ruck q u a r k c a r r i e s m o m e n t u m a n d e n e r g y .r/v,, (/?,,) in I lie i n c o m i n g ( o u t g o i n g ) s t a t e , a n d c o r r e s p o n d i n g l y t h e n o n - i n t e r a c t i n g s p e c t a t o r q u a r k s (1 — x)Ev. S i n c e s p e c t a t o r q u a r k s d o n o t pick u p a n y t r a n s v e r s e m o m e n t u m , t h e energy- m o m e n t i n n c o n s t r a i n t s t u r n out t o he xEv xEp
-)-(l-
+ (1 - x)Ev
•')£„
+ & = E(l + E'c + (1 - x)Ev
- Ev = E
E'Qsin0C. = — E()sin0,t
(F. 11) ( F . 12)
.
(F.l.'l)
C o i n h i n i n g ec|ii ( F . I I) a n d c q n ( F . 1 2 ) . a n d t a k i n g t h e d e f i n i t i o n of E f r o m e q n (F.ti) w e a r r i v e at £ = £•<, (1 - cosfl,,) .
(F.ll)
F r o m e q n ( F . L I ) we o b t a i n
"
sill 2 »,,
sill 2 «,,
V
B y c o m b i n i n g e(|iis ( F . 14) a n d ( F . I">) we a r r i v e at. a q u a d r a t i c e q u a t i o n for // COS0q. v2 (f + ri)r/ - 2r/ + ! - « = <), a = — j - . (F.lti) T h i s e q u a t i o n h a s t w o s o l u t i o n s , //i
- 1 and
;/•> = cos0 < t -
P~ , t
E2 ^
(F.17)
which is a l s o d e f i n e d a s t h e h a d r o n i c a n g l e c o s 0 | , .
I
Let us u s e t h e n o t a t i o n of Fig. 2.1. T h e i n c o m i n g l e p t o n . a p o s i t r o n in F i g . F.ti, h a s f o u r - m o m e n t u m /. t h e i n c o m i n g a n d s t r u c k q u a r k , a d - (ii-) q u a r k in Fig. F.ti left ( r i g h t ) , h a s t h e f o u r - m o m e n t u m p,,. a n d t h e o u t g o i n g l e p t o n . a n ant ¡electron n e u t r i n o iu Fig. F.ti. c a r r i e s t h e f o u r - m o m e n t u m I'. W e h a v e l e a r n e d t h a t t h e deep inelastic lepton proton scattering can he described as an incoherent sum o v e r e l a s t i c l e p t o n q u a r k s c a t t e r i n g s . Since we will neglect all m a s s e s , we h a v e t o c o n s i d e r a n elastic s c a t t e r i n g of t w o m a s s l e s s p a r t i c l e s . F u r t h e r m o r e , let us c o n s i d e r t h e s c a t t e r i n g in t h e C . o . M . f r a m e w h e r e I = p ( ) . a n d t h u s E = E,t
SOLUTIONS
F i e . F.(>. Configurai.ion of m o m e n t a a n d spins in a positron q u a r k s c a t t e r i n g process
a n d E = E'. T h e C.o.M. s c a t t e r i n g angle 0'. of t h e Icptou is then given by t h e following r a t i o of l.orcntz sealars: p„ • I'
E,.E'
p,,-/
E,,E-pn-l E2(\ =
- p „ • I'
EE'
+
EE+
Teos0*) 1 og->— = 2(1+
11' 11 /JM
0= ' " f • 2
( (,s
T h e m o m e n t u m t r a n s f e r is defined as --I). where p is t h e p r o t o n m o m e n t um. T h e i n c o m i n g q u a r k carries a fract ion r of t h e p r o t o n ' s m o m e n t u m , t hat is. p ( , = .17». Using t h e s e relations we find COS — = -¡—r
2
y,, • /
= J
—j = I p,, • I
r = 1 - y . up • I
Let us now consider t h e specific case of a charged current i n t e r a c t i o n as depicted in Fig. F.G. which o c c u r s via t h e e x c h a n g e of a \Y + . R e m e m b e r t hat only (right.)left-handed ( a n t i ) p a r t i c l e s p a r t i c i p a t e in a weak i n t e r a c t i o n . T h i s is indicated by t h e small arrows showing the spin vectors which a r e ant ¡parallel t o t h e m o m e n t u m for particles a n d parallel for ant ¡part icles. T h e left, figure shows a positron s c a t t e r i n g off a d o w n q u a r k , leading t o an o u t g o i n g a n t i e l e e t r o n neut r i n o a n d a u - q u a r k . T h e t o t a l spin of t h e incoming s y s t e m is ./ = -1-1. A n g u l a r m o m e n t u m conservation requires tin 1 s a m e t o t a l spin for t h e o u t g o i n g s y s t e m . From spin a l g e b r a we know t hat t h e a m p l i t u d e ( p r o j e c t i o n ) of t w o spin-1 s t a t e s at a n angle 0 is given by (1 + c o s f l ) / 2 = c o s - ( 0 / 2 ) . Therefore* t h e cross section for c + d s c a t t e r i n g is p r o p o r t i o n a l t o cos' , (tf < "/2) = (1 — y)~. O n t h e o t h e r h a n d , if t h e p o s i t r o n s c a t t e r s olf a u - q u a r k . Fig. F.ti (right ), t h e n t h e t o t a l spin of t h e
I'll)
HOI.H I IONS
s y s t e m is ./ distribution.
7 2
For the the scat
0, a m i t h e o u t g o i n g p a r t i c l e is e m i t t e r ! w i t h a n i s o t r o p i c a n g u l a r
simplicity, lei us n e g l e c t t h e p r o t o n m a s s a n d a l s o t h e c o n t r i b u t i o n f r o m l o n g i t u d i n a l s t r u c t u r e f u n c t i o n , t h a t is, /?/, = (I. F u r t h e r m o r e , we will use n o t a t i o n k - ( M \ y / ( Q ~ + M f v ) ) 2 . T h e n t h e n e u t r i n o p r o t o n d e e p inelastic t e r i n g c r o s s sect ion a c c o r d i n g t o e q n (7.1-1) is given by ,\2„l']>
r '2
w h e r e we h a v e used t h e d e f i n i t i o n Y~ = I ± (I i/)~. F r o m s i m p l e c r o s s i n g s y m m e t r y we e x p e c t t h a t t h e c h a r g e d c u r r e n t c r o s s s e c t i o n for e + p s c a t t e r i n g is of t h e s a m e f o r m . I h a t is,
=
d.r
-17T.7*
fv+ f f
I
+
Y
-
*
^
1
•
We a s s u m e t h a t t h e l e p t o n b e a m is fully p o l a r i z e d , t h a t is. all p o s i t r o n s a r e r i g h t - h a n d e d . Now we shall find out how t hese s t r u c t u r e f u n c t i o n s a r e r e l a t e d t o each o t h e r . First , we insert t h e l e a d i n g o r d e r e x p r e s s i o n s for /-.' IJ in t e r m s of t h e p a r t o n d i s t r i b u t i o n f u n c t i o n s , given by e q n (7.1(i): ,|2_I'|>
_____
=
/
'"J
\(
(u(x)
+ c(.r))]
.
T h e n , we c o m p a r e t h i s t o t h e e x p r e s s i o n for t h e e + p c r o s s s e c t i o n in t e r m s of t h e p a r t o n d i s t r i b u t i o n f u n c t i o n s , given in e q n s (7.ti) a n d (7.7):
da'
AQ-
= —>>-[ 1 («(*)
+ Hx))
+ (i - 1 i f (
] .
F r o m t h i s , t h e r e l a t i o n s F'l '' = F a n d :r/;':'t '' = —xF'i'' c a n b e d e d u c e d . T h e i n v e r t e d (1 //)- b e h a v i o u r w i t h res|)ect. t o q u a r k s a n d a u t i q u n r k s c a n be u n d e r s t o o d a l o n g e x a c t l y t h e s a m e lines ¡is d i s c u s s e d in E x . (7-1). T h e e + p s c a t t e r i n g is discussed t here. Iu n e u t r i n o p r o t o n scat t e r i n g t h e i n c o m i n g n e u t r i n o is l e f t - h a n d e d . T h e r e f o r e , w h e n s c a t t e r i n g olf a r i g h t - h a n d e d i i - q u a r k . t h e t o t a l spin is . 7 = 1 . l e a d i n g t o a c o s ' ( 0 / 2 ) = (1 - //)-' a n g u l a r d e p e n d e n c e for t h e o u t g o i n g p a r t i c l e s . H o w e v e r , w h e n s c a t t e r i n g olf a l e f t - h a n d e d d o w n - t y p e q u a r k , t h e n e t s p i n is ./ = 0. which r e s u l t s in a flat, a n g u l a r d e p e n d e n c e .
7 3 T o s t a r t , we w r i t e d o w n t h e F, s t r u c t u r e f u n c t i o n s for n e u t r a l c u r r e n t e l e c t r o n proton and electron neutron scattering: K
"(•<•) = x
I ^ (u + u + r + c) + - (il + il + .s- + s )
s u i . i i HONS
(il I il I I' T
I'M
-I- - (« + II I
C)
I .s)
.S
Wo o m i t t h e a r g u m e n t of t.lir p a r t o u d i s t r i b u t i o n f u n c t i o n s . T h e second equality is o b t a i n e d a s s u m i n g isospin invariaiice, t h a t is. u = M1' = il" a n d il = < 1 = n". T h e nucleoli s t r u c t u r e f u n c t i o n is defined as 1
2 = x
J ^ ( " + W + ' / +
()
(•" + S) + I (c -1- c) (l
xvhcrc we h a v e i n t r o d u c e d t h e s h o r t - h a n d n o t a t i o n (q T q) — (u + i7 1- il T il + s i s I- c + c). T h e s t r u c t u r e f u n c t i o n for neut rino nucleoli s c a t t e r i n g h a s been given iu eqn (7.IS). A s s u m i n g s = s a n d using t h e a b o v e s h o r t - h a n d n o t a t i o n , il can b e cast in t h e simple forili F>*(x) x (q q). Now. it is easy t o see t h a t t h e r a t i o is given bv N
Fi F.';
N
(7 + g) IS
2 („ + s } + I (,. + e)
3 s -I- .s - (c -f c)
1
18
5
18
q + q
1
s i- s ;
> q
T h e lasl a p p r o x i m a t i o n is valid b e c a u s e of t h e small c h a r m c o n t r i b u t i o n t o t h e sea. T h i s result can also b e r e w r i t t e n a s an expression for x\s -f s). by simply inserting F." = x(q -f q) a n d inverting t h e last ec|iiation: x(s + s) = - F-f
- (i r-z
N
I lu Ex. (7-2) we have seen t h a t t h e n e u t r i n o nucleoli D1S cross section is given I iv d-y""> ci Y+F^r'\x.Q2) ± V'_.rF:;"",(.,-,(/-')] . 2 d.r clQ 4irx J SXIJ a n d
1 di7 , , ( " ) d;v
Gì 2tr
Ms
v , F**\Q'2)
:I: V .
f ./a
t\XXF:;{,/)(X.C?
= J,' da: F^'~'\x, Q2). Now let us s t u d y t h e limit // - 0. We h a v e with F^"\Q2) Hit .o y + = 2. Ii111?/ .0 V = 0, l i i n H _ o i ^ J = " a n d l i i n y _ o k. = 1. T h e r e f o r e .
S< »1,11 I IONS
0) = C , w h e r e t h e c o n s t a n t C is i n d e p e n d e n t of e n e r g y a n d t h e s a m e for n e u t r i n o s a n d nut ¡neut rinos. T h e neut r i n o flux "("> drr"'"Vd.
T h e c o n s t r a i n t s for t h e p r o t o n of t o t a l electric c h a r g e 1. h a r y o n n u m b e r I. a n d z e r o net. s t r a n g e n e s s a n d c h a r m c a n be f o r m u l a t e d a s i n t e g r a l s o v e r t h e full .1 r a n g e of t h e part.on d i s t r i b u t i o n funct ions, w e i g h t e d by t h e relevant c h a r g e : jf
d.r | jj (u - ft +c-
c) - - ( < / - d -I -s - s)
f 1 1 f1 / d./- - (« - H + c - r) + T (d-d .'o I •» •»
[
da: ( » — » ) =
[
./a
-
+ s - «)
d:i.:(c-c) =
= 1.
= 1 ,
0.
Jo
Now s i m p l y insert t h e c o n s t r a i n t s of t he last line i n t o t h e first t wo e q u a t i o n s a n d solve t h e r e m a i n i n g e q u a t i o n s y s t e m . T h i s r e s u l t s in
[ da- ( i / — f / ) = f d.r i/v = 2 ./() ./o
,
[
Jo
d.r (d - <1) = f .Ai
W i t h /I. a n a r b i t r a r y unit v e c t o r , o n e f i n d s
n
r
„, Hp(n'PHl>' If 77 = ^ „ ^ Zp7>-
n
)
=
E/jl>\. , E/W'T ' 2 = 1- ^ — 7 . E p /'" E,,/'2
w h e r e pi. a n d p|- a r e t h e l o n g i t u d i n a l a n d t r a n s v e r s e c o i n p o n e u t s of p with respect t o 7i. M i n i m i z a t i o n of t h e s u m of t h e s q u a r e s of t h e t r a n s v e r s e m o m e n t a , which is t h e d e f i n i n g p r o p e r t y of S p h e r i c i t y , t h e n c o r r e s p o n d s t o (hiding t h e d i r e c t i o n n which m a x i m i z e s n ' II n . W i t h t h e c o n s t r a i n t t h a t n be a unit vector o n e o b t a i n s V
n
( n r l l ' n + A(1 - n ' n ) ) = 0 .
w h e r e A is a L a g r a n g e m u l t i p l i e r t a k i n g c a r e of t h e n o r m a l i z a t i o n c o n s t r a i n t . T h e s o l u t i o n is given by I I ' M = A Y7 .
SOI,l> I IONS
l!i:t
I liai is. u is a n cigcuvcctoi <>l II wit li e i g e n v a l u e A. Since IF is a s y m m e t r i c 3 x 3 m a t r i x , t h e r e a r e t h r e e e i g e n v e c t o r s n , . i = 1 . 2 . 3 . which f o r m an o r t h o g o n a l basis. It follows
i=l anil
thus
V n'j'Wn, irr
=
= A, + A 2 + A;I = ZpP
S o r t i n g t h e e i g e n v a l u e s a c c o r d i n g t o Aj > A_> > A.t. t h e S p h e r i c i t y S c a n b e written as S = ' L ( 1 -
A,) =
|(A
+
2
A3).
T h e e i g e n v e c t o r t t | t o A| d e f i n e s t h e Splu-ricMy axis. For a n ideal two-jet. event o n e h a s A| I a n d S = 0. For a p e r f e c t l y s p h e r i c a l event all e i g e n v a l u e s h a v e t h e s a m e v a l u e A, 1 / 3 a n d t h e S p h e r i c i t y a t t a i n s t h e value S = 1.
2 For a e o l l i n e a r s a f e e v e n t - s h a p e v a r i a b l e t h e value must r e m a i n u n c h a n g e d if a n y of t h e c o n t r i b u t i n g m o m e n t u m v e c t o r s is split i n t o t w o e o l l i n e a r o n e s . S i n c e S p h e r i c i t y is q u a d r a t i c in t h e m o m e n t a , t h i s r e q u i r e m e n t is not m e t . T o illust r a t e t h i s , c o n s i d e r t h e c a s e that o n e m o m e n t u m v e c t o r w h i c h is p a r a l l e l t o t h e S p h e r i c i t y a x i s is split like P
,
2
+
P
a"
Since p is parallel t o t h e event, axis, t h e d i r e c t i o n which m i n i m i z e s t h e s u m of t h e t r a n s v e r s e m o m e n t a d o e s not. c h a n g e u n d e r t h i s o p e r a t i o n . T h e value of S. however, is a f f e c t e d . Since
•
-
»
(
f
M
.
t h e d e n o m i n a t o r of t h e d e f i n i n g e x p r e s s i o n e q n (8.22) for S b e c o m e s s m a l l e r , t h a t is. t h e S p h e r i c i t y is i n c r e a s e d .
3 T h e C ' - p a r a i n e t e r is e o l l i n e a r s a f e if t h e linear m o m e n t u m t e n s o r (-) w i t h c o m p o nents e
,J
=
1
y E p IpI ¿ r
^ IPI
a l s o h a s t h i s p r o p e r t y . S p l i t t i n g a n y of t h e m o m e n t a p i n t o a s u m of e o l l i n e a r ones. p = £
okP h
with
y , Qfc = 1 • <.
ok > 0 V/r,
ST >1 I ' L L ! I N N
o n e sees i m m e d i a t e l y that t h e n o r m a l i z a t i o n in t h e d e n o m i n a t o r has tin- desired p r o p e r t y . It r e m a i n s t o he cheeked w h a t h a p p e n s to t h e individual t e r m s of t h e s u m in t h e n u m e r a t o r . O n e finds PH'j
=
y-
fc
\P\
"ÍPiPj
=
ViPi y-
\P\
a U
¿
" '
a n d since t h e s u m of t h e coefficients n/. is normalized hv c o n s t r u c t i o n , it follow, t hat B is collinear safe.
S I f o r a n ideal two-jet. event t h e m o m e n t u m flow is only along o n e direction in space. Since in t h e C . o . M . s y s t e m t h e t o t a l m o m e n t u m vanishes a n d since the ( ' - p a r a m e t e r is collinear safe, it can h e a s s u m e d w i t h o u t loss of generality that one h a s only a two-particle event, one particle with m o m e n t u m p a n d t h e otliei one with m o m e n t u m —p. T h e linear m o m e n t u m t e n s o r t h e n becomes 1 2 ppr
,, e
=
r =
2 b f F
n
n
"
t h a t is. t lie direct p r o d u c t of t h e u n i t - v e c t o r « a l o n g t he direction of p with itself. O n e verifies i m m e d i a t e l y that t h e first eigenvector of B is n with eigenvalue A| = I. T h e r e m a i n i n g t w o eigenvectors then have t o he o r t h o g o n a l t o n . t h a t is. they have eigenvalues A j A.j = 0. Since t h e C'- p a r a m e t e r is t h e s u m of all p r o d u c t s of t w o different, eigenvalues it will he zero. For a n isotropic event , t he m o m e n t u m flow into every element of solid angle is t h e s a m e . W i t h a t o t a l m o m e n t u m Y2\P\
p
=
P
one t hus has <1» ~ -1-" W i t h 0 t he polar angle a n d <;> t h e a z i i n u t h . o n e o b t a i n s t h e C a r t e s i a n c o m p o n e n t s of t lie m o m e n t u m flow a s d/>, P . .. —— = - — s i n f / c o s cfr. r/ii 4tt
. and
a n d t h e linear m o m e n t u m t e n s o r b e c o m e s
d/'/d<> S u b s t i t u t i n g t h e a b o v e expressions t h e total m o m e n t u m P cancels a n d all that, r e m a i n s a r e integrals over t he solid angle. T h e off-diagonal elements of W, 7 vanish, t h e diagonal elements, t h a t is. t h e eigenvalues of t h e m o m e n t u m tensor, are B„ 1/:J. i = 1 . 2 , 3 . It follows that C = 1 / 3 .
:;< >|,ii l it >NS
A s s u m e t h a t in t h e C.o.M. s y s t e m t h e electron a n d p o s i t r o n m o v e a l o n g t h e •axis. II m a s s effects can he neglected, t h e y a r e most, conveniently described t h r o u g h helicity eigenstat.es. t h a t is, a s s t a t e s with t h e spin e i t h e r parallel o r ant ¡parallel to t he m o m e n t u m vector. A s s u m e t h a t t h e a n n i h i l a t i o n p r o c e e d s parity conserving via a v i r t u a l p h o t o n . Since t h e coupling of t h e ferniions t o t h e p h o t o n is helicity conserving, t h e i n t e r m e d i a t e s t a t e which then decays into a pair of s e c o n d a r y particles can be described I h r o u g h t h e a n g u l a r m o m e n t u m s t a t e |/) = |./. in) 11. ± 1 ) . In t h e C.o.M. s y s t e m t he final s t a t e f r o m t h e decay consists of a pair of back-to-back particles, which a r c e m i t t e d a l o n g a new direct ion T h e a n g l e between z a n d z' is d e n o t e d by B . If t h e final s t a t e p a r t i c l e s a r e s p i n - 1 / 2 ferniions, t h e a n g u l a r m o m e n t u m s t a t e is \ f j ) = |./. m') = | 1 . ± 1 ) . for spin-U particles it has t o b e |fi,) = |./. m ' ) | 1 , 0 ) . Note that t h e final s t a t e h a s a r o t a t e d q u a n t i z a t i o n axis c o m p a r e d t o t h e initial s t a t e . T h e a n g u l a r d i s t r i b u t i o n is then o b t a i n e d by t h e q u a n t u m mechanical overlap of t he initial a n d t h e final state, d ^ e - D « ' ) ! ' . if where t h e s u m r u n s over all possible initial a n d final stat.es. From t h e spin algebra t h e s c a l a r p r o d u c t s ( i \ f ) a r e given by t h e so-called il-functions, <1,'„,„•• O n e obtains -
I d ! , I 2 + IrfL,, I 2 + | d j - , I 2 + |rfL, _ , I 2 = 1 + cos 2 O
for s p i n - 1 / 2 a n d ~ \ < III2 + IdL, ,,| 2 = s i n 2 e = 1 - cos 2 e for spin-0 particles.
T h e differential cross sections for three-jet. p r o d u c t i o n a r e given by eqn (9.-ri) for t he case of a vector gluon a n d eqn (9.6) for t h e scalar gliion. Lines of c o n s t a n t cross section o in t h e a-,, r, { p l a n e for b o t h cases a r e shown hi Fig. F.7. T h e lines a r e equidistant, in Inrr. O n l y t he t r i a n g l e a b o v e = 1 is kineniat ieallv allowed. Note t h a t energy liioinentum conservation gives t h e constraint. .i:(| + x < ( + .rK = 2. For e n e r g y o r d e r e d configurat ions r t > .?••> > .r :( . t here a r e six regions with different m a p p i n g s f r o m {.< i. .r-_>,.r:j) to {j',,..i:«,,:r B }. T h e y . t o o . a r e p i c t u r e d in Fig. F.7. 1
2
.1
-1
•r
,T<( ^<1
av,
(i 5 a:fI •1:, a:R
S o l , I I I IONS lei IMI I|u;inliiy lines
,t(q)
.»• li| i
F i c . F . 7 . T h r e e - j e t k i n e m a t i c s a s a f u n c t i o n of t h e s c a l e d q u a r k a n d a n t i q u a r k e n e r g i e s . T h e u p p e r row s h o w s lines of c o n s t a n t c r o s s s e c t i o n tor t h e v e c t o r (loft ) a n d t h e s c a l a r g l u o n ( l i g h t ) . T h e lower left plot s h o w s t h e r e g i o n s c o r r e s p o n d i n g t o a fixed m a p p i n g bet ween e n e r g i e s a n d p a r t o n t y p e s w h e n p e r f o r i n i n g e n e r g y o r d e r i n g of t h e j e t s , a n d t h e lower right finally d i s p l a y s lines of c o n s t a n t T h r u s t .
T h e b o r d e r lines b e t w e e n dilferent r e g i o n s a r e given by = .r ( | . ./'<, = I O, = 1 - r , , / 2 a n d x ( | = 2( 1 - a ,,). For a given throe-jot c o n f i g u r a i ion t h e v a r i a b l e T h r u s t . / ' is given bv Epli) • H T = m a x —r^—¡—¡— = .i'i . " E p \P\ T h e Thrust.-dist.rihut.ion is o b t a i n e d by i n t e g r a t i o n over t h e d i f f e r e n t i a l c r o s s section i\a dT
r _
./
d.i.„d ' «
? W "
- T(x,t..»•„))
.
w h e r e t lie ()- fit net ion a s s u r e s t h a t o n l y t h o s e r e g i o n s of t h e p h a s e s p a c e c o n t r i b u t e w h e r e t h e T h r u s t h a s a fixed value 7". S i n c e t h e f u n c t i o n T(x,,../,,) c h a n g e s for different p a r t s of I he p h a s e s p a c e , t he i n t e g r a l h a s t o b e d o n e s e p a r a t e l y for I hose
SOI-H I IONS
regions, fiocinisi' ol t he letry of /'(.'«,. •'•,,) a s well a s t h a t of t h e d i f f e r e n t i a l cross section a b o u t t h e line x,t xu o n e o n l y h a s t o c o n s i d e r region (1). w i t h /' = xK, a n d regions (2) a n d (.'i) w i t h T = ./>,. For d o i n g t h e i n t e g r a t i o n it. is m o s t c o n v e n i e n t t o s t a r t w i t h t h e c,,. U s i n g t h e e n e r g y c o n s t r a i n t x,, + x,t I .i:g 2 o n e finds •1-772 2(1 -T)
da
d-'c dxvjdav,
o f I + 2 /
S u b s t i t u t i n g t h e e x p r e s s i o n s e<|,i (9.5) a n d e q n (!l.(i) for t h e v e c t o r anil s c a l a r g l u o n . respectively, t h e final r e s u l t s i n c l u d i n g c o u p l i n g const a n t a n d c r o s s s e c t i o n normalization are 1 <1(71 (T(,
asCF 2n
(IT
1
and 1
Q S'r
2 In
2 T l -
Thrust distributions
T
- 9 ( 1 — T ) - 10
(cl
vector
F i t ; . F . 8 . T h r u s t d i s t r i b u t i o n s for v e c t o r a n d s c a l a r g l u o n s . T h e left plot, s h o w s t h e d i f f e r e n t i a l c r o s s s e c t i o n s for b o t h eases, t h e right f r a m e is t h e r a t i o of t h e t w o . T h e c r o s s s e c t i o n s a r e n o r m a l i z e d s u c h t h a t t hey a r e t h e s a m e for small values of T.
T h e d i f f e r e n t i a l c r o s s s e c t i o n s a r e d i s p l a y e d iu Fig. d i c t i o n of t h e v e c t o r g l u o n for T • 2/.'i. w h e r e t h e s 1 / 2 - "> E approaches the limiting value
F.S. n o r m a l i z e d t o t h e p a s r a t i o of t h e t w o p r e d i c t i o n s " I " , / 1 ' - E ( " ! i + "¡j)- T h e prein t h e t w o - j e t region T — I
soi,in
|!IK
IONS
I :S T h i s is L a n i y ' s t h e o r e m . In tin* C . o . M . f r a m e o n e lias / ' p , + p 2 I p., 0, which implies t h a i all m o m e n t a a r c w i t h i n a p l a n e . A s s u m i n g inassless particles, t h e cross p r o d u c t s of p, a n d p., w i t h / ' give Ex E> sin >I'| 2 + E, E:t sin >I':|| = 0
p , x P = (I p, x P - 0
=>
E2EI
sin ^i.» + E,EAsin
'I'j.t = (J,
w h o r e >1',^ is t h e o p e n i n g a n g l e b e t w e e n p, a n d p y . C o m b i n i n g t h e s e t w o e q u a t i o n s yields ( E i + E - i ) s i n 'I',-.. -I £:t(sin *I'Kt + sin >I'j :t ) = ( I . a n d w i t h t h e e n e r g y - e o n s e r v a t i o n c o n s t r a i n t E\ 4- E•> E
- J7-
y/s — E-.i we find
sin sin i ' i2 + sin lI'2:j 4- sin tf'm
T h e c o r r e s p o n d i n g e x p r e s s i o n s for E\ a n d E> a r e o b t a i n e d by cyclic p o r i n u t a t i o n of t h e indices.
II I First we w r i t e d o w n t h e e x p a n s i o n o( a ^ p 2 ) in t e r m s of a j s ) . I n l r o d i i c i n g L In //*/.-• a n d t l i u s u ; = 1 I- « s (.s)/i 0 L. we find u p t o 0()
2
«*(/') =
u,'
/
/,,«„(,s) \ 1- 7 hi u,1 \ <>o u> /
« , ( « ) [I - «,/><•£, + a2Ml<Similarly, we o b t a i n for n2
- l>I ¿ ) ] +
(•-')) •
(p2)
<>;(ir) = <<:(s) [1 - 2/;„« s (.s)I] + <9(nJ(*)) . Now. i n s e r t i n g t h e s e e x p r e s s i o n s i n t o c q n (1 ().(!). — ^ = <>MMU) T|l!l(l d// + n2(s)B(y)
' <>2(s)(b„L)A(!,) 4-
n*(s)2(b()L)B(y)
+ a*(»)(l%L2
-
„2(s)(b()L)A(y) - af(s)2(l>2L2)A(y)
+ 0(uj(s))
we see t h a t t h e d e p e n d e n c e on (b»L) c a n c e l s out in 0(if(.i)). r e m a i n s a d e p e n d e n c e in t h e next h i g h e r o r d e r , n a m e l y 1
<]n
= ns(s)A(y) <(s)b„L
f
bxL)A(y)
. However. t h e r e
n2(.s)B(y)
b„LA(y)
+ 2B(y)
I
~A(y) «0
+
0(a!t(s))
111(1
SOLUTIONS
II ' Willi four |»article's in t h e final s l a t e we have Hi variables t o s t a r t w i t h , n a m e l y I'onr t i m e s t h e f o u r c o m p o n e n t s of t h e f o u r - m o m e n t a . If t h e p a r t i c l e m a s s e s a r e k n o w n , t h e n t h e r e a r e f o u r mass-shell c o n s t r a i n t s . Ef = + inf. i = I I. which r e d u c e t h e n u m b e r of i n d e p e n d e n t v a r i a b l e s to 12. Next we h a v e f o u r e q u a t i o n s f r o m e n e r g y m o m e n t u m c o n s e r v a t i o n , which gives a f u r t h e r r e d u c t i o n t o e i g h t . T h e overall e v e n t o r i e n t a t i o n c a n b e d e s c r i b e d by t h r e e angles, s u c h a s t h e E u l e r a n g l e s , a n d if we i n t e g r a t e over t h e m , we e n d u p with live i n d e p e n d e n t kinematic variables. In o r d e r t o get a list of such variables, we w r i t e d o w n e n e r g y c o n s e r v a t i o n in t e r m s of t h e f o u r - m o m e n t a . ii i P\
with for e x a m p l e pi„i
+
(Ec„,.0)
i' i PI
+
i' />'( +
ii PI
i* =
PI„I
I
at LEI*. T a k i n g t h e s q u a r e of b o t h sides yields
2pi • i>2 -I- 2pi • /»;( + 2pi • i> i 4- 2p> • p 3 + 2 p 2 • m + 2p ; j • p i + £ i=l If we now i n t r o d u c e .s-
momentum
mj = pfM
¡>jtil a n d t h e L o r e n t / - i n v a r i a n t q u a n t i t i e s
.'/<> = (P. + P . ; ) 2 / * = (2Pi • Pi + ntf + //;;)/«
= m f j / s.
we c a n r e w r i t e t h e p r e v i o u s result a s 'I 2/12 + 2/13 + 2/11 + 2/23 + 2/21 + 2/31 = 1 T 2 £ m 2 / « . ¡=1 W e s e e t h a t we have six variables, w h e n ; o n e c a n b e e x p r e s s e d by t h e o t h e r five, which t h u s can b e t a k e n a s t h e r e q u i r e d set of i n d e p e n d e n t variables. O n e possible choice is y\>, y|3, 2/M- 2/23* 2/21-
17 I A p p l y i n g eqn (fJ.10) t o t h e onefold s y m m e t r i c e v e n t s , t h e h a r d e s t jet h a s E x = My. • s i n 6 0 ° / ( s i n 6 0 ° + 2 s i n 150°) « 4 2 . 3 G e V a n d = E, • siii(l. r >()°/2) « 40.» G e V . T h e t w o lower e n e r g y j e t s h a v e E = E> = E3 = M y / s i n ir>0°/(sinG0°-f 2 s i n 150°) = ( M z - 2Ei)/2 « 2 4 . 5 G e V a n d Qy2 = Qt:» = E • siii((i()°/2) % ~ 1 2 . 2 G o V . In t h e t h r e e f o l d s y m m e t r i c e v e n t s . E = E\ = E-> = E3 = MyJ'i 30.4 G e V a n d Qyi = Q r 2 = Q n = E s i n ( 1 2 0 ° / 2 ) ss 2 6 . 3 G e V .
! 2 T h e <|iiark At high x. these only eqs.(3.268)
a n d gluon f r a g m e n t a t i o n f u n c t i o n s evolve a c c o r d i n g t o ecin (3.285). only t h e z • I p a r t s of t h e s p l i t t i n g f u n c t i o n s c o n t r i b u t e a n d of t h e s i n g u l a r t e r m s in i ' ^ z ) a n d /',,,,(;) a r e i m p o r t a n t . R e f e r r i n g t o a n d (3.50) we h a v e j—1
! <1*1(2)
4* 5 * 0
.
s» >I,N H O N S
W
I-I —
_ 1 ^ ( T - y ;
»
2 =
cA
(1 - z )
+
IICA ' „
ìniTr *<» - c)
Oa =
/
— 'A
3 . + -<5(1 2
The case CA = :5 is a p p r o p r i a t e t o colour S U ( 3 ) , whilst n/ = 3 is justified sin« « heavy flavours essentially d o not c o n t r i b u t e in t he evolution of t h e part,on showet T h u s , in ( Ik* limit x — 1. we have
where C / Cr for / = <| a n d Cj = C'A for / = g. If we a p p r o x i m a t e D / ( : r / i , / « ) in t h e integrand by D / ( . r . / i 2 ) we o b t a i n
where t he p r o p o r t i o n a l i t y c o n s t a n t is independent, of Dj. T h i s result allows the t o be e x t r a c t e d by c o m p a r i n g scaling violations colour c h a r g e ratio. CA/Ci.. seen for leading p a r t i c l e s in q u a r k a n d gluon jets: for m o r e details see (Nason a n d W e b b e r . 1994: DKLPIII Collab.. 19«),S/;).
i I T h e result follows i m m e d i a t e l y if t h e convolution is w r i t t e n as D(x) = (x) = I i\y I d~ 6(x ./<) J{ i
yz)(lvr(y)'lNv(-
I n t e g r a t i n g over x a n d g e t t i n g rid of t h e ¿ - f u n c t i o n . t h e r e m a i n i n g integrals faetori/.e, thereby proving the result: (x) = [ <\xxD(x)= 7(i
[ d y f d* f <\x6(x./(i ./(i J» rt [ D;/ [ c\z(yz)dvr(y)
yz)dM'j)dw(z) = (*>I>T<*>NP •
5 2 T a k i n g x = x,, (x = .(/.; gives t h e s a m e result.) a n d using rclativistic k i n e m a t i c s tlie c h a n g e iu e n e r g y is given by: A E = sjv2
+ >"ìì - s f i w + •>
"TI
JUO„
- ^ ( 1 - .<'„)V
+ '"r, 2
SOLUTIONS
rj
r.oi
f t
(»',|/mg)-\
2jp| V
•'•'/.
/ '
By a p p r o x i m a t i n g t h e mass of t h e h a d r o n with t h e m a s s of t h e heavy q u a r k . \lQ,, % //IQ, a n d e q u a t i n g t h e m a s s of the; light q u a r k with t h e inverse of t h e typical h a d r o n size. in,. [<~1. t h e required form for t h e f r a g m e n t a t i o n f u n c t i o n follows.
I l :i We work in t h e .r y plane a n d use t h e explicit foi n - v e c t o r s (/'' = ( 1 . 1 / 2 . I \ / 3 / 2 . 0 ) .
qqg :
Ì a ì C f - CA : 2CA -I- CF : C,t/2
+
C,,}
7 .2
22
3 ' 3
171 :
¥ J
M I Iu o r d e r t o simplify t h e following discussion we describe pions using t h e p l a n e wave a p p r o x i m a t i o n . Hence t h e wave f u n c t i o n or a m p l i t u d e .4(1) for a pion (spin zero) with m o m e n t u m k.\ at s o m e s p a c e point x„ a t lime I I) is given by / l ( l ) = <\xp(i A:i • x,,), u p t o a n o r m a l i z a t i o n a n d s o m e a r b i t r a r y p h a s e <1>„. For a s y s t e m of two d i s t i n g u i s h a b l e pions (for e x a m p l e , pions of o p p o s i t e c h a r g e ) t he joint, amplit u d e is .4 + - ( 1 . 2 ) = <«'
e
i K A - X I . (,i«fci
•!•(.
=
/l(i)/l(2)
.
where t h e wave f u n c t i o n of t h e second pion is c h a r a c t e r i z e d by a m o m e i i l i u n A-j a n d s p a c e c o o r d i n a t e s xi,- For i n d i s t i n g u i s h a b l e (like-sign) pions t h e joint a m p l i t u d e has to be s y m m e t r i c in a:,, a n d xi, b e c a u s e of Bose Einstein s t a t i s t i c s : therefore. —
o) -
^
[e'^'1 x„Qik.x,,
| pik7x„{jk,
Xi. ,,1'K.pi'h. _
\/2 Lot us now introduci' a c o r r e l a t i o n f u n c t i o n ( ' ( 1 . 2 ) , defined a c c o r d i n g t o C(I
21 =
=
|.4+-(1.2)r-
M++ —(l,2)la |A(1)4(2)|2 •
Inserting t h e expressions for t h e various amplit udes, t h e correlation function is found t o lie C(1.2) = 1 + -
¿(k,-kJ){x.,-xl.)
+ 0-i(fc,-fc,).(a:..-x,.>j
SOLUTIONS
= I I cos(AA: • Ax)
,
w h e r e we h a v e d e f i n e d Ah: = k\ a n d Ax = x„ - x./,. Already, a t t h i s s t a g e , it. is e l c a r t h a i for a n a r b i t r a r y Ax ^ 0 t h e c o r r e l a t i o n g e t s e n h a n c e d w h e n Ak (I. T h e s e c o n c l u s i o n s a r e not d r a s t ¡rally a l t e r e d if we inl r o d i i c e w a v e (jackets i n s t e a d of p l a n e waves, a s long a s t h e s p a t i a l s m e a r i n g is very n a r r o w , t h a t is. t he w a v e f u n c t i o n in m o m e n t u m s p a c e b e c o m e s (t>{kt) =
y
a n d t h e s p a t i a l w a v e f u n c t i o n (x) is c e n t r e d a r o u n d Xn w i t h very s m a l l s p r e a d a r o u n d t hat p o i n t . T h e n it is e a s y t o s e e t h a t a n e w l y d e f i n e d e o r r e l a t i o n funct ion C'(1.2) = h a s a g a i n t he b e h a v i o u r C ( l , 2 ) — 1 + cos(Afc •
Ax)
for n a r r o w s p a t i a l w a v e (jackets c e n t r e d a r o u n d x„ a n d xHere +~.o++'' a r e t h e w a v e f u n c t i o n s in m o m e n t u m s p a c e for unlike-sign a n d like-sign (nous, o b t a i n e d by s m e a r i n g t h e a m p l i t u d e s .-l + _ a n d w i t h t h e |>roduct of s|)atial wave f u n c t i o n s . For t h e r e m a i n i n g d i s c u s s i o n w e r e t u r n t o t h e p l a n e w a v e a p p r o x i m a t i o n . Now we m a k e t h e a d d i t i o n a l a s s u m p t i o n t h a t t h e p r o b a b i l i t y or c r o s s s e c t i o n for p r o d u c t i o n of a pion at a point x„ is p r o p o r t i o n a l t o t h e a m p l i t u d e s q u a r e d for t h e s i m p l e f r e e p l a n e wave, t h a t is. P( 1) ex |.4(1)|*. a n d s i m i l a r l y / ' ( 1 . 2 ) oc |.1( 1 . 2 ) | " . which is t h e p r o b a b i l i t y for t h e p r o d u c t i o n of two p i o u s at ( j o i n t s x„ a n d xi,. If t h e s o u r c e for pion p r o d u c t i o n h a s s o m e s p a t i a l d e n s i t y d i s t r i b u t i o n l>(x). t h e n we o b t a i n t h e t o t a l c r o s s s e c t i o n a for pion p r o d u c t i o n by i n t e g r a t i n g o v e r t h e full s o u r c e e x t e n s i o n , a n d t h e c o r r e l a t i o n f u n c t i o n is r e w r i t t e n a s
where » " ( f c i . f c a ) 'X f j d 3 a ; ( 1 d 3 x h p ( a : „ ) p ( x < , ) | / i M ( l . 2 ) | 2
.
T h e i n d e x s.s s t a n d s for like- (ss = + + . — • ) o r unlike-sign p i o u s (.s.s = Filling in th<> e x p r e s s i o n s for t h e a m p l i t u d e s w e get C(1.2)= 1 +
+—).
T
( ,2) =
= J d\r p(x) (>
p/N.
p(x)
= p'(x).
.
T
4
(A
)
2
,
( .18)
I <\:i.r p(x)
p(Ak)
s,
.
s
s
s
s
s s
s s
s
,
. s
s
ss s Ak — ).
2
,
s
,
ss
s s
, (
s
s .
) . ). s
s
. A
s s
s.
s ,
s
ss
, s
s s
s ( ) = . (,
s
+ s
s
A
T
4 fbeik,Xl-) (f„e
s
s
.
ss s
. T .4(1.2) = (
T
,
s
,
i k
'
x
s
" + f,,cik*x*)
s
s
s
.
s
.
s [m(,c-''k'
( ,2) , s
s s. T s
,
s
. . )
+
. .
.
s
,
s
s
F.
( 1) {2))
s
s
ss ,
s s s
s
.
. . (1.2) ( s
= s
Ak • Ax.
s . ,
+
s. ( ,2)
s s s
s
)
f;jhv-'k-"{X"-Xh)
4- , -I-
s
--
s s s ( . ))
s
, . . (P( 1 . 2 ) ) .
s s (
s.
',(11
SOLUTIONS
average out to zero, and only t hose t e r m s with q u a d r a t i c or quartic dependence on the emission a m p l i t u d e will remain. Hence, the result must he (P( 1,2)) cc (Si + Si f
+ 2 Si Si cos(AA: • A x ) .
Since {P( 11) = (P{ 2)) a Si + Si- l ' u > t o r relation function becomes
In the special case of two sources of equal s t r e n g t h . / „ = //,. we finally find (7(1.2) = 1 + \ cos(Afc • A x ) — ^ for Ak
0 .
It can he shown t h a t in t he case of II r a n d o m sources of equal st rengt h t he above expression generalizes to C(1.2) — 1 +
-
for Ak -
0 .
Thus, in the case of a large n u m b e r of coherent sources t h e correlation function approaches a m a x i m u m of 2 for AA; — 0. just as we have found previously for th<' simplest case of incoherent emission from t wo sources and neglecting any time dependence. It is r a t h e r obvious that any realistic description of a physical source should take account of bot h coherence effects and time dependence such as random fluctuations. T h e net el feet should be t h a t the effective s t r e n g t h of the Hose Einstein enhancement lies somewhere between 0 and 1. C o m p a r i s o n s to m e a s u r e m e n t s should t herefore be based on a p a r a m e t e r i z a t i o n such as G ( 1 . 2 ) oc 1 + A|/i(Afc)| 2
.
(F.20)
where p has been defined in the previous exercise, a n d 0 < A < 1.
13 (i T h e density distribution for a Gaussian source in t h r e e space dimensions, with different widt hs = 1 . 2 . 3 . for every dimension is given by 3
'
. - A
The s satisfy / d :J .i:p(x) = 1. In order to get t h e two-pion correlation function as defined in eqn (F.18). we have to c o m p u t e the Fourier t r a n s f o r m of this density: p(Ak)
= f &\r,,(x)i^k
x
= f i ( / t e i P , ( . r , ) e i A t " - ) = J~[ /»¿(AA,) .
:.
HONS
I he Foiu ier transl'oi tu ol a Ganssian ilistribution can be (bund in every s t a n d a r d ci illect ii in of integrals,
p1(AA1)=e"5fc---
r>(Ak)=vK
V
^^(Ak,fo-f]
.
When scpiaring tliis. we linally linci a parameterization l'or the correlaiion fuiiclion. as |»roposed in cqn (F.20).
C(l,2) = 1 + Aexp
.
l'Ile special case of a sphcrically syininetric source is obtained for cri = a-, = T,
(7(1.2) = I + A e - ( A f c ) V .
(Afc)2 =
Y+Aki? . i=i
F
C S
Abbawo. . et. al. (1998). Tlio S Heavy F l a v o u r Working G r o u p , HF 98-0 A b b a n e o , . et al. (2000). Tlie 1. 1 Collab.s A IMI. 1)HI.l'Ili. 3 a n d A , t h e IS l e k t r o w e a k W o r k i n g G r o u p a n d S I Heavy F l a v o u r a n d lect, roweak Groups. C 2000-010. A b b a n e o . . et al. (2001«). T h e Collab.s A II. H I . 1.3 a n d A , t h e I. l e k t r o w e a k W o r k i n g G r o u p a n d S I) Heavy F l a v o u r a n d lectroweak Groups, C 2001-021. A b b a n e o , . et. al. (2001 b). T h e Collab.s A II. C F. H I . 3. A a n d S I.'). C - -2001-050. A b b a s . A. (2000). arXiv hep-ph 0009242. Abraniovskii. V. A. et al. (1!)72). In X nt. Conf. ou lli li n cr h sics, Chicago atavia (ed. .1. . . ackson. A. o b e r t s , a n d . o n a l d s o n ) . Volume I, pp. 389. A (Batavia). Adler. S. . (1903). h s. ev., 1 4 3 . 1144. A F S Collab.. Akesson. T . et al. (1983). h s. ett., B 1 2 3 . 133. Aitchison. I. .1. . a n d lley. A. .1. ( i . (1989). Gauge theories in particle ph sics a practical introduction. Hilger. Bristol. A I I Collab., Busknlic, . et al. (1992«). Z. h s.. C 5 5 , 209. A IMI Collab.. Busknlic. . et al. (1992 ;). h s. ett.. B 2 9 5 . 390. A I I Collab.. Busknlic. . et al. (1993). Z. h s.. C 5 7 . 17. A I I Collab.. Busknlic. . et al. (1994«). Z. h s.. C 0 4 . 301. A I I C'ollab.. Busknlic. . et al. (1994 )). Z. h s.. C 0 2 . 1. A II Collab.. Busknlic. . et al. (1995«), uel. nstr. and eth.. A 3 0 0 . 481. A I'II Collab.. Busknlic. . et al. (1995 )). h s. ett.. B 3 5 7 . 187. r r a t u m : (1995). ibid.. B 3 0 4 . 217. A I I Collab.. Busknlic. . et al. (1995c). h s. ett.. B 3 5 5 . 381. A I I Collab.. Busknlic. . et. al. (1995< ). Z. h s.. C 0 9 . 15. A IMI Collab., Busknlic. . el al. (1990«), Z. h s.. C 7 1 . 357. A I I Collab.. Busknlic. . et al. (19066). h s. ett.. 13384. 35:1. A IMI Collab.. Busknlic. . et al. (1990c). Z. h s., C 0 9 . :579. A IMI Collab.. Busknlic. . et al. (1997«). h s.. C 7 3 . 409. A IMI Collab.. B a r a t e . . et al. (1997 )). Z. h s.. C 7 0 . 1. A IMI Collab.. B a r a t e . . et al. (1997c). Z. h s.. C 7 0 . 191. A IMI Collab.. B a r a t e . I et al. (1997 ). Z. li s.. C 7 4 . 151. A I I Collab.. B a r a t e . . et al. (1998«). h s. ept.. 2 9 4 . 1. A I I Collab., B a r a t e . . et al. (1998 ;). A H 98-025. coni , note C F 98-014.
HEFEUIÏNOES
51)7
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>23
INDEX
a b c l i a n Q C D . .<",(1. 3 5 2 353. 355, 358, 3(13 a c c e l e r a t o r s . vi. 5 (i. 8. 2 0 0 213. set also: H E H A , ISH. l . R P , U I C , P S , S L C . S I ' S . TIC V A T I {ON B-facloruiS, 3!)(i b e a m e n e r g y , 220. IT!) b e a m optics, 210 213 d i p o l e m a g n e t s , 21 I Iiha-se f o c o s i o n . 212 <|iiadrtipolc m a g n e t s . 2 1 0 212. 479 180 s t r o n g f o c u s i n g . 211 2 1 2 . 2 2 0 • I Ti t 180 t r a n s f e r m a t r i x . 2 2 0 2 2 7 . 179 182 b e a m s ! raliliing. 3 8 brcinssl r n h l u n g . 37 38 l u m i n o s i t y , 38. 50. (12. 207. 2(1!' 21(5. 223 221. 230. 2 18 251. 262. 267. 27!» 280. 301. 15«, 192 s e c o n d a r y lieanis, 20!) 210. 2 8 0 unions, 207 n e u t r i n o s . 2 0 0 . 277. 27!) 280 s y n c h r o t r o n r a d i a t i o n . 2 1 0 211 s y n c h r o t r o n s . 210 s y s t e m s . 207 2 1 0 a c c e p t / r e j e c t a l g o r i l I n n . ISO a c t i o n , 25. 2 7 30. 102 Ailler Bell .lackiw a n o m a l y , see
18!), I'M 190, 205. 270. 317. 112. 151. 103 101. 107, 10!). 171 172. 178 at N'l.O. 135 137. I 10. 171. 180. 205. •155, 177 178 lieliavionr at l a r g e 205. 177 178, 19!) 500 lieliavionr at s m a l l I 10. I 18 n o n - s i n g l e t . 130. 155. 107 108 at I.O, 130 131 N I . O . 1 13 •singlet. 130. 155 N I . O . 143 115 spaci'-like. 15. 130 t i m e - l i k e . 15 a n a l y t i c c o n t i n u a t i o n , 57. 69, 87, 131 133. 100 a n g u l a r o r d e r i n g . 112. I!)l l!>0. 1 9 8 . 2 0 2 . 2 0 5 . 3 2 5 . 3 0 5 3 0 0 . 308. 3 7 2 . 3 8 0 , 3 8 8 . 3 9 ! ) 101. 177 p h y s i c a l p i c t u r e . Hill. 205. 175 170 ¡ummulies g a u g e . l(i 17 g l o b a l . 17 a n o m a l o u s d i m e n s i o n s . 137
227, 18(1,
210. 252, 295.
I )(.•!. A P, 137 139. 177. 107 170 g a u g e p a r a m e t e r . < \ . '.13 il l m a s s . - . „ , . 9 3 98. 100 101. 175 170, 130 137. 101 102 w a v e f u n c i ion. 7 , . 9 3 9 8 a s y m p t o t i c f r e e d o m . 3. 03. 95. 9!) 100.
anomalies A G K cut till); rules. 201 o decay. 210
118
atoms, 1 a u x i l i a r y lields. 20 Avagadro's n u m b e r . 280 avalanche. 215 a x i a l - v e c t o r c o u p l i n g . <j,\. 2!H 2!)2. 3 1 3 a x i o n , 27 15 h a d r o n s . 182. 220. 38!) decay multiplicity. 378. 390 B — \ l / v + X . Illl b a r y o n n u m b e r . 7 N. 108. 102 b a r v o n p r o d u c t ion. 390, 102. 100 b e a m p i p e . 19. 213. 223. 231. 232. 231. 2111. 215, 2 1 0 . 2 5 0 . 252. 202. 200. 207. 271. 2 7 3
52-1
INDKX
be;
••
antII, 37, 17, 18, 01 02. 181. 189. I!)!) 201. 202. 200. 271, 3 2 0 . 329 b e a m g a s .scattering, 231. 23 1 • i-decay. 1. 291 292. 3 1 3 ,1-filiicl ion. 9 3 - 9 8 , 100 101, 103 105. 170. 300. 326, 317, 359 301. 3 0 1 . 130 137, -113. 101 103. 177. •198 Bet h e Bloc!) f o r m u l a . 2 2 0 Bet,ho l l e i U c r p r o c e s s , 218 24!) Bliablia s c a t t e r i n g . 2 2 3 b i n d i n g e n e r g y . 2 0 0 207 B j o r k e n s c a l i n g . 12 13. 1!). 12. Hi). 150 B j o r k e n .r. 10-11, II 12, 121. 123. 172 173. 215 217. 258, 271. 277. 280. 403. 105 B o s e K i n s l c i n s t a t i s t i c s , 75. 7(i B o s c - K i n s t e i n c o r r e l a t i o n s , see corrclat ions b o s o n g l u o n f u s i o n , 18. 122. 132 133. 170. 200. 271. 270. 101 100 Breit f r a m e . 3 7 5 Breit W i g n o r p e a k . 3 9 3 b r o m s s l r a l i h m g s p e c t r u m , 36, 155. 158 100, 237, 352. 355, 303. 378. 392 b r o k e n s c a l e i n v a r i a n t 9 5 !)|i. !)!». Illl B U S s y m m e t r y . 90, 108
cliii ality. 101, 133 < ' l u i d a k o v e l l e c t . 100, 192 1 9 3 . 2 0 5 . 475 470 ( ' K M m a t r i x , 205, 280 287, 427 C a b b i b o a n g l e , 173. 4 5 1 ('lebscli G o t ' d a n coellicienls. I0(i 107 C l i l l o r d a l g e b r a , 05. 433. 101 c o l l i n e a r s i n g u l a r i t y , sic d i v e r g e n c e . col l i n e a r f o l o i i m b r e g i o n . 53, 55 c o l o u r . 15 21
C ' - p a r a m e t e r . 3 2 3 321. 327, 329. 333, 193-491 C a b b i b o s u p p r e s s i o n . 280 C a l l a n ( i r o s s r e l a t i o n . 13. 121. 127. 278. 451. 103 C a l l a n S y u i a n z i k e q u a t i o n . 90 c a u s a l i t y . 30. 425 ceut.re-of-iiiomenliiin s y s t e m , 34 C K U N . 57. 02. 200. 208 209. 250. 279. 280. 283, 292. 321. 4 1 3 c h a i n r u l e . 9 3 , 90, 461 c h a r g e c o n j u g a t ion. 27. 31. 85. 120. 131. 130. 150
n u m b e r of. 15-19, 21. 25. 01. 358. 140 c o l o u r a l g e b r a . 0 4 . 7!) 80. 123. 128. 311, 440, 159 t r i c k s , 80. 155, 4 2 2 - 4 2 3 c o l o u r a n t e n n a e . 17. 155 150. 190. 195 190. 198. 199, 201. 205. 398. 4 7 5 177 c o l o m c h a r g e o p e r a t o r s . 154. 178. 171 c o l o u r c h a r g e s . 2. 23 24. 01. 80. 9 2 . 98, 155. 160, 178, 193. 344. 119 120, 123 124. 130 437, 442, 4 5 0 152. 471 r a t i o s . 21, 61. 3 4 5 310, 319, 370 377. 382. 384, 386, .589. 3 9 1 . 423 124. 430, 142, 458. 499-500 c o l o u r c o h e r e n c e , vi. 80. 156, 100, 177. 190. 191 190, 205, 302. 305. 366. 308. 372. 37!) 380. 3 9 8 102. 112. 170 471. 4 7 5 477, 501 c o l o u r d i p o l e m o d e l . 157. 15)0 200. 202 linked d i p o l e c h a i n m o d e l . 200 soft r a d i a t i o n m o d e l . 197, 199 2(H) c o l o u r dipoles. 17. 107. I II. 143. 154 150. 190. 190. 3 7 9 . 110 111 b r a n c h i n g p r o b a b i l i t ies, 198 d e c a y o r i e n t a t i o n . 198 p h a s e s p a c e . 196-199, 205, 178 IT!» s p l i t t i n g p r o b a b i l i t i e s . 190 197, 2 0 5 . 178
c h a r g e d c u r r e n t i n t e r a c t i o n s . 38. 48, 173. 219, 2 3 0 . 201 20!I. 2 7 7 288. 303. 304. 427, 154. 18!» 191 c h a r g i i l l e p t o u nucleoli s c a t t e r i n g . 8 I I. I!) 21. 3 8 40, 48. 2 0 0 277. 281. 285, 290 292. 299. 3 0 3 305. 320. 188 491 chemical elements. I C'lierenkov r a d i a t i o n . 2 1 7 c h i r a l p e r t u r b a t i o n t h i o l v. vi
c o l o u r recoiinoction. 110 112 combinatorial background. 393 c o m p l e x a n g u l a r m o m e n t iini. 57 computer algebra. 80 c o m p u t e r p r o g r a m s . 30. 17!). 237. 301 c o n f i g u r a t i o n s p a c e . 158 c o n f i n e m e n t . 10. 32. 99. 111. 228. 4 1 3 c o n f o r n u i l i n v a r i a n c e . 173 c o n s e r v a t i o n laws, 100, 105, 171. 172, 181, 183. 189, 288. 102
Mt i
INDKX
a n g u l a r m o m e n t 11111. 107, 267. 313.
•180 l!K) b n r y o u n u m b e r , 102 c h a r g e d weak c u r r e n t . 28!) (i >!• m i . (>.'{. I.VI, Ift.1».158 Kid. 1 9 3 . 3 1 1 . •177 e l e c t r i c clinrge, I I. 105. I l l , 28:5 e n e r g y m o m e n t u m , '16. 1 .'IK. 139, 1 12, 152. 162. 176, 197. 370. 377. 383, 386, HIS. 126. 167 169, 171 llavotir, 13, 138, 139. 170. 177. 180. 102. •168
lielieity. 19ft st r a n g e n c s s . 111 I eonsl ¡1 uent (pmrk m o d e l , 6 8, 15 17. 13, 16. S3, 55. 111. 118. 107. 189. 258. 120 121 U i r y o n d e c u p l e ) , 8. 15. 121 m e s o n o c t e t , 121 conloiir i n t e g r a l s . 137. 205. 130. I I I . 169 171. 176 177 correct ions. 239 21 I. 219 a c c e p t a n c e , 210, 218 bin-by-biii. 211 2 12. 218. 25-1 d e t e c t o r , 210 212. 255 256, 3 2 6 , 351. 353. 356 357. 359. 379, 3 8 0 hadronizaiion. 172,211 211.326, 319 351. 353, 356, 358, 359, 379, 381, 386 M o n t e C a r l o i n d e p e n d e n c e , 211. 256 t r a n s i t i o n m a t r i x , 211 212 c o r r e l a t i o n s , vi. 157. 393, 101 lit) a z i m n t h a l angle. 191. 201. 3 9 3 b n r y o u n u m b e r . 168. 201. 393. 102 105 aziiiml lial angle. 103 p o l a r angle. 103 101 r a p i d i t y . 103 Hose Kiuslciu. 167 168. 393, 102. 106 111). 112. 501 5 0 5 (¡nussial) s o u r c e . 107 109. 112. 501 505 r e f e r e n c e .sample. 108 109 s o u r c e clinoticity. 106-107. 109. 112. 503 501 s o u r c e size. 100 109. 112. 502 5 0 3 F e r m i Dirac, 102 polar angle. 393 rapidity, 57. 201. 3 9 3 s p i n . 352 s t r a n g e n e s s . 393. 101 106 r a p i d i t y . 101 -106 c o s m i c rays. 192. 267
covai iiiltt d e i i v a l i v e . 21 22, 2 5 26, 29, 172. 152 C P violation. 27 cross sections
fli — ih, 211
INDKX
.lac(|llet Hloudel m e t h o d , 2-16 2-18, 257. -181, 187 signta m e t h o d , 217 219. 257. 187 188 k i n e m a t i c s . 9 10, 18 1 9 , 6 7 , 120 121. 172 173. 229 230, 233. 215 218, 2 5 7 - 2 5 9 , 267, 271. 276 277. 279, 286, 3 0 3 301, 329. 153. <163 165. 181 189. •191 192 l e p l o n t e n s o r . I 19 physical p i c t u r e , 20. 17 18 s i n g u l a r i t y s i r u c l lire, 122 s t r u c t u r e timet ions, sec s t r u c t u r e functions target region. -17 50, 239, 186 Demokritos. I d e n s i t y effect, 220 D K S Y . 38, 22!), 260 d e t e c t o r a c c e p t a n c e . 61. 2-10 211. 2-13, 218. 255. 256. 309. 32 I. 3 3 5 . 399. I l l d e t e c t o r ellicienc.y, 321. 156 d e t e c t o r r e s o l u t i o n . 107. 110. 230. 2 1 0 - 2 1 1 . 2-13. 217 218. 251- 251, 256. I l l (.¡aussiau p e a k . 3 9 3 d e t e c t o r s i m u l a t i o n , 210. 2-13. 217 2-18. 252. 25 1 -'255, 3 3 5 d e t e c t o r s , vi, 5. 2 1 3 226, 228, 210. 211. 392 A I . R P l l . 223 226 c a l o r i m e t e r s , 2 1 5 220. 228 231. 2 3 3 235, 238, 250 252. 255.
286. 101 c o m p e n s a t i n g , 2lit e l e c t r o m a g n e t i c , 2 1 6 219. 223, 226, 253, 296 297, 392 e n e r g y scale. 230. 213, 252 251. 262, 275 h a d r o n i c . 216 219. 2 2 3 - 2 2 1 . 226, 253. 296, 3 9 2 invisible energy. 21!) m o n o l i t h i c , 21 7 218 p r o j e c t i v e g e o m e t r y . 216. 2 3 8 s a m p l i n g . 217. 223. 227. 182 ( ' I ) I I S . 219 220 IV s. 276, 312 e l e c t r o n s . 223. 226. 230. 233 235. 267. 295, 392, '103 r/. 3 9 3 fat f o r w a r d . -19. 56, 271 273 g r a n u l a r i t y . 107. 1 16, 221 225, 229. 236 high / . ! leprous, 22!). 231 235, 291. 312
h v p e r o n s . 392 j e t s . 2311, 235, 213, 2 5 0 251. 296 . 1 / 0 . 393 K - s , 222, 227, 3 9 2 . 393. 103. 182 K'j' s, 392 K¡is, 312. 392. -101 K ( 9 8 2 ) ° \ 393 As. '101 m i n i m u m ionizing p a r t i c l e , 2 2 0 missing /•;_ . 230, 231. 235, 2 6 6 267, 291 u n i o n s . 219 220. 2 2 3 22-1, 226. 231, 235. 286. 292, 295. 392 n e u t r i n o t a r g e t s . 279 2 8 0 n e u t r i n o s . 21!) 220, 223, 226, 230. 231. 235. 210. 266, 291, 3 9 2 n e u t r o n s . 392 f a s t , 219 slow. 21!) p a r t i c l e i d e n t i f i c a t i o n , 215. 221 223. 228 230, 262. 393. 102 103 Olierenkov r a d i a t i o n . 221 2 2 2 . 227. 22!), 182 183 d / i / ( I t . 221. 223, 229. 102 KICI I, 222 223 t h r e s h o l d c o u n t e r s , 222 time-of-l light. 221. 223. 227. 229. Is2 photoiilillt¡pliers. 217 p h o t o n s . 226. 230. 231. 235, 296 297. 392 - - s . 222. 226 227. 3 9 2 . 393, 103,
181 182 - ° s , 273, 2 9 7 p r o t o n s . 222. 392. 103 s c i n t i l l a t o r , 217 slow n e u t r o n s . 21!) t a u s . 392 t r a c k i n g devices. 215 210. 21!). 223 221. 227 231. 251. 297. 392. 180-182 b u b b l e c h a m b e r s . 215 cloud c h a m b e r s , 2 1 5 multi-wire proportional chambers, 2 1 5 216. 2 2 3 m u l t i p l e s c a t t e r i n g , 220 221, 227,
•181 182 p h o t o g r a p h i c e m u l s i o n s , 215 r a t e of ionization, d/v'/d-r. 220. 2 2 3 silicon vertex d e t e c t o r s , 216, 223. 392 tin»- p r o j e c t i o n c h a m b e r s . 2 1 5 216. 221. 223. 226. 255 d e u t e r i u m . 260 d i f f r a c t i o n region. 53. 55
V2H
INI H X
dllfriielivc d e e p i n e l a s t i c s c a t t e r i n g . •19-52.
200
f a c t o r i z a t i o n , 51 g e n e r a l f o r m a l i s m . .M k i n e m a t i c s . 49 51 physical p i c t u r e . 5 0 51 si i n c l u r e f u n c t i o n s , sec s t r u c t u r e functions, ililfraclive d i l l u s i o n , I 12 d i l o g u r i t l i m , 117, 112 d i h i t i o n f a c t o r . 291 iliinensional c o u n t i n g tuli-s. 5 3 5 5 d i m e n s i o n a l r é g u l a r i s a t i o n . «1. 9 5 . 107 111. 125, 128. 129 131 a n i s o t r o p i c i n t e g r a l s . 131 lumie Integral. I 13. 131 •y-niiil rices. 175. 131 133. Kit) 101 i s o t r o p i c i n t e g r a l s . 112 113. 175, 130 131, .1I»0 M'alcless i n t e g r a l s . 82. I l l 112 I m n s l a t i o n a l i n v a r i a n c e . 112. 130 unii m a s s . //. 87. 89 91, 93 98. 103. III!). 175. 129. 132. Kill d i m e n s i o n a l t r a n s m u t a t i o n . 99 d i p h u m p ' s t r u c t u r e , 55 5 6 dipoli- f o r m u l a , 11 12 di<|iiarks. 161. 165. 166. 168 I l i r a c e q u a t i o n . 71. I I I . 121, 131. 151. 159. 163. 165 direct p h o t o n s . 37, 312, 3 7 2 3 7 3 d i s t r i b u t i o n f u n c t i o n s , 15, 121, 125. 129 130. 176. 183. 161 lleaviside s t e p function. 1 9 1 , 3 2 1 325 p l u s p r e s c r i p t i o n , 15, 125. 170, 113, 115. 163 10 I. 166 108 divergences c o l l i n e a r . 3 3 31. 00 01. 72. 100 110. 116. 117. 120. 123 127. 130. 139. I l l 115. 117. 150 151. 153. 155. 150. 1 5 9 - 1 6 0 . 177. 179 181. 190 191. 193. 191. 190. 198. 235. 323. 132. 105, 171 172 g a u g e . 2 8 29 soft . 33 31. 30. 72. 82. 106 110. 117. 123 125. 130. 132. 139. I l l , 117. 153 150. 159 100. 178 181. 183. 190. I ' l l 195. 198, 235, 323, 337. 132. 165. 173-171 u l t r a v i o l e t . 28. 33. 8 1 . 8 2 . 87 8 9 . Illl 107. 111. 122. 130, 150, 170. J29, 132. 102 103
d o u b l e a s y m p t o t i c s c a l i n g . I II d o u b l e c o u n t i n g . 80. 198 d o u b l e leading l o g a r i t h m a p p r o x i m a t i o n . su l e a d i n g login it Inns, d o u b l e d o u b l y resolved -y-y i n t e r a c t i o n s , 37 3 8 Drell Van l.cvy c r o s s i n g . 173. 15 I d u a l t o p o l o g i c a l unit at'¡nation m o d e l . 201 D y s o n r e s u m m a t i o u , 8 0 87 early universe. I l l elicci ive s l o p e , 50 e i k o n a l a p p r o x i m a t i o n . 153 151. 178, 173 171 e l e c t r i c f o r m f a c t o r . I I . 53 elect r o m a g w l ic force, 1 3 eliH'l rouiagiiet ic s h o w e r s . 2 1 7 219. 221. 223, 230. 273, 182 r a d i a t i o n l e n g t h . 2 1 7 219. 221. 223 e l e c t r o n s t r u c t u r e f u n d ¡on. 30 e l e c t r o n p o s i t r o n a n n i h i l a t i o n , v. vi, 31. 31 3 8 . 198. 2 0 8 2 0 9 , 223. 228 232. 239, 305. 3 3 3 3 12. 3 1 0 . 393 398. 100. 101. 113 at L O . 02 70. 100. 109. 123 at N I . O . 02. 70 71. 106 II I. 121 123.
128. 161 f o r w a r d b a c k w a r d a s y m m e t r y . 35. 6 9 . 203 201. 3 3 5 f r a g m e n t a t i o n f u n c t i o n s . 397 3 9 8 h u d r o u t e n s o r . 00. 0 9 72. 108 109. 103 invariant m a s s - m u l t i p l i c i t y d i s t r i b u t i o n . 231 232 kinematics, 07 l e p l o n t e n s o r , liti. 09 73. 108, 15,s 159 s. 31. 228. 323. 37!), 391-390 r a d i a t i v i ' r e t u r n s . 37 r e s o n a n c e r e g i o n , 31 /•• -,. 18 19. 103 105. 30!I 311, 3 3 0 333, 138 -I II ! ! , . 230. 309 31 I. 3 3 0 333, 3 0 0 302, 301. 130. 138 111 s i n g u l a r i t y s t r u c t u r e . 71 72. 71. 107 100, I 10 l o f o u r f e r i n i o n s . 231. 3 5 5 l o f o u r j e t s . 31. 195. 318. 351 359. 301 3 0 1 n . u . 353. 355 350 B e n g t s s o n Zervvas a n g l e , 352 353. 355, 350 Isomer Schierholz Willrodt angle, 355 N n c l u m a u n Ucilcr angle. 353 350
INDEX
l u t a i r u l e , 319, .152 l o h a i l l o n s , 17 I!), 31 37. 17, 02, 100 117. 151, 157 101. 109, 2 3 0 238, 210. 255. 309 310, 3 1 0 329, 310. 3 6 5 375, 377, 3 8 3 . 381. 102 III). 138 139 m o m e n t u m s p e c t r u m , 203. 3 6 0 - 3 6 9 m u l t i p l i c i t y d i s t r i b u t i o n . 365. 369 372 t o h e a v y q u a r k s . 378, 39-1. 3 9 0 l o j e t s , 31. 110 117. 203. 225 220, 230, 237 239. 2 1 2 213, 3 3 0 332. 3 7 3 375. 3 7 7 - 3 9 1 r a t e . 251 257. 3 2 5 3 2 0 t o l e p t o n s , 231, 13!) e l e c t r o n s , 231 m u o n s . 18 1 9 , 3 1 . 2 2 5 , 2 3 1 . 3 0 9 . 138 lati l e p t o n s . 2 2 5 2 2 6 tans. 255 t o p h o t o n p l u s j e t s , 372 373 t o p h o t o n p l u s t wo j e t s . 1 5 5 - 1 5 6 . 377. '100, 112, 501 t o t h r e e j o t s , 3 3 31. 71. 110 117. 155 156, 178. 195. 30!), 321, 3 3 0 311. 310 351. 358 300. 303, 377 391, 171. 195 c r o s s s e c t i o n . 175, 3 3 0 337, 158 159 d i f f e r e n t i a l r a t e . 110 117. 3 2 5 320 e n e r g y llows. 112. 501 event p l a n e o r i e n t a t i o n , 73. 175, 330. 3 3 9 310. 158 159 inter-jet e n e r g y llow. 3!)!) 100 jet e n e r g y r e c o n s t r u c t i o n . 338, 313. •198 k i n e m a t i c s , 330, 3 9 1 . 49!) A|.;K d i s t r i b u t i o n . 337 338 m u l t i p l i c i t y , 384 t o t a l r a l e . 34, 110 117, 3 2 2 323. 3 4 9 . 351 j':i d i s t r i b u t i o n . 3 3 7 3 3 8 Z d i s t r i b u t i o n , 338 3 3 9 t o t w o j e t s . 34 35, 74, 110 117. 187. 30!). 3 2 2 325. 333. 377. 103. 493-494 a n g u l a r d i s t r i b u t i o n , 310, 343, 195 different ini rate, 242. 307. 3 4 7 318. 359 t o t a l r a l e , 110 117. 3 0 7 . 3 2 2 - 3 2 3 . 3 2 5 320. 349, 351 l o VV 1 VV-, 35. III!) 112 m u l t i p l i c i t y . 110 III t o v;/.. n o
wo
t o t a l c r o s s s e c t i o n , 3 3 35. 08, 108 I Iti. 123. 235 236. 317, 3 5 7 elect i oweak i n t e r a c t i o n s , 7. 14, 27. 32. 38. 02. (¡8, 120, 122, 203. 2 7 7 , 313. 317. 127. 133. 139 4 4 0 . I 10. 189 elect roweak u n i f i c a t i o n , 2 3, 48. 2 0 7 2 6 9 Kills K a r l i n e r a n g l e . 3 3 7 338 K M C cited, 280 e n e r g y t r a n s f e r . ;/, !) 10. 3 8 39. 12. 172 173, 277. 4 5 3 e n e r g y e n e r g y c o r r e l a t i o n . 324 3 2 5 a s y m m e t r y , 325 il decay, 321 Kuelidoan s p a c e , 4 3 0 Killer a n g l e s . 116. 4 9 9 Killer / ¿ - f u n c t i o n , 110. 113. 129, 4 3 1 . 135. 465 Killer I - f u n c t i o n . Ill), 12!). 130. 131. 135. 460, 465 Killer-Mascheroni constant. 9 1 . 132. 435 event d i s p l a y s . 225, 251 252 ALICI Ml. 225. 228 2 2 9 . 2 3 7 C D F , 251-252 DO, 251 111. 260. 262, 264. 2 0 0 e v e n t m i x i n g , 408 e v e n t r e c o n s t r u c t i o n efficiency. 250. 2 7 6 event s e l e c t i o n . 2 3 0 235. 248. 2 5 0 252. 2 5 5 250. 262, 2 6 6 - 2 0 7 . 294 295. 335. 3 8 6 efficiency, 230 232. 231 235, '¿48, 251 252 p u r i t y , 2 3 0 231. 2 3 3 event s h a p e v a r i a b l e s , 157. 161, 102. 203. 2 3 0 237. 239. 211. 243. 305, 307. 313, 321 333, 3 1 1 . 3 4 6 351 event s i m u l a t i o n . 210. 241, 255 2 5 6 e v o l u t i o n e q u a t i o n s . 3 0 . II 17.0(1. 135 I 13. 157. 249, 288. 3 1 7 318 l i F K L , 139 143. 149. I!ll. 273 p r o b l e m s . 142 DCîl.AP. 118. 135 I 10. 142 143. 148. 170 177. 182 183. 190. 194, 2 6 2 263. 20!) 270. 2 7 3 274, 280. 301. 3 1 0 318. 167. 109 h i g h e r o r d e r . 135 130. 174 . 4 5 5 l e a d i n g l o g a r i t h m s . I 18 Monte Carlo implementation. 181 189. I ' l l 195, 175
I N I II;.\
MO
non-singlet. 138, 171, 177, 455, Ili* physical i n t e r p r e t a t i o n . 2(1. 1.1 17. 118, 137, l ili 119, 183 181. 181» 187. 189 singlet, 171, 177. 15ft. '168-469 space-like. 1 I 17. 51. 118, 135 137, 171. 182 181. 187 189. 155 lime-like. 15. 136 137. 182 183. 185 187. 388. 389, 391. 199 500 f r a g m e n t a t i o n f u n c t i o n s . 317 319 generalized. I K). I 12 C C I ' M , I 12, 200 C I . H . I 12 1 13. 1 19 non-singlet. 285 polarized, 291 singlet, 285 exotics, 389 experiments 12866,
290
AI.KI'II. 228 229. 232. 255 257, 307. 301 361. 380 381, 381 388. 390, 395. 390. 399 UK). 103 100. 108-109 A KG US. 390 al DKSY. 113 al MURA. 211 250. 258. 260. 320 al IS II. 55, 250 at LICK 251 257. 31(1, 362, 363. 378. 382. 388, 102. 106. I l l at Ì.IIC, 260 al S I . A C . 299 al S I ' S , 250 a l T E V A T R O N . 250 251. 260 Ì K ' D M S , 281. 285, 299 BICIÌC. 280. 283. 289 C C F R , 279 280. 285 286. 289. 313. 320 C D F , 55, 251 252. 251. 295. 297 298. 321. ;IOO 101 C D I I S . 286 C D I I S W . 279. 281. 285 C U A R M . 280 C H A R M II. 280
CIISW. 280 ("LlCO. 315. 3 8 3 C O M I ' A S S . 292 IXI. 250 251. 297. 100 101 D E L P H I . 232. 327 328. 3 8 3 381. 388 391, 395. 396. 100 Ivi 10. 281 IÎ554, 273 IC665. 271. 273. 285. 299 li706. 296 297
E866, 292 291 K M C . 281 I I I . 217 250, 258. 261 262. 265 208. 271, 271 277. 285. 299. 329. 397 11 ICR M ICS. 291 JADIC, 237. 399, 100 1,3, 232. 395. 396. 399 100 M A R K II. 395 NA51, 292 N M C . 262, 290. 299 N u T e V , 279, 3 1 3 O l ' A L . 232 233. 382 381. 390 391. 395 396, 3 9 9 - 1 0 0 . 103, 406 S L A C - H C D M S . 320 S I . A C - M IT. 1 2 - 1 3 SUD, 232. 395. 396 TASSO. 399. 100 TOPAS!. 139 I J ' C . 399. 100 U A 5 . 200 U A 6 . 297. 321 VVA7. 55 VVA70, 297 W i l l ' F O R , 280 ZICUS, 258. 262, 266 267, 271. 299. 329, 397 e x p o n e n t i a l integral f u n c t i o n . 165 f a c t o r i a l m o m e n t s , 365 f a c t o r i z a t i o n . 32. 3 5 36. 12. H . 17. 59 01 118. 126 128. 132 136, I 12. I 19. 151 152. 177. 181. 299. 172 173 cocllicicnl f u n c t i o n s . 118. 127 for DIS. M . 120 128. 133 135. 166 for Droll Yan. 151. 177. 172 173 for e l e c t r o n p o s i t r o n a n n i h i l a t i o n . 36. 115 116 k'f f a c t o r i z a t i o n . I ll r e g u l a t o r . 121 126. 128. 132 scale. 35 36. I I 15, 51, 60. 118. 126 127. 135-136. 139. 152. 275. 297. 317 319. 115 I 16 s c h e m e . 36, I I. 60. 126 128. 135. 151 152, 259. 3 0 1 - 3 0 2 . 106 DIS. I I . 127 128. 132. 131 135. 152. 177. 279. 281. 166. 172 173 MS. 132 MS. 127 128. 132 135. 151 152. 177 301. 106. 172 173 fail d i a g r a m s , I 19 Fastest A p p a r e n t Convergence. 105
IN DUX
n u m b e r of. 358 359 polarization, 352 .<51 p r o p a g a t o r . 23. 2!) 31. 77. 78. «1. K(i 87. 92, 125 polnriy.nl ion t e n s o r . 158 tensor s t r u c t u r e , 87 88. 175, -125 self-energy, 78. 81 81, 87. 1(12 tensor s t r u c t u r e , 81 spin, 3 I. 31. 75. 100. 305. 30(1. 331 310. 313. 370, 377, 128, 131, 112. 195 197 g r a n d unificntion. 3 ( ¡ r a s s n m n variables. 29 graviton, 3 gravity, 1, 3 ( ¡ t e e n ' s f u n c t i o n s . 90. 91. 93 90 g r o u p theory, 115 121 abelian g r o u p s . 21. 310. 115. 118 axioms, 115 compact groups. 118 c o n n e c t e d g r o u p s . 110 exceptional g r o u p s , 123 121 l.ie g r o u p s , 7 8. 25, 311. 115 121. 130 liaker-('ampbell-1 lausdorlf f o r m u l a . •117 C a s i m i r o p e r a t o r s . 193. 311. 119 120. 123 completeness relation. 79 generators. 22. 21 25. 82. 311. •110-123 J a c o b i identity. 70. 195. 118 11!). •123, 171 l.ie algebras, 22. 7!), 117 119, 422 s l r i i c t u r e c o n s t a n t s . 22 24, 78 80, 82. 344, 117 119. 422 423, 451 o r t h o g o n a l g r o u p s . 410. 423 424 S O ( 2 ) , 346 S O ( 3 ) . 346, 116 S O ( N ) , 418. 124 r e p r e s e n t a t i o n s . 1. 7. 75. 345, 415, 118-421. 423 a d d i t i o n . 420 421 a d j o i n t , 29, 75, 78, 1(H), 173, 176. 344, 358. 419 421. 423 Clebsch G o r d a n coefficients, 420 c o m p l e x . 119 Dynkiti index. 417. 120. 423 equivalence. 119 f u n d a m e n t a l . 7. 22. 79. 100. 176, 344. 358. 415. 419 421. 423, 424 irreducible. 420 real, 419 root d i a g r a m s . 420 121
s e x t e t . KHI Schiir's lirst lemma. 4 l!l s e m i - s i m p l e g r o u p s , 418 syniplcctic g r o u p s . 123 124 u n i t a r y g r o u p s . 25. 416, 121 424 U ( l ) . 2. 25 S U ( 2 ) . 2, 8, 346, 419. 422 colour SU(3), 2, 4. 15 10. 21 24. 75. 98. KHI. 101, 305. 341. 340. 350, 300, 302 361. 413. 422. 436, 442. 451. 4 9 9 - 5 0 0 llnvotir St 1(3), 7 8, 21. 136. 138. 174. 287 288, 419 122 S U ( 6 ) , 167 Sl.!(N). 25. 100, 136. 34 I. 118, 421-424
h a d r o n c h a r g e d i s t r i b u t i o n . II 42 liadron shrinkage. 56. 58 h a d r o n h a d r o n s c a t t e r i n g , v, 31, 38. 51 62. 118. 128. 149. 198. 199. 230. 234 235. 237 239, 260. 292 299. 305. 320 321. 333. 400-401, 400 dim-active. 52-5-1. 17 I. 200. 155 456 central dilfrnction. 57 d o u b l e d i l f r a c t i v c dissociation. 54. 56 57 h a r d , 52 57 single d i l f r a c t i v c dissociation, 54. 56-59 direct p h o t o n s . 6(1, 230, 234 235, 253. 200, 292, 295 299, 302, 303. 321. 330 333 a s y m m e t r y . 321 p r o b l e m s , 297 d o u b l e s c a t t e r i n g . 61 02. 201 Droit Van. 52. 02 03. 09. 106, I 18. 122. 1 19-153. 2(H), 206, 230, 234 235. 253. 260. 292 295. 298. 299. 301 at I.O. I 19 150, 152. 177, 472 at NIX). 150-152, 177. 472 173 kinematics, 149 150, 177, 472 plus j e t s , 239 t r a n s v e r s e m o m e n t u m , 152 153 elastic. 52 59, 174, 456 h a r d inelastic, 52 53. 59 02. 201, 250 heavy q u a r k s . 52. 60 61, 299. 321. 330 333 plus j e t s . 321
INDEX
l'ermi Constant, <.',.. 38, 1«, 265 l 'ermi mot ioli. 201. 280. 281 intrinsic k - r . I l i , 152, 189. 296 -297 Fermi D i r a c s t a i istics, 29. 78 FICUMILAD, 61. 208. 250, 273, 279, 293. 296 F e y n m a n p a r a i n e t e r s . 81. 87. 112 I 13. •129. 132 F e y n m a n rules. 63. 75, 76. 125 127. 129 s v i n i n e ! r y factors, 8-1. 126 libres. 28 lield st rengt.il t e n s o r , 21 23, 26. 1 13. 172, 152 (Inai, 27 linai s t a l e i n t e r a c t i o n s . 102 line si rilettile Constant, 2, 20. 31. 38. 03. li«. 215. 260. 265, 296. 31 I. 315. 321. 150 llavour ragging, 32, 311 342. 356. 359 lltix faci o r . 67. 72. 119. 1 16. -127 128, 117. 159 llux t u b e . 162 I N ' A L C o l l a b o r a i ioli. 315 four-jet variatile, 352 Fourier t r a n s f o r m a t i o n , 2, 29. Il 12, 70, 152. 158, 168, 106, -112. 502-505 Fourier lìessel l r a u s f o r i n a l i o t i , 50 f r a c t a l s . 197 f r a e l u r e fuilct IOIIS, 51 f r a g m e i i t a t i o n f i m c t i o n s . 35 36. 12, 15. 60. 116. 136 137, 157, 181. 183, 276, 285 286, 310 320. 330 333. 376 377. 388 389. 391 393. 396 398. 112. 112—MG, 199 501 al large ,r. 388 389. 391. 199 500 at Minali .e, 177. 379. 388 389. 1711 171 C a u s s i a u p e a k . 177. 319. 377. 388 389. 3 9 1 , 1 7 0 171 heavy q u a r k liailrons. 397 398, 112. 500 501 lougittidiual. 316 317 n o n - p e r t u r b a t i ve. 162. 161 105. 171. 398. -112. 500 501 left t ight, s y m u i e t r i c . 165. 171. 172 Potersi ili. 276, 398. 112. 500 501 non-singlet, 317 318 p a r a m e l e r i z a l i o n s . 319 320 p e r t u r b a i i v o . 398, 412. 500 s i n g l e t , 317 318 t r a n s v e r s e . 316 317 w a r n i u g . 397
531
I'Yoissarl b o u n d , 53 f u n c t i o n a l integrals. 28 29 F u r r y ' s t h e o r e m , 85 86 7 - i n a t r i c e s . 132 133 a l g e b r a . 65 66, 109 7;,, 61 65. 4 3 3 tricks. 66. 71. 131. 144-1-15, 175. 132. 448. 459-461. 465 gauge axial. 3 0 31. 77. 182. 125 coupling, 25. 31. 33. 90. I 13, 1611. 31 I. 429, 160 («.variant. 29 31. 79. 81. 91. 125 F e y n m a n , 30, 63, 75. 88. 154 L a n d a u , 92. 91 fields, see gl n o u s fixing, 27 -30. 84. 87. 911, 108. 125 C r i b o v a m b i g u i t y . 28 g r o u p , 340. 341 invariance, I. 25 27. 10. 62. 69- 72. 71 77. 81. 82. 84, 91). 1118. I 11, 119. 172. 199. 310, 125. 152.
-159 p a r a m e t e r , i . 28 30. 69. 81. 82. 92. 9 I. 9 6 - 9 7 , I 12. 125. 127 physical, 79. 117 t r a n s f o r m a t i o n s . 25 29. 70. 172. 152 g a u g e theory, v. 1 3. 21 31. 71. 100. 161). 305. 331. 341. 413. 419. 121. •136 C e l l - M a n n matrices, 22 21. 150 152 g h o s t gluon v e r t e x , 29 31. 78. 88. 125 126 corrections, 92 gliosis. 2 8 - 2 9 , 76 79 colour charge. 29. 78 p r o p a g a t o r . 29 31. 88. 125 s p i n . 29. 78 global s y m m e t r y t r a n s f o r m a i ion, 25 glue balls. 59. 389 glners, 159 160 gl i linos, 176. 316. 3 19. 351. 358. 361 361. -162
bound states. 363 g l u o n r e c o m b i n a t i o n , 112 143. 119, 28(1 gluon gluon s c a t t e r i n g . 33. 76. 171 175. •157-458 glilons. 3 I. 21 25. 32. 17. 100. 1 18. 292 charges, 1. 19, 21. 12. 75. 118. 122. 258 colour charge. 23 25, 64. 197. 314. 334. 376 m a s s . I. 26 27. 77, 82. 108
INDIA
jell», 52, I 10, 230, 231 235, 239, 250 251. 292. 294, 299. 332. 378 angulai d i s t r i b u t i o n s , 239 di-jets. 01. 03. 171 175, 198 199. 239. 251. 298 299. 157 458 inclusive rate. 250 251. 260. 297 298. 321, 3 3 0 - 3 3 2 t h r e e jels. 4(H) 401 t r a n s v e r s e energy s p e c t r u m . 239 kinematics. 00 62, 171 175. 257. 2 9 2 - 2 9 3 . 295. 298 299. 455 158. 484 pedestal effect. 01 02. 200 201 - " s . 290 297 resonance region. 53 soft inelastic, 52, 57 58. 01 02. 2(H), 201. 250 t o t a l cross section, 52 54. 58 59. 171. 252. 450 W p r o d u c t i o n . 294 295 plus j e t . 401 r a p i d i t y a s y m m e t r y , 294- 295 h a d r o n i c showers. 218 219. 223 221 interaction length. 218 219, 280 h a d r o n i z a t i o n . vi. 5, 17 18. 31 33. 17, 150 172. 181. 182. 202. 237. 239. 241. 314. 322. 339, 302. 373 374, 384. 387, 389, 394. 390, 1(H). 411 b a r y o n s u p p r e s s i o n . 101 102. 100 107, 171. 396 hadron selection rules. 102. 100. 171 inside-out p a t t e r n . 159. 103 104 /»/- suppression. 102. 166 107. 171 spin d e p e n d e n c e . 106, 17(1 1 7 1 . 3 9 0 . 3 9 8 strangeness. 405 si rangeness suppression. 101. 100 107, 171. 394 390 l ime scales, 158 Kid h a d r o n i z a t i o n models, 31. 157 158. 101 172. 181. 203. 228. 242 244. 322. 389. 391 393 cluster. 101. 108 172.201 2 0 3 . 3 7 5 . 391. 402 104, 400 cluster mnss s p e c t r u m , 169 170 problem cases. 170 172 c o m p a r i s o n . 171 172 i n d e p e n d e n t . 101 102, 171 172. 201, 202, 375, 399, 401 conservation issues. 162. 171, 172 t e r m i n a t i o n criterion, 162
533
s t r i n g , 150, 102 108. 171 172, 190. 201 203. 375, 399, 102 403 a c t i o n , 162 area decay law. 103 165, 107 A r t r u Mennessier. 104. 167 CalTech II. 164 165, 167 classical d y n a m i c s , 162 103. 105. 107 decay t o clusters, 104 105. 107 108 decay t o h a d r o n s . 104 107 d i q u a r k iiicclmiiisin. 108 family tree. 107 gluoiis. 103 104. 190. I l l light s t r i n g s . 171 I . S I T . 105-107 b u n d . 162. 164 167. 171 1 7 2 , 2 0 2 . 239. 399. 403. 100. 111 p o p c o r n m e c h a n i s m . 108. 103 101. 400 spin-/).;- correlations. 107
UCLA, 104, 100 107, 171 172 t u b e , Kit heated filament. 209 heavy ions, vi. 201, 207 heavy (¡narks, 14. 91. 101. 102. 319. 320. 378. 390. 392 p r o d u c t i o n . 52. 149, 100, 200. 38!). 391. 390 weak decays, 100. 182. 321. 405 helicitv. I I. 01 05, 08. 101. 303. 301. 353 354. 189 490, 495 I IKK A. 38. 49. 52. 179. 203, 229. 233. 239. 244. 257. 258. 200 262, 265 207. 271. 273. 291. 375. 397 b e a m energies, 245 Hermit icily. 418 119. 121 422. 121 Iliggs particle. 3. 311), 301, 110. 139 hot. s p o t s . I 13 livpercharge strong. 7 weak. 22 hyperons. 392, 402, 403 idenl¡lied pnrlicle.s, 389 191. .191 III« correlations, .in correlation» m o m e n t u m «peclin. I', III, ' i n • 1.1 391 IU| mu lun u n mullipllellIn. I V .(III J,|ll, li.n m. lini volili Itili IDI III I. mil convelli lull III I Ili'iiVV HlliH'h limili,li. Inn IIII Ulti Immilliti»
lumini
Hill
Hill IDI
Hill
liti
Hi.
UNI
r,:t i
o r b l t n l l y e x c i t e d I m d r o n s , 39-1 3 9 5 p h o t o n s , 391. 395 p i o u s . 390, 391. 395, 400 i m p o r t a n c e s a m p l i n g , 179 index of r.•fraction. 222. 183 inelasticity, ;/, 11. 172 173. 215. 258. 277. •153 i n f r a r e d c a n c e l l a t i o n s . 33. 106 117. 121 122, 130, 183 iufriired c a t a s t r o p h e . 11 1 i n f r a r e d cnt-olTs. 112. 157, 162, 171, 183 185. 203. 201 i n f r a r e d s a f e o b s c r v a b l e s , 33. 101. I l l 117. 120, 125, 127. 180. 3 2 3 326, 333. 193 191 i n f r a r e d safety, 90. 105 117. 123. 153, ISO 181. 235 236. 239. 257, 323 326. 183 181 i n f r a r e d sensitive o b s e r v a b l e s , 33. 110. 121 128. 153 I n f r a r e d u n s a f e observaliles, 1 to. 323. 333. 193 i n s e r t i o n c u r r e n t , 151, 178. 193. 205. 100. •173-474, 477 integral e q u a t i o n s . 185 i n t e r f e r e n c e region, 53. 55 inlorinittoncy. 365 366 inlet-quark p o t e n t i a l , 314 ionized h y d r o g e n . 209 i s o m o r p h i s m . 115 isoscalar t a r g e t s . 2 7 8 - 2 8 0 , 286, 28!) isospiu s t r o n g . 6 - 8 . 13 I I. 278, 286. 290. 191 w e a k . 2. 3. 13. 21. 22. 173. 127. 4 1 0 . 446. 454 ISH, 297 •laeobian. 28 29. 153 jet a l g o r i t h m s , 237 239. 271, 296. 322. 323. 3 2 5 - 3 2 6 . 382. 386, 388-391 c l u s t e r . 237 238, 255, 3 2 5 . 351. 3 7 3 374. 384 D u r h a m . 238. 242. 255. 275, 337. 35», 377. 380, 386. 387, 3 9 9 JADI5, 116 117. 237- 239, 255, 372 m o m e n t u m c o m b i n a t i o n , 237 2 3 8 resolution p a r a m e t e r , 116 c o n e b a s e d , 116, 2 3 7 - 2 3 9 . 250, 3 7 8 dilliculties, 239 resolution p a r a m e t e r . 115, 237 239, 255 250. 356, 372 375, 399 S t e r n u m W e i n b e r g . 110 1 1 7 . 3 7 7
IN DUX
jet jet jet jet jet
b r o a d e n i n g . 347. 351, 387 38S c a l c u l u s . 182 c h a r g e s , 2 0 3 204 f r a g m e n t a t i o n , 254. 255, 392 m a s s , 347 heavy, 307. 324. 327 328, 351 m e a n value, 3 2 8 j e t s , 31 31. 00 61. 156, 229. 2 3 7 - 2 3 » , 241. 305, 316, 322, 325. 332. 398. 409 e n e r g y scale. 362, 3 6 3 m u l t i p l i c i t y , 3 6 2 363. 374 s u b j e t s , see subjet.s .1 /V>. 3 1 5 k i n e m a t i c f u n c t i o n , 174 kinks, sec hadronization models, siring, ghions KI.N t h e o r e m , 114 KN'O s c a l i n g . 201 k u r t o s i s . 383. 171 l a d d e r d i a g r a m s , 143 149, 182 L a g r a n g e m u l t i p l i e r . 492 I . a g r a n g i a n , v, 1. 4 . 0, 21 32. 74 7!). 86. 228, 315, 334 b a r e , 81. 8 9 90. 9 3 classical. 26 27. 2!) c o u n t e r t e r m , 88 89 f o r b i d d e n t e r m s , 26 27 g a u g e fixing a n d ghost t e r m s . 2!) 31 g a u g e t e r m , 26 27. 3 0 31. 76 in D d i m e n s i o n s . 175, 129. Kill q u a r k t e r m , 25 27. 30 31 r e n o r i n a l i z e d . 8 9 90. 9 3 fl-lerm. 27 AQCD- 2 3. 31 32. 99. 101 102. 153. 170. 197. 202. 203, 205, 308. 320. 162, 477 178 l . a i n v ' s t h e o r e m . 198 L a n d a u Vang t h e o r e m , 314, 377 laser i o n i z a t i o n . 209 l a t t i c e g a u g e t h e o r y , vi, 17. 31 I 315. 323, 329 c o n t i n u u m limit. 315 fertnions, 315 L a u r e n t e x p a n s i o n . 87. 5)6, 175. 401 lead-glass. 217 l e a d i n g l o g a r i t h m s . 139 149. 157, 181. 182. 300, 3 6 5 collinear. 139. 142. I 14 148. I'M) d o u b l e . 139 141. 143, 147 l is, 177. 469-470
INDEX
I t C I v 104 105 s o f t . 139, 1 11. I l l , 149, 153 189 196 l e a d i n g p a r t i c l e ellects, 201, 384, 389. 394, 405 least s q u a r e s lit. v. 203, 301, 358 359. 361 b i n n e d , 350 I.KI\ v. 179, 203. 2 0 8 210, 223. 2 2 8 230. 232. 238. 242. 255, 2 5 6 . 3 2 2 . 326. 327. 333. 335, 342. 345. 346. 369. 372. 376 378. 3 8 2 . 394, 3 9 8 - 4 0 2 , 109. 113, 199 b e a m e n e r g y . 212 213 lepto-quark. 268-269 leplon universality, 312 lepton- liadron s c a t t e r i n g , v. 8 15, 31, 3 8 52. 173. 198. 199, 299. 154, see. see also c h a r g e d l e p t o n nucleoli s c a t t e r i n g , n e u t r i n o nucleoli s c a t t e r i n g diffract ive. 233. sec d i l f r a c t i v e d e e p inelastic s c a t t e r i n g elastic, 10. 11-42 g e n e r a l f o r m a l i s m , 3 9 II. 118 liadrol) t e n s o r . 3» 40. 172. 452 153 inelastic, .sec d e e p inelastic s c a t t e r i n g k i n e m a t i c s , 3 8 3», 41. 172 173. 453 lepton t e n s o r . 3 9 - 4 0 p h o t o - p r o d u c t ion, 5 2 q u a s i - c l a s t i c , 42 s t r u c t u r e f u n c t i o n s , see s t r u c t u r e functions I d l e . 57. 62. 179. 234. 413 light q u a r k s , 71. 101, 314. 319, 320. 378. 396 likelihood f u n c t i o n . 357 linear collider, 4 13 linear c o n f i n i n g p o t e n t i a l . 162 liquid A r g o n , 218 liquid K r y p t o n . 218 local s y m m e t r y t r a n s f o r m a t i o n . 25 log-normal distribution. 365 loop d i a g r a m s , 78 87, 89. 101. lllli. 429 432 L o r e n t z i n v a r i a u c e . 81 L o w ' s t h e o r e m . 153 L P I I I ) . 365. 368. 369. 371. 400. 402 l n ( Q 2 ) - l n ( l / x ) p l a n e , 13!) 140 m a g n e t i c f o r m f a c t o r . 41, 5 3 magnetic moments. 4 I M a i i d e l s l a m v a r i a b l e s . 67. 69. 174 175. •157- 458 M a r k o v p r o c e s s , 182. 186 187. 18!)
m a s s s i n g u l a r i t y . 106. sec d i v e r g e n c e . collinear u n i t l i x element m a t c h i n g . 1 8 7 . 2 0 0 . 202 m a x i m u m likelihood m e t h o d , 3 5 6 - 3 5 8 Mellin t r a n s f o r m a t i o n . 137 138. I II. 147 148, 177, 107 471 M e r c e d e s e v e n t s , 3 7 8 . 391. 4 9 9 m e t h o d of c h a r a c t e r i s t i c s , 9 4 - 9 5 m e t h o d of m o m e n t s . 137 139 m i n i m u m b i a s e v e n t s . 01 02. 200 m i n i m u m ionizing p a n i c l e s , 225 Minkowski s p a c e . 430 M l . L A . 190. 196. 319, 362, 365. 368-36!). 372, 3 7 5 modilicd liessel f u n c t i o n . 148 m o m e n t u m s p a c e , 158 C a r l o m e t h o d s . 17!) 1 8 1 . 3 1 5 p h a s e s p a c e slicing. 180 181 s u b t r a c t i o n m e t h o d . 180 u n w e i g h t e d e v e n t s . 180 w e i g h t e d e v e n t s , 17!) 181 M o n t e C a r l o p r o g r a m s , vi. 158. 172, 182. 201 204, 242. 2 5 6 - 2 5 7 . 322, 3 7 2 - 3 7 3 , .375. 3 8 0 . 383. 380 387. 391. 39!) 4111. 111). •Ill AHIAONE. 196. 202. 242. 256. 372. 374. 385, 387
COJKTS. 162. 201. 374. 101 PTIUET, 201 Kitri'ioF, 201
«UNCI., 200 lit.UNCI. 201 HKIUVK;, 169, 200, 202 203. 372. 374. 382, 385. 401. 403 101 1SAJET. 162. 201 202. 401 JK'rSKT. 202. 203, 256. 372. 385, 387. 391. 39!). p a r a m e t e r t u n i n g . 172. 202 322. 373
242. 256, 387. 3 9 1 .
371. 382. 4 0 3 104 204. 257.
I'VTIIIA. 201 202. 242, 3!)!). 401 w a r n i n g , 204 .Vlonte C a r l o s i m u l a t i o n s . 179. 241 242. 275, 276. 280. 283. 287. 301. 356, 3 7 8 - 3 7 9 , 408 M o n t e C a r l o s . 179 204 all o r d e r s , 147. 157. 181 204 lixed o r d e r . 17!) 181. 202, 321 mult ¡ p e r i p h e r a l m o d e l s . 168 natural units, v n e g a t i v e b i n o m i a l d i s t r i b u t i o n , 201
r>:tu
N e u m a n n scries. 185, 187 m ill rnl ciirrent interact ions, 3 8 3!). 18, (¡3. ¿1«), 2 « ) -264, 200 209, 274 277. 127. 4 9 0 n e u t r i n o nucleoli s c a t t e r i n g . 13 15. 19, 38. 40. 48, 173, 277 290. 299. 301, 3 0 3 - 3 0 5 . 320, 454. 490-492 n e u t r o n electric d i p o l c m o m e n t . 27 new physics. 3, 53, 100, 234. 250. 254, 2 6 8 - 2 0 9 . 297, 299, 341. 348 319. 303 304 N e w t o n ' s c o n s t a n t , C>'y, 2 n e x t - t o - l e a d h i g l o g a r i t h m s , 139 140 n o n - p e r l u r b a t i v e physics, vi. 4. 27 28. 31 32. 35, 38. 44. 47. 5 2 - 5 3 , 59, 97, 9 9 - 1 0 0 , 10«. I l l , 11«, 118. 135. 142 143. 157. 158. 108. 181, 180. 189, 200. 202. 204. 228. 239. 270 273. 287 290. 292. 294. 299. 302. 309 311. 313. 314, 310. 318 320. 322. 3 2 6 - 3 2 9 , 334, 335, 301. 302. 373, 375. 392. 397. 109. 410, 413, 441 n o n - P o i s s o t u a u l l u c t n a t i o n s . 305 non-relativist ic Q C I ) . vi N l t Q C D C o l l a b o r a t i o n . 315 nuclear fission. 219 nuclear physics. 6, 20« nucleus. 1. I n u m b e r of a c t i v e llavours. n j , 18. 85. 87. 98. 100 102. 32(1. 344 345. 3 0 3 - 3 6 4 . 436 438. 462. 499 51X1
sr. 7 operator mixing. 93 o p e r a t o r p r o d u c t e x p a n s i o n , II. 305, 309, 311. 322 o p t i c a l t h e o r e m , 5«, 79. 174. 150 I ' a d e a p p r o x i m a t i o n , 306, 313. 315 pails. 215 p a r a m e t e r free ( ¿ C I ) . 99 parity, 7 8. 34, 68. 121. 334. 343. 495 violation, 14. 27. 39 40. 48. 68 69. 267. 489. 490 p a r t i c l e multiplicities, vi p a r t i c l e p r o p e r t y ' s . 1. 213 214 part icle zoo, 7 p a r t i t i o n f u n c t i o n . 28 29
INIH'.X
partoii b r a n c h i n g s . Ill, 71 72, 139. 144 140, 177, 182 183. 19«. 471-472 j d i s t r i b u t i o n s . 45 46. 183, 18« 187. 189 k i n e m a t i c s . I l l 145, 183 181. l i l l . 201, 471 172, 4 7 4 - 1 7 5 p r o b a b i l i t i e s . 183. 185 187. 189 190, 475 resolved, 183 189 space-like. 184, 187 189. 194. 204. 474-475 time-like, 184, 18«. 194 unresolved, 183, 185 187. 189. 475 p a r t o n d e n s i t y f u n c t i o n s , II I I. 19. 32. 35. 42 47. « 0 61. 118-119. 120—128. 134 130, 151, 157, 170. 181 183. 188 189. 201. 250. 259 260. 265 267, 278 279. 292. 304. 320 321. 400, 463, 467—169, 490 192 al large x. 16. 48. 61. 188 189. 262, 264, 267. 283 285. 295 299. 301-303 a t s m a l l .r. 46, 59, 61, 110 143, 174. 177. 189. 2 6 3 204, 269 273. 281. 301. 303, 455. 409 17(1 dilfractive, 51 effective, «1. 174 175, 457 458 global lils, 260. 285. 290 291. 294 295. 299-303 a m b i g u i t i e s . 301 302 B o t j e , 299 302 CTICQ. 206, 285. 292. 294 299. 301 302 CJKV, 285, 296. 299 M R S , 285. 292. 295 296 M U S T , 299 gltions. 13. 40 17. «1, 131. 138, 140 141, 143, 171, 177, 189, 2 6 3 2«4, 269 270. 273 277. 279. 281. 285. 290 291. 295 303. 321. 455. 408 170 heavy q u a r k s , 43, 302, 303, 454 c h a r m , 279, 299. 303. 454, 491 m o m e n t s , 47, 137 138 non-singlet. 1 7 4 . 2 8 5 . 1 5 5 , 4 0 8 p a r a u i e t e r i / . a t i o u s , 285. 299 .501 p h o t o n s . 37 38. 52 p o l a r i z e d , 291 P o n i c r o n . 52 power law b e h a v i o u r . 1 II 142. 470
INDEX
scale d e p e n d e n c e , SU •'Volution <-<|i ml ions s c h e m e do|>ond 127 12H, 134 135 sen q u a r k s , 43, Hi. 1H. 130. Mi). 189. 203, 267, 209. 273, 271. 278. 281. 281. 286, 290. 293 291. 2 9 6 . 299 301, 303, 321. 191 192 llavoitr a s y m m e t r y , 13. 290. 292 294. 300 301, 3 0 3 p o l a r i z a t i o n , 292 s i n g l e t . 174, 285, 4 5 5 s t r a n g e q u a r k s . 279. 285 288. 299. 301 uncertainties, 302-303 u n i n t e g r a t c d , 141 142 v a l e n c e q u a r k s , 43. 46. 18, (¡I, 136. 143. 262 264. 269, 274, 279. 281. 283. 286. 2 8 8 289, 299 301, 3 0 3 304, 313, 321, 492 p a r t o n e x c h a n g e s y m m e t r y . 173. 154 p a r t o n m o d e l . 6. 8 - 2 1 . 32. 34 35. 38. 12 43. 17, 5 9 - 6 1 . 117 118. 120 121. 143. 149 150. 174 175, 177. 182, 250, 257 258. 273. 2 7 8 279. 281. 2 8 8 - 2 8 9 , 291 294. 313, 457 458, 472, 485 186. 488. 490 p a r t o n r e c o m b i n a t i o n , 140, 273 p a r l ó n s a t u r a t i o n , 143. 273 p a r t o n s h o w e r s . 33. 37. 147. 157. Kit). 161. 163 164, 169 170. 172, 181 184. 198. 203. 263. 316. 3 2 5 - 3 2 6 , 3 6 5 - 3 6 6 , 368. 372 373. 376. 380. 389. 392. 3 9 4 . 397, 402 b a c k w a r d s e v o l u t i o n , 183, 187 189 c o h e r e n t , 170. 172. 1 9 3 1 9 0 , 199, 202, 375, 382 383, 385 3 8 7 . 399 101. 413, 4 7 8 e v o l u t i o n v a r i a b l e s . 184, 194, 197 198, 202. 205. 3 8 0 , 178 179 f o r w a r d s e v o l u t i o n . 183, 180 187 i n c o h e r e n t . 102. 172. 202, 308 309. 375.
401 self-similar s t r u c t u r e , 3 6 5 3 6 6 . 372 space-like, 17. 01, 181 182, 184. 187 189. 194-190. 199 202. 400 spin c o r r e l a t i o n s . 191. 198. 202 time-like, 01. 181 182. 184 187. 189. 194. 190, 199, 2(H). 202. 4 0 0 p a t h integrals, v. 28 29
r
.:tv
I'auli exclusion p r i n c i p l e , 15 I 'anli m a t r i c e s . 422 p e r i p h e r a l h a d r o n i c collisions. 57 p e r t u r b a t i o n t h e o r y . 31 33, 3 0 5 300. 313. 310 all o r d e r s , 103 105 a s y m p t o t i c series, 3 3 at l,C). 3 2 - 3 3 . 00. 3 0 7 at NIX). 32 33. 00. 102. 305 308. 3 2 0 at N N I . O , 305, 3 0 8 finite o r d e r . 103 fixed o r d e r . 3 2 - 3 4 . 105. 179, 21 I. 3 0 6 . 305 r e s u m m e d . 32 33. 3 0 5 - 3 0 8 , 311. 313. 321 - 3 2 2 , 326, 329, 3 6 5 m a t c h i n g s c h e m e . 300. 322 p h a s e s p a c e . 34. 19. 100, 139, 179, 237. 309. 310. 322. 362. 365. 379. 389, 4(H). 402. 412. 426 4 2 8 b r a n c h i n g , 146, 177. 171-472 f o u r - b o d y . 356. 364, 199 in I) d i m e n s i o n s . 107 108. 433 134 o n e - b o d y , 120 121. 433. 403 s p i n f a c t o r , 100. 170 171. 3 9 6 t h r e e - b o d y , 72 74. 109, 3 3 0 . 3 3 8 . 434, 459 t w o - b o d y , 6 7 68, 109, 113. 123 121. 128, 131, 170. 4 3 3 - 4 3 4 , 447. 449 450, 465 466 p h a s e t r a n s i t i o n , 101 p h o t o - p r o d u c t i o n . 207, 273, 270. 277 p h o t o m u l t iplier, 218 p h o t o n b r e m s s l r a h l u n g . 200. 231 p h o t o n conversions. 229, 231 p h o t o n initial s t a t e r a d i a t i o n . 30 37. 52. 231 -232, 247 p h o t o n p h o t o n i n t e r a c t i o n s , vi. 231. 2 5 5 p h o t o n - p h o t o n s c a t t e r i n g , 37 38 p h y s i c a l g a u g e , see g a u g e , axial p i l e - u p events, 02, 253 pious. 0 JT" decay, 17. 253. 297, 321, 394 p l a n a r a p p r o x i m a t i o n , 80. 143, 156, I (if) I'oissou process, 17. 201, 482 polarized b e a m s , 291 polarized d e e p inelastic s c a t t e r i n g . 264 -265, 291 292, 313. 4 9 0 p o l e m a s s , 4 3 7 438. 440 I'onierauchuk theorem, 53 5 I I ' o m e r o u . 50 52, 50- 57. 59, 201. 2 7 0 h a r d , 59 s o i l , 59 s u p e r - c r i t i c a l . 59
, 2 , 27 s, , 102 10 , 1 7. 211. 288. 0 0( , 10. 10, 18. 20. 22. 20 2 ), ( , 10 ) 170. 180 , 187 . 2 1. 2 . 272, 02 s , 10 . 208 20 ) s . ). 171. 2 8 2 , 2 0, 101. 1 0 )
ss, 18. 121 120. 128 1 0, 170. 271, 2 2 0. 2 s . 12 12 , 128 s s . 12 12 s . 12 . ) , 0. 2. , 1 18. 0 02. 118, 121 1 7. 1 0 1 2. 177. 2 8. 27 . 2 1 2 2. 172 17 ) . 20. 22, 21. 0. 7. 10. . , 71 7 . 77. 82. 8 . 8 8 . 0. ). 108. 111. 120, 121, 1 0. 1 2 1 , 1 0, 228, 10, 1 , , . 1 , 21. 27. 28, 47, . 121 s. . 10. 1 . 1 . 107. 1 1. 102. . 01 . 1 s, 1 s. 2 , 02 , 7 1. s s. 77 78. 87 88 s. 1 . 2. 70 78. 81. 88 8 . 1. 4 00 ss . 77 . 2 s, 8 1 s , 1 s, 81 s. 8 1 s s, 80, 82 8 s, 2 . 78. 80 81 s s , 82 8 s. 80 81. 8 . 80 s. 7 . 81 81. 8 s. 7 . 8 s s. 81 87 . 81 87
s s
s
s. 8 8 s . 81 87 s . 81 87 . . 77 7 ) . 78, 101 s, , 7 . 82. s, 78 s, . 78 80 , 78. 80, 0 . 7 ) 81 s s, 77, . 7 81. 8 84. 88 1, 1 s , , 207. s , 02. 71 80, 171 17 . 1 1 , 40. 1 7 4 8 . 2 24. 0 1, 7 , 78, 88. 40, 42 42 . 4 0 1 2 s. 81, 84 8 . 87. . 170, 4 2 40 . 0 . 10 . ) . 17 . 127 . 1 0 s , 78, 171 17 , 4 7 4 8 200. 14 1 s . 14 1 . 0 2 s. 1 1 . 0 2 s, . 8. 12 1 , 2. 100 . . 21. 2 2 . 04. 2 0, 4, 40. 4 , 72. 7 s. 4. 7 8. 1 . 17 ). 21. 12. 1 . 120, 122. 12 , 1 . 222, 2 . 2 8. 27 , 27 ). 284. 2 1. 72. 427. 440. 1 1 4 , 4 2 ss s, . . 22. . 8, 71. 7 , 1. . , ). 101, 10 , 1 1. 202. 287. 0 . 1 ). 4. 41. 40. , 1, 1 410 , 7 . 2 , 0 1, 70 71. 81, 0. 88. 1. 100, 12 , 81 82. 8 . 112, 12 4 2, 4 2 . . 7. 1 . 21, 24. . 08. ). 121, 0 , 0 , 17, 4 , 4 . 4, 7 . 77. 428. 4 4. 442. 4 s . 08. 2 4. 2 . 2 0. 2 7 s. 4. 42. 1 . 122, 2 8, 277, 427. 4 440. 4 4 , . 2 . 0 1. 4. 70. 88, 147. 42 420
s. 1 0 ) , 170 477 . ) 1 470 477 s. 2 1 2 s . 20 . 1. 180 s . 240 . 142 . 48 4 . ( . 174. 177. 2 8 2 . 2 7. .
20 .
s s
, 20 . s s
1 4
. . 472.
484
. s
ss. 8 s
s s
.
s . ss ss ss . . s
s. 47, 4 0, 7. 2 . 7 8. 2. 1 ) . 1 . 201 , 1 2 s. 2 , 7. , 1, 7 . ), 270 271, 27 . 8 8, . 270 s . 220. 221 . 21. 2 . 1, 8 ) 0 . . 28. . 81, 87 2. 112. 110. 120. 42 81. 88 8 ). 1 . 101, 104. 4 , 8 ) 4. 0 7 4. 7. 17 170, 4 1 1 2 . 0, 14, 1. 10 10 . 12 . 2 7, 00 07. 40. 1 . 18 1 . 21 22, 20. 18. 0 2. 4 , 442 44 . 10 . 244, 0 07 . 44. 8 1 , 104, 01 . 7 8 , 1 , 2. 4. 0 8, 70 1. 1 , 7 8. 101. 102. 4 0 . 1 . 8 4. 7, . 1 2 . . 1 s , 1 10 , 17 170, 401 402 s , 101 10 s. 0 . 1
, 2 , 42 s s s s s
s s s
. . 177. 40 171 , 480 181 s . 12 . 1 4, 17 . 10, 4 , 4 8 4 s . 1 4. 4 . 4 . 4 s. 1 21. 47. 0 , 1 20, 0 . 81, 88 8 . 442 11 s s. 217 218 . 1 2 s . 14. see s s, s
s
s
. 14 s
s . 11 s 77 s, 7 8 ss, 8 . 471 ,A , 2 2 T s, )(), . . 2 2. 4 s s , 280 s , 1 1 . 172. 18 1 0. 8 401, 410 s s , s , s s , 0 1 2 . 181. 200, 2 4 2 , 2 8 2 . 2 . 2 , s s
3 2 0 . 401
s. 181. 200 201
s
, 201) 201
. 102 s
s, 7 8 s, 4. 1 7. 100, 170, 181, 202. 77, 402. 108 s s , 7. ) . 142. 1 8 1 , , 1 2. 4 , 8 100, 102 10 , 1 7, 142. 147, 14 , 1 . 1 1, 17 . 184. 2 7. 2 , 28 . 2 0, 0 08. 20 28. 2. . 47. 4 . ) 2. 4, 71. 72. 7 . 41 , 4 4 8. 4 2, 40 . 477. 4 8 s s, 101 102. 08. 1 7 4 8 s. 27 28 , 4 ss. 4 0, 8. 100 101, 10 . 4 0 4 8. 440, 402 s . . 221. 4 7
. 21
, 20 201 , 200 201
Mil
INDEX
string. '.'ini .idi spini- limi, pici,lire, 172. III). 1.12 How KinNt.oili correlai ions, 107 108, 107 d e e p incinsi I. s c a t t e r i n g , 17. 120 127 election positron a n n i h i l a t i o n t o h a i h o n s , 158 101
to W + W " . 4 t p hndronizutlon. 157 101, 103, 107 108. Ill Q ( ' | i ( ' o i n p t o n process. 123 singularities, Ulti S p e m e (linciion. MI: dilogaritlun Sphericity, 110. 323. 333. 108. 492 11)3 axis. 323, 103, l!)3 spherocity, I 10 spin a s y m m e t r y . 2!>1 spin p o l a r i / a t ion, vi in l> dilueiisions. 131. 105 in electron positron a n n i h i l a t i o n , 31, 05. lit,. 08 09. 232 in lepioii hadron s c a t t e r i n g , 39 10, 173, 113. <151 s u m . 30. 01 05. 71. 77 78. 108. 131. I I I I 15. 151. 175, 178. 117 118. 158. 105. 171 171 spin-slatistics problem. 15 Iti spin-statistics tlicorcm. 78 SI'S. 200. 208 210, 321 «quarks. 170. 102 s i a n d a r d model. 2 1, 22. 100. 152. 170. 277. 299. 113. 121 statistical b o o t s t r a p model. 108 statistical errors v. 213 statistical mechanics, I, 28 s t r a n g e n e s s , 0 ,v 192 s i r i n g effect, 155 1 5 0 , 3 9 8 100. 112.501 «Itili)', tension, K. 102 103. 100 107 s t r i n g theory, 3 -irong coupling. 22. 305. 31 I s t r o n g CI" p r o b l e m . 27 s t r o n g Ibrce. v. 1 I. 159 s t r o n g o r d e r i n g . 139 I 10. 112. I 17 I 18. 171, 175 s t r u c t u r e f u n c t i o n s , vi. 5. 10 15. 19 20. •IO I I. 17. 110. I 18 122. 125 127. 133 135. I l l , 173. 230, 211 250. 258 292. 299. 301. 310. 320. 332, 113. 153 dillractive. 51 52 /•'i, II 13. 10 13. 120 121. 127. 132 131. 170. 215, 278, 291. I53 151. 103
/•j, 11 I I. 19 21. 125 127, 173 171, 200 20 I. 278 281. 302. 301. 153 155, /•',. II 15. 10 13. 170. 200 285 289, •153 151. /•'l. 10. 153 /•;,. 10. 4 5 3 /•'r„ 10. 153
HI 13. 12(1 121. 132 135, I II) I 12. 170, 211. 219 250. 209 271. 273. 283 290. 299. 301. 320. 330 332. 103. 100 191 18. 121, 127, 133 131, 202. 278 281, 320, 330 332. 103, 100. 190
/• /., 120, 127. 131. 21 I 245, 219, 260-202 II - -I2
f \ . 200 201 291 292 II,.. 119 121. 125. 131 132. 170. •105 100 II 173. 151 //>;. 119 125. 128 132, 170. 101 100 power law b e h a v i o r . 209 271. 273 K,,. 278. 281 283, 190 scaling violations, 273 271, 270. 277. 279, 280. 285. 299 warning, 42 HI.
W,. 10 s u b j e t s , 239 mult iplicilics. 372 3 7 5 . 3 8 4 387 Sudakov form factors. 181. 198 space-like, 188 189. 201 205. 175 time-like. 185-189 S u d a k o v variables. 145. 177. 171 172 s u m rules. 200. 288 292. 312 313. 331. 333 Adler. 173. 288 290. 454 lljorken. 288. 291 292. 313, 330 332 corrections. 313 1011 is Jaffe. 288, 292 p r o b l e m s . 292 e x a c t . 288 llavour. 43. 40, 289. 301. 304. 192 G o t t f r i e d , 288, 290 G r o s s l.lcwellvn-Smit h. 14 15. 288 290. 313. 330 332 correct ions. 170. 289 290, 313. 400 m o m e n t u m . 19. 21. 40 17, 118, 288. 290 291. 301 s u m m a t i o n of large logaritInns. 32 33. 117. 142. 181. 182. 305 differential two-jet r a t e . 359
INDEX
event s h a p e variabiles, 318, 351 e x p o n e n t i a l ion, 152 jet rales, 238 III(.N/I). 59 l l l ( l / . r ) . 59. 142 l n ( y i / ( / o ) , 380 387 mass corrections. 140 m a t c h i n g , 153
hiiM~./l>'r), 152 153 s u b j e t multiplicity, 374 375 superficial degree of divergence. 81. 429 supersym try. 173. 170. 34(i. 319. 303. 402 s y s t e m a t i c errors, v. 230. 213 211. 250. 254. 250. 207. 280. 293. 298, 320, 378. 389 390. 108 e x p e r i m e n t a l . 243. 202. 275 270, 280. 291, 313, 315. 324 statistical t r e a t m e n t . 211. 290, 307 308. 328. 333 theoretical. 172. 202 201. 243 211. 288. 300 308. 312 313. 315, 319, 322, 329. 379. 390 391. 109,
III t a d p o l e s . 81 83. I I'J tail leptotls b r a n c h i n g r a t i o s . 310, 312. 332. I l l decays, 182. 31 I K T . 309 312, 330 332, 300 302, 304. 130. 438, 440 III li-sl beam m e a s u r e m e n t s . 253, 254 TKVATKO.W (¡1. 208. 231, 238. 250, 293 295, 297. 298. 100, 101 T h r u s t . 115. 117. 203. 230 237. 257. 307, 3 2 3 324. 320 328, 330 332, 330. 340. 343. 34«. 351. 483 484 axis. 23«. 243. 255. 32 I. 335. 342. 377. 380. 403, 105 m e a n value, 328 t ime reversal, 27. Id t o p q u a r k . 01. 101. 160, 234, 310, 345, III). 439 4 4 0 t r a n s v e r s e energy. 2 3 8 - 2 3 9 trigger bias. «2 I rigger sealing. 235 triple-gluon vertex, vi. 23 24. 3 0 31. 34, 74 7«. 79, 88. 91. 340. 347, 351. 352. 354. 355, 358, 425 120. 451 correct ions. 81. 85 87 I wist. I I
higher I wist. I 1. 280. 302 ult raviolet. cut-off, 80 81 u n c e r t a i n t y principle. 31 32. 123, 158. 172. 452. 175 unitaritv. 53, 59. 78. 79. 143. 183. 125 universal clusters. 57 T . 18. 31. 149, 305, 315. 398 decay to gg-). 315. 330 332. 380 decay to ggg. 315. 330 332. 390 level splitting. 314 315. 330 333 U r a n i u m . 219 v a c u u m . 2«. 59. 158. 101. 310 c o n d e n s a t e s , 311 Unci nat ions. 14. 19 2(1 s t r u c t u r e , 27 v a g a b o n d c u r r e n t s . 213 valence q u a r k s , SIT part on d e n s i t y f u n c t i o n s , valence q u a r k s van-dor-\Vnals force. I vector coupling, i/y. 291 292, 313 vector meson d o m i n a n c e , 38 generalized. 270 \V bosons. 149, 150 mass, 152. 299, 409 112 w i d t h , 410 W a r d ident ities. 81, 9(1 weak decays, 7, 202 weak force, I 3 W e i n b e r g angle, 204, 2«1, 277, 127, I HI, 44« W e i n b e r g ' s t h e o r e m , 8«, 104 Wcyl s p i n e l s , KM), 170 Wick r o t a t i o n , 43(1 W o l f r a m ansai/.. 104. 109 X-ray, 372 ( - d i s t r i b u t i o n . 300 309 charged part i d e s . 300, 391 n e u t r a l particles, 300 peak position, 300 309, 389, 391 q u a r k s a n d gluoiis, 388 389. 391 Y - s h a p e d events. 378 379. 382. 391. 199 Yang Mills theory. s i c g a u g e theory yo-yo modes, 102 103 Yukawa force, « '/. resonance, 35, 149. 150 X variable. 338 339
UNIVERSII > \ D A U T O N O * ' \ DE
