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Prof. Dr. rer. nat. Manfred BOhm Born 1940 in Memmingen. studied physics in Wllnburg und Munich. doctotate in physics from University ofWUrzburg 1966; resean:h in elementary parLicle physics in WUn.burg. at the Deutsches Elektroncl1 Syncltrotton (DESY) in Hamburg and at the European Organization for Nuclear Research (CERN) in Gelleva; since 19n Professor for Theoretical Physics at the University of WUrzburg.
Privatdou:nt Dr. rer. nat. Ansgar Denner Born 1%0 in Schwemfun, studied physics in Wllnburg, doctorate in physics from University of WUrzburg 1986, habilitation 1991 ; research in elementary particle physics in WUrzburg, Karlsruhe. Leipzig, at the Max- Planck-Institut for Physics and Ascrophysics in Munich and at the European Organization for Nuclear Research (CERN) in Geneva; since 1996 staff member at the Paul Scherrer Inslitul in Villigen/switteTtand and Privatdozcnl at the ETH Zurich.
Prof. Dr. ret. nat Hans Joos Born 1926 in SnLttgan, studied physics in TtJbingen, doctorate in physics 1961. research in elementary particle in Sao PaulolBrasil, at the Universi ty of Hamburg, aI the Institu te for Ad vanced Study in Princeton. at the University of Minnesota, at the European Organization for Nuclear Research (CERN) in Geneva, and at the Deutsches Elektroncn Synchrotron (DESY) in Hamburg; from 1963 until retirement in 1990 Senior Scientist at DESY in Hamburg and since 1965 Adjoint Professor an tbe Uni versity of Hambu rg.
O le Ocutoo;he BibliQth<:Jr. _ C IP _EJnbc:i ... urnah ....
Ein TItc ldatcnsatz fUr diese Publikation i51 bei Ocr Deut£hen Bib~othekerhillticb.
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Oaa Werk dnsch lie8Jich aller seiner Tcilc iSI urbcben'echtlieb geschliW. Jede Verwertun8 ",Bemalb der engen Grenzen des UrhebcrrcchtsgC5CUU ist ohne ZUlti mmu~g da Ver\agcl unzulksig und strafbar. [)as gill bcsoodcn fUr Vervielflltigungen, Obersctzungen. Mikrovcrfilmungc.n und die Einspcicherung und Verarbeilung in elektroni$Cnen Sy,temcn.
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1.2 Elements of relativistic quantum field theory
5
construction, general pr()perties, and application of these gauge theories to elementary particle physics are the main concern of this book. 1.1.2.2
One-particle-exchange model
Ideas about the dynamics of the strong and electroweak interactio n were taken over from Quantum Electrodynamics. There, in lowes1-ordcr approx~ imation, the forces between charged elementary particles are generated by the exchange of the qua.nta of the electromagnetic field , the photOfl8 , which
are the gauge bosons of the electromagnetic interaction. In the static limit, the Coulomb force is reproduced in this way. The ooson-euhange model has been generalized to the other interactions.
The exchange of the massless gauge bosons of QeD, the
gltJoRS,
yields the
simplest dC5Cription of the strong force between the quarks. In contrast to the photon, the gluans are not neutral but carry colour charges. For theoretical reasons (cf. Sect. 1.2.2) , eight gluollil must exist for the case of three colours. Like the coloured quarks, the coloured gluons cannot be observed directly as free particles. There is, however, indirect experimental evidence for their existence from deep-inelastic scattering (cf. Sect. 1.5) , the observation of the corresponding jets, and other experiments [Br79J. In contrast to the ele<:tromagnetic interaction! the weak interaction has a very short range. Therefore, the corresponding gauge bosons have to be massive. The charged weak 9auge ooson, the W boson, and the n eutral weak gauge boson, the Z boson, were discovered in 19B3 [ArB3]. Their properties have been measured with high accuracy. The W-boson mass is Mw = 80.419 ± 0.056 GeV a.nd the Z·0080n mass is Mz = 91.1882 ± 0.0022 GeV [PDGOOj. The Wand Z bosons are unstable and decay within about 3 x 1O -2~ S aJ~ most exclusively into fermion~antifermion pairs (d. Sect. 1.4). They can be experimentally detected only via an analysis of their decay products.
1.2
Elements of relativistic quantum field theory
In our introductory remarks we presented the quantum numbeNl and basic
properties of the fundamental constituents of matter and their interactions, i.e. of the elementary particles. The corresponding theory must satisfy the approved physical principles of quantum mechanics and relativistic invariance. These arc guaranteed by the description of the elementary particles
6
1 Phenomenological basis of gauge t heories
and their interactions by quantized, relativistic, locally interacting fields. The connection between the structure of conserved charges and the symmetry groups of t he fields is of great importan<:e. Therefore, some of the relevant field- and group-theoretic concepts are coJlected in this section [FT, GT].
1.2.1 1.2.1.1
Basic concepts of relativistic quantum field theory Quantum-mechanical states of a free, relativistic particle
We begin our introduction to relativistic q uantum mechanics by describing the I]tlantum-mechanical states of a free mllSsive particle M , ,;,h) [Wi391 with mas..'i M , four-momentum pi' = (Ep , p ), Ep = M"l + p2 , spin ; , and third component of spin ja, normalized as (1.2.1)
These free one-particle states are eigenstates of the relativistic energy~mo mentum operator PI' and thus transform under space-time tra.nslations x" - t x" + a D.CCordiDg to " U(a)IM,p,;, iJ) = eiJ'OIM,p,;, ja).
(1.2.2)
Homogeneous Lorentz transformations r~ = A~X" , gJU' = Al'pA"agPU , transform the one-particle state into one with four-momentum Ap and with a changed orientation of the third component h of the particle spin; described by the rotation matrix nUl U(A)IM,p, j , ja) = IM, Ap, j, j~)n u,,). (R(A ,p)).
'"
(1.2.3)
The Wigner rotation R(A ,p) = L - I(Ap)AL(P ) is composed of the boost L(P) which transforms the rest momentum ji = (M, O) into the actual fourmomentum of the particle p = L(P)p, the Lorentz transformation A, and t he boost L - I (Ap) which brings Ap back into the rest frame. Equations (1.2.2) and (1.2.3) define an irreducible representation of the Poincare group, i.e. the group of space-time translations and homogeneous Lorentz transformations. For massless particles, no rest frame exists, and t he one-particle states ha.ve to be constructed somewhat differently [Wi39, We95] . Here, the boost L(P) has to be chosen such that it transforms a. standard light-like momentum,
1.2 Elements of relativistic quantum field theory
7
e.g. p "" (1 , 0, 0, I), into t he actua) four-momentum of.the particle. Since this vector is not invariant under SU(2) but rather under the euclidean group E(2) , the states cannot be classified according to spin quantum numbers but only according to helicity quantum numbers. As a. consequence, Ul8.S8less particles can have only two different h elicities.
1.2.1.2
C reation and annihilation operators
One- and multi-particle states, which are symmetric (for bosons) or antisymmetric (for fermions ) under permutation of identical particles, are described by creation operators a},(p) and annihilation operators ahIP). These are characterized by their (a nti)commutation relations
[aj3(P) , aj;(p')J± = !ah(P),aj;(p'))± = 0,
[ah (P) , a;; (P')I± = 2Ep (21r)3 6(p - p') 0M3'
+ for fermions, -
for bosons.
(1.2.4)
The operators aJs(p) annihilate t he relativistically invariant ground state,
the vacuum 10),
(010)
~
1.
T he one-particle states (1.2. 1) a re created by aj, (p}
IM,p,;,;,) ~ .),(P)IO),
( 1.2.5)
frOUI
t he vacuum
(1.2.6)
and free multi-particle states by repeated application of at . Creation and annihilation operators of the associated antiparticles are denoted by bj,(P) and bi3 (P), respectively.
8 1 Phenomenological basis of gauge theories 1.2.1.3
Free, quantized fields
A complementary description of free particles, which empbasizes their causal propagation, uses free quantized fields ,p(x). Tbese are defined by field equa..tions and commutation relations. For complex scalar fields these are the Klein - Gordon equation (1.2.7)
and the commuta!ion relations
["(.) ,,,(.')1_ ~ ["I(x),,,I(x')I_ = 0, [,p(x),,pt(x')l _ = id (x- x', M ),
(1.2.8)
where (1.2.9)
Commutativity of the field operators for space-like distances, i.e. d(x , M) = o for ':];2 < 0, guarantees relativistic causality. The free fields ,p.,(x) are connected to creation and annihilation operators of particles with arbitrary spin by a Fourier transformation
,p.,(x) = (2!)3
~/
"
:;: [u.,(P,h) ah(P) e- ip:E
'
+ Va (P, hl bj3 (P) e+ip:E] , pz = pOx D _ px,
pO = Ep.
(1.2. 10)
Equation (1.2.10) incorporates the wave-particle dualism: particles are quanta of fields. The particle and antiparticle wave functions ua(P,h) and v",(p, jJ) describe the polarization degrees of freedom. They are constructed in such a way that ,p.,(x) transforms locally under Lorentz transformations. For scalar fields,
1.2 Elements of relativistic quantum field theory
9
u(P) = v(P) = L In the spin.! case, U,.(P,j3) and va(P,h) are t he well· known four-component Dirac spinors. The corresponding four·component Dirac field ¢a(z) satisfies the Dirac equation (written here with all indices)
•
L)iJ.:,a. - M'.,)~,(x) ~ 0,
(1.2 .11 )
.8=1
together with the ant-icommutation relatiof1.5 i~.(x),~.,(x')l+ ~ i¢.(x),¢o'(x')l+ ~ 0,
which involve the conjugate Dirac field ~a = ¢k-fp,.. The Dirac matrices "II' are defined by the Dirac algebra
(1.2.13) witb "10 bermitian and "Ii antibermitian. A useful short-band notation for products of four-vectors with Dirac matrices is the Feynman dagger ~ =
"Il'kl' . For spin·l fields , the wave functions correspond to the polarization vectors E,.(P, h) and E~(P,h).
A collection of formulas for Dirac matrices, spinors, and polarization vectors can be found in App. A.l. 1.2.1.4
Lagrangian formalism for relativistic fields
Rela.tivistic field equations can be obtained from an invariant Lagrangian
£(¢,8I'tP) with belp of Hamilton's principle, (1.2.14) aB
Euler- Lagrange equatiol18 of motion
(1.2.1 5)
10
I Phenomenological basis of gauge theories
The use of the Lagrangian formalism is important for the formulation of theories with interaction and their quantization. This can be illustrated by Quantum Electrodynamics (QED), which describes- in its simplest version- the interaction of photons, electrons, and positrons [QED]. The Lagrangian of QED reads (1.2.16)
for an electron with charge - Qe (Q = -1, e > 0), where the electromagnetic field-strength tenso!" F,..,,(x) is derived from t he four~potentia1 Dr photon field A,.. (x) ( 1.2. 17)
Variation of CQED with respect to AI' gives the inhomogeneous Ma:r:well
equations (1.2.1 8)
The homogeneow Maxwell equatio1l8 8,..eJW P'T FP'T = 0 follow directly from (1.2. 17). Variation of CQED with respect to ¢a(x) yields the Dirac equation for the electron field (1.2.19)
The elementary charge e or Sommerfeld 's fine-structure constant Q = e2/ 41f ::::: 1/131.036 determines the strength of the coupling between the electromagnetic field and the Dirac field. For vanishing coupling, i.e. e = 0, (1.2.18) is the free Maxwell equation, and (1.2.19) the free Dirac equation. The electromagnetic current i!m = Qf/rr,..!/J, as the source of F,..,,(x) in (1.2.18), as well as the potential term in the Dirac equation result from the variation of the trilinear expression, i.e. the interaction term (1.2.20)
1.2 Elements of relativistic quantum field theory
1.2.1.5
11
S matrix
Cross sectioJl.9 for scattering processes are obtained from the matrix clements of the .scattering matrix or S matrix. This transforms incoming states into outgoing states. It can be obtained as the infinite-time limit of the timeevolution operator in the interaction picture and is directly related to the interaction Lagrangia.n 1FT]
(1.2.21)
This expression has to be understood as a formal power series in L" and T denotes the time-ordered product. OperatoI'll occurring under the T symbol are ordered from right to left with increasing times. The time-ordered product of two factors reads explicitly (t = XO)
with + for bosonic a.nd - for fermionic field operators. This definition is relativistically invariant owing to the commutativity of field operators at space-like distances (1.2.8) . The absolute square of the S-matrix element (fISli) is related to tbe probability for an incoming state Ii) to evolve into an outgoing state If). If Ii) and If) are momentum eigenstates witb momenta Pi and PI> respectively, One obtains, upon subtracting the unit matrix and extracting tbe o-functioll of four-momentum conservation , the invariant matrix element M/i
(1.2.23)
1.2.1.6
Differential cross section and decay width
The differential cross .section for the scattering of two particles with momeutap! and 1'2 and masses M\ and M'l into n particles with momenta ql ," . , qn is calculated by squaring the invariant matrix element M and multiplying
1.2 Elements of relativistic quantum field theory
13
~+
,Fig. 1.1 Fe)'nman graph for the one--photoll approximation of the reaction e+e-
1.2. 1. 7
-t
p+ p -
Feynman graphs
The quantization of the non-linear field equations (1.2.18) and (1.2.19) is oonsidered in Sect . 2.4. For small values of the coupling constant e, the quantized theory can be evaluated by a perturbative expansion. This perturbative expansion can be expressed by Feyoman graphs. These intuitively describe the interaction in terms of scattering processes between the particles (quanta) of the free fields. For example, the reaction e+e- --t IJ+IJ- is described in lowest order of the electromagnetic interaction by the FeYllman graph shown in Fig. 1.1. The in- and outgoing particles of the reactions are represent.ed by external lines. The internal lines describe the field propagation between the vertices (interaction points). In this sense, the Feynman graph in Fig. 1.1 visualizes the above reaction as the annihilation of an e+e- pair into a photon at space-time point 1, the propagation of the virtual photon from point 1 to point 2, and the creation of a. IJ+IJ- pair from the photon at point 2. 1.2.1.8
Feynman rules
Together with the Feyuman rules, the Feynman graphs yield analytical expressions for S-matrix elements, or more precisely, for i times the invariant matrix elements M/I' The momentum-space rules described here are for QED, i.e. the interaction of elementary charged fermions, namely leptons and quarks with the photon; 1.
The analytical expressions for the ext ernal lines are given by the wave functions, i.e. spinors and polarization vectors, of t he fields (1.2.10).
1.2 Elements of relativistic quantum field t heory
15
In Feynman graphs, tbe propagation is symbolized by an intemallinej for this, the Fourier transforms iSp(p, Mlo8 a.nd i.a.~" (p, 0) are inserted as factors in the analytical expression
The arrow in the fermion propagator indicates the fermion-number flow, the momentum Hows from left to right. The pholon propagator (1.2.34) is given in the Feynman gauge. A systematic treatment is given in Sects. 2.3 and 2.4. 4.
The composition of the elements of the graph, i.e. the factors of the analytical expression for the invariant matrix element iM , follows the structure of tbe reaction. This is most evident for the diagrams without loops, Le. tree gruplu. If the Dirac spinors and Dirac: matrices are written down in the order that is obtained by rollowing the fermion lines through the diagram oppositely to the direction of fermion-number Bow, the order of the Dirac matrices is appropriate for matrix multiplication and all Dirac indices can be omitted. The momentum flux from the incoming to the outgoing particles can be chosen freely, provided the conservation of momentum at each vertex is respected. Fl-ee internal momenta (loop momenta) q" have to be integrated over using (211")-4 f d4q.
5.
The relative signs between Feynman graphs are determined by the an· ticommutativity of fermiollic operators: every closed fermion liJle gets a. factor -1 , and every interchange of two external fermion lines gives a. factor -1 . If the spinors and Dirac matrices axe ordered oppositely to the fermion number flow, the latter sign can be easily determined as the signature of the permutation by which the actual order of all external spinors tI, U, lI, and ii in the analytic expression can be obtained from a given standard order.
6.
The graph contributions have to be divided by the symmetry factor. It is defined by t he number of possibilities of mapping the graph on itself
1.2 Elements of relativistic quantum field theory
17
Therefore, M can also be written in current-current form
(1.2.40) The differential cross section in the eM system for unpolarized incoming and outgoing particles is obtained from (1.2.26)
1 ~~" "
d. _
dflc M - 6411" 28 P 4 ~
~
IM I'
0+,0 _ .\+ ,.\_
e4.
q 1
= 641f 2sp48 2
x
LL:
ii(P+,u+h"u(P_, u_) fi CP_,,,"_h,,V(P+, U+)]
+-x[
L:
u(q_. " _h" v(q+. "+) v(q+> "+)i'"(q-. L)
1
.\ +,.\-
(1.2.41) Using (A.1.40) to work out the spin summations, the lepton ic tensQrs 4~ and are calculated with the trace formulas (A.l.30) ,
With the kinematical variables in the eM system defined in (1.2.35) and (1.2.37) one gets for t.he contraction of the leptonic tensors
18
1 Phenomenological basis of gauge theories
and for thc differcntial cross section of e+e- -+ 11+11 da
dncM
For E
~
m", this simplifies to (1.2.45)
Integration over the solid angle then gives the total cross section
at')
~
f
2
dO dO" = 411"u dO 3s'
(1.2.46)
Other lowest-order QED cross sections, e.g. e+e- -+ in a similar way.
1.2.2
f j , can be calculated
Lie algebras and Lie groups
As we have seen in Sect. 1.1.1 , t be elementary particles can be characterized by charge quantum numbers. The corresponding multiplet structures can formally be described by symmetry groups [GT J. Well-known examples are the isospin doublets with the associated group SU(2) and the quark colour triplets of the SU(3) colour symmetry. 1.2.2. 1
Lie algebra
In the language of group theory the charge operators correspond to the generators qa, a = 1, .. . ,dimG, of the Lie algebra of a symmetry group G of dimension dimG (= number of group parameters). For instance, for the group of unitary N x N matrices with determinant one, SU(N), there are N 2 ~ 1 generators, i.e. dim G = N 2 ~ 1. The algebraic properties of the generators can be fully specified by their commutation relations a, b, cE {l , .. . , dimG}.
Mo""'" ,
(1.2.47)
20
1 Phenomenological basis of gauge theories
Examples: N = 2: SU(2) (isospin group) , q':::: [ II = TII/2, a = 1, 2,3, with the Pauli matrices T
I
=
(01 0') '
N = 3: SU(3), QII =
).1
A'
2 T
).11(2 ,
~
(0 -i) i
0
(\.2.50)
'
a = 1, ... , 8, with the Gel/-Mann matrices
C0000) A2~ 0-i 0)00 . A'~ U
=
1 0 1 0
o
•
o
1) . oo 0 o0
0
-
COO) 0 0 -i
).7 =
o
i
).5
,
=
A'
0
CO -i) 0 0 i 0
0 0
•
A'~
('001 00) . 3 0 0 -2
~~
o 0) ,
-1 0
o0
001100) . o
(1.2.51)
Extensions to higher N are straightforward. The N 2 -1 Gell-Mann matrices ).11 have the property Tr().II) = O. The Dynkin index TR of an irreducible re presentation R is defined as
(1.2.52) For the fundamental representation of SU (N) , the Gell-Mann matrices give TF = 1(2. In an irreducible rcprcscntation, the Casimir operator C ofSU (N) is diagonal, N1 _ 1
L L (T;')'1j,) ~ 6;.CR. 11=1
(1.2.53)
j
and we denote its value with CR. In the fundamental representation, its value is given by
CF =
N'
- I
2N
.
(1.2.54)
1.2 Elements of relativistic quantum field theory 21
r
The structure constants bt can be used to define matrices that satisfy the commutation relations (1.2.47) because of the Jacobi identity (1.2.48) according to (\,2.55)
(T.:!i)'" = -if""·
This representation of the Lie algebra which has the dimension of the group is called the adjoint representation. In this representation, the Casimir o~ erator C of SU(N) is given by
C=
L:rlJcfabd = CAO
cd
= N6 cd •
(\'2.56)
""
The value of the Dynkin index in the ildjoint. representation equals tbe value of the Casimir operator TA = CA = N.
1.2.2 .3
Lie groups
Finite transformations, depending on tbe parameters 8 = (iP') , are obtained by exponentiating infinitesimal transformations, i.e. by exponential functions of the generators
g(8 ) = exp(i8"Q") = 1 + ;!8"Q" +
(~2 B"Q" B'Q' + ....
(\,2.57)
The finite transformations g(8 ) form tbe Lie group G corresponding to tbe Lie alg:ebra of the generators Q"
Vice versa., the generators Q" are obtained from the finite transformations g(O) as derivatives at the origin Q"=_i
8g {)(J4 8=0
(1.2.59)
24
1 Phenomcnologica1 basis of gauge theories
1.2.2.4
The flavour group SU(NF)
The group-theoretical concepts just described are widely used in elementary particle physics. The arrangement of the particles in flavour multiplets is associated with representations of the flavour group SU(NF), i.e. the particles in these multiplets are associated with basis vectors in the representation space. Electric charge Q, strong isospin 1, strong hypercharge Y , charm C, et c., are closely connected to the flavour·group generators. For details we refer to standard text books [EP].
1.2.2.5
The colour group SU(3)c
The colour degrees of freedom of strongly interacting particles are described by the group SU(3)c. The Quarks are classified in the three-dimensional fundamental representation of SU(3)c, the gluons in the eight-dimensional adjoint representation. The colour charges Q of hadrons are zero
c
Q~ l had) ~
o.
(1.2.73)
This is the group-theoretical formalization of the rule: hadrons are colourless.
1.2.2.6
Weak isospin SU(2)w and weak bypercharge U(l)y
In Sect. 1.4 we show how leptons and quarks are classified, and how the structure of the electromagnetic and weak interaction is group-theoretical1y described by the weak-isospin and weak-hwercharge group SU(2)w x U(l}y.
1.2.3
Conserved currents and charges
In this section we describe t he implementation of symmetries in Lagrangian field theories.
26
1 Phenomenological basis of gauge tbeories
This together with (1.2.76) allows to define the Nodher current
(1.2.79)
that is conserved,
(1.2.80) Current conservation, i.e. the continuity equation, implies conservation of the corresponding charge
Q(t) =
J
io(t , x)d 3 x ,
(1.2.81)
namely
~~ = 1~: (t, X) d3x = ( div j (t , x ) d 3x = (
iv
i{}V j (t, x) dS = O.
(1.2.82)
Here we applied Gauss's law to transform the integral over the whole threedimensional space into one over a surface at infinity, and used the boundary condition that tbe CUlTents vanish sufficiently fast at infinity. The Lagrangian of QED (1.2.16) is invariant under one-parameter phase tnu15fotD)
1.2 Elements of relativistic quantum field theory
1.2.3.2
27
Examples
We illustrate the application of Noether's theorem for the case of a Lagrangian for fermions which is invariant under the SU(N) transformations
1MX) -+
(e t"l");j Wj(x), i
T" : T"I ,
(1.2.83)
of infinitesimally (1.2.84)
Since the o(J4 are arbitrary, we obtain from (1.2.79) the conserved currents
(1.2 .•5)
where we used the anticommutativity of the fermionic operators. In a free fermion field theory, the currents (1.2.85) have the form
1"'"(x}: ;P;(x}TI)""v>;(x}.
1.2.3.3
(1.2.86)
Operators for the conserved charges
The operators for the conserved charges can be represented by the field operators
(1.2.87)
As a consequence of the field commutation relations (1.2.12), which give at equal t imes (xo = x~) (1.2.88)
28
I Phenomenological basis of gauge theories
they satisfy the commutation relations
(1.2.89)
r
where k are the structure constants of SU(N) according to (1.2.47). The charges Q'" generate the symmetry transformations (1.2.84)
(1.2.90) In this way, the symmetries considered above are implemented as infinitesi· mal unitary transformations on the quantized fields. Besides the direct phy&ical interpretation of currents and charges, this shows their importance in connection with symmetries in quantum field theories. Apart from exact symmetries also broken symmetries are relevant. A symmetry can either be broken explicitly, i.e. by adding explicit symmetry~violating terms to the Lagrangian, or it can be broken spontaneously (cf. Sect. 4.1), i.e. by the very existence of degenerate vacuum states that. are not invariant. Moreover, it can happen that a classical symmetry is broken in the quantized theory. This is called an anomaly (ct Sect. 2.7). If a symmetry is only valid approximately, such as the flavour symmetry, the
associated currents are not exactly conserved, and the corresponding charges are time-dependellt. Despite this, (1.2.89) are satisfied as equal-time commutation relation. The investigation of such structures in tbe framework of current algebra gave interesting results [Ad68J. This was developed further leading to the non-linear q model and dural perturbation th.eory. Its predic. tions and its relation to spontaneous breaking of flavour symmetry in QeD are discussed in Sect. 3.5. We end this section with a general remark on the close connection between symmetry la.ws and field theory. In field theory, dynamics is described by means of local interaction of fields. Global changes of the fields- for example of their phase by global gauge transformations (1.2.83)- are not in the spirit of local quantum theory. This observation suggests that global symmetry transformations as described in (1.2.83) should be generalized to transformations in which the fie lds are transformed locally. For phase transformations (1.2.83) , this would mean that the parameters fP become space-time dependent: fP = fP(x ) . The elaboration of this idea of local symmetry leads to the concept of gauge theories [We29] .
1.3 The quark model of hadrons
1.3
29
The quark model of hadrons
The quark model was propo5e(i by G. Zweig [Zw64] and M. Gell-Mann [GeM] in 1964. In spite of an intensive search, quarks have not been found as free particles. But point-like scattering centres- partons with quark propertieshave been seen in deep-inelastic lepton- nucleon scattering. This apparent contradiction is explained within QeD by the confinement of quarks in hadrons. In tms section, a simple explanation of the properties of hadrons derived from their quark composition is given.
1.3.1
1.3.1.1
Quantum numbers and wave functions of hadrons in the quark model Quantum numbers
It is well·known that the many meson and baryon resonances observed can be claasified by meaD..3 o f quan/um number. . (EPJ. The elcctrk charge Q, the
baryon number, and the geometric quantum numbers angular momentum i ,j3 , parity P, and charge-conjugation parity C are conserved by the strong interaction. This is also the case for the flavour quantum numbers strong isospin f , f3 , strangeness S, charm C, bottomness B , and topness T . The remarkable experimental fact that only certain vaJues of these quantum numbers occur in nature can be explained within the quark model by th.e following hypothesis: mesons consist of one quark and one antiquark , baryons consist of three quarks. The quantum numbers and wave functions of hadrons are formed fTom the quark degrees of freedom according to the rules for two- and three-particle bound states of quantum mechanics. The quark states /u), Id), Is) , ... can be understood as basis vectors of t he fundamental representation of an approximate flavour symmetry group SU(Np) . The meson and the baryon flavour states are constructed by considering product representations and t heir reduction to irreducible compo.nents. Since SU(Np) is only an approximate symmetry, mixing between state vectors with equal quantum numbers can occur.
30 1 Phenomenological basis of gauge theories 1.3.1.2
Wave functions of mesons as quark-antiquark systems
The composition of meson watle fu nctions from flavour, spin, and orbital
parts is sketched in the following [Li78, C179J. Flatlour part 11, 13 ; 5, C): in the case of four quark flavours it is constructed. from the quark state vectors lu), Id), Is) , Ie) and the antiquark states Iii) , Id) , Iii), Ie) in the following way: Il , l ;O, O} = -I ud},
The state Id} is always accompanied by a minus sign, because the doublet (-Id) , ]u» transforms in the same way as the doublet (l u),ld» under SU(2). The structure of multiplets of two (u, d), three (u, d ,s) and four quark flavours (u, d, !I, c) can be read off directly; an extension to five and six quarks is straightforward. There are several states with quantum numbers 10, 0,0, 0) which can mix. For example, pseudoscaJar mesons do Dot follow the scheme above, but show roughly SU(3) mixing 1
I.) '" - ",,(luil)
+ Idd) -
21"n ,
Spin part 18, 83): quark and antiquark spins have two possible orientations characterized by the state vectors It ) and 1.1.). They can be combined to fonn a spin triplet and a spin singlet
11 , 1) ~ lit) ,
11, 0) ~ (It<)
+ l.l.tn/v'2,
10, 0) ~ (lW -IW )/v'2.
11 , -1)
~
1>.1.),
(1.3.2)
32
I
Phenomenological basis of gauge theories
well-known symmetric ISy) and antisymmetric IAu) states 1
( 1.3.4) corresponding to the two two-dimensional mixed symmetric representations M.i± of the permutation group of three elements. Here E = exp(27ri/3) =: (- 1 + iV3)/2, E· = exp(-21fi/3) = E2 . The states IAn), IML), and IML) vanish for a = b.
Flavour part: the states of the flavour multiplets can be obtained by combining the quark produc t states into those with definite symmetry according t.o (1.3.3) and (1.3.4). For instance for baryons made of two up and one dowll quarks this gives the mixed symmetric states
_ 1 IP) ~ v'J(l uud) + ' Iudu) IP)
+ , Olduu)),
~ ~(luUd) + , Oludu) + ' Idun)),
(L3.5)
and for baryons made of three up quarks the symmetric state
(L3.6)
1.6.++ ) = luuu).
Spin part: the s pins of the three quarks can be combined to form the total spin (3/2)5)' and (1/2)MI by the same method
Orbital part the relative motion of the three quarks is a function of two relative coordinates z\ = (X2 - xd/..f2, Z2 := (2X3 - X I - X2)/../6. Thus, there are many different combinations of internal angular momenta leading to the total orbiLaI momentum l. SchrOdinger wave functions of all symmetry types (Sy, An, Mi+ , ML) can be systematically constructed using the mixedsymmetric, complex relative coordinates Z = (ZI + iz 2 )/.J2, z· = (Z\ iZ2)/../2. The details of these wave functions are determined by the quarkquark interaction potentials (d. (Gr76, BoSO)) . Composition of the baryon walle functions: when flavour , spin, and orbital parts are combined to give a total wave function , the following symmetry types result Sy®Sy = Sy, Sy0Mi = Mi,
An®An = Sy, An0Mi = Mi ,
Sy ® An ~ An, (1.3 .8) Mi®Mi=Sy $ Mi$An.
In this way, the flavour and spin parts combine to give the multiplets with definite symmetry, as shown in Table 1.5. As an example we present the ftavour-spin part of the wave function of a proton with spin up:
If the interaction shows omy a slight flavour and spin dependence, it is advantageous to combine multiplets of equal symmetry to form supermultiplets of the higber approximate symmetry group SU(2NF) {C179J. The final stage consists in the combination with the orbital wave function to give states with definite tota] angular momentum. If the tot8ol wave function is to be symmetric under permutations, then the classification shown in Table 1.6 follows . This is a. good description oC the experimenta.l spectrum Cor baryon resonances with strangeness 0 and -1. The many blanks which occur in the case of resonances with strangeness - 2 or -3 are due to experimental difficulties in the production of these resonances. Baryons with open charm or open bottom, i.e. with non-vanishing charm or bottom number, ha.ve been observed at the expected masses.
The fact that a symmetric total wave func tion must be chosen to explain the experimental baryon spectrum was a. problem for the naive, phenomenological quark model: quarks with spin 1/2 should obey Fermi statistics; therefore
1.3 The quark model of hadrons
35
quark statistics was solved by introducing an additional degree of freedom for the quarks, colour [Ge72]. This was an impor tant step on the way to
QeD.
1.3.2
Quark model with colour
The colour degree of freedom of the quarks can assume three values (e.g. red, green, and hlue). It spans a three--dimensional complex space with basis vee·
Is}, and Ib}. Hadtons atO constructed accOfding to tho rule "hadrons are colourless n. This means that tbe mesons and baryons are colour singlets tors If ),
is the Lcvi- Civita tensor , and r = I , 9 = 2, b = 3.
For illustration we give two simple examples of wave functions including t he colour degree of freedom. We choose the p+ meson and the tJ. ++ resonance since these have vanishing orbital angular momenta. The wave function can be approximated by a gaussian
( 1.3.11 )
1.4 Basics of the elcctrowca.k interaction
39
where the minus sign for mesons results from the presence of an antiparticle. potential model, quark amfinemenl is enforced by introducing an infinitely rising potential. A simple linear ansatz In
Ii
(1.3.17)
gives already a good description. The derivation of the form of Vc is an important problem ofQCD. The value for the strong fine-structure constant 0: 8 and the string constant K have to be determined by comparing the calculated spectrum with experiment. Figure 1.5 gives a comparison between the results thus obtained for the charmonium system and experiment. The model parameters were determined from the masses of Jjtp(3097) and 1,b'(3686) and the leptonic width of Jjtp , r(Jjtp-+ e+e-) = 6.02 ± 0.19keV. Typical values for these parameters are
a. ~ 0.25--0.35,
K ~
GoV
0.8- 1.0 fro '
me
~
1.2- 1.6 GeV .
( 1.3.18)
The agreement between the general structure of the experimental spectrum i:Wd UlI:: c.;&cuJah:d
W~
ill rea!SOliabie. HuweV\!r, the lIpedrulII
(;
be weU
reproduced up to masses of 4 GeV with other potentials, too. Therefore, these results alone cannot be regarded as an experimental confirmation of the potentials (1.3.16) and (1.3. 17). As non-relativistic approximations are the basis of the quarkonium model, a. short remark on the quark velocity in the bound state is appropriate. The expectation value of the velocity in tbe charmonium ground state is calculated as (v) ~ 0.4 c. This roughly jWJtifies the non-relativistic calculations. However, relativistic effects, like spin- spin and spin-orbit interactions, are of importance also for heavy-quark bound states (cf. Sect. 3.4) .
1.4
Basics of the electroweak interaction
Electromagnetic and weak processes between leptons and quarks are successfully described phenomenologically by photon , W*-boson, and Z-boson exchange [EW]. At low energies (E 4:: Mw) or small momeutum transfers (Ill «: Mar) the W- and Z-exchange diagrams can be approximated by a four-fermion contact interaction, i.e. by the Fermi model. As the example of e+e- ann ihilation into /-1 pairs in Sect. 1.2.1.9 shows, S-matrix elements of
1.4
Basics of the electroweak interaction
11
currents. It. has to provide a consistent treatment of the massive W and Z bosons. This can only be formulated if electromagnetic and weak interactions are combined in the form of a non-abelian gauge theory with spontaneous symmetry breaking. This is discussed in detail in Chapt. 4.
1.4.1
E l e c t r o w e a k i n t e r a c t i o n of l e p t o n s
1.4.1.1
Fermion-gauge-boson couplings
Experiment tells us that the couplings between the elementary fermions and the electroweak gauge bosons, i.e. the 7, the W * boson, and the Z boson are of vector and axial-vector type. No experimental indications for scalar, pseudoscalar, tensor, as well as derivative couplings have been found. The 7 couplings are purely vector, the W-boson couplings are vector minus axial vector, i.e. left-handed, and the Z-boson couplings arc a mixture of vcctor and axial vector. VVe write the interaction Lagrangian in the general form £
> = i E ^ W t v S - W ^ ^ ^ ^ W , V i,j
(1.4.1)
where V = {7, W + , W ~ , Z } . The sum over i,j extends over all elementary fermions. The vector and axial-vector couplings are denoted by v]- and o-'-, respectively. The couplings are specified later. In particular, they include no mixing between leptons and quarks. An important property of the electromagnetic and weak interaction is their universality. On a naive level it means that all fcrinion families interact in a similar way with the gauge bosons. More precisely, the electromagnetic and weak interactions of all fermions are described by equally structured contributions to the interaction Lagrangian with universal coupling constants. We begin our discussion with the leptons. Because of universality their couplings are of the same type and strengths. They can be expressed by two parameters: the elementary charge e and the weak mixing angle Qw. We explain in a first step the coupling structure for the lightest lepton doublet, e), made of the electron neutrino //e and the electron e. Restricting the coupling matrices to this case of two fermions, we have
42
I Phcnnmcnological basis of gauge theories
v
w+
=
flW+
=
2V2. Sw V» V W- =
Hw- =
1
fO
2N/2SW V =
a
2 ^ ; (0
z_
1
/è
2 . s w c w \0
Oj
2V2sw
0\
0 X
t
2\/2.S w =
- i + 2 4 ) - \ )
1
=
1
'
2 ^ ;
_ T
( y ~
_ 1 _ t 3
=
)
• (1.4.2)
2swCw 2 '
where s w = sin 0 W , c w = cos0 W , and t + , t Paidi matrices (A. 1.11).
Q
, and r ;i are combinations of the
Introducing the doublet of fields $ = (i/v> V-0 and defining the electromagnetic. current the charged weak current and the neutral weak current V Ç = fi
f-r^Qy,
= 3 2 n - sw7/XQ #* = /'/i - s°W '/J!
r = i ' 7/iL y
(1.4.3)
with _ 1 ~ 7s 7/iL — 7/i 2 '
(1.4.4)
the interaction Lagrangian can be written as 2 A = -ellA»
+
" ( / ¿ ^
+
+
(L4-5)
The coupling matrices and consequently the currents of the leptons of the second and third lepton family are constructed in the same way. Altogether we have the three lepton doublets <- Q= <- Q=
0 -1 '
(1.4.6)
and the total lepton current is the sum of the corresponding three contributions. "'The factor \/2 in the denominator is a consequence of the different normalization of neutral and charged fields.
I.'t Basics of the electroweak interaction
1.4.1.2
4.'}
Charges a n d S U ( 2 ) W x U ( l ) y s y m m e t r y
'L'he charges of the left-handed currents /."[, = I d3xl±(t,x),
lt(t)
and
I*{t) = J d 3 x / g ( t , x )
(1.4.7)
have the commutators [ / + , / - ] = 21*
[J®,1*] = ±J ±
(1.4.8)
owing to the equal-time commutation relations of the Dirac fields (1.2.88). This is the Lie algebra of the weak isospin group SU(2) W . The weak hyperchargc, which is defined via the Gell-Mann -Nishijima relation iy
w
=Q-/3=
id3x(r0(t,x)-l30(t,x))
(1.4.9)
commutes with all components of the weak isospin. Together, they generate the group SU(2)W x U ( l ) y . This group plays an important ròle for the construction of a gauge theory of the unified electroweak interaction, thus leading to a theoretical explanation of universality. The SU(2)W x U(l)y symmetry of the currents and charges seems to be broken by the prefactors e / \ / 2 s w and e / s w c w in the interaction Lagrangian (1.4.5). Introducing orthogonal linear combinations of the neutral gauge fields Ay and Ay,
B"=
Cy,A* + Sy/Az,
(1.4.10)
gives
where lj* = tf, - l f r Now the SU(2)W x U(l)y- symmetry of the interaction is manifest. Introducing left- and rigid-handed Dirac fields VL=
=
(1A12)
'I'l
I Plienomenological basis of gauge theories
fi
*
?
fj
V
/to
"
*
*
fl
Fig. 1.6 Boson-exchange diagram for four-fennion processes
il, is possible to arrange the left-handed u c and e into a doublet and the right-handed e into a singlet V'eR (there are no right-handed couplings for the neutrino) and similarly for the other lepton families. The left-handed doublet is coupled to the weak isospin triplet (W"h, W~, Wz) with strength e/sw and to the singlet D with e/cw times the weak hypercharge YL = —1. The right-handed electron is only coupled to B with e/cw times its hypercharge YK = -2. 1.4.1.3
Four-fermion processes
We consider the invariant matrix element M of an elementary four-fermion process (Fig. 1.6). The Feynman rules together with the propagator for a massive vector boson, V :
= iA-(p,My) =
i
(
1
-
4
.
1
3
)
give 3 M = -C2
£
-(Pi) (Vvji - l ^ a j i )
V=7,W,Z
niPi)~9'^ktiM2v K
~
M
V
(7UVhn - 7"75Otm) U(P™) = M1 + A4 W + M7',
(1.4.14)
where k = pi — pj. This matrix element can be rewritten using the fermionic current matrix elements (1.2.40) —n'1" M-* = —e 0 ' | ^ ( 0 ) | i ) ( / | / ^ ( 0 ) | m ) — , ''in the photon propagator the k,,kv term is absent.
1.4 Hasics of (.lie electroweak interaction 47
—n>"' \ h^ky IM"2 9 2 = - ^ w l m m t m r n ) . k;_MJ
d.4.15)
The matrix element for photon exchange is identical to (1.2.40). For |A;2| < A/2V, M | and for light fermions, where the k/tkv term may be neglected since its contribution is proportional to the product of the fermion masses, the expressions for _MW and M 7 ' simplify to
z
M
e2 = - c2 2 M'-i( w j t' fI'i m ^ r n ) . r
(1.4.16)
These matrix elements are of current-current form and can be directly compared with those in the Fermi model of the charged and neutral weak interaction. In the Fermi model the interaction Lagrangian reads
AC C f e ™ = ^ [ l p - > ' + lJf<>'].
(1.4.17)
The corresponding matrix elements for four-fermion processes are identical to (1.4.16) if we identify
^
=
M W M i .
(1.4,8)
These are the lowest-order relations between the gauge-boson masses Mw and Mz, the muon decay constant (Fermi constant) Gfl, and the weak mixing angle 0V..
'1(1
1 Phenomenological basis of gauge theories
1.4.1.4
Comparison with experiment
As an application, and in order to determine the parameters in the ansatz for the couplings (1.4.1) and (1.4.2), we apply (1.4..15) to a charged-current process, the fi decay. In this case we have indeed |k 2 1 = |(p f L -;>e)''| •C Myj and 77i0, trip Mw- Therefore, the Fermi matrix element (1.4.16) for tW w is appropriate. The decay probability per unit time, F^, is obtained from M u using (1.2.25). The phase-space integration gives
(1.4.19)
for the width F;1 or the lifetime r ;j of the muon. From the experimental value of the muon lifetime TjiCXp — 2.19703(4) x 10" 6 s the following value for the muon decay constant is obtained 4 [PDG00] G,L = 1.16639(1) x 1 0 - 5 G e V ~ 2 = (292.8GeV)~ 2 .
(1.4.20)
The measured decay spectrum as well as polarization correlations arc in excellent agreement with the results obtained from the left-handed matrix element (1.4.16). This is also true for r decays, thus supporting the universality of the weak interaction. The interaction Lagrangian C\ (1.4.1) describes not only the weak decays of leptons but also their scattering processes. Since only electrons are available as target, the reactions shown in Table 1.7 are possible. The experimental data can be used to determine the weak mixing angle from purely leptonic reactions. The result is s 2 = 0.221 ± 0.008 [PDG00]. Together with the fine-structure constant a = e 2 /'\n = 1/137.036, this allows for the determination of the W-boson and Z-boson masses in lowest order from (1.4.18) and (1.4.20). The result is M w « 79GeV, and My, « 89GeV. These values are roughly 1 2GeV lower than the experimental ones. This indicates that higher-order elfects are not small and needed for an adequate comparison between theory and experiment. In fact, after taking into account the radiative corrections within the Electroweak Standard Model, which is described in Chapt. 4, the predicted values agree with the measured values within the experimental errors, which arc at the per-mill level. 'including QI3D corrections iw given in (•!.<» 7)
1.4 Hasics of (.lie electroweak interaction
47
Contributions reaction
from
total cross section
Vlte~ -> F /t e~
i/t ,(Z)
a - 4 + K > o i ( i - 4 + 44)a0
U^cT -> ISeH~
i/x
UeG~ -> V,Lix~
t/i
i/Me~ ->
5^0
l
vee~ —> vce~
/l 1 lll |(Z) ,(±) ) 'il
(T+ 4 I (i + 4
+14) + 4 4 ) a0
Tab. 1.7 Cross sections for neutrino-electron scattering (oo = Gpm c B v /T: = 1.72 x IO"41 (£„/GeV) cm 2 ) 1.4.2 1.4.2.1
=
E l e c t r o w e a k i n t e r a c t i o n of h a d r o n s Hadronic currents
Since the quarks arc the fundamental constituents of matter rather than the hadrons, the hadronic currents are expressed in terms of the quark fields. In analogy to the leptonic current (1.4.3), the electromagnetic hadronic current for three generations is given by (suppressing the colour degree of freedom) h]t(x) = V ; ( * ) 7 M i x )
Experiments on the weak interaction show that the S U ( 2 ) w x U ( l ) y structure formulated for leptons applies to hadronic weak processes as well. Thus, one extends the definition (1.4.7) of the generators of weak isospin with commutation relations (1.4.8) by adding the hadronic current component /±(i)=y
= yd 3 ®(Zg(i,x) + Ag(i,x)).
tr=z<
As with leptons, the left-handed quarks are arranged into doublets
c o , « , /
( a
(1.4.22)
IN
I I'henomenological basis of gauge theories
and the right-handed ones into singlets UR, <1R, CR, SR, IR, and 1>R. The composition of the doublets is based on t.he quark mass: (u, d) are the light quarks, (c, s) the moderately heavy ones, and (t, b) the heavy ones. The quarks u,...,1> arc defined as flavour eigenstates, i.e. eigenstates of the conserved quantum numbers of the strong interaction and therefore mass eigenstates. Since the electromagnetic and weak interactions do not conserve these quantum numbers, the eigenstates of the weak interactions are different. As a consequence the charged current for the weak interaction of hadrons involves a 3 x 3 quark-mixing matrix V and has the form [Ko73, Ja79]
/»+(:x) = (u{x),c(x),t{x))-yllL
[VudVu8Vub Vcd ycs Vcb
(1.4.24)
VKd Ks Kb The matrix V describes the transformation from the mass eigenstates to the weak-interaction eigenstates of the quarks. In the corresponding construction for leptons, V was chosen diagonal because of the separate conservation of lepton numbers and equal to the unit matrix because of universality [cf. (1.4.3)]. The non-electromagnetic component of the neutral current h'^ (x) is constructed as commutator of and h~ (.'/;) as the zeroth component of weak isospin (1.4.22) 2kt(x)
=
[i:,h;(x)]
(1.4.25)
VVt 0
-vtv
fu{x)\ c(x) t(x) d{.x) J s{x) \b(x) /
The commutator is calculated using the equal-time commutation relations (1.2.88). Inserting h* from (1.4.25) into the relation (1.4.26) implies V V ^ V = V and thus V V 1 = V+V = 1, i.e. the quark-mixing matrix is unitary and the neutral current li'ft(x) is flavour-diagonal. This
1.4 Basics of the electroweak interaction
53
In a similar way, the decay widths of the W boson into fermion-antifcrmion pairs can be calculated
r(W
--i
-
,aMw IVijl
!;lj) = Nc 128;'
2 [
1-
ml + mj (mf - m;)2] 2Ma,
-
2M~
J[M~ - (m. + mj)2)[M?,. - (fnj - mjp] x M'w f erMw
::::: Nc 2
IVijl 2
'w N'C G,.Milv ,,'12 • 61rV2 lv0 I
=
2
for
m,« Mw (1440) ..
Again, the total width is the sum over aU possible decay channels
rw = I:r(w.., Ii!;) '" 2.0G,V .
(1.4.4 1)
ij
The values for the partial and total widths of the Z and W bosons obtained from the expressions involving G,. are in good agreement with the experimental results [pDGOO] rz = 2.4952 ± 0.0026 GeV,
rw =
2.12 ± 0.05 GeV.
(1.4.42)
The radiative corrections to the decay widths (if parametrized with G,,), which are discussed in Sect. 4.6.3 , are sma.l.l.
Z-boson production in e+e- annihilation: purely electromagnetic e+e- annihilation into fermion pairs was calculated in Sect. 1.2.1. In the complete electroweak model, the Z-exchange diagram has to be taken into account betlideH the photon-exchWlge diagram. Neglecting terrIl8 proportional to the electron mass, which arise e.g. from the k"kjJ terms in (1.4.13) , the invariant matrix e1ement reads
(1.4.43)
60
I Phenomenological basis of gauge theories
(or t he unpolarized differential cross section, where we have introduced the iep!onic tensor L~Y' [second line of (1.5.9)J and the hadronic lensor W~V'''' [third line of (1.5.9)]. The leptonic tensor L:;'y' can be evaluated with help of the trace formulas for Dirac matrices (A.1.30), LVV ' JU' = "'=
L u(k')Y'(vY - 'Y5av)u(k) U(kb'"(V( - 75anu(k')
"" [~'T' (vY - 'YsanW(vY' - 'Ysat)]
Tr
= 4
((k'1" k" + kill It'" - !I'" kk'}dJ vV' -
i elWPU k~kaG~VV').
(1.5.10) In the casc of antilepton scattering the sign of the term involving c.PllfJ
.....,j VV' and Cd.I VV' represent combina.tions of to be reversed. The constants tit! the lepton couplings,
(1.5.11) The hadronic tensor W,!;,v'Af cannot be calculated from first principles. Because of Lorentz covaria:nce it can be decomposed into tensors and invariant structure functions which only depend on t he invariant variables Q'l and v defined in (1.5.1). Using current conservation and neglecting tenru! of the order o( the lepton masses yields (or the hadronic tensor the structure T T
WVV'N = _gT WVV'N(Q' u) I'"
I'"
.
I
,
purl
+ Pl"p" pq
pq WVV'N(Q' u)
M2
pq WVV''''( Q' , v ) , 3
- leIWU(j-pq- M2
2
,
(15 2) . .1
where g'Jv = gl"" - ql"qi' /q2 and pJ = 'PI" - q,.pq/q2. Instead of the functions Wi, often the slightly differellt ones (1.5.13) are used.
60
I Phenomenological basis of gauge theories
(or t he unpolarized differential cross section, where we have introduced the iep!onic tensor L~Y' [second line of (1.5.9)J and the hadronic lensor W~V'''' [third line of (1.5.9)]. The leptonic tensor L:;'y' can be evaluated with help of the trace formulas for Dirac matrices (A.1.30), LVV ' JU' = "'=
L u(k')Y'(vY - 'Y5av)u(k) U(kb'"(V( - 75anu(k')
"" [~'T' (vY - 'YsanW(vY' - 'Ysat)]
Tr
= 4
((k'1" k" + kill It'" - !I'" kk'}dJ vV' -
i elWPU k~kaG~VV').
(1.5.10) In the casc of antilepton scattering the sign of the term involving c.PllfJ
.....,j VV' and Cd.I VV' represent combina.tions of to be reversed. The constants tit! the lepton couplings,
(1.5.11) The hadronic tensor W,!;,v'Af cannot be calculated from first principles. Because of Lorentz covaria:nce it can be decomposed into tensors and invariant structure functions which only depend on t he invariant variables Q'l and v defined in (1.5.1). Using current conservation and neglecting tenru! of the order o( the lepton masses yields (or the hadronic tensor the structure T T
WVV'N = _gT WVV'N(Q' u) I'"
I'"
.
I
,
purl
+ Pl"p" pq
pq WVV'N(Q' u)
M2
pq WVV''''( Q' , v ) , 3
- leIWU(j-pq- M2
2
,
(15 2) . .1
where g'Jv = gl"" - ql"qi' /q2 and pJ = 'PI" - q,.pq/q2. Instead of the functions Wi, often the slightly differellt ones (1.5.13) are used.
62
1 Phenomenological basis of gauge theories
-
1
0 .1 0.01 [ >ZGeY
0. 001 I~+-~,--~+-+-~---r-+~
Il.~, t.•j.,. .... T 'r 'otI'\ 0.1
.... ~ -. ,
... ' . ..~ '."
..."
0 .00 1 OL_-J.,-~--,OL,~~,L-~---,L8~-~ 0.2 .. 0.6 O. 1.0 x
Fig. 1.11 The BUUc;t un func;tioll8 ~P (QI. JI) Q£ dee:p-inelll.'Jtic eP lC&f.t.ering accOTding to ReI. {Ta15J
1.5.1.1
Bjorken scaling
Figure 1.11 shows the classical experimental results for the structure runc· tions W;nr aud W;np. The data are plotted as functions of the vatiable !1; for different values of Q2. One observes that for Q2 > 1 Gey2 and v > 1 GeY the structure functions do not depend 0 11 the two variables Q2 and v but approximately only on the dimensionless "eating variable x = Q2f2Mv. T hey
1.5 The Quark- parton model 65 The calculation of the lepton-parton cross Beclion in the boson--excbange model proceeds according to that of elementary particles (compare c.g. thc treatment of e+e- annihilation into IJ pairs in Sect. 1.2.1.9). The Mandelstarn variables for the parton kinematics are related to t he hadronic. variables as 8; = (k + Pi)2 = (k + (ip}2 ~ 2{ik" = (is , Hi = (k' ~ Pi)2 (k' - ~iP)2:::::: - 2~;kp' {iU , = (k - k')2 = t.
=
4.
=
( 1.5.22)
Assuming that the partons are point-like objects with spin 1/ 2, the unpolarized parton cross section becomes '· d ", dO
1
1
!LVV',PIII i.v~
'l
" jM j2 a ~ 2 "1 11/.1 ,1 = 64'/1'2 8 .4 L.. = 4 §· L (Q'+M')(Q'+M') ' 1 ' VV' V V' poI • (1.5.23)
where LrY' is the lep ronic tensor (1.5.10) , and i~::; a similar tensor for the parton i. Contraction of the tensors and use of Malldelstam variables leads to
L
(SI " d (' 1· , _ 2'/1'0' dl VV' •
T
+ U") d.",VY'G"VY· ± (-2 Si " i (t
M~)(t
-2)d.IJ.VY ·G"VV' Q
Ui
M3,) ( 1.5.24)
with + for lepton- parton or antilepton- antiparton scattering and - for antilepton- parton or lepton- antipartoll scattering, respectively. The produets of quark couplings G,,'I VY' and G!'' VV' are defined as in (1.5.11 ). The Mandelstam condition for vanishing parton masses, Si + it; + t = 0, is made explicit. ill ( 1.5.24) by int.roducing a 6-Cunction
d'&; (' ' ) dtdit; si, t,ut = x [(§~I
2.0'"
§~ ;,v,(t
1
M~)(t
(1.5.25)
M t, )
+ u.~)Gj,VV' (j ,vv' ± (s?I ~l H~) a GI,VV' (j,vv'j 6(; ,· + u'• +. t) '''" a
Use of the kinematical re13,tioull (1.5.22) allows to transform to variables sand u.
~he
hadronic
66 1 Phenomenological basis of gauge theories The deep-inelastic lepton- nucleon scattering cross section can now be built up from the parton distribution functions and these parton cross sections
t d & dtdu (" t, u) ~ L 1, d(of;",";) ( ; dtd;;; ((;" t, (;u) 2
d 2 (1W
,
21ra2
~".- V,V' LLi
J,'
X
(11 2 + u2)d,;VV' G~ V V' ± (S2 _ u2)d.,:VV' G~VV' (t M')(t- M ') V
dE,; !;,N((;)(; 0((;(, + u) + t).
V'
(1.5.26)
e;
Because of the J-function is identical to z = -tj(s + u) = Q2 j2M lI, i.e. the Bjorken scaling variable. Hence :t determines the longitudinal part of the parton momentum. Carrying out the integration yields d 2 u'" 21ra 2 1 dtdu = 82 s + u
as final result for the lepton- nucleon cross !;eCtion. Indeed the lepton- nucleon cross section calculated in the parton model shows scaling behaviour. By comparing the cross section formulas for lepton- nucleon (1.5.19) and lepton- parton scattering (1.5.27) , the connect ion between the structure functions P and the parton distribution functions Ii can be derived, •
\lV'}/
(v,(v F1
(x) =
'1"", 2 L...J1i,.N(z)G,;iVV' , ;
• (1.5.28)
The coefficienUl (v are given in (1.!l.R) . From (U i.2R), an important result of the parton model, namely the Callan- Gross relation [Ca69j, Fz(z) = 2xFdz)
(1.5.29)
68
I Phenomenological basis of gauge theories
and similarly for ant.iquarks. According to (1.5.32), the structure function for electron- proton scattering for four quark fiavours becomes6 P Fr (x) =
In the case of charged-current scattering, the quark-mixing matrix must be taken into account. At energies high enough to create the various quark doublets with nearly equal probability, however, the effects of the quarkmixing matrix drop out. Consequently, beyond the charm threshold we have
for the neutrino stru cture fun ctions. These results correspond to our normalization of the weak currents. In the literature it is common to use a different normalization such that the factor 1/2 in (1.5.35) is replaced by 2. The latter convention is used in the following. Then the sum rules take their usual form . Explicit expressions for the different structure functions of PH and vN scattering via neutral currents (photon- and Z-boson exchange) and charged currents (W-boson exchange) can be found in Ref. [PDGOOJ .
1.5.3
Applications of the simple parton model
In this section t he theoretical predictions of the quark- parton model are compared with experimental results. Special emphasis is put on the question as to what extent the partons carry the quark quantum numbers. SWe do not inclu de the heavy quarks b and presently available ellergiell.
j
since their contributioo is small at
1.5 The quark- parton model
1.5.3.1
69
Parton spin
If partOllS are identical to quarks they must be spin· l j2 particles. This demands the validity of the C41tan- Gross relation (1.5.29). As a. consequence, the ratio R of the cross sections resulting from the exchange of longitudinally polarized virtual photons O"L and t ransversely polarized ODes O"f is expected to disappear in the Bjorken-Iimit (up to QeD corrections) ,
R= ~ = (1 +v2jQ2)W2 - W , O"T
WI
..
--->
Q~.~ _oo ~
F,(x) - 2xF1(x) ~ 2xFdx)
o.
(1.5.36)
In contrast to this, R should diverge for spin-D partons where F, = O. Thus, R is a sensitive measure of the spin-l j2 property of the partons. The result of the CDHS vP-scattering experilOent [Ab83] , R = 0.039 ± O.014(stat) ± 0.025(syst) for 0.4 < x :::; 0.7, Q2 == 38GeV 2 , strongly confirms that the charged. parlons are spino! /2 particles. However, the nucleons contain also spino! particles, the gluons. These give a contribution to tbe longitudinal strudure functions which is discussed in Sect. 3.2.4. 1.5.3.2
Sum rules for internal quantum numbers
The electric charges of proton and neutron can be calculated. using (1 .5.33),
r dx [2a(U(X) 1 ~ Jo
1
1
t[2
o ~ Jo dx
- 1 1
. (x)) - a(d(x) - d(x)) ,
(1.5 .37)
-
(1.5.38)
1 a(d(x) - d(x)) - a(u(x) - ii(x)) .
Strangeness and charm of the nucleon are zero, consequently
(1.5.39) In the context of the parton model, those quarks that detcnnine the quantum numbers of the relevant hadron, e.g. II a.nd d for proton and neutron, are called valence quarks, the others, which appear only as quark- antiquark
pairs, are caUcd
BOO
quarJc.,.
70
1 Phenomenological basis of gauge theories
From (1.5.37), (1.5.38), and (1.5.39) sum rules for the structure functiollR can be derived. Subtracting (1.5.37) and (1.5.38) leads to the Adler 8um
This was confirmed from neutrino-scattering data using hydrogen and deuterium targets, where a value of 1.01 ± 0.20 was measured [AI8S]. Similarly, a weighted sum of (1.5.37), (1.5.38), and (1.5.39) yields the GrossLlewellyn Smith sum rule !Gr69J. It measures the number of valence quarks in the nucleon, since the parity-violating structure function F3 is equal to the difference between quark and antiquark densities, which is just the valence quark density,
~
l1dx [Ft - W- P(x) +FJV+W+P(:z;)] = 3.
(1.5.41)
This parton-model result gets perturbative and non-perturbative QeD corrections [Br87, 8h79]. These corrections shift the 3 on the right-hand side of (1.5.41) to 2.55 ± 0.03 at Q2 = 3 Gey2. The experimental value of 2.50 ± 0.018±0.078 [Le93] confirms that the proton consists of three valence quarks, and hence the partons carry baryon number 1/3. Neglecting the sea-quark contributions, the values of the electric charges of the quarks can be derived from the GoUfried 8um rule !Goo7] 1
10 ~
[Fr'P(x) - F[YN(xl] =
Q~ - Q~ =
;.
(1.5.42)
The experimental result including valucs in the region 0.004 < x < 0.8 at Q2 = 4 Gcy2 is smaller, namely 0.2281 ± 0.0065 !Ar95] , indicating a Havour asymmetry in tbe quark-antiquark sea. Averaging the experimental results over neutrons and protons and using the ratio of electron and neutrino structure functions for large x, where the sea contribution is small, gives
(1.5.43)
72
1 Phenomenological basis of gauge theories
1.5.3.4
Experimental parton distribution functions
The experiments on deep-inelastic electron- nucleon and (anti)neutrincrnucleon scattering contain enough information to determine the distribution functions of the valence quarks u(x) and d(x) , the sea quarks sex), c(x), u(x) , d(x), s(xl , and c(x) , and the gluons Cram the measured values of the structure functions (d. Sect. 3.2.4). They are presented in Fig. 3.5.
1.5.4
Universality of the parton model
The naive parton model can also be used for a phenomenological description of other inclusive hard-scattering processes, such as the e+e- annihilation into hadrons and inclusive badron production in electron- nucleon scattering. Just as in lepton- nucleon scattering, the cross section is calculated from tbe following ingredients:
al
the parton cross sections (for free , point-like particles),
b)
the probabilities of detecting the partons i with momentum fraction x in the hadron h, the parton distribution junctions li,h(X) ,
c)
the probabilities that a parton i is convcrted to a hadron h, the fragmentation /unctions Dh,;(X).
We give several examples to illustrate the applications of the parton model.
1.5.4.1
Electron-positron annihilation into hadrons
The total cross section for the electron-positron annihilation into hadrons is calculated according to the above concepts from the point-like e+e- - t qq cross section similar to (1.5.23) and from the probability that quarkB are converted to hadronic states. According to the confinement hypothesis the latter is equal to one. For energies that are small with re5pect to Mz only the one-photon-exchange diagram has to be considered, and the cross section is given by (1.5 .45)
76
I Phenomenological basis of gauge theories
the fragmentation , but at low values of x a significant amount of sea quarks is present. The parton model has been successfully applied to many other hardscattering processes, e.g. inclusive p+ JJ - creation in nucleon- nucleon or hadron- hadron scattering with high transverse momentum, jet production, etc. The validity of the simple parton model confirms that - investigated with high resolution- hadrOnB contain point-like, in good approximation free scattering centres carrying quark-flavour and colour quantum numbers. Moreover, flavour-neutral partons, the gluons, exist inside hadrons. Details of this are discussed in the framework of perturbative QeD in Sect. 3.3.
1.6
Higher-order field-theoretical effects in QED
We based our phenomenological discussion of dynamics in elementary particle physics essentially on t he one-particle exchange model (Sect. 1.1.2). This model is a generalization of the simplest approximation of QED. The experimental verification of hil;her-order effects with great precision further strengthens the role of QED as a prototype for theories of tbe other interactions. The following discussion of the anomalous magnetic moment of the muon in Sect . 1.6.2 illustrates this point.
1.6.1
QED as a quantum field theory
In Sect. 1.2.1 we already got some insight into the physical content of QED
as described by the Lagrangian (1.2.16). The most important points are: •
The quanta of the free basic fields ¢ and Ali describe the observed fundamental fermions (electrons, muons, .. . ) and photons, respectively.
•
The non-linear terms in the field equations are responsible for the interaction between these particle! and can be described by Feynman graphs representing a perturbation theory with expansion parameter a = e 2/ 41r.
•
In the simplest case, a Feynman graph describes a reaction by an exchange of particles (tree graph) .
78
1 Phenomenological basis of gauge theories
Fig. 1.15 Correctioos to the photon-electron vertex
The experimental evidence of radiative corrections is the decisive teat for the quantum.field·theoretical nature of a dynamics. The next section shows that this has been achieved for QED with extremely high precision.
1.6.2
A test of QED: the magnetic moment of the muon
We consider one of the precision experiments carried out to test the validity of QED, the measurement of the magnetic moment of the muon.
1.6.2.1
Description of the experiment
If the magnetic moment of the muon ,, ~
-
e
2m
(1.6.2)
9'
deviates from its Dirac value (g = 2), i.e. if it has a Pauli coupling, then it.s spin s rotates with the frequency
eB Wa =a-, m
a
~
I
-(9 - 2) 2
(1.6.3)
1.7 Towards gauge theories of strong and eloctroweak interactions 81
1. 7
Towards gauge theories of strong and electroweak interactions
Toda.y, the Standard Model of elementary particles describes physics by quarks and leptons interacting through fields of vector particles (Sect. 1.1 ). In the preceding secti(;ms we collected the basic: experimcnl.8.i facts which led to the following picture. •
Hadrollll consist of constituents, the quarks. Because of confinement, these do not exist as free particles but a.s panons which in interactions ha.dronize into particle jets.
•
The charges of the strong interaction, called colour, are related to an SU(3)c symmetry. The strong interaction is mediated by gluons. Like the quarks, the gluons do not exist as free particles but only as partons. The gluons have been verified experimentally as jets.
•
The disentanglement of the involved current- current structure of the electrowcak interaction revealed a global SU(2)w x U(l)y symmetry. The associated charges are weak isospin and weak hypercharge.
•
The carriers of the weak interaction, the weak vector OOSOM have been found experimentally with masses of 80 GeV and 91 GeV.
•
The discovery of the top quark, which was required by the SU(2)w symmetry, has completed the spectrum of fundamental fermions.
•
For the successful precision tests of QED the inclusion of radiative corrections into the theoretical predictions is crucial.
Relativistic quantum field theory and , in particular, well-established QED provided guidance for the phenomenological disctlSSion. Symmetry considerlLtioWl IItruc tured the dilfer-ent experimental result.,. lUI in QED , cUJ"r-ents
and vector particles play an important role in the description of all interac-tiotl8. It turllB out that these concept8 are naturally embedded in quantized gauge theories, Yang- MiUs theories, in which interactions are essentially determined by local symmetries. The description of such t heories of the strong and electroweak interactions is the main theme of the following chapters.
1.7 Towards gauge theories of strong and eloctroweak interactions 81
1. 7
Towards gauge theories of strong and electroweak interactions
Toda.y, the Standard Model of elementary particles describes physics by quarks and leptons interacting through fields of vector particles (Sect. 1.1 ). In the preceding secti(;ms we collected the basic: experimcnl.8.i facts which led to the following picture. •
Hadrollll consist of constituents, the quarks. Because of confinement, these do not exist as free particles but a.s panons which in interactions ha.dronize into particle jets.
•
The charges of the strong interaction, called colour, are related to an SU(3)c symmetry. The strong interaction is mediated by gluons. Like the quarks, the gluons do not exist as free particles but only as partons. The gluons have been verified experimentally as jets.
•
The disentanglement of the involved current- current structure of the electrowcak interaction revealed a global SU(2)w x U(l)y symmetry. The associated charges are weak isospin and weak hypercharge.
•
The carriers of the weak interaction, the weak vector OOSOM have been found experimentally with masses of 80 GeV and 91 GeV.
•
The discovery of the top quark, which was required by the SU(2)w symmetry, has completed the spectrum of fundamental fermions.
•
For the successful precision tests of QED the inclusion of radiative corrections into the theoretical predictions is crucial.
Relativistic quantum field theory and , in particular, well-established QED provided guidance for the phenomenological disctlSSion. Symmetry considerlLtioWl IItruc tured the dilfer-ent experimental result.,. lUI in QED , cUJ"r-ents
and vector particles play an important role in the description of all interac-tiotl8. It turllB out that these concept8 are naturally embedded in quantized gauge theories, Yang- MiUs theories, in which interactions are essentially determined by local symmetries. The description of such t heories of the strong and electroweak interactions is the main theme of the following chapters.
2
Quantum theory of Yang-Mills fields
The phenomenological discussion of Chapt. 1 suggests that the tbeory of the strong, the electromagnetic, and the weak interactions takes the form of a quantized Yang- Mills theory. Therefore, the structure and properties of relativistic quantum field theories in general and Yang- Mills theories in particular are investiga.ted in t his chapter. In Sect. 1.2, we have t reated the quantum aspect of fields with help of the wave-particle dualism for linear field equations and with the recipe-like use of Feynman rules (cr. Sed. 1.2.1 ). Although many applications can be discussed on this basis, tills procedure is insufficient for the general treatment of gauge theories. In the following, we need the general connection between fields and physical particles. In Sect. 2.1 we show bow the expectation values of field operators, in particular Green functi ons, describe particles and their interactiolls in fieln theory. Thi!l i!l of part.k lliar import.an r.e for Qcn !linr.e quarks and giuolls do not appear as free particles. One possibility to describe the full content of a gauge theory, especially the consequences of gauge invariance, is the path-integral representation (Sect . 2.2) of quantum field theory. It provides a closed expression for all Green functions of the theory directly in terms of the Lagrangian . This formulatioll is the starting point for a deductive treatment of both QeD and the theory of the electroweak interaction. The principle of local gauge invariance and the construction of Yang- Mills theories are discussed in Sect. 2.3. Section 2.4 is devoted to the quantization of gauge theories in the path-integral representation. Perturbat ive evaluation of tlte path integral results in the familiar Feynman rules for cal culating Smatrix elements. The path-integral representation a lso allows for a simple derivation of the Ward identities resulting from gauge invariance. In order to evaluate quantum field theories in higher orders, renormalization has to be performed. The principles and the technical details of Tcnormalization are discussed in Sect. 2.5 and applied to QED and non-ab€lian gauge theories at one-loop order (Sect . 2.5.3). In Sect. 2.5.2 the methods for the calculation of higher-order corrections are presented. In the case of gauge
86
2 Quantum theory of Yang- Mills fi elds
theories, the Ward identities play the decisive role to proof that the renormalized theory gives a gauge-invariant, unitary S matrix in the space of the physical state vectors. This is discussed in Sect. 2.5 .4. The relations between different renormalization prescriptions a.re rela.ted by the rcnormalization group (Sect. 2.6). The rellormaJizability can be spoiled by anomalies. The origin and t he physical consequences of anomalies are discussed in Sect. 2.7. Besides the ultraviolet divergences, quantum field theories contain also infrared divergences. Their origin, and the methods to extract and to treat these divergences are explained in Sect. 2.8. Besides perturbative techniques, the F'eynman path integral allows to apply also non-perturbative techniques to derive results for physical observables (Sect . 2.9). We consider the classical solutions of the field equations and discuss the evaluation oC the path integral in the semi-classical approximation. A different non-pertnrbative treatment of quantum field theories consists in the lattice approx.imation which is discussed in Sect. 2.10. This allows, in particular, to approach the problem of confinement (Sect. 3.7) and other non-perturbative phenomena.
2.1
Green functions and S-matrix elements
This section extends the field-theoretical vocabulary which was summarized in its simplest form in Sect. 1. 2.1. We show how in a genera.l field theory the S matrix for relativistic particle reactions call be obtained from quantummechanical expectation values of field operators. At the same time, some general properties of the S matrix are discussed. Our presentation is predominantly descriptive; for more detailed treatments we refer to the literature
1FT].
2.1.1
The principles of quantum field theory
The main concepts in quantum mechanics are states and observables. In field theory we met already the vacuum state !O), the free one-particle states IM ,p,j,j3), and those of many free particles (d. Sect. 1.2.1.2). The latter play an important ro le in theories with interaction for the description of
incoming and outgoing particle configurations.
2.1 Green fUllctions and S-matrix elements
87
Quantum-mechanical operators occurred in the form of quantized basic fields , e.g. free fields in Sect. 1.2.1.3, or as pbysicaJ observables, e.g. currents and charges composed of them AS in (1.2.79) and (1.2.8 1). The basic physical principles can be formulated with help of these concepts in a rather general way. They arc summarized in the Wightman axioms
[W;56] , 1.
The principle of special relativity for closed systems requires a representation of the Poincare group (the group of translatious by a vector a and Lorentz transfonnations A) by unitary operators Uta) and UtA) that leaves the vacuum state invariant,
U(aIlO) = ]0),
U(A)]O) = ]0),
(2.1.1 )
and transforms the particle states according to (1.2.2) a.nd (1.2.3). Accordingly, field operators "'o(z) must transform covariantiy, U(a)¢o(x)U - 1(a) = ¢o(x + a) , U(A)"o(x)U - 1(A) = S~~,(A)¢o'(Ax).
(2 .1 .2)
Here S';~(A) is the representation matrix of an irreducible representa..tion of the homogeneous Lorentz group spanned by the 1/Jo [Ca77] . The translation operators Ural = r}P"o,. and the Lorentz-transformation operators UtA) = exp(-iMJI",B,..,) are generated infinitesimally by the four-momentum operator P'" and the generalized angular-momentum operator MIW . The antisymmetric tensor fJ",,, is composed of the rotation and boost parameters. 2.
The eigenvalues of the energy and mass-square operator must be nonnegative,
pO ~ 0,
(2.1.3)
The physical vacuum 10) is the only state with the lowest energy pOlO) = o and is non-degenerate. All other states have positive energy pO > 0 and mass squared p2 > O. 3.
Causality is implemented by the requirement of locality. Observablcs must be quantum-mechanically independent at space-like distances, i.e. they have to commute. Consequently, the field operators of hosolls (fermions) have to commule (anticommute) for space-like distances (observables must be even in anticommuting fermionic operators),
IWo(z),1/Jo,(x' )h, = 0 for {z - x ' )2 < O.
Mo"",,' ,
(2.1.4)
88
2 Quantum theory of Yang- Milts fields
4.
General charyc-conscrvation laws follow from the existence of charge operators Q. These generate infinitesimal symmetry transformations of the fields according to (1.2.89) and (1.2.90) and annihilate the vacuum. QIO) ~ O.
(2.1.5)
These Wightman w oms represent the general framework of relativistk quantum field theory. We note that the Wightman axioms 2 and 4 are not fulfilled for realistic theories, because these contain massless particles (p 2 = 0) and degenerate gr ound states. Massless gauge bosons are typical for gauge theories. The cOl1litruction of gauge theories is discussed in Sects. 2.3 and 2.4 .
2.1.2
Green functions
The physical content of a quantum fi eld theory is contained ill the quantummechanical vacuum expectation values (vevs) of products of field operators. Consider a multiplet of Gelds 'I/I,, (x) that t ransforms according to an irreducible representation of the Lorentz group as in (2.1.2). The vacuum expecta tion values of products of Geld operators are called Wightman jUflctioru (W ;56J, W0 1. . .a"
2.1 Green functions and S-matrix elements 89 This argument illustrates how properties of Wightman functions are derived from the general principles. The analyticity properties which follow from (2. 1.3) and (2. 1.4) are discussed in the literature (FTJ. Besides the Wightman fUllctions another type of vevs, the Green /unction6, plays an important role in field theory. Green functions are the vevs of timeordered products of field operators [cf. (1.2.22)j
(2. 1.9) Just like the Wightman functions these are relativistically invariant fuuctions of the coordinate differences and (anti)symmetric in the coordinates because of the (anti)symmetry of the T product for (fermions) bosons.
2.1.2.1
Green functions of free fields
The representation of the free fields by creation and annihilation operators allows for a simple calculation of the vevs. Starting from the representa.tion of scalar hermitian (neutral) fields 1/;(x) by creation and annihilation operators (1.2.10),
tJ>(x) "'"
~ j (211")3
d'p [a(p)e-iPZ + af(p)e+iP:Z:] 2Ep ,
(2. 1.1 0)
and using a(p)IO) "'" 0, (Olat(p) = 0, and la(p) ,a'(p')] = 2Ep(211")lJ(p - p I) gives
W(x"x,)
~
~ (~(x,)~(x,))
_ 1_ {21T)6
j
= (2:)31 = ~ (21T)3
3 3 d p d p' (a(p)at(p'))c-ipzleiP':E, 2£1' 2Ep'
~~ e- i p{:EI- :E3)
(2.1 .11 )
jd'p '(P2 _ M2)8(p')e- P{:E I-:E 2) = A +Ix i
I
-
x 2, M) .
This and (1.2.3 1) imply
G(x"x,)
~ (T~(x,)~(x,))
= 9(x~ - xg)(tJ>(Xd1/l(X2)}
(2.1.12)
+ 9(xg -
xYHVI(X2)1/I(xtl}
90
2 Quantum theory or Yang- Mills fields
for the two-point Green runction. Using the Fourier transrorm of the ()function
O(x) = lim -
1+
i
~ ..... o+ 211"
00
dy
-00
e- i.2:U
.,
(2.1.13)
Y + Ie
this gives
(2.l.1.)
Thus, the two-point G reen function of the free fie ld is i times the Feynman propaga!or .6. F . In coordinate space, the Feynman propagator can be written in terms of tbe modified Bessel fun ction of first order K J [Ab70]
(2. 1.15)
For large distances, M
JiX2f ~ I, this behaves as
(2. 1.16)
i.e. for large space-like distances it drops exponentially. The behaviour close to the light cone, M M « 1, is
6(x'2). 1 1 + Isgn(x'2)-4 " . 11" :z; + If"
L),F(X) = - -.- 1T o
'2 M 2 (
- lsgn(x)S1T2
log
+ 0 (M IOglx'l)
Mv-x2+ i£ 2
+2"1) (2.1.17)
2.1 Green functions and S-matrix elements 91 The four-p oint Green function for the free scalar field can be calculated in the same way,
G(Xl ' ... , x~) = i 2 ( 6p (Xi
+ 6 p(X i
-
-
x3)6P(X2 -
xZ )8p(X3 - X4) Xi)
+ 8p (X i
-
(2.1.1 8)
x .. )8.P (X2 - X3 ))-
Since 6F (X - y) = 8.F (Y - x ), the four-point Grcen function is in fact symmetric in the four space-time points X; . The general expression for the symmetric n-point Green fUllction of the free scalar field vanishes for n odd and is given by a sum of products of two-poillt functions for n even,
where the sum extends over all nOli-equivalent partitions of the X i . Similarly, the Green functions of the free Dirac field are sums of products of the Dirac propagator (T1/Io (X)~ti'(Y)) = i5F (x - Y)o/.l = (i" + M}o pi8. p(x - y). 111 view uf the impur~am:;e of !Spi n- l partidC!:l ill gauge tlil.-'Orie:;, t he t wv-vuillt fUll ction of massive vector bosons is considered explicitly. Inserting the field operator
3p _711-)3 - • "L..J Jd . (p) e - ;" + h .c. , (2.1.20) A!' (X ) -_ (2. 2E [e,. (p.) ,J3 aJ~
1
1;J= O,±1
P
where el'(P,h) is the polarization vector defined ill App. A.1. 1.6, into the vacuum expectation value leads to
(2.1.21)
M,,""'" ,
92 2 Quantum theory of Yang- Mills fields where we have used the polarization sum (A.1.45). By time ordering in (2.1.21), the Green function of a massive vector field
G
(x x) =
I'"
I,
2
~
(211' )4
= .I (
/d'
-0"." -
p -gil"
A 8M2
p2
+ ppp"IM e-ip{~I -~2) 2
M2
) tl.F (XI
+ if;
- X2, M
)
(2. 1.22)
is obtained. For M2 -+ 0, (2.1.22) gives a divergent result. In order to derive the vectorboson propagator in the case of zero mass, gauge invariance is crucial. As shown in Sect. 2.4, tbe Fourier transform of a massless vector-boson propagator in gauge theories is given by G",,,,(p) = _ig",,,/(P2 +i£) in the Fellnman gauge.
Then, the Wightman function of the free massless vector field reads ( )) _ _ ~/ (A ,. ( X I )A "X2 ~ g".., (2.,.-)3
21pl e-;~ .. -., ) . 3
dp
(2.1.23)
Now consider a state created from the vacuum by application of AI)(z)
I~) = /
(2. 1.24)
d· x](x) A,(x)IO).
Then, with the Fourier representation of l(x)
f(p ) =
I (211')3/2
/d
4x
ei(lp!:rD- px) f(x)
'
(2.1.25)
the norm of this state is
J J
1I ~112 = d4 xI d4 x2 (Ao(xd A o(x2))f"(xdf(x2) d3p •
= -
/
•
12p ( (P)] (p) < 0,
(2. 1.26)
i.e. the vector-boson propagator of the form used above in QED implies the appearance of unphysical states with nega.tive norm. This is not the case for a propagator of the form (2. 1.22). The possible occurrence of negative-norm states is a frequent phenomenou in the relativistically covariant treatment of quaotized gauge-field theories. Care must be taken to prevent these contributions from baving any physical signifi cance and violating the unitarity of the S matrix (d. Sects. 2.5.4.3 and 2.5.4.5).
2.1 Green functious and S-maLcix elements 95
2.1.3
S-matrix elements and the LSZ formula
As already mentioned , Green funct ions determine S-matrix clements. This can be seell by comparing t he Feynlllan rules for S-matrix ele ments and Green fu nctions. The S-matrix elements are obtained in the following way: first , take tbe Fourier t ransform GWI , " _,Pn) of the corresponding Green function G(x\, . . . , x,,) ,
- (p C
t, . .. ,
Pn )
_jd
~
=:
4 ". •• . d'... \ .... n
( 21r)4J{~)(PI
e- i (Pl:1' l +" '+Pn:l'II ) C(_ - n) ... I . .. · ' ...
+ ... + Pn)G(PI>' "
, Pn)'
(2.1.32)
Then, remove the poles of those internal lines that end in fi eld points by multiplying by - i (p~ - Ml) . This operation is called truncation. The rc-suit ing expressions at P~ = Ml a nd P~ > 0 for i = 1, ... ,3 and - P~ > 0 for i = s + 1, .. . ,n ("on-shell") give the S-matrix elements for s incoming particles with the momenta Pi, i = 1, ... ,s and n - s outgoing particles with the momenta P: = - Pi , i = s + 1, . .. , n , (- 1'.+1,· .. - PnlS[PI , . .. ,PI) I
This is the reduction formula for scalar fields , also called LSZ formula. It expresses the S-matrix element as the mul tiple residue of t he Green function with respect to the particle poles corresponding to the external lines. For particles with spin, the correspond ing wave fu nctions have to be attached . An extended vcr~ioll o r (2. 1.33) incorporating the descriptio n of bound states is given in Sect. 2.1.5. The wave-function renormalization. coflstants n- I / 2• also called LSZ fact ors, are required for a correct normalization of t he S-matrix clements. They are determined from the onc--particJe parts of the field operators or equivalently by the residues or the propagators at the physical mass (2.1.34)
Thus, the S-matrix elements are given by the truncated Green junctiona [Zi59, Sy60, Sy67j,
taken on-shell. Conversely, the truncated Green functions represent S-matrix elements which are continued to off-sheU momenta Pi. A particle or an externalline is called on shell if its momentum squared equals its mass (P2 = m~) and off shell otherwise. The truncation can also be performed ill configuration space. Here, a Green function is truncated in a field point (indicated by underlining the space coordinate) , (2.l.37)
by convolution with the inverse two-point function C - 1 (Xlo X2) defined by
(2.1.38) Truncated Green fundions can be represented perturbatively by truncated Fe!l1lman graphs. These are obtained from tbe usual Feynman graphs by removing all parts that are connected to a field point and can be separated from the other field points by cutting a. single internal line. This is illustrated in Fig. 2.1 , where the graph is truncated at the field point XI by removing the subgraph C. In the following we visualize truncated graphs by omitting the field points at the external legs.
2.1 Green function« and .S matrix elements
!I7
FIR, 2.1 Graphical ropremutation of the truncation procedure
2.1.4
Connected Green functions and vertex functions
The representation of Green functions and S-matrix elements by graphs describes also some structural properties. The; vacuum and one-particle structure is physically of great important;« beyond perturbation theory and leads to t he concepts of connected Green functions and vertex functions. '2.1.4.1
Connected Green functions
The graphs for a Green function consist of cmmcctcd. subgruphs. A graph is said to be connected if any two of its points ;ire connected to eaeb other via internal lines. In a free theory only the graph of the two-point function (2.1.28) is connected owing to the lack of interaction vertices. The contributions of disconnected graphs to a Green function factorize according to the Feynman rules. Trivial examples are the Green functions of free fields (2.1. i 8) and (2.1.19). As another example consider the four-point function with interaction. The graphs which contribute 1 can be classified as x{
o
*X'2 C3 l
x:i
~X*
X\
*X2 C^)
X3
VVc omit graphs containing vacuum subgraphs and tadpoles, which do not «Hitribute in tin; S matrix.
98
2 Quantum theory of Yang Mills fields
where the open circles represent the sum of all connected graphs with two or four field points, i.e. the connected two- and four-point, functions. If we denote the corresponding analytic expressions by Gr(;C[,x-i) and (>\ [x\.x2, :'•':(, the decomposition of the general two- and four-point functions into connected parts reads (assuming that the one-point function vanishes, i.e. Gc(a:) = 0) G{xux2) G[X\,X2,X3,X^)
=
(2.1.39)
Gc{XUX-2),
= Gc(x\,x2,
X
+ Gc{xx,
X3)Gc(X2,
GC(X\,X2)GC(X3,X4)
x.,) + Gc
, . ' i )G'c(X2,
X3).
In general, if there is a unique vacuum [IIa92] we get the following decomposition of n-point Green functions into connected parts, the cinstcr decomposition G(x t,...,.T„)
The sums extend over all partitions of the indices l . . . . , / t . This equation can be inverted iteratively to give the general definition of connected Green functions Gc(xu...,
*„) = G(x!,...
,xn) - 1! £
G(...)G(.
•.)
+ 2!J3G(...)0(...)G(...):F--
(2.1.41)
The decomposition of the Green function G(:/:| x„) into connected parts, called vacuum structure, of Green function* [Wa53, Sy54, Zif>9], describes an important, physical point. Independently of the representation by Feynmau diagrams, the following properties of Green functions can be derived in the framework of general field theory [Jo65] ( 7 W * , ) • • • i>(x„)T/,(xn+] + a) • • • rf>(xn+m + a)) |o|-+oo
(2.1.42) ii (-m
2.1
s
s
)
I-ig. 2.2 Example of a onoparticle reducible Feymnan graph
Gc(xi,.. . .
.,xn,xn+i
+ a,.. .,xn+m
. A
(2.1.
+ ) = 0,
(2.1.4 )
ss s ), s,
S ,
s
s
ss s. T
ss s
s s
s s,
.
,
s
2.1.4.2 A
.T s s 2.2, .
s
ss s
s
s
ss s A, 1 ,
s s
ss s
one-particle reducible . s s, ( .
, proper vertex graph. T s s . s s s . F\(x],... f'\ (z, z') ss s
s s s,
s x, . T s
. 2.1. ). s onc-particlc. irreducible, s A, . 2.2 FD(z,xu+\,... s,
,xn),
s
100
2
s s
ss
s Fq(Xu.. j As
. ,i6) l
z
V i'\(xU£2,x3,z)
s ss s.
T s s ,
=
,
s
F^(z,z')Fn(z',x,u
,
xT>, , ).
(2.1.44)
,
s
s
s s
0, T s vertex function
s
s
s
s
s
s
(2.1.4 )
T
s s. T s
s
s s
structural graphs. T s ,
x2 G {XuX2,XZ) ,
= 2
(2.1.4 ) GZ{X\,
)
(x2, Z'i)Gc(
, z-i)\Y{z\, z2, z,3).
With the help of the combinatories of Feynman graphs, the relation of Green functions with n Held points to those with (71— 1) field points is the following: the additional lield point .T„ is connected via a propagator to either an mvertex (n — 1 > rn > 3) or to a propagator (via a three-vertex) of a structural graph. This procedure has to be carried out in all possible ways on the
2.1
lìn*cii f u n c t i o n s and ¿»'-matrix e l e m e n t s
diirerent (n — 1) structural graphs. In the case of the connected function this leads to the result
101
four-point
X\
X3
OT]
X\
X3
X2
X4
X2
X2
x<\
+ ( x 3 f > I i ) + ( l j H 2:2)-
This recursive construction can be inverted to a recursive definition of the vertex functions T(a;j,...,:;:„) from the connected Green functions G c ( x i , . . . ® „ ) (cf. Sect. 2.2). The representation of Green functions by general propagators and vertex functions describes a general collision process as made up of individual processes connected by propagating particles. The individual processes are described by the vertex functions, and this is the reason for t heir great physical importance. Important physical parameters, such as effective coupling constants. can be expressed in terms of the vertex functions as described later. Because of energy-momentum conservation, the particle propagation between individual processes might be only virt ual. In any case, the mass pole of the Fourier transform of the two-point function determines the long-range component oc e x p ( — M x ) of the particle propagation [cf. (2.1.16)].
2.1.5
S c a t t e r i n g of c o m p o s i t e p a r t i c l e s
Poles in the invariant-mass variables of Green functions point to the existence of free particles and the corresponding one-particle structure of their reactions. This was illustrated with help of a simple example of a field theory in which the field quanta describe directly the free particles. However, even particles that are bound states and their reactions can be described by Green functions. In order to formulate this, the fieldtheoretical amplitudes for one-particle states composed of field quanta, Lhe Bethe Salpeter (BS) amplitudes X'> are introduced. For a i/J"VJ bound
.
2
s
s
, , , ), . . j, , s
s
s
s
a-( i> 2) *( )( i + 2 ~
ss
A,
,
)
(2.1.47) T
s
s
s ,
s
s. s
s
. T
s
,
s
s
s s. 1,
1,
s 0,
T
s 7 . S
s
s 1.
T
T
. s G{p\,..., pn) = , +... + i s 2 2 = , ( = 2) ss s s s.
s s , . .
s
=
2
, (
s, s
2.
poles ss s (2.1. 1). s r
s
s
t,
r =
+... +
=
, =
ss s s
s,
s
s s
f= X , s,
t+.
X r{- s+i,---,- r\S\ ,..., s)x
X
i
s (2.1. 4). T
s s s
(2.1.48)
general
SZ
2.1 (
formula
s
,
s
s s (2.1.
. s.
).
X (lh)6 ( i
~
ipi
k) =
J d*x(mxme ,
10
s . s s
(2.1. )
e. Xhi l) = u( ).
s s G(p,
2
( p)
s
s
(2.1.48), G(p\,... 2 ). A s
,pn)
s 1,
.
s
s
s
s
. 2.1. .
s
s s
As
s
+
s
s c ij s
s
(2.1.48) s .
+ s,
s
A 2 = ( \Tipqi (xi)il>q3(x2)\ i2).
T
.
s
s
s + 2
)
, . . s ( . T
s
2
s 2
n
(2.1. 0)
(2.1.48) s 2 )(4, )(A 2 s
s
, 2
s )
1
,
n
=
.
2
s
s
X2
X-i
2.2
T
s
s s
s
s
s
s
s
s,
.
s
s
s,
s
s s
s Fe nrnan path integral . s s s s
s s
s s
2.2.1
generating unctional. ss s s s s s ss s.
T
s s G(x\,...,
s
xn )
s
x), r T{
=
,
G(x 1 , . . . ,* ). (*,)
( x n)
(2.2.1)
2.2 Path-integral representation o f q u a n t n n i lield theory 10!) defines the generating functional2 T{J} from which—as we will see—the Green functions can be reconstructed by functional differentiation. In order to profit from this approach, several rules for the manipulation of functionals are needed. Therefore in the following, concepts from the analysis nf functions of several variables, such ¿us partial differentiation, power series expansion, and integration, arc extended to functionals, i.e. to functions of an innumerable infinite number of variables [BeC6, FV72]. A systematic treatment of these methods is beyond the scope of this book; we only give a pragmatic orientation and refer to Ref. [Ro94] for more details. Another approach to functional calculus is by discretization, e.g. by the lattice approximation (Sect. 2.10).
2.2.1.1
Functional differentiation
If we restrict our analysis of functions of several variables y = { y \ , . . . ,y„) to polynomials and power scries, then the partial differentiation operator d/diji can be formally defined by
#-1-0, oVi
~y oiH
k
= kk,
(2.2.2)
and the product rule
»
( w c H )
=
Analogously, the functional the rules
6 y(x)
2
(^)
c ( i l +f H
differentiation
6/Sy(x)
(?|!).
(„3)
is a linear operation with
6y(x)
We enclose the ;irgiiincnts of functionals in curly brackets.
I()(j 2 Quantum theory of Yang Mills fields
2.2.1.2
2.2 Path-integral representation of quantum field theory
Volterra series
'2.2.1.3
The generalization of the power-series expansion of a function F ( y ) of k variables
oo
Functional integration
The methods for functional on the gaussian integral f / J-oo
k
= E £ •••£ ¿ »1=0 ¿1=1 ¿n=l
,. • •, *„) », • • • yin,
(2.2.5)
If (y, Ay) = then
F
{y}
~ r = £ d --d J Xl ri—O
integration, as used in the following, are based
d
V / -7= exp (—ay ) =
1
(2.2.9) \Ai
j JJiAiji/j is a positive-definite, quadratic form in k variables,
+00 leads via the formal transition to innumerable infinitely (continuously) many variables, i x, yt y{x), -> fdx, to the representation of a functional F{y} by the Voil erra series
r+OO
K
A... (2.2.10)
/
... / r r % e - ^ y ) = (detA)-1/2, -oo J-oo L, ILS can be proven by diagonalization of A by an orthogonal transformation under which the volume element f j <1?A ' s invariant. This formula can be generalized to the space of functions y(x)
l
Xn-T(x ... x )xj(xl)---y(xn). n u 1 n -
(2.2.C)
(y, Ay) = J
dxdx'y(x)A(x,x')y(x'),
j V[y] e-(y'A<j) = (det/1)- 1 / 2 , The symmetric coefficients T(xu...,xn) of the Volterra scries (2.2.(5) can be obtained as functional derivatives at the origin,
T(xu...,xn)
=
/ " ày(xi) • • • 5y(xn)
. ly=o
(2.2.7)
(2.2.11)
where the volume element V\y] is an infinite product of ordinary volume Similarly, elements dy(a:) at all space time points, 'D[y\ = \\xdy(x)lsfn. the determinant det A is defined as the (infinite) product of the eigenvalues of the operator /1(:c,.t'). The formula det A = exp[Tr(log A)}
Using this formula it can be easily seen that the results (2.1.19) for the Green functions G{xi,... ,x„) of the free scalar field can be reproduced from the functional
7frce{-/}
= exp
(-¡¿J
The generating functional man propagator.
107
d4x
j t f y J(x)AF(x
- y)J{y)^
.
(2.2.8)
(2.2.12)
often allows for an approximate calculation of det A. For the precise mathematical definition of the functional integral in quantum field theory we refer to Ref. [Ro94]. .lust like in the case of conventional integrals, expressions of the type f'D[y]F{y} exp(—(?/, Ay)) with a power series /•'{?/} can be evaluated by algebraic manipulations of the gaussian integral. This is extensively used in the next section. Thereby, the (infinite-dimensional) differential V[y] is regarded as translationally invariant
oj free Green functions is determined by the Feynfv[y]F{y
- y'}e-«y-y">>A(y-y'»
= f V[y)F{y)Q-^Ay\
(2.2.13)
.
2
s
, determinant
s
s
jv ]
s
s
(
)
s
e~^' z^
F{z{
=
acobi
v z]F{z e~^ (2.2.14)
s
z{ (x)
fdx'z(x,
s
( ,
=
2.2.1.4
))
s
(2 2.1 )
s
ss s
. A
s
s
s
s.
2n generators s , , i = 1,... , ,
of a
Grassmann s ,
= i) van + VjVi = ViVj + VjV* = v*yj
T
s
s, s Grassmann-algebra-v luud s s
s
s
T s algebra, . . r
ss
s
s
x') (x'), z(x,x')
s
s,
T
, f =
s
ss s T(l\,...,
2
+ y'jVi =
.
(2.2.1
= 0. s
lr\ \,...,
a)
s
, i
, n
n
,s= l,, j = 1
(2.2.17) ss
s ,
s
s.
2.2 Path-integral representation o f q u a n t n n i lield theory
10!)
Differentiation with respect to Grassmann variables is a linear algebraic operation defined by o
Q
7T-1=0,
-TT-yi = (Sjfc,
TTTJ/i = 0, ciy;.
(2.2.18)
and the product rule
¿
M
-
(
2
'
2
1
9
)
with ± if Fi is even/odd in the number of anticommuting variables. These rules actually define the left derivative which we use throughout this book. Similarly, integration is defined as a linear algebraic operation by
Idyi
= 0,
Jdjn
yj
= Sij,
etc.
(2.2.20)
The differentials obey the anticommutation relations [dy„ dyk)+ = [dyj, dy£] + = [dy„ y , ] + = [dy„ y£] + = 0.
(2.2.21)
The conjugate quantities yt* are treated in the same way as y, in all these calculations. The above definitions can be used to derive the usual rules of calculus. However, one has to take care of the various signs resulting from the anticommutativity, and of the fact that yj = 0. For example, for z, = A^-Vh the substitution rule of integration reads
Idzn
• • • dz, F{z) = (det A)~l Jdyn
• • • dy, F(z(y)).
(2.2.22)
In contrast to the ordinary integration this rule corresponds to a transformawith the inverse matrix tion rule for the differentials dz,- = X^jti-^ - ' 1 A instead of the matrix A. The substitution rule (2.2.22) can be derived by expanding F into a power series in the y, and using the anticommutation relations together with the basic formulas (2.2.20).
I .'IS 2 Quantum theory of Yang Mills fields
A formula similar to the gaussian integral (2.2.9) plays an important role in field theory, namely the Fulrini formula
J d y , • • • dy„dy* • • • di/J exp ^
y\AikVk
j = det A.
(2.2.23)
It follows directly by applying the substitution rule (2.2.22) to the integral ion with respect to the y,. The formulas of the analysis of a finite number of variables (2.2.16) (2.2.23) for the definition of power-series expansion, differentiation, and integration are extended to Grassmann functionals: 1.
Generating elements of Grassmann algebra y(x), y* (x1): [y(:r.),y(* 2 )] + = [j/*
IV[y]V[y*] exp ( I dar'dx y ^ ' M s ' s M s ) ) = det A.
(2.2.27)
The formula for the gaussian integral can be easily obtained by expansion of the exponential and use of the anticommutativity of the Grassmann variables.
2.2
s
.s s s functions of a free Dirac field ip(x) (2.2.8)
T
4
dx
s f](x), T (X) ss
,
)
> *n) (x'n)
fj{x)SF(x ,
(
- x')r](x)j . .
.
T
(
) 0)
(2.2.28)
s s T{.
4
~
r,(x\ )
s
s
= £
)
.
2.2.2
T{
of the Green (2.2.1)
,
ixxmT xr
)
= s s
generating functional s 0( )
4
= (
T
10 )
d'lxn G(xixn) (
s
(2.2.1)
)
(xn).
=0 T
s G(x\,...,
xn)
T{
,
-W u . 2.1
s
,
s
rnifs s
s,
s, s s s s
s, s. s.
s
(2 2 2 )
s
2.2.2.1 T
s
generating functional of connected Green functions s s T{ , TC{
= £
. .. -T d xi
-d*x
Gc(xl ..
Tc{
.,x ) (xj)-
) s
(xn). (2.2. 0)
I.'IS2
Quantum theory of Yang Mills fields
The relation between T{.7} and T c {./} is obtained from the relation between G(x¡,...,xn) and G c ( x ¡ , . . . , x n ) which was given in (2.1.41),
T{ J } = exp[T c {«/}],
T c {./} = log[7'{./}].
(2.2.31)
For the free field we obtain from (2.2.8)
îfree,c{./} =
1 d4x d\J J(X)Af(X
- y)J(y).
(2.2.32)
For an illustration of (2.2.31) we consider the four-point function. Applying the rules (2.2.4) and (2.2.29) yields
ath-integral representation o f q u a n t n n i lield theory
10!)
T h e vertex functional
The relation between the generating functional
of vertex
functions
oo
r a
d4xl---dixnrn{xu...,xn)a{xl)---a{xn)
= 5 J -
71=0 ^
and that of connected Green functions T c . transformation
ir a
= - i J d4
J(x)a{x)
(2.2.35)
takes the form of a
+ T c J}.
The factor i has been introduced for later convenience. of r is a and differs from that of T c , namely J.
egendre
(2.2.37) ote that the variable
For the free scalar field we obtain from (2.2.32) a(s)
dSjAP(x
- y)J{y)
(2.2.38)
and thus J(x) = TfrceM = -
(m-M2)«^-), d 4 a ( x ) ( + l 2 ) a ( s ) = 5free a ,
(2.2.39)
i.e. the generating functional of vertex functions for the free theory is ust the action (cf. also Sect. 2.2.5.3). For this reason, the vertex functional is also called effective action. The inversion of (2.2.36) is obtained by differentiation of (2.2.37) using the chain rule ST{a} . . f ,4 SJ(y) 6Te{J} SJ(y) - lay . . . a(y) + (1 .. . , . . . c . . = -J[x) v m ' J óa(x) J 15J(y) 6a(x) An(x) = -J{x).
(2.2.40)
I .'IS 2 Quantum theory of Yang Mills fields
As an example we calculate the two-point vertex function. For this we differentiate (2.2.36) using again the chain rule
jttVfrV
"
., 6 STC{J} r 4 S T {J} Sa(y) 26a(y) iSJ(x) J ' \8J[x)iU{z) -i^rja} f . S TC{J}
v)-Mx) J
2
C
i6J{x)\6J{z)6a{z)Sa{y)'
i5J(z)
5a{y) ^-¿-n)
If in this expression the integration with respect to z is formally considered and i52R/<Sa{z)6a{y), then ;us multiplication of the kernels S2TJ\5J{X)\5J{Z) one is the negative inverse of the other
\SJ(x)i6J(z)
- Z & v 4 J, ' '
(2.2.J
Sa(z)Sa(y)
As X(x,y)\j=o = {Tij>(x)ip(y)) is the usual propagator, X(x,y) is generally called the propagator in the presence of an external source [see the explanation after (2.2.55)]. In this sense the two-point vertex function is the inverse propagator. We continue by considering the three-point vertex function. According to (2.2.41) and (2.2.42),
Sa{z\)Sa(z2) (2.2.43) is an identity. Differentiation with respect to ./(.T3) using the chain rule, combined with
For J = a = 0 (2.2.45) agrees with (2.1.46). Further differentiation of (2.2.45) with respect to .J(x) gives, with help of (2.2.44), the inductive definition of vertex functions as discussed after (2.1.46). This shows that 1{«} is its generating functional.
2.2 Path-integral representation o f q u a n t n n i lield theory
2.2.3
F u n c t i o n a l - i n t e g r a l r e p r e s e n t a t i o n of t h e S
10!)
matrix
The LSZ formula (2.1.35) relates S-matrix elements to Green functions. This relation can also be c;ist into a functional form. For scalar fields the S matrix can be concisely written as
S = : exp [ d*x d *y ijj(:r.)G~1 (X, y)R1'2
^
:T{J} J=o (2.2.46)
where ip(x) is the free asymptotic field, G~l(x,y) its inverse propagator, and T{J} the corresponding generating functional of Green functions. The symbol : . . . : denotes normal ordering, i.e. the creation operators are left of the annihilation operators, e.g. :a f (p)a(ç): = :a{q)a](p): =
(2.2.47)
a){p)a{q).
For general fields (2.2.46) is generalized in an obvious way. In order to show t h a t (2.2.46) is equivalent to the reduction formulas (2.1.35) we expand the exponential and replace the derivatives of the generating functional by the Green functions, SO OO J
..
= £ ~ / d 4 ®i d 'y! • • • d'1*,, d 'y„ G{y\,--•,».) n=0 71' 7 xR^G-'ix,,?/,)•••
Rl/*G~l(xn,y„)
^, K4S. ) • • • tf(*„):. (2.2.48)
It remains to calculate the matrix element between general incoming and outgoing states |Pi, • • • »Pk) = (Pi ) • • • (Pk) |0), (p\,...,p[\ = (0\a(])[) • • • a{p[).
(2.2.49)
If we assume that, all final-state momenta differ from the initial-state momenta, as it is usually the case in experiments, only the ?ith term in (2.2.48)
^
I .'IS 2 Quantum theory of Yang Mills fields
contributes, i.e. there are no disconnected terms, and the corresponding matrix element is easily evaluated with help of (1.2.10) and (1.2.4),
I . . . C'P'lxk+l
xG{yi,...,yk+l)
Pi=p'/=m2
= G - 1 (pi, - p . ) • • • G~x{pk, -pk)G-l(p[,-p\)xRW2G(Pl,
• • • ,pk, -p\, ..., -p\)
• • G~](p'h
/2 nr2 —nr=rn-
,
-p\)
(2.2.50) (2.2.
where the factor 1/n! is cancelled by the n! pairings of field operators and one-particle states. The result is just the reduction formula (2.1.33). 2.2.4
The
field-theoretical
p a t h integral
The representation of Green functions by the Feynman path integral is by far the most important application of functional calculus in field theory [Fe48, Fe65, Sc51, Ro94j. Despite the fact that the field-theoretical path integral lacks a mathematically rigorous definition, such as a measure theory, it is of great heuristic value. It is a very convenient tool for studying perturbation theory and using lattice discretization; it serves as the starting point for the investigation of non-perturbative phenomena. 2.2.4.1
P a t h i n t e g r a l f o r f r e e fields
Let us first consider the free field with the Lagrangian ¿r r c c = -[d"4'>{x)d^{x) 2
(2.2.51)
- MV(!)].
We define Zhce{J)
= Jv{4>] exp ( i J(\4x
[Chce + A^Ht.)]
) ,
(2.2.52)
2.2
s
10 )
ss , ss
ss =
.
s
,
. T
s
s Ztree{ Z rcc{ 0
=
(2.2.
s
s
T
s
,
)
s
{.
s
.
ss
s (d'ld
2
+
T
)A (
x') = -
(x
-
Fe nman propagators
(2.2. 4)
x').
(2.2. 4) s
2
s
s (2.2. 2).
s
, d =
i
i
'
Fd x
(X){U+
= jd x
(j>(x)
x {-d> d, - ^
d x~d x'
s
s ss Ztm{
2 2
^ (x)
+
)T (X))
+
( a -
s
2
+ 2
A(
) ( {x) 7( )A (
4
A (s
x') {x')
x') {.x'),
ip(x) 4 ( s
{0 =
(X)T {.
x') (x')
+ jd x'
J d xd x'
=
(x)if>(x)
4
A (
x') (x')
> i >'(x)
s)
(x) (x
- x ' ) ( x ' f ) Zit C{ ,
o'^'(x)d,^'(x)
-
2
ipt2(x (2.2.
T
s
ss T
{.
Ziri i{
Z{rc(.{() (2.2.8).
s
)
I .'IS 2 Quantum theory of Yang Mills fields The field equation resulting from C(rcc+. /> is (d,1dlt+M2)i/t{x) = J{x). Thin is the reason why the argument of the f u n c t i o n a l 7f r e e . and Tf rcC)C . id called a source. 2.2.4.2
Interacting
fields
The representation of the generating functional of Green functions can he extended to interacting fields, T{J} [
'
=
Z{0}'
Z{ J} = JV[il> exp ^i Jd4x where the
[C +
,
(2.2.5(i)
agrangian C is given by
C = Ct
e
+ gCi.
(2.2.57)
Typical examples for the interaction agrangian arc C = — I/J3(X)/ or 1 C = — :i (:i:) 4!:. These formulas are the basic equations of the functionalintegral representation of quantum field theory with arbitrary agrangian. The 7J functional (2.2.56) can also be written as Z{.J} — JV{i>) e i 5 M exp ^i Jd x J(x)ip{x)
= fvty
ciSj^
J
(2.2.58) i.e. as a functional Fourier transform of the exponential of the action S{ip} of the field, or as a functional average over the exponential of the action with external source .S'. ? >,. = S{IJ)} + F d1X J(X)I/J(X). By expanding ' ' . functions
one obtains the pat -integral
Z)[V>]eiiW
representation
of
reen
. (2.2.59) . (x)=0
The formulation of a quantum field theory with help of this Feynman pathintegral formula is an equivalent alternative to the uanti ation with help of the canonical formalism [Ab73], i.e. use of the Hamiltonian and Heisenberg commutation relations. It forms the starting point for the non-perturbative calculation of Green functions (Sects. 2.9 and 2.10.1). In Sect. 2.4, gauge theories are defined in this compact and complete way.
2.2 Path-integral representation o f q u a n t n n i lield theory
2.2.4.3
10!)
I n t e g r a t i o n of f e r m i o n i c p a t h i n t e g r a l s
In renormali/able theories, the Lagrangian is bilinear in the fermionic fields. Consider a Lagrangian of the form C = i>Qi>,
(2.2.GO)
where Q is an arbitrary operator that may depend on other fields, as for instance on the photon field in QED (1.2.16). After adding fermion sources and ./,/,, we can write £ -
+ •h'i' =
Q~
= $ + J*Q~l)QW
H
- g-1^) +
We can integrate the fermionic gaussian integral (2.2.27): =
+ J<ji'ip
I
(2.2.61)
fields in the path integral with help of the
cx
p (• /
[c - ^ + j ^ f j
= det(iQ) exp ^ J d{x J^Q^J^j
.
(2.2.62)
The operator Q is related to the generalized fermion propagator S(x,x') QS{x, x') = ¿ ( 4 ) (x - x'),
by
(2.2.63)
and therefore Q~lil>{x)=
J d*x'S(x,x')ip(x').
(2.2.64)
Differentiating (2.2.62) with respect to the fermion sources and putting those to zero we find fv[i>]v$]i>(xl)${yl)'--1>(xn)$(yn)vv
(i | d
<7pCrni i5(xi, y„) • • • \S[xn) y I n ),
= det(iQ)
(2.2.65)
perm
where the sum runs over all permutations of the y,, and er porin is the sign corresponding to this permutation.
I.'IS2
2.2.4.4
Quantum theory of Yang Mills fields
E u c l i d e a n field t h e o r y a n d s t a t i s t i c a l m e c h a n i c s
The formulation of quantum field theory with help of the Feynman path integral formula brings field theory close to statistical mechanics [Sy6(i, Wi75]. We mentioned already that the gaussian functional (2.2.55) require« a damping factor. From a mathematical point of view, one particularly satisfactory solution of this problem is obtained by considering the fields ip(x) at imaginary times: xo = -i.T,i with X4 real. 3 This makes the action in (2.2.58) imaginary and the exponent in the path integral negative-definite iS{il>} = \Jdx0d3xi^
(¿V»)
- J dx,
=
- N )
2
[(¿V)* + W )
- m V 2
+
+ M V ' 2 ] -
< 0.
(2.2.66)
Fields in space-time points with imaginary time are called euclidean fields; accordingly Sp,{iJ)} is the euclidean action. The generating functional of euclidean Green functions can now be obtained by analytic continuation of (2.2.58) Ze{J} z m '
TE{J}
=
ZE{J}
= ¡Vty)
exp (-SE{i>}
+ Jd4xJ(x)iP(x)^j
.
(2.2.67)
As S*ii{V} > 0, d/i{>!)} = 'D[tj>\ exp(-.S'[.:{V;}) approximately a gaussian measure. From a physical point of view, the analytic continuation of Green functions to imaginary times may seem somewhat artificial. However, the exact conditions under which euclidean Green functions can be analytically continued back to physical times are known [Os75]. The pole structure which is important for the physical interpretation of Green functions remains essentially preserved in the euclidean region. The formulas of euclidean field theory for the calculation of Green functions formally correspond to the basic equations of statistical mechanics. In order to show this, put 5u{V'} = :{•>!'}• If the functional differentiation <5/J./(ar) Formulas for the euclidean Dirac matrices are collected in A pp. A. 1.2.
2.2
ath-integral representation o f q u a n t n n i lield theory
10!)
for the calculation of the Green functions is carried out under the integral, it follows that 1
J
[i> c x p
{-piI {ip}) 2.2.
Hence, euclidean Green functions are statistical expectation values of prodin ucts of field operators with the canonical distribution e.xp(-ftHpi{ip}) four dimensions [Hu87]. T h e action is interpreted as Hamilton function of a four-dimensional field theory. The coupling constant g ~ l ? 1 2 (cf. Sect. 2.10.1.2) gets the meaning of temperature, T = 1 3, and Z{0 = Z{[5) = f T>[tp exp(—/ Hp1{ip}) that of a partition function. The notions — log Z(/ )// , internal energy U ~ 5F/S( , etc., are used free energy F accordingly. The close formal relation between relativistic quantum field theory and classical statistical mechanics is of great, practical importance since concepts and techniques from one of them can be taken over to the other and vice versa. This formal analogy of euclidean field theory with statistical mechanics should be strictly distinguished from a quantum theory at finite physical temperature. It. is well-known [StM] that the partition function of a finitetemperature system Z(pp) and the thermal expectation value ( I) can be represented in terms of a functional integral over a finite euclidean time extent, Z{Pt) {A)Pr=
= i
^ e ^ W ,
J V[tP)A{i>}e-s^rW^
(2.2.69)
where S b 1> = r J
J
I
d
(
2
.
2
.
7
0
)
Here the fields have to satisfy periodic boundary conditions in euclidean time directions I/J(X+PT T) — V;(:C)i and ? is related to the physical temperature fir = l/[ T) with the Bolt ,maun constant . In this sense, one considers euclidean field theory as the zero-temperature limit. ' uclidean Green functions involve no time ordering.
I.'IS2
2.2.5
Quantum theory of Yang Mills fields
Feynman rules and path integral
In order to establish the connection between the perturbative treatment of theories with interaction, as discussed in the preceding sections, and the path-integral representation of quantum field theory, we indicate the derivation of the Feynman rules from the path-integral formula (2.2.56). The general procedure works as follows. First, the field variable in the interaction term C\ is replaced by a differentiation with respect to the source J(x). Then, the result of gaussian integration over free fields (2.2.55) is used. Finally, the exponential of C\ is expanded in powers of the coupling constant. This leads in the example of scalar fields to the following sequence of equations
Z{J}=
\g Jd4x
Jv[1>)exp
= exp
xC\[
1 4I- ig
Ci(ip(x))
— i SJ{x) S Jd4x\: C\ r i(5J(xi)
[¿free + Jif>] )
J V[i>] exp (i Jd*x[ctne + J1>f) +
(j iSJ(Xl) X exp
exp (ifd*x
d ' x d V J(x)AF(x
A
: i SJ(x2)
+ ...
- x') J { x ' f j Z frcc {0}. (2.2.71)
Here Zf r c c {0} is the Z functional of the free field for vanishing sources. The evaluation of the functional differentiations leads to the perturbative expansion of the Green functions which can be formulated with the Feynman rules.
2.2.5.1
F e y n m a n r u l e s in c o n f i g u r a t i o n s p a c e
We illustrate this procedure by a simple example. For this purpose we consider £] = —y/'(x)/4! as interaction Lagrangian of the scalar field and calculate the four-point Green function up to first order in g. Because of
£1 (
6
I \U(x)
V I
1
*
4! <5./(.r)'1 '
2.2
ath-integral representation o f q u a n t n n i lield theory
10!)
t he corresponding expansion (2.2.71) reads 1 Zfree{0}
SJ(xl)--6J(xA) 6'1
= (a) + ( ) -f (c) + (d) + . . .
(2.2.72)
up to the contributing order in g and J(x), («) = -
part.
~ xj)Ar(x
~
where x
l),
((>) = —ui j d''.r Ai.-(x - Xx)Ap(a; — x-2)A (x - x )Ap(x (c) = - f
-
part.
(CL) =
d4
- Xj)AF(x
Al7
(x -
)Af(
— x, ),
-
- X,) j d4x Af(X
x
l),
- X)Af(X
-
x),
part.
(2.2.73) and Z 0
Zrrae 0
= 1+
d4x A,,(.r - x)AF(x
- x) + . . . . (2.2.74)
The suins in (2.2.73) extend over all non-equivalent partitions of the x,-. It follows
= (a) + (b) + (c) + . . . .
(2.2.75)
The term (d) in (2.2.73) is cancelled by the product of g/2 fd4x A (0) 2 in the expansion of 1 /Z{()} in powers of g with the term (a). Thus, the vacuum loops yield no contribution. The general role of the factor 1 ? ()
I.'IS2
Quantum theory of Yang Mills fields
, V 1>< (o) •
, n P1 (b) k
PJ •
P2
®I
Vk
/ X
Pi
• Xfc
pa/
• Xi
(c)
s
r x3
V 7 2
X,
P*
> x4
Pi • Xj
PI •
I
J
Pj * ' Xi
(d)
Pi
• — — • Xt x
Pj • x}
Pk, O 0 . , Pi
• X*
Fig. 2.3 Feynman diagrams for the four-point Green function up to first order in
• aj
tj
in the definition of 7'{./} is to cancel all vacuum subgraphs in Z{.J}. On the level of Feynman diagrams this is easily taken into account by omitting all graphs containing vacuum subdiagrams, i.e. subdiagrams without external lines (as in Fig. 2.3d). The results (2.2.73) can be obtained from Feynman rules in configuration space. The corresponding graphs are shown in Fig. 2.3. The analytic expressions result from these graphs by attaching to each internal line joining the points Xj and Xj a space-time propagator iAp(x,- — xj), to each vertex associated with coordinate x a factor —ig f d x, and to each field point a factor factors, e.g. 1/2 for (c) and 1/8 for (d), have 1. In addition the symmetry to be multiplied and, in case of fermions, also the factors —1 as discussed in Sect. 1.2.1.8. Note that the correct symmetry factors are obtained by simply differentiating Z { J } with respect to its sources.
2.2.5.2
F e y n m a n r u l e s in m o m e n t u m s p a c e
In order to derive the Feynman rules in momentum space we have to perform a Fourier transformation. To this end, we substitute the Fourier representations for the propagators (1.2.31) into (2.2.73) and obtain
(«) =
"i Pk(xic-Xl)_ i E / d V a d V e - ^ - ^ - ' pf - m2 pj. — m2 ^ ' part. r d V i d 2 p 2 d-y-i d-'/M J
1
[2T:Y (27T)' (2TT)'' (27r)'
¡(P.S.+^.K,,,^.,,,*.,•)
1
pf — mz pi - m* '1
part. 2
3
(1)
2.2 Path-integral representation o f q u a n t n n i lield theory
xe-'(7>Hi+P23:2+P3i3+P4aM)
-
-I
/
1
Pj — m 2 p |
!
-
rn2
! ' v \ ~~ m 2 p'Î ~
10!)
7ft2
d ?J
' ' d2j>2 d 3 P 3 C]ip4 r-ifPi (2n)< (2Tr)'1 ( 2 ^ ) ' ' ( 2 t t ) " °
x(-i) i ;(27r)' 1 (5 ( ' ,) (/>, + p 2 + p 3 + p 4 ) i i i i 2 2 2 2 2 2 — m p — m p — m2 ' p —m (c) = - ¡ ^ ( ¿ p
E part.
4 ) t r*, . + -LPn/.)) I/ ¿,lK, / d V i d V A i ^¿ (i P V ^ T TL _i2 771
pj — m2 p2. - m 2 pf — m2 d V i d2/>2 d 3 p 3 dV,i (2Tr)4 (27T)-1 (2TT)'> (2TT)4
-I
-¡( P l x,+ P 2 x 2 + P 3 x3+ P .^)
part. X
2~^ r - r/r
P
z
pj. - m pf -
2/ m- J
(¿7r) '
T *-»ryz - 7«/
f2-2 ™)
These results can be directly obtained from the momentum space Feynman rules introduced in Sects. 1.2.1.8 and 2.1.2.2, and the vertex
(2.2.77)
Since Ap(x) is singular at x = 0, our treatment is completely formal. Regularization methods which make this procedure more rigorous are discussed in Sect. 2.5.
2.2.5.3
Loop expansion and q u a n t u m effects
We conclude the discussion of the derivation of the Feynman rules from the path-integral representation with a short remark [NaGCj. For this, the
I.'IS2
Quantum theory of Yang Mills fields
Planck constant /i., which has been used so far as a unit, is written explicitly in (2.2.71) Z{.J} = J 2>Mexp Q Jd4x
[£ +
llJ(x)ip[x)]
= exp x exp
(2.2.78) ^ / J ( x ) & r { x - x')J(x')J
Z(rcc{0}.
This shows that each vertex is linked with a factor / i - 1 and each propagator with a factor h in the perturbation series. T h e number of loops for any connected vertex graph consisting of I internal lines and V vertex points is L = I - { V - 1 ) = I - V + \,
(2.2.79)
as V — 1 lines are required to connect the V points simply by a tree graph and any further line generates a loop. Thus, the above counting of powers of li gives the following result for the perturbative expansion of the generating functional of vertex functions i f » = /ilog(Z{J}/Z{0}) - i h J d i x J { x ) a { x ) = i^VT/„ l,
(2.2.80)
where I'/. is the contribution of the graphs with L loops. This result indicates a close relation between the order of quantum corrections and the number of loops in Feynman diagrams. In particular, one expects that the trec-grapli approximation without loops is closely related to the perturbative solutions of the classical field equations. Actually, for the most simple, semi-classical approximation, where the path-integral is evaluated using the method of stationary phase [Dc61], we have
TC{J}
=
fclogfB
« i Jd4x
[¿(acOc^M*))
+ AJ(®)oc,(®)], (2.2.81)
2.2
ath-integral representation o f q u a n t n n i lield theory
10!)
where o c (:c) is a solution of the classical equation of motion 6 5ad{x)
I d 4 y (a, i(y), d,tac {x))
=
- J{x)
2.2. 2
If both these equations are compared with the introduction of the functional of vertex functions (2.2.37) and (2.2.40) one concludes that in this approximation the vertices are generated by (2.2.83) i.e. by t h e action. ifferentiation of (2.2.83) with respect to «ci(a.) gives the vertices of the tree-graph approximation.
2.2.6
W a r d i d e n t i t i e s a n d e q u a t i o n s of m o t i o n f o r G r e e n functions
In Sect. 1.2.3 we treated the relations between a symmetry of the classical agrangian, currents, and charge conservation. These considerations are now extended to the path-integral representation of quantum field theory. The results are Ward identities for Green functions, i.e. relations between Green functions resulting from a symmetry of the action [Wa50, Ta57]. They are of great, importance for the treatment of gauge-field theories. The equations of motion for the classical fields can be generalized to equations of motion for Green functions. The derivation of these equations and of the Ward identities proceeds along similar lines. The starting point, is the invariance of the path integral under field transformations, which simply corresponds to the invariance of a usual integral with respect to transformations of the integration variables.
2.2.6.1
I n v a r i a n c e of t h e p a t h i n t e g r a l u n d e r field transformations
We consider a theory with fields ipi{x), i = l , . . . , n , which are collectively , i >,l), a agrangian (t >, ^xf)), the vector denoted by a vector i> = (V>i,
I.'IS2
Quantum theory of Yang Mills fields
of sources
= ( i , . . . , . „), and the generating functional
= y V[ii> exp (i d 4 x [ c ( i f > , d , t r j > ) +
Z{
= s .
s
s
s
s
s i>i
fpi + ttyi,
s
s
s
s s s.
ss
s,
,
, s s
x.
(2.2.8 ) of s
S{ip + 5ip} = S{i/>} + 6S{if>},
) /'
\J>L [dMx)
J
SS
did^Tpkix)) d c
(•)
{"d(d,Mr))
),
(2 2.87) s derivative
variational dC
SMx)
(2.2.8G)
Sipk(x) +
s s (2.2.87)
s.
(2.2.84)
Hi(®) = e/,{V; :'••},
s
S{i{>}
.
s
s
s s s
+ Jd4xJ{x)4>[x)J
( { ss s ,
(x)
dC
tyk{x)
2.2.
"didMx))'
As
s 6Z{
0 = SZ{
s
Z{
,
s
= Jv[4>]ex
( SW + J d A y J ( y ) ^ ( y ) .
(2.2.8 )
2.2 Path-integral representation o f q u a n t n n i lield theory
10!)
The "trace" term bJk/W'k originates from the Jacobian of the infinitesimal transformation (2.2.85). Differentiating (2.2.89) with respect to the sources and putting these equal to zero gives the desired relations between Green functions
+ E ( T U l M * / ) / u { l M f c } E[ ^ . f e ) ) ' k lk 2.2.6.2
(2.2.90)
E q u a t i o n s of m o t i o n f o r G r e e n f u n c t i o n s
Let us consider the case where fi{ip;x} 5I/H(X) = efi{ip;x}
are ordinary functions of x
= efi(x).
(2.2.91)
The functional measure is obviously invariant under this transformation. Since fi(x) are arbitrary functions of :r we find from (2.2.89) 0= J
+
j ^ f j
cxp
(-¡sty}
+ i Jd4yJ{y)i/>{y)^J
• (2.2.92)
As 6S/6ipi(x) = 0 are just the classical equations of motion, the identities for Green functions resulting from (2.2.92) and corresponding to (2.2.90) are called equations of motion for Green functions
X
/
V,V
''
k - <
r
Ijtk i s b n * . i * > > -
(2.2.93)
The ¿-functions in (2.2.93) result from the differentiation of the explicit sources ./¿(x) in (2.2.92). If we restrict ourselves to one field type and one explicit field in the Green function, this simplifies to
130
2 Quantum theory of Yang Mills lipids
This provides a relation between the full i/; propagator and the higher Green functions, as can for instance be seen by considering a scalar field theory, where 6S
6iJ>(x)
= - ( D + m 2 ) ^ - V'(IFJ),
(2.2.95)
and V' is the derivative of the scalar potential with respect to its argument.
2.2.G.3
W a r d i d e n t i t i e s f r o m local s y m m e t r i e s
In order to study the consequences of symmetries of the action we consider transformations of the type Wi(x)
= sfi{ip;x}
= e<,?{xp-,x}0"(x),
(2.2.96)
where 0n(x) are the parameters of the local symmetry transformation. The invariance of the action under the local transformation (2.2.96) implies _ SS^+egW} ° ~
S0*(X)
_ 6
[
]
¿Sty)
~ 6M*Y)K{
}
(
)
(2 2 97)
- -
If in addition the functional measure is invariant under (2.2.96), as it is the case form many symmetries, the first term in (2.2.89) drops out, and we obtain, using that the 0"(x) are arbitrary functions of .t, 0 = j X > [ i / ) ] J ( . r ) g " { - 0 ; .t} exp ^iS'{t/>} + i J d
4
y J(i/)<%))
(2.2.98)
from (2.2.89). Since we are only interested in T { J } , it is in fact sufficient for (2.2.98) that the Jacobian of t h e transformation (2.2.85) is non-zero and field-independent. If there exists n o régularisation where this is the case, the contribution of the Jacobian gives rise to anomalies [Fu80] (cf. Sect. 2.9.3.8).
2.2
s
2.2. .4
10 )
s
s
s
s WiW
s,
,x} = eg?{il>;x}Ga,
= efiW
dlt9n(x)
(2.2.
= 0,
)
s 0 =
dS{j> + eg 0 } 00" s
0 = jv[ip
s
, (2.2.100)
1 J d 4 x J { x ) s a { ^ x } e x p (\S{*I>} +
J
s
( )
( )) (2.2.101)
(2.2.101) s
Ward identities for Green
functions
Kk
i n>k 2.2.1 2
(2.2.10 ) (2.2.101) s .
s
s
(2.2. 8) s
s
s
x,
s s
x. s
s
s
s
, s
s
dC
= h - /
[DM*) " dC(4> + Egboh,dllw
+
J\ egrer))
-Qn
0°(x),
(2.2.104)
\'.V2 2 Quantum theory of Yang Mills fields
where dew
+ egW.cW edOa
+
egc6'))
(2.2.105)
is the change of the Lagrangian under the global transformations (2.2.99). Invariance of the action under these transformations implies that this change is at most a total derivative, i.e. ¿ K W + g g W . W + eg c0')) = d"KaM;x}e 00" " "»
(2.2.107)
with some vectors K"r Inserting this into (2.2.104) yields Jd*x
= Jd*x
with the classical Noether
{d>%{i>-,x}) 0a(x),
(2.2.108)
currants
which can be considered as a local functional of the fields il>i(x). This definition of the Noether currents constitutes a generalization of (1.2.79). The conservation of these currents for classical on-shell fields follows directly from (2.2.108). T h e classical equations of motion imply SS{iJ>}/6ipk{-c) = " and thus = 0,
(2.2.110)
since 0n(x) is an arbitrary function of x. Inserting (2.2.108) into (2.2.89) and using the invariance of the functional measure and the arbitrariness of the 0a(x) gives the result 0 = JVty) x exp
{d'%{^x}
+
Jk(x)ti{^x})
+ i j d 4 » J(3;)V>(*)) •
(2.2.111)
2.
s s
.{.{
j"{ip;x}
s
g?{ij>',x} s
s, T{
(2.2.111)
s
s
s
(2.2.112)
= 0.
Ward identities
for
Green
functions (T
,
))
(2.2.11 )
i
k T
s
\l
s
s
)
s
m>k
s
s s
) s s
s composite operators, s s. (2.2.111) (2.2.11 s (2.2.101) (2.2.102), s
s 2.4. . .
s
0''jfl s ) .
. s s
s
, s s s. 2.4. .
2. T
s s ,
s
s
s
ss s
s
s
s s
s ,
s
s
s
s
ss s s. s. s
. 1.2. s.
s . T s s
2
s
s
principle of local gauge invariance. s . A
s,
s
s
. 2. .1
s
,
, s
.T s
s
s
ss
s
. 2. .2. T s s.
T
s s
. 2. .
s s s
s
ss
ss
s. s
s s. T s s s
s
.
s.
,
s s s s ( .. 2. ). . 2.10.1) s s
s
s s .
.
s s
, s
ocal gauge invariance in
s
, s s
s
s ( .
2. .1
s
E
Dime equation
Q/e, - m)ip(x) = 0.
(2. .1)
s C =
)
( ).
(2. .2)
(2. .1) global abelian transformations i>{x) T
, s
i>'{x) = c-iQ'e0i/)(x),
, x, 0 — 0(x),
(ip - m)if>'(x) =
(2. .2) {d,t0 = 0), 4>'(x) = j>(x)eiQfe0.
(2. . )
local gauge transformations s d,t0(x) s ,
0
-
= e-^Wfi
i>{x)
m)e~'Qic0^\l>{x) -m
+ Qfe(fl9{x))]i/>(x).
(2. .4)
2.3
Local gauge invariance I.'{.'{
Local gauge invariance can be achieved by introducing a gauge field Atl{x) which transforms in such a way that the term
(2.3.5)
The resulting Dirac equation behaves under local gauge transformations as [lip - Qjefi{x) = c-'lQlc°M[vj)
- m\il>'{x) - Qfe4'(x)
- m + Qfe{^9(x))]ip(x).
(2.3.6)
It, becomes invariant if the gauge field transforms as A'll(x) = Ali(x) + dll0(x).
(2.3.7)
The essential property of the covariant derivative is D'^'IX) =
D'ILE-IQJC°WJ>{X) = E - ^ F ^ D ^ I X ) ,
(2.3.8)
or D'tl = e - ' QfO(*)DiiCiQ,cO(x) The electromagnetic field-strength tensor - d^A^x) F,w(x) = d,Mx)
(2.3.9) (2.3.10)
is gauge-invariant. T h e gauge field itself becomes a dynamical quantity by adding a gauge-field term to the Lagrangian. The simplest. Lorentz- and gauge-invariant, choice is 5 ¿SAUGc
= -\FIW(x)F'"'(X).
(2.3.11)
Thus, we obtain the Lagrangian of QED £QED =
-l-Flw(x)F>w(x)
+ ï>{x)(u/) - m)if}{x).
(2.3.12)
' T h e term F'"'{x)F1"' is a total derivative and therefore does not enter the equation of motion. All other gauge-invariant terms involving only the gauge field are nonrenormali/able (cf. Sect. 2.5.1.3).
I .'IS 2 Quantum theory of Yang Mills fields
The corresponding field equations read (\ip - m)if>{x) = {i
- m)ip(x) = 0,
= Qje^(x)
(2.3.13) (2.3.M)
By construction these equations and the Lagrangian (2.3.12) are invariant under the abelian gauge transformations Atl(x)
-> A,l{X) +
dlt0{x),
i>{x) -> e-iQf':°Wi/>(x),
i>{x) -4 ¿>{x)e>Qic°(xl
(2.3.15)
The equations above are valid for QED with one fermion with charge Qre. In general, one has to sum over all elementary charged fermions / with masses nij and charges Qj. Thus, the principle of minimal gauge-invariant coupling to a dynamical gauge field introduces the interaction into a free theory with charge symmetry (global gauge invariance). This principle is highly successful for the electromagnetic interaction described in QED. 2.3.2
G e o m e t r y of n o n - a b e l i a n g a u g e s y m m e t r y
In this section we define gauge transformations, the covariant derivative, gauge fields, and the field-strength tensor of non-abelian gauge fields, and give their geometric interpretation. 2.3.2.1
Non-abelian symmetry transformations
We consider a multiplet of fields, ip(x) = (//;,(:c)), i = l , . . . , n , e.g. Dirac fields, which transforms according to a representation V 3 of a symmetry group G with elements g G G. T h e infinitesimal generators T°, a = 1 , . . . ,dim<7, the charges of G, are represented by n x n matrices, e.g. r " / 2 for SU(2) or A tt /2 for SU(3) (cf. Sect. 1.2.2.2). Then, the local gauge transformations read (*) -> rj>'(x) = VgrJ;(x),
Vg = e
or in components h f o ) = T>gyljlJ)j(x).
,
(2.3.16)
2.
U{x +
.{.{
,
\
+
)
/di>{x
+
)
+ . 2.
(
T
s
s s 0"(x), s s
s s <j{0(x)) = g(x) s
2. .2.2 T x+
s
)
s
s
s, 0" =
. . s s 'Dg(0(x))
s
= Dn{x) .
s
s ip(x) s parallel displacement
s U(x +
, ss
s s s
. T
s
, )
ip(x), (l
tJ>(x
) = U{x +
s s A''(x),
, x)ip(x)
connection,
(2. .17)
s s
gauge field
gauge potential
G, U{x +
, ) = 1+
n U, 1, Alt s ip'{x) = T>i)ip(x) (
+
,
4 1( ) n
s. T s
) = U'(x
= Vg(x + dx)dt()(x
(2. .18)
, ) +
( ) = U'(x +
s U
s ( . , )
) = Vg(x + dx)U ( +
( )
. 2.4) ( )
dx,x)ip(x), (2. .1 )
.
2
s
. .
s
s
U'{x + dx,x) s
s
s s
= Vg(x + x)U{x + dx,x)Vg\x).
(2. .18)
s,
UJ,( x) = \Va{x)A^x)Vg\x) = Vg{x) ( s
s
(2. .20) . ,
-£J-V~l(x)
+
( ) - Olt)V^(x),
d,l[D(J{x)Vg s s
1
( ) = d,tl
(2. .21)
= 0 s
s
. T s
s.
2. .2. T
covariant
derivative.
Dtl = dl$
(2. .22)
s
s •DgD^V;1
s
s
= Vg{d^D~1 ) iVgA,tV-x l A< - T FAv T) Vg{d = vg{d,lv-x) —1 - ;\A'„ tl g )
+ dtl
= dtl - \A'{• = D' As s derivatives
,
+
(2. .2 )
Dirac or Klein Gordon equations
covariant
) 0 ( ) = 0,
(2. .24) s
s
with
s. s s
ss
L
s s.
D' s
s s
s A''\x),
s
2.
2. .2.4
s
T
1
s
s
(
+
, )
s
curvature. s + dy +
s .
x +
x + ),
( U[x +
+
+
+
,
+
- U(x +
x)U ( +
+
,
= 1 + iA,t(x + 1+ =
T
s
)
+ -
) =
A
+
, )
\A„{I )
[A T L ( ), A , ( ) )
1
+ ....
)
+ ...
(2. .2 )
field-strength tensor
T
s
1 Av{x)Axw)
- DVA^X)
= iTlw{x)dx'ldy"
,
, )
+ dy)U(
)
4 ( +
s
T,,„(x)
4 ( )
(2. .2 )
s s
.
A s
s
s
ss 0 ,,
s
s Flll,(x) T
s
= \ [D,t,D„\.
(2. .27)
s
s
^u(x)
= Vg(x)F„„(x)V;l(x),
. .
s
s s gauge transformations A ST^x)
s
) = (
s (2. .21)
s .
T , ( )
(2. .28)
+ T
= \[Ta,Tl,l/ (x)]60" ( ).
)
s (2. .28) s
infinitesimal
0 ( ), (2. .2 )
.
2
s
s
s
s, s
Bianchi
identity
s
[£>„, Tvp ] + \Du,Tmi
] + [Dp, Ttt„\ = 0
(2. .27). T s
(2. . 0)
s
s
s
s .
2. .2.
s Aft, Dt, ,
T[lv
s
s
s
Aft,tj(x)
= )=
s
\<jAl{x)T?j,
{ )T .
(2. . 1)
s g s gauge transformations
s
6A;(x) SF^u(x)
s T ,
gA°(x)T,
Dft,i3(x) = dftSij -
T
s
s
+ ~d,t50a(x)
= r^AfaW'ix)
s s A"t(x) l
=
s. T F^u(x),
-D;bseb(x),
= rb'FbJx)60c(x),
(2. . 2)
s
s s s
s
f s b
,
D"
s s
abc
G. T . (1.2.47)
s s s
(1.2. ) s
s
.
s
s
s .
A
= s ,
s.
+
) s
( ). s
(2. . s
)
2.
2. .2.
s ,
s
s
. s x x y
rp(x)
U(y) s )
y
s
s parallel transporter Uc{y,x) . s
s s
s
,
,
s
s
s
s s
(2. . C
s
Uc(y,x) Atl{x). s,
s
0
s
,
s
(0) =
s U(x + (\x,x) C s s
s ,
,,
s
(1) = L
s
s
x)V~\x).
s
s,
. s
U'c{y, x) = Vg(y)Uc(y,
C = { (s)
s
(2. . 4)
s T>;l(x) . (2. .20) ,
T . .
s
C
= Uc(y,x)iJ>(x).
s
T
. {. {
1
s
) ,
. s
s
(2. .17),
1=0
1=0 s
+
) )
A
(
) .
(2. .
, s
s s
. T
)
s ,
(2. . s
) path-ordering
I.r>8 2
Quantum theory of Yang Mills lields
operator P [Fe51]. From the last line of (2.3.3(>), one concludes that Uc is a solution of the differential equation
- i ±U(s) as with Uc(y,x)
=
=
U(0) = 1,
d.s
(2.3.37)
(/(I).
The parallel transporter Uc can be used to perform gauge transformations so that Alt(x) has a simpler form. As an example we show the existence of a transformation for which ./loC1') = 0 (temporal gauge). For tliis purpose we choose a parallel displacement Uo along a straight line from y = (t, x) to x = (0,x), i.e. parallel to the time axis,
A0(t',: UQ = Pexp ( i j/ \ t 'di'A)(/.',x)
.
(2.3.38)
This parallel displacement Uq can be interpreted as a gauge transformation of i/>(/.,x) and A/^t, x)
V>(z)->
rJ>'(x) = U0il>(x)1
Alt(x) -> A'lt(x) = U0(A,t(x)
+ idjU^1.
(2.3.39)
It describes the transformation to the temporal gauge. This is confirmed by the following calculation: A'0(x) = Uq(Ao + =
¡Wo"1 U0(A0-iU0-^U0->
= U0(A0 - A0)U^
= 0.
(2.3.40)
The last step follows from dUo/dt. = — iL>roAo(L, x), the an alogue 0 of (2.3.37) for U0. °The interchange of A and U aiul the different sign with respect to (2.3.37) result from the fact that I. is the lower limit of the integral in (2.3.38).
2..'{
'2.3.2.7
Local gauge ¡»variance
14.'}
R e m a r k s on g a u g e t h e o r i e s and fibre b u n d l e s
We complete t his section by extending our discussion of t.lie relations beUveen t,he concepts of differential geometry and physics with some remarks on fibre bundles [Ch82, Go87]. In this framework, one considers the mapping of points x of a base space M 011 fields, i.e. vectors i/>{x) of a local Hilbert space X'H at each point, :c, together with local transformations g of a gauge group G which map XH into itself. The set of all elements {.r,ip, g} = {:/; € M, tp € XH, g € G'} is a vector fibre bundle T with the fibre X'H. The base space is, for instance, a Minkowski space or a euclidean space, and the fields can be matter fields or gauge fields. The symmetry transformations at different points are connected by finite parallel transports, or infinitesimally by covariant derivatives defined by the gauge potentials yl/t(:r). The different spaces X'H arc isomorphic. If and only if the vector bundle is equivalent to the direct product of the space time with a universal representation space 'H of the symmetry group, M x H, it. is called trivial. There are non-trivial fibre bundles that have such a product structure only 011 covering subsets M.i C M . The mappings, M —> xrH, are continuous only locally but not globally. On the overlap of M i with M n the fields are connected by gauge transformations. For three overlapping subsets, these gauge transformations have to fulfill the cocycle condition (2.9.36). Such field configurations play an important role for the non-perturbative structure of gauge theories. We elaborate more on these aspects of gauge theories in Sects. 2.9 and 2.10.1. 2.3.3
Y a n g - M i l l s field t h e o r i e s
The extension of local gauge transformations to non-abelian symmetry groups makes it possible to transfer the principle of minimal gauge-invariant coupling to fields with non-abelian charge algebra. These field theories are called Yang Mills field theories [Ya54, Ut.56], 2.3.3.1
The Yang-Mills Lagrangian
Above we already mentioned the gauge-covariant Dirac equation (2.3.24) with the Lagrangian £ = -0(.T)(i//> - m)ip(x)
(2.3.41)
114
2 Quant,urn theory of Yang- Mills ileitis
;us an example of a gauge-invariant coupling of a gauge field to a relativist»: spin-1 / 2 field. However, in a self-contained physical theory, Afl cannot be an external gauge field but must itself be a dynamical quantity. This is achieved by adding a gauge-invariant term depending only on the gauge field to the Dirac part of the Lagrangian,
¿gauge =
=
x)J^{X)).
(2.3.42)
This form is motivated, apart from gauge invariance, by simplicity and similarity to QED [cf. (2.3.12)]. 7 Altogether we obtain the following Yang Mills Lagrangian for the gauge-invariant interactions of fermions with non-abelian gauge bosons C = -l-F;u(x)F^'"(x)
+ $(x)(\ip
- m)tp(x).
(2.3.43)
Variation with respect to ip(x) yields the Dirac equation (2.3.24). The variation with respect to the gauge field A^(x) gives the generalized Maxwell equation D"F^(x)
= -g^{x)Talvti>{x).
(2.3.44)
In these equations all indices associated with the fermion multiplets of the group G have been suimpressed in order to emphasize the formal similarity to QED.
2.3.3.2
N o n - a b e l i a n g a u g e i n t e r a c t i o n s of scalars
The restriction to Dirac fields is not necessary. By similar considerations an interaction mediated by the dynamical gauge field A,,(x) is introduced for a multiplet of free scalar fields = ((/>,). To this end, the free Lagrangian ¿free = (d^xtf
d'1 (x) - M'ty (x)(x)
(2.3.45)
An £„UPA F""F*° term is Lorentz- and gauge-invariant but violates CP invariance. Moreover, it is a total derivative and does not contribute in perturbation theory. Therefore, it is not taken into account here. For non-pcrturbativc effects see Sect. 2.9..3.-1.
2.. i
ocal gauge invariance
14
is extended in gauge-invariant form and is supplemented by the gauge-field part resulting in C = -l-F u(x)Fa'> (x)
+ (Drfixj)1
D» >(x) - M2 >Hx){x). 2. .4G
From tliis
agrangian the iic;ld e uations
D^D'^ix)
+ M'2(x) =
,
2. .4
= —\g[(x)
- (Du(x)^Ta{x))
2. .4
follow by variation with respect to (^{x) or (x) and A t(x). hile 2. .4 is the gauge-covariant Klein-Gordon equation, 2. .4 shows the conserved current of the scalar fields that couples to the gauge field. The gauge couplings of fermions and scalars exhaust all possible renormali able gauge couplings. In addition, general gauge theories contain Yukawa couplings scalar-fermion couplings and scalar self-couplings up to the fourth order. These additional couplings do not contain derivatives and must respect the global gauge invariance. This fixes the general framewor of renormali ablc gauge theories cf. ect. 2. .1. .
2. . .
T h e generali ed
axwell e uation
more detailed analysis of the generalized Maxwell equation 2. .44 for a non-abelian gauge theory with fermions and its comparison with electrodynamics is useful. For this purpose it is written with the linear field-strength tensor G'l {x) = dpA x)
- 0 A (x)
2. .4
resulting in i'G^ = -gSabc{2Ablld'LAc -gzinbcrlr
- Ab d'1 Actl -
Ab' A'lAl
= (AJ (x) +tl2(x))
-
A^'d^)
g ^ T ^
= J',:(x).
2. .
I.r>8 2 Quantum theory of Yang Mills lields Equation (2.3.49) implies d" d'lG"Ll,{x) = 0 and therefore conservation of the currents ./"(:;:) 0"J^x)
= 0.
(2.3.51)
Two aspects are important here: in non-abelian gauge theories, not only the matter fields but also the gauge fields are charged. Accordingly, there is a self-coupling of the gauge fields A"(x) expressed by the non-linear terms in the Maxwell equations (2.3.50), i.e. by the charged gauge-field current A J*{x). The conserved currents ./"(x) (2.3.50), which form the sources of the gauge field, are proportional to the Noether currents which are derived from the global symmetry of the Lagrangian (cf. Sect. 1.2.3). The proportionality constant g which was introduced in (2.3.31), represents the gauge coupling constant and determines the strength of the universal interaction of all particles which bear the charge of the local symmetry group 6*. This is known from QED (T* = -Q,fabc = 0). where e = g > 0 is the elementary charge. Several coupling constants can appear in the case of a non-seini-simple symmetry group G which consists of a direct product of simple factors, e.g. G = G¿ x G\ with generators T" and Sa for G¿ and G'i, respectively. In this case, the Lie-algebra-valued potential of G can be written as A„(x) = g^A'lixyr"
+ gxBbtl{x)Sb
(2.3.52)
with different coupling constants g¿ and g\. Because [T",5 ] — 0, the gauge potentials A'^x) and Bbt(x) are independent of each other. This case is of importance in the Glashow Salain Weinberg theory of the electroweak Interaction, where G = SU(2) W x U ( l ) y . 2.3.3.4
Q C D , t h e Y a n g - M i l l s t h e o r y of t h e s t r o n g i n t e r a c t i o n
For the foundation of QCD [Fr73, We73, Gr73b], the field-theoretical formulation of hadron dynamics, several aspects played a role: (1) the result of the plienomenological discussion; (2) the success of QED; and (3) the geometric beauty of Yang Mills gauge theories.
2.3
Local gauge ¡»variance
147
Assuming that (2) and (3) provide sufficient motivation for the formulation of hadron dynamics as a Yang Mills theory, the phcnomenological discussion suggests to start with coloured quark fields as matter fields. As colour charges play an important part in the strong interaction of quarks (Sect. 1.3.1), they should be the source of the gauge fields, which mediate the interaction. Consequently, QCD is based on the colour symmetry group SU(3)c- According to the dimension of SU(3)c, eight gauge fields A"t(x), a = 1 , . . . , 8, the gluon fields, mediate the interaction between the quarks. Deep-inelastic scattering and jet production in high-energy collisions provide direct evidence for these types of flavour-neutral constituents. According to (2.3.43), the Lagrangian of QCD takes the form
a
^qA X )( U Pccf ~ TLq,c'(x),
+
(2.3.53)
U,ctd
where ^q^X'x) are the quark fields with masses m q , flavour index q, and colour index c, and A"t(x) are the gluon fields. The covariant derivative Dti CC> = d^Scc' — i(7 s (A a ) cc '/l / "(.r)/2 induces the minimal gauge-invariant interaction of strength 5 s , and F«„(x) = O.A'^x) - O.A^x) + gJ^A^A^x) is nl c the QCD ficld-strengtli tensor. The definitions of f ' and A" are given in Sect. 1.2.2; colour and flavour indices are summed up explicitly in £QCDThe Lagrangian (2.3.53) allows for the formulation of QCD as a quantum field theory. The non-abelian nature of the SU(3)c gauge transformations makes it necessary to treat gauge invariance and quantization in more detail. This is done in Sect. 2.4. QCD has three important general properties: (1) QCD is renorrnalizable. This guarantees its consistency, the validity of general physical principles, and its predictive power. (2) QCD is asymptotically free. This establishes perturbative QCD, i.e. the parton model with many experimentally tested applications. (3) In the non-perturbative regime, lattice QCD leads to an understanding of quark confinement and the properties of the hadrons. These properties and the impressive agreement with experiment establish QCD as the theory of the strong interaction.
I.r
2
2. . .
uantum theory of
ang-
ang
ills lields
i l l s t h e o r y for t h e e l e c t r o w e a
interaction
There are similar arguments in favour of a ang- ills theory of the electrowea interaction. In ect. 1.4 we saw that the electrowea interaction can he described by the exchange of the vector bosons 7 , , and . The charges corresponding to the electrowea currents generate the group 2 x l y . This symmetry turned out to be manifest after the neutral vector fields were replaced by suitable linear combinations 1.4.1 . This suggests to formulate the theory of the electrowea interaction is a non-abelian gauge theory with the gauge group 2 x l y. The resulting Electrowea tandard odel E was constructed for leptons by . . Glashow G1 1 , . einberg eC , and . alam a and extended to include the hadronic degrees of freedom by . . Glashow, . Iliopoulos and . aiani G1 , It is a minimal model as far as the number of re uired fields is concerned G . However, the non- ero masses of the wea gauge bosons seem to be in conflict with exact gauge in variance. In order to incorporate massive gauge bosons in a gauge theory one needs to introduce scalar fields that brea the gauge invariance spontaneously and provide masses for the gauge bosons via the so-called Iliggs mechanism. Therefore, we postpone a detailed formulation of this gauge theory to Chapt. 4.
2.4
s
The uanti ation of systems with finite degrees of freedom usually is performed by constructing the Ilamiltonian and postulating the Heisenberg commutation relations between canonically con ugate observables. lternatively, an e uivalent uanti ation using Feynman s path integral is possible. The latter is the preferred method for relativistic field theories, especially gauge theories. The reasons are two-fold. orent invariance is not manifest in the Ilamiltonian formulation since time is used as a distinguished variable. The four-potential of a gauge field has unphysical degrees of freedom which have to be eliminated before canonical uanti ation.
2.4
2.4.1
ath-integral
ath-integral formulation of gauge theoriesICiI
u a n t i a t i o n of g a u g e t h e o r i e s
ince the path-integral uanti ation is based on the relativistically invariant agrangian, or the action, it is manifestly orent -invariant. oreover, the treatment of the unphysical degrees of freedom is performed in a very elegant way. 2.4.1.1
r o b l e m s in t h e
u a n t i a t i o n of g a u g e t h e o r i e s
The straightforward application of the Feynman path-integral formulation of uantum field theory to non-abelian gauge theories leads to the ansat T{J = {J / {0 , J
exp i
iJ d
x
,
d
2.4.1
for the generating functional of Green functions, where , and , . The measure involves at each space-time point a product over all group and vector components of the field Aft(x). The ansat 2.4.1 using the four-potential with its unphysical degrees of freedom contains mathematical inconsistencies that ma e even a perturbative evaluation of impossible. This problem can already be seen in the case of the free photon field. The evaluation of the corresponding functional integral {
J
exp i J
^ A ^ U - d ^ A u + J^A,,
in analogy to 2.2. , fails, because the operator K'w g't n- ltdv) has eigenvalues ero. Thus, the necessary condition for the existence of the gaussian integral, (A^, KiWAv) , is not satisfied and the result diverges, ;is does not exist. For the correct uanti ation of any gauge theory it is necessary to solve this problem, which is caused by the gauge invariance of the agrangian C. The singularity in 2.4.1 results from the integration over physically e uivalent configurations A(x). These are related to .c by a gauge transformation g(x) according to 2. .21 . s C and the measure are gauge-invariant,
I.r>8 2 Quantum theory of Yang Mills lields
the integration over the physically equivalent configurations results for each space time point in the volume of the gauge group and thus in total in an infinite unique gauge-invariant factor. When forming the ratio Z{J}/Z{()} this Infinite factor drops out, and a meaningful result emerges. In the following we describe the method proposed by L.D. Faddeev and V.N. Popov |l''a()7] to separate off this factor. The Faddeev-Popov method can also be applied in the presence of zero eigenvalues resulting from other symmetries (cf. Sect. 2.9.3.2). 2.4.1.2
The Faddeev-Popov procedure
We consider the path integral without source terms, Zj = j V[A] / { A } exp(iS{ A } ) ,
(2.4.2)
where / { A } is an arbitrary gauge-invariant functional. Let C = (G"1! A; x}) be a set of local Junctionals of the. gauge fields that fixes the gauge in the sense that the condition Ca{f)A\x} = 0 selects for any initial 8 field A one solution A out of the set of gauge-transformed, physically equivalent potentials flA. In order to enforce the gauge-fixing condition in the functional integral we use the ¿-functional ¿{C{"A}} = []*,«¿(C"{ ? 'A;x}), which is an infinite product of Dirac ¿-functions. Now consider the functional A""1 {A}, defined by A - ' { A } = [ £[,*(,,)] ¿ { C f A } } , where D[fi(g)] = f ] r t\/i(g(x)) stands for the infinite product of invariant measures dfi(g(x)) (1.2.64) of the gauge group G at every space time point x. The invariance property of the group measure (1.2.65) implies V[n(g)} = V[fi(gg')] = V[fi[g'g)], and thus the gauge invariance of the functional A ~ ' { A } , A - ' { » A } = Jv[,,.(g)}S{
C{™'a}}
= [
V[fi(gg'))6{C{^A}}
= J V W ) ) 6 { C{
(2.4.3)
In general, there may be more than one solution. These are called Gribov copies [Gr78]. For perturbation theory, however, these arc irrelevant, and the method described here applies.
2.2
Path-integral formulation of gauge theoriesICiI
We insert 1 = A { A } J T)\n(g)\ <S{C{ffA}},
(2.4.4)
into the path integral (2.4.2), interchange the order of integrations, and use the gauge invariance of the functional A{ A } , of the measure f [ A ] , of / { A } , and of the action 5 { A } , Zj = Jv[A]A{A} = J V\p(g)) J
% ( f f ) | i { C [ " A ! ) / { A ) exp(iS{A}) V[°A}A{»A}6{C{9A}}f{°A}exp(iS{»A})
= Jv[,i(g))Jv[A]A{A}6{C{A}}f{A}exp(\S{A}).
(2.4.5)
Tlie last line is obtained by the substitution !'A —> A in the functional integral. The expression under the V[/i(g)] integral is independent of <7, and the integration over the gauge group gives a multiplicative divergent factor. This factor contributes only to the irrelevant normalization of Zj and we can write Zj oc J V\A] A { A } 5 { C { A } } / { A } e x p ( i S { A } )
(2.4.6)
This expression contains the ¿-functional and therefore defines Zf in the C a { A ; x } = 0 gauge. We still have to evaluate A { A } . We use the gauge invariance of A { A } to choose A in such a way that it satisfies the gauge condition C"{ A ; x } = 0. Then because of the ¿-functional, we need to consider only infinitesimal gauge transformations g = 1 + \0aTa, 0a <§; 1, and the invariant group measure takes the simple form V[n(g)] oc V{6\. A change of integration variables yields
(2.4.7)
152
2 Quant.uni theory of Yang Mills Holds
where "A stands for "A with g = l + \OnTa. Since the condition C'"{°A,,; re} = 0 is fulfilled for 0 — 0, we finally have SCb{°A;x} A { A } = det S0c(y)
= det Mbc{x,
y)
(2.4.8)
0=0
We have to calculate the determinant of a matrix Mbc(x,y), I'opov determinant, in both space-time and group indices,
= o=o j J
S0c{y)
the Fadd.eev-
6Cb{°A-,x}5At(z) 6A'l{z) 60r{y)
,4 (1 z
0=0
(2.4.9) As SA/60
is proportional to the covariant derivative [cf. (2.3.32)], we find
bc
M (x,y)=
I (Pz
SCb{A;xj 6A*(z)
SCb{A-x}
1
tMtv)
»
1 dc -D u {z)6^{y-z) g
d
(2.4.10)
Choosing for instance as linear local gauge-fixing
functional
Ca{A;x}=d»AaJx),
(2.4.11)
we obtain Mbc{x, y) = J d A z d$S{A){x - z
)
(
y
- z) (2.4.12)
Note that (2.4.8) implicitly assume? that C"{A;rc} = 0. Since this is enforced by the ¿-functional. A { A } can be replaced by det M under the path iiite.oral (2.4.(0,
2.4 Path-integral formulation of gauge theoriesICiI
2.4.1.3
T h e effective Lagrangian for a g a u g e theory
It is convenient to rewrite the path integral (2.4.6) in such a way that all contributions appear in the form of an effective Lagrangian. To this end, we first note that the above construction works in the same way for gauge-fixing conditions of the form x
x
0,
(2.4.13)
where c(.-;:) = (c"(.-e)) are arbitrary functions of x resulting in -c}/{A}exp(iS{A}).
(2.4.14)
Because x is not affected by gauge transformations, x is the same for all gauge conditions of the class (2.4.13). Moreover, by construction is gauge-invariant and consequently the same for different gauge-fixing conditions, i.e. independent of x Carrying out the functional integral over with any weight function will thus only change the (irrelevant) normalization of It is common to use a gaussian weight function resulting in a
cjexp
dx
x
x
x
- c } / { A } exp(iS{A}) d e t M / { A } exp(i«S{A})
x exp
y d^rcC a {A; x
x}^ .
(2.4.15)
Finally, according to the Fubini formula (2.2.27), the functional determinant «let can be expressed as a functional integral over anticommuting scalar fields u(®) = x and ü(z) = x dot
oc
exp ^-iry
d4a;d
x
x
154
2 Quantum theory of Yang- Mills fields
where a factor —g has been introduced for later convenience. Note that the fields ua(x) and ua(x) are independent of each other and, in particular, not related by herinitian conjugation. The auxiliary Grassmann-valued fields u"(x) and u"(x) are called Faddeev Popov (/host fields. These scalar anticommuting fields violate the spin-statistics theorem. This is unproblematic as they are unphysical degrees of freedom which occur in perturbative calculations only inside loops but never as external fields or particles. Thus, we finally arrive at the following representation for the path integral ZS = J
P[A]D[u]Z>[u]/{A}exp(iS{A}) d4.T C a { A ; a:}C fl { A; a:
xexp x exp (-ig
J d ':cd'y u"(x)Mab(x,
= J P[A]P[u]P[u]exp ^i j
y)ub{y)^
d4xCettJ (2.4.16)
with the effective Lagrangian for a quantized gauge theory =
¿gauge
¿gauge "I" ¿fix
=
¿ghost i ''
¿ghost = -9 J d
(•''))
*yua(x)Mab{x,y)ub(y)
In the Lorentz gauge (2.4.11), the Faddeev Popov-ghost Lagrangian reads ¿ghost = - f d4zua(x)d2DaxbjM(x =
-
z)ub(z)
-ua{x)d'lDabu\x)
= -ua(x)d>l{d,l5ab
- gfabrAclt{x))ub(x).
(2.4.18)
2.2
Path-integral formulation of gauge theoriesICiI
As in this case, the Faddeev-Popov Lagrangian is local as long as the gaugefixing functional Ca{A;x} is local in the fields. Note that (2.4.16) differs from (2.4.2) only by a constant normalization factor which drops out in 7 n {J}. The path integral (2.4.16) is suitable for perturbative evaluation. The Feyniiiaii rules can be derived from the effective Lagrangian (2.4.17) as usual (cf. Sect. 2.2.5). They depend on the gauge via £[-lx and £ghost> but the final physical results, such
of Green
(2.4.19) where we have introduced besides the sources for the gauge fields also anticommuting sources J" for the ghost fields. Since tin; source terms are not gauge-invariant, (2.4.19) differs from (2.4.1) and depends on the gaugefixing functional. The same applies to the Green functions that are obtained as functional derivatives. Nevertheless, physical observables, as for instance 5-matrix elements, that are calculated via these Green functions are unique and independent of 6'"{A;.7;}. The generalization to gauge theories including matter fields is straightforward. One uses the appropriate classical Lagrangian £ and introduces sources for the matter fields. Gauge-fixing and ghost terms remain unchanged. In the following we consider a gauge theory with fermions i¡> = (%/;,) and •tft = (V'i) that transform according to the representation Ta of the gauge group. If we denote the corresponding anticommuting sources by .1,,, and J,;, the generating functional reads Z{J, J u , Jfi.J*, J 9 } = I x exp I( ii J/ d4x[c x cn
V[A\V[n]V{ñ]V[i¡>]V{i¡,)
+ J"'aA° + ./,>" ü" j" J~ ++ ya -- u"
ai
(2.4.20)
I.r>8 2 Quantum theory of Yang Mills lields
with
¿off
=
¿gauge
¿form = Mx)(Jpij
¿ferm "I" ¿fix "t" ¿ghost)
- miSij)lf>j(x),
(2.4.21)
and ¿gauge» ¿fix, and ¿ g i,ost as in (2.4.17).
2.4.1.4
T h e e f f e c t i v e Lagrangian of Q E D
In QED the situation is simplified by the fact that all structure constants jabc v a n j s | , a r i ( ] the representation matrices T" are trivial: T" = — Q, g = e. (In the following we put Q = —1.) As a consequence the Faddeev Popovghost fields are free fields and the generating functional factorizes into a part containing the ghost fields and a ghost-less part. The ghost part gives only an irrelevant normalization factor and can be omitted. Considering only one fermion field if), we can write for the generating Junctional oj QED in the gauge (2.4.11)
Z{J} = I V[A]V[^]V[^}exp
( i I d 4 a;¿totJ
(2.4.22)
with
¿tot — ¿gauge
¿fix
+ J"(x)Au(.r) ¿gauge-
¿ferni
- i>(x)J^x)
^ Fiw{ x )I'' [ x ) i
+
¿fix—
¿form = i>(x) [nfc(d'1 - ieA") - m ] •!/;(*).
MxW(x), O'^Wx) (2.4.23)
2.4
2.4.2 2.4.2.1
Path-integral formulation of gauge theoriesICiI
F e y n m a n rules for g a u g e t h e o r i e s Derivation
11 the gauge coupling is small, gauge theories can be evaluated perturbatively. The Feynman rules can be derived directly from the effective Lagrangian. For a gauge theory with a multiplet of fermionic matter fields ifr = (i/>i) in a representation Ta of the gauge group and in the gauge (2.4.11) the effective Lagrangian reads ¿cr = -l-F^v(x)Fa'"v{x) - üa(x)d'íDfub(x)
For the gauge group SU(3) and the fermions in the fundamental representation (T" = A"/2) this is the effective Lagrangian of QCD. The Feynman rules can be derived most easily by using the fact that the vertices are the tree-level vertex functions. Their generating functional is, according to (2.2.83), directly given by the space-time integral of the Lagrangian. As an example, we derive the three-gauge-boson vertex. From the tree-level vertex functional (2.4.25)
I.r>8
2 Quantum theory of Yang Mills lields
the corresponding vertex function in configuration space is obtained as irabc
x _
(
SAa'ii{x)SAb^(y)6Ac'P(z) A=u=fi=l/>=l/>=0
y> z>
iW*. a
l
3 b6
SA <> (x)6A <"(y)ôAc>i>(z) ( - i I d V gf*" = "i9fa'b'c' d
[x')Ab:,
Jd4x'[6aa'9fill.SW(x
{x')d>W - x')6wgv^\y
- x')
i (àcc'<)';', ¿ w { z - x')) + 5 permutations
= "i 9fabc gp„6W(z
x
~ x)0^\z
-x)~
+ 9P,JW{z
- y)dȉW(x
- y) - glipS^(x
- y)d^A\z
- y)
+
- z)d*/4\y
- z ) - 9u,^\y
- z)0^\x
-
9lJjW(x
- x ) ^ ( y - x)
z)].
(2.4.26)
Fourier transformation to momentum space yields
i f o% P (P, r) = j d4x d A y d"z W ^ x , y, z) e x p ( - i ( x p + yq + zr)) = 9 f a b c I d'.T \pi>P(P +
exp ( - i x ( p + q + r))
where p,q,r denote the incoming momenta. In order to integrate over the 6funetions we performed a partial integration and dropped the surface terms. The momentum-conservation ¿-function is a consequence of translation invariance and can be split off by defining
)*SW(P + 9 "I' r)ra0bfwp(p, q, r).
(2.4.28)
2.2 Path-integral formulation of gauge theoriesICiI
Now \rtff (p,q,r) is the vertex factor to be used in the Feynman rules. The I'eynman rules for all other vertices can be obtained in the same way. The propagators can be directly evaluated by performing the path integral over the free part of the effective Lagrangian as in Sect. 2.2.4.1. Alternatively they are given by the inverse of the corresponding two-point functions [cf. (2.2.42)]. For the gauge-boson propagator we first determine the two-gauge-boson vertex function
=
S
A
- l- (d^A^x')) = i f
'
J
S
( ¿ v ^ ' V ) -
A
M
*
4
M
)
¿^'[-(d'S^g^ix-x'))
x (d^'g^S^Hy ~
- x') - d»'6ba'g^,(y
-
- x')) (d-'S^'g^Hy
x')) -
*'))]
= i<5
(2.4.29)
where we again performed a partial integration. In momentum space this yields i f f e ^ P ,
-i6ah ?>
(i -
\
(2tt)AóW(p
+ q).
(2.4.30)
According to (2.2.42), the propagator is given by the negative inverse of the two-point function. Splitting off the ¿-functions, the gauge-boson Feynman propagator in momentum space is obtained as ¡A t , M = -[irjj ¡ U ( P , — 7 ' ) ] ' = i
p + 1£
(2.4.31)
The \e results from an infinitesimal imaginary bilinear term added to the Lagrangian that is required for the convergence of the path Integral
I.r>8 2 Quantum theory of Yang Mills lields
(t:f. Sect. 2.2.4.1). As a consequence a causal propagator results. The case £ = I is called the Feynman gauge and the case £ = 1 the Laiulau gauge. The propagators for the fermions and ghost fields can be calculated in the same way. 2.4.2.2
F e y n m a n rules
Altogether, the Lagrangian (2.4.24) gives rise to the following Feynman rules for a non-abelian gauge theory (in particular QCD): Propagators fermion: (quark)
I
P —>
/
gauge boson: (gluon)
P < V W \ A A A ^ a, P b, i/
¡i — 7 n + ie '
a
(
=
PpPuX
^V^iuiP) cPpPu'
6" b
p2 + ie u
ghost:
u
p
»
iSab p2 + ie '
a Vertices (all momenta incoming) f,i, a a, ¡I, pi
=
"Al„Wlfp
a\,fi\,p\ =
2.4
Path-integral formulation of gauge theories
X
®2i PhPT.
«<1, AM, P'\
I Ci I
-i£T 1 '' (f)intl[i (j¡¡o/i.i — Hit i /M 9 mm) x [/" /"1" ' ''/" '•"2'' {9imi9mm ~ 'hL\m-)mi'\) + /"1" '^/"' "''(9m/i291 ¡.¡m ~ 9/t\m^mi1^^
U,'l2,P2 gf^^p2fl.
(2.4.32)
The use of the Feynman rules is not very different from that in QED. The self-interaction of gauge bosons gives graphs with closed gauge-boson loops which have to be weighted with appropriate statistical factors. In the composition rules, Faddeev Popov ghosts arc treated as fermions, in particular each closed ghost loop gets a factor - 1 . The gauge-fixing part of the Lagrangian gives rise to the gauge parameter £ in the gauge-boson propagator. It cancels out in physical matrix elements. The Faddeev Popov ghost fields ua(x) and iia(x) lead to the appearance of ghost propagators and ghost gauge-boson vertices. However, the ghost states have no physical significance (cf. Sect. 2.5.4.3). Their purpose is to cancel the unphysical contributions caused by the gauge-fixing term. External ferrnion and gauge-boson lines are treated as in QED (cf. Sect. 1.2.1.8).
2.4.3
B R S invariance and Slavnov-Taylor
identities
The manifest gauge invariance of the non-abelian gauge theory is lost by the introduction of the gauge-fixing term Ca{A;.T}. However, a symmetry of the effective Lagrangian CK\\ with all the consequences of gauge invariance for all physical results can be defined again by an extension of the gauge transformation to the ghost fields. As generalizations of the Ward identities of gauge symmetry the Slavnov Taylor identities arc obtained.
I.r>8 2
2.4.3.1
Quantum theory of Yang Mills lields
B R S invariance
The extended gauge transformation is the Becchi-Rouet Stora or BRStransformation [Be74]. For gauge-boson and fermion fields, the BUS transformation is a gauge transformation (2.3.32) with <50" (x) = gfiXua(x), = SXid.S1"' -
¿BUSA';t(x) = 5\Dfu"(x) SbrsM*)
=
SXsA^(x),
=
fi\ua(x)\g(TaTl>(x))i
Stt RS&(S) = - ¿ A ua(xyig(Mx)Ta)i
=
A^(x))ub(x) S\sA(x),
= 6X
(2.4.33)
where ¿A is an infinitesimal, Grassmann-valued constant which anticominutes with the ghost fields ua(x) and ¡¿"(a:).9 The BRS operator s is defined as the left derivative with respect to ¿A of the BRS transformed fields and consequently the product rule reads s{FG)
= {sF)G ± FsG,
(2.4.34)
where the minus sign occurs, if F contains an odd number of Grassmann variables. Note that the BRS symmetry is a global symmetry in contrast to the local gauge symmetry. The Lagrangian ¿gauge (2.4.17) and the fermionic Lagrangian £ f e r m are evI he transformation idently BRS- invariant, <$BRS ¿gauge — '^BUS£fcrin — behaviour of the ghost fields, ¿BRS"A(Z) = -6X1-gfabcub(x)uc(x)
=
=
A; a;} = ¿A sua(x),
6Xsua(x), (2.4.35)
is chosen so that £fix + ¿ghost i-s a l s o BRS-invariant although both terms are transformed non-trivially, ¿BRS (¿fix + ¿ghost) = o. 9
(2.4.36)
If a gliost number is introduced in analogy to fermion number, then u"(x), fi"(a:), and <5A have ghost numbers 4-1, —1. and —1 respectively. Consequently, S0a(x) ' fiu"(x)6\ has ghost and fermion number zero.
2.2 Path-integral formulation of gauge theoriesICiI
This shows the importance of the Kaddeev-Popov gliost. fields which reestablish the gauge invariance of a theory destroyed by £r,x. In order to prove (2.4.36), we notice that in view of the anticommutativity of the ghost fields
= I
i < a i , z
« T S ) ^ '
1
' ' ^ ' ' '
1
(
-
2
'
4
'
3
7
)
and because of t he Jacobi identity of the structure c onstants f'lbc (1.2.48) SmsDfu'Xx)
- gfabcA<;t(x)ub(x))
=
This implies ¿BKS£GHOST = - ¿ B R S I
d\jûa(x)âCsajbA{^}
= SXl-Ca{ A ; * } J d«y
= 0.
(2.4.38)
Dbfuc(y)
Db^{y).
(2.4.39)
On the other hand, we obtain ¿BRS£fix = -7Ca{A;x} <
I J
= -l-C«{A;x}
f
d4ydC°}b^*}6miiiAi(y) °K\y) (2.4.40)
This proves (2.4.36). The BRS operator is nilpotent, i.e. s2A* = .s2V = s2ï> = s2ua = s2Ca = 0,
(2.4.41)
except for the antighost , V =
A;*} = - I |
* 0,(2.4.42)
I.r>8 2 Quantum theory of Yang Mills lields
but s V = 0.
(2.4.43)
Equation (2.4.42) entails that the Faddeev-Popov-ghost written as ¿ghost = -ua(x)sC"{A-,
x}.
Lagrangian can be
(2.4.44)
Finally we note that the BRS transformation leaves the measure of the functional integral invariant. The BRS symmetry can be formulated in a more symmetric form by introducing an auxiliary (commuting) scalar field Ba, the Nakanishi- Lautrup field, together with the corresponding path integral over Ba, and the gaugelixing Lagrangian [Na66a, La67] Aix = |BaBa
+ BaC°{A-x}.
(2.4.45)
Because the field B" appears only in quadratic terms without derivatives, it is an auxiliary field without an own dynamics. Upon performing the functional integral over B'1 by completing the square one recovers the original Lagrangian and generating functional. The BRS transformations are given by (2.4.33) and (2.4.35) with the antighost transformation replaced by ¿BRS«a(z) = SXB" =
SXsu"{x),
¿BRS Ba(x) = 0 = 6XsBa{x).
(2.4.46)
In this formulation the BRS transformations are independent of the gaugefixing term and thus independent of the effective Lagrangian and strictly nilpotent s2u° = sBa = 0.
(2.4.47)
Moreover, the sum of the gauge-fixing and ghost Lagrangian can be written in a concise form ¿fix + ¿ghost = * (ua
(^Ba
+ Ca{ A ; .T}) ) .
(2.4.48)
2.4
Path-integral formulation of gauge theoriesICiI
Thus, these terms can l»e written as a pure BRS transformation. In this formalism, other gauges can be obtained as
¿fix + ¿ghost = S (uaFa) ,
(2.4.49)
where Fa{ A, u, u, B; a:} is an arbitrary functional of the fields with vanishing ghost number. According to the Noether theorem (cf. Sect. 1.2.3.1), the invariance of £efr under the BRS transformations implies the existence of a conserved current
j™5
=
{¡ruT + i
Iq/^ttV 2
d{d"A™)
d C
DC Ba— l d{d' û")
+
d(d»ua) DC \gua{i/;Ta)i d(d"A)
DC gua(TarP)i—
(2.4.50)
Using the equations of motion and the fact that the gauge field appears in the matter Lagrangian only via the covariant derivative, the matter contribution can be eliminated. The explicit form of the conserved BRS current (2.4.50) depends on the gauge-fixing term. In the gauge C°{A;a;} = V A f ^ x ) , it can be brought into the form [Ku78]
jBRS = Bu(D„uYl
- (¿yj>°
+
l
-grbc{d^a)ubuc
-
d"{F°wua). (2.4.51)
The corresponding conserved charge, the BRS charge, reads
QBRS = I d3ar ^Ba(D0u)a
- (d0Ba)u"
+ \gJabc{dQûa)ubuc
. (2.4.52)
106
2 Quantum theory of Yang Mills ileitis
2.4.3.2
S l a v n o v - T a y l o r identities
In Sect. 2.2.6, a functional differential equation for the generating functional of the Green functions (2.2.112) was derived from the global symmetry of the action 5{-j/>} of a multiplet of fields ip{x) and the invariance of the measure 'D{x¡>] in the functional integral (2.2.84). Carrying these considerations over to BUS symmetry gives the Ward identities for the generating functional of the Green functions of a non-abelian gauge theory, tin; Slavnov-Taylor identities [Ta71, S172, tH7l, tH72, Le72]. These summarize all relationships between the Green functions of the theory resulting from local gauge symmetry. The Slavnov-Taylor identities are essential for the proof of renormalizability of gauge theories (cf. Sect. 2.5.4). For definiteness we consider the generating functional of Green functions for a gauge theory with fermions (2.4.20). In order to get linear identities we introduce sources K, K„, K,„, and K ^ for the BRS-transformed fields, too, Z{Ji J u , J s , J^,
K, K„, K^,, K ^ } =
f V[A}V[u}V[u}V[i/>}V$)GxV(i ¿tot = ¿oír + Ja'M£
I
dAxCtol\
+ ./>" - üaJg + 4 A - 1>iJi
+ Ka>»sA«t + K*sua + K^sih + Wi)Ki.
(2.4.53)
All sources are assumed to be invariant under a BRS transformation, and ./,?, J", J'-, and Ka are Grassmann variables. The vanishing of the ghost number for £ t o i implies ghost number - 1 for K", K^, and Kx- and - 2 for In the following we assume that the gauge-fixing functional C n {A;:?;} is linear in the field A^, i.e. C°{ A;:c} = lbAb'
(2.4.54)
where < 1 i s field-independent but may contain a derivative as in (2.4.11). The corresponding ghost Lagrangian reads ¿ghost = -üa{x)^bDb^uc{x)
= -ü'(x)*fsAb"{x).
(2.4.55)
The use of a non-linear functional C"{A;:i:} would require the introduction of an extra source coupled to it.
2.4 Path-integral formulation of gauge theories
ICiI
Changing the variables of the functional integration according to the I3RS transformation (2.4.33) and (2.4.35), using BRS invariance of the effective Lagrangian and the functional measure together with the relations (2.4.41), one obtains (cf. Sect. 2.2.6.4)
0 = J V[A]V[u]V[û]V[i/>]V[ï>]exp(^ x J d4x [j**sA% - JUV
J
d4xCto^j
- (sû").J£ - J^h
(2.4.56)
Expressing the fields in the last integral by derivatives with respect, to the corresponding sources, we obtain the Slavnov-Taylor identities for the generating functional T,
r
-
J^
Jsi Jv»
(2.4.57)
In terms of the generating functional Tc = log? 1 of connected Green functions this reads
~ JjçT^iï
J«' Jfii
J^,» K, K„. K^, K^},
(2.4.58)
V'
where we used the linearity of C"{A; J;} in A to derive (2.4.58) from (2.4.57). Similarly, the invariance of T under the transformation ûa —» ûa -(- Sua gives rise to the relation 0 = J P[A]P[u]P[ûp[V>]î>$]exp(i | d"a:Aot) x J d4x[sCa{A-,x}+J£],
(2.4.59)
I.r
2
uantum theory of
ang
ills lields
where we used 2.4.44 . This is nothing but the e uation of motion for the antighost field cf. ect. 2.2.G.2 . It results in T
,
,
fl,
.
u.
, 2.4.
or for connected functionals
2.4. .
I d e n t i t i e s for G r e e n f u n c t i o n s
s discussed in ect. 2.2. , Slavnov Taylor identities between Green functions can be directly obtained by differentiating 2.4. with respect to the external sources and putting the sources to ero afterwards. e write the v v Green functions genet ically as (T ; I , where i represents anyone of the fields l t, u a , u , ipi, or i i, and the index comprises all indices of the actual field including the space time coordinates. The resulting identities can be written as i
k
r
n
n i
,
r
r
« l
i 2 4
n m>k
- -
2
where a stands for the sign that results from anticonmiuting the R operator with Grassmann-valued fields. The sum extends over all fields These relations can also be obtained directly from the invariance in I, of the action under R transformations by performing a R variation in the generating functional of Green functions cf. 2.2.1 2 . imilarly 2.4.
gives rise to
.)=-<{nsA = -(T(sC { ;
,
^)) t
,,) , ,
2-4.
2.4
ath-integral rornmlalion of gauge theories
wliic.li is nothing but a conse uence of the e uation of motion 2.2. antighost field u . s an example let us consider the Green function (Tu (x)Cb{ tion 2.4. 2 yields Q=
1
of the
; y ) . E ua-
s(Tiia(x)Cb{A;y )
= (r(sua(x))Cb{ ; ,
Tfia
-
= - i (TGa{A-x Cb{A-,y )
aC
;y
+ (T[sCb{A;y )ua(x))
.
2.4. 4
If the gauge-fixing term involving the field 1 2.4.4 is used, the same identity is obtained with the help of the e uation of motion cf. 2.2. for the field 1 , i
r
x
ic-
i,
n 2.4.
sing 2.4.
we obtain from 2.4. 4 (TCa{A-x Cb{A-,y )
= i
r-^u
a
(x)^
i
ab
8W(x
- y). 2.4.
In the covariant gauge 2.4.11 this yields %% (TAl{x)AbM)
= d dyGt(x
- y) =
- y), 2.4.
or, in momentum space G
-i t
ab
.
2.4.
This is the Slavnov identity 1 2, Ta 1 , the ard identity for the gaugeboson propagator of a non-abelian gauge theory. It implies that tin; longitudinal part of the gauge-boson propagator is not changed by interactions and that this propagator is transverse in the andau gauge - . It is evidently fulfilled in the tree-graph approximation, i.e. for the free propagator given
I.r>8 2 Quantum theory of Yang Mills l i e l d s
n (2.4.32). Its validity in higher orders implies that the gauge-fixing term iced not be renormalized. \ s a further example we apply a BRS transformation to (Tii" (x)il>(y)-tl>(z)) ind obtain
s{Tüa(x)iPi(v)4>j(z))
0 =
= (T(süa(x)my)^(z)) + =
(Tüa{x)ipi(y)siJjj{z)) A-,xMy)$j(z))
-
(Tüa(x)(sMy))&(*)>
-
-
uj(Tüa[x)iJ)i(y)ub(z)(ip(z)Tb)j).
ig{Tüa(x)ub(y){Tb1,{3i))iij{z)) (2.4.69)
In diagrams this can be written as
£ i'j f Sljji (2.4.70) U" i>3
where we use a double line with a circle to indicate the gauge-fixing term. The ghost lines go through the diagrams owing to ghost-number conservation. Equation (2.4.69) is the generalization of the QED Ward identity (2.4.86) that relates the electron-photon vertex function to the electron propagator to non-abelian gauge theories.
2.2 Path-integral formulation of gauge theoriesICiI
2.4.3.4
Lee identities
The Slavnov Taylor identities (2.4.58) and (2.4.61) can be converted into identities for the generating functional of vertex functions T, the Lee identities. The functional T is defined as follows: iP{A, u, u, tp, V>, K . K „ , K^, K ^ } = T C {J, J u , J,-,, J^, J^, K, K„, K^, K v> } - i J d 4 x {.J'^A'l + J > Q - u"J? + J ^ i - j a J i ) . (2.4.71) Note that the sources K", K", K'~, and K^ are not transformed, and that we denote the other sources of F with the same symbols as the integration variables of the functional integral. These sources are given by 5TCc "
.in
STCc
.a
A =
* =
STC
r-a
(2-4.72)
uUi
or ra _ _ i L " <5/1«'
,« 6
v
SiJ>i
=
i
I
l ¿u«'
r
u
iL
=
oipi
In addition we have ¿T C
IIR
JT C
5K°
6K"'
SK-
¡¿R
STC
SKS'
SKI
isr
SRC
\sr
SKI'
SK^
SI^
(2.4.74) Inserting this into (2.4.58) and (2.4.61), we find the Lee identities r ° ~ ./
[ ¿r
4
*
sr i'
MC«*
<5F ¿r +
° =
"TTT;—+
SKI +
sr +
sv
1
<S/C£
£c
<sr 1A ;
3:1
¿r <sr fyiSKl (2-4.75)
I.r>8 2
Quantum theory of Yang Mills lields
If wc extract the gauge-fixing term from the vertex functional and define d4xCa{A-,x}Ca{A;x},
f = T
(2.4.76)
the Lee identities read
-/
d"x
0
=
sr [SA* SI<
sr sr sr sr sr sr + T T T ^ : + TTTTT^ + —777T , (2.4.77)
sr 6ua '
(2.4.78)
\sr
ab sr
" SK\[
+
In this form the Lee identities are universal, i.e. independent of the parameters of the theory (masses, coupling constants, and gauge parameters). Among other things they are useful to proof renormalizability of gauge theories.
2.4.3.5
Ward i d e n t i t i e s in Q E D
The Slavnov-Taylor identities are valid in any gauge theory. However, in an abelian gauge theory like: QED the situation is simplified (cf. Sect. 2.4.1.4). The infinitesimal gauge transformations SA» = d,LS(),
Sip = i eipSO,
6j> = - i ej>60
(2.4.79)
leave ¿gauge) ¿form» and the measure of the functional integral invariant. Thus, the invariance of the functional integral under a change of variables according to (2.4.79) yields (cf. Sect. 2.2.G)
0 = J V[A]V[1>]V[i>] exp ^i y d 'x^tot X J d*X
+ ic.ip (x)SQ(x) J^ (:/;) +
ieJ^(x)ip(x)60{x)
(2.4.80)
2.4 Path-integral formulation of gauge theoriesICiI
After integrating by parts, cancelling 60(x), and expressing the explicit fields by derivatives with respect to their sources we find the Ward identities
For connected Green functions these read
0
=i1
+
v ' w
-
(2.4.82)
and in terms of vertex functions ¿r . - , . ¿r . ¿r , . . f . , k n - ic^(*)-=— \e-—'il>(x). o = - • 0"Alt(x) - d,,—— 4 oA/^x) Sip{x) Oip{x) (2.4.83) As applications, we derive some relations for special Green functions. Differentiating (2.4.82) with respect to J„(y) and putting all sources to zero yields -[ld'lGAA
(t V) = -Ud>' £ = \OjW(x
— \SJ'1(x)i6J'/{y) J=Jz=J*=Q - y) (2.4.84)
or in momentum space
-
-k) = ik„.
(2.4.85)
This is the (JED version of the Ward identity (2.4.68). It implies that the longitudinal part of the photon propagator gets no higher-order corrections.
I.r
2
uantum theory of
ang
ills lields
ifferentiating 2.4. 2 with respect to J-(z)
and . , .
results in
e
2.4. iSMxYMMiSfyz) W
= e
{x
- y)T
Tr
-eS{ ){x-z)-
STC
or diagrammatically after putting the sources to ero Mz) = e W(x-y)
I D *
1>(x)
AJx)
. . i>{x)
- e >{y)
This
— —— ^ i>
w
{x-z).
2.4.
ard identity can be also obtained by applying x to 2.4. with ;x d ftfe and using the e uation of motion for the free ghost
field. Transforming to momentum space yields the Ward-Takahashi --k2k»G?f(k,p,
= e [< *(- ?,?))
- Gf(p,-p)]
identity ,
2.4.
where k,p, q denote the incoming momenta corresponding to the outgoing fields A,t/),tJ>. In diagrams this reads
k kn ~ ?
.
AJk)
e
>{?)
I
2.4. il>{-p)
2.2
Path-integral formulation of gauge theoriesICiI
liquation (2.4.88) provides a relation between the electron photon Green function and the electron propagator. Splitting the photon propagator oil" in the vertex function G?f(k,p,q)
= GA;l(k,-k)G^(k,p,q),
(2.4.90)
and using (2.4.85) yields -k»G*f(k,p,q)
= e [Gf(~q,q)
- G^(p,
-p)] .
(2.4.91)
as well By differentiating (2.4.82) with respect to {./ ; ''(xi)}, i = 1,...,NA as {J^(zj)} and {./,;,(/;,)}, j = 1 , . . . , Nwe can derive in the same way the identity ~k'LG~ftltf e
'' (k, ki > Vj J (lj)
E
+
l
=
P j
,
q j
+ Sjlk)
.
(2.4.92) When truncating all external lines on the left-hand side and putting them on mass shell, all terms on the right-hand side vanish because they have no pole in Pi or qf, and we find the identity
-k'1 G c , w ( k , k i , P j ' J l j )
= 0,
(2.4.93)
*?=0 ,P?=9?=m?
which holds also for ¿"-matrix elements and states that these vanish if one of the photon polarization vectors is replaced by the photon momentum. If we truncate all external lines apart from the one corresponding to tpj, we obtain for k'f = 0, pj = q'j^l = rn2:
= -eG^'r-"
- "'(ki,Pj,qj
+ 5jik)Gf^(—qi
- k, qx + k).
(2.4.94)
176
2 Quantum theory of Yang-Mills fields
By substituting the fermion propagator on the right-hand side using (2.4.91), this can be written as '-"(/c, ki,pj, (¡j) Gf{-qu
G;
=
+ Sjik) eGfi-quq^+k^fi^-q.-k^i)]
and after multiplication with
(—
(2.4.95)
1
ki,pj, (¡j) = G i ^ r ''~ ' {ki,pj,qj + ô^k) e + k' l G^Jr{k, -<71 - A:,(/,)j . (2.4.96) Substituting r/i
r/i - Arj, this results in
(^G^r-
' - 'ik, ki,pj,qj
-G'c,^,
= e Gc,/i,
-
Sj\k)
{ki,Pj,qj)Gc^-{k,
(ki,pj,qj)
-qi
- M l )
(2.4.97)
,
or in diagrams (suppressing all on-shell lines) rf,{q - k)
V;(r/ - fc)
it"
«
e
-
^
-
Q
.
L^(fc) (2.4.98) Similar relations can be derived for vertex functions. Differentiating (2.4.83) with respect to / M y ) and putting /l /t = iji = -îj> = 0 we obtain, for example, ¿r f
M"(x)W(y)
2.4
ath-integral formulation of gauge theoriesICiI
This is a condition on the photon two-point function. to momentum space it reads k'TfJ(k,
fter transformation
1 -k) = —-k2k .
V
2.4.1
This identity is e uivalent to 2.4. and ensures that the higher-order contributions to the photon two-point, function are transverse. ifferentiating 2.4.
with respect to ij>(y) and ij>(z) gives rise to
i il)(y)
-ie utting A,l
i/j
k>T^(k,q,p)
fr-y ;
= ie W(x - z)- Sip(x) ip{y) f oip(z)oip(x)
2.4.1 1
and transforming to momentum space yields = e r**(-p,p) - r - q ) ]
,
2.4.1 2
where now k.p,q denote the incoming momenta corresponding to the incoining fields A,ip,i ;, or in diagrams
, v i>(p)
—e
V p
2.4.1 Tj'(l)
This is the Ward -Takahashi identity for the electron photon vertex Ta . I Hlferent.iating with respect to k,t for fixed q, using p = -q - k, and putting k , results in 2.4.1 4
1
2
2.4.4
uantum theory of
ang-
ills fields
T h e bac ground-field
method
The classical agrangian of a gauge theory is gauge-invariant. The same holds for any physical uantity. ut uantities with no direct physical mean ingsuch as off-shell Green functions or counterterms are not gauge-invariant. oreover, the uanti ation of gauge theories, as described in ect. 2.4.1, reuires the introduction of a gauge-fixing and a Faddeev opov ghost term, The resulting effective agrangian and the effective action are not gaugeinvariant but only invariant under the non-linear R transformations. s a conse uence, the Green functions in the conventional formalism do not directly reflect the underlying gauge invariance, but rather obey complicated lavnov Taylor identities resulting from R invariance. In addition, the Green functions depend on the particular gauge fixing chosen, and only physical uantities are gauge-independent order by order in perturbation theory. In the bac ground-field approach e , tH a, 1 , o 1, Ha , b 1 , the uanti ation is performed in such a way that an explicit gauge invariance is still present once gauge-fixing and ghost terms have been added. s a result, Green functions obey the naive ard identities, i.e. the ard identities corresponding to the classical action without gauge-fixing and ghost terms , and even unphysical uantities such as divergent counterterms ta e a gauge-invariant form. The bac ground-field approach does not only ma e the structure of gauge theories more transparent and easier to understand but can also greatly simplify practical calculations.
2.4.4.1
T h e gauge-invariant e f f e c t i v e a c t i o n
e follow the treatment of lief. b 1 and consider a gauge theory with a multiplet of fermion fields ip, as in ect. 2.4.1. The classical agrangian reads cl
- I V
1
m i
ix
2.4.1
with
Dn = du - \gTaA .
2.4.1
'2.'I Path-integral formulation of gauge theories
17!)
We split the gauge field into an arbitrary background field and a fluctuating quantum field A"t (here and in the following we denote all background fields with a caret), A * ¿ J
+
(2.4.107)
The background field is treated as an external source and the quantum field as the integration variable of the functional integral. After this split the classical Lagrangian can be written as
+ d;cAcu
¿ci.BFM =
+ fi[x)(il/)
+gfabcAbllAl)2
- D?A%
+ gjfinTa - m^jipjix),
(2.4.108)
where
=
gfabcAbtAl,
~ d^A; +
D«c = dlt6ac +
gfabcAbt,
b» = d,L - igTaAl
(2.4.109)
are the field strength of and the covariant derivative with respect to the background field. For fixed background field, the Lagrangian (2.4.108) is invariant under the infinitesimal gauge transformations
A"h-
c c A'" /* = yl"f4 + -qD"/* S0 +
= A;t +
l
b c fabc /AL 50
-6acdfl60c + fabc{Al
ipi -»• Vi = i>i + i ( T » , < 5 0 ° .
+
Abt)60c, (2.4.110)
In order to define the functional integral we must fix a gauge for A"L using the Faddeev Popov procedure. We choose a gauge-fixing functional that is covariant with respect to the background gauge field CaBVM{A,A-x}
=
frb^Ab(x).
(2.4.111)
I.r>8 2 Quantum theory of Yang Mills lields
The corresponding Faddcev -Popov Lagrangian is determined as described in Sect. 2.4.1.3 from the variation of the gauge-fixing operator (2.4.111) with respect to the gauge transformations (2.4.110), r _= - J Cl zu (x) ¿-ghost.,BFM
i gf „ ecdbA )u (lJfl -tfl
i\lizua(x)Dac'll(D^
= -J
+ gfc(lbA'{1)ub{z).
(z) (2.4.112)
The complete Lagrangian £ B F M = £cl,BKM -
A ; x } C f 5 F M { A , A;:c}
(2.4.113) is gauge-fixed but invariant under the local transformations _> A'« =
+ yDah5eb,
ua -> u'a = ua + fabcub6dc, i>i
$ = i>i + i(r a il>)i6d",
A'£ = Al + fabcA^SO'1,
A"
ua ->• il'a = ua +
fabcub66c,
& -> # = ft - i(i>T a ) l 6d n . (2.4.114)
Under this transformation the background field transforms inhomogeneously as a gauge field, while the quantum field and the Faddcev Popov ghost field transform (homogeneously) as a matter field in the adjoint representation. This background-field gauge invariance follows directly from the global invariance of t he Lagrangian (2.4.113) and from the fact that the background field y\", appears in (2.4.113) only within the covariant derivative and the field strength. The gc.ne.rati.ng functional
of Green functions is defined as
TBFM{J,A1JII,J
+ J^A*
JD[A)V[u}V[U\V[IJ>}V$] + j y
a
- uaJ£ + 4 i n -
ftjj]) . (2.4.115)
It depends both on the conventional sources and on the background field A. The functional measure is invariant under the transformation (2.4.114).
2.4
Path-integral formulation of gauge ilicorkis
IHI
My performing a change of integration variables of the form (2.4.114) it is easy to sec that the generating functional TBFM is invariant under the transformation A(;t -> /i;;1 = i ; i + K
l
-Dab6Qb,
./;; ->
Ju = Ju + fabc4soc,
4
=4
-
=
-n -» J? =
¡Wa)w\
./j -» 4
fabcjbtwc,
+ + =
fobc4ßci
4 +
i{Taj^yö6a. (2.4.1 IG)
As in the conventional formalism the generating functional Green functions is given by
of connected
T C ,BFM { J > A , J „, J ft, J,/,, J ^ , } = l o g TUKM { J , A , J „ , J ft, J $ , J ^ } .
(2.4.117) The corresponding generating functional of vertex functions is obtained after a Legendre transform as i r B F M { A , A , u , ü , ip,ij>}
- i J d4x
= T C I B F M { J , A , J „ , J,x, J^,, J ^ }
- üaJ? +
+ j y
- Äjj),
(2.4.118)
where the sources of TBFM J
C.BPM, l6J
i>
= 777-TCBFM lSJ
(2.4.119)
<,
are denoted with the same symbols as the integration variables of the functional integral. Note that the background field is not transformed. Since the sources of FBFM are just the conjugate variables to the sources of 7'UFM it follows immediately from the invariance of TBFM that the effective action (2.4.118) is invariant under the transformations (2.4.114). In contrast to the BRS transformations these transformations are linear in the fields that correspond to the sources of the effective action.
1 2
2
uantum theory of
ung
e can define a gauge-invariant f
,t
,-
r F
ills fields
effective action V via ,
, , ,
.
2.4.12
ll. is invariant under ordinary gauge transformations of the bac ground gauge field and the fermion fields, i.e.
B f - S - + igi>( )Ta-^' SAftx)
S 0a(x)
ig*jLT*M*). tl>{x) 2.4.121
s a result, the vertex functions generated by differentiating f with respect to , - , and if obey the naive ard identities of gauge invariance. These ard identities are similar to those of E 2.4. and lead to similar conclusions. s an example we mention that the ard identities 2.4.121 imply that the product gAb,(x) needs not. to be renormali ed cf. ect. 2. .1. . esides the bac ground-field ard identities 2.4.121 , also lavnov Taylor identities related to the uantum gauge transformations 2.4.11 are valid.
2.4.4.2
Relation to the conventional effective action
In order to establish the usefulness of the bac ground-field effective action we have to relate it to the conventional effective action. erforming a shift A* A* - Aatl in T F 2 . 4 . 1 1 , one finds Tbfm
,
,
T
,
, fi,
l
fi
, ,
;
exp
-i
d4x
a
,
2.4.122
where T is the normali ed generating functional of the conventional formalism 2.4.2 evaluated with the unconventional gauge-fixing term ,
;x
d A'^x)
- d A it{x) + gf^Ab (x)A%{x).
2.4.12
The ghost agrangian of TRF he bac ground-field gauge goes over into the correct ghost agrangian of in the gauge 2.4.12 . ote that T is still a functional of because of the presence of this field in the gauge-fixing
2.1
ath-integral formulation of gauge theoriesICiI
term 2.4.12 . s a conse uence of 2.4.122 the generating functional of connected Green functions are related by C,HF
,
TC
,
,
,,,
,-,,
,
,
,
,
;
- iI d x
ifferentiating with respect to
=
2.4.124
yields
>< i4
a
l.
~ '
(2A125)
Tc,eFM= A%=A 1 ~ ^
where t denotes the gauge-field source of the conventional vertex functional. erforming a egendre transformation we find the relation between the effective action in the bac ground-field formalism and the conventional formalism F
,
,
, u, T
V
T
, u,
,- , V ;
Finally, the gauge-invariant effective action 2.4.12 f
,t
,-
r
,
,
,
;
.
I I
. .
2.4.12
-
is given by 2.4.12
The bac ground field A t enters F both through its gauge-fixing term and through its field argument. It. is instructive to consider the bac ground-field method for the case of a non-gauge field theory with field i . plitting this field into a uantum part t > and bac ground part ip and performing the same manipulations as above we find r F
V ,V
F
-
2.4.12
or t t f )
R
,
R V4-
2.4.12
The bac ground-field effective action F F V ; V ; is ust the conventional effective action computed in the presence of the bac ground field -ip. In particular, FRF , ip is the vacuum-to-vacuum amplitude in the presence of
I.r
2
uantum theory of
ang
ills lields
the bac ground field. Thus, according to 2.4.12 the conventional effective action is e ual to the vacuum-to-vacuum amplitude in the bac ground-field method. This statement can be generali ed and implies that for a non-gauge theory the bac ground-field effective action is e uivalent to the conventional effective action. If we split in 2.4.1 part T >-
also the fermion fields into a bac ground and uantum
> + t >,
ip->ij)
+ jj>
2.4.1
we obtain r F i
,
r
, u, u, ip, ip, ip, ip
f
. -. . - T a A=A+A,1p=1p+1p,1p=1p+1p
,u,u,- ,ft
instead of 2.4.12
2.4.1 1
and 2.4.12 , and
, ft. ip r
=
T
, , ,ftft
,
, , , , ft
,
.
2.4.1 2
The difference between the gauge-invariant effective action and the conventional effective action is only due to the gauge-fixing term. Thus, for fields that do not appear in the gauge-fixing term, as for example the fermion fields, the split-up into bac ground and uantum fields is often not relevant.
2.4.4.
T h e S m a t r i x in t h e bac ground-field m e t h o d
nce the bac ground-field effective T action has been determined, the connected bac ground-field Green functions and the S matrix are constructed in the same way as in the conventional formalism by forming trees of vertex functions. To this end one has to introduce propagators for the bac ground fields. In this respect, we note that in this application of the bac groundfield method, the bac ground fields are purely formal ob ects and have no nhvsical meaning.
'¿.'1 Path-integral fonmilation of gauge LhtHjrios
185
In order to define the background-field propagators that connect the vertex functions one lias to add a gauge-fixing term for the background fields to P resulting in r full = f + J d « * £ g f .
(2.4.133)
The gauge-fixing term is independent, of the gauge-fixing term (2.4.111) for the quantum fields and can be chosen suitably. The generating functional of connected Green functions Tc can be obtained from r f u " via a Legendre transformation
(2.4.134) with J"''1 = ~ t - f r u 1 1 ,
A =
./i = - l f f u " .
(2.4.135)
We note that the background-field gauge invariance gives rise to Ward identities for f f u " and fc that resemble those of QED [De96]. From the generating functional of connected Green functions Tc the S matrix is constructed as usual (cf. Sect. 2.2.3). In order to justify the backgroundfield method, one has to show that the resulting S-matrix is equal to the conventional one [Ab83a, Re85]. In non-gauge theories this follows trivially from (2.4.129). In gauge theories the proof from the relation (2.4.127) is complicated by the extra dependence of P { A , 0,0, ip, V>; A } on A via the gauge-fixing term. While the conventional vertex functions are obtained by first differentiating P { A , 0 , 0 , A } with respect to A for fixed A and then putting A = A, the background-field vertex functions are obtained by first putting A — A and then differentiating R{A.(),0,i/*,^; A } with respect to A. This is equivalent to taking derivatives with respect to A and A and then putting A = A. In order to prove that the background-field method yields the correct S matrix one therefore has to proof that the extra derivatives with respect to A do not affect the S matrix. This is plausible because the dependence of F { A , 0 , 0 , A } on A is entirely due to the gauge-fixing term. In fact, the proof of this statement resembles closely tin; proof of the gauge independence of the S matrix.
I.r
2
2.4.4.4
uantum theory of
ang
ills lields
F e y n m a n rules
I Vom the bac ground-field agrangiau F Feynman rules can be derived as usual; one merely has to distinguish between uantum and bac ground fields. The gauge-invariant effective action I can be computed by summing all oneparticle irreducible Feynman graphs. ince we have put u ii in the gauge-invariant effective action 2.4.12 , no uantum fields or ghosts appear on the external lines. i ewise no bac ground fields appeal inside loops since the functional integration is performed only over the uantum fields, i.e. no bac gound-field propagators appear within the loops. The fermions and other fields that are not split into a bac ground and uantum part appear both inside loops and on external lines. nce the bac ground-field vertex functions have been calculated, the connected Green functions are composed by oining the one-particle irreducible graphs with bac ground-field propagators. t this point a gauge for the bac ground field must be chosen. ote that in particular the lowest-order diagrams contain only bac ground fields. The Feynman rules can be obtained from the agrangiau 2.4.11 in the usual way. ll vertices that result, only from the classical agrangiau are e ual to those in the conventional formalism, no matter whether they contain bac ground fields and or uantum fields. The difference between both formalisms results only from the gauge-fixing term. ince the gauge-fixing term is strictly uadratic in the uantum fields, the only vertices that are different from the conventional ones are those involving exactly two uantum fields or Faddeev opov ghosts. Ta ing all momenta as incoming and indicating the bac ground fields by hats over the group indices, these differing vertices are listed in Fig. 2. . ote that in contrast to the conventional formalism vertices involving two Faddeev- opov ghosts and two gauge-bosons appear. ll vertices involving fermion fields coincide with those of the conventional formalism. Therefore, a split of the fermion fields into bac ground and uantum fields would not matter. imilarly, in non-gauge theories the Feynman rules in the bac ground-field formalism and the conventional formalism are identical. For connected Green functions one needs in addition to the above Feynman rules the propagator of the bac ground fields. If is chosen using a gaugefixing condition as in 2.4.11 , the bac ground-field propagator ta es the usual form given in 2.4. 2 but with a new gauge parameter n important
2.1 PnUi-integral formulation of gauge theories
187
a2.M2.p2 fl/d,02a3LW2(2i>l + P 3 ( 1 - 1/0),1 a + Wa(P2 - < W i ( 2 Pi + P2U - 1 / 0 ) , J
2.5 Extra vertices in the background-field formalism involving gauge-boson and Faddeev-Popov fields
I.r
2
uantum theory of
ang
ills lields
advantage of the bac ground-field method is that a way that actual calculations simplify.
can be chosen in such
t first sight the use of the bac ground-field method seems to be more complicated than conventional calculations. However, once the Feynman rules have been determined, calculations within the bac ground-field method are in fact often simpler. In addition, the bac ground-field Green functions have many desirable properties that the conventional Green functions are lac ing. In this section, the bac ground field was used as a technical device to simplify calculations. The bac ground-field formalism can also be used for bac ground fields with physical significance. n example for this can be found in the discussion of the one-instanton calculation in ect. 2. . .2.
2.
s
The precision of many experiments re uires the evaluation of the theories for the strong and electrowea interaction beyond the tree-graph approximation. The higher-order corrections to Green functions and -matrix elements, i.e. the uantum effects, result from Feynman graphs containing loops. ince these corrections change the relations between the parameters in the agrangiau and the observables, the original parameters of the theory, the bare parameters, are no longer directly related to physical uantities. oreover, the bare parameters become divergent. To allow for a proper treatment of divergent uantities, a regularization procedure is needed. This amounts to a modification of the theory so that the possibly divergent expressions become well-defined, and that in a suitable limit the original divergent theory is recovered. Conse uently, a redefinition of the basic parameters, a renormalization of the theory, is needed. s is well- nown from second-order uantum-mechanical perturbation theory also the normali ation of the state vectors is modified. This has to be corrected for by a renormali ation of the fields and or external wave-functions states .
2. .1
ivergences and renormali ation
technical and conceptual complication arises in relativistic uantum field theories because divergences occur in the evaluation of loop diagrams.
2.5 llenormulization of quantum field theories
195
Therefore, regularization prescriptions [Pa49] that are compatible with the structure of gauge theories are required. The divergences in all Green functions and 5-matrix elements must be eliminated in the renormalization procedure by absorbing them in renormalization constants for parameters and fields. A theory in which this is possible in all orders of perturbation theory is called renormalizable. The proof of the renormalizability of gauge theories [Le72, tII7'2] was one of the most important steps in the development of QCD and the Glashow Salam-Weinberg (GSW) theory. A detailed and complete treatment of the theory of renormalization is beyond the scope of this book (cf. [Co84]). We restrict our survey to the principles of the renormalization program, address the crucial points, and illustrate the renormalization procedure by presenting the techniques needed for the evaluation and renormalization of the one-loop corrections in gauge theories. The requirement that divergences arc compensated does not fix the finite parts of the renormalization constants. Indeed, these can be defined in many different ways leading to different parametrizations of the theory. As a consequence, calculations in finite orders of perturbation theory performed in different renormalization schemes differ by higher-order contributions (renormalization-scheme, dependence:). In an exact (all-order) calculation all different renormalization schemes would lead to equivalent relations between physical quantities. Their independence of the choice of the renormalization scheme and the consequences of this fact are studied with the help of the renormalization-group equation (Sect. 2.6).
2.5.1.1
C o n c e p t of r e n o r m a l i z a t i o n
In order to clarify the meaning of renormalization we begin with a simple example: a theory described by a field ipa, some parameter go, and a Lagrangian £(V;o, ¿^¡V-'Oii/o)- For the moment we assume that one can exactly calculate n measurable quantities o\(go),... ,nn(go). From an exact measurement E\ of ai(go) it is possible to determine go and thus to obtain predictions for These can be compared to the measured values E2, • • •, En yielding n — 1 tests of the theory. Let us introduce a new parameter g by substituting go = goid)- Then, <J\ — °i(i?o(<7)) = Under this reparametrization which may be extended to act also 011 the field V;o we have exact equivalence of the relations between the physical results. In this sense the exact results are strictly parametrization-independent. This is of importance since by this procedure
!!)()
2 Quantum theory of Yang Mills iielcts
the original parameters in the Lagrangian can get replaced by others which have a more direct physical meaning. For example, it is possible to introduce well-defined quantities that have been measured with high precision as parameters of the Lagrangian. Rclativistic quantum field theories involve additional complications. Since higher-order calculations lead to divergent expressions, a regularization scheme is required. This introduces an effective cut-off parameter A , where A —> Ao corresponds to the absence of regularization, i.e. to the original but possibly divergent expression. Consequently, we have a, = <7,(<70, A ) . If we determine the bare coupling 90 via a\ from E\, _r/o = a[ ''(E\, A ) , we obtain a value for the bare coupling that is cut-off-dependent, possibly ultraviolet-divergent, and therefore evidently has 110 direct physical meaning. However, in renormalizable theories, elimination of 90 in CR2(FTO> A ) = ' ^ ( I ? I , A ) , A ) gives a result which is finite for A —> A Q . As a result, bare quantities are regularization-scheme-dependent, but the relations between physical quantities are regularization-scheme-independent and finite. A convenient way to perform the reuormalization is to introduce rcnormalized parameters by a suitable reparametrization 9o = 9o(.9, A ) = Zg(g, A ) <7,
1/2
(2.5.1)
1])0 = Z j (g, A ) IJ>.
The reuormalization of the fields allows to introduce appropriately normalized fields and Green functions. The renormalization transformation (2.5.1) applied to C gives ¿ftfo,90) = C(Zl/2Z~1/2^,
ZgZ~lgo)
= C(ZlJ%
Zgg).
(2.5.2)
Zg{g, A) = 1 + SZ9{g, A),
(2.5.3)
In perturbation theory we write Z^g, A) = 1 + SZ^g, A), yielding m>o,9o)
= ¿((1 + = £{ip,g) + Cct
SZ+Y^Al+SZJg) (i),9]SZ^,SZg).
(2.5.4)
2.
Rcnorntali .nfion of
uantum licit 1 theories
l!)l
The functional dependence of C(ip,g) on ip ;U1s t'H3 same as that of £( ;O>!7O) on Vo.ffo, and (ip,g) yields the same Feynman rules for the renormalized Held and parameter as C(i/>o,go) does for the bare ones. The conn terterm Lagrangian £,.t summarizes all terms containing the renormalization constants SZ and generates the countertenn Feynman rules. These allow to calculate the Green functions and from them the S-matrix elements of the theory. For this purpose we define the renormalized Green functions R. They are obtained from the corresponding unrenormalized ones G by multiplying them with field renormalization constants Gn(ki,...,kn)
=
^'1 2G(ki,...
,kn).
(2.5.5)
They depend on the renormalized coupling constant g and the renormalization constant 6Zrj. The renormalized truncated Green functions G t n m c , K , are obtained by truncating all external legs with renormalized propagators, Girunc,\i(k\,...,
kTl) =
^
GtTunc(ki,...,kn),
(2.5.0)
and the 5-matrix elements arc according to (2.1.35) given by (~Ps+1, = i? /2
> ~Pn| |pi,..., /,2
Gtrunc(pi,.
ps) = . . , pn)
ft2Gtmnc,K(Pi» ,Pn) ,
P?=m
(2.5.7)
Pi=Mf
where r is determined analogously to (2.1.34) from the renormalized twopoint function. The product = \{ ,j,, the unrenormalized LSZ factor, is independent of the renormalization scheme. The renormalized vertex functions FR, the one-particle irreducible components of (7R, are obtained from the renormalized Green functions by a Legendre transformation as in Sect. 2.2.2.2. They are related to the unrenormalized ones by FK(/C, , . . . , kn)
= Z J ^ R I A , , . . . , kn).
(2.5.8)
The requirement that the renormalization constants absorb the divergences fixes those only up to finite parts. The latter are fixed by renormalization conditions for the renormalized vertex functions. This allows to
I.r>8 2
Quantum theory of Yang Mills lields
relate them directly to measurable quantities at suitable points in momentum space. The renormalization constants are fixed by prescribing values for certain renormalized vertex functions at chosen values of the momenta. The choice of the renormalization conditions fixes a renormalization scheme. The renormalization procedure has to guarantee the absorption of divergences, the definition of the physical parameters, and a convenient normalization of the field ij>. More complicated field theories involve more complicated renormalization procedures, but the basic ideas are the same as in the simple example discussed here. 2.5.1.2
T h e superficial d e g r e e of d i v e r g e n c e
The renormalization of gauge theories has to handle the divergences. As shown in Sect. 2.1.4 all Feynman graphs can be composed by multiplication of one-particle-irreducible (1PI) graphs. Moreover, all ultraviolet (UV) divergences result from loop integrations in 1P1 graphs or subgraphs. Therefore, it is sufficient to consider the UV divergences of 1PI graphs. One of the simplest divergent Feynman diagrams is the one-loop self-energy diagram for a scalar field theory with 3 interaction,
From the Feynman rules in App. A. 1.3.3 we obtain for the self-energy S (p) =
(-ig)2
f
dAk
i
i
(2tt) 4 k2 - m\ + i e { k - p) 2 - m*2 + it'
. (
' '
}
For large values of the integration momentum (|A:| m\,m.2,\p\) the inte] grand behaves as k dk/k' . The integral is logarithmically divergent. As this divergence originates from the large momentum region it is called an UV divergence. It is interesting to observe that the integral is convergent for dimensions D < 4. This is the basis of the dimensional regularization scheme [tII72a,
2.5 llenormulization of quantum field theories 195
Bo72, Co84] in which the Feymnan integrals are calculated in 4 - 26 dimensions, where is the regularization parameter. Tliis scheme is presented in Sect. 2.5.2. In order to study the general structure of the UV divergences and to obtain a criterion for renormalizability we perform a systematic power counting of I lie Feynman integrals in D dimensions. Consider a graph G with E external lines with momenta pi,... , p e and L internal loop momenta ki,..., A/, which must be integrated out. The integrand ¡ a is a product of vertices and propagators which depend on P I , . . . ,PE and k\,..., kiThe expression for the graph has the form
FA{pi
>E)=
I
h:{P\»• • • ,PE\
• • •, &/,) dDfei • • •
d°kL. (2.5.10)
The possibly divergent integration is assumed to be cut ofT at momentum A. Simple counting of powers provides a definition of the superficial degree of divergence of sub-integrations. For this purpose consider the behaviour of ¡c when a subset S = {/,:,<} of M loop momenta in a general configuration (i.e. no partial sum of the momenta p\,... A ^ , . . . , ki must disappear) is scaled by a factor A, A> —> A T h e r e is an upper bound for Iq as a function of A |/G(A)| <
A c < 5 > x constant,
(2.5.11)
[in our example (2.5.9), M = L = 1, S = {A:} and c = - 4 ] . The integral over the momenta k? G S is bounded by
IGu dDki'
ir
'l
•••
d% < A U ' D ^ / ( A ) , M
(2.5.12)
where the function / grows slower than any power of A for A -» oo, i.e. at most logarithmic ( Weinberg's theorem [WcGO]). The quantity w/>(.S') = DM + c(.S') is called the superficial degree of divergence [Dy49] of the integration with respect to the momenta k? 6 S in D space-time dimensions. The integration over the momenta in S is convergent if w/j(5) < 0. The superficial degree of divergence with respect to the set of all loop momenta of a graph G is called the superficial degree of divergence of the graph un(G).
15)4
2 Quantum theory of Yang Mills Holds
A graph G is called primitively divergent [Dy49] if only the final integration causes a divergence, i.e. if cjp(S) < 0 for all m(S) < L but uip(G) > 0. We can define a subgraph G of G by contracting all lines in G that do not depend on one of the momenta in S to a point. It can be shown that all divergences of an arbitrary divergent graph G originate front the primitive divergences associated with its subgraphs Gs [We60]. In particular, any graph G is convergent, if G and all its subgraphs are superficially convergent. On the other hand, a superficially divergent graph can nevertheless be convergent. For more details we refer to [Co84]. 2.5.1.3
Criteria for r e n o r m a l i z a b i l i t y
Since renormalization proceeds order by order, and the inductive proof assumes that all subdivergences are already absorbed, it is sufficient to consider the primitive divergences of one-particle irreducible diagrams. The set of primitively divergent graphs can be deduced from dimensional analysis. Let us consider a general quantum field theory with scalars Svi>f Aa\
(2.5.13)
where dv, sv, fv, and av denote the number of derivatives, scalar fields, fermion fields, and vector fields, respectively, the dimension of the coupling gv is obtained as r
1
,
foj = D - dv
D —2 —sv
D —1 . —/„
D-2 —av.
(2.5.14)
Some examples are listed in the second part of Table 2.1. Let us consider an arbitrary one-particle irreducible Feynman diagram G with ,\ ( I I , / , , and Ia) external (internal) scalar, fermion, %i,. and
Tab. 2.1 Canonical mass dimensions of field operators and coupling constants in D and 4 dimensions and vector-boson lines, respectively, vertices, and loops. For large momentum Ak the fermion propagators (1.2.33) behave as A 1 and the scalar propagators (1.2.31) as A - 2 . The behaviour of the vector-boson propagators is of the form A _ ° and depends on the theory and in gauge theories also on the gauge. In general a < 2. Using the Feynman gauge propagator (1.2.34), we have a = 2 in QED. Finally, each vertex v scales as A''", where dv denotes the number of derivatives associated with v and acting on internal lines. Collecting all contributions, we find for the superficial degree of divergence of G in D dimensions
u)0(G) = DL - 21 -I*-
aIA + ^
dv.
(2.5.15)
+ 1 = I , + I , + IA ~ ( E 0 +
(2.5.16)
Using Euler's loop equation
L = I,j, + 7V, + IA-V
196
2 Quantum theory of Yang-Mills fields
the relations sv = 2 +
] T f" = 2/V> +
E4,
V
V
= 21A + EAl
(2.5.17)
V
and (2.5.14), we obtain u>d(G) = D + (D= L>
2)1* + (D - 1 )Ii, + (D - a) Ia + £ ( d „ _
D-2 ^—^
+ (2 - a ) / , , -
D-
1 2— ^
D)
D-2 2—
- d,
'2.5.18)
where d = ^2v{dv - dv) is the total number of derivatives acting on lines or in momentum space the total number of external mora(, nla four dimensions this reduces to «4(G) = 4 -
- -
-
Ea + (2 - a)IA -
- d.
^
5
jn
19)
V
From this result we can derive the set of primitively divergent Fc graphs and vertex functions.
man
In general, a field theory is called renormalizablc if it involves only finj(,e number of primitively divergent vertex functions. Inspecting (2.5. ^ a n ( j (2.5.18), we see that this implies [¡7,] > 0, a = 2 ( a > 2 does not <\pp,,ar) and D > 2. In this case the superficial degree of divergence of any F k y n m a n graph contributing to a given vertex function is bounded from abo )Ve ^ we can define the superficial degree of divergence for the vertex func.j.jon this upper bound. In four dimensions we thus find w 4 (r) = 4-Et-EA-
- d(D,
(2.5.20)
where ri(T) is the minimum number of derivatives acting 011 exterr ia j jj n e s in all graphs contributing to f . If, on the other hand, [<7,] < 0 for t j o a s ( one coupling or a < 2, any vertex function gets divergent contributions
2.5 llenormulization of quantum field theories 195
associated with couplings of negative mass dimension, and the theory is non-renormalizable according to power counting. These considerations can be summarized as follows: a Lagrangian field theory is renortnalizablc if all coupling constants have non-negative mass dimensions and if all boson propagators behave as k~'2 and all fermion propagators as A:-1 for large momenta k. 2.5.1.4
R e n o r m a l i z a b i l i t y of specific theories
We start with the D Lagrangian specified in (1.2.16). It involves only fermion fields, vector fields with propagators that behave as A - 2 because of gauge invariance, i.e. a = 2, and one interaction vertex without derivative. The corresponding coupling constant has dimension [e] = (4 — D) 2, i.e. it is dimensionless in four dimensions. Thus, (2.5.19) and (2.5.20) reduce to W
D
-
ua{G)=A-^E+-EA.
(2.5.21)
For the physical space-time dimension D = 4, the superficial degree of divergence of any Feynman graph depends only on the number of external lines and thus is a property of the vertex functions w.i(F) = u\(G). The number of primitively divergent vertex functions of QED, i.e. those in which w.|(F) > 0, is limited to the following cases (because of fermion-number conservation E$ must be even) • • •
E^i = 2, E/[ = 0: electron self-energy, ^ = 2,
E^, = 0, Ea = 1: photon tadpole, vanishes because of Lorentz invariance, ^, = 0,
•
..\ = 1: electron-photon vertex,
,\ = 2: photon self-energy,
^ = 0, t\ = 3: triple photon vertex, vanishes in QED because of charge-conjugation invariance (Furry's theorem), ,p = 0,
a = 4: photon-photon scattering, finite owing to gauge
I.r>8 2 Quantum theory of Yang Mills lields
Gauge invariauce makes QED renormalizable and reduces the number of primitively divergent vertex functions to three. In the Fermi model mentioned in Sect. 1.4.1, only fermions propagate and have a four-fermion contact interaction with a coupling constant Gtl of dimension [G,,] = - 2 (cf. Table 2.1). This implies w4(<3) = 4 L - / , / , = 4 - 3E\/,/2 + 2V for the superficial degree of divergence. The number of primitively divergent vertex functions increases ad infinitum with incre;ising number of vertices V; therefore the Fermi model is not renormalizable. The W and Z propagators in the intermediate-vector-boson model for the weak interaction have—¿is can be seen from (1.4.13)—an asymptotic power behaviour of zeroth order in the momentum k ( a = 0). Therefore, ^(G) = 4 - 3 / ^ / 2 — 2 ( E \ V + ^ Z ) + 2 ( / W + IZ) represents the superficial degree of divergence of primitively divergent graphs. The number of primitively divergent vertex functions grows with the number of internal vector-boson propagators and the intermediate-vector-boson theory of the weak interaction is not renormalizable. The difference between interacting, massive vector particles, as encountered in the previous example, and QED is gauge invariance. It allows to bring the in which (JED is photon propagator into the Feynman-gauge form -g,,v/k2 renormalizable. This suggests a formulation of the weak interaction theory as a gauge-field theory (cf. Chapt. 4), which is an extension of the intermediatevector-boson model. The decisive step in this program is ( lie Higgs-Kibble mechanism (Sect. 4.1). It allows to express massive vector fields as gauge fields. A suitable choice of gauge leads to propagators that behave as l/k2 at large k2 and thus—exactly as in QED to a renormalizable theory. Non-abelian gauge theories, e.g. QCD, can be evaluated in various gauges. Because physical quantities are gauge-invariant it is sufficient to prove renormalizability within one particular gauge. We consider the primitively divergent graphs in the Feynman gauge where the gauge-boson propagators and l a , (Ea, E^, and Ea) be the numbers behave as A - 2 . Let Ia, of internal (external) lines of gauge bosons, fermions, and Faddeev Popov ghosts, respectively. With respect to power counting, the ghosts behave just as scalars. According to Table 2.1 all couplings arc dimensionless and we obtain from (2.5.20)
w4(r) = 4 - E A - E c - ' ^ - d ,
(2.5.22)
2.5
Reiioi uiulizatiou of quantum field theories
I!)!)
where d is the minimal number of external momenta present in all graphs contributing to the vertex function T. The number of primitively divergent vertex functions is finite in four dimensions. Since gauge-theories involve unphysical degrees of freedom one still has to show that these decouple from the S matrix. To this end, one must show that the renormalized theory is invariant under a renormalized gauge transformation. This is done in Sect. 2.5.4.1.
2.5.1.5
S t r u c t u r e of t h e c o u n t e r t e r m s
We add a few remarks on the structure of the counterterms. If a theory has been renormalized to L - 1 loops, then at the L-loop level all subdivergences are already cancelled and only the primitive divergences are left. If these can be compensated by //-loop counterterms the theory can be renormalized at the L-loop level. Therefore, the counterterms are directly linked to the priinitive d ivergences. I )ilferentiating a Feynman graph with respect to one of its external momenta reduces its superficial degree of divergence by one [Co84], and all superficial divergences disappear by differentiating sufficiently often. Consequently, the primitive divergence of a vertex function must be a polynomial in the external momenta with degree w(r). and the counterterms, which have to cancel these divergences, must have the same structure [We60]. Such local counterterms can be generated from terms in the Lagrangian which are polynomial in space time derivatives, d, with degree o/(F). Thus, the counterterms that compensate the primitive divergences of a vertex function r" 5 , n / , n / 1 with n s > 7if, and n,\ external scalar, fermion, and vector-boson lines, respectively, have the general structure 8C = ckdlk 'il)n A'1
A
with
7* <
w{rn>'n''nA).
(2.5.23)
In particular, we find for the dimensions of the coupling constants of the counterterms using (2.5.20)
= w(r»">»/**) -7k
> 0.
Thus, the counterterms themselves are renormalizable.
(2.5.24)
200
2 Qtianl.mii theory of Yang Mills fields
2.5.1.6
R e n o r m a l i z a t i o n of Q E D
As an example, we perform the renormalization of QED. Since all types of possible countertenns arc already contained in the unrenormalized Lagrangian, the subtraction of divergences can be carried out by multiplicative renormalization. The bare QED Lagrangian for the electron-photon system,
£QED =
- - M I M , » ~ DVA 0 I L L ) 2 -
+
-L(D'M„,„)2
- m0)ij>o + e o A o ^ o l ' ^ o ,
(2.5.25)
where A,t)q, v/'o, «o, "¡.q, and £o are the unrenormalized fields, the bare charge, mass, and gauge-fixing parameter, respectively, is rewritten in terms ofmultiplicatively renormalized quantities by the substitutions A,i ,o = Z1a/ZA^,
ip0 =
eo = Zcc,
mo = Zmm,
This renormalization
transformation
Cqfa, = - \ z
a
^ A
v
Z]J2ip, £o = Z^.
(2.5.26)
yields
-
±ZAZ-l(d"Alt)2
-
— + Z^Tpiflij) - Z^Zmipmip
+ Z
c
1/2 zj
— Z^eAfSpy'ip.
Writing Z, = I + SZ,, the Lagrangian is split into the original one with renormalized fields and parameters and into the counterterm Lagrangian
¿QED =
- D„ARF
+ ?/»(i
-
eAu^y^tJ;
- \hZA{d{lAv - d v A t f - ^ ( Z A Z ~ l - 1)(0'MP)2 + 5Z^Tpi
- 1 )i}>mi/>
- 1 )eA,frf1>.
(2.5.27)
The counterterm Lagrangian gives rise to counterterm vertices which have to be added to the ordinary Feynman rules and lead to counterterm graphs, which vanish for Z, —> 1. Before we determine the QED renormalization
2.5
Kenormalizatiou of quantum lielcl theories
201
constants '¿i, we first consider relations between them that follow from gauge invariance, i.e. from the Ward identities. The Ward identity
for the renormalizcd
-^tW'SZaG^Jci,
photon propagator
[cf. (2.4.85)]
(2.5.28)
~q) = i qv
implies that Z aJZ^ must be finite. We can fix the finite part by choosing Zf. = ZA.
(2.5.29)
The Ward identity
for
the renormalizcd
electron
photon
and the electron propagator Of{'''t. (2.4.88) reads with
vertex =
function Z\
- ^ V Z ^ z f G i f ^ p ' )= = Zee
- Cg(p, -P)).
(2.5.30)
Therefore, Z e Z^[ 2 must bo finite, and we can choose = Z~X'2.
(2.5.31)
Thus, charge renormalization can be related to the renormalization of the photon field. The renormalization constants Z/1, Z^,, and Zm are sufficient to cancel all UV divergences of QED (cf. Sect. 2.5.1.4). In the on-shell renormalization scheme, they are determined from the conditions that the photon propagator and the electron propagator have residue one, and that m is the physical electron mass. The photon-field
lim 0
renormalization
1
t
2
is fixed by requiring
=
(2.5.32)
Since — q) gets contributions of ordinary and counterterm Feynman diagrams this condition fixes the renormalization constant Zj\. Equation
2 2
2
tianl.mii theory of
ang
ills fields
2. . 2 enforces that the pole at 2 of the renormali ed photon propagato- has residue 1. ith this normali ation the explicit wave-function renormali ation constant R in the reduction formula 2.1. can be omitl L - 1 , pro ect in 2. . 2 ted. The polari ation vectors ' ( ), e' ,, = , s''e onto the physical degrees of freedom of an on-shell photon. In oiler to analyse the ard identity 2.4.1 , which is e uivalent to 2. .2 , and the renormali ation condition 2. . 2 in more detail we deinto orent tensors and scalar functions, compose T a r1AA
R,ii1
The
(„
- (n - ) 9iif n
ard identity 2. .2
(
W^ VAA(n21 J R.T
lli(
rAA
2
2. .
imposes
i or 1
(
2\
R.LW ) —
1 2
(2.5.34)
£<7 1
i.e. tie longitudinal part gets no higher-order corrections. The Ward identity and the absence of poles in vertex functions, in particular in riH„(<7» — q), enforce that the transverse part of the photon vertex function vanishes at
r
, it/rC°) = I R £ ( 0 ) = 0,
(2.5.35)
i.e. no photon mass term is generated by higher-order corrections, inserting (2.5.53) into (2.5.32) yields the renormalization condition i 2\ Inn
dq2
=
9
-1.
=0
The constants Ztn and Z^, for the re,normalization field ire fixed by
lim
+ m)r* (p,-p)u(p) o 9 p2 in2
=
(2.5.36)
U
. \ )
of the electron mass
and
(2.5.37)
2.5 Kenormalizatiou of quantum lielcl theories
203
11« re w(p) denotes the electron spinor. Equation (2.5.37) implies in particular Ir 0 (P> ~ p M P ) | p 2 = m 2 = 0,
(2.5.38)
i.e. m is the physical electron mass. Moreover, it guarantees the usual normalization of the electron propagator, i.e. the renormalized electron propagator has residue one. Using again Lorentz covariance, r - P )
= ¿rtytP2)
+ mrg(P2)
(2.5.39)
= 0-m)r^v(p2) + m(rg,(p2) + rg(p2)), expanding around p2 =
m 2,
< % y } ( P 2 ) = r ? i s , v } ( - 2 ) + (P2 ~ m2K%y](rn2)
+ ... ,
and inserting into (2.5.37) yields the two conditions r g , ( m 2 ) + r g ( m 2 ) = 0, r g , ( m 2 ) + 2m 2 ( ^ ( m
2
) + r g ( m 2 ) ) = 1.
(2.5.40) (2.5.41)
Tlie mass renormalization constant Z,„ is determined from the first, the field renormalization constant Z^ from the second of these equations. We have fixed the charge renormalization constant by putting Z,, = Z ^ This is equivalent to the following condition fi(p)rijf
( 0 , p , - p ) u ( p ) ' = cu(p)7 M tt(p)
for
p2 = m 2 .
2
.
(2.5.42)
This can be seen by using the Ward identity (2.4.104), the decomposition (2.5.39), the condition (2.5.41), the identity 2p,t = {^,7,,}, and the Dirac equation. Thus, the theory is fitted to physical parameters in such a way that a photon scattered with momentum transfer zero on a real, physical electron (p2 = m2) feels a coupling strength e, (e 2 /47r = 1 / 1 3 7 . 0 3 6 . . . ) . Then, according to Thirring's theorem [Th53] all radiative corrections to Compton scattering vanish in the low-energy limit and the Thomson cross section acquires its usual form.
204
2 Qtianl.mii theory of Yang Mills fields
For a ferinion with relative charge -p)u(p)
, one obtains in the same way
= -eQjuiph/Mp)
for
p2 = m 2 . (2.5.43)
Equation (2.5.43) expresses the charge universality in QED: the renormalized charge, i.e. the on shell coupling to the photon, is independent of the fermion species. Since r ^ , r ^ , and Fjj v v are the only primitively divergent vertex functions of QED, and since the renormalization conditions guarantee that the divergences can be compensated by the couuterterms, the multiplicative renormalization constants Z{ of fields and parameters, which become infinite in the limit A —> Ao or within dimensional regularization for 26 = 4 - D -> 0, are sufficient to absorb all divergences. Therefore, all renormalized Green functions and vertex functions are UV-finite. The one-loop results for the renormalization constants are calculated in Sect. 2.5.3.2.
2.5.1.7
R e n o r m a l i z a t i o n of a n o i i - a b e l i a n g a u g e t h e o r y
The renormalization of a non-abelian gauge theory with fermions proceeds along the same lines as that of QED. The Lagrangian C = -\{dtl lv
+
0(
- d
- 7
llt
)0+
- ua (6abn- *rbcd
g o f ^
iio
l
U
2
~
'oY^)
ltll)ub0
(2.5.44)
can be multiplicatjvely renormalized (cf. Sect. 2.5.4.2). Again, no gaugefixing counterterm is needed, and we choose Z^ = Z,\. Because of the gauge symmetry it is possible to have only one renormalized coupling constant. Therefore, the renormalization transformation reads [cf. (2.5.149) and (2.5.150)] = = Z,j(j,
tpo = m
= Zmm,
= Zuu\ = Z £,
u
=
u\ (2.5.45)
2.5 Kenormalizatiou of quantum lielcl theories
205
.Mid the Lagrangian becomes £ = -\{
ti
l
- d
- und (6at>dlt + S Z
-{Z2gZ\
- 6Zuua ua
- d -
f
lt)u
innij)
l)g ^ Tai>D
+ { z . z f z ^ -
-
(d
c
- gf
i p - {Z, ,Zm -
- Uza ^
-
g lin> rrai>
+ m)i> -1-
+
cjf^ ^ l)2
l+
*f
l)g2fal,c ' + {ZgZ\ 2Zu
- {ZgZf i
f
a,le d
\)gd,l Uabc ^ r'
>' e>'
- 1 )guafabcd'l
c
tlu
b
.
(2.5.4(1)
From this Lagrangian the complete Feynman rules including the counterterms follow. The renormalization constants are determined from the primitively divergent vertex functions. Because of the Ward identities it is sufficient to consider the gauge-boson propagator, the Faddeev Popov ghost propagator, the ferrnion propagator, and the fermion gauge-boson vertex. A particularly important non-abelian gauge theory is QCD. Since the fundamental degrees of freedom of QCD, quarks and gluons, are confined in the hadrons and do not exist as free particles, an on-shell renormalization does not make sense. Commonly used rcnormalization schemes are the minimal-subtraction scheme (MS scheme), where just the divergences are subtracted, or the modified minimal-subtraction scheme (MS scheme), where in addition remnants of the regularization procedure are subtracted. Both are defined after (2.5.63) and in Sects. 2.5.3.3 and 2.6.3. Here we present the so-called momentum-subtraction scheme (MOM scheme). In this scheme, the renormalized vertex functions are fixed at the renormalization 2 , which is situated at space-like momenta 2 > 0 in point 2 = p2 = order to avoid possible infrared divergences,
T ^
-*>
=
+ ,
i -
i
,
206
2 Qtianl.mii theory of Yang Mills fields
r
i5'°(Pi>7>2,P3)
(2.5.47)
= 5y7,,
These conditions determine the vector-boson-field renormalization constant the Faddeev Popov-ghost-field renormalization constant Zu, the the renorfermion mass and field renormalization constants Zm and Zand malization Zg of the gauge coupling constant. The one-loop results for the renormalization constants and the renormalized self-energies are derived in the following sections. Other renormalization schemes are discussed in Sect. 2.5.3.3. 2.5.1.8
R e n o r m a l i z a t i o n of a n o n - a b e l i a n g a u g e t h e o r y in t h e background-field m e t h o d
We consider the renormalization of the non-abelian gauge theory with fermions discussed in the previous section within the background-field method introduced in Sect. 2.4.4. The corresponding Lagrangian is given in (2.4.108) (2.4.113). It. involves fermion fields ip. background gauge fields (l, quantum gauge fields fl, and Faddeev Popov ghost fields u. In order to render all Green functions, including those involving quantum fields, finite all these fields have to be renormalized. We introduce the following renormalization constants , = zf a tfl
= z\ 2
f)o = zgg,
t,
,
=
Z
g = Zuu\ m 0 = Zmrn,
ug = ua, =
Z^.
(2.5.48)
We note that owing to the non-renorinalization of the gauge-fixing term the renormalization of the gauge-parameter is actually related to the normalization of the quantum gauge fields (Z^ = Z \). For the gauge-invariant effective action only background-field Green functions are needed. As far as these Green functions are concerned the renormalization of the quantum gauge fields and Faddeev Popov fields is irrelevant; and need not to be considered, i.e. we can choose Z,\ = Zu = . Because these fields appear in background-field Green functions only inside loops their renormalization constants cancel exactly between propagators and vertices. The renormalization of the gauge parameter £ does not cancel in Green functions, but enters only beyond the one-loop level.
2.5 Kenormalizatiou of quantum lielcl theories
207
Fig. 2.G The tadpole graph
Because explicit gauge invariauce is retained in the background-field method, the renormalization constants Z ^ and Z,, are related. The infinities appearing in the gauge-invariant effective action r{A,t/>,i/>} must be proportional to the gauge-invariant operators Fa,liu and -ij>(x)(\ ) - m)iji(x). This requires that the renormalization constants of the background-gauge field and the coupling constant are related by Z
= Z X'2.
(2.5.49)
This relation is analogous to (2.5.31) in QED and can also be derived from the Ward identities (2.4.121). 2.5.2
C a l c u l a t i o n of o n e - l o o p
corrections
Higher-order corrections to Green functions or S-matrix elements are obtained from Feynman diagrams with loops. Starting from the Feynman rules, all diagrams of a given order in perturbation theory, together with the corresponding counterterm diagrams, have to be generated and translated into analytical expressions. This can be done by hand or especially for more complicated problems—-using computer-algebraic programs (for a review see [Ha99] and references therein). The result contains integrals over products of propagators and coupling terms. In this section we describe the systematic evaluation of one-loop diagrams. 2.5.2.1
T h e t a d p o l e i n t e g r a l in d i m e n s i o n a l r e g u l a r i z a t i o n
Let us start with the treatment of the simplest one-loop diagram, the scalar tadpole diagram (Fig. 2.G). It, already shows typical features of one-loop integrals, but is free of the technical complications that occur for diagrams with more external legs. The tadpole diagram involves the following integral iy—-. A ) ( « i ) = X , i d A k -r2 17T - 17?/ + if
(2.5.50)
208
2 Qtianl.mii theory of Yang Mills fields
The factor {\ir2) 1 is added for convenience. According to power counting, this integral is quadratically divergent in four dimensions. Therefore, it has to l»e regularized. We use the method of dimensional regularization [tH72a, Bo72] which has the advantage of respecting Lorentz and gauge invariance. In this regularization scheme calculations are performed in D instead of four dimensions. Since the loop integrals converge for small enough D (D < 2 for /1o) the usual calculational rules for integrals, such as linearity, translational and rotational invariance, and the usual scaling can be used. The analytic structure of these integrals allows for an analytic continuation to arbitrary complex values of D. The UV divergences manifest themselves as poles at integer values of D. Changing the dimension of the integral also changes the dimension of Ao(m). This effect gets compensated by multiplying the integral with the factor (27r//)'1_/:>, where /t has the dimension of a mass. A precise definition of dimensional regularization can be found in Ref. [Co84]. Especially, it implies that integrals that do not depend on any scale vanish in dimensional regularization,
J d D k ( k ? ) Q = 0,
(2.5.51)
although being formally infinite for all a and integer D. For later applications we consider /In with an arbitrary power of the propagator and define ( ^ 4 1V )
ux~D =
i7r
dDk 1 /J T ^ u- n h1 j n l (¿tt) \k - m* -1- ie)r r
^
L
(2.5.52)
1 The momentum-space integration over kft — (k°, kl,..., k ), k2 = (k())J k', contains a k° integration along the real axis from - o o to +oo. The integrand has poles in k° at
k° = ±V^
+ m2-ie
V*-2 + m2
= ±( V
. f , -I- 0(e2)) . 7 2%/k2 + m 2
These singularities lie in the second and fourth quadrant of the complex k° plane (cf. the crosses in Fig. 2.7). Therefore, the integral over the closed contour in Fig. 2.7 vanishes owing to Cauchy's theorem. For D < 2r, where
2.5 Kenormalizatiou of quantum l i e l c l theories
209
In» k°
Rck°
Fig. 2.7 Singularities and integration contour for the tadpole integral in the k" plane.
A1,)'1 is convergent, the integrand vanishes fast enough so that the contribution of the arcs drops out for |A.°| —» oo. Therefore, we can replace the integral over the real axis by one over the imaginary axis 4 V )
i* 2
J-loo
K* 0 ) 2 - k 2 - m 2 + fe]r
J
Substituting k° = ifc®. and k = kp, gives Ar),
>
(2?T/j)
-
. f°°
J
d
f
)l0
%
d"-'kE
(
2
j
-
j
3
)
where ¿E = (fcj^krc) is a euclidean vector, and the euclidean volume element. This procedure, the WtcA: rotation, transforms integrals over Minkowski space to integrals over euclidean space. It is possible to use the rotational symmetry of the euclidean integral to decompose it into a radial and an angular part by introducing spherical coordinates d f t o = ^ ( ¿ i ) ( i ; - 2 ) / 2 dkl (2.5.54) c\vkE = k{?-\\kK
210
2 Qtianl.mii theory of Yang Mills fields
The integral over the surface of the ¿^-dimensional unit sphere yields 2nn'2 I Here V is Euler's
Gamma
function r r (z) = / Jo
tz~1e~tdt,
which has the basic properties r ( z + i ) = zr(z),
r(n + i ) = n ! ,
r(i) =
%A
:
,
(2.5.56)
and first-order poles for z E {(), —1, - 2 , . . . } . Putting everything together, we have l(r)
_ (2,/x)«-» 2 i t D i 2 1
»0
-
„2
r ( D / 2 )
= (47r/z 2 )^- p )/ 2
F(r
~ °/ r(r)
p ( ~ i r ( 4 ) ( ^ -
2 y0 2 )
( 4 +
m
2
2 _
) /
2
fejr
( ~ l ) r ( m 2 - i £ )<°/2-r)
(2.5.57)
for D < 2r, where we used the fact that the integral in the first, line is a representation of Euler's Beta function B(x,y) = r(x)V(y)/T(x+y). T h e Gamma function r(?- - D/2) in (2.5.57) contains poles for r - D/2 = 0 , - 1 , - 2 , . . . , which can be identified with the UV singularities. The analyticity of Aq as a function of D allows to calculate the integral in a region of the complex D plane where it. is convergent and to define an analytic continuation to other dimensions. All UV singularities show up as poles in D. In order to work this out in more detail for the physical dimension four, we write D = 4-26 and obtain = (4V)-5
r ( r
r
(
y ~ V l )
r
K
- icf-r~6) •
(2.5.58)
The original tadpole integral is obtained by putting r = 1,
Ao{m) = - f ^ r )
r(—1 + <5)m2.
(2.5.59)
2.
enormali atiou of
I localise 6 is small in the vicinity of 6
log )
exp
1
= - - 7
r
uantum lielcl theories
4, we expand 0{62 ,
6 log 2
Euler constant 7
,
211
.
... ,
2. .
and obtain 2. . 1 with -
log
7
log4ir The term 1 shows the singularity for 6 {
- 4 4
m
. s
2. . 2 typical for dimensional regular i ation. It clearly , i.e. 4. s a conse uence we have
= -2m2
+ 0{
- 4 .
2. .
In the minimal-su traction sc eme the terms with 1/5, in the log4 r are subtracted leaving a finite exsc eme the terms with 1 pression for the renormali ed uantities.
2. .2.2
calar F e y n m a n integrals
The tadpole integral or scalar one-point, integral is the simplest of the scalar n-point integrals. These correspond to scalar one-loop diagrams with n external legs oined to the loop via three-vert ices, schematically depicted in Fig. 2. . These integrals are orent -invariant and depend 11 the scalar products of the external momenta and on the masses of the inner propagators. The two-point, function, the scalar self-energy n0(pt B
2
mi,m2) V = ——,—
17TZ
/
integral, is defined as d
[
2
- m [(J + p)2 - rn221
where we have suppressed the ie terms. s discussed in logarithmically divergent in four dimensions.
2. . 4
ect. 2. .1.2,
o is
212
2 Qtianl.mii theory of Yang Mills fields
k + p\ + p2
Fig. 2.8 Conventions for the n-point integral
The higher scalar n-point functions are finite in four dimensions. The triangle integral is defined as Co (p'f, p\, Pi; ™ l , m2, m3) _L ¡7T2 J
=
I [k* - mf][(*; + p, )2 - m2][(jfc + p, + p 2 ) 2 -
mj)' (2.5.65)
where the incoming momenta are pi,
and p3 = —p\ — p2-
The box integral reads
A)(PI,P2»P3>/M;(P2+P3) ,(PI+P2) ;M 1 ,7N 2 ,M3,M 4 ) = [ d '/c— , - mi ./ [A:'* —
1 + p i ) 2 _ m 2][ ( A .
' [(£ + pi + pi + p 3 ) 2 - m | ] '
+ p i + p2)2
_
m2]
(2.5.66)
The pentagon, hexagon, . . . integrals (with five, six, . . . propagators) are defined in a similar way. In four dimensions, they can he algebraically reduced to linear combinations of box integrals Do by using the linear dependence of more than four momenta [Me65, vN84]. Therefore, only Ao, Bo, Co, and Do have to be calculated directly.
2.5 Kenormalizatiou of quantum lielcl theories
213
The momentum-space integrals for Bo, Co, and Do can be reduced to those of the tadpole integral (2.5.52) using the Feynman parametrization of the integrand _ f ( a t + a-i + . . . + o„)
1 D ? • • • Dn"
r(a,)r(a2)---r(a„) ¿(1
/ dxi • • • / Jo Jo
r n ) x »>-«
da;,, [xlD,+...
+
xnDnF+-+a" (2.5.G7)
In the case of the Bo function we apply this formula to the product of two propagators, i rl dx — — = / Jo [(1 - x)Di + XD2]2 D\D2
(2.5.G8)
to obtain \A-D r 1 Bo = ^ f t — I d , I d^Jk I7TZ
L
r 2 ~2 2 2 x I k + 2 xkp + x(p - m j ) - { I - x)m i .
(2.5.69)
Using translational invariance of the momentum-space integral (k —> k—xp) gives a tadpole integral (2.5.52) with r = 2 and thus using (2.5.58)
=
>7T2
^ J [k2 - x2p2 + x{p2 - m22) - (1 -
Jo
= (4 ir/L2)6 r ( i ) f
d
x [ x V - x(p2 - m\ + m'f) + m\
x)m2f - 6
(2.5.70) ! B0(p'\mU7n2) Jo
= A
P-
(2.5.71)
2
2
tianl.mii theory of
ang
ills fields
where we have restored the ie term via m 2 rnf ie and used from 2. . 2 . The terms in the first line reflect the logarithmic singularity of the momentum-space integral and lead to (D -
)
-2
(D
- 4 .
2. . 2
The parameter integral is elementary and yields .
2 - log 11
mimo I1
1 , log
1
2 2
]r
mi
11 1 - - V 1
where r and 1 r are the solutions of the 2
r2
-m]-ml
\r
1 - r
)
, log r,
2. .
uadratic e uation
er 1
i2
namely p2
i2
i2 - ie
p2
m'i - Tii2 + ie
2
vi^m^
2 11 2 2. . esgn 1 . The For real, negative -, the ie prescription yields I mi two-point, function o is an analytic function of p with a cut from the ra a 2 2 to 00. bove threshold it gets for real physical threshold p 2 p 2 an imaginary part which is related to the decay of a particle with mass f p- into two particles with masses 11 and m 2 . The triangle and box integrals can be calculated in a similar way. For illustration we present the Feynman parameter integral for Co Co pi, pi, pI ; rn 1. m 2 , m -
I
d
' d \p2x2 + p2 l o mi - m -
2
+ p 2 - p 2 - p\)x m - m2
pi - p y
2. . m
- ie
.
2.
enormali atiou of
uantum l i e l c l theories
Twofold integration yields an expression which contains Enler s \ (z) e also called Spence unction)
i2{z) = -
\\t o
g 1
Zt)
,
arg l
For the explicit expressions of Co and c 1, e .
2. .2.
z)\
n.
21
dilogarithm
2. .
o we refer to the literature tH
,
irac algebra and tensor integrals
The consideration of scalar integrals is sufficient for calculations in scalar field theories. Gauge theories contain also gauge bosons and fermions. Therefore, techni ues for the treatment of the irac algebra and tensor integrals in D dimensions are needed when dimensional regulari ation is used cf. Ref. Co 4 . For positive integer D dimensions the metric tensor has the following form r
diag l,-l,...,-l v s -
and obeys for arbitrary D > = D. The Dirac algebra can be extended to D dimensions. The spinor representation of the pseudo-orthogonal group has anticommutation relations which are formally the same as in four dimensions ,
2 T 1,
ince (i runs from 1,
to D
,i
,1,..., D - 1 .
2. .
1 we have ,i
2-
,
etc.
2. . 1
The trace of the unit matrix can be chosen as in the four-dimensional case, TV 1 4 .
21
2
tianl.mii theory of
ang
ills fields
roblems arise with the definition of the matrix in D dimensions. ince this is related to chiral anomalies, and since is only needed in theories with parity brea ing, we postpone this topic t.o ect. 2. .2.2. The propagators for gauge bosons and fermions as well as the couplings that contain derivatives generate tensors of the integration momenta in the numerators of the Feynman integrals. systematic method to reduce these tensor integrals to the set of basic scalar integrals lo, Bo, Co, and Do was wor ed out in Refs. a , 1 , e . e illustrate this techni ue here for the case of the self-energy tensor integrals. The vector two-point integral reads
f
p;m,,m2 iirz
.^V1?^
m
2- - 2
J [k* - rn\\[{k + pY -
\
and the tensor integral n m i
(>
(27T/ ) -0 rn,,m) = - r ^ *
f 2
k»k p
d Dk p 2
m2
2. .
These integrals are orent -covariant. Therefore, they can be built up from a complete set of orent tensors and invariant scalar coefficient functions. The orent tensors are constructed from the external momenta and the -dimensional metric tensor g'1 , =p'' i, B> ' = g> 'Boo+ p'YBn.
2. . 4
These linear e uations can be inverted = p
\p,t 'l
and the expressions 11 the right-hand side directly reduced to scalar integrals as follows. riting 2kp
p)2 - ml] - [k2 - m]] - p2 + m\ -
m\
2.5
Henormali/ation of quantum field theories
217
yields _ (27r/t)''- p 1
i^ 2
~
1
f
2pk (l°k [k2 - m?][(ib + v)2 -
V i
ml]
i) ~ M ™ * ) - ( p 2 + m f - m | ) j 5 0 ( p 2 ; "M , m 2 ) ) ,
=
(2.5.8«) and in a similar way i) (/1°(T"'2)
Boo =
+
2m2£o(p2;mi,m2)
+ (p 2 + m 2 - m \ ) B \ (p 2 ; m i , 7 n 2 ) ) , Bn =
1 2 { D
1)/)2((^
- 2)/lo(m,) - 2m2JB0(p2;m,)m2)
- D(p2 + m\ - m ^ ) S i ( p 2 ; m i , m 2 ) V
(2.5.87)
These equations, inserted in (2.5.84), give finally v 1' i B'1 =
•
\
(Mmi)-Mm)-(p2+m?-mi)£0(P2;rn\,m2)J,
A"" = ^ [ - ¡ y2 — [ (y»» -2 ^ r ) 2 ( ^ o ( m 2 ) + 2 m 2 £ 0 ( p 2 ; m , , m 2 ) 2 + (p + m - m )Bx (p ; m , , m 2 ) ) (2.5.88) +
^ T " ( ^ o ( m 2 ) - (p 2 + m 2 - 7 n 2 ) i ? i ( p 2 ; m i , m 2 ) ) ] .
The treatment of the triangle and box diagrams is basically the same. The corresponding results can be found in [Pa79, De93]. In this way every one-loop diagram is expressible by the scalar integrals Ao,Bo,C'o, and Do and thus finally by elementary functions and dilogaritluns. From the inversion of the linear equations negative powers of determinants of the scalar products of the external momenta (Grain determinants) occur. For the two-point functions considered above these are the factors l / p 2 . For those values of the variables where these determinants vanish, i.e. p 2 = 0 for the two-point case, the linear equations cannot be inverted. However, in this case the linear equations yield relations between
21
2
tianl.mii theory of
ang
ills fields
the coefficient functions corresponding to tensor integrals with fewer indices. These can be solved for these tensor integrals t . For instance for p , the second e uation 2. . determines \ in terms of o and q. Thus, for vanishing determinants the reduction to scalar integrals can be performed by investigating the tensor integrals with one additional orent index, i.e. ,, for the reduction of tl. In this case the final results become actually simpler. The evaluation of the irac and tensor structures of the integrals and the reduction to scalar basic integrals follows clear algebraic rules. They can be implemented in algebraic computer programs Ha which then perform these calculations automatically.
2. .2.4
R e m a r s o n t h e c a l c u l a t i o n of h i g h e r - o r d e r c o r r e c t i o n s
The preceding pages show that the calculation of one-loop Feynman diagrams is based on a well-wor ed-out procedure. This includes the generation of the diagrams, their reduction to standard integrals, and their numerical evaluation. In contrast to this, the situation in the two-loop case is much more involved. Complications arise because of the large number of diagrams which ma es a treatment by hand almost impossible. oreover, with the exception of self-energy diagrams, no general reduction procedure to scalar two-loop integrals with arbitrary masses is available, and the scalar integrals are in general not expressible in terms of well- nown functions of mathematical physics. ome results on the reduction of two-loop tensor integrals can be found in Ref. e 4, Ta , Gh . umerical methods for the evaluation of massive two-loop two-point, three-point, and four-point integrals have been given in Refs. a , a a, a b , C a, Fr , and s , respectively. Two-loop calculations exist for many problems in massless theories li e C . However, in the electrowea case we only have results for special problems such as the calculation of the anomalous magnetic moments of leptons, contributions to decay, or certain limits small momenta, large momenta, etc. . In view of this, we restrict ourselves to some remar s on the calculation of higher-order contributions and their renormali ation. The leading parts of the higher-order corrections can in many cases be obtained from the renormali ation-group e uation cf. ect. 2.G . The many Feynman diagrams can be generated automatically using algebraic computer programs including the counterterm diagrams. Renormali ation
2.5
Renormalization of quant um field theories
21!)
has to be performed in an iterative way. In the ¿-loop case it is assumed that the theory is renormalized up to L— 1 loops. The countertenu diagrams up to ¿—1 loops are sufficient to cancel all non-local divergent terms. This allows to compensate the remaining local ones in a way similar to the one-loop case, i.e. by the terms that have been generated by the ordinary reuormalization transformation.
2.5.3
O n e - l o o p r e u o r m a l i z a t i o n of g a u g e theories
As an application and illustration of the renormalization program discussed in Sect. 2.5.1 we perform the explicit renormalization of an abelian gauge t heory, QED, and of a non-abelian gauge theory with the gauge group S U (yV) at the one-loop level. Because of the Ward identities, in QED all renormalizn tion constants can be determined from the photon and electron propagators (cf. Sect. 2.5.1.6). Consequently, we calculate the one-loop corrections to these propagators for QED, using the methods of Sect. 2.5.2, and perform their oil-shell renormalization. The renormalization of the non-abelian gauge theory, which for instance applies to QCD, proceeds along similar lines. It is presented in Sect. 2.5.3.3.
2.5.3.1
QED self-energies
We decompose the photon-photon vertex function and the pliolon self-energy
into its free part
E'1,,, (2.5.8!))
The photon self-energy is calculated from the one-loop diagram
Using the Feynman rules of QED this translates to
220
2 Qtianl.mii theory of Yang Mills fields
Using (A.1.30) the Dirac trace gives
T r ^ t f + mfr„(Jf + ff + m ) ] = = 4 ^2kltkr
+ k,Lqu +
+ qk - m 2 ) J .
Writing 2(A;2 + ^ - m 2 ) = [fc2 - m 2 ] + [(A: + r/)2 - m 2 ] -
2
»
= -4i e V -
O
/ < 0 > ( (A:2 - m 2 ) + ((A;+q) 2 - m 2 ) - q2J
2k/tku+A^ry,,+ry,, A;„ X
[A;2 - m2][{k + ,7)2 - m 2 ] 2
= ¿ 2 ^ „ - W y ^ - f S ^ - 2f-(A0{m)
+ A0(m)
where the arguments of the B functions are (q2;m,m). formulas for the tensor integrals, (2.5.84), we find
= ¿ {
0«" -
+ Mil
q
-f)
(W™
-
(2/ioo + 2q2(B] i+B^
+
J
- r/ 2 Z? 0 )) ,
Using the reduction
B o
+ ^Bo-
) ,!„)}.
(2.5.90)
With help of (2.5.88) this can be expressed in terms of the basic scalar integrals Aq and Z?o,
In agreement with (2.4.100) the one-loop photon self-energy is purely transverse. Introducing the fine-structure using the relation
B0(0;rn,m) = ^
M
constant
= ( l -
a = e 2 ¡ \ t x , putting D = 4-2
5
) ^ ,
(2.5.92)
2.
Rciiormali ution of
which can be read off from 2. .
Sn )
l , 2. . 2 , and 2. .
S)q2 (q2-,m,m)
= ^ 3 - r ^
+ 2m2^ (q2\m,Tn) = ^
{q2 {q2-, ~ j
uantum lield theories
m, m)
+
-
i
2m2
, we obtain
+ ; n, n o(q 2 ; m, n) -
o ; m, m
(6),
2
2. .
for the transverse and longitudinal self-energies defined as in 2. . . I 2 2 shows that indeed ^(q ) q~ for small q in accordance with 2. . . conse uence of gauge invariance no photon mass renormali ation is nrc The electron self-energ E ; is the higher-order contribution to the responding vertex function and is decomposed into a vector and a sc; part,
r
p,-p
f-m
E
= (t-m)
+
p E
p2
m
2
.
2. .
t the one-loop order it. results from the Fey 11 man diagram
Introducing a small photon mass in order to regulari e possible infra III divergences and ta ing the photon propagator in the Feynman gau gives 1
L 1
(P}
~lC''
J
2
- m
- p 2 - X 2]
T h e introduction of the photon mass brea s gauge invariance. However, the gnu invariance-brea ing terms are proportional to and thus vanish for physical finite ii tities in the limit .
222
2 Qtianl.mii theory of Yang Mills fields
Working out the algebra and the integration with the methods of Sect. 2.5.2 and neglecting terms proportional to A2 (which vanish for A —> 0, since the IR divergence is only logarithmic) yields A0(7n) - (p2 4- m2)B0{p2; p2
m, A)
2 m, A) j},, f m ( 4 - 25)B0(p2; ;m,A)
(2.5.96)
and for the scalar and vector part of the self-energy v,«, 2| E v(p ) = S
a
/lp(wi) - (?r + m2)B0{p2; - <5) -2
n
s(P2) = - £ (
n
4
m, A)
- 2<5)/?o(p2;m, A).
Next, we verify the Ward identity vertex
r¿+*{q,p,p) =
' (2.5.97)
(2.4.102) between the photon-electron
+ cA;i(flf ,p,p)
(2.5.98)
and the fermion self-energy by explicit calculation. T h e vertex correction given by the Feynman diagram
and reads
A„() = -ieV ,lVhy,i'J ~ H
D
[ J
d pk J^yj
K --pn r+ ip — m T » pi + I 7f — m7* k* TT-vi— Az
is
2.
ultiplying by - p indeed yields
1
Ronoruiali ation of
- p ' and writing -p -
- Pll)h,l(
(-p
=
-
i
, -p
-
p+ fi
2
-
m
22
T
d fc
J
,
1 - m E
uantum f i « 1 1 theories
T. —p + j/i — m.
k2 -
2
;
p ,
2. .1
i.e. the one-loop contribution to 2.4.1 2 .
2. . .2
Renormali ation constants of
E
The E agrangian 2. .2 contains the counterterms which are needed for the renormali ation procedure. In one-loop order, we obtain for the it-normali ed photon and electron vertex functions (Z^ = Z )
rf
p, -p
r
- SZmrn
p, - p
sing the decompositions 2. . 2. this can be written as Tr,T n 2 = r
R,v
rls(q2)
2
1
l2 + X
and 2. . 4
2
- m).
together with
2 i
2. .1 1 2. .
and
2. .1 2
Z
)
zt(n2)-szm-sz,l,.
= -i +
The photon-field renormalization ing 2. .1 2 gives q2 +
+ 6Z+
constant
^{q2)+q2SZ )
is obtained from 2. .
2. .1 . Insert-
l<72=0
or
szA = - v>-4(
2. .1 4 o
224
2 Qtianl.mii theory of Yang Mills fields
Taking the result (2.5.93) for the photon self-energy yields s z
t =
» 0 - W ( 0 ; m, m ) + 2m 2 7^(0; m, m) ^ 7T 3 — 2(5
where B'0(q2;m,m)
— d/dq2Bo{q2',m,in).
B„(0;m,m) -
(^jjj-)
(2.5.105)
Using (2.5.70) we obtain
r(i),
and thus for the photon-field renormalization constant
" " 5
( ^ ) "
r
<
{
+
> — 5
' « à )
+
O W P.5.207)
The renormalized photon self-energy is finite for (5 —> 0, £R,T(<72) = £
[(q2+2m2)(B0(q2)-B0(0))-2q2m2B'0(Q)]
, (2.5.108)
and behaves like {q2)2 for small q2. The conditions (2.5.40) and (2.5.41) fix the electron muss and field renormalization in the on-shell scheme. Inserting (2.5.103) gives SZm = — m
= S c ( m 2 ) + Xt(m2),
(2.5.109)
SZy = — S y ( m 2 ) - 2m 2 ( s ? ( m 2 ) + S ' ^ ( w 2 ) ) .
(2.5.110)
The explicit result for the electron mass renormalization constant is obtained by inserting (2.5.97) in (2.5.109) and using (2.5.59), (2.5.70), and (2.5.G2), (5m
in
a
=
(
47T \
«
/lo(m)
,
or>
,
2
(1 — A ) — ^ + 2 t f o ( m 2 ; m , 0 )
f4n,,2\S
,
3-26
2.5 Renormalization of quantum lielcl t heories
225
The dec Iron-field renormalization constant is both UV- and Ill-divergent. in tin* 011-shell scheme. Equation (2.5.110) gives Aü{m)
' s)
m'
-
+ 4m2B'0(m2;
211
m, A) j . (2.5.112)
The UV-singular term simplifies to a .
A0(m) - 213, in-
.
ct (Airp?\S ~47T )
=
1 i ^ 2 S -
/[X m
The Ill-singular part :c - x 7r \ m¿
7T
J
l+<5
J0
can be evaluated in two ways: i) keep the small photon mass A, (0 < A2 -C m 2 ) and let 6 -* 0: ,.2 2
-m B'0=n
n Jo
[\\x
x2 + % ( l - x ) A2
a f =
~2tt
\
m
2
+ 0(AZ),
+ 2
(2.5.113)
ii) regularize the IR-diyergent integral dimensionally (A = 0, 6 = <5ik < 0):
=
+
0(5,R).
+
(2.5.114)
This allows to compare the two schemes 9
1
logA «
r+
9
log 47T/X .
OlR Using the photon-mass regularizaron we obtain .2 =
- ¿
(
A + , o g
^
+ 2
, (
I o g
A2 ^
+
2
) )
+
A
'
2
)
(2
-51
15)
with A from (2.5.62). This completes the determination of the one-loop renormalization constants Zx¡i, Zm, and Z,\ = Z~~ of QED in the on-shell scheme. The IR singularities are treated in Sect. 2.8.2.
22
2
uantum theory of
ang
..........
Fig. 2.
2. . .
ills fields
d
Gra hs contributing to the gauge-boson self-energy in one-loop approximation
ne-loop renormali ation of a non-abelian gauge theory
e consider the one-loop renormalization of a non-abelian gunge, theor with gauge group V and p multiplets of fermions in the fundamental representation. The masses of the fermions of multiple , are denoted by m . For this corresponds to C with p uar flavours. For the renormali ation we need the one-loop contributions to the gauge-boson and fermion propagators and to the gauge-boson fermion vertex. e do not discuss the field renormali ation of the Faddeev- opov ghost, which can be determined from the ghost -gauge-boson vertex, because this is not relevant for the calculation of physical uantities at the one-loop level. e present the calculations for the gauge-boson self-energ in some detail, for the other uantities we list only the results. e use the Feynman gauge in this section. The graphs contributing to the gauge-boson self-energy in one-loop approximation are given in Fig. 2. . e start by calculating the vacuum polarization induced by fermion pairs graph a of Fig. 2. . This differs from the E photon self-energy only by a group-theoretical factor the colour factor in C of the fermions Tp 1 2 for cf. 1.2. 2 and the sum over the p fermion multiplets. The results are s~, ,ab
,
2
a,
pTp6
b
I
2
, 2
x
.
2.5
Itenoruuüization of quantum Held theories
2m E
M//)('?2)
227
B0(
=
( 2
-511(I)
This contribution to the gauge-boson self-energy is again transverse. It is diagonal in the group indices and gets equivalent contributions from all ferniion multiplets. We have introduced the shorthand as = g2/'\tt, which is the strong fine-structure, constant in the case of QG'D. In many cases it is sufficient to treat the fermions as massless. Using (2.5.70), we find r2(i-rf) •2 2 £o(g ;0,0)= ( z j r ^ ) r(i) r ( 2 - 26) ' and (2.5.116) simplifies to vA,ab
(n2\ -
a
'X°bKr
= -6abNFTF^-
r
( a + log
7r
r2(l - ¿ )
2 (_JV_Y
«> \
!! —
. + l ) + 0{6). -
te
1-6
(2.5.1J7)
3/
The other graphs must be calculated in a similar manner. T h e vacuum polarization by gauge-boson-pair creation (AA) (Fig. 2.9b) is particularly interesting. It results from the self-interaction of coloured gauge bosons and is not present in an abelian gauge theory with neutral gauge bosons such as QED. In Feynman gauge, the contribution of this graph to the self-energy is v*A,ab W '
/ _2 \ ^ S
a
s tab f ~Ca\ 4 [ — J '
xF(6)
1
2(
V Te)
r2(l - 6 ) 19-125 r(2-2¿)
3-2(5 ' <5
z t u *
- % r
( t ) <2
( ^ c )
This result was obtained with the help of (1.2.56), £ f c d a f a l b = CA6,lb, C'A = 3 for SU(3).
with
The gauge-boson contribution (2.5.118) is not transverse. The contribution containing the quartic coupling of the gauge bosons (Fig. 2.9d) gives zero in
228
2 Quantum theory of Yung Mills (¡elds
dimensional regularization and therefore does not compensate the longitudinal part. T h e transversality of the gauge-boson self-energy which is guaranteed by the Ward identity (2.4.68) is restored by the Faddeev-Popov-ghost contribution (uii) (Fig. 2.9c) which leads to
vyMI-
{a
2 , _ <** 5 «l> ( /
4ir
{
xr(6)
4
1
J
f
{-q
r2(l-5)
2
V
-ie)
1
(2.5.119)
r ( 2 - 25) 3 - 25'
-5) S l W « ) - ^
{ — j
9
r ( 2 ~ - 2d")'
l ^ k j
Adding (2.5.118) and (2.5.119) indeed gives a vanishing longitudinal gaugeboson self-energy,
xT{6)
r 2 ( l - (5) 5 - 35 r ( 2 - 25) 3 - 25' (2.5.120)
Collecting all contributions, we finally find 2
" — 5) 25)
3-25
3 - 25
= 0 for the self-energy
of a non-abelian
(2.5.121) gauge
boson.
The renormalized gauge-boson self-energy is obtained by adding the counterterrn to E-p. The gauge-boson-field renormalization constant has to be fixed by a suitable renormalization condition. In contrast to the photon, which exists as a real physical particle, the non-abelian gauge bosons of QCD, the gluons, are screened inside hadrons. This renders an on-shell renormalization of QCD impossible. Instead, the following schemes arc used.
2.
enormali atiou of
In the minimal-sub traction schema, sists only of the V-singular term, cy
a
uantum lielcl theories
, the renormali at.ion constant con-
1
S
4
22
2. .122
i
The modified minimal-subtraction ing
scheme differs from this by replac- -
d
7
-f log 4tt.
third possibility is the momentum subtractioji scheme where the z . The renormali ation is performed for space-li e momenta q~ corresponding result for the gauge-boson renormali ation constant reads
r 2 i - 6) a
ii
{'m^')
2 )
- 26
The fermion self-energ is calculated as in the group-theoretical factor T
CFSi
2. .12
- 26
with
C
-
E , the only difference being
for
.
2. .124
«.
For inassless ferinions in the Feynman gauge, the result, reads
g2, ,
gcF lE
V
2. .12
,
.
For the fermion-field renormali ation constant this implies 1
xy
cy
,
r
a
s
r
(
i
.2 2
y
2 r C r i- - y
- i
2. .12
2
2
tianl.mii theory of
ang
ills fields
in the different schemes. Finally, we need the renormali ation of the coupling constant g. It is obtained from the vertex corrections, e.g. the fermion gaugebosun vertex. There are two diagrams, the first one is similar to that of E , the second one contains the non-abelian triple gauge-boson couplings
and within dimensional reguFor massless on-shell fermions {p2 = p2 lari ation the following expressions for the renormali ation of the fermiongauge-boson vertex are obtained, t
i
( 6Zn + -6Za
+ hZ
= - g (C
+ C F)
I
.
( + -SZ +SZ^
-g
x T(S)
C
p
2
-
finite terms,
2. .12
where we have not explicitly given the finite terms in the second expression. From this and the results 2. .122 , 2. .12 , and 2. .12 one finds S 1 1 fTf V 11C ' 4
sz,
r., r
ni 2
n t x I ,
-I- finite terms
r2 i-a r 2
2T)
- pTp
2. .12
for the renormali ation of the coupling constant. ote that the finite terms are relevant when calculating physical observables. This completes the determination of the renormali ation constants in the physical sector of a non-abelian gauge theory. They are sufficient to obtain
2.
enormali atiou of
uantum lielcl theories
2 1
V-finite results in one-loop order. The calculation of one-loop corrections to observables in C is discussed in ect. . . The arbitrariness of the renormali ation prescription does not lead to physically different results. This can be formulated using the renormali ationgroup e uation, which is investigated in ect. 2. i.
2. .4
etch of t h e proof of renormali ability of gauge theories
In the previous sections we used E and C in one-loop approximation as examples to show the essential steps of the perturbative evaluation of a uanti ed gauge-field theory 1.
gauge-invariant, e.g. dimensional, regulari ation;
2.
subtraction of divergences by introducing counterterms into the agrangian e uivalent to multiplicative renormali ation of fields and parameters in the examples in ect. 2. . ;
.
ad ustment of the renormali ed finite parameters of the theory to their measured values renormali ation conditions .
In this section, we discuss the general proof of the renormali ability of gauge theories. e do not give a complete treatment but rather illustrate the main ingredients and, in particular, the role of gauge invariance. ince in non-abelian gauge theories with mass less gauge bosons the S matrix is not well-defined owing to severe III divergences, our considerations are formal. They can, however, be directly transferred to spontaneously bro en gauge theories where, if all gauge bosons become massive, this problem is absent, and our arguments become ustified. y introducing gauge-lixing terms and associated ghost terms, gauge theories are defined in such a way that they can be evaluated in perturbation theory. s the gauge couplings are dimension less, gauge theories are renormali able according to power counting cf. ect. 2. .1. in suitably chosen renormali able gauges. The generating functional T of vertex functions is well-defined in D 4 26 dimensions, and the addition of a finite number of counterterms to the agrangian results in a renormali ed vertex functional . which remains finite for
2 2
2
tianl.mii theory of
ang
ills fields
However, gauge theories involve unphysical negative-norm states and arbitrary gauge parameters. ne has to show that the unphysical states do not contribute to the physical S matrix, and that the latter is independent of the gauge parameters, i.e. one must prove that the physical S matrix is unitary and gauge-independent. To this end, one needs the gauge invariance, or more precisely the R invariance, of the theory and the associated ard identities. Therefore, one has to show that the gauge theory can be renormali ed in such a way that a renormali ed R invariance still holds. This is proven by induction in the number of loops in the perturbative expansion. The expansion in powers of the loop parameter, i.e. the lanc constant i, h;us the advantage that it preserves any symmetry of the agrangian order by order, because the loop parameter multiplies the whole agrangian cf. 2.2. . n the other hand, splitting up the agrangian into a free -independent part and an interaction part is not gauge-invariant as the non-abelian gauge transformations explicitly contain the coupling constant cf. 2. . 2 .
2. .4.1
Recursive construction of gauge-invariant counterterms
e consider a pure gauge theory without matter fields and assume that it is regulari ed in such a way that the R invariance is not bro en. practical regulari ation scheme with these properties is dimensional regulari ation. Then, the theory is well-defined in dimensions, and the ee identities 2.4. and 2.4. hold in the regulari ed theory. ccording to power counting cf. ect. 2. .1. the theory is renormali able, ami the counterterms are a local polynomial of degree four in the fields ect. 2. .1. . e determine the counterterms in the scheme, i.e. the counterterms are chosen in such a way that they cancel exactly the divergences of the loop integrals but contain no finite contributions, apart from those re uired by gauge invariance. This is sufficient to prove renormali ability. ther renormali ation schemes can be obtained by adding appropriate finite counterterms that respect the symmetry of the renormali ed action. e restrict ourselves to linear gauges of the form 2.4. 4 . e have to show that by adding a well-defined set of counterterms to the agrangian, the resulting generating functional is finite for D 4 and satisfies the ee identities.
2.5 Kenormalizatiou of quantum lielcl theories
233
To this end, we follow lief. [Zi75] and consider the loop expansion of the hare vertex Junctional [without gauge fixing terms (2.4.70)] oo f
o = Efo/'). L=0
(2.5.129)
where L is the number of loops. The lowest-order contribution consists of the classical action, the ghost contribution, and the terms involving the sources K and K u of the BRS-transformed gauge and ghost fields, respectively [cf. (2.4.17), (2.4.53), (2.4.55) and (2.4.76)]
f<0)
S„
=
= j
d*x
[ £ g a „ S c ( A ) + (K«<" -
ü^'^sA*
-
I
The linear relation (2.4.78) implies that also in higher orders the vertex functional depends on ü and K only through the combination K'1'1' - ü
r = J d'x
r t *1 2
STj
sr2 a
sr, a
sr2
5/1« SK >i' " 6u óK£ J
(2.5.131)
for arbitrary functionals T) and r 2 , the non-linear Lee identities (2.4.77) can be written as f o * To = 0.
(2.5.132)
and the contribution of order L reads ]£
f ^ * f ^ = 0.
(2.5.133)
p+q=L
We seek counterterms so that the renormalized" vertex functional f ZZo satisfies Y ^ f ( p ) * f1;
=
(2.5.134)
p+q=L 11
We do not use an index R to denote the renormalized functional in litis section in order to avoid too many indices.
2 4
2
tianl.mii theory of
ang
ills fields
ecause the tree-level action is ll -invariant and does not need to be renormali ed f f , both relations are fulfilled in the tree-graph approximation o
o
f
r,
f
f
et us first consider the one-loop level. a for D
4 divergent part, f
.
2. .1
e split f
l
which satisfy the
identities in the one-loop approximation separately. and ff,
o into the
into a finite part and
ee identities
2. .1
ubstituting
ee
2. .12
we find that the one-loop
divergent part satisfies o
f
f
.
i v
2. .1
In order to obtain a finite renormali ed one-loop vertex functional we add counterterms to the action that cancel the one-loop divergences. This results in the one-loop renormali ed action o- f ;
iv.
2. .1
E uation 2. .1 guarantees that S satisfies the ee identity to one-loop order, i.e. S C r . However, the recursive construction wor s only if the one-loop renormali ed action satisfies the ee identity i exactly. This can be achieved by adding appropriate higher-order counterterms , which do not modify the functional i at one-loop order, i.e. we define ,
o-fg
,.
2. .1
Here is a local polynomial in the fields of degree four of order i 2 and determined such that i i exactly. The explicit construction of i is performed in ect. 2. .4.2. Then, i yields a finite one-loop vertex functional F that satisfies the ee identities. e now proceed by induction. e assume that we have constructed a renormali ed action that satisfies the ee identities and generates a finite perturbation series up to order n. Then, the corresponding vertex functional r satisfies the ee identities 2. .1 2 as well and has a loop expansion similar to 2. .12 . The contributions f in this expansion are finite for n 1 and the divergences of f originate only from overall divergences. The
2.5
Ileuormalization ol c uantiiiu liciti theories
divergent pari, of rj," 1 is therefore an integral over a local p o l y n o m i a l " degree four in A, u, u, , and n that satisfies the Lee identities. At or«1 n l I these identities yield cf. (2.5.13:5)
5 0
^
^ , 5 0
= 0
(2.5.13^)
for the divergent part, since all other contributions are finite. ow we c« " continue as in the one-loop case. The divergences of f ," 1 c a n b e remove^ by adding a counterterm to S so that the renormalized action at ord a I 1, 5 i , still satisfies the Lee identities This concludes the induct^® proof.
2.5.4.2
Structure of the renormalized action
According to the preceding section, the renormalized action in the MS scheme is obtained from the bare action by adding appropriate local polyii inials of degree four so that the Lee identities hold in each order of pertm" is the most genera hation theory. Consequently, the renormalized action S local polynomial of degree four that satisfies the identities
0 =S0 S
Sn
S0 =BsoSr.
(2.5.140)
ere we introduced the operator B ^ , which is defined as jj
f l'i J ' X
SSp 6 a 5A 5 >1
SS0 5 -I 5 So 6 « ' SA 6ua 6
j.
SSn 6
6 ( ia (2.5.141)
sing the Lee identity for the tree-level action, B ^ S o = 0, and the anticommutativity of the rassmann variables it can be easily shown that Bg0 is nilpotent It 0 BSoBs
= 0.
(2.5.142)
Any gauge-invariant functional 6 { A } of /1 is B S-invariant and thus satisfies B, Su {1 A } = f ' J
5A
6 a>n
=
f J
6A«
= 0.
(2.5.143)
230
2
Qua»!,mil theory of Yang Mills fields
Therefore, any expression of the form S R = G{ A } + BâoF{ A, u, Û, K, K u }
(2.5.144)
with an arbitrary functional F is a solution of (2.5.140). In fact this is the general solution as has been first proven in Ref. [Jo76]. Since the only local gauge-invariant functional of dimension 4 is proportional to the action, 12 we have G { A } = O f d 4 X £ g a u g c ( A ) . Because S'R is a local functional of dimension 4 and ghost number 0, and because 13§ has dimension 1 and ghost number 1, F must be a local functional of dimension 3 and ghost number —1. Requiring that the global invariance is not broken by the choice of gauge (Z?Vf) is globally invariant) and using (2.4.78), the most general form is 5 r = a Jd4s£gauge(A)
+ % Jd4x[p(I<;
- ûa$b;)Aa<»
-
7/jÇu
a
]
where a, /3, and 7 are arbitrary dimension less constants that can be used to absorb the divergences that appear when the régularisation is removed. Using the relations f ,1 /a\ tn SS , SSo 2 J d z £rg a u g e ( A ) = A, — 0 + Kua— ~
0
U
*SA Sua
=
U
+ 6K"
,l
SK;;
OSo g—, 0, ôû ' a
(2.5.146) '
v
which can be easily verified from the explicit form of the lowest-order action (2.5.130), we find the general solution to (2.5.140)
x
«5o{A, u, 0, K, K„; <7}.
'"There is actually a further gauge-invariant functional, e'",pa J<1 ]xFllt,Fpa. Because this term is a topological invariant and irrelevant in perturbation theory, it is dropped.
2.5 Ucnormalization of q u a n t u m lit;!«! theories
237
All counterterms arise from renormalization of the parameters, the sources, and the fields of the effective action. Moreover, A'^ and K"t as well as u", (t", and K't\ are renormalized in the same way. If we identify =
z y =
= -f,
(2.5.147)
this implies 5k{ A, u, u, K , K u ;
ZXJ2U, Z)J2U, 2ry2K,
^/2KU;
^ f f }
= 5o{Ao,uo,fio,Ko,K„,o;i7o}-
(2.5.148)
By adding the lowest-order gauge-fixing term we finally obtain the complete renormalized action. S. The non-renorutilization of the gauge-fixing term in linear gauges can be accounted for by choosing the renormalization constant of the gauge parameter £ (2.5.45) as Z^ = Z,\. Thus, the counterterms can be generated by the following renormalization transformations of the fields and parameters: =
"0 =
Z\/2A;, zlJ2u\
K$U = z f i < i fiS
= zlJ2u\
io =
multiplicative
zAt,
Kftn = z ^ k ; ,
Zgg.
(2.5.149)
Because of ghost-number conservation, which is expressed by the second line of (2.5.14G), there is a freedom in the renormalization of the fields with nonvanishing ghost number. This can be used in such a way that the antighost field is not renormalized and (2.5.149) is changed to „.a r, a "o = Zuu ,
-a -a u0 = u ,
r/-a isa K0 = K ,
71/2 i/a A isa K0 u = —— Ku. ¿U (2.5.150)
The renormalized action and the renormalized vertex functional are obtained from the bare quantities as S r { A , u, u, K, K u ; g,£} = 5 0 { A 0 , u 0 , u 0 , K 0 , K u , 0 ; i/o,£o}> f r { A, u, u, K, K „ ; 9 , ( } = T 0 { A„, u 0 , u 0 , K 0 , K„ )0 ;
(2.5.151)
230
2 Q u a » ! , m i l theory o f Y a n g Mills
fields
These relations imply that the renormalized action is invariant under renormali/ed BRS transformations. Since A K , u K „ , and u K are renormalized in the same way, the renormalized Lee identities are obtained by multiplying the bare ones by the corresponding renormalization constants. The renormalization constants have the loop expansion oo ZA = I+
Y ,
/,=1
oo .
ZU = 1 + 22
oo «5ZLL>,
ZG = 1 + 22
¿=1
6Z L)
I .
L=\ (2.5.152)
and can be determined order by order in perturbation theory. Note that the action, renormalized to order L, S[J} fulfils the Lee identities to all orders as required in the inductive proof in Sect. 2.5.4.1. The required terms of order hl, i > L arc automatically generated by the non-linear appearance of the fields in S. After having performed the proof we can set
= K" = 0.
The proof can be extended to gauge theories involving matter fields with gauge-invariant couplings of dimension less than or equal to 4. The relations between the renormalization constants of the various fields stay intact and, in particular, the renormalization of the gauge coupling remains universal. As the proof required the existence of a gauge-invariant regularization scheme, it breaks down for chiral symmetries, in particular for gauge theories involving the matrix 75. But also for chiral gauge theories, such as the Electroweak Standard Model, the existence of a gauge-invariant renormalized action can be proven as long as 110 anomalies occur [Be74, Be76b, Pi81, Kr98]. Anomalies are discussed in Sect. 2.7.
2.5.4.3
Physical s t a t e s and physical s o u r c e s
The covariant quantization of a gauge-field theory in a renormalizable gauge, as described in Sect. 2.4.1, requires the introduction of unphysical degrees of freedom: 1.
(zero-mass) gauge fields give rise to negative-norm states (2.1.26), and
2.
fictitious ghost degrees of freedom have to be introduced to provide the formal gauge covariance of the theory (Sects. 2.4.1.3 and 2.4.3.1).
2.
Renoruuili atiou of
u a n t u m Hold llicori .s
2
Thi physical S matrix transforms physical initial states into physical final states. Thus, a definition of ph sical states must be made before any further investigation of the S matrix is carried out til 2, e 4, Co , u . uite apart from the uestion as to whether such states can actually 1 dynamically reali ed, the definition of physical states is phrased ;us follows h sical degrees of freedom comprise all degrees of freedom invariant the. renormalized gauge group.
under
The definition of physical states is most conveniently done in the Iiamiltoniau picture after canonical uant i ation of the effective agrangian 2.4.24 u . In gauges that do not involve ghosts, e.g. in Coulomb-li e gauges, this amounts to the re uirement that physical states are invariant under renormali ed gauge transformations E
phy.
-
2. .1
In general renormali able gauges this condition is generali ed to the re uirement that physical states are R -invariant u , i.e. annihilated by the S charge uits 2.4. 2 , IV
,
2. .1 4
which acts on anticouiiiiuting commuting fields as R , fc
-
2. .1
The R operator s was defined in 2.4. . e implicitly assume here the a formalism with the auxiliary field cf. end of ect.. 2.4. .1 . wing to the nilpotency of the R transformation, the R charge is nilpotent and, as a conserved charge, commutes with the effective Hamiltonian H and thus with the S matrix brs
2
,
R ,H
,
R ,
.
2. .1 G
oreover, it is hermitian provided that the herrniticity properties of the Faddeev- opov ghosts arc chosen appropriately, i.e. ( )* = ua,
(u )
=
-u .
2. .1
230
2
ua
, m i l theory of
ang Mills fields
The minus sign appearing in the hermitian conjugation of the antighost is required because of the anticommuting nature of the ghost fields u7 . The B S charge can be used to divide the eigenstates of spaces: 1.
the subspace of states that are not annihilated by 2
into three subi3us:
= {tya) I b sI «) ^ 0},
2.
the subspace of states that are obtained by applying thus are annihilated by brs: 0 = { ;« I I -n) = QnnsW'a)M) 2 },
3.
the subspace of states that are annihilated by , = { 0 ffl ) n sM a) = 0, iPa) t 0 }.
brs to
2 and
B S but are not in
In the following we denote the states in the subspaces 4 0 ) , hpil)), and \ipa']), respectively.
o,
i, and
o:
by
The states in are characterized as unphysical by (2.5.15 4). The other . wing two subspaces form the space of physical states p lvs = o to (2.5.154) and (2.5.156), pi,ys is invariant under the S matrix, S p i, ys C phys- For gauge theories the inner product in this space is positive semidefinite u7 . It, is easily seen that all states in o have zero norm and are orthogonal to all states in i, and thus also in p ,ys w40)lvr) = ^2)i (40)lvi ) =
B I { S v i 0 ) ) = o,
4 2) l 2B s
i l) ) = 0.
(2.5.15 )
As a consequence these states decouple, i.e. they disappear from any physical matrix element, and physical states that differ only by zero-norm states are physically equivalent. The physical I filbert space pi,ys can be defined ;is the quotient space of p )ys with respect to the subspace o ftphys =
p i, ys /
o,
(2.5.15 )
i.e. the distinguishable physical states are defined as equivalence classes of states with strictly positive norm. Besides, physical states are required to have vanishing ghost number. This immediately characterizes all FaddeevPopov ghosts as unphysical.
2.5
R e n o r m a l i z a t i o i i of q u a n t u m field t h e o r i e s
241
In quantum field theory, within perturbation theory, the space of states is the Fock space of asymptotic fields. This Fock space is generated by the creation operators appearing in the Fourier representation of the asymptotic licld operators (L.2.L0). According to our definition of physical states (2.5.154) and the relation between states and asymptotic fields (cf. Sect. 1.2.1), the physical components of asymptotic, fields must have vanishing BRS variation >sypas,phys
_
()
(2.5.160)
As discussed in the following, the BRS variation of asymptotic fields is essentially obtained by discarding all terms that involve composite operators (products of fields) from the BRS variation of the fields (2.4.33) and (2.4.35), i.e. the BRS variation of t he asymptotic fields is linear. This implies, in particular, that fields transforming in homogeneously under gauge transformations, and therefore have contributions involving only a ghost field in their BRS transforms, e.g. gauge fields or Higgs fields (cf. Sect. 4.1.2), involve unphysical components. In order to determine the physical degrees of freedom of the fields we have to derive the BRS transformations of the asymptotic fields [Ku78]. The asymptotic fields are related to the discrete poles of Green functions via the reduction formula (2.2.46). Consequently, the properties of the asymptotic fields can be determined by investigating the Green functions. To this end, we consider an arbitrary Green function with one of the external fields
r x
. =
/ J t f g X r ^ i c * / , . . . >£/„) + non-pole terms, K
or in diagrams
I
(2.5.161)
230
2
ua
, m i l theory of
a n g Mills fields
non-pole terms,
(2.5.162)
where we indicated the B S-transformed fields by a double line. ote that in the preceding equations includes also the integration over the corresponding space lime coordinates rc; . quation (2.5.161) suggests to define the B S transformation of the asymptotic fields as s /
on-shell
s
f = CI ^
(2.5.163)
with c
= m-*,)hs)
,
=
j
j
^
on-shell (2.5.164)
s^i
v iy
vft
and .,4 , is a source coupled to By construction the B S transformation of asymptotic fields is linear, and a field i /i can only contribute if its mass is equal to the one of P/. In practice the field I/ - is an antighost field since styi involves a ghost field, and the ghost number is conserved. e define ph sical sotirces j hvs as those sources that couple only to physical fields. Because of (2.5.160) we immediately find that these satisfy /phys <
xT(as,phys
-
rphys
= 0,
<
(2.5.165)
on-slicll which is equivalent to t 72 f h y s 0 s^/j^/f
= 0,
(2.5.166)
where vI / -) is the state belonging to the asymptotic field fy . Let us consider gauge fields. wing to ghost-number conservation the (linear) B S transformation of the asymptotic gauge fields must have the form 11 sA^(x) =
l j f{x,y)u^b{y),
ote our conventions in App. A. 1.3.2.
(2.5.167)
2.5
cnormalization of q u a n t u m lit; « theories
243
where b
{x,
) =
T (.syt;f n(.r)) ub( )\ \ ' f ^ on-shdl
= S
tjjr n
i'(x) hib( )
ll.shc
(2.5.16 ) because of (2.5.163) and (2.5.164). The fact that the ghost masses coincide with those of the gauge-boson fields follows from B S invariance. quations (2.4.62), (2.4.33), (2.4.35) and (2.4.54) imply (T(D u)aub)
= (T(sAl)ub) = (TAlA
-(TA sub)
=
j =
^/l^1-.
(2.5.16 )
The poles in the expression on the left-hand side are determined by the ghost propagator, those on the right-hand side by the gauge-boson propagator. quality of poles on both sides yields equality of masses for those components of the gauge-boson fields that are not orthogonal to 1 ":1/. Considering diagrams, (2.5.167) implies
2
2
ua
, m i l theory of
ang
ills fields
non-pole terms non-pole terms
2. .1
with
i
x
x
acd u ,U 2. .1 1
orent invariance and global gauge invariance restrict x
x
to the form 2. .1 2
because ab is the only gauge-invariant tensor with two group indices. The linear ee identity 2.4. implies Ia
wr Su (x) uc(y)
wr (x)6uc{y)
5
a
2. .1
'
and the renormali ation condition for the renormali ed ghost two-point function can be fixed in such a way that this two-point function is on shell e ual to its tree-level contribution i 6u (x)Suc(y) a
=
c ,l
d 5W{x - y).
2. .1 4
ll-shcll
utting all this together yields f
^ i x , y) =
1>«c0
{x, y) = V tcd 6 {x
- y),
2. .1
or (x,y)
=
W(x-y),
2. .1
and thus sA
a
(x) = d ^ ' a ( x ) .
2. .1
2.5 U c n o r m a l i z a t i o n of q u a n t u m lit;!«! t h e o r i e s
245
The IiltS transformation of the asymptotic ghost field vanislies because there is no single particle state with ghost number two, and the BRS transformation of the asymptotic antighost field follows directly from the linear BRS transformation of the antighost field, so that we find sA™'a( x) = d,tuie'a
(x)
su a s , n ( x) = 0, su^ix)
=
Bas'a(x).
(2.5.178)
Note that these have exactly the form of the BRS transformations of the free fields. The preceding relations allow to read off the physical fields of a pure gauge theory. The antighost ûas,a and the longitudinal component of the asymptotic gauge-boson field (e (l = k /t ) are not annihilated 14 by the BRS operator and are thus unphysical fields (€ V^). The ghost field u38'® and the auxiliary field /jfas>" are BRS transformations and consequently zero-norm states (6 VQ). As the equation of motion (2.5.179) identifies the auxiliary field with the component Aas'b'11 of the gaugeboson field, also this component is a zero-norm state. Consequently, the only physical states (€ V|) are the two remaining (transverse) components of the gauge field. The corresponding polarization vectors are orthogonal to the momentum and to the gauge-fixing function. There are exactly four unphysical modes (including zero-norm modes), the antighost, the longitudinal gauge-boson mode, the ghost and the B mode. This is an example of the so-called quartet mechanism [Ku78, Ku97], According to this, unphysical states appear always as quartets, and only the combinations of the quartet states with zero norm can appear in the physical space. The quartet mechanism can be derived from the BRS algebra. Let us specialize to the Lorentz gauge, ) = k tl = (|k|, k), MThis can be easily seen by transforming (2.5.178) to momentum space.
(2.5.180)
230
2 Q u a » ! , m i l theory of Yang Mills fields
the zero-norm (scalar) polarization vector (e^k^ ^ 0) reads (2.5.181)
e ^ k , 0) = ( | k | , - k ) > and the physical polarization vectors arc given by < h y s ( f c , l ) = (0,k! L 1) ) ] k ( ; } k = o,
(fc,2) = (0,k ( x 2) ), =
f , i = i,2.
(2.5.182)
All physical polarization vectors are transverse and have non-vanishing norm. All vectors e'l(k,i) = e'' liys (/c,i) +ce''(fc,0), i — 1,2, with arbitrary c describe the same physical state and are members of the equivalence class characterized by £t\xys(k,i). While the space Vi spanned by £phys{k,i), i = 1,2, is not Lorentz-invariant, the space of physical states Vpi,yS spanned by £p\iys{k, i), i = 0 , 1 , 2 , and consequently the physical Hilbert space %,,hyS is. Note that the time-like excitation e,L = (1,0,0,0), which introduces negative norm-states (2.1.26) is a linear combination of the unphysical and the zero-norm polarization and thus does not contribute to physical matrix elements. In the case of fermion fields, the BRS transformation (2.4.33) in combination with ghost-number and fermion-number conservation exclude one-particle intermediate states in {(.sV-O^/i - - - 'J*/„)• Consequently, the BRS transformation of the asymptotic fermion fields vanishes and the fermion fields involve only physical degrees of freedom. In gauge theories with spontaneous symmetry breaking, scalar fields occur that transform inhomogeneously under gauge transformations. The corresponding BRS transformation involves a linear (ghost-field) term. Owing to the mass degeneracy of the scalar field and the ghost field this term survives in the BRS transformation of the asymptotic field. Consequently, these scalar fields are unphysical degrees of freedom (cf. Sect. 4.4.2).
2.5.4.4
G a u g e i n d e p e n d e n c e of t h e physical S m a t r i x
The introduction of gauge-fixing terms along with the corresponding terms introduces a gauge dependence in the theory. Therefore, the ¡»variance of physical results has to be demonstrated by proving the independence of the S matrix. This can be done with help of the
ghost gauge gauge Ward
2.5
cnormalization of q u a n t u m lit; « theories
247
identities that follow from B S invariauce. It, is essential that the reuormalization has been carried out in such a way that the renormalized theory is invariant under (renormalized) B S transformations. Consider the gauge theory defined by (2.4.20) and (2.4.21). Since we are only interested in the S matrix and thus in reen functions involving only physical fields, we put the sources , , , u . ^, and ^ to zero. e want to study the gauge dependence of the resulting generating functional Z{ , 0, -,.}. e assume that the theory is renormalized and invariant under B S transformations. Let us perform an infinitesimal local change of the gauge-fixing functional Cb -4 Ch + Cb. To first order in AC, we have
,
Se+ Cp, '-l-Cb{ xexpfi
} - Zc{ , Cb{
-
,
J
} =
)
] u] [um] )
j d
/ d 4 x Cc,r + J ' A;
i>i -
faji])
D .
]
u*{z) (2.5.1 3)
In order to simplify this expression we use the Slavnov Taylor identities (2.4.56). ifferentiating (2.4.56) with respect to b{ ) and putting = ,j = 0 afterwards yields
] u] u]D\tl)] ^\ x exp (i J d x [Cetf x{-ati f c (y)
J^A'l
iub( ) j d z . ^s *
perating on this equation with
Cb{ \ 3\
-
J^i
;t- i )
-
, using
(2.5.1 4)
- ( ), j j } = 0.
'¿'18
2 Quantum theory of Yang Mills fields
and sub{y) = - C 6 { A; ?/}/£, leads to
I D[A}V[u]V[ri\V[jJ>}V[iJ)} x exp ( i y <\*x [ce„ + J£Al
+ J^h
- ÏH J i ] )
(2.5.185)
xlic^AjyjAC-nAîyJ+û^) / ¿ z ^ F ^ + i A C 6 { A ; î , } û 6 ( y ) J c\*z
sAfr)
- J ^ , - ( # ) i J j ] } = 0.
Adding (2.5.183) results in
~ exp
= J
j d4x [£ ef r + J,l'aA°
+ J^h
V[A]V[u]V[u]V[if,}V$} - 4>ji] )
x | i A C " { A ; y } u 6 ( y ) J d 4 * [j a '"sA^ -
(2.5.186) - (s^ji] }
or to first order in A C
ZC+AC{J,
J*, J*} = y
i>[A]X>[u]X>[Qph/»W]
x exp | i y d ^ j ^ r + J"'a (a^ + 4
- s Al j"
dAyûb{y)ACb{A;y}
( ^ i + srPi y d 4 y Û'(J/)AC°{A; y}^
- (% + s $ i f dV6(j/)ACa{A;j,}) Jj | .
(2.5.187)
As a result, the generating functional Z c + a c differs from Z c only in the source terms. An infinitesimal change in the gauge condition corresponds to a change in the source terms by an infinitesimal amount.
2.5
e n o r m u l i z a t i o n of q u a n t u m field t h e o r i e s
24
To first order in A C , the difference of Green functions calculated with the two different gauge-fixing terms reads
where xh/ represents anyone of the fields r ?/;,, or ipi, and the multi-index /( includes also the space time coordinate X[. According to Sect. 2.1.3, the 5-matrix elements are obtained from the Green functions by truncating the external legs and multiplying with the appropriate physical polarization vectors and wave-function renormalization constants. Thus, only terms that involve poles in each of the external lines con tribute. These terms correspond to diagrams that are one-particle reducible in each of the external lines, i.e. in which each external line is connected to the rest of the diagram via a single line. This must hold, in particular, for the line with the BRS variation at the end. All diagrams in which the single line is located between the BRS insertion ) and the composite operator a fd u' ( ) C{ j do not contribute to the S matrix because such contributions are only non-zero for unphysical field components [cf. (2.5.1G5)]
J
7pliys
= 0.
(2.5.190)
A C 6 •••ub
Therefore, the only contributions to the S matrix arise from diagrams in which the product of operators ( . s ^ ) f d i ub( ) C {A is separated
230
2 Q u a » ! , m i l theory of Yang Mills fields
from the rest of the diagram by a single line, i.e. from diagrams of the structure
ub
l Cb
(2.5.191) 4' ub
n
l Cb
which factorize into the original Green function and a factor
r, k =
(2.5.192)
m. i i u
/AC6
We have assumed that the mass of the field ^ ik does not coincide with any other physical mass so that only ipi contributes. On shell, the factor rj is a constant. Thus, we arrive at
+ on-shell-irrelevant terms.
(2.5.193)
The wave-function renormalization constants arc determined from the twopoint, functions according to (2.1.34) and are related by rc+
C
=
(
+ r
f
c
(2.5.194)
2.5
U c n o r m a l i z a t i o n of q u a n t u m lit;!«! theories
251
Thus, when going to the S matrix [cf. (2.1.33)) the extra (gauge-dependent) constants 77 cancel between the Green functions and the wave-function »•normalization constants and the S matrix is gauge-independent [Le7Cj.
2.5.4.5
U n i t a r i t y of t h e S m a t r i x
Formally the unitarity of the S matrix can be proven very easily once; the physical states have been defined [Ku78]. If the hermiticity assignment of the Faddeev Popov ghosts is done appropriately (2.5.157), the effective l.ngrangian is hermitian and the full S matrix is unitary, i.e. SS' = I. The physical S matrix is defined 15 as the full S matrix projected onto physi cal states, represented by states £ Vi. Since the S charge uilt) commutes with S, we have C?BRs5|Vi 1) >= 0,
i.e.
St
)
6 Vphys-
(2.5.195)
Thus, we have, owing to (2.5.158),
c
(2.5.190) The states V'(0) do not contribute in (2.5.196) owing to (2.5.195) and (2.5.158). Therefore, (2.5.196) expresses the unitarity of the S matrix projected to the subspace of physical states, i.e. of the physical S matrix. Unitarity can also be proven directly by exploiting the Ward identities [tII71, Ta71, Co75]. Having established the gauge independence of the S matrix, this can be done by considering a convenient gauge. Of course, also in this approach the definition of physical states or sources is inevitable. For gauge theories in which all gauge fields acquire a mass by spontaneous breaking of the gauge symmetry (cf. Sect. 4.1), a simple proof of unitarity is possible [tII72]. In this case, a gauge exists in which the masses of all corresponding unphysical degrees of freedom reside at infinity. Therefore, they do not. contribute to the S matrix and the latter is unitary. This so-called unitary gauge is non-renormalizable according to power counting because ,s
More rigorously, the physical S matrix S p i, y , is defined as the quotient mapping of the full S matrix into the physical Hilbert space (2.5.159) [Ku78].
230
2 Q u a » ! , m i l theory of Yang Mills fields
the corresponding propagators (1.4.13) do not behave as 1 / 2 for —> oo. However, this special gauge can be continuously transformed into a renortnalizable gauge in such a way that in all intermediate gauges the theory is well-defined. Then, the gauge independence of the S matrix ensures that the theory is unitary also in renormalizable gauges.
2. Relativistic quantum field theories require, as we have seen, regularization and renornialization. Renornialization is a reparametrization of the theory which at least has to compensate possible divergences. This reparametrization is not unique. It can be fixed by renormalization conditions in different ways. Physical results are independent of the choice of rcuormalizcd parameters, i.e. a reparametrization of the coupling constant g and mass m, —> g', 2 2 m —> m' , only changes the explicit expressions for physical quantities but not the relations between them. This fact is the basis of the re normalization group (RG) which quite generally describes the composition law of the renormalization transformations. For simple transformations like changes in the renormalization point, —> +d , and corresponding changes of the 2 2 2 parameters, g > g + dry, m —> m + dm , etc., one can derive a differential equation, the renormalization-group equation [St53, Ge54] which describes the effects of this change on Green functions or vertex functions.
2.6.1
Renormalization-group equation
In this section we derive and discuss the equation which describes the behaviour of vertex functions under a shift M —> M + <\M of the renormalization point. First we consider a massive scalar field theory. Starting point is the independence of physical quantities under reparametrizations of the theory (cf. Sect. 2.5.1). This condition reads for 5-matrix elements
•M-7^-T = 0.
(2. .1)
2.6
2.6.1.1
Renormalization group
253
Derivation of the renormalization-group equation
The connected parts of the S-inatrix elements arc obtained from the corre) mulsponding truncated renormalized Green functions G|^| r u n c (p,; g,m\ 2 according to tiplied witli the wave-function renormalization constants (2.1.35). The truncated Green functions are products of vertex functions and propagators. As a consequence, the vertex functions behave exactly ;is the truncated Green functions under infinitesimal changes of the renormalization point .. The invariance of the S-matrix elements (2.G.1) implies n2
S
=
+ d ),m{
+ d )-,
{ )Y ^p \g{ ),m{ )\ x r ^{p ] (
j
n2
'( +
)
+ d m \
where g = g (gon6,
,
m)
,
m = m (go
S
,
,
(2. .2)
with <5 = 2 - D 2. The bare parameters go, mo and the parameters for dimensional regularization /t, are kept fixed. 10 Expanding (2.6.2) gives .
d
dq
d
dm
n
\
T,tn)
i
, (2. . )
Defining the ( -function,
3(g, m
also called renormalization-group
) = lim <S->0
the renormalization-group 7m(g,m
)
and the anomalous -Y(g,M
)
m l//., ( vI
function.
m) ,
(2. . )
\
coefficient for the mass
= - lim — -^- in(gofi6,m , < ->o m UM V
term n,S m)
/
(2. . )
dimension = - lim ^
( G
^
,
T U Q ( 2 . 6 . 6 )
"'In other regularization schemes they have to he replaced by the corresponding cut-olf quantities.
230
2 Q u a » ! , m i l theory of Yang Mills fields
we obtain the renomuilization f r
d
group
equation
d
\ J
d
"
f
\ 0. (2.6.7)
The functions f3(g,m\M), 7m(fl>ro>.M)> and y(g,m;M) are the same for all vertex functions and thus a characteristic of the theory. The anomalous dimension 7 { ( j , m \ M ) reflects the anomaly connected with the dilatation transformation of massless theories (cf. Sect. 2.7). This anomalous breaking is controlled by the Callan--Symanzik equation (2.6.13). The renormalization-group (KG) equation can alternatively be obtained from the fact that the unrenormalized vertex functions rn(pk;9olJ-S,m0,n,S) Z~n/2
=
(
xru (pk;
rn0, ¡i, S\ M)\M^j
(2.6.8)
are independent of the renormalization point M. Applying Md/dM (2.6.7) with (2.6.4), (2.6.5), and
yields
7 ( g , m - , M ) = Jim \ z ~ x M - ^ Z ^ ^ ^ S - m ) -
(2.6.9)
Using the definition of R, (2.1.34), and the relation between the renormalized and unrenormalized propagator, we find R = -i(p2
- m2)GR{p,
= ~ i(p 2 - m2)Z~lG(p,
-p)\pt=m2 —p)|pj_m2.
(2.6.10)
The independence of the bare propagator G(p, -p) on M. implies the RG invariance of R x Z and thus the equality of (2.6.6) and (2.6.9). This shows that the RG invariance of the S matrix and of the bare Green functions are equivalent. We have derived the RG equation for the case of a scalar theory with one coupling and one mass parameter. Generalization to several couplings
2.(5
Reuormalizatiou group
2.r».ri
masses m rr , n/j bosonic fields, and n j fermionic fields is straightforward and yields
0
o
xrlB'n'
0
d
- niaii
(pk;9i,ma;M)
- npj
=0.
(2.6.11)
In the case of gauge theories the vertex functions also depend on the gang«1 fixing parameter and an additional term l ^ d / d ^ has to be incorporated in (2.6.11) [Gr73a]. An elementary dimensional consideration can be used to reshape (2.6.11). The usual mass dimensions for fermion fields, dimif> = M 3 ' 2 , and boson fields, d i m B = M \ given in Table 2.1, imply that dimF 1 »'"/ (the ¿-function for energy-momentum conservation is split oil"). Euler's differential equation for homogeneous functions gives
'A d K OX
| m
d + M 8M '' dmp
xTlB'n,(Xpk,-,gi,mQ-,M)
+ «/i + h
~
+
i7m
>e+
) (2.6.12)
= 0.
Thus, if MO ¡DM in (2.6.11) is replaced by Xd/d\ the Callan Symanzik aquation [Ca70, Sy70]
('xix
4
l ) r n
^
+ nw(1
using (2.6.12), we obtain
+ 7/i)
+ n j Q + 7 / ) - 4 j r ^ ' n / ( A P A : ; 9 i , m a ; > i ) = 0.
(2.6.13)
Here, the renonnalization point M. is kept, fixed. The Callan Symanzik equation (2.6.13) describes the behaviour of the vertex functions under scale transformations of the four-momenta. The breaking of the naive scaling behaviour is contained in the anomalous dimensions and 7 / . This becomes clear when (2.6.13) is solved.
230
2 Q u a » ! , m i l theory of Yang Mills fields
2.6.1.2
S o l u t i o n of t h e R G e q u a t i o n
The RG equation (2.6.13) is a linear, partial differential equation of first order which can be solved by the method of characteristic curves [Co68]. The characteristic curves <j,(£) and fnQ{t), where t = log A, are determined from the system of ordinary differential equations
^
=
0,
= —Ym,a[9j>™P>t) - 1,
together with the initial conditions
(2.6.14)
ma.
The explicit solution of (2.6.13) reads with t. = log A C
n /
(*Pk-,9i,rnn-, M) = r£ B> "'(?*;&•(«), ™a (t);M)
x exp
J dr
+ npf
A4""»~3"//2
(.7j(t),m^r), t )
(ffj(r), mp{T), r ) ] ^.
(2.6.15)
The solution of the RG equation shows that a scale change in the momenta by a factor A can be expressed by a change in the coupling constant and in the mass, and by a multiplicative factor. This factor differs from that of homogeneous functions with canonical dimension, i.e. A 1 - "' , ~ 3 n ' / ' 2 . The contribution coming from the dynamical dimensions 7 b and 7 / violate naïve scaling. The RG equation is very similar to the continuity equation between a density p(x, /,) and a current density j(x, t) — />(x, i)v(x, /.),
m
+ V(pv) =
%
+
+ p V v
=
(2 G 1(i)
- -
In this case the characteristic curves are the flow lines. As an example we solve the RG equation for the simple case of one field with one coupling constant g and one mass m, where the ft- and 7-functions depend only 011 g, ft = ft(
f)/-\
d log m ~dT
=
~7m{9)
~1
2.6
Renormaliy.ation group
257
decouple in this case and can easily be solved rm
dg' (2.0.17) iu'
rm
dg'
implicitly yielding the running coupling g(t) and running massm{t). ing tliis in (2.0.15) we finally obtain
HUA Pk\ ,m)
=
f
rn (pk-,g(t),fn(t))\ 'nexp
Insert,-
^
m
(2.0.19) The typical behaviour of the RG functions in one-loop approximation is
(2.6.20) leading to
or 2
(t)
=
(2-0.22)
1+ Th* and a2m\ym-o/2fio rn(l) = TO ( ^ )
e
•
(2.0.23)
The vertex functions have the scaling behaviour r^(XPk]g,m) = n ( P k , g { t ) , m ( m 4 - n i ^ f
2
)
•
(2-0.24)
The scaling behaviour (2.0.24) holds also for truncated Green functions. Note that on the right-hand side of (2.0.24) the possibly large terms involving log A are re-summed [cf. (2.6.22)]. Because of this re-summation, the RG can be used to improve finite-order predictions.
230
2 Q u a » ! , m i l theory of Yang Mills fields
2.6.1.3
Fixed points
The structure of quantum field theories for large or small momenta is given by the limits of the running coupling constant g(X,g) for large or small A. Concerning large momenta, it is important to note that mass terms are insignificant for the behaviour of the vertex functions for large euclidean values of the momenta, p 2 —oo. This has been confirmed in perturbation theory [We60, Sy70]. Therefore, the behaviour for large momenta can be studied neglecting the mass terms in the Lagrangian and in the IIG equation. Thus, for large A we can put 7m(<7) = 0. Then the ^-function and anomalous dimension of the corresponding massless theory, ft(g) and 7 ( g ) , provide the leading terms for the massive theory [We73a, Sy70]. The running coupling constant g and the anomalous dimension 7(
(2.6.25)
where in perturbation theory r > I. Equation (2.6.17) gives g-g'
= (g-g*)^\
r = i;
(2. .26)
for the behaviour of the running coupling constant around the fixed point. Depending on the sign of ft' the fixed point <7* is approached in the following way - \ \ g(\,g)->g
c for
|
f A -» 00, q,
if ft' <0, if ft' > 0.
In the first case, ft' 0, g is called an ullraviolet-stable ft' > 0 it. is an infrared-stable fixed point.
(/o«o<7\ 2 (, 27)
"
fixed
point, for
The vertex functions are homogeneous functions of the order 4 — (3/2 + 7 j W ) ) n ! ~ (' +7»(.(/*))"« at the stable fixed point, in agreement with the scale-change equation (2.6.15). Thus, the shape of the RG function ft(g),
2.6
Renormaliy.ation g r o u p
259
Pig. 2.10 Examples for fixed points of the IIG equation
illustrated in Fig. 2.10, the renormalization eral properties of a field theory.
group flo , determines the gen-
Theory (a) has an UV-stable fixed point g*2 and an IR-stable one g\ with r — 1. If the renormalized coupling constant g lies between and then the theory approaches the corresponding limits with homogeneous vertex functions for large and small distances, respectively. Theory (b) —for which as we shall see QCD is an example—has an UV stable fixed point at g = 0 with ft' 0. This type of theory is called as mptoticall free since for large momenta, i.e. small distances, its Green functions have the scaling behaviour of a free theory, which is modified only by logarithmic terms for r > 1 as (2.6.26) shows (in perturbation theory r = 3). Equation (2.6.27) implies that g( ,g) -> oo for A 0. Theory (b) is IR-unstable. The important special cases of QED and QCD are treated in Sects. 2.6.2.1 and 2.6.2.2. If 3(g) and 7 ( g ) are calculated in one-loop approximation, the RG equation provides the asymptotic summation of the leading logarithms to all orders of g and thus improves the perturbation series in the asymptotic region. For small momenta, mass terms can have a considerable effect on the behaviour of Green functions. 2.6.1.4
R e n o r m a l i z a t i o n invariance of physical q u a n t i t i e s
The starting point of our considerations was the fact that S-matrix elements are invariant under RG transformations (2.6.1). This holds for all physical quantities. So, if a measurable physical quantity O, e.g. a mass, is calculated
260
2
Q u a n t u m theory of Yang Mills Holds
in gauge t heories, it. depends on g, m, and M, i.e. 0 = 0(g,m, M), but only in a combination that is invariant under renormalization [Gr74, Ca78]
+
"
7 m m
^ )
0 ( g , m
'
M ) = ( 2 - 6
-
2 8 )
;us a change in M must be compensated by a change in the meaning of g and m. In a massless theory like QCD, any mass M must be proportional to M. for dimensional reasons. Therefore, it must have the renormalization-invariant form M = Mexp
( - / ' ^ y ) -
(2.6.29)
The specific value of M is determined by the integration constant. 2.6.2
Renormalization-group function and anomalous d i m e n s i o n s of massless gauge theories
As an example we consider massless gauge theories with fermions. In order to determine the RG function ¡5(g) and the anomalous dimensions 7^ and 7 / l of fermion and gauge-boson fields in one-loop approximation, we need the renormalization constants Zg, Z^, and ZA. For QCD in Feyntnan gauge these were calculated in Sect. 2.5.3.3. All three renormalization constants We depend on t.lie renormalization point M in the form f{S)(4nfi2/M2)s. need the logarithmic derivative of this quantity M
d OM
m
( ' i v \M2
v
(2.6.30)
)
for J -> 0, and consequently only the 1/5 term of f{5). In a general massless gauge theory with fermions in a representation li the relevant terms read ZG =
2 / V ^ 9 1 2 l(i7T ^ M2
zA = 1 zi>
(4 TTf? ^ 1 2 16tt2 i, M
t
c
{\ «- >
(4tt,X2 V r = 1 16Tr2 M 22 )
f1
(2.6.31)
2.1)
Renorninli'/.atiou g r o u p
2(>l
for arbitrary values oi the gauge parameter The Casimir operators C'A, C'/f, and the Dynkin index Tu are defined in Sect. 1.2.2.2. The rcnor1uali7.fition of the coupling constant g is, as it should be, independent of the gauge parameter Using (2.6.31) and the independence of the bare coupling QQ = Zgg of we obtain
«9) =
for the (3-function
-gM
=
of a massless gauge theory, and in a similar way
Ca - ¡Tn)
1*<S) =
,
7jig) = for the anomalous
M,
(2.6.33) dimensions
of the gauge-boson and fermion fields.
According to the criterion for the /3-function formulated in Sect. 2.6.1.3, g 0 is a fixed point. Non-abelian, massless gauge theories are asymptotically free, provided the number of fermions is not too large. The gauge-boson selfcoupling, which gives the term proportional to Ca and thus the negative sign to the RG function, is responsible for the asymptotic freedom of non-abelian gauge theories. The /3-function of non-abelian, massless gauge theories has been calculated up to the four-loop approximation. The corresponding expansions of the R( I function /3(g) .i m
=
ifo "
5 A
(ife? ~
7 02
+
' -'
' £ = 1 .nul £ = 0 correspond to the Fcynman and Landau gauge, respectively.
230
2 Q u a » ! , m i l theory of Yang Mills fields
and of the anomalous dimensions
defines the coefficients fa, f j \ , fa,... and 7 0 , 7 1 , . . . . The one-loop coefficients fa and 70 read
fa
—
^
~
_
5
1-Ç
7/1,0 —
2~
4
A
3
/f
'
7/,o =
(2.6.36)
The two-loop coefficient A
=
j ( C ' A )
2
is given by [Ca74, Jo74] jCATR
-
-
ACrTr.
(2.6.37)
The three-loop coefficient fa is scheme-dependent. Its value in the MSscheme is [Ta80] _ 2857 - —CA
1415
3
2
CrTR + jCR(Tjt)2
158
2
+ 2 (CR)2TR.
(2.6.38)
The four-loop coefficient fa has been calculated in Ref. [Ri97].
2.6.2.1
QED
Since (¿ED is abelian we have Ca = 0, and the /^-function has a positive sign. This means that QED is not asymptotically free. The effective coupling constant "(32) =
a
(2.6.39)
increases as distance becomes smaller, i.e. as momenta become larger. This leads to a screening of the unrenormalized (bare) charge [La55, Be76, No78). Vacuum polarization by virtual electron-positron pairs screens the bare
2.6
Renormaliy.ation g r o u p
263
charge iis the latter attracts particles with opposite charge and repels those with equal charge. So, from a given distance, the bare charge is seen surrounded by a cloud of charges with opposite sign. The physical electrical charge measured in a sphere with this radius is smaller than the bare point charge. The one-loop result for the running coupling constant of QE1) diverges for 2 as mf, exp(lin a). This Landau pole is an indication of the breakdown of perturbation theory of pure QED at very high energies. On the other hand QED gets free for large distances, i.e. small energies. This and Thirring's theorem [Th53] guarantee that the value of the electric charge e can be determined from the Thomson cross section without any corrections.
2.G.2.2
QCD
The gauge group of QCD is SU(3)c- The coloured quark fields occur in iVf fundamental representations of SU(3)c. i.e. in Nf flavours. This means A = 3, CR = CV = 4/3, and Tn = N(TF = Nf/2. Thus, we obtain for the one-loop RG coefficients
= 11 7/1,0
=
_
2
3
/ '
7q,o =
(2.6.40)
and the coefficients fi\, fa 38,, Pi = 102 - —N(,
2857 0 (i2 = —
5033 _r 325 / A r x 2 —Nf + — {Nf) .
(2.6.41)
The sign of the one-loop /?-function of QCD changes for Nf = 33/2. If the number of quarks is below 33/2, then QCD is asymptotically free at one-loop order. Provided that quarks are not too numerous, /3(g) remains negative in higher orders. The negative sign of the /^-function of QCD may be explained by the following physical mechanism. Also in non-abelian gauge theories vacuum polarization by the Coulomb interaction of fcrrnions and charged, transverse, i.e.
264
2
Q u a n t u m theory of Yung Mills liciti»
physically polarized, gauge-boson pairs causes colour-charge screening. However, the self-interaction of the gauge bosons generates an additional dipole term between a transverse and a longitudinal gauge-boson. This dipole is oriented so that it makes an antiscreening contribution which is larger by a factor of 12 [Ap77]. 2.6.2.3
T h e o r i e s w i t h scalars
All known rononnalizable theories which are asymptotically free are nonabelian gauge theories. Gauge theories for the electroweak interaction and the unified strong and electroweak interaction contain scalar fields in addition to ferraion fields. Their gauge-invariant coupling with the gauge bosons makes a positive contribution to the RG function which is similar in structure to that of the fermions and which destroys asymptotic freedom when the number of fermions and scalars is too high. Moreover, the electroweak theories require scalar-field self-interactions. This implies that they are no longer asymptotically free [Co73]. 2.6.2.4
C a l c u l a t i o n of t h e ^ - f u n c t i o n in t h e background-field method
The /^-functions of gauge couplings can be calculated very efficiently within the background-field method [Ab81]. As discussed in Sect. 2.5.1.8, in the background-field method the renormalization constant of the gauge coupling is related to the renormalization constant of the background gauge field according to - 1 / 2
As a consequence, the /^-function is directly linked to the anomalous sion of the background gauge field
(2.6.42) dimen-
(2.6.43) The renormalization constant for the background field 7 , (
2.6
Renormaliy.ation
group
265
i.Ik; conventional formalism (cf. Sect. 2.5.3.3). The non-vanishing Feynman graphs are given by Fig. 2.9a), b), and c) on page 226 witli external background fields. While graph (a) gives the same result as in the conventional formalism, graphs (b) and (c) yield different results owing to the modified Feynman rules. It turns out that each of the graphs (b) and (c) yields a transverse contribution. We finally obtain for the background-field renormalization constant
(2.6.4'l) and for the corresponding anomalous dimension
(2.6.45)
From this the (1-function (2.6.32) is directly obtained using (2.6/13). Note that only the background-field two-point function had to be computed. This is in contrast to the conventional approach where the gauge-field twopoint function, the fermion two-point, function and the fermion gauge-boson vertex function (or the ghost two-point function and the ghost-gauge-boson vertex function) have to be calculated. Using the background-field method also the two-loop /^-function has been calculated [Ab81, Ab83a]. In this calculation the technical advantages of the background-field method become even more obvious.
2.6.3
Relation between different renormalization schemes
The various definitions of the running coupling constant in asymptotically free theories often lead to different second-order formulas. The importance of controlling this problem in the framework of QCD is discussed in Sect. 3.1.2. Therefore, a systematic review of the transformation of coupling constants is necessary [Sy73, Gr73a]. If the Lagrangian contains a set of independent, dimensionless running coupling constants gi(M). these satisfy differential equations of the form
(2.6.46)
230
2 Q u a » ! , m i l t h e o r y of Yang Mills
fields
If a different renormalization procedure leads to a different set of coupling constants <)i(M) which satisfy (2.6.46) with $i(gj), then within the theory a transformation formula between these coupling constants exists which for dimensional reasons must be of the form
This yields the transformation formula for t he RG functions fti
Pi(9k) = £
j
p-Pj(9k). °9j
(2.6.48)
Consequently, the transformation (2.6.47) maps fixed points into fixed points. As a result of (2.6.48), in perturbative transformations of coupling constants in non-abelian gauge-theories, 93 9 = 9 + 0 ^ + ..., -p = 1 + ag
4TT
+ - -•5
(2.6.49)
the first two coefficients fa and fa of fi(g) in (2.6.34) remain invariant [Sy73], fa —
fat
fa=fa-
(2-6.50)
These transformations are needed when results from different schemes are compared. So far, we have used the MS and MS schemes and the MOM scheme defined by the renormalization conditions (2.5.47). For both practical and historical reasons, these and other schemes arc used in the literature. Let. us summarize these schemes here for convenience. In the minimal-subtraction scheme (MS) o f ' t Hooft [tH73a], only the singular terms, e.g. the terms ~ 1/6, are subtracted. This simplifies higher-order calculations. However, as the transition to a dimensionless coupling constant in the form g2 —> g2n& is not unique, the MS method can be generalized [Co74]. The substitution g2 -> g2/iA f(S,g2) with f{6,g2) = 1 - ¿(7 - log47r),
2.6
Renormaliy.ation group
(7: Euler's constant) yields the modified minimal-subtraction where to one-loop order
7 a
7 - 7 + l°g47r.
267
scheme (MS)
(2.6.51)
The determination of finite renormalization terms in the momentum-subtraction procedure (MOM) can of course also be done by fixing the three-gaugeboson or four-gauge-boson vertex normalization instead of that of the quark gauge-boson vertex at symmetric momenta. All these various ways of renor malization constitute different renormalization schemes but give the same relations between physical quantities. In perturbation theory, the relevant renormalized coupling constants are linked by transformations as in (2.6/18). These leave the /^-functions invariant up to the second order.
2.6.4
Running unrenormalized coupling constant
The RG equations allow for the renormalization procedure of a quantum lield theory to be viewed from a different angle [G176]. We assume that the singular expressions for Green functions can be made finite by any regularization procedure with a cut-off parameter e. Physical results then follow after subtraction, i.e. renormalization, in the limit e —> 0. This is of importance for the lattice-approximation of gauge theories (Sect. 2.10), where the cut-off parameter e is given by the lattice constant. Then, instead of the running renormalized coupling constant f j an unrenormalized coupling constant .70(e) and its dependence on the cut-off parameter e is considered [Wi75, Sy77, Sy80]
+
(2 G 52)
- -
For finite 1/e, the renormalization procedure relates renormalized and 1111renormalized coupling constants by a perturbativc expansion. Thus, (2.6.50) implies that the lowest-order coefficients of the /3-function agree with those given in (2.6.36) and (2.6.37) for non-abelian gauge theories.
2
2.7
2
ua
, m i l theory of
ang
ills fields
Anomalies
e have seen in ects. 2.2. i and 2.4. that a symmetry manifests itself in relations between the Green functions of the theory, the ard identities. e derived these relations in a purely formal way not caring about the possibility that they could get destroyed or modified by the V divergences which occur in higher-order calculations. If a regulari ation scheme exists that, is compatible with the symmetry relations, then the ard identities are valid also in the uanti ed theory. For example, we have seen in 2. .1 that the ard identities for the vector currents of E arc exactly fulfilled in dimensional rcgulari ation in one-loop order, so far ustifying the applied renormali ation procedure. If a symmetry is bro en by the rcgulari ation scheme, two cases are possible. In the first case, the symmetry is restored after renormali ation and switching off the cut-off. n example is lattice rcgulari ation of pure gauge theories. oincar invariance is bro en to a discrete symmetry but restored in the continuum limit. If the symmetry remains bro en after renormali ation, i.e. if the uantum fluctuations violate the classical symmetry relations, this is called an anomaly for an exhausting review see e . This happens, for instance, for axial or chiral currents of global symmetries. The corresponding anomalies may have physical meaning. For instance, they determine the matrix element for the decay of the neutral n meson ttu . further example is dilatation symmetry > x which is bro en by all rcgulari ation schemes since they introduce a dimensionful parameter e.g. a momentum cut-off . The corresponding anomalies lead to scaling violations. These can be formulated in renormali ation-group e uations, the Callan- yman i e uations, and lead to important effects in hard-scattering processes. If anomalies occur in the oether currents of local gauge symmetries, i.e. internal currents, they destroy the validity of the classical ard identities. The ard identities have to be modified by including extra anomalous, 11011renormali able terms. I11 gauge theories, ard identities are crucial to prove rcnormali ability. Conse uently in the presence of anomalies, unitarity and gauge invariance of the S matrix are 11 more guaranteed. uch anomalous theories are physically not acceptable. The following result of Ref. a is important in this context. Theories that are invariant under space reflection or charge con ugation are free of anomalies of the ard identities related to gauge invariance. This class of gauge theories includes pure ang ills
2.7
Anomalies
27!)
theories, QED, and QCD, but in general no theories in which the gauge bosons are coupled to axial or chiral fermionic currents.
2.7.1 2.7.1.1
T h e triangle-graph a n o m a l y C u r r e n t c o n s e r v a t i o n in n o n - a b e l i a n g a u g e t h e o r i e s
Noether's theorem for current conservation in classical field theory was delived in Sect. 1.2.3.1. As preparation for the study of anomalies, we investigate the conservation of the currents of a multiple! of charged fermion fields ip(x) = (ifti(x)) which transforms under gauge transformations as a Si/>, = i(r°V),<M , 5i>i = - i ( i > T a ) i S e a and has vector couplings to nonabelian gauge fields. The fermionic part of the Lagrangian, C = Mx)(\J/> - m)(x),
Dlt = 0,, - igT'Afa),
(2.7.1)
yields the Dirac equation + gTafla(x)
(i\IP - in)il>{x) =
- m)ip(x)
= 0
(2.7.2)
and its adjoint $(x)(ifl-
(jTn4a
+ 7n) = 0.
We define the vector and axial-vector j«(x)
= Ì W t ^ W ,
and the singlet j°(x)
(2.7.3) non-singlet
currents
j Z J x ) = V>(x)7/j75T"ifi(x),
(2.7.4)
currents
= ^(x)7^(x),
j g j x ) = $(x)7/l75i/;(x).
(2.7.5)
Using the Dirac equation (2.7.2) and its adjoint (2.7.3), we calculate the divergence of the non-singlet vector current (Tj° = d"($7/lT"-i/>)
=
+
= - i $ ( g T b 4 b - m)ratp = iry^[Tn,T6]/V
- hj>Ta(-gTb4ib
= -gfabc
Ab^7>lT'^
+ m)ip =
(2.7.6)
-gf"bcA"%
Ml
v
11,, , , l ) m „ ilit'ory of Yang Mills fields
The 11011-singlet fcrmion current is not conserved. This is not surprising since Noether's theorem guarantees only the conservation of the total current, the sum of the fermionic and the gauge-boson current. The latter does not vanish in the noii-abelian case because the gauge bosons are charged. Since the structure constants fnbc vanish in the abelian case, the vector current is conserved in QED and similar models. However, t he non-abelian fermion current is conserved covariantl . This can be seen by moving the term on the right-hand side of (2.7.6) to the left and identifying the resulting expression as the covariant derivative, (D jll)a(x)
= 0.
(2.7.7)
A similar calculation yields (D'ljr)>lt)n (x) = 2\mtj (x)^Taip(x)
= 2impa(x)
(2.7.8)
for the divergence of the non-singlet axial-vector current with the pseudoscalar density pa(x). The singlet vector current is conserved, dft (x)
= 0.
(2.7.9)
The divergence of the singlet axial-vector current is given by the pseudoscalar density, d'lj%t(x)
= 2im (x)To >(x)
= 2i mp (x),
(2.7.10)
and vanishes in a inassless theory. For theories with parity violation like the Electroweak Standard Model, we have to consider chiral couplings between the fermions and the nonabelian gauge bosons. For this purpose we use left- and right-handed fields and currents,
/
1
" 75 .
,
1
+75 .
(2.7.11)
2.7
Anomalies
27!)
In the masslcss case, the chiral fennion Lagrangian C = ^ ¡ M l
+
faM*.
(2.7.12)
implies the covariant conservation of the non-singlet currents, (D>'jltllT(x)
= 0,
(D£jKili)a(x)
= 0,
for m = 0,
(2.7.13)
and the ordinary conservation of the singlet currents, = 0, 2.7.1.2
d"&Jx)
= 0,
for m = 0.
(2.7.14)
W a r d i d e n t i t i e s of c u r r e n t G r e e n f u n c t i o n s
Anomalies were found first, in the Green functions of axial-vector currents in abclian theories, e.g. QED, [St49, Sc51a, SuGG, Ad69, Be69]. Let us consider Green functions of abclian local current operators, i.e. vacuum expectation values of time-ordered products of the currents. From (2.7.7) and (2.7.8), Ward identities for the current Green functions follow just like those discussed in Sect. 2.2.G.4. The simplest Green function in which anomalies occur is the three-point function of two vector and one axial-vector currents, 18 T™A(xi,
X2, x3) = ({)\Tj/l(x\)j„{x2)j^p{x3)
|0).
(2.7.15)
This Green function is symmetric under the exchange of the two vector currents. Its naive Ward identities read d ^ T ™ A ( X l , x 2 , x 3 ) = % 2 T ™ ( x l t x 2 , x * ) = 0, d"xzT^(xux2,x3)
= 2mi <0|
Tjli(xl)jv(x2)p(x3)\0)
= 2m\T^vp{xux2,x3).
(2.7.16)
The equivalent relations in momentum space are p ' i r ^ { p u p 2 l V 3 ) = p v 2 T ^ [ p u p . h p z ) = o, p W p A ( P ' - P 2 , P 3 ) = 2mT^p(pup2,p3)
(2.7.17)
with pi + P 2 + P : i = 0. Similar naive Ward identities may be derived for other Green functions. 1H
As will bo seen in the following, the chir;d anomalies involve the tensor e,tvpa. Consequently, Green functions of two currents are anomaly-free since they have only two Lorentz indices and depend only on one momentum.
2
2
ua
, m i l theory of
ang
l '-i
ills fields
ii it
/1
, -
p
;
Fig. 2.11 Feynman diagrams for the triangle anomaly
2. .1.
C a l c u l a t i o n of t h e t r i a n g l e - g r a p h a n o m a l y
The first-order uantum corrections to these relations are obtained from the one-loop diagrams of Fig. 2.11. The two diagrams for T pi,p2, are linearly V-divergent. Conseuently, they must be regulari ed. e use the auli- illars regularization, i.e. we subtract the same diagrams with mass m. From the Feynman rules we get each current contributes a factor ,,, i or ,
ti + T
-
1
7
(
- h )
7, (
1
fa - m )
+
1
A
2. .1 fter subtraction, the integral is finite and can be treated in the usual way. I11 order to test the vector ard identities we multiply with p\l. decompose fa = (t + fa ~ m) -( i-
m) =
and obtain
' ^
1
For dimensional regulari ation see ect. 2. .1. .
m) + ( -
m),
2.7
+ 'IV [(^ + fa, - m r - {fi + h -
1
^
Anomalie»
~ ™) _1 7p75
- fa -
m _1
)
7p75
27:*
(2.7.19) - (m -> M ) j .
Since the integral is finite we are allowed to perform the shifts k —> k -I P2, k —> k + p'2 — p\ and find a vanishing integrand. Consequently, the vector Ward identity is fulfilled. This is the case for all correlation functions involving only vector currents. For the study of the axial-vector case, we use fans = - {fa -h ^2)75 = {fi-fal
- ™<)75 + 7 5 + i>\ - " 0 + 277175
= {fi - fa -
7n
hr> + JrM + fa - m) + 277175,
which follows from the anticommutativity of 75 in four dimensions. With help of translation invariance and Dirac algebra we find
7 «
A
=
2i
/
(0f
(2.7.20)
j m T V ^ - m r S ^ - j k
-
m
) - ^
+
fa - m ) ~ l ]
+ m l ï - ^ d i - m ) " ^ ^ - f a - m)~l 75 (J£ + fa - m ) " 1 ] - M T t ^ - M r ^ - l h - MTr[7„(#
~ My^W+ti
- M ) - 1 7 m ( ^ - f a - M)~lTM
~
M)~l]
-I- fa - M ) - 1 ] j .
This has to be compared with The corresponding diagrams are those of Fig. 2.11 with 7 p j s replaced by 75. They give an UV-finite expression which does not require régularisation, rp
V 'n,
/ll/
. f = ,
d 4k (2 721)
./(W
|iv[7/t(^ - m ) -
-
1
+ Tr[ 7 t /(£ - m ) "
^ - fa - m ) - I 7 5 ^ + fa - m ) " 1 ] 1
^ - y , - m)~xrM
+ h ~ m)" 1 ] } ,
230
2 Q u a » ! , m i l theory of Yang Mills fields
and is identical to the first two lines of (2.7.20). Thus, we arrive at = 2"li T ™ P +
i
(2.7.22)
The integral d k
I
(2tr)4 (ff + - h + M) 75 (*f + ¿1 + M)) 2 2 2 2 2 ( ¿ 2 - M )[(k - p2) - M ][(fc + p,) ~ M ] + ( M ) *+(*/, 2) j
(2-7.23)
involves the regulator terms. It is UV-finite but non-vanishing. We have an anomaly. The Dirac trace yields AiMe^papIp^, independent of k. The second term gives the same contribution. In the remaining scalar integral, p\ and P2 can be neglected relative to M leaving a special case of the tadpole integral (2.5.57), namely A § \ M ) = - 1 / ( 2 M 2 ) . Thus, we find A,i
"
=
=
^ w P i r f -
(2-7.24)
This anomaly is finite and independent of the masses in the triangle diagram. A similar anomaly occurs in the triangle diagram with three axial-vector currents. The modified (anomalous) tex functions read P'W™ = P P Z tfTZ*
Ward identities
A
for the regularized one-loop ver-
=
= 2 ™\T™> +
(2-7.25) ¿eMVfit,pW.
As they stand, they may be interpreted in the following way: the vector currents are anomaly-free whereas the axial-vector current acquires an anomaly from the quantum fluctuations. In spite of the fact that the anomaly is finite one might wonder whether it can be compensated by suitable renormalization counterterms for the composite operator T ^ / V These counterterms must have the appropriate Lorentz and
2.7
Anomalies
27 )
symmetry structure and must be renormalizable (cf. Sect. 2.5.1.5). The only choice is = Ce,wpa(p\
(2.7.26)
- vzY-
Since this term contributes to all divergences p t T ^ f , p ^T^ ^, and p ^T^^, also the vector ard identities get anomalies for C ^ 0, but for a specific choice of C the ard identity for the axial-vector current can be made anomaly-free. The counterterm redistributes the anomaly but cannot compensate it. For chiral currents, the renormalization can be used to distribute the anomalies equally over all currents. There is no renormalization scheme that is anomaly-free for this simple model of a charged spinor field and its currents. In the case of non-abelian currents, the coupling matrices T a enter. The general expression for the triangle-graph anomaly of the non-singlet axial vector current reads Aabc =
T
({Ty }7«)
(2.7.27)
Also in the general case, the anomaly appears as an extra term in the identities of Sect. 2.7.1.2.
2.7.1.4
An application: 7r
ard
77
As an example for the physical relevance of the anomaly of the axial-vector current we calculate the decay width of the neutral TT meson. (A discussion of chiral symmetry in C can be found in Sect. 3.5.1.) The invariant Smatrix element is obtained from the corresponding reen function using the LS formalism (cf. Sect. 2.1.3), (7,PI;7,P2 S 7
,P
=
P
PL(P
2
-
M ;
( P .
( P 2 )
(2.7.2 )
This reen function can be written as a three-current reen function using the Maxwell equation = e f and the C C (partiall conserved
230
2 Q u a » ! , m i l theory of Yang Mills fields
axial-vector current) hypothesis. This relates the divergence of the SU(2)flavour-isotriplet axial-vector current, i.e. a pseudoscalar density, to the corresponding 7T-meson field operator [Na60], di'fijx) = ^
Mfrix),
with the pion decay constant (A.1.11). We obtain <7,Pi;7,P2|S|Tr°,p) =
jIil =
(2.7.29)
= 130 MeV and the Pauli matrices r"
JnM*
~ MÎ)e»( P x )e»( V 2 )
x i / T ^ v ^ A ' 0 ( - P l , - p 2 , p ) ( 2 7 r ) 4 r f ( 4 ) ( p - pi - p 2 ),
(2.7.30)
where we have used the fact that the commutator terms vanish when extracting the derivative of the axial-vector current from the time ordering. Neglecting the anomaly, this expression can be evaluated in the soft-pion approximation, i.e. for p 2 = 0, as follows: we write the most general decomposition of the tensor ' ' ,A '°(—pi, -p2>p) taking into account Bose symmetry, electromagnetic current conservation [the first two identities of (2.7.25)], and the negative parity of the axial-vector current. Contracting this expression with pp yields PPT^Jy'r'K\-Pu-P2,p)
= £fivpoPiP2P2f(P2)
(2.7.31)
with some coefficient function f(P~)- Since no strongly interacting zero-mass particle is around (m 2 ^ 0), T / ^'p' V ' 7,A ' 0 and thus / ( p 2 ) does not contain any poles for p 2 0. Therefore, ppT^l;Jy,r,A'° has to vanish for p 2 -> 0 and the pion decay would be strongly suppressed. This is the Sutherland- Veltman theorem [Su67, Ve67]. However, in the presence of gauge fields, the PCAC relation has to be modified by the anomaly (cf. Sect. 2.7.2.1), d"jllt(x)
=
- JLeW°FliU{x)Fpa{x)TT
( V y ) , (2.7.32)
2.7
Anomalies
27!)
where ¡')„, is t,he electromagnetic Held strength. The trace corresponds to the one in (2.7.27) and results from the charges in the electromagnetic currents and the SU(2) matrix in the isospin current. As a consequence, we obtain a contribution of the anomaly in (2.7.30) which gives a non-vanishing result in l he soft-pion limit. Thus, the amplitude is given in this limit by the anomaly, .IIKI we find for the invariant decay matrix element, M(n°
->
77
) = ^
Tr ( V y ) eliupae"(pl)e"(p2)p'y2.
(2.7.33)
The trace gets contributions from the constituents of the pion, the up and down quarks with charge Q,,, third component of isospin / 3 = T 3 /2, and Nc colours,
V
q
'
The total decay width is given by [cf. (1.3.12)]
pols.
(The factor 1/2 accounts for the two identical particles in the final state). The polarization sum is performed with ^ x E , l ( \ ) e " ( \ ) = -g1'", leading to /1
i
£ I^P
2
¡If t = - f .
The resulting distribution is isotropic, and the angular integral gives -1 tt. Putting everything together, we finally obtain =
f '
(2.7.35)
Inserting / T = 130 MeV and m„ = 135 MeV, this yields P(7r° -> 77) 7.81 eV in good agreement with the experimental result [PDG00] rfiXp(7r0 77) = 8.3 ±0.GeV. Depending 011 the attitude, this agreement is used an argument for three colours, the PCAC hypothesis, or the importance axial anomalies.
= -> as of
2 «
2
u a n t u m theory
2. .1.
f
ang
ills fields
in dimensional regulari ation
The existence of anomalies of axial or chiral currents shows that no chirally invariant regulari ation scheme exists. Chirally invariant theories arc massless. The auli-Villars scheme brea s chiral symmetry by introducing heavy fermions. In the dimensional regulari ation scheme, the anomaly of the axial-vector current is related to the properties of the matrix 75. This is defined in four dimensions as the completely antisymmetric product of all four irac matrices, 4
T
V
,
2. .
where the totally antisymmetric tensor e upa is given in nition implies the properties 2
tT TV T V
V T V
i,
V
,
Tv
-4ie
T
. 1. . This defi-
, fV
,
,
2. .
which we have used in the calculation of the triangle-graph anomaly. If the -dimensional matrix is defined in such a way that it anticommutes with , -
-f
2. .
the following trace relations can be derived see e.g. Ref. Co 4 1
(
,
- 4 Tt- t V t
(
- 2 T-
= ,
i ,
etc.
2. .
The last e uation shows that this is not a suitable analytic continuation of the four-dimensional ob ect since it enforces a vanishing trace for the product of four irac matrices in clear contradiction to the four-dimensional case 2. . . everal ways out of this dilemma have been proposed. lowing ones
e mention the fol-
2.
nomalies
2
The four-dimensional is used also in dimensions. T h i s t Hooft Veltman or reilenlohner aison scheme can be handled by splitting -dimensional space into ordinary four-dimensional physical space plus an extra ( - 4 -dimensional space tH 2a, r , Co 4 . The cyclicity of the trace of
irac matrices is given up
o 2.
In one-loop calculations in the anomaly-free Electrowea tandard odel it is possible to use with certain care the anticommuting , in -dimensional calculations together with the last relation in 2. . e , despite the fact that this scheme is in general inconsistent. The existence of axial anomalies can easily be seen in the t Ilooft Veltman dimensional r gularisation scheme. The divergence of the axial-vector current. contains a term of the form ip(x){,yIJ,-y^}d,lip(x). The anticommutator is effectively proportional to 4. Therefore, it does not contribute in lowest order but gives a finite result, if it gets multiplied with V singular cx ( — 4 - 1 expressions. In general, operators that vanish in 4 dimensions but are non-vanishing in 4 dimensions are called evanescent, operators. et us finish this section with a general remar . The non-existence of a chirally invariant r gularisation scheme is the formal origin of the axial or chiral anomalies. I11 the path-integral formulation, the anomalies result from the non-invariance of the integration measure under chiral transformations F11 . The anomalies can be obtained directly from the path integral and coincide with those resulting from one-loop calculations. s an important by-product of this calculation one obtains the relation of the anomaly to topological charges and to the number of ero modes of the covariant irac operator ore details are discussed in ect. 2. . . .
2. .2
n o m a l i e s in g a u g e theories
In the last section we have calculated Green functions of fermionic currents. those fermionic currents which are the group can be expressed in terms of the clearly in the path-integral formalism techni ue c 2 .
the triangle-graph anomalies of the I11 gauge theories, the anomalies of oether currents of the local gauge gauge fields. This can be seen most Fu or within the point-splitting
230
2 Q u a » ! , m i l theory of Yang Mills fields
2.7.2.1
Anomalies and gauge fields
In the non-abelian case, anomalies do not only arise in the triangle diagrams with an odd number of axial-vector currents. They occur also in box and pentagon one-loop diagrams with an odd number of of axial couplings. Explicit calculations [Ba69, We71] show that the total anomaly can be expressed in terms of an operator composed of gauge fields. For the singlet axial-vector current in mass less gauge theories, where the fermions are coupled by their vector currents to the gauge fields, it reads = -^-e^F;w(x)Fba(x)
d»jllt(x)
Tr(TaTb)
(2.7.40)
with the non-abelian field strengths F^(x)
= dtlAl(x)
- d„A1{x) + g f ^ A ^ A K x ) .
Using !Fllty(x) = gTaF°„(x),
(2.7.41)
this can be written in the form
= - [ ¿ 2 ^ TriF^WFpAx))-
(2-7.42)
This expression is proportional to the Pontryagin density indicating a close relation between anomalies and topological properties of gauge theories. This is considered explicitly in Sect. 2.9.3. The result (2.7.42) can be rewritten in terms of the potentials A,L(x) — gTaA't\(x), d
'liL = -
TT
- fiAuA^Aaj .
(2.7.43)
For the non-singlet current, assuming covariant conservation of the vector currents, Bardeen [Ba69] found the result (D'\75,„r
= -
T
r
[TaTtli/{x)Tpa(x)).
(2.7.-14)
For chiral theories, assuming that the classical conservation laws are broken in a symmetric way, Bardeen [Ba69] obtained for the non-singlet currents: (D"ju,„)a = 1
(2-7.45) T" (AL,vdpA^a
-
^AL^AI^AL,^
2.7
A n o m a l ¡os
281
l'br right-handed currents, L has to be replaced by R and the total expression multiplied by - 1 . In the abelian case, (2.7.40) remains valid but the field strength is linear in (:/;). Expressed in terms of the gauge field, the anomaly reads
= - ¿ ^ ( ^ M * W
p
A
a
{ x ) .
(2.7.40)
The Fcynman rules for this local operator give V
=
(2-7.47)
i.e. the same result (2.7.24) as for the one-loop triangle diagram.
2.7.2.2
Cancellation of anomalies
If anomalies occur in the internal currents of gauge theories, i.e. in those currents that constitute the fundamental interaction between fermions and gauge bosons, they enter the basic Ward identities (2.4.75) and destroy the renormalizability of the theory. Therefore, it is important to find criteria for the absence of these anomalies in gauge theories. For this purpose it is sufficient to consider the algebraic prefactors of the anomalies (2.7.27) of the left- and right-handed internal currents of the Dirac fields. These contain the hermitian representation matrices 7'," and Tj" of the generators of the gauge group, i.e. of the couplings of the fermions to the gauge bosons, in the form A"bc oc Tr(lt
{ t £ , T £ } ) - Tr(T£ { t ^ } ) ,
(2.7.48)
where the trace extends over all fermionic degrees of freedom. The anomaly vanishes if the trace vanishes, Tr (r L a { î t T ï } ) - T r { t ^ T ^
= 0.
(2.7.49)
The algebraic anomaly factor is zero if these traces vanish separately, or if the contributions of the left-handed and the right-handed couplings compensate each other. In the first, case the representations are called safe. The safe
230
2
ua
, m i l t h e o r y of
ang Mills fields
representations include, in particular, all real representations. A representation is called real if (T a )* = -UTaU with a unitary matrix U, i.e. if the negative of the conjugate-complex representation is unitary equivalent to the representation matrix. 20 Hermiticity implies that T = (T") 1 = (T")* r and consequently for real representations
Tr(r° { T 6 , T C } ) = Tr((T a )* T { ( T ' T T , ( T c ) ' t } ) = Tr((Ta)' { ( T ^ C H ' } ) = - T i - ( r " { T 6 , T C } ) = 0 . Algebras that have only safe representation are called safe algebras. These include, in particular, those that have only real representations. The simple Lie algebras have been classified by Cartan [GT]. The following ones are safe [Ge72a]: SU(2) ~ SO(3), SO (AT), > 4, Sp(27V),
f- C,
0(2), F(4), B(6), E(7), E(8).
(2.7.50)
Direct sums of these algebras are also safe. Consequently, anomalies can only occur for gauge algebras that involve abclian factors U(l) (our example with one fermion field was an U ( l ) model) or unitary algebras SU(A0> > 3 [SO(6) ~ SU(4)]. The symmetry groups of the Standard Model are the non-safe SU(2) x U ( l ) and SU(3). Therefore, cancellation of anomalies must be obtained in another way. The second possibility, the cancellation of anomalies between left-handed and right-handed contributions, is guaranteed if the left- and right-handed couplings are unitary equivalent, T" = VT^V 1. Theories of these type are called vector-li e. The internal currents that arc coupled to the gluons in QCD are covariantly conserved vector currents. Therefore, the Lee identities of QCD are anomaly-free. On the other hand, the gluons can be coupled to external axial-vector currents, hi this case, the external, singlet axial-vector current j j t has an anomaly owing to Tr(T a T") ^ 0. But. the renormalizability of QCD is not destroyed by this external current. Physical effects of this anomaly are discussed in Sect. 3.5. The lectro ea Standard odel is built on the non-safe group SU(2) x U(l) and contains left- and right-handed currents which are coupled to the ""The notion comcs from mathematics where iT
is used instead of T .
2.7
Anomalies
27!)
gauge bosons. Compensation of anomalies can be realized only with special representations for the leptons and quarks. Since these are identical for the different families it is sufficient to discuss cancellation in one family. The generators of the group are weak isospin /";, hypercharge Yv,, and the electric charge = '. + Y 2. While the electric charge is the same for lefthanded and right-handed fermions, i.e. vector-like, the isospin I" is only non-vanishing for left-handed fermions. The following combinations occur in l.lic triangle diagrams: T>( £{ *, £ ),
T\(
{C,rl ),
Tr (C
2
),
1V(Q 3 ).
Since the group SU(2) is sale, the first term vanishes. The last term is purely vectorial and therefore gives a vanishing contribution. The second term gives
\( {C,lt )
= VTr(<2) =
- ' ^
f -
(2.7.51)
Because the charges of the members of all isospin doublets differ by one, the third term gives no independent constraint. Altogether, the condition for cancellation of the anomalie.it in the Electroweak Standard Model reads £ Q , = 0. f
(2.7.52)
This requires for one family, e.g. ( f e , e , u,d),
(0-l) +
A f c ( | - i ) = - l + ^ = 0 .
Thus, three quark colours are needed for the compensation of anomalies. In this context also the non-renormalization theorem [Ad69a, BaG9] is important. If the one-loop diagrams are anomaly-free, all higher-order diagrams are anomaly-free, i.e. the chiral anomalies are determined by the one-loop diagrams. This is a consequence of the fact that a chirally invariant regularization (method of higher covariant derivatives) exists for all diagrams beyond one loop.
230
2 Q u a » ! , m i l theory of Yang Mills fields
2.7.2.3
Structure of anomalies in non-abelian gauge theories
Our discussion of anomalies was based on the evaluation of one-loop diagrams. The direct treatment of the regularized path-integral formula gives the same results. An even more general approach considers the Lee identity (2.4.77) for the generating functional of vertex functions P. This can be written in the form [cf. (2.5.140)] 0,% f = 0
(2.7.53)
with ¿Bp defined as in (2.5.141). In the presence of anomalies the right-hand side does not vanish, S f f = A.
(2.7.54)
Applying £?,-. and using ByBy = 0 (2.5.142), we find that the anomalous term A must obey
Bp A = 0.
(2.7.55)
Power counting implies that A consists of polynomials in the fields with dimension 5 and ghost number +1. An analysis similar to the one in Sect. 2.5.4.2 of all possible polynomials reveals that A must contain the ghost field ua(x) and be of the form [Be76a] A = J i\*xpaA(x)ua{x),
(2.7.56)
where paA(x) depends only on the gauge fields Aft(x). erator of gauge transformations of the gauge field
The infinitesimal gen-
has the property w"{x)w\y)
- wb(x)wa{y)
= gfabcwc{x)SW{x
- y).
(2.7.58)
2.8
Infrared a n d col I ¡near s i n g u l a r i t i e s
28!)
Insert ing this into (2.7.55) and using the definition of jBp. as in (2.5.141) and (2.7.5G), yields the Wess Zumino consistency condition [We71] for p'\(x), wa(x)pbA(y)
- wb(x)paA(y)
= yfabcpCA(x)SW(x
~ ?/)•
(2.7.59)
'Phis condition fixes the anomaly up to normalization. The result for the anomaly is identical with the symmetric anomaly (2.7.45) obtained by the evaluation of Feynman diagrams. More results on the relation between anomalies and the topological properties of gauge theories using methods of differential forms or fibre bundles are presented in Sect. 2.9.3. An almost complete treatment is contained in [Be96].
2.8
I n f r a r e d and collinear singularities
Besides UV singularities, which result from infinite loop momenta and are removed by renormalization, quantum field theories contain singularities that are connected with finite momenta. The singularities of the Feynman integrals are called Lamlau singularities [La59] and occur in general only for special values of the external momenta. Some of these singularities occur, however, independently of the external momenta. Since they are always related to vanishing masses they are generically called mass singularities. There are two types of mass singularities. The infrared (III) or soft singularities are connected with vanishing momenta and are thus long-distance effects. The collinear singularities, on the other hand, are related to collinear light-like momenta. Soft and collinear singularities result not only from virtual soft or collinear massless particles in the loop integrals. The real emission of soft, or collinear massless particles gives rise to similar singularities when integrating over the phase space of the emitted particles. The singularities resulting from real and virtual particles cancel each other in suitably defined observables. Starting from the general singularity structure of Feynman integrals we discuss the origin of these singularities in loop integrals and their form in Sect.. 2.8.1. In Sect. 2.8.2 we show that the infrared singularities cancel between virtual and real contributions and that the remaining terms exponentiate. The treatment of collinear singularities in QED is discussed in Sect. 2.8.3.
2M
2
2.8.1
Q u a n t u m theory <>f Y a n g Mills fields
T h e origin of m a s s singularities
Mass singularities of Feynman amplitudes have been investigated extensively in Ref. [Ki62]. In order to classify these singularities it is useful to consider the general singularities of an arbitrary Feynman graph G. The corresponding amplitude has the generic structure
Here pj denote the external momenta, i the loop momenta, and 7»j the internal masses. The momenta <ji that (low through the I internal propagators are linear combinations of pj and . The numerator (pj, i,mn), which results from vertices and fermion propagators (in gauge theories we work in the Feynman gauge), is assumed to be a regular function of the momenta and masses.
2.8.1.1
Singularities of complex integrals
The possible singularities of Feynman amplitudes have been discussed in Ref. [Ed66]. Leaving aside the UV singularities at infinity, i.e. k[ —> oo, the function FQ can only become singular if the integrand has a singularity on the integration path. This requires that one or more propagators develop a pole. Since the integrals in (2.8.1) are actually contour integrals in the complex plane, these poles can in general be avoided by deforming the integration contour. For any set of external momenta and masses for which this is the case, the function Fa is analytic in all these variables. There are two situations where the poles cannot be circumvented by contour deformation. This can be best illustrated by a simple example. Consider a general function (z) of a complex variable z defined by an integral along a contour C in the complex plane with fixed end points a and , (2.8.2)
The integrand f(iu,z) is assumed to be analytic in the variables and z, except, for singularities located at = ius(z). Evidently, (z) is analytic for any 2 for which the integration contour encounters no singularities. Even if
2.8
Infrared and col I ¡near singularities
28!)
one of the singularities ws(z) happens to lie on C, I(z) remains analytic as long as the contour can be deformed to avoid it. Therefore, singularities of l(z) can only occur in two cases. 1.
One of the singularities ws(z) coincides with one of the endpoints of the contour C, ws(z) wa or WF, for z —> ZQ. Then, zo is called an end-point singularity.
2.
The contour is trapped between at least two singularities w\ and w»2, i.e. as z —> zo, u>\ and w2 approach t he contour from opposite sides and singularity. coincide; in this case ZQ is called a pinch
The singularities produced by the above mechanisms are usually branch points. bet 11s now turn to a function of several complex variables w = ( »>,) and z = (zj), 1(7.) = J/ydli c ' " ; i ) / ( w i z ) [Ed66]. The boundary of the integration domain, the hyper-contour H, is specified by a set of analytic functions S r ( w , z) = 0, and the singularities of the integrand take place on analytic manifolds <Ss(w, z) = 0. The integral becomes singular if the hyper-contour II is pinched between two or more surfaces of singularities or when a singularity surface meets a boundary surface. More precisely, it can be shown that a necessary condition for a singularity is the existence of a set of complex parameters A.s- and A,, which are not all equal to zero so that simultaneously A.s.<S.s(w(), zq) = 0
for all s,
d
Ar
for all r, = 0
for all i. (2.8.3)
The last condition requires that at the pinching point (wo, Zo)> the normals to the hyper-surfaces are linearly-dependent, i.e. tangent for two hypersurfaces. Whether there is actually a singularity at this point requires a detailed study.
2.8.1.2
The Landau equations
Let us now apply these results to the Feynman integral (2.8.1). The boundaries of the integrand are at infinity. In the following, we disregard the endpoint singularities, which are actually UV singularities, and introduce
230
2 Q u a » ! , m i l theory of Yang Mills fields
no term <S r (w,z). The singularities of the integrand are given by the equations S{ = qf - mf ~ 0, i.e. the mass-shell conditions, and the necessary conditions (2.8.3) read A ¿{qf - mf) = 0
for all i = 1 , . . . , 7,
(2.8.4)
and
=0
i = 1 , . . . , L.
(2.8.5a)
These are the Landau equations [La59]. Using the fact that the internal momenta are linear combinations of the external and loop momenta with coefficients ±1 and that those internal momenta that depend on the loop momentum hi lie on a single loop the second equation can be written as £ ±Xiqi iec,
= 0.
(2.8.5b)
Equation (2.8.4) implies that each internal line is either on its mass shell or the corresponding A, vanishes. In the latter case this line does not appear in the other Landau equations. Thus, each singularity can be associated with a reduced diagram where a subset of lines has been contracted to points and all remaining ones are on their mass shell. Landau equations can also be derived for other representations of the Feynman integrals, their solutions being of course imique. The solution corresponding to Aj ^ 0 for all i is called the leading singularity, while all others are non-leading. The non-leading ones are leading in the corresponding reduced diagrams. The Landau equations determine the possible locations of all singularities of Feynman integrals, the Landau singularities, as a function of the external momenta 7;,-. In general, the singularities are branch points, such as threshold singularities, but do not necessarily give rise to divergences in transition amplitudes.
2.8
2.8.1.3
Infrared a n d col I ¡near s i n g u l a r i t i e s
28!)
S i n g u l a r i t i e s in physical regions
We now restrict ourselves to singularities that occur in the physical regions. P'or these singularities, it can be shown that the loop momenta ki must be real and the variables A, real and positive at the solutions of the Landau equations. Since the external momenta pj are real in the physical regions and each qt is a linear combination of the ki and pj, also the 0) with the proper time of flight, of particle i between the two vertices at the endpoints of line i. Then, the space-time displacement between these two vertices equals Ax — A and (2.8.5) guarantees that the total space-time displacement along a closed loop is zero. For non-leading singularities, the corresponding reduced iliagrams provide physical pictures in the same way.
2.8.1.4
M a s s singularities
A special kind of Landau singularities are the mass singularities. Firstly, they give rise to divergences and not only to branch points. Secondly, they appear for arbitrary directions of the external momenta, and their appearance depends only on the masses of the external and internal particles. An extensive discussion of mass singularities of Feynman amplitudes has been given in Ref. [KiC2], We sketch the main results in the following. We assume that the amplitude contains no identical propagators. 21 Let us consider the ease of one-loop diagrams, where the integrals have the general structure (cf. Fig. 2.8)
FG(pj,rn„) 21
f = / d ' f c N ( P j , k,mn) ./
' i TT ' . ^, qf ~ mf + le
(2.8.6)
Amplitudes with identical propagators can be obtained from the considered ones by differentiation with respect to appropriate masses.
230
2 Q u a » ! , m i l theory of Yang Mills fields
with (¡i = Aj+^JJJ, pn = k+Pj. If the singularities have to be independent of the directions of the external momenta, the corresponding Landau equations
Ai{qf - m 2 ) = 0 l ] T V / i = 0, i=i
for all t = 1 , . . . , I,
(2.8.7a) (2.8.7b)
must not involve scalar products of different external momenta. Equation (2.8.7a) implies that each (/, for which A, ^ 0 must have a norm independent of the orientation of the external momenta, i.e. it must be either light-like (for nii = 0) or proportional to one of the external on-sliell momenta. Since this can be fulfilled for at most three neighbouring qi - qj = Y^nJjPni propagators, provided the loop momentum is chosen appropriately. Since (2.8.7b) relates the three corresponding momenta
2.8
Infrared and col I ¡near s i n g u l a r i t i e s
28!)
Fig. 2.12 Diagrams leading to IR (left) and collinear (right) singularities
called infrared singularity m r = 0,
and defined by the conditions
p 2 _ , = (Pr - P r _ ! ) 2 =
rn2_,,
p2 = (Pr - P r + 1 ) 2 = m 2 + 1 .
(2.8.8)
Typically, this singularity occurs for photon or gluon exchange between external particles. It gives rise to divergences proportional to logm r for
mr
0.
The IR singularity is absent, if the pole is cancelled by a zero in the numerator TV, i.e. by a momentum k in the numerator. This may typically arise from the numerator of the propagator r or the two vertices at its endpoints. Therefore, in particular, massless neutrinos do not lead to IR singularities. If the Landau singularity results from two propagators, i.e. Ar ^ 0 and A., ^ 0 but A, = 0 for i ^ r, s, the Landau equations imply q2 - m 2 = 0,
q2 - m2 = 0,
A r q r + Asq„ = 0.
(2.8.9)
The corresponding subdiagram is shown in Fig. 2.12b. The momenta qr and qs have to be proportional to each other, i.e. collinear, and qr = ±—qs, ms
qf = ™2,
{qr - qs)2 = (m r ± m , ) 2 .
(2.8.10)
The last condition is only angular-independent if r and s are neighbouring propagators, i.e. s = r ± 1. Then (qr - qs)2 is the square of the external momentum pT-\ or pr, i.e. an external mass squared. In general, these singularities are the well-known branch points resulting from two-particle thresholds.
2!)2
2
Qiuuil.mil theory of Yang Mills fields
In order to determine whether a divergence actually occurs, we set s = 7 -1-1, parametrize the loop integral as qr — k and c/ r+ | = k + p r , and use the Sudakov decomposition for the integration momentum [Su56], k = ypr + zn -f kx, v2
= 0,
p r Ti i
0,
prkjL
= 0 = nk±,
k±
= (0, k j . ) ,
(2.8.11)
where n is an appropriate vector. Using I d4k = |p r n| I" d2kxdydz
= 7r|prn| J dk^dydz,
(2.8.12)
the relevant part of the integral reads
/
n\prn\
J
+
pr)^-m2r+x)
2 d kj_dydz
1 \p2y2 + 2 p r n y z - k2, - rn2] 1 b2( 1 + y)2 + 2pTn{ 1 + y)z - k'i - m2+I]'
(2.8.13)
For Pr = (Qr+I ~ qr)2 = o,
m 2 = 0,
m 2 + 1 = 0,
(2.8.14)
this integral diverges logarithmically. The divergence originates from the part of the integration region where k', —> 0 and z —> 0, i.e. k —» ypr. Since this singularity is associated with collinear momenta, k oc pr, and massless particles mr = m r + ¡ = 0, it is called collinear singularity. It yields singular terms of the form log7n = {log77ir, logm^, log \f\p 2 ] }, depending on the variable that is sent to zero at the end. Condition (2.8.14) implies that all three lines meeting at a three vertex must be massless to produce a mass singularity. The mass singularity cannot be enhanced by other propagators, since the corresponding requirements q2 = mf depend on the external momenta. In practice, III and collinear singularities can overlap, leading to divergences of the form (c log A + log rn) log m. 2 2 The mass singularities of any one-loop ""Here A regularizes the IR singularities [as m r in (2.8.8)] and m the collinear singularities.
2.8
Infrared and col I ¡near singularities
28!)
diagram can be written as a sum of terms of this kind and single logarithms log A and log m. This analysis can be extended to the case of more loops [Ki(52]. The requirement, that the singularity should be independent of the relative orientation of the external momenta implies immediately that only one vertex with an external line may exist between any pair of internal propagators that define the Landau singularity. Thus, the singularity is localized to a neighbourhood of one external vertex. Without enhancement, only the IR and collinear singularities discussed above are found in multi-loop diagrams. However, if there is strong enough enhancement, new types of mass singularities may occur. At a mass singularity, a Feynman amplitude can be reduced to a product of irreducible components corresponding to reduced diagrams. For example, the reduced diagrams of Fig. 2.12 are those where the bubbles are contracted to a vertex. The strongest mass singularities are found in those reduced diagrams that involve only a single loop. Loops with two lines contribute primarily a logm factor, loops with three lines may contribute a factor (clog A + logm) log in. The parts of Feynman diagrams that give rise to these singularities are associated with one or two external lines corresponding to collinear or IR singularities. The relevant diagrams are shown in Fig. 2.12 and become singular if the conditions (2.8.8) or (2.8.14) are fulfilled. Mass singularities do not only result from loop integrals but also from the integration of the phase space of physical particles created in some reaction. The mass singularities in the phase-space integrals can be related to the absorptive (imaginary) parts of loop integrals via the optical theorem. It can be shown, that the mass singularity of the absorptive part may not be as strong as that of the dispersive (real) part. Furthermore, the relation bet ween mass singularities in virtual and real diagrams leads to cancellations when these contributions arc added (cf. Sects. 2.8.2.3 and 2.8.3.9). The physical origin of mass singularities is the presence of degeneracies in initial and/or final states. In the case of IR singularities, the states of a charged particle with an arbitrary number of soft photons are nearly degenerate. They are indistinguishable for a detector with finite energy resolution. In the case of a collinear singularity, the state of a light-like particle with momentum p is degenerate with the states of an arbitrary number of lightlike particles with the same overall quantum numbers with momenta a,p,
230
2 Q u a » ! , m i l theory of Yang Mills fields
if p = Ylin 'i- These states are indistinguishable for a detector with linite angular resolution. In physical observables all these states contribute simultaneously and the singularities cancel. A general theorem concerning mass singularities is the inoshita ee auenberg ( ) theorem [Ki62, LeG'l]. It states that, as a consequence of unitarity, transition probabilities arc finite when summed over all degenerate states in initial and final states. This is true order by order in perturbation theory for unrenormalizcd quantities or in renormalization schemes that do not introduce mass singularities via the renormalization constants. For the cancellation of IR singularities in QED with finite fermion masses it is sufficient to sum over degenerate final states, i.e. all states with arbitrary numbers of soft photons in the final state. This is the loch ordaiee theorem [B137]. In general one has to include contributions involving real radiation of soft and collinear particles in order to obtain observables that are free of mass singularities. The mass singularities typically cancel between real and virtual contributions. The cancellation of infrared and collinear singularities is discussed in the Sects. 2.8.2.3 and 2.8.3.9.
2.8.2
Infrared singularities
In the previous section we have seen [cf. Fig. 2.12a] that 1R singularities in loop diagrams result from the exchange of a massless particle between two on-shell external legs, provided that the emission of this massless particle does not change the masses of the emitting particles. In physically relevant theories these massless particles are ghions or photons. In this section, we study the IR singularities in QED resulting from both virtual and real corrections. We give their general form and show how they are re-summed to all orders. We demonstrate that in QED the IR singularities resulting from virtual and real corrections cancel in physical observables. The complete treatment of IR singularities has been given by Yennie, Frautschi and Suura [YeGl]. We follow the simplified analysis of Weinberg [WeG5].
2.8.2.1
Amplitudes for soft-photon emission
Consider a process i > f involving any number of charged (and neutral) particles of any types. The corresponding amplitude is denoted by (pj),
2.8
Infrared a n d col I ¡near s i n g u l a r i t i e s
28!)
incoming (right) chargcd particle
where the pj denote the momenta of the external particles. We want to derive a universal formula that gives the amplitude for the processes i —> /+wy where 7t is any number of soft photons, i.e. photons with energies small coinpared to all energy scales in the hard process. Since we do not assume that the photon lines are on-shell and contracted with polarization vectors we can use the resulting formula to obtain directly the virtual IR singularities in Sect. 2.8.2.2 and the real singularities in Sect. 2.8.2.3. Let us start with the emission of a single soft photon with momentum k and Lorentz index /i from an outgoing charged particle with momentum p (Fig. 2.13a). The corresponding amplitude is obtained from the one of the hard process i: —> f by inserting an additional charged particle propagator with momentum k + p and an additional charged-particle photon vertex. For an outgoing fermion with mass m and charge — Qa, the amplitude of the hard process can be written as (we single out the momentum p from the Pj for the time being) M"f(
P
,
P j
) = u(p)A(p,pj),
(2.8.15)
where u(p) is the spinor of the fermion and A the remaining part of the amplitude. The amplitude with photon emission reads Ml^(
P
,
P j
,k)
=n(p)[-\eQ%1)
Í(|f + # + 7«) (p + k)2 - m2 + ie
A{p +
k,pj). (2.8.16)
If we assume that A(p,pj) is sufficiently smooth 2 3 in p, this becomes in the limit k —> 0, i.e. in the soft-photon approximation,
(2.8.17) 23
This result has, for instance, to he modified in the presence of a resonance unless the energy of the soft photon is small compared to the width of the resonance.
230
2 Q u a » ! , m i l theory of Yang Mills fields
The vanishing denominator leads to IR singularities for emitted from an outgoing scalar particle, we find Mi?'->ip,pi,k) =
-\e {2p
-> 0. For a photon
= +
)il]
2
(p + k) - 7n + ie
r n T r S k ^ t o P i ) ,
fc > Zp + i£
2
^
f
( p+
,pj) (2.8.18)
i.e. the same result. In fact, independently of the spin of the charged particle the emission of a soft photon from an outgoing particle yields a factor 2CQP
" 2p + ie
(2.8.19)
in the soft-photon approximation. Any spin-dependent, terms are IR-regular. The emission of a photon from an incoming charged particle can be treated in the same way. Since after the emission of the photon the charged particle has momentum p we find a factor
-2p
+ ie
(2.8.20)
instead of (2.8.19). If the soft photon is emitted from an (off-shell) internal line, no IR-singular factor results, since no IR-singular propagator occurs. Summing over all external particles, we obtain the amplitude for the emission of a single soft photon from the considered process
where i and p/ are the relative charges and momenta of the Zth external particle and /// is a sign factor that equals + 1 for outgoing and —1 for incoming particles. Note that the charges of antiparticles have to be taken opposite to the charges of particles in (2.8.21). Let us now consider the emission of two photons with momenta ki and k? and Lorentz indices //1 and /t 2 (Fig. 2.14). If the two photons are emitted from different external lines, each of them gives a factor like in (2.8.19) or (2.8.20) to the matrix element of the hard process i —> / . I f the two
2.8
p+a2
Infrared a n d c o l I ¡near s i n g u l a r i t i e s
p
v
p + ^2 + ^1
y
28!)
p + p
p + ki + k>
Fig. 2.11 Graphs for the emission of two photons from the same outgoing charged particle
photons are emitted from the same external line we have two contributions, one where photon 1 is emitted first and one where photon 2 is emitted first. Adding the factors from these two contributions yields in the limit kt —> 0 iJ2cQp^_\ ( 2v.Qp,i2 \ \2rfpk\ + ieJ \2rip(k\ + k2) + ie) 2eQPll, \ / 2eQPlll + ( V2i}pk2 + ie J \2rjp(k\ _ r-Mhw \ (2coft, \2ripk2 + ieJ \2?ipk\
-{- k2) + \e/
\ +\£J
i.e. the product of two factors (2.8.1!)). The corresponding result for the emission of an arbitrary number of photons can be derived by induction. As a consequence, soft photons are emitted independently. Thus, we find the amplitude for the emission of n soft photons with momenta k\,...,kn and Lorentz indices / ¿ i , . . . , / i „ in the process i —I f in the limit ki —> 0 by multiplying the matrix element of the hard process by a product of factors like the one in (2.8.21),
,~0 n
2.8.2.2
( E a H ^ f c )
(2.8.23)
Virtual soft-photon amplitudes
Let us use the results of the previous section to calculate the virtual radiative corrections originating from soft photons for the process i —> / . To this end, we have to consider the amplitude (2.8.23) for an even number of soft photons, join pairs of soft-photon lines to form virtual photons, and sum over all different pairings. For each virtual photon we must multiply with a
;{()(» 2
Q u a n t u m theory of Yang Mills fields
propagator factor -uj'l ( 2-l-ie) and integrate Over the photon momentum. In this way we get the virtual corrections owing to n soft photons to the process i > ,
M*~*F (PJ)
(2.8.24)
with
(27T)'1 (
2
+ \e)(2in
l
- - ie){-2 )m
m
+ ie)'
(2.8.25)
The factor 1/n! cancels the 71! equivalent permutations of one of the ends of the photon lines and the factor 1/2 in the bracket the 2 equivalent contributions resulting from the interchanges of the two ends of each of the photon lines. The sign of p has been changed in one of the denominators of (2.8.25) because the photon momentum must be incoming in one of the photon lines of each pair. The sum in (2.8.24) involves also terms where both ends of the photon lines are attached to the same external leg (/ = m). It has been shown in Ref. [Ye61] that (2.8.25) is also valid for this case. These terms originate from the wave-function renormalization constants that appear in the LSZ formula relating truncated Green functions and S-matrix elements and/or in the field renormalization constants of the external fields [cf. Sect. 2.1.3 and (2.5.7)]. The factor 1/2 in (2.8.24) results in this case from the fact that these renormalization constants are determined from the square-roots of the residues of the propagators (cf. Sect. 2.1.3). Summing over the number of virtual photons n in (2.8.24), we find for the radiative corrections owing to any number of virtual photons
= eX
P \ X/ L
e2
QlQm-hm M ^ ' f a ) .
(2.8.26)
l,m
The integral (2.8.25) is both UV- and IR-divergent. The UV divergence can be eliminated by restoring the 2 terms, which are irrelevant in the IR.
2.8
Infrared and collinear singularities .'11)1
limit, in the denominator, and the IR divergences can be regularized by an infinitesimal photon mass A. Then, the integral (2.8.25) becomes /
Jh
d fc
f "~
1
_ (2tt)< (A; — A2 + is) i 2
\/
4pipm x
p
2
(2 8 27)
f 2fji/p/Ar + i e ) ( F - 2i]mpm
+ ie)'
This is just an ordinary scalar three-point function (2.5.65). Using standard techniques it can be evaluated to ,
m
VlVm =
1
2
,
log
A2
log
tt m mimm Ill-finite terms VlVm
1 2
.
log
\
2
~ 1)
7ll[Tllm
1 -
mim n 1 T im IR-finite terms for
(VlVmPlPm{Plm
, .
\ J
im
log , , a 1 h.
2i T
, , 2. .2
to, where n2m2
is the relative velocity of the particles and m in the rest frame of either one, and we used the fact that the scalar product of two physical momenta is positive, pipm . The imaginary part in 2. .2 leads to an IR-divergent phase factor in 2. .2 , which drops out when ta ing the absolute s uare of the matrix element to calculate the cross section for the process i > . This phase factor is the relativistic counterpart of the well- nown Coulomb phase. The integral can be obtained from 2. .2 by putting ta ing the limit m . e thus find for the real part, Re u
2 1 - - log Sit* mf
IR-finite terms,
while the irrelevant imaginary part diverges for
VlVm
1 and
2. .
,
.
;{()(» 2
Q u a n t u m t h e o r y of Yang Mills fields
For the study of the IR-singular terms it is more convenient to regularize the IJV and IR divergences by introducing an upper and lower cut-off on the momenta of the virtual photons, i.e. km¡„ < |k| < fcmax. The integral over ko in (2.8.25) can be done by the method of residues. The integrand is analytic in ko except for the poles at k° = - |k| + ie,
k° = |k| - ie, k0
_ 2p;k - i\i'\£
f0
_ 2p,„k + i]m ie
(2.8.31)
2pt,o The first two poles are called photon poles and the last two particle poles. The integral can be evaluated by closing the integration contour at infinity. If particle in is outgoing (r/„, = +1) and particle I is incoming (i]i = —1) or vice versa the particle poles can be avoided by closing the contour in the upper or lower half plane, respectively. Consequently, the integral is determined by one of the photon poles only and turns out to be purely real. If particles I and m are both incoming or both outgoing, the particle poles lie on opposite sides of the real ko axis and one of them contributes to the integral. Thus, we find d Ak Jim ~ ~
4pipm
hm,n<\k\
+ ie)
PlPmVlVm fcn,i„ ojk| - p„,k) PlPm K
" l m J [plo\k\ 2 - ( P i k ) 2 ] bm,o(p/k) - p/,o(Pmk) - iejj • (2.8.32)
Integration over |k| makes the IR singularity explicit, J m =
'
7{2iry ^3
1o
PlPmVlVrn £• TT^ fcmin /J
Jim = ¿T2 7T~ Wl"11()8 ™ Pim
(1Q
><
-2(p/,0 - P/k)(Pm,0 - P m k )
•m I 1 - p( m
2?ri
WmVrn) log
«min
(2.8.34)
2.8
Infrared and collinear singularities
.'11)1
for I ^ in, and the real part o f c a n easily be found from the limit fti,n > 0. for high energies, B in, (2.8.34) contains a product of two large logarithms, called Sudakov double logarithms. Note that the terms proportional to log A:Inj„ in (2.8.34) are equal to the terms proportional to log A in (2.8.28). In fact, the effect of virtual soft photons on the absolute square of the matrix element is obtained from (2.8.26) and (2.8.34) as
0
K i s o r t l ^ f r ^ ) \ "-max /
^ ! ^ !
2
(2.8-35)
with the exponent B^f = l7r
fe
J
ft% 7 7 (a0-p/k)(pm,0-pmk)
= - ¿ £ QlQmVlVm-^2?r f ^ ftlm
log
1 - Plm
(2.8.36)
where the contributions for I = in have to be obtained from the limit /?/„, > 0. For instance, for a process with a single charged particle with charge Q iu the initial and final state we find B =
Q2 2 . — l '¿Ti p
o g
1 + ft . r r p ~ \
(2.8.37)
from (2.8.36). The logarithmic term originates from the virtual photon exchange between the initial-state and final-state charged particle (ftim = ft and 7/(7/,,, = - 1 ) and the constant from the virtual photon attached to the linai-state particle and the initial-state particle only [fti m = 0 and r/;r/m = 1 ). The exponent is positive for all 0 < ft < 1. For ft = 0, B vanishes, and for ft —> 1, it is given by B =
2 Q2 7T
2 • o g ^ - . in
(2.8.38)
where p and p' arc the momenta of the incoming and outgoing charged particles. Also in the general case B is always positive. As a consequence, the effect of the 1R divergences summed to all orders makes the rate for any charged particle process i -> / vanish in the limit A 0.
;{()(»
2.8.2.3
2
Q u a n t u m theory of Y a n g Mills fields
Real soft-photon contributions and cancellation of IR divergences
The IR divergences lead to divergent decay rates and cross sections for processes involving charged particles and definite numbers of soft photons in finite orders of perturbation theory. When summed to all orders, the results vanish. However, processes with a fixed number of photons are unobservable, since they cannot be distinguished experimentally from those involving additional soft external photons. Because photons are massless, their energies can be arbitrarily small and thus less than the resolution of any detector. Therefore, the only processes that can be observed are those where in addition to a definite number of charged particles and observed photons with energies greater than some minimal photon energy A E , an arbitrary number of soft photons with energies less than A E , and some small total energy Eioi are produced. The ¿'-matrix element for the emission of n real soft photons in a process i —> f is obtained by contracting each of the n Lorentz indices fi\,... ,/i„ of the amplitude (2.8.23) with a photon polarization vector £*(&£, Aj), where A, denotes the helicity of the corresponding photon,
.(2.8.39)
Since pf 0, the denominators are always non-zero for kinematical reasons, and \E can be omitted. The factor
appearing in t his matrix element is sometimes called the eikonal or convcction current. The corresponding term for a single charged particle in the initial and final state,
(2.8.41)
equals the Fourier transform of the classical current resulting from a classical charge that is instantaneously accelerated from momentum p to momentum
2.8
Infrared and collinear singularities .'11)1
// Because of the factorization of the soft-photon contributions discussed in Sect. 2.8.2.1, (2.8.39) holds also in the presence of virtual soft photons. The cross section is obtained by squaring the matrix element [cf. (1.2.24)). Since the soft photons are not detected, we must sum over their polariza I inns and integrate their momenta over the available phase space. Owing to i liargeconservation, rnQi = 0, the terms in the polarization sum (A.I .-III) involving the photon momenta do not contribute, and the polarization sum reduces to (2.8.12)
M ) = ~9iw-
Cutting everything together, we arrive at the differential cross section for the process i —> f with n undetected real soft, photons, d3k, i=l
—e2QiQm l,m
ptpm
WlmPlkiPmki
(2.8.43)
where E, is the energy of photon i, J lm
=
d'$k d3k _ f J (27r) 3 2E
VOhnPlPm Win (p(th)(p ,k) mk)'
(2.8.44)
P
and the factor 1/n! originates from the n identical photons in the linal state. Note that in the soft-photon approximation the photon momenta are neglected in the overall momentum-conservation ¿-function. For simplicity we disregard the fact that the sum of the energies of the soft, photons is bounded by Eloi. The general case can be found in [Ye61, We65]. Summing over n we obtain the differential cross section for the process i —> f with an arbitrary number of undetected real soft photons
=
dai
>f
(Pj)<*P
5>2QiQTO/in
(2.8.45)
l,m
Because the photon energy is restricted by E < A E in a certain reference frame, the integral (2.8.44) is not Lorentz-invariant. Moreover, it is
;{()(» 2
Q u a n t u m t h e o r y of Y a n g Mills fields
Ill-divergent. For a finite photon mass as III regulator it has been worked out, for instance, by 't Hooft and Veltman [tH79],
=
7ftrm ITT oTT
,0
S
¿AE 1 l o 1 -fiim _ „ x o S , , a + IR-finite terms. A film 1 + ft{m (2.8.46)
Note that the IR singularity contained in this integral is just the negative of the IR singularity contained in (2.8.28). If we regularize the III divergence instead with a photon mass with a lower cut-off on the photon energy, E > km\„, we find [cf. (2.8.33) and (2.8.34)] d3k fkmil,<|k|
mVmPlPm
(2vr)3 2|k|(p,, 0 |k| - pik)(p m , 0 |k| - p m k )
1
= ~
1 -.AE Id VlVmPlPm , , log /f di2 x — k {¿irr ¿min ./ 2(p; o ~ P(k)(p m , 0 - P,„k) 1 1 , 1 + Pirn , A E -nirim logl o g 8K2Pl,n ' ' -Plm *
(2.8.17)
with the same angular integral as in (2.8.33). Evidently, /<„, is just the negative of the photon-pole contribution to ./;„, if both the maximal and the minimal photon energies are chosen equal. In any case, t he 111 singularity in /¡„, equals the one in J;„, up to a minus sign if both integrals arc regularized in the same way. 24 The contribution of real soft photons to a hard process can be written as /
A/?
s off—'
=
(2.8.48) \ "•mm /
with i r ~ > f from (2.8.3G). Because of the factorization discussed at the end of Sect. 2.8.2.1, real and virtual soft-photon corrections yield independent factors to the hard-scattering matrix element. Consequently, the differential cross section for the process "Note that k,„i„ lias to bo defined in the same reference frame for both virtual and real photons.
2.
Infrared a n d collinear s i n g u l a r i t i e s
.i .r
I wilh an arbitrary number of virtual and undetected real soft photons ronds d
rfi
do*~*f pj) exp
e2g l,m
, Re.
m
4-
lm)
=
min
max
/APN
2. .4
max
Tims, the correction factor is independent of the IR cut-off mm and in general independent of the way the IR singularities arc regulari ed. ince nmx can be chosen arbitrarily large, t,he correction factor is always between and 1 and decreases with the soft-photon cut-off . If photons with energies larger than max are ta en into account in the hard cross section da' ' ^(pj) the dependence on c max cancels exactly. The fact that IR singularities cancel between virtual and real soft-photonic corrections in E is nown as the locli ordsiec theorem 1 .
2. .
Collinear singularities in
E
Radiative corrections by soft real and virtual photons were discussed in lowest order and higher orders in the last section. further source of mass singularities is the emission of collinear but not soft photons. These singularities are regulari ed by the mass in of the emitting particles which is non- ero in E . However, for scattering processes at high energies, > m, the corresponding contributions arc important since they are of the order of 0 l o g ( 2 m 2 ) . They are usually denoted as leading-logarithmic contributions. The logarithmic enhancement may re uire the summation of the multiple radiation of collinear photons.
2. . .1
Radiation of a collinear photon
e start by considering the emission of a photon from an incoming electron in a general scattering process, i.e. e i , where i and denote sets of other incoming and outgoing particles, respectively cf. Fig. 2.1 b . e want to extract the contribution that results from the emission of
;
2
u a n t u m theory of
ang
ills fields
col linear photons, i.e. photons that are radiated almost parallel to the highenergetic electron and remain undetected. To this end we have to integrate over the phase space of these collinear photons. For the invariant matrix element we use a notation similar to 2. .1
and
2. .1
=
(
, )
( , A)
. ), 2. .
where p, , and denote the momenta of the electron, the photon, and the other particles, respectively, and K and the helicities of the incoming electron and outgoing photon, respectively. hen extracting the contributions at high energies, T , we can neglect the mass of the incoming electron in the matrix element, 2 p 2 m2 . The mass singularity then appears as a logarithmically divergent integral and the electron mass has to be reintroduced in the phase-space integral as lower cut-off. Then, the four-momenta of the incoming electron, the photon, and the internal electron can be parametri ed is ;
, , ,
((1
l
)
,
xE
.
,
+
. cos
. sin
and obey p
-4 ( 1
0 0 11
+4
-{ -
-h
-""
,
E )
^
+
) -
,
2. . 1
2
,
2
= ,
p'2 =
~ 1
a
2
.
.
2
The variable x determines the energy of the emitted photon ~ 1 - x) and of the intermediate fermion ' sa x , i.e. the energy for the scattering 2
Terms involving the electron mass lea l to integrals of the form c l i 2 m i , 2i i and thus do not give rise to large logarithms. They are, however, relevant, for the non-logarithmic finite terms.
2.8
Infrared and collinear singularities
.'11)1
process e + i —> f . The soft-photon limit corresponds to x —> I and k | —> 0. 'The transverse momentum kp is a measure of the acollinearity of the photon and determines the off-shellness of the intermediate fermion p'2 = — k 2 / ( I — r) (instead of the on-shell value p'2 = 0). The LL contributions come from small values of the denominator p' 2 ~ k-j ~ 0, i.e. from the emission of collinear photons, |kp| (1 — x)E. Up to terms of the order of p' 2 = 0 ( k 2 ), the numerator of the fermion propagator can be replaced by the polarization sum of on-shell fermions with momentum p' = p — k,
(2.8.53)
resulting in Me£if(p,k,pj)
= -Aei~>'(p
~
k,Pj)
x Tu{p',k')—l-—eu(p',K')f ^ (v - *o
(k,\)u(p,K) + Odkxl0).
=
+<9(|k±|°) (2.8.54)
This matrix element consists of the matrix element with reduced momentum xp, the scalar propagator, and the electromagnetic vertex
= eu(p', K')f (k, A ) u ( p , k) with helicity spinors u(p,n)
(2.8.55)
and u(p',/s').
Neglecting terms of the order of |kjJ 3 , the electromagnetic vertex can be calculated, for instance, using the explicit expressions for the circular polarization vectors,
± { kJ )
1 ( IMe** = ' ~" "V2\2(l -x)E
£¿4
= -!,
11
e'±k = 0,
^
_
Ikxlc*' \ 2(1 x)E) (2.8.56)
308
2
Q u a n t u m theory of Yang Mills fi<.'l
and the helicity Dime spinors [in the VVeyl representation (A.1.25)],
u(p', - ) = (o, 0, u(p',+)=
v / 2 ^
(SteE,
,
u(p,-)
= (0,0,0,
,
u(p,+) =
V2E) "",
(v/2^,0,0,0)'.
(2.8.57) This yields W A , a = ^ l ^ j t ^ A + x5- KX \6 K , K e- iX + + 0 ( | k x | 3 ) ,
(2.8.58)
and thus
A=±
*
'
(2.8.59) independent of <j>. Inserting this into (2.8.51), the absolute square of the matrix element summed over photon polarizations is obtained as \M:rf\2 =
E A=±
E A,K',K"
= 2e 2
M * r f \ 2 + 0(\kx\~l),
(2.8.60)
i.e. it is proportional to the matrix element squared for the subproccss ei —> /• As a consequence, after integrating over the phase space of the invisible photon and summing over its polarizations, the cross section resulting from (2.8.54) is composed of the cross section for e + i -» / at momentum xp,
d
'
[xdc*rf(xp,pj)
-¡kj" T 3 7 + 0(|kx|)] .
(2.8.61)
2.8
Infrared and collinear singularities . 11 1
Here several comments are in order the cross sections d and da 1 are to be ta en in the same reference frame, i.e. if (]>,pj) is in the C1 C frame, dcr 'f(xp,pj) is not in its C frame for x 1. The factor x in front of corrects the flux factor in the definition of the cross section cf. 1.2.24 since the incoming electron in the non-radiative process has inoinentum xp instead of p. hen integrating over the photon momentum not all momenta p,pj can be fixed. ince we arc only interested in the eollinear singularity, which comes from the region « 1 x)p, we can c l neglect in da and replace p by xp. lthough this is incorrect Ibr large j , it still yields the correct coefficient of the eollinear singularity. e can either integrate over the complete photon phase space or over the pari of this phase space that contains the singular region. The volume element of the photon phase space can be decomposed as
^
dxd^d j
2* 2
4 2tt
1
x)
+
(
(
2
ince da ^ xp, pj) is independent of f>, the integration over a factor 2-n and the cross section is given by
(
>
2
)
j) yields simp
d
« .
i,max x ,mui f non-logarithmic terms.
(2.8.(»:i)
The upper limit of the transverse-momentum integral is determined by the phase space boundaries experimental cuts and given by some scale ,max- If the cuts are not too strict, is of the order of the energy of the process. ince we are only interested in the contributions, the precise definition of the scale is not relevant. For massless electrons, the lower limit is ero, reflecting the eollinear singularity. hen calculating with finite electron mass, this regulari es the singularity, and the effective lower limit is actually of the order of the electron mass and in approximation can
;
2
Fig. 2.1
u a n t u m theory of
ang
ills fields
Graph with virtual collinear photon
replaced by the electron mass,
tm i
in. Thus, we find
a ' ^ i b ) =
The collinear singularity manifests itself in the logarithm log 2 m2 ~ log 2 m2. umming over K yields the same relation for the unpolari ed cross section. The soft-photon singularity at x - 1 in 2. . 4 is of the type that was discussed in ect. 2. .2. s was shown there, this singularity is compensated by the virtual radiative corrections. 2 . . .2
V i r t u a l collinear p h o t o n s
eading-logarithmic corrections from virtual collinear photons result from field and or wave-function renormali ation constants as well as diagrams of the form shown in Fig. 2.1 . The corresponding amplitude reads
i
I i
f- f m . t;; - l y2 i i il m 2 4-
.
.ft
2. .
—iN1'" i 21 i e
where N'"' denotes the numerator of the photon propagator, e.g. g1'" in the Feynman gauge, and n f represents the amplitude with the on-shell electron replaced by an off-shell electron and photon. ecause the matrix does not contain contributions element Ml'l~^ is fully truncated, which are one-particle reducible between the electron photon pair and the rest of the diagram, i.e. it is given by the diagrams in the brac ets on the left-hand side of 2.4. .
2.
Infrared and collinear s i n g u l a r i t i e s
. i .r
ince in this section we arc only interested in mass singularities, we calculate all loop integrals in four dimensions and regulari e the V singularities hy a simple cut-off if necessary. s the V singularities are cancelled by other V-divergent but non-mass-singular contributions, we can set the cut-off e ual to the scale of order of the energy , ~ . bi e for the real collincar singularities, we can put the electron mass to ero for the extraction of the virtual collincar singularities. The collinear singularity can be extracted by using the uda ov variables 2. .11 . In the collinearsingular part, we can use . p in the numerator and obtain v\nf
i j)
XV
(l2
=
xd
dz
f
(d -y)p>yp>n){k2+i£)[{p_k)2 d 2 k x d y dz (1 -
= 2ie-^f pn
1 - r/)p,yp,p (p,
+ i£] (2. .66)
) )(fe2
N>'"p .g) (p -fc)2
. ,
where we applied the massless irac equation in the last line. If we choose the gauge in such a way that the numerator of the photon propagator is transverse, i.e. ,l u 0, this vanishes owing to p. In such a gauge the only collinear-singular virtual contributions originate from the wavefunction renormalization constants and/or field renormalization constants of the external fields cf. (2.5.7) , where the above does not apply. In the following we assume the on-shell scheme of Sect. 2.5.1.6, where the field renormalization is chosen in such a way that no wave-function renormalization constants are needed. An appropriate 20 transverse gauge is the axial gauge, where n
^ n
^
n2 =
0
(n )
In order to calculate the fennion-field renormalization constant for massless fermions in this gauge we consider the electron self-energ cf. (2.5. 5) on page 221 ie 2 i
d
, 2 o ficn
" The Landau gauge, A"1" = /" k''k" /k2, introduces further singularities owing to 2 the l/k term, so that the approximation k = xp used in (2. .C ) is not valid.
;{()(» 2
Q u a n t u m theory of Yang Mills fields
with the covariant decomposition (2.8.69) Then, the electron-field dition (2.5.37) as
renormalizalion
constant is obtained using the con-
= - S Ì ( 0 ) - S
Inserting (2.8.68) with (2.8.07) and calculating the traces yields SZ$ = lini 4ie i e 2 J p2->0
d'1 k 1 2 (2ny (k + ie)[(p - k)2 -f ie]
(2.8.71)
x Terms that involve (p - k)2 or k2 in I,he numerator cancel one of the zeros of the denominator and therefore give not rise to logarithmic singularities. After substituting pk = (p 2 + k2 - (p - k)2)/2 -> p2 /2 in the numerator, we find
2 d fc
6Z^ = 4ie J ^ 4
1
(27r)" (A:2 + k)[(p _ k)2 + ie]
+ non-logarithmic terms.
kn 2 pn
1+
^ nk p2= o (2.8.72)
We evaluate this integral using Sudakov variables (2.8.11), where kn/pn = y. After integrating over 2 by closing the integration contour in the upper half plane, we obtain
--iw/fi'Mi-'^} + non-logarithmic terms
(2.8.73)
As in (2.8.63), the lower limit of the transverse-momentum integral is given by the electron mass. At the upper limit this integral diverges owing to the
2.
Infrared a n d eollinear .singularities
.{M5
singularity. Since we are only interested in the mass-singular contributions, we set the upper limit equal to the scale , as explained above, and arrive at a
5Zj, =
27T
(
2
\
log
' ^ ) /
70
+ drc
2
h non-logarithmic terms,
1-
(2. .74) where we substituted I singularity.
(1
re). The pole at x = 1 results again from the
Finally, we find
non-logarithmic terms
(2. .75)
for the contributions of one-loop eollinear singularities to the cross section in the axial gauge. For comparison we calculate these contributions in the Feynman gauge. Then, the electron-field renormalization constant is obtained as = -
2tt
log
\m1)
I
da: (1
x)
non-logarithmic terms. (2. .76)
n the other hand, in this gauge the diagrams in Fig. 2.15 yield a nonvanishing contribution given by (2. .66) with ,iupv = p '. After setting p = , which holds in the eollinear limit, we can use the ard identity (2.4. 7) on page 176, which translates to ' ^ f ( p-
, ,pj)
= eAei->'(p,pj)
(2. .77)
because the second term in (2.4. 7) subtracts the one-particle reducible terms that are not present in . Thus, (2. .66) turns into lhli( ' j) =
2ie
=
o 1 , TTT^i
,/",,, , , 1-1/ 2 / d kxdydz -
et
;
->f(p,pj)u(p, ) . , , , . ,
;{()(» 2
Q u a n t u m theory of Yang Mills fields
r
f f ^ « ^
j
(2.8.78)
up to non-logarithmic terms. Adding the contribution of the field renormali/ation constant, M " ^SZ^/2, and calculating the cross section yields the same result as in the axial gauge (2.8.75).
2.8.3.3
Electron-electron splitting function
Summing the contributions of real and virtual collinear photons, (2.8.63) and (2.8.75), the IR singularities at x = 1 cancel, and we find the one-loop LL corrections originating from an incoming electron,
= Jo
[¡y-r^ i —x
Mere we have introduced the (+)-distribution which is defined by jf' d x f ( x ) ( h ( x ) ) + = J1 dx ( / ( * ) - / ( ! ) ) / » ( * ) ,
(2.8.80)
or r = Hm 0{l-x-p)h{x)-S(l-x-/3) /?->o
/ Jo
i h{y)dy (2.8.81)
and IR-finite by construction. Defining the lowest order electron splitting function
electron
2.
I ll ,. 2.16
Infrared 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 . 11)1
raph for collinear electron emission
we linally obtain the leading-logarithmic of the radiative cross section
(LL approximation)
approximation
(2. . 3) In the collinear approximation the radiative cross section factorizes into the logarithm of the collinear singularity and the convolution of the lowestorder cross section at momentum xp with the splitting function ^\x). The splitting function is related to the probability of finding a fermion with momentum xp after the emission of a collinear photon with momentum (1 - x)p from the incoming fermion with momentum p.
2. .3.4
Collinear e l e c t r o n e m i s s i o n
A second possible reaction mechanism is the one where the collinear photon undergoes a hard scattering 7 i / and the collinear electron remains undetected (Fig. 2.16). This is part of the process e i e / . sing the conventions for the momenta as defined in Fig. 2.16 and (2. .51) and denoting the helicity of the outgoing electron by K , the corresponding invariant matrix element reads r
*£e (p,
-
,pj)
=
eu( , n'h,
,
a
>
,
^ ~ (2. . 4)
in the Feynman gauge. As in the previous case we can neglect the electron mass and terms of order k j when extracting the LL contributions. In particular, in the numerator only the leading terms in kj are needed, and the
;{()(
2
u a n t u m theory of
ang Mills fields
metric tensor in the numerator of the photon propagator can be expressed as -ST
= £
e> (p',\)e (p',\)
-
,
pi,pi
(2. . 5)
1
A =
where p' = ( p , p ) for p' ( p ,P )- The contraction of p'v with the amplitude A„ vanishes owing to a ard identity (2.4. 3), and the contraction of p ; with the electromagnetic vertex gives zero. In other gauges, the numerator of the photon propagator contains additional terms involving p' ' or p' which also drop out. Therefore, independently of the gauge, the numerator of the photon propagator can be replaced by -ST-
2
,A)
,A),
(2. . 6)
A =
i.e. the polarization sum for the physical transverse polarization vectors only, and the matrix element turns into c f
(p,
,
j
) = -e u(k,
)f(j/,A)u(p, )
A
+ o(|kx|0)
x e V , A)4r'(p = -
em^Mj^ip -
2
(lkil )-
(2- . 7)
X The corresponding electromagnetic vertex ,«
A
A
= eu( ,
N')F
(p , A)u(p, K)
(2.
.
)
differs from (2. .55) only by the interchange p - k, i.e. by the substitutions x 4 (1 - x) and cf. (2. .51) . Therefore, from (2. .5 ) we immediately find
,
^A
,A = C ( kx 3),
(1
x t f - ^ e -
* (2. .
)
2.8
Infrared and collinear s i n g u l a r i t i e s .'11)1
and thus k
E
K-m>,k',k,AKm,K',K,X'
+
=
2e
l
x)26-K\
8k\ + (1 -
_
X
+
— S -
¿A A'
(2.8.90)
X
When integrating over <-/>, the second term averages to zero27 and the nonvanishing term is proportional to S\y. Therefore, we effectively obtain for the absolute square of the matrix element
E \
K'=±
K',A,A'
w ^ w ^ ^ r
(1 - x ) 5k\ + (1 = 2C E x K 2
A
7
^
7
) (p - -? k)<^
i j~ri—>f i2 \Mi^j\2+o(\k±ri)}
* L
(2.8.91)
i.e. a contribution proportional to the absolute squares of the photonic matrix element. Performing the same steps as in Sect. 2.8.3.1, we find the cross section in the LL approximation
d<7
LL,K!/(P>Pi) = =
£
l 0 g
( S )
with the polarized electron photon splitting
P
WAx)
=
+
(2.8.92)
function
(2 g 93)
The l/x behaviour of this splitting function does not lead to a singularity since the cross section d v a n i s h e s for x —> 0, i.e. for zero photon energy. 27
We have chosen ( lie phases of the spinors and polarization vectors in such a way that the matrix element (p — k,Pj) is independent of in the collinear limit k = xp.
;{()(» 2
Q u a n t u m t h e o r y of Yang Mills
fields
Fig. 2 17 Graphs lor collinear fermion emission
The case of photon splitting from an incoming positron gives the same result. Summing over the electron or positron helicitics yields the corresponding results for unpolarized cross sections with the unpolarized splitting Jmiction
The cross section (2.8.!)2) is the product of the logarithmic collinear singularity and the convolution of the splitting function P $ { x ) with the cross section of a real, transversely polarized photon of momentum xp. Since is calculated from the photonic cross section d a 7 ' - ^ , the result (2.8.92) is called the equivalent-photon approximation or Weizsacker Williams approximation.
2.8.3.5
Photon splitting
Scattering processes 7 + i —> c + / with incoming high-energetic photons involve collincar-singular contributions originating from the splitting of the photon into fermion-antifermion pairs (Fig. 2.17). One of the collinear fermions is taken inclusively, i.e. its phase space is integrated out, the other one takes part, in a hard-scattering process. For the case of an inclusive collinear electron (Fig. 2.17a), the matrix element (2.8.16) reads Mt^ef(p,k,pi) = cû(k,K)t(p,\)
= A°'~*f(p - k,pj), (p — k)£ — m£ + \e
(2.8.95)
where the momenta are defined in Fig. 2.17 and (2.8.51). As in Sect. 2.8.3.1, we neglect the electron mass and terms of order k~i and write the numerator of the fermion propagator in terms of fermion spinors [cf. (2.8.53)]. The
2.8
Infrared 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
.'11)1
resulting electromagnetic vertex involves an incoming photon and lakes the form A = eu{k, K)i{p, \)u(p',n').
(2.8.96)
Using the explicit polarization vectors 4 ( * ) = ^(0.1.±«>0),
(2.8.97)
and the helicity Dirac spinors 7
N
-|kx|e~^
«(<:,+) - i 7 2 ( 1 -x)E,
as well ;us u(p',±)
V&S^x
'
,
(2.8.98)
from (2.8.57), we find
[(1 - X)8kX
= e
-
X S - ^ S ^ I P *
+ C?(|kj.|3),
V®(1 - x) (2.8.99) and
£
[ W ^ l
2
= 2c 2 ~7T~
K' = ±
X
\ [^kA(* - X? + ¿-«A*2] + 0 ( | k x | 4 ) .
> (2.8.100)
The calculation of the cross section in the collinear approximation proceeds as in Sect. 2.8.3.1 and leads to
= £
log ( $ ) j f d* E
p£l(x)dcrr'(xp,pj) (2.8.101)
with the photon
electron splitting
(x) = 5kX( 1 - xf
junction
+ 6-kXX2,
(2.8.102)
;{()(
2
Fig. 2.1
u a n t u m theory of
ang Mills fields
raph with virtual collinear electron -positron pair
or in the unpolarized case
Pi70)(*) = (l -x)2
+ x2.
(2.8.103)
For the case where the positron escapes (Fig. 2.17b) we find the same result with the same splitting function.
2.8.3.6
Virtual collinear e l e c t r o n - p o s i t r o n pairs
Unlike in the non-abelian case, where the splitting of a gauge-boson in two gauge-bosons is an important reaction mechanism, the splitting of a photon in two photons is not possible as an elementary process in abelian gauge theories. Therefore in QED only virtual diagrams like those in Fig. 2.18 contribute to the splitting function • By evaluating the diagram Fig. 2.18, one finds that all collinear-singular terras arc either proportional to k 2 or e( ) and therefore drop out for physical transverse photons. Thus, the only collinear-singular terms result from the photon-field renormalization. According to (2.5.101) and (2.5.93), this is given by
(2.8.104)
Evaluating the
SZ
= ~
function as in (2.8.72) (2.8.74), yields (i
log
+ non-logarithmic terms,
and the corresponding photon
/>#(*) =
1-.T).
photon splitting
unction
(2.8.105)
reads (2.8.106)
2,8
2.8.3.7
Infrared ami colliriear singularities
321
Probabilistic interpretation and distribution functions
Tlii! results for the QED cross sections in the LL approximation (2.8.83), (2.8.92), (2.8.101), and the one corresponding to (2.8.10G) have the following physical interpretation [A177] which is closely related to the parton piclure discussed in Sect. 1.5. The cross section factorizes into the probability flt,(x, Q2) of finding a parton j (an electron, positron, or photon) with longiludinal momentum fraction x in the incoming particle i, taking into account the radiation of col linear partons with transverse momentum k\ < Q2. The parton j enters the hard-scattering process. Up to order a, the probabilities are given by
/e,eM2) =
Q 2 ) = S(l - x) + £
log ( ^ j
pW(:r),
/ c . e M 2 ) = / c . s M 2 ) = 0, A,e(*,Q2) fy,tixì
= A A*>Q2)
Q2) = 6(l-x)
Uy(x,Q2)
= hJx,Q2)
= £
+ ^
log ( § )
/*?{*),
log ( J p ) pW(a:),
= ~
log i ^ j P W ( x ) .
(2.8.107)
The integration constant in the logarithm was chosen in such a way that for Q2 = 7/r no splitting occurs. A simple integration, using (2.8.80), shows functions that the probabilities fj,i[x, Q2), which are also called distribution respect the conservation of electron number and of the longitudinal momentum.
(2.8.108)
;{()(» 2
Q u a n t u m t h e o r y of Yang Mills
fields A
Fig. 2.19 Graphs for the emission of a two collinear photons from an incoming electron
Finally, we write down the expression for the cross section in a very condensed form,
dx f j i ( x ,
d a ' - Y i f
2
)daj{x).
(2.8.109)
o
i
This equation holds also in higher orders and summarizes the factorization of mass singularities in QED. In this context, 2 is called factorization scale. The generalization of (2.8.109) to non-abelian gauge theories, e.g. QCD, and more detailed calculations including the non-leading contributions is performed in Sect. 3.3.
2.8.3.8
Multiple splittings and evolution equations
The subprocesses with soft and collinear photons give important contributions to high-energy cross sections in QED since they are enhanced by large logarithms. Therefore, the investigation of higher-order soft and collinear processes is also of practical importance. The general results for the softphoton case were derived and discussed in Sect. 2.8.2. Now we consider the higher-order collinear singularities. The double-photon emission from an initial charged particle (Fig. 2.19) leads to large contributions if it causes two singular propagators. If the first emitted photon has a small acollinearity |kx,i| |k ,2 , then the second one still can give an enhanced contribution. In this case, the first virtual electron is very close to its mass shell compared to k 2 2 and we can ignore its virtuality in computing the emission of the second photon. The appearance of the leading logarithms is illustrated by the following simple calculation rQ 2 d k 2
, a U
L
n
g
<j..2|2 d k 2 .
2
L
v
g
2
/ o. -
t
i
*
;
^
)
2 (2
-8-110)
2..S
Infrared and col I ¡near s i n g u l a r i t i e s
323
In I,In; opposite limit, |kx,2l |ki.,i|i there is no denominator of order k 2 and thus no double logarithm. A more detailed calculation supporting this result is performed in Sect. 3.3. The generalization to the LL contribution of the emission of n photons reads i f f log 2 ! ) " , n! \27r 77i//
(2.8.111)
resulting from integration over the photon phase space corresponding to the strong ordering |k X l i| < |kjLi2| « |k ± , 3 | < . . . < |k L i „|.
(2.8.112)
The summation of these important multi-photon contributions leads to an integral equation which was first obtained by Gribov and Lipatov [Gr72j. We derive this equation using the probabilistic interpretation. Consider the which describes the probability of finding a distribution function f{x,Q2) certain particle in a given particle taking into account collinear photon emissions with transverse momenta k 2 < Q2. Then, the distribution function f{x,Q2 + AQ ) at momentum transfer squared Q2 + dQ must take into account the possibility of an additional emission of a parton with transverse momentum Q2 < k 2 < Q2 -I- dQ 2 . The differential probability to split off a parton that carries away the part 1 — x of the energy is
<
2
-
8
»
3
>
and we can write / (x, Q2 + dQ2) = f ( x , Q2) + f dy I' dz 6(x - yz) Jo Jo
^ f w . « " )
This leads to the intcgro-differential equation
m
m
n
*
<
n
m
*
J!
T
F
ii;)
(2 8 114
- -
>
;{()(» 2
Q u a n t u m theory of Yang Mills fields
It describes the Q2 dependence of the distribution function if its initial value, = ¿(1 - x), is given. The evolution equation (2.8.114) contains e.g. f(x,m2) on the right-hand side a multiplicative convolution of the evolution kernel and the distribution function. If we write for the multiplicative convolution of two functions f ( x ) and
[f ®<J}{*) = [ dy f dz 6{x Jo Jo
yz)f(y)g{z)
(2.8.114) reads
j g ^ M
2
) ^ !
PefiM
1
)-
(28.116)
For the electron-positron-photon system, i.e. the simplest case of QED, the evolution equations are a system of three coupled equations for the fs{x,Q2), electron, photon, and positron distribution functions fc{x,Q2), and / 7 (.r, Q2). The kernels are the splitting functions
p £ ) w = p£)(x) = o, p f f w = p $ \ * ) = pM(x)
1 + { 1
= pM(x)=x2
+
;
x )
\
(l-x)2,
=
(2.8.117)
Using the symmetry properties of the splitting functions, the evolution equations read
0)
e o ^ m ^ g(iogQ*)Mx'
^ = £K q2) =
£
y A ( x , Q2) = £
®• +
+
® a ] <«•
^
® f-'}{x' q2)'
(2
"8"118)
[p<°> e (/e + h) + Piff ® /,] (*, Q2)-
2.8
Infrared and collinear s i n g u l a r i t i e s .'11)1
l''if{. 2.2(1 Graph for collincar photon emission in the final state
These evolution equations of QED summarize the multi-photon-em ission effects in the LL approximation. For given initial conditions, i.e. / c ( s , m 2 ) = ¿(1 - x),
Mx,,m2)
= / 7 ( x , m 2 ) = 0,
(2.8.119)
they yield the electron, positron, and photon distribution functions at arbitrary ( f . The evolution equations of QED are an important tool for the calculation of electromagnetic radiative corrections in high-energy scattering processes. Similar equations are valid in non-abelian gauge theories, e.g. QCD, where they play a basic role in determining the higher-order corrections to the parton model (cf. Sect. 3.2.3).
2.8.3.9
C o l l i n e a r particles in t h e final s t a t e
We now turn to the discussion of collinear particles in the final state. The virtual collinear corrections are given by the same factors as in the initial state, e.g. (2.8.75) for electrons in the final state. For the emission of real collinear particles the situation is different. For initial-state particles, the incoming momentum is kept fixed and the momentum entering the hard-scattering process varies owing to the momentum carried away by the collinear emission. In the case of collinear emission from a final-state particle, the detector usually cannot distinguish the two collinear particles, and these are detected as one. Therefore, the momentum participating in the hard scattering, i.e. the sum of the momenta of the two collinear particles, is measured and has to be kept fixed when integrating over the momentum of the collinearly emitted particle. For definiteness we consider an outgoing electron emitting a collinear photon, as depicted in Fig. 2.20, where also the momenta are defined. Using
;{()(» 2
Q u a n t u m theory of Yang Mills fields
momentum conservation, the phase-space volume element of the two massless, collinear final-state particles can be rewritten as f d 3 k f d 3 p'
1
fd3
1
f d 3 p' f
,
,
(4)
(27T) _ J _ /•
= (1
V
• '
}
x)p, the extra factor becomes | p | / | p - k| =
The diagram of Fig. 2.20 gives rise to the matrix element
= —eu(p;, n ' ) f ( , X) (p>
(2-8.121)
As above we omit electron-mass terms. For the momenta we use the parametrization [neglecting terms of order (k-; )~]
4s(l
7,2 =
*2
= 0,
7/2 =
l)
(2 8 122)
-
- -
This parametrization differs from (2.8.51) because now p' is on-shell and p is off-shell. Replacing the numerator of the fermion propagator by the polarization sum [cf. (2.8.53)], results in j) = ~e £
"(p'l
A)"(p, K)fi(p, k)
K +
(2 8 123)
2.8
Infrared a n d collinear s i n g u l a r i t i e s
.'11)1
Kij;. 2.21 Graph for collinear electron emission in the final state
Up to terms of order k 2 the electromagnetic vertex is the same as in (2.8.55). The resulting cross section then reads
d (p,pj) = 2e j2ir)^2 * E \
-r
L;c
cf
(xp, j)
+
kT T ~x (2.8.12-1)
o(\ x\)
where the factor l x results from the transformation of the phase space (2.8.120). Since we lix p but not p' when integrating over , we can pull the lowest-order cross section out of the integration and obtain instead of (2.8.63) the result
da^f(p',pj) = £
log ( g )
d * r
J
M
jf d*
+ non-logarithmic terms.
(2.8.125)
The logarithmic terms cancel exactly against those of the virtual contribution (2.8.75). For final-state photons we similarly find front Fig. 2.21 a contribution proportional to the integral over the splitting function 1
2 dxP^ix) = -, 3 o
(2.8.126)
which cancels exactly against the virtual contributions from (2.8.106). Since the final-state collinear fermions are not detected individually, we have to sum over their polarizations, and the unpolarized splitting function shows up. The exact cancellation of the final-state collinear singularities in QED is a special case of the inoshita- ce- aucnberg ( ) theorem (cf. page 294 in
328
2
Q u a n t u m theory of Yang Mills fi<.'l
Sect. 2.8.1.4). According to this theorem, the mass singularities associated with the final state must cancel, when we integrate over all degenerate final states.
2.
s
s
s
We mentioned at different places that there are important properties of gauge theories which cannot be treated by conventional perturbation theory. In this section we discuss some of these features. •
Classical gauge-field configurations decompose into different topological classes. These features become most transparent if we consider them on closed manifolds in the euclidean region. These different Chern classes are distinguished by topological quantum numbers. Perturbation theory considers only the trivial class.
•
There is a particular relationship between modes with zero eigenvalues, the zero modes of the Dirac operator in a background gauge field, and the topological quantum numbers. This is expressed by the index theorem.
•
It is not yet finally clarified how far the different topological aspects survive the quantization procedure. In the semi-classical approximation of the path-integral formulation, the decomposition of gauge-field configurations into Chern classes introduces an additional summation over the topological quantum numbers. The appearance of the zero modes requires a modification of the path integral.
The first item is well established in mathematics [Bo(j5, At68, C!i95] and theoretical physics [Be75a, Go87]. We treat some examples in Sects. 2.9.1 and 2.9.2. In Sect. 2.9.3 we discuss non-perturbative methods of quantum gauge theories in semi-classical approximation. As an example, we consider't, Hooft's one-instanton calculation [t.II7G] in Sect. 2.9.3.2. Renormalization is treated analogously to perturbation theory in the background-field method. The range of validity of the semi-classical approximation within such a treatment of ultraviolet quantum fluctuations is not well-known. Maybe, this approach to non-perturbative problems of quantum gauge theories gives only qualitative insight.
2.9
N o n - p o i t u r b n t i v e a s p e c t s of gang« theories
32!)
However, the semi-classical approximation sheds some light on very important questions. For instance, the cluster properties of the vacuum lead to the introduction of the 0-vacuum (Sect. 2.9.3.4) and the strong CP problem (Sect. 2.9.3.9). New processes like vacuum tunnelling (Sect. 2.9.3.5) cannot he described by perturbation theory. It leads to new types of Green functions describing vacuum condensates and topological excitations (Sect. 2.9.3.7). These help to understand spontaneous breaking of chiral symmetry and the U(l) problem (Sect. 2.9.3.10), i.e. the approximately zero mass of the pion and the high mass of the r/' (cf. Sect. 3.5.1). Index theorems exhibit the geometrical origin of the anomalies (Sect. 2.9.3.8). It is of great importance that the lattice approximation allows another approach to the non-perturbative problems of gauge theories [Wi74]. Basically, it starts from a more systematic approach yielding a mathematically better defined path-integral formula. We treat this method in Sect. 2.10. However, compared with lattice approximation, in many cases the semi-classical approximation gives an intuitive picture of the physical problem. Thus, wo consider lattice approximation supplemented with semi-classical approxima lion as the appropriate method for non-perturbative problems. Needless to say, this method has not yet reached the maturity of perturbation theory. Non-perturbative physics is mainly discussed for problems of QCD. We treat the problem of confinement in Sect. 3.7 in the spirit just explained. There we use SU(2) symmetry as a simple example. Concerning the electrowcak interaction, the perturbative calculations agree with experimental data at the pcr-inill level. There are presently no indications for non-perturbative effects.
2.9.1 2.9.1.1
Topological q u a n t u m n u m b e r s Flux quantization on the torus
We begin with one of the simplest examples of a topological quantum number. To this end, we consider an abelian gauge field on a two-dimensional torus T of lengths \ and 2- On the torus we introduce cartesian coordinates (:/;i,.T2). The two sides a\ X\ = 0 and x\ — \ as well as the two sides a2 x2 = 0 and x2 = 2 must be identified, and therefore the points 3 (cf. Fig. 2.22). The U ( l ) gauge potential ,t{x), //. = 1,2, and the field
;{()(» 2
Q u a n t u m t h e o r y of Yang Mills fields
L2 > B
O
;
aa,2
;
»
»
M 6
B B
i
\ \ \ \ \ ' \ A / / // I 4 \ \ \ \ \ \ , / / /
U : u r c,
A 2
'
a,
/
!
U
c,
a
a,i
:
B o 0
a2 O 1>
; 1 x,
/
/
/
i
B B L, 0
. 1
* 2
J 3
0 4
Fig. 2.22 Topology of the torus as explained in the text with Li = 4 units and ¿2 = G units (left) and the phase of exp(—ieL\A(x)) (right)
strength l')tl, (x) are related by F ^ x ) = dtlAu(x)
- dvA^x).
(2.9.1)
The torus topology, the identification of sides and points, requires boundary conditions. In Sect. 2.3.2 we introduced Alt(x) as the components of an infinitesimal parallel transport. This implies that should be doubleperiodic up to a gauge transformation,
A,t{x + l x ) = Atl(x)
-
l
-9^(x)d,l9i(xl
i = 1,2.
(2.9.2)
Here L, denotes the vectors L\ = (L\,0) and L2 = (0, L2) and m{x) continuous gauge transformations describing a parallel transport Uc along a path C traversing the torus parallel to a side at (cf. Fig. 2.22). Continuity around the point B requires
91 (x)92(x + ¿1) = 92{x)g\ (x + ¿2).
(2.9.3)
2.9
Non poilui'l>ative aspects of gauge theories
331
I'o lie specific, we take the gauge <j\{x) = 1,
(x) = exp
(-««fr) •
(2.9.4)
Then, periodicity (2.9.3) requires exp(—2irik) = 1, and therefore k = integer. The number k is called the topological quantum number of the gauge Hold on the torus. We may express this quantum number by an integral over the field strength. Stokes' theorem transforms the integral over the torus T into one over the lines of the boundary. Using the boundary conditions (2.9.2)-(2.9.4) results in
We note that k does not depend on a specific gauge potential. There is a special solution with constant field strength and topological quantum number k, the instanton solution of the two-dimensional torus, which satisfies the Maxwell equation, (e\ 2 — 1, e t w = —£i//») (2.9.6) This example describes the well-known flux quantization of a magnetic monopole enclosed by a torus. 2.9.1.2
The SU(2) instantons
As an example of a field configuration with non-trivial topological quantum number in four euclidean dimensions we discuss the instanton solutions of an SU(2) gauge theory on a sphere [Be75a]. The SU(2) gauge potential is described with help of the Pauli matrices r", Ah{x) = gA"tTa/2, /1 = 1,2,3,4. The definitions of the field strength, the action, and the field equations follow from the scheme in Sects. 2.3.2 and 2.3.3. •Fm/ — d / I A V
SE{A}
= ~
j
di/AP
i (^4/! • A,,},
^ X ' ï x i V " ) =
(2.9.7)
;{()(» 2
Q u a n t u m t h e o r y of Yang Mills
fields
with the field equation D' T^{x)
=
~ i H ",
= 0
(2.9.8)
and the shorthand FuT-z)
(2-9-9)
= 2J d ^ T r ^ ^ r ) -
We are interested in solutions of the classical field equations with finite action 5];;{yt} < oo. This means that Fltu(x) —> 0 for re2 oo, or that ,t approaches a pure gauge, for a;2 -» oo.
,t = ig{x)dltg~'(.'/;)
We consider the dual field strength f ^ x ) = - ^
^
{ x ) ,
(2.9.10)
Ttlv T,lt,{x)
=
(2.9.11)
with £z-,fivpa the euclidean totally antisymmetric Levi-Civita tensor 28 defined in (A. 1.48). The Bianchi identity [cf. (2.3.30)] DpFl (x)
+ D,tTup(x)
+ DvTpn (x) = 0,
(2.9.12)
or
implies that self-dual (anti-self-dual) £
(*). =
fields
Tflv{x)
(2.9.14)
are solutions of the field equations D^liU(x) 28
= 0=
D Flll,(x).
If it. is evident that we are in the euclidean region, we omit the index "E".
(2.9.15)
2.
o n - p e r t u r b a t i v e a s p e c t s of g a u g e theories
.i.t.t
In order to solve the self-duality e uation 2. .14 we ta e as the asymptotic gauge in 2. .1 x'LTU
,
ix r
:c,t
(2. .1 li)
=
T
where (r,t) = (ir, 1),
(2. .17)
f/1 = ( - i r , l )
with T
/I //
r r
i iL
2 6FW,
T T, T
(2. .1 )
2ÓIIV.
TVTFT
e make the ansatz fl(x)
= i f{x2)g(x)d^ ~l(x)
f{x2)rl ^
= 2
(2. .1 )
with the self-dual (anti-self-dual) matrix tensor T,,„ (TIW) =
T,
~
T f
f
i'),
i
~ f " T / )-
=
(2. .20)
Then, the corresponding field strength
uv(x) = dMAv +
'(i2)o
2
9i/A/i - f(x2)(
=
\[Ap,A„] -
( x
2
4
7i x2
) ) ] ^ ^ - ^ x
T/\iv ^
(2. .21)
is self-dual if the term in the second line disappears, f'(x2)x2
- f(x2)(\
- f(x2))
The solution of this equation is f(x2) ,4j; (1) (x) =
x2
= 0. = x2 (x2
(2. .22) + A2), which leads to
= -
2
2
t , w . (2. .23)
;{()(» 2
Q u a n t u m theory of Yang Mills fields
The instanton potential A ^ f \ x ) is regular in the whole euelidean space but vanishes only as ]x| _ 1 for x —> oo. Using the gauge transformation (2.9.16), we can change this feature. The resulting potential and field-strength, A\P\x)=g-\x)A^\x)g{x) T
1,(2)/ X "" { '
=
"
v
+ \g-\x)d,lg(x) "V"J '
2 _ 4A (x 2 + A 2 ) 2
{ >"> x 2
o\2 Tu = ^ ~ * " x 2 -(- A2 "" x 2 ' " p x 2 )_
, (2.9.24)
are singular at x = 0, but vanish as |x|~ 3 and |x| - ' 1 , respectively, at infinity. In the language of Jlbre bandies (Sect. 2.3.2.7), we can express this fact in the following form: we have covered the euelidean space H?1 by two coordinate systems. In both systems, the gauge potential as connection can be separated into the dependence on the base space x G K'1 - {oo} = IR^ or x G R'1 - {0} = M(2) and the SU(2) adjoint representation space. The •4j,'^(x) describe trivial bundles on these sub-manifolds. In the overlapping space the connections are related by a gauge transformation. Thus, on the whole IR4 (or compactified to a sphere by a stereo-graphic transformation), our solutions describe a non-trivial bundle. We characterize this non-trivial bundle by a topological quantum number, the Pontryagin number. It is obtained from the topological charge density or Pontryagin density (cf. Sect,. 2.7.2.1) Vk{x) =
16^ n ^ M ^ i * )
= ¡ ¿ C C ^ "
(2.9.25)
as k=
I dAx'PE{x).
(2.9.26)
The Pontryagin density can be written as a divergence (cf. Sect. 2.7.2.1), VE(x)
=
Gm(X) = -
d.C'ix), ¿
W
Ti- (
^
W
+ iA"d"A^ .
(2.9.27)
This follows by direct calculation from the definition of the field strength and its dual.
2.
o n - p e r t u r b a t i v e a s p e c t s of gauge theories . i. t. t
I br illustration, we calculate the topological uantum number using Gauss s a x2 1 and theorem. e ta e the regular solution .4 1 , , x in 2 in x x 1 . The integral 2. .2 then turns into one over the unit sphere,
f A VV{ ) A
=
=
J
I
JsM
d4xV {x)
+I
- Tr r
JsW
i
ci pv
d4xV {x)
;
p
f
= - JdS^^TVlb-^J^aEjJIff-HxJ^Wlb-H«)^*)!} 1.
2. .2
The final result was obtained using 2. .1 , 2. .24 , g~x (x)dftg(x) and the symmetries of e' pa and of the trace. —(d)lg~l(x))g(x),
=
This derivation contains some remar able features. In the last integral of 2. .2 only the gauge transformation g(x) = (x. i x r x that connects the two regular solutions appears. Further, g(x) describes a mapping of the -sphere on the group 2 . In order to see this, we introduce four-dimensional polar coordinates p, x, and , x1
psin x sin
x
psinxcostf, x (Sdi
d x
vr,
x2 — psin x sin 0 sin c/>
cos ;, x4 d
pcosx, vr,
f
2tt,
p sin 2 x s i n i dx dd dcj> dp.
p
oo, 2. .2
Considering g(x) in these coordinates and comparing it with the standard parametri ,at ion of g(0', f/) 2 1.2. , we see that the mapping can be described by X
-9'
The sin 2 x s
t> = >
2. .
2 Ilaar measure 1.2. 4 reads in the polar coordinates 1 2TT 2 n fdxdi d , i.e. it is normali ed to the volume of . pplying the
;{()(
2
u a n t u m theory of
ang Mills fields
coordinate transformation (2. .2 ) to the: integral (2. .2 ), it becomes transformed into the aar measure of S (2). The expression (2. .2 ) counts how often the image of S3 under g(x) covers S (2), it is the inding number o g(x) Be 6, o 7 . The concept of winding number is more familiar from the mapping of a circle S1 : { / 0 p 27r} 011 the group ( l ) with parameters {exp(ir//) 0 f>' 2ir}. The map M: p > 4 = n f) covers (l) n times, its winding number is v/;(M) = n. ur description of M, (2. .30), is not complete. The spherical coordinates do not cover the "south pole" x 1 = p, the S (2) coordinates do not reach g = 1. So we should describe the tangent spaces of , or the infinitesimal transformations of S (2) by overlapping coordinate systems, i.e. by vector bundles. e have described this procedure by the discussion of the instanton potential (2. .24). A complete definition of the winding number refers to the map of the corresponding vector bundles. From these considerations we get a geometric interpretation of the topological quantum number 1 of the instanton: it. is the winding number of a map of the sphere into the gauge group S (2). ther similar types of solutions, the so-called antiinstanton solutions, are obtained from the condition for anti-self-dual fields f t i v = T^,,. The simplest antiinstanton solution with Pontryagin number 1 can be written in the regular gauge as 2x2
x2
A2
f
^ x2 '
r
-
x
=
2
tx2 + X2)2
f
tl,,
(2
31) '
and similarly to (2. .2 1) in the singular gauge. The definition of duality is translation-invariant. Therefore we may substitute x by x-z in the expressions (2. .23), (2. .24), and (2. .31) for getting other solutions of the field equations with = . Self-dual solutions for higher Pontryagin numbers from the ansatz a77 \ (x)
=-f^d
logp(x)
1 can be obtained
(2. .32)
in the singular gauge as a solution of the self-duality equation d,td'' p(x) = in the form (2. .33)
2.1)
Non-pertnrbative a s p e c t s of gauge theories
337
with 5A: free parameters A, and Z{. They can be interpreted as k instantons which are located at positions z; and have sizes A,. The most general selfdual solution with topological charge k is characterized by 8k—3 parameters a*. No simple explicit representations have been found for these solutions. However, there are algebraic methods which allow for the construction of all solutions of the self-duality equation. This also applies for self-dual SU(iV) gauge lields [At78, Ch78, Co78). For self-dual fields T til , = tum number up to a factor 5e =
=
the action agrees with the topological quan-
=
^
( 2 9 3 4 )
Moreover, Schwarz's inequality, (T, T ) < y (T, T)(ÌF, T ) , and the relation = ( T ^ ) imply > for arbitrary fields. Consequently, within the class of fields with topological quantum numbers k, self-dual fields give the minimal value of SE. Thus, in agreement with Hamilton's principle of minimal action, self-dual fields arc in fact solutions of the field equations.
2.9.1.3
T h e C h e r n classification of g a u g e fields
The Chern classification of classical gauge fields by topological quantum numbers is most straightforward for compact space time manifolds. (In certain cases these can be replaced by boundary conditions at infinity [cf. (2.9.10) and (2.9.23)]). Such manifolds M require a covering by different local coordinate systems U l . A well-known example is the sphere where we have for instance two coordinate systems, one continuous at the north pole and one continuous at the south pole, respectively. If we consider the gauge fields as infinitesimal connections, then the representations of the gauge fields in the different coordinate systems are related by gauge transformations, (2.3.21), -4° =
fl^V2)"1
- i(cW/2)(i/12)-1
(2-9.35)
in the overlapping region. The are the connections in the coordinate systems W. In order to be consistent in the case of three overlapping U \ the transition transformation g'k G G must satisfy the cocycle conditions cf k = ( / i } l k .
(2.9.36)
;{()(
2
u a n t u m theory of
ang Mills fields
The systems of coordinates in the gauge fields .4j, and the transition l transformation g satisfying the cocycle conditions are called a principle, fi re undle of t e gauge group - It, is a concretization of classical gauge fields that includes the possibly twisted boundary conditions Ch 2, a 0). In Sects. 2. .1.1 and 2. .1.2 we gave some illustrative examples of such fibre bundles. ur discussion contained also the main elements of their classification by topological quantum numbers. In the following we give a definition of topological uantum num ers for general gauge fields in arbitrary even euclidean dimensions = 2 j with complex gauge groups. Such considerations play a role in theories beyond the standard model. e consider the unitary group S (/ ) and denote the hermitian generators by T'l , i„ = 1 ,..., . auge potentials A / 2 (x) and the field strength F lJ2{x) are constructed according to the general scheme explained in Sect. 2.3. The generalization of the Pontryagin density for such cases in dimensions reads i 5, o 7 1 V
'
J
iU...,lf=X
x F ^ ( x )
T (2. .37)
This formula (2. .37) contains a sum over the permutations ir of the indices i ,i-2, ,ij with signature sgn(7r) = 1. The topological quantum number j (F ) is defined as cf. (2. .26) {Fj)=
[
c (FJ{x))d x.
(2. .3 )
JM
ithout proof we add the following remarks on (2. .37) and (2. .3 ). 1.
For general manifolds M , e , l l , l p must be interpreted as the covariant volume tensor. nly in flat space, e , n ,L is the constant antisymmetric tensor.
2.
The coefficients in the definition of Cj are chosen such that the topological quantum numbers become integer in a compact manifold. Similar to (2. .27) and (2. .2 ), the proof of this fact, follows from showing that C j ( F ) is a generalized divergence of some functional , [A}.
2.9
N o n poilui'l>ative a s p e c t s of g a u g e t h e o r i e s
339
The topological quantum number k(F:') is invariant under a continuous transformation of the gauge potential. All gauge potentials with the same topological quantum number belong by definition to the same Chern class.
!l
Finally we want to show that. (2.9.37) and (2.9.38) really cover our two examples. Restricting (2.9.37) to the two-dimensional SU(1) case (Sect. 2.9.1.1) gives directly c ^ F 1 ) = l/(47r)F /II/ e' i ", i.e. (2.9.5). For the SU(2) case in 4 dimensions, we put Fftl„ = J2„ F ^ / 2 . The evaluation of the sums in (2.9.37) gives 2
T°
Th
E E ^ w ¿1,12=1 n
li
1
t2) = \ ^
coil T a T r T
" ~
=
- T '
and therefore c 2 ( F 2 ) = -l/((iATT 2 )F^F^£' Wf " T corresponding to (2.9.25). The instanton has 29 k(F2) = - 1 . These two examples are good illustrations of the general theory of topological quantum numbers.
2.9.2
T h e index t h e o r e m
There is a characteristic relationship between the numbers and n of right.- and left-handed solutions of the inassless Dirac equation in a background gauge field and the topological quantum number of this gauge field. Index theorems express this fact [At68, Eg80, Gi95]. Dirac equations on gen eral manifolds have their own involved features. Therefore, we restrict ourselves to demonstrate the index theorem for the two- and four-dimensional cases where n+—n~ is equal to the topological quantum number k as defined in (2.9.5) and (2.9.2G), i.e. 7i+ - n~ = k.
(2.9.39)
We give some explicit examples and add some hints on the proof of the index theorem. 20
In this scction \vc use the mathematical notation where k(F2) instanton with k = 1.
= —1 for a physical
;{()(» 2
2.9.2.1
Q u a n t u m theory of Yang Mills fields
T h e zero m o d e s of t h e Dirac equation on the torus
We consider again the solution of the two-dimensional euclidean Dirac equation on the torus T [Jo90, Jo94], l/hf>(x)
= Y{dlt
- ieA^(x) = 0
(2.9.40)
T h e normalizable solutions of with 7 1 = T\, 7 2 = r2, 7 5 = —¡7 l 7 2 = (2.9.40) are called zero modes, because they belong to the eigenvalue zero of the Dirac operator ¡¡). In order to have an example of the index theorem, we consider the potential Atl(x) (2.9.6) with a constant field strength F,tl/(x),
= (--2L-X2, v eL\L2
o) , J
Ftl„ = epu-^r, eL\L2
(2.9.41)
as a physical background field. The flux is quantized and belongs according to (2.9.5) to the topological quantum number k = 1. The solutions must be continuous on the U(1) fibre bundle on the torus, i.e. they must be consistent with the periodicity conditions (2.9.2) and (2.9.4). They must satisfy
tfx
+ Li) = gi(x)i>{x)
with
= exp(—2nix\/L\).
* ( x ) = exp ( - ¡ » f ^ ) exp
(2.9.42) The ansatz
+
V>(0)(*), (2.9.43)
where
is a solution of the free Dirac equation
f ,
(2.9.44)
solves the Dirac equation locally. The solution of the free Dirac equation has to be chosen in such a way that the boundary conditions (2.9.42) are
2.9
N o n - p e r t u r b a t i v e a s p e c t s of gauge theories
.'i.'t.'t
mil islicul. This amounts to il>\0)(x + L i) = exp
:
rH).
V;j0)(.-/; + ¿2) = exp - \ n ( z + ^.¿''(x+Z/i ) = exp
^
,-RH).
>l>.^(x-\-L)) = exp
(2.9.46)
with z =
X\ + 1X2
X\ - \x2 L
X=
II
(2.9.46)
Since Y'ip2(x) = - i h ( x ) , the solution with (re) ^ 0 is exponentially growing and has to be discarded. Thus, the only consistent solution obeys V4 0) (z) = 0 or j5ip(x)
= ij>{x).
A more explicit argument follows from the theory of elliptic functions. The function V'i^O1') >s analytic in 2 and periodic up to an exponential factor in the real and imaginary direction. This leads to elliptic functions. Therefore, it is no surprise that the solutions of (2.9.40) can be expressed by Jacob!'« 0-function 0:i(z, r) = ]T n exp(27rinz - |T|7m2), namely
•ip(x) =
7T . X\X2' HI-" exp exp u L\L,2 2M i 1 | - l / ' l n=+oo L
E
ex
p
-
\z?) 0i{z,T)
(g+n)
(1)
(¿). (2.9.47)
This is an example of the index theorem with n+ = 1 and n~ = 0. In Fig. 2.23 we give the absolute value p and the phase of the zero mode. The size of the torus is that of Fig. 2.22. The physical interpretation of this zero mode is the following [Va84]: the four-dimensional Dirac equation with a constant magnetic field Mz with energy E = 0 and zero momentum in z direction can essentially be reduced to the two-dimensional equation (2.9.40). In this sense the wave functions (2.9.47) describe the lowest Landau
;
2
u a n t u m t h e o r y of
ang
ills fields
.2 f
r
.1
'
4
0 .
2 2
f
\
( )
\
\
I
2 x,
t
—
,
1
I
Fig. 2.2
2
x,
,
4
ero mode of the irac e uation on t.he torus in the instanton bac ground field lines of constant absolute value p left and phase right
level of a charged irac particle in a constant magnetic field with aligned spin. The exact compensation of the orbital ero-point energy by the spinalignment energy renders this a dynamically preferred mode. This effect is not described by perturbation theory. e discuss the significance of 2. .4 in ect. 2. . . .
2. .2.2
irac
and of Fig. 2.2
for the path integral
e r o m o d e in t h e i n s t a n t o n b a c g r o u n d
e consider the solution of the four-dimensional euclidean in the instanton bac ground field tII (t)
{x)
=
(d,t - Uj«
i>{i){x)
.
field
irac e uation
2. .4
Here x are the instanton fields 2. .2 and 2. .24 , which are regular 2 in and , respectively cf. discussion after 2. .24 . In order that i x is continuous in the 2 instanton bundle, the two solutions have
2.9
Non-perturbative aspects of gauge theories
to lie related by the gauge transformation g(x) 0 < |.T| < oo, 1>W{x) = g ( x ) ^ ( x ) ,
g(x) =
Xi
\x\
.'i.'t.'t
in the intermediate region
(2.9.49)
llecause of the topological quantum number k = 1 of the instanton, the index theorem states that there is at least one normalizable solution of the massless Dirac equation with j^rp = ip. (There is only one such solution because in our case there is no solution for yr'ip = —i/J). In order to describe explicitly the solution we should have in mind that 0
^•
m
( (x)=l
K a
(')<m \ j, o \ 3/2
In order to prove (2.9.48) for (2.9.50) by direct calculations it. is useful to rewrite (2.9.23) as
and make use of (2.9.18) and of the completeness relation of the Pauli matrices =
= 28nJf.
(2.9.53)
Equation (2.9.51) follows from (2.9.50) by using the gauge transformation (2.9.49). Thus, the validity of the Dirac equation (2.9.48) results for i = 2
;{()(
2
u a n t u m t h e o r y of
ang Mills fields
directly from (2. .4 ) and from the transformation property of the covariant derivative (2.3.23). For the antiinstantons we get similarly a solution ^(a;)=
2. .2.3
(2. .54)
e m a r k on t h e p r o o f of t h e i n d e x t h e o r e m
e derive the index theorem with help of the heat-kernel equation i 5. The heat ernel G of a hermitian operator A with non-negative eigenvalues A , 0 and corresponding eigenfunctions frm(x, a), where a denotes discrete indices, A 0 m ( x , « ) = \'fn fiul(x, «), satisfies the heat- ernel equation ^ +
G (x,ct, , r)
X
= ,
A(.T, , , f3 0) =
f(.r -
(2. .55)
)Saf
with the solution G (x,a, ,fcT)
( , \er*T\x,a)
=
=
x,a),
for r
0.
(2. .56)
m
The orthonormality of the eigenfunctions implies = 2C~X'
d xG (x,a,x,a-,T)
T
= rft{e~ T)
(2. .57)
e consider now a differential A operator which acts on a vector field. The components of the vector are labelled by «. ere a contains besides the spinor index also the S (/ ) index. In our application, A is the product of the antihermitian euclidean Dirac operator ) = j''( lt i.4 ;i ) cf. (2. .40) or (2. .4 ) with its adjoint A = pjp* = -lpip =
= -Dtlir
- iA tf"
M")
+
7" ^
^ 7",7" ^-
(2. .5 )
2.9 Non-perturbative aspects of gauge theories .'i.'t.'t
I'ln'ii- is ¡i well-known asymptotic expansion in r for G'&(x, ;/]T). If the mling differential operator of A is the Laplacian -WO^ in D dimensions, lll r m (2.9.58), this heat-kernel can be expanded in the asymptotic series ,
00
,
s >
(2.9.59)
with Uo(x,y) — 1 owing to the initial condition (2.9.55) and Uj(x,y) 0, is a J (/y, :i:) from henniticity. For Ao = —d,ld,t already the first term, j ii nolntion of (2.9.55). In the following, the chiralities of the eigenfunctions i/'m of the Dirac operator, i # m = il' l (d„ ~ iAJlpm = A m Vm,
(2.9.00)
play an important role. The relation { 7 5 , 7 ' ' } = 0 implies i#>7'Vm = - ¡ 7 r ' W m = - A m 7 ' V m -
(2.9.61)
Hence, for A m ^ 0, A m and — A m are both eigenvalues which belong to independent orthonormal eigenfunctions. For A,„ = 0, we can distinguish between left-liaTided mid right-handed lions VC±=
i(l
sola-
±75WA("),
= ^ W L "
= 0.
(2.9.62)
The index of I f ) ind //> = n+ - n_
(2.9.63)
counts the difference of the multiplicity of the left- and right-handed eigenvalues A„, = 0 of //>. It appears on the left-hand side of the index theorem (2.9.39).
;{()( 2
u a n t u m theory of
ang Mills fields
ith this information on the chirality dependence of the eigcnfunctions if),, of If), and therefore of A = // // , we calculate dDxTr(-r G (x,x->T)) d Dx
=
=
d Dx
^ ( rn
^ ( x l e -
£ M,AM = 0
d0 rfftl75ai(«,«) .
(2. .64)
The trace refers to the summation over the spin and S (/ ) indices of the matrices G and aj(x,x). In the transition from the first line to the sec0, /;, and 7 r 'ip m are orthogonal ond line we used the fact that for A2, degenerate eigenfunctions of A. Alternatively, if one uses the basis with eigcnfunctions of 7 , (l these contribute to the sum with opposite signs, and their contribution cancels. The remaining contribution with Af = 0 is independent of r. Therefore, we can apply (2. .5 ) in the last line for arbitrary small r. The proof of the index theorem is now reduced to the evaluation of Cj
= jdDxTv ^aj(x,x)\.
ne can prove cj For aj(x,x)
0 for j
(2. .65) D 2.
we have the recursion relations (ij i(x,
x) = ja,j(x, x),
ao(®,x) = 1.
(2. .66)
This follows from (2. .55) and (2. .5 ). It is immediately evident from the ) for r . For A we fact that the gaussian in (2. .5 ) approaches t)(:c use the expression (2. .5 ). Because of the trace formulas of 7 s with the irac matrices only the commutator term in (2. .5 ) contributes, and we obtain e, =
d2xepvFltv(x)
(2. .67)
for the two-dimensional case, and ( 2=
d e^ F^Faptx
(2. .6 )
2.
on pcrlurbativc s p o o l s of g a u g e theories
347
lor SI 1(2) in four dimensions. e add some remarks to this short sketch of the proof of the index theorem. I iillation (2. .5 ) is mathematically proven only for some bounded region in r,y. But for all the assertions made here, there are precise mathematical proofs. The heat-kernel expansion is used here as a regularization scheme for not well-defined sums like T)m ij n{x) y5^m(x). This regularization is used for many other applications in perturbative and non-pert,urbative gauge theories (el. Sect. 2. .3.2 and Sect. 2. .3. ).
2. .3
P a t h integral and t o p o l o g y
In this section, we investigate the consequences of these topological properties by evaluating the euclidean path integral for the quantum-mechanical expectation value of an operator i2, e
The significance of the classical solutions, like the instantons, for the quantized theory might be inferred from the semi-classical evaluation of the path integral. Tn this approximation, the integral / A over the gauge fields must be supplemented by a sum over all Chern classes. ero modes for fermions appear as a consequence of the index theorems. These afford a more sophisticated consideration of the rassmann integrals of the fermion integration. e first, treat these two problems independently. In this section we discuss several physical consequences of this procedure. These are not yet firmly established since most of these considerations depend on the validity of the semi-classical approximation.
2. .3.1
Semi-classical a p p r o x i m a t i o n
The semi-classical approximation is based on the saddle-point method. This method approximates integrals of the form (2. .70)
:i l
2
u a n t u m theory ol
ang M ills fields
for small g by gaussiau integrals. ere 4 (x) and f ( x ) are real functions of x. In general, the most important contribution to such an integral comes from the region in which / ( x ) assumes its minimal value (a more precise statement must be based on specific assumptions). So if f ( x ) is expanded around its minimum.
f ( x )=
f(xo)+f (xo)
(x-.ro)2
f'(xo)
= 0,
f (xo)
0, (2. .71)
this leads to r+oo ( )
t
dxexp
x) -oo I 2 rg f
7
/( o)
/"( o)
(x - x 0 ) 2 1
(2. .72)
I (x0) exp
If the function / ( x ) has several minima, the contributions of all these minima have to be summed. This method can be generalized to the evaluation of the path-integral formula (2. .0 ). To this end one has to find the minima of the euclidean action Se 0. According to amilton s principle, field configurations which minimize S are solutions of the classical field equations. This is, in particular, true for the self-dual or anti-self-dual solutions. Moreover, these yield the absolute minimum of S for field configurations with a specific topological quantum number , in agreement with the discussion after (2. .34). Similarly to (2. .71), we expand field configurations around classical solutions A® , here self-dual solutions with winding number , A
A cl 4 A q
S e { A } = S { A C A "} = 5 { A C I } d
S { A } =
7T*
AqlIAAqu
0({A
}3), (2. .73)
The fluctuation operator A is defined by the second-order expansion of .S p,(A) in A q ". The partition sum Z and the expectation values are sums of
2.
on poilui l ative a s p e c t s of gauge theories
34
gaussian integrals,
The integration extends over a volume M k in the ( fc - 3)-dimensional space of parameters a of the solutions belonging to the minimal action cf. remarks following (2. .33) . In general, the fluctuation operator A depends on these ( A- - 3) instanton parameters. An eigenvalue zero of A is linked with each instanton parameter and must be split off when det A is evaluated. This can be done by introducing collective instanton parameters cf. Sect. 2. .3.2,4) . 2. .3.2
T h e one-instanton calculation
As an illustration of the rather general procedure of Sect. 2. .3.1, we calculate the one-loop contribution of one instanton to the vacuum-to-vacuum transition. To this end we use the bac ground-field method of Sect. 2.4.4 with the classical instanton field A 1 as background field A. In this approach, the renormalized semi-classical approximation can be considered as an approximation in the background-field method where the background field has the physical meaning of a classical instanton field. The background-field method is presented in Sect. 2.4.4.1. ere, however, all quantities refer to the euclidean metric. The vacuum-to-vacuum amplitude about a single instanton can be represented by a path integral, { -=T{A } =
i p A,u,u e-^A,A.}j (2. .7«)
where S (A A 1 ) is the euclidean version of the action corresponding to the Lagrangian (2.4.113). e want to calculate (2. .77)
;{()(» 2
Q u a n t u m theory of Yang Mills fields
We follow't. llooft's paper [tll7(i] but cannot treat all the details. However, we sketch the basic ideas, i.e. 1)
the definition of the fluctuation matrix,
2)
the calculation of the determinant,
3)
the regularization of the determinant,
4)
the treatment of the zero modes, and
5)
the results.
1) For the discussion of the fluctuation matrix we take as classical solution the instanton potential A^1' (2.9.23) and its field strength,
AHx)
Alf{x-z?i+AF+{x)
=
(2.9.78)
where t ;( „ of (2.9.20) is decomposed as t,w — r/",yr"/2. The five free parameters z1' and A are associated with translation and scale invariance. The splitting of Aft = + /lyl"'" into its classical and its quantum part, is formally equivalent to (2.4.107) in the background-field method (cf. Sect. 2.4.4) with the background field equal to the classical instanton field ÂfL = à]',". The path integral (2.9.76) requires gauge fixing. For this the backgroundfield gauge fixing (2.4.111) is used, Ca{A(iU;
x) = D
a c
' '
l
A \ D ™ = d,t5ac + geabcAJ'1.
(2.9.79)
According to (2.9.73), the Lagrangian including gauge-fixing (with £ = 1) and the ghost term is expanded up to second order. The result reads
onp o i l u i l a t i v ea s p e c t s of g a u g e t h e o r i e s
351
rite terms linear in the quantum field A 1" are proportional to the equations i l motion of the classical fields S ) l and vanish because A 1 is a classical nnliition. From (2. . 0) we can read oil the fluctuation matrices, Aa ^a
= - D
+ 2geahcFjl,b
2
qxi c,l
(2. . 1)
' ,
A g h ii = -D2u,
(2. . 2)
= b2i> - Y ^ Y i ' i
(2. . 3)
e have chosen the instanton solution A d ) , which is regular in 1 l bi a complete discussion we should study A ^ and A ^ and their connection (2. .24) including the transformations for the ghost fields. This is beyond l.lie scope of this book.
2) In this paragraph, we discuss the one-loop vacuum-to-vacuum ampli tude using the Lagrangian (2. . 0). e restrict ourselves in the follow 30 ing to a pure gauge theory. sing (2.2.11) and (2.2.27), we get formally
(
A
(
detAo ,AJ where det
detA 0 ) B h /
a and det A,, , are the determinants for the instanton background
a n d d e t Ao,/i a n d d e t A 0 ) g i , t h o s e for A 1 = 0.
The determinants of the fluctuation operators A can be calculated as a product of their eigenvalues, AI
= A2
,
det
=
A2.
(2. . 5)
ero modes in A, lead to infinite fluctuations. In the path integral their (non-gaussian) integration must be treated separately. Therefore we start without the zero modes and consider detA
A2,
A=
A2 0 30
log A2
log det A^ = A, 0
The contributions of fcrmions are considered in Sect. 2. .3.3.
(2. . «)
;{()(» 2
Q u a n t u m theory of Yang Mills fields
and
- 1 / 2
det A'A ' det A 0,A
w>0) =
det A'Rl> det A; o.Ri>.
(2.9.87)
G. 't Hooft determined the non-vanishing eigenvalues, A2 ^ 0, by considering (2.9.85) as a type of Schrodinger equation in four dimensions including spin and isospin. Later it was discovered [Or77] that A can be treated by group-theoretical methods if one transforms the euclidcan space Ri by a stereographic transformation to S.i, the unit sphere in K5. In this compact space A h a s discrete eigenvalues. Without going into further details, we give the results [Or77, Ch77] for the eigenvalues A2 for A^ and A g |, together with their multiplicities dn,
Aa : A2 = 7l(7i. + 3) - 4, dn = ¡(2 n + 3 ) ( n + l ) ( n + 2), n > 1, A2 = n(n + 3 ) - 2 , dn = A\2
n = 71(71 + 3) + 2, dn
=
+ 3)n(7i + 3),
n > 2,
f ( 2 » + 3) (71- 1 )(n + 4), n > 1,
2 Ao,,t : A = n(ii + 3) + 2, dn =
+ 3)71(7?, + 3),
n > 1,
2 Ag h : A = n(n + 3) - 2, dn =
-1- 3)n(ri. + 3),
71 > 1,
Ao,gh :
\2 = 71(71 + 3), 71
dn =
+ 3 ) ( n + l ) ( n + 2), 71 > 0. (2.9.88)
The determinant Ao,/i contains five zero modes and the determinant Aoi(.i, contains three. The eigenfunctions are composed of spherical harmonics, vector spherical harmonics, and tensor spherical harmonics of ¿V Since d„ ~ n 3 and log A2 ~ log n, the product of these eigenvalues (2.9.80), even when divided by del, Aq, badly diverges, so that we must regularize.
2.9
N o n poilui'l>ative a s p e c t s of g a u g e t h e o r i e s 118
3) Instead of dimensional regularization, it is more convenient to regulari/«' I lie determinants by the Pauli Villars method. 31 In this scheme (2.9.8G) in substituted by it (det A')re g = det A' J J det(A' + Mf)Ci i— 1
(2.9.89)
with sufficiently large R. The Pauli Villars regulators with masses Mt arc of alternating bosonic and fermionic character e, = ±1. They play the role of effective cut-off parameters (cf. Sect. 2.5.1.1). The product (2.9.89) converges rapidly if the regulator conditions
¿ e j = 0, ¿ > M ? = 0, ¿=0 ¿=0 li ^ d log M? = - log M2 = i—1
¿ 0 ^ i=0
= 0, . . . ,
finite
(2.9.90)
arc satisfied. 32 For the evaluation of (2.9.89) it is helpful to introduce the zctu function 0 [Lu82a],
CAW = E
TT^ys •
(2.9.91)
The sum includes all eigenvalues with their degeneracies. As a consequence of an estimate on the growth of A2 of differential operators, this series is convergent for large enough Re(.s). Using i
i
r°°
(A2) 31
The rather involved dimensional regularization was discussed by M. Liischer [Lu82, Lu82a], Wc have included the original term in the product by defining co = I and Mo = 0.
354
2
u a n t u m theory of
ang Mills fi . l ls
where r(S) is uler s T-function, (A(S) can be related to the lieat G(x, ,r) cf. (2. .56) ,
=
i -1
^ d t
dDxG (x,a,x,a-,t).
ernel
(2. . 3)
If A is a second-order differential operator in /^-dimensional euclidean space-time the following results can be proven i 5: can be analytically continued into the entire complex plane with the exception of possible simple poles at s = D 2 - j, j = 0,1, A(-S)
The residues of (A ( ) at these poles are esCA( )
=
f
(2. . 4)
ere Aj is obtained from the heat-kernel coefficient defined in (2. .5 ), A
i =
^l
t jTr(aj(x,x))d x.
(2. . 5)
If s is a non-positive integer, i.e. s = 0, 1 , . . . , then these points with the value /Ij/ es F(.s).
A(-s)
s
regular at
Thus, the singularities of the (-functions are closely related to the heatkernel expansion. e can use the regular (-function in order to define the finite part of log det A a77 , logdet A = log 1 1 4 A
0
L
s=
=
- f c
s=0 (2. . 6)
In order to evaluate the regularized expression (2. . log(det A )reg = Tr
=
c
T
lo
) for large
t
we write
«(A
(
A,
))
(2
7
)
2 . 9 NOn poilui'l>ative a s p e c t s of g a u g e t h e o r i e s
355
wliere we used / °°(di/£)exp(—/.) ~ - 7 — log a; and the rcgularization x->0 '1 c condition J^ILo ' ~ Equation (2.9.93) gives r(.s)( A <(.s) as Mellin transformation of TV , exp(-A't). The inverse Mellin transformation [Er53a] reads / \ 1 /-c+ioo A r(s)Cu(s)rsds, IV ( e " *) = — J
(2.9.98)
where c has to be chosen in such a way that F(.s)(,y (.s) is regular on the right-hand side of the straight line from c - ioo to c + ioo. Since P(.s) has poles at s = 0 , - 1 , . . . , and CA'(S) at s = 2, 1 for D = 4, we must choose c > 2. Now we put (2.9.98) into (2.9.97), log(det A')reg = roo (R-
-
1
t
1
(
/-c+ioo
\
<2M»>
L
and shift the path to c = e > 0, e infinitesimally small, by taking into account the contributions of the poles at s = 2,1. Next we shift the path for the part of the integral that comes from the regulamation terms (i > 0) to - 1 < c < 0 by taking into account the contributions of the pole at ,s = 0. At. c < 0 the integral becomes of order 0(M~2 log Mt) for large M2. The result of these operations is .
log(dct A')reg = - CA'(0) log M2
R
R e M
' >" logMf-AxY,
- ^ C A ' ( ^ ) | s = O + J2
log Mf °(
M
R
2
!og Mi).
(2.9.100)
¿>i Equation (2.9.100) expresses the logarithm of the regularized determinant in terms of contributions of the Pauli Villars cut-off parameters and a finite part. We want to add some more details 011 the derivation of (2.9.100). The first two terms come from the residue (2.9.94) of the ^-function, and the third term from the residue of T(.s) at .s = 0 for the regularization fields i = 1 , . . . , / £ . Since the contribution of i — 0 is not included in the residue
;{()(» 2
Q u a n t u m theory of Yang Mills fields
at s = 0, a term — 7ÎA' (0) remains. We used for the t integration the for= r > -I- 1, e) [Er53]. The remaining integral for i = 0 mula J(°° t"exp(—t)dt can be evaluated as follows. First, we transform the inversion of the Mellin transformation into one of the Fourier transformation by the substitutions /, = e x p ( - r ) and •> = ix. This is allowed by the régularisation of the ultraviolet divergences at I. = 0. Because of the resulting ¿-function at x = 0 we may expand F(i:c)(^y (ix) around zero. Thus, we find r°° dt l / T F JO 1 i
ff+ioo +i
r °°
Je-ioo r OO
= ~2-J
r(sKA/(s)rsds =
r oo
& j dxeixrr(\xKà.(\x)
oo
/ •oo dxJ(x)r(ix)CA'(i-c) = - J " ds<J{x)
( J ^
-
7 )
[CA'(0)
+ ixCk'(O)]
= 7CA'(0)-Ca'(0)-
(2-9.101)
For the Fourier inversion we have to use the principle value. The term 7(A'(0) gels compensated by a corresponding term of the regulator fields mentioned above. So we finally find the fourth term of (2.9.100). The first term in (2.9.100) describes a divergent contribution of 4th order. In the difference log(det A')rog - log(dct. A(,) reg this term disappears. In a renormalizable theory, the second term in this difference drops out, too [t.H76, Be77]. Thus, the final result for the regularized determinant of a renormalizable theory without zero modes reads [Ch77]
- ¿ ( C A ' ( « ) - C A ; M ] | , = O .
(2.9.102)
The log M2 term describes the cut-off dependence in the Pauli Villars regulator scheme with respect to the eigenvalues of the fluctuation matrix, 't. Hooft relates M to a fixed-mass regulator po [tH76], l o g p 0 = log M + log 2 - - . 0
(2.9.103)
2.9
Non poilui'l>ative a s p e c t s of g a u g e theories
357
I'lie formula (2.9.102) is the starting point for the calculation of the de tei minant of the fluctuation matrix in an instanton background field. The (, functions £(0) and £'(0) for eigenvalues of the type (2.9.88) were discussed by |(!h77, Or77a]. Putting everything together, one obtains't Ilooft's final result for the determinant without zero modes W'M = e~3 log(/ioA)-a(i)
(2.9.104)
with <*(1) = - 8 C R ( - 1 ) + \ log 2 O
J
=0.443307... ,
(2.9.105)
and the derivative of the Riemann ^-function Ca(-l) = -0.165421... .
(2.9.106)
The A dependence follows from straightforward dimensional analysis. 4) There are 8 linearly independent zero modes Ao of the operator A,.|. These are related to the symmetry transformations of the instanton parain eters: four translations, one dilatation, and three global gauge rotations. The zero modes do not satisfy the necessary condition for gaussian intégra tion, A'^AA' 1 " > 0. The functional integration D[A] into the directions of the zero mode must be performed explicitly. Following a general procedure, we transform these integrations into integrals over the collective instanton parameters. This is achieved by adopting the Faddeev Popov procedure (Sect. 2.4.1.2) for the finite symmetry groups. i)
Translations:
the gauge fields
IMS,/I(®; -)LL2 = I
J
are zero modes of A^. Now Aq (x — z\v)
=
~
(2.9.107)
is up to a gauge transformation,
358
2
Q u a n t u m theory o f Yang Mills fi<.'l
equal to the spatial derivative of the instanton field A|,^(X), A^(x
- z; u) = -
—
2
KAx-z)"
g (x - z)2 + A2 (2.9.108)
The gauge transformation appears because the gauge-fixing term depends on the background field, and the derivative of the classical field with respect to the collective coordinates, in general, is not in the background gauge [Be79]. After separating off the zero modes, the quantum-mechanical fluctuations should be orthogonal to the manifold of gauge fields connected by symmetry transformations. The separation can be done by the Faddeev-Popov procedure discussed in Sect. 2.4.1.2 (cf. Ref. [Mo85]). Like in (2.4.5), we insert the identity 1 = A{A} x J d 4z
(2.9.109) n
* ( / <1AXA^(X
- z,„)
\A'l(x)
- A)f(x
into the path integral. Using the well-known formula 6(f(x)) the Jacobian can be evaluated as
x [A—(x) -
- z)] |
A ( l ) = A
-
*)])
= J(x)/|/'|,
c ( 2 - 9 . 1 1 0 )
After a change of the order of integration, we can apply the semi-classical approximation (2.9.75) for the evaluation of W^1), (2.9.87), w^
= | y d4z j
D [ A ] A { A } e ~ J d 'xC
1 r x J"] ¿ ( / d*xAa0t/l(x
- *;«/)
- Al™{:x
-
z)])
,8x 2
j d4zA{A'}
j V\A""] — ~ A'^'A/j A'1" (2.9.111)
2.9
Non poilui'l>ative a s p e c t s of gauge theories
359
We 11sv expanding A<|U in eigenfunctions and considering only the noti-vanishing eigenvalues. The evaluation of the Jacobian A { A ' } results in
A
^
/ d're Aq /t(x — y; is)Aci'n',l(x
— z) y-z
= J ]
I d*xAa0ill(x-Z]v)Aa0"(x-z-,v)
v=.\ J A = II K ^ ^ l l
2
-
(2.9.112)
I/-Ì
The expansion that leads to W ' ^ is based on normalized eigenfunctions. Therefore, we have to add a factor ( \ / 2 7 r | | / ' ) | | ) ~ ' for each zero mode in order to be consistent with the evaluation of the factor 1/Z. The factor \/27r results from the gaussian integration of the zero mode of A in Z. Moreover, we have disregarded the fact that we should use the regularized determinant (2.9.89) in the evaluation of (2.9.111). In the regularization factors det(A.4 + M'f), zero modes are not omitted. A detailed analysis shows that these contribute a factor /¿o for each zero mode. Hence, the transnational zero modes contribute a factor
- nu=;I y/2n
'"Oralis
1J
(2.9.113) 9'
to The other zero modes have to be treated in the same manner.
.'i(¡'I 2
ii)
Q u a n t u m theory of Yang Mills fields
Dilatations:
/10
'"
the configuration
(X A) =
'
4 AT" X" q [(.-£ - z) 1 + A 2 ] 2 '
47r H^o,/!(a^; A)|| = - , V
(2.9.114)
is a zero mode of A^. Il is a pure infinitesimal dilatation of the classical solution
and yields the factor 'on = -%=\\Aao,tt(x-, A)|| = V27T "
23/27r /2/i0
' <7
(2.9.116)
to in)
Global gauge rotations: Aa0Jx;b)
the fields a
= -2
+
= !'2'3'
(2-9-117)
are zero modes of A,i related to global SU(2) rotations. As 't Hooft showed, the relation of these zero modes to the global SU(2) rotations is somewhat hidden by the complexity of the gauge-fixing procedure. It was shown by Bernard [Be79] that these considerations could be simplified if one considers the zero modes in the singular gauge (2.9.24) with H^g^a;; fr)|| = 4n\/g. These zero modes contribute a factor
/Ga =
ri
=
2
°/Vi?:Vt°
(2-9.118)
to W ^ and a factor 7r2 from the integration over the gauge group SU(2). Combining all contributions we find
WW = Jd*zJ
IO8
"0A_O(1)
(2.9.119)
2.9
Non-porl.tubalive a s p e c t s of gauge theories .'301
wil.li I'/.nro = ^'iVans^Dii^Ga = 2l07r''<7 H. We associated gl with g2{(io). Note dial the coefficient 22/3 multiplying logpoA coincides with the first-order coollicicnt of the /3-function (2.6.36) of the renormalization-group equation. Thus, (2.9.119) becomes independent of the subtraction point if we choose (/•'(/Jo) to obey the renormalization-group equation [tII76]. We remark that //(i 22/3 results from //q of the zero modes and from /¿0 2
22
1
— log no
— log/z + - ,
a s (po) -> a s {n),
(2.9.120)
where /t is the mass scale of dimensional regularization. 5)
The final result for SU(2) in the MS scheme has the form
r = / d4z / JRI
JO
dA A
(2.9.121)
5
with 22 c = /30 = y ,
= 0.145873....
N = 2,
(2.9.122)
A direct calculation in the MS scheme was performed in Ref. [Lu82a] and confirmed this result. Comparing (2.9.121) with the semi-classical approximation (2.9.74) with S[.; = 8n2/g2, we see that g2 = A-na is replaced by the running coupling constant [cf. (3.1.3) and (3.1.16)]. According to 't Hooft, the interpretation of (2.9.121) is best given in the language of path integrals: if |0) is the vacuum, and |0) is the gauge-rotated
.'i(¡'I 2
Q u a n t u m theory of Yang Mills fields
vacuum, then the density D\(z, A, a(/io)) in (2.9.121) is the total contribution to (0|0) from all fields in euclidean space that have an instanton located at z within d' l z and a scale within A and A + dA. We give more details of this interpretation in Sect. 2.9.3.5. The result (2.9.121) is also true for general SU(JV), N >2 [Be79] and NP fermion multiplets in the fundamental representation [tH76]. In this case the constants c and d are in one-loop accuracy given by
c = ß0 =
j N 2E5/C
d =
t:2(N
(2.9.123) -
1)!(AT -
2)!
The constant d is scheme-dependent. The above result holds in the MS scheme. In the Pauli Villars scheme, the term N/(i in the exponent is missing. The two-loop contributions are also known [Mo85]. Including these via
c= A +
47T
(2.9.124)
where the coefficient (i\ and the running coupling (vs(po) were defined in (3.1.13) and (3.1.15), respectively, renders the instanton density two-loop RG-invariant, i.e. Dj" '
2.1
2.1 .It.
Fermionic
on perturbativo a s p e c t s of gauge theories
. ili
ero modes
ero modes for massless ferniions exist in the topological non-trivial gauge in i l ors as a conse uence of the index theorems. The Fubini formula 2.2.2 m the euclidean version allows to perform the fermion integration in the path integral for the partition sum , =
V exp
d .T
m V
det
m .
2. .12
ccording to 2. . , a ero mode leads for m to a vanishing Z,/,. Thus, only the topologically trivial sector, which is free of ero modes contributes to the partition sum. However, we get a non-vanishing result for the Green functions, i.e. for the path integral including fermion sources. This follows from the integration rules for Grassmann variables, 2.2.2 and 2.2.21 , as the following simple example shows I dadoe -caa
a
-fl
d a d « 1 - eaa = e + r]f)-> rpi
or - ai) - arjfja) for
e
.
2. .12
The fermionic Green functions in an external gauge field A with k modes i{x) an(' j(y) are obtained from cf. 2.2. x,
y, ---V
=
/
^
=
(
± 1 L
y
t
illi)
ero
2. .12 i>{x7l)i>(yu) exp
) A i ) • • • X1MXi(Vj,)
cl .c ^Jpiji
• • • XlilJj,)
perm
x
(xil+l,yjl+i\A)---S{xin,yjn; l
exp
for n > and are ero for n . The signs 1 result, from anticommutations of the fermionic operators. The ingredients of this formula are the ero2 mode wave functions Xiix) a n l Xj( )i the effective action e x p - F det , and the Green function S(x, ) of the irac operator in the space orthogonal to the ero modes. This formula can be derived in a way similar
.'i(¡'I
2
Q u a n t u m theory of Yang Mills fields
to (2.9.126) by expanding ip(x) and ij}(y) in an orthogonal system which contains the zero-mode wave functions as base vectors. Equation (2.9.127) is a euclidean generalization of (2.2.65) with interaction that takes into account the existence of zero modes. Finally, the calculation of the fermionic Green functions requires the averaging over the gauge fields, = Yo J V[A,u,u](iP(xMy0
(2.9.128) •••
)$(Vn))A
e"/«1^^
with £„if = £gauge + £fix + £gi10st as in (2.4.17). With the general results (2.9.127) and (2.9.128) 't Hooft's calculation of the one-instanton contribution can be completed. We calculated the righthanded fermionic zero modes in an SU(2) background field in Sect. 2.9.2.2, ,,,
(
(l),m \ J.
\3/2 4"-"
(2.9.129)
+
This zero mode is iVp-fold degenerate for fermions that carry a flavour index / = 1 , . . . , Np. Therefore, we gel. r oo fi \ d ( 2 . 9 . 1 3 0 )
_ _ r (Xn)1>(?/»))Zero = / x nx(z,
JR., jV|.- A'f 1 ,
JO
A,«(/,)) E ( ± l ) I i n W ( ® i perm i= I j— 1
A
- z ) ^ \
y j
- z )
for the zero-mode contribution of the fermionic Green functions and a corresponding expression for the antiinstanton sector. For Np = 1, (2.9.130) becomes (2.9.131) 1
r
2ir2 h , '
,
r ° ° dA ^ Jo
A 3 (l +
5 7
)<S mm '
A5 [(x - zY + A2]3/2[(y - z f + A 2 ] 3 / 2 '
This expression is formally similar to the introduction of a chiral-symmetryviolating one-point vertex with C\ ~ G'A(1 -I- y}) and a fermion propagator
2.
o n - p e r t u r b a t i v e a s p e c t s of g a u g e t h e o r i e s
365
v (rr - z ) 2 A 2 -3 / 2 in configuration space (cf. Sect. 2.2.5.1). Tlie Feynman i uliM of instanton perturbation theory t 76, Mo 5, Mo 7 are based on this Interpretation of (2. .131). 2.0.3.4
C l u s t e r i n g and
-vacuum
In iliis section we investigate the qualitative implications of the topological properties of classical gauge theories for the evaluation of the quantumincchanical path integral. The result of the semi-classical approximation (2. .74) and (2. .75) violates the cluster-decomposition condition (2.1.40), mi important property of reen functions. The cluster decomposition of reen functions requires the uniqueness of the vacuum a 2 . Therefore, the; semi-classical approximation must be reconciled with the uniqueness of tlu vacuum. This leads to the introduction of the 0-vacuum. The fact, that the semi-classical approximation (2. .73), (2. .74), and (2. .75) leads to a violation of the cluster property may be seen by the following argument Lu 0 . The instanton contributions (F /il /(0)F " / (0)) 1 = p ^ 0 violate; parity. Parity may be restored by adding the antiinstanton contribution (l ) t , y (0)F 1 " (0)) 1 . If the cluster property were valid, one would expect. lim
x j/ - oo
where (.. .) I " =
F
h
I
(Flll (x)F (x)Fp ( )Fp(r( ))l+]
= ((...) cf
(.. .) )/2.
= p2
0,
owever, because of Ff
= Fjtu and
" (2- .14) , we get
(^(0)^(0)),
I
= 0 5ip.
(2. .132)
This means a violation of the cluster property. Apparently, in the definition of the vacuum 0)14 mixed instanton antiinstanton states are missing. It is not yet known how to incorporate these. The instanton-liquid model mentioned above Sc is an attempt in this direction. Formally, gauge theories in temporal gauge allow for canonical quantization with canonical commutation relations, amilton function, and wave functions. Let us reconsider the problem of clustering in the semi-classical approximation in this more general quantum-mechanical context. To this end we consider the gauge field tl enclosed in a large box with volume . e consider the corresponding quantum-mechanical wave function as a
.'i(¡'I 2
Q u a n t u m t h e o r y of Y a n g Mills f i e l d s
functional $[.4,,] of the gauge potential. In order to make the quantummechanical scheme gauge-invariant, we proceed by the following steps. First we consider 4'[.4;i] for a temporal gauge A\ = 0 [cf. (2.3.38)]. Then, we apply subsidiary conditions which render ^[.4,,] invariant against infinitesimal, time-independent, local gauge transformations x) = j0lt60"(x)
+ rbcAb,L(xW(x),
dx,80(x)
= 0.
(2.9.133)
Accordingly, these gauge transformations respect the condition A<\ = 0. However, there are gauge transformations that cannot be generated infinitesimally. Such gauge transformations have a winding number. We have discussed this fact for the example g(x) in (2.9.16) in Sect. 2.9.1.2. The winding number of g(x) is w(g) = k = 1 [cf. (2.9.28)]. Let v(x) be a gauge transformation 53 consistent with the temporal gauge of winding number w(v) = 1. The gauge transformation v(x) acts on the wave function ^[.4,,] as U(v)*[Aft]
=
+idjv-1(x)).
(2.9.134)
The gauge transformation [v(s)] n = vn has winding number w(v") = n. Gauge transformations like v", n = 1 , 2 , . . . , are called large gauge transformations, in contrast to the infinitesimally generated small gauge transformations. The space of wave functions {^[4,,]} invariant against small gauge transformations decomposes into sections { v J'[4 ;1 ,n]} labelled by the winding number n. Large gauge transformations Un = U"(v) = U(vn) connect the different sections, tf[4/t,n
+ n'] = U ^ i v M A ^ n ' ) .
(2.9.135)
Several of these intuitive arguments need more mathematical justification. These are based on the homotopy theory of the gauge group [Co79, Be96]. The postulate of invariance under large gauge transformations leads to some new features of the quantum theory of gauge fields. In each section, there is a state |n), the n-vacuum, with minimal classical energy En. The Hamiltonian commutes with large gauge transformations. Therefore, E„ is independent of n. On the other hand, the cluster condition requires the uniqueness of the quantum-mechanical vacuum |fi) [Ha92]. Therefore |i2) must be quite distinct from the 7i-vacua. '"We give an explicit example in Sett. 2.9.3.5.
2.9
Non-perturbativo «.spools of gauge theories
.'{(¡7
In the following discussion of clustering we follow an argument by S. Coleman [Co78]. Let us consider a system in a box with volume V. The transition matrix element from some initial state with winding number 0 to some final :ii at,c with winding number ?i is described by the euclidean path integral H e - " T | 0 ) =
=
f{V,T,n),
D t f C13.T£(A), Jv
(2.9.136)
where in semi-classical approximation only fields contribute that interpolate between a field configuration with winding number 0 and one with winding number n. This is indicated by the index n in 2?[A,n]. It is not important for the transition matrix element. It is, however, that we take the n-vacuum important that the equation f(V,ri+r2,n)=
£ f(V, T\, n\)f(V, rii +na=n
T2, n2)
(2.9.137)
holds for large times T\ and T2. This follows from the fact that the winding number n is related via Stokes' theorem to the integral of a local quantity. We have shown this for the classical instanton solution on a sphere in (2.9.2(>) and (2.9.28). In the example above, it tells us that the way to put total winding number n in a large box is to put winding number ni in one part of the box and winding number n2 = n - n\ in the remainder of the b o x . " In this interpretation we may consider (2.9.137) as a special form of the cluster property. Formula (2.9.137) is not what we expect from the transition matrix element of a single non-degenerate energy eigenstate like a vacuum. Such an object would be a simple exponential and would for large times obey a multiplicative composition law rather than the convolutive composition law (2.9.137). However, by forming a Fourier series f(V,T,
9) = ^ e "
1 0
/ ^ » ,
(2.9.138)
71
31
Such counting misses field configurations with significant action density on the boundary between the two sub-boxes. For our rather qualitative argument, we assume this surface effect to be small for large boxes.
This is the desired composition law. Thus, we identify / ( V , T, 0 ) as being (up to a normalization constant) the expectation value of exp(—7/T) in a vacuum state, in which the cluster property is valid, the Q-vacuum
|0) =
¿
e in ®|n).
(2.9.140)
Following (2.9.13G) and (2.9.140), we write formally
(0|e-//r|0) = ^
î?[A,n]e-5E
(2.9.141)
where Z' is a new normalization factor. Replacing the topological quantum number n by (2.9.26), n = 1/(16tt 2 ) $ dAxTr TtlvPw, we get
n x exp ( / D
4
*
T r ^ n
+ Í ^ T V ^ H ) )
•
(2.9.142)
Some far-reaching conclusions [Ca76, Ca78] follow from this treatment of the topological properties of a pure gauge theory: •
o n p e r t u r b a t i v o a s p e c t s of g a u g e t h e o r i e s
. ili )
In a pure gauge theory the path integral 35 (2.4.1 ) or (2.4.1 ), has to lir extended by summation over topological sectors (Chcrn classes) ZE{J}
^[A,n]ein®e-5B
=
(2.9.144)
Tl liquation (2.9.144) is based on the fact that the Hamiltonian is invariant under small gauge transformations. Therefore, all other energy eigenstates can be chosen so that they change at most by this constant phase. •
The inclusion of the topological properties (e.g. winding numbers) in the semi-classical approximation causes super-selection rules for gauge-invariant quantities, labelled by a parameter 0 . Only non-gaugeinvariant operators transform states of different sectors into each other.
•
In a complete theory one lias to distinguish between a perturbativc vacuum |0), which disregards topological excitations, and the nonperturbative, physical vacuum | 0 ) .
2.9.3.5
Vacuum tunnelling
Large gauge transformations connect fields belonging to sectors with different winding numbers. In order to see this, we transform the instanton solution (2.9.23) into the temporal gauge AQ = 0 [Ca78, Hu92], using <7(x,£\i) = exp
17TXT
$(x,x4) V x 2 + A2 _. . 1 1 a,\i $ ( x , x 4 ) = - + - arctan . I 7T \ J x l + A1
(2.9.146)
Note that 3r
{i 0
for for
x^ —> oo, J\ —> - o o ,
'We omit gauge-fixing and gliost terms like those in (2.4.17).
(2.9.147)
.'i(¡'I 2
Q u a n t u m theory of Yang Mills fields
and therefore
{
(
¡TTXT
A
u(x) = exp = for XA —» oo, _ , V W x 2 -I- A V 1 ' for —> — oo. (2.9.148) Like g(x) in (2.9.16), v(x) has winding number iu(v) = 1. This can be seen by straightforward calculation. Alternatively, one can express x in g(x) in polar coordinates (2.9.29) and finds that x = 7r|x(/(x" 4- A 2 )'/ 2 is a one-toone map between 0 < x < ^ and 0 < |x| < oo. Hence, i;(x) can be smoothly deformed into g{x). Now we apply the gauge transformation g(x,x^)
to the instanton solution,
2r xv - 1 A ' ^ x t ) = g ( x , z 4 ) ^ ^ 5 f f ( x , $ 4 ) -1- i oo, which implies
M *v , * * ) - * '
\ 0
[° r for
—> - o o .
(2.9.150)
As in (2.9.10), A,L(x) approaches a pure gauge. Thus, this calculation shows that the instanton solutions interpolate between gauge field configurations with different winding numbers (k = 0 for x4 —» - o o and k = 1 for xA —» oo in this case). The transition amplitude between neighbouring ra-vacua is in semi-classical approximation [cf. (2.9.73), (2.9.74) and (2.9.136)] given by
(n + l\exp(-HT)\n)
= (l|exp(-//T)|0) ~ 4 = ^ = , V (let A
(2.9.151)
or less formal b y ' t Hooft's result (2.9.121). The formula (2.9.151) reminds us of the energy splitting between the lowest energy states of a double-well potential V(x) = g2(x2 - a 2 ) 2 / 2 . It is known from elementary quantum mechanics that the tunnelling between the classical ground state at x = —a
2.9
Non-porl.tubalive a s p e c t s of gauge theories
.'571
In I,lie classical ground state at x l a leads to a splitting between the Hyiumetnc and antisymmetric ground-state solutions with [Sc98] =
The picture of vacuum tunnelling in a pure gauge theory changes if one includes massless fermions [Ca78]. The zero modes of the Dirac equation in an (anti)instanton background field with topological quantum numbers k = ±1 (2.9.63) render the tunnelling amplitude zero in the absence of external fermion sources. Therefore, Green functions which contain at least one pair of Dirac fields i/j(x)iJj(y) have to be considered. Moreover, we have to include in the «-vacuum the occupation number of fermions and antifermions. In Dirac's hole theory, these numbers count the occupied or empty fermion states, respectively. A perturbation-theoretical 7i-stat,e is thus one for which the gauge field belongs to a winding number n, and all negative energy fermion levels are occupied. One can think of tunnelling as being an adiabatic process as far as the fermions are concerned. The time-independent Dirac equation in an instanton field has an eigenvalue e,(/) and eigenfunctions il'i{t, x) for each value of the euclidean time I. Since the net effect of tunnelling is just a gauge transformation, the original eigenvalues of £¿(0) must be in one-to-one correspondence with the final eigenvalues. For the massless Dirac equation in an instanton background field this mapping £¡(0) —> e;(T) is non-trivial £¿(0) ^ ei(T). In particular, the highest right-handed negative-energy eigenstate crosses zero and becomes the lowest right-handed positive-energy eigenstate while the lowest left-handed positive-energy state becomes the highest left-handed negative-energy state. This is a special consequence of the spectral-flow theorem [Va84] which can be derived from the index theorem. Thus, if in the initial configuration all negative-energy eigenstates are occupied and all positive-energy eigenstates empty, then the final configuration contains one right-handed positive-energy eigenstate that is occupied and one left-handed negativeenergy eigenstate that is empty. In the instanton case, the tunnelling process is n —>??.+ ! + qi/]\\ rather than n —>• n + 1 as in (2.9.151). Strictly speaking this is only a virtual vacuum tunnelling n —» n + 1 + q\,q\{ —> n. This has important consequences for the strong CP problem. As shown in Sect. 2.9.3.9, the strong CP problem is absent for massless fermions. It is beyond-the scope of this book to go into the details of the spectrumflow theorem. However, Fig. 2.23 on page 342 shows as an example of this theorem the zero mode (2.9.48) of the two-dimensional Dirac equation in a two-dimensional instanton background field (cf. Sect. 2.9.2.2). The U(l) phase is constant on the line x2 = 0. At this line we have an n-vacuum with
2.1)
N o n p e r t u r b a t i v e a s p e c t s of g a u g e theories
ri 0. At the line x2 = 2, the phase is e x p ( — 2 i r x \ / n o mode interpolates between these two states.
'.I.0.3.7
.'i7.'t
\ ) and n = 1. The
Vacuum condensates
I'lio considerations above show that non-perturbative effects change the vacuum structure of gauge theories considerably. As a consequence, new types of Green functions must be discussed. We consider the two most important examples. The condensate. {g2FF) = (g2F^{ )Fa'f { )) has its origin in the nonperturbative part of the physical vacuum."' Since it is most important for QCI), it is called generally luon condensate. The formal definition of ((/'' F F ) contains the product of field operators at the same space-time point, which needs regularization. Only the product g2FF is renormalizationgroup-invariant, i.e. independent of the subtraction point [Va82]. The condensate ( g 2 F F ) plays an important role in the sum-ride technique introduced by Shifman, Vainshtein and Zhakharov [Sh79, Na89, Sh92]. Its physical results depend on some skillful combination of phenomenological data with perturbative and non-perturbative properties of two-point functions. We do not treat this technique here. A theoretically more satisfying calculation of (g2FF) is based on the lattice approximation of gauge theories. Within large errors the results of the sum-rule technique [Na96] and of the lattice approximation [DE97] agree: (g2FF)
« (0.89 ± 0.11 GeV)'1
from sum rules,
(g2FF)
~ (1 GeV) 4
from lattice.
(2.9.156)
One of the problems in the determination of {g2FF) is the not well-defined separation of perturbative and non-perturbative contributions. In the definition of (g2FF) we put (-) = 0, However, the dependence on 0 might be helpful for such a separation [Ha98]. In gauge theories with massless fermions the cliiral condensate {ip{ )ip( )) is the order parameter of cliiral symmetry breaking (cf. Sect. 3.5.1.1). This is a non-perturbative effect. The connection with the vacuum picture is seen by the following qualitative argument. "'In perturbation theory we have (g2:FF:) = 0.
.'i(¡'I 2
Q u a n t u m theory of Yang Mills fields
We consider {ip{0)ip(0)} for a finite euclidean volume V and a finite fermion mass in. Then we have (•H>) = ^ Tr
1 m + If)
(2.9.157)
The density of the eigenvalues A of the Dirac operator lßil> = iAn?/; is Vp(A) with p(A) = p(—A) = (£n f>{\ - A„)). It becomes continuous for V -» oo and leads to
J-OO
M
+ iA
./o
ra2
+ A2
(2.9.158)
Using lim
— = i7T(5(A) + P P Q j , m + iA
(2.9.159)
we get in the limit in —> 0 = lim [ dA p { " X \ = Tr I dAp(A)i(A) = 7rp(0). j-oo m + iX 7-oo (2.9.160) This relation [Ba80] states that for chiral symmetry breaking one must have an accumulation of eigenvalues at A = 0. Now we know from the index theorem (Sect. 2.9.2) that we have such zero modes in an instanton background field. Thus, under the assumption of the dilute-gas approximation (2.9.153), the separated instantons can be a source of eigenfunctions at A = 0. Numerical values of (ipip) come from the sum-rule technique and from lattice approximation. We quote « - ( 2 3 0 MeV) 3
(2.9.161)
from sum rules [Do98], and $il>) « - ( 2 4 5 MeV) 3 from quenched lattice approximation [Gi99],
(2.9.162)
2.9
2.0.3.8
N o n - p e r t u r b a t i v e a s p e c t s of g a u g e t h e o r i e s
375
Anomaly and tbe index theorem
In Sect. 2.7.1 we found that anomalies of axial-vector currents are related to the lack of a consistent regularization scheme for operators containing yr' matrices. However, it is not only the high-energy behaviour of the triangle I'.i.tph (Fig. 2.11 on page 272) which is responsible for the anomalous nonconservation of axial-vector currents in quantum field theory. It is a problem of the evaluation of the path integral which is related to the topological zero modes of the Dirac equation. We can see this from the equation for the singlet anomaly (2.7.40) in Minkowski space, d^ljx) =
(2:9.163)
My integrating over R 3 we get
t
- I t L * * * ™ = ¿ / ^ " ( M ^ M ) , (2.9.164)
and by integration over R 1 , Jpi d'xd'%(x)
= -
f
(1
":cFa^(x)F^(x).
(2.9.165)
Comparing (2.9.165) with (2.9.63), (2.9.64), (2.9.65), and (2.9.68), the right hand side of (2.9.163) might be considered as an index density [Ja76, Ni77], or the anomaly equation as a local form of the index theorem. The following description makes this considerations more precise. According to Fujikawa [Fu80], the anomalous non-conservation of the axialvector current can be traced back to the non-trivial transformation of the euclidean fermionic path-integral measure under chiral transformations. This measure was introduced in Sect. 2.2.1.4 as a generalization of the integration over a finite number of Grassmann variables (2.2.20)-(2.2.23) to an integration over functional (2.2.27). In this spirit we expand the fermion fields ij)(x) and ip(x) = i/J-y'1 in orthonormalized eigenfunctions of the cuclidean Dirac operator, 1>(x)
i J/>ipm{x) = A m^m(x),
(2.9.166)
i i>l{xW
(2.9.167)
in i>{x) = X
il>Ux)bm,
= AmTpl{x),
.'i(¡'I 2
Q u a n t u m theory of Yang Mills fields
and define the measure by Vty{x))V\i,{x)}
= H damdbm, m
(2.9.168)
where d a m and dbm satisfy (2.2.20) and (2.2.21). The fields ip(x) and ij>(x) transform under local cliiral transformations as ij>\x) = eiß^6ifj(x),
i>'(x) =
.
(2.9.169)
This corresponds to a linear transformation C'(ß) = (c,„„(/?)) for the expansion coefficients am and bm,
(I'm =
'
(2 <J-170)
= XI
n
n
or for an infinitesimal transformation G = 1 + SC, 5cnm(ß)
= i J d4xß(x)ij>l(xh5il>m(x).
(2.9.171)
For the measure (2.9.168) this implies [cf. (2.2.22)] H da' m d6^ = (det C)~2 J] damdbm, m
(2.9.172)
m
or formally V[ip\x)]V[i>'{x)]
= (detC)-2V[ip{x)]V[i>{x)].
(2.9.173)
We use det 6' = exp(TrlogC) = exp(Tr<56') to get the Jacobian J(ß) oi the cliiral transformation (2.9.171),
J(/3) = ( d e t C ) - 2 = e x p ^ - 2 i J d4x ß ( x ) ^ l ( x h ^
m
( x ) Y (2.9.174}
2.9
Non-perturbative a s p e c t s of gauge theories
377
()l course this formal expression needs regularization. Fujikawa proposed the lnuil ernel regularization (cf. Sect. 2.9.2.3),
(E
rinWr'I'mW)
\ m
/
, = Inn TV [ T
= lim ( E \ m
reg
*^] =
) /
„/it/ o/3
(2.9.175)
The last equality results from (2.9.64)-(2.9.G8), the proof of the index theorem. The result is therefore ./(/J) = e x p
J d
'
x
^
e
^
l
•
(2.9.170)
The non-invariance of the Jacobian leads according to (2.2.90) to extra con tributions in the Ward identities, resulting in anomalous Ward identities.
2.9.3.9
O n t h e p r o b l e m of s t r o n g C P v i o l a t i o n
The inclusion of non-trivial topological sectors in the path integral requires the 0-terin of (2.9.145) in the euclidean Lagrangian
For massless fermions, the chiral invariance of a theory implies that the physics is the same in all 0-super-selection sectors. In the path integral we can always transform (-) to zero by a global chiral transformation (2.9.109) with ¡3 = 9 / 2 . This follows from a comparison of the Jacobian for an infinitesimal chiral transformation (2.9.176) with an infinitesimal 0 in in (2.9.177). Therefore, the CP problem is absent for massless fermions. Maybe, the CP problem can be solved by a more fundamental theory which starts from massless fermions.
2.9.3.10
The U ( l ) problem
Because of the anomaly, the charge Q h of the singlet U(1)A axial-vector current is not conserved. However, the two-point function of the divergence of the corresponding current [cf. (2.7.42)], = p(x) = - ¿ J Tr[F»v(x)Fllt/(x)},
(2.9.178)
obeys an interesting relation, the Wit-ten Vencziano relation [Wi79, Ve79], originally derived in the l/N approximation, where N is the number of quark flavours. It states that the topological susceptibility Xtopi which measures the fluctuation of the topological charge density p(x), is proportional to the square of the pseudo-scalar flavour-singlet mass mo and its decay constant /o, xiS = j <14* = 2 / 0 2 mg.
(2.9.179)
Here, the left-hand side refers to xlop ' n a pure gauge theory. 1 ' A sensible determination of m^ requires a theory including fermions. Thus, the WittenVeneziano relation compares quantities of different theories. Equation (2.9.179) has important consequences for the pseudo-scalar meson spectrum (cf. Sect. 3.5.1.2). The main point is that the mass of the i] meson in the theory with two flavours (the mass of the 7/ meson for 3 flavours) is large compared to the pion mass (m- and m/c). This U ( l ) problem is solved by the instantons [1.H7G], i.e. by the Witten Vencziano relation. If we take a value for Xtop = (ISOMcV)'1, coming from lattice calculation [DE97], we get a reasonable agreement with experiment. ,7
Iti a gauge theory with iVf massless fermions, we have
' = 0.
2.10
Lattice approximation of gauge theories."li>I
2.10
s
In order to treat reliably the non-perturbative aspects of gauge theories, it is necessary to find an approach for the evaluation of the basic gaugeinvariant path-integral formula that is different from perturbation theory (Sect. 2.4.1.3) [YVi74]. Lattice approximation of gauge theory is such an attempt. [Se82]. In this scheme, the (D - l)-dirnensional configuration space or the ^-dimensional euclidean space-time is approximated by discrete points of a lattice I\ The approximation of a finite euclidean space-time by a finite lat tice appears most promising. There are quite a few limiting processes necessary in order to get from such a field theory on T to the Green functions in the four-dimensional continuum. These limiting processes define the gauge-invariant, path-integral formula. In this section, we sketch the main features of the lattice approximation of gauge theories [LAT]. Based on the geometry of local gauge symmetry, we describe the gauge-invariant field theory on F in Sect. 2.10.1. The transition to the continuum is treated in Sect. 2.10.4. The infinite-volume limit is considered in Sect. 2.10.5. Because fermions pose problems which have not yet been solved completely, we postpone a short overview of these questions to Sect. 2.10.6. The computational procedures are strong-coupling approximation (Sect. 2.10.2), and above all, numerical methods (Sect. 2.10.3). Results for QCD are given in Sect. 3.G.
2.10.1 2.10.1.1
B a s i c s of lattice a p p r o x i m a t i o n G e o m e t r i c a l e l e m e n t s of pure g a u g e t h e o r i e s
The geometry of local gauge symmetry described in Sect. 2.3.2 can be transferred point by point to the lattice [Ma81]. First, we describe pure Ddimensional gauge theory. The description of Dirac particles has some particular difficulties about which we report on at the end of this section. For the sake of simplicity, a cubic lattice with lattice constant a and lattice; points, where is finite or infinite, is used. The various lattice elements are p o i n t s x with coordinates x = (ii , 2,...,
nD),
nl = 1 , 2 , . . . ,
,
links I) = [a;;/i] connect neighbouring points with coordinates ( x \ , x ) (x + clt,x), WK;re c t — a/7 and //. is oriented from x to x\,
=
.'i(¡'I 2
Q u a n t u m theory of Yang Mills
fields
Fig. 2.24 Objects on the lattice
p l a q u e t t e s V = [.-c; p, v] are the smallest closed loops with coordinates (xz,x2,xi,x) = (x + e^^ + e/t + e^^x + eftjx), {ft ^ v), oriented by this sense of rotation, c u b e s K, with corresponding orientation, and s u p e r - c u b e s with corresponding orientation. A path C\J on the lattice is a sum of connected links C\J = h> a surface is a sum of plaquettes 13 = ^2'Pi- The boundary of a path is denoted by dC\„ and that of a surface by OB. Figure 2.24 illustrates these concepts. The basic geometrical concepts of lattice gauge theory can be summarized as follows: •
Gauge fields describe the infinitesimal parallel displacement of charged fields [cf. (2.3.17) and (2.3.18)]. Hence, parallel displacements U(b) (2.3.36) along links b 6 F correspond to gauge fields on Ike lattice,
Pexp (i Jdt^A^t))
U(b) = U([x-,{i]) =
(2.10.1)
2.10
Lattice approximation of gauge theories."li>I
Lattice gauge fields U(b) = (/([:;:; /i]) are therefore functions that are defined on the links of the lattice and their values arc group elements of G. They possess the property U\-b)
= U~x{b)
Parallel displacement proximation
with - ( x i , x 2 ) = (*2,*i).
(2.10.2)
along a curve C [cf. (2.3.36)] reads in lattice ap-
U{C) = Pexp (\ [ df'A^t)] V Jc
U{CL) = TT U{b,), ¿cL
)
(2.10.3)
where C\, = Yl, bj C F is a path that approximates the continuous curve C'. In agreement with (2.3.25), the field strength describes the parallel displacement around a plaquette. Parallel displacement around the boundary dVx of a plaquette with the links b.\,b3,b2,bi corresponds to U(&PX) = P [ ] U(bi) = U(bA)U(bz)U(b2)U(M
=
U([x-,,1,»)).
bieav
(2.10.4)
As a result of path ordering, the initial point x is distinguished on tinboundary dV. The notions used in Fig. 2.24b are also used here. The Baker I lausdorff formula for the multiplication of exponential functions of non-commuting factors, e M c \B
_ eiA+iB-±[A,B}+...^
(2.10.5)
implies U(dVx)
=
=
(2.10.6)
The lattice analogue of the field strength is Flw(x)
•-> T(PX)
= ^-([ar;p,,/]).
(2.10.7)
Local gauge transformations g(x) G G as in (2.3.35) are defined at lattice points x e T. The geometric quantities then transform in a way similar to the continuum U(b) -> g(x2)U(b)g~l(x,) U(CL)->g(x')U(CL)g-l(x),
with
b = (x2y X] ), (2.10.8)
2
2
u a n t u m theory o f
ang
ills fields
where C is a curve from x to x', and g(x)U(d x)g-l(x),
U(D X) ^ x)
2.1 .1.2
The
g{x)T{ x)g~l(v).
^
2.1 .
ilson action
The dynamics of a lattice gauge theory is determined by t he definition of an action. ccording to . ilson i 4 , the following expression is considered for a pure euclidean gauge theory with gauge group V w
-
Y, P
U-\d )
- 2.
2.1 .1
1 1
The sum extends over all non-oriented pla ucttes of the euclidean space time lattice, i.e. each pla uette appears in the sum only with one orientation,
E
x
E
p u
x
pu
The trace 'l\ U{D' x)] permits a eyelie permutation of the factors in 2.1 .1 and is therefore independent of x. The gauge invariance of w follows from 2.1 . and the trace property. In order to investigate the correspondence with the continuum case we use 2.1 .1 - 2.1 . , especially ( x,v )
f a n
)
i
y ,
2.1 .11
where g{) is the bare gauge coupling. Together with 2.1 .G , 1.2. 2 , and ~ f d x a i one can show that w is an approximation of the action S = f d 1 ; ;a g 2. .42 for small lattice constants, w
~
p
r 4
^
) + U- (drp) - 2
^ T
x
= -
^
^
h
d xF^(x)F ' (x),
fo
^
2 for
,
2.1 .12
2.1
attice approximation of gauge theories. li I
i.e. the eontimuun action. For abcliau gauge groups
2.1 .1.
1 we have f
.
a t h integral o n t h e l a t t i c e
The uantum dynamics on a lattice can be derived from the path integral. II can he formulated with help of the action defined above. If 2 is an observable which is given as a gauge-invariant functional of the lattice gauge lields, then its vacuum expectation value can be calculated from the general ruclidean palli-integral formula cf. ect. 2.2.4.4 ,
The measure V[U given as
must be defined in the space of gauge fields. It can lie
2.1 .14
in which d / i ( g ) is the Haar measure 1.2. 4 - 1.2. for the symmetry group = ( ) . The invariance of the Haar measure implies that a gaugeinvariant uantum theory is described by the basic e uation 2.1 .1 . The gauge invariance of the expectation values follows from the gauge invariance of i i . Gauge-fixing terms fis in 2.4.1 are not necessary for the integration with the finite Haar measure.
2.1 .1.4
a t t i c e a n a l o g u e of physical
uantities
In ects. 2.1 and 2.2 we have discussed in general how to derive in the continuum case dynamically relevant uantities li e masses and cross sections from the vacuum expectation values of the corresponding operators. This can be translated in an appropriate way to the lattice. s examples, we indicate this for the determination of the ilson loop and the lattice masses. The relation to the continuum uantities is discussed in ect. 2.1 .4. e report about the applications in C in ect. . .
.'i(¡'I 2
Q u a n t u m theory of Yang Mills fields
In Sect. 3.7.1 we show that a gauge-invariant definition of astatic potential can be derived from the Wilson loop [Wi74, We7l] W(C) = (Tr [U(C)}) = | | U(C)=
Pexp
P[A]'IY[i/(C)]exp(-Sw{A})
d.s'M»(.s)y^ .
(2.10.15)
This expression can be treated in lattice approximation. To this end, we choose a rectangular path on the lattice which can be approximated by a Sequence of lattice links U[C]^U[CL\=
J ] U(bi).
(2.10.16)
Then, the lattice approximation for the Wilson loop W(C) reads W{C) -> W{CL) = ||V[U]
r
I\(i/[C,J)e-' 9 w l , ; ],
Z = Jv[U]e-s™M.
(2.10.17)
Poles of Green functions in an invariant mass variable P2 at P2 = m2 point to the existence of free particles with mass m (Sect. 2.1.5). In coordinate space, these poles correspond to an exponential fall-off in space-like directions, and also in the euclidean time-like direction [cf. (2.1.16)]. This sets the frame for the calculations of masses in lattice approximations. Let us take the simplest case of a two-point function (F(x)F(y)) of a gaugeinvariant, field F(x) exciting the vacuum to a one particle state. We assume (F(x)) = 0 and (0\F(x)\M,p, j, jz) 0. We consider the lattice approximation F(x) i-> ii(x,*). For a large lattice extent in the euclidean time direction, we expect the behaviour
~ Ae-'"al'l+..., t-y oo
(2.10.18)
where the euclidean time t is measured in lattice units. Because of the finitesize effects, t should be large but much smaller than N, the length of the periodic lattice. The quantity £ = ma is called the correlation length. Measuring the correlation length of the two-point function determines the mass
2.10
Lattice approximation of g a u g e theories."li I
In lattice uuits. lt is the inaili problem of the transition to the continuum lo deterrnine the lattice Constant and the limit. a 0. l Iiis is only a first exainple of the translation of the structure of general liciti t.heory (Sect. 2.1) into the lattice approximation and the extraction of pliysical quantities.
2.10.2
Strong-coupling a p p r o x i m a t i o n
ext we discuss the strong-coupling approximation g\\ > oo, i.e. the limit /) I of the ilson action (2.10.10) and the vacuum expectation value (2.10.13). According to the notion of Sect. 2.2.4.4 this may be called a high temperature expansion. This method played an important role for getting insight into the essential features of lattice field theory. owever, as the continuum limit requires ft oo (cf. Sect. 2.10.4.3), high temperature expansion is of limited value for physical problems.
2.10.2.1
A simple model
In lattice approximation it is possible to have gauge theories with finite gauge groups. e consider the Z2 gauge theory on a two-dimensional lattice with 2 points. The gauge group is the finite group with two elements {c r = 1 } , o2 = 1. Lattice gauge fields U(b) = o{b) are functions on Z2 links b r 2 , which have values 1 . In view of the commutativity of group is multiplication, the field strength (2.10.4), o(d' ) = o(b^)a(b3)a(l)2)( (bi), a pure plaquette function. The following definitions correspond to (2.10.10), (2.10.13), (2.10.17), and (2.10.14):
Action:
6 v = ft
Partition sum: ilson loop:
v(l Z =
~ \a) exp( . A
(C) = \ f
)
a) a(C) e x p ( - 5 w a )
aar measure: j d i i ( a ) f ( a ) = with 'd i(cr) = 1, f d/x(
f{ )
.'i(¡'I 2
Q u a n t u m theory of Yang Mills fields
The integrals for the partition function Z and the Wilson loop W(C) for small ft, i.e. the high-temperature expansion, are best evaluated by the following steps. First, the Boltzmann factor for an individual plaquette is calculated, g -Q{\-O)
_
cosh ft + oe~P sinh ft = C(1 + av)
(2.10.19)
with C -- e x p ( — f t ) cosh ft and v = tanh ft. This result is an example of a Fourier representation of the Boltzmann factor. The first term C is the contribution of the identity representation of the Z2 group, the second aCv that of the defining representation. Since v ~ ft for ft 1, a power series in v can be considered instead of one in ft. The expansion of the Boltzmann factor reads e x p f - / 3 Y ^ [J \ 7>er2
= '
II c[l+t/a(0P)] -per2 /
=c»
2
n Te i'2
[
i
+
»
*
( n
« =o
>
(2.10.20) \
)
] n peG(">
After evaluation of the product Opec*")' coefficient of v" emerges as a sum of the contributions of all possible lattice graphs A lattice graph is a set of n non-oriented plaquettes. Its contribution to the Boltzmann factor is the product of all o(dV) of these plaquettes. The high-temperature (strong-coupling) expansion for the partition sum is obtained by inserting (2.10.20) in its expression above,
Z = CN~ E »
z
E
(G'(n))
\G(">
>
(2.10.21)
/
in which Z(6'^"') is the contribution of a lattice graph to the partition sum, Z(GW)=
i v [ a ]
¡"J a{0V), •pec(")
V[a) =
f j dp(6). Per 2
(2.10.22)
The properties of the Haar measure imply that Z ( G ^ ) is only different from zero if a(l>) in flpeGi") appears in even powers. In view of cr(d'P) =
2.1
attice approximation of gauge theories. li I
I i, -trpa b , this means that each lin must be the boundary of at least two pla uettes of , i.e. Z( ^) only when the boundary d of G vanishes. In a two-dimensional finite lattice this only applies for i.e. the empty lattice graph, and hence 2.1 .21 implies Z = C '. Ta ing a periodic lattice in which opposite lin s of r 2 are identified, then is also e ual to , and one gets the result C
v
i
C
for
TV2
1,
v
.
2.1 .2
This shows that in this limit there is no difference between a lattice with no boundary condition and one with periodic boundary condition. The in llucnce of the boundary disappears for large . The corresponding strong-coupling approximation for the ilson integral is obtained by substituting the olt. mann factor 2.1 .2 into the expression of the ilson loop,
W(C) = J2V'1
( X G
W(C-,
{n)
)
) ,
1,
2.1 .24
with W[C-,
{n)
)=
V[o o{C)
a(d V).
2.1 .2
pect The conclusion is similar to the one from the discussion of the contributions to the partition sum, i.e. only those contribute in which ^ C applies. In two dimensions and with no periodic boundary conditions, this is a single graph consisting of r C pla uettes, where r C is the number of pla uettes which are enclosed by C, i.e. the area enclosed by C. The result obtained is W C) =
r C e
r C logv
2.1 .2
Thus, the area law of the ilson confinement condition cf. ect. . .1.1 is satisfied with a string constant logr; in lattice-constant units for a two-dimensional , lattice gauge theory.
.'i(¡'I 2
Q u a n t u m theory of Yang Mills
2.10.2.2
fields
G e n e r a l i t i e s of t h e s t r o n g - c o u p l i n g a p p r o x i m a t i o n of S U (TV)
This example of a Z 2 gauge theory shows the important steps for the general strong-coupling approximation: •
group-theoretical expansion of the Boltzmann factor for the individual plaquette, i.e. its character expansion (2.10.20);
•
expansion of the total Boltzmann factor into lattice-graph contributions (2.10.21);
•
determination and calculation of graphs which give a non-vanishing contribution;
•
graph summation.
The expansion of the Boltzmann factor (2.10.19) is a special case of an expansion in group characters x ^ ( t f ) = Tr D ^ { g ) of the irreducible, unitary representations (k) of dimension dim(Ar) of the gauge group G. For this expansion the orthogonality and completeness relations of the group characters are important. We write the action as a sum over the contributions of the plaquettes S[f7] = (i^-p S
= ^ c iim(fc)Ajb {P)xik){U)
(2.10.27)
W with AjbC/3) = - - J —
J v [ U } e ~ ^ ( 2 . 1 0 . 2 8 )
The expansion of the contributions of all plaquettes in effective couplings v(k) leads to lattice graphs as in the simple example of the Zi model. Since in the modern treatment of lattice gauge theories the strong-coupling approximation no longer plays a dominant role, we only give some short hints for QCD. For details we refer to the literature [Ba75, Dr78]. For an SU(3) group, the effective coupling constant u(3) associated with the fundamental three-dimensional representation (3) is approximately ¡;(3) ~
2.10
Lattice approximation of gauge theories."li I
/ . Therefore, it is often used ;is a general expansion parameter for the strongcoupling series. As in (2.10.20), the expansion in lattice graphs is carried out with the Boltzmann factor exp( S w) written as a product. The fact that the strong-coupling approximation of the S (3) gauge theory on a lour-dimensional lattice is more complex than the above example is due to the greater number of S (3) representations with associated coupling constants v( ), and to the geometric diversity of simple lattice graphs. It follows from the orthogonality relations that { ^ ; k ( i ) } ^ 0 only if ilic boundary vanishes, i.e. is closed. Two plaquettes of ^ are called connected if they have a common link. A general graph can be decomposed into connected components. Connected graphs without boundary form closed surfaces which may have branch lines. The simplest example is a graph in the form of an elementary cube in a four dimensional lattice with the fundamental representation (3). It has 12 links and (j plaquettes. Its contribution Z{ C,(3)} to the partition sum can be evaluated according to the general method described above. The corresponding geometrical notations are given in Fig. 2.24d. The result is
{/C;(3)} =t/ dhn(3) = u 6 dim(3)
d/^) 2
=
d/i(M
u6.
(3)
(3Pi)
(3)
(
fl)
(2.10.2 )
If the contribution of all graphs of a certain order v is considered in a finite, four-dimensional lattice F ^ with periodic boundary conditions, the summation must be taken with respect to graphs in the various positions which follow from each other through AT1 translations. For example, Z is given in lowest order by the cubic graph described by (2.10.2 ). Multiplicity by translation and four-fold orientation of the three-dimensional cube in four-dimensional space results in Z = (l+ 36 v6
...)c.
ere c is an irrelevant constant, corresponding to C" drops out in the calculation of expectation values.
(2.10.30) in (2.10.21), which
Already the AT dependence of the lowest-order term shows that the convergence of the series for , i.e. (2.10.30), and the transition to the infinite lattice oo, the thermod namic limit, causes non-trivial problems.
.'i(¡'I 2
Q u a n t u m theory of Yang Mills fields
These can be solved by considering the strong-coupling expansion for the free energy
F =
~ w
X o g Z
(210
'31)
instead of the one for the partition sum Z. In this case, the thermodynamic limit N —> oo exists in any order, and the resulting power series in [i converges for sufficiently small ft. The transition from Z to F leads to a representation of the strong-coupling approximation for F by means of graph clusters. If the local variables depend only on gauge fields U(l>) on a finite number of links, a theorem guarantees the existence of the thermodynamic limit and the convergence of the strong-coupling approximation for the expectation values of local observables (íí[í/]). Roughly speaking, the N dependence in the numerator and the denominator in the path-integral representation of (ü[í/]) is compensated in the thermodynamic limit. 2.10.3
Numerical methods
The calculation of the euclidean expectation value of Í2[Í7], {Sl[U]) = ^ JV[U]
n\U]
Z = Jv[U]
(2.10.32)
on a finite lattice f permits the application of numerical methods developed in the context of statistical mechanics. By now, these methods have reached a degree of refinement that is far beyond the scope of this book. We give only a short introduction into the main concepts. For more details we have to refer to the literature [Cr83, Mo94, FrOO]. Some of our description we owe to the book by I. Montvay and G. Münstcr [Mo94].
2.10.3.1
Monte Carlo integration
In order to work out the similarities of (2.10.32) with statistical mechanics, we write Sii[{7] = fiE[U] > 0. Then, ft = 2N/g% corresponds to the inverse temperature 1/kT, E\U] is the energy, and (2.10.32) describes the expectation value of an observable in the canonical ensemble. The integration with respect to the measure 'D\U\ involves a large number of variables. For instance, on a 101 lattice with the gauge group SU(3) we have typically
2.1
attice approximation of gauge theories
. li I
H - l x 1 -dimensional integrations. In this case, the only general integration procedure is a onte Curio integration. onte Carlo integration in the simplest case consists in calculating the statistical average of randomly generated gauge-field configurations . Howe v e r , this is very inefficient, because the olt mann factor exp E is sharply pea ed at some specific configurations. Therefore an importance sampling with the weight U] oc exp E of the configurations f is needed. e consider an ensemble, i.e. an infinite number of field configurations i with density fV i , defined with respect to the measure U\. For om purpose U] should be proportional to the density of the canonical ensemble,
U] oc
2.1 . .2
C
U]
-
.
umerical simulation and
2.1 .
ar ov processes
The tas in numerical simulation is therefore to generate samples consisting of a large number of configurations , n = 1 ,..., , in such a way that its distribution approximates the desired distribution of the canonical ensemble uantity i2 , c. Then, the sample average of a 1
2.1 . 4 n 1
approximates the canonical ensemble average which according to is e ual to the euclidean expectation value 2t , lim - X)
rif 1
ii i l .
The difference between cf. ect. 2.1 . . .
2.1 . 2
2.1 . an
s
given by an error estimate
The configuration samples are generated in numerical simulations as a seuence i , n = 1 ,..., . This se uence is prepared by updating, i.e. repeated application of an algorithm which creates from a configuration the next one, t
-
,.
2.1 .
. 3 2 2
u a n t u m theory of
ang Mills l i p i d s
pdating might be considered as a stochastic sition probability <- i/ ),
P([U'
|V[U']
P([U']
process
\U'\ with tran-
- t/ ) = 1.
An updating step changes the ensemble densities W'[U'
t/
(2.10.37)
U] as
= Jv[U]P([U']
(2.10.3 )
or in short matrix notation ' = . The label n of the sequence of updatings, i.e. of the configurations, is sometimes called (computer) time. The following conditions on P and W are requested: very configuration can be reached with finite probability from any other, i.e. strong ergodicit holds, P([U'
- 17 )
The ensemble density jv[U]W[U)
0,
for all U and U',
(2.10.3 )
\U] is normalized, (2.10.40)
= l.
quations (2.10.37), (2.10.3 ), and (2.10.40) define a sequence of configurations is a ar ov chain.
ar ov process, the
An important condition for the updating process is that for any initial distribution Wo (having an overlap with \VC) the repeated application of updating steps P leads to the canonical ensemble, lim Ph W0 = Wc. ->oo
(2.10.41)
This implies that Wc is an eigenvector of P with eigenvalue 1, (2.10.42)
PWC = WC, i.e. a fixed point.
e require that this fixed point is unique.
A sufficient condition for (2.10.42) is detailed P([U') <— [U]) WC[U] = P( f/
balance,
i/ ) WC[U'\.
(2.10.43)
2.10
L a t t i c e a p p r o x i m a t ion of g a u g e t h e o r i e s
393
1'ogether with (2.10.37) this implies I D[U\ P(\U'}
[C/]) WC[U] = J V[U] P([U]
[U']) WC[U'} (2.10.44)
= WC[U%
i r. (2.10.42). According to theorems on positive matrices [Bh88], the eigenvaluc 1 of the matrix P is non-degenerate and all other eigenvalues are smaller than 1. Therefore, the repeated updating process damps all other components and converges to the canonical ensemble, and (2.10.41) holds. Examples of updating procedures satisfying detailed balance are the Metropolis algorithm and the heat-bath algorithm discussed below.
2.10.3.3
Error estimates
The sample average converges for large samples (N -> oo) to the expectation value (i2[<7]). A reliable result requests that N is large compared to the degrees of freedom of the system. In order that a finite sample average S2;v[i7] with not unreasonably large N is a good estimator of (ii[(/]) we must request smoothness properties 38 of ii[i/]. For reasonable ii[£/] one can estimate the error. In the ideal case, when the configurations Un contained in a sample are all statistically independent, the sample average is normally distributed around the mean value. We get in the standard manner the error estimate by the variance an, /emm
n
.
2
- (Î2AQ2
(2.10.45)
However, the assumption that the configurations [i/ n ] generated by the usual updating process are independent is unrealistic. Quite often one step in the updating consists only of a single local change of the configuration or of a sweep of local changes over the whole lattice. There are autocorrelations. For a (computer-)time-independent updating procedure and infinitely large samples, the autocorrelations depend only on the time difference r. In this case, the true variance of il^ is for large N given by (2.10.46) ,s
An example for a non-smooth
is one where iî(ï/) ^ 0 only for a single plaquette.
3 4
2
uuiilnin theory of
ang Mills fields
with the effective autocorrelation
time
-ttN)(iin.lT-2 S2/y)) - (fi*)
(2.10/17)
where S2 = ii t/j, . Comparing (2.10.45) with (2.10.46) one can see that owing to autocorrelations the effective number of independent measurements is {2rinl,n). ne can get a numerical estimate of Tinti by binning. This means one divides large ensembles in sub-ensembles which can be considered statistically independent and evaluates the results in the sub-ensembles.
2.10.3.4
Metropolis algorithm
e end this short review of numerical methods by describing the main features of two of the most popular updating procedures, i.e. the Metropolis algorithm and the heat-bath algorithm. The transition probability matrix for the etropolis algorithm for a system with A/" discrete possible configurations is defined for / f- (t/ by
-
min { l , exp(S [U)
- SE[U'})}
.
(2.10.4 )
This transition matrix is realized by the following iterative procedure: 1.
choose a trial configuration from the M configurations;
2.
accept it as the next configuration if the action decreases;
3.
if the action increases, then accept the change with a probability of the ratio of the Boltzmann factors.
The accept reject step can be implemented in a computer program by comparing the ratio of the Boltzmann factors to a pseudo-random number between 0 and 1. The detailed balance condition (2.10.43) can be verified immediately. In the form above, strong ergodicity (2.10.3 ) is also fulfilled. In practice it is not convenient to consider all possible changes of a configuration simultaneously. ften one considers only local changes of a single
2.10
Lattice approxiinat.ion of gauge theories
3!>r>
link variable. In order to have ergodieity one has to sweep through all links «it the lattice. The transition probability P can be generated as a two-step process P = P,\Pc consisting of a proposed change and an accept reject step, where •
Pc{[U'} <— [(/]) is an arbitrary probability distribution for the proposed configuration change [U] [i/'j, and
•
the acceptance probability P,\([U'\ [(/]) is defined a.s a generalization of (2.10.48) in such a way that the detailed balance condition (2.10.43) holds for P = PAPC, i.e.
2.10.3.5
Heat-bath algorithm
The aim of an updating procedure is to create a canonical (equilibrium) distribution. In the heat-bath algorithm this goal is approached by local steps. In each step one link variable U([x, /z]) is distributed canonically, while all other variables are kept fixed. In thermodynamical language, U([x, /¿]) is in contact with a heat bath provided by all the other variables. This elementary step has to sweep consecutively over all link variables of the lattice, and the whole process has to be repeated many times. We want to put the heat-bath procedure into the general form of an updating process (Sect. 2.10.3.2). Let Ux = U([x,//.]) be the link variable to be updated and Ux all the other variables forming the heat bath. By WC\UX\ Ux] we denote the conditional probability distribution of Ux in the canonical ensemble for fixed Ux. The whole canonical distribution can be written as WC[U) =
WC[UX-UX]WC[UX).
.
(2.10.50)
Similarly, the conditional transition probability for the updated variables is denoted by PX{U!X <— UX;UX). The transition probability matrix for the whole system then is PAW']
<- [U]) = PX(U'X <- UX-UX)6(U'X
-
Ux).
(2.10.51)
.'i(¡'I 2
Q u a n t u m theory of Yang Mills fields
It is assumed that PX(\U'] [£/]) satisfies the conditions (2.10.37) and (2.10.42). A sufficient, condition for (2.10.42) is local detailed balance, Px(i4
The local updating step described by Pj:([i/'] [i/]) is of course not ergodic because it acts only on Ux. One does require local ergodicity, Pr\U'x
Ux; Ûx) > 0
for all Ux and U'x.
(2.10.53)
In order to achieve ergodicity for the whole configuration one can perform a cycle of local updates, called sweep, over all links consecutively, P(lU'}^lU})
=
(2.10.54)
l[Px([U'}<-lU}).
The local heat-bath algorithm corresponds to the conditional probability matrix PX{U'X f - Ux, Ux) = WC(UX;UX). The main task in a heat-bath updating is to generate the distribution WC(UX] Ux). For the SU(2) gauge group the simple properties of SU(2) matrices help. Here Ux € SU(2) may be represented by the Pauli matrices (A.1.11), Ux = a x o + ¡Ta x , Uxl = « x o ~ ¡7"ax with a 2 0 + a 2 = 1 and = (o.xi). A sum of SU(2) elements Ux is up to a normalization factor k proportional to an SU(2) element Sx. This allows us to write the Wilson action (2.10.10) as 5\v[U] =
(2.10.55)
ReTr[U x S t ] + 5w[i>],
where Sx represents the sum of the products of link variables other than Ux which are contained in the boundaries of all the plaquettes that have the link \x,n] in common. Then, the conditional probability distribution of the link variable Ux can be written as \Yr(Ux; Ùx) dU x <x exp ~ R è T r { U X S X ) dUx it
= exp ft E(lxiSx . i=0
d Ux,
(2.10.56)
2.10
Lattice approximation of gauge theories."li I
where .sx, are the components of ST = sxo - i r s x . The lieat-bath algorithms based on (2.10.56) may be further simplified by using the invariance of the aar measure. ere we present only the principle of this algorithm, for details we have to refer to the literature Cr7 , Cr 0 . My using sufficient S (2) subgroups of S (y ), one can generalize the S (2) heat-bath formalism to a heat-bath algorithm for S (i ) a 2. The canonical distribution hides often conjectured long-range structures like instantons (Sect. 2. .1.2) under short-range fluctuations. It is a widespread practice to suppress the short-range fluctuations by cooling. This means one puts between the different updating sweeps cooling sweeps. These are heat-bath sweeps with very high / corresponding to low temperature Be 1, Ilo 7 .
2.10.4
Transition to the continuum
The lattice approximation of a field theory corresponds to a regularization by a momentum cut-off, as described in the discussions on divergent Feynman graphs in Sect. 2.5.1.1. Before discussing the continuum limit within the renormalization program, we illustrate this aspect of lattice approximation using the amiltonian of a free, scalar field on a infinite lattice as an example.
2.10.4.1 The
L a t t i c e a p p r o x i m a t i o n of t h e
amilton operator of a free scalar field
=-
d3x
vr2(x, /.)
a m i l t o n i a n of a f r e e field (x)
is
( / l ( x , t))2 -I- 7n 2 yl 2 (x, 0
(2.10.57)
The commutation relations for the canonically conjugate quantities and 7r(x, / ) = d (x.t) dt. read
7r(x,/,),/l(x ,i)
= -i (x-x ),
/l(x,I),^(x ,0
=
TT( , / ), 7 (x , /.)
=0.
(x,t.)
(2.10.5 )
.'i(¡'I 2
Q u a n t u m theory of Yang Mills fields
We approximate these equations on a three-dimensional lattice, putting x = en, n = («1,712,713) with lattice constant e. Using 7r(n) = e 2 7r(x, i)|x=£n,
4 ( n ) = eA(x, /.)|x=£n,
( V } ^ ) ( n ) = .4(71, + l , n 2 , n 3 ) - A(nun2,7i3),
etc.,
(2.10.59)
the equations (2.10.57) and (2.10.58) can he translated to [*2(n)
+ (V^(»))2
+ p2A2(n)]
,
n = em,
n [7r(n),>l(n')] = - i < W .
(2.10.60)
As in the continuum, the free Hamilton operator on the lattice is diagonally,ed using the lattice Fourier transformation
A{n)=(2TT)-3/2 r
[
J-It J-IT J -IT TT(n)=(2TT)-V 2 R R R ^ ^ D ^ , J —5T J — 5T J —TT ÂH0) = A ( - 0 ) ,
= ir(-jS),
and introducing creation and annihilation
n»(/3) = f ^ f
(2.10.61)
operators
(M-B) - ^ ¿ ( / 3 i j .
(2.10.62)
A direct calculation, in which the essential steps are based 011 the orthogonality and completeness relations f * dfljl"-"1» = Snm,
^
. (00 _L y /TT
'
J
H——OO
n , m = 0, ±1 , ± 2 , . . . ,
= S((3 _ jg/)i
-n
+7T,
(2.10.63)
2. JO Lattice approximation of gauge theories
.19!!
that the canonical commutation relations (2.10.58) are equivalent to the commutation relations for the creation and annihilation operators,
MIIOWS
(«(/3),at(/3')l = 6(0 - p%
[a((3),a(P')\
= [«•(/?), a t'(/9')] = 0. (2.10.64)
Moreover, using yfc ci"/3
=
e i(n»+i)A
_ e in k fi k
2ie
=
in
^ sin ( ^ j
(2.10.65)
the form //
- 7J ifti<* £
d3 p u ( p ) ( j ( p ) a ( p ) + constant
(2.10.66)
with 1/2
(2.10.67)
w(0) = 1
V
'
follows for the lattice Hamilton operator. Writing exp(in/3) = exp(ipx) in physical quantities x = en and p = /3/e, the periodicity in P, i.e. |/?j| < n, corresponds to a maximal physical momentum on the lattice |p;| < n / e . Using the physical quantities which approximate the continuum quantities on the lattice, a(p)=e3'2a(p),
u(p) =
(2.10.68)
W (j9)/e f
the Hamiltonian operator becomes 1/2
•I
H
=
[
d3p
m
8 1
+¿2L »
(^-j
af(p)o(p). (2.10.69)
This shows that also the energy is bounded. The lattice acts as cut-oil" to all quantities derived from creation and annihilation operators and the Hamiltonian operator, e.g. to the Feynman propagator as described in Sect. 2.1.2.1. There is also a latticc perturbation theory [Rc88] which is, however, more tedious than the perturbation theory in the continuum. Finally, (2.10.69) shows how the Hamilton operator of the continuum is approximated formally for decreasing lattice constant e —> 0 and lixed in2.
.'i(¡'I 2
2.10.4.2
Q u a n t u m theory of Yang Mills fields
Critical p o i n t s in s t a t i s t i c a l m e c h a n i c s
For the general discussion of the continuum limit, the analogy between the lattice approximation of euclidean quantum field theory and statistical mechanics has turned out to be very fruitful. In Sect. 2.10.1.4 we have indicated that the correlation length which governs the asymptotic large-distance behaviour of the lattice approximation of the two-point function, is related to the lowest particle mass by £ = 1 /ma. Therefore, in order to extract a finite (renormalized) mass from lattice approximation, the correlation length must become infinite. In statistical mechanics, a point in the space of parameters where the correlation length diverges is called a critical point corresponding to a second-order phase transition. Thus, a continuum quantum field theory is obtained from lattice approximation at the critical point of the lattice coupling constants and lattice masses. Near critical points the lattice spacing can be neglected, and one expects a restoration of euclidean symmetry. In statistical mechanics, various methods exist to find critical points and to discuss the corresponding physics. A general approach is the KadanoffWilson renormalization-group theory [Ka65, Ka77, YVi71, Wi74a]. This theory embeds the cut-off dependence of the action in an infinite parameter space of actions. It explains why in the neighbourhood of critical points different statistical systems behave similarly. They fall into a relatively small number of universality classes. The renormalization-group approach which we discussed in field theory (Sect. 2.6) may be considered as a reduction of the infinite space of actions to a small number of relevant parameters, like the renormalized coupling constants and masses. We restrict our discussion of the continuum limit of lattice gauge theory to t his more special frame.
2.10.4.3
R e n o r m a l i z a t i o n g r o u p on t h e l a t t i c e
In contrast to Sect. 2.G. 1 we start with the renormalization-group equation for the bare vertex functions. We explain the main features by considering a scalar theory with a renormalized mass m a and a renormalized coupling constant gn, defined by the renormalized two-point vertex function, m 2 { = — r^(p 2 = 0), and the renormalized four-point vertex function, 9u = — Pf\(0,0,0,0), respectively. The renormalized massTORjust fixes the scale of all dimensionful quantities. In particular, the cut-off is characterized by 1/mna, a being the lattice constant. Fixed gn = mil«) describes
2.10
I jfiUice approximat ion of g a u g e theories
101
( iirvi' in t he plane of the bare lattice coupling constant <70 and the cutoil «, the curve of constant physics. One can show in k-loop order lattice perturbation theory that the renonnalized vertex function obeys '\\(Vi;yR,rnR,mRa)
= r\{(Pi; gKt m R , 0 ) + (9 ( « 2 ( l o g « ) * )
(2.10.70)
or a ^ r & ( p i ; < m , m R , m R a ) = O (a 2 (loga)*) .
(2.10.71)
Neglecting the scaling violation term O (« 2 (loga)*), using the relation between the renonnalized and unrenormalized vertex function, Hi =
0, mRa)ro {go, Tn r , mRo),
(2.10.72)
we derive the renornialization-group equation for the bare coupling constant similar to Sect. 2.6.1.1 with the help of the chain rule of differentiation. The independence of f ^ on a implies
1
0
(•¿-«•¿+»»-) *- ' where the lattice )3-function by
21073
< >
and the anomalous dimension 7 l are given
A,(.9o, "Mi") = -
Sll 7. . . 1 L d\ogZ 7i.(iyo,"tR«) = + rJ —^ 2 da 9R
(2.10.74)
The derivatives have to be taken for fixed g\i. Equation (2.10.73) describes the change of the bare vertex function under the change of go while gu is kept, fixed. The equation is similar to the renormalization-group equation of Sect. 2.6.1. The discussion of the fixed points follows that of Sect. 2.6.1.3. Neglecting the scaling-violation terms, the functions and 71, depend only 011 go- The funct ion AX'/o) determines the change of go with the cut-off, i.e. the lattice constant a is related to a change of go in order to stay 011 the curve of constant physics.
.'i(¡'I 2
Quantum
theory of Yang Mills fields
Wc; discuss the; transition to the continuum with help of the renormalization group for the bare coupling constant. More refined improvement methods are mentioned in Sect. 2.10.6.6. The derivation of the continuum limit of a pure SU(7V) gauge theory follows similar lines. The renormalized coupling constant in the momentum subtraction scheme s defined by the threegluon vertex. Lattice perturbation theory in one-loop approximation gives the result ( ^ f log(a2M2)
fMOM = So (l "
+ Cj ) (1 +
0(a2M2)) (2.10.75)
with C = 67r2/(117V2) - 9.44598. The renormalization point M plays the rôle of WR in the scalar theory. Equation (2.10.75) gives the first term of the expansion of ft^ for a ss 0,
A> = J^V,
Pi = jH'2-
(2.10.76)
These coefficients agree with the expansion of the /^-function in the continuum (2.6.34)-(2.6.37) since N = CA for S U ( N ) . fn Sect. 2.6.3 we have given arguments for the regulaiization-procedure independence of the first two coefficients of the /3-function in asymptotically free theories. Integrating the differential equation (2.10.74), results in a = cxp
(if/
(-R ihXiï'). To id^J
or
Ï - m
+
1
LOSLG
"
+
(2 IA78)
° tss^ù • -
This equation, which is equivalent to (3.1.16), describes how the lattice constant a has to be changed as a function of the lattice coupling constant
2.1
attice approximation of gauge theories. li I
i o in order to stay 11 the curve of constant physics. The integration constant i with the dimension of a mass,
can l e related to similar constants of different regulari ation schemes 11 .1.14 . Comparing with the continuum definition of ,
(V "fl )
-
\
1 (2.10.80)
we get, using 2.1 .
1,
,
lim a FF
. ,
-
F
expi I
T
-
-
e
2
.
2.1 . 1
Wo
Thus, t he numerical constant C in 2.1 . determines the relation between the parameters in one-loop approximation. For C , , the relation to g is more interesting. In the case of ero flavours it is found as IIa , c 1 ag
2 .
.
2.1 . 2
The agrangian of a pure gauge theory contains 11 parameter with dimension of a mass. Therefore, every physical uantity with the dimension of a mass must be proportional to dimensional transmutation) m = Cm . In 2.1 .1 we described how the mass of particles can be calculated in lattice units ma. If go is sufficiently close to the continuum limit, i.e. go is in the as mptotic scaling region, the leading term of 2.1 . is a good approximation, and cf. 2.1 .
(
)
Hence, the ratio of two different masses m i and m2 is independent of go in the asymptotic scaling region, and thus according to 2.1 . independent of the lattice constant.
'KM
2
Q u a n t u m theory of Yung Mills fields
In order to calculate the continuum values of physical quantities from lattice quantities, it is necessary to reach the asymptotic scaling region for 0. On the other hand, the strong-coupling approximation is an expansion iu /S = 2 N / ( J Q . SO a high-order strong-coupling expansion allows for a reliable calculation of continuum quantities only for exceptional cases. For numerical methods, the transition to the continuum causes some difficulties because of autocorrelations. The correlation length £ = 1 ¡ma diverges near critical points. The autocorrelation time (2.10.47) depends on the only relevant length scale TjUtin cx Here z(i2) is the dynamical critical exponent, which is characteristic for the updating process and may also depend on the observed quantity fi. In local updatings one has usually 2 ~ 2. Obviously, for z(£L) ^ 0 we have a deterioration of the efficiency of the updating process near critical points. This is called critical slowing down. Let us repeat the essential points of this subsection. Lattice approximation of field theory is a method to calculate physical continuum quantities. In the asymptotic scaling region, the ratio of mass values is independent of the lattice constant [up to terms of O ((aA|,) - 2 )]. More complicated Green functions with a possibly non-perturbative long-range behaviour may be calculated according to (2.10.72). In principle, the continuum limit always follows the line described here. In view of the various difficulties, large computers and refined methods, like improvement (cf. Sect. 2.10.6.6), arc necessary for getting reliable results. We mention some results for QCD in Sect. 3.6.
2.10.5
Finite-size effects
Numerical simulations of quantum field theories on a lattice refer to the approximation of the fields in a finite volume. In order to get small deviations from the continuum limit, one has to choose a small lattice constant a, i.e. according to (2.10.78) a large l/ry^ and thus a large ¡3. For a lattice with an extension of Ns points in space direction and Aft points in time direction, this gives a small physical three-space volume Vs = L { and a small time interval 7', Vs = a3iVs3,
T = aNt.
(2.10.84)
However, generally one is interested in the large-volume limit —> 00, and in large times T [cf. (2.10.18)]. Since the computer power limits Ns
2.10
Lfttticr approximation of gauge theories
405
it ltd , one lias to compromise between a reasonable approximation of the continuum and a suificieiitly large space time volume. Ill«- optimal choice for s and depends on the physical problem. The i haracteristic quantity is the correlation length e have seen how = ma drier mines the masses in lattice approximation. Similarly, it determines the iputial extension of a state. ood approximation of the continuum on the mi hand and of the large-volume limit on the other hand requires a £ . e mention as an example the determination of a mass m 1 e , i.e. mp 3 in HI//, in an .f x = 4 x 64 lattice in Sect. 3.6.1.1. For much smaller iniisses, like the pion mass m , finite-size effects might become large. M. biischer studied the behaviour of finite-size effects for asymptotically large Lu 4 . For massive fields he finds an exponential fall off for the mass-correction term A m = m( ) m, A m oc L - z
2
e-mL,
(2.10. 5)
where the details depend on the particular dynamics. Spontaneous symmetry breaking and tunnelling near first-order phase transitions is another source of finite-size effects. The resulting energy splitting is due to tunnelling between vacuum states in finite volumes (Sect. 2. .3.5). The analysis is based on a picture of domains in space and time as discussed in Sects. 2. .3.4 and 2. .3.5. From this picture a prediction for the energy splitting of the form oc e~a
3
(2.10.86)
is obtained, where a is the interface tension associated with the domain walls Fi6 , Pr 3, Mo 4 . ne can get a rough estimate on the errors caused by finite volumes by comparing the values of different numerical lattice approximations with the same lattice constant and different physical volumes. In this context we only mention that there is a method which uses controlled finite-size effects, finitesize scaling. It uses a finite-volume renormalization scheme for matching a sequence of lattices of different sizes. By this procedure one can study the evolution of physical quantities over large length scales (Lu j.
.'i(¡'I 2
Q u a n t u m theory of Yang Mills fields
2.10.G
Lattice a p p r o x i m a t i o n of fermionic interactions
In order to treat QCD in lattice approximation one has to include quarks on the lattice. Surprisingly, the lattice approximation of fermions leads to conceptional and practical difficulties which get solved only slowly. We mention some of the problems in the following. This should explain why important problems of non-perturbative QCD, like the calculation of the hadron spectrum (Sect. 3.6.1), are still not completely solved.
2.10.G.1
Naïve lattice approximation of the Dirac equation
Let us first follow formally the scheme of lattice approximation for gauge fields explained above. We consider coloured Dirac fields on a four-dimensional lattice as spinors at a lattice point ri = (n^,ri2,713,714) (cf. Sect. 2.10.1.1), Ì>aA n ) =
= ^ ^ VW(a')ls=<m5 (2.10.87)
V>a,c(-T)U=an,
where a = 1 , . . . , 4 is the spinor index, c = r, g, b the SU(3)-colour index, and we suppress the flavour index. The lattice spinors are made dimensionless (cf. Table 2.1 on page 195). According to the geometrical interpretation of gauge transformations (Fig. 2.4 011 page 137), the forward and backward covariant derivatives of Dirac fields 011 the lattice (L) correspond to
(£>L>)a,c =
-
Vcd{[n -
A;//.])V«c'(n -
fi)
(2.10.88)
with aft = e,t (cf. Sect. 2.10.1.1). The coupling of the quarks to the gluon fields is described by the naïve lattice Dirac equation
n
(Ql»a,c =
n
7ai}{DL,,il!>)f}A ) + ™fl>a,c{ ) = °>
0 ^ = 1(0^ + 0^),
(2.10.89)
where m — a m is the mass in lattice units. The corresponding part of the action reads sq[u,ip,ip)
•i 1 / _ = Y ^ - Î 0 ( n ) 7 , i i / - 1 ( [ n ; / i ] ) 1 / > ( n + ,ì) ner L//=i v
2.10
-
I,ai tici' approximation of gauge theories
0 ( 7 1 ) 7 " C ( [ T » - / M V ' (n
fi^j +
-
407
mip(n)i>{n)
±-1
= E
-
^
[ ^ ( N + / I ) Y ' I 7 ( [ N ; / I ] ) V ' ( N ) ] + MI>{N)IJF{N)
ner
(2.10.90)
= (>Ì>QLÌ>)-
I''or the transition from the first to the second line we used U~1 ([n; /v.]) I'([n-\fi\ -//.]) [cf. (2.10.2)] and introduced t h c 7 matrix in opposite direction •y *» = - 7 ' ' . We suppress the sum over the spin and SU(3) indices. Now we Can extend the path integral to include the quark fields,
= J J
(2.10.91)
hike in the continuum, the. gaussian integral over the quark fields has to be understood as integration over Grassmann variables (Sect. 2.2.1.4). Thus, Jcrmion integration as in (2.2.65) leads to the final result for gauge-invariant Green functions of quark fields i>(7it) and ip{n.j) connected by a parallel displacement U(Cij),
(UMniWiCiMnj)}^
=
|
(
2
.
1
0
.
9
2
)
In order to evaluate the quark Green functions 011 the lattice, one has to calculate the propagator Q{ni,ny,U) in the background field U(n), G(nunj\U)
= (ni, a, c\QL
1
\n'p or', c'),
(2.10.93)
and the determinant erint/lsdeK^-.^clQLln^^y)).
(2.10.94)
of the massive Dirac operator (¿¡_. Finally, one has to perform the gauge-field integration.
.'i(¡'I 2
Q u a n t u m theory of Yang Mills fields
2.10.6.2
The spectrum-doubling problem
What are the specific problems of this scheme? There is the problem of spectrum doubling. In order to investigate this, we consider the free Dirac equation in momentum space,
¡ ^ y ' s i n t p ^ + m ] i>{p) = 0, / M=i
(2.10.95)
where ip(p) is the Fourier transform of i/>(n) on the lattice (2.10.61). We used dK^pna =
= isin(p i i a)e i p "°,
+
(2.10.96)
with d£ l t and d f ^ given by (2.10.88) with [7 = 1. The propagator of S q becomes
(
t — i
\
'
m — V , , 7''-; sin p w a
^ y i r i n p ^ + m = 1 • \2 (2.10.97) T' « ) ™ 2 + £ / t ( j snip,,«) 2 with p,, varying in the first. Brillouin zone \p,t\ < ix/a. For lattice spacing a -> 0, Q{p) approaches the continuum expression G c o n l (p) « ($ -f m ) - 1 not only for « () but for 16 different regions in the Brillouin zone where either aplt « 0 or apfl ~ 7r for p = 1,2,3, A independently. The lattice propagator Q(jp) has poles at pl — m2 at these 16 different regions. Thus, the naive action describes 16 Dirac particles instead of one. In the free case, this might he not so serious, however in an interacting theory, the additional fermions influence the physical content of the theory. 2.10.6.3
The Wilson fermions
There are two proposals to overcome this problem. K. Wilson suggested to add to the fermion action an additional term [Wi77],
Sqw = S„ + r £ (- X ner \ n=±i =
+
+4^(n)^(n) (2.10.98)
2.10
In the free case,
Lattice approximation of g a u g e theories."li I
( n,ii ) = 1, the propagator of (2.10.
) becomes
2
(2.10. ) For small p, this propagator has the form of the free continuum propagatoi, and the doubler poles at p -/- 0 disappear. owever, the ilson expression L has some shortcomings. For m l.vv is no longer invariant under cliiral transformations, 7
A
7
0,
0,
(2.10.100)
asymmetry which for examj le plays an important role in the understand)nr, of the pion as a oldstonc particle (Sect. 3.5.1.1). 2.10.6.4
T h e staggered fermions
ne may try to find another geometrical interpretation of the irac equation which is more appropriate for the lattice approximation. In Sect. 2.10.1 we introduced the geometrical elements o f a lattice: points, links, plaque ttes, cubes, and super-cubes, for n = (),... ,4. The lattice approximations ol the gauge potential and the field strength are defined as an average of the continuum quantities over such elements, i.e. (2.10.1) and (2.10.7), respectively. The lattice approximation of fermions described by the irac abler equation follows this line of approach Be 2 . e use the formulation of differential calculus o 7 . Antisymmetric covari ant tensors are described by linear 7),-forms, I = I /M.../liidx/ 1 A . . . A da; ", where the wedge product obeys da; 1 Ada; 2 = da; 2 Ada; 1 . Therefore, s .../i completely antisymmetric. The gauge potential .4/tda: and the field strength (l/2)jf"/11/da; Ada;" are examples. In differential calculus the operator d of exterior differentiation and its adjoint 6 are the natural generalization of the curl and the divergence to higher dimensions,
(2.10.101)
.'i(¡'I 2
Q u a n t u m theory of Yang Mills fields
These definitions and antisymmetry imply d 2 = 6 2 = 0,
d<5 + <5d = -d^d' 1 = - • .
(2.10.102)
The Dirac Kahler fields are described by inhomogeneous differential forms,
(2.10.103)
For fermions we get the free Dime- Kiihler equation [Ka62, Be82], (d - 6 + m) = 0,
(2.10.104)
with (d — <5 + 7/T)(d — (5 — 7/I) = ( • — 7/I2)«I>. This means, the operator (d — 5) shares the property (d — <5)2 = • with the operator if). But this was the original motivation by Dirac to introduce the Dirac matrices. With the formal complex conjugate <1>, the Dirac-Kahler equation is related to the action
5
¿ ^ " ' " ' " » [ ( d - <5 + m ) * ] , ,
,„„(*) J . (2.10.105)
This action can be made gauge-covariant by replacing the partial derivative On by the covariant derivative D,t in the definition of <1 and ó, (2.10.101). The advantage of differential forms is that they provide a straightforward lattice approximation which maps onto lattice elements
X(V) = [ <*>,„ Jpo '%)
„„ dV< B \
/ / . ! < . . . < //„•
(2.10.106)
The n-dimensional volume element is denoted by d V ^ . The central points of the lattice elements y = {x + l / 2 ( e „ + . . . -I- e , J } , (points, links, . . . i n the notation of Sect. 2.10.1) form a lattice with the lattice constant a/2. Thus, on this finer lattice the staggered fermion fields x(v) represent the average of the Dirac Kahler fields over In (2.10.106) and in the
2.1
a t t i c e a p p r o x i m a t i o n of g a u g e t h e o r i e s. li I
following we set a = 1. The definition 2.1 .1 imation of the gauge-invariant action 2.1 .1 fields, Sqs\u, X, x) = m
leads to a lattice approxof the staggered fennion
(2.10.1
x( )x( ) y
E
f
x
lvw
xfo
«
- x(
+ «v
fow
x
,
where the sign factor pXill c o m e s from the ordering in 2.1 .1 . If one does not use the language of algebraic topology, the derivation of this formula is straightforward but clumsy e 2. The propagator of the lattice approximation of the irac ahler action has no doublcrs. However, the free irac ahler e uation represents a four-fold degeneracy of the irac field in the continuum and on the lattice. Furthermore, a reduction with respect to the flavour degeneracy of the naive lattice approximation of the irac e uation ect. 2.1 . .2 leads also to 2.1 .1 , u , I1 I . The action 2.1 .1 has interesting symmetries which can be understood as restriction of the symmetries of the irac- ahler e uation o . The natural four-fold flavour is sometimes used to study the flavour dependence of lattice approximation. physical interpretation of this flavour degeneracy is not yet nown.
2.1 . .
Chiral symmetry and the
ielson- inomiya theorem
There is no simple and completely satisfactory lattice approximation to the irac e uation. This is a conse uence of the ielson inomiya theorem i 1 , which states that the following properties cannot hold simultaneously for the Fourier transform of the free massless irac operator on the lattice D\X ) a
D\,{p) s a n analytic periodic function of the momentum p t with the period 2ir/a. This is necessary for . to be an essentially local operator.
b
For momenta p,t l
we have
It is customary to denote the
p
i up to terms of order up2.
irac operator Q\, by D\. for massless fcrmions.
.'412 2
(c)
Q u a n t u m theory of Yang Mills lipids
D\XP) >s invertible at all non-zero momenta in the first Brillouin zone. This ensures, together with (b), that the correct continuum limit is obtained.
(d) Di, anticommutes with 7 5 : 75Z?I, + JDI.75 = 0. This guarantees that the fermion action is invariant under continuous chiral transformations. The Nielsoii Ninomiya theorem was considered as prohibiting a lattice approximation of chiral fermions. However, an old suggestion by Ginspargand Wilson [Gi82] to substitute (d) on the lattice by 75 A, + D\rth = aD\rfD\.
(2.10.108)
regained interest recently [Ha98a, Lu98, Ne98]. Examples of such a free Dirac operator can be constructed. The construction of gauge-invariant might be complicated [Lu99a].
2.10.6.6
R e m a r k on the i m p r o v e m e n t program
In order to approach the continuum limit more rapidly, one can add terms 5 S to the action that depend on the lattice constant a, Simp =
Sw [U] + Sq [U, Va i>] + SS.
(2.10.109)
Based on ideas for perturbation theory by K. Symanzik [Sy83], M. Liischer started such an improvement program for the lattice approximation of QCD [Lu85]. To improve the action for the Wilson fermions S q w> (2.10.98), one should add the Sheikholeslarni-Wohlert term [Sh85] SS = a 5 E
c
sw
(2.10.110)
X
In (2.10.110), J'),,, is a lattice approximation of the gluon field-strength tensor involving four plaquettes. The additional «.-dependent term (2.10.110) compensates the linear term in a in the corresponding improved Green functions of field operators. The coupling csw {go) depends on the bare coupling constant go- A one-loop calculation gives csw(.9o) = 1 + 0.265#o + 0(g¡j) [Wo87]. There are schemes to determine it uon-perturbatively.
2.1
2.10.6.7
Lattice approximation of gauge theories
413
uenched and unquenched approximation
Alter choosing an appropriate lattice approximation of the irac operator i one follows the scheme described in (2.10. 2), (2.10. 3), and (2.10. 4) for the calculation of quark reen functions. In quenched approximation, the fermion determinant is set equal to one, det( i,) = 1. This means that the effects of fermion loops are neglected. For Home problems this might be a good approximation (cf. Sect. 3.6), however, the corrections are essentially unknown. hat remains is the inversion of the the equation QiM> = .
irac operator
}
i.e. the solution of
(2.10.111)
n a finite lattice this is a finite-dimensional linear equation. It is certainly beyond the scope of this book to describe the methods of numerical mathematics like acobi iteration or conjugate-gradient method which are used and refined for the purpose of lattice approximation Mo 4 . The numerical simulation of the lattice approximation of the complete gauge theor ith d namical fermions requests a procedure for the evaluation of the fermion determinant in equation (2.10. 4). Many different procedures are developed for this difficult task. e sketch the main ideas of the h brid onte Carlo procedure which is frequently used. The fermion determinant of two degenerate fermions can be described with help of a complex bosonic pseudo-fermion field I ( .'), 1^ ( ) , (2.10.112) ere we used the different expressions for the gaussian integrals with bosonic variables (2.2.11) and fermionic variables (2.2.27). The lattice approximation of the bosonic gaussian integral over the pseudo-ferrnion fields can be evaluated by the Metropolis algorithm (Sect. 2.10.3.4) or the heat-bath method (Sect. 2.10.3.5). The transition to the effective action for the pseudo-fermion fields requires again the computer-time consuming inversion of , like in (2.10.111). This favours the integration over the pseudo-fermions by the heat-bath method. The effective action of the pseudo-fermions is non-local.
.'i(¡'I 2
Q u a n t u m theory of Yang Mills fields
This makes a straightforward Metropolis algorithm for the remaining integration over the gauge fields U very slow. One has to evaluate the fermion determinant many times. Therefore one introduces a further step. The update of the gauge fields U in the Metropolis step is produced by molecular dynamics. Roughly speaking this method has the advantage to propose comparatively uncorrelated configurations without a breakdown of the acceptance rate in the Metropolis decision. This very short, description of the hybrid Monte Carlo algorithm shows that a numerical treatment of the lattice approximation of gauge theories with fermions is possible. However, this method requires much more computer time. Thus, only relatively small lattices can be treated up to now. A nontrivial numerical simulation of QCD with three light quarks at the physical value of quark masses needs to perform at least 1020 floating-point operations. With the availability of teraflop computers in the coming years this seems to be a feasible task [SaOO].
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uantum Chromodynamics The theory of the strong interaction of elementary particles, uantum Chromodynamics C , is a non-abelian gauge theory with as gauge group. The degrees of freedom corresponding to this are called colour. The uar s have besides the flavour also colour degrees of freedom and transform according to the fundamental representation of colour . The eight coloured gauge bosons arc called gluons. The colour degree of freedom was introduced in Refs. Gr 4, Ha , Ce 2 , Coloured gluons were proposed by II. Frit sch, . Cell- ann, and H. eutwyler Fr . The colour degree of freedom solved the following three pu les of the uar model the problems with the lifetime of the neutral ir meson, the total hadronie cross section in e e annihilation, and the wrong uar statistics in the baryon bound states. C is renorinali able, as was shown in ect. 2. .1.4, and therefore treatable by perturbative methods. erturbation theory is appropriate for C at high energy or high momentum transfers as was shown already in 1 by .. . Cross and F. ilc e Gr a , H. . olit er o , and G. t Ilooft unpublished . They discovered1 the asymptotic freedom of C ect. .1 . This opened the way to the successful applications of perturbative C to deep-inelastic scattering ect. .2 , hard hadronie scattering processes ect. . , et physics ect. . . , and many other problems in particle physics. n the other hand, C as the theory of the strong interaction becomes strongly interacting at low energies. In this regime, the non-perturbative effects and the topological properties of a non-abelian gauge theory ect. 2. are essential. on-perturbative techni ues for the; evaluation of C include the strong-coupling approximation ect. 2.1 .2 , the semi-classical approximation ect. 2. . .1 , and, above all, lattice methods ect. 2.1 . They enable the calculation of the masses and of the inner structure of the hadrons ect. . and lead to an understanding of confinement of uar s and gluons ect. . . For tlic history of the discovery of asymptotic freedom see t 11
.
3.1
A s y m p t o t i c
f r e e d o m
o f
Q C
'127
n important issue is the connection between low-energy physics and the region of asymptotic freedom. This gap is bridged by lattice gauge theory, ect. . . , factori ation theorems ect. . .2 , and s illful application of the renormali ation group. Indeed, the present status of C is characteri ed by the consistency and success of all these different methods and their results ect. . . odels based on C li e the heavy- uar effective theory for uar onia ect. .4 , chiral perturbation theory for low-energy physics ect. . , or sum rules are the basis for further applications in phenomenology.
3.1
A s y m p t o t i c f r e e d o m of Q C
In ect. 2. .2 we have seen that C as a non-abelian gauge theory is asymptotically free if the number of uar s is not too big. This allows the application of perturbation theory for hard-scattering processes. This section deals with physical conse uences of the renormali ation-group-improved scaling behaviour and introduces the concepts of the running coupling and the running uar masses of C . .1.1
T h e r u n n i n g c o u p l i n g c o n s t a n t of
C
The renorinali ation-group e uation for the vertex functions in C can be solved with the help of 2. .22 and 2. .24 . First of all, 2. .22 and 2. . yield the one-loop approximation of the scale dependence of the running coupling constant of massless C tf V s with cf. 2. .
l
yf
log
.1.1
2
and 2. .4 y C , , - lTF
V -
11 -
.1.2
where denotes the number of colours corresponding to the gauge group f and ( the number of effective massless uar flavours in the fundamental representation . The running coupling constant gs vanishes for oo. This is the asymptotic, freedom of C .
4 2
s
The value of the strong fine-structure constant a s = 72/47r is linked to the rcnorrnalization mass parameter M by the renormalization condition (2.5.47). sing the fact that A2 = Q2/M2 is the scale factor with which the four-momentum transfer at the vertex is multiplied, (3.1.1) yields
with the initial condition a s =
(.M2). This allows
to calculate in oneloop approximation the running coupling âs(Q ) from the experimentally determined value of t s at M 2 . efining 4TT (3.1.4) A2 = M2 cxp 2 2
[
0Qâa(M )\
the momentum dependence of the running strong fine-structure constant a s ( 2 ) can be brought into the simple form
Po log f r The asymptotic scale parameter A of C is a renormalization-group invariant quantity, as described in Sect. 2.(5.1.4. The introduction of a dimensionful parameter into a dimensionless field theory like massless C by the renorrnalization procedure is known as dimensional transmutation Co73 . The dimensional parameter A is closely related to the dimensionless coupling constant and determines like a s ( M 2 ) the coupling strength of the strong interaction. The determination of the running coupling constant from the first-order formula (3.1.3) from two different asymptotic scale parameters A and A gives two coupling constants that differ by a second-order term, =
A) log 7A
2
A)(log Q2/A2 + log A 2 /A 2 )
l 2 ^ log ( ^ I âj(Q2) + ....
(3.1.6)
Thus, different values of A lead to the same running coupling constant in first order. A second-order calculation is needed to fix A.
3.1
A s y m p t o t i c
f r e e d o m
of
Q C D
'127
Upon replacing the: renormalizcd coupling constant as(.A42) by the running coupling constant a s (Q 2 ) i" the quark-gluon vertex, contributions of arbitrary order in aB are partially summed up, as it is obvious from (3.1.3). Iliese (logQ 2 /A 2 )" contributions come from higher-loop corrections to the elementary vertex. In this sense, application of the renormalization group (RC!) leads beyond perturbation theory of finite order since it sums up the leading-logarithmic (LL) terms. Methods for partial summation of logarithmic contributions to the perturbation series are described in Sects. 2.8.3.8 and 3.3.3.2. According to (2.6.34) and (2.6.35) the (1- and 7-functions of QCD have the structure of (2.6.20) in one-loop order. Therefore, the explicit solution of the RG equation of a vertex function with 1 ¿,. gluon and n q quark fields is according to (2.6.24) given by r;r"(A/c t , Apy; as;M)
= r^'"" (k^pj; a3(A, a s ); M) i n \ ("KTg.0 + "q7q.0)/2/?0 x A'l-nK-3«q/2 f j
= T^'n''(ki,pj;as(X,
as);
x A ,-« g -3, 1
fj V
+
M)
^ / 3 0 ] O gA 2 ) _ ( n B 7 g , 0 + n " 7 " , 0 ) / 2 A 47r / (3.1.7)
in one-loop approximation. The QCD vertex functions show an asymptotic scaling behaviour in which the dynamical dimensions occur in the form (log A2 ) _ ("b7b,o+nq7q,o)/2j9o We discuss as an example the two-point vertex function of rnassless quarks, the inverse of the quark propagator
I f V , -/>'; cts-M) = r°' 2 (p, - p ; cvs(A, as): M)A ( f ^ )
, (3.1.8)
where p' = Ap, 7 q ) 0 = Crf = 4£/3, and /50 = 11 - 2N(/'S [cf. (2.6.36)]. Introducing the anomalous dimension of the quark propagator d
q
„ =M =
'°
/9b
UCa
_
4TpN[
=
^
33 - 2ATf'
,, , (n
4, (()
.
s
which depends on the gauge parameter (vanishing only in the Landau gauge), we obtain for the quark propagator
l o
The asymptotic behaviour (p2
K,R(P)
3.1.2
i ( v2 OC j (tog
(3-1.10)
6
A2) then is 33 i
{
.
(3.1.11)
I m p o r t a n c e of h i g h e r - o r d e r c o r r e c t i o n s
In Sect. 3.1.1, we outlined asymptotic freedom under the most simple conditions, i.e. lowest order in perturbation theory and a mass-independent renormalization scheme like the MS or MS scheme. There are many results available in the literature which go beyond this. At the two-loop level the running coupling is obtained from (2. .17) and (2.6.34) as 32tt^
k r
16tt2 / " I T
1
1
i m
W m 2 ) I
0
1 h
2
- y C
T
c /
, l rfyo
with cf. (2.6.37) and (2.6.41) Pi = j ( C A )
/
- 4C T
I6 2/ 0
3.1
A s y m p t o t i c
f r e e d o m
o f
Q C
'127
I li llning I lie asymptotic scale parameter A in t.his approximation a.s
*
I
=
M
M
= M cx
-
2
f
'
m
("«cm 5 ) " M106 TTSggp J'
(3 W4)
'
t o-loop equation for the running coupling constant I
A
.
ska"W2>
ft. =
Q2
, , . ,r. <3U5>
AN approximate solution of this transcendental equation can be obtained by expanding into powers of (logQ 2 /A 2 ) - 1 , <MQ2) 4*
=
1 log Q 2 /A 2
/j.logQogflVA 2 ) 01 (log Q2IA2)2
Bi /?03 (log Q 2 /A 2 ) 2
Taking only the first two terms means neglecting ( l o g Q 2 / A 2 ) - 2 terms, i.e. squares of the first-order result, a convention which was often used in the literature. The values of the asymptotic scale parameter determined from the full equation (3.1.15), A, and from (3.1.16) omitting the ( l o g Q 2 / A 2 ) - 2 term, A', are related by
For five quark flavours this is a 10% difference. The scale parameter A depends on the regularization (dimensional, Lattice, . . . ) and renormalization scheme (MS, MS, MOM,...) (cf. Sects. 2.5.3.3 and 2.10.4.3). Since the /^-function is scheme-invariant up to the second order, the transition to another scheme can be compensated up to this order by rescaling A , e.g. A ^ = AJA.MSThe asymptotic scale parameter A of QCD determines the strength of the strong interaction and defines the energy scale where an evaluation of QCD
4 3 2
3
Q u a n t u m
G h r o m o d y t i a i n i c s
based on perturbative calculations is possible. Therefore, the higher-order coefficients of the loop expansion of the /¡-function of QCD are important quantities. They are known in the MS scheme analytically in the third [Ta81], (2.G.41), and the fourth order [Ri97]. Precise data come from experiments at the Z-boson resonance. Therefore, it has become customary to use the value of the running QCD coupling constant at Q2 = M2 instead of A . The p r e s e n t value for « ( M | ) is [ P D G O O ] S
a s ( M | ) = 0.1181 ± 0.0020.
(3.1.18)
This corresponds to A
S
= 208
-23
McV
(3 L 1 9 )
'
in the MS scheme for live active flavours [PDG00]. In Fig. 3.1 we present (3(aa) (left figure) and the running as(Q2) of QCD for a s ( M | ) = 0.115 (right figure) in one- to four-loop approximation taking into account the quark thresholds (variable N() [vN00]. The two-loop result (NLO) for /3(«s) still gives a sensible correction to the one-loop result (LO). The three-loop corrections (NNLO) are already small, and the curve which includes the four-loop corrections (N3LO) is indistinguishable from the one including the three-loop contributions. The situation is similar for o^i? 2 )3.1.3
T h e r u n n i n g quark m a s s e s of Q C D
The renormalization-group equation of QCD determines besides the running coupling constant also the running quark masses. Their behaviour is governed according to (2.6.23) by the /J-function and the renormalizationgroup coefficient 7 m of the mass term. The latter follows from the mass renonnalization constant ZM by
The mass renonnalization constant ZM is gauge-independent; the one-loop result was derived for QED in (2.5.111). In QCD it is given by =
+
(3.1.21)
. {.
A s
Fig. 3.1 /^-function and running of
Writing
7m =
lt7m'°
+
27m 1 +
'
• •' '
(3
"L22)
(3.1.20) and (3.1.21) yield 7m,o = 6C f ^
8.
(3.1.23)
Inserting this in (2.6.23) finally gives for the running mass in one-loop approximation m(Q2) m
( _
\2
^
dm
-
2
A )
f
V / W (3.1.24)
4 2
s
with the anomalous dimension of the mass ,
_ 7m,0 _ 9G> sutt) '° "2fh~ 11 CJ,\ — 4T,,/Vr
dm
12 33 — 2N(
[i l
Introducing a rcnornializat,ion-group-invariant integration constant m (similar to A), (3.1.24) can be written as
The anomalous dimension d mi0 is flavour-independent in leading order (cf. (3.1.25)]. Consequently in this approximation, ratios of running quark masses are scale- and scheme-independent. The running mass gets smaller for Q2 -» oc. The t.wo-l(jop residt for the anomalous mass dimension in the MS scheme and the running mass is also known [Ta81, Na81]. As a typical result we give = 2.30 ±0.05.
(3.1.27)
The treatment of the quark thresholds and the corresponding matching conditions can be found in Refs. [I3e83, Ma84]. The three-loop coefficient 7,(li2 has been calculated in ltef. [Ta82], and the four-loop coefficient 7,M,3 in Refs. [Ve97, Ch97]. The values of the quark masses are not only scale-dependent but, since quarks do not exist as free particles, also scheme-dependent. Following Ref. [PDG00] we give the MS quark masses for the light quarks (u, d, s) at the scale 2 CeV and for the heavy quarks at the scale of the mass, m u = 1 to 5 McV,
m
mc = 1.15 to 1.35 CeV,
ms = 75 to 170 MeV,
m t = 174.3 dh 5.1 GeV,
in b = 4.0 to 4.4 GeV.
(3.1.28)
The masses of the u, d, s, c, and b quarks at other scales arc obtained in oneloop order by using the scaling factor of (3.1.26). There is some experimental evidence for the running of the b-quark mass [Ab98, Br99].
Zo)
. .,
s
s
1
The fcrinion propagator and other Green functions are, as we have seen, gunge-dependent. Therefore, it is necessary to introduce a running gauge parameter £(Q 2 ). 2 The result is [Na82] ( I |OLr Qi)->'t.O • S
13A'-'1/V(
c> A ? t
where £ is a RG-invariant parameter and the anomalous gauge dimension reads <
k
137V - 4N { _ 39 - 4JVf o = = z 2(11 N - 2JVf) _ n2(33 - 2N() ,
.2
s
(3.1.30)
s
The renormalizability of QCD allows to work out a consistent perturbation theory. Getting asymptotically free at high momentum transfers, QCD can be applied to hard-scattering processes. This is carried out for deep-inelastic lepton nucleon scattering in this section. The basic concept of the partem model (cf. Sect. 1.5) is visualized once again in Fig. 3.2a. Accordingly, the inclusive cross section is obtained by calculating the parton cross sections for the scattering of leptons off free, massless quarks and multiplying it with the probability of finding these quarks with momentum fraction x in the nucleon f. In the framework of QCD this model has to be extended in order rvs by gluon bremsstrahlung (Fig. 3.2b), quarkpair creation (Fig. 3.2c), and internal radiative corrections (Fig. 3.2d). In higher orders more and more diagrams contribute. In this section we first present the field-theoretical formulation of the parton model (Sect. 3.2.1). Then the operator-product expansion together with the renormalization-group is used for a partial summation of the perturbation series in Sect. 3.2.2. Finally, the evolution equations for the Q2 dependence of the structure functions are discussed in Sect. 3.2.3 and applied to the experimental results in Sect. 3.2.4. 2
See the remarks after (2.0.11).
432
3
Q u a n t u m
G h r o m o d y t i a i n i c s
a)
b)
b)
d)
Fig. 3.2 Deep-inelastic lepton nuclcon scattering in the naive parton model (a) and QCD corrections in order a s (l>,c,d)
3.2.1
The
field-theoretical
approach to the parton model
3.2.1.1
Properties of the hadronic tensor
As in Sect. 1.5.1, the starting point, is the spin-averaged hadronic tensor Wliu(p, q) expressed by the current matrix elements between the incoming hadron A), p 2 = M, and an arbitrary hadronic final state | X , p x ) , W,iV[p>q) = ^Y/{2itY6W{q
+
p-px)
A
Z
A
Using the Fourier representation of the ¿-function, (27r)4(5^(p - px +q)=
j d 4 y e1™ e^-**)»,
(3.2.2)
3.2
QCM )
in
d e e p - i n e l a s t i c
s c a t t e r i n g
137
1i anslation invariance,
= e^-^^pl/iliOJIpx}, mid the completeness of the states \X,px),
(3.2.3)
this can be written as-5
V V G h i ) = J $ e i w ( p l [ ¿ ¡ M M o)] |p).
(3.2.4)
'I'he mirent.- current commutator instead of the product is obtained by subtracting from (3.2.1) a similar expression where the ¿-function (3.2.2) is replaced by (/; — px — ( ) and 0 p0—P°X < 0 imply q°—p +p°x 0. In the following we assume p 2 = M 2 pq and neglect p~ where appropriate. The next step is the decomposition of the hadronic tensor into Lorentz covariants and structure functions [cf. (1.5.12) and (1.5.13)]
Wfil/(p,q)
= -g^Fx
+ PJ^F2 pq
- i
F
pq
Ppq° „ _ ( <)Ju pJ,Pu\ „ 9jw „ --z 1 Pi + i r - t h - i£,u>p
(3.2.5)
with T
9,w
_
— .9/
_
W»
1
T _
Pp
~
Pf-
_
P'l 2'
(3.2.6)
and
.
2pq
- f . 2pq
'The average over the hadron spin 1/2
(3.2.7, is not written explicitly in the following.
'138
3
Q u a n t u m
C h r o m o d y n a m i c s
In D dimensions, the inversion of this decomposition reads
=
( - £ + ID - 1)2/-|f ) T
FL(x,Q2)
= —2xF\ (x, r/2) +
= 4.r2^iy""(p,
F3(X,Q2) =
3.2.1.2
T
q), (3.2.8)
Light-cone behaviour of the free current commutator
We arc interested in the behaviour of Wlt„, (3.2.4), in the Bjorken limit, i.e. for Q2 -» oo, // = pq/M —> oo, x — Q2/2Mv fixed. According to the method of stationary phase this comes from that part of the integration region where qy remains finite. In the laboratory frame, where q'1 = (f, 0,0, \Ji>'1 + Q 2 ), we obtain
Iwl =
7A)
2/3 \J 1 + 2Mxjv ~ i/\yo -1/3 v—>oo
(3.2.!))
In order to keep |i/?/| finite for // —> oo this requires |j/o — ?/:í| = 0. Hence the essential contribution to the current commutator in the Bjorken limit originates from the vicinity of the light cone. Next we rederive the results of the quark-parton model for the hadronic tensor W^}' A of deep-inelastic lcplon nucleón scattering (cf. Sect. 1.5.2) in a field-theoretical manner using the commutation relations for free, massless quark fields. Here we consider general vector and axial-vector quark currents
<11 ,'/2,C
>ii,Ay)=
E
:
$xqmip
(3.2.10)
3.2
(.¿CI)
in
d e e p - i n e l a s t i c
s c a t t e r i n g
'If»!)
whore X''nq2 denotes the Cell-Mann matrix (ef. Sect. 1.2.2.2) in the llavour »pace of the quarks.'1 The electromagnetic, charged, and neutral weak currents (cf. Sect. 1.4.2.1) are linear combinations of these general flavour currents h'iAv) = YL^nivh^QMv) <1 ll
?Ay) = Y <1
=
i'iyhi'Q^iv)'
- 75aq)ii>q{y),
hi(y) = J^Myh^-^-Vij^iy). ij
(3.2.11)
For illustration we give the relation between the Gell-Mann-matrix notation in (3.2.10) and the one in (3.2.11),
Qq = diag I
I=
,
(3.2.12)
for the electromagnetic current in the case of three quark flavours. The Bjorken limit of the structure functions is obtained from the commutator of the quark currents5 (:c), ht(y)] = ]T fan WlvKmW« Qi
<*)>
Mfr^«^«(»)
X
- (vH'^vi, (?/)> i>qi (x)hli*qiq2M )) •
(3.2
Inserting the anticommutator of free massless quark fields, (1.2.12), U>
= <W(i7A^)iA(rc - y),
(3.2.14)
and using the relation XaXb = (i fabc + dabc) Ac + -6"b 1 3
(3.2.15)
'In the following we suppress the symbols : . . . : for normal ordering and the colour index c. We here restrict the treatment to vector currents.
'I'll)
3
Q u a n t
u m
C h r o m o d y n a m i c s
for the product of Cell-Mann matrices yields K](x),hi(y)]
=
(3.2.16)
- r6c(^bM7A7.AcV>(y)
i^A(x
- y) + (x
+ ¡c/atfi(V;(*b,i7A7i>Acy>(?/) \dxA(x +
(yWwxlA
y,,i <-> „))
- y) - (z
jy))
i>{y) id*A(® - y) ~(x y, fi
Decomposing the product of Dirac matrices with help of the Chisholm identity 7/i7a7i/ = {g,L\(j„f, + g\>,g,,p - y,wy\p) Y - ie/,A757p - ie^vplsY,
=
(3.2.17)
finally gives for the current commutator [/»?(*),/£(*)]
=^A(x-y)
x { - i / a 6 c [ ^ A , p {i>(x)Yxc->P(y) + - ie^Aup {M*hpvAci>(y) c
- dabc[o,lXup
i>(y) -
i'(yhpmx)) - i%)7"75A c V(*))]
$(y)-f\e1>{x)) Ac v(y))]
- \&ab[
- V>(y)7pV>(z))
(^(x)7p75^(?y) + ^(2/)7"75V>(tf))] }•
(3.2.18)
This provides an example, which already shows the typical features, of an operator-product expansion (OPE). The expansion consists of singular functions, in this case the derivative of the free scalar commutator function (1.2.9) A(.r - y ) = - ( ¿ y j / A = ~sgn(x°
-iq(x-y) sgn(q0)6(q2)exp~>
- y°)S {(x - y) 2 ),
(3.2.19)
3.2
(.¿CI)
in
d e e p - i n e l a s t i c
s c a t t e r i n g 'If»!)
llavour tensors f"bc, dabc, 6ab, Lorentz tensors cr,,\„p, e,,\„/(, and regular 1 >ilond operators. I'lie bilocal operators are of the vector- and axial-vector type, symmetrized and antisymmetrized, Of^{x,y)
= ipq(x)lpipq.(y)
± il>q(yhP4>q-(x),
?/) = i><,'{v) ± i'q[vYiplr^'q1 (x), HIICII
(3.2.20)
that the total expression (3.2.18) is antisymmetric.
The electroweak currents transform according to the adjoint representation of the flavour group [cf. (3.2.10)]. Their commutators, i.e. the bilocal operators, lie in the adjoint representation (Ac terms) with symmetric (d ) or antisymmetric {fabc) couplings, or in the singlet representation (1 terms) with coupling Sab and are organized in expressions with definite chargeconjugation parity. The corresponding contributions are denoted flavour non-singlet (NS) CjNS.c.d: = J 2 X '
respectively, with charge-conjugation parity ±. Inserting these expressions into (3.2.18) gives
- \*ah [cr^O*«'-
(x, y) - i
(
x
,
j/)] }.
(3.2.23)
This decomposition into irreducible representations of the flavour group with definite charge-conjugation parity is important when higher-order QCD corrections are calculated (cf. Sect. 3.2.3).
432
3
Q u a n t u m
G h r o m o d y t i a i n i c s
Similar results are obtained when commutators with axial-vector currents or chirai currents are calculated. This arid a more detailed discussion of (3.2.18) is postponed to Sect. 3.2.1.3 where also the generalization to the interacting case in QCD is considered. We are interested in the contribution of the free current commutator to the nucléon structure functions. In order to work out the essential points we evaluate the contribution of the electromagnetic current x) = x i )liLQqtf'q(•'')• To extract the Bjorken limit of the unpolarized structure functions, the expression (3.2.18) has to be substituted into (3.2.4). For the case of the electromagnetic current, the terms with the structure constants /"'"" and with 75 are absent in the unpolarized case, and using translational invariance (x = 0) the hadronic tensor reads
Id
= J
<1
x{p\O^~(y,0)\p),
(3-2.24)
where A(y) = - A ( - y ) . Written in the group-theoretic notation of (3.2.18), (3.2.21), and (3.2.22), this reads
= ~ J dV
W
< W (¿tfA(y))
x ^(p|C?f' 3 '-(y,0)|p) + ^ + f
(3.2.25)
This representation of shows that in the electromagnetic case both non-singlet and singlet operators contribute. The same occurs in neutraland charged-current reactions. The spin-averaged nucléon matrix element can be decomposed into Lorentz covariants //', yp and invariant functions of py, p2, and y2, (p\0«;h (y,0)|p) = 2 p P g q M p y , y 2 , P 2 )
+Vl>M%tM(py,V\p2). (3.2.261
3.2
(.¿CI)
Mocause of the J(y2)-function in d*A(y) this elfectively reduces to (p\O^'-(y,0)\p)
in
d e e p - i n e l a s t i c
s c a t t e r i n g
[cf. (3.2.19)] and p 2 = M 2 «
= 2PpOgMipy),
'If»!)
Q'\
(3.2.27)
since the term hqjj gives a contribution which is suppressed by a factor M /pq in momentum space and therefore can be neglected in the Bjorken limit. The form (3.2.27) of the matrix element of the bilocal operator has I lie consequence that the longitudinal structure function [cf. (3.2.8)] Fu =
(3.2.28)
4X2LJ—WIW
pq
is proportional to p 2 = M 2
^gqMO^{py\
-oo
(3.2.29)
yields WtfjV(p,q)=
= 7-n E
(3.2.30) [
J
(
t
f
A(y)) ,
where the sum over q runs over all quarks. Equation (3.2.8) then yields for the structure function
(3.2.31) and by using after partial integration again the integral (3.2.19), ^,77AVy,
= pq £ tf
[
J
^
+
s [ l+
{'
tofyiMt)(3.2.32)
4 4 4
3
Q u a n U m i
C h r o i n o d y n a m i c s
Neglecting the nucleoli mass, we have in the Bjorken limit, since 0, sgn(<7° + &>°)J((<7 + i v f ) ~ ¿(if 2 + 2 ^ 9 ) =
- a-)
(3.2.33)
with the scaling variable x. Kinematics implies 0 < x < 1 [cf. (1.5.3)]. Thus, performing the £ integration with help of the ¿-function, the Bjorken limit of the structure functions F ? 1 ^ is obtained as F r
«
[ x )
=
1£
= 2 3 ; F i 77^ ( x )
(3.2.34)
<7 The functions gqjtf(x) are the nucleoli expectation values of the bilocal operators 0'ph~ [cf. (3.2.27)]. Writing these operators in terms of quark creation and annihilation operators the physical meaning of g,hM{X) becomes obvious: gqjj(x) is the density of quarks and antiquarks with momentum fraction x in the nucleoli and thus identical with the quark distribution functions, 9 = Sqjs{ x ) + J()M{X)- Therefore the final result is FTV(*)
= \ E QKfnMx)
+ /wrW),
n
F ^ ( x ) = x J 2 Q q U i M * ) + f'lM*))•
(3-2.35)
<1
Altogether we found: the light-cone singularity of a free quark field gives Bjorken scaling (and vice versa, see [Bj69]). The Call an Gross relation (1.5.29) is found to be valid. In addition we obtained a field-theoretical definition of the quark and antiquark distribution functions f,!r\f(x) and /,/,.y (•'<") • These are the Fourier transforms of the reduced hadron expectation values of the bilocal operators 011 the light cone. In a similar way the parton model results for neutral and charged weak current processes (1.5.28) can be obtained field theoretically. 3.2.1.3
Wilson expansion
Expanding the bilocal operators of (3.2.20), e.g. O%'~(yt0)
= TpqilMqiO)
~
TpqiOhMy)
(3.2.36)
3.2
(.¿CI)
in
d e e p - i n e l a s t i c
s c a t t e r i n g
'If»!)
into a power series in y, oo n=0
(0) =
• • • OMv)7pV-,(0)
y=o,
(3.2.37)
substituting tliis into (3.2.18), one obtains the Wilson expansion for tin* free current commutator, i.e. its representation by singular functions, Lorentz tensors, and local operators, the Wilson operators [Wi69].
JIIUI
The mass dimension of the operators O wll ,.. (lll (0) can be calculated as ditn[C7] = 71 + 3 according to the above equation. On the other hand, the Opin...)in are tensors of rank (71 + 1) and thus possess the maximum angular momentum lmax = n + 1. The difference between dimension and angular momentum of an operator is called twist, T = dim [O] - I.
(3.2.38)
Hence the operators O have minimal twist 2. The spin-averaged nucleón matrix element of the Wilson operators can be decomposed into covariants and reduced matrix elements,
=
( » 1 • --PvMWfl
-1- M2ppgfiut2p,l3.
••P,i„Bj/
n+l
(3-2.39)
+ terms with more fl^'sY
The terms containing g,w have higher twist, T > 4. In the Bjorken limit their contribution to the structure functions is suppressed by the factor ( M 2 / Q ¿ ) . Therefore, their effect is small compared with the leading twist-2 terms. Again we discuss the electromagnetic case in more detail. Comparing (3.2.27) with (3.2.37) and (3.2.39) shows that the coefficients A\ r < n + i are obtained from the invariant functions gqw\r[py) by power-series expansion, (
d
\
n
py=o
1
s
sing the Fourier transform of
gqj^(py)
yields
71-1 .V
(3.2.40)
The x integral extends from to I only, since gqrrf(x) vanishes for x < and x > 1 for inematical reasons cf. 1. . . The reduced matrix elements ilson operators ft ..tln are r n of the r the sums of the Mellin moments fqjj)U and f,],A ,n of the quark and antiquark distribution functions. They determine the moments of the structure Junctions in the or en limit,
F
f
The inversion of formation.
o
dxx- F
ix
2
,f
,n
.2.41 is obtained by performing an inverse
.2.41
ellin trans-
utting everything together, we can write
(3.2,12)
for the electromagnetic structure function in the case of free masslcss uar s as partons. imilar expressions result for the case of neutral and charged wea -current deep-inelastic processes.
3.2
.'{.2.2
(.¿CI)
in
d e e p - i n e l a s t i c
s c a t t e r i n g 'If»!)
Q C D corrections t o the parton model
In Sect. 3.2.1, the parton-model results for the structure functions in deepinelastic scattering were derived for free quarks using the Wilson expansion of the current commutators. In this section we generalize this procedure to 111«' case of interacting quarks in the one-loop approximation. We consider the quark currents (3.2.10), which contain as special cases the elect romagnetic as well as the charged and neutral weak currents (3.2.11). Then, all three structure functions F^ v > j V , F%V>J^, and occur. Moreover, hesides the quarks also the gluons have to be taken into account as partons. As a consequence, the Callan-Gross relation is violated. Therefore, we consider — 2xF\ instead of F,v V"A' and the longitudinal structure function F\T = introduce the index a = {2, L, 3} corresponding to the projectors (3.2.8) on the structure functions FL,, and the parity-violating part F3. 3.2.2.1
Operator-product expansion in higher orders
The operator-product expansion for the general case has been introduced in Ilef. [Wi69, Ka69]. Using this and the facts discussed above, the moments of the hadronic tensor can be in general expressed as =
k , T NVV'k
Ak
•
1)P(la R'W'k Ak
(3.2.43)
a s , M) describe the hard-scattering proThe Wilson coefficients C^'k{Q2, cess between the virtual elcctroweak gauge bosons and the partons, and can be calculated in perturbation theory [Gr73a]. They depend 011 the flavour and parity of the currents and 011 the parton species but not 011 the properties of the external hadrons. In the electromagnetic case they are given by the quark charges, e.g. CQ™ (Q2, as = 0, M) = Q2, in lowest order. The coefficients A ^ n ( a s , M ) are the reduced hadronic matrix elements of the Wilson operators. They are given by the moments of the parton distribution functions and cannot, be calculated in perturbation theory. In one-loop order of QCD it is necessary to use the decomposition into non-singlet (NS) and singlet (S) contributions (3.2.25) with definite symmetry under charge conjugation. The singlet quark contribution can mix
4 4 8
3
Q u a n t u m
C l u o i n o d y n a u i i c s
with gluonic contributions with the same quantum numbers. The index /.• = {NS,r:; S,q; S,g} discriminates between all these contributions: quark contributions are denoted in the group-theoretic notation by NS,c and S,q in the non-singlet and singlet case, respectively. The gluonic contributions, which occur in one-loop order, are labelled by S,g. The following Wilson operators of QCD with twist two generalize the expressions (3.2.20), (3.2.21), (3.2.22), and (3.2.37) of the free quark case: O™;c.,ln(0)
= ï>q{0)S{'ypDfil
... D,ln}A^VY
(0),
(°) = ^ q ( 0 ) S { l p D , i l . . . D l l n } r P q ( 0),
^,..^(0) =
•••DMM-
(3-2-44)
Here S denotes symmetrization of the Lorentz indices and taking out traces in order to obtain, with respect to the Lorentz group, irreducible tensor operators. Tlu; covariant derivative with respect to the colour SU(3) group D,t occurs because of gauge invariance. In QCD also the gluon fields contribute to the singlet Wilson operators with twist two, ...^(0)=«S{1VV(0)^I
F tt/irl (0)},
(3.2.45)
where the trace refers to the colour indices. The operators 0 S,
= i n PpP>n - P , 1 „ 2 ^ 1 ( a
8 l
M).
(3.2.46)
A comparison of (3.2.43) with the general decomposition of the hadronic tensor (3.2.5) provides relations between the moments of the structure
3.2
(.¿CI)
in
d e e p - i n e l a s t i c
s c a t t e r i n g
'If»!)
functions and the coefficient functions,
F i r m
= [
l
dxx»-*Fr'"(Q2,x)
Jo
FCn"(Q2)
k = f Jo
= E
dxx»-*Fr'"(Q?,x) C?Xk{Q\as,
M)AiiMin(as,
M),
k F$r«[
= /*' Jo
dxxn~lF^v^(Q2,x)
= ^ EC3Tfc(QVs>^MW(«s,M). k
(3.2.47)
Thus, the moments of the structure functions are determined by the Fouriertransformed Wilson coefficients and the nucleoli matrix elements of the Wilson operators. In lowest order of the strong coupling constant a s , the covariant derivative Dtl is replaced by the partial one 0in and there is 110 contribution of the Ilavour-ncutral gluon fields. Consequently, we have only aspin-1/2 contribution, and F\, vanishes. The matrix elements of the Wilson operators A'.^ n and A^^f n arc the sums and differences of the moments f q j^, n ± Jqjf,n °f the quark and antiquark distribution functions [cf. (3.2.42)]. The Wilson coefficients reduce to the products of the quark couplings (1.5.11). I11 this way the results of the parton model are reproduced.
3.2.2.2
Renormalization-group equation
The quark currents h,L(x) and the Wilson operators (0) = Op,„.../Jn(0) are local composite operators built up from quark and gluon fields to which the multiplicative renormalization procedure of QCD (cf. Sect. 2.5.1.7) can be generalized [Zi73, Di74, Jo76, Co84]. However, mixing of operators of the same quantum numbers and the same or lower dimensions may occur. In our case the flavour-singlet Wilson operators C^''1 and Of,'K mix. Ilence in
432
3
Q u a n t u m
G h r o m o d y t i a i n i c s
general, a renormalizat ion-constant matrix (Zo n ) kl relates the renormalized Wilson operators ;1 (« s , .M) to the unrenormalized ones, h%L{x,aa,M)
=
Z£\aa,M)h»(x),
0& i f l (O,a„M) = '¿[Z£{at1M))k,Oln(
0).
(3.2.48)
From the corresponding Green functions, = (0| Ti}>{xx) [ f c £ ( s ) , M 0 ) ] i ( i 2 ) |0), (3.2.49)
= (0| Ti>{Xl) 0*(0) i>{x2) |0),
the renormalized vertex functions and arc derived by truncation of the quark and antiquark legs. These vertex functions satisfy the renonnalizat,ion-group (RG) equations0 M
E
M
-
(1M
027"' + 2 7 " rfhUl(as,M)
d M -fry*)
kl
S"' + In
rf°'«{aa,M)
= 0, = 0.
(3.2.50)
The coefficients 7' 1 and 7* are the anomalous dimensions of the current operator and the Wilson operator, respectively, [cf. (2.G.9)]
(3.2.51) The charges of the quark currents h'1 are the generators of the flavour symmetry group. Because h'1 is a conserved current, it does not need renormalization, i.e. Z/t = 1, and its anomalous dimension vanishes, i.e. 7'' = 0. The current current and Wilson-operator vertex functions are linked together by the Wilson expansion, written in a condensed form as (3.2.52) k,n c
The different signs in front of truncated.
and 7*' result from the fact that the if/ legs are
.'{.'2
(.¿CI)
in
d e c p - i n c l a s t i c
s c a t t e r i n g
'151
Ni is allows to derive a RG equation for the Wilson coefficients C b y substituting (3.2.52) into (3.2.50), 0 = M
AM
- £k,n = E
E Cn ( Q21 «s, M) r
-2^
(aa,M)
k,n
M^ci, + ci ( « dJL M rR°'n(<*s,M)
-
7?
k,l,n
) rr!
) C*(Q\as,M).
(3.2.53)
for irreducible Lorentz tensors this equation must be fulfilled for each n and I separately,
0
= E
( ' ^ d ^
-
(3.2.54)
Ckn(Q2,<*s,M).
As the Wilson coelficients are diinensionlcss, the associated scale-transformation equation reads (in the massless case)
E 6'k(XA~P{9s)^)+'Y"{9s)
C
n{Mf,<*s)=0.
(3.2.55)
This general RG equation is a coupled system of linear, partial differential equations. It allows for the application of perturbation theory to the calculation of the momentum dependence of the Wilson coefficients, because QCD is asymptotically free. No mixing of operators occurs in the flavour non-singlet case, where ONS.C and O f ' c contribute. The corresponding one-loop anomalous dimensions are flavour- and parity-independent in massless QCD. Therefore, we denote them by 7^S(ffs)- The RG equation is a scalar equation and (suppressing flavour indices) solved by
C™((\Q)\as)
= C„ ns (Q 2 , a s ((AQ) 2 )) exp
| V
, (3.2.50)
4 5 2
3
Q u a n t u m
C l i r o m o d y n a m i c s
where ots is the running coupling constant. With (3.2.47) this implies F„ns ((AQ)2, ots) = F„NS (Q2, â s ( ( A Q f ) ) exp ( - j T dg'
. (3.2.57)
for the Q 2 dependence of the non-singlet moments. The breaking of scaling in the moments of the structure functions is determined by the anomalous dimensions 7^'s of the Wilson operators and the function of QCD. Now we treat the flavour singlet case, e.g. contributions to neutral current scattering, at one loop. Because of mixing among the operators and S sa 0 'S in the singlet case, 7®(<7S) > constant 2 x 2 matrix, 7« - I 7r,q
7r,g
J »
(¿.2.o8)
which can be diagonalized
with 7,f = I (xT 1 + 7 , f ± y / h i f ' - 7n 8 ) 2 + s w r r M " j ,
Here Nf denotes the number of flavours. This procedure decouples the two differential equations for C«''1 and C^'8. The linear combinations Cn'+ and c C'„ then behave ¡11 the usual way, C;^((A
Q)\as)
= Csn±(Q\as((\Qf))
exp ( - j \ g '
.
(3.2.61) This treatment of the singlet case can be generalized to higher orders if a decomposition into irreducible representations of the flavour and chargeconjugation symmetry is performed. This will be treated in Sect. 3.2.3.
3.2
3,2.2.3
(.¿CI)
in
d e e p - i n e l a s t i c
s c a t t e r i n g
'If»!)
Structure of QCD corrections
Evaluation of (3.2.55) and (3.2.61) by means of perturbation theory starts from a power-series expansion of the relevant quantities, explicitly
The expansion of the Wilson coefficients C ^ ' f c ( a s ) starts with the partoninodel values =
+
+ ...
(3.2.64)
with _
r
°2,n,0
— u
O
= GI!'1"".
t>
i
vvg _
n
°2,n,0 ~ u> C ?
= 0.
O = 0. < » = 0 (3.2.65) in the product base and similar expressions in the decomposition into the adjoint and singlet representation of the flavour group. The coupling strengths Gv and Ga of the quarks are given in (1.5.11). Since Fi, = 0 in the parton model, and since the gluons do not couple directly to the flavour currents, the expansion of the corresponding quantities starts with the a s terms. From (3.2.64) (3.2.65), the calculation of the behaviour of the moments of the structure function proceeds. Using only the first terms of the expansions above to work out (3.2.57), one gets the leading-order results. There is a difference between the non-singlet case, where only one Wilson operator of order n exists, and the singlet case, where mixing of operators occurs. From the corresponding RG equations one obtains
<
W S
( Q
2
) =
(3.2.66)
4
4
,
s
for the Q2 dependence of the non-singlet moments and the parityviolating non-singlet moments F^, where Qo is the reference scale and
C
=
(3.2.67,
The Q2 dependence of the singlet moments F 2 jf is given by
C'
S , ±
W
2
) = (Ull)"3"'"
F
lnKS^Q2o)
(3-2.68)
with
f = " g -
(3.2.69)
The anomalous dimensions i/„,o are calculated in the following section.
3.2.2.4
Anomalous dimensions of Wilson operators in Q C D
In order to calculate the anomalous dimensions of the Wilson operators O in order o s , the QCD Feynman rules must be extended to the case of local composite operators. The following field points are assigned to the Wilson operators (3.2.44) and (3.2.45) (cf. [Gr73a] for further details): i > q A x ) S { > 1 >>{<,'dm • • • V,>n } V y , c ( * )
^ { T / m ^ ^
i!>q>c(x)S
}<W
j 7 / 1 , \ f q q , d n • • • fkAl
X a, r s-f-^S
(x) ^
q\d n,f J I ^ ®
i ,c
. . . d,in }
fyjix)
(3.2.71)
n,f,a
q,c
. , , { (ii ,,, ... £fl, ... ,ln ->
> (3-2.70)
,
'
3 . 2
^liZ/i/z^/ij • • • klht +
_>
( . ¿ C I )
in
d e e p - i n e l a s t i c
9fit191'HI
s c a t t e r i n g 'If»!)
•••
~~ kti9viii • • • k,ln — ku9wn k/12 • • • kti„ k -> g,« n g, b 'x/N/XgW/x,
.
(3.2.72)
Similar rules apply for axial operators. The diagrams representing the oneloop corrections to the operator (3.2.70) are shown in Fig. 3.3. The calculation is similar to that for the one-loop diagrams in Sect. 2.5.2. It was worked out initially by Gross and Wilczek [Gr73a] and Politzer [Po74]. Symmetrization and taking out traces can be most easily achieved by contracting the operator with the tensor A/M • • • A/Ifl formed by light-like vector A;1. In Feynman gauge, using dimensional regularization, the Feynman rules give
for diagram (a). In the last step we used the Feynman parametrization of the integrand (2.5.(i7). For the anomalous dimension of the Wilson operator
432
3
Q u a n t u m
G h r o m o d y t i a i n i c s
only the residue of the pole at D — 4 is needed. This is determined by the: k,tkt, terms in the numerator.7 These act in the integral like glwk2/I). Since A 2 = 0, there is only one contribution, M[a)
= -ig2CFA^(Ap)"-l(/J>
2)
dz (1 — z)zn~
2 [
V
Jo
I x „_D r d»k 2 M /j J J (2TT) 2[k + z(l - z ) p 2 ] 2 as (D - 2) = — Gf.vvlo D 4?r /•l x 2 / dz(l -«)2n"142)(2(l -2)p2),
(3.2.74)
Jo
where we have inserted the lowest-order expression M(0a)
(3.2.75)
= A'£(Ap)n~l.
The tadpole integral /l[|'!) was calculated in Sect. 2.5.2.1. Writing as usual D = 4- 26, it reads [cf. (2.5.58)] 42)(z( 1 -
z)p2)
= ( ^ p )
r(i)(z(l -
z))-s.
(3.2.76)
The z integration gives Euler's /^-function, i.e. a ratio of T functions. The result of diagram (a) finally is A„(•)
a
V
4?r
, j(a) o (2 ~ 26)2 r ( n - ¿ ) r ( 2 - S) f4ir,i2 0
2
4-2*
r(n + 2 - 2 J )
( ^ J
^ *w
0
P(<5)
(3.2.77)
For the anomalous dimension we need only the singular part of this expression [cf. (2.6.30)], { a) M^ , 2 S1,,l = ^C T .M 0 e 4?r 7i(?i+l)i 7
(3.2.78)
The contributions of the terms with more k's in the numerator vanish owing to A 2 = 0.
.2
(
.s .
s
1
7
The second and the third graph of Fig. 3.3 yield
Adding these contributions and the renorinalization constants for the ex ternal fields (2.6.31) yields the renorinalization constant of the non-singlet operators in the MS scheme, =
I n(n + l)
4TT
sj
*
/
(3.2.80) and with (3.2.51) and (2.6.30) the one-loop coefficient of the anomalous dimension,
Hence for the colour group SU(3) the exponent in (3.2.66) is given by
The elements of the 2 x 2 matrix for the singlet case have also been worked out in [Gr73a],
n2 + n + 2
qg _ n, o -
_ 7 »-°"
e i F n
(
n
+
) (
+2)'
a
iC
n + n+ 2 n(n 2 — 1) '
2 2 ^ee = _ 4 C , I I 7 A "-° I »(« - 1) r (n + l)(n + 2)
- ^T f N { ,
6
j (3.2.83)
4 2
s
where Cv = 4/3, C A = 3 and T = 1/2 for SU(3). The exponents of the logarithms that determine the scaling violations in asymptotically free gauge theories are given by (3.2.82) and (3.2.69) together with (3.2.60) and (3.2.83). For light quarks they are flavour-independent. Since the Q 2 dependence of the moments is only logarithmic, it is weak compared with that of non-gauge theories. The anomalous dimensions of the Wilson operators at two loops have been calculated in Refs. [F177, F179, Go79, Ha92a]. For the three-loop case some specific moments are known [La97]. 3.2.3
Evolution equations
Using the Wilson expansion and the renormalization group we have derived QCD predictions for the moments of structure functions. In particular, logarithmic corrections to the scaling behaviour of the parton model have been found. In the naive parton model the structure functions are independent of the momentum transfer 2, i.e. the probability of finding a parton with momentum fraction x in the nucleoli does not depend on the resolution the nucleoli is probed with.8 The scaling violations in QCD emerge from t he fact that the interaction at short distances may convert a quark into a qUark and a gluon (cf. Fig. 3.2 on page 436), and a gluon into two gluons or into a quark-antiquark pair. The observation of these fluctuations depends on the resolution. Increasing the momentum transfer reveals the structure at shorter distances. More virtual quar s and gluons are found, each of them carrying a smaller fraction of the original valence-quar momentum. This is the parton picture of scaling violations. The 2 dependence of the structure functions can be controlled by evolution equations derived by Dokshitzer, Gribov and Lipatov, and Altarelli and Parisi [Gr72, Li75, A177, Do77]. We start from the solution (3.2.66) of the RG equation for the flavour non-singlet, moments '^i 2)They satisfy the differential equation ^2 8
N
n
S
(0=<
S
o^
with
t = log
2
A 2.
(3.2.84)
Thc three-momentum of the virtual photon in the nucleoli rest frame is (Q 2 + i/"')"'• According to I lie uncertainty principle it. probes the nucleoli structure at the corresponding length scale.
3.2
Inserting
(.¿CI)
= - 7 l ^ / ( 2 / i ( ) ) and «,(/.) =
in
d e e p - i n e l a s t i c
4ir/f)0t
s c a t t e r i n g
'If»!)
yields
= - ^ - f F ^ ( t ) .
(3.2.85)
In the quark parton model the structure functions F, v '^(x, t) arc products of tlio coupling strengths GqVV and the quark distribution functions U M ' ^ t ) and J a M ^ t ) [cf. (3.2.35)], F ^
N S
( x , t) = £ x (G«vv' <7
0 +^
0) . (3.2.85)
This implies for the moments of the quark distribution
functions
This set of equations is equivalent to an integro-differential equation, the evolution equation for the quark distribution functions fqjf(x,t), also called Dokshitzer-Gribov-Lipatov Altarelli -Parisi (DGLAP) equation, dfq^rjx, t.) _ dt
as(t) dy / ' dzS(x - yz)pW(z)fq,Kr(y, 2tt Jo Jo _ <*B [t) q<1 J f g j A ( M ) , 27r
t) (3.2.88)
provided
7«,o = - 4
f d Jo
zzn-'P$»{z).
(3.2.89)
The structure of (3.2.88) is identical to the one of QED for t he emission of hard, collinear photons (2.8.11G). This is no surprise since both equations describe the same physical situation, collinear emission of masslcss gauge bosons.
4 2
s
The evolution ernel also called splitting function, is calculated in one-loop approximation in Sect. '2.8.3.1 for QED and in Sect. 3.3 for QCD. The result for QCD is [cf. (3.3.1)]
+ (3.2.90) with the {+)-distribution defined in (2.8.80) or (2.8.81). Multiplicative convolutions [cf. (2.8.115)] h{x) =
] (s) = f
d
Jo
can be factorized by taking
f
dz6(x -
z)f{ )g(z)
Jo
ellin
moments
hji = [ d « " - ' / » ( i ) , Jo
(3.2.92)
as the following calculation shows: hn =
dxxn~*h{x)= Jo
=
drc Jo
d
n
d
dz 6{x -
Jo
J0
f{ )g(z)
Jo
f dzz ~lg(z)
~ f( )
z)xn~
= fn n.
(3.2.93)
Jo
The relation (3.2.89) between the anomalous dimension (3.2.81) and the splitting function (3.2.90) can be verified as follows:
(3.2.94) The first integral is elementary, ri Jo
„2z»-'
.+ + 2 „ + 1I -_ 2 2 _ i - *
U+l
r,I Jo
n
(z -
+ *»"' + 2 £ £ 0 V )
dz
jr; J i
1 -z
n(n+l)
.'{.12
Q C I )
in
d e e p - i n e l a s t i c
.scattering
4 6 1
Adding the contribution of the ¿-function term finally gives
(3.2.%) in agreement with (3.2.81). In general, the evolution equation couples the gluon distribution fy„\r(x,t) with the quark and antiquark distribution functions9 d/ff.yV _ as{t) At 2ir <*s (t) 2ir
di
®
®
+ A.g ®
j P'li'lj
Miff
j
II
<*s(t)
® /
2?r
dt
+ pwi
function
+ /gg ® /g,A/' (3.2.96)
This coupled system of 2N[ 4- 1 equations is reduced by using chargeconjugation and flavour symmetry. The splitting functions are written in terms of flavour-singlet. (S) and flavour-non-singlet (V) contributions, V s +1 1 P S 1Pqq P =
"g
=
"gq-
Furthermore, we introduce distribution functions with definite charge-conjugation parity,
/ t v = UirM ± 9
This corresponds to operator mixing in tlie operator-product expansion.
(3-2.98)
1( 2
.{
s
and flavour-synnnetry properties, namely non-singlet quark distributions
k
h,qi = E t=i
~
»
A; = 1 , . . . ,ATf — 1,
(3.2.99)
and singlet quark distributions t
(3.2.100) i=i Then, the evolution equations decompose into
sCr - ^ [ « ,+
• O " + M. - >4) • E ,
5 #
= ^
< +
1
<
O
,+
+ * (/* + ^ ) } e y j f
+ 2JVfPqg / g(A r],
i M
=
+ PgB
^r
(3.2.101)
®V
There are two separate equations for the non-singlet quark-distribution functions f ^ y and /^'y-- The quark-singlet distribution function f^Jj- is coupled to the gluon-distribution function /R>Jv. The splitting functions Pij(x) are related to the probabilities of finding parton i with a momentum fraction x in parton j. They are (-f)-distributions and fulfill the sum rules for quark-number conservation (3.2.102) Jo
;
and momentum conservation I '
dx
(
j f dxx
( ) + PQ*i
+ PW ( * ) ) = 0.
[/>„(*) + P l J x ) ] +
= 0.
(3.2.103)
3.2
Q C I )
ill d e e p - i n e l a s t i c
s c a t t e r i n g
163
The integrals have to vanish since the lowest-order contribution, i.e. the iiiinple parton-model contribution, saturates the sum rules. In the one-loop approximation the corresponding kernels are flavour- and ncherne-indepeudent and have the form P X ( , (*) = .
<^(x) i&
= T,
= o,
p^ \x)
= 0,
x2 + ( - x)2
o(0
gq (x) gg
= 2 ,
Li
1
r i i
—
11
+
2
+
+ { j CrA - ^N(Tv) ~ 3
1-
x x
/, \ +x(l - x )
(1 - x).
(3.2.104)
Consequently, the non-singlet evolution equations are diagonal and have the same kernel /^(.-E). This kernel also enters the singlet equation. The QCD splitting functions differ from the QED case (2.8.117) only by the group-theoretical factors and the contribution of the gluon self-couplings. At two-loop order the singlet contributions Pq(f1' and P^f 1 ^ are non-zero but obey the relation Pqq 1 ^ = equations still are diagonal.
''. Therefore, the non-singlet evolution
As the strong fine-structure constant a s is not small in the complete region, higher-order approximations give a significant contribution. The t oloop splitting functions are known [Cu80, Fu80a, Fu82j. A complete secondorder calculation for deep-inelastic scattering was performed in Refs. [Ma90, vN91, Zi91, Zi92] both in the MS and the DIS (deep-inelastic scattering) scheme including non-leading terms.10 In this calculation also the one-loop corrections to the Wilson coefficients [Ba78, FI79] enter. The evolution equations (3.2.88), (3.2.96) [or (3.2.101)] and (3.2.104) describe the QCD corrections to deep-inelastic lcpton nucleon scattering in a l0
T h c DIS scheme is defined such that the simple parton model is exact at the renormali'/ation point. The conversion formulas between the diiTerent schemes are given in Sect. 3.3.2.
4 6 4
3
Q u a n t u m
C h r o t n o d y n a m i c s
concentrated form. The solution of the evolution equations can be approximated by suitable expressions with free parameters for fqrtf[x,t), Jq,A/{x,t), and / g ( jv(i, i). These have the typical form J(x, t) = C(t)xa^{i
- x ) W [ l -1- c(/.)v/i + d(t)x], (3.2.105)
where C, a, b, c, and d are expanded in power series in t. Fitting these parameters to experiment, while respecting the QCD sum rules, one obtains phenomenologically useful, (¿"'-dependent parton distribution Junctions (cf. [G198, LaOO, MaOO] and references therein). 3.2.4
E x p e r i m e n t a l t e s t s of Q C D
A huge amount of experimental results for deep-inelastic lepton-nucleon scattering is available from SLAC, CERN, Fermilab, and HERA. They cover a wide range of x and Q2 values, namely 0.00002 < re < 0.85,
1 GeV2 < Q2 < 30000 GeV2,
and have attained a very high degree of accuracy. A detailed discussion of these results and the corresponding theoretical QCD calculations is not possible in this book. Instead, we restrict ourselves to the discussion of some important results in order to give an impression of the quality of the applications of QCD in this field. A comparison between the experimental data of the nucleón structure Junction F 2 ( X , Q 2 ) and the solution of the evolution equation in next-toleading order with parton distribution functions fitted at the initial scale Q2 = 4GeV 2 , and using a fixed value of the strong fine-structure constant as(M2) = 0.118 is shown in Fig. 3.4 [F199]. The data agree impressively with the QCD prediction. This is also the case for high energies in the small x region where no deviation of the next-to-leading order QCD prediction is seen. Note, however, that the figure does not show the scaling violations and the comparably large experimental errors at large x. For a review see Ref. [St95], for a compilation of recent data Ref. [PDG00]. Combining the results of these experiments in a suitable way, it is possible to determine the parlón distribution Junctions Jijj{x,Q2). Results are available for the valence-quark densities Ju,v{x, Q2) and Jd,v(x,Q2), but also for
3.2
( . ¿ C I )
in
d e e p - i n e l a s t i c
s c a t t e r i n g
'If»!)
(f/cW
Fig. 3.4 The proton structure function F 2 ( x , Q 2 ) for 1 GeV 2 < Q 2 < 30000 GeV 2 and 0.00002 < x < 0.65. The line is a next-to-leading order QCD fit (from Ref. [F199]). For display purposes c(x) = O.G(«(x) — O.-l), i(x) — 1 , . . . , 24 is added to Fj each time the value of i is increased.
antiquark densities and the distribution of strange and charmed quarks in the proton. The gluon distribution Junction fgj>(x, Q2) dominates at small x although it enters the experimental results only in next-to-leading order. As an example, the results on the parton distribution functions from the MRST collaboration [Ma98] are shown in Fig. 3.5. The valence quarks u and d are most important at x > 0.1. The sea quarks contribute at smaller values of x, and the gluon dominates for x < 0.1. The probability of carrying
432
3
Q u a n t u m
G h r o m o d y t i a i n i c s
X
Fig. 3.5 The parton distribution functions f,,p(x, 2), denoted by the parton symbols (u,d, g), for Q a = 20 GeV 2 (from Ref. (Ma98])
a large momentum fraction x decreases for the single parton. Therefore, the parton distribution functions must decrease at large x with increasing 2. Because of momentum conservation they increase for small x. 2xF\ vanishes in the quark The longitudinal structure unction F\, = parton model (cf. Sect. 1.5.3.1). Owing to higher-order corrections it is no longer equal to zero in QCD but is given by
+ 4 E ^ 0 ~ ; ) zhMz>Q2)
the at the one-loop level [A178, G178].
(3.2.106)
3.2
(.¿CI)
in
d e e p - i n e l a s t i c
s c a t t e r i n g
'If»!)
Tim C>(a;*) corrections have been calculated in R.ef. [vN!)l, Zi91, Zi92, MoOO]. Tin- measurements of the ratio R = F\j2xF\ are in agreement with the QCD |.i«diction [Ar97, YaOO].
3,2.4.1
The Q C D scale parameter A
I'hu evolution of the structure functions with QA depends on the scale pauimeter AQCD- R s value in the MS scheme, taking into account four quark llavours and three-loop corrections, is [Sa99] A ^ = 282.7 ± 25.1(stat.) ± 24.5(syst.) MeV,
(3.2.107)
corresponding to a s ( M | ) = 0.1172 ± 0.0017(stat.) ± 0.0017(syst.).
3.2.4.2
(3.2.108)
Sum rules
We have presented the different parton-model sum rules and their tests already in Sect. 1.5. The two-loop QCD corrections have been calculated to most of them and are necessary to find agreement between the predictions and experiment.
3.2.4.3
Polarized deep-inelastic scattering
Results for polarized deep-inelastic electron nucleoli scattering have become available recently. The measured quantity is the nucleoli spin asymmetry"
which depends on the spin-dependent, structure functions g\;i{x,Q2). These are defined by decomposing the hadronic tensor for polarized deep-inelastic 'The first arrow denotes the lepton spin, the second the nucléon spin.
4 5 8
.'5
Q u a n t u m
C h r o m o d y u a m i c s
lepton- liucleon scattering as (we restrict ourselves here to the photon contribution) wv(P>
Vj
i uupoQ
~f2[x,Q2) \
(3.2.110) (
J
where s denotes the spin vector of the nucleoli with the properties sp = 0 and s 2 = — 1. The nucleoli spin is composed of the spins of the valence quarks, the sea quarks, the gluons, and the orbital angular momenta. In the naive parton model only the contribution of the valence quark spin is taken into account. Then, the structure functions g\ depend on the difference of the distribu|
tion functions for quarks with spin up and spin down A = For electromagnetic deep-inelastic scattering this gives 9V"{*) = \
+
|
— fqr\r-
=
(3.2.111)
q
The QCD corrections to this parton-model result can be obtained from 2 evolution equations for A q j s f ( x , 2 ) and jf(x, ) of the form (3.2.88) with the lo est-order polarized splitting functions [A177] if(x)
=
qq(x)
=C
^(x)=TF(2x-l), APff(x) = C F ( 2 - x ) ,
(3.2.112) Taking input data and an ansatz as in (3.2.105), predictions for the 2 dependence of gi(x, 2) can be made (again CF = 4/3, TF 1/2, and C,\ - 3 for QCD). The t o-loop polarized splitting functions are also known [Me96, Vo96]. From the experimental data it turns out. that approximately only one third of the nucleoli spin is carried by the valence quarks [Kr99] (spin crisis [Le88]).
3.3
crl u r l m l i v c
Q u a n t u m
C h r o m o d y n a m i c s
4 6 9
However, because of the axial anomaly (cf. Sect. 2.7), the spin densities of the corresponding constituent and parton quarks do not have to he equal. Hie possible contribution of the gluons to the quark spin function may cure this defect [A197]. The spin structure of the nucleons is also tested by the lor the spin asymmetry [I3j66], 5bj = f
dx [9f{x,
=
jor en sum rule
1 — —^ { 7T / <7v | ^
—
l
-
where g\j and
3.3
erturbative Quantum Chromodynamics
The operator-product expansion is a method suitable for dealing with deepinelastic lcpton nucleoli scattering. However, it cannot be used for many
432
3
Q u a n t u m
G h r o m o d y t i a i n i c s
other hard-scattering processes such as the inclusive production of hadrons, the Droll- Yan process, or jet production in e + e~ annihilation. Therefore, it is important to provide the techniques for a direct application of the QCD perturbation series to these processes. An important property of QCD is factorization of processes into shortdistance and long-distance parts. The short-distance part is a hard partonscattering function, which is calculable in perturbation theory. It does not depend on the external hadrons. It is IR-safe, i.e. it does not involve IR or mass singularities. After renormalization it is UV-finite and depends on the renormalization scale. The long-distance parts, the parton distribution functions and the fragmentation functions, are independent of the hardscattering process, IR-sensitive, and contain the properties of the external hadrons. Both parts are put together at the factorization scale like in the parton model (for a review see e.g. [St95]). The detailed rules needed are derived for deep-inelastic lepton nucleón scattering in Sect. 3.3.1. A prescription for the factorization of hard-scattering processes is formulated in Sect. 3.3.2. This approach is compared with the operator-product expansion in Sect. 3.3.3. As applications, fermion-pair production (Sect. 3.3.4) and jet production (Sect. 3.3.5) are considered.
3.3.1
One-loop corrections to the parton model
First, we perform a perturbative calculation of the QCD corrections of order a s to the parton model for the case of lepton-nucleoli scattering. The relevant diagrams have been shown in Fig. 3.2 on page 436. As can be read oil" from (1.5.30), the calculation of the quark lepton scattering in lowest order («!?) yields
PF{x, Q2) = Gfx5(
1 - x)
with
i =
¿pq
F[>V(x, Q2) = F f ( x , Q2) - 2 x F f v ( x , Q2) = 0
(3.3.1)
for the parity-conserving quark structure functions. Variables referring to partons are indicated by a hat, and G'C = G'„ 1 is the product of coupling constants of the vector boson (1.5.11) for V = V', the case that we are considering here.
3.3
Fig
.
P e r t u r b » live
Feynman diagrams for
The Feynman diagrams V(q) + qi(p) qi(p') + l.ieal with those of E invariant matrix element gluons reads Mti
,
Q u a n t u m
C h r o m o d y n a m i c s
q, -I- g
of Fig. .G for gluon bremsstrahlung off uar s g(k) q2 -Q2, p2 = p12 k2 are idenup to the colour matrix . The corresponding for the vector coupling v f of massless uar s and
= ev^u(p')fh~
+
u(p)eQ(k),
where ea is the polari ation vector of the gluon e2
=
'171
q)2 =
t = ( p - k)2
1, ek =
= -
1
- C
. .2
, and
T ,
2
u = (q - k)
cos
,
s
t
u = -Q2.
. .
ur interest lies in the unpolari ed, parity-conserving part of the cross section summed over gluon colours and integrated over the gluon momentum. sing the polari ation sum in Feynman gauge pol ea£ = the following expression results for the invariant matrix element s uared
2 spins
colour
ss2 + I'M
+
p - lb"?
si
+ 7"(if - PHJ'-YaW
+ 4H
. .4
'172
3
Q u a n t u m
C h r o m o d y n a m i c s
We regularize possible IR divergences by calculating in D = 4 + 2c dimensions. Then, we have to use the D-diincnsional two-particle massless phase space, dif =
+ ,
q
-
^
Y
^
^
i
f
t
,
and the relation between the differential "cross section" and the invariant matrix element squared, 1i
da^t*
p' w
/p' / y 22 sin s i n2 *i) a y
1i
',a/r(l + 6) V 47TP2 y '
dQ
^
^
where p and p' are the initial- and final-state three-momenta in Vq CM system, i) the scattering angle, and fi the scale parameter of dimensional rcgularization (cf. Sect. 2.5.2.1). We have the relations * + Q2
,
V»
1 P'
£
In order to obtain the contribution to the structure functions we relate the "total cross section" cr,it, to the hadronic tensor [cf. (1.2.24), (1.5.9) and (3.2.1)], 1 QP -P . 1 Q2 2xî/ e W,u, = —o, lv = —s—a, tl/ = - — o , t „ ,
71
¿7T p
/Q o (3.3.7)
7T ¿X
and project out the structure functions like in (3.2.8), \ h =
x
(-9,IV
1 +é \
+ 2 x 7 - ^ ( 3 + 2e)) W ^ ,
pq
J
K = 4x 2 ^ - W , w . pq
(3.3.8)
The contractions of (3.3.4) give, adding the contribution of the axial coupling with (1.5.11) g^M"" P^uM'1"
2
=-4e Gf = -4c2Gfg2Cv(
g*Cv(l+e) 1 + e)
\S
t
2Q2ÏI
st
T
+ T +e s
(S 4 - 1 ) 2
st
Û 2
(3.3.9)
.13
P e r t u r h a t i v c
Q u a n t u m
C h r o m o d y n a m i c s
'173
The D-dimensional integral for the total cross section reads (j/ 2 = (1 x)Q2/
n)
,i>»
>=
2 \ x_ J _ / ( I - i ) Ql2\< 2 2 — 647r 3Q2 rr(i ( I + C ) V IOTT,t2x J
x 2tt [
dt?(sintf) 1+2t J
(3.3.10)
Jo
Rewriting this integral in terms of the momentum transfer I. = —Q2( 1 cos ?9)/2x instead of the scattering angle gives Q2) =
r ( l + E) ( i ^
(QZ)
x J
dt(-ty
(l +
2
) M^(t).
(3.3.11)
From the ¿-pole terms of M,iv(i), (3.3.9), the mass singularity, i.e. logarithmically divergent contributions to the quark structure functions, emerges. Since t. = —2EqEg(l - cos??), small I corresponds either to ER « 0, i.e. radiation of soft gluons, or to 0 ~ 0, i.e. collinear radiation of gluons. Contracting (3.3.11) with g'"', performing the integral, and expressing everything by the scaling variable x leads to _rdVq-u,H, 9
Q2V =
4naGqV
( (l-£)Q2V
C
v
'
2
4ttp .t 2 _. , 1 + X 3 1 „ . 7 1 r(e)— r + 3 - x - -e-. 2 l-x l-x 21 — x
=
2e) J r ( l + 2e
+ 0(e).
Similarly the contraction with p^p" yields l'Yàlïï™
T(l+e)
+
(3.3.12)
O(c).
(3.3.13) Using (3.3.7) and (3.3.8) we obtain for the gluon-bremsstrahlung contribution to the quark structure functions 111 f 2
'
n M
< r I fi. )real
1 —x
"
2n
/II
\
— :i.
47rp2i
21 - x
F ^ M ( x , Q 2 ) r c a I = G f ^ C r f x 2 4- 0(e).
^
J T(1 + 2c) 2 1 - x
+ 0(e), (3.3.14)
4 7 4
3
Q u a n t u m
C l i r o m o d y n a m i c s
where we have omitted terms that give a vanishing contribution for t. -» 0. This "real" part of !''•)'' is Q2-dependent and IR-singular. For the complete expansion of (3.3.14) in e, the following identity for distributions is useful: x~c{l
- x) - l + < =_ -¿(1 ~.t) + 1 - ® )
/ l o ^ ) \
+
\
1— ®
+
_JoE[x1 /
+ 0{(2)
(3315)
1— x
The contribution of real-gluon emission graplis finally reads
2 r l e £^9,(1)/n \ riv z(J?LY ( + F (x, Q ) _ Gl -C nx j 2
rcal
v
2c)
+
3 /
1 \
7
2
..
1+i ,
+
"Ur^J f 2 ~
t^J
. , g x+2x+3
(3.3.16)
The contribution of the internal radiative corrections depicted in Fig. 3.2d on page 436 is also IR-singular but proportional to tf((p-W/)2) = S(]—x)/2pq. The detailed calculation proceeds like in Sect. 2.8.3.2 and shows indeed that the singularities at x = 1 of the real and virtual corrections cancel. The total result for the corrections in order « s for the quark structure /miction in the dimensional regularaation scheme reads [A 179, Hu81]
1) 4(1 - *> - T=X
_ /-yqV Qs
-G{:
+
r
,
2
-OPx
1 + X-2 1
- \ l + x
-X
log
1- x X
f l
+ Q2
¡
3 ~ 4,
9 + 5:c +
Q2
" 2tt
\
(3.3.17)
3.3
l'eri, u r i n i t i v e
Q u a n t u m
C h r o m o d y n a m i c s
4 7 5
with the finite, scheme-dependent coefficient function <1>.1 ''(x) and the splittin;) function [cf. (3.2.90)] P^(rc)
(3.3.18)
= c\
l'ho quark structure functions are not directly measurable quantities reflecting the fact that the quarks are not free particles but confined in the hadrons. On this level we cannot expect cancellation of the collinear sin gularities. Therefore in one-loop order, Q2) still contains the collinear singularity P$(x)(l/f. + 7 + log(Q 2 //' 2 )). Moreover, it is not expressed by ordinary functions but by (4- ^distributions which were defined in (2.8.81). A second reaction mechanism in C(a s ) is quark-autiquark production in gluon vector-boson scattering, V + g —> q, + q%• The calculation gives tf «">(*. Q2)
- 2 E Gf i g k > ( x ) ( i + 1 V (I
+ P®Hx) log ^
7
+ log *
)
- Tp(6x2 - 6x)
=2 E
(7 + -r + log
+ (3.3.19)
for the gluon structure function in the dimensional régularisation scheiuo [A179, Hu81] with the splitting function [cf. (3.2.104)] P$(z)
= TF(z2 +
(l-z)2).
(3.3.20)
and the coefficient function ^¿^(i) [I3a78]. The QCD corrections to the quark and gluon structure functions are flavour-independent at one-loop order. The physical process is inclusive lepton hadron scattering. The hadron structure functions are built up from the parton structure functions and the parton distribution functions. The singularities of both have to compensate each other in order to give finite and unique hadron structure functions. The mechanism of this compensation is discussed in the next section.
432
3
Q u a n t u m
3.3.2
G h r o m o d y t i a i n i c s
Factorization
In order to obtain the nucleón structure function F^ (x,Q¿) we generalize the parton-model procedure (1.5.31). There, the parlón structure function (3.3.1) F 2 V '' (0) (i) = GlVxS{ 1 - £ ) is multiplied with the parton distribution function f,)^f(z), the probability of finding the corresponding parton with four-momentum p = zp in the nucleón. The condition x — Q2/2pq = Q¿/2zpq = x/z is imposed with help of the ¿-function 8(x — xz). Finally, the resulting expression is integrated over x and z. This factorization into a short-distance and a long-distance part was proven, for instance, for deep-inelastic scattering and the Drell-Yan process [Co89j and is the basis of perturbative QCD. The first factor can be calculated in perturbation theory, whereas the second one has to be taken from experiment. In perturbative QCD, like in QED [cf. (2.8.115)], factorization is a multiplicative convolution,
-FX*\x,
Q2) = I 'dz I Xd¿ 6{x - xz) Jo
X x
Jo
fe\F^{x,Q2)[fqM{z)
+ J^Z))
+
+
jF^(x,Q2)fgM(z)
® / g X I (x, Q 2 ),
(3.3.21)
L 1 where the sum runs over all quark flavours, and ^¿(x) = Fi{x)lx. An important property of factorization is that the hard-scattering parts, the parton-scattering cross sections - in deep-inelastic scattering the parton structure functions parton!structure functionF^ '(x,QJ)—depend only on the interacting vector boson V and the parton i but not on the external hadrons. They are calculated in form of loop corrections to the Born approximation and depend on the strong coupling constant i.e. a s , the renormalization scale A4, and the hard-scattering variable Q'. They arc IR-safe, i.e. free of soft-gluon singularities. On the other hand, the long-distance parts, the parton distribution functions fij\f(z)i depend on the external hadron and the interacting parton but not on the specific interacting vector boson, i.e. they are universal. They are
3.3
P e r t u r b a t i v o
Q u a n t u m
C h r o i n o d y n a i n i c s
177
independent of the hard-scattering variable Q¿. Since they contain the longdistance cffects they cannot be calculated in perturbation theory and contain .ill the 111 sensitivity of the original cross section. The separation of the cross section (3.3.21) into a long-distance part and a ühort-distance part is not unique. It is possible to shift finite contributions from the long-distance to the short-distance part and vice versa. Therefore, the separation requires the definition of a soft hard scale parameter or factorization scale Msu relative to which soft and hard is defined and a prescription for the distribution of the finite parts. These two points have to be fixed by a factorization scheme. In order to make this more explicit, we rewrite the 0 ( a s ) expression for / f M ( x , Q 2 ) as
(3.3.22) where fj^-(z) are bare parton distribution functions. The mass-singular term oc l/e has to be, the other parts may be absorbed into fj°^(z) leaving finite but scheme-dependent, "renormalized" parton structure and parton distribution functions. The absorption of the mass-singular term can be considered as a kind of renormalization of the parton distribution functions. In this way the measured nucleón structure function leads to scheme-dependent parton distribution functions. Therefore, all manipulations have to be performed in the same scheme. The following factorization schemes arc mainly used in the applications of perturbative QCD. In the MS scheme, the leading-logarithmic term is shifted to the parton distribution function. The hadron structure function of the MS scheme is
432
3
Q u a n t u m
G h r o m o d y t i a i n i c s
obtained as convolution of these par ton structure functions with parton distribution functions / ¿ ^ just like in (3.3.21),
J«
2ir
= E1
® ( « + /») + Km • /»•
(3.3.23)
The MS parton distribution functions obey the equation f ^ ^ l n )
= /$(*) + g
( j + 7 + log
^¡H
^
(3.3.24)
and similarly for fj^-{z,Q ). From (3.3.23) it is possible to determine the parton distribution functions | H ) experimentally. QCD then pre2 dicts the Q dependence of the nucleoli structure functions in C9(cvs). Often the soft-hard scale parameter .Msn is identified with the renormalization scale parameter M , i.e. .Msn = M . In the DIS scheme, all correc tions to F^q(x, Q2) are absorbed into the quark distribution functions at Q2 = A1gH, Q2) = £
Gf
(/$?(s,
Q'2) + f $ ( x , Q2))
Q2
+ ST nnV <*s ,
C (f) +
f]
(iz
+
(I) &(*><&
(3.3.25)
.
with the
47 )
I parton distribution functions
f
,
R I
fil
ln
and similarly for f
at Q2 =
s
F
- f sf
dy
C
. .2«
y. The relation between the two schemes reads ,
F 2 r
it, 2
,
, i
, ,f,i g - f m 2 r V
. .2
and analogously for the anti uar distribution function.
The hadron structure functions Fi{x,Q'2) and F3(x,Q2) including the finite parts can be treated similarly. n , E1 . The parton distribution functions arc determined from experiment using . .21 . ince the parton structure functions have to be fixed within a factori ation scheme, the parton distribution functions are factori ationscheme-dependent. . . . . .1
F a c t o r i a t i o n and t h e o p e r a t o r - p r o d u c t e x p a n s i o n oments of the nucleoli structure function
The multiplicative convolution . .21 of the uar structure functions with the uar distribution functions can be split up into factors by a ellin transformation. This yields for the moments of the structure functions
T d««- E tf
C
C)
&]
«2 (
.
.28)
4 8 0
s
e considered in Sect. 3.2.2 the P for the leading-logarithmic (LL) approximation of the moments of the structure function. Therefore, we take the corresponding approximation of the one-loop parton structure functions cf. (3.3.17) and (3.3.1 )
SijS( 1 - x)
(3.3.2 )
An explicit expression for the moments can be calculated from (3.3.2 ) by using the connection (3.2. 5) between splitting function -j\x) and anomalous dimensions 7 ^. Factorizing at .Ms 11 as described in Sect. 3.3.2 yields
i?vAf 2,
,
" r« v Sij - C*S(M) lo S ;; 7n,0 S
2
fjMn(MsH)-
(3.3.30)
3.3.3.2
Summation of leading higher-order terms
The order-as calculations already reproduce some of the resvdts previously obtained using operator-product expansion and renormalization-group formalism, summarized by the evolution equations. orking in higher orders of a s , quite a lot of Feynman diagrams contribute. Thus, the interesting question is which diagrams dominate in the Bjorken limit. An answer was given by . . ribov and L. . Lipatov r72 for abelian vectorgluon theories, and later extended to non-abelian gauge theories like C L17 , Am7 , Fo 7 . In an appropriate gauge, generalized ladder diagrams with renormalized propagators and vertices give the leading-logarithmic contributions l nitarity relates the hadronic tensor ,n, to the forward Comp12
e present only a few details for the photonic case; for further information we refer to the quoted literature.
3 . 3
( ' ( » t u r b a t i v e
Q u a n t u m
-i 17
C l i r o m o d y n a m i c s
181
7 ro
l
Pj-i
Vi
Pi
Kig. 3.7 Ladder diagram for photon-quark scattering
ton scattering amplitude of virtual photons off quarks,
WV =
£ / p (3.3.31)
where the dashed line indicates cutting, i.e. the corresponding propagators are taken oil-shell. In formulas, the relation reads W^ixyQ2)
= ¿Im
{Tlt„(x,Q2)}.
(3.3.32)
The ladder-like gluon-exchange diagrams with bare propagators and vertices (an example is shown in Fig. 3.7) can be investigated by iteration. The relevant quantity is the square of the amplitude T for forward-scattering of virtual photons off quarks. The change in the corresponding expression occurring when enlarging the ladder from j rungs to j+l rungs by the emission of an additional gluon can be calculated in a similar way as in Sect. 3.3.1 for the emission of one gluon. In the leading-logarithmic approximation only the terms with the ¿-channel pole are relevant. A similar calculation has been performed in Sect. 2.8.3.1. One obtains the following recursion formula for
4 8 2
.
s
the square of the amplitude T for forward-scattering of virtual photons oil quarks: l
= f
J-Q1
f
IIT
tj Jo
^-P^^m^q). Zj
(3.3.3 )
I hire, for simplicity we regularize the mass singularities by a cut-off parameter o- ressed vertices and propagators are included by taking the running The variables are defined coupling constant as(tj) = 4 T T / ( / ? O log (-ij/A2)). as b =
j ,
(3.3.34)
j = Zj j+l
and obey -Q2 = q2 < t,
t2
...
(3.3.35)
tj < -Ql
The I singularities of the real and virtual corrections compensate each other. Inserting the explicit expression for the running coupling constant yields =
[ J-Q2
—j^ fio
f ^ - P ^ t j i p y q ) . Jo J
(3.3.36) This is the desired iteration formula for the dominant terms. wing to the relations (3.3.32) and (3.2. ) it also holds for the corresponding structure functions. The term of zerotli order is the contribution of the naive parton model, (3.3.1) . sing this expression as starting point for P io = v the iteration, an explicit representation of F^ can be achieved,
r Q o dti -
x
2
1
I log
r
di2
I
h log
(3.3.37)
3.3
P e r t u r b a t i v e
}
2piq
Q u a n t u m
C ' h r o m o d y n a m i c s
47!)
with
Q2 2;)<7
:/: =
Q'2 2pjq
—
ZjZj-1
. . . Zj.
(3.3.38)
rinv i integrals can be evaluated in the following way: Q
r ~^ do d
fr - Q o ( \ t 2
I1
h log ^
•/(, J'U
-<5odi,
1
1
¿2 log
^
log log ^
- log log
2
<1 l o g ^ 21 1 Q2 , , Qo , , Q2 i = o l ^ l 0 g | f - l o g log l o g l o g p - + - log log
log(Qg/A 2 ) log log(i? 2 /A 2 )J • The result of the j'-fold integration is 1
I
log(Q 2 /A 2 )
log
iog(Q;i/A 2 )j '
The z integrals again can be decoupled by considering Mellin moments,
^UQ')
= f d Jo =
xsr-*F]fj(x,Q*
Gv f jl
- r''v—
d NS.
log(Q 2 /A 2 )l J l°g(Qo/A 2 )J '
(3.3.39)
where (3.2.89) and = —7„,o/2/io have been inserted. Summation over the number of rungs finally leads to
2
EKln(Q ) j=0
=
Gf
log(Q 2 /A 2 ) l°g(Qo/A 2 ) J
ti,0
(3.3.40)
4 2
s
for the ladder diagrams. This expression can be factorized into the mass singularity and a regular part. The LL result for the moments of the nonsinglet nueleon structure function follows with (3.3.2 ),
(3.3.41)
sing (3.1.5), it can be seen that this result is the same as the one obtained with the equation and the ilson expansion in (3.2.66). It consists of the products of the moments of the quark structure functions and the moments of the quark distribution functions. Consequently, from the summation of the leading-logarithmic terms of generalized ladder diagrams using the running strong coupling constant the results of the P arise for deep-inelastic lepton nueleon scattering. Let us summarize the most important results. The dominant C process is the collinear radiation of hard gluons. The associated mass singularities factorize. Formally, this corresponds in (3.3.41) to the replacement of the bare ilson coefficients C = 2, occurring also in (3.2.42), by the renormalized, mass-singularity-free ones C'n(as) from (3.2.64). This picture of hard C processes can be extended to cover other reactions and higher-order calculations.
3.3.4
Lepton-pair production in h a d r o n - h a d r o n scattering
The ideas and methods presented in the preceding section for deep-inelastic lepton nucleoli scattering are the starting point for applying C to other hard-scattering processes. Perturbative C justifies the universality of the parton model and permits calculations of radiative corrections for each particular process o 0 . The rules for dealing with hard-scattering processes in higher orders of C can be formulated as follows. Based on factorization and the parton model, the cross section is composed of the parton-model cross sections b for parton lepton or parton- parton scattering, convoluted with universal 2 -dependent quark distribution functions fq,h(x, 2), antiquark distribution functions fq t h{ x i 2) gl ion distribution functions
.
l'eri
s
Fig. 3.
T h e parton-niodel diagram for the
/g h(x,
), or parton fragmentation
rell
48
an process
functions
D)t tl(z,
2
2
),
), and
This prescription is successfully applied to lepton-pair and jet production in hadron hadron scattering, jet production and inclusive hadron production in e + e ~ annihilation and lepton hadron scattering, and others. For illustration we present some typical examples: the Droll Yan process iu Sect. 3.3.4.1, jet production in Sect. 3.3.5, and the total cross section in e + e ~ annihilation in Sect. 3.3.G.
3.3.4.1
QCD corrections to the Drell-Yan cross section
The first example is lepton-pair ( e + e ~ , / i + / i ~ , e t c . ) production in hadron hadron scattering, the so-called Drell-Yan process (cf. Fig. 3.8) [Dr70, Po77, Sa78] hi (Pi) + h2{p2) ->
+ X -> Z, (h) + h( 2)
The parton momenta are x\p\ delstam variables s = (pi + p2)2 = 2
= { i +
2)
2
+ X,
and x2p2 with p2 = p2
= 7 , Z, W. 0. With the Man-
2p\p2,
= x\pi + Xip2]2 = xix22pip2
= x\x2 s,
(3.3.42)
the parton-model cross section reads
x
Squhii^fq^x-i)
+ fq2,iu(x\)fqt,h2{x2)].
(3.3.43)
4 8 6
.'5
Q u a n t u m
C h r ó m o d y n a m i c s
In the electromagnetic case, qq —> 7* (q\ = 72), we have
d£jí(¿iPi,Í2P2)
47ra2
- -
x
f><>A^
yielding
dQ2
_
=
E<1
I <1*11 ^ ' « ~ *«*•)
2) + /i,/ M (®i )/ 9 ,/« 2 ( £ 2)],
x UgM
(3.3.45)
where the factor l/Afc = NC/NQ results from the colour factor /Ve and the average over the colours in the initial state. The Z and W bosons were discovered in P P scattering. In the Z-boson case, the quark cross section is diagonal in the quark flavours and reads dtfri{xlPuX2P2)
_ JTTO2
2 2 2 (vf + aj)(vt + iM " 3NcQ2" 1
(1Q2
a¿g)5(Q¿-XiX2s)
(Q2)2
(Q2 - M2)2 + r|M|
(3.3.46)
with the couplings (1.4.38) and the total width Fz given in (1.4.37) and (1.4.39). A more detailed discussion has to include the 7-Z-interference terms. The cross section for W production is built up in a similar way, 47m 2 j V ^ j 2
d^'(xlPux2P2) d^
=
3 A ^ l 6 4 T
í ( Q
_ _ "
X,X25)
{Q2 ~ M§,)2 + TI/M2,'
[Ó ÓA{)
'
with the total width I \ v from (1.4.40) and (1.4.41). For hi and h-¿ being nucleons, the quark distribution functions measured in deep-inelastic lepton nucleón scattering can be used to make absolute, predictions [G178] based on these parton-model formulas. However, these are by a factor K — 1.5-2 lower than the experimental results [CoS2, Be85, A187, An87]. This difference is due to the higher-order corrections, which are important since a s is not small.
.
s
1 1
already mentioned in Sect. 3 . 3 . 2 , factorization was proven to be valid for tin; Drell Yan process (cf. Ref. [Co89] and references therein). The hardKc.ittering part is q + q —> V, the long-distance part is hidden in the quark distribution functions f q j , and /*/,. The virtual one-loop corrections to q I q > V are vertex and self-energy diagrams, the real ones are due to qlnon brcmsstrahlung, q + q -> V + g. The calculation of these contributions is closely connected to those for deep-inelastic scattering, the difference is that the momentum of the virtual boson V is time-like for lepton-pair production, whereas it is space-like in deep-inelastic scattering. In one-loop order, there is also a contribution from the gluons in the hadrons, namely the processes <1 -f g —» q + V and q + g —» q + V. Therefore, also the gluon distribution function f R i h enters the result. AH
As in deep-inelastic scattering, dimensional regularization with D = 4 2< is used to handle the collinear singularities. Then, the parton distribution functions of the corresponding results for deep-inelastic scattering can be used. With z = Q 2 /X\X2S, the one-loop result for the hard part of the electromagnetic production reads [A179, Hti81] ita2 d Q2
.2as 1
3NCQ
do^\z,Cj2)
2
' 7r .s
2
WCQ2
dip
f*s 1 2tt S (3.3.48)
+ nv ,K(z)
where and are given again by (3.2.104). In the MS scheme, the finite, scheme-dependent parts read
, = CV
1-z
- 2(1 + z) log(l - z) 2
= Tv ( 1 - 2 z + 2z ) log
(1-z)2
lo
-z
+
J
S2 +
2z- z 2 (
.
.00)
'188
3
Q u a n t u m
C h r o m o d y
m u n i e s
These equations together with the corresponding parton distribution functions provide one-loop-improved absolute predictions for the lepton-pairproduction cross section. It is important to use the same scheme for the parton cross sections and for the parton distribution functions, e.g. in both cases the MS or the DIS scheme. The one-loop corrections are not small and therefore essential for a good agreement with the experimental results. This is even more the case in the DIS scheme which is related to the MS scheme by
î'dyU-W =
=
-
(3-3.50)
with 0>2<x(z) from (3.3.17) and similarly for 'î'ny I'1 this scheme one obtains very large vertex corrections, i.e. a large factor 1 -I- 27TO:SCF/3 multiplying the ¿(1 - z) function. For a typical value of c*s « 0.3, this gives a correction factor of « 1.8. It. was proposed to cure the effect of these large vertex corrections by exponentiation of the soft-photon contribution [Pa80, Cu80a]. The remaining correction then is small. Therefore exponentiation of the soft-radiation contribution is essential in the DIS scheme. Full two-loop calculations for deep-inelastic scattering and lepton-pair production exist (cf. [Ma90, Ha91, vN92] and references therein). They are smaller than the one-loop corrections and lead to a very good agreement with the experimental results. This is important for W and Z production and the determination of the scale parameter A of QCD.
3.3.5
J e t cross sections
3.3.5.1
e + e ~ —> j e t s in t h e p a r t o n m o d e l
Production of jet pairs in e + e _ annihilation [E176, De78, Wu84, Kr84] was discussed in the framework of the naïve parton model in Sect. 1.5.4. The basic process is the production of a quark antiquark pair, e + e - —> (7*, Z) —> (¡(j. Taking into account, for simplicity only the annihilation via an off-shell photon 7*, we get the parton-model cross section [cf. (1.2.45)]
=
(3.3.51)
.
s
18
hihI the total cross section [cf. (1.2.46)]
(3.3.52)
3s
Kadi primary quark hadronizes completely into a bunch of hadrons, a jet.
3.3.5.2
Jets in QCD
In QCD, this simple picture has to be extended and refined to include besides quark jets also gluon jets, radiative corrections, details of hadronization, quark decays, etc. Moreover, a jet definition is needed which leads to lR-safe observables and allows for factorization into hard, perturbatively treatable parts and soft parts including, in particular, hadronization. In lowest order (c*j?) of QCD, i.e. in the par ton model, two-jet production results from the quark vector-boson vertex * > qq. In order a s , besides one-loop virtual QCD corrections to the quark-vector-boson vertex, real three-jet events occur. The primary process in this case is gluon bremsstralilunq * > qqg followed by hadronization, leading to three jets if the primary quarks and gluon have high energies. We discuss the primary process of the final state, the decay of the virtual vector boson into a quark pair and a gluon. The matrix element of this process is given by the same Feynman diagrams as gluon bremsstrahlung in deep-inelastic scattering or the Drell Yan process. However, the virtual s > 0), and the kinematics vector boson is time-like in the jet case (q 2 > q{pq)q{pq)g(^), where the momenta are is that of a decay process, *(q) indicated in parentheses. Using the
andelstam
«1 = ( •S2 = q ~
~
variables
= (* + rf a
= ( + pq
= 5 - 2 ~s = s - 2 fi
q
= 2
, = 2
•S3 = q ~ j2 = (pq + pq)2 = s - 2 V s £ g = 2 s = SI + S 2 + .S3,
0 < .SI I2 ,3 < S,
si + s2
g q{\ g
q(
q q(\
S,
- cos 0 gfJ ), 1 - cos - cos
q), qq),
(3.3.53)
490
.'I Q u a n t u m Cliromoclynainics
one: obtains, alter performing the angular integrations of the three-particle phase space, the differential gluon-bremsstrahlung cross section 2 2 . j d g ( e + c " ->• qqg) 4TTa2 2 a s C p (.s - .sQ + (a - .s 2 ) x ^ s :—: = — — N c > Qn 7 T dsid.59 3.s ' ' 2ir s\S-> q
- -¿jet/^ «s r (* - ai) 2 + {s- < M ^C* —
S2)2 .
,, „ r (3.3.o4)
We omit the contribution of the Z boson for simplicity. For .si = 0 or .s2 = 0, III divergences occur. They correspond, ¿us can be seen from (3.3.53), to soft and collinear gluon emission and can be regularized diinensionally. Because of the III singularity, the differential cross section for e+e" —> q(/g rises steeply for s i ) 2 —> 0, becoming eventually bigger than the one of the basic parton process e + e _ —> qq. This region of phase space corresponds to soft and collinear gluons which do not hadronize into observable jets. Therefore, that part of the cross section does not lead to three-jet events. A cut-ofT in phase space is needed to define a finite, measurable jet cross section. This is free of singularities but cut-off dependent. Under the assumption that hadronization is a soft process one identifies the cut-off in the parton cross section with a corresponding cut-off for the hadrons which form the jets. From the several existing definitions of jet observables we discuss here only two: the JADE algorithm [Be88, Be92] and the Sterman-Weinberg approach [St77].
3.3.5.3
The J A D E algorithm
In (.lie JADE
yij
algorithm
one defines variables
OFF= — ^ ( 1 " cos Bij),
(3.3.55)
where E{ are the energies of the observed hadrons and Ojj the angles between the pairs (ij). Since pairs with small yij have either small energies or are almost collinear, the hadrons are grouped into jets according to the following iterative procedure: the pair of hadrons with the smallest y t j is combined into one object with four-momentum p1' — pf + Pj- Then a new list of y's is calculated and the combining procedure of pairs of ob jects is iterated until every remaining y is bigger than a cut-off value yc > 0. Thus, the hadronic
.
s
1 1
final state is decomposed into jets depending on ;/c. This procedure is IR-safe since soft and col linear hadrons are combined directly into jets. Introducing the cut-off 1,2 . — > yc, s
c
in the parton phase space by
S - Si s
s2
2p,,pq = — — > ?/c, s
1 rt, 3
c
3.3.50
yields the total cross section for e + e ~ —> c/c/g [Be92] ^(«,|fc) = cr8*(3)gcF rs-zy rs-2yces
(3.3.57)
rs-si-ycs rs-S]
7JyJ IcS s
J JCS rS
^
_
S j
)2
(
+
s
_
s>2
ds 2
c
+ | ( 1 - 2»c)log ( j T ^ )
s s
\2
+ < 1 " 3 »c) ( j + jVc)^
where Li2(z) is the dilogarithm (2.5.77). For not too small values of i/ c , this defines an acceptable three-jet cross section. The .JADE algorithm has the tendency to create spurious jets. This is avoided by the so-called Durham algorithm [Ca91] where instead of ,j, (3.3.55), the variable r
y!j
_ 2min( f, V = 3'
])
J
(1 - coefly)
(3.3.58)
is used. The two-jet cross section to order a s is obtained by adding the virtual corrections and the gluon-brcmsstrahlung cross section with the cuts «1,2,3 < -V/c to the parton-model result r2jct(s,Z c) =
(l +
- a3*l(s, c).
(3.3.59)
The first term is taken from the result for the total cross section of the annihilation into hadrons [cf. (3.3.07)], which is calculated in Sect. 3.3.0, cr(e + e
—» hadrons) = a p m ( e + e
—> hadrons)
4-
^-CV^ . (
.
.00)
4
2
.
s
ALEPH prelim (189Ge Pythia 5.7 I lerwig 5.9
A • o T
n=3 n=4 n=5 IliO
Fig. 3.9 y c d e p e n d e n c e of t h e classification into multijet ( n = 2, 3, 4, 5, > 6) e v e n t s (from Ref. [AL99])
In lowest order, the ratio of three-jet to two-jet. events is given by a s . Therefore, it allows dependent on the renormalization scheme and the soft-hard scale—for the determination of Aj^g. In a similar way, multijet cross sections, for example in order a 2 four-jet cross sections resulting from qqqq and f/cygg final states, have been calculated [AI79a, A180, Kr84]. The four-jet cross section depends directly on the triplegluon vertex. Therefore, it allows to test the non-abelian character of QCD. The experimental results clearly rule out an abelian gauge theory of the strong interaction. A QCD analysis of the c dependence of the classification of the events into multijets by the ALEPH collaboration at 189 GeV shows good agreement between experiment and theory as described by the Monte-Carlo generators PYTHIA and HERWIG (Fig. 3.9).
3.3
I V r t u r h a t i v e
Q u a n t u m
C h r o m o d y n a n lies
4 9 3
I'ig. 3.10 Stcrman-VVeinberg definition of a jet
At large values of the cut-off yc most events arc interpreted as two broad jets, for smaller yc these split, into narrower multijet topologies. For yc < 0.005 t he multijet events become as frequent as the two-jet, ones, indicating a breakdown of this classification. A more detailed analysis compares the shapes of the jets with the QCD predictions. For this purpose, IR-safe, calculable shape variables such as thrust, sphericity, moments, etc. have been defined. For details we refer to [E196]. ¡1.3.5.4
T h e S t e r m a n - W e i n b e r g approach
Following the Bloch Nordsieck procedure, in the Sterman Weinberg approach jet cross sections are defined as inclusive cross sections like in QED (cf. Sects. 2.8.2.3 and Sect. 2.8.3.9), where the bremsstrahlung process is added to the virtual corrections in order to obtain IR-finite, masssingularity-free quantities. The final-state hadrons are considered as soft if their energy El is small compared with the total energy y/s, e = Ei/y/s -C 1. All particles that lie in a cone with half-angle aperture <5 < 1 are collected into a jet (see Fig. 3.10). The jet cross section then depends on these cuts e and S. Including the QCD corrections the Sterman -Weinberg two-jet cross section [St77] reads firT
phys _
¿a
~
drTpj'n
<m
1
~
(4
log 6 log 2e + 3 log 6
(3.3.61) The Sterman Weinberg approach is based on the simple picture of cones but not suited to analyse details of multijet events. Therefore, nowadays the
4 2
3
Quantum
hromody ai
cs
(modified) JADE algorithm is usually used in the experimental analysis and in theoretical calculations.
3.3.5.5
Parton showers and hadronization
Up to now, we have considered only the primary steps in the formation of jets, namely the production of a quark antiqUark pair including the effects of gluon bremsstrahlung and virtual radiative corrections. A more realistic approach has to consider the successive production of parton sho ers and, after the soft energy scale is reached, hadronization. The full calculation of parton cascades is, although it is a hard-scattering process, too complicated to be performed analytically. But it is possible to treat the most important contributions: collinear emission and soft radiation. The collinear emission is formulated with evolution equations with kernels like in deep-inelastic scattering. Multiple soft-gluon radiation leads, like in QED, to exponentiation, yielding the so-called Sudakov form factors. Hadronization is a soft process and is described by models such as independent fragmentation, string models, and cluster models. All these calculations are implemented in computer simulations, i.e. Monte Carlo programs, finally producing QCD event generators. For details we refer to the literature [E196].
3.3.6
Total hadronic cross section
The parton-model cross section for the annihilation of electron positron pairs into hadrons was discussed in Sect. 1.5.4.1. The basic mechanism is annihilation into a virtual photon or Z boson followed by the production of quark antiquark pairs. These pairs radiate gluons and further quark pairs, and finally hadronize. Here, we are mainly interested in the QCD effects and therefore treat only the photonic contribution to the cross section. It is similar to the point-like cross section of rr(e+e~ —> Therefore, the ratio (1.5.46), r r i e ' e - —> hadrons)
. -
,W->>.+/•
)'
« • -
E
*».
(:ur,2
is given in lowest order, i.e. in the parton model with massless quarks, by
>
.
r
l or urbative
u a n t u m Clii
iiodynainics
1 .r>
e' '
I *|. .J. 11 F e y n m a n diagrams for t h e o n e - l o o p corrections to
(s)
winvre only those quarks contribute which can lie produced at the energy considered. The QCD calculation of (s) can proceed in two ways. In the first way, the jet cross sections are integrated to obtain the total hadronic cross section. The second way is based on the fact that the photonic cross section is determined by the matrix elements of the electromagnetic current operator between the vacuum and hadron states (cf. Sect. 1.2.1.9). Therefore, it is related to the hadronic tensor i . e . the vacuum expectation value of the current- current commutator, =
Id4xe^{0|^m(x),ir(0)]|0)
(3.3.G'l)
where q2 = s now is time-like, in contrast to deep-inelastic scattering. The tensor can be treated like , with renormalization-group techniques. The diagrams for the QCD corrections to (s) in order cvs are shown in Fig. 3.11. The virtual corrections are UV-finite after renormalization but 111and mass-divergent. The total cross section is the most inclusive cross section and therefore, according to the theorem, free of mass divergences. 13 Indeed, adding the real gluon bremsstrahlung contribution, the result is infinite. The calculation of these contributions is the same as in QED; only a colour factor Cp has to be added. The detailed calculations for both cases can be found in Rcf. [Fi89]. The results in the dimensional régularisation scheme read
T h e r e are no Q C D corrections to t h e initial s t a t e .
'196
3
Q u a n t u m
C h r o i n o d y n a m i c s
/ Rptti
vr
v
y
' A W
i r ( l + e)
L
_A
?
~
^
1£ _ üí1 +
t
~
+
0(e).
y
(3.3.65) The sum of these contribution is finite and very simple, SRW
3a,
=
Rpm
(3.3.66)
47T
The application of the renormalization-group equation leads to the replacement of cvs by the running coupling o s . The reason is that the anomalous dimensions of the conserved electromagnetic current vanish. Therefore, the final result, including the first-order leading corrections, reads [Ap73, Ze73]
ncJ2Q*
R(s) =
SU(3) ,
1+
^ C K + 47T
1 +
<*s(s) 7T
o(c4)
+otf).
(3.3.67)
The two-loop contributions in the MS scheme have been calculated in Ref. [Ch79],
-
SU(3)
m
[ - F R R + V i -
+ Í - Y
+ C(3)^
1
* * ) ™
CFTFN{
(3.3.68) with £(3) = ]CHLI n [Go91, Su91].
( 1 . 9 8 6 - 0.115JVr) ~ 1-202. Also the three-loop corrections are known
When comparing with experimental results, the threshold regions have to be omitted. The number of quark flavours Nf in d s has to correspond to those quarks that can be pair-produced at the corresponding energy. Under
.
(
T )
)
these conditions good agreement between theory and experiment is found an can be seen from Fig. 1.1 . illi these results we conclude this short survey of perturbative C . ethods for dealing with hard-scattering processes beyond leading order are irmly established. The same applies for the treatment of the effects of even higher orders and of the processes that occur during the hadroni ation of uar and gluon ets. 14 consistent picture of these processes, including higher orders in « s exists and is tested in many details by experiments. ll llie.se show good agreement between experiment and theory. ne possibility to get an impression of the validity of C is to compare the values of tH obtained from different experiments. This is shown in ect. . .
.4
(
T)
The values of the effective uar masses range from several eV for the light uar s to 4 GeV for the bottom uar and 1 4 GeV for the top uar G . The masses of the heavy uar s are large compared with the C scale parameter « 2 eV, and in many cases with the energies of the considered process. Conse uently, in these applications the effects of heavy uar s Q can formally be described in the limit - oo, i.e. in the heavyuar effective theory H ET Is 2, c 4, a , e a . In atomic and molecular physics the properties of bound states of light with heavy particles arc studied. In atomic physics, the nucleus may be replaced in very good approximation by a static source, the Coulomb potential. The effects of the liniteness of the nuclear mass, the Rydberg corrections, are very small. The orn ppenlieimer method in molecular physics is based on the fact that the movement of the nuclei is much slower than the one of the electrons. s a conse uence of the small velocity of the heavy particles, the spin-dependent effects are suppressed since spin interactions are relativistic effects and proportional to the inverse mass. The situation in H ET is very similar. In lowest order, the heavy uar acts li e a static source of the strong interaction with a strength independent of the mass of the heavy uar Therefore, there is no difference between the interactions of different heavy uar s. This leads to an approximate symmetry, the eavy-Jlavour symmetry, with the corresponding 11
HI
n overview of t h e application of .
C
to high-energy s c a t t e r i n g p r o c e s s e s is given in
1
8
s
heavy-flavour group S (i i,f)j,f flavours.
Is
, wliere
,( is the number of heavy
Moreover, because heavy quarks move slowly, a non-relativistic treatment is possible in a lirst approximation. The interaction is independent of the spin orientation of the heavy quark. This spin symmetry r 0, e 0 gets corrections from rclativistic spin-dependent effects which arc proportional to 1 /rii and thus small. Both symmetries may be combined into the flavour associated symmetry group S (27 hf)hf.
3.4.1
T h e Lagrangian of
spin symmetry
with
T
ur starting point of the treatment of heavy quarks is the path-integral representation of C (2.4.16). The T approximation is based on an idea similar to the Foldy Wout uysen expansion of the irac equation. The small components of the irac field of the (slow) heavy quark arc integrated out in the path integral. The resulting non-local effective theory is expanded into inverse powers of the heavy-quark mass i yielding terms of the order 1 /ni ,
1/m. , . . . .
T has been applied to exclusive and inclusive weak decays of heavy flavours. As in all applications of perturbative C , a separation of shortand long-distance parts is necessary. Physical quantities have to be independent of the corresponding scale parameter. This is guaranteed by the renormalization-group equations of C .
3.4.1.1
Path-integral derivation
e write the generating functional of the reen functions of C in a form in which the heavy-quark field (x) with mass and its source r/ are given explicitly, whereas all other fields arc collected into I . with action S and source A,
Z{ri, V, A} = I V[
V[
.VI
I l e a v y - q u a r k
effective
t h e o r y
( M Q E T )
I'M)
where
5{7/,r;} = J
[ Q W P ~ mq)Q
D'1 = d'L - \gsA>1,
+ ftQ - Qi,] ,
A'L = A
a
^ .
(3.4.1)
For the applications to heavy hadrons the momentum p'q of the heavy <|uark is decomposed as p'Q — rriQv'1 -I-k'1, where v,L is the four-velocity of the quark (v° > 0, v 2 = 1), and k,L describes the off-shell part of the momentum of the heavy quark inside the hadron. In order to derive the heavy-quark approximation, the heavy-quark field Q(x) is decomposed into its large (x) and small h'v (x) components using the projectors P,f = ( l ± ^ ) / 2 with ( P ± ) 2 = P ± and P+P~ = 0. In addition, the phase factor of the free movement of the heavy component is separated off h+ = GimQvxp+Q^
h-
= 0imQvxp-Q
(343)
The covariant derivative D'1 is split into a longitudinal and transverse part, D" = v"(vD)
+ D'l,
vltD'l=
0,
{I[)\_J}=
0.
(3.4.3)
Inserting this into (3.4.1) and denoting the sources for the large and small components by p+ and p~, respectively, yields S{p+,p-,p+,¡r}
= J d4x[h+ivDht
+ h f i i p j i - + h~ilp±ht
- h-{2mQ
+ p+h+
+ ivD)h~
- h+p+ +p~K
~
KP~ (3.4.4)
for the action of the heavy quark. The small component h~ of the heavyquark field with the "mass term" 'InIQ is now integrated out using the gaussian fermionic functional-integral formula (2.2.(j2). After switching off the sources, p~ — p~ — 0, and performing the algebraic completion of the quadratic form, this results in Z{p+,
p+, A} = JV[h+] x exp
V[h+] P[A] det(2m
+ Sx + J
c\4x$x\
(3.4.5)
5 0 0
3
Q u a n t u m
C h r o m o d y n a m i c s
with Sh{p+,p+)=
f d" x k f w D h t - h t i p ^ } n .Pxht J L ¿i riQ + ivIJ — le + p+h+ - h + p +
(3.4.6)
Tlie relation between the small and large components is
= 2viq 5 • WtK.. +J ivDn - íe
(3.4.7)
Equations (3.4.5) and (3.4.6) are still exact. However, they contain the fermion determinant and a non-local interaction. The fermion determinant comprises the effects of closed loops of the small components of the heavy-quark fields. As it stands, it is highly divergent. The constant part of the divergent determinant can be absorbed by dividing Z through the determinant of free quark fields det(2mg -I- wd) resulting in det (2 mQ + ivD) det [2mQ + ivd)
=
^
t
/
+
\
gsvA 2mg + ivd,
- » " ( ^ ( ^ s ^ a ) ) -
(3A8)
Since the generators of the colour group SU(3)c are traceless, Tr(yl) = 0, the physical effects are of order I/VIQ. In a first approximation they are usually neglected.
3.4.1.2
Heavy-mass expansion
The non-locality of the interaction has its origin in the fact that we have integrated out structures with an extension of the Compton wavelength 1/M.Q. At tree level, the higher-order terms in an expansion in 1 /TTIQ are obtained by expanding the heavy quark propagator in (3.4.6) into a geometrical series, 1 2m q + ivD — ie
1 2m,q
ivD (2 mq)"'
+
. . . ,
(3.4.9)
.VI
I l e a v y - q u a r k
e f f e c t i v e
t h e o r y
( M Q E T )
I'M)
resulting in the Lagrangian £efr = hfivDht
- J-h+(l[)Lfht
+ ^ThtJf)1vDJp1h+
-I-... . (3.4.10)
Tin! physical content of the leading terras of Cc\\ can be seen by going into Hie rest frame of the quark, v'1 = (1,0), £efr = h+\D°ht
-
+ ....
(3.4.11)
Varying C,.\\ with respect to /ijj" gives the Schrodinger equation for the heavyquark field i d ° h i = --—(V Ztiiq
- \gsA)2h+
- gsA°ht
+ -^-EBh+, ¿VIQ
(3.4.12)
where B = V x A is the chromo-magnetic field and S is the Dirac spin matrix [cf. (A.1.21)]. Equation (3.4.12) is the correct non-relativistic approximation including the interaction of the chromo-magnetic field with the quark spin. lu order l / m q , the 1IQET and the Foldy Wouthuysen transformation are identical. They deviate formally at order 1 /TII'Q because they are based on different expansions of the heavy-quark fields. Taking (3.4.2) and (3.4.7) yields «,) =
(. +
2mQ +
l„D _;•"/>,)
tfw.
(3.4.13)
i.e. a geometric series in 1/zJIQ after expansion. On the other hand, the Foldy-Wouthuysen transformation works with products of unitary operators Q(x) = e-',mQvx
• • eiSa/8mQ
iS 2/Am e
'
Q
iS e
»/2m«)
h+{x).
(3.4.14)
They arc constructed such that the small terms of the Dirac fields decouple from the large ones. Only the leading term .Si = ip_L is identical with the loading term of the HQET expansion. The different representations of the quark field generate different, expansions of the Lagrangian. The first term of the HQET Lagrangian Cc\r (3.4.10), the dimensionfour operator, is unique and renormalizable. It gives rise to the following momentum-space Feynman rules:
)02
s
heavy- uar
propagator i
v
(v
ie
1-
v
2,
heavy- uar -gluon vertex \ jsv'lT^c,. ith tliese Feynmau rules field-theoretical corrections li e self-energies, vertex functions, renormali ation constants, and anomalous dimensions can be calculated in the usual way. The higher-order terms in the 1 TIIQ expansion consist of operators with dimension , , . . . and therefore are not renormali able. They occur in a finite number of insertions only and can be treated by matching the regulari ed, cut-off-dependent expression at a certain scale to the full theory.
.4.2
y m m e t r i e s of H
ET
The leading term of the effective agrangian .4.1 , the static approximation of H ET, is invariant under flavour and spin transformations. These approximate symmetries lead to many useful relations between the masses, or the electromagnetic and wea properties of the heavy hadrons.
.4.2.1
Heavy-flavour and spin s y m m e t r y
The agrangian of C approaches in the infinite-mass limit, TIIQ oo, the static agrangian, i.e. the first term in .4.1 . This leads to symmetry relations for matrix elements involving hadrons with charm and bottom uar s. 1 The static agrangian for the c, b - uar system with spin orientations t , l , T = ci > Del + blivDbl + ciivDci is invariant under spin rotations tions
2
sp
+ and
HwDbj,
.4.1
2 ,r flavour transforma-
.4.1G e consider mesons with one heavy uar Q and one light anti uar q (q u , . . . and denote the flavour part of their state vector by \Qv,q)- The 2 ,f group then transforms a -meson state into generator , f of the 1
T h e top
uar
does not live long enough to form b o u n d states.
.11
eavy-quark effective theory (
T)
503
i charmed one, / hf b ,r/) = c ,r/), and similarly for the other generators. This leads to symmetry relations for matrix elements of the bilinear, currentl.ypr operators h'v,(x)riiv(x), where F = {7 / t ,7,,75}. Thus, there are relations between the matrix elements { ^^j lh ^^hvl vi i)-
1.4.2.2
S p i n s y m m e t r y and m e s o n s t a t e s
In contrast to full C , the static Lagrangian (3.4.15) is invariant under , tpin rotations. Consequently, the mesons with the same flavour content and in the same orbital state are degenerate in mass. In this approximation the pscudoscalar and the vector ground-state mesons differ only in the spin part of the wave function and are related by a spin-flip operation. These spin parts x S and x of the state vectors of the pscudoscalar and the vector mesons are in irac-inatrix notation given by
ps x
= n/M(7 P-),
V
(e) =
/MtfP"),
P~ = i ^ i ,
(3.4.17)
where e (A) (e 2 = 1, ev = 0) is the polarization vector of the vector meson with mass M. The spin amplitudes Aa carry a heavy-quark spinor index /I and a light-quark spinor index a and are normalized according to
T (xx) = 2 M, The operator Sv(e) Sv(e)x S = X (e).
3.4.2.3
x = 7
t
7 -
(3-4.1 )
= 7 5 ^ , 5 2 ( e ) = 1, flips the spin of the heavy quark
Flavour-spin symmetry
The static Lagrangian (3.4.15) is invariant under the flavour-spin group S (2Arhf) ,f which is generated by simultaneous flavour and spin transformations. Thereby pscudoscalar and vector mesons with charm and beauty are members of the adjoint representation of this group. This large approximate symmetry leads to many relations between matrix elements, some of which are discussed in Sect. 3.4.3.
0 4
s
3.4.2.4
eparametrization invariance
Tlie representation of the quark four-momentum by the four-velocity of the quark and ' is not unique. ther decompositions, i.e. replacements v>' > Sv ' must lead to the same physical results. Moreover, a reparametrizaTUQ -I- MQ, does not change the tion of the effective quark mass, M,Q predictions of II T since the quark masses are, in contrast to the meson masses, no physical quantities. The invariance under these reparametrization transformations leads to exact relations in T Fa 2 .
3.4.3
A p p l i c a t i o n s of
T
3.4.3.1
Masses of heavy mesons
As a consequence of the spin symmetry of II T, one expects equal masses for the heavy pseudoscalar mesons and the vector mesons with the same flavour and orbital quantum numbers, M M y and M i Mu-. This is Mlr = 2007 Me indeed approximately true P 00 : M0 = 1 6 Me and Mb = 527 Me « Af B . = 5325 Me . Moreover, since the spin interaction is proportional to Av,ps =
- Mp S = ( ss 2mn(Mv
+
ps){ v
1/TTIQ
we expect
- Aips)
- Mp S ) « 2mnco"stant 2rri
=
constant. (3.4.1 )
This relation is fulfilled experimentally: AB-,B 0.4 e 2 , AI .) « 2 2 2 0.55 e , and even A P « 0.57 e . Similar results 0.53 c , A -, hold for heavy baryons.
3.4.3.2
elations between current matrix elements
lectromagnetic and weak transitions of hadrons are obtained from the had(0) /7I, I,AI), ronic matrix elements of current operators, i.e. ( I / 2 ^ 2 , A 2 and ( / / 2 , 2, A2jf 5 ( 0 ) i / i , v\, Ai), where / / , v, A) is the state vector of a heavy hadron with four-velocity v' and lielicity A, and j' and ji' are the vector and axial-vector parts of the hadronic charged currents (1.4.24). 10 "
e here use the symbol j in order to avoid confusion with the heavy quark field hv-
.
(
T )
0
For simplicity we discuss the current matrix elements between pscudoscalar ,uid vector-meson states. Because of Lorentz invariance, they have the form
{PS2, 2 i t(0) PSi,t;1) = (vi v2yF (q2) («, - t ^ F - ) , ( 2) A2, 2 j"(0) PS1, i) = ie^el^p i^Fviq 2 ), ( 2,\2,v2\jp( )\ Sl,vl)=eptF l(q2) (3.4.20) + V'I( 2VI )Fj\2{q2) with q2 = (pi - p 2 ) 2 = M2 + M - 1M\M2V\V2. contain the dynamics of the strong interaction.
+
v£{efa)F s(
)
The form factors Fi(q2)
e investigate the current matrix elements in the heavy-quark limit. liecause of flavour spin symmetry, they can be decomposed as (
2,V2,
q2\ i 2Th
l
//,, v,, q~i) = Tr(x// 2 r x//l ) (y),
(3.4.21)
where = v\v2. The sgurise unction (,(viv'i) Is corresponds to the form factor. It. is a universal dynamical quantity which group-theoretically was obtained os can be seen from above after performing the spin- and llavour-syrnmetry decompositions by applying the igner- ckart theorem and separating off Clebsch ordan coefficients. In our case we take the amplitudes (3.4.17) for the pscudoscalar and vector mesons with f = 7 / and F = 7 75 for the vector and axial-vector currents, respectively. sing (3.4.21) yields for instance for the PS PS transition ( P S 2 , v 2 b / i ( 0 ) P S , , t ; 1 ) = x/A7 2 M, Tr { = - ^
~ 7 5 7
Tr {(1 - hh,
^
}
( /)
(1 - h)} i (v)
= / M , M 2 ( 7,} + v'i) i{ ).
(3.4.22)
This gives the two relations F+{q2)
= \/M\M2
F-(q2)
(?/),
(3.4.23)
= 0.
The results for the other form factors read Fv = Fm = 0,
i -^i ), FA
F = -
i
= v^m,M2( 1 jd
2
( /).
vxv2)
a ), (3-4.24)
0
s
The six form factors are reduced to one quantity, the Isgur ise function. 2 For v = v', i.e. vv' = 1 or q = 0, the left-hand side of (3.4.22) is the matrix element of a conserved current associated with the heavy-flavour symmetry. The corresponding conserved charges simply transform the heavy quarks into each other and the diagonal charges count the number of heavy quarks. This fixes the normalization of the form factor F in (3.4.20), which in turn implies that the Isgur ise function is normalized at the kinematic point t; = v' as (1) = 1. In this way one obtains symmetry relations between different decays which can be compared with experiment. For 13 decays into and mesons plus a lepton pair it is possible to compare the following ratios of form factors with experiment u 6:
(1
v v 2 ) ^ - = 1.1 F Ai (l
viv2)-FA3
0.32 i I
i B / M ^ ) F ^ = 0.71
0.23.
(3.4.25)
These ratios should be equal to one in the infinite-mass limit. The first ratio is in good agreement with theory; the second shows sensible deviations from 1. The reason is that those form factors that do not vanish in lowest order get only second-order corrections in m.Q, whereas F.\., and F get first-order corrections. This is the content of u e's theorem Lu 0 . II T was successfully applied also to purely leptonic decays of heavy mesons, semi-leptonic decays into light mesons, baryon decays, inclusive decays, etc. For details we refer to reviews, e.g. e 4, Ma 7, e a ,
3.4.3.3
C o r r e c t i o n s t o t h e l e a d i n g - o r d e r i-esults
For an adequate comparison with precise experiments, it is necessary to calculate the corrections to the results of the static Lagrangian C ^ , (3.4.15). There are two types of corrections. The first class are the recoil corrections. They originate from the heavy-mass expansion of the irac operator in (x) into (.ir (3.4.10), and the expansion (3.4.13) of the heavy-quark fields the large component fields. The expansion of the field operator and the corresponding corrections to the state vector of the heavy meson may be interpreted as corrections to tin; current operator. The expansion of ,.u in
3.5
L i g h t
q u a r k s
a n d
cliiral
p e r t u r b a t i o n
t h e o r y
M)7
l/mQ leads to non-rcnormalizable operators. The matrix elements of these new operators give new reduced amplitudes. The second class consists of the field-theoretical loop corrections. They can he calculated using the Feynman rules of QCD for the gluons and light quarks and those for the heavy quarks given in Sect. 3.4.1.2. In this way the i rnonnalization constants and anomalous dimensions are determined, which enter the renormalization-group-improved results. Moreover, in some cases it may be necessary to include electromagnetic, radiative corrections. For details on the many applications of HQET we refer to the reviews [Is92, Ne94, Ma97, Ne98a].
3.5
Light q u a r k s a n d cliiral p e r t u r b a t i o n t h e o r y
The running coupling constant of QCD grows with decreasing energies. This leads to the confinement of quarks and gluons in hadrons and prohibits the application of perturbation theory in the region below lGeV. Low-energy hadron dynamics can be investigated by lattice QCD (cf. Sect. 2.10) or by the method of effective field theories. The latter is based on the fact that at low energies a description of the strong interaction directly in terms of the physical degrees of freedom, the light hadrons, e.g. the pscudoscalar mesons, is possible. A lot of phenomenologically successful work in low-energy hadron physics was done in the framework of current algebra [AdG8, A173] long before QCD was established. This led to the development of effective field theories for low-energy hadron physics, the non-linear a model, and cliiral perturbation theory [We79, Ga84], Among the many reviews of this field we mention only Refs. [Le91; Le94, Pi95, Le96].
3.5.1
Chiral s y m m e t r y of m a s s l e s s Q C D
The bridge between the QCD Lagrangian (2.3.53) and the low-energy effective theories is built by the symmetries of the light-quark sector which appear if the masses of the N( light quarks (u, d, s, . . . ) vanish, i.e. in the cliiral limit. We start by studying the possible symmetries of the quark mass term ¿QCD.in = - i p M q i p = -
y ]
<1
(3.5.1)
432
3
Q u a n t u m
G h r o m o d y t i a i n i c s
The quark mass matrix A/(| = diag(m u , m ( j, m s , . . . ) in general allows for a global (U(])v) / V f symmetry ipq exp^or 9 )^, q = u , d , s , . . . , with the conserved vector currents ji'ix)
= i>q(x)^i>q(x),
d„ji;{x) = 0.
(3.5.2)
If the N( quark masses are equal, £QCD,HI IS invariant under a SU(7Vf)v symmetry i> exp(ia°A a /2)^ [A" are the SU(A0 Gell-Mann matrices (1.2.51)], leading to the conserved vector currents j;(x)
=
d%(x)
= 0.
(3.5.3)
In the case of vanishing quark masses also chiral transformations if> —> exp(i/3a75An/2)^> become a symmetry. As a consequence, we have in addition conserved axial-vector currents, jijx) = ¿>(x)yin5^(x),
d " j l l t ( x ) = 0.
(3.5.4)
The flavour currents (3.5.3) and (3.5.4) as well as their charges form the basis of current algebra. The charges of these currents, together with those of the singlet vector and axial-vector currents (1 is the unit matrix in flavour space), il(x)
= $(x)lltlrp(x),
jlfl(x)
= ^(.E)7;x75l^(x),
(3.5.5)
generate the global group SU(ATf)v x SU(JV f ) A x U ( l ) v x U(1) A with dimension 27Vf2. In this context it is very convenient to consider besides the vector and axialvector currents also chiral currents jlAx)
= \tit<(x)
~ J5t/z(a;)],
JnAx)
= \[j)^x)
+JaAx)](3.5.C)
3.5 Light quark and cliiral perturbation theory
3.5.1.1
50 )
S p o n t a n e o u s b r e a k i n g of ebiral s y m m e t r y
This cliiral symmetry is spontaneously broken, 17
S (/ f)v x S (
f)A
x
(l)v x
(1) A
S
f
)v x
(l)v,
to the subgroup of vector symmetries with dimension 2. The breaking of S (A f)A generates a multiplet of 2 1 pseudoscalar Goldstone bosons. For the case of three massless quarks, these are eight massless pseudoscalar mesons. They are identified with the lightest pseudoscalars, 7r , 7r , 7r , 1 , , , , and r)s- The breaking of chiral symmetry by the C quarkmass term generates their experimentally observed masses cf. Sect. 3.5. l . The global group (1)A is not anomaly-free and therefore gives no massless oldstone boson, cf. Sect. 3.5.1.2 The transition amplitude from the vacuum 0) to a pseudoscalar-meson state \ ,b),
0 b ^ ( 0 ) p , 6 ) = (0 ^(0)7,«75 y ^ ( 0 ) p , 6) = a,b =
,...
, f
1,
(3.5.7)
contains the order parameter, the pseudoscalar-meson decay constant /" -, which is related to the quar condensate (01^(0)^(0)10) cf. (3.5.25) . In strict S (A r f)v symmetry f ,s is independent of a. The current matrix element (3.5.7) obeys as a consequence of (3.5.4) the ard identity cf. the C C relation (2.7.2 )
6 = \p2fpSSab
This implies p2 = 0, i.e. the
= 0.
oldstone bosons are massless.
" e t a i l s of s p o n t a n e o u s s y m m e t r y breaking are presented in Sect. 4.1.
(3.5. )
5 1 0
•!
3.5.1.2
Q u a n t
u m
C h r o m o d y n a m i c s
The U(1)A anomaly
Tlie Ward identity of the flavour-singlet axial-vector current (3.5.5), the generator of £/(1)A, contains an anomaly 1 8 [cf. (2.7.40)]
= - ^
W
Tr(F^F^),
(3.5.9)
where i ;v
3.5.2
P i o n - p o l e d o m i n a n c e and effective low-energy t h e o r y
Low-energy physics deals with the S matrix and current matrix elements of the pseudoscalar mesons | p , a ) . In order to derive the relation between QCI) and the effective low-energy theory we consider the axial-vector current Green functions
{ ° I
T
( ® 0 ¿5?w 0*2)''" J5,M„ (*») 1°)
and their generating functional
e x p ( r c { j } ) = E Jn / d ' 1 x i • • • d " 3 : » j a u l l i to) • • • j a ' l ' , l n to.) 71 ><(or^I1(^t)---^;,lto.)io)>
(3.5.10)
where J",(x) denotes the sources of the currents. The current two-point function is dominated at low energies by the poles of the pseudoscalar mesons (pion-pole dominance). The contribution of the continuum states begins at the two-particle threshold p2 = 4Mp S , i.e. at 1
"Gauge theories with a n o m a l i e s are discussed in S e c t . 2.7.2.
. .
s
11
higher energies. e investigate the one-particle contribution by inserting a complete set of meson states r/) and using (3.5.7),
d4x
eipx(01Tjr-tli(x)jif^(0)10)
xi0(® )( b
= fd
(/i(
{
d xe^
=
) r 7 )( i / jt( )
)e-^
(oix0^»*
+
J p
2
s)
2
For the last step see (2.1.12)-(2.1.14). In a similar way, other reen functions can be analysed, and vertices may be constructed that describe the effective low-energy interactions. Based on pion-pole dominance, the axial-vector currents can be replaced by the corresponding pscudoscalar meson fields,
= -fp
s
d
,
(
3
-
5
.
1
2
)
In fact, the derivative d,,iTa(x) of the pseudoscalar fields transforms like an octet of axial-vector currents, i.e. it has the same quantum numbers as jt , and can therefore be used as an interpolating field for the axial-vector currents. In this way, the effective low-energy interactions can be collected in an effective agrangian Cc\\(- ,dir) for pseudoscalar meson fields. The effective Lagrangian must fulfill the condition exp(7 c { }) =
irjj
^"-0"-^^,
(3.5.13)
and has to reproduce the symmetry properties of the current reen functions, ard identities, and pion-pole dominance. The effective Lagrangian e ir can be expanded in a series in the number of the IT fields, efT =
4ff
(3-5-14)
T h e m e s o n m a s s e s vanish in the c a s e of strict chiral invariance. e write t h e following e q u a t i o n for finite m a s s e s of t h e pseudoscalar m e s o n s since chiral invariance is o b v i o u s l y broken in nature.
r>12
.'5
Q u a n t u m
C h r o m o d y u a m i c s
wliicli corresponds to an expansion in powers of the (small) momenta. As a consequence of current conservation, this expansion starts with [We79, Le91, Le94]
4ff = \ W d " n a .
(3.5.15)
No Lorentz- and flavour-invariant 7r:i terms can be constructed. Therefore, the interaction terms start with the quartic expressions JC^J/ in the TT fields. A detailed analysis [Ga85] shows that the effective Lagrangian is best expressed in terms of the exponentials of the 7r fields. This leads to the nonlinear a model. The linear and non-linear a models were discussed as effective low-energy models for the theory of the strong interaction already before QCD was proposed.
3.5.3
T h e non-linear cr m o d e l
In the preceding sections we have seen that at low energies chiral massless QCD is governed by axial-vector-current Green functions and pion dominance and further structured by Ward identities and analyticity. This structure can be formulated as an effective theor in the form of the non-linear o model [Ga85]. The fields are written in the form of an exponential,
U (:c) = exp
Í(.T)
(3.5.16)
/PS
where the octet of the light pseudoscalar mesons is arranged in a 3 x 3 matrix, ( >(x) = na(x)Xa
=
+
2n+
2TT- - T3 + ^ s2 ~
2 '
s2
+\
2 -^f
(3.5.17) j
The exponential U transforms linearly under chiral transformations. With the unitary SU(Aff) matrices V\. and VR this transformation reads U'(x) =
U{x) l
(3.5.18)
.'t.r>
L i g h t
q u a r k s
a n d
chiral
p e r t u r b a t i o n
t h e o r y
5 1 3
I'lii! Lagrangian of the non-linear ÍT model is constructed from U(x) in such
¿o = f f - m w w u i ) ] .
(3.5.19)
Expanding the exponential functions in the Lagrangian Ca we get, besides (2) the bilinear term C a (3.5.15), higher-order terms. The quartic coupling
4 ° = t À T T r { [ $ ( ^ í > ) - (¿yi>)][<£>(c)") - ( $ , $ ) $ ] }
(3.5.20)
as well as all other couplings are of derivative type. The effective Lagrangian (3.5.19) gives the correct expression M{nir -> 7r7r) = i / / p S for the elastic low-energy irn scattering amplitude (t is the momentum transfer squared) [We66]. Expanding further, similar results can be calculated for multi-pion scattering. Noether's theorem leads to the following conserved vector and axial-vector currents [Le91]:
jìAx) = jlv(x)
3.5.4
=
iffTr(\a[dtlU,U% \!fTv(\a{dllU,Ul}).
(3.5.21)
Breaking of chiral invariance
Chiral invariance is the basis of the connection of massless QCD with the non-linear a model. However in reality, the masses of the light quarks do not. vanish. Chiral symmetry is broken by the QCD quark-mass term (3.5.1) which induces a corresponding term in the non-linear a model. A suitable ansatss is
CatSh = ^2B0Tr[Mq(U
+ Ui)}
(3.5.22)
with a free constant Bo and the mass matrix A/(| of the light, quarks.
432
3
Q u a n t u m
G h r o m o d y t i a i n i c s
The complete Lagrangian of the non-linear [Ga85]
Ca>m =
J
f
massive
+ 2A, rl\[Mq(U
(lV[(^t/)(9"C/*)]
a model now reads
+ i;')]) .
(3.5.23)
It contains two parameters, the decay constant /PS and the constant DQ of the Symmetry-breaking term. Comparing the vacuum expectation values of (3.5.1) and (3.5.22) yields m u <0|$ tt tf„|0) + m(l(0|^sips\0) = -fps&a
K
+ md + m s ) .
(3.5.24)
This relates BQ to the value of the (¡nark condensate. Assuming approximate flavour symmetry of the quark condensate, (3.5.24) simplifies to (Q\fqif>q\0) ~ -IPSBO
"o sum over q.
(3.5.25)
The symmetry-breaking quark-mass terms contribute not only to the meson masses but also to decay and scattering amplitudes. The low-energy expression for the 7T7r scattering amplitude, for instance, is changed to M(TTX —> 7T7T) = ( t - M 2 ) / / ^ . The non-linear o model is the simplest low-energy approximation of QCD. Instead of quarks and gluons it contains the pseudoscalar mesons which indeed dominate hadron physics at energies below the p-meson mass. The tree graphs resulting from (3.5.23) give the simplest approximation to current matrix elements, scattering amplitudes, and masses of the pseudoscalar mesons. A general, complete set of terms with four derivatives respecting chiral in variance and Lorentz in variance and including the quark masses was constructed by Gasser and Leutwyler [Ga85], 8 »=i where the L, are free parameters and = (TVK^i/XcWt)])2, P-2 = Tr\(d,JJMAJ*)}
'[Y\(()»U)(d»V%
.'t.r>
P-i =
L i g h t
q u a r k s
= 2Z? 0
chiral
p e r t u r b a t i o n
t h e o r y
5 1 5
Tc[(dpU)(&,Ut)(dl,U){&'U*)\i
p.\ = 2BQ'\\\(dtlU){d>lU^)] Pr>
a n d
Tr[M q (U + £/*)]
Tr[(d,tU)(dl'U^)(MqU
+ U*Mq)),
P 6 = 4flg(TY[M q (i/ + t/ t )]) 2 , P7=4BUTr{Mq(U-Ul)})2, P 8 = 4BlTt[MqU]
MqU] +
UMqUMq\.
(3.5.27)
If there are less than three flavours, only P\, P2, Pi, P<>, and P-, are independent. In order p'1, the axial anomaly also contributes to the effective Lagrangian [Le91], Cw z =
f(U)eflt,PAd>iU)(d"U)(dpU)(d<'U),
(3.5.28)
where the function f(U) is determined by the Wess-Zumino consistency condition (cf. Sect. 2.7.2.3). Physical effects of this term can be seen, for instance, in the two-photon decays of the neutral pseudoscalar mesons, 7R° -> 77, 7] —» 7 7 [Bi93]. The non-linear a model is non-renormalizable. Since each loop integration gives two additional powers of the momenta, the one-loop corrections can be written as higher-derivative terms. Therefore, the divergences can be absorbed in the unknown coefficients Ll leaving renorinalized couplings L\en which can be fitted to experiment. Besides this, the loops give rise to logarithmic. tadpole terms of the type
(3.5.29)
where ¡i is the parameter of dimensional rcgularization (cf. Sect. 2.5.2.1 on page 208). As in every non-renormalizable theory the number of undetermined parameters increases with the number of loops. This reduces the predictive power of the higher-order corrections considerably. The coefficients L\cn were determined from experiment, for instance, in Ref. [Bi94]. They are typically of the order of 1 0 _ i and produce effects of 2% at t = Ml and of 20% at I. = Ml.
s
. .
p p l i c a t i o n s of chiral p e r t u r b a t i o n theory
. . .1
asses of the pseudoscalar mesons
Expanding the exponential of the pseudoscalar fields a in the symmetrybrea ing term . .22 to second order and leaving out the constant term, gives the mass term of the pseudoscalar mesons,
p2 -
x
Tr xi
Tr
ms
. 1
mixing term, we find for the masses
= m u + ms)B0,
=
Ml
mu
o = md +
2(2ms — m„ rt,1
V
ms)B0,
+ md - - . - ( m u " T " j ) 2
V 2
. .
m( )Bo,
m
,2
x
-
fter diagonali ation of the Ml
x
m d 4- 4m s
rn d m
-
-
md
,
V nd
2 2m s - m u
m u - md
o
Bo + o({mu 2
md . . 1
with the
mixing angle
tan 2 From
27
I
.
. . 2
. . 1 the sum rule (M'f,
Mio) - 2 ( M l ,
2
Ml+)
=
. .
3 . 5
L i g h t
(|iia.rkH
a m i
chiral
p e r t u r b a t i o n
t h e o r y
5 1 7
follows, or, neglecting isospin breaking, the Gell-Mann Okubo mass formula 3M 2 + M l - 4 M,2< = 0,
(3.5.34)
which is in good agreement with experiment. For the ratios of the quark masses one obtains from (3.5.31) TOy _ M'l+ + M2 RN
<\
fflii rns
M =
2 +
- M
+
2 +
-
M2o
+ M
2
O'
+ - Mlo -M2+-fM2++M2 0
(3.5.35)
up to terms of order (m u — m^) 2 . The value of Do can be determined only if one of the quark masses is known. A detailed analysis including higherorder terms in the effective Lagrangian and loop corrections was performed by Gasser and Leutwyler [Ga85, Le96a]. They obtain for the ratios of the quark masses the values 7/7
777
— = 0.553 ± 0.043, md
— = 18.9 ± 0.8. m({
(3.5.3G)
Taking from QCD sum rules m s ( l GeV) = 175 ± 25MeV [Bi95] for the running mass of the strange quark at 1 GeV, gives for the running masses ?n.u(l GeV) = 5.1 ± 0.9MeV and m d ( l G e V ) = 9.3 ± 1.4 MeV. The value of the quark condensate follows from (3.5.25) as (0|i/;^|0)(l GeV) = - ( 2 3 5 ± 2 0 MeV) 3 . This value is in agreement with the results obtained from sum rules (2.9.161) and quenched lattice approximation (2.9.162) given in Sect. 2.9.3.7.
3.5.5.2
Decay constants
Many precise experimental results have been derived in low-energy liadron physics for decay constants, electromagnetic and weak form factors, scattering amplitudes, non-leptonic decays, etc. On the one hand they can be used to determine the couplings on the other hand they allow for tests of chiral perturbation theory including contributions of loops and of CÌ[ìn-
5 1 8
3
Chroinodynamics
Q u a n t u m
As a typical case we present the expressions for the n and K decay constants in the isospin limit (m„ = m j ) [Ga85],
= / P S (, + \
+
ü S + ü f i u - - 2 AM) - MIL •
J PS
/ PS
J
/ K = /p., ( l + i ^ L ? " +
V
Jps
" \ m l ) - \ m l )
/ps -
'¡MM^,
where A(M2) is the tadpole term (3.5.29). Taking the ratio JtJJk L\jcn, giving up to logarithmic terms
¡
(3.5.37) eliminates
,3.5.38)
JK
J PS
Performing a complete analysis and using the value /K//TT = 1-22 ± 0.01 [Le84] yields L"'n = (1.4 ± 0.5) x 10~ 3 [Bi94]. This value is very close to the result of phenomenological considerations, i.e. polo dominance of scalar mesons, = / p S / 4 M | , where Ms = 980 MeV is the mass of the corresponding scalar meson, the /o(980) meson. The value for f,, can be predicted from a formula similar to (3.5.37) as / , , / / _ = 1.30 ± 0.05 [Ga85]. As already mentioned, many results have been obtained for observables in meson physics. All of them are in agreement with experiment. Extensions including assumptions like vector-meson dominance have been successfully applied not only to low-energy meson physics but also to baryon physics.
3.6
Results of lattice a p p r o x i m a t i o n of Q C D
In Sect. 2.10 we discussed the lattice approximation of gauge theories. Applied to QCD, the main results are due to numerical methods which make use of powerful computers. In order to get physically relevant results, one has to overcome several problems. In spite of the remarkable progress in the theoretical understanding and great computational efforts, these are onlysolved incompletely. We repeat some of the sources of these problems: •
extrapolation to the continuum limit,
.
s
1
extrapolation to the infinite-volume limit, nenehed approximation of full
C
with dynamical
uar s,
extrapolation to the chiral limit, and algorithmic sources of systematic errors. The related uncertainties lead to si able systematic errors. Thus, we have results which in the best cases have errors of 1 -2 . ith this precaution, we discuss some topics which are typical for non-perturbative C the spectrum of the light hadrons ect. . .1 , the glue balls ect. . .2 , and the relation between non-perturbative and perturbative C ect. . . . e refer to the literature for lattice results on matrix elements u and structure functions c .
. .1
T h e hadron s p e c t r u m
ince the uar model of the hadrons ect. 1. played an important role in establishing C , the calculation of hadron masses from first principles was always a main goal of lattice C Ha 1a, e 2 . However, the available computer power allows for large enough lattices only in the quenchal approximation where the effects of uar loops are neglected. evertheless, it is hoped that these results reproduce some ualitative features of the physical hadron spectrum. ut the error of this approximation is not really understood.
. .1.1
The parameters of the simulations
For the uenched V x t = 1 1 x 1 Go , o ft . and ft
approximation, the biggest lattices used have typically 2 to 4 x 4 points. In the recent calculations 1 4 , the coupling constant ft f g l 2.1 .12 varies between . .
In Table .1 we collect the results of some simulations with a variety o f values. The lattice constant a is determined from the mass of the p meson mp eV in calculations with ero uar mass. e compare the input ft with the calculated value ftca c following from formula 2.1 . . The agreement of both ft values gives an impression of perturbative as mptotic scaling. For the calculation g 1 eV was assumed, which corresponds according to 2.1 . 2 to , eV. ith all the uncertainties about this
2 0
s
'i x Aft
p
size (fm)
A: ale
5.70
24 3 x 4
0.13
3.4
5.6
6.00
24 3 x 32
0.10
2.0
5.
6.20
S
0.075
6.20
24 3 x 4
6.25
4
3
x 4
6.50 6.50 Tab. 3.1
a (fm)
6.1
0.070
1.6
6.1
0.063
3.03
6.27 6.53
0.05
s
4
3
x 64
0.047
2.6
6.53
uenched lattice simulations of C for ilson fermions ( ) and staggered ferrnions (S). Further notations are explained in the text.
fictitious quenched A value with quark number q = 0, this seems to be a reasonable value (cf. the physical A^r = 20 2 5 - 23 Mc of Sect. 3.1.2). The finite-size effect (cf. Sect. 2.10.5) was checked by comparing the results in volumes of different physical sizes. For the "measurement" of the masses of the p meson and of the nuclcon the corrections are less than 2 for spatial sizes a = 2fin, and negligible for sizes above a = 3fm (cf. Table 3.1). The finite-size effects become severe for light particles (Sect. 2.10.5). Besides a large spatial volume, the measurement of small masses m /a requires a large extension in time direction cf. (2.10.1 ) . The requirement of a large physical volume with small lattice constants exceeds the power of present-day computers. In the present calculations the quarks get a mass which makes the pion massive, and therefore m^/nip 0.5. This renders the extrapolation to the chiral limit m,, 0, and therefore to 0, particularly unreliable (cf. Sect. 3.6.1.4).
3.6.1.2
S p e c t r u m of h a d r o n m a s s e s
adron masses calculated in the quenched approximation with ilson fermions are shown in Table 3.2. e took the recent, values of the CP-PACS collaboration Ao00 . They are extrapolated to the continuum. In order to fix the bare quark masses m i o, m di o, and mS)o, and the lattice scale A ,, the values of rn . m p , and either m or m are taken as an input.
.0
UOHUIIH
21
7T exp.
1
140
76
input
input
input
input
input
553
A exp.
6
4
40
1116
7
101
11 3 1117
7
1060
1176
5
101 57 input —•
n
13 4
1532
1672
1257
135
145
1561
1257
13
1517
1647
a,
A
1315
1232
1201 12
Tab. 3.2 Lattice results for the hadron spectrum using besides rn , rnp, and alternatively M.P (line 3) or JTIK (line 4) as input (from Ao )
T
The errors of the whole procedure arc estimated to 5 10 for the mesons and around 12 for the baryons. ithin these errors, the baryon masses are the same for either 7«k or m , j . as input. A comparison with the experimental results (Table 3.2) indicates that the quenched lattice approximation can describe the experimental spectrum with errors in the 10 region.
3.6.1.3
yperfine splitting
Lattice C allows in principle for the determination of the masses of light quarks k 3 . The renormalization-group invariant masses belong to the fundamental parameters of the standard model. Therefore, they are of great importance. The hyperfine splitting in the system, i.e. the mass difference TTI 7/i , may be used to determine the mass m s of the strange quark since in this difference an additive quark-mass renormalization drops out. The calculation proceeds by fixing the bare parameters from physics and then transforming them into the running mass of the MS scheme. e have shown in Table 3.2 how numerical simulations describe the hyperfine splitting. From this we gain the bare mass 7ris oFor the transition from the bare lattice perturbation theory to the MS scheme in the continuum, we have to use arguments similar to those for rvs in Sect. 2.10.4.3. Similarly to (2.10.75), we get Ma 7a ms(p) = m j l
g
log(ap)
logCm) j .
(3.6.1)
432
3
Q u a n t u m
G h r o m o d y t i a i n i c s
1.5
>«
1.3
S 11
IS
0.9
a [GcV-1 ]
Fig. 3.12 Different, simulations for the determination of m 3 ( / i ) for ¡t = 2 G e V (based on R.ef. [Yo98]). Circles refer to the perturbative scheme, triangles to the P C A C scheme.
Here a is the lattice constant, ¡i the MS scale parameter, and Cm a- calculable constant which depends on the details of the lattice regularization. For the Wilson action, Cm = 1.G7. The mass m is given by m = r»o in one-loop lattice perturbation theory and by rn = r??o/(l — .
mlf
K
= Z,,(mu
+ m s )(0|n 7 55|K+),
(3.6.2)
where is taken from experiment, and the renormalization constant Z/> can be determined by lattice perturbation theory as above or by non-perturbative methods [Ca99]. We represent this determination also in Fig. 3.12.
.0
2
z
m Fig. 3.13
n
/ mp
m 71 / inp
dinburgh plot: q u e n c h e d a p p r o x i m a t i o n w i t h s t a g g e r e d fermions, / = .l/i ( o p e n large circles on the left-hand side); full C w i t h s t a g g e r e d fermions, two flavours, (3 = 5.60 (small circles with error bars on t h e l e f t - h a n d side); q u e n c h e d a p p r o x i m a t i o n with i l s o n fermions, 0 = 6.17 (open circles on t h e right-hand side); d y n a m i c a l ilson fermions, ft = 5.5 (small circles with error bars on the right-hand side) (based on lief. o 7 )
The result as derived from Fig. 3.12 is rTis(2 e ) = (110-130) Me in quenched approximation. A determination based on (3.6.2) including an estimate of all errors gives m s (2 e ) = 7(4) Me a (cf. Sect. 3.1.3).
3.6.1.4
Tbe
d i n b u r g h plot
The dinburgh plot shows the mass ratio of the pion and the p meson in^/iiip versus the mass ratio of the nucleoli and the p meson m.^/m p . This plot is particularly well-suited to demonstrate the progress and the difficulties of the lattice approximation of the hadron spectrum. In Fig. 3.13 we present the dinburgh plot extracted from o 7 . The different points of the simulations refer to different bare quark masses. The filled square is the result of a strong-coupling approximation yielding m^/nip = 1.5 771-tt jnip. This means that all mesons consist of two quarks and all baryons of three quarks. The filled circles represent the experimental values mn/rrip = 0.1 and m^/mp = 1.22.
5 2 1
:i
Q u a n t
inn
C l u o m o d y n a m i c s
As a relation between mass ratios, the Edinburgh plot does not depend on a determination of the lattice constant. For quenched staggered fermions, simulations with different ft allow to perform the continuum limit. The result is given by the line through the points for ft = 6.15. The dashed line is a theoretical continuation based on a phenomenological quark model. Figure 3.13 also shows that the simulations connect the regions of the strong-coupling approximation with the physical point to which the quenched approximation allows to extrapolate. The simulations with dynamical quarks follow in general the line of the quenched approximation. However, the large computational errors do not allow for definite conclusions. (For quenched approximations, the errors lie within the size of the symbols.) The approach to the chiral limit, and hence to the physical ¡joint, is still poor. For Wilson fermions we have m^/iiip > 0.6. For staggered fermions it is somewhat better. There are some indications that improvement of Wilson fermions will make some progress in this respect.
3.6.2
Glue balls
Glue balls are a new feature of SU(3) gauge theory [Pr72a]. In contrast to the linear Maxwell equations of the electromagnetic field, the gluon fields have self-interaction. This leads to bound states of gluons, the glue balls. Their masses can be determined by the lattice approximation of SU(3) gauge theories [Is82]. However, in full QCD the coupling to quark fields introduces mixing with quarkonium states. This is one of the reasons why this important prediction of QCD is not yet definitely confirmed by experiment.
3.6.2.1
Q u a n t u m n u m b e r s of g l u e balls
In the limit where the glue balls may be considered as free relativistic particles, they possess quantum numbers [Mi89] like mass, spin, parity, charge parity, and internal quantum numbers (cf. Sect. 1.2.1.1). In QCD, the internal quantum numbers are determined by the quark content. Since glue balls are essentially bound states of pure gluons, they have the flavour quantum numbers of the vacuum. Lattice approximation violates rotational symmetry. Therefore, the definition of spin by the irreducible representations of the rotation group SO(3)
3.6
P e n u l t s
of
lattice
a p p r o x i m a t i o n
J PC
Tll/y/o
77l(; ( lYIeV)
A++
0++
3.52 (12)
1550 ± 50
£++ V++
2++
5.25 (25)
2270 ± 100
On
of
Q C D
5 2 5
5.07 (17)
2
0-+
5.3 (6)
2330 ± 250
Tab. 3.3 The glue-ball spectrum (from Ref. [Ba93]) fails, and SO(3) has to bo substituted by the cubic group. There are live irreducible representations Oh of the finite symmetry group of the cube, A\, /\-2i E, T\, and T2 with dimensions 1, 1, 2, 3, and 3, respectively. If we restrict the rotations to those of the cubic group, the spin-0 representation of SO(3) becomes A\, spin I gets T\, spin 2 corresponds to E®T2, and spin 3 contains A2- Rotation symmetry is restored in the continuum limit of lattice approximation. Consequently, for a spin-2 glue ball there are two mass-degenerate states ¡11 the lattice representations E and T2. This example indicates tinprocedure for the determination of spins in lattice approximation. Parity P and charge-conjugation parity C can be determined 011 the lattice <us in the continuum. There are combinations of quantum numbers that cannot be formed by pure qq states: () + ~, 1 2 + _ , . . . (cf. Sect. 1.3.1.2). Such spin exotics are particularly interesting candidates for glue balls. However, there might be mixing of glue balls with qq states with vacuum quantum numbers, leading to hybrids, which show similar features. 3.6.2.2
M a s s e s of g l u e balls
Simulations of the type discussed in Sect. 3.6.1.1 allow to calculate the mass spectrum of the glue balls in quenched approximation. We give in Table 3.3 the results for glue balls with a mass below 3GeV (from Ref. [Ba93]). The simulations range from (5 = 5.7 to (5 — 6.4. They allow for a good extrapolation to the continuum for the spin states 0 + + and 2 + + . I11 particular, the lattice states E++ and T2 + are represented by mass-degenerate states. As we have discussed above, this is a signal for the restoration of rotational invariance. The string constant K = (440 MeV) 2 (cf. Sects. 3.7.1.1 and 3.7.3) is used for the transition to the physical mass scale. This value is in good agreement with the scale determined by m p .
4 2
s
According to ef. Ba 3 , there are indications for states above 3 there are no exotic states below 3 c .
e . But
o glueballs as a new feature of gauge theories shed some light on their non-perturbative structure as discussed in Sect. 2. .3 It might be worthwhile to mention that there exists a treatment of glueballs by an effective amiltonian which is based on the vacuum tunnelling picture vB 6, vII 7 . A result of this method for the mass ratio is m ( 2 ) / m ( 0 ) 1.5, in good agreement with the lattice result of Table 3.3.
3.6.2.3
Comparison with experiment
uenched lattice approximation gives increasing evidence for a glue-hall mass of m e 1600 Me . Mow does this compare with experiments There are three 0 resonances in this mass region, /o(1370), /o(1500), and 7,(1710) with j = 0 or 2. In this energy region one expects besides the glue-ball state ), the quark states qq( )) = uu dd - 2 s s ) / / 0 and |<77(1)) = uu dd -f s s ) / / 3 (cf. Sect. 1.3.1.1). For the o mesons one expects mixing, My)
= a(v) ) - f c 0 ( y ) w ( l ) ) y = 1370, 1500, 1710.
7( )l
( ) , (3.6.3)
So far, there seems to be no agreement on the mixing pattern. owever, there is good experimental evidence that the glue ball fits in such a scheme C1 , e 7.
3.6.3
T h e c o n n e c t i o n b e t w e e n long- and short-distance physics: n o n - p e r t u r b a t i v e renormalization group
A fundamental problem of C is the relation between non-perturbative physics at long distances and asymptotic freedom at short distances. f course, lattice approximation and perturbation theory approximate the same formal Lagrangian. But having in mind the different steps in lattice approximation (Sect. 2.10), including the transition to the continuum on the one hand and the unsolved problem of convergence of renormalized perturbation theory on the other (cf. Sect. 1.6.2.2), one would like to see that both schemes match smoothly.
.
. (,6.3.1
s
s
)
2 7
T h e a s program
I ll I problem is pursued by the a s program Lu 3 . The aim of this pro c,ram is to calculate the running coupling C\S( 2) with good precision from ome low-energy data at the hadronic scale. Since the perturbative coupling "H(Q ) is not. a good choice for the description of low-energy hadron data, other definitions of the strong coupling are used. The determination of (x ( 2) from the lattice approximation involves three steps: (1)
efine a suitable coupling constant rv, together with quark masses 7fi , and measure their values on the lattice.
(2) Set the scale at which « , and mi, take their values. (3) Convert
to as(
2
)
and mi, to ms.
In the following we discuss these different points. 3.6.3.2
T h e input from long-distance physics
As low-energy input data we may take, for instance, the values of different masses, e.g. the mass of the lightest glue ball, the spin splitting of quarkonia, or the potential between heavy quarks. The corresponding lattice approximation sets the scale Aj, at which «h is determined from these data. ne possibility for the low energy input is to take the force Fqq(r) between a qq pair at a certain distance r(). The reference scale r() has been chosen such that r
-^ (ro) = 1-65,
(3.6.4)
giving 7o 0.5 fm for typical phenomenological quarkonium potentials (cf. Sect. 1.3.3.2) So 4 . If we put Fq(-, = K at 7-0 = 0.5 fm for a pure confinement potential (1.3.17), then we get k = 1.65 x r^ 2 = 1 . 3 e f m 1 . The difference to a realistic value K — 1 e f m - 1 can be attributed to the gluon-exchange part (1.3.16). Since there are not yet good lattice data with dynamical quarks, we have to restrict ourselves essentially to the quenched approximation. In this case we do not have to consider quark masses and their scale dependence. For the transition to the continuum, asymptotic scaling of the bare coupling constant /o, or / l = 6/floi is used. In particular, we recall Table 3.1 and
4 2
s
Sect. 3.6.1.1. In order to get a s with good precision, we have to improve perturbation theory. ne of these attempts goes back to Parisi Pa 1 , who noted that the poor convergence of bare perturbation theory may be due to the appearance of tadpole diagrams. These diagrams can be re-summed by replacing the bare coupling go by gp = go/{\ - QQ/6 . . . ) . This is one example of the choice of a suitable coupling oi, which possibly leads to a better approach in lattice constant a to the physical continuum value than those of in Table 3.1. Another possibility is the Symanzik Liischer improvement program (Sect. 2.10.6.6).
3.6.3.3
on-perturbative evolution
The conversion of orp, or in general cv n to drs cannot be approached directly in a controlled manner with high precision. As we have seen in Sect. 3.6.1.1, the typical lattice approach to the continuum involves a scale of order 0.1 fin (2 e ) - 1 whereas o s is reliably measured at the -boson mass, as(M ) 0.12. Such a large scale difference cannot be covered by numerical simulation on a single lattice, since this would require a size of 1000 1 lattice points. There are schemes to overcome this problem. These are based on finite-size techniques and the non-perturbative renormalization group. It is beyond the scope of this book to describe these methods in all technical details. Instead we refer to the work of M. Liischer Lu and present directly the results obtained for a s {My and A^g from low-energy data.
3.6.3.4
esults
Starting from ( 3 . 6 . 4 ) as low-energy input, the non-perturbative evolution for the quenched approximation results in Aj^g = 0 . 6 0 2 ( 4 ) / r o . As explained above, one takes ro = 0.5 fm from charmonium physics giving A
MS = ( 2 3
1
)
M
(3-6-5)
or by the two-loop formula (3.1.16) for NQ = 0 a s ( M l ) = 0.07 .
(3.6.6)
e may compare this with the result from asymptotic scaling of the bare coupling constant in Sect. 3.6.1.1, Ajjg = 170 Mc or a s ( M ) = 0.075.
. 7
Prom heavy quark physics Sh 7 0.0 1. This result is based 011 the uplifting, and the relation aa(Q2) where C2( q) lies between 0 and not used.
.
one gcl,s A^q = 350 Me or M.j) determination of rvp( .2 e ) from spin = ap(e 3 2)(i 2«/ /7t C2(JVt,) dj,), 1. on-perturbative evolution to , is
The quenched value T (MJ) = 0.07 is significantly lower than the measured perturbative one, a s ( A f ) = 0.11 5 0.0020 P 00 . owever, sea quarks affect the evolution of the coupling, and they also influence the low-energy reference scale. As an example, the quenched value A^g = (23 1 ) Me yields rv s (M ) = 0.11 for five flavours. A result like (3.6.5), (3.6.6) for an unquenched simulation is still missing. evertheless, there are estimates in full C using the incomplete data with dynamical quarks to extrapolate to the physical flavours and masses. sing heavy quark data as input one gets a s ( M ) = 0.11 or a s ( M ) = 0.116 for N, = 5 Sh 7 . owever, these values have not the quality of the quenched result (3.6.5). For the determination of d s ( M ) in full C one has to take into account the running of the masses rh(Q2). Similar methods as explained above lead to some recent progress (cf. Sect. 3.6.1.3). Finally, we want to emphasize again the significance of this section. If unquenched results would reach the same controlled precision of the quenched ones, this woidd put potentially the treatment of non-perturbative effects 011 the same level as perturbative C .
.7
.
The simplest physical explanation of quark confinement can be given by describing the quark interaction by means of an attractive potential which increases as distances become larger, the confinement potential. Such a model was discussed in the treatment of quarkonia in Sect. 1.3.3.1.111 this section we investigate the question whether the concept of a confinement potential has any significance in C . Most probably, the discussion of this basic question will determine the treatment of the long range structure of C . 3.7.1
The
ilson criterion
hen defining a potential within C , the relativistic nature of the interaction together with gauge invariance gives rise to several effects which
0
s
must be dealt with first of all. In a relativistic field theory and this includes C the interaction is always linked with the creation of virtual and real particles Ak65 . This type of physics is not described by a potential. Therefore, the first step in formulating the potential concept for the confinement problem is to consider infinite-mass quarks in C . This freezes all kinetic degrees of freedom of the quarks, and there is no quark-pair creation. The colour charges of the S (3)c triplet quarks are merely static sources and the fmiteness of the propagation of the gluon interaction is not significant. In this approximation, the existence of a confinement potential between static triplet and antitriplet colour charges is a property of pure gluon dynamics. hen considering the existence of a confinement potential in pure gauge theory as evidence for confinement, it should always be remembered that this potential refers to the interaction of static quarks. The physical confinement problem is only fully solved if it can be shown that the existence of a confinement potential guarantees that low-inass dynamical quarks cannot be produced as free particles. e start this discussion with a two-dimensional model which shows the effects of such an interaction. 3.7.1.1
e f i n i t i o n of t h e c o n f i n i n g p o t e n t i a l
.
ilson i74 defined a potential between a quark q and an antiquark 1 in a colour-singlet state by considering the quantum-mechanical vacuum expectation value (C) for the path C of a heavy static qq pair from x to
(C) = {TvU(C)) ere the
=
/
A T i/(C)cxp(-S {A}).
(3.7.1)
ilson loop. (3.7.2)
represents t he finite parallel displacement along a path C as in (2.3.36). As the antiquark q at has the opposite colour charge of the quark q at x, the colour displacement connecting these sources is a closed line (cf. Fig. 3.14). The trace of U(C) occurs since the qq pair is in a colour-singlet state. The quantum-mechanical expectation value of U(C) is calculated using the euclidean path-integral formula (2/2.67).
.7
.
The gauge iuvarianee of (C) follows directly from the gauge dependence of the colour displacement (2.3.35) along a closed path,
T U{C) -> Tr a(x)U{C)g~l{x)
= Tr U(C).
(3.7.3)
To calculate the potential, C is taken as a planar, closed curve like in Fig. 3.14. Then,
T oo
m
(3.7,)
can be interpreted as a static potential (r) between a quark-antiquark pair in a colour-singlet state. This gauge-invariant definition of a potential is justified in the following section. The interpretation of the ilson integral (3.7.1) as a potential, (3.7.4), provides the ilson criterion for quar confinement. If the euclidean expectation value (C) for similarly shaped paths C decreases exponentially with the minimal surface Ar(C) = rT enclosed by C,
{C)
x exp(-«Ar(C))
for large paths C,
(3.7.5)
this is regarded as the criterion for confinement, because a confinement potential
(r)
KT,
(3.7.6)
follows according to (3.7.4) for large rectangular paths C-. The constant K is called string constant. The reasons for this are described in Sect. 3.7.3. The string constant K has dimension (mass) 2 and is a renonnalization-groupinvariant quantity which cannot be calculated by perturbation theory.
432
3
Q u a n t u m
G h r o m o d y t i a i n i c s
q
Fig. 3.14 Wilson loop C(r, T) for ihc definition of a gauge-invariant s t a t i c quark antiquark potential
3.7.1.2
The Wilson definition of the Coulomb potential
In order to give a better idea of the details of the gauge-invariant potential definition by K. Wilson, we consider (3.7.1) and (3.7.2) in the framework of QED. For this purpose, the pure electromagnetic gauge-field Lagrangian £gauge (2.3.11) is supplemented by a gauge-fixing term £f, x = (d , l A, t ) z /2. After Wick rotation, i.e. continuation to the euclidean time .T.I = i.xo, A\ — iAo, we obtain the action
5
E
{A} =
Jd4x
^{d,LAv d4x(OtlA
=
- dvA^A"
- du A'1) +
u)(d"A").
¿(¿JM,,)2
(3.7.7)
Hence, the Wilson integral (3.7.1) becomes
=|J
V[A]exp
(ie JdafApis)
x e x p (-l-jdix(d"A")(dllA„)^
.
(3.7.8)
This is a gaussian integral as (2.2.11). It can be evaluated like the example (2.2.51)-(2.2.55) but with J'D 'x J(X)I/J(X) replaced by e d.s'M,,(s). The result is W(C) = exp
f ds'1
dt" S,iV{s - i ) ) •
(3.7.9)
.'{.7
( . ¿ n a r k
c o n f i n e m e n t
53!)
Here, S,IIJ(x) is the euclidean Green function of the electromagnetic field, HIMO called the Schwinger function, in Feynman gauge, 1
C { \ _
1
x
_ V f ,4 " (27r)'./ d
• $„„(* - y) =
^^
- y).
(3.7.10)
The two-point Green function [cf. (2.1.22)] G,i„{x,y)
= -glw\AF(x
-
y),
and the Schwinger function Sl„,(x-y) continuation,
= 6,„,S(x — y) are related by analytic
S(x - y) = \AP(x -
S(p) =
-AF(p)|Po=-iW, (3.7.11)
In order to evaluate the Wilson integral (3.7.9) further, we choose a rectangle with sides of lengths r and T (cf. Fig. 3.14) for C. Hence, with (3.7.10), we have
M
*
"
*
-
«
-
"
-
-
¿
i
R
r
I r a Jo ' V
i + (r' - r»)»
Because of the factor ¿/J(/ in 5';ii/, only integrations between parallel sides give a contribution. Those integrals in which and s run over opposite (same) sides are shown in the first (last) two lines. Elementary integration gives fT
f
r
dt'dt"
IT
T
(
T2\
The integrals over the same side diverge. Systematic discussion of these divergences would involve gauge-invariant rcgularization as in Sect. 2.5.1.
432
3
Q u a n t u m
G h r o m o d y t i a i n i c s
For tlie following consideration a simpler regularization is sufficient, which deforms the contour around the singularity, rT rT RCS
X
J.
dt'dt" j
t
w
r 3
J
=
T
~ ,
. 2
-(T-e)
e U
r L
T
, „ A
I
l W - t")ï
+ 2\og^.
(3.7.13)
Adding all contributions gives
W(0«ex
T T r r — arctan —H — arctan — r r 7 I
(3.7.14)
for the Wilson integral. Hence, according to (3.7.4), the electric potential between opposite charges is
i.e. the known Coulomb potential. The singular constant term « e _ 1 represents the self-energy of a regularized point charge, ft is subtracted by the renormalization process. The logarithmically divergent term ~ log(e /rT) does not appear for paths C with a continuous tangent [Do80a]. In a general gauge, a term ((£— 1)/16tt2)c^
Quark c o n f i n e m e n t in strong-coupling approximation
In Sect. 2.10.2 we introduced the basics of the strong-coupling approximation on the lattice. The most important example is the application to the Wilson loop 1 V(C). Based on the Osterwaldcr-Seiler theorem [Os78], there is a convergent strong-coupling expansion of W(C) on the infinite lattice for small /?. This factorizes the Boltzmann factor exp(—5|r{A}) by character expansion leading to effective coupling constants v(h), where k denotes the
3 . 7
Q u a r k
c o n f i n e m e n t r>.'t I
Irreducible representations of SU(3). The expectation value of an observable in expanded in strong-coupling lattice graphs G with a given order in v(k). We have shown this for the Wilson loop W(C) in a simple model, the Zi gauge theory in two dimensions, in Sect. 2.10.2.1. For quark triplets, triality-llux conservation can be used to show that the lowest-order, non-vanishing contribution to the Wilson loop W(C) comes from the graph G'm) which has C as boundary, d G = C, and which consists of the smallest number m of plaquettes which can fill C [Wi74). According to the Wilson criterion (3.7.5) we have quark confinement in lowest-order strong-coupling approximation. In this case v(k) = v('.i) n m and the contribut ion of G^ ' is
W(C) = constant x vm = exp f ^ A r [ C ] logu^ .
(3.7.16)
The leading approximation of the string constant can be read off from this: with the lattice constant a, the expansion v •• a2ti = — log v « log (nô 2 +
= 6¡g\.
Terms of higher order of the strong-coupling approximation, described in Sect. 2.10.2, give the result
Details and the relevant graphs can be found in the literature [Dr83, Mu81]. We emphasize again, that the strong-coupling expansion gives a convergent, series expansion for the Wilson loop on the lattice if the coupling constant is sufficiently large, i.e. for small /i. Quark confinement is proven in this way under the conditions given. This gave rise to the hope that a consistent application of QCD can describe the phenomenon of quark confinement. However, for a complete proof of confinement, the non-trivial transition to the continuum must be analysed. This does not yet exist for the strongcoupling approximation (cf. Sect. 2.10.4.3).
0
s
. .
Tlie string picture
The validity of the ilson criterion . . would imply that pure gluon dynamics describes a potential between a static uar -anti uar pair that increases linearly with increasing distance. This type of potential is very different from the Coulomb potential, which appears in E and arises also in perturbative C from one-gluon exchange. Thus, there must be a dynamical process which cannot be described by perturbation theory and which causes confinement in C . ne physical picture for this stems from the theory of superconductivity and is called the chromo-electric eissner effect tH b, a i .
. . .1
The
bri osov- ielson- lesen string
The bri osov ielsen- lesen string b , i is a field-theoretical model for a linear potential between two magnetic monopoles with opposite magnetic charges in a superconductor of second ind. e start with the relativistic Ginzburg- andau agrangian Gi for a superconductor of second ind,
C
I E
-
-
-
-
I I2 -
. .1 with and r . This agrangian describes the gauge-invariant coupling of the electromagnetic field ,t with a self-interacting, scalar, complex field I . fn superconductivity, I describes an order parameter which is proportional to the number of superconducting electrons, i.e. Cooper pairs. The e uations of motion for . .1 , the Gin burg andau e uations
d Full =
ejlt = -ie I
«I -
-
2
.
. 1
give a macroscopic approximation Go , a , a of the atomistic C theory a . They arc applicable in situations where the variations in I are small compared with the coherence length cHl i.e. predominantly for temperatures in the critical range.
.'{.7
( . ¿ n a r k
c o n f i n e m e n t
53!)
Non-abelian generalizations of such gauge theories play an important role in the description of the weak interaction (ef. Sect. 4.1). In this section static, cylindrically-symmetric solutions of (3.7.19) are considered. We use cylindrical coordinates p,0,z, and t: x\ = pcos 0, x2 psin0, x-i = xz, x<\ = t and correspondingly for Afl: A,„ Ao, Az, and A/, in these coordinates we make the ansatz ein0G(p),
(.T) =
7i =
/l„(a:) = A(p),
0, ± 1 , ± 2 , . . . ,
At = Ap = Az = 0.
(3.7.20)
This implies that the non-vanishing element of the field-strength tensor, the magnetic induction in z direction B(p), has the form
-Fn{x)
= B(p) =
A|\pA(p)],
(3.7.21)
and the field equations reduce to Id d — -TPT+ p dp dp
n\2
( .. . [eA(p)--] y
PJ
u2
-Lf+-GÀ (p) G{p) = 0, 2 2
= ejo (p) = 2e
£-\pA(p)] dp P dp
V
(eA(p) - - W ( p ) . PJ
(3.7.22) (3.7.23)
The energy of such a field configuration is obtained from the Lagrangian (3.7.18). In temporal gauge, At = 0, the Hainiltonian has the form
+ |a,*i2 + i(v - i,A).if + £ ( n f -
= Jd*x
-
2
* I
D
*
= j dzK{A,G},
[
2
?
+
( È
0
' " ' )
Ç )
2
(3.7.24)
8
s
where represents the energy per cylinder length. The solution of the field equations (3.7.22) and (3.7.23) with lowest energy, i.e. with = 0, the classical vacuum solution, is p) =
(3.7.25)
G(p) =
Transformation of this solution into cartesian coordinates shows that the associated potential
(x)
arccos
= c
^
. f1 s A
. ) A )
(3.7.26)
is a pure gauge. Accordingly, 4 (:i;) is obtained by the corresponding gauge transformation of the constant p / A , l (-i:) =
(3.7.27)
vA
ow, we consider other solutions with finite energy. As is made up of oo. quation positive expressions, these must disappear individually for p (3.7.24) shows that (p) and G{p) thus approach the vacuum solution for oo. Therefore, these solutions can be written as p {p) = - + ep
(p),
(3.7.2 )
G(p) = -£= + G(p). vA
Inserting this info (3.7.22) and (3.7.23), we can read off the asymptotic oo, behaviour for p
{p) ~ p~ '2 exp
epp\
,
G{p)
p- /
" .
(3.7.2 )
Similarly as in (2. .5), we apply Stokes theorem to the magnetic flux 4 jg(/C) through a large circular surface . with boundary C,
4 b(/ ) =
JK
Bdf =
J ok
/l, i (.s)d.s
1
= lim >0 p
P~ °
JO
d =
e
n. (
.7.
0)
.'{.7
(.¿nark
c o n f i n e m e n t
53!)
Because of the strong decrease of A(p), only the asymptotic vacuum solution contributes at the boundary. The fact that n. is an integer follows from the uniqueness of (:/;) according to (3.7.20). This means that the magnetic, flux i,i quantized by the integer topological quantum number n [Lo50, Al>57]. As an integer valued functional of A and , the flux B(/C) can be deformed continuously without changing its value. The boundary condition for p > 0, A(p)
~
p->o
constant,
—
or
A(p)
~
71
p->o
,
ep
(3.7.31)
follows from fK: B d f = —27r(pA)l/,.,o- This condition establishes the form of a solution by a power series around p = 0. As A(p) ~ constant according to ( 3 . 7 . 3 1 ) , and since the repulsive potential ~ n2/p¿ in ( 3 . 7 . 2 2 ) acts like the centrifugal potential in the Schródinger equation, the form of A(p) and G(p) for small p is A(p)
= C + ...,
G(p)
= Cnp"
+... .
(3.7.32)
This implies for the magnetic field strength, using ( 3 . 7 . 2 1 ) and ( 3 . 7 . 2 3 )
B(p) = B{ 0) + l"dp'±B(p') Jo dp' = B{0) +
J\lp'2e
= B( 0) +
\ cA -
l"dp'±^±[p'A(,/)] Jo dp' p' dp'
G2(p').
(3.7.33)
Inserting the expansions (3.7.32), this gives B{p)
= j5(0) - eClp2n
+ ....
(3.7.34)
In general, the complete solution of the non-linear equation ( 3 . 7 . 2 2 ) cannot be given analytically. However, there is a limiting case, \Jl¡Xep 2/i, for 2 which (.c) gets constant, o = y/2p/X = finite, with 2//./A -> 0, the London limit. In this case ( 3 . 7 . 2 3 ) can be reduced to Bessel's differential equation. The final result contains the modified Bessel function Ko,
B{p) =
2
-^KO
(e p y f \ p j
(3.7.35)
0
s
P
Fig. 3.15 T h e ielsen- lcsen string: (p) 3( ) (dashed line) a n d G{p) G{oo) as a function of p in lattice units 0.0 fin
The string constant,
(solid line)
i.e. the energy per cylinder length, becomes
/ co
« = 27T
pdp- 2(p)
= -p
2
(3.7.36)
in this limit. A numerical solution is given in Fig. 3.15. The parameters are taken from a lattice simulation Ba a which we discuss in detail in Sect. 3.7.3.3. It incorporates the features discussed here. 3.7.3.2
T h e c h r o m o - e l e c t r i c M e i s s n e r effect
The classical calculation of the magnetic string of Sect. 3.7.3.1 is an interesting intuitive model for quark confinement. The following features are worthwhile to mention: 20 " Some of t h e m like the order parameter and the m a s s generation occur also in the weakly interacting lcctroweak Standard Model.
3.7
uark confinement .
The undisturbed superconducting medium is characterized by the value of the order parameter = i A, the classical vacuum value of f>(x). •
In the medium, a magnetic llux tube, i.e. a magnetic string, is formed between very distant magnetic poles. The property of a superconductor of expelling a magnetic field (Meissner effect) leads in this case to a magnetic induction B(p) being concentrated in a string. As a magnetic field cannot penetrate a superconductor, the latter is regarded as a perfect diamagnet with permeability pinagn — 0.
•
The constant, finite energy per string length [K in (3.7.24)] gives a linear potential when the monopoles are very distant.
•
The magnetic llux is quantized, (3.7.30), and so arc the magnetic charges of the monopoles. Quantization of the classical magnetic flux in units of 2-rr/e shows the importance of topological quantum numbers for the non-perturbative structure of this mechanism described here.
•
The magnetic flux is stabilized by a circular current jg = [cf. (3.7.21) and (3.7.23].
•
The medium resumes its behaviour as a normal conductor, \ j>\ ~ 0, at the centre of the magnetic string.
•
The penetration depth of the magnetic field is \ Xf2 {ep) [cf. (3.7.29)]. In dynamical solutions, this corresponds to the Compton wave length of a massive "photon" (cf. Iliggs mechanism in Sect. 4.1.2.1).
•
The displaced scalar field is restored to the vacuum value with a coherence length p, 1 (3.7.29). In perturbation theory of the Higgs model this is related to the Higgs-boson mass M = \/2\i (Sect. 4.1.2.1).
AB(p)/Ap
For QCD we consider the Abrikosov-Nielsen- Olesen string as a classical model for the Wilson criterion of quark confinement. Because now we have a string between electric quark charges and not between magnetic monopoles, we have to exchange the electromagnetic fields by a duality transformation, E B , B —• - E . The gluon field between the colour charges of a separated static quark antiquark pair is compressed to a string in the QCD vacuum. Similar to the Meissner effect, the QCD vacuum acts as a confinement medium (cf. Sect. 3.7.4) in the Abrikosov Nielsen-Olesen string. This confinement mechanism is called the chromo-electric Meissner effect [tH81, tII82], The constant field energy per unit length of the string gives the
0
s
magnetic Abrikosov Nielsen-Olesen string magnetic monopolcs with opposite magnetic charges magnetic flux between magnetic charges perfect diamagnet
sources string of embedded in
Mmagn
produced by internal phases
=
0
(superconductor) magnetic Mcissner effect normal conducting
chromo-electric gluon string non-coi 111 nu tative charges of quarks and antiquarks chromo-electric flux between colour charges confinement vacuum Pel = 0, e = 1 / P c l (dielectric constant) chromo-electric Mcissner effect asymptotically free
Tab. 3.4 The analogy between magnetic and chroino-electric strings linear potential required by the Wilson confinement criterion. In Table 3.4 the analogy is conceptionally carried further. In order to be more explicit, we discuss the modifications of Sect. 3.7.3.1 for the description of a confining string. We consider the string between qij sources on the 3 axis at 2 -4 ±oo. In the scheme described in Table 3.4, magnetic and electric fields are exchanged like in a duality transformation. Therefore, we have to substitute: (a)
> -
(b) < M £ )
j , the electric component in 3 direction; el = !K £ 3 d f , (3.7.30);
(c)
27r/e —> 1, because we give unit charge to the quarks. This substitution corresponds to the Dirac condition e m g = 2-KN, N integer, in a theory with electric and magnetic charges [Di31, Di48, tH81];
(d)
y / \ / 2 / ( e / i ) ->
(e)
V2/11 —>
(f)
Ao —> Ao. In order to get dynamical equations, one has to introduce dual potentials Ffiv = dlt - i)u lt\
(g) jo
the dual penetration length, (3.7.29);
the dual coherence length, (3.7.29);
^o, the magnetic current;
(h) G'(p) -> G(p) = |$(2;)I, (3.7.20), where (.T) is an effective description of the confining vacuum.
3.7
Q u a r k
c o n f i n e m e n t r>.'t I
Up to now we have assumed uncritically that we can use the duality concept in non-ahciian gauge theories like in abelian theories. One hypothesis could justify this assumption, the conjecture of abelian dominance. This asniiines that there is a gauge in which fields related to an abelian subgroup ofSU(A^) dominate the long-distance physics. We make some comments on this assumption in Sect. 3.7.4.1. We want to mention two effects not described by this picture of confinement: i)
The string thickness increases with increasing string length [Lu80b] in quantum mechanics.
ii)
The centre of the gauge group plays an important role in a non-abclian gauge theory. The colour charges of static quarks have a triality which differs from zero. In a pure gauge theory, these cannot be screened by the triality-zero colour charges of the gluons. A quark charge is always a source of a chromo-electric flux which can end only in a colour charge of opposite triality, e.g. an antiquark. As mentioned above, the chromo-electric Meissner effect concentrates this flux into a string and thus produces a confinement potential. Hence, the representation of the centre of the colour group, characterized by triality, is important in the dynamics of quark confinement.
3.7.3.3
L a t t i c e a p p r o x i m a t i o n of t h e s t r i n g p i c t u r e
There arc first attempts to demonstrate string formation between static qq pairs in lattice approximation of pure gauge theories [Ba95c], Figure 3.16 shows such a string-like configuration of the action density in a pure SU(2) gauge theory. For the string picture, only the difference of the vacuum expectation values (vevs) of observables calculated from the action with and without static sources are relevant. The lattice approximation of the action density, i.e. the Lagrangian at a fixed time, follows the line of derivation of the Wilson action in Sect. 2.10.1.2. We consider two sources q and q at spatial distance R. The action density on the lattice c / i ( n ) consists of a chromo-electric part £/f(n) and a chromo-magnetic part Bn(n),
(3.7.37)
5 4 4
3
Q u a n t i m i
C h r o m o d y n a m i c s
0.4
1.0 li'ldlivi) prior
Fig. 3.16 A S U ( 2 ) string in lattice a p p r o x i m a t i o n [Ba95c]: b e t w e e n t h e s t a t i c q<) sources at a d i s t a n c e R = 1.2 fin t h e action density develops a string-like configuration. Close to t h e qq sources t h e action density b e c o m e s singular.
with £„(n) = (0,7?|E 2 (n)|0, R) - (0|E 2 (n)|0> = (|E 2 (n)|)| 0 ,*)_!<», BR(n) = ( 0 , i 2 | B 2 ( n ) | 0 , R ) - (0|B 2 (n)|0) = (|B 2 (n)|),o,«)_|o)(3.7.38) Tliesign conventions in (3.7.37) correspond to the action density C = ^(E 2 — B 2 ) in the continuum witli Minkowski metric. The lowest energy state with these sources, i.e. the vacuum in the presence of the qq sources, is denoted by |0,R). We have seen that vevs of static quantities, like masses m in Sect. 2.10.1.4, or V(r) in Sect. 3.7.1.1, can be obtained as a T oo limit. In this sense, vevs of operators (0, /?.|0(n)|O, R) can be obtained as a T —> oo limit of correlations of C(n) with Wilson loops [Ba95c] ,m T\\ - (Q(n,T/2)YV(R,T)) (C(n, i ))vv =
- (ö(n, T/2))(W(R, (W(R.T))
T)) (3.7.39)
3.7
Q u a r k
c o n f i n e m e n t r>.'t I
The Wilson loop W(R,T) stands for the string of the qq pairs. The correlation (3.7.39) yields directly the difference of the vevs with and without the (/(/ sources. As an example, we consider the calculation of ¿'u(n) and Bn(ix). Similar as in (2.10.12), we get i V = l- Tr[U(dV)}
= 1- |
for the oriented plaquette V = in the limit T —> co 2B -^(Un(n,r))w ^ 20 -J^(Uij(n,T))w
+
...
no sum
(3.7.40)
v\. Inserting UIU, in (3.7.39), we obtain
{*?(n)>|o,fl)-|o>,
(3.7.41) i,j,k
cyclic.
(3.7.42)
The expressions for the field [Note that Ef = E?E? = 21\(E?TaE!?Tb)]. operators Ei(n) and / ^ ( n ) can be found in a similar way, based on the geometric interpretation of Sect. 2.10.1.1. The lattice approximation of such extended structures encounters many difficult problems, which we suppressed in our cursory presentation above. •
One has to compromise between a large physical lattice extension tiN^ and a small a for the transition to the continuum ( N s is given by the computer power).
•
In order that W(R,T) describes the qq string well, the ground state overlap must be increased by suitably smeared operators. Corrections for finite T in the limit T —> oo have to be introduced.
•
One has to reduce the noise in the combined heat-bath and overrelaxation procedure in the algorithm for updating the gauge fields (cf. Sect. 2.10.3.5).
•
There are the general lattice problems like the transition to the continuum, finite size effects, etc. The treatment of a string in an N s x N t = 16'1, 32 4 , or 48 3 x 64 lattice for [i = 2.50, 2.65, or 2.74 requires in the limit T —> oo ten thousands of measurements [Ba95c].
All these efforts lead to the picture shown in Fig. 3.16 together with the following results.
4
1)
s
There is a static potential proximated by
{r) which for r > 0.5 fm can he well ap-
(3.7.43) This result is obtained by fitting the potential in lattice units for different values of ft, and transforming the different fits with help of the canonical string tension S = 440 MeV into physical units. For different ft, asymptotic scaling is approached reasonably well. A constant term is omitted in the lattice fits. Because of the infinite self-energy of the static sources [cf. the hist term in (3.7.15)], the lattice approximation is not valid in these regions. 2)
The universal constant with the value 7r/12 follows from the solution of a classical string equation [Lu80b]. It is the second term of an asymptotic solution for large r. The lattice approximation confirms this result. [For r
3)
According to (3.6.4), the reference scale is 7*0 ~ 0.5fm. Under the assumptions (i) that the quenched approximation leads approximately to the same potential (3.7.43) for SU(2) and SU(3) (this is confirmed by lattice calculation) and (ii) that the relation between quenched and unquenched approximation described in Sect. 3.6.3.4 is correct, the potential (3.7.43) is in agreement with the perturbative coupling constant o^g « 0.12.
4)
The transverse width of the flux distribution in the plane perpendicular to the string halfway between the sources is described by Fig. 3.16 with arguments transformed by duality [Ba98]. The result of the "lattice measurement" of the penetration length <5 and the coherence length £ is 6 = 0.16(2) fm,
£ = 0.27(3) fm.
(
These results arc valid for separations /?. between 1 fm and 2 fin.
.7.44)
.7
.
Fig. 3.17 Screened p o t e n t i a l of the Sclnvinger model
3.7.3.4
Quark confinement with dynamical quarks
The Wilson criterion for confinement applies only for pure gauge theories. In QCD with dynamical quarks, colour charges of quarks and antiquarks can be screened by quark-pair creation. The physical states should be the conventional, colourless hadrons which can be considered as bound states of (/(] and qqq in the simple quark model. In the string picture there is shin where the string of the source brea ing. This occurs at those distances has enough energy so that it can decay into a pair of q and q mesons. String breaking is not yet established in the lattice approximation of SU(2) and SU(3) gauge theories with dynamical quarks. All the problems of observing extended structures discussed in the last section occur even more in unquenched approximations [ScOO]. We show string breaking in a (1+1)dimensional QED model with "photons" coupled to massless fermions (Sch inger model [ScG2]). Since in (1 + 1) dimensions the Coulomb potential is {r) = e 2 |r|/2, we have confinement according to the Wilson criterion.
18
s
For large r > 0, the Coulomb potential is screened by the qq pairs. Figure 3.17 is based on an exact calculation of the model in a finite volume [Di92, Di94]. Its unquenched lattice approximation is based on staggered fermions treated by a hybrid Monte Carlo algorithm. Figure 3.17 shows the different steps: (i)
Coulomb potential in a finite volume (dot-dashed line);
(ii) Coulomb potential in a finite volume with topological sectors omitted (dotted line); (iii) Screened potential in an infinite volume (dashed line); (iv) Screened potential in a finite volume (solid line); (v) Lattice approximation with [5 — (ea)~2 = 10, where a is the lattice constant and e the coupling constant in the (H-l)-dimensional Coulomb potential with dimension of a mass (squares). Lattice approximations of such low-dimensional models may give valuable hints on what one expects in more realistic situations (cf. Sect. 2.9.1.1 and Sect. 2.9.2.1). 3.7.4
Long-range correlations of the Q C D v a c u u m
As an outlook on recent developments of some old ideas [tH81, tH82] on nonpcrturbative properties of gauge theories we discuss again the dual Meissner effect (Sect. 3.7.3.2). As already stated, we consider the combination of lattice approximation with the intuitive pictures of topological properties derived from the semi-classical approximation as the appropriate method for non-perturbative problems. The progress in this program is due to the fact that computers now allow for the treatment of volumes of sizes ~ (23fm) 4 with a resolution 0.05-0.1 fm (cf. Sect. 3.6.1.1). However, there are still many unsolved shortcomings of this approach. We mention only two. •
It is not yet finally established how much the intuitive picture depends on the chosen gauge.
•
In lattice approximation the study of long-range effects depends on the cooling (Sect. 2.10.3.5). The effect of cooling is not yet completely understood.
Thus, the non-perturbative theory of QCD is not yet firmly established.
3 . 7
3.7.4.1
Q u u r k
c o n f i n e m e n t
5 4 9
Maximal abelian gauge and the chromo-electric Meissner effect
A|>plying the duality concept of Sect. 3.7.3.2 to the Ginzburg Landau effective Lagrangian (3.7.18), we get the (euclidea») Lagrangian of the dual gauge theory
C=
+ \ (||2 -
+
(3.7.45)
with {D'l$)
= {dlt + igingcll)
G,lu = dtlcu - duclt.
(3.7.4G)
How can one understand dual gauge theory as an effective gauge theory? This question is approached by the conjecture of abelian dominance and the condensation of magnetic mono-poles [tH81, tH82]. We illustrate the main concepts of this procedure for SU(2). The meaning of maximal abelian gauge is simple: a regular SU(2) potential A,t(x) = g /1^(.T)T',/2 is diagonalized by a gauge transformation as far as possible. To this end, the expression R{A} = I d4rc [¿¿(xM 1 ^®) + Al{x)A2'"{x)}
(3.7.47)
with the components of the gauge field corresponding to non-diagonal Pauli matrices is minimized by a gauge transformation w(x), A^{x) = w M ^ W w - ' f i ) - iuj(x)dliij~l(x)
= A,t + M„,
(3.7.48)
where A,t is defined as the homogeneous transformation of A,t and M,L as the inhomogeneous part. The condition for a local extremum of the functional /?.{A} is [d,t ± igAl(x))AÌ(x)
= 0,
Aft{x)
= A)t{x) ± iA f a ) .
(3.7.49)
5 < 5 5 0 3
Q u a n t u m
C h r o m o d y n a m i c s
In this gauge, we define the abelian field a,,(x) and its field strength as « /t (x) = iTY[r 3 ^( 3 :)],
f/n,(x)
(3.7.50)
f,w[x) = d,la„{x) - dvatl{x)
- d,,Al - d„Al + d,tMl
-
(3.7.51) The definition of the abelian potential a,L(x) still admits gauge transformations of an U(l) subgroup of SU(2), <)(x) = cxp(-i£>(x) T3/2), a'tl{x) = a^x)
+ -dtl6(x).
(3.7.52)
Thus, cij,(x) plays the role of an abelian gauge field. The fields AP'J:(x) transform as charged fields (cf. Sect. 2.3.2.5), A'^ix) = e ^ A ^ i x ) .
(3.7.53)
According to (3.7.48), the first term in (3.7.51) originates from a gaugerotated regular vector potential J43. By the abelian Bianchi identity, this part of the dual field strength f,u/ = spupa Jl>a ¡2 is divergence-less. Only the term involving Mp might be singular and contribute to a non-vanishing, magnetic, conserved current [I
k'1 = -¿-ft,/' 1 ",
= o,
47T
(3.7.54)
as kti = -±-elll/p„Tr(T3d,'(d"Ma
o7r<7
=
W
Tr(r3r[A4^,
-
M°]).
d"Mp))
(3.7.55)
In the last step we used the explicit form of AAp = — icjc^u;-1. A detailed analysis shows, that M p becomes singular at the position of a magnetic charge, i.e. of a magnetic monopole [tII81, Su95, Br97].
3.7
Q u a r k
c o n f i n e m e n t r>.'t I
The conserved current k,,(x) generates a symmetry U(l)t„B. However, one assumes that this symmetry is spontaneously broken (cf. Sect. 4.1.2) by a condensation of the magnetic monopoles, <M*)My)>
= <4WV
- y)f(x)
+
(3.7.56)
In a translation-invariant field theory, J(x) is constant and describes a mass in tree-graph approximation. Phenomcnologically, one can describe the condensed monopoles by a field (.T). Then, the conjecture of abelian dominance and monopole condensation suggests a possible interpretation of the basic fields of dual SU(2), Gtiu{*), c;i(a;), and (x) [cf. (3.7.46)] by the identification Glu,[x) = flw{x),
c,i{x) = a,1( x),
(a;) = 4>(rr).
(3.7.57)
The coupling constants of dual SU(2) are A, p, and g Ing according to (3.7.45) and (3.7.46). The magnetic coupling <7mg satisfies together with the ("electric") coupling constant g the Dirac quantization condition [Di31, Di48], !l
¡i = - j - 5 . 2 3 f m - 1 , ()2 =
.,2 A
« 0.5 fm - 2 .
A = 8tt 2 (5V ~ 55.28, (3.7.58)
In a more systematic derivation of dual SU(2) one integrates the path integral over the massive charged fields A*-(x) [Ko98]. If one takes into account the gauge fixing by the BRS procedure (Sect. 2.4.3.1), one can show that a m g = (/2,g/47r satisfies the renormalization-group equation in one-loop approximation. Thus, a m g (Q ) should run consistently with the Dirac quantization. Unfortunately, there arc no Q2 regions where both o:nig and a are small. The maximal abelian gauge can be approximated on the lattice [Kr87]. Lattice calculations for the string tension show evidence for the importance of the diagonal gauge-field elements, i.e. for abelian dominance. They indicate also the existence of condensates of magnetic monopoles [Ha98b].
2
s
1
2
ft
1 -2
Fig.
.1
. .4.2
I .1
0.2
.4
.
.
1
Instanton distribution the p o i n t s represent lattice results, t h e straight line to t h e left gives the result of p e r t u r b a t i v e semi-classical a p p r o x i m a t i o n , and t h e line to the right gives a it to dual g a u g e theory from Ref. Ri .
attice results for instantons
Topological configurations seem to play an important role in the understanding of non-perturbative aspects of gauge theories. Figure .1 illustrates, how instantons fit in this picture Ri . It shows the density distribution of instantons D\(p, a po 2. .121 of the si e p called in ect. 2. . measured by a cooled lattice approximation m . For small instantons, i.e. p . fm, t Hooft s semi-classical approximation ect. 2. . .2 wor s well. The straight line gives the result of the twoloop approximation of 2. .121 . It is an absolute prediction of Di(p,a(po)) without any fitting parameters. The g parameter is determined by the non-perturbative evolution of ect. . . .4 2 1 eV. The pa. fm was introduced in this context. rameter e speculate that the large instantons are screened with a penetration length of the dual eissner effect. n extension of t Hooft s calculation
3.7
Q u a r k
c o n f i n e m e n t r>.'t I
I,'¡« I t. 2.9.3.2) contains such an cffcct. The result is an additional contribullnn in the action, A S = 47ra/r()-. We include such a contribution in the density distribution Di(p) [Sh99]. A fit for p > 0.6 fm with the ansatz Dx(p) =
= constant x p V ^ W
(3.7.59)
gives ()2 = 0.39 fin - 2 . It is included in Fig. 3.18 as a line interpolating the lattice points. It. might be compared with the result (3.7.58) from the SU(2) parameters (3.7.44), ()2 = p 2 /A « 0.5fm" 2 . In view of the comparison of Nl 1(2) with SU(3), and all the other approximations mentioned above, this .•.reins to support these considerations. With this interpretation, Fig. 3.18 shows how the instanton interpolates between the perturbative (quenched) SU(3) QCD and the long-range part described by the dual gauge theory. 3.7.4.3
Experimental consequences
What are the experimental consequences of this complicated picture of the non-perturbative QCD vacuum? We mention only three examples. There is the prediction that for high physical temperatures (cf. Sect. 2.2.4.4) a deconfining phase transition takes place [Co75a]. This means that quarks are no longer confined at high temperatures but form a plasma of approximately free quarks and gluons interacting according to asymptotically free QCD. Lattice calculation gives Tc « 270 MeV for the transition temperature in quenched approximation and Tc ss 170— 190 MeV for 3-flavour QCD [Ka00]. According to the picture above, the magnetic-monopole condensation is responsible for quark confinement. Indeed, lattice calculations show a sharp decrease of the order parameter of magnetic-monopole condensation at the transition temperature [Di99]. Experimentally, one tries to generate a quark-gluon plasma by high-energy heavy-ion collisions [Ap99, SaOOa]. In the theories about the development of the early universe, this phase transition plays also an important role [Ko90]. Many "QCD-inspired" phenomenological models on non-perturbative features of elementary particle physics are based on dual gauge theories, i.e. on dual QCD. We mention only the model of the stochastic vacuum [Do88]. This model tries to explain the main features of high-energy hadron-hadron scattering and photon hadron scattering by non-perturbative QCD [Do94,
5
4
s
I3e . It relates the long-distance correlation function of the field strength to dual C Ba . Since instantons and their derivates play such an important role in 11011perturhative gauge theories, it would be of utmost importance to see experimental signals of their actions i .
.8
A
s
s
Twenty-five years after it was proposed, C has become the generally accepted theory of the strong interaction. Since then, effective methods for the calculation of physical quantities have been worked out. For hard-scattering processes a consistent perturbation theory is available. In most cases twoloop calculations exist, sometimes there are even three-loop results, and for the ^-function a fourth-order calculation was performed. Methods for the treatment of C in the strong-interaction region have been developed. These include the strong-coupling expansion and the lattice approximation of C which has found many applications. The investigation of the topological properties of C has let to a deeper understanding of this non-abelian gauge theory. The physical meaning of the corresponding nonperturbative instanton solutions still has to be investigated. The accuracy of these perturbative and non-perturbative calculations is in most cases adequate for a comparison with the experimental results. e have shown this for the perturbative calculations for deep-inelastic scattering, lepton-pair production, the total hadronic cross section in e e annihilation, and jet production. The non-perturbative methods were applied to low-energy hadron physics. In this way the masses and other properties of mesons, glue balls, and baryons were obtained from the description of these states as bound states of quarks, antiquarks, and gluons. Based on C , models for heavy-quark physics on the one hand, low-energy effective theories in the form of chiral perturbation theory on the other hand, and many more were developed. A huge amount of detailed experimental and theoretical results is available. e conclude this discussion of C by considering the results for one of the fundamental constants of nature, the strong coupling constant gs, or « s = y/47r. This enters all C results for physical quantities and in this sense has to be universal, i.e. the same in hard-scattering processes, bound state properties, and low-energy scattering. sing the renorinalization group
)
--
Theor
L
Data
s
eep Inelastic Scallcring c Annihilation adron Collisions eavy uarkoma
Lanice
as(
2
A
L
.8
A
• o
•
:•!
I" 251 Mc
0.3
21 17 Me
0.2
0.1
10
Fig. 3.1
100
Q
S u m m a r y of e x p e r i m e n t a l d e t e r m i n a t i o n s of a a ( = ( n e x t - t o - ) n e x t - t o - l e a d i n g order
2
) (from
ef. Be
)
( ) L
the value of strong coupling is usually given at since the most precise determinations come from this region. Instead of a s ( M ) in many cases the as mptotic
scale parameter A of
C
(cf. Sect. 3.1) is given.
In Fig. 3.1 , a summary of the measurements of the running coupling
as(
2
)
together with a C fit in the MS scheme for five flavours is shown. ver a region from 2 e up to 200 e an impressive agreement between theory and experiment is found. The best fit gives the values Be A. I -- 2 1 3 ^ Me ,
a s ( M ) = 0.11 4
0.0031.
As a by-product, the number of colours is determined as via the dependence of the ^-function on NQ.
(3. .1)
c = 3.03 0.12,
5
21
3
uantum Chromodynamics
IS
pol. sirct. fctn.l
IS
Bj-S
IS
LS-S
T-dccays
(L P
xF-,
IS)
F2
e-, p -
jets
ISI
shapes
lattice decays
A C
e e" c had e e e e
jets jets
shapes 22 shapes 35
e e
shapes 44 shapes 5
e e )
^Itflwdl c e jets e C" jets
pp bb pp.pp- y CT(pp-- jcts) F ( « - - had.) IL P e e s c a l i n g . viol. jets shapes 1.2 e jets shapes 133 e jets shapes 161 e jets shapes 172 e jets shapes 1 3 e jets shapes 1 e 0.0
L P
0.10
0.12
0.14
s(Mj) Fig. 3.20 S u m m a r y of e x p e r i m e n t a l d e t e r m i n a t i o n s of a s ( A / ) (from
cf. Be
)
A comparison between different measurements of a s (My) is shown in Fig. 3.20. The dashed line represents the statistical expectation value of a 3 (M ), the grey band corresponds to the one-standard deviation range. This illustrates the impressive agreement of C with data from many different experiments.
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h s.
C
Coll.),
So 4
. Sommer, ucl. h s. B411 (1 4) 3 .
St77
. Sterman and S.
St 5
. Sterman et al., ev.
Su 1
5
(1
)
einberg, h s. ev. ett. 3 (1 77) 1436. od. h s. 67 (1 5) 67.
L. . Surguladze and M.A. Samuel, h s. ev. ett. 66 (1 1) 560.
T. Yoshie, Nucl. Phys. D (Proc. Suppl.) G3 (1998) 3.
|/,c73]
A. Zee, Phys. Rev. D8 (1973) 4038.
[Zi73]
W. Zimmermann, Ann. Phys. 77 (1973) 536 and 570.
[Z191]
E.B. Zijlstra and W.L. van Neerven, Phys. Lett. B 2 9 7 (1992) 377.
[Zi92]
E.B. Zijlstra and W.L. van Neerven, Nucl. Phys.
B273
B383
(1991)
476
(1992) 525.
and
4
auge theories of the electroweak interaction
As discussed in Sect. 1.4, the electroweak interactions of fermions exhibit a symmetry under the group S (2) x (l)y. Promoting this symmetry to a gauge symmetry yields a gauge theory of the electroweak interaction. owever, in contrast to the gauge bosons of the electromagnetic and of the strong interaction, the electroweak gauge bosons are massive. Therefore, the formulation of a gauge theory for the electroweak interaction requires a new concept, spontaneous symmetry brea ing. By applying this concept to gauge theories we arrive at the only known form for a renormalizable quantum field theory for the weak interaction with massive gauge bosons. pon combining the S (2) x ( l ) y gauge theory with spontaneous symmetry breaking we construct the Standard Model of the electroweak interaction e67, Sa6 , 170 . This theory is in accordance with all known experimental facts about the electroweak interaction.1 As a remnant of spontaneous symmetry breaking it predicts a physical scalar boson, the iggs boson i64, n64, u64, i67 , which has not been detected experimentally so far. owever, its mass can be confined to a limited range by comparing the lectroweak Standard Model with precision data. In Sect. 4.1 we explain spontaneous breaking of global and local symmetries as well as the quantization and renormalization of quantum field theories with spontaneously broken symmetries. In Sect. 4.2 we introduce the Lagrangian of the Standard Model of the electroweak interaction. Simple applications of this model are given in Sect. 4.3. Its quantization and renormalization are performed in Sects. 4.4 and 4.5, respectively. Finally, the electroweak radiative corrections for various processes are discussed in Sect. 4.6. e u t r i n o oscillations can in principle b e described w i t h i n t h e S t a n d a r d Model by adding right-handed neutrinos (cf. Sect. 5).
.
4.1
s
(7
s
s s
xact pure gauge symmetry demands that all gauge fields are massless. n the other hand, we know from experiment that the electroweak interaction is mediated, apart from the photon, by massive vector particles. xplicit mass terms 2 ,t ' 2 must not be added to the Lagrangian as they violate gauge invariance. This, however, is crucial for the Slavnov-Taylor identities (Sect. 2.4.3) which .are needed for the proof of renormalizability and unitarity as outlined in Sect. 2.5.4. The only known possibility for introducing masses without violating gauge invariance is the iggs mechanism which is based on spontaneous symmetry breaking. Thereby, gauge fields are coupled with additional, postulated scalar fields which, owing to their selfinteractions, acquire asymmetric non-vanishing vacuum expectation values (vevs), despite the gauge symmetry of the Lagrangian. These scalar fields are also necessary to guarantee tree-level unitarity in a theory with massive vector bosons L173, Co74a . Some of these scalar fields, the so-called iggs fields, give rise to physically observable scalar particles. In order to define the meaning of spontaneous s mmetr brca imj2 we consider a Lagrangian that is invariant under a group of transformations G. There are two alternatives for the ground states 0), i.e. the states with lowest energy: 1.
nly one unique ground state 0) exists; in this case, 0) must be invariant under transformations with respect to G.
2.
There is a set of degenerate ground states which transform into one another under the group G. All these ground states are physically equivalent. owever, each of them selects a direction in representation space. Choosing one of them as the ground state of the theory, the G symmetry is said to be spontaneously broken.
Systems with a finite number of degrees of freedom usually possess a unique ground state. The possibility of spontaneous symmetry breaking is typical for theories involving an infinite number of degrees of freedom. The phenomenon is, independently of its application for mass generation in gauge field theories, of distinct physical significance. For instance, the spontaneous breaking of chiral symmetry has been discussed in Sect. 3.5.1.1 2
T h e p h r a s e w a s c o i n e d by M. Baker and S.L.
lashow
BaC2 .
)()
I
s
A well-known example is the infinitely extended ferromagnet below the Curie temperature e2 : although the interaction of the magnetic moments is symmetric under rotations, in the ground state all moments are aligned in an arbitrary but fixed direction, which causes a spontaneous magnetization M ^ 0. Conversely, rotation symmetry is not spontaneously broken in the phase above the Curie temperature, and thus M = 0. This example exhibits two characteristic features which occur in all the models described below: 1.
A symmetric phase and a phase with spontaneous symmetry breaking can (separated by a phase transition) occur in the same system depending on the physical boundary conditions (the temperature of the ferromagnet, in the example).
2.
In contrast to the symmetric phase, the phase with spontaneous symmetry breaking is characterized by a non-vanishing order parameter for the ferromagnet this is the magnetization M. In quantized models these are vacuum expectation values.
The iggs mechanism has already been discussed on the classical level using the example of the ielsen lesen model during discussions on the chromoelectric Meissner effect in C (Sects. 3.7.3.1 and 3.7.3.2). After a preparatory section on the breaking of a global symmetry (Sect. 4.1.1), the concept will be extended to include non-abelian gauge symmetry (Sect. 4.1.2). The iggs mechanism is used to explain dynamical mass generation in the lashow-Salam einberg model for the electroweak interaction in Sect. 4.2. An experimentally interesting question is whether iggs fields are fundamental fields corresponding to physically detectable particles or not (cf. Sect. 4.3.2). 4.1.1
S p o n t a n e o u s breaking of a global s y m m e t r y
Because of the importance of spontaneous symmetry breaking its dynamics is first described using a global symmetry as an example. 4.1.1.1
oldstone model
The Goldstone model j)(x). Its Lagrangian C = (d'^^d^
o61 describes the self-interaction of a complex field
-
( f>)
(4.1.1)
I.I
s
( 7
s
(4.1.2) is invariant under global phase transformations f>(x) -> ei j>{x),
f>\x) -> p](x)e~ ,
= constant.
(4.1.3)
The coupling constant A is positive, otherwise the energy would not be bounded from below. n the other hand, there are two possibilities for the sign of p.2 in (4.1.2): .
2.
p2 0: the term involving p2 is a mass term, and the field / has the mass M2 = p 2 ; the potential V((f>) has a unique, classical vacuum (field configuration with lowest energy), f>o{x) = 0. p 2 0: in this case, (4.1.1) describes a phase with spontaneously broken symmetry. The constant fields f> o(x) = ve ,
v =
= constant,
0
0
2n, (4.1.4)
minimize the potential (cf. Fig. 4.1) and therefore describe configurations of lowest energy. The modulus v of r/ o is the order parameter, the phase 0 is arbitrary and can be changed by a global gauge transformation (4.1.3). In order to formulate a perturbation theory for p 2 0 we have to expand the scalar field about one of the ground states (4.1.4). e use the parametrization (4.1.5)
in which 7/(2:) and (o;) are real fields describing small deviations of the modulus and the phase of the field f> from the ground state values (4.1.4). The choice of one of the ground states, i.e. of a particular phase , leads
fi!)()
I
G a u g e
t h e o r i e s
of
t h e
e l e c t r o w e a k
i n t e r a c t i o n
Fig. 4.1 Potential of t h e G o l d s t o n e m o d e l in t h e phase with s p o n t a n e o u s s y m m e t r y breaking /j 2 > 0
to spontaneous symmetry breaking. Inserting (4.1.5), the Lagrangian (4.1.1) reads £ - \<MWt
+
- i „ V -
-
and we find: 1.
//(:/:) is a scalar field with mass /¿; it describes the radial oscillations of (j> around the vcv v (cf. Fig. 4.1). The finite mass results from the non-vanishing radial curvature of the potential.
2.
£(x) is a massless, scalar field, the Goldstone field. It is massless, as a result of the vanishing curvature along the potential minima. The Goldstone field £(.T) describes the associated normal mode with eigenfrequency zero (cf. Fig. 4.1).
The Goldstone model exhibits a general property of theories with spontaneous breaking of a continuous global symmetry, namely the occurrence of massless, scalar field excitations, the Goldstone bosons.
I.I
1.1.1.2
s
( 7
s
eneralization to non-abelian symmetry groups and the oldstone theorem
I he gauge group of the oldstone model is the commutative group of phase transformations (4.1.3), the group (l). Its extension to non-abelian groups G is obvious. e consider a multiplet of n real scalar fields / (:c) = ( / , (a:)) ((/ i ( x ) , . . . , i(x) -4 [exp{WaTa)]ij(f>j{x),
0n = constant,
M*) = \T%j{x)80a
(4.1.7)
cf. (1.2. 3) and (1.2. 4) . The potential V{ ) of the Lagrangian C=l-(d i)dl^i-V(ct>)
(4.1. )
is required to be invariant under the gauge transformations (4.1.7), 0
=
S
dV
6
= T T Ufa
dV
1
=
7ttrT
a
-
(41.0)
d >j
Since 60a is arbitrary, this implies dV 2 ^
= 0, a= 1,...,dim . ifferentiation with respect to fk yields d2V
rT i dkd(f>i
dV irrTl d(f>j
1in
(4.1.10)
= o.
(4.1.11)
As the example of the oldstone model showed, spontaneous symmetry breaking occurs when ( / ) has a minimum for >{x) = v = (v,) ^ 0, i.e. dv d h (f)=v
= 0,
'
d2V dfadffii
= (M2)fci,
M2
0.
(4.1.12)
=\
The symmetric matrix M2 formed from the second derivatives of 1 I pirn itive semi-definite in the minimum, i.e. it has non-negative eigenvul
)()
I
s
Because of the invariance of ( f ) under the transformations (4.1.7), also exp(i0nTa) jj , is a field configuration with minimal energy. In perturbation theory, deviations from the vacuum field configuration j>{x) = v constant are considered in analogy to the oldstone model, and hence fields i up to second order yields c = \ d * U*)
- \(
2
) i x) x)
... .
(4.1.13)
The appearance of (M 2 ),^ in the bilinear terms provides the physical interpretation of (M2)ik as the matrix of mass squares for the fields i(x). In order to obtain information on the eigenvalues of 2 we put i(x) = Vi in (4.1.11) resulting in (
2
) iTtj j
=0,
a = 1... .,dim 7.
(4.1.14)
In general, v is invariant under a subgroup of , the stability group. If we choose an appropriate basis for the generators Ta of G we can write the generators of the Lie algebra of as T = T , a = 1 , . . . , dim , and the invariance of v under transformations from means ffj j = 0,
o = 1 , . . . , dim .
(4.1.15)
The remaining linearly independent generators, T" = T", a. = dim 1 , . . . ,dini , which span the coset, space G , obey T^ j
0,
a = dim 7/
1,... ,dim ,
+
(4.1.16)
As a result, dim/7 of the dim equations (4.1.14) are trivially satisfied. owever, the remaining dim dim// equations force eigenvalues zero 2 for . The result of this consideration is the Goldstone theorem o l, 2 o62, B163 : the matrix of mass scpiares has dim d i m / / eigenvalues zero, one for each broken generator. The associated field excitations are called Goldstone fields and result from spontaneous symmetry breaking. According to (4.1.14), the oldstone fields are linear combinations of Tfj j The remaining eigenvalues of M 2 are positive. This result can be directly generalized to complex unitary representations by noting the relation bet een complex and real representations. n multiplet
I.I
S p o n t a n e o u s
s y m m e t r y
r>(»7
b r e a k i n g
«»I complex fields ip that transforms according to some complex representation of G with hermitian generators T" can be rewritten in terms of a mnlliplct of real fields
that transforms according to a real representation with the generators / — Iin T"
.ja lt
4.1.1.3
s s
{
ReT a
- Re T" \ - ImT° ) '
U8
>
Linear a model
An important example for a spontaneously broken non-abelian symmetry is provided by the linear a model. The fields
£ = ¿ ( ¿ W d ' V i - V(4>),
V(<j>) = £ ( f o b - y )
•
(4-1.19)
The classical configuration with lowest energy is a constant field ^>0(:c) = v that minimizes the potential V(<j>) Li'2
Mio
= vm = '—=v2>
0,
(4.1.20)
i.e. the fields that minimize V form a (JV - l)-dimensional sphere with radius v in ^/-dimensional cuclidean <j> space. It is conventional to choose the field that points into the Nth direction 0 o = v = (O,...,O,t>).
(4.1.21)
as ground state. This is invariant under the orthogonal transformations of the first N - 1 coordinates, i.e. under the subgroup // = SO(N - 1). Hence dim (7 = N(N—l)/2,
d i m / / = [N-[){N-2)/2,
and dim G
dim// =
N-\.
fi!)()
I
G a u g e
t h e o r i e s
of
t h e
e l e c t r o w e a k
i n t e r a c t i o n
According to the Goldstone theorem, there are N — 1 rnassless fields TT = (ttj,..., 7rjv_j) and one massive field a. Writing (f> = (7r,er + v), the Lagrangian (4.1.19) is expressed as C- =
- ¿(2pV + -
r2 + <72)2 - \vcj(n2 + a2).
(4.1.22)
The mass of the field a is \[2fi. For N = 4, the group SO(4) is locally isomorphic to the group SU(2) x SU(2). The chiral SU(2) x SU(2) symmetry is the classical example of a broken symmetry in elementary particle physics (cf. Sect. 3.5.1). The quanta of the three Goldstone fields can be interpreted as pions and those of the massive field as o meson. This can be formulated within the so-called a model (cf. Sect. 3.5.3). 4.1.1.4
Goldstone theorem for quantized theories
So far, the models were considered as classical field theories. They represent the lowest order of a semi-classical approximation for the corresponding quantized theories according to Sect. 2.2.5.3. We now show that the Goldstone theorem is also valid for the quantized model. This could be proven by analysing the matrix elements of the charge operator which generates the internal, spontaneously broken symmetry [Gi64, Be74a]. Instead we use the path-integral quantization (Sect. 2.4) [Jo64] and consider the generating functional of general Green functions [J = (./,-), (f) = (<&)], T{J} = Z{J}/Z{0}, Z{J j = J V{4>\ exp ^i J d 4 s [ £ + ./£(a;)^(3;)]^
(4.1.23)
with L from (4.1.8). We perform an infinitesimal gauge transformation in the path integral Z{J} in the same way as for the derivation of the Ward identities in Sects. 2.2.6 and 2.4.3.2, i(x) —> i(x) 6(j>i(x) with 5[(p], we obtain
/
I.I S p o n t a n e o u s s y m m e t r y breaking r>(»7
or, in terms of the generating functional of connected Green functions 7|,{J} = logT{J} [cf. (2.2.31)], (4.1.25) The Legendre transformation to the generating functional of the vertex func I,ions [cf. (2.2.35) (2.2.36)] has the form
ir{0} = -i
J
dlxMx)Mx)+
8TC{3}
Mx)
4>i\X) = - r j , V,
=
Tc{3}, (4.I.26)
-
6i(x)'
i5Ji(x) and (4.1.25) results in (4.1.27) For spontaneous symmetry breaking, the field remains constant but differs from zero in the ground state, i.e. STC{ J}
i<5J,(.r)
(4.1.28)
7^0
tij = (0|fc(a:)|0) = r
J=O
for some i, or equivalently
SMx)
= o.
(4.1.29)
The last equation is a consequence of (4.1.26) and (4.1.28) because for J : I), (f> takes the value v and vice versa. The inverse propagator of the field then follows from (2.2.41), ¿2rc{j} JhO
8i{x)(j)k{y)
-(G~l)ik(x-y).
0=> (4.1.30)
()
I
s
ifferentiating 4.1.2 with respect to , putting e , e ual to Vi, and performing the transformation to momentum space yields with 4.1.2 and 4.1. {G~l)u{p
= 0)T^kVk
4.1. 1
,
i.e. the same formula as in the classical approximation 4.1.14 which was used as the starting point for the proof of the Goldstone theorem. This means that massless Goldstone fields occur also in uanti ed theories with spontaneously bro en global symmetry.
4.1.1.
Renormali ation of the a model
The agrangian of the a model is renormali able in the sense of power counting. It turns out that the renormali ation of the spontaneously bro en case is related to the renormali ation of the symmetric case cf. e 2b and references therein . Consider the potential in 4.1.1 . The case with spontaneous symmetry brea ing, , can be obtained from the symmetric case, r , by a 2 continuous variation of p . nfortunately this procedure involves a transi. in which physical uantities are not tion through the singular point p 2 2 analytic in x . This problem can be circumvented by introducing a linear symmetry-brea ing term c,r , in the agrangian. fter the inclusion of this term, the Goldstone phase can be reached from the symmetric phase by without passing through singular points. varying p 2 and c Consider now the C = C +
agrangian 4.1.1
plus a linear brea ing term 4.1. 2
Cifaix).
In order to generate the correct perturbation theory we introduce the shifted fields - v,
v
.
4.1.
I.I
S p o n t a n e o u s
s y m m e t r y
b r e a k i n g
r>(»7
The generating functional of Green functions for the fields (f>j in the theory with synnrietry breaking reads T<{J} = exp(Tcc{J}) = f Z>[0] exp (i / dlx [£(0 + v) + (Ji +
Ci )&])
(4.1.34) A
/ Z>[0] exp (i f d x [£(0 + v) + c, where Tcc {J} is the generating functional of connected Green functions, and the denominator ensures Trc{0} = 0. Performing a substitution 0 —> 0 - v yields / P[0] exp (i / dl-r [£(0) + (Ji + Ci )(0i - v,)]) exp(Tcc{J}) / X?[0] exp f i / d * s [£(0) +
-
V i )])
(4.1.35) This allows us to express T£{J) in terms of the corresponding functional for c = 0, i.e. Tc{ J} = logT{J} with T{J} from (4.1.23), TCC{J} = TC{J + c} - T c {c} - i |d'xJiVi.
(4.1.36)
Note that Vi is determined by the condition that the derivative of Tcc with respect to J vanishes at J = 0, (q\M*M
6T?{ J}
= 7i8J,(x)
= 0,
(4.1.37)
j=o
i.e. Vi —
STC{3} i 5Ji(x) J=o
(4.1.38)
A Legendre transformation yields ir c {0} = TCC{J} - i J D 4 xJiii = TC{J + c} - T c {c} - i IdAxJiÜ>i + VI)
(4.1.39)
5 7 8
4
G a u g e
t h e o r i e s
o f
t h e
c l e c t r o w e a k
i n t e r a c t i o n
for the corresponding vertex functional with
'
4tc<={j>
>5rc{j + c}
i6Ji(x)
iSJi(x)
^ (4.1.40)
Using (4.1.26) we find r c { 0 } = r { 0 + v} - r{v} 4- J d ^ a f r
(4.1.41)
for the vertex functional with c¡ =
-
(4.1.42)
6i(x) 0(a;)=
Expanding (4.1.41) in 0 leads to (n > 1) ^¿i ...t„ (xi > • • • > xn) — OO J - / 771 Z ) ^T / I n d4yjk to=0 ' ^ U-=l
(4.1.43) \ r
/
" -inji -jm (x 11 • • • » » 1 , •. . ym),
i.e. the vertex functions for the case with broken symmetry are obtained by re-summing vertex functions in the symmetric theory with external legs contracted with vevs. In this way, in particular, mass terms in the two-point functions result from the vertex functions with more external legs. Note that r c are Green functions for fluctuations around the asymmetric vacuum. In terms of Fc the identity (4.1.27) reads
1Í d''x dx
'¿r c {0} 8(}>i{x)
T?k [4>k(x) + vk] = 0.
(4.1.44)
Differentiating with respect to <¡>j, putting
= o, 0=0
(4.1.45)
I.I S p o n t a n e o u s s y m m e t r y breaking r>(»7
i e. the vanishing of the (/»-field one-point function or tadpole., and transforming to momentum space yields T*k
-
*6kj = T?k U r ^ ( O ) + ck8ji
for the inverse propagator r a the antisymmetry of the matrices
t
=0
(4.1.46)
zero momentum, where we used in the last equality. This generalizes
(4.1.31) to c ^ 0. Because is proportional to ¿¿_, in the space transverse to c, where the symmetry is unbroken, (4.1.46) implies that c and v are collinear. One can define a mass parameter for the (with respect to c) transverse states according to r ^ ( 0 ) = -m? r ,
c = m\\.
(4.1.47)
— 0, i.e. the Goldstone theorem, which is thus For c = 0 this implies proven to all orders in perturbation theory. liquation (4.1.41) expresses the vertex functional of the theory with broken symmetry in terms of the vertex functional of the symmetric theory. Thus, the renormalization of the symmetric theory implies the renormalization of the broken one. The renormalized functional of the symmetric theory is related to the bare one as r R ( 0 , V , A) = r ( 0 o , - p 2 , A0) = T(Zl'2cf,,
- p 2 - ¿/i 2 , ZxX).
(4.1.48) Then (4.1.41) implies for the renormalized functional of the broken case
r&(0, - / i 2 , A, c, v)
=
rc(0o, - / i g ,
Ao, Co, v 0 )
= rc(Z1/20, -/i2 -
ZXA, Z-1'2c,
(4.1.49) Z^v).
In addition to the renormalization of the parameters of the symmetric theory a renormalization of v and c is necessary. The rescaling of c ensures the invariance of the term ct, under renormalization. Equation (4.1.49) allows us to discuss the renormalization of the case with spontaneous symmetry breaking. By starting with p 2 < 0, c = 0, we first continue in c to c ^ 0, then for constant c ^ 0 in p.2 to p2 > 0 and finally in c to c = 0. In this way (4.1.49) is analytically continued from p 2 < 0 to p2 > 0 without hitting the singularity at p2 = 0. Thus, we arrive at
fi! ()
I
G a u g e
t h e o r i e s
of
t h e
e l e c t r o w e a k
i n t e r a c t i o n
i.hc result that the spontaneously bro en theory can be renormali ed by the same mass, coupling, and field renormali ation constants as the symmetric theory supplemented by a renormali ation of the vev. power-counting argument shows that even the values of the coupling and field renormali ation constants can be ta en over from the symmetric theory, and only the mass counterterm has to be ad usted to renormali e the bro en theory e 4 . In order to ensure the validity of 4.1.4 the counterterm p.2 has to be fixed appropriately. y introducing finite renormali ations one can relax the relations on the renormali ation constants in 4.1.4 while eeping 4.1.4 . This is, for instance, useful if one wants to fix the renormali ation of p 2 at the mass shell of a physical particle. e note that the validity of 4.1.4 is not necessary. However, if it does not hold, the presence of tadpole contributions considerably increases the number of graphs contributing to a given Green function. There is a different approach where the vev v is not renormali ed Co 4 . Then, the vertex functions are no longer finite, but the connected Green functions apart from the one-point, function still are. This can be directly seen from 4.1. G by putting c in the spontaneously bro en phase (v?É0) (4.1.50) ith the exception of the tadpole, the connected Green functions in the spontaneously bro en phase are e ual to those in the symmetric phase. Thus, the counterterms of the symmetric phase render also the connected Green functions in the spontaneously bro en phase finite. ince the tadpole does not enter -matrix elements, this approach can be used e ually well for the calculation of physical observables. ote that this approach re uires to ta e into account all tadpoles explicitly. e finally mention that in the same way one could impose any renormali ation condition for the tadpole without changing the S matrix. 4.1.2
S p o n t a n e o u s breaking of a g a u g e s y m m e t r y
4.1.2.1
The Higgs mechanism
In theories with local symmetry, in which the gauge fields are minimally coupled to scalar fields 0 = (,-), spontaneous symmetry breaking implies that
I.I
S p o n t a n e o u s
s y m m e t r y
b r e a k i n g
r>(»7
I.IK; massless Goldstone bosons disappear from the physical spectrum and Home of the gauge fields become massive. This mass-generation mechanism ¡H called Higgs mechanism [Hi64, EnG4, Gu64, Ki67, Be74a]. The massive scalar fields that remain in the physical spectrum are called Higgs fields. We investigate the Higgs mechanism using the gauged version of the Lagrangian (4.1.8). It describes a non-abelian gauge theory with a multiplct of n scalar fields coupled to the gauge fields, C =
+ ^(DMDKfr)
Dp = dp - igA« t T a .
- V(cf>),
(4.1.51)
The Lagrangian £ is invariant under local gauge transformations. The gauge (ields transform according to the adjoint representation of the gauge group G [cf. (2.3.32)], the scalar multiple! (f> according to a real representation [cf. (4.1.7)] with generators Ta. The results can straightforwardly be generalized to complex Higgs representations. A specific example is provided by the gauged linear a model, where the gauge group is SO(Ar), the scalar fields transform according to the defining representation of this group (77. = N), and the scalar potential is given in (4.1.19). We assume that the potential is chosen in such a way that the symmetry is spontaneously broken, i.e. the potential is minimized by the constant field = v / 0 with v 2 = v 2 . In the a model for /.i2 > 0 the minimum is given by (4.1.21). In this case, the stability group, i.e. the subgroup H the symmetry of which is preserved, is SO(N - 1). As at the end of Sect. 4.1.1.2, the generators of G involve the dim H generators of H, T", a = 1 , . . . , d i m / / , which leave the ground state invariant (4.1.15), and the dimG — dimH generators of the coset space G/H, Ta, a = dimH + l , . . . , d i m ( 7 , which obey (4.1.16). Similarly to the Goldstone model (4.1.5), small excitations from the ground state are parametrized by /ifif'l(x)\
(x) = exp (
J (v + Tf(x)),
d = l , . . . ,dim(7 — dim //. (4.1.52)
The dim(7 - dimII Goldstone fields £"(a;) appear in the exponential, and the vector ri(x) involves only n — [dimG* — dim H] non-vanishing components. Since the Goldstone modes are linear combinations of T^Vj
()
I
s
cf. ect. 4.1.1.2 , the condition that rj(x) involves no Goldstone modes reads r,l x f ;v
.
4.1.
For the group { ) , dirnG - dim = ( - l 2 - ( - l)( - 2 2 - 1, and T](x) involves only one finite component, r (x) = ,... , , r](x)). In contrast to the models considered so far, in gauged models the Goldstone fields £a(x) can be regarded as parameters of local gauge transformations exp iT u ; ; and thus have no physical significance. s a result of the gauge symmetry of the agrangian 4.1. 1 , the Goldstone degrees of freedom can be eliminated by carrying out the gauge transformation cx >(-\Ta£a(x) v). In this gauge, the Higgs field (f>(x in 4.1. 2 is given by (f,(x) = v
; .
4.1. 4
o far, gauges have always been fixed by specifying the form of the gauge field, e.g. by re uiring d'' x . However, any other fields that do not transform trivially can be used for fixing the gauge as well. The re uirement that f x has the form 4.1. 4 , or e uivalently the condition 4.1. , fixes a gauge, which is in fact a unitar gauge because the unphysical degrees of freedom £(x) tlo not occur, and therefore the unitarity of the S matrix is evident cf. ect. 2. .4. . In the gauge 4.1. 4 , the agrangian 4.1. 1 reads =
+ -
-b i h>
l
+
-(
lim)(
'\)
mass terms for rj(x) terms involving
or 4
fields.
4.1.
The gauge-boson mass term results from inserting the vev v into the gaugeinvariant inetic-energy term of the scalar fields,
i Vi
oi
\g ntjvj)(Yr*iv ) g2 v;rpiv
i
i l
l
>> l
-
2 b
i
h
,
4.1.
.
s
s
( 7
where we used (4.1.15) ami the antisymmetry of the matrices Tu for real representations. For complex representations we obtain, using (4.1.1 ), the matrix of gauge-boson masses squared
M?h = g2(fiv)l(fbv)i
= ^vt T
T
.
(4.1.57)
From (4.1.57) it can be easily seen that Mfib is real and positive-definite. Its rank is dini dim . Consequently, all gauge bosons associated with generators in / / / become massive. In the a model, the gauge-boson mass matrix is given by
h
= g2v2{f Th)nn
= g2v2
(lh
(no sum over n).
(4.1.5 )
From this one concludes: 1.
The dim G dim = - 1 for S (j ) oldstone fields ^(.r) have disappeared. o oldstone bosons appear in spontaneous symmetry breaking of gauge theories.
2.
The n - dim - dim// scalar fields in T]( ) remain as in general massive fields; their quanta are called iggs particles. In the a model, one finds one iggs field i)(x) with mass = p. 2.
3.
dim - dim// = - 1 for S ( ) vector fields have become massive x gv = gv in the a model and thus have longitudinal, physwith ical polarization states, in addition to the two transverse polarization states of massless vector fields cf. (2.1.20)-(2.1.22) .
4.
The remaining dim have zero mass.
= (
- 1 ){
- 2)/2 for S ( ) vector fields
The comparison of the degrees of freedom described by C according to (4.1.51), or gauge-equivalent (4.1.55), gives the picture shown in Tible 4.1 for the general case, and for the a model in brackets. The transmutation of oldstone modes into longitudinal polarization states of massive vector bosons is known as the Higgs mechanism.
fi!)()
I G a u g e theories of th e e l e c t r o w e a k i n t e r a c t i o n
C as in (4.1.51)
massless gauge fields
dim G N(N-l)]
r physical states
C as in (4.1.55)
2J
dim H F
(¿V—1)(JV—2)1
2 dim (7
2 dim II
massive gauge fields
0
dim G - dim H 1] [=N-
physical states
0
3 (dim G - dim H)
Higgs fields
11
n — [dim G — dim //]
[= 1] all physical states
2 dim G +
2 dim G + n
Tab. 4.1 Degrees of freedom in the symmetric and spontaneously broken Lagrangians [G = SO(JV), H = S O ( N - 1)]
4.1.2.2
Quantization of spontaneously broken gauge theories
The gauge theory defined by (4.1.51) can be quantized using the pathintegral formalism according to the procedure outlined in Sect. 2.4. This requires the fixing of a gauge (Sect. 2.4.1.2) and the introduction of FaddeevPopov fields (Sects. 2.4.1.3). The unitary gauge used in (4.1.55) gives the canonical form (2.1.22) of the vector-boson propagators. As unphysical scalar bosons are not present, the unitarity of the S matrix is manifest. However, Green functions are not renormalizable in this gauge. Divergences do not cancel until the S matrix is calculated [We72, Le72a, Ap72]. Moreover, the definition of the finite part of the S matrix may not be unique under certain circumstances [Ja72, Ba72a]. Thus, the choice of a renormalizable gauge (in the sense of Sects. 2.5.1.3 and 2.5.1.4) is preferable, just as it was for the case without symmetry breaking. In principle, (2.4.11) and (2.4.17) can be used as gauge-fixing term, but this is often unsuitable. Instead one prefers generalized forms, e.g. the FradkinTyutin gauges [Fr74] or the 't Hooft gauges [tH71].
I.I
S p o n t a n e o u s
s y m m e t r y
b r e a k i n g
r>(»7
As a ' t Hooft gauge is employed in the next section, we discuss it in some detail. Consider first the gauged a model (4.1.51) with the scalar potential of (4.1.19) for the abelian gauge group SO(2). With <j> = (<j>i,v + ij) and '/' T-i [ef. (A.1.11)] the gauge-invariant Lagrangian takes the form C
M2 + — A ^
1 =
1 + ^dltv)(d'lr,)
-
p V
1 2l
4- terms involving 3 or 4
fields
(4.1.59)
with M = gv. In the't Hooft gauge, the gauge-fixing part of the Lagrangian, £r,x, is chosen in such a way that all bilinear terms that involve two different fields, i.e. —MA'l(dIL(f>\) in the example, cancel in the Lagrangian (up to irrelevant total derivatives). For the Lagrangian (4.1.59) this is achieved by choosing C[Afi(x),M*)}
= dPAp{x)+ZMMx)>
¿fix =
(4-1-60)
= —(rArW
+
ZMMx))*-
Thus, the mass matrix of C + £r,x is diagonal in the fields. The would-be Goldstone field 4>\ acquires a non-vanishing mass \/£M, and its propagator is given by iA^1 =
—-—A-2 - £M 2 + ie
1(4.1.61)
'
The gauge-field propagator has the form
_ ¡-V +W - O / i * 2 - ^ ) "" k2 — M2 + ie
U1 '
1
'
The Lagrangian for the Faddeev-Popov ghost fields £ fi h 0 st' s calculated from Ca[A'i{x), ¡(x) = !l(v + f])S0.
(4.1.63)
fi!)()
I
G a u g e
t h e o r i e s
of
t h e
e l e c t r o w e a k
interaction
We obtain ¿ghost
= - J d4® u{x) [dltd" + £Mg(v
+
77)]
u(x).
(4.1.64)!
Thus, also the Faddeev- Popov ghost acquires a mass y/^M and is no longer free in the abelian theory. Note that in the't Hooft gauge, all boson propagators behave as l/kA for large k2, and the theory is renormalizable according to power counting. Consider now a more general gauge theory with spontaneous symmetry breaking as given for example by (4.1.51). We assume that the gauge group is simple and hence has a single coupling constant and that the scalar fields 0 belong to a real representation. Scalar fields with vanishing vev are obtained by the shift 0 = 0 + v with (0) = 0. The shifted fields 0, involve Goldstone fields Xi> he. linear combinations of the Tav, and other linearly independent scalar fields characterized by (4.1.53). By introducing the fields 0 in the part of the Lagrangian (4.1.51) that describes the interaction of the scalar fields with the gauge field, we find the following bilinear mixing terms between gauge and scalar fields - i g i d ^ A ^ T ^ i = -igid^xMiif^i.
(4.1.65)
Because of (4.1.53), the mixing terms (4.1.65) involve only would-be Goldstone fields Xi and arc thus absent in the unitary gauge. The mixing terms (4.1.65) can be eliminated by introducing a ' t Hooft gaugefixing term of the form
Aix =
(3M;i + itgxiftjVj)2.
(4.1.66)
The term quadratic in the scalar fields Xi yields a mass matrix for the would-be Goldstone fields, \{Ml)ijXiXj
= -\i92{XittkVk){vit^Xj).
(4.1.67)
Note that the gauge-fixing term (4.1.66) does not modify the masses of the physical scalar fields. The masses of the would-be Goldstone bosons are
I.I S p o n t a n e o u s s y m m e t r y breaking r>(»7
l imes the non-zero vector-boson masses. This can be seen by inspecting an eigenvector xa of (M 2 ) i b with eigenvalue M 2 ,
( M
2
) ^ = -g2{fav)i(fbv)ixh
= M2xh.
(4.1.58)
Then, x''(Thv), is an eigenvector of (M2)ij with eigenvalue \/£M2,
(M$)ijxi(fhv)j
=
-£g2(fiv)i(f&v)jxb(fiv)j
= Z(M\bxb(f*v)i
= ^M2xd(f%)i.
(4.1.69)
The Faddeev Popov Lagrangian is given by
¿ghost = ~J
dW(*)
(cTDf
- £g2ViT!jTbk{vk
+ <j>k)) U\X).
(4.1.70) The ghost-mass matrix reads (Mg2host)ai = Zg 2 viTffi k v k = Z ( M \ b ,
(4.1.71)
and thus is proportional to the gauge-boson mass matrix (4.1.57). The fact that the masses of the Faddeev- Popov ghosts and would-be Goldstone bosons are equal to those of the longitudinal gauge-bosons is crucial for the cancellation of the poles of these unphysical fields in the S matrix. For finite all boson propagators behave as l/k2 for large k2, i.e. the 't Ilooft gauge is renormalizable. For £ —> oo, the gauge-boson propagators agree with the propagator (1.4.13) in the unitary gauge. Consequently, the unitary gauge is a limiting case of the't Ilooft gauge (4.1.66). The masses of the would-be Goldstone bosons and of the Faddeev -Popov ghosts associated with massive gauge bosons diverge like for large Hence, in this limit these fields decouple from the S matrix. As a consequence, the proof of the unitarity of the S matrix simplifies as discussed in Sect. 2.5.4.5.
fi!)()
I
4.1.2.3
G a u g e
t h e o r i e s
of
t h e
e l e c t r o w e a k
i n t e r a c t i o n
Renormalization of spontaneously broken gauge theories
The renormalizability of spontaneously broken gauge theories [tH71, Le74] can be proven in a way similar to the renormalizability of the a model (cf. Sect. 4.1.1.5). We restrict ourselves to the discussion of the gauged a model. The complete Lagrangian is given by (4.1.51) and (4.1.19) supplemented by gauge-fixing and Faddeev-Popov ghost terms. We assume that the gauge-fixing functional is linear in the fields, as for example in the 't. Hooft gauge (4.1.66). We set ft2 > 0, so that the field 0 acquires a non-vanishing vev (0) — v. We want to show that the counterterms for the symmetric theory (p2 < 0) plus a renormalization of the vev are sufficient to render the broken theory (p2 > 0) finite. Since the point p 2 = 0 is not analytic (a phase transition occurs at this point) we cannot simply continue p 2 from positive to negative values. We use the same strategy as in Sect. 4.1.1.5 and introduce a constant external source coupled to the field 0 as in (4.1.32). Then, (4.1.41) holds here as well. From Sect. 2.5.4.2 we know how to renormalize the vertex functional of the symmetric theory:
r R ( 0 , A , . . . , <7, - p 2 , A) = r ( 0 o , A 0 ,... ,go, -p«, -V>)
= r(Z'/ 2 0, Z\/2A,...,
Zgg, V
- ¿p 2 , Z\\).
(4.1.72)
The dots represent unphysical fields and gauge parameters. The renormalization of the gauge parameters is determined (in linear gauges) in such a way that the complete gauge-fixing term is not renormalized. Equation (4.1.41) tells us that the broken theory is renormalized as TfJ^, A . . . . ,g, - p 2 , A, c, v) = r c ( 0 o , A n , . . . , go, = r c { Z x ' 2 t , Z f A , . . . , Zgg, V
A0, c 0 .v 0 )
- 6fi2, ZxA, Z~x'2c,
Zx'2w).
(4.1.73) Upon analytic continuation this result holds also for the case of purely spontaneous symmetry breaking, i.e. c = 0 and p 2 > 0. The corresponding normalization conditions for the various vertex functions can be deduced from the identity (4.1.41) and the normalization conditions of the symmetric functions. Once we know how to renormalize spontaneously
I.I
S p o n t a n e o u s s y m m e t r y breaking
( 7
broken theories in terms of symmetric counterterins we can perform additional finite rcnormalizations in order to conform with more physical renormalization conditions. ne can, for example, demand that the coupling count ant is fixed as the value of a vertex function at some on-shell point, or that the renormalized particle masses are equal to the physical masses. According to (4.1.73) the vev is renormalized in the same way as the iggs field itself. It is, however, convenient to perform a further finite renormalization of the vev in order to introduce the true vev of the interacting iggs field even if the other counterterms are already fixed by physical renormalization conditions. equiring that the renormalized vev is the true one, i.e. given by the actual minimum of the effective iggs potential, is equivalent to the requirement that the renormalized. tadpole, i.e. the term in the effective potential linear in the iggs field, vanishes. In this scheme, tadpole contributions do not have to be taken into account in practical calculations. Instead of a counterterm for the vev one can directly introduce a tadpole counterterm. Similarly to the a model (Sect. 4.1.1.5), also spontaneously broken gauge theories can be renormalized without renormalizing the vevs Ap73a . hile the S matrix and the connected Green functions, apart from the tadpole, are finite in this approach, the vertex functions are still divergent. 4.1.2.4
General theory with spontaneous symmetry breaking
A non-abelian gauge theory for spinor and scalar fields interacting with each other via massive intermediate vector bosons is a suitable dynamical concept for describing the electroweak phenomena. The Higgs mechanism can be used for constructing a renormalizable theory of this kind: the vevs of some of the scalar fields generate the masses of the gauge bosons by spontaneous symmetry breaking. The construction of the general, gauge-invariant part of the Lagrangian of such a theory proceeds as follows: 1.
Choose a gauge group G. As the gauge fields always transform according to the adjoint representation of G, the number of gauge fields Wa — (H7"1", W~,Z°, 7,...) is fixed. The gauge-field couplings are given by the structure constants fabc.
2.
Choose representations La, R", and Pa of G under which left- and isT, T ... ,uc,dc,cc,sc, tc, bc...) right-handed fermions Vi = ("e,
)()
s
(c denotes the colour index) and, for definiteness real, scalars <pr transform, respectively, [La, Lb] = i fabeLe, [Pa,Pb]
[ a, .b] = i fabc c,
= ifabcPc.
(4.1.74)
When choosing the fermion representations, care must be taken that no 75 anomalies can occur (cf. Sect. 2.7.2.2). 3.
The most general form of the gauge-invariant, local, renormalizable Lagrangian, which also includes self-couplings V(
+
n ^
-
^
with F% = W
gfabcWbW ,
- d W* +
Dt 'j
= M " - i g **
D?JS
= [
rsd
+
- igW**P?s]
i>3>
(4.1.76)
The gauge-invariant potential V(0) is a polynomial of the fourth degree at most in (p [cf. (4.1.19)]. Higher powers in the fields must not occur in the other coupling terms, too. Otherwise coupling constants with negative mass dimension would destroy the renormalizability of the theory (cf. Sect. 2.5.1.3 and Table 2.1). The G symmetry can be spontaneously broken and a Iliggs Kibble mechanism introduced by a suitable choice C G, which leave of ( j>). The gauge bosons of the symmetry group a classical vacuum vector v invariant, remain massless, and the other gauge bosons become massive (cf. Sect. 4.1.2.1). The gauge symmetry is said to be broken spontaneously from G > . The ijxfnp coupling term is gauge-invariant if (in matrix notation) [La, 3
L r
'] =
h s
> P?r,
[
a
, X Il ' r ] = X R ' s P° r
(4.1.77)
For the choice of the coupling constants, see the note given at the end of Sect. 2.3.3.3, (2.3.52).
'1.2 The Standard Model of the Slectroweak Interaction 5!):!
applies^ I'he phenomenological analysis of the electroweak interaction in Sect. 1.4 has shown that the appropriate gauge group of the ea isospin and ea ^', h percharge is G = SU(2)W x U(1)Y, and hence four gauge fields +, ' f,, and ,t are needed as the basis for a "minimal" theory. More complicate models based on the above scheme must contain this as a substructure.
We summarize the important points of the concept of spontaneous s mmetr brea ing, introduced in this section. The Lagrangian (4.1.55) is gaugeinvariant. The currents coupled to vector fields are conserved. Only the necessity to choose one of the physically equivalent ground states as basis for the perturbation theory hides the gauge invariance. The gauge transformation law, expressed in the shifted fields (p. becomes inhomogeneous. Spontaneous symmetry breaking introduces mass terms for gauge bosons. Since gauge invariance is not broken but only hidden, the Iliggs mechanism can be used to formulate a renormalizable
4.2
theor
ith massive vector
ields.
T
In Sect. 2.3.3.5 we argued that the theory of the electroweak interaction is a non-abelian gauge theory with the gauge group SU(2)W x U(1)K- The masses of the gauge bosons can be introduced by spontaneous symmetry breaking as described in Sect. 4.1.2. In this section we describe the lectro ea Standard odel (EWSM) [We67, SaG8, G170]. It realizes in a minimal way the unification of the electromagnetic with the charged and neutral weak interactions by postulating the S U ( 2 ) W x U ( l ) y gauge group of weak isospin and hypercharge and mass generation by a complex doublet of scalar fields. We formulate the classical Lagrangian of the EWSM in its symmetric form and then pass to the representation in terms of physical fields and parameters so that its physical content becomes visible.
fi!)()
I
Gauge theories of the electroweak interaction
4.2.1
T h e Lagrangian of t h e Electroweak Standard Model
We start by formulating the Lagrangian of the EWSM in a form in which the symmetry structure is manifest. As the gauge group is not simple but rather a product of SU(2) and U(l), electromagnetic and weak interactions arc combined by two gauge coupling constants,
for the weak isospin SU(2) W and g\ for the weak hypercharge
U(l)y. We denote the three generators of weak isospin by the generator of weak hypercharge by Yw. The generator of the electric charge is defined via the Gell-Mann-Nishijima
relation
i4-2-1)
Q = il + Y4.2.1.1
Gauge part
According to the dimension of the gauge group SU(2)W xU(l)y there are four gauge fields which transform according to the adjoint representation. The gauge fields belonging to the weak-isospin group SU(2)W are called W",(x), a = 1,2,3, and the one belonging to the weak-hypercharge group U(l)y is called Blt(x). The Lagrangian for the gauge fields reads (cf. Sect. 2.3.3) ¿gauge = -l-FUi„(x)Fr(x)
~ ^FlfW(x)F^(x).
(4.2.2)
The gauge field strengths are defined as Fhllv(x)
= d,LBv(x) = dpW^x)
-
duBp{x), - duW^x)
+ <72e a h c W';(x)WZ(x),
(4.2.3)
where eabc are the totally antisymmetric structure constants of SU(2) (cf. Sect. 1.2.2). The gauge interaction of the matter fields is determined by the covariant derivative Dp = (dp - \g2I«w;
+ i7l ^ B p ) .
(4.2.4)
'1.2 T h e S t a n d a r d M o d e l of t h e ISlectroweak Interaction
4.2.1.2
5!):!
Fermionic part
According to the structure of the fennion-gauge-boson interaction discussed iu Sect. 1.4, the fermions are arranged into isospin doublets (V'i.t-iV'i, )i where the indices + and - (represented by a in the following) distinguish Ihc two members of the doublets. The index i runs over the three lepton doublets (1.4.6) and the three quark doublets (1.4.23) including the three different colours so that there are in fact nine quark doublets. The colour Index is suppressed in the following. The prime at the fermion fields indicates eigenstates of the electroweak interaction, i.e. the covariant derivatives are diagonal with respect to the generation indices in this basis. These states are not necessarily mass eigenstates. The left-handed fields ip'L(x) = ~(1 - 75)ip'(x) in these doublets are associated with the fundamental two-dimensional representation of the weakisospin group SU(2)W, ^-''(.r) = V;i!-)> w i t l 1 representation matrices /*{, = r a /2, where r a are the Pauli matrices (A.1.11). Each right-handed field ij/R(x) = 5(1 4- 75)V;'(-''-') ' s P u t into a one-dimensional trivial representation of SU(2)W (/" = 0). Consequently, the term involving <72 in the covariant derivative is absent, for right-handed fields. Note that in the minimal Standard Model no right-handed neutrinos are included.4 The weak hypercharge of the fermions is fixed in such a way that the correct electric charge results from the Gell-Mann-Nishijima formula (4.2.1), leading to YiR+ = Yt + 1 = 2Q1)+,
= Y,L - 1 = 2Qif_
(4.2.5)
for the weak hypercharges of right- and left-handed fermion fields. The fermionic part of the Lagrangian reads (4.2.6)
Because left- and right-handed fields transform according to different representations of the symmetry group SU(2)W x U(l)y, the theory is chiral, and explicit mass terms for the fermions are forbidden. 4
Itight-handed neutrinos can be easily included ¡11 the Standard Model without affecting its basic structure. This becomes necessary in order to allow for the description of the recent experimental results on neutrino oscillations.
)()
I
s
It is often convenient to denote the left-handed lepton and quark doublets by
respectively, and the right-handed fennion singlets by l\ , u\ , and d f . Here v, , it, and d represent neutrino, charged-lepton, up-type-quark, and downtype-quark fields, respectively, and i is the generation index (i = 1,2,3). In this notation the fermionic part of the Lagrangian has the form ,, =
{ 't\ f) 't
+
'h 't)
i
+ £
4.2.1.3
+ u *x]pu
+df\]pdf).
(4.2.8)
Higgs part
In order to introduce mass terms for the weak gauge bosons we use the iggs mechanism to break the SU(2)W x U(l)y symmetry in such a way that the electromagnetic symmetry U(l) cm is the remaining symmetry (cf. Sect. 4.1.2). To this end, we postulate a wcak-isospin doublet (/" = r"/2) of two complex scalars fields (a;) = (cf)+(x), j> (x)). The choice of a weak-isospin doublet with hypercharge = ±1 also allows for Yu a a couplings, i.e. couplings between scalar and fennion fields, that are necessary for the generation of fennion masses via spontaneous symmetry breaking. The Higgs part of the Lagrangian reads £
h
= ( D ^ ) H D ^ )
-
"
+
+ h.c.),
(4.2.9)
where V() = A
2
_
2 /i
$t < P )
^ x> ,
(4.2.10)
'1.2 T h e Standard Model of the ISlectroweak Interaction
5!):!
i< the gauge-invariant self-interaction of the scalar doublet with coupling A and mass parameter /¿2, and (:(x) is the charge-conjugate lliggs lield 4'c(x) = ir2«I>* = {<j>°'(x), -(¡>-{x)). Note that the Higgs-doublet field allows for Yukawa interactions to both up-type and down-type right-handed fermion fields. The Higgs self-interaction is chosen in such a way that the lliggs field acquires a non-vanishing vacuum expectation value. This vacuum expectation value leads to mass terms for the gauge bosons and arising from the Yukawa couplings—also for the fermions. This is explained in detail In Sect. 4.2.2.2. In the notation of (4.2.8) the Higgs Lagrangian reads £„ =
- ^ (*)
(4.2.11)
- Y , (i>'tG l a 1 ?® + Q ' i ' G ^ u f ^ + Q'tGijdf® + h.c.) , ij where G* denote Yukawa coupling matrices. The classical Lagrangian Cc\ of the GSW theory is the sum of the individual terms (4.2.2), (4.2.8) and (4.2.9) [or (4.2.11)], ¿cl = ¿gauge + ¿F + ¿H-
(4.2.12)
It is invariant under the infinitesimal gauge W?(x)
1 W"{x) + -dlt60a
+
transformations,
eabcW"SOc,
92
BJx) < I > ( x ) —>
Btt(x) +
91
-dMseY,
1 - i l-80Y(x) + i y < W * ( x ) Hx), l _ i YLS0Y(X)
+
\^-80"(X)
vp'Hr),
y.H 1-
i-^60vy(x) 2 '
(4.2.13)
where 0" and eY are the parameters of the gauge transformations corresponding to the groups SU(2)W and U(l)y, respectively. The Lagrangian Cc\ in the above form depends on the two gauge couplings g2 and g\, the Yukawa couplings G{J, and the parameters p 2 and A of the scalar potential.
)()
I
4.2.2
s
T h e Lagx*angian in t h e physical basis
The gauge symmetry forbids explicit mass terms in the Lagrangian £ c j, apart from a mass term for the scalar doublet 4>. In order to introduce gauge-boson and fermion masses in the EWSM we use spontaneous symmetry breaking, i.e. the Higgs-Kibble mechanism [Hi64]. In order to extract the physical content of the Lagrangian in the spontaneously broken form, we rewrite it in terms of physical parameters and fields corresponding to mass and charge eigenstates. 4.2.2.1
Spontaneous symmetry breaking in the EWSM
The scalar-field self-interaction is chosen in such a way that spontaneous symmetry breaking and Higgs-Kibble mechanism take place as specified in Sect. 4.1.2. The scalar potential K(), (4.2.10), has a minimum for
)2=
2
=
.
(4.2.14)
Out of this three-dimensional subspacc we choose a vector 4>o that is annihilated by the electric charge operator Q, (4.2.15) so that the remaining (unbroken) symmetry is the one of electromagnetic gauge transformations U(l) crn . Note that 4>0 is neither invariant under SU(2)W nor under U(l)i'. The conditions (4.2.14) and (4.2.15) have (up to a phase) a unique solution, (4.2.16) In order to expand the Lagrangian about this classical ground state we write the Higgs-doublet field as
4.2
Tin; Standard Model of t.lio ISIectroweak Interaction
597
where i](x), x(-c)i and <> / ' (x) have vanishing vacuum expectation value. The llelds
j>~ (x), and x(x),
f>' (:/),
'he
ould-be Golds tone fields, are unphysical
degrees of freedom and can be eliminated by transition to the unitar gauge in cording to (4.1.54). We will see that rj(x) is a massive physical iggs field, l hat gives rise to the physical iggs boson. 1.2.2.2
Physical fields and parameters
The physical content of the EWSM can be extracted most easily in the unitar gauge, where = x — 0 (cf. Sect. 4.1.2.1). In this gauge no nilphysical degrees of freedom occur and the physical degrees of freedom can be classified as eigenstates of electric charge and mass. Since the gauge bosons have vanishing hypercharge, the corresponding charge eigenstates can be found as eigenstates of the (adjoint-representation) matrix (/£)„*, = -i
= ^ [ W » T i l(x)l
(x)
= il
f(x) =
(4.2.18)
(x),
*{x)
Thus, we have one positively charged gauge boson, charged one, ~, and two neutral ones, * and .
= +
fl{x)
= 0.
, one negatively
In order to determine the mass eigenstates we have to inspect the Higgs part of the Lagrangian. Using (4.2.17) and (4.2.18) and putting (p* = x = 0, it reads £
=
l
-v2g
~
+
+ lv2(g2^+gl li)(g2
3ti
+ gx
)
-f -{dtiii){d,l'n) - li 2 !? + trilinear and quadrilinear terms -
+
)£
(j'
i f + u'\G
jU'f
+ d'lGfjd'f
+ h.c.) .
(4.2.19) From (4.2.19) we can deduce the following properties of the physical states: 1.
The scalar, neutral Higgs field r](x) has the mass \\ =
2 i.
(4.2.20)
)()
2.
I G a u g e theories of the electroweak interaction
Tiie masses of the vector fields result from the bilinear form
The mass of the charged vector boson W ± can be directly read off as Mw = j v .
(4.2.21)
In order to determine the masses of the neutral vector bosons we have to diagonal ize the corresponding mass matrix )2=l-(
(
l
4 \gi 2
v\
2
(4.2.22)
As the determinant of (M 0 ) 2 vanishes, one of the two eigenvalues is zero. This reflects the fact that owing to the unbroken U(l) e m symmetry one of the neutral vector fields—the photon—remains inassless. The second (non-vanishing) eigenvalue is directly obtained from the trace of (M 0 ) 2 and provides the mass for the heav
neutral vector boson, the Z boson,
Mz = \ J g 1 + 9 l v . 3.
(4.2.23)
The mass eigenstates of the neutral vector bosons are obtained by an orthogonal transformation
with =
,
0
2
=
\v
g+ i 5w
= sin 0W =
. f
(4.2.25)
oi + 2 and the
1.
ea
mixing angle or
einberg angle
.
The Yukawa interaction term of the Lagrangian (4.2.11) provides mass terms for the fermions with the following mass matrices for leptons, up-type quarks, and down-type quarks A ^ = ~G\ v,
M
=
Gf
j V
,
M?j = - GfjV.
(4.2.26)
4 . 2
T h e ;
S t a n d a r d
M o d e l
o f
t h e
E l e c t r o w e a k
I n l e r n c t i o n
5 9 9
They can be diagonalized by a bi-unitary transformation with and ujf]]i for left-handed and right-handed fermion fields, respectively, resulting in the fermion mass eigenstates, ft = E
^
k
and the fermion m
= E
U
i
K
f'l
(4.2.27)
k
masses,
U = 4 E
U
*
G
LU^v.
(4.2.28)
k,m
Owing to the absence of right-handed neutrinos in Cci, no gaugeinvariant Yukawa interaction term exists that can provide masses for the neutrinos. Consequently, all neutrinos remain massless. 5.
After diagonalization of the fermion mass matrices also the interaction of the Higgs field i](x) with fermions is diagonal and the associated couplings mj/v are proportional to the fermion masses.
In terms of the physical fields (4.2.18), (4.2.24), and (4.2.27), the fermion gauge-boson interaction reads E
£F,int=
E
[-^QjIiYf,^
¡—1,11,11,(1 i
+
E
I-l,v,u,d
E
i
w
'fKift^ft
-
s i Q ^ m
u + E ^ h ^ v ^ r f w ;
+ ilYiu^u^pv-}, (4.2.29)
where we used
By requiring that the photon fermion couplings equal those in QED,
f=l,u,u,d
i
(¡00
-1
G a u g e
t h e o r i e s
o f
t h e
c l e c t r o w c a k
i n t e r a c t i o n
we can relate the coupling constants t/\ and (¡2 to the elementary charge, s/Ana = e = g2sw
= gicw = — a V Hi+92 fI
2.
ss — . 16 '
(4.2.30)
fR
Owing to their unitarity, the matrices U'f and u;f drop out of the interaction terms between fermions and neutral gauge bosons. Consequently, there are no flavour-changing neutral currents at tree level. However, in the quark-'W-boson interaction a non-trivial matrix remains, the unitary quark-mixing matrix (cf. Sect. 1.4.2.2): V = uu'hUdM.
(4.2.31)
Since all neutrinos are massless and thus degenerate in mass with each other, we can choose freely and eliminate the mixing matrix in the lepton-W-boson interaction by a suitable choice of the neutrino fields, "
N
E
W
(4.2.32)
k
Thus, the physical neutrino states are fixed in such a way that their charged-current interaction is diagonal. Note that massive neutrinos with non-degenerate masses would give rise to a non-trivial mixing matrix for leptons. On the other hand, the quark-mixing matrix would be trivial for degenerate masses of the uptype or down-type quarks. As a consequence of spontaneous symmetry breaking we got the masses of the physical Higgs boson M\\, of the charged and neutral gauge bosons M\\ and Mz, of the fermions mj, and the quark-mixing matrix V. In addition we could identify the massless gauge boson with the photon, having the correct coupling to fermions proportional to e. This is the only remaining coupling parameter. 4.2.2.3
The Standard Model Lagrangian in terms of physical fields and parameters
We can now rewrite the complete classical Lagrangian Cc\ of the EWSM in terms of the physical fields, i.e. the mass and charge eigenstates Aln Z,n
'1.2
T h e
S t a n d a r d
M o d e l
of
t h e
I S l e c t r o w e a k
I n t e r a c t i o n
U'(I', //, I, v, u, and d, the would-be Goldslone fields (p1 and x> physical parameters e, Mw, My,, M\\, rrij, and V,
£ci=
E E ^ J—l,i>,u,d i
-
m
;ill(
5!):!
l the
e
Qffrt'lfiAßl
f)f< -
+
E r ^.'W- Fl-W^ w J t Y f t - s l Q / f a " / ^ . E . /=í,t/,u,d ¿ e + E - s - l f i S V " 4 - dh^v^u^-] . . V «<*>\v I,J + E. -y kZs- t â ^ ï K + ^7'V t L H/-] w - \ Id lt A„ - dvA,t - i e ( l V - W + - W-W+) I2 _1 ~ 4 - <W+ - k { W +
- r,
+ie—(W^Zl/
-
A
» -
Wj'Zf,)
S\y
+
1 d ß (v + ix) - i - W ~ 4 > + + 2
+ ir
- ieC;v ~ ^ Z,^
+ dtL+ +
-
Z/t(r/ + i X ) iMwW+
- i ^ - ^ + ix)' - 2 M H '' —e -
E E J=l,v,u,d t
^
^
-
^
.
/
M
()
s
m
d,j
fw
£
2s
(
+ f
The physical content of 4.2. 1.
V
i
)) r
.
4.2.
is the following
The first, four lines constitute the free ferraion agrangian and the couplings of the fermions with the gauge bosons. These couplings agree with the ones given in ect. 1.4.2. . Thus, the muon deca constant is in lowest order given by cf. 1.4.1
i.e. it is directly related to the vacuum expectation value v. The experimental value for G yields v 24 GeV. 2.
ines - contain besides the inetic energies of the gauge bosons their non-abelian gauge couplings, which depend only on e and s .
.
ines -11 originate from the gauge-invariant Higgs gauge-boson interaction. esides the inetic terms for the Higgs bosons they contain the gauge-boson mass terms and bilinear terms that give rise to tree-level mixing between gauge bosons and would-be Goldstone bosons.
4.
ines 12 and 1 describe the Higgs-boson self-interactions. This part depends on the Higgs-boson mass, a parameter which is presently not yet nown experimentally.
.
The u awa couplings in the last 4 lines, however, are fixed by v = 2s \ \ e and my oidy.
The agrangian 4.2. is invariant under gauge transformations that follow from 4.2.1 by inserting the physical fields and parameters, ,t
+d S
+ ic
+S ~ -
Zp -> Z,t + dtl6dz - ie—[W+Se~ s
f
(x)
->
*{x) s
+
~S +],
W~69+],
,^ *
[w± (swseA - c^se2) - (s^Ap -
c^z^e*],
'1.2
T h e
S t a n d a r d
V~>V + ^—XS9Z
M o d e l
of
t h e
ISlectroweak
+ ^ 6 0 -
z
1Pi,±
<j>+se- +
\e ic^ 60a + — — ^ - M 7 ' ±—[v 2 cwsw ¿Sy/
i>\,± ~
5!):!
- 4>-60+),
X->X- ^—[v + rj\60 + T
I n t e r a c t i o n
<j>-66+),
+ n±i x]S0±1
le
(4.2.35) where S0 ± = — (¿01 q=W02),
(4.2.36)
02
S0A = —cwS0Y 01
- — sw<503, 92
¿0 Z = — cw<503 + —S w 60 k , 02
01
= V , j , v{- = V- ; for quarks, and vf- = 8tJ for leptons. After adding a gauge-fixing term for the photon as in QED [ef. (2.4.11) and (2.4.23)], the Lagrangian (4.2.33) is suitable for tree-level calculations. The would-be Goldstone fields and x c a n be eliminated by a gauge transformation. This yields Cc\ in the unitary gauge which is directly obtained from (4.2.33) by putting = x = (). The corresponding Feynman rules can be obtained from those given in the App. A.2 by leaving out all terms involving the would-be Goldstone fields (/>* and x and the Faddcev-Popov ghosts u , and u*, omitting in the remaining terms all counterterms SZ, and using the massive gauge-boson propagators in the unitary gauge (1.4.13) (V = W,Z),
vlt
Vu
. -9,w + kpku/Mlr =i k2 - M2
(4.2.37)
If we choose the parameter v in (4.2.17) in such a way that it does not satisfy (4.2.14), we get additional contributions to the Lagrangian (4.2.33) arising
)()
I
a u g e theories of t h e electroweak interaction
from tlie
iggs potential,
(4.2.3 ) with the tadpole (4.2.3 )
t = v
From this equation it is manifest that the correct, vacuum expectation value v corresponds to the vanishing of the tadpole t.
4.
s
e have already discussed some applications of the SM in Sect. 1.4. e considered, in particular, niuon decay, neutrino-electron scattering, and decays, and -boson production in e fe annihilation. In all these reactions only the fermion-gauge-boson interaction of the SM enters in lowest-order perturbation theory, which is exactly the one considered in Sect. 1.4. In this section we add examples that are sensitive to other sectors of the SM. In order to illustrate the effects of the non-abelian gauge couplings, we consider -pair production in e e collisions. As examples for reactions involving the iggs boson and its interactions we discuss decays of the iggs boson and production of the iggs boson in e e collisions via e e . 4.3.1
- p a i r p r o d u c t i o n in e e
annihilation
allows to study directly the properties of the The process e e -boson, i.e. its mass, its total and partial widths, and its couplings in the
4.3
(¡(l.r>
Simple application« of tin; Electroweak Standard Model
« lean environment ofe'e" colliders. One important aspect is the investigation of the non-abclian gauge interaction. On the other hand, the W-pair production process in the LEP2 region can be used for an independent direct measurement of the W-boson mass at the level of 0.05% [Ku96). The present experimental error is 46MeV [GuOO]. This in turn gives rise to a stringent precision test of the EWSM (cf. Sect. 4.6.5). We first settle our conventions. Because W-pair production requires a centreof-mass (CM) energy greater than twice the W-boson mass, we can safely neglect the electron mass. The momenta and helicities of the incoming e 1 are denoted by p± and a±, respectively, those of the outgoing W 1 by k± and A±. In the CM system of the e + e~ pair, the momenta read p'± = £(1,0,0,±1),
k£ = E{ l,±/3sini9,0,±/3cosi9)
(4.3.1)
with E denoting the beam energy, the scattering angle between the e' and the W + , and /? = y l - M ^ J E 2 the velocity of the W bosons in the CM frame. The Mandelslam variables are given by s = (p++p-)2
k-)2=4E2,
= (k+ +
t = (p+ - k+)2 = (p_ - k-)2
- 2 E 2 + 2E2f) cos
=
u = (p+ - fc_)2 = (p_ - A;+)'2 = M2W - 2 E 2 - 2E2(3 cosfl.
(4.3.2)
In order to define helicity amplitudes, we need to introduce the corresponding polarization vectors for the W bosons,5 4"(/c±,+l) = -^(0,±cost?,-i, T sinT?), 4 " ( f c ± , - l ) = -J=(0,±cost?,i,Tsint9), e?{k±,
0) = -r^-03, ± sin 0,0, ± cost?) Mw -
i
k
^
-
m
(
4
.
3
.
3
)
''Note t h a t the helicity of m a s s i v e W b o s o n s is n o t Lorentz-invariant. W e define it in t h e CM s y s t e m .
(i()(i
4
G a u g e
t h e o r i e s
of
t h e
e l e c t r o w o a k
i n t e r a c t i o n
W+
e
W+
e
Fig. 4.2 Lowest-order d i a g r a m s for e + c ~ —> W + W ~
Since we neglect the electron mass, the invariant matrix element is only non-zero for <x_ = — o + = a. Right-handed electrons are labelled by a = + and left-handed ones by a — —. At tree level only the three diagrams shown in Fig. 4.2 contribute, if we neglect a Higgs-exchange diagram which is suppressed by a factor m e /Mw originating from the Higgs-electron coupling.6 The ¿-channel /^-exchange diagram contributes only for left-handed electrons, whereas the s-channel diagrams containing the non-abelian triple gauge coupling contribute for both helicities. Introducing the basic matrix elements
Mf = v(p+)f+(U -^-^^«(P-), Mf = v(P+)[2U{e-k+) - 2Me+fc_) (4.3.4) we can write the lowest-order amplitude as
with a" defined in (A.2.22). In the second line we have inserted the explicit form of the Z-bosou-fermion couplings and rearranged the contributions of the individual diagrams into two gauge-invariant subsets, one of which 6
A c t u a l l y , the suppression factor is (HI,./J\/W) 2 , w h e r e t h e s e c o n d factor results from
4.3
S i m p l e
v/i [GeV] 165.0 176.0 190.0 250.0 500.0 1000.0
a p p l i c a t i o n «
_Born 10.66 16.47 18.12 15.40 6.71 2.51
of
tin;
E l e c t r o w e a k S t a n d a r d
„Horn
-.Horn
21.27 32.74 35.92 30.52 13.37 5.00
0.05 0.21 0.32 0.27 0.05 0.01
„Born T 5.19 8.84 10.76 11.77 6.34 2.46
M o d e l
(¡(l.r>
„Born M 4.13 5.84 5.70 2.87 0.25 0.02
„Born \j
1.34 1.79 1.65 0.75 0.11 0.03
Tab. 4.2 Integrated lowest-order cross section in pb for different polarizations and different CM energies involves the charged-current coupling constant e/(v/2^w) and the other the electromagnetic coupling constant e. According to (1.2.26) the unpolarizcd cross section is given by (4.3.6) \ M J B orn
Integration over the solid angle yields the total cross section [A177a] Elioni -=
—Ê.Î 1 +
.5
+
2 Ml,
1
MU 1 ~ 2 4 ) s-Ml
+
2 Mwi
2Mk(2
1
Mk
+
5
1 + p
V.
l+
P
^v1 12 M&
A^(l-4s2+84)/32 + 48(.s - M2)2
.s2 K
20.s io + TTT- + 12
<
}
(4.3.7)
In Table 4.2 we list the cross section a for different CM energies and different polarizations of the electrons and W bosons for Mw = 80.30 GeV and My, = 91.187 GeV. The cases of two transverse, one transverse and one longitudinal, and two longitudinal W bosons are denoted by T, M, and L, respectively. Positrons are assumed to be unpolarizcd.
(¡08
4
G a n g «
t h e o r i e s
o f
t h e
c l e c t r o w e a k
i n t e r a c t i o n
Since the longitudinal polarization vectors are proportional to E/Myj [cf. (4.3.3)], the contributions of individual Feynman diagrams to the invariant matrix element grow with energy for longitudinal W bosons. Using e *± ~ A£/M W) E > Mvv, we obtain
=
(4.3.8)
for the matrix elements involving two longitudinal gauge bosons. The absolute square of the spinor chain w(p + )^ + (l ± 75)7i(p~) can be evaluated as a trace, ^(P-f)ff-i-1 s2 2
= jp sm2ti,
(4.3.9)
resulting in 7 } ( p + ) ^ + ^ p 7 z ( p _ ) = ¿/Jsintfe*
(4.3.10)
where <j> is a convention-dependent phase which we set to zero. Putting everything together we obtain for two longitudinal W bosons in the highenergy limit Ma
KA"
t
(4.3.11)
Consequently, the cross section resulting from the ¿-channel diagram alone grows with s in the high-energy limit LL
¿ t o )
a2
.s2
sin2 d,
(4.3.12)
Born/"
and thus eventually violates unitarity. In the EWSM, the gauge symmetry guarantees that the s-channel diagrams exactly cancel the bad high-energy behaviour of the ¿-channel diagram so
4.3
S i m p l e
a p p l i c a t i o n «
of
tin;
E l e c t r o w e a k
S t a n d a r d
M o d e l
(¡(l.r>
y/s [GeV] Fig. 4.3 Gauge cancellations in t h e total cross section for W-pair production: the individual curves correspond to the Born cross sections arising from the 5-channel (<7norn,.«) an ( l ¿-channel (crisorn.i) diagrams alone, and to the complete EWSM cross section (<7u<>rn.SM)-
that the complete lowest-order matrix element for longitudinal W bosons approaches a constant at high energies. The cross section in the EWSM
\
d i
V Born
16s
16s„<4
drops proportional to 1/s in the high-energy limit (s The cancellations between the individual Feynman diagrams are illustrated in Fig. 4.3 for the total cross section. They reach one order of magnitude at 400 GeV and two orders at 1 TeV. We have found that the cross section tr(e+e~ -* W^W,~) within the EWSM respects unitarity at lowest-order. This is an example of a general property of gauge theories: while tree-level unitarity is violated in processes involving longitudinal gauge bosons in general theories, it is guaranteed in gauge theories. In fact, the requirement of tree-level unitarity of all physical processes applied to a general quantum field theory involving vector fields enforces a gauge theory [L173, Co74a].
fi!)()
I
G a u g e theories of the electroweak interaction
0 Io] Fig. 4.4 Lowest-order differential cross section a t y / s = 5 0 0 GeV ( T h e m e a n i n g of t h e curves is explained in t h e t e x t )
After the gauge cancellations, the ¿-channel contributions dominate the EWSM cross section at high energies owing to a large coupling constant and the l/t pole. However, the ¿-channel contributions are absent or suppressed for all polarized cross sections except for transverse W bosons and lefthanded electrons. Thus, this polarization dominates the total cross section at high energies and all others are smaller by at least a factor 30 log(.s/Mv2v). These features are illustrated by Fig. 4.4, which shows the differential cross section for various polarizations at yfs — 500 GeV. The solid curve corresponds to unpolarized W bosons and unpolarized electrons, the dotted one to unpolarized W bosons and right-handed electrons. The dashed curves represent unpolarized electrons and, in order of increasing dash length, the cases of two transverse, one transverse and one longitudinal, and two transverse W bosons, respectively. The situation at y/s = 190 GeV is illustrated in Fig. 4.5. Even at this energy a ¿-channel enhancement in the forward direction of about a factor 20 is present.
The threshold behaviour is important for the determination of the W-
4.3
Simple application«
of
tin;
E l e c t r o w e a k
Fig. 4.5 Lowest-order differential cross section at Fig. 4.4)
S t a n d a r d
M o d e l
(¡(l.r>
= 190 GeV ( s a m e conventions a.s in
boson mass [Ku96] from the measurement of the W-pair-production cross section close to threshold. For small the matrix elements behave as
M
—
°s «
OC ß,
Ml t
(4.3.14)
a 1
for fixed scattering angles. Consequently, the ¿¡-channel matrix elements vanish at threshold and the ¿-channel graph dominates in the threshold region. For P 1 the differential cross section for unpolarized beams and W bosons is given by da dfì
a* 1 Born
T44
34-1 + ß 1 +4/3 cost? 4c?„ - 1
ö(ß2)
(4.3.15)
where the (angular-independent) leading term oc (i originates only from the ¿-channel diagram. In the total cross section 7TO2 1 "Born
s 0
r1 4
4s'° w
ß +
0(ß3)
(4.3.15)
fi!)()
I
G a u g e
t h e o r i e s
of
t h e
e l e c t r o w e a k
i n t e r a c t i o n
/l h h h Fig. 4.6 D o u b l y - r e s o n a n t lowest-order d i a g r a m s for e + e
W
>i/c w
^
fx
^ ^
h h
^
h
—> W W — » 4 /
all terms oc ft2, drop out and the ¿¡-channel and the s-i-interference contribution are proportional to ft3. These properties can be derived from elect ronhelicity conservation, CP invariance, and the orthogonality of different partial waves [Be94]. The total cross section close to threshold is completely determined by the tchannel diagram and thus practically independent of possible non-standard physics contributions, such as anomalous gauge-boson couplings, which affect mainly the s channel. The normalization of the cross section in the threshold region is determined by the well known W-boson-fermion coupling. Its energy dependence is governed by the linear rise in ft. This is even true after the inclusion of finite-width effects because the next-to-leading ft3 term becomes sizeable only several Tw above threshold. This fact allows for a largely model-independent determination of the W-boson mass from the W-pair-production threshold. In fact, the W-boson mass has been measured in this way at LEP with the result Af w = 80.40 ± 0.22 GcV [LEP97]. As discussed already in Sect. 1.4.2.4, the W bosons are unstable and decay rapidly. The approximation of stable W bosons is not adequate, and one has to include the effects of the finite width of the W bosons. Thus, instead of the process e + e~ —> W + W~ one has to consider e + e~ -» W + W~-> 4/, i.e. a four-fermion final state [Bc94]. The corresponding Feynman diagrams are shown in Fig. 4.6. The decay of the W bosons allows for an alternative method for the measurement of the W-boson mass: one measures the four-momenta of the final-state fermions and reconstructs from those the invariant masses of the fermion pairs. Then, the W-boson mass can be obtained by a fit to the invariant-mass distributions of the fermion pairs. The present experimental result from LEP2 for the W-boson mass reads M w = 80.427 ± 0.046 GcV [Gu00]. The cross section resulting from the Feynman diagrams in Fig. 4.6 is not gauge-invariant. This is due to the fact that subprocesses other than W-
4.3
S i m p l e a p p l i c a t i o n « of tin; E l e c t r o w e a k
S t a n d a r d
M o d e l
(¡(l.r>
pair production contribute: to the four-fcrinion final state. Therefore, all Foymnan diagrams for e + e~ —> 4/ have to be included in order to obtain the gauge-invariant cross section that is observed experimentally. However, the W-pair-production diagrams dominate the cross section in regions of phase space where on-shell W-pair-production is allowed owing to the enhancement by the two resonant W-boson propagators. By subtracting the small contribution of the non-W-pair-production diagrams from the experimentally measured cross section one can define an off-shell W-pairproduction cross section, conventionally called CC03 cross section [Ba96]. This cross section is theoretically defined from the W-pair-production or CC03 diagrams in Fig. 4.6, calculated in the Feynman gauge. Figure 4.7 shows the experimental data points for this cross section as given by the LEP Electroweak Working Group [LEPOO] for the Summer 2000 conferences together with theoretical calculations. There is good agreement between data and theoretical predictions at the per-cent level. To reach this agreement, the complete O (a) radiative corrections to e + e -> WW - » 4 / had to be included in the theoretical prediction. In order to obtain a gauge-invariant result these corrections have been calculated in the double-pole approximation. In this approximation only the leading term in an expansion of the matrix elements about the poles of the two resonant propagators is taken into account. The theoretical prediction including these and further relevant higher-order effects are provided by the Monte Carlo programs YFSWW3 [Ja98] and R A C O O N W W [DeOOa]. Below a CM energy of 170 GeV, where the double-pole approximation is no more applicable, the prediction in Figure 4.7 is continued by the program G E N T L E [Ba93a], which does not include the non-universal electroweak corrections. The version of G E N T L E used in this figure is tuned to reproduce the prediction of R A C O O N W W and YFSWW3 on the total cross section at LEP2 within a few per mill (see Ref. [GrOO]). While the treatment of the four-fermion process is necessary for a theoretically adequate description, the essential features of W-pair production discussed above in the on-shell approximation are not affected by these refinements. 4.3.2
P r o d u c t i o n a n d decay of the Higgs boson
In the EWSM, the fundamental particles (leptons, quarks, and weak gauge bosons) acquire imisses through their interactions with a scalar field with
()
s
21
LEP 20
.
1
Preliminary
Racoon Gentle 2.1
2
F
1.14
.
-
10
-
1
1
210
GeVl Fig. 4.
Total
production cross section at
E 2;
given by the
E E
G
E
non- ero vacuum expectation value. The experimentally observable conse uence of this mechanism is the neutral scalar Higgs particle with its couplings to fermions and gauge bosons. o far, however, there is no experimental evidence for its existence. ithin the E the mass of the Higgs boson is a free parameter that is not directly related to nown experimental uantities. 4. .2.1
Higgs-boson mass
For a large Higgs-boson mass, the Higgs sector becomes strongly interacting and instead of the Higgs boson many resonances should show up in the TeV range. The physical content of the E changes drastically. Therefore, it is interesting to derive limits on the mass of the Higgs boson by re uiring
4 . 3
S i m p l e
a p p l i c a t i o n «
of
tin;
E l e c t r o w e a k
S t a n d a r d
M o d e l
(¡(l.r>
that the EWSM docs not become strongly interacting. These limits do not apply to the model in general but oidy to its perturbative realization. The strength of the Higgs self-interaction is determined by the Higgs-boson mass itself. For instance, the coupling between four Higgs bosons is given hy i/iiiiim = 3e 2 M, 2 ,/4s 2 M'fa = 3M^/2v2. Requiring <7HHHH/47t < 1 implies |Ve77, Le77] Mn £ \Z&n/3v ~ 700 GeV.
(4.3.17)
A similar limit can be obtained by considering the running of the quartic Higgs coupling A(p) = 2M2i(n)/v2 [Ca79, Li86]. Solving the renormali/.ation group equation for the quartic coupling
d i ^
=
¿
( a 2 + Xal
-
gt) +
°{e2)
+ o{02
(4:U8)
where
(31G
4
G a u g e
t h e o r i e s
o f
t h e
e l e c t r o w c a k
i n t e r a c t i o n
H
Fig. 4.
4. .2.2
owest-order diagram for e e
-
II
Higgs-boson production
e have seen in ect. 4.2 that the couplings of the Higgs boson to other particles are proportional to the masses of those particles. Therefore, light particles essentially play no r le in reactions involving the Higgs boson. In particular, the direct production of the Higgs boson via e e or qq annihilation is practically excluded. The Higgs boson can be most easily produced via its couplings to the wea gauge bosons or to the top uar . The present lower limit on the Higgs-boson mass \\ of the order of 11 GeV I I . The originates from direct searches at E using the process e e the cross relevant Feynman diagram is shown in Fig. 4. . For QM section for this process is enhanced, because the boson that couples to e e - becomes resonant. This allowed to search for a Higgs boson with a mass below GeV. For larger Higgs-boson masses the Higgs signal cannot be separated from the bac ground. However, for Ecu y, the second -boson can be real, and the corresponding cross section becomes again large. In fact, this process would be the dominating source of Higgs bosons for energies below GeV, should the Higgs-boson mass fall into this range. e consider the Higgs-strahlung or e p.,.,a
e p ,or
or en process c,
H fcH
in some detail. Here o 1 2 , - 1 2 and licities of the incoming leptons and the outgoing
4. .1 1 , , - 1 denote the heboson. eglecting the
n e x c e p t i o n is given by the gluons, which c o u p l e to t h e Higgs field v i a fermion loops involving t h e strong coupling.
4.3
S i m p l e applications of t h e Electrowcak Standard Model
FIR. 4.9 Lowest-order diagram for c + c
(j!7
—> HZ
electron mass, which is small with respect to the energies involved, the momenta read p£ = £ ( l , 0 , 0 , ± l ) , \ / Jtf2 _ V/2 ¿z,h = E ( 1 ± ———, ±(3 sin t?, 0, ±/3 costiJ
(4.3.20)
in the centre-of-mass (CM) system, where E denotes the beam energy, s • '\E2 the CM energy squared, -0 the scattering angle between the e"1' and the Z boson, and P = ^ y/[s - {Mz + M„)2][s - (Mn - M z ) 2 ]
(4.3.21)
the velocity of the outgoing particles in the CM system. The helicities of the Z boson are defined via the transverse (±) and longitudinal (0) polarization vectors e*tt{±) = 4=(0,cosi9,Ti,-sini?), v2 E
(o) = — My,
(4.3.22)
s M M . Q1?, s + My -- M,2, 51 cos i9 \ I. n 0, I(a¡3, + l ^ ~ u & sill
\
S
S
J
Since we neglect the electron mass we can omit all Feynman graphs containing a coupling of a (physical or unphysical) Higgs particle to an electron, as they contain an explicit suppression factor m,./Mw- Then, only one Feynman diagram is left at lowest order (Fig. 4.9). The corresponding invariant
fi! ()
I
G a u g e
t h e o r i e s
of
t h e
e l e c t r o w e a k
i n t e r a c t i o n
amplitude reads
nor,
=eg*
2
1
^v(p+)f(\)
u p
,
i
wCw
4. .2 where the coupling of the given in .2.22 .
-boson to left- and right-handed electrons fl f is
From the amplitude 4. .2 its da df
the differential polari ed cross sections result
4. .24
(s - M*)*
orn
1
cos2 -
for
2
x
l ( ^
+ 1 si 2
for
)
1 A= a
The unpolari ed cross section is obtained by summing over the polari ations of the boson and averaging over the fermion polari ations da d f 2 g orn
a2
m l
t ffs 2,
t f 2
s -
4 4
m
2 ,
2
V
4. .2 For high energies and well above threshold, the longitudinal gauge bosons dominate the cross section owing to the factor s originating from the longitudinal polari ation vector. hile the cross section for transverse bosons drops li e 1 s 2 for high energies, the one for longitudinal bosons behaves as 1 s. The sin2 angular behaviour of the longitudinal cross sections is characteristic for the production of a scalar particle. y integrating 4. .24 and 4. .2 the total polari ed cross sections
a orn
2
m l
over the production angle
ffe
we obtain
1
for
2
2
s
2
« I
r
,
f r
4. .2
4..'í
S i m p l e
a p p l i c a t i o n s
o f
t h e
í i l e c t r o w e a k
S t a n d a r d
M o d e l
(¡I!)
v/i/GeV Fig. 4.10 Lowest-order cross section for different H i g g s - b o s o n m a s s e s MH = 60, 100, 200, 300, and 500 GeV
and the total unpolarized cross section
_ oftr
m
l
to?)2+
(<>.-)' ( . , sD2 \
The total cross section for different Higgs-boson masses is shown in Fig. 4.10. For small Higgs-boson masses it is about 0.5 1 pb at \fs = 200 GeV, and about 50fb at s/s = 500GeV. The unpolarized cross section drops proportional to 1/s at high energies. H, At energies above about 500GeV, the WYV-fusion process, e + e~ —> becomes more important for Higgs-boson production since the corresponding cross section, cr(e+e~
f c P e H) =
fi! ()
I
G a u g e
theories
o f
t h e
e l e c t r o w e a k
i n t e r a c t i o n
H
e
Fig. 4.11
owest-order diagram for c c
1 oivH
rises logarithmically with the energy. The cross section for fusion is about one order of magnitude smaller owing to the smaller couplings to the boson. 4. .2.
Higgs-boson decays
ince the couplings of the Higgs boson to other particles are proportional to the masses of those particles, the Higgs boson will predominantly decay into the heaviest particle that is inematically allowed. If M\\ 14 GeV, the Higgs-boson decays mainly into fermions, in particular, into bottom uar s. The corresponding partial decay width is
r ( H
^
=
with the colour factor N[.
(4.3.29)
(
1 for leptons and
for uar s.
For larger Higgs-boson masses the decays into one real and one virtual gauge boson become important. If M\\ > 2M\\, the Higgs boson can decay into a pair of real bosons and for H 2 My also into real bosons. The corresponding partial decay widths
4. .
4 . 3
S i m p l e
a p p l i c a t i o n «
o f
tin;
E l e c t r o w e a k
S t a n d a r d
M o d e l
(¡(l.r>
Fig. 4.12 Branching ratios of the important Higgs-boson decay m o d e s and total Higgsboson decay width (from Ilef. [Dj96])
dominate for M\\ > 2M\\ even if the decay channel H —> tt opens. For M\\ M\\ the dominant contribution to the decay comes from longitudinal gauge bosons (with polarization vector proportional to A/n/Mw), and the total decay width is approximately given by r „ « r(H -> w w ) + r(H
zz)»
M2
(4.3.31)
For Mn a ^
^ v6o!
=
V 3
» 1.4 TeV
(4.3.32)
the width becomes larger than the mass, and the physical interpretation of the Iliggs boson as a particle becomes doubtful. The total decay width of the Higgs boson and the branching ratios are shown in Fig. 4.12 as a function of the Higgs-boson mass. Apart from the decay modes discussed above, also the decays into a pair of gluons, H —> g g , or photons, II —» 77, are relevant for intermediate Higgs-boson masses.
fi!)()
I G a u g e theories of t h e electroweak interaction
Although these decays proceed via loop diagrams, the corresponding rates arc non-negligible because the coupling of the Iliggs boson to the virtual particles is not suppressed and, moreover, the gluons couple with the strong coupling constant.
4.4
Q u a n t i z a t i o n of t h e Eleetroweak S t a n d a r d Model
4.4.1
G a u g e fixing a n d F a d d e e v - P o p o v
fields
The quantization of the EWSM requires, as the quantization of all nonabelian gauge theories (cf. Sect. 2.4), the introduction of a gauge-fixing term and of Faddeev-Popov fields. For higher-order calculations and renormalization it is preferable to choose a renorraalizable gauge, i.e. a gauge in which the gauge-boson propagators behave as l/k~ for large k2. Spontaneous symmetry breaking introduces mixing terms VfA d,,<j> between the gauge-boson lields and the would-be Goldstone fields [cf. (4.2.33)], which give rise to non-diagonal propagators. As explained in Sect. 4.1.2.2, one can choose a renorraalizable gauge in which these mixing terras are absent. To this end, we introduce a gauge-fixing terra of the form £fi* = " ¿ ( C A ) 2 - ¿ ( C Z ) 2 -
C-
(4.4.1)
with linear gauge-fixing operators C± = d"Wf Cz
= d"Zlt -
CA = d^Afi.
T
iMw^*, Mrf'zx,
(4.4.2)
This general linear gauge contains five independent gauge parameters £a, a = Z, ±, where ^ = a = A, Zy ±, and For = ^iv and = the terras involving the would-be Goldstone fields in (4.4.2) cancel the mixing terms Vn d''4> in the classical Lagrangian (4.2.33) up to irrelevant total derivatives. This gauge is called '/. HoofI gauge and is used in the following if not stated otherwise.
4.4
Q u a n t i z a t i o n
o f
t h eI S I e c t r o w e a kS t a n d a r d
M o d e l
(¡2!)
I lie tonus quadratic in the gauge fields allow for well-defined gauge-boson propagators [cf. (4.1.62)], \AV"(k)
'
{
)
- j ~a'"/
~ '
+ k
Hk"lkl
_ ¡^^
& - Ml
(A A
(4,4 3)
' k* k2 - iaMl'
-
where « = A,Z,±, V7' = Z,V± = W±, and M± = M w . The transverse parts, which describe the physical degrees of freedom, are independent of ( lie gauge parameters whereas the longitudinal parts are gauge-dependent.. The terms quadratic in the would-be Goldstone fields generate mass terms for those fields, - Z w M U + r ~ \tzMlx\
(4.4.4)
so that the corresponding propagators read
with a — Z,±, <j>7j = xAccording to the general results of Sect. 2.4.1, the Faddeev-Popov ghostfield Lagrangian is obtained as follows (a,6, c G {/I, Z, ±})
(4.4.6) Inserting the gauge-fixing terms (4.4.2) and using the gauge transformations (4.2.35), we find up to total derivative terms ¿ghost
+ i w M ^ ) u + + ie(0"« + ) {A tl -
=
- ie(d"u + )W+ L A - eM^wu+
ZSy/
- (u+ -> u~, W+
+ \X)u+
- + (uA \ £Cy/S W
YV~, + -> <j>~, i
-i)|
C
/
4—^uZ)
6 2 4
4
G a u g e
t h e o r i e s
of
t h e
e l e c t r o w e a k
u '{d dlt+Zz 'i)u -
e ' X' u-z
- u d'ld,tu
2 Cv s
- ie TMZ -
+ ie(d'lu ) (
s
i n t e r a c t i o n
(S^u ) ( +u~ ((f>+u~ + tT -
~u+)
~u+)
f)~u+) .
(4.4.7)
The ghosts are coupled to the gauge fields and to the would-be oldstone fields. From the bilinear terms we obtain the Fa.dd.cev- opov ghost propagators cf. (4.1.62) ,
The Faddeev-Popov ghost field u is massless. The masses of the other ghost fields coincide with those of the scalar gauge bosons e(/c) oc ] and the woidd-be oldstone bosons (cf. Sect. 4.1.2.2). This is necessary to allow for the cancellation of the corresponding unphysical poles in S-matrix elements. The gauge dependence of these masses reflects the unphysical nature of the corresponding fields. The form of the propagators (4.4.3), (4.4.5), and (4.4. ) ensures that the t ooft gauge leads to a renormalizable theory for finite values of the gauge parameters For a - oo the masses of the unphysical degrees of freedom (except for u ) tend to infinity. As a consequence, the unphysical degrees of freedom decouple (uA becomes a free field) and we are left with the theory in the (non-renormalizable) unitary gauge. For = 1, the 't ooft-Fe nman gauge, pole positions and residues of the longitudinal parts of the gauge-boson propagators coincide with those of the transverse parts, so that the gauge-boson propagators take the simple form
=
(4-4. )
In this gauge, the masses of the unphysical fields equal those of the corresponding physical fields and the k^ki, parts of the gauge-boson propagators are absent. Therefore, the t ooft-Feynman gauge is most convenient for practical higher-order calculations.
4 .4
Q u a n t i z a t i o n
o f
t h e
E l e c t r o w e a k
S t a n d a r d
M o d e l
i>2f>
Adding up all terms (4.2.33), (4.4.1) and (4.4.7) we obtain the complete Lngrangian of the EWSM suitable for higher-order calculations, (4.4.1(1)
¿GSW = ¿cl + ¿fix + ¿ghostThe corresponding Feynman rules are given in App. A.2. 4.4.2
B R S s y m m e t r y and physical fields
The gauge-fixing and Faddeev-Popov terms destroy the ordinary gauge invariance of Cc\. However, as discussed in Sect. 2.4.3 for a general gauge theo(BRS) transformations ry, ¿GSW is invariant under the Becchi• RouetStora [I3e74]. The transformations for the gauge-boson, scalar, and fermion fields A,Z,±, are obtained from (4.2.35) by the replacement 59a = ¿A?za, a = where u"(x) are Faddeev-Popov ghost fields, and f5A is an infinitesimal anticommuting constant. Using the BRS operator s, introduced in Sect. 2.4.3.1, these transformations read sAfl = dpUA + ic[W+u~ sZ,t = dpuz sWf
- i
ec.
W~u+),
-
= d^u^ie
sr/=-
W~u+],
-
W* y
u
A, ~
IC . . . _
+ ^—\4> »
^
I,
-<}> u+J,
e
— [v + rj\uz + ^—[4>+u~ + ~u+], ¿sw wcw w Cw S^
5
= Tfietf,* u + w A
c..2
wu
Cww z.
,c
7, ±I ^-[v
+ i/iix]^,
(4.4.11)
with vf, defined after (4.2.36).
fi!)()
I
G a u g e
t h e o r i e s
of
t h e
e l e c t r o w e a k
i n t e r a c t i o n
The BRS transformation of the ghost fields in the symmetric basis is directly obtained from (2.4.35), <W*a
= -6\l-geabcv.buc,
a,b,ce
{1,2,3},
¿BRS«B = 0.
(4.4.12)
Translating this into the basis of mass and charge eigenstates
uz = c w u 3 + swuB,
uA = - s w u 3 + cwuB,
(4.4.13)
results in ^brs«*
=
i — ^ ( s w ^ - cwuz) sw
¿BRS«2 = —<5Ai e—u~u+ sw ¿BRSuA
=
=
¿Xsu*,
SXsuz,
= SX ie«"« + = SXsuA.
(4.4.14)
The BRS transformation of the antighost fields is determined by the gaugefixing terms [«^ = (it1 ± m2)/%/2], ¿BRS«* = -SX-^-C* w
= ¿As«*,
¿BRSUz = -SX^-CZ iz
=
¿BRS« / 1 =
SX^-C
A
U
6Xsuz,
= 6Xsua.
(4.4.15)
In order to get BRS transformations that arc independent of the gaugefixing term and strictly nilpotent, one introduces auxiliary fields I3a, a = A, Z, ±, the so-called Nakanishi- Lautrup fields [NaGGa, La67], and rewrites the gauge-fixing Lagrangian as (cf. Sect. 2.4.3.1) £fix = D+C~ + B~C+ + ZWB+B~
+ BZCZ
+ ^-BZBZ
+
BACA
+ ^BABA
(4.4.1G)
4.4
Q u a n t i z a t i o n
o f
t h e
B l e c t r o w c a k
S t a n d a r d
M o d e l
(127
with the gauge-fixing operators from (4.4.2). The BRS transformations of the antighost fields are replaced by ¿BRS«rt = SXBa = S\sua, SBnsBa = 0 = SXsBa.
a=
±,Z,A (4.4.17)
As argued in Sect. 2.5.4.3, the physical degrees of freedom of a gauge theory can be extracted by considering the BRS transformations of the asymptotic lields which are related to those of the free fields. We do not go through the formalism but consider directly the BRS transformations of the free fields, sA,t = ¿V»A, sq = 0,
srpif = 0, sua = 0,
sZfi = d(iu7\ sx = —M%uz,
s ^
sW*
=
scf>± =
d^u*, iitfw«1,
= 0,
sua = Ba,
sBa = 0,
a = ±, Z, A.
(4.4.18)
The physical degrees of freedom of the photon field are as discussed in Sect. 2.5.4.3 (pages 245 246) for the general non-abelian inassless gauge field. For massive gauge fields, the situation is as follows. The scalar component of the massive gauge fields, the would-be-Goldstone fields x and , and the antighost fields u a are not annihilated and thus correspond to unphysical degrees of freedom. The ghost fields and the auxiliary field Ba are BRS transformations and consequently involve only zero-norm states that decouple from the physical S matrix. Thus, the physical states are the three degrees of freedom of the massive gauge-boson fields, the two transverse photon polarizations, the Higgs field, and the fermion fields. Here we see again the quartet mechanism at work. The unphysical fields Z z Z form quartets, ( u * , ^ , « * , ^ ) , and (UA,A£,Ua,BA). The quartet involving the longitudinal mode of the photon field A\' is equivalent to the quartets in an unbroken gauge theory and has already been discussed in Sect.. 2.5.4.3 on page 245. For the quartets of the broken gauge theory, the equations of motion for the B fields, BA = —C'L/£A, tell us that also
(U ,x,u ,B ),
"The scalar polarization of vcctor licltls is generally defined by c(k)k / 0. While for inassless vcctor fields this implies e(k) oc (|fc|,— k), for massive vector fields it entails e(k) oc k. Note that for inassless vector fields £•(&) oc k corresponds to the longitudinal polarization.
fi!)()
I
G a u g e
t h e o r i e s
of
t h e
e l e c t r o w e a k
i n t e r a c t i o n
the C°, which are linear combinations of the would-be Goldstone fields and the scalar components of the associated massive gauge fields, arc zero-norm states. The linearly independent combinations of these fields are unphysical owing to their non-vanishing I3RS transformation. Instead of this linear combinations we have written for simplicity the would-be Goldstone fields in the quartets. 4.4.3
S l a v n o v - T a y l o r identities
As shown in Sect. 2.4.3, the BRS symmetry gives rise to relations between different Green functions, the Ward identities or Slavnov-Taylor identities. The generic form of these identities (2.4.62) can be directly transferred to the Glashow-Salam-Weinberg theory: O . ^ n * / , ) ,
(4.4.19)
/ represents anyone of the fields Vfi, ua, ua (a = A, Z,±), " (a =
where Z,±), r/,
or
Here, we use the formalism without B fields.
Consider the identities that follow from the BRS invariance of Green functions involving one antighost field and arbitrary other fields. Using (4.4.19) we obtain
o - T T ^ n » , , )
k
l
m>k
where a¡. stands for the sign that results from auticonnnuting 6\ to the left and the sum Y^k extends over all fields in [7; If all the fields VI' ¡, are physical and on shell, their BRS variations vanish [cf. (2.5.160)], and we find (omitting terms that vanish after truncation) 0 = (rc"
n I
(4.4.21)
4.4
Q u a n t i z a t i o n
of
t h e
ISIectroweak
S t a n d a r d
M o d e l
(¡2!)
For the ease with additional gauge-fixing terms we obtain in the same way (no sum over a) 0 =
¿BRS /rr-a
(rī(nc-)n*rys)
5X
^a
m
I
m
I
- E(Tüa(SCa*)( n k
° a m ) n ^r , P h y S ) =
mj&k
(4.4.22)
I
Using the equation of motion for the antighost Heids [cf. (2.4.63)]
where X stands for any product of fields, and the fact that neither Ca"' nor ^as.phys j n (4 4 2 2) involve antighost fields, we arrive at ( 4 c ( i ) ( n c a m (®m)) n * r p l , y s ) 7n
(4-4-24)
I
fc
ni/fc
I
The terms 011 the right-hand side involve ¿-functions and thus correspond to disconnected parts. Thus, we obtain for connected Green functions
(Tic»n^(xra)n^phys)c=o-
(4-4-25)
Without physical fields 4^' phy3 and for one extra gauge-fixing term, equation (4.4.24) reduces to (TCa{x)Cb(y))
so *
'
= -iSabS^(x
- y).
(4.4.26)
(¡30
4
G a u g e
t h e o r i e s
of
t h e
c l e c t r o w c a k
i n t e r a c t i o n
Inserting the gauge-fixing functions (4.4.2) and going to momentum space we find the following non-trivial relations for the propagators of the charged bosons
WkTG%,+W~
(k) - ZwMvi&Gy+r
- ZwMvt&Gfw-
(k)
(k) +
(k) =
(4.4.27)
and for the neutral boson propagators k"k»G™(k)
- 2\izMzk^Gz^(k)+ezMlG^k) k"k"G^(k)
= -ife,
- i l z M z k r G * * { k ) = 0,
k»k"G^(k)
= -iU
(4.4.28)
According to the conventions of App. A. 1.3.2 the field labels indicate the outgoing fields. The momentum arguments denote the incoming momentum of the particle corresponding to the first upper index. These relations between the longitudinal parts of the gauge-boson propagators, the gaugefield would-be-Goldstoue-field mixing propagators, and the propagators of the would-be-Goldstone bosons are exact and of course fulfilled in lowest order. They show that the unphysical parts of these propagators are not independent and, in particular, guarantee that their poles coincide. The relation for the photon propagator ensures that its longitudinal part gets no higher-order corrections.
4.4.4
Lee identities
Just as in the Yang Mills theory discussed in Sect. 2.4.3.4 also in the Standard Model of the electroweak interaction Lee identities can be derived from the BRS invariance of the effective action. For linear gauge-fixing Junctionals of the form (a = A, Z, ±) Ca{V.;x} =
Y, b=A,Z,±
+ £ b=Z,±
(4.4.29)
4.4
Q u a n t i z a t i o n
of
t h e
B l e c t r o w c a k
S t a n d a r d
M o d e l
(127
like (4.4.2), the identities corresponding to (2.4.77) and (2.4.78) read
°' J
J
6r
k
¿T
b=A,Z,±X
dK
+
^ ^ : J
sr
¿r
sr
'
b=Z,±K
+
V
+
J f A w m J
¿r
(4.4.30)
°n/
where T is the vertex functional with the gauge-fixing terms subtracted (cf. (2.4.76)], and i runs over all fermion fields. The terms involving the sources K of the BRS-transformed fields [cf. (2.4.53)] read ¿BRS-sources =
£
{K^sA% + K ™ " ) +
a=A,Z,±
£ a=Z,±
+ KvST) + K^I
+ (sfc)ffj,
(4.4.32)
and the sources for the charged gauge-boson, ghost, and would-be Goldstone fields arc defined as iC± = ( K l ± \K'2)/V2. Classically, the physical fields and their properties can be directly derived from the Lagrangian. In the quantized gauge theory these properties can be extracted from the Lee identities as follows. We investigate the consequences of the Lee identities for the two-point, functions in the charged-boson sector following Ref. [Ao80]. The relevant fields are the charged gauge-bosons fields W^, the charged would-be Goldstoncboson fields and the charged ghost and antighost fields u ± and u*1. Differentiating (4.4.30) with respect to and uT", putting the sources equal to zero, and transforming to momentum space yields f ^ ^
(k, -k)rKvu*'"(k,
- k ) + f j f * ^ * , -k)t'K?u*(k,
- k ) = 0,
(4.4.33) where for instance T K v u:f ' v (k, —k) is the Fourier transform of =
SF
5K$v{x)5irf{y)
(4.4.34)
()
I
s
ll other terms vanish because of ghost-number and charge conservation. imilarly, differentiation of 4.4. with respect to p and u T yields T
( ,-
+
)t ^( ,- )
)
>*( ,- ) = . 4.4.
In order to proceed we have to decompose the vertex functions into covariants and invariant functions. For the and vertex functions we write
rr
ifc, -fc
v f
fc, -fc
r
T ,v 4
-if
fc 2 fc, -fc
fc2 ,
fc, i f
4.4.
with the pro ectors on transverse and longitudinal parts T
( =
tl u\
l
m> ~
In addition we have f u fc, -fc
,
=
u
fc f
fc,jfci 2
.
,
.
(4.4. 7)
fc2 .
4.4.
In the absence of absorptive parts imaginary parts that arise from loop integrals , the effective action is hermitian and we obtain the further relations f r - fc, - f c
-
, -fc f
f c , -fc
fr
fc, - f c fc, -fc
-
f *u~ fc, -fc
f f
f c , -fc , u
f c , -fc ,
f * u ( , -fc .
4.4.
If we re uire C invariance in addition, these vertex functions are purely real. ince the C violation resulting from the uar mixing matrix in the E does not affect the two-point functions at the one-loop level, these are real in the general case. Therefore, we assume no C violation in twopoint. functions in the following discussion. in t h e presence of absorptive parts these relations still hold if t h e c o m p l e x c o n ugation is a c c o m p a n i e d by a multiplication of t h e a b s o r p t i v e parts with I.
4.4
Q u a n t i z a t i o n
o f
t h e I S I e c t r o w e a k S t a n d a r d
M o d e l
(¡2!)
Inserting everything into (4.4.33) and (4.4.35), we arrive at pJVWp/furti _ pVVc/ipK'^u _ Q ^pjv^p^u _ p^p^u
(4.4.40)
= 0)
and by eliminating the ghost contributions pVVWp^ = fc2(p VV0J2
(4.4.41)
The inverse propagators T' 7 are obtained by adding the contributions of I he gauge-fixing Lagrangian to r / J . Using the gauge-fixing terms (4.4.1), we obtain for the inverse propagator matrix r±
-
U"') ~ I
plV±lVT plV^TX /"' ±w
J
rt *
( < j ^ r
v
+ & (RRH/ -
^
( ± F R T M
W
|K) \
(4.4.42) By choosing the gauge parameter10 eiv = | ^ ( A : 0 2 ) , this matrix becomes diagonal for k2 =
(4.4.43) Selecting kfi so that
r? W (k 2 0 ) = r ? w ( k * ) - j—k'o = 0,
(4.4.44)
and using (4.4.41), we obtain r*(A§) = f**(A02) - M ^ ^
= f^(A: 2 ) ( l -
= - ^ r * * ( k 2 ) r ? w ( k 2 ) = 0. K
(4.4.45)
o
10
I n o t h e r g a u g e s t h e matrix can be diagoaalized by a redefinition of t h e fields of the form W,, —> aW,, + bd,,<j> In the presence of absorptive parts only t h e real part of T * can be m a d e diagonal with real (,[ v .
4
s
Thus, the inverse propagators for the (f field and the longitudinal component of the -boson field vanish at the same position, i.e. the poles of the corresponding propagators coincide. ext we determine the pole of the propagator of the charged Faddeev Popov ghost fields. In the gauge (4.4.2) the identity (4.4.31) for a = dr reads
ifferentiating with respect to and putting the sources to zero implies after transforming to momentum space o = \ f
u
( ,
- ) + \e
r
u
- ) + ra
( ,
u
( ,
- ).
(4.4.47) Inserting (4.4.3 ) and using (4.4.40) and (4.4.43), we find ru
u
{ 2)
= ^i 2r£ u( 2)
= Ttt^fn
\(
) (l -
v
r'^u( 2)
frf yi)
^
g
)
(4.4.4 )
for the ghost propagator and thus owing to (4.4.44) rc
u
(fco) =0
(4.4.4 )
i.e. also the poles of the charged Faddeev-Popov ghost propagators coincide with the one of the longitudinal part of the -boson propagator. Thus, the propagators of the four states, the scalar component of the ^ boson, the would-be oldstone boson (f)^, the ghost u , and the antighost m , have a pole in a common position. The fact that all these poles coincide is necessary for the decoupling of these unphysical excitations from the S matrix. These quartet particles can appear in the physical subspace only in a zero-norm combination and hence cannot be observed. e now turn to the two-point functions of the neutral bosons. In the absence of CP-violating effects, the two-point functions of the iggs field do not mix
4.4
Q u a n t i z a t i o n
of
t h e
B l e c t r o w c a k
S t a n d a r d
M o d e l
(127
with those of the other bosons. Equation (4.4.30) gives rise to the following relations [corresponding to (4.4.40)]: £
r f t p « +
= 0,
(4.4.50)
k2 Y , f j ^ f f ^ - f**f K *« e = 0,
(4.4.51)
a=A,Z
a=A,Z
where b,c G {A,Z}. Interpreting f*/''v" and f | A v " as matrices with indices b and a, (4.4.50) implies det f £ v det f '¿vu = - det(f
** u ) = 0.
(4.4.52)
Since detf[ M/ " ^ 0 at tree level and thus in perturbation theory, this means det f \[ v = 0,
(4.4.53)
i.e. ff; 1 has one vanishing eigenvalue. This is just a consequence of the fact that there is only one neutral Goldstone boson that can give mass to the neutral gauge bosons. The Lee identities involve only the longitudinal parts of the gauge-boson propagators. In order to gain information on the physical, transverse parts we need further input. At least within perturbation theory, one-particle irreducible Green functions do not involve any poles. Together with the covariant decomposition of gauge-boson two-point functions
rr :
v
\ k , -k)
= 9Jwrrvb
+ ifcrr*
(4.4.54)
this entails
f r * ( 0 ) = f i T (0).
(4.4.55)
and consequently det f'pV (0) = d e t f £ v ( 0 ) = 0.
(4.4.56)
Therefore, also the matrix of the transverse parts of the neutral-boson twopoint-functions has one eigenvalue zero. The corresponding massless gauge boson is identified with the photon. The physical, transverse photon field is
i. iH 1 Gauge theories of the electrowea
interaction
defined in such a way that it does not mix with the -boson field at ero momentum transfer. Then, the transverse parts and owing to 4.4. also the longitudinal parts of the photon two-point function and of the photon -boson mixing vanish for , fv
f
,
f4
f
.
4.4.
Inserting this into 4.4. for and and observing that u r in lowest order and thus in perturbation theory, we find f
.
4.4.
Thus, the on-shell photon decouples from all the other fields. y analysing the identities 4.4. and 4.4. 1 further, one can deduce in a similar way as for the charged bosons that the eros of the -boson and -boson two-point functions coincide. To this end, it is convenient to choose the gauge-fixing function C
-
x
and to ix the gauge parameters
and
and r l x
l
4.4. in such a way that r
at the value of ;g for which r
fcg
4
.
Finally, we consider the two-point functions of the neutral ghosts. from the identity 4.4. 1 in the gauge 4.4.1 for where are the physical photon and -boson fields,
e start and
tl i
ifferentiating this with respect to in, ifc T i
u
,
4 4
-
- -
gives rise to
r i
4.4. 1 r
4.4. 2
4.4 Quantization of the Electrowcak Standard Model
637
After inserting the covariant decomposition
rf'"" (k, -k) = k^u\k2)y
(4.4.0!})
(4.4.61) turns to pfi4ti° ( ¿ 0 =
- ¡ ^ " V ) ,
(4.4.64)
i.e. pfl'V 1
(o) = o = r f I / , , , ; i (o).
(4.4.65)
The photon ghost is massless. We can define the photon ghost field uA in such a way that it decouples from the Z ghost at k2 = 0, i.e. rfiZu/,(0)=0.
(4.4.66)
Then, as a consequence of (4.4.62) and (4.4.50) for b = Z and c = A
r K * u A (0) = 0 = rH* u A (0),
(4.4.67)
which is used later. It can further be shown that the Z-ghost two-point function has a zero at the same location as the longitudinal Z-boson propagator. All this is consistent with the fact that the contributions of scalar Z bosons (e1' oc A*''), x bosons, u , and uz as well as the contributions of longitudinal photons (e11 oc k'1), scalar photons [e^ oc (k0,— k)], uA, and uA cancel in 5-matrix elements (<:f. Sects. 2.5.4.3 and 4.4.2). As discussed above, there are two quartets of unphysical modes in the neutral sector.
(i.'iH
1 Gauge theories of the electroweak interaction
4.4.5
T h e b a c k g r o u n d - f i e l d m e t h o d for t h e E l e c t r o w e a k Standard Model
4.4.5.1
T h e gauge-invariant effective action
The background-field method discussed in Sect. 2.4.4 can be directly applied to the Cdashow Salatn Weinberg model [De95]. To this end, the gauge fields are split as in (2.4.107) into background fields (denoted with a caret) and quantum fields, K
K
B
+ K>
(4.4.68)
13
1' + 13 >"
»
For the fermion fields no splitting is necessary, as discussed at the end of Sect. 2.4.4.2. On the other hand, we have to split the Higgs-doublet field since we want to include it in the gauge-fixing term in order to eliminate tree-level mixing between the gauge bosons and the corresponding would-be Goldstone bosons. This is done in such a way that the background Higgs field has the usual non-vanishing vacuum expectation value v while the one of the quantum Higgs field is zero,
(4.4.6!))
In order to retain background-field gauge invariance, we must introduce a gauge-fixing functional that is covariant with respect to gauge transformations of the background fields. A suitable ansatz is [Ei89] ¿GF.BFM =
2Çw ¿w L
(ô"cdlt+g7eabcWÏ)Wc>"
- ^ ^ ( ¿ ¡ T ^ - j Ti - r
+
i
2
î 2.
(4.4.70)
Background-field gauge invariance requires that the background gauge fields appear only within a covariant derivative in the gauge-fixing term and that
1.4
uanti at ion « 1 the Electrowea
tandard
odel
G
l lie terms in brac ets transform according to the ad oint representation of I,lie gauge group. oreover, only four independent gauge parameters arc al lowed, and for 2 and and for l y. ixing between gauge bosons and the corresponding would-be Goldstone bosons is eliminat cd by choosing and resulting in the for the bac ground-field formalism. In order to avoid tree-level mixing between the uantum photon and -boson fields, we set b- ote that 4.4. translates to the conventional gauge-fixing term 4.4.1 for , it upon replacing the bac ground Higgs field by its vacuum expectation value and omitting the bac ground 2 triplet field The Faddeev- opov agrangian can be derived as usual, and physical bac ground and uantum fields can be introduced as in ects. 4.2.2.2 and 4.4.2. ore details and a complete list of Feynman rules can be found in Ref. e .
4.4. .2
ac ground-field
ard identities
The most important property of the bac ground-field formalism is the invari ance of the bac ground-field effective action under bac ground-field gauge transformations with associated group parameters cf. 2.4.121 ,
i
4.4. 1
a
This invariance implies linear identities for the vertex functions that are precisely the i ii related to the classical agrangian. For the two-point functions involving neutral bosons these identities read
,
,
,
-
fcT
i
p
w
rf
,
x
.s w c w
T
4.4. 2
rr
,
, -r
4.4. 4.4. 4
4.4. .
4.4. G
This follows also in the conventional formalism by demanding rigid, i.e. global, 2 x l y invariance of the complete agrangian g s w (4.4.10).
(i.'iH 1 Gauge theories of the electroweak interaction
Since the vertex functions are one-particle irreducible, they contain no tadpole contributions, and the tadpoles appear explicitly as r 7 '(0) in tin; Ward identities. For the photon-fermion and the photon-W-boson vertices, QED-like Ward identities are found,
k»rj (k,p,r)
= —eQj [ r % ) - r " ( - P ) ] , = e [r%+*-(k+)
lk,k+1k.)
r £ + * - < - * _ ) - ri*{k)
-
,
^ri*(k)
+
(4.4.77)
(4.4.78)
From (4.4.77) the universality of the electric charge in the EWSM can be derived in the same way as for QED (cf. Sect. 2.5.1.6). 4.4.5.3
Connected Green functions
The construction of connected background-field Green functions and of the S matrix has been described in Sect. 2.4.4.3. In order to define f r u " = f + J d 4 ®£(3p,
(4.4.79)
we choose the usual 't Hooft gauge-fixing term ^
= - 4 - (o,M - J- \d,lW+'tt (.w '
2
- 4 -
-
- \iwMw+ ^
izMy.x)2 (4.4.80)
for the background fields with the background-gauge-fixing parameters £4, and Then, the generating functional of connected background-field Green functions, T c , is obtained from r f u " by a Legendre transformation [De96], Tc{Jj?,J/,Jf
= iffu"{F,/,/} + i J d*x
(4.4.81)
4.4 Quantization of the Elec.trowoak Standard Model
(541
where F = A,Z, W+, W~,T),x,+,(f>~, and <Sfh,n
¿f f ""
T
¿ffu"
As a consequence, the one-particle reducible Green functions and S-matrix elements are composed as in the conventional formalism from a tree structure of vertex functions. While the vertices in these trees arc directly given by the background-field vertex functions, the propagators are determined as the inverse of the two-point vertex functions resulting from rr>l". Consequently, these propagators are exact to the given order of perturbation t heory, i.e. the Dyson re-summation of self-energy corrections is included by construction. Of course, one can expand the propagators and recover the ordinary perturbative expansion. The Ward identities (4.4.71) for the effective action T, which generates l.lie vertex functions, translate into Ward identities for the functional T c , which generates the connected Green functions. These Ward identities were explic itly derived in Ref. [De96]. The two-point functions involving neutral gauge bosons, for instance, obey, ^Giiik) =
k » G ? l ( k ) = 0,
l ^ i ( k ) + i £zMzG**(k)
=
k , l G^ t (k) + ii x MzG**(k) =
(4.4.83)
;
k2 -
For the photon-fermion vertex we find - j-k2k"G^{/\k,p,p)
= -eQj
[G{i{p)
- G'J(-p)]
,
(4.4.84)
i.e. the same Ward identity as in QED (2.4.88). After truncating the Green functions and putting fields on shell, many terms drop out in the Ward identities. For example, we find the following identities k"G^ninCi„ = 0,
k
^Gt?unc> = ± M w G ^ i n c ,
„ = iMzG* runc , (4.4.85)
(i.'iH 1 Gauge theories of the electroweak interaction
where the ellipses stand for any on-shell fields. The first of these identities expresses electromagnetic current conservation, the others imply the Goldstone-boson equivalence theorem (cf. Sect. 4.4.7), as discussed in Ref. [De96]. The background-field Ward identities for connected Green functions hold exactly even in finite orders of perturbation theory if the re-summed propagators are used to connect the vertex functions [De96]. It is oidy necessary to calculate the inverse propagators in the same order of perturbation theory as all the other vertex functions. This property is due to the linearity of the background-field Ward identities for vertex functions. In contrast, the Slavnov-Taylor identities in general hold for connected Green functions in a given order of perturbation theory oidy if all contributions, including the propagators, are expanded up to this order. The background-field method can be used to simplify calculations in the EWSM. This is mainly due to the fact that the gauge fixing of the background fields is totally unrelated to the gauge fixing of the quantum fields. This freedom can be used to choose a particularly suitable background gauge, e.g. the unitary gauge, in order to reduce the number of Feynman diagrams considerably. Moreover, in the't Hooft Feynman gauge (£ = 1) for the quantum fields, many vertices simplify with respect to the conventional formalism [De95].
4.4.6
C h a r g e universality in t h e E l e c t r o w e a k S t a n d a r d Model
In QED, the charges of the elementary fields are products of the charge quantum numbers and the elementary charge e. This fact is built into the Lagrangian and guaranteed by gauge invariance. While higher-order corrections affect the elementary charge, i.e. the electromagnetic coupling constant, they do not modify the ratios of the physical charges of the particles, which are defined as the on-shell couplings of the charged particles to the photon. The fact that (apart from the charge quantum numbers) the onshell coupling constant of charged particles to the photon is independent of the species of the charged particles is called charge universality. It turns out that it holds in the same way in the EWSM. In QED, charge universality is easily derived from the identities (2.4.102) or (2.4.104), which hold not only for the electron but for arbitrary charged
1.4 Quantization of tlu; IOlectroweak Standard Model
643
fermions. Similar identities are valid for electrically charged scalar bosons and vector bosons that couple gauge-invariantly to the photon. Sandwiching (2.4.104) between two spinors and using the on-shell renormalization condition (2.5.37), which can be written as [note that r * * = - r * * ] d_ dp,i
r
(4.4.86)
u{p) = 7"tt(p),
yields u ( p ) r £ * * ( 0 , P ) -p)u(p)
= -eQfû(p)ru(p)
= -2eQfPp,
(4.4.87)
i.e. the on-shell coupling e is independent of the fermion species (and more generally of the matter species). The identity (2.4.102) is a direct consequence of the fact that the photon couples to the conserved current that generates the U ( l ) gauge transformations. In the EWSM the situation is different, as the current to which the photon couples is only conserved up to unphysical terms. This renders the proof of charge universality non-trivial. Following [Ao80], we start by differentiating the Lee identity (4.4.30) with respect to ipi, ipit, and u A , where the charges of the fermion fields are equal Qk = Ql• After transformation to momentum spacc we obtain
A=A,Z
- rxî","k(k,p,p)r'<xvA + f* ^
(p,p,
{-k,k) (-P, p) + rt<*< (p, -P) r W *
(4.4.88) (p, p, k).
We insert the covariant decomposition (4.4.38), differentiate with respect to k'1 for p'1 fixed and p11 = -p'1 - k1', put k,t = 0, and sandwich everything between spinors w ( p ) . . . u(p). From the first term of (4.4.88) only the part in which the derivative acts on the explicit k'1 in (4.4.38) contributes. Moreover, the AZ mixing contribution vanishes owing to (4.4.67). The second term drops out, because YKxu vanishes at k2 = 0 [cf. (4.4.67)], and its derivative with respect to k'1 is proportional to k1'. The third term drops out because p * * * ( - p t p ) vanishes when projected on u(p), i.e. on the mass shell. For
(i.'iH 1 Gauge theories of the electroweak interaction
the same reason, the fourth term contributes only if the derivative acts on VM'<(p, -p). Thus, we arrive at the following result fi(p)fii,^(0,-p,p)«(p)fJf<jM>l(0) rK*^A(-p,p,o)u(p)
= -«(p)
(4.4.89)
mi
where we used (4.4.86) and the fact that owing to Lorentz invariance the expression in the last but one line must be proportional to pti. From (4.4.64)
we obtain [rfiu = -r , i f i ] f
L
^(o) =
i ^ raAuA(k2)
2
(4.4.90)
Jt =0 where we assumed, without affecting the argument, that the residue of the photon ghost propagator is equal to one. Inserting this into (4.4.89) we find i(p)r^(o,-p,p)u(p)
= -i^tt(p)f/f>*ttVp,p,o)u(p). rrii (4.4.91)
By fixing p instead of p in the above derivation one obtains instead
û{p)rf^(0,-p,p)u(p)
K uA = i—H{p)r^' ^ (-p,p,0)u(p). rn k
(4.4.92)
- K'-ti>kUA In QED, the ghosts are free and the factor P * (— p,p,0) is given by its tree-level contribution which can be read ofr directly from the BRS transformation lc,.A f * * * « V P l P l 0 ) = - ^ ' ^ ^ ( - p . p . O ) = ieQiSik, (4.4.93) and (4.4.91) reduces to (4.4.87). In the Electroweak Standard Model the vertex function r K *^ k U (—p,p, 0) plays the role of the fermion charge.
4.1 Quantization of tl«> Electroweak Standard Model
64I>
For the WW A vertex function we can derive a similar relation
= 2\klle*',/(k)rpj'W+uA
(,k, -k,
0)e (k)
= -2iktl *' (k)r^W~uA{-k,k,0)eP(k),
(4.4.94)
\y uA and obviously F , , / { k, ^k, 0) is related to the charge of the W boson. A relation between t^ ^(-p,p,0) and J ± f c , =|=M) can be obu ' > and tained by differentiating (4.4.30) with respect to V;/> where the charges of the fermion fields fulfill Qk - Qt 1 = 0 . After transforming to momentum space, setting the momentum corresponding to uA equal to zero, putting the other external fields on shell, and multiplying the fermion legs with spinors and the W leg with a polarization vector, we find the following non-vanishing terms
W
+ f + f^
u A
^(k ,p,p)rK^uA(-p,p,0)
( p ,-pt0)t
w
^(k
, p , p)]ttip)eW(*i). (4.4.95)
The contributions of dropped out when projecting on the polarization vector e {k ). Equation (4.4.95) supports the interpretation of the quantities r * ^ f c u / l ( - p 5 p , 0 ) and rKvw uA- (-k ,k ,()) as charges and is consistent with
f*^(-p
> p i
0 ) = - f i l ^ " " ( - p , p > 0 ) = ie'QiSlk,
(4.4.96)
with some constant e'. Unlike in the case of a symmetric gauge theory with a simple gauge group, in the EWSM the Lee identities are not sufficient to prove this result. Fortunately, the functions appearing in (4.4.96) can be easily evaluated in the Landau gauge, i.e. for = = 0. In this gauge, all gauge-boson propagators are purely transverse, the photon ghost uA couples only to gauge
(i.'iH 1 Gauge theories of the electroweak interaction
bosons, and the corresponding coupling is proportional to the momentum of the antighost u . Therefore, any diagram of the form q- k k H (4.4.97) vanishes for k = 0, because the momentum q is contracted with the purely transverse propagator. As a consequence, the photon ghost with zero momentum does not interact and the vertex functions (4.4.96) are given by their tree-level (and counterterm) contributions. These are directly obtained from the BRS transformations of the fields (4.4.11),
f ^ « V p , P , 0 ) = - f * ' * K ( - P , P , 0 ) = i eQtSlk, y.v
(~P,P,0) =
\eg,w.
(4.4.98)
This shows that the coupling of the photon to charged particles is described by a universal coupling, i.e. the universality of the electric charge.
4.4.7
The Goldstone-boson equivalence theorem
An important consequence of the Slavnov Taylor identities in spontaneously broken gauge theories is the Goldstone-boson equivalence theoir.jn [Co74a, Ch85, Go86]. It states that the amplitudes for reactions involving highenergetic, longitudinal vector bosons are asymptotically proportional to the amplitudes where these are replaced by their associated would-be Goldstone bosons. The existence of non-trivial proportionality factors resulting from higher-order corrections lias been realized in Ref. [Ya88]. The practical virtue of the Goldstone-boson equivalence theorem is twofold. First of all, it facilitates the calculation of cross sections for reactions with longitudinal vector bosons at high energies, as the amplitudes for external scalars are much easier to evaluate. On the other hand, it might allow to derive information on the mechanism of spontaneous symmetry breaking from the experimental study of longitudinal vector bosons.
•M Quant ization of the Elcctrowoak Standard Model
(»17
We demonstrate the derivation of the equivalence theorem for the case of one longitudinal W boson as an example. We start with the Ward identity (4.4.21), i.e. 0 =
(4.4.99)
In order to obtain a relation for ¿»-matrix elements, we have to truncate the external legs, put them on shell, and restrict to physical fields. This is most conveniently done in momentum space, where (4.4.99) reads 0 = (T(V'W*(k)
± Mwdw^ih))
n
*rhyS(*'))>
(4A,0())
and k and ki denote incoming momenta. The truncation of the physical legs is straightforward. The truncation of the gauge-fixing leg is complicated by the fact that the W-boson field is contracted with a derivative. Owing to the mixing between the W and (f> bosons we have to introduce the (full) propagator matrix
(4.4.101)
where the projectors on transverse and longitudinal parts arc defined in (4.4.37), and k is the incoming momentum of fields corresponding to the first indices. Making the propagators in the gauge-fixing legs explicit, (4.4.100) can be written as (underlining indicates truncation)
0 = (A
(i.'iH 1 Gauge theories of the electroweak interaction
This yields k»(rw
[]
MwA {k2)(r
=
J]
(4.4.102)
with
Equation (4.4.102) provides a relation between a truncated Green function involving an unphysical scalar W-boson with the one involving the associated would-be Goldstone boson instead. By using the Ward identity (4.4.27) and the relation [cf. (2.2.41)]
0
1
between the propagator matrix G and the matrix of the corresponding vertex functions (4.4.42), the factor A (k2) can be expressed as
(
)
" M„(Mw + I f * ) " M w f ; r *
(4
'4105)
and thus involves no explicit gauge dependence. In lowest order it reduces to 1. In one-loop approximation it can be expressed in terms of self-energies as
^
-
.
-
a
g
The self-energies S} v w and contributions, i.e. r\vw = M l -
p
-
™
.
are defined by splitting oir the tree-level
f f * = Mw +
(4.4.107)
In the background-field method (A:2) is equal to one in all orders of perturbation theory owing to the Ward identity (4.4.75) [De96].
I. I Quantization of the Electroweak Standard Model
(¡'I!)
In order to derive the equivalence theorem from (4.4.102), we have to consider the longitudinal polarization vector of a massive vector boson. For a boson with momentum k'1 = (k , k) and mass M it reads
For energies large compared with the vector-boson mass it becomes parallel to k'1, i.e.
This relation allows to replace the unphysical scalar polarization vector klt/M\\ in (4.4.102) by the physical longitudinal one. According to the reduction formula (cf. Sect. 2.1.3) the 5-matrix element is obtained from the truncated Green function as ( . . . | S | W ±( f c ) . . . ) = R\l2e£(TW±(k)...),
(4.4.110)
where the dots stand for the physical fields, and is the wave-function normalization constant of the W boson. Inserting (4.4.109) and using (4.4.102) this results in (... \S\W± (k) ...) = ±R\lr2A±(Ml)(T^(k)
...) + 0 ( j ^ j . (4.4.111)
In the case of the Z boson one has to take care of the mixing with the photon. Nevertheless, via the use of the Ward identities (4.4.21) for the photon and the Z boson and (4.4.28) one can derive a relation equivalent to (4.4.111), (... \S\ZL(k) ...) = i R f A ° ( M ' i ) ( T x ( k ) ...) + 0
, (4.4.112)
with - tzM7Axx
_ A"{k ) = 2
M'/XAp
- \i-/M'/Ay'x)
+ r p
=
Mz{Mz
- il^x)
=
ifp M z ' (4.4.113)
(i.'iH 1 Gauge theories of the electroweak interaction
The invariant functions related to the % x propagator matrix and two-point functions are defined analogously to those of the W and x bosons. In low( 2 ) equals 1. In one-loop approximation it can be est order the factor expressed in terms of self-energies as
with r l x = \M
f p = M| - S P ,
+ Hz*.
(4.4.115)
In the background-field formalism this holds in all orders owing to (4.4.73). For more external longitudinal vector bosons similar relations can be derived. For the case of several longitudinal W bosons, (4.4.25) yields
.
s
)
n
) =
( ^ (4.4.116)
I
after truncation of the external legs, putting them oil-shell, and replacing their momenta by the corresponding longitudinal polarization vectors. By starting from Green functions in which some of the gauge-fixing functions are replaced by physical W bosons we arrive at relations that differ from (4.4.116) only by the absence of some of the fields. By combining all these relations we finally obtain
n
n
( 7i
I
s
)
n
) (\
I
)
n /
or for 5-matrix elements ({n} = { n i } U {712})
(
)
"1
)
"2
=(
(4.4. 8) +
n
»1
(
) \
71 /
4.5 Keuonnalizatioii of tin; Electroweak Standard Model
651
II wo include longitudinal Z bosons, on the right-hand side for each of them 11 factor \RXJ2AQ[k2)
and the corresponding x held appear.
Equation (4.4.118) and the corresponding generalization involving Z bosons constitutes the Goldstone-boson equivalence theorem. It relates ¿"-matrix elements involving longitudinal gauge bosons to matrix elements involving the corresponding would-be-Goldstone-boson fields in the high-energy limit.
4.5
R e n o r m a l i z a t i o n of t h e Electroweak Standard Model
The EWSM is a non-abelian gauge theory with spontaneous symmetry breaking and as such a renormalizable theory. All infinities can be absorbed into counterterms that are generated by a redefinition of the free parameters and a renormalization of the fields and/or the external wave functions. In order to define a renormalization scheme one hits to choose a set of independent parameters. One possibility is to start from the Lagrangiau in its symmetric form (4.2.2) and (4.2.8)-(4.2.12). As discussed in Sects. 4.1.1.5 and 4.1.2.3, one needs the counterterms of the symmetric theory plus a renormalization of the vacuum expectation value (vev) of the Higgs field. For practical calculations it is more convenient to fix the counterterms by renormalization conditions on the physical parameters. Since there is a oneto-one correspondence between the physical parameters and the parameters of the symmetric Lagrangiau, this approach also yields finite quantities. It has become customary to require that the renormalized masses of the gauge bosons, of the Higgs boson, and of the fermions equal the physical masses and that the renormalized electric charge equals the one measured in the Thomson limit, i.e. in the limit of low-energy Compton scattering of on-shell particles. This is the so-called on-shell renormalization scheme which was originally proposed by Ross and Taylor [Ro73], then worked out and used by many authors [Si80, Ba80a, F181, Sa81, Ao80, Bo86, De93]. The on-shell scheme is distinguished by the fact that all parameters have a direct physical meaning and can be measured directly in suitable experiments. This is evident for the particle masses. 12 It is also true for the definition of the electric 13
This is not the case for the masses of the light quarks owing to the presence of the strong interaction. However, their contributions can often be expressed in terms of directly measurable quantities.
(i.'iH 1 Gauge theories of the electroweak interaction
charge, because in the Thomson limit of Compton scattering all higher-order corrections vanish, and the lowest-order QED cross section becomes exact in all orders of perturbation theory [Th41, Di97]. The renormalization of the parameters together with the unambiguous renormalization of the external wave functions [cf. (2.1.34)] is sufficient to obtain finite S-matrix elements. In order to obtain finite Green functions, field renormalization is required in addition. When renormalizing directly the Lagrangian in its symmetric form, it is sufficient to introduce only one field renormalization constant for each multiplet. This minimal field renormalization 08 requires one renormalization constant for the triplet of SU(2) W gauge fields, one for the U ( l ) y gauge field, one for the complex Higgs-doublet field, one for every lefthanded fermion doublet, and one for every right-handed fermion singlet. In this scheme, additional wave-function renormalizations have to be applied when calculating S-matrix elements. Another possibility is to renormalize directly the physical fields A 0 8 , De93]. Because fields that carry the same charge and spin in general mix, the field renormalization constants take the form of matrices. This allows us to perform the field renormalization such that the renormalized fields do not mix 011 shell and that no further wave-function renormalization is required. In this section we sometimes make use of CP conservation. We note that the CP violation resulting from the quark-mixing matrix in the EWSM does not affect the two-point functions and three-point functions at the one-loop level, because at least four quark-mixing-matrix elements are required to construct CP violating quantities (cf. Sect. 1.4.2.3). Since the renormalization relies only 011 the two-point functions and three-point functions, our discussion is general at the one-loop level.
4.5.1
T h e r e n o r m a l i z a t i o n t r a n s f o r m a t i o n i n t h e 011-shell scheme
We perform the renormalization directly for the physical parameters and fields, i.e. the fields that correspond to definite charge and mass. The bare parameters are split into renormalized parameters and counterterms as follows (bare quantities are denoted by an index 0): = M'i, + SM'i,
Ml0
= Ml+
8 Ml
4.5 llfliiormalization of the
Ml{)
=
M?i+5Ml ii>
o = rnfii + Vlho
lOlectroweak .Standard Mode] (¡55
=Vij
6mfti,
+ SVij,
eo = Zce = (1 + SZc)e
(4.5.1)
— e + 5e.
Here Mw, Mz, Mh, and ro/ti are the physical masses of the corresponding particles defined by the poles of the propagators. In addition we perform a renormalization of the tadpole (4.5.2)
t.0 = t + St,
which corresponds to a renormalization of the vev of the Higgs-doublet Held (cf. Sect. 4.1.2.3). We introduce field renormalization of a general form using renormalization matrices for fields with equal quantum numbers. We assume that the Iliggs field does not mix with other fields, which holds if CP conservation is required. The bare physical fields are split as
( Z
0
U<J
\ _ (Zz'zZ^) f Z \ \ z f
z
z f j W
=
( \ + \5Zzz V
\5Zza 1+ ^ Z
A A
J [ a J '
(4.5.3) where the last expressions in each line are valid in first-order approximation. This general transformation allows us to define the fields in such a way that their quanta are mass eigenstates also in the presence of higher-order corrections. The counterterms introduced so far are sufficient to render physical S-matrix elements and Green functions of physical particles finite. For a complete renormalization of the EWSM we need the renormalization of the unphysical
(i.'iH 1 Gauge theories of the electroweak interaction
sector in addition. This consists of the field renormalization of the would-be Goldstone fields,
=
K '
2
=
^
( *
+
\ *
Z
(
* )
4
-
5
-
4
)
of the Faddeev-Popov ghost fields,
Wn =
u ( Z Z
= (l + z z Z
Z Z
A
(4.5.5)
\ ( u z \ J \uAJ
=
( 1+ \ Z
a z
SZZ +SZ
\ ( J ^
l l Z u
\ a ) '
and of the renormalization of the gauge parameters,
= Ztf&l
=
Co,a =
(4.5.6)
The Faddeev-Popov antighost fields u are not renormalized [cf. (2.5.150)]. By choosing = and £'z = the would-be Goldstone bosons decouple from the scalar gauge bosons, and the poles of their propagators coincide with the physical masses in lowest order. In order to have these features at higher orders, one has to renormalize and £ independently. By inserting the renormalization transformations into the bare Lagrangian Co and writing = 1 + S for the multiplicative renormalization constants (matrices) we can split Co [cf. (2.5.4)] Co = C + C c l -
(4.5.7)
The basic Lagrangian C has the same form as C o but with unrenormalized parameters and fields replaced by renormalized ones. The counterterm Lagrangian £ t t yields the counterterms. If not stated otherwise, we restrict ourselves to the one-loop corrections and consistently neglect terms of order {SZ) 2 in the following.
4.5 llfliiormalization of the lOlectroweak .Standard Mode]
1.5.2
(¡55
Renormalization conditions
The renormalization constants are fixed by imposing renormalization conditions. These consist of three sets: the conditions that define the renormalized physical parameters, those that define the renormalized physical fields, and those that fix the renormalization in the unpliysical sector. While only the first set is relevant for physical quantities (S-matrix elements), a clever choice of the second set allows us not only to eliminate the explicit wavelunction renormalization of the external particles, but also to simplify t he; explicit form of the renormalization conditions for the physical parameters considerably. The choice of the third set determines the form of the renormalized Slavnov-Taylor identities.
4.5.2.1
Tadpole renormalization
As discussed in Sect. 4.1.2.3, it its convenient to require the vanishing of the renormalized tadpole, i.e. the vanishing of the renormalized Higgs-field one-point vertex function, r^ = v
+ St = 0.
(4.5.8)
As a consequence of this condition no Feynman diagrams involving tadpoles as subdiagrams need to be considered in actual calculations. However, St enters some of the counterterms. We assume in the following that the tadpole renormalization (4.5.8) is performed. Otherwise, extra tadpole terms would appear in the renormalization conditions discussed below.
4.5.2.2
M a s s r e n o r m a l i z a t i o n and r e n o r m a l i z a t i o n of physical fields
The renormalized mass parameters of the physical particles are fixed by the requirement that they are equal to the physical masses, i.e. to the real parts of the poles of the corresponding propagators. These poles are equivalent to the zeros of the one-particle irreducible two-point functions projected on physical states, i.e. on polarization vectors elt(k) for gauge bosons and on spinors u(p) and v(p) for fermions. In case of mass matrices these conditions have to be fulfilled by the corresponding eigenvalues resulting in complicated expressions. These can be considerably simplified by requiring
(i.'iH 1 Gauge theories of the electroweak interaction
simultaneously the on-shell conditions for the (ield renormalization matrices A 0 8 , De93]. The latter conditions require that close to its pole each propagator is given by its lowest-order expression with the bare mass replaced by the renormalized mass. As a consequence, an on-shell particle does not mix with others, and its propagator has residue one, i.e. the quanta of the renormalized fields are mass eigenstates. Thus, we arrive at the following renormalization conditions for the two-point functions for on-shell external physical fields:
Rer
n.
0,
(*)*"(*)
,
=
, Z,
,
iviy
-
0, 1, 0,
P ~*m f,, P
171
Ui(p).
(4.5.9)
f,i
Here Re takes the real part of the loop integrals appearing in the self-energies but not of the elements of the quark-mixing matrix appearing there. Its purpose is to remove the absorptive parts. Owing to the required hermiticity of the renormalized Lagrangian, the counterterms can only affect the nonabsorptive parts. In order to proceed, we decompose the two-point functions into covariants. For the fermionic two-point functions we write,
r&
r § V ) +
V ) +
(4.5.10)
If we neglect absorptive parts, which are irrelevant for the renormalization, the hermiticity of the Lagrangian implies the hermiticity of the effective
4.5
({"normalization of I.IUT Electroweak Standard Model
(¡57
action. Therefore, the ferniionic two-point functions must have the following Hyminetry r & ( P ) = 7°r '
R
>b
0
(4.5.11)
,
which implies
r £ V ) = r£RV),
r¿V) = r£V),
r £ V ) = r jr(p2)-
(4.5.12)
Together with the covariant decomposition (4.4.36) for the gauge-boson twopoint functions we obtain from (4.5.9)
R e r ^ ( M ^ ) = 0,
Re
2
R e r ^ ( M ) = 0,
= -1,
dk2
dr'H'(k:2) Re ^
= 1,
+ R e P ^ .(m 2 ^.) = 0, ^ / j R ^ a S K i ) + R e { r g K
+
¿
) + r
+
2A
(rfe(P2) + r £ >
R e r
fe(
m
L)
[m?. ( r / . > 2
))] |
J
= 2
) + rgi(p2))
= 2.
(4.5.13)
Note that the condition = 0 is guaranteed by the Lee identities and gives no constraint on the counterterms. 4.5.2.3
R e n o r m a l i z a t i o n of t h e q u a r k - m i x i n g m a t r i x
The mixing angles of the quark-mixing matrix V¡j are free parameters of the Electroweak Standard Model and thus have to be defined by renormalization conditions in higher-order calculations. The definition of the mixing angles depends on the parametrization (cf. Sect. 1.4.2.2). One could use definite physical observables to fix these parameters in a unique way. This, however, would destroy the symmetry of the renormalized quark-mixing matrix with respect to the quark generations.
(i.'iH 1 Gauge theories of the electroweak interaction
In order to formulate a renormalization condition for the quark-mixing matrix that is independent of the parametrization and respects the symmetry, we reconsider the origin of the quark-mixing matrix. The bare quark-mixing matrix Vb)tj is given by [cf. (4.2.31)]
KM; =
(4-5-14)
>
where the matrices i/o'1' transform the bare weak-interaction eigenstates to the bare mass eigenstates /o,
^/yLt/j;0 = f%•
(4.5.15)
In the on-shell renormalization scheme, the higher-order mass eigenstates arc related to the bare mass eigenstates through the field renormalization constants of the fermions,
ft
=
zfJ'Lf]fi-
(4.5.16)
This suggests to define the renormalized quark-mixing matrix in analogy to the unrenormalized one through the rotation from the weak-interaction eigenstates to the renormalized mass eigenstates [De90]. Since the renormalized quark-mixing matrix should be unitary just as the bare one, only the unitary part of the field renormalization matrices can enter. In the one-loop approximation the rotation contained in the fermion wavefunction renormalization 1 + ^SZ1' is simply given by its antihermitian part ^/,L,AH
=
1
_
SZF,LT)
(4
5 1 ? )
This leads us to define the renormalized quark-mixing matrix as
% = ( S i k 4-
t)
(5nj +
= vo,ij + \ ( s Z ^ ' M 1 % k j + Vo,ik frAH)
\ S Z ^
.
(4.5.18)
4.5 llfliiormalization of the
lOlectroweak .Standard Mode] (¡55
ll has been shown that this condition correctly cancels all one-loop divergences and that V{j = VQ¿j in the limit of degenerate up- or down-type quark masses [De90j. Using the ferinion wave-function renormalization constants defined by the oil-shell conditions (4.5.13), yields a gauge-dependent counterterm for the <|iiark-mixing matrix and thus gauge-dependent 5-matrix elements [Ga99]. Gauge-invariant results can be obtained by using a different set of ferinion wave-function renormalization constants and for the renormalization of the quark-mixing matrix. These renormalization constants result from requiring the renormalization conditions (4.5.9) for the off-diagonal ferinion two-point functions not on the mass shell, but at the symmetric point p = 0,
R e r ^ . ( O ) = R e r ( 4 ( 0 ) ^ + Re 1 ^ . ( 0 ) ^ d ~ „ t f , ^Ref / »
v
= 0,
_ ^ /,L , m 1 ~ 75 , p ; r / , R , m 1 + i s = R e F f 4 ( 0 ) ^ + R ^ O ) ^ = 0. (4.5.19)
Since the right-handed off-diagonal ferinion self-energies are finite and proportional to the ferinion masses, one can also choose to disregard the renormalization condition for
and set the right-handed off-diagonal renor-
malization constants
to zero. Note that while the renormalization
constants z } { 2 h a v e to be used for the renormalization of the quarkmixing matrix, they must not be used for wave-function renormalization in S-matrix elements. There, the usual oil-shell wave-function renormalization constants enter.
4.5.2.4
Charge Renormalization
The electrical charge is defined as the full ecy coupling for on-shell external particles in the Thomson limit, i.e. for vanishing photon momentum. The full ee7 coupling consists of the corresponding vertex function and the wavefunction renormalization constants. Owing to the photon-Z-boson mixing, also the eeZ coupling occurs in general. In the complete on-shell renormalization scheme the field renormalization is chosen in such a way that the on-shell photon does not mix with the Z boson and that all wave-function
(i.'iH 1 Gauge theories of the electroweak interaction
renormalization constants are trivial. Then, the charge renormalization dition takes the simple form
con-
(4.5.20) for the (truncated) vertex function
(4.5.21)
irgfop') = e
,p
Owing to charge universality we could impose the above renormalization condition on any charged particle to obtain the same renormalized charge.
4.5.2.5
R e n o r m a l i z a t i o n of t h e unphysical s e c t o r
The renormalization of the unphysical fields and of the gauge parameters does not affect S-matrix elements. Therefore, the choice of the corresponding renormalization conditions is a pure matter of convenience. Wc fix the renormalization of the unphysical sector such that the Lee identities (4.4.30) and (4.4.31), i.e. (4.4.46) and (4.4.60), still hold in terms of renormalized fields. We do not give the explicit formulas but only sketch the procedure. Wc have seen in Sect. 2.5.4.2 that in linear gauges the gauge-fixing term need not be renormalized. We therefore require that the gauge-fixing Lagrangian has the form (4.4.1) and (4.4.2) in terms of renormalized fields. This fixes the renormalization of the gauge parameters. The corresponding bare gauge-fixing functional can be obtained with the help of the renormalization transformations in Sect. 4.5.1. From these bare gauge-fixing functionals the Faddeev Popov Lagrangian is derived as usual and expressed in terms of renormalized fields. In order to ensure the validity of (4.4.46) and (4.4.60) for renormalized fields, the renormalization of the sources Ky,L must be equal to that of the Faddeev-Popov antighost fields, i.e. in our conventions Ky' 1 is not renormalized. Moreover, the renormalization of the sources K^ and Kx has to be fixed appropriately.
4.5 llfliiormalization of the
lOlectroweak.StandardMode](¡55
The validity of (4.4.30) for renormalized fields can be guaranteed by renormalizing the other sources K of the BRS variations appropriately. The field renormalization of the would-be Goldstone-boson fields and the Faddeev Popov fields is still free and can be chosen in such a way that the corresponding propagators have residue equal to one. 4.5.3
E x p l i c i t f o r m of r e n o r m a l i z a t i o n c o n s t a n t s
The renormalized vertex functions consist of the unrenormalized ones and the counterterins. The renormalization conditions allow to express the counterterms by the unrenormalized vertex functions at specific external momenta. We perform this task for the renormalization constants for the physical fields and parameters at the one-loop level. The tadpole counterterm is fixed by
6t = -r"(0).
(4.5.22)
The renormalized two-point functions entering the renormalization conditions for the physical fields are decomposed into lowest-order contributions, unrenormalized self-energies, and counterterins as r i & V ) = Svv(k2 -
- Ml) -
Uk2 - My)ôZvv ¿i
S r ' V )
+ \{k2 ¿J
- Ml,)5Zv,v
- 8vv,8Ml
rW(fc2) = (k2 - M 2 ,) + £"(A 2 ) + (k 2 - Mfosz,, r f e V )=
+ z{jL(p2)
r'R>2) =
+ E§V) +
+ \ {szff- +
,
- ¿M,2„
szff>•),
\ (szp + sz{*t),
r(4(p2) = - m / A + s £ V )
r{(ij(p2)
= -mfii6ij
+
V{f(p2)
- \ (™SiiSz!jK where VV' = {WW, AA, AZ,
ZZ).
+ rrifjSZW)
- S^Sm^,
(4.5.23)
(i.'iH 1 Gauge theories of the electroweak interaction
Inserting these equations into the renormalization conditions, we find for the mass counterterms dM'i, = Re
8MI = Re E f * ( M | ) ,
¿M,2, = ReE''(M,2,), Sm
f,i
=
2Ile m f t i (
S
S/;- R (m 2 ,))
+
(4.5.2'1) The field renormalization constants of the boson fields are obtained as a s y v (ib 2 )| 8Zyy = —Re dk2 SZAZ =
-2Re
Ml dE"(fc2)
SZr,= -K*
V = A, Z, W,
ÔZza = 2
'
S-^(O) M| ' (4.5.25)
dk2
and those of the fermion fields read
SZ
!i* =
=
m2
2
^
K ' f K i )
+ m/.iEf/imJj)
+ m/jS^rn^)]
m2
,
2
m2
+ mftjSff
+ m^.rn^Efim2,) ,
m
Re h L - S f ^ / j ) + (mjj)
+ m/.iSf/(m^)]
8Z{?' = - R e E ^ K i ) - m ^ R e
i ^ j,
,
f,'Tnf
t # j,
[m/. ( ^ V ) + s£ V ) )
P2=MH
8Z{*
= -RcS^KJ
d - m/^Re
+ s£V) + E£V)
P2=mj .
( e £ V ) + S ¿ V ) ) (4.5.26)
4.5 llfliiormalization of the
lOlectroweak .Standard Mode] (¡55
Owing to (4.5.12), 6Zjj = 5Zij{mf
<-> m]),
(4.5.27)
and, in particular, SZl = SZu.
(4.5.28)
In the: lepton sector we have V{j = 6ij. Consequently, all (one-loop) lepton self-energies are diagonal and the off-diagonal lepton wave-function renorinalization constants are zero. The same holds for the quark sector if one replaces the quark-mixing matrix by a unit matrix, as is often done in calculations of radiative corrections for high-energy processes. The renormalization constant for the quark-mixing matrix Vij can be di rectly read off from (4.5.18),
= \
- SZ^)Vkj
- Vlk{8Zif
,dM - 6Z\kj
(4.5.29)
The non-diagonal fermion-field renormalization constants determined from (4.5.19) read S Z
*'M{ =
2 \(™h + ™ M ) 4 L ( 0 ) + 2 m / i t m / ) , s i R ( 0 ) f,k l
m* m
f,i ~
i ± j.
(4.5.30)
Inserting these fermion-field renormalization constants yields rm 2
2
2
u,k ^ mu,iK^u,L/ri\ . m mm E VU,R/n()\ —Z (°) + ",i »,* ifc ( )
+ m« l i S^ 1 (0)+m l l , f c S^ r (0) 1
\™jj
+
m
Vkj
lk^d,Ltn-
604
4 Gauge; theories of the electroweak interaction
If one chooses to set the right-handed off-diagonal renormalization constants to zero, the terms involving £ R have to be omitted. It remains to fix the charge renormalization constant SZC. To be more general, we investigate the / / 7 vertex for arbitrary fermions / . The renormalized vertex function reads MI
(4.5.32)
For on-shell external fermions it can be decomposed as (k = p' — p) K ! L M
= h {l,Miy{k'¿)
- WA^A(k2)
(4.5.33)
+ Ex})rcssing the renormalized quantities by the unrenormalized ones and the counterterins, and inserting this in the analogue of the renormalization condition (4.5.20) for arbitrary fermions, we find, using the Gordon identities (A.1.42), 0
where T{ J J ^ } is the generating Junctional of Green functions and J,/, and are the sources corresponding to and (fr, respectively. Because the field
752
A
Appendix
operator creates antiparticles and annihilates particles, the particles and • • •) have to be considered as outgoing thus the fields in {T(f>(x) • • • [cf. (A. 1.67)]. The order of the field indices equals the order of fields in the vacuum expectation value and the order of corresponding derivatives. This is crucial for anticommuting fields, where the exchange of two field indices yields a minus sign. In the diagrams, the arrows indicate the flow of the particles, which is opposite to the flow of antiparticles. The transformation to momentum space is defined as
x exp {i (px + ... +
(]y+
...)}
G^'ip,...,
q,...),
where the momenta p, q are incoming. In momentum-space Green we usually split off the momentum conservation ¿-function
= (2n)UW(p
+ ...
+ q
+ .. .)G*~+U~(p,...,
g,...).
functions
(A.1.61)
The generating functional for connected. Green functions is defined as Tc{-U, J* t } = log T{J}, J $ i } .
(A. 1.62)
The corresponding Green functions are denoted as in (A.1.59) with an additional index "c". Truncated Green functions are denoted as
G
t r • • • » » • • • • )
= Gt
— {x,...,
= ( T 4 > ( x ) > • • • r *(?/) • • -Kruno
y , . . . ) = (T£(x),...,
¿(y)...)
(A.1.63) 0
A.l
Convention»
7i).'l
Underlining is also used to indicate truncation of single lines. Because of the relation G + (x,...)
= J d yG+*( ty)G-* (y,...)t
(A.1.64)
(' denotes the field that appears in the propagator together with ; usually (p' = (ft) the field indices in truncated Green functions denote incoming particlcs or fields. The generating functional of vertex functions is defined by
^
=
\T{cl>, *}
isMx) -i
+
'
H x )
=
isj^x)
•
d4a; [Mx)cf>(x) + J^ (x)
(
A
L
6
5
)
},
and from this the vertex functions result as
-
«
ote that in the vertex functions, as in truncated Green functions, the field arguments denote incoming uanta. The Fourier transformation of truncated functions or vertex functions is as in . I.G , i.e. all momenta are incoming. For Grassmann fields the above formula hold as well. The order of labels corresponds to the order of left derivatives. Interchange of labels of the Green functions leads to signs. From the truncated Green functions, the via the reduction formula
-matrix elements are obtained
Mfcx)... +(k2)... \s\(k3)... 4>+(k,) ...) = ?n
= RnJ\Tct>{k,)... V
...
... (j>{k\)...
.1.G
For Grassmann fields the above formula hold as well. The order of labels corresponds to the order of left derivatives. Interchange of labels of the Green functions leads to signs.
7i)4
A
Appendix
In particular, in our conventions the lowest-order two-point functions for fermions read
R £ Û > . -P)
= - R $ ( - P , P ) = TF -
«$<"»»> =
"•),.»,
-P> = ¡ ^ É f e -
(A.1.68)
Note that G** has to be used for the truncation of a ij> line but G ^ for the truncation of a ij> line in (A.1.64). The relation corresponding to (2.2.41) is
<">(,-,)-
U ,
r
f
<*»>
(A.1.69)
and the one corresponding to (2.2.43) reads S2TC{J}
S2Tc{J^J^}
f . . ' d z\ d z2
—
iSJ^(x2)\6J^(z2)
5ip{zi )(hp{z2) '
(A.1.70)
Acting on this relation with f (x) S
J =
4 (
f ,4
d'TcjJ^Jj,}
S
Z
\SJ^{x)i8J^z)Si>{zy *
,A,
71s
gives the decomposition of the connected Green functions in terms of vertex functions as discussed in Sect. 2.2.2.2 for the case of commuting lields. The sign conventions of the vertex functions agree with those of the truncated Green functions as defined by (A.1.64) (with the order of fields given there). The conventions for Faddeev- Popov ghosts are equivalent to those for the fermions.
A.I
A. 1.3.3
Convention»
755
F e y n i n a n rules
The Feynman rules for the calculation of Green functions or scattering amplitudes in momentum space read as follows: 1.
Draw all possible topologically distinct diagrams (connected or disconnected but without vacuum subdiagrams) contributing to the process under consideration in the desired order of perturbation theory.
For each diagram the following steps have to be performed: 2.
Associate external momenta to all external lines and L loop momenta to the internal loops. Determine the momenta of the internal line» so that four-momentum conservation holds at each vertex.
3.
For each internal line attach a propagator, ••—
= —x —7t——, pi — M* + le •
= (
p
1 -"T 2_M2
+ ie
for a scalar boson
(A. 1.72)
,
(A. 1.73)
for a fermion
'(1 ~ Qp'Y ( p 2 _ M 2 + ie)(p2 - £ M 2 + ie)
..cr-p'Y/M2 ' p 2 - M2 + \e
.
p'Y/M2 P2 - p p + ie '
1
' '
;
for a vector boson in a gauge with the kinetic Lagrangian I
2I
1
C = —A{d,Lvu - d„v„) - Y^W ) 4.
M2
+t W -
(A-L75)
For each vertex attach a factor derived from the relevant monomial of the interaction Lagrangian. It can be obtained by acting with functional derivatives with respect to the external fields corresponding to the considered vertex on f d 4 a;£j n t, transforming to momentum space and omitting the momentum conservation ¿-function ^Tr)' 1 ^ 1 '(£]/>) (cf. Sect. 2.4.2). In this process each field derivative dtl results in a factor - i p , n where p is the corresponding incoming momentum. The vertex factors for QED and a non-abelian gauge theory are given in (1.2.30) and (2.4.32), respectively. Further examples are given below.
ppendix
.
Integrate over the L loop momenta q)L using the measure f d lr 2 r 1, possibly after a regulari ation.
.
ultiply the contribution of each diagram by i.
a factor - 1 for each closed loop of anticommuting fields,
ii.
a global sign for external lines of anticommuting fields, coming from their permutation in the actual graph as compared to the argument of the considered Green function, simple rule for external fermion lines is given in item of ect. 1.2.1. .
iii.
a symmetry factor 1 , where e uals the number of possibilities of mapping the graph onto itself by permutation of lines and vertices.
.
ummation over the contribution of all diagrams yields truncated functions with no factor on the external lines. Contributions to one-particle irreducible functions are obtained from one-particle-irreducible diagrams. on-truncated Green functions are obtained by putting propagators on the external lines. Connected Green functions are obtained by retaining only connected diagrams.
.
The scattering amplitude is obtained by multiplying with i, putting the external lines on their mass shell, i.e. pf = Mf, and multiplying with the appropriate wave-function factors, i.e. for incoming scalars, fermions, antifermions, and vector bosons S :
f '• —•—• 1
/
:
u<*{p,h)
—«—•
il •
VaiVih)
'v/x/x» i fc
, .1. G
and for outgoing scalars, fermions, antifermions, and vector bosons S : •
f '• •—• 1
««
,i
Finally we give some examples for vertices. The vertex in the (j>'] theory,
/
:
•—*— vQ(p,j3)
n' e*n(k,j3). .1.
I .I
Conventions
reads
- i g.
.1.
The vertex in the ft theory,
is given by y• /
- i g. v
s
. 1.
. \
n example for a derivative coupling is given by ,
g
dpfa
.1.
which leads to the vertex factor Pi,' =<)(PI+P2),,
.1. 1
\
The appearance of the sum p\ + p2 in the Feynman rule results from the I ose symmetry of the two scalar fields in . 1. . If the two scalar fields would be different, only the momentum corresponding to the field on which the derivative acts would be present in the Feynman rule.
ppendix
A.2
s
In this appendix we list the Feynman rules of the in an arbitrary t Hooft gauge. e write down generic Feynman rules for all propagators and vertices and give the possible actual insertions. The rules for the external lines are as in .1. and .1. . For brevity we introduce the shorthand notation c
cw
cos
,
s = sw = sin
.
.2.1
In the vertices all momenta and fields are considered as incoming. The propagators read as follows for gauge bosons V = A, Z, and W
V,t r^Jk^j»
V„
Ma
,
~UJ>1" V - Ml
i l - (.y)^};,, ' (k* - M*)(k* -ivM^y .2.2
Faddeev- opov ghosts G = uA, uz, and u MU±
=
(Mua
, Muz = \ ^ , M Y N and
Y/£WMW),
scalar fields S
-JQ-
rj, x, and {Mx
\fizMy
.2. and M,p =
y/£wMw),
and fermion fields F = fi,
F
F
jr - mf-
.2.
.2 Fbyiunan rules for the Electrowea
tandard
odel
The vertices are given by the following expressions WW
vertex
= \e2c\2(j,wgap V2,u
9Pti9vo\
(A.2.6)
V4i(r
with the actual values of V\, V2, V ,
ViV2V3V.w{ +w~zz
and C
W+
W'AZ
2
c
- g„p(jli(r -
c - 2-
W+W~AA
c s
-1
W+W+W-W1 C 7! V
.2.
vertex
9nv(k\ - k2)p + g„p(k2 - k3)IL + 9piiih ,pi
~ .2.
with the actual values of V , V2, V3, and C
ViV2V3AW+W- zw+w~ c c 1 s
.2.
i
SSSS
ppendix
.2 Feynman rules for the
vertex
VVSS
s
lectrowea
tandard
odel
,1.1
vertex
y i e 2C
x
( A .2.14)
i e*9vn>C
.2.1
S4
2
with the actual values of Vi, V2, S\, S2, and C with the actual values of Si, S2, S3, S,1, and C V\V2S\S2 Si S2 S3 S.\ WW,
xxxx wxx, fi
c
w~ ,xx+4> (/>+- <j>+~
ZZrpi
W+W-J]7)
ZZxx
W+W~xx
AA(/>+(f)~ ZA- ZZ*
w+w~4>+c
.2.11
1 2F
1 2c'sl
W±A*x 1 2s
C
SSS vertex
F4
2
W±Z4FT] 1 2c
C2-
c2-
2
W±Z*x
2
2c
.
.2.1
F
VSS vertex 2
S\,k\
Si ieC
s
.2.12
\
.2.1
= ieC{ki - k2),l
a S2,k2 with the actual values of Si, S2, S3, and C
with the actual values of V, S\, S2, and C S\S2Si c
wv vxx, . . I 2s w 2s v
A(j>+(jr
2
.2.1 C
i 2ca
-1
z+c 2 -s 2 2cs
r i 2s
.2.1
7G2
A Appendix
SVV
vertex:
(A.2.18)
= 1e g ^ C V2,„
with tlie actual values of S, Vj, V2, and C VZZ
V2
(¡¿W^A
C
-Mw
^W^Z
(A.2.19)
-§Mw
VFF vertex:
with the actual values of V, F\, F>, Cn, and C\,
A/i/j cr,
W ,
W+xiidj
i K -t. v/2s J'
-Q/^tj
CR
W'ljWi
ly-JjUi
(JfSij
0
0
0
0 (A.2.21)
where
c
3]
sc
(A.2.22)
A.2
Feynirian rule for Lho
The vector and axial-vector
Vf = a
f
SFF
=
2 ( ,+ p (»/
leetroweak Standard Model
couplings of the
7M
-boson are given by
2sc
=
(A.2.2. )
— j ) = 2 sc
vertex:
=
(A.2.24)
with the actual values of S, F\, /' >, C'a, and C\,
S FyF2 CL
6
Vfifj
I "/..x 2.5 A/w
fifj 3 L o / '^LLS 2s w,/ M v i 2/3 'JLLiS0 2* w,f My, '
+ ilj
Ijtti
0
1 "" ) A/w
JLHILL
2s Aiw -
A/w
.
0
+Uidj 1 T/ 1
1/
A/w "«
(f> djUi J L ^ I /
/2.s + / 21s
1
Aiw "iiii u t/d Aiw 1
(A.2.25)
il
ppendix
V GG vertex G\,k\
x
ie i tC
\
.2.2
G 2, k-2 with the actual values of V, G\, G2, and C VG,G2
Au u
u Tfi «
c
1
1
TfiTu 2
.2.2
s
SGG vertex
G\ S
eCix
.2.2
v.
G* with 1 G , G 2 , and C . G1 2
for
1
. ux, respectively, and the actual values of S,
Huzuz
C (fy^U^V? 2
G,G2
C
i
w
.2.2 ab w
s [EP]
Elementary particle physics: D. Griffiths, Introduction to lementary Particle Physics, (John Wiley & Sons, New York, etc., 1987); E. Lohnnann, llochenergiephysik, (Teubner, Stuttgart, 1992); D.H. Perkins, Introduction to High- nergy Physics, (Addison-Wesley, Menlo Park, 1987); YV.B. Rolnick, The Fundamental Particles and their Interactions, (Addison-Wesley, Reading, Massachusetts, 1994).
[E\V]
Electroweak interaction: E.D. Commins and P.H. Bucksbauin, Weak Interactions of Leptons and Quarks, (Cambridge University Press, Cambridge, 1983); K. Grotz and U.V. Klapdor, The Weak Interaction in Nuclear, Particle and Astrophysics, (Hilger, Bristol, etc., 1990); P. Renton, lectroweak Interactions: an Introduction to the Physics of Quarksand Leptons, (Cambridge University Press, Cambridge, 1990).
[FT]
Field theory: J.D. Bjorken and S. Drell, elativistic Quantum Mech anics/ ela t i vis tic Quantum Fields, (McGraw-Ilill, New York, 1964); C. Itzykson and J.B. Zuber, Quantum Field Theory, (McGraw- Hill, New York, 1980); C. Kaku, Quantum Field Theory, (Oxford University Press, New York, Oxford, 1993); T. Kugo, ichtheorie, (Springer, Berlin, etc., 1997); M.E. Peskin and D.V. Schroedcr, An Introduction to Quantum Field Theory, (Addison-Wesley, Reading, Massachusetts, 1995); S. Pokorski, Gauge Field Theories, (Cambridge University Press, Cambridge, 2000); S. Weinberg, The Quantum Theory of Fields, Vol. 1 and 2, (Cambridge University Press, Cambridge, 1995).
[GSW]
Glashow-Salam- Weinberg theory: E.S. Abers and B.W. Lec, Phys. ept. 9 (1973) 1; J.C. Taylor, Gauge Theories of Weak interactions, (Cambridge University Press, Cambridge, 1976); D. Bardin and G. Passarino, The Standard Model in the Making, (Oxford University Press, Oxford, 1999).
766
eneral references
T
roup theory: . eorgi, i i i i : i ii i (Benjamin/Cumnnngs, eading, Massachusetts, 1 2); M. amcrmesh, i i i i (Addison esley, eading, Massachusetts, 1 62); .B. Lichtenberg, i i (Academic Press, ew ork, 1 7 ); B. . ybourne, i i i ( ohn iley Sons, ew ork, 1 74). T
rand nified Theories: P. Langacker, i 72 (1 . . oss, ii Massachusetts, 1 4).
1) 1 5; i (Benjamin-Cummings,
eading,
uantum lectrodynamics: . . Bjorken and S. rcll, i i i i i i i uantum Fields, (Mc raw- ill, ew ork, 1 64); .M. auch and F. ohrlich, (Springer, Berlin, 1 76); T. inoshita, i ( orld Scientific, Singapore, 1 0). StM
Statistical Mechanics: .P. Feyiunan, i i i : ( .A. Benjamin Inc., eading, Massachusetts, 1 72); . . Popov, i i i i i i i ( . eidel Publishing Company, ordrecht, etc., 1 3); . inn-. ustin, i ii ( xford niversity Press, xford, 1 ).
S S
Supcrsymmetry: P. Fayet and S. Ferrara, 32 (1 77) 24 ; . . II aber and .L. ane, (1 5) 75; II.P. illes, 110 (1 4) 1; . ess and . Bagger, i niversity Press, Princeton, ew ersey, 1 2).
LAT
(Princeton
Lattice gauge theory: M. Creutz, i (Cambridge niversity Press, Cambridge, 1 3); I. Montvay and . Münster, i i (Cambridge niversity Press, Cambridge, 1 4); II..I. othe, i i : an i ( orld Scientific, Singapore, etc., 1 7).
Ultraviolet lIlSeiMHlH .., |i|,l unification of coupling«, ; lm t )\ , ii. of fermion uuu • , 11, f m unitarity of S matrix in gnii|>ti theory, 251 tarity violation, 608 unitary gauge, 582, 597
UMlvtniiallty of i Inline In I VY MM, l| l',» (¡40,660 in <jl i i ' i n , tit,', 0 4 3 III IIHlllHI llltllll'l, 711, IH I ill Willi ltd' • III I Inn. 11
VIUIIMtlll tllllllll'il 'Mi. vai'iiiiin • I|I>" I ill' ii
