Hadronic Structure, Procs of the 14th Annual Hugs at Cebaf

£2 - i p ' V • drf - mfo • tp] + i + ^ 2 [iff • # V ) ( v • d^) £ 4 = -pf [16L4 + sL5]-d^

- (if •
• d»ip

,1 - ^ | [ 3 2 L 6 + 1 6 L 8 ] ^ m ^ . ^ + 0(^4).

(95)

Performing the loop integrations on the <j)A (x) component of the above yields £

eff

1 -d^ipd^ +

1 5m 2 2 - -m 0

2

)ip • tp

0F2 (siksH ~ ^ijhi)I(ml)(Sijd^(pkdlj,ipi + Skimlvupj)

1„ „.. ml + 2d»

• ip

+

1 + 8L5)] - ^mlip •ml
m1 1 (16L4 + 8 X 5 ) - = f - ^ = J ( m » ) m2 1

l + (32L6 + 1 6 L 8 ) - ^ - ^ j / K )

(96)

from which we can now read off the wavefunction renormalization term Z^.

82

When this is done we find

Z„F
l - ^ ( 2 L ml

4 + i 5

IH

+ 24TT F2 2

1+

" ^

+ 7 - 1 - In 47r + In

+1 1 ln47r

l4\

M2 J

mt

+ln

2^F2\--e - -

2L, „ 2 9

+

q

^) Jf

(97)

while from the loop diagrams given earlier

5ml

Fi(q2)

2

(5a)

f 2

'48TT F

2

j

: - - + 7 - l - l n 4 7 r + l nm ^ /i-

1

Fi(q2)

2

(5b)

16TT F

m

2

^ - ^

2 m2 --+7-l-ln47r + ln-f

+ itf-4^(^)-^}

(98)

Adding everything together we have the final result, which when written in terms of the renormalized value Lg is finitel V l n m* ^

(q2-4ml)H mi

(99) Expanding to lowest order in q2 we find (r)

F1(q2) = l + q2

2L, Fl

967r2F2

* ^

+

1

+

(100)

which can be compared with the phenomenological description in terms of the pion charge radius F1(q2) = l + -(rl)q2

+

(101)

83 2 Abs [V]

0.6

Figure 6: Calculations of the modulus of the pion form factor squared compared to experimental results. Here the solid line gives the result of the inverse amplitude method, while the dashed line gives the one loop chiral perturbation theory prediction. The dotted line shows an empirical simulation of the inelastic uin contribution obtained by multiplying the inverse amplitude result by the factor 1 + 0.15s/s u 4 6 .

By equating these two expressions and using the experimental value of the pion charge radius—(r£) exp = (0.44 ± O.OlJfm^16—we determine the value of (r)

the counterterm Lg ' shown in Table 2. As seen in Figure 6 this form gives a reasonable representation of the experimental pion form factor near thereshold but deviates substantially from the empirical result as the p resonance is approached. This is not surprising as any perturbative approach will be unable to reproduce resonant behavior. This failure should not be considered a failure of chiral perturbative techniques per se—just that as one approaches higher energy the importance of two-loop (0(p6)) and higher terms become important. Although for simple processes such two loop studies have been performed, the number of p6 counterterms is well over a hundred and a general chiral analysis at two loop level is not feasible. Nevertheless things are certainly not hopeless, and in the closing chapter of these lectures we present some approaches to extend the validity of chiral predictions to higher energy. More relevant at this point is to stay near threshold and ask if chiral

84

pertubation methods are predictive. Can they be used as a test of QCD, for example? The answer is definitely yes! We do not have time in these lectures to give a detailed presentation of the status of such tests-a simple example will have to suffice.17 We have seen above how the pion charge radius enables the determination of one of the chiral parameters—Lg. A second—Lj 0 can be found from measurement of the axial structure constant—h A —in radiative pion decay—7r+ —> e+t/ej or ir+ -» e+vee+e", for which the decay amplitudes can be written Mn+^e+Vel

=

M^+^e+^e+e-

=

eGF

^- cos01Mllv{P,q)e''*(q)u{pv)j'/(l

+

j5)v(pe)

j=f- COS
y.u(pi)^v{px)u{pv)-iv{\

+ J5)v(pe),

(102)

ans the hadronic component of MA„ has the structure M^(p,q)

= f dixeiqx

< 0\T(J*m(x)Jl-i2(0)\-!r(p)

> = Born terms

- hA({p - q)u.qv - g^q • (p - q)) - rA(q^qu + ihve^a^p3

9^q2) (103)

where HA, rA, hy are unknown structure functions. (Note that TA can be measured only via the rare Dalitz decay TV+ -» e+vee+e~.) We also note that the related amplitude for Compton scattering can be written in the form ~iT^(p,p',q)

< Tr+(if)\T(J™(x)Jtm(0)\*+(p)

= -ijfxe*-*

= Born terms + a(q2^qi^ - g^qi • q-z) H

> (104)

The 77r+ —• 77r+ reaction is often analyzed in terms of the pion electric and magnetic polarizabilities OLE and /3M which describe the response of the pion to external electric and magnetizing fields.18 In the static limit such fields induce electric and magnetic dipole moments p = 4naBE,

/Z=47r/?M-ff

(105)

which correspond to an interaction energy U = ~

(47raEE2 + 4 7 r / 3 M # 2 )

(106)

85

Reaction 7T+ —> e+uery ix+ —> ervee+e~ 77T + - 4 77T +

Quantity

hvim-1) ry/hy

( a £ + /?M)(10- 4 fm 3 ) ajB(10- 4 fin 8 )

Theory 0.027 2.6 0 2.8

Experiment 0.029 ± 0.017 iy 2.3 ± 0.6 19 1.4±3.120 6.8±1.421 12 ± 2d22 2.1±1.123

Table 3: Chiral Predictions and data in radiative pion processes.

Use of chiral perturbation theory yields the results h

11 2 r = = 0.027m0.027m; n , , M^ = 327r (LS + L 10) 12V2n F„ hv

IT =

327f2

¥—2 v = ,» m\„

2

t-T^Pf*1.

&E + @M = 0

Use of the experimental result j ^ - fc 0.46 ± 0.08

gives

LTw{n = m„) = -0.0056(3)

(108)

and once this is determined chiral symmetry makes four predictions among these parameters! As shown in Table 3, three of the four are found to be in good agreement with experiment. The possible exception involves a relation between the charged pion polarizability and the axial structure constant HA measured in radiative pion decay. In this case there exist three conflicting experimental results, one of which agrees and one of which does not agree with the theoretical prediction. It is important to resolve this potential discrepancy, since such chiral predictions are firm ones. There is no way (other than introducing perversely large higher order effects) to bring things into agreement were some large violation of a chiral prediction to be verified, since the only ingredient which goes into such predictions is the (broken) chiral symmetry of QCD itself! 4

Baryon Chiral Perturbation Theory

Our discussion of chiral methods given above was limited to the study of the interactions of the pseudoscalar mesons (would-be Goldstone bosons) with leptons and with each other. In the real world, of course, interactions with baryons

86

also take place and it is an important problem to develop a useful predictive scheme based on chiral invariance for such processes. Again much work has been done in this regard,24 but there remain important problems.25 Writing down the lowest order chiral Lagrangian at the SU(2) level is straightforward— l

C*N = N(i P - mN + Y^)N

(109)

where QA is the usual nucleon axial coupling in the chiral limit, the covariant derivative D^ = d^+ TM is given by

rv = \W,d,u] - i«t (V/i + Aii)u _ i u ( y M _ Aii)utt

(110)

and uM represents the axial structure u„ = m t V M C/u +

(111)

Expanding to lowest order we find C„N = N(ip - mN)N - -^N^TN

+ gAN^lb\fN •ivxdfin

• (-^d^

+ 2AM)

+ ...

(112)

which yields the Goldberger-Treiman relation, connecting strong and axial couplings of the nucleon system26 F„g„NN = mNgA

(113)

Using the present best values for these quantities, we find 92.4MeV x 13.05 = 1206MeV

vs.

1189MeV = 939MeV x 1.266

(114)

and the agreement to better than two percent strongly confirms the validity of chiral symmetry in the nucleon sector. Actually the Goldberger-Treiman relation is only strictly true at the unphysical point gnNN(q2 = 0) and one expects about a 1% discrepancy to exist. An interesting "wrinkle" in this regard is the use of the so-called Dashen-Weinstein relation which uses simple SU(3) symmetry breaking to predict this discrepancy in terms of corresponding numbers in the strangeness changing sector.27 A second prediction of the lowest order chiral Lagrangian deals with charged pion photoproduction. As emphasized previously, chiral symmetry requires any pion coupling to be in terms of a (co-variant) derivative. Hence there exists a

87

iVA/'7r±7 contact interaction (the Kroll-Ruderman termf8 which contributes to threshold charged pion photoproduction. Here what is measured is the s-wave or Eo+ multipole, denned via Amp = 47r(l+ n)E0+a • e + . . .

(115)

where /J, — mv/M. In addition to the Kroll-Ruderman piece there exists, at the two derivative level, a second contact term which arises from

4 % = mk0^^1^)

4\^]P+ = ^.S-cvq^+S")

(116)

Adding these two contributions yields the result?9 1 e F - | 9A n M ^ = egA ( 1 - \n ^ 0+ :t * 47r(l + /i) V2Fn( + 2 ; 4V2Fn \ - l + ^^~ _ f +26.3 x 10-3/mn TT+n ~ \ -31.3 x 1 0 _ 3 / m T n-p

(117)

and the numerical predictions are found to be in excellent agreement with the present experimental results,

Etx+P

4-1

(+27.9 ± 0.5) (+28.8 ± 0.7) (+27.6 ± 0.3) (-31.4 ± 1.3) (-32.2 ±1.2) (-31.5 ± 0.8)

x lQ-3/mn x 10~3/mn x lQ-s/mK x 10- 3 /m^ xlO^/m^ xlO^/m^

30

w+n

31 34 30

n-p

(118)

33 32

Heavy Baryon Methods

Extension to SU(3) gives additional successful predictions—the linear GellMann-Okubo relation as well as the generalized Goldbeger-Treiman relation. However, difficulties arise when one attempts to include higher order corrections to this formalism. The difference from the Goldstone case is that there now exist two dimensionful parameters—mjy and F„—in the problem rather than one—Fn. Thus loop effects can be of order (mAr/47rF7r)2 ~ 1 and we no longer have a reliable perturbative scheme. A consistent power counting mechanism can be constructed provided that we eliminate the nucleon mass from the leading order Lagrangian. This is done by considering the nucleon to be very heavy. Then we can write its four-momentum as35 pM = Mi)„ + k„

(119)

88

where v,, is the four-velocity and satisfies v2 = 1, while &M is a small off-shell momentum, with v • k « M. One can construct eigenstates of the projection operators P± = | ( 1 ± f), which in the rest frame project out upper, lower components of the Dirac wavefunction, so that? 6 $

iMv

= e-

-*{Hv+hv)

(120)

where Hv = P+ip,

hv = P-ip

(121)

The effective Lagrangian can then be written in terms of N, h as CnN = HVAHV + hvBHv + HvjoB^ohv

- hvChv

(122)

where the operators A, B, C have the low energy expansions A = iv • D + gAU • S + ... B = iipx- - -gAv • «7 5 + . . . C = 1M + iv • D + gAu • S + ...

(123)

Here D^; = (gM„ —v^v^D" is the transverse component of the covariant derivative and S^ = §7507t„u" is the Pauli-Lubanski spin vector and satisfies the relations 3 S2 = - - ,

1 = -{vltvv-gltv),

[Sli,S„]=ieliua0VaS0 (124) We see that the two components H,h are coupled in this expression for the effective action. However, the system may be diagonalized by use of the field transformation h' = h-C~1BH (125) S-v = 0,

{Slt,S„}

in which case the Langrangian becomes CnN = HV(A+

{~ioB^0)C-lB)Hv

- h'vCh'v

(126)

The piece of the Lagrangian involving H only contains the mass in the operator C~l and is the effective Lagrangian that we desire. The remaining piece involving h'v can be thrown away, as it does not couple to the Hv physics. (In path integral language we simply integrate out this component yielding an uninteresting overall constant.) Of course, when loops are calculated a set of

89

counterterms will be required and these are given at leading (two-derivative) order by A{2) = -^•(c 1 Trx+ + c2(v • u)2 + c3u-u + ci[S>1,s',]ultuv + c5(X+ - T r X + ) - J L [ S " , S"]((l + c6)F^

+ c 7 Tr/+,))

C

( |+ M c s ^ ^ T r a V ) - -fiTru»u Tru"u vw - (-£

(127)

Expanding C * and the other terms in terms of a power series in 1/M leads to an effective heavy nucleon Lagrangian of the form (to 0(q3)) £„N

=

HV{AW+AW+AW

+ (7o#(1)t7o)2^(1)

( 7 o g ( 1 ) t 7o)g ( 2 ) + (7og ( 2 ) t 7o)g ( 1 ) 2M - hoB^0)i{v-D)^MfU-S)B^}Hv

+ Otf)

(128)

A set of Feynman rules can now be written down and a consistent power counting scheme developed, as shown by Meissner and his collaborators.25 4-2

Applications

As an example of the use of this formalism, called heavy baryon chiral perturbation theory (HBxpt) consider the nucleon-photon interaction. To lowest (one derivative) order we have from A^ CyNN = ieN±(l

+ T3)e»vN

(129)

while at two-derivative level we find 4 A W =N ^ ( l

+ r 3 )e • ( P l +p2) + JjjL[S . e,S • k](l + KS + 73,(1 + K V ) \ N (130)

90

where we have made the identifications c% = KV, C7 = \(KS - /sy). We can now reproduce the low energy theorems for Compton scattering. Consider the case of the proton. At the two derivative level, we have the tree level prediction

(7o0(1)t7o)2^(1)U = ^ A

(131)

which yields the familiar Thomson amplitude e2 Amp 7 p p = - — e'-e

(132)

On the other hand at order q3 we find a contribution from Born diagrams with two-derivative terms at each vertex, yielding 2 1

e

Amp 7 p p = (—- ) -p[(e' -kS-exk-e-

k'S • e' x fe')(l + KP)

+ iS • (e x k) x (e' x fe')(l + np)2}

(133) z

The full result must also include contact terms at order q from the last piece of Eq.(128)

~eP+ ^ ( S F H ± P + = -^-IxX

(134)

and from the third ^ { 6 ^ , ^ ^ } ? +

= «

p

^ 5 - l x i

(135)

When added to the Born contributions the result can be expressed in the general form25 Amp = e • i'Ai + e' •fee• k'A2 + ia • (e1 x i)A3 + ia • (fe' x fe)e' • eA4 + ia • [(e' x fe)e -ft -(ex

k')e' • k]A5

+ ia • [(£' x fc')e •fe'- ( e x fe)e' • k]A6 (136) with c

* = -£• A - A c

M

A4--A5--

= W~« ^3 = ^ d + 2 « - ( l + ^ - f c ' ) e 2 1 + /t ( ) A ( ) mn

e2 1 + K 2 2M2L}

,

A e - - ^ ^

(137)

Here we have used the identity 3 • (e' x fe') x (e x fe) = S • (fe' x k)i • e' + a • (e' x e)fc' -k + ff-(ex k')i' • k - a • (e' xfc)e• fe'

91

(a)

(b)

(c)

(d)

Figure 7: Loop diagrams for Compton scattering. Figures b,c,d must, of course, be augmented by appropriate cross diagrams.

which agrees with the usual result derived in this order via Low's theorem.37 A full calculation at order q3 must also, of course, include loop contributions. Using the lowest order (one-derivative) pion-nucleon interactions CNN

= ^-NraS

C^NN

= -^v

C^NN

=

l

• qN • (ft +

-^-ea3bNe

q2)eabcNTcN • SrbN

(138)

these can be calculated using the diagrams shown in Figure 7. Of course, from Eq.(128) the propagator for the nucleon must have the form 1/iv • k where k is the off-shell momentum. Thus, for example, the seagull diagram, Figure 7a, is of the form

A-P - 4 ' 2( fe* -glw?«-m-

-»5« " 2- «)• - -»

<139)

Since there are no additional counterterms at this order q3, the sum of loop diagrams must be finite and yields, to lowest order in energy and after considerable calculation

and { =

gyiXFi.

92 The experimental implications of these results may be seen by first considering the case of an unpolarized proton target. Writing e2 AmPunpol = (e • e ' ( - jtf + ^aEcj2)

-. + (e x k) • (e' x k')4ir0M}

(141)

where as, PM are the proton electric and magnetic polarizabilities, we identify the one loop chiral predictions38 d%eo = 10/3^° =

= 12.2 x 10- 4 fm 3

°agg OOVK

(142)

r^m^

which are in reasonable agreement with the recently measured values39 o4*p = (12.1±0.8±0.5)xl(T 4 fm 3 ,

/ 3 ^ p = (2.1^0.8^0.5)xHT 4 fm 3 (143)

For the case of spin-dependent forward scattering, we find, in general —-Amp = /i(w 2 )e • e' + iojf2(u2)a • e' x e

(144)

47T

with / l ( ^ 2 ) = -—^

h{u2)

+ (<*E + PMW + 0(W 4 )

= -sSfe + 7sw2 + 0{uji)

(145)

where 75 is a sort of " spin-polarizability" and is related to the classical Faraday effect. Assuming that the amplitudes / i , / 2 obey once-subtracted and unsubtracted dispersion relations respectively we find the sum rules (*E+PM

2

7re

=

4

^5

I

^(o-+(w)+<7_(w))

7s=

r°°du. .

^r

S

...

[
--

(M6)

where here a±(ui) denote the photoabsorption cross sections for scattering circularly polarized photons on polarized nucleons invovling total 7./V helicity 3/2 and 1/2 respectively. Here the first is the well-known Baldin sum rule for the sum of the electric and magnetic polarizabilities, while the second is

93

\

/

\

\

\

'

*

(a)

(b)

Figure 8: Diagrams for neutral pion photoproduction. Each should be accompanied by an appropriate cross term.

the equally familiar Drell-Hearn-Gerasimov sum rule.40 The third is less well known, but follows from that of DHG and offeres a new check of the chiral predictions. Another venue where there has been a great deal of work is that of neutral pion photoproduction, for which the one-derivative contribution vanishes. In this case the leading contribution arises from the two derivative term given in Eq.(128), augmented by the three derivative contribution from Born diagrams. The net result is

w-s^

•»*{;&

d47)

for the contact term and Amp(3) =

-^[s^s-k]{1+Kp)^rq2MFs-{2p-q)m-=

" i f Sfi+xp)**

(148) for the pole terms (Note: only the cross term is nonvanishing at threshhold.) Finally we must append the contribution of the loop contributions which arise from the graphs shown in Figure 8

W~—»£*»»• r

(149)

The result is the prediction41

Eo+ =

!SfM{1 ~ [l{3+Kp)

+ { )2](1+0<Ji2))

ik

(150)

However, comparison with experiment is tricky because of the existence of isotopic spin breaking in the pion and nucleon masses, so that there are two thresholds—one for 7r°p and the second for 7r+n—only 7 MeV apart. When

94

E0+(Tr°p)(xlO-3/mn)

theory -1.2

E0+(n0n){xl0-3/mn) Pi/\q\(np){xGeV-2)

2.1 0.48

expt. -I.31i0.0842 -1.32±0.1143 1.9 ± 0.344 0.47±0.01 4 2 0.41 ± 0.0343

Table 4: Threshold parameters for neutral pion photoproduction.

the physical masses of the pions are used recent data from both Mainz and from Saskatoon agree with the chiral prediction. However, there are concerns about the convergence of the chiral expansion, which reads E0+ = C(l —1.26 + 0.59 + . . . ) . There also exist chiral predictions for threshold p-wave amplitudes which are in good agreement with experiment, as shown in Table 4, and for which the convergence is calculated to be rapid. Finally exists a chiral symmetry prediction for the reaction •yn —»7r°n Eo+ =

~^h^{\Kn

+ {

w/}

+

••• =

2 1 3 x 10_3/m7r

(151)

However, the experimental measurement of such an amplitude involves considerable challenge, and must be accomplished either by use of a deuterium target with the difficult subtraction of the proton contribution and of meson exchange contributions or by use of a 3 He target. Neither of these are straightforward although some limited data already exist.44 Other areas wherein chiral predictions can be confronted with experiment include the electric dipole amplitude in electroproduction as well as in weak interactions such as muon capture. However, we do not have the space here to cover this work. We can succinctly summarize the situation by stating that at the present time there exist no significant disagreements with experimental findings, but the predictive power in the baryon sector is only in the near threshold region, due among other things to the feature that the expansion is in terms of p/m rather than p 2 / m 2 which occurs in the meson case. 5

On to Higher Energy: Dispersion Relations

Above we have seen the power of chiral perturbation theory in addressing near-threshold phenomenology. However, we have also observed its associated weakness—loss of predictive ability in the meson sector once E,p > ~ mp/2 and in the baryon-meson sector even sooner. In this closing chapter, we try to address the question of whether one can somehow extend the success of xpt

95

Figure 9: Contour for form factor integration.

to higher energy without losing its model-independent connections to QCD. Strictly speaking the answer is no—as the energy-momentum increases one must go to higher and higher order in the chiral expansion, which means increasing the number of loops and of unknown counterterms. Any other approach must be model dependent at some level and hence use more than simply the (broken) chiral symmetry of QCD as an input. Nevertheless, there are some techniques which keep such model dependence at a minimum. One such procedure is to marry the results of xpt with the strictures of dispersion relations,7 the validity of which, given that they rely only on the causality properties of the theory, are not in question. In the case of the pion form factor studied above in one loop xpt the causality condition asserts that the function F(q2) is analytic in the entire complex q2-plane except for a cut along the real axis which extends from Am\ < q2 < oo. Along this cut F(q2) has an imaginary part which is given by the unitarity condition

(P2 -P1)MF^)

= \J

jSmiwm^6^1+P2~kl-k2)

x < 7T7r|T|7r7T >*< 7T7r|jM|0 >

(152)

Use of Cauchy's theorem involving the contour as shown in Figure 9 and

96

the assumption that F(s) -> 0 as s ->• oo then leads to the dispersion relation

* hml s-ql

-it

If the asymptotic condition is not satisfied one can use various numbers of subtractions in order to guarantee convergence. For example, provided that Fi(s)/s -> 0 as s -»• oo we have 1 f°° F1(g2)-F1(0) = - /

JmFtWda

s — q2 — ie

s — ie

ImFl(s)ds

(154)

7!" J4mJ * S(« - ^ - «e)

and further subtractions may be performed if necessary. Even if such subtractions are not required it may be advantageous to utilize them. The point is that in order to get a prediction one requires the value of the imaginary component of the form factor at all values of s. However, the more subtractions the less the sensitivity to input from large s, where in general the form factor is not as well determined. In the case of the pion form factor it is convenient to perform two subtractions, whereby

Here by current conservation we may set Fi (0) = 1 while from experiment we have F{(0) = < r 2 > / 6 = (0.073 ± 0.002) fm2. In order to find Im F(s) we begin by projecting onto the p-wave TTTT channel via T?(s) = - i - / d(cos0)Pi(cos0) < 7T7r|T|7r7r > = J 647T 7_i V

S s -

A ,e"' **m£

sin<5

i (156)

whereby the unitarity condtion Eq.(152) reads ImFi(s) = e - " * sin6\F t (s)

(157)

Of course, the solution of this equation is simply the Fermi-Watson theorem F1(s) = \F1(s)\expiS11(s)

(158)

97 which states that the phase of the form factor is the p-wave nit phase shift. Now at lowest order we have the chiral prediction

and if this is substituted into Eq.(157) along with the lowest order result F i ( s ) « F i ( 0 ) = 1 we find

Fi(q2) = 1 + ^ - q

2

+ ^Tfl

((*2 - A<)Rtf)

+ I**)

(16°)

where we have used the integral

Comparison with the chiral perturbative result—Eq.(99)—reveals that the results are identical provided that one identifies

_2L$

m2

1 +

—6~ -~F?

{1 + ln

m^Fl

-W]

(162)

In this form then we see that the chiral perturbative predictions are simply a result of use of the lowest order chiral forms in the input to the dispersion integral, while a real world calculation would utilize experimental data. In this context it is clear that the renormalized counterterms simply play the role of subtraction constants. So far our dispersive calculation has simply reproduced the chiral perturbative results. How can one do better? Clearly by using experimental data rather then the simple lowest order chiral forms.45 For example, if we divide Eq.(155) by q2 and then take the limit as q2 —> oo we find a sum rule for Lg(/x)

which can be evaluated from experiment. Actual data on ImF(s) are available only from the -Kit threshold up to the middle of the resonance region so other methods must be found in order to generate the high energy component required in order to perform the dispersive integral. For very high energies we can use the prediction of perturbative QCD 64TT2

F2

98

Im f„(s)

V« [GeV]

Figure 10: Input data for the pion form factor dispersion relation. 46

which corresponds to the asymptotic form 64TT3

ImFi(s)

9

Fl

sin 2

(165) A*

Knowing the low and high energy forms of the function, we can generate a smooth matching which joins them in the intermediate energy region, as shown in Figure 10 Obviously the most striking part of this function is the strong p peak, but note also the long negative tail at high energy. That the high energy component must be negative is easy to see from the fact that Eq.(165) guarantees that the dispersion integral converges even without any subtractions. Then, taking the q2 —> oo limit produces two additional sum rules

-i/ 0

ds 4mJ

s

ImFi(s)

1 f°° - / dalmJFi(s)

(166)

n Jiml

In particular the latter sum rule requires that the large positive contribution generated by integrating over the rho peak be cancelled by a significant negative result from higher energy. Now in fact both sum rules are satisfied by the spectral function shown in Figure 10, as they were used in its construction

99 and are responsible for the small bump to the right of the rho tail. Having generated a consistent form for Im F(s) we can now use Eq.(163) in order to generate a prediction for Lr9. We find Lg(m 7 ,)| disprel - = 0.0074 45 compared to L£(m,,)| expt - — 0.0071(3) obtained from analysis of low energy phenomenology (specifically the pion charge radius). Obviously there is good agreement here, confirming the validity of the dispersive approach (and therefore causality!) However, while we have gained a new understanding of the chiral perturbative results we have not yet succeeded in extending their validity to higher energy. Indeed, Figure 6 shows that, while the form factor F(q2) matches onto the experimental result at low q2, it is strongly at variance by the time q2 * 400 MeV2 due to the stron rise of the rho. In fact the prominance of the rho peak suggests an oft-used approximation—replacement of the spectral density by a simple delta function ImFi (s) = nm2p5{s - m2p)

(167)

This drastic approximation—called "vector dominance"—does not satisfy the barely convergent sum rule Eq.(166) a but explicitly satisfies the better damped Eq.(166b). With respect to that for LQ we find < r 2 > = 6m2 ~ 0.40 fm2

(168)

in reasonable agreement with experiment. An approach which does allow us to move to higher energy is the "inverse amplitude method," wherein one looks not at the form factor but rather at its inverse.46,47 The unitarity condition in this case reads 1

lm

_

e a s i n g } (a)

Ws~)-

fiw

la-4m%T}{a)

--V

.

*i(.)

(169)

and a doubly subtracted dispersion relation reads

W)

1 _ <^> r 2 _ I [°° da._ irj4m2s2(s-q2-ie)y

. /a - Ami ??(*) s F^a)

K

'

Using the leading chiral forms, the integration can be performed using Eq.(161) and yields F l i 9 ) =

1- ^ 9

2

- ^

((9 2 - *n%)H(q2) + \q2)

^

Expanding in q2 we observe that this form is consistent with the C(p 4 ) chiral result and represents summation of the Lippman-Scwinger equation in terms of

100

a geometric series. (Sometimes this is also called the Pade (1,1) approximant form.46) Phenomenologically it has the right stuff. Indeed, as shown in Figure 6, the rho resonance arises very naturally as a pole a s s - n n ^ and the phase shift, given by 6\ = argF(s) is in reasonable agreement with experiment. We stress that it is not a simple result of chiral symmetry—rather it is a chirally motivated form which reduces to the chiral result at low energy. It is interesting to note that there is an exact solution to the problem of finding a function whose phase is 5\ along the real axis from Am\ < s < oo. This is the Omnes solution and has the general form48 F

Onme,(g3)=p(ga)expgi

f°°

gf

fl(«)

(172)

7r j 4 m 2 s s — q* — le Here P(s) is an arbitrary polynomial which for our case is set equal to unity. Using experimental TT-K phase shifts one can construct the function Fo(q2) and compare with experiment and with the inverse amplitude form. Agreement with experiment is basically quite good. Note that the Omnes solution assumes the validity of the Fermi-Watson theorem along the entire real axis and hence ignores the substantial inelasticity which arises above ~ 1.2 GeV. Thus this form should presumably only be trusted up to energies where such inelasticity takes over. One can in similar fashion construct a form for the partial wave scattering amplitude which is unitary 1

• s-Ami

.,„.

lm

(173)

Wi) = -V - r

and agrees with the -K-K phase shifts found from the form factor as 1 Tl{S) =

(s — 4m 2 ) 4m )H(s) +

****? i - * P — w&y ((- "

*

(174)

H

In fact, such a form was written down long before the development of chiral perturbative techniques as a simple unitary generalization of the lowest order (Weinberg) scattering amplitude. 49 It is sometimes called the N/D form, where D is the inverse form factor and satisfies the Fermi-Watson theorm along the right-hand cut while N is the lowest order chiral amplitude and is analytic along this discontinuity. Of course, Eq.(174) violates crossing symmetry in that it does not have the proper discontinuity along the left-hand (u-channel) cut, but we may hope that this does not matter along the physical (right-hand) cut as it is far away.

101

We see then that by the use of chiral-based methods, one can construct analytic forms for observables which satisfy the strictures of unitarity exactly (not perturbatively as in the case of chiral perturbation theory itself) and which reduce to the forms demanded by chiral symmetry at low energy. They are admittedly no longer model-independent, nor do they satisfy all the strictures of field theory (such as crossing symmetry). However, any reasonable model must assume a similar form and they can be used with some confidence to construct a successful phenomenology. 6

Closing Comments

In the preceding lectures we have covered a lot of ground—effective field theory, basic chiral models, chiral perturbation theory in the meson and baryon sectors, and high energy extensions based upon dispersion theory. Nevertheless the discussion has essentially been at an introductory level and there is lots more to learn for those who are interested. The state of the art both for experiment and for theory can be found in the proceedings of the first two chiral dynamics workshops (which took place at MIT and Mainz respectively) which were published by Springer-Verlag.50 The next such meeting, by the way, will be held next summer right here at JLab, and you are certainly invited to attend. Despite all the work which has taken place there is still much to do for those who wish to take up the challenge. This lies in many areas: i) calculating to yet higher order (two loop) in both the meson and baryon sectors; ii) making the connection with fundamental theory by calculating the chiral coefficients directly from QCD by lattice or other methods; iii) development of reasonably model-independent methods to extend chiral results to higher energy; iv) identifying and performing the experiments which most sensitively probe such theories. My basic message, which I hope you take with you, is that chiral perturbative methods are an important and vital component of contemporary particle/nuclear physics and that at present they allow perhaps the best way to confront low energy phenomenology with the fundamental QCD Lagrangian which presumably underlies it.

102 Acknowledgement It is a pleasure to acknowledge the support of the Alexander von Humboldt Foundation and the hospitality of the Institut fiir Kernphysik at Forschungszentrum Jiilich. This work was also supported in part by the National Science Foundation.

References

1. See, e.g. A. Manohar, "Effective Field Theories," in Schladming 1966: Perturbative and Nonperturbative Aspects of Quantum Field Theory, hep-ph/9606222; D. Kaplan, "Effective Field Theories," in Proc. 7th Summer School in Nuclear Physics, nucl-th/9506035,; H. Georgi, "Effective Field Theory," in Ann. Rev. Nucl Sci. 43, 209 (1995). 2. J. Gasser and H. Leutwyler, Ann. Phys. (NY) 158, 142 (1984); Nucl. Phys. B250, 465 (1985). 3. For a pedagogical introduction see J.F. Donoghue, E. Golowich and B.R. Holstein, Dynamics of the Standard Model, Cambridge University Press, New York (1992). 4. B.R. Holstein, Am. J. Phys. 67, 422 (1999). 5. A corresponding classical physics discussion is given in R.P Feynman, R.B. Leighton, and M. Sands, The Feynman Lecures on Physics, Addison-Wesley, Reading, MA, (1963) Vol. I, Ch. 32. 6. H. Euler, Ann. Phys. (Leipzig) 26, 398 (1936); W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936). 7. See, e.g. J.F. Donoghue, "Dispersion Relations and Effective Field Theory," hep-ph/9607351. 8. J. Svardiere, Am. J. Phys. 51, 1016 (1983). 9. J. Goldstone, Nuovo Cim. 19, 154 (1961); J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127, 965 (1962). 10. C.P. Burgess, "Goldstone and Pseudo-Goldstone Bosons in Nuclear, Particle, and Condensed Matter Physics," hep-ph/9808176. 11. M. Gell-Mann, CalTech Rept. CTSL-20 (1961); S. Okubo, Prog. Theo. Phys. 27, 949 (1962). 12. B.R. Holstein, Phys. Lett. B244, 83 (1990). 13. S. Gasiorowicz and D.A. Geffen, Rev. Mod. Phys. 41, 531 (1969). 14. S. Weinberg, Physica A96, 327 (1979).

103

15. A. Manohar and H. Georgi, Nucl. Phys. B234, 189 (1984); J.F. Donoghue, E. Golowich and B.R. Holstein, Phys. Rev. D30, 587 (1984). 16. E.B. Dally et al., Phys. Rev. Lett. 48, 375 (1982). 17. J.F. Donoghue and B.R. Holstein, Phys. Rev. D40, 2378 (1989). 18. B.R. Holstein, Coram. Nucl. Part. Phys. 19, 221 (1990). 19. Particle Data Group, Phys. Rev. D54, 1 (1996). 20. Yu. M. Antipov et al., Z. Phys. C26, 495 (1985). 21. Yu. M. Antipov et al., Phys. Lett. B121, 445 (1983). 22. T.A. Aibergenov et al., Czech. J. Phys. 36, 948 (1986). 23. D. Babusci et al., Phys. Lett. B277, 158 (1992). 24. J. Gasser, M. Sainio, and A. Svarc, Nucl. Phys. B307, 779 (1988). 25. V. Bernard, N. Kaiser, and U.G. Meissner, Int. J. Mod. Phys. E4, 193 (1995). 26. M. Goldberger and S.B. Treiman, Phys. Rev. 110, 1478 (1958). 27. R. Dashen and M. Weinstein, Phys. Rev. 188, 2330 (1969); B.R. Holstein, "Nucleon Axial Matrix Elements," nucl-th/9806036; J.L. Goity, R. Lewis, M. Schvellinger and L. Zhang, Phys. Lett. B454, 115 (1999). 28. N. Kroll and M.A. Ruderman, Phys. Rev. 93, 233 (1954). 29. P. deBaenst, Nucl. Phys. B24, 613 (1970). 30. J.P. Burg, Ann. De Phys. (Paris) 10, 363 (1965). 31. M.J. Adamovitch et al., Sov. J. Nucl. Phys. 2, 95 (1966). 32. J. Bergstrom, private communication. 33. E.L. Goldwasser et al., Proc. XII Int. Conf. on High Energy Physics, Dubna, 1964, ed. Ya.-A Smorodinsky, Atomizdat, Moscow (1966). 34. M. Kovash, nN Newsletter 12, 51 (1997). 35. E. Jenkins and A.V. Manohar, in Effective Theories of the Standard Model, ed. U.-G. Meifiner, World Scientific, Singapore (1992). 36. Here we follow V. Bernard, N. Kaiser, J. Kambor and Ulf-G. Meifiner, Nucl. Phys. B388, 315 (1992). 37. F. Low, Phys. Rev. 96, 1428 (1954); M. Gell-Mann and M.L. Goldberger, Phys. Rev. 96, 1433 (1954). 38. V. Bernard, N. Kaiser and Ulf-G. Meifiner, Phys. Rev. Lett. 67, 1515 (1991); Nucl. Phys. B373, 364 (1992). 39. F.J. Federspiel et al., Phys. Rev. Lett. 67, 1511 (1991); E.L. Hallin et al., Phys. Rev. C48, 1497 (1993); A. Zeiger et al., Phys. Lett. B278, 34 (1992); B.E. MacGibben et al., Phys. Rev. C52, 2097 (1995). 40. S. Drell and A.C. Hearn, Phys. Rev. Lett. 16, 908 (1966); S. Gerasimov, Sov. J. Nucl. Phys. 2, 430 (1966). 41. V. Bernard, J. Gasser, N. Kaiser and Ulf-G. Meissner, Phys. Lett. B268, 291 (1991).

104

42. 43. 44. 45. 46. 47. 48. 49. 50.

M. Fuchs et al., Phys. Lett. B368, 20 (1996). J.C. Bergstrom et al., Phys. Rev. C53, R1052 (1996). P. Argan et al., Phys. Lett. B206, 4 (1988). J.F. Donoghue and E.S. Na, Phys. Rev. D56, 7073 (1997). T.N. Truong, Phys. Rev. Lett. 6 1 , 2526 (1988). T. Hannah, Phys. Rev. D59, 057502 (1999). R. Omnes, Nuovo Cim. 8, 1244 (1958). L.S. Brown and R.L. Goble, Phys. Rev. Lett. 20, 346 (1968). Chiral Dynamics: Theory and Experiment 1., ed. A. Bernstein and B.R. Holstein, Springer-Verlag, New York (1996); Chiral Dynamics: Theory and Experiment 2., ed. A. Bernstein and Th. Walcher, Springer-Verlag, New York (1998).

105

LATTICE G A U G E THEORY - QCD FROM Q U A R K S TO HADRONS0 D. G. RICHARDS Jefferson Laboratory, MS 12H2, 12000 Jefferson Avenue, Newport News, VA 23606, USA Department of Physics, Old Dominion University, VA 23529, USA E-mail: [email protected]

Norfolk,

Lattice Gauge Theory enables an ab initio study of the low-energy properties of Quantum Chromodynamics, the theory of the strong interaction. I begin these lectures by presenting the lattice formulation of QCD, and then outline the benchmark calculation of lattice QCD, the light-hadron spectrum. I then proceed to explore the predictive power of lattice QCD, in particular as it pertains to hadronic physics. I will discuss the spectrum of glueballs, exotics and excited states, before investigating the study of form factors and structure functions. I will conclude by showing how lattice QCD can be used to study multi-hadron systems, and in particular provide insight into the nucleon-nucleon interaction.

1 1.1

Lattice QCD: the Basics Introduction

The fundamental forces of nature can by characterised by the strength of the interaction: gravity, the weak interaction, responsible for /3-decay, the electromagnetic interaction, and finally the strong nuclear force. All but the weakest of these, gravity, are incorporated in the Standard Model of particle interactions. The Standard Model describes interactions through gauge theories, characterised by a local symmetry, or gauge invariance. The simplest is the electromagnetic interaction, with the Abelian symmetry of the gauge group U(l). The model of Glashow, Weinberg and Salam unified the electromagnetic and weak interactions through the a broken symmetry group SU(2)U(l). The strong interaction is associated with the unbroken non-Abelian symmetry group SU(3), and is accorded the name Quantum Chromodynamics (QCD). The strength of the electromagnetic interaction is characterised by the dimensionless fine-structure constant ae ~ 1/137. A very powerful calculational technique is to expand as a series in ae - perturbation theory. QCD "Lectures given at the 14th. Annual Hampton University Graduate Studies at CEBAF, 1st. to 18th. June, 1999

106 is characterised by a strong coupling constant as ~ 0{1). QCD, however, is asymptotically free, with an effective running coupling a3(Q2) decreasing logarithmically with increasing Q2. Thus processes with an energy scale large compared with the natural scale of the strong interaction, of the order of the proton mass, are often amenable to the techniques of perturbation theory. At energy scales of the order of the proton mass, perturbation theory fails. Yet a quantitative understanding of QCD is crucial both for the study of the strong interaction, and for the study of the other forces which are masked by the strong interaction. In this energy regime, we can either employ lowenergy effective models of QCD, or seek some way of performing a quantitative calculation directly within QCD. Lattice QCD is the only means we have of performing such an 06 initio calculation. Before proceeding to a description of lattice QCD, it is useful to make a comparison between the properties of QCD and of QED: QED Gauge particle Coupling to Charged particles

vs.

Photon, 7 Electric charge, Q e,(j,,u,d,s... Photon is neutral

QCD Gluon, G Colour charge Quarks, u,d,s...,G Gluon has colour charge

The gluon self-coupling reflects the non-Abelian, and highly non-linear, nature of QCD. Where are the quarks? They are bound into the colour-singlet hadrons we observe in nature. Lattice gauge theory provides the means to relate the quark degrees of freedom with the observed hadronic degrees of freedom. 1.2

Lattice Gauge Theory

Lattice gauge theory was proposed by Ken Wilson in 1974.1 Because of the gluon self-coupling, we have a sensible pure-gauge theory of interacting gluons, even without quark, or matter, fields. We will consider this theory first. We begin by formulating QCD in Euclidean space, which we accomplish by a Wick rotation from Minkowski space, t-*r

= it.

(1)

= A%(x)Ta,

(2)

The gauge fields are defined through A^x) where the Ta,a=

1 , . . . , 8 are the generators of SU(3), satisfying [Ta,Tb] =iffTc

(3)

107

Figure 1: A schematic of a lattice showing the association of the SU(3) matrices U^(x) with the links of the lattice.

TrTaTb

= ^6ab.

(4)

We now introduce the field-strength tensor F;„ = d»Al - dvAl + gf^A^Al,

(5)

in terms of which the Euclidean continuum action is S = \jdixsF^F^.

(6)

As we will see later, the crucial property of Euclidean space QCD for the formulation of lattice gauge theories is that the action is real. Gauge invariance is manifest through invariance under the transformation A^x)

->• A(x)All(x)A-1(x)

- -^(cVO^A-1^).

(7)

We proceed to the lattice formulation of QCD by replacing a finite region of continuum space-time by a discrete four-dimensional lattice, or grid, of points. The gluon degrees of freedom are represented by SU(3) matrices U^x) associated with the links connecting the grid points, as shown in Figure 1. We work with the elements of the group, rather than elements of the algebra, and the SU(3) matrices U^x) are related to the usual continuum gauge fields

108 through dtA^ Un{x) = expigaia / dtA^x

+ tafi),

(8)

Jo

where g is the coupling constant, and a the lattice spacing. Under a gauge transformation A(x), the link variables transform as U^x) -»• A ( x ) t y z + AJA-Har),

(9)

in analogy with Eq. 7. Wilson's form of the lattice gauge action is constructed from the elementary plaquettes l UD^)

= Uli(x)UI/(x + fi)Ul(x + i>)Ut(x).

(10)

The plaquettes are clearly gauge invariant, and the action is then written

50 = ^ E E f 1 - ^ ^ ( » ) l s - | - E E * ^ ^ . (ID where we have ignored the constant term, and introduced

with, for QCD, Nc = 3. It is straightforward to show that the Wilson lattice gauge action is related to the continuum counterpart, Eq. (6), by

SG = \JdixF^F^

+ 0(a2),

(12)

so that the lattice gauge action has 0(a2) discretisation errors. 1.3

Observables and Lattice Gauge Simulations

Within lattice gauge theory, the expectation value of an observable O is given by the path integral (O) = i

fvUO(U)e-SGW

(13)

where VU = Y[dUll(x) X,fi

(14)

109 and Z is the generating functional Z=

j VUe-SG{u^;VU

= J[dU^{x).

(15)

Before proceeding further, we need to define what we mean by the integration over a group variable dU. We do this through the Haar measure, which for a compact group is the unique measure having the following properties: 1. f dUf(U)

= f dUf(VU)

JG

JG

= f dUf(UV)

VV e G.

JG

2.

f dU = 1. JG

This choice of measure respects gauge invariance. Note that, because we are employing the compact variables, U^x), rather than the elements of the algebra, we do not need to fix the gauge, and indeed in most circumstances we do not do so. However, there are cases where working in a fixed gauge is useful, most notably in lattice perturbation theory, where gauge fixing is essential, in the definition of hadronic wave functions, where it is often useful to work in Coulomb gauge, and most directly in the study of the fundamental gluon and quark Green functions of the theory. On a finite lattice, the calculation of observables is equivalent to the evaluation of a very high, though finite, dimensional integral. In principle, we could estimate this integral by evaluating the integrand at uniformly distributed points. This, however, would be hopelessly inefficient; the exponential behaviour ensures that the integral is dominated by regions where the action is small. Instead we use importance sampling, and generate gauge fields with a probability distribution e~s°(u). (16) The interpretation of this exponential in terms of a probability distribution requires that the action be real, and hence the need to work in Euclidean space. The formulation follows that of many systems in statistical physics. 1.4

Statistical and Systematic

Uncertainties

Statistical Uncertainties Observables in lattice QCD calculations arise from a Monte Carlo procedure, and thus have statistical uncertainties. Once we have reached thermalisation,

110

these uncertainties decrease as the square root of the number of configurations, providing successive configurations are sufficiently widely separated to be statistically independent. Systematic Uncertainties Of even greater delicacy than the statistical uncertainties are the systematic uncertainties that enter our computations. These arise from a variety of sources, including: • Finite Volume: Our box must be sufficiently large that finite volume effects are under control. For light hadron spectroscopy, box sizes of at least 2 fm are necessary to ensure that the hadron is not "squeezed", but for excited states even larger volumes may be required. In addition, the requirement that the spatial extent of the lattice be large compared with the correlation length, set by the pseudoscalar mass, sets a still more stringent constraint at the physical pion mass. • Discretisation Effects: Increasing the inverse coupling /? corresponds to progressing to weaker coupling, and hence smaller lattice spacing a. We must ensure that /? is sufficiently large that the scale-breaking discretisation errors are under control, and in practice we perform calculations at several values of a and extrapolate to the limit a = 0. We will encounter several other potential sources of systematic errors when we discuss the inclusion of the quarks. 1.5

Including the Quarks

The full generating functional for lattice QCD with a single flavour of quark is Z = fvUV^e-SGiU)+Z*^{x)M{x'y'Umy\ where M(x,y,U) M(x,y,U)

(17)

is the fermion matrix which, in its "naive" form, is =mSX:V

+- ^ 7

M

([7M(a;)(5y,x+A -Ufa-

A)^^-^)

(18)

with m the quark mass. Because the fermion fields are represented by Grassman variables, we can integrate out the fermion degrees of freedom, to obtain Z=

[VU

detM(U)e-Sa(u).

(19)

111

The determinant fluctuates rapidly between configurations. Thus it is not sufficient for a Monte-Carlo procedure to generate configurations with probability e-sa(u)} a n c i o n iy include the determinant in the calculation of observables; det M(U) must be included in the measure. Furthermore, whilst multiplication by the fermion matrix M involves only nearest neighbour communication, the evaluation of det M(U) is essentially a global operation. Thus detM(C) must be re-evaluated every time we update even a single value link variable U^x). The most efficient algorithms for the simulation of QCD with dynamical quarks, such as the Hybrid Monte Carlo algorithm,2 involve a non-local updating procedure. Nevertheless, calculations with dynamical quarks are at least 1000 times as expensive as pure gauge calculations. The computational overhead of completely including the quark degrees of freedom has encouraged many calculations to be performed in the quenched approximation to QCD, in which we set d e t M = 1, in Eq. (19). This corresponds to suppressing the contribution of closed quark loops in the path integral. There are two justifications for this seemly radical approximation. Firstly, the phenomenological observation that the neglect of the quark loops corresponds to the neglect of OZI-suppressed processes. Secondly, the quenched approximation emerges in the large Nc limit of QCD, where Nc is the number of colours.

1.6

Fermion doubling and chiral symmetry

Unfortunately, the lattice formulation of fermions presents a further challenge. To illustrate how this arises, let us consider the momentum-space fermion propagator M_1 = /

—i-

-

(201

In the massless limit, the propagator has a pole not only at pM = 0, but also at the edges of the Brillouin zone pM = n/a. Thus in four dimensions, we have a theory with 2 4 non-interacting, equal mass fermions. This situation is a consequence of the Dirac equation being first order. Historically, there have been two solutions to this problem.

112

Wilson Fermions Wilson proposed the addition of a second derivative term, or momentumdependent mass term, to the action: S™ = Y, {(m + 4r)${x)rl>(x)X

\ Y, $(*)(r - iMixMx

+ A) + $(x + A)(r + iJUfeMx)] 1 .(21)

In the continuum limit, we find Sf = fdixi>(x)(D+m-^~)xP(x)+0(a2).

(22)

The addition of the second derivatives lifts the mass of the unwanted doublers, but at a price. We have added 0(a) discretisation errors to the fermion matrix, and furthermore the additional term explicitly breaks chiral symmetry at any non-zero value of the lattice spacing, though it is important to remember that chiral symmetry is restored in the continuum limit. The breaking of chiral symmetry has the unfortunate consequence that the fermion masses are subject to an additive mass renormalisation. In simulations, it is conventional to reparameterise the fermion matrix as MXiV = (m + 4r) {Sx,yl - K X "hopping term"}

(23)

where the hopping parameter

« = -^A

r

(24)

y 2(4r + m) ' is now a tunable parameter, reflecting the additive quark-mass renormalisation. The critical value of the hopping parameter, KCHt> is that value for which the pion mass vanishes.

Kogut-Susskind Fermions The second approach, due to Kogut and Susskind, regards the complete loss of chiral symmetry at finite a as too great a sacrifice, and therefore aims to preserve a remnant of chiral symmetry whilst reducing the flavour-doubling problem. This formulation leads to four flavours of quark, but the different flavour and spin components are assembled from fields at the sixteen corners of

113

a 2 4 hypercube. While the Kogut-Susskind formulation has been extensively employed in the calculation of quantities where chiral symmetry is crucial, the problematic flavour identification means that I will concentrate on the results using the Wilson formulation in these lectures. Chiral Fermions and the Ginsparg-Wilson Relation Is it possible to construct a formulation that does indeed possess a symmetry analogous to chiral symmetry at a finite lattice spacing, whilst admitting the correct spectrum of quark states? Let us list four properties desired of the free fermion action, which we write in the form

X

1. D(x,y) is local 2. Far below the cut-off, D(p) ~ «7MpM + 0(p2). 3. D(p) is invertible at all non-zero momenta. 4. y5D + £>75 = 0. The last two requirements require explanation; 3 demands that the only poles occur at zero momentum, and hence that there be no doublers, whilst 4 is just a statement of chiral symmetry. The famous Nielsen-Ninomiya "no-go" theorem 3 states that it is not possible to find a Dirac operator that allows all four requirements to be satisfied simultaneously, and since the first two requirements were deemed sacrosanct either chiral symmetry had to be broken, or flavour-doubling had to be accepted. The recent revolution in our understanding of chiral fermions are related to the rediscovery of the Ginsparg-Wilson relation 4 75£>

+ £>75 = aDj5D,

(25)

as a replacement for requirement 4. At non-zero distances, the Dirac operator does indeed commute with 75, and a symmetry, reducing to chiral symmetry in the continuum limit, is preserved even at non-zero values of the lattice spacing. The problem is finding a formulation of the Dirac operator that does indeed satisfy the relation of Eq. 25. Recently, two formulations satisfying this relation have been discovered. In the case of Domain Wall fermions 5,6 (DWF), an auxiliary fifth dimension,

114 • Wilson m0=1.65 8x32 • Wilson m0=1.65 163x32 > Iwasaki m0=1.40 83x32 • Iwasaki m0=1.65 83x32 A Iwasaki m0=1.90 83x32 < Iwasaki mn=1.65 163x32

0.20

0.15 O

t

E"

^ w 0.10 IS

E 0.05

0.00 10

20

30

40

50

Ls Figure 2: The residual pion mass m j as a function of the extent Ls of the fifth dimension in the D W F formulation. 7 The calculation is performed in the quenched approximation to QCD, and Wilson and Iwasaki refer t o the standard Wilson gauge action, and an improved gauge action respectively.

with coordinate s and extent Ns, is introduced. The action is essentially a five-dimensional Wilson fermion action 5

DWF=

J2

$(x) {D(x,y)6s,s,

+ D5(s,s')6x,y)iP(y),

(26)

x,y,s,s'

where D(x,y) is the usual Wilson-Dirac operator introduced in Eq. 21, but with a negative mass term M. The operator in the fifth dimension, D5(s, s'), couples the boundaries through a parameter —m, which is proportional to the usual four-dimensional quark mass; note that no gauge links are introduced in the fifth dimension. The chiral limit corresponds to Ls —> oo, and then m —>• 0; following the former of these limits, the quark mass is only multiplicatively renormalised. A crucial issue is how small a value of Ls is sufficient to maintain good chiral properties whilst minimising the computational cost. For the case of hadronic physics, perhaps more important than having good chiral properties is being able to perform simulations at sufficiently small quark mass for the

115

pion cloud to emerge. Fig. 2 shows the residual pion mass in the chiral limit as a function of function of LSJ The second approach is the Overlap formalism, introduced by Narayanan and Neuberger.8'9 Here the overlap-Dirac operator satisfying the property 4 is

"

•

^

"

^

^

'

-

"

^

(27)

where H(m) is the Hermitian Wilson-Dirac fermion operator, with negative mass m, defined by H = j5D where D is the usual Wilson-Dirac operator. The parameter [i is related to the physical quark mass. In this case, the extra computational cost comes not from computing in five dimensions, but rather from evaluating the step function

t(H)

=J l -

(28

»

The relative computational overheads of the two implementations is the subject of intense investigation,10 but in any case the overhead compared to the standard Wilson fermion action is considerable. Whilst this overhead is justified for chiral gauge theories, the situation in the case of hadronic physics is less clear; perhaps there are more efficient ways of approaching physical values of the light-quark masses. 1.7

Improvement

The addition of the Wilson term to the fermion action has introduced 0(a) discretisation errors; in contrast the gauge action has only 0(a?) discretisation errors. Thus there has been an emphasis on reducing the errors in the fermion sector through the addition of higher dimensional terms to the action, the improvement programme of Symanzik.11 In the case of the Wilson fermion action, the leading 0{a) errors can be removed through the addition of a single dimension-five operator, the magnetic moment, or clover term, proposed by Sheikholeslami and Wohlert 12 SF = Sf

- i^p

£

i,{x)FliV{x)a^^)-

(29)

The name "clover" is clear from the natural lattice discretisation of F^v illustrated in Figure 3. Using tree-level perturbation theory, the clover coefficient csw is unity, and the discretisation errors on hadron masses and, with an appropriate discretisation of operators, on-shell matrix elements are formally 0(ag2).13 UKQCD

116

M-v

X

MFigure 3: The lattice discretisation of the field-strength Fjlll{x) in terms of the plaquettes with corners at x

performed an extensive investigation of the hadron spectrum and hadronic matrix elements using this value of csw, and the discretisation errors, particularly for systems containing heavy quarks, can be substantial. 14 More recently, two prescriptions for determining csw have been proposed with the aim of reducing discretisation errors still further. In the first, the clover coefficient is constrained to its mean-field-improved, or tadpole, value 15 csw

TAD = -=•

(30)

where u0 = (-TrUa)

(31)

is an estimate of the mean-value of the link variable U^. Though formally the discretisation errors remain 0(ag2), this prescription recognises the poor behaviour of naive lattice perturbation theory arising from the "tadpole" contributions, and attempts to resum the dominant higher-order contributions through the use of a more physical expansion parameter. The second prescription 16 ' 17 determines csw non-perturbatively in such a way as to remove all calO(a) discretisation errors from hadron masses, and, with an appropriate choice of operators, from all on-shell matrix elements 18 c s w = NP =

1 - 0.656gg - 0.152gg - 0.054gg 1 - 0.922^


(32)

117

C(R,T) R

T Figure 4: The construction of the RxT

wilson loop in a space-time plane.

where g$ — 6/(3. We end this section by remarking that the chiral-fermion formulations are already O(a)-improved; the 0(a) discretisation are introduced through the chiral-symmetry-breaking Wilson term. 2

The light-hadron spectrum

The calculation of the spectrum of hadrons containing the light quarks (u,d,s) is the benchmark calculation of lattice QCD; we know many of the results! It also provides a useful theatre for discussion of some of the issues I raised in the introduction. However, let us begin with one of the simplest observations we can make in lattice QCD, that of the linear confining potential. 2.1

The Static Quark Potential and Quark Confinement

The simplest observable we can obtain from a lattice simulation is the potential between two (infinitely heavy) static quarks. We construct the Wilson loops W(R,T)

= (TTU(C(R,T)))

(33)

where U(C(R,T)) is the product of gauge links around a R x T space-time loop. At large times, we can extract the potential V(R) between two static quarks Q at separation R, using

W(R,T)Ttfxe-TV^.

(34)

118

An area-law decay of the large Wilson loops is characteristic of a linear, confining potential in QCD, giving rise to a constant force with increasing separations r. We parameterise this force F(r) by

*Xr)rS = 1.65+£(;f-l)

(35)

where r0 ~ 0.5 fm is a phenomenological parameter 1 9 which we use to determine the scale. We show this force for the pure-gauge theory, corresponding to the quenched approximation, in Figure 5. At large separations we see the constant force indicative of confinement. But there is a further important observation. The results from all three calculations lie on a single curve, even though the calculations span a factor of two in lattice spacing a, from about 0.05 fm to 0.1 fm. Do we expect the same picture of a rising, linear static-quark potential in full QCD, with dynamical quarks? As the two heavy quarks are separated, the energy stored in the string increases. Eventually, the string can "break" to form a quark-antiquark pair, a process that does not occur in the pure-gauge theory. This should lead to a flattening of the potential at some distance corresponding to an energy in the flux tube of 2MB, twice the binding energy of static-heavy-light meson. In practice, there has been no clear observation of such a feature at zero temperature from the simple Wilson loop operator; the behaviour of the potential in full QCD from a calculation using an 0(a)improved fermion action 21 is shown in Figure 6. As the string breaks, the QQ system crosses to a system of two heavylight Qq mesons. The lack of observation of the string breaking from the Wilson loop is ascribed to the poor overlap of the Wilson loop operator with two such heavy-light mesons. The mixing between these two states has been successfully investigated in simpler systems 2 2 ' 2 3 ' 2 4 , and recently a study has been made within QCD.25 Both the ground state and first excited state energies are extracted by a variational calculation using the Wilson loop and QqQq operators, as shown in Figure 7. We will study a related problem when we discuss the nucleon-nucleon interaction in Section 4. 2.2

Spectrum recipe

The elemental building blocks of a spectrum calculation are the quark propagators G%(x,y) = (0\ra(x)my)\0). (36) The quark propagator to every point x on the lattice from a fixed source point y, or linear combination of source points, is obtained by inverting the fermion

119

Figure 5: The force between two static quarks, in units of Sommer's ro parameter,19 is shown at three values of the lattice spacing, corresponding to /3 = 6.0,6.2 and 6.4.20

120

o

# o IH

Figure 6: The scaled and normalised potential as a function of r/ro, where ro is the Sommer scale discussed earlier, as obtained on a 12 3 x 24 lattice, using O(o)-improved dynamical fermions. 2 1 The expected region of string breaking is shown by the horizontal lines.

121

3.0

A-

AGround

V T 1st excited Q- — D Wilson

2.8 2.6

.;£

^5 2.4 O 2.2

if

2.0

^3T

* ' 0.8

(fmy

1.4

Figure 7: The ground and excited state energies obtained from a variational calculation including the QQ and QqQq operators.25 The horizontal line is 2MB, the static binding energy shown also in Figure 6

122 matrix for a fixed source vector. There are a variety of linear-solver methods used to accomplish this. In principle, the recipe for determining the mass of a ground-state hadron P is straightforward: 1. Choose an interpolating operator O that has a good overlap with P (0 | O | P) * 0, and ideally a small overlap with other states having the same quantum numbers. 2. Form the time-sliced correlation function

X

This is expressed using a Wick expansion in terms of the quark propagators of Eq. 36 3. Insert a complete set of states between O and O^. The time-sliced sum puts the intermediate states at rest, and we find C

® =EE/

( 2 J3^

^(0\O(x,t)\P(k))(P(k)\O^6,0)\0)

(2n) 2E(k) 2

E" n(0\O\P)\ "2M

riMPt

P

4. Continue to Euclidean space t -> it, and we find

At large times, the lightest state dominates the spectral sum in Eq. (37), and we can extract the ground state mass. However there are many considerations that complicate this picture. Firstly, the temporal separation between the hadrons must be sufficiently large that the ground state can be identified in Eq. (37); we aim to accomplish this by choosing an operator having a large overlap with the ground state relative to the excited states, and by fitting to several interpolating operators. Secondly, the correlation lengths in our calculation must be small compared with the

123

size of the box in which we are working. This correlation length is simply the inverse of the pion mass, and we require m^L ~ 5, where L is the spatial extent of the lattice. Most, though not all, simulations to date have restricted consideration to quark masses in the region of the strange-quark mass. The benchmark calculation of the quenched light-hadron spectrum using the unimproved fermion action has been performed by the CP-PACS collaboration.26 They perform their calculation on a variety of lattice sizes, to control finite size effects, and at a variety of lattice spacings, to enable an extrapolation to the continuum limit. Recently, there has been a similar calculation using the non-perturbatively improved clover fermion action by the UKQCD Collaboration.27 They also perform an extrapolation to the continuum limit, but in their case the discretisation uncertainties are 0(a) rather than 0(a2), and I will describe some of the details of this calculation. UKQCD generated 163 x 32 lattices at /? = 5.7, 163 x 48 lattices at /3 = 6.0 and 24 3 x 48 lattices at j3 = 6.2, corresponding to approximately a factor two span of lattice spacings but at roughly the same spatial volumes. In addition, a calculation was performed on a larger 32 3 x 64 lattice at /? = 6.0 to enable a study of finite-volume effects. Quark propagators were computed with the clover coefficient having both its tadpole-improved value csw — TAD (/3 = 5.7,6.0,6.2), and its non-perturbatively determined value csw = NP 03 = 6.0,6.2). Since the quark propagators were computed at values of the quark mass in the region of the strange quark mass, it is necessary to extrapolate in the quark mass to obtain results at the u- and d-quark masses, and interpolate to obtain results at the s-quark mass. The bare, or unrenormalised quark mass, is related to the hopping parameter K through mq = ^ - ( - -

—)

.

(38)

The bare quark mass must be rescaled in the 0(a)-improved theory so that spectral quantities approach the continuum limit with 0{a2)}7 fhg = m i ( l + bmamq),

(39)

where the perturbative one-loop value of bm is used.28 To extrapolate the results to the physical quark masses, the following ansatz is employed: mp S = B(rhqti + m, | 2 my = Ay + Cy{mq
(40) (41)

124

m 0 c t = A0ct + Cbct("Vi + m,, 2 + mq<3)

(42)

m D e c = A D e c + C De c(m,,i + fhqa + m,,3).

(43)

where the subscripts PS, V, Oct and Dec refer to the pseudoscalar meson, vector meson, octet baryons (£- and A-like) and decuplet baryons (A-like) respectively. The critical hopping parameter corresponds to the value of K for which mps vanishes. The quality of the chiral extrapolations of the vector, £ and A masses for the Csw = NP data is shown in Figure 8. The cgw = TAD data has in principle a remnant 0{a) discretisation error, whilst the csw = NP data is fully (9(a 2 )-improved. In the continuum extrapolations, UKQCD performs a simultaneous fit to both the NP and TAD data. In order to investigate the approach to the continuum limit, it is not necessary to study the chirally extrapolated values. Indeed, a clear demonstration of the efficacy of improvement can be seen by looking at the lattice-spacing dependence of hadron masses at a fixed m^/mp ratio, 29 shown in Figure 9. The final UKQCD results for the quenched light-hadron spectrum, together with the CP-PACS results using the standard Wilson fermion action, are shown in Figure 10. The different UKQCD plotting symbols correspond to determining the lattice spacing by requiring that either K* or N have its physical value. Both calculations support the assertion that the quenched light-hadron spectrum agrees with experiment at the 10% level. Now that we have established the reliability with which we can compute the known hadron masses, it is time to investigate the predictive power of lattice QCD. We begin by considering the glueball spectrum. 2.3

Glueball Spectrum

The gluon self-coupling that distinguishes QCD from the Abelian QED admits the existence of purely gluonic bound states, or glueballs. Indeed, glueballs are the only true states in quenched QCD! The good agreement of the quenched light-hadron spectrum with the experimental value was perhaps not surprising; Zweig's rule tells us that hadronic decays involving the annihilation of the initial quarks are highly suppressed. Zweig's rule also suggests that glueball mixing with quark states should be similarly suppressed. Thus a quenched calculation of the glueball spectrum is very important, and can yield crucial information to guide experiment. The calculation of the glueball spectrum has been plagued by two interrelated problems. Firstly, the glueballs are relatively heavy and thus the correlation functions die rapidly at increasing temporal separations. Secondly the glueball correlators are subject to large fluctuations independent of separation. In consequence, the signal-to-noise ratio for glueball correlators is very poor.

125

2-

I

I

I

I

. ( 1 1 1

r n m„

i

I

I

I

I

I

I

I

I

I

1—t

1

1

1

1

1

1

1 i^-**" 1

I

I

I

I

I

1

1

1 -

4

\

^

*

*

^

)

3^

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

i—i—i—i—I—i—r-^r-

r0mA

5

4--

I

0.1

I

I

I

0.2

0.3

r0(ma)

Figure 8: Data 2 7 for (a): vector mesons, (b): E-like baryons and (c): A-like baryons is plotted against the average value of the masses of the component quarks mq = (m 9 ] i + m g ,2)/2 (vector meson) and mq = (m 9 ,i + rhQt2 + »Tig>3)/3 (baryons). Squares and circles denote the NP data at /? = 6.2 and p = 6.0 respectively, and the lines correspond to the fits of Eqs. (41)-(43)

126

I

I

I

I

I

I

i

I

I

I

I

I

I

|

-'

-

-

—m

CD

\

-

Nucleon

3 — —

—

^

"
—e

v*?

_ Vector meson

—

—

-

x

Wilson

-

-

O

Improved

-

" I

i

0.00

i

i

i

I

i

I

I

0.05

I

0.10

i

i

i

i

| 0.15

a2a

Figure 9: The hadron spectrum using the Wilson and non-perturbatively improved fermion actions is shown against a2 at fixed mps/mv- 29 The scale is set from the string tension.

127

I : UKQCD, a„. I : UKQCD. e„ i: CP-PACS. K-input, ap

\

H

1.5

T IT

% *

> N IN T T

w m




S

|—

V}*

1

i:i

r

E

*

A

1

K 0.5

Figure 10: The quenched light hadron spectrum computed in the £>(a)-improved theory : and the comparison to results obtained using the unimproved Wilson action (full circles).1 The levels of the experimental points are denoted by solid lines

128

Glueball

Sites Figure 11: Construction of improved glueball operators, with the aim of increasing the overlap with the ground state.

Calculations of the spectrum using the standard Wilson gluon action, Eq. (11), have emphasised the construction of improved gluonic operators which more correctly describe the ground-state glueball wave function, as illustrated in Figure 11. In the case of the determination of the spectrum of states at rest, the rotation group used in the construction of the continuum glueball operators is reduced to the cubic group of the lattice. The different components of, for example, the JPC = 2 + + glueball lie in the E++ and T 2 + + representations of the cubic group. As the continuum limit is approached, the masses obtained from the different representations of the cubic group should become degenerate, signifying the restoration of rotation symmetry. Despite the discretisation errors of the standard Wilson gauge action being 0(a2), the lightest glueball state is subject to much larger 0{a2) discretisation errors than the spectrum for quark states, as shown in Figure 12.30 To improve upon these calculations, Morningstar and Peardon 31 chose to employ an C(o 2 )-improved gluon action, having discretisation errors of C(a 4 );note that it is only necessary to consider operators of even dimension in the construction of the gluonic part of the action. Unfortunately, the relatively large mass of the glueball states in lattice units allows only a couple of time slices to be used in extracting the glueball masses. They then observed that one could employ a relatively coarse lattice in the spatial directions to produce a reasonable approximation to the glueball wave function, whilst employing a finer lattice in the temporal direction to enable the isolation of the ground state and excited state masses in each channel.32 The anisotropic lattice is implemented through the choice of different coupling constants for the space-space and space-time plaquettes in the gluonic

129

0.00

0.05 (a/r0)2

0.10

Figure 12: The continuum extrapolation of the masses of the JPC = 0++ and 2++ states.30 The different plotting symbols for the 2++ states correspond to the lattice operators T2 (octagons) and E (diamonds). The lines represent linear continuum extrapolations in a2.

130

Figure 13: The quenched glueball spectrum is shown after extrapolation to the continuum limit.33

131

action Eq. 11; this involves a non-trivial tuning, since the bare couplings are renormalised. The resulting glueball spectrum, 33 after extrapolation to the continuum limit, is shown in Figure 13. 2.4

Exotic Hadrons

The search for hadrons with excited glue is one of the primary goals of the iV* programme at CEBAF. There has been a flurry of recent activity looking for exotic states, and in particular exotic mesons, in the lattice community; lattice gauge theory aficionados generally try to comprehensively understand the mesonic sector before venturing into the realm of baryons. Exotics and Hybrids Within the quark model, the charge conjugation C and parity P of a meson are related to the spin S and orbital angular momentum L through P = ( - 1 ) L + 1 ; C = (-1)L+5. States not conforming to these relations are called exotics, and examples are the states with JFC = l-+,0+-,2+-. An exotic can be formed in two ways. Firstly, as a quark-antiquark-glue state, which we call a hybrid. Secondly as a bound state of two quarks and two antiquarks. In this section I will discuss lattice studies of hybrid states. These studies have been given increased impetus by experimental observations of 1 _ + resonance states in the region of 1.4 GeV.34'35 Hybrid interpolating operators The usual interpolating operator for the pion is Ow(x,t)=iP(x,t)j5iP(x,t).

(44)

In the case of the hybrid state 1 _ + , a possible interpolating operator would be £>!-+ {x, t) = $(x, thiFij (x, t)xP(x, t)

(45)

where F^ is a lattice discretisation of the field-strength tensor constructed in Figure 3. In practice to get any sort of signal for hybrid mesons, it is necessary to use interpolating operators in which the quark and antiquark are separated in space. The signals are inevitably much noisier than for the pseudoscalar and

132

vector meson states. To see why, we note that the hybrid correlator Chyb(t) decays exponentially with the mass of the hybrid, Chyb(t) ~ e~m^bt.

(46)

In contrast, the correlator for the variance is that of the square of the interpolating operator CAt)

= ^(|Ohyb(^i)| 2 |Ohyb(0,0)| 2 ).

(47)

X

Typically, |Ohyb|2 is an interpolating operator for two pions, and therefore the signal-to-noise ratio with increasing temporal separations increases as si

S n a l ~ e -(m h y b -m„)t

/48)

noise Since the masses of the hybrids are relatively large, the signal quickly is lost in the noise. In the case of the glueball calculations, the situation is particularly severe since the square of the glueball operator has a non-zero vacuum expectation value, leading to the constant noise alluded to earlier. There have been recent calculations of the light-quark hybrid spectrum, and in particular of the 1 ^ state, in both the quenched approximation, 36 ' 37 and in full QCD. 38 ' 39 Indeed, there has even been a measurement of the Coulomb-gauge wave function for the 1 _ + state, shown in Figure 14, in which the separation between the quark and anti-quark is clearly apparent. These calculations all find a 1 _ + mass around 2 GeV, far larger than that of the experimental candidates. A possible resolution is that this resonance is actually a four-quark state; we will return to this interpretation in Section 4. 2.5

The N* Spectrum

The measurement of the excited nucleon spectrum reveals the full SU(3) nature of QCD, and is a critical part of the experimental programme at CEBAF. The observed N* spectrum is shown schematically in Figure 15. The nucleon iV(938) has positive parity; its parity partner, with negative parity, is the 7V(1535). The usual nucleon interpolating operators employed in lattice calculations are Na=tijk{uiCl5dj)uka. (49) On forming the time-sliced correlator, both positive and negative parity states can contribute to the correlation function. However, we can perform a parity projection C(t) = £ > | Na(x,t)(l ± 7 o ) Q / 3 ^ ( 0 ) | 0) (50) X

133

1.5

1

1

1

1

- M: "vl%

1

1

1

1

1

1

1

1

I

1

-

Wave function

— _

1.0

- *

1.

-^ 1

-

" ^. \ x% 1 ;, x \ 0.5

CD

a

-

O.Oix-

-0.5

1

1

1

i

1

i

i

i

i

1

1

i

i

i

i

2

Radius (fm) Figure 14: The wave function in Coulomb gauge of the 7r (crosses), p (diamonds) and 1" state (bursts) obtained from quenched QCD at /3 = 5.85. 37 N(2200) N(2080)

NO090)

A(1830)

A(1800) 2(1750)

NO650)

A(1670)

N(1700)

A(1690)

NO520)

A(1520)

A(1670)

N(1675)

2(1620) £(1580)

N(1535) N(1440) AQ405) N(938)

1/2

+

1/2'

3/2

5/2

Figure 15: Schematic showing the observed N* spectrum, labelled by

Jp.

134 1.6 preliminary 1.5 -

*

N* (exp.)

1.4 _ * N (exp.)

1.3h 1.2 1.1 1.0

f

0.9 0.8

-*

0.7

B-|+ (nucleon) D B^o B2" x

0.6 0.5 0.4

0.00

_L 0.02

0.04

_L 0.06

J_ 0.08

0.10

0.12

0.14

m: quark mass Figure 16: The masses of the nucleon (N) and its negative-parity partner in lattice units ( a - 1 ~ 1.9 GeV using the p mass to set the scale). 42

to project out the forward-propagating positive (+ sign) or negative (- sign) parity states, with states of the opposite parity propagating in the negative direction. On a periodic lattice, our correlator contains both the forwardand backward-propagating states, and we rely on a sufficiently long temporal extent to our lattice to delineate the two parities. Recently, there have been two lattice calculation of this mass splitting. The first 40 employs a highly improved fermion action, the £>x34 action of Hamber and Wu.41 Not only do the authors extract the mass of the J = l / 2 ~ state, but also find a signal for the J = 3/2~ state. The second calculation 42 employs domain-wall fermions; here the authors argue that, since the J = 1/2+ and J = 1/2 - state are degenerate in an unbroken chirally symmetric theory, the use of an action having a possessing exact chiral symmetry, even at non-zero lattice spacing, is crucial in correctly extracting the splitting. Both calculations find mass splittings between the positive- and negativeparity states in accord with experiment, though with still substantial system-

135

atic and statistical uncertainties. The masses of the J = l / 2 + (nucleon) and J — l/2~ states using DWF at a fixed value of the lattice spacing are shown in Figure 16. 3

Hadron Structure

As well as enabling the calculation of the hadron spectrum, lattice QCD enables the study of the distribution of the quarks and gluons within hadrons. Information about these is contained in the form factors, and in the quark and gluon structure functions. These calculations have in common the determination of some (local) hadronic matrix element, so we will begin this section with a discussion of the lattice technology of determining matrix elements. 3.1

Hadronic Matrix Elements

The paradigm calculation is that of f„, the pion decay constant defined through <0|i4/1|7r)=ipM/W)

(51)

where AM = V>7M75?/> is the axial vector current. This matrix element we can obtain as a by-product of the determination of mn, and our analysis will follow that of Section 2.2. We construct the correlator C(t) = £ < 0 | Al?(x,t)Al?(0)\0.),

(52)

X

where Alat is a lattice discretisation of the continuum axial-vector current. Inserting a complete set of states between the two interpolating operators, we obtain C

^

=

EE(2^3/2^<°l^ t ^*)|i'®>(i , tf)l^ a t t (0)|0>

= E E (2^)3 / = E / ^§je~im p

^

\ m^flf2

e~ m"l

^c~""+i**<°l4at(0)\P(P))(P(p)\Alf\0)\0) J(3)

® (Q\Aat\P(p))(P(p)\A^\o)

+ excited states

(53)

The lattice discretisation has provided both a cut-off, and a renormalisation scheme. We, of course, want f„ is some familiar continuum scheme, such as

136 MS. To provide us with that, we need to compute the matching coefficient ZA for the axial vector current, such that A* = ZAAla\

(54)

The determination of the matching coefficient Z is one of the most delicate, and generally onerous, tasks of any calculation of hadronic matrix elements. The lattice formulation reproduces continuum QCD as the lattice spacing approaches zero, and thus the anomalous dimensions of the operators in the lattice formulation matches those of the continuum operators. However, the replacement of the continuum Lorentz symmetry by the hypercubic symmetry of the lattice, and the lack of chiral symmetry in most fermion formulations, complicates the calculation of the matching coefficients enormously, even for such a simple operator as the axial vector current. In particular, the lattice allows mixing with higher dimension operators, combined with appropriate powers of the lattice spacing a. An important element of the improvement programme is finding a combination of lattice operators such that matrix elements are free of 0(a), or higher, discretisation errors. In principle, we can compute Z in perturbation theory. Perturbation theory using the bare lattice coupling g as an expansion parameter apparently fails, resulting in very large perturbative corrections. The bulk of these large corrections can be identified as lattice artifacts, the "tadpole" contributions arising from the 0(g2) term in the expansion of the link variable, Eq. 8. These terms can effectively be resummed through the expansion in terms of a renormalised coupling constant. 15 This prescription is precisely that used in the specification of the tadpole-improved clover coefficient of Eq. 30. An alternative route is to attempt to determine the matching coefficients, and improvement coefficients, non-perturbatively through the imposition of chiral Ward identities.16 Indeed, this prescription enables, in principle, the elimination of all 0(a) discretisation effects from on-shell quantities, providing appropriate renormalisation conditions can be found. Both these routes require considerable effort, and uncertainty in the calculation of matching and improvement coefficients is a major uncertainty in the calculation of hadronic matrix elements. Let me conclude this subsection by noting that the chiral fermion actions of Section 1.6 are automatically 0(a)improved, and admit a smaller degree of operator mixing. This may be prove a substantial advantage for these formulations. 3.2

Nucleoli Form Factors

The electric and magnetic form factors of the nucleon are among the simplest quantities that contain information about the structure of the nucleon, and

137

are measured in electron proton scattering. They are related to the matrix elements of the vector current JM through (p'.*' I Jn(q) \P>S)

=u(p',s') i M r t + u r ^ F t f )

up(p,s),

(55)

where q — p — p' is the momentum of the photon probe. Note that F\ and F2 satisfy Fi(0) = l; F2{0)=n-1 where the former result expresses current conservation, and /u, is the magnetic moment of the nucleon. For point-like particles, both quantities would be constant, and therefore they are a measure of the spatial extent of the nucleon. Rather than quoting these quantities directly, it is usual to form the Sach's form factors GE{q2)=F1{q2)

+ ^rF2{q2)

(56)

GM(q2)=F1(q2)+F2(q2),

(57)

where we note that q2 is space-like. Phenomenologically, it is usual to parameterise the form factors through the vector dominance model by a dipole fit ~ GpM(q2)/n2

GW)

~ GNM{
GnE(q2) ~ 0.

^ 7 2 q2lm2v) (58)

A recent lattice determination of the proton electromagnetic form factors 43 is shown in Figure 3.2, together with the experimental data. 44 The lines are fits to the data using the dipole forms of Eq. 58. 3.3

Hadronic Structure Functions

The hadronic structure functions, describing the distribution of quarks and gluons inside, say, a nucleon in inclusive processes, are related to the hadronic tensor W v

»

=

h Jd4xei9'X(N(P>s)\Mx)M0)W(p,s)),

(59)

where JM is the electroweak current, and p, s are the nucleon momentum and spin respectively. Decomposing W/M„ according to the possible Lorentz structures yields four structure functions, two spin-averaged, Fi<2(x,Q2) and two

138

-q 2 [GeV2] Figure 17: The solid points are lattice determinations of the electric and magnetic form factors of the proton in the quenched approximation to QCD.43 The open circles are the experimental measurements.44 The lines represent the dipole fits of Eq. 58 to the experimental and lattice data.

139

spin-dependent, 31,2(2;, Q2), where x is the Bjorken variable, and Q2 — -q2. Phenomenologically, these are determined in Deep Inelastic Scattering, and parameterised at some reference energy scale. Near the light cone, x2 ~ 0, the structure functions can be expanded using the operator product expansion (OPE) in terms of the matrix elements of certain local operators, of twist (dimension - spin) two, together with Wilson coefficients calculable in perturbation theory. Historically, it is these matrix elements that are computed in lattice simulations, since the non-zero lattice spacing in our calculations precludes the measurement of the currents in Eq. 59 at sufficiently small separations, and more fundamentally it is unclear how to extract this quantity at light-like separations in Euclidean space. The local matrix elements are related to the x-moments of the structure functions. In principle, we can recover the full x-dependence of the structure functions by measuring increasing moments. The simplest operators are the non-singlet operators for both the unpolarised and polarised structure functions,

0£O...„B = $ 7 W ^ / . i • • • iD^rip,

(60)

where symmetrisation of indices and removal of traces is understood, and the T is an SU(2) flavour matrix. The first few moments moments for the nucleon have been measured by several groups. 46,47 ' 49 ' 50 The lowest moment of the unpolarised quark distribution has a particularly simple interpretation in terms of the momentum fraction carried by the quarks. Figure 18 shows the first moment of the u and d quark distributions in the quenched approximationf 1 both distributions are somewhat higher than phenomenological expectations. Unfortunately, it is unclear the extent to which the calculation of the first few moments of the structure functions enables a useful picture of the x-dependence of the structure functions, and a means of performing a direct computation of the hadronic tensor, Eq. 59, would be invaluable. The study of the flavour-singlet sector involves consideration of both the quark and gluon distributions, which mix. Computationally, these studies are much more demanding, but of tremendous interest since they venture beyond the simple valence picture of the nucleon. More recently, there have been attempts to investigate the role of higher-twist contributions,52 to which upcoming experiments at JLAB should be particularly sensitive.

140

I

0.6

'

1

'

1

'

1

1

I

u

-

-*1-

-1

-

^

1-

>c

J-

'

0.4

X -

"* 0.2

-

^

—

•

—B~I

d

-* 1

-*——T^ ~^~_

1

0.02

,

1

1

0.04

0.06

,

!

1

0.08

0.1

0.12

Figure 18: The first moment of the unpolarised u and d quark distributions in the nucleon as a function of the quark mass at 0 = 6.O.51 The circles and crosses denote results with the Wilson and clover fermion actions respectively. The bursts denote the CTEQ results.48

141

4

The Nucleon-Nucleon Interaction

In the preceding, we have used lattice gauge theory to acquire a fundamental understanding of the internal structure of an isolated hadron from first principles. It is natural to aim at a similar understanding of the interactions between hadrons, and in particular of the nucleon-nucleon interaction, the very foundation of nuclear physics. Understanding the strong interaction in multi-hadron systems from lattice QCD is a notoriously difficult problem. Multi-hadron states involve the computation of a four-point function and are relatively massive, and therefore the corresponding correlation functions quickly vanish into noise at increasing temporal separations. Furthermore, multi-hadron systems are large, and therefore the spatial extent of the lattice needs to be correspondingly larger than that used in hadronic spectroscopy. Finally, the use of a Euclidean lattice obscures the the extraction of the phase information of the full scattering matrix. 45 Despite these difficulties, the problem is fundamental and compelling. Historically, there have been two approaches to this problem within lattice QCD. The first aimed to extract certain quantitative parameters of the hadronhadron interaction by direct lattice simulation. Liischer53'54'55 exploited the finite-size dependence to extract a discrete set of s-wave scattering lengths. The was thoroughly tested within an 0(4)-symmetric <j)A model 5 6 , and scattering lengths have been computed within QCD for pions, 57 ' 58 and for nucleons 57 Fiebig et al. 59 explored the I = 2 n — n system by extracting a residual interaction potential, and then proceeded to compute the scattering phase shifts which were compared with experiment. The extraction of the s-wave scattering lengths of the ir — n interaction has been encouraging, as illustrated in Figure 19 where the s-wave scattering length for the isospin 1 = 2 pion-pion interaction is shown.58 The investigation of the N — N system is more problematical. Here the scattering lengths are of the order of 10 fm, rather than less than 1 fm as is the case for the TT — TT interaction. Therefore, while lattice calculations do indeed find scattering lengths for the N — N interaction considerably larger than those for the -K — IT and -K — N interaction, 57 this approach is limited by our present inability to simulate on lattice sizes of the order of 10—20 fm, and at physical values of the pseudoscalar mass. The second approach is motivated by the realisation that important insight into the nucleon-nucleon interaction can be gleaned by studying the interactions of a much simpler system, that of two heavy-light mesons, with static heavy quarks. 60 ' 61 Such a system exhibits most of the salient features of the nucleon-nucleon system, namely quark exchange, flavour exchange and colour

142 ao/m, 0.0

(1/GeV*;

— i — i — i — | — i — i — i — | — i — i — i —

-------Ours o CHPT-Tree(atm K =140MeV; -0.5

•

Expt.

.•+*'"

-1.0 >-

A'"

-1.5

Ours =-1.91(25) 1/GeV2 ; Expt. =-1.43(61) 1/GeV2 CHPT-Tree =-2.3 1/GeV2

-2.0

_i

-2.5 0.0

i

i_

0.6

0.4 a(1/GeV)

0.2

0.8

Figure 19: The 1 = 2 pion scattering length obtained in the quenched approximation to QCD is shown as a function of the lattice spacing, and in the continuum limit. 58 Also shown as the open circle is the current-algebra prediction for mw = 140 MeV.

O/x)

Oj(y)

r1

R^

T f~~^ 02(r)

Op)

Oj(x)

MAAAN

^VL 02(s)

^NAA/I^ 62(r)

02(s)

Figure 20: The connected (flavour-exchange) and disconnected contributions to the interaction between two heavy-light mesons are shown. The solid lines denote the heavy quarks, whilst the curly lines denote the propagation of the light-quarks.

143

polarization. The interaction between the mesons can be understood by the study of the four-point functions, shown in Figure 20: C^(x,r;y,s)

= .

(61)

Here each operator can either be that for a pseudoscalar (P) or for a vector (V). The range of the interaction can be assessed by forming the z- and ^-sliced sum, C(R,T)= ^ C(x,r;y,s), (62) where

x = (6,t1),y = (6,t1+T) r = (a±,R,t2),

s = (a±,R,t2

+T).

At large distances R and times T, the correlation function is dominated by the lightest state | n), of mass Mn, that can be exchanged between the mesons, with C(R,T)

~e- M " f l (0|Oi(0,T)(5I(0,0)|n) x (n|O 2 (0,T)O|(0,0)|0).

(63)

Can these correlation functions be constructed from the elemental quark propagators of Eq. 36? The connected diagram of Figure 20 requires the evaluation of the propagator from one point on every time slice to every point on the lattice; this is manageable. Unfortunately, the computation of the disconnected diagram requires the evaluation of all-to-all propagators. Nonetheless, the study of the connected diagram alone is valuable. It describes the flavourexchange interaction, and lattice simulations have shown that the interaction is indeed mediated by meson exchange at large distances, and furthermore that the quantum numbers of the exchanged particle are in accord with naive expectations; for the process PP —> PP a vector meson is exchanged, whilst for the process PV -»• VP a pseudoscalar particle is exchanged.61 The especially attractive feature of heavy-light systems is that the heavy quarks admit the definition of a relative coordinate, and thereby a local adiabatic potential. Recently, exploratory studies have been made of this potential,62'63 and evidence for nuclear binding sought, as illustrated in Figure 21. The investigation of this potential is important and more feasible than the measurements of the scattering lengths, because the large scattering lengths for the N — N system come from a short-range potential. Furthermore, by exploring the potential, we can discover the relative importance of gluon and meson exchange contributions at various distances. Understanding the nature of this potential is also crucial to spectroscopy; are exotic mesons predominantly quarkantiquark-glue hybrid states, or four-quark states?

144

-i

T

> o >>

o.o

" <

[Jr-

i

r

<>

-cr

-J<>

60 U
C H

60

g g

-0.2

o 16i?3 BB 1 = 0 S = 0 • 12 BB 1=0 g=0

s -0.4

1 R/R 0 Figure 21: Adiabatic potential between two heavy-light mesons, with the light quarks in an isospin / = 0 and spin S = 0 configuration, from reference63. Measurements are obtained in the quenched approximation to QCD, on lattices at /3 = 5.7. The separations are measured in units of Ro = 0.53 fm, and the different plotting symbols correspond to different lattice sizes.

145

5

Conclusions

In these lectures I have tried to convey the power of lattice gauge calculations as a means of understanding hadronic physics. There are very many areas that I have not addressed - topology and the role of instantons, finite-temperature and finite-density phase transitions, and, reaching beyond hadronic physics, the calculation of weak-interaction matrix elements and even supersymmetry and gravity. I trust that I have convinced you that lattice gauge theory provides not only an ab initio tool for obtaining quantitative results (the spectrum, form factors etc.), but also provides a means of increasing our conceptual understanding of the strong interaction, for example through the study of the nuclear-nuclear force. Acknowledgements I am grateful for helpful discussions with Stefano Capitani, Robert Edwards, Rudolf Fiebig, Nathan Isgur, Xiandong Ji, Frank Lee, and Stephen Wallace. I also thank Jose Goity and staff for their excellent HUGS workshop. This work was supported by DOE contract DE-AC05-84ER40150 under which the Southeastern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator Facility. References 1. K. Wilson, Phys. Rev. D10 (1974) 2445. 2. S. Duane, A.D. Kennedy, B.J. Pendleton and D. Roweth, Phys. Lett. B195 (1987) 216. 3. H.B. Nielsen and M. Ninomiya, Nucl. Phys. B185 (1981) 20; Nucl Phys. B195 (1982) E541; Nucl Phys. B193 (1981) 173. 4. P.H. Ginsparg and K.G. Wilson, Phys. Rev. D25 (1982) 2649. 5. D.B. Kaplan, Phys. Lett. B288 (1992) 342. 6. Y. Shamir, Nucl. Phys. B406 (1993) 90. 7. Nucl. Phys. (Proc. Suppl.) B83 (2000) 224. 8. R. Narayanan and H. Neuberger, Phys. Lett. B302 (1993) 62; Nucl. Phys. B412 (1994) 574. 9. R. Narayanan and H. Neuberger, Nucl. Phys. B443 (1995) 305. 10. R.G. Edwards and U.M. Heller, hep-lat/0005002. 11. K. Symanzik, Nucl. Phys. B226 (1983) 187; Nucl. Phys. B226 (1983) 205. 12. B. Sheikholeslami and R. Wohlert, Nucl. Phys. B259 (1985) 572. 13. G. Heatlie et a/., Nucl. Phys. B352 (1991) 266.

146

14. C.R. Allton et al. (UKQCD Collaboration), Phys. Lett. B292 (1992) 408. 15. G.P. Lepage and P.B. Mackenzie, Phys. Rev. D48 (1993) 2250. 16. K. Jansen et al, Phys. Lett. B372 (1996) 275; 17. M. Liischer et al, Nucl. Phys. B478 (1996) 365. 18. M. Liischer et al, Nucl. Phys. B491 (1997) 323. 19. R. Sommer, Nucl. Phys. B411 (1994) 839. 20. H. Wittig (UKQCD Collaboration), Nucl. Phys. (Proc. Suppl.) 42 (1995) 288. 21. C.R. Allton et al. (UKQCD Collaboration), Phys. Rev. D60 (1999) 034507. 22. O. Philipsen and H. Wittig, Phys. Rev. Lett. 81 (1998) 4056. 23. F. Knechtli and R. Wommer (Alpha Collaboration), Phys. Lett. B440 (1998) 345. 24. P.W. Stephenson, Nucl. Phys. B550 (1999) 427. 25. P. Pennanen and C. Michael, hep-lat/0001015 26. CP-PACS Collaboration (S. Aoki et al), Phys. Rev. Lett. 84 (2000) 238. 27. UKQCD Collaboration (K.C. Bowler et al), hep-lat/9910022, Phys. Rev. D (to appear). 28. S. Sint and P. Weisz, Nucl. Phys. B502 (1997) 251. 29. R.G. Edwards, U.M. Heller and T.R. Klassen, Phys. Rev. Lett. 80 (1998) 3448. 30. C. Michael, hep-ph/9710249, in Proc. of Advanced Study Institute on Confinement, Duality and Non-Perturbative Aspects of QCD, Cambridge, England, 23rd. June to 4th. July, 1997. 31. C. Morningstar and M. Peardon, Nucl. Phys. (Proc. Suppl.) 42 (1995) 258. 32. C. Morningstar and M. Peardon, Nucl. Phys. (Proc. Suppl.) 53 (1997) 917; Phys. Rev. D56 (1997) 4043. 33. C. Morningstar and M. Peardon, Phys. Rev. D60 (1999) 34509 34. E852 Collaboration (D.R. Thompson et al.A Phys. Rev. Lett. 79 (1997) 1630; Phys. Rev. Lett. 81 (1998) 5760. 35. G.M. Beliadze et al, Phys. Lett. B313 (1993) 276; A. Zaitsev, Proc. of HADRON-97, Brookhaven National Laboratory, August 1997. 36. P. Lacock, C. Michael, P. Boyle and P. Rowland (UKQCD Collaboration), Phys. Rev. D54 (1996) 37. MILC Collaboration (C. Bernard et al.), Nucl. Phys. (Proc. Suppl.) 73 (1999) 264. 38. C. Bernard et al. Nucl. Phys. (Proc. Suppl.) 53 228. 39. P. Lacock and K. Schilling (SESAM Collaboration), Nucl. Phys. (Proc.

147

Suppl.) 73 (1999) 261. 40. F.X. Lee and D.B. Leinweber, Nucl. Phys. (Proc. Suppl.) B73 (1999) 258. 41. H. Hamber and C M . Wu, Phys. Lett. B133 (1983) 351; Phys. Lett. B136 (1984) 255. 42. S. Sasaki, invited talk at Jefferson Laboratory Workshop on the Physics of Excited Nucleons, Feb. 16-19, 2000, hep-ph/0004252. 43. Nucl. Phys. (Proc. Suppl.) 73 (1999) 294. 44. C.V. Christov et al, Prog. Part. Nucl. Phys. 37 (1996) 91. 45. L. Maiani and M. Testa, Phys. Lett. B245 (1990) 585. 46. G. Beccarini et al, Nucl. Phys. B456 (1995) 271. 47. M. Gockeler et al, Phys. Rev. D 5 3 (1996) 2317; Prog. Theor. Phys. Suppl. 122 (1996) 145; Phys. Rev. D54 (1996) 5705. 48. H.L. Lai, J. Botts, J. Huston, J.G. Morfin, J.F. Owens, J.W. Qiu, W.K. Tung and H. Weerts, Phys. Rev. D51 (1995) 4763. 49. R. Brower et al, Nucl. Phys. (Proc. Suppl.) 53 (1997) 318. 50. D. Dolgov et al, Nucl. Phys. (Proc. Suppl.) 73 (1999) 300. 51. M. Gockeler et al, Minireview given at DIS97, Chicago, April 1997, hepph/9706502. 52. S. Capitani et al, Nucl. Phys. (Proc. Suppl.) 79 (1999) 173; Nucl. Phys. B570 (2000) 393. 53. M. Liischer, Commun. Math. Phys. 105 (1986) 153. 54. M Liischer, Nucl. Phys. B354 (1991) 531. 55. M. Liischer, Nucl. Phys. B364 (1991) 237. 56. M. Gockeler, H.A. Kastrup, J. Westfalen and F. Zimmermann, Nucl. Phys. B425 (1994) 413. 57. M. Fukugita, Y. Kuramashi, M. Okawa, H. Mino and A. Ukawa, Phys. Rev. D52 (1995) 3003. 58. S. Aoki et al (JLQCD Collaboration), Nucl. Phys. (Proc. Suppl.) B83 (2000) 241. hep-lat/9911025, to appear in the Proceedings of Lattice 99, Pisa, Italy. 59. H.R. Fiebig, H. Markum, K. Rabitsch, and A. Mihaly, Few Body Systems (Suppl.) 10 (1999) 467. 60. D.G. Richards, Nucl. Phys. (Proc. Suppl.) B9 (1989) 181. 61. D.G. Richards, D.K. Sinclair, and D. Sivers, Phys. Rev. D42 (1990) 3191. 62. A. Mihaly, H.R. Fiebig, H. Markum and K. Rabitsch, Phys. Rev. D55 (1997) 3077. 63. C. Michael, and P. Pennanen (UKQCD Collaboration), Phys. Rev. D60 (1999) 054012.

148

LIGHT AND EXOTIC MESONS Curtis A. Meyer Carnegie Mellon University, Pittsburgh, PA 15213

1

Introduction

In this series of lectures I want to provide an overview of the field of light-quark meson spectroscopy. What do we understand about mesons? What does studying mesons tell us about QCD? How do we study mesons? Why do we study meson; what is exciting about this? The area of light-quark meson spectroscopy deals with mesons built up from u, d and s quarks. Typically, these systems have masses below 2.5GeV/c2. 2

Mesons in the Quark Model

To do this, I want to start with the very basics of the strong interaction, namely the conserved quantities, J, P, C, • • •. With this, I want to look at spectroscopy within one specific model, the constituent quark model. This model is by no means perfect. It provides no explanation for confinement, and the role of gluons is not obvious. It also makes no absolute mass predictions, and no absolute rate predictions for decays. However it does make a rather large number of very good predictions. It also provides a very natural framework within which to classify mesons. It provides a natural handle to address issues such as structure and decays, and even makes some rather nice predictions for relative decay rates. The strong interaction conserves a number of quantities, some of which are listed here. B Baryon number. Q Electric charge. J Angular momentum. S Strangeness. I Strong isospin. P Parity. C Charge conjugation. G G-parity.

149 B

Q

J

S

1 3

2 3

1 2

0u

d

1 3

1 3

1 2

0 i

-I

2

2

s

1 3

1 3

1 2

Quark u

u

I

Iz

i

+1

2

^2

- 1 0

0

Table 1: Quantum numbers of the quarks. B is baryon number, Q is electric charge, J is the spin, S is strangeness, I is the strong isospin and I , is the projection of I along the quntization axis, (usually defined as z).

Those that are used will be explained as we go along. However, a number of these are carried by the quarks themselves. In table 1 are given the quantum numbers of the three lightest quarks. In the constituent quark model, we treat a meson as a bound quark-antiquark pair, qq, and then draw an analogy to the positronium system, e + e~ to understand what we are seeing. In this picture the q and the q both have spin \. These can combine to either total spin S = 0, or total spin S = 1. -. S = 0 -/=(ti4-2 - 4-ita)

titi S = 1 ^ ( t i | 2 + 4-ita) 4-i4-2

In addition to the total spin, we can have orbital angular momentum L between the qq pair. Then, the L and S can combine to total angular momentum J = L © S, where J = | L — S |,| L — S + l |, • • •, | L + S |. The states can be written in spectroscopic notation as 2 S + 1 L j , and are shown for positronium in table 2. Using the quarks as given in table 1, we are then able to use L, S and J to construct the J P C quantum numbers of the mesons. Let us start with parity, P . Mathematically, State l S0 "Sl 'Pi 3

Po

*Pi 3 P2

s

0 1 0 1 1 1

L 0 0 1 1 1 1

J 0 0 1 0 1 2

P -

c

+ -

+ + + + + + +

jPC

o-+ 1— 1+0++ 1++ 2++

ai

Mesons r) rf K u> K* hi h\ Kr h /o K5 h f'x Ki

02

/2

7T

p

h a0

/2

Kv

Name pseudoscalar vector pseudo-vector scalar axial vector tensor

Table 2: The positronium states as a function of L, S and J. These then correspond to the named mesons of the specified J p c .

parity is a reflection operator, and if the wave functions are eigenstates of the parity operator, then P ( ^ ( r ) ) = V ( - r ) = Vpi>(?).

150 Since applying parity twice should return us to the original state, the eigenvalues of parity, rjp can only be ± 1 . We can normally separate ip into a radial and an angular piece,

V(f)=HWim(M). In this case, the operation of parity leaves R unchanged, but transforms the angular piece to Yim(ir — 6,4> + ir), and it can be shown that: Ylm(n - e, = T)c | 7T° >

where r/c = ± 1 . If we imagine a meson built from a quark and its antiquark, say uu, with some total wave function of both its position and spin, $ . ¥(»*5)=R(r)yWM)x(S) The charge conjugation operator acting on this state reverses the meaning of u and u. This has the effect of mapping f which points to the quark into —r so that it continues to point at the quark. Under the same arguments that we used in parity, this leads to a factor (—1) L+1 . This also flips the spin wave functions, leading to a factor of (—1) for the S = 0 case and a factor of ) + 1) for the S = 1 case. This is a factor of (—l) s + 1 , which when combined with the L factor leads to: C(qq) = ( - 1 ) L + S

(3) +

Clearly charged particles cannot be eigenstates of C, C | 7r > = 77 | 7r~ >. However, if we were to apply the C operator followed by a rotation in isospin, R = exp(i7rJr2) such that | / , Iz >—>| / , —Iz >, then charged particles could be eigenstates of this operator. We define the G parity operator as G = C R , and from this it is easy to show that for a qq system, G = C • (-1) 1 . These then lead to the following formulas. J = L0 S

(4)

P = (-1)L+1 C = (-1)L+S G = (-1)L+S+I

(5) (6) (7)

151 Using these relationships to build up possible J P C ' s for mesons, we find that the following numbers are allowed: 0 - + , 0 + + , 1—, 1+-, 1—, 2 ~ , 2-+, 2 + + , 3 — , 3 + " , 3—, • • • and looking carefully at these, we find that there is a sequence of J P C ' s which are not allowed for a simple qq system. 0-,0+-,l-+,2+-,3-+,--These latter quantum numbers are known as explicitly exotic quantum numbers. If a state with these quantum numbers is found, we know that it must be something other than a normal, qq meson. Following the positronium analogy as in table 2, we can now assign the J p ( c ' quantum numbers to the listed atomic states. In the case of mesons, we have three quarks, u, d and s which can be combined with three antiquarks. This leads to nine possible qq combinations with the same J p ( c ) , rather than the one positronium state. If we now assume that the three quarks are flavor symmetric, then we can use the SU(3)-flavor group to build up the nominal nine mesons, (a nonet). 3®3=188 The nine members of the nonet are going to be broken into two groups, eight members of an octet, | 8 > and a single member of a singlet | 1 >. Under the SU(3) flavor assumption, all the members of the octet have the same basic coupling constants to similar reactions, while the singlet member could have a different coupling. The nominal qq combinations for the pseudoscalar mesons are shown below. The three 7r's are isospin 1=1, while the K's are all isospin 5. The | 1 > and | 8 > state are isospin 0. K°

K+ 7T°, T}, T)'

7T~

K°

7T+

K~

(ds) (du)

(us) A~(uu — dd)

(sd)

(ud)

(8)

(su)

I 8 > = -^(uu + dd- 2ss) I 1 > = -j$(uu + dd+ ss)

(9)

There is also a well prescribed naming scheme for the mesons as given in [caso98] which is summarized in table 3. This of course leads to an entire zoo of particles, but the name itself gives you all the quantum numbers of the state. If we put all of this together, we obtain an entire expected spectrum of mesons as shown in Fig. 1. Where no state is indicated, the meson has not been observed, while the dark names indicate well established states.

152 Isospin M ud,uu — dd,du 1 = 1 ss,uu + dd 1=0 us,ds /=!

\lJeven)j

1

(L0dd)j

(,-^even ) J

(Lodd)j

Tj

h

pj

aj

r)J,rfj

hj,tij

UJ,4>J

J"-= 0 - , l + , 2 - , - -

Kj

fjJ'j K}

Jp = 0+,l-,2+,-- •

Table 3: Naming Scheme of the light-quark mesons.

Because the SU(3) flavor symmetry is not exact, the | 8 > and | 1 > states discussed above, (equation 9), are not necessarily the physical states. The two isospin zero states can mix to form the observed states. This is usually parametrized using a nonet mixing angle 9n as in equation 10. There is one exception to this rule, which is the pseudoscalar mesons. For historical reasons, the r/ and rj' are interchanged with respect to all the other mesons, (see equation 11). (f\_(

cos0n sin0 n \ / | 8 > \

{f')-\-smOna»Oa)\\l>) (i\_(

cos9p sin0 p \ / | 8 > \

(W)

. .

If we put the qq content from 9 into the mixing equation, 10, we arrive at the quark content of the physical mesons. For the choice of 6n = 35.26°, (cos#„ = «/|, sin 9n = J j ) , we find that the physical states have the quark content as in equation 12. This is known as ideal mixing, and separates the s quark from the lighter u and d quarks.

±(uu + dd)\

(12)

* ) " ( * Finally, it is possible to use these SU(3) wave functions to predict mass relations between members of a meson nonet. For a pure nonet, one can derive a generalized linear mass formula, (equation 13). This formula is useful in predicting the masses of nonet members, and also verifying that a set of states can actually form a nonet. (m,f + mji){AmK — rna) — 3mfmp = 8m2K — 8m^m a + 3m^

(13)

In addition to the linear mass formula, it is also possible to predict the nonet mixing angle, 9n purely from the masses. Equation 14 can be used to determine the mixing angles, and when applied to three well established nonets, we find the mixing angles given in table 4. What is particularly interesting is that the three nonets are all reasonably close to ideally mixed. It appears that in many situation, nature wants to separate the light quarks, (uu and dd) from the heavier s-quarks, (ss). In fact, there are only two clear situation where this angle appears not to be ideally mixed. The ground state pseudoscalar mesons, where other effects are important, and the scalar

153

mesons, where a glueball may be mixed into the nonet. If there are other nonets which are not ideally mixed is an open and important question. ,

, .

3m// — ArriK + m0

t a n ' 0„ = •—!•

ArriK — ma — 3m/ jPU

1— 2++ 3—

a f f K On p(770) w(782) ^(1020) iiT*(892) 36.6° a2(1320) /2(1270) $(1525) tf|(1430) 29.3° ,03(1690) w3(1670) ^(1850) tf3*(1780) 31.0°

Table 4: Mixing Angles for well established nonets as computed using equation 14.

(14)

154

Light Quark Mesons (2S+1)

^

PC 3

H.

U U

a.

f," U:

K.:

'H.

2.5 GeV/c2

*.

'G, ft-:'

iftPs

'Gt 2.3 GeV/c2 3

F.

% 3

°«.

K ' * : • "

»-" :

25F4

2% 2%

KJ::

F2

«».::•

«:>

2'F 3

'F,

2.0 GeV/c2

U

«>: *»

ft

3

D2

3

D, ft."

'P,

"t:

TV 1 .

P*

3

U

K,

P, Pi

fi

Po o»:

•f.

u a

:K.- 1 " K. 0 "

3

h. h,' K 1.3 GeV/c2

•p, b,

p

£d

f

TV

n

V

0.6 GeV/c'

K K

2"

fi:

Oj

•1~

3~ 2" 2 3 D, Pi : 1~ 2'D 2 " i KV 2"* 2.1 GeV/c2 2>D2

K\:: 1 " _ 1.' K V 2 *

1.7 GeV/c2 3

2 3 D 3 P>:

KV 3 " KV 2"

25P2

fi

My

*

*

•

2 3 P, o, 2 3 P„

f.::

.«•••

2'P,

1.7GeV/c2

23S, 2'S„

1~ O"* iJGeV/c

Oj

K'i,

1

3 3 S, P.: 3'S 0 K : :t? : 1.8GeV/c'

1~

".

Figure 1: The expected meson spectrum showing the 2 S + 1 L j representation, the J P C of the nonet, and the names of the states. Along the vertical axis are plotted nonets for increasing values of L, while along the horizontal are plotted radial excitations. The average masses are indicated under the boxes. Dark names indicate well established states, while the lighter names are tentative assignments. All other states have not yet been observed.

155 3

D e c a y s of M e s o n s

While the quark model does not make absolute decay predictions, we can use the conservation laws of the strong interaction to determine if a decay is possible. We can also use the SU(3) flavor symmetry to make predictions for relative strengths of decays. We will first take up the use of conservation laws. Consider the decay a£ -¥ T)n°. The a2 has ( I G ) J P C = (l-)2++, the TT has ( I G ) J P C = ( l - ) 0 " + and the r] has (IG)JPC

=

( 0 +)o-+.

a2 -v TTjT ( r ) 2 + + -j- (o + )rr+ © ( r ) f r + >

«

'

The only way for this reaction to conserve angular momentum, J, is to have Ln7t = 2. Under this assumption we can now check the remaining quantum numbers: P : (+1) = ( - 1 ) ( - 1 ) ( - 1 ) 2 = (+1) — Parity is OK. C : (+1) = (—1)(—1) — Charge conjugation is OK. G : (-1) = ( + 1 ) ( - 1 ) — G-Parity is OK. I :
h,h,hz,hz

> = < 1,0 | 1,0,0,0 > = 1 — Isospin is OK.

The reaction is not prevented by any conservation laws. We now look at a second reaction involving an / i decay to two 7r°'s. The f\ has ( I G ) J P C = ( 0 + ) l + + , and both pions are (1~)0 _ + states. h -> 7r°7r° ( 0 + ) l + + -»• ( l - ) 0 - + © ( l - ) 0 " + The only way for this reaction to conserve angular momentum, J, is to have Lnw = 1. Under this assumption we can now check the remaining quantum numbers: P

:

(+1) = ( _ i ) ( _ i ) ( _ i ) i = ( _ i ) — Parity fails.

C : (+1) = (—1)(—1) — Charge conjugation is OK. G : (+1) = ( - 1 ) ( - 1 ) — G-Parity is OK. I : < I, Iz | h, I2, hz, hz >=< 0,0 j 1,1,0,0 > = ^ — Isospin is OK. This reaction is prevented by parity conservation. These exercises can tell us if a particular reaction is allowed. However they don't tell us anything about the rate of the reaction. In order to try and say more, we will invoke our SU(3) flavor symmetry. Under this symmetry, all members of a particular representation should have the same decay rates, modulo some sort of SU(3) ClebschGordon coefficients. 8®8 = 27©10©10©8©8©1

156 In particular, there is one coupling constant for each type of allowed SU(3) transition. 8>-> 8>-> 1 >-> 1 >->

8> 8> 8> 1 >

|8>

9T

|1> |8>

ffl8 9i

U>

9u

Allowed under SU(3)

(15)

We will use the SU(3) flavor symmetry to compute decay amplitudes, 7. However, in order to compare to measured branching fractions, we need to turn these into decay rates, T as given in equation 16. r = 7 2 -/i(
(16)

If we consider the reaction A —• BC as shown in Fig 2, then the quantity q is the momentum of B and C as seen in the rest frame of A. (mB + mc)2){m2A

sftnK

- (mB -

mc)2)

2mA q is related to the available phase-space via p = 2q/m. In addition, there is an angular momentum barrier factor /r,(g) which depends on the relative L between B and C, and their momentum q. For small q, we expect this to scale like q2L. An empirical form for this factor is given as in equation 17 where f3 ~ 0.4to0.5GeV/c. •• q2L

h(Q)

exp

(17)

"8/3 2

The form of //,(?) is shown in Fig. 2 for L = 0,1,2. A rule of thumb is that a decay needs about 200 MeV/c of momentum for each unit of L. In order to use these, we need the SU(3) Clebsch-Gordon coefficients. The ones which are applicable to meson decays are given as follows. |1>-4|8>®|8> ( % ) - > ( (K+, K°)K (7T+, 7r°, ir-)ir°

W

(K~,

-1(23-1-2)* >® |8>

|8>

(K\

(

KIT KK

KT)TTK 7T7T 7J7T 7T77

KKnn \ •nK f

JJK\ KK

7777 KK qK Kir Kr)J

9 -1 -9 - 1 \ 6 0 4 4-6 2-12-4 -2 ^(20 V 9 -1-9-1/

K°))

157 1 0.9

1

^Tie^

0.9 0.8

0.8 0.7

S °-6

^

-""0.5 0.4 0.3

L=J/

0.4

/

0.3

r

0.2

0.2

//\-=1

0.1

°0

0.7

•~io.6 "fro-s

T

0.25

0.5

0.75

0.1 0

1

[GeV/c]

q

0.25

0.5 q

0.75 [GeV/c]

1

Figure 2: a The form factor, /z,(g), for the Decay A -> BC as a function of the q of the reaction. b The form factor multiplied by the momentum q for the same decay, 9/1(9). The three curves are for different orbital angular momentum, L and the form factor is given in equation 17.

We can now use these coefficients in conjunction with the four decay constants to compute decay rates. As an example, let us consider the decay / -> inr. We will ignore the J of the / for the moment, and only assume that the reaction is allowed by our basic conservation laws, (this is true for J even). We also need to break the / into its octet, / 8 , and singlet, /1 pieces. Recall that / = sin#/ 8 + cos6f\. Prom this, we can write the amplitude for our decay as: 7(/ ->7r7r) = 7 ([sin 0f& + cos 9f\] —• irn) • sm0y(f& —> 7T7r) + cos9y(fi —> nn)

j(f

-¥

7T7T) •

-3 T sin0 + W-0iCos0

(18)

Similarly, we can examine the decay / —• KK. The only difference is the ClebschGordon coefficients, which when putting it together yields equation 19. 7(/ -¥ KK) = J—gTsin6

+ -gxcos0

(19)

Similarly, we can examine the decays of-/' by writing it in terms of / x and /g as in equation 10. For the TTTT decay, this yields equation 20. j(f

—> TTTT) ••

4

gTcos0-

W-5! sinfl

(20)

158 If we had considered ideal mixing, (see equation 12), then we could compute the rate for ss into mr as given in equation 21. j(ss

•

T):

Vl 5r vl

(21)

At this point we want to invoke something called the Zweig rule, or OZI suppression. This basically says that diagrams that destroy the initial quark and antiquark are strongly suppressed with respect to those that do not. In Fig. 3 are shown examples of these, where a shows the initial quarks destroyed and b shows them preserved. This observation comes from cc decays where the \P states below the open charm threshold have very narrow widths, but after the threshold for D production is crossed, the widths become much larger. Our reaction ss —• 7T7T is an example of one of the suppressed diagrams, and we are going to set the rate for this to zero. Doing this, equation 21 gives us that g\ = —-jt9r-

(a)

Forbidden

Allowed

Figure 3: a) OZI forbidden and b OZI allowed decays. Two other reactions, 22 and 23, should also be OZI suppressed and we will set their rates to be zero. [uu/dd] -> [ss] -n

(22)

[uu/dd] ->• [ss] [ss]

(23)

For reaction 22, an example of such a decay is a —> [ss] n, where we find the decay rate as: 7(0 -> [ss]n) = -J-_7(a8

-> Tj6Ka)+J-y(as

-» Tftfl-g^.

Setting this rate equal to 0 yields the relation: gi& — \QT- Doing the same thing for reaction 23, we can write that the isospin 0 light quark mixture is:

159 We then examine its decay amplitude to two ss pairs as follows. 0= 7

0

-ft +

0 = ,/-

[Jim - f^mj \Jlvs - Jl Vi

J/8 + \/|/l

g7(/8 -> mm) + g7(/s -> vim) — ^ ^ ~> ^m) 2 1 o7(/i -> W s ) + - 7 ( / i ~> m»?i) 2 , 1 , ( 9T

3 "W

2^2

2\/2 3 ~ 7 ( ^ "*• 7?1% )

2 . 1 .

9w

(

~ ~T .

3 ~7I

1 )5l +

3511

Which yields that <7n = — Aggr- We now have sufficient information to express all the decay amplitudes of a given nonet in terms of one unknown decay constant, grThese are given as a function of both the nonet mixing angle, 0 and the pseudoscalar angle, Op. If we take dp = —17°, then we can plot 7 2 as a function of 0\ these are shown in Fig. 4. 7 ( / —• 7T7T) = - y - (sin# + \/2cos0J gr 7 ( / ' —> 7T7r) = —»/- (cos0 — \/2sini9j py 7(/

7

-> M ) = J ^ (sine -

2>/5COB0) 5 T

( / ' -> M ) = J ^ - (cos0 + 2\/2sin0)

5T

7 ( / -> w ) = y - { \/2 cos 0 - sin 9 (cos 2 0 P + 2 v^2 cos 0 P sin 0P} } # r 7 ( / ' ->W) = J - { - c o s 0 ( c o s 2 0 P + 2\/2cos0Psin0P) -

\/2smO}gT

7 ( / ->• W ) = —7= { 2 ^ 0 0 5 0 s i n 2 0 P + sin0 (2\/2cos20 P - sin20 P )} gT 7 l 7 ' ->• W ) = —;p { c o s # (2-y/2cos20P - sin20 P ) - 2A/2sin0sin20 P } gT

This simple prediction does a remarkably good job in describing the tensor mesons, JPC = 2 + + . From [caso98] we find the decay rates to two pseudoscalars as given in table 5. In addition, using the masses as given in table 4 and the mass formula from equation 14, we find an optimum mixing angle of (29.3 ± 1.6)°. We can also use the decay information to fit for the mixing angle as well. In Figure 5 is shown the results of such a fit, where the optimum value comes out as 32.8°. This is in remarkably good agreement with the simple mass prediction.

160

Figure 4: Decay amplitudes, 7 2 , as a function of nonet mixing angle 0. (a) is for / decays while (b) is for / ' decays.

State /2(1270)

Decay —¥ 7T7T

-+KK -*rm /J(1525)

—> 7T7T

-+KK -*nn a2(1320)

—> 7/7T

—> rj'ir

-+KK

q [GeV/c] Rate 0.622 0.846 ± 0.02 0.046 ± 0.004 0.327 0.402 0.0045 ± 0.0015 0.0082 ± 0.0015 0.749 0.888 ± 0.031 0.580 0.531 0.103 ±0.031 0.145 ±0.012 0.535 0.0053 ± 0.0009 0.287 0.049 ± 0.008 0.437

9/2(9)

72

0.118 11.54 ±0.27 0.0236 4.84 ± 0.42 0.0107 1.29 ±0.43 0.223 0.049 ± 0.009 0.0919 16.65 ± 0.58 0.0668 2.90 ± 0.87 0.0686 3.95 ± 0.33 0.0064 2.86 ± 0.49 0.0324 3.46 ± 0.56

Table 5: Experimental decay rates for the tensor mesons decaying to pairs of pseudoscalar mesons. The factor 7 2 is the rate corrected for both phase space and barrier factors as in equation 16.

161

o t 26

i 28 6

30

32

34 36 38 [Degrees]

i 40

Figure 5: The x 2 as a function of nonet mixing angle for the tensor mesons. The combined decays optimizes for a mixing angle of about 32.8°.

4

Exotic Mesons

If it were just for normal qq mesons, one could argue that there is not really a lot to do in light-quark meson spectroscopy. The quark model does a nice job of explaining things, and when extended to the flux tube model, with the 3 P 0 model for decays, a very nice picture appears. In fact, a picture which is quite consistent with known meson phenomenology. To this, we can add lattice QCD calculations, and the picture improves. A good picture of masses and decays emerges which is reasonably consistent with data. So, why are we continuing to study this? What is there that we can still learn? The quark model has no confinement and in fact we don't even need gluons in the picture. However, things like the lattice QCD or flux tube model say that glue has an extremely important role in QCD. In fact when any model with glue makes predictions about the meson spectrum, a consistent prediction of gluonic excitations emerges. Not only do we get the normal qq spectrum, but we get additional states which directly involve the gluons. Ones involving only gluons are called glueballs, while those that involve gluonic excitations of a qq system are known as hybrids. 4-1

Glueballs

Naively what is going on? The gluons carry the color charges of QCD, in fact a gluon carries both a color and an anti-color, and are members of an SU(3)-color octet. This leads to eight different gluons. Because these gluons carry color charge, it is possible for them to bind into color singlet objects. In the bag model picture, the simplest glueballs are either two or three gluons confined together as shown in Fig. 6. Currently, the best predictions for the glueball spectrum comes from the lattice. A

162

RG

GR Figure 6: Two and three gluons bound into color singlet glueballs.

recent calculation using and anisotropic lattice [morningstar99] is shown in Fig 7. From this figure, we see that the lightest glueball is expected to have J P C = 0 + + , followed by a 2 + + state and then a 0~+ state. Unfortunately, all of these quantum numbers are also the quantum numbers of normal mesons. In fact the lightest glueball states with exotic or non-qq quantum numbers are the 2 + _ near 4GeV/c2 and the 0+~ state near 4.5GeV/c2. Both well beyond the mass regime that we consider for light-quark mesons. This means that as far as quantum numbers go, the lightest glueballs will appear to be / 0 ) ft and rj states. The main difference is that we expect one additional state beyond the nominal nonets. Worse still, as we have already seen mixing between the two iso-singlet states in a nonet, we should expect that the glueball will also mix into these states as well. If we first consider the scalar glueball, ( J p c = 0 ++ ), we find that the lattice prediction for the pure glueball state is m = (1.6 ± 0.3) GeV/c2. Unfortunately, this is extremely close to the nonet of scalar mesons, a0(1450), /0(1370), and ^(1430). This means that it is going to be difficult to establish such a state as a glueball. We will first need to find a 10'th scalar state in the same mass regime. We can also look at the naive predictions for the glueball decay to pairs of pseudoscalar mesons. Under the assumption that the glueball coupling to all pairs of octet mesons are the same, then we obtain that the following relationships. -y(G-urir)

j(G^r]8r)g)

=

g(d-

= -gg1\l-

7 (G

- • KK) = g\

7 (G

-+ KK) = -g{

The singlet glueball can also couple to two singlet rfs as follows: 7 ( G -+ T/iJfc) = g9n.

163

Figure 7: The predicted gluebali spectrum from a lattice calculation [morningstar99].

164 We can now expand the possible pairs of physical 77 and rf states in terms of | % > and | % > states as follows. | 77 > | r)' > = (cosOp 17?8 > -smOp

I rji >) • (sin0 P | T?8 > + c o s 0 P | % >)

= sin dp cos 6P{\ % > | % > - | 771 > | 77! >) + (cos2 6P - sin 2 9P) \ % > | 771 > 177 > | 77 > = (cos0p 1778 > — sindp 1771 >) • (cos0p | Tjs > —smOP 1771 >) = cos20P 17?8 > | 778 > + s i n 2 0 p 1771 > | 771 > - 2 s i n 0 p c o s 0 p 1778 > | 771 > 177' > | 77' > = (sin#p 1778 > +cos0p 177! >) • (sinflp 1771 > -costfp 177! >) = cos 2 0p I 7?i > | 7?i > + sin 2 0p) I 778 > | 77s > +2sin0pcos0p | 778 > | 771 > The glueball state is an SU(3) singlet, so it can only couple to | 771 > | 771 > and I V8 > | V& >• This leads to the following rates for glueball decay into the physical states:

j(G -> 777?) = -g{ cos2 9p J- + g9u sin2 Op •y(G -^ 7777') = sm6Pcos6p{~J-gl y(G -»• 77'77') = ff?i cos2 BP - g{ sin2

- g9n) 9P^

If we assume that gd = — y gflf, which is equivalent to: 7(| G >->| ?7i77i >) = 7(| G > - H 778778 >) then we can can simplify the above rates. This assumption is really quite reasonable, and one would need good reasons for not choosing this, (e.g. the 77' has a large glueball component). Simplifying, we obtain that rate for 7777' is zero for any choice of Op, and that that rates for 7777 and 77'77' are the same for any choice of Op. Putting all of this together, we obtain the predictions in 24 as given in [close88]. These are what are typically quoted as the expected flavor independent glueball decays. T(G -+irir:KK:rir):

7777': 77V) = 3 : 4 : 1 : 0 : 1

(24)

There is also one lattice calculation which computes glueball decays [sexton95]. They find the mass of the 0 + + glueball to be 1.740 ± 0.071 GeV/c 2 and a total width to pairs of pseudoscalar mesons of about 0.1 GeV/c 2 . So we expect to find a 0 + + glueball near 1.6 GeV/c 2 , which is unfortunately rather close to the normal scalar mesons. Where should we look for this object? There are certain production reactions which are considered glue rich. Such reactions have a

165

Figure 8: a) J/V> Decay, (b) 77 fusion.

lot of glue, and are considered prime sources of glueballs. There are other reactions which are glue poor. In these, some other production mechanism is at play which would suppress the pure glue signal. The best glue rich reaction is considered to be radiative J/ip decays, (Fig. 8a). In this reaction the only mechanism to get from the initial cc state to the final state is through intermediate gluons. Typical radiative rates are on the order of 10~4 to 10~ 3 with the total radiative width being on order of 6% of all decays. Current existing event samples consist of a few million events, which leads to at most a few thousand events in any one channel. Currently, the only running experiment is BES in Beijing. Their plans are to accumulate on the order of 107 Jftp's within the next couple of years. However, to make significant progress would require a sample of 108 to 109 events, and would require the construction of a r-charm factory. Somewhat related to J/ip decays is the two-photon fusion process, 7 7 —>• X and is considered to be glue poor. The photons only couple to electric charge, of which the gluons have none. Both of these reactions are done at e+e~~ machines, so historically they are reactions that could be looked at in the same detector. The basic reaction is shown in Fig. 8b. The idea is that the q2 of a radiated photon is q2 =

-4EbeamEiSm2-±.

A real photon has q2 of zero, so by selecting $i as close to zero as possible, the process involves two real photons. This is done by n o t seeing the scattered electrons. Currently, there is some effort in the LEP experiments as well as in CLEO to look at two-photon production of mesons. The most recent review [cooper88] is fairly old. The two-photon production couples to the electric charge of a meson, while the radiative J/ip decays couple to the color charge of a meson. One can define a quantity known as the stickiness, (equation 25), which is essentially the ratio of the color charge to the electric charge of a state [chanowitz84]. S = N-

mx (ml - ml)l2mi,

r(tp -> 7 X ) ' T{X -+ 77)

(25)

166

This quantity is normalized to be one for the /2(1270) which is believed to be a pure qq state. One would expect that S would be large for states which are gluonic in nature.

Figure 9: Proton-antiproton annihilation into (a)meson pairs and (b) gluonic final states.

A second reaction which is considered glue-rich is proton-antiproton annihilation. Typical annihilation to mesons proceeds via a quark rearrangement as shown in Fig. 9a. However, as there is expected to be a lot of gluons, a reaction such as Fig. 9b. is also expected to play an important role. A large amount of data has been recently accumulated at the Low Energy Antiproton Ring, LEAR, at CERN. Particularly with the Crystal Barrel experiment, see the very recent review by Amsler [amsler98]. For pp annihilations at rest, which experimentally are a very good source of scalars, one is limited to i/s = 2mp - m„, or about 1.74GeV/c2. Afinalplace which is considered to be glue rich are in central production reactions, (see Fig. 10a). Essentially the two initial state particles leave the reaction as the same state, while the meson, X, is created in the exchange of two virtual particles of momentum transfer q\ and q2. For large enough energies, this reaction tends to be dominated by diffractive processes, which in turn appear to be dominated by so called pomeron exchange. The nature of the pomeron is not clear, but it is believed to have a significant gluonic nature. This means that a double pomeron exchange would be a very good place to look for gluonic excitations. Finally, another nominally glue poor reaction is photoproduction, (see Fig. 10b). This is suppressed due the fact that the photon couples to electric charge. However, it may not be as suppressed as the 77 reactions. 4.2 Hybrids Not only can we consider purely glue states, but one could imagine the gluons contributing directly to the quantum numbers of the system, valance glue. These states, (qqg) are know as hybrid mesons or hybrids. The basic quantum numbers are slightly more difficult to get at, as we do not think of the g as a simple 1~ object, but rather as some coherent excitation of the gluon fields. One particular model, the flux

167

Figure 10: (a)Central Production, (b) Photo Production.

Figure 11: Hybrid Meson

tube model [isgur85], describes mesons as qq states bound together with a flux-tube of glue. The normal mesons have the flux in its ground state, but it is possible to excite oscillations in the flux-tube. In the model, the flux-tube carries angular momentum, m, which then leads to specific C P predictions. For m — 0, C P = (—l) s+1 , while for the first excited states, m = 1, CP = (—l)5. The excitations are then built on top of the S-wave mesons, (L = 0), where the total spin can be either S = 0 or S = 1. These lead to the expected quantum numbers for m = 0 and m = 1; in this picture the m = 0 are the normal quark-model mesons, while the m = 1 are the lowest lying hybrid mesons.

(m = 0)

S

oo-+\

s~°

(_1)£+1(_1)S+L

=

(_!) s+i

Normal Mesons

= oo-+\ + + = 1 1 " } 0-+, 0 - , l - + , l + - , 2 - + , 2 We also note that for the two S = 0 nonets in the quark model, we have eight hybrid nonets, (72 new mesons!), and that three of the eight nonets (indicated in bold) have non-qq, or exotic quantum numbers. In this picture, these hybrids are no different than the excitations of the qq states, we just need to consider Orbital, Radial, and Gluonic excitations as the natural degrees of freedom. In Fig 12 are shown the approximate expectations for the quark model states, qq, the glueballs, the lightest hybrids, and where some two-meson thresholds are.

168 Because these hybrid excitations are perceived as normal excitation, the natural places to look for these are in reactions which populate the excited qq spectrum. These states should in principal be produced as strongly as other states. However, there are several difficulties in observing them. 1. Most of these states are predicted to be very broad. 2. Most of these states have non-exotic quantum numbers. 3. The expected decay modes of these states are complicated. In particular, point 3 can be used as a guideline when looking for these states. Almost all models of hybrid mesons predict that the ground state ones will not decay to identical pairs of mesons, and that the decays to an (L = 0)(L = 1) pair is the favored decay mode. Essentially, the one unit of angular momentum in the flux-tube has to go into internal orbital angular momentum of a qq pair. Examples of such decays are given below. The important thing to note is the rather large number of final state pions — ranging from four to six. This makes it difficult to fully reconstruct such states. H -¥ /i(1285)7r (/i -» rprx) -»• (-Mnr)irMt H -> 6i(1235)7r (&! -» um) ->• (i:+TT~n0)i:ir H -¥ a2(1235)7r (a 2 -)• pw) -* (Tnr)nir

169

•qq Mesons

o

x

s f

s CO -OL

CO CD W

O

6"

3

:&?•::•.

:5^V.

w

CD O ;3**.,::

c_

•3*-- : -:

CD CO

•i"=3y

:-2* :it'

2" L=0

Lit.: L=2

2«

&

1" 0" 1*L=t

KK pp/coor

KT< E3T

I-

KK

cr L=0

Figure 12: The expected mesons spectrum. For the qq

170 5

Partial Wave Analysis

Partial wave analysis is a technique which attempts to fit the production and subsequent decay of a meson by examining not only its mass distribution, but also all angular distributions of the system. The technique is very powerful, but requires rather large statistics to be able to identify things. 5.1

The Resonance Shape

Strongly decaying particles have lifetimes on the order of 10~ 23 seconds which, through the uncertainty principle, leads to widths on the order of 100MeV/c 2 . If we had an isolated state of mass mo and width r 0 ) then we would describe the resonance in terms of a Breit-Wigner amplitude as in 26. r

BW(m) =

° / 2 „ .„

(26)

This form is the non-relativistic form, and is valid when To < < mo, and the mass mo is far from the threshold for the decay. This can be extended to the so called relativistic form as given in 27. In this form, the resonance shape depends on the relative angular momentum, L, with which the resonance is produced. In addition, the width, T(m) depends on the orbital angular momentum / between the daughter products, as well as phase space available to them. BWi(m) =

, "f(m) _, , mj) — mz — tmol (m)

rH =r „ ^ ^ mp0

(27)

(28)

Ff(po)

The angular momentum barrier factors are computed as a function of z = (pR = 197MeV/c), and are given as follows:

(P/PR)2,

F0{p) = 1

13z2 (z - 3) 2 + 9z F3(P) -

277z3 \ z(z - 15)2 + 9(2z - 5) 2

The shapes are quite similar if both the width of the resonance and the mass of its daughter particles are small compared to its mass. In Fig. 13 are shown a comparison of equation 26 and 27 for several different situations. In a is shown the normal p mesons, where mo = 0.770 GeV/c 2 , T 0 = 0.150 GeV/c 2 , and both daughter particles

171

0.4

0.6

0.8

Mass

[GeV/c2]

0.4

0.6

0.8

Mass

1

1.2-

[GeV/c2]

1.4

0.4

0.6

0.8

Mass

1

1.2,

[GeV/c2]

1.4

Figure 13: A comparison of the relativistic and non relativistic resonance shapes for a p meson under different assumptions, a mf = 0.770GeV/c 2 , T p = 0.150GeV/c 2 and mx = m 2 = 0.140GeV/c 2 . b m , = 0.770 GeV/c 2 , r , = 0.150 GeV/c 2 and mx = m 2 = 0.350 GeV/c 2 . c m„ = 0.770 GeV/c 2 , r„ = 0.350GeV/c 2 and m 1 = m 2 = 0.140GeV/c 2 .

are pions with mass m = 0.140 GeV/c 2 . In b, we have let the daughter particles have mass m = 0.350 GeV/c 2 , and one can easily see the threshold effect of the two daughters. Finally, in c, the width has been changed to r 0 = 0.350 GeV/c 2 . A second issue is what are quoted as the mass and width of a resonance? Usually, one will quote a complex value of m, (m = IUR — iTR/2), such that the amplitude has a pole at that value. For the case of 26, it is easy to see that if TRR = m0 and Tfi = r 0 , that there is indeed a pole. However, in the relativistic form in 27, it is rather obvious that m 0 and F 0 will not produce a pole, though they are not very far off for many cases. There are also other things which are quoted. The mass which makes the amplitude purely complex is one such possibility. Another is the mass that yields the maximum rate of change in the amplitude. Because of this, the values which are quoted in literature tend to have a wide variety of meanings. In my mind, the most logical value to quote is the so called T-matrix poles. In scattering theory, one considers an S-matrix which takes an initial state to a final state, S/i = < / | S | i > such that the S matrix is unitary, S 5 f = / . We can rewrite the 5 in terms of the T-matrix as S — I + 2iT where the T matrix can be written in terms of a scattering phase as T = e's sin 6. This is discussed in more detail in section 5.3. 5.2

The Angular

Distributions

By fitting Breit-Wigner forms to the data, we can in principle learn the masses and widths of a state. However, we are unable to determine the J P C of a state. In order to do this, we have to look at the angular distributions of the decay products with respect to some initial state. It is the fitting of these distributions which is referred to as a partial wave analysis. There are two forms by which one normally constructs

172

• 160

Non Relativists Relatlvlstic

-

[Degree

co o ro oe e

,-,140 W

-

®

60

-

^

(b)

40 20 -0.2

0

).3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

0.2

Real Part

Mass

[GeV/c2]

Figure 14: a The imaginary versus the real part of the Breit-Wigner amplitude for the p(770). b The phase S as a function of mass for the p(770). Non relativistic corresponds to equation 26 while relativistic corresponds to equation 27.

these angular distributions. The Zemach Tensors [zemach64] and the Helicity formalism [jacobs59]. They both work equally well for many problems, but we will only discuss the latter in this work. Let us consider the decay of particle A with mass m^ and spin JA into two daughter particles, B and C. These have masses mB and m,c, and spins SB and Sc respectively. We will look at the system from the rest frame of A. If there is no preferred direction in this system, then we are free to choose the direction of the z axis to be along the direction of one of the daughter particles. (Actually, even if there is some preferred direction, we can always rotate the system such that the z axis is aligned so.) We will also allow the two daughter particles to have relative orbital angular momentum L. A(mA, J A) -> B(mB, SB) + C(mc, Sc) Using z as our quantization axis, then there are 13\ + 1 initial states and (25B +

AGW C(SC)

•

• B(SB)

* z

l)(25c7 +1) final states possible. Our job is to consider the transitions from the initial to the final states. If we now write these in terms of momentum helicity states, then the final states can be referenced as | p, \B, \ c > where p is the momentum of particle B, and the A's are the helicities of the two daughters. The helicity is defined as the projection of the total spin, J, along the direction of the particle. Using the fact that

173

L and p are normal to each other, we arrive at: A = f ^ = [ ^ © f ^ = 0 + ms, (29) \p\ \P\ \P\ where ms is just the projection of the particles spin along the z axis. This means that \B = ms(B) and Ac = —ms(C). Now we can generalize this to B being emitted in some arbitrary direction, (9, ), rather than simply along the z axis. We can get from this new frame back to the frame where B is moving along the z axis via a rotation in three space, R(9,<j>) — Ry2(9)Rzl(). We may recall that when doing classical rotations, we used three Euler angles to accomplish the rotation of a reference frame. The third would be a rotation about the direction of particle B, which in the case of a spin-less particle is not needed, (only two angles axe needed to rotate a vector). It should however be noted that if the final state particle has non-zero spin, it is necessary to perform a third rotation to align the polarization vectors. Rotations can be expressed as a unitary operator, U(0,(j>) = e'^e1,7"6, which yields the following: (m'\U\m)=

eim'*(m' | eiJ'e | m)

The d-functions are elements of a rotation matrix, and can be looked up in several sources [caso98]. For a given j , there are relations between the elements given as in equation 30. <&m = £™»' = (-l) m - m '«*L.' (30) These can also be written in terms of the V matrices, VJmml(e>^) = eim^dJmm,(0)

(31)

So, applying these to our particle, we can rotate our helicity state from system 3 where B is moving along the z axis to system 1 where it is moving in some direction

(M). | p, 9, cj>, Xb, Ae, M) = VJMX(-9, -) | p, \b, Ac) We are now interested in the general transition amplitude from some initial state | J, M') to our final state. This can be written as a transition matrix, / as follows. /A.,A2,M(0, <j>) = (p, 6, , Ai, Aa, M | T | M')

= Z^A(-0,-0) = DlM(-0,~)(\i\i\T\M'>.

(32)

(33) (34)

Ai,Aj

This is a matrix with (2J+1) columns corresponding to the initial states, and (2Si + 1)(252 +1) rows corresponding to the final states. The TXu\2 are formed by summing over all possible / and s values with an unknown complex coefficient, a is for each one. TxlM = Ea"(J>X Is

I J,s,0,A)<8A | SUS2, \u-\2)

(35)

174

This is non-zero only if A = Ai — A2. If the initial state now has a density matrix, pit ( (2J + 1) x (2J + 1)), then the density matrix of the final state, pf is computed as: Pf = fPiP The angular distribution of the final state is then obtained by taking the trace of pf, wD(tf,fl = Tr(p,) = 1V(/ f t /t).

(36)

What we have done allows us to handle simple decays into two daughter particles, but it can easily be extended to more complicated decay chains. Let us consider the decay: A-+[B-> BiB2] [C -»• CiC2]. We can write a transition matrix, / for each of the three decays, f(A—> BC), f(B —> BiB2) and f(C —*• C1C2). The total transition matrix, fa can then be written as a tensor product of the individual transition matrices. fT =

{f(B)®f(C)]®f(A) /A(BI)A(B2),A(B) ®

A(B),A(C)

/A(CI)A(C2),A(C)

/A(B)A(C),A(/1)

Bc,
»,*

Let us now look at a simple example. We will take proton-antiproton annihilation from an initial atomic 1So state into pn. The p will then decay into two pions. The decay chain is as shown below, we see that in order to conserve angular momentum, the orbital angular momentum between the p and the 7r must be L = 1. 1

So(pp)-+

p± ^

0"+->lT, L=l

->!r±ir°

We now identify A as the pp state, B as the p* and C as the 7rT. We have J A = 0, SB = 1, Sc = 0, and 5 = 1 . This means that our transition matrix will have one column (corresponding to J) and three rows, (corresponding to (2SB + l)(25c + 1))The transition elements are now given as in equation 35. We see that since J = 0, the only non-zero elements will have A = 0, in addition A^ = 1,0, —1 and X^ = 0. From this, we see that Tw = T_i0 = 0 and that Too = (00 I 1100) (10 I 1000) -i/V5

so the amplitude for the transition to pn is

1

175

Next, we need to consider the decay of p -*• 7T7T. Here we have J = 1 for the p, and S = Si = 52 = 0 for the 7r's. Both Ai and A2 are zero, which means that we only A = 0 contributes. This means that L = 1, and leads to: TAlA2 = (1A | 100A) < <0A | 00A! - A2) Too = <10 | 1000) (00 | 0000) = 1

f(p ->• TTTT) = {V^{9P, „) 2>i,(0„ „) 2 ^ ( 0 , , , P))

-^Poo(0p>tfv) = - ^ c o s 0 p

P/ = / r f t / r = 3 cos2 0„ Next, we consider the example 3Si(pp) —¥ p±7rT. Here the initial state has Jpc = 1 , so in order to couple a 1~ and a 01 particle to get total J = 1, we must have L = 1. Similarly, Sp = 1, Sr = 0 BO S = 1. For the transition to pit, we get: T10 <1 + 1 | 110 + 1><1 + 1 | 10 + 10> Too = (10 I 1100)(10 I 1000) = T_10 <1 - 1 j 110 - 1)(1 - 1 I 10 - 10)

"&D\x{0A) -^2>io(M) /A,O,,M

'•

0

0

-72 0 ^

-feDl-AW 0

If there is no preferred direction in the problem, such as the decay of an unpolarized state produced at rest, then we are able to choose the direction of the coordinate axis such that both 0 and are zero. Under this assumption, the matrix simplifies to:

fx,o,M —

0 0 0 0

°7S,

176 The subsequent decay for p —¥ irir can now be expressed as: ( P k C p , P) W P , <S>P) Z»O-I(Op, P))

h = X) fo,O\(0p, <j>p)f\u0m{0, 4>)

(-&>II{0PAP)*

°

0

0

0

Pf = frPifr

'WI)2(8P,^P)0

Pf

oK^-im,^,), wD = Tr(P/) = i [{vme,, h) + {pltfip,, 4>p)} 1 rl "6L2

,„ p +

2

l . •>„ 1 "

= g sin2 0,

/a00N p< = 0 6 0 VOOa, wD = sin2 0P [sin2 <j>p (b sin 2 0 + a cos2 0) + a cos2 0 J 5.5

Putting it all together

We have now looked at both resonance shapes, and the angular distributions of all final state particles. We now want to combine these, to produce a total amplitude as given in equation 37. The sum is over all possible transition amplitudes, M, while the product is over all resonance chains within that amplitude, R.

wD = Tr<

Ml

R

\

*=i \

j=i

/

ikh n MM)

(37)

177

The fk are the transition amplitudes from above, while the BW terms parameterize the resonance shape. The 7* are a priori unknown complex coefficients. As mentioned earlier, relative decay rates are important, and we need to be able to pull these out of the data. Unfortunately, it is not obvious how we are going to do this with what we currently have. The simple Breit Wigner forms provide no natural mechanism for this. We may also have thresholds that occur in the middle of our resonance. One way to treat this is using a Flatte form [flatte76]. Here the <& are related to the partial widths in each of two final states, while the Pi are the phase space available for each final state. b91

T(mod ei ) =

m\-m2

T(mode2) =

ml-m2

- i{p\gl + p-ig2) bjto - i(pig2 + p2g2)

X9? = m0r0

2pi

pi (m) =

, -. p2(m)

2p 2

m Note that at threshold, Pi becomes 0, and Pi goes to zero. When we are under threshold, then pi becomes imaginary. We can actually handle things a bit better by returning to scattering theory, and writing down the transition from some initial state to a final state via an SMatrix, which in turn can be written as a T-matrix. For a good reference on this, see [chung96]. S = I + 1iT The T-matrix describes the transition from the initial to final state, and it can itself be written in terms of a K-matrix, where K = KV K~l = T'1 + iT This can be inverted to yield: T = K(I-

iK)-1 = (/ -

iK)~lK.

From this, it is possible to get to the familiar form for a resonance, T = ei,ssin(5 and K = tan S. If we consider a process where we have several different initial states each with the same quantum numbers, a, and each potentially decaying to many different final

178

states. The states have K-matrix masses ma, and K-matrix widths F'a. The elements of the n by n K-matrix are given as: KiAm) =

£ 7^^aBai{m)Ba.{m)

+ Cih

where the i and j index correspond to decays final states of the states, a. The 7^ are real numbers which are the coupling constants for the initial state a to the final state i, such that Y^lti = 1. The partial width into some final state i, rQ* = 7a j r„. The K-matrix total width, r Q = J2% ra»- Finally, there are barrier factors are given as the ratio:

„ _ MPi) •Dni

—

FL(Pai)

where pa« is evaluated at the K-matrix mass. This formalism now allows us to account for multiple resonances with the same quantum numbers with multiple decay modes. In the case of one resonance with one decay mode, we can easily transform the Kmatrix into a T-matrix: T--

m0r0B2(m)/p(m0) ml — m2 — imoTm

where

^

<"•» = ^ • S i ' ^ ( " ' ,

What we have at this point describes the decay of an already created resonance. In order to be complete, we need to describe the production of this resonance, through some presumably unknown mechanism. One way to do this is with a P-vector, Pj(m).

qw=EA!T^ The 0a are unknown complex production strengths for each K-matrix pole. The production vector can be combined with the subsequent decay to define an F-Vector, T={\-iKp)-lP,

(38)

where the matrix p contains the phase space factors, p(m) = 2pi/m along its diagonal, and zero elsewhere. Multiple resonances with multiple decays can now be handled within this prescription. There is one additional complication in that if the daughter products of the resonance subsequently decay as well, then the F-vector in equation 38 is modified by multiplying by the T-matrices of the daughter's decays. F=[(l-iKp)-1p]j[Ti

(39)

179 In fitting data with these amplitudes, it is natural to use the K-matrix parameters, ma and T'a as the fitting parameters. These quantities should not be quoted as the resonance parameters, rather one should use these to determine the T-matrix, and then search for its poles in the complex energy plane. Depending on how the experiment is performed, it is often possible to decouple the helicity pieces form the resonance parameterization. In these approaches, one uses the helicity forms to define partial waves, and then fits the data to obtain intensities and phases of these waves as a function of the mass of the system. In a subsequent fit, these first distributions are fit with the resonance parameterizations to extract information on the states themselves. In other experiments, everything must be fit at once. In both of these cases, understanding the acceptance of the detector system and having good detector resolution is crucial. In fact, the closer the acceptance is to perfect, the better the measurement will be. It is key in designing detectors for this type of physics that as uniform an acceptance as possible be built into the highest resolution system possible. 6

Overview of The Current D a t a

I now want to proceed with data a results from a number of recent and current experiments. These experiments provide the best evident to date that we have seen observed gluonic excitations. 6.1

The Status of Glueballs

As discussed in section 4.1, the lightest glueball is expected to be a scalar state, J p c = 0 + + with a mass in the range of 1.5 to 1.7GeV/c 2 . In addition, we want to look for this in so called glue-rich reactions, such as J/tfi decays, pp annihilation, and central production. pp Annihilations at Rest The most significant data on pp annihilations has come from the Crystal Barrel experiment running at the Low Energy Antiproton Ring at CERN [aker92]. This experiment has collected a huge statistics of data on pp annihilations both at rest and in flight. A very good review of this experiment can be found in [amsler98]. Here we wish to concentrate on a small subset of that experiment's results. In particular, the reactions pp at rest goes to 7r°7r°7r°, TT°T]T], -K°T]TI' and n°KLKL. All of these final states could be formed via the reaction: pp -» TT°X -»• 7r°M1M2.

in addition, the two identical pseudoscalar mesons can combine to make ( I G ) J P C = (0+)0 + + and ( 0 + ) 2 + + , or fa and / 2 states. Of course there can and will be other things in these data by pairing up other mesons, but the key point is that we can look for /o states decaying to many different final states.

180

The data in these analyses are presented in the form of a Dalitz plot. A three body system from an unpolarized initial state at rest, the final state can be uniquely described by two variables. One possible choice is any pair of invariant masses squared. For the reaction X -¥ a + h + c, there are three possible two-invariant masses, m^, m^ and m*c. These are actually related to each other via ml + ml +ml + ml

[

mlb + mta + mlc

In Fig. 15 are shown the Dalitz plots for pp annihilation at rest into T T W and ?r°TO- While the analyses of these channels involve many intermediate resonances [amsler95a, amsler95c], there is one new state which stands out in both, the /0(1500). This state has a mass of 1505 MeV/c2 and a width of 110 MeV/c2. In later analysis [amsler95b, amsler94a], the /o(1500) has also been observed in the r\rj and KiKi final states, (see Fig. 16). Examining all these data, it is possible to extract many different annihilation-decay rates for the /o(1500) as given in table 6. If we recall equation 16 from section 3, we can convert the numbers in this table into rela-, tive decay amplitudes squared, -y2, which if normalized to the TO mode are given in table 7.

11.25 m a (7tV)

m2(n°ij)

Figure 15: Dalitz plots for \ i annihilation into three pseudoscalar mesons. The left is jr°?r0ir0, and the right is TT°TP).

Decay /o(1500) -»• ww /o(1500) -* KR /o(1500) -+ TO /o(1500) -> TO' /o(1500) -»• 4TT

Rate 0.290 ± 0.035 ± 0.046 ± 0.012 ± 0.617 ±

0.075 0.003 0.013 0.003 0.096

g[GeV/c] 0.740 0.567 0.516 0.0889 0.5

Table 6: Measured branching fractions for pp -4 /o(1500)jr°

The rates in table 7 can now be compared to the SU(3) predictions given in Fig. 2 to see if the /0(1500) can be identified as a normal meson. In Fig 17 are plotted both

181

Figure 16: Dalitz Plots for pp annihilation into three pseudoscalar mesons. The left is v'KtKiaad the right is it"r)rf'.

the Crystal Barrel data and the relative rates as computed from SU(3) for an /Q as a function of the nonet mixing angle. While a normal meson with a mixing angle of about 150° can accommodate the KK, r)T) and i]rf modes, it is a factor of ~ 4 too small with respect to the observed ww modes. In fact, 150° corresponds to a mostly ufi and dd state, not and s3 state. The best comparison for a meson is shown in table 7, and at least in this model, it is clear that the /0(1500) is not a normal meson. One can also compare to the decay rates of a pure glueball, and again it is clear that the /o(1500) is not a pure glueball. At this point, it's exact nature is unclear. KK WW Decay Rate m 5.13 ±1.95 0.708 ± 0.209 1.00 /o(1500) I Meson 20 0.7 I Glueball 3 4

vv'

1.64 ±0.62 1.6 0

Aw 13.7 ±4.4 Large

Table 7: Relative decay amplitudes squared, 72 normalized to the rfr) rate for the /0(1500). These are compared to the SU(3) prediction for an ss meson with mixing angle of 150°, as well as for a pure glueball.

182

20 15 10 ' %" ' %"' - \ ^ '* **, "'

5 ' '*.** 0

:

rfV. ...1

"b^«

50

0

_—~r~-~~~~~~~~

100

12*<*<16*

150

92'<*<123*

(f->WA->'w) ( f - ^ W / f - ^ ) -*'

_

"nonet

- ,

-

IdegreesJ

150

U8 , <«154*

164*<1><166'

Figure 17: The experimental decay rates of the /o(1500) compared to the SU(3) predictions for an /o and /o state decaying into pairs of pseudoscalar mesons as a function of the scalar nonet mixing angle. There is no angle which is consistent with the measured data.

183

Central Production Experiments Another glue-rich channel is that of central production, and a great deal of analysis has been done recently by the WA102 collaboration at CERN. WA102 have looked at central production of 7r+7r~ [barberis95, barberis99b], 7r°7r° [barberis99c], KK [barberis99a], and ir+ir'n+n~ [barberis95, barberis97b] in 450GeV/c pp collisions. In all of these analyses, they observe two scalar states, the /o(1500) and the /o(1710). In addition, in the 4ir data, they observe the /0(1370). They also find that by kinematically selecting on their data, they were able to enhance the scalar signals.

State /o(980) /o(980) /o(1370) /o(1370) /o(1500) /o(1500) /o(1500) /o(1710) /o(2000)

Mass GeV/c'2 0.985 ± 0.010 0.982 ± 0.003 1.290 ±0.015 1.308 ±0.010 1.502 ±0.010 1.497 ±0.010 1.510 ±0.020 1.700 ±0.015 2.020 ± 0.035

Width GeV/c 2 0.065 ± 0.020 0.080 ± 0.010 0.290 ± 0.030 0.222 ± 0.020 0.131 ±0.015 0.104 ±0.025 0.120 ±0.035 0.100 ± 0.025 0.410 ±0.050

Decay K+K7r+7T~ 7r 7r"7r + 7r _ 7r+7T~ 7r+7T~ +

K+K7r+7r~7r+7r~ K+K7r+7r_7r+7r_

Reference [barberis99a] [barberis99b] [barberis97b] [barberis99b] [barberis99b] [barberis99a] [barberis97b] [barberis99a] [barberis97b]

Table 8: Observed scalar mesons in various final states in WA102.

In Fig. 10a, the quantities qi and 92 represent the four-momentum transfer from each proton to the produced system, X. WA102 observed a striking difference between ratio of events with |
Mass GeV/c 2 0.987±6±6 1.312 ± 2 5 ± 1 0 1.502 ± 12 ± 10 1.727±12±11

Width GeV/c 2 0.096 ± 0.024 ± 0.016 0.218 ± 0.044 ± 0.030 0.098 ± 0.018 ± 0.016 0.126 ±0.016 ±0.018

Table 9: T-Matrix pole positions from a WA102 coupled channel analysis [barberis99d].

184 State KK-.TTT:

/o(1710) 5.0 ± 0 . 6 ± 0 . 9

/o(1500) /o(1370) 0.46 ±0.15 ± 0 . 1 1 0.33 ± 0.03 ± 0.07

Table 10: Relative KK to itir decay rates from WA102 [barberis99d]. Isospin corrections have been performed. Radiative J/ip Decays In radiative J/ip decays, the results are hampered by the finite statistics that have been collected to date. There are plans to increase the sample from the BES detector to about 20 x 106 by 2001, but even this increase by a factor of 3 over current samples is still going to leave things ambiguous. There are reports of all three /o states in these decays, as given in table 11. However, there are some inconsistencies. Both [bai96] and [bugg95] find both a 0 + + and a 2 + + signal near 1.7GeV/c 2 . However, [dunwoodie97] finds only a 0 + + signal. In addition, [dunwoodie97] sees the / 0 (1370) in radiative J/ip decays, while [bai96] and [bugg95] do not. Finally, both [bugg95] and [bai96] see the / 0 (1500), while [dunwoodie97] does not. State /o(1370) /o(1370) /o(1500) /o(1710) /o(1710) /o(1710) /o(1710)

Mass GeV/c 2 1 4OQ+U043

Width GeV/c 2 0.169^:076

Decay

Rate (10- 4 )

o.i69i8;J7J

KK

(3.715ES) (o.6±8-|)

1.505+ 1.704lg5g

0.120+ 0.124±°;°g

7r+7r~7r+7T~

(2.5 ± 0.4)

WIT

1.7041JJ.-SS 1 7Q 1+0.018 I.l0i_o.o39

0-124±85g o.o85i83g

KK K+K-

(2-o±8:I) (7.5±?;g) (o.8±8:l)

1.750 ±0.015

0.160 ± 0.040

1 4OQ+0-043

7T7T

(4.3 ± 0.6)

Reference [dunwoodie97] [dunwoodie97] [bugg95] [dunwoodie97] [dunwoodie97] [bai96] [bugg95]

Table 11: Observed signals for J/i/i -> 7/0

State KK-.TTIT

/o(1370) /o(1710) 0.149 + 0.215-0.133 3 . 7 3 + 1 . 6 7 - 2 . 3 5

Table 12: Relative KK to TTJT decay rates from Mark III [dunwoodie97]. Isospin corrections have been performed.

Other D a t a Interpretation If we examine the scalar data as a whole, there appears to be three states, /o(1370), /o(1500) and /o(1710). The decay pattern of the /o(1500) seem to exclude it being simply a meson. In addition, its strong production in pp annihilation at rest seem to exclude it being a mostly ss state. All three states are observed in central production, while the / 0 (1710) is clearly observed in J/^l decays, with some conflicting information on the other two states. The simplest explanation of what we observe appears to be

185 i—i—i—i—i—i—i—i—i—r

(a)

1500

1000

<£

500

(D »n o

*i

i

i

i

I

i

i

i—i—L

* |

(b)

1500

LU 1

1000

><

500

K-

•f J

.0

I i 1.5

•*/• i

Mass (GeV)

i

i

L 2.0

7236A8

Figure 18: Data taken from the Mark III experiment. These have been acceptance and isospin corrected, (a) Events from J/tp -> -y(K+K~), (b) Events from J/ip -t f(iT+ir~).

that the two scalar mesons, /o and /Q along with the scalar glueball exist, and that they have mixed to form the three observed states, / 0 (1370), /o(1500) and / 0 (1710). In order to try and understand this picture, we will extend the earlier mixing picture to allow the glueball to mix with the isoscalar states. To do this, we need to define two additional mixing angles, which for convenience we will call a and f. In this scheme, we can write the three observed states in terms of the three SU(3) basis states as follows: | /) = + | /') = | /") = +

[cos £ cos 9 cos a — sinasin£] | 1) + [cosf sin0] | 8) .Rfcosf cos 0 sin a + sin £ cos a] \Q) - [sinflcosa] | 1) + [cos0] | 8) - R [sin 0 sin a] | Q) — [sin £ cos 9 cos a + cos f sin a] | 1) - [sin £ sin 0} | 8) R [cos a cos f — sin £ cos 0 sin a] \Q)

186

80 70

Jl

6050 40 30

fll

20 10

1.5 2 Mas8(Tt°jt0 )

2.5 [GeV/c 2 1

_L—U

1.2

1.5

2.0

Figure 19: Data taken from the BES experiment. (a)Events from J/tp -> •y(w+ir~) (b) Events from j/y> -> 7(i<:+i(r-).

It is also convenient to rewrite the SU(3) states in terms of their light quark, nn and strange quark, ss content. Most of the following will express the mixing in terms of these states, while the decays are computed in terms of the SU(3) states. [2 (uu + dd\

/T

[l fuu + dd\

[2

= nn

u -frs

In addition to the three mixing angles, simple SU(3) makes no prediction on the relative decay rate of the glueball with respect to the normal mesons, (gf versus gx of section 4.1). We can now take all the data on scalar meson decays, -and try to fit it to obtain 0, a, £ and R. If we do so, we find three different solutions as given in table 13. It should be noted that there are several mathematical ambiguities in the the formulation we have written down. It is fairly strait forward to show that without loss of generality, we can restrict R to be negative, and the three mixing angles to be in the range of 0 to TT. All solutions outside this range are mathematically equivalent to ones inside the range.

187

Solution

Ratio

a

-Vf -V!

b c

0

4V2

a

x2

£

81.5°

154.5°

133.5°

1.32

103.5°

128.5°

132.5°

1.29

88.5°

100.5°

110.5°

2.04

Table 13: The three minima to the fit of branching ratio data. Fig. 20a shows a plot of \2 against the ratio R. Even though there is a clear minimum around — 4 i / | , in fact all the values that fall below the 65% line are within one a errors. It is particularly tricky fitting the angles with only these data. Also, around the deepest minimum, there are actually two distinct solutions as given as a and b in table 13. However, there is also a second minimum dip around —\J\ as given as c in the table. A particular solution can be represented as a fraction of nn, ss and Q in each of the three physical states, / 0 (1370), / 0 (1500) and / 0 (1710). These are represented for the three solutions in Fig. 21. It is clear that even though we cannot pin down the exact mixtures, there are a few common features. First, the / 0 (1500) has at most about 30% ss, and is best described as either largely nn or Q. The current data favor a large ss component and a small nn component in the /j(1710). Finally, it is difficult to say where the Q state is.

-10 -9

-6

-7

-6

-5

-4

-3

-2

-1 *0

RC/KM

°0

20

40

60

00 100 120 140 160 1BO

\

[Degrees]

°0

20

40

60

B o " 100 120 140 160 160

8

[Degrees]

Figure 20: (a) shows x2 of the fit as a function of the Relative decay strength of the glueball to mesons in the SU(3) picture. The two lines indicate the 65% confidence level and 85% confidence level curves, (b) shows a plot of a versus £ for all solutions that have, 91%, 85% and 65% confidence level, (c) shows £ versus 6 for all solutions that have 91%, 85% and 65% confidence level.

188

f0(1370)

f0(1500)

Gluebal!

¥ /111t\\

f0(1500)

Glueball

f0(1710)

Glueball

Glueball

f0(1370)

%(1710)

f„(1500)

umo)

o

Figure 21: Row 1 to solution a, row 2 is solution b , and row 3 is solution c. The three solutions are from from table 13.

189 6.2 The Status of Hybrids The most striking evidence of a hybrid meson would the observation of states with non-gg quantum numbers, e.g. 0~~, 0 + _ , 1~+, 2 + _ , • • •. However, this observation by itself would not be sufficient. There are models of qqqq states which can also have exotic quantum numbers. Such an observation would unequivocally indicate something beyond the normal qq structure of mesons.

•np Peripheral production Brookhaven experiment E852 has reported the observation of two states with 1~+ quantum numbers. Thefirsthas been observed in the reaction ir~p —> r)w~p [thompson97, chung99]. A partial wave analysis of the ryn system shows evidence of a Jpc = 1~+ state at a mass of (1.370 ± 0.016 + 0.050 - 0.030) GeV/c2 and a width of (0.385 ± 0.040) The final state is dominated by the production of the Jpc = 2 + + , a2(1320) and the exotic state is seen mainly through its interference with the a^. Figure 22 shows the partial wave results of this analysis. The 02(1320) is seen to dominate the 2 + + wave as shown in a. The other allowed wave is the 1~+ exotic wave as shown in b. While there does appear to be a peak in this wave, the real evidence for resonant behavior comes from the phase difference between the two waves, (c). The lines on top of the data are the best fit. The three contributions to the phase difference are shown in d where 4 is the fit phase difference from c. 1 is the Breit-Wigner contribution from an 02 resonance, 2 is the contribution from the exotic wave, and 3 shows the assumed flat background phase. It is under this latter reasonable assumption that two interfering Breit-Wigners give a very good description of the data. This leads to the conclusion that there is a 1~+ resonant state. It should be noted that the peak of the exotic wave contains about 400 counts, while the peak of the 02 contains about 13,000 events. The exotic signal is on the order of 1% of the dominant signal, and it is only through its interference is it extracted. A earlier analysis was performed by the VES collaboration [beladidze93] on the same reaction at different incident n~ energy. They see exactly the same intensity and phase distributions as E852, but due to more limited statistics, did not claim the presence of an exotic signal. A second analysis by E852 examines it~p —• Tt+n~ir~p [adams98]. The three-pion system is much richer than the ryn system seen before. In particular, 7r7r pairs could form (irir)„ p(770) or /2(1270) intermediate states. In fact, due to the presence of non resonant effects at low energy, this analysis is not able to completely understand the data in the 1.4GeV/c2 region. However, the 1.6GeV/c2 region is dominated by two partial waves. The 7r2(1670) in the 2 _ + wave, and a second isospin 1, 1~+ exotic state a t m = (1.593 ±0.008+ 0.029-0.047) GeV/c2 and T = (0.168 ±0.020+ 0.1500.012) GeV/c2. Fig. 23 shows plots of the the intensity of these two waves in a and b, and their phase difference in c. In d is shown the individual phase motion of these two waves under the Breit-Wigner assumption, and the relative production phase between them.

190 x 10 3

d)

f-« ^

yj U;

« 0.0 1.4 M(TIJI) GeV

1.8

1.0

3

1.4 Mftiit) GeV

1.8

Figure 22: Results from E852 on the partial wave analysis of 7r~p -+ r)ir~p. Plot a shows the intensity of the 2 + + wave as a function of mass. Plot b shows the intensity of the exotic wave as a function of mass. Plot c shows the phase difference between a and b and d correspond to the different elements in the fit to the phase differences.

pp Annihilations The Crystal Barrel experiment has studied the reaction pd —> 7r~rr°7jp [abele98] and pp —> 7r°7r°77 [abele98]. The deuterium annihilation shows the most striking evidence for an exotic 1~+ state decaying to »j7r. The observed mass is m = (1.400 ± 0.020 ± 0.020) GeV/c2 and width of V = (0.310 ± 0.050 + 0.050 - 0.030) GeV/c 2 . Annihilation on the neutron has a different set of allowed initial states than from the proton. Table 14 gives the annihilation rates into the possible final states. The (rpr)p entry corresponds to the 1~ + rpr system. What is interesting is that the exotic state is produced nearly as strongly as the a2(1320) from both allowed initial states. This is unlike the peripheral production where it is on order 1% of the a2(1320). Secondly, there is a crossing p(770) band which provides far more complicated interference. Fig. 24a shows the Dalitz plot for this final state. When this is fit with all final states except the exotic wave, the x 2 / n d f = 1233/(411 - 12) = 3.07. Addition of the exotic wave to the fit mixture reduces this x 2 / n d f = 506/(411 — 20) = 1.29. It is clear that the exotic wave is a critical component in explaining these data, even though it is not directly visible in the Dalitz plot. Fig. 24b shows where both the intensity and the interference terms from the 1 _ + wave contribute to the Dalitz plot. The phase motion of the exotic wave is clearly necessary because of both the interference with both the o 2 (1320), and the crossing p~(770).

191

13000 -(b) MOD •

A \

/

/ 4000 IS

•

U& 1.7 1JB

1J

Mass (GeV/c2)

1

I

U

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V

2*

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f

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15 2i Mass (GeV/c^)

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Figure 23: Results from E852 on the partial wave analysis of ir~p -> Tr+ir~ir~p. Plot a shows the intensity of the exotic 1 - + wave. Plot b shows the intensity of the 2~+ wave as a function of mass. Plot c shows the phase difference between a and b and d correspond to the different elements in the fit to the phase differences.

Initial State 3 5!(66.4%) (IG)JPC

= (1+)1-

^1(33.6%) (I°)jrc = (!+)!+-

Intermediate State Rate (%) Production Phase 0 (fixed) P~(770)r, 30.0 ± 3.5 (7 ± 8)° a2(1320)7r 11.1 ±1.0 (210 ±10)° (r}ir)PTT 7.9 ±1.0 0 (fixed) p-(770)»7(L = 0) 10.3 ±3.0 (145 ± 10)° p-(770)V(L = 2) 17.3 ±1.2 (315 ± 25)° a2(1320)7r 3.8 ±0.8 (70 ± 35)° (r?7r)p7r(L = 0) 2.8 ±1.3 (110 ± 50)° iw)Pn{L = 2) 0.5 ±0.5

Table 14: Crystal Barrel results on the reaction pd -> r)ir~ir° showing the contribution from each intermediate state to the total reaction. It is interesting to note that the exotic T/TT-P wave is of similar strength to the 02(1320) in both initial states.

Interpretation The obvious question is Have we found hybrid mesons? The data seem to indicate two 1 = 1, Jpc = 1~+ states. One near m = (1.4)GeV/c2 with a width of about T = (0.3) GeV/c2, and a second near m = (1.6) GeV/c2 with a width of T = (0.17) GeV/c2.

192 x 10

n Amplitude and fnfprfenpiice

Figure 24: Results from Crystal Barrel on the reaction pd -> r)ir°w^p. a:The Dalitz plot for the reaction pn -»• !pr^ir°. The dominant diagonal band is the p"(770). One can also see a vertical and horizontal band for the a2(1320). b: The contribution of the exotic JHJ wave to the Dalitz plot. While this is not directly observable in the Dalitz plot, its interference with both the a2 and the p are critical in explaining the data.

Neither of these states has been observed in decay modes favored by hybrid mesons. The lighter state is seen in T)ir, while the heavier one is seen in pn. Both states are lighter than the expected lowest hybrid mass of around m - (1.8)GeV/c 2 . Finally, only one (/ = 1) state with these quantum numbers is expected, though we do expect two additional I — 0 states, and the corresponding strange states to fill out a nonet of hybrids. It is possible to explain the E852 data on the lower mass state as a threshold effect from the higher mass state interfering with a background precess that is typical in pion production [donnachie98]. However, this explanation fails for the Crystal Barrel result. In order for it to work here, there would have to be a different background in pp annihilation that conspired in exactly the same way to produce exactly the same signal. This seems highly unlikely, and relies on the statistically more significant Crystal Barrel results being wrong. Other explanations with threshold effects have also been proposed most relying on the fact that both the /I(1285)TT and &i(1235)jr thresholds are near the observed mass of 1.4. Actually, a better understanding of this state will only come from studies of the / 1 (1285)JT and &i(1235)'/r final states. The 1.6GeV/c 2 state is closer in mass to the expected hybrids, but again, its decay to pw is not expected to be a large hybrid decay mode. Additional information is needed on different decay modes of this state. Also, to demonstrate that a hybrid meson has been seen, it is important to find more than one member of the nonet. Currently, an I = 1 object has been seen, but we expect two 7 = 0 objects as well.

193 7

T h e Future of Spectroscopy

So given where we stand, what is the future of light quark spectroscopy? Over the last few years we have seen tremendous advances in theory, and with the current rate of improvements, we can only expect this to continue. We have also seen the first hard evidence of non-qq states with exotic quantum numbers, and there is good evidence that that ground state glueball has been found, but that it is mixed into three states. In order to see what it next, we would like to look at Fig. 12, and focus in on the mass region of 1.5to2.5GeV/c 2 . This region encompasses the lowest lying hybrid nonets and glue ball states. It unfortunately also covers several orbital and radial excitations of qq states. Identifying and untangling the nature of the states in this region is key in fully understanding the bound states of QCD. There are several areas where progress could be made and one in particular where almost no data exist. This latter case is the photoproduction of mesons. Due mostly to beam intensity limits, typical earlier experiments are limited to a few thousand events in any specific channel. In addition, the photon is a fundamentally different probe than a 7r or K. The current experimental situation in photoproduction is sparse at best and a unique opportunity now exists using high intensity polarized photon beams available at a 12 GeV CEBAF to radically change this. With tagged photon intensities between 107 and 108 per second, it will be possible to generate data samples comparable or larger than all other meson production reactions. This will allow a full partial wave analysis, which in turn will allow us to identify both normal qq mesons, as well as non-qq states. The addition of linear polarization will both simplify the partial wave analysis, and improve our understanding of the production mechanisms. Improving the situation in photoproduction is a primary goal of the proposed HALL D experiment at Jefferson Lab [halld99]. A cut-away view of the proposed HALL D detector is shown in Fig. 25. The detector is a nearly 4TT hermetic device for both charged particles and photons. This will allow complete reconstruction of most final states, and when coupled with the linearly polarized photon beams, will allow for high statistics partial wave analyses of many different final states. Current photoproduction data Table 15 is a partial compilation of known photoproduction cross sections and the numbers of events from the existing experiments. The typical cross sections range from of order 0.1 /jb up to of order 10 fib, with most measurements involving rather small numbers of events, typically on the order of a few thousand. These experiments were able to perform careful spin analyses, from which much was learned. However, the statistics were insufficient for a full partial wave analysis. This situation can be radically improved in photoproduction experiments at CEBAF, where a full partial wave analysis involving both production and many decay channels will be possible. Typically, these analyses divide the photoproduction data into bins of invariant mass and | t |. A reasonable grid might be 5 bins in | t | between 0.0 and 1.0 (GeV/c2)2, and 0.01 GeV/c2 wide bins in mass. Making the following conservative assumptions

194

Reaction 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P 7P

—y pn+ir~ —¥ prr+TT—• pTr+Tr~Tr0 —f pTT+TT~Tr° —¥ pn+Tr~ir° —V prr + Tr - Tr° —• pn+TT~Tr° —• nir+7r+Tr~ - » riTT+TT+TT—> nTr+Tr+Tr"" —>• pn+ir~ir° —• pTr+TT~Tr+Tr_ —> pTr+Tr_Tr+Tr~ —• prr + Tr _ Tr + Tr _ —• pw+Tr~Tr+/jr~ —> p7r+Tr-TT+TT—• pTr+Tr~Tr°Tr0 —> p7r+Tr+Tr""Tr"Tr° -+ A++Tr""Tr+TT_ —» A + + Tr"Tr + Tr _ —> A++Tr~Tr+TT~ —>• A ++ Tr _ Tr + Tr~ —• pw —> pw —> put —> pu> —> p<£ —> p^> -> n a + -> n a +

7P -» nat

g 7 GeV 9.3 19.3 2.8 4.7 9.3 4.7-5.8 6.8-8.2 4.7-5.8 6.8-8.2 16.5-20 20-70 4-6 6-8 8-12 12-18 15-20 20-70 19.5 4-6 6-8 8-12 12-18 4.7-5.8 6.8-8.2 4.7 9.3 4.7 9.3 4.7-5.8 6.8-8.2 19.5

a (fib)

13.5 ± 1.5 fib 11.8 ± 1.2 fib 4.6 ±1.4 fib 4.0 ± 1 . 2 fib

4.0 ± 0.5 fib 4.8±0.5pfc 4.5 ± 0.6 fib 4.4 ± 0 . 6 fib

1.65 ± 0.2 fib 1.8 ± 0 . 2 fib 1.1 ± 0 . 2 fib 1.15 ± 0.2 fib 2.3 ± 0 . 4 fib 2.0 ± 0 . 3 fib 3.0±0.3/z& 1.9 ± 0 . 3 fib 0.41 ± 0.09 fib 0.55 ± 0 . 0 7 ^ 6 1.7 ± 0.9 fib 0.9 ± 0 . 9 fib 0.29 ± 0.06 fib

Events 3500 20908 2159 1606 1195 3001 7297 1723 4401 3781 14236 ~ 330 ~ 470 ~ 470 ~ 380 6468 8100 2553 ~ 200 ~ 200 ~ 200 ~ 200 < 1600 < 1200 1354 1377 136 224

~ 100

Ref. [ballam73] [abe84] [ballam73] [ballam73] [ballam73] [eisenberg72] [eisenberg72] [eisenberg72] [eisenberg72] [condo93] [atkinson84] [davier73] [davier73] [davier73] [davier73] [abe85] [atkinson84a] [blackett97] [davier73] [davier73] [davier73] [davier73] [eisenberg72] [eisenberg72] ballam73] ballam73] ballam73] ballam73] [eisenberg72] [eisenberg72] [condo93]

Table 15: A sample of measured photoproduction cross sections from several references. Note the small numbers of events in any given channel.

195

Figure 25: A cut-away view of the proposed HALL D detector.

allow one to estimate a typical year's worth of data. s The tagged photon flux is $ = 10 V" 1 . The tagger sees a 3 GeV energy bite which tops out at 95% of the electron beam energy, Ee, (Hall B can currently tag five times this rate.) * The total photoproduction cross section for a reaction jp —> NX is a = l.O/jh. This is independent of the photon energy, and distributed over the m , vs t plane using a independent weight, e~8W. It should be noted that the slope parameter of 8 (GeV/c2)"2 is at the high end of expected slopes, which range from 4 to 8 (GeV/c2)"2. • The experiment runs for 300 days per year with 33% live time. » The overall reconstruction efficiency is 10% In Fig. 26 are shown the expected number of events collected under the above assumptions for Ee=6.0, 8.0, 10.0 and 12.0 GeV. Fig. 26(a) is a sum over all | t | between 0 and 1.0 (GeV/
196

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Figure 26: Expected numbers of events from the reaction jp -* Xp for 0.010 GeV wide bins in m x .The data assume an energy independent total cross section of a = 1.00/L»6. The four curves are from top to bottom for Ee = 6 GeV, Ee=8 GeV, Ee=W GeV and Ee=12 GeV.

197

spectra. Unfortunately, the photon data is so limited in statistics, that a full partial wave analysis is not possible. With the proposed HALL D experiment, we hope to completely change the landscape as far as photo production data goes, and plan to unravel the spectrum of gluonic excitations which is just starting to emerge from other experiments. The future of light-quark spectroscopy will likely be one of photon beams.

> 80
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Invariant Mass of K~n+n System (GeV) Figure 27: A comparison of pion peripheral production and photoproduction data.

198 References [abe84] K. Abe et al, Phys. Rev. Lett. 53, 751, (1984). [abe85] K. Abe et al, Phys. Rev. D32, 2288, (1985). [abele98] A. Abele, et al, The Crystal Barrel Collaboration, Exotic rjir state in pd annihilation at rest into 7r~7r°jjp, Phys. Lett. B423, 175, (1998). [abele98] A. Abele, et al, The Crystal Barrel Collaboration, Evidence for a irrfP-wave in pp-annihilation at rest into r)ir°-jr°, Phys. Lett. B446, 349, (1999). [adams98] G. S. Adams, et al, The E852 Collaboration, Observation of a new J p c = 1~+ exotic state in 7r~p -> 7r+ff~7r"p at 18 GeV/c, Phys. Rev. Lett. x, x, (1998). [afanasev98] A. Afanasev and P. R. Page, Photoproduction and electroproduction of JPC = 1 _ + exotics, Phys. Rev. D57, 6771, (1998). [aker92] E. Aker, et al, The Crystal Barrel Collaboration. The Crystal Barrel Detector at LEAR, Nucl lustrum. Methods A321, 69, (1992). [amsler83] C. Amsler and J. C. Bizot, Simulation of Angular Distributions and Correlations in The Decay of Particles with Spin, Comp. Phys. Coram. 30, 21, (1983). [amsler94a] C. Amsler, et al, The Crystal Barrel Collaboration, rjrf threshold enhancement in pp annihilation into n°i]r)' at rest. Phys. Lett. B340, 259, (1994). [amsler95a] C. Amsler, et al, The Crystal Barrel Collaboration. A high statistics study of the /0(1500) decay into n°n°. Phys. Lett. B34&, 433, (1995). [amsler95b] C. Amsler, et al, The Crystal Barrel Collaboration. Coupled channel analysis of pp annihilation into iraTr0ir0, n°ir0r) and -K°T]T). Phys. Lett. B355, 425, (1995). [amsler95c] C. Amsler, et al, The Crystal Barrel Collaboration. A high statistics study of the /0(1500) decay into T)T). Phys. Lett. B3S8, 389, (1995). [amsler96a] C. Amsler, et al, The Crystal Barrel Collaboration. Observation of the /0(1500) decay into KLKL. Phys. Lett. B385, 425, (1996). [amsler96] C. Amsler and F. E. Close, Is the f0(1500) a scalar glueball?, Phys. Rev. D53, 295, (1996). [amsler98] Claude Amsler, Proton-antiproton annihilation and meson spectroscopy with the Crystal Barrel, Rev. Mod. Phys. 70, 1293, (1998). [atkinson84] M. Atkinson et al, The Omega Collaboration. Nucl. Phys. B231, 15, (1984). [atkinson84a] M. Atkinson et al, The Omega Collaboration. Nucl Phys. B243, 1, (1984). [bai96] J. Z. Bai, et al, The BES Collaboration. Structure Analysis of the /j(1710) in the Radiative Decay J / * -> yK+K~, Phys. Rev. Lett. 77, 3959, (1997). [bali93] G. S. Bali, et al, The UKQCD Colaboration, A comprehensive lattice study of SU(3) glueballs, Phys. Lett. B309, 378, (1993). [ballam73] J. Ballam et al, Vector-meson production by polarized photons from 2.8, 4.7, and 9.3 GeV, Phys. Rev. D7, 3150, (1973).

199 [barberis95] D. Barberis, et al, The WA102 Collaboration, A further study of the centrally produced TT+TT~ and Tr+n~ir+n~ channels in pp interactions at 300 and 450GeV/c, Phys. Lett. B353, 589, (1995). [barberis96] D. Barberis, et al, The WA102 Collaboration, Observation of vertex factorisation breaking in central pp interactions, Phys. Lett. B388, 853, (1996). [barberis97a] D. Barberis, et al, The WA102 Collaboration, A kinematical selection of glueball candidates in central production, Phys. Lett. B397, 339, (1997). [barberis97b] D. Barberis, et al, The WA102 Collaboration, A study of the centrally produced ir+n~n+ir~ channel in pp interactions at 450 GeV/c, Phys. Lett. B413, 217, (1997). [barberis99a] D. Barberis, et al, The WA102 Collaboration, A partial wave analysis of centrally produced K+K~~ and KsKs systems in pp interactions at 450GeV/c and new information on the spin of the /j(1710), Phys. Lett. B453, 305, (1999). [barberis99b] D. Barberis, et al, The WA102 Collaboration, A partial wave analysis of centrally produced ir+n~ system in pp interactions at 450 GeV/c, Phys. Lett. B453, 316, (1999). [barberis99c] D. Barberis, et al, The WA102 Collaboration, A partial wave analysis of centrally produced ir°ir° system in pp interactions at 450 GeV/c, Phys. Lett. B453, 325, (1999). [barberis99d] D. Barberis, et al, The WA102 Collaboration, coupled channel analysis of the centrally produced K+K~ and w+n~ system in pp interactions at 450 GeV/c. Submitted to Phys. Lett. B, (1999), (hep-ex/9907055). [barnes97] T. Barnes, F. E. Close, P. R. Page and E. S. Swanson, Higher Quarkonia, Phys. Rev. D55, 4157, (1997). [beladidze93] G. M. Beladidze, et al, The VES Collaboration, Study of -rr-N -» n^N and 71-AT - • rfir'N reactions at 37GeV/c, Phys. Lett. B313, 276, (1993). [blackett97] G. R. Blackett et al, The Photoproduction of the 6i(1235)7r System, hep-ex/9708032. [bugg95] D. Bugg, et al, Further amplitude analysis of J / * -» 7(7r+7r~7r+7T'), Phys. Lett. B353, 378, (1995). [burakovsky98] L. Burakovsky and P. R. Page, Scalar Glueball Mixing and Decay, Phys. Rev. D59, 014022, (1999). [caso98] C. Caso, et oi., the Particle Data Group, Eur. J. Phys. C3, 1, (1998). [chanowitz84] M. Chanowitz, Proceedings of the VI International Workshop on Photon-Photon Collisions, Lake Tahoe, CA, Sept 10-13, World Scientific, 1985, R. L. Lander ed.. [chung96] S. U. Chung, J. Brose, R. Hackmann, E. Klempt, S. Spanier and C. Strassburger, Partial wave analysis in K-matrix formalism, Ann. Phys. (Leipzig) 4, 404, (1995). [chung99] S. U. Chung, et al, The E852 Collaboration,Evidence for exotic

200

J p c = 1~ + meson production in the reaction ir~p —• rjir~p at 18 GeV/c, Phys. Rev. D 60, 092001, (1999). [close79] F. E.Close, An Introduction to Quarks and Partons, (1979), Academic Press, Harcourt Brace Jovanovich. [close88] F. E. Close, Gluonic Hadrons Rep. Prog. Phys. 51, (1988). [close95] F. E. Close and P. R. Page, Photoproduction of hybrid mesons from CEBAF to DESY HERA, Phys. Rev. D52, 1706, (1995). [close97] F. E. Close, G. R. Farrar and Z. Li, Determining the gluonic content of isoscalar mesons, Phys. Rev. D.55, 5749, (1997). [condo91] G. T. Condo, et al., Photoproduction of an isovector pn state at 1775 MeV, Phys. Rev. D43, 2787, (1991). [condo93] G. T. Condo et al., Phys. Rev. D^«,3045, (1993). [cooper88] S. Cooper, Meson Production in Two-Photon Collisions, Ann. Rev. Nucl. Part Phys. 38, 705, (1988). [davier73] M. Davier et al., The reaction 7p —¥ ir+Tr~ir+n~p at high-energy and 7 dissociation into 4TT, Nucl. Phys. B58, 31, (1973). [donnachie98] Alexander Donnachie and Philip R. Page, Interpretation of experimental J p c exotic signals, Phys. Rev. D58, 114012, (1998). [dunwoodie97] W. Dunwoodie, J/'t! Radiative Decay to Two Pseudoscalar Mesons from Mark III (1997), SLAC-PUB-7163. [eisenberg72] Y. Eisenberg et al., Phys. Rev. D5, 15, (1972). [flatte76] S. M. Flatte, Phys. Lett B63, 224, (1976). [godfrey99] Stephen Godfrey and Jim Napolitano, Light Meson Spectroscopy, submitted to Rev. Mod. Phys., (1999), hep-ph/9811410. [halld99] The Hall D Collaboration, Photoproduction of Unusual Mesons, Design Report, Version2, August 1999, http://www.phys.emu.edu/halld/cdr/. [herndon75] D. Herndon, P. Soding and R. J. Cashmore, Phys. Rev. Dll, 3165, (1975). [isgur85] N. Isgur and J. Paton, Flux-tube model for hadrons in QCD, Phys. Rev. D31, 2910, (1985). |jacobs59] M. Jacobs and G. C. Wick, Ann. Phys. 7, 404, (1959). [Iee98] W. Lee and D. Weingarten, Scalar Quarkonium Masses and Mixing with the Lightest Scalar Glueball, (1998), (hep-lat/9805029) [morningstar99] C. Morningstar and M. Peardon, The Glueball Spectrum from and Anisotropic Lattice, Phys. Rev. D60, 034509, (1999). [page97] F. E. Close and P. R. Page, Distinguishing hybrids from radial quarkonia, Phys. Rev. D56, 1584, (1997). [sexton95] J. Sexton, A. Vaccarino and D. Weingarten Numerical Evidence for the Observation of a Scalar Glueball, Phys. Rev. Lett. 75, 4563, (1995). [thompson97] D. R. Thompson, et al. The E852 Collaboration, Evidence for Exotic Meson Production in The Reaction w~p —• »j7r_p at 18 GeV/c, Phys. Rev. Lett. 79, 1630, (1997). [weingarten97] Don Weingarten, Scalar Quarkonium and the Scalar Glueball,

201 Nucl. Phys. Proc. Suppl. 53, 232, (1997). [zemach64] Charles Zemach, Three-Pion Decays of Unstable Particles, Phys. Rev. 140, B1201, (1964).

202

QCD and the Structure of the Nucleon in Electron Scattering W. MELNITCHOUK Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, and Special Research Centre for the Subatomic Structure of Matter, University of Adelaide, Adelaide 5005, Australia The internal structure of the nucleon is discussed within the context of QCD. Recent progress in understanding the distribution of flavor and spin in the nucleon is reviewed, and prospects for extending our knowledge of nucleon structure in electron scattering experiments at modern facilities such as Jefferson Lab are outlined.

1

Introduction

The internal structure of the nucleon is the most fundamental problem of strong interaction physics. Understanding this structure in terms of the elementary quark and gluon degrees of freedom of the underlying theory, quantum chromodynamics (QCD), remains the greatest unsolved problem of the Standard Model of nuclear and particle physics. Historically, the basic strong interaction which we have sought to explain has been that between protons and neutrons in the atomic nucleus. The original idea of massive particle exchange of Yukawa 1 has been a guiding principle according to which later theories have been developed. It was pointed out by Wick 2 that this idea was consistent with the Heisenberg Uncertainty principle, whereby the interaction range of the nuclear force is inversely proportional to the mass of the exchanged meson. Over the years a phenomenological description of the forces acting between nucleons has been developed within a meson-exchange picture. Following the experimental discovery of the pion in 1947, the 1950s and 1960s saw an explosion of newly discovered mesons and baryons, as particle accelerators pushed to higher energies. To bring some sense of order to the profusion of new particles, Gell-Mann and Zweig introduced the idea of quarks 3 , which enabled much of the hadronic spectrum to be organized in terms of just a few elementary constituents. Soon after, however, it was realized that a serious problem existed with the simple quark classifications, namely the A + + isobar. The quark model wave function for the A + + was predicted to be totally symmetric, however the A + + obeyed Fermi-Dirac statistics. A solution to this problem was found by assigning extra internal color4 quantum numbers to the quarks, in which baryons would have in addition an antisymmetric color wave function.

203

The discovery of scaling in deep-inelastic electron-nucleon scattering in the late 1960s at SLAC 5 confirmed that the nucleon contained point-like constituents, which were soon identified with the quarks of the quark model. Imposing local gauge invariance on the color fields, and introducing vector gluon exchange to mediate the inter-quark interaction, led naturally to the development of QCD as the fundamental theory of strong interactions 6 . Because QCD is an asymptotically free theory — the effective strong coupling constant decreases at short distances — processes involving large momentum transfers can be calculated reliably within perturbation theory. Yet despite the successes of perturbative QCD, we are still unable to extract from QCD sufficient details regarding its long-distance properties. This is because in the infra-red region the strong coupling constant grows and perturbation theory breaks down, and the available non-perturbative tools are not yet sufficiently developed to allow quantitative predictions. In a sense it is ironic that the theory which arose out of the desire to understand nuclear forces is able to explain backgrounds in hadronic jets produced in high energy collisions, yet is unable to describe the properties of the ground state of the theory. Although one can argue that QCD in principle explains all hadronic and nuclear phenomena, without understanding the consequences of QCD for hadron phenomenology one may as well argue that the entire physics of atoms and molecules can in principle be explained from QED 7 . Understanding how the transition from the quarks and gluons of QCD to the physical mesons and baryons takes place remains the holy grail of modern nuclear physics. In the next Section some basic elements of QCD relevant for later application to nucleon structure are reviewed. This is followed in Section 3 by the basic definitions and kinematics of electron-nucleon scattering, including elastic, deep-inelastic and semi-inclusive scattering. In Section 4 we focus more closely on the flavor and spin content of the nucleon, and outline some recent highlights in the study of valence and sea quark distributions. Finally, some concluding remarks are made in Section 5.

2

Elements of QCD

Quantum Chromodynamics is a non-Abelian gauge field theory based on the gauge group SU(3)coi0r> and defined in terms of the Lagrange density 6 ' 8 - 1 0 : £QCD =

+

^-gauge

i *^ghost j

(1)

204

where Cim is the classical Lagrangian, invariant under local gauge transformations of the SU(3)coior group: 4nv = ^ , / ( * 7 ^ M - mfWij

-

\F«VF<"

a

.

(2)

The quark fields ipij (for a particular quark flavor f = u,d,s,- • • , with mass mf) are labeled by color indices i,j = 1,2,3. The covariant derivative is Dfi = dfi — igTaA^, where g is the QCD coupling constant and Ta are the generators of the SU(3) group, with a — 1, • • •, 8. In terms of the gluon field A" the gluon field strength tensor is F°v = d^A^ - dvA^ + gf^A^Af,, with fabc the SU(3) structure constants. The major differences between QCD and quantum electrodynamics is the appearance, due to the non-Abelian structure of the theory, of gluon self-couplings in the F • F term. This gives rise to 3and 4-point gluon interactions which make the theory highly non-linear, but also leads to the property of asymptotic freedom (Section 3.2). Under local gauge rotations the quark fields transform (dropping color and flavor indices) according to: ip(x) -> ip'(x) = U{x) V>(z) ,

(3)

where U is an SU(3) unitary matrix: U(x) =exp(ida(x)Ta)

,

(4)

with 9a(x) real. The gluon fields transform according to: A^x)

-> A'^x)

= U(x)All(x)U-1(x)

+ - (dpUix)) U'^x)

.

(5)

One can easily show that £i n v is then invariant under these transformations. The gauge invariance of the classical Lagrangian introduces some difficulties when quantizing the gauge theory. This problem is avoided with the introduction of an addition term, £ ga uge, which fixes a specific gauge. In the Lorentz (covariant) gauge, one has £gauge = — (l/2a)(9 M A^) 2 , where a is an arbitrary gauge parameter. Of course observables cannot depend on the choice of a, and some common choices are a — 1 (Feynman gauge) and a —> 0 (Landau gauge). Other, non-covariant, gauge choices are the Coulomb gauge (diAf = 0), the axial gauge {A% — 0), and the temporal gauge (Aft = 0). The Faddeev-Popov ghost density, £ g h o s t = (d^xa)(Sabd„ 9fabcA^x0, a an a are where x d x scalar anii-commuting ghost fields, ensures that the gauge fixing does not spoil the unitarity of the 5-matrix. Further discussion about the gauge fixing problem can be found in Refs. 1 0 > n .

205

In renormalizable field theories such as QCD the strength of the interaction depends on the energy scale. A property almost unique to QCD is that the renormalized coupling constant decreases with energy — known as asymptotic freedom. The running of the QCD coupling with energy allows one to compute cross sections for any quark-gluon process using a perturbative expansion if the coupling is small. Writing the full QCD Lagrangian as CQCD = Co + CI, where C0 = jHyfdp

- m)rp - -^d^

- ^ ( ^ )

2

- duAl){d»Aa

" - dv Aa ")

+ (d"x a )(^X°)

(6)

is the free Lagrangian, and Ci = g 'frfTail>Aa

- §/ ai,c (d M A£ -

~fabefcdeAlAbvAc^Ahv

dvAl)Ah>iAcv

- gfabc(d>1xa)xbAcll

(7)

the interaction part, one can derive from C\ a complete set of Feynman rules for computing any scattering amplitude involving quarks and gluons 8 _ n . Perturbative QCD has been enormously successful in calculating hard processes in high energy lepton-lepton, lepton-hadron and hadron-hadron scattering. However, even at high energies one can never avoid the fact that the physical states from which the quarks and gluons emerge to undergo the hard scattering are hadrons, so that one always encounters soft scales in any strongly interacting system. While operator product expansions usually allow one to factorize the short and long distance dynamics, understanding the complete physical process necessarily requires going beyond perturbation theory. Over the years the problem of non-perturbative QCD has been tackled on several fronts. The most direct way is to solve the QCD equations of motion numerically on a discretized space-time lattice 1 2 . Recent advances in lattice gauge field theory and computing power has made quantitative comparison of full lattice QCD calculations with observables within reach. Alternative methods of tackling non-perturbative QCD involve the building of soluble, low-energy QCD-inspired models, which incorporate some, but not all, of the elements of QCD. Phenomenological input is then used to constrain the model parameters, and identify circumstances where various approximations may be appropriate. These approaches often exploit specific symmetries of QCD, which for some observables may bring out the essential aspects of the physics independent of the approximations used elsewhere. A good ex-

206

ample of this, which has had extensive applications in low energy physics, is chiral symmetry. Consider the classical quark Lagrange density in Eq.(2) in the limit where the mass of the quarks is zero: A?nv = ^i-fD^

= ^Lt^D^L

+ ^Rl-fD^R

,

(8)

where ipL,R = (1/2)(1 ± 75)^ are left- and right-handed projections of the Dirac fields. Under independent global left- and right-handed rotations £? nv remains unchanged. For Nf massless quarks, the classical QCD Lagrangian is then said to have a chiral SU(iV/)£® SU(Nf)R symmetry. Of course non-zero quark masses break this symmetry explicitly by mixing left- and right-handed quark fields, however, for the u and d quarks, and to some extent the s, the masses are small enough for the chiral symmetry to be approximately valid. If chiral symmetry were exact, a natural consequence would be parity doubling. The nucleon would have a negative parity partner with the same mass, and the pseudoscalar pion would have the same mass as the scalar meson. In nature, the lightest negative parity spin-1/2 baryon is the Su resonance, which is several hundred MeV heavier than the nucleon. Moreover, the pion has an exceptionally small mass, while the lightest candidate for a scalar meson is several times heavier than the pion, so that chiral symmetry in nature is clearly broken. The way to reconcile a symmetry which is respected by the Lagrangian but broken by the physical ground state is if the symmetry is broken spontaneously. According to Goldstone's theorem, a consequence of a spontaneously broken chiral symmetry is the appearance of massless pseudoscalar bosons. For Nf =2 (namely, for u and d flavors), these correspond to the pseudoscalar pions; for Nf = 3 (counting the strange quark as light), these include in addition the kaons and the 77 meson. The physical mesons are of course not massless, but on the scale of typical hadronic masses (~ 1 GeV), they can be considered light — their masses arising from the small but non-zero quark masses. A perturbative expansion in terms of the small pseudoscalar boson masses can be developed 13 , and applied systematically to describe hadron interactions at very low energies. As will be demonstrated in Sections 4.2 and 4.3 the chiral properties of QCD are in fact critical to understanding many aspects of nucleon structure, from low energy form factors to deep-inelastic structure functions. Before proceedings with further discussion about QCD and nucleon structure, however, in the next Section we first define the observables through which one can study the internal structure of the nucleon in electron scattering.

207

3

Electron-Nucleon Scattering

Because the electromagnetic interaction of leptons is perhaps the best understood part of the Standard Model, the cleanest way to probe the internal structure of hadrons is through lepton scattering. This applies to both charged leptons and neutrinos, although to be concrete we shall consider the scattering of electrons. In the one-photon approximation, the scattering of an electron with fourmomentum / from a nucleon with momentum p is illustrated in Fig. 1, where the outgoing electron momentum is V and the hadronic final state is denoted by X: eN —¥ e'X. The energies of the incident and scattered electrons are E and E', and the electron scattering angle is 9. The energy transfer to the nucleon in the target rest frame is v = E—E', and the four-momentum transfer is q2 = {l- I')2 ~ -4EE' sin2 9 for me «.E,E'. In the nucleon rest frame the differential cross section is given by: „,2

J2_

dfldE'

rv WtiV

2MQ4 E

W

'

where aem is the electromagnetic fine structure constant, M is the nucleon mass, and Q2 = —q2. The normalization of states in Eq.(9) is such that v {P\P ') = (2TT)3 2p 0 S{p— p'). The lepton tensor L^ is given by: If

= 2 m,v + 2 l'Hv + g^q2

T 2ietivX('hllp

,

(10)

corresponding to an electron with helicity ± 1 / 2 . Note that for unpolarized scattering only the part of L^v which is symmetric under the interchange /x •*->• v is relevant, while for polarized only the antisymmetric part enters. The hadronic tensor, WW = l ^ i ^ S ' i p + q-px) 1 x

(JV|J M (0)|X)(X|J„(0)|JV) .

(11)

contains all of the information on the structure of the target. For inclusive scattering one sums over all final states X, while for exclusive scattering X denotes a specific hadron. The most general form for the hadronic tensor consistent with Lorentz and gauge invariance, as well as invariance under time reversal and parity, is:

i«v = (-*.

q +

-f) w + (» - ^ * ) (P. - ^ V ) w,

ax

+ Vx^(/M

2

G

1

+(p.

9

/-

S

.(;/)G

2

)

,

(12)

208

X(px) Figure 1: Electron-nucleon scattering in the one-photon exchange approximation: X = N for elastic scattering, while X is a sum over hadrons for inclusive inelastic scattering.

where sp is the nucleon spin vector, denned by sp{\) = (2\/M)(\p \;pop) for nucleon helicity A, so that s2 = — 1, s -p = 0. The structure functions W\, W2, G\ and G2 are in general functions of two variables. Usually one chooses these to be Q2, and the Bjorken x variable, defined as x = Q2 /2p • q. Since W\ and W2 are coefficients of Lorentz tensors symmetric in fiu, they can be measured in unpolarized electron-nucleon scattering. The unpolarized differential cross section can be written in the target rest frame as 14 * 15 : 2

d2an+^

Sa

dSldE'

E'2 Q4

„ „ , o2. . 2 ^(x^^sin^^^ + ^^Q

r

2

)]

,

(13)

where flT (4-1r) refers to the polarization of the electron parallel (antiparallel) to that of the target nucleon. The structure functions G% and Gi can be measured by taking the difference of cross sections with electron and nucleon polarizations parallel and antiparallel 14 ' 15 : LdE<

=

| F ¥

[(E + E'cose)MG1(xJQ2)-Q2G2(x,Q2)]

. (14)

It will be convenient later to introduce dimensionless structure functions •Fi,2, 01,21 defined as: MWi = F i , M2vG1=g1,

uW2 = F2 Mu2G2 = g2 .

(15)

As we will see in Section 3.2, some of these dimensionless structure functions have very simple interpretations in terms of quark densities in deep-inelastic scattering.

209 2.5

p-1—•—•—r

-i—i—i—i—I—i—i—i—i—I—i—i—i—r-

C-2.25 I-

£

2

«L75

I

1.5 |

5*1.25 =-

Ig 1

ir 0.75 £-

."V/VVN,, .

0.5

* 0.25 0 0.5

_L

1.5

2 W (GeV)

2.5

3.5

Figure 2: Electron-nucleon differential cross section 16 as a function of hadronic final state mass, W.

The structure functions describe scattering to final states whose spectrum depends on the amount of energy and momentum transferred to the target nucleon. In Fig. 2 the differential cross section for unpolarized scattering is plotted as a function of the invariant mass, W, of the hadronic final state, where W2 = (p + qf = M2 + IMv -Q2. At low energy, since v > Q2/2M only elastic scattering is kinematically allowed (represented by the spike in Fig. 2 at W = M). As the energy increases above the pion production threshold, W th = M+m„ (left vertical line in Fig. 2), inelastic scattering to nucleon + multi-pion states can occur, as well as excitation of nucleon resonances. The first peak corresponds to the spin- and isospin-3/2 A resonance at W = 1232 GeV. The second peak is predominantly due to the negative parity partner of the nucleon, the S n resonance, and the third to the Fi5. In the region between W = Wth and W = 2 GeV many resonances contribute, most of whose contributions are buried underneath the background. The vertical line in Fig. 2 at W - 2 GeV corresponds to the approximate boundary between the resonance and deepinelastic scattering (DIS) regions. In the next Section we shall examine electron scattering to the simplest final state, namely the elastic. 3.1

Elastic Form Factors

The most basic observables which reflect the composite nature of the nucleon are its electromagnetic form factors. Historically, the first indication that the nucleon is not elementary came from measurements of the form factors in elastic electron-proton scattering 17 .

210

The nucleon form factors are defined through matrix elements of the electromagnetic current, JM = ipj^xp, with rp the quark field, as: = u(P') ( 7 ^ 1 (Q2) +

(N(P')\MO)WP))

i

-^f^Q2))


where P and P' are the initial and final nucleon momenta, and q = P - P'. The functions T\ and Ti are the Dirac and Pauli form factors, respectively. In terms of T\ and Ti the Sachs electric and magnetic form factors are defined as: GE(Q2) = Ti{Q2) - (Q2/4M2) 2

2

F2(Q2) ,

(17)

2

GM(Q ) = ^I(Q ) + ^ 2 ( Q ) .

(18)

Squaring the amplitude in Eq.(16) and comparing with the cross section in Eq.(13), one can write the structure functions for elastic scattering in terms of the electromagnetic form factors: Ff

= MrG2M(Q2)6^-^j (G2E(Q2)

Ff = H i

,

(19)

+ rG2M(Q2)) S(u-^j

,

(20)

where T = Q2/AM2. For the spin-dependent structure functions one has:

at = rh°M(Q2)

(GE(Q2)

+ rGM(Q2)) 8 (v - ^ )

9t = f^GM{Q2)

(GE(Q2) - GM(Q2)) S (V - J ^ ) .

,

(21) (22)

At the quark level, the form factors can be decomposed as:

GEMQ2) = T,eiGhM(Q2)>

(23)

g

so that contributions from specific quark flavors can be identified by considering different hadrons. For the proton one has: GE,M

2 1 1 ~ ^GEM - -GEM - -G%,M

>

(24)

,

(25)

while for the neutron (using isospin symmetry): 2 1 G E , M - ^GEM - -GEyM

1 - -GSE>M

211

where GqE M by definition refers to the quark flavor in the proton. At low Q2 the distance scales on which the electromagnetic scattering takes place are large, so that the resolving power of the probe is only sufficient to measure the static properties of the nucleon. In this region the form factors reflect completely the non-perturbative, long-distance structure of the nucleon. At Q2 = 0 the electric form factor is equal to the charge of the nucleon, while the magnetic form factor gives the magnetic moment: Gg(0) = eN ,

G&(0) =

HN

,

(26)

where ejy = 1(0) for the proton (neutron), and in units of the Bohr magneton, Up = 2.79/xo and p,n = — 1.91/xoAt small Q2 away from zero, in a frame of reference where the energy transfer to the nucleon is zero (namely, the Breit frame, u = 0, Q2 = q2), the electric and magnetic form factors measure the Fourier transforms of the distributions of charge and magnetization, pE and PM, in the nucleon: GE,M(.Q2)

= / d3r e-^pEMr)

•

(27)

Expanding the form factors about Q2 = 0, - \Q2{r2)E,M + 0(Q4) , (28) o enables one to define the charge and magnetization radii of the nucleon in terms of the slope of the form factors at Q2 = 0: (r2)E,M = - 6 dGE,M/dQ2\o2=0. Empirically, the form factors at low Q2 are well described by a dipole form, GEMQ2)

Gp(02)

„ GpM(Q2)

=

GEMW

„ G"M(Q2) ^

2

/

1

V

(29)

where QQ = 0.71 GeV 2 , in which case the electric and magnetic r.m.s. radii of the proton, and the magnetic radius of the neutron, are (r 2 ) 1 / 2 = 0.81 fm. Because the neutron has zero charge, the neutron electric form factor, although non-zero, is very small. The dipole form can be qualitatively understood within a vector meson dominance picture, in which the photon at low Q2 fluctuates into a Jp = 1~ meson (such as the p), which then interacts with the nucleon. The Q2 dependence of the form factor is then given by the vector meson propagator, with the mass of the p corresponding approximately to QQ. However, while providing a reasonable approximation to the form factors at low Q2, deviations

212

from the dipole form have been observed, and it is important to understand the nature of the deviations at larger Q2. At the other extreme of asymptotically large Q2, the elastic form factors can be described in terms of perturbative QCD 1 8 . Here the short wavelength of the highly virtual photon enables the quark substructure of the nucleon to be cleanly resolved. By counting the minimal number of hard gluons exchanged between quarks, the Q2 behavior of the form factors is predicted from perturbative QCD to be GE,M(Q2) ~ 1/Q 4 at large Q2. Just where the perturbative behavior sets in, however, is still an open question which must be resolved experimentally. Evidence from recent experiments at Jefferson Lab and elsewhere suggests that non-perturbative effects are still quite important for Q2 at least w 10 GeV 2 . Understanding the transition from the low to high Q2 regions is vital not only for determining the onset of perturbative behavior. Form factors in the transition region at intermediate Q2 are very sensitive to mechanisms of spin-flavor symmetry breaking, which cannot be described within perturbation theory. In Section 4 we give several examples where form factors reflect important aspects of the non-perturbative structure of the nucleon. 3.2

Deep-Inelastic Structure Functions

Because of the 1/Q 4 dependence in the elastic form factor in Eq.(29), the elastic cross section dies out very rapidly with Q2. It was therefore expected that the inelastic cross section would behave in a similar fashion at large Q2. Contrary to the expectation, the observation 5 in the late 1960s that the inelastic structure function does not vanish with Q2, but remains approximately constant beyond Q2 ~ 2 GeV 2 , provided the first evidence of point-like constituents of the nucleon 19 ' 20 and led to the development of the parton model and later QCD. For inclusive scattering the sum in Eq.(ll) is taken over all hadronic final states X. Using the completeness relation ^ x \X)(X\ = 1 and translational invariance, the hadronic tensor W^v can be written: w

»"

=

hfd^

^'HN\MOM0)\N) ,

(30) 2

where £ is the space-time coordinate. In the limit wherep-q and Q —> oo, but the ratio of these fixed (Bjorken limit), WpV receives its dominant contributions from the light-cone region. This is clear if one writes the argument of the exponential in light-cone coordinates, Q • ? = —Y~

H

J~

~

(31)

213

where £± = £o ± €z- In the target rest frame the photon momentum can be taken as q^ — (v,0±,-\/v2 + Q2) — (v,0±,-v - Mx), so that q • £ = -Mx{£o ~ 6 ) / 2 + ( 2 y + MX)(£Q + 6 0 / 2 , where in this frame x Q2/2Mv. The largest contributions to the integral are those for which the exponent oscillates least, namely q • £ ~ 0. In the Bjorken limit q • £ behaves like J/(£O + £ 2 ), so that only when £o — — £z will there be non-negligible contributions to Wfi,,. Therefore the DIS cross sections is controlled by the product of currents J M (£)J„(0) near the light cone, £2 ~ 0. Parton Model The connection between the deep-inelastic structure functions and the quark structure of the nucleon was first provided by Feynman's parton model 19 . The hypothesis of the parton model is that the inelastic scattering is described by incoherent elastic scattering from point-like, spin 1/2 constituents (partons) in the nucleon. The validity of the parton picture relies on the treatment of the interactions of the virtual photon with the partons in the impulse approximation. The legitimacy of the impulse approximation rests on two assumptions: (i) final state interactions can be neglected, and (ii) the interaction time is less than the lifetime of the virtual state of the nucleon as a sum of its on-shell constituents. The first assumption seems reasonable since in DIS the energy transferred to the parton is much greater than the binding energy, so that the partons can be viewed as quasi-free. The second assumption can be verified in the infinite momentum frame (IMF), where the momentum of the nucleon is Pti — {Pz + M2/2pz; Ox, — Pz), with pz -» oo (or v/c —> 1), in which case the photon four-momentum is q^ = (—xpz(l - M2x2/Q2) + Mv/2pz; 0j_, xpz(l — M2x2IQ2) + Mu/2pz). Since the dominant contributions to WM„ are those for which q • £ £0/Pz, so that the interaction time is ~ £o < Pz/^Mv. The lifetime of the virtual state can be obtained by observing that the energy of the virtual nucleon state consisting of on-shell partons with momenta xipz and mass m; is w J2i(xtPz + Tn2/2xipz), so that the difference between the energies of the virtual and on-shell nucleons is « {J2im2/Xi ~ M2)/2pz. Therefore the lifetime of this virtual state is proportional to pz, and in the Bjorken limit the ratio of interaction time to virtual state lifetime ~ 1/v -» 0. The parton picture is then of a quark with momentum fraction Xi absorbing a photon with xt = x, since S((q + Xip)2) -> S(x - Xi)/2p • q. One can then relate the structure functions to the parton densities (quark and antiquark

214

momentum distribution functions) in the nucleon as 1 9 ' 2 0 : F2{x) = Y,el

*(«(*) +«(*)) = 2xFl{x) ,

(32)

9

9i(x) = ^ Yl

e

\ x(A

(33)

9

where q(x) = q^(x) + q^(x) and Ag(x) = q^(x) - q^(x) are the spin-averaged and spin-dependent quark densities. The consequence of point-like partons is the non-vanishing of the inelastic structure functions at large momentum transfers, since the structure functions in Eqs.(32) and (33) are independent ofQ 2 . Although providing a simple, intuitive language in which to interpret the qualitative features of the deep-inelastic data, the parton model is not a field theory. The formal basis for the parton model is provided by the operator product expansion and the renormalization group equations in QCD, which actually gives rise to small violations of scaling through QCD radiative corrections. Operator Product Expansion In quantum field theory products of operators at the same space-time point (composite operators) are not well defined 10 . The short distance operator product expansion (OPE) of Wilson 21 , in which the composite operators are expanded in a series of finite local operators multiplied by singular coefficient functions, provides a way of obtaining meaningful results. Because deep-inelastic scattering probes the £2 ~ 0 region, rather than the £ ~ 0, one needs an expansion of the product of currents in Eq.(30) which is valid near the light-cone (this is because at short distances Q —> oo and P • q/Q2 -^ 0, while in DIS the light-cone region corresponds to the Bjorken limit, Q2 -¥ oo and p • q/Q2 = 0(1))- The general form of the light-cone operator product expansion is 21 : ^ ) J ( 0 ) ~ £ C f ( a ^ - - - U


(34)

i,N

where the sum is over different types of operators with spin N (i.e. those what transform as tensors of rank N under Lorentz transformations). In DIS the spin-iV operators Of1'",J'N represent the soft, non-perturbative, physics, while the coefficient functions Cf describe the hard photon-quark interaction, and are calculable within perturbative QCD.

215

It is useful to categorize the operators according to their flavor properties, namely those that are invariant under SU(iV/) flavor transformations (singlet) and those that are not (non-singlet). For unpolarized scattering (the extension to spin-dependent scattering is straightforward) the operators must be completely symmetric with respect to interchange of indices fii • • • HN, SO that one can construct at most 3 kinds of composite operators 8 , 1 0 . The non-singlet operators must be bilinear in the quark fields: ,-AT-l

Ofts'""

_

= olvT ^

(T"1-0"2

• • • D"N

+ Mi

permutations) A V , (35)

where A are the eight Gell-Mann matrices of the flavor SU(iV/) group. The singlet operators are: iN~l — e>^-'MJV = ——ip('yiilD^2---D'iN Og'"'iN

+ fiifij permutations) V ,

(36)

;JV-2

= ——(FliiaD^---DiiN-1F^N

+ HiH permutations), (37)

corresponding to the quark and gluon fields, respectively, and color indices have been suppressed. Equations (35)-(37) represent operators with the lowest 'twist', defined as the difference between the mass dimension and the spin, N, of the operator. Whereas the leading twist terms involve free quark fields, operators with higher twist involve both quark-gluon interactions 22 , for example ipF'"''yvtl), and are suppressed by powers of l/Q2. The matrix elements of the operators contain information about the longdistance, non-perturbative structure of the nucleon. They can in general be written as: <J\r(p)|0* ,1 -" Ar |W(p)) = A? p" 1 • • -p"* -

( f l «w terms) ,

(38)

where -4^ represents the soft physics, and the 'trace terms' containing the giaiij a r e necessary to ensure that the matrix elements are traceless (i.e. so that the composite operator has definite spin, N). When contracted with the Qin %i these give rise to terms that contain smaller powers of v2 (i.e. Q2 = 0(v) instead of (p • q)2 = 0(u2)). The OPE analysis allows one to factorize the moments of the structure functions into short and long distance contributions, where the latter are target-dependent (and <92-independent). For the F2 structure function, for example, one has:

M?{Q2) = C dx xN~2 F2(x,Q2) = TC?(Q2) Jo

,

A? .

(39)

216

The target-independent (and <32-dependent) coefficient functions C^ can be calculated at a given order in perturbation theory directly from the renormalization group equations, which will introduce logarithmic Q2 violations of scaling compared with the simple parton model. The structure function can be obtained from the moments via the inverse Mellin transform: 1

F2(x,Q2)

pNo+ioo

= -±-1 dN x1-" lm Jjvo-ioo

Mf(Q2) ,

(40)

by fixing the contour of integration to lie to the right of all singularities of M2V(<52) in the complex-N plane. In terms of the quark and gluon distributions, i<2 is given by a convolution of the parton densities with the coefficient functions 8 - 10 ' 23 describing the hard photon-parton interaction:

F%{x,Q2) = \ £ + ^Cq(x,Q2)

^(cNS(x,Q2)

xqNS(^,Q2^

s £ ^ , Q 2 ) + \cG{x,Q2)

^(^,g2))

, (41)

where qNs{x, Q2) = (u + u — d — d — s — s)(x, Q2) is the flavor non-singlet combination (for three flavors), while the singlet combination E(x,Q2) = J2q(l + q){xiQ2)-> a n d G{x,Q2) is the gluon distribution. The gluon coefficient CG enters only at order as (since the photon can only couple to the gluon via a quark loop). The challenge to understanding the quark and gluon structure of hadrons is to calculate the soft matrix elements . 4 ^ in Eq.(39), or equivalently the parton distributions in Eq.(41), from QCD. At present this can be only be done numerically through lattice QCD, or in QCD-inspired quark models of the nucleon. Considerable progress has been made over the last two decades in attempting to establish a connection between the high energy parton picture of DIS on the one hand, and the valence quark models at low energy on the other 2 4 ~ 2 6 . The underlying philosophy 27,28 has been that if the nucleon behaves like three valence quarks at some low momentum scale ~ fj?, a purely valence quark model may yield reliable twist-two structure functions. Comparison with experiment at DIS scales, where a description in terms of valence quarks will no longer be accurate, can then be made by evolving the structure function to higher Q2 via the renormalization group equations.

217

Renormalization Group Equations In an interacting field theory like QCD quantities such as coupling constants, masses, as well as wave functions (operators), must be renormalized. The renormalization procedure introduces an arbitrary renormalization scale, fi2, into the theory, although of course the physics itself cannot depend on /j,2. In the following we will consider the renormalization of the non-singlet operators corresponding to the F2 structure function. The generalization to singlet operators in straightforward, although one needs to take into account mixing between the singlet quark and gluon operators 10 . If the unrenormalized matrix elements of the OPE are independent of the renormalization scale /i, then ^(N(p)\MOM0)\N(p))

=0•

(42)

Defining the wave function renormalization of the spin-iV non-singlet operator ^ ^bare = %N 0^,n, where ZN is the renormalization constant, Eq.(42) can be rewritten as: "ik

+l3{9)

k

~ ^ )

CJV

W2MS) =0 ,

(43)

for each spin TV. This is the well-known renormalization group equation for the coefficient functions. In Eq.(43), the strong coupling constant g is renormalized at the scale fj,2, and 7 ^ is the anomalous dimension of the twist-two operator jN=^(\nZN),

(44)

and the /3-function is given by: />(*) = / * ! •

(45)

The solution to Eq.(43) is: CN(Q2/n2,g2)

=

CN(l,g2)exp

~9iQ) dg>Y W

L>)

M)

(46)

where g is the effective (running) coupling constant, defined by dg2/dt = g /3(g), with t = ln(Q 2 //i 2 ) and g(t = 0) = g. The coefficients CN, anomalous dimension ~yN(g) and the /3-function can all be calculated in perturbation

218

theory by expanding in powers of the coupling, g: 1N(9) = 7 (0)JV ^

+ 0(g2) ,

r>3

JL 0(9) = - ^ 1^2 + °(95) , N

2

C (l,g )

16 +

N

= C^

(47)

2

0{g ).

It is then straightforward to derive the equation governing the Q2 evolution of the non-singlet moments of the structure functions, which to lowest order in g is

29,30.

Q"(Q2)=

N 1 = / Cdxx ~ dxaP-iqrfsix)

=

Jo

aa{Q2)^°)N'^ .a*(»2).

Q*V) ,

(48)

where the lowest order non-singlet anomalous dimension is:

and fa = 11 —2iV//3 for Nf active flavors in the evolution. The strong coupling constant has been rewritten in Eq.(48) as:

by putting the arbitrariness of the renormalization scale into the parameter AQCD, known as the QCD scale parameter, In A Q C D = ln/z 2 -167r 2 /(A) g2(fi2)). Once the moments of the structure function or parton distribution are known at /J,2, Eq.(48) can be used to give the moments at any other value of Q2. An intuitive and mathematically equivalent picture for this Q2 evolution is provided by the DGLAP evolution equations 3 1 . For the non-singlet quark distribution one has: dqNs{x,Q2) jt

as(Q2) =

27r

/

fldy — lNs(y,t)

Pqq(x/y)

,

(51)

where Pqq(x/y) is the q -¥ q + g splitting function, which gives the probability of finding a quark with momentum fraction x inside a parent quark with momentum fraction y, after it has radiated a gluon. The splitting function is

219

related to the anomalous dimension by 7 ^ ~ / dzzN~1P(z). The generalization to singlet evolution is again straightforward, but involves a set of coupled quark and gluon equations 8 , 1 0 , 2 3 . The physical interpretation of the DGLAP equations is that as Q2 increases the quarks radiate more and more gluons, which subsequently split into quark and antiquark pairs, which themselves then radiate more gluons, and so on. In this manner the quark-antiquark sea can be generated from a pure valence component. This process modifies the population density of quarks as a function of x, so that the momentum carried by quarks is no longer a static property of the nucleon, but now depends on the resolving power of the probe, Q2. In general, the larger the Q2, the better the resolution, and the more substructure seen in the hadron. It is a remarkable success of perturbative QCD that it can provide a quantitative description of the scaling violations of structure functions 3 2 ~ 3 4 for a large range of x and over many orders of magnitude of Q2. 3.3

Semi-Inclusive

Scattering

Inclusive electron-nucleon scattering is a well-established tool which has been used to study nucleon structure for many years. Somewhat less exploited, but potentially more powerful, is semi-inclusive scattering 3 5 , in which a specific hadron, h, is observed in coincidence with the scattered electron, eN -> e'hX. This process offers considerably more freedom to explore the individual quark content of the nucleon than is possible through inclusive scattering 3 6 . A central assumption in semi-inclusive DIS is that at high energy the interaction and production processes factorize. Namely, the interaction of the virtual photon with a parton takes place on a much shorter space-time scale than the fragmentation of the struck quark, and the spectator quarks, into final state hadrons. Furthermore, the hadronic products of the scattered quark (in the current fragmentation region, along the direction of the current in the photon-nucleon center of mass frame) should be clearly separated from the hadronic remnants of the target (in the target fragmentation region). The cleanest way to study fragmentation is in the current fragmentation region, where the scattered quark fragments into hadrons by picking up qq pairs from the vacuum. The production of a specific final state hadron, h, is parameterized by a fragmentation function, Dg(z), which gives the probability of quark q fragmenting into a hadron h with a fraction z of the quark's (or, at high energy, the photon's) center of mass energy. Because it requires only a single qq pair, the leading hadrons in this region are predominantly mesons. At large z, where the knocked out quark is most likely to be contained in

220

the produced meson, one can obtain direct information on the momentum distribution of the scattered quark in the target. At small z this information becomes diluted by additional qq pairs from the vacuum which contribute to secondary fragmentation. In the QCD-improved parton model the number of hadrons, h, produced at a given x, z and Q2 can be written (in leading order) as: Nh(x,z,Q2)

<x J2 e? q(x,Q2) Dhq{z,Q2) .

(52)

Although factorization of the x and z dependence is generally true only at high energy, recent data from HERMES 37 suggests that at v ~ 10-20 GeV the fragmentation functions are still independent of x, and agree with previous measurements by the EMC 38 at somewhat larger energies. Where the factorization hypothesis breaks down is not known, and the proposed semi-inclusive program at an energy upgraded CEBAF at Jefferson Lab, with v typically ~ 5-10 GeV, will test the limits of the parton interpretation of meson electroproduction. 3-4

Off-Forward Parton

Distributions

The nucleon's deep-inelastic structure functions and elastic form factors parameterize fundamental information about its quark substructure. Both reflect dynamics of the internal quark wave functions describing the same physical ground state, albeit in different kinematic regions. An example of how in certain cases these are closely related is provided by the phenomenon of quark-hadron duality (Section 4.1). Recently it has been realized that form factors and structure functions can be simultaneously embedded within the general framework of off-forward (sometimes also referred to as non-forward, or skewed) parton distributions 39 ' 40 . The off-forward parton distributions (OFPDs) generalize and interpolate between the ordinary parton distributions in deep-inelastic scattering and the elastic form factors. As illustrated in Fig. 3, the OFPD is the amplitude (in the infinite momentum frame) to remove a parton with momentum k^ from a nucleon of momentum P M , and insert it back into the nucleon with momentum k'^ = k^+P^ — Pfj,, where P^ is the final state nucleon momentum. The simplest physical process in which the OFPD can be measured is deeply-virtual Compton scattering 39 (DVCS). For the spin-averaged case, the OFPDs can be defined as matrix elements of bilocal operators ip(—\n/2)C'jflip(\n/2), where A is a scalar parameter, and

221

Figure 3: Leading twist off-forward parton distribution, as seen in deeply-virtual Compton scattering. The nucleon, parton and photon initial (final) momenta are labeled P (P1), k (fc') and q (q1), respectively. Deep-inelastic scattering corresponds to q = q', P = P' •

Ufj, is a light-like vector proportional to (1;0,0, - 1 ) . The gauge link C, which is along a straight line segment extending from one quark field to the other, makes the bilocal operator gauge invariant. In the light-like gauge AMnM = 0, so that the gauge link is unity. The most general expression for the leading contributions at large Q2 can then be written 3 9 : dX /

2TT

eiXx(P'\i>{-\n/2)^iP{\n/2)\P)

= H(x,£,t)

+ E{x,S,t) u{P')i^{K~Jv)u{P)

u{P')^u{P)

+ ••• , (53)

where t = (P' — P)2 and £ = — n • (P' — P), with u(P) the nucleon spinor, and the dots (• • •) denote higher-twist contributions. A similar expression can be derived for the axial vector current. The structures in Eqs.(53) are identical to those in the definition of the nucleon's elastic form factors, Eq.(16). The chiral-even distribution, H, survives in the forward limit in which the nucleon helicity is conserved, while the chiral-odd distribution, E, arises from the nucleon helicity flip associated with a finite momentum transfer 3 9 . The off-forward parton distributions display characteristics of both the forward parton distributions and nucleon form factors. In the limit of P' —> P M , one finds 39 : H(x,0,0) = q{x) , (54) where q(x) is the forward quark distribution, defined through similar light-cone correlations 41 (the dependence on the scale, Q2, of H and q is suppressed). On

222

the other hand, the first moment of the off-forward distributions are related to the nucleon form factors by the following sum rules 3 9 : /

dxH(x,£,t)

J-l

= -t—(GE(t)

+ rGM(t))

,

(55)

i- + T

r1 i J dxE(x,{,t) = Y^(GM(t)

- GE(t)) ,

(56)

where the ^-integrated distributions are in fact independent of £. These sum rules provide important constraints on any model calculation of the OFPDs 4 2 . Higher moments of the OFPDs can also be related to matrix elements of the QCD energy-momentum tensor. Because the form factors of the energymomentum tensor contain information about the quark and gluon contributions to the nucleon angular momentum, the OFPDs can therefore provide information on the fraction of the nucleon spin carried by quarks and gluons, which has been a subject of intense interest now for more than a decade (see Section 4.4). Having introduced the tools necessary to study nucleon substructure in inclusive and exclusive reactions, in the next Section we examine more closely the dependence on flavor and spin of the quark momentum distributions. 4

Flavor and Spin Content of the Nucleon

The distribution of quarks in the nucleon is perhaps the most fundamental problem in hadron physics. Knowing the total structure functions and form factors as a function of x and Q2 is important for determining the scaling properties and global characteristics of the nucleon, however, understanding the relative contributions from different quark flavors gives us deeper understanding of the nucleon's internal structure and dynamics. In this Section we first explore the flavor dependence of the valence quark distributions at large x, then discuss the flavor structure of the quark-antiquark sea. Finally, we review several topical issues concerning the spin structure of the nucleon. 4-1

Valence Quarks

Much of the emphasis in recent years has been placed on exploring the region of small Bjorken-a; at high-energy colliders such as HERA. Delving into the smallx region is necessary in order to determine integrals of structure functions and minimize x —> 0 extrapolation errors when testing various integral sum rules. At small x (x < 0.2) most of the strength of the structure function is due to the quark-antiquark sea generated through perturbative gluon radiation

223

and subsequent splitting into quark-antiquark pairs, g -¥ qq. Genuine nonperturbative effects associated with the nucleon ground state structure are therefore more difficult to disentangle from the perturbative background. Valence quark distributions, on the other hand, reflect essentially longdistance, or non-perturbative, aspects of nucleon structure, and can be more directly connected with low energy phenomenology 2 4 _ 2 6 associated with form factors and the nucleon's static properties. After many years of structure function measurements over a range of energies and kinematical conditions, the valence quark structure has for some time now been thought to be understood. However, there is one major exception — the deep valence region, at x > 0.7. Knowledge of quark distributions at large x is essential for a number of reasons. Not least of these is the necessity of understanding backgrounds in collider experiments, such as in searches for new physics beyond the standard model 4 3 . Furthermore, the behavior of the ratio of valence d to u quark distributions in the limit x —> 1 provides a critical test of the mechanism of spin-flavor symmetry breaking in the nucleon, and a test of the onset of perturbative behavior in large-x structure functions 44 . SU(6) Symmetry Breaking and the d/u Ratio The precise mechanism for the breaking of the spin-flavor SU(6) symmetry is a basic question for hadron structure physics. In a world of exact SU(6) symmetry, the wave function of a proton, polarized say in the +z direction, would be given by 4 5 : Pt=

- 5 " t (ud)s=o + —?==u t (ud)s=i -

l

o u + (ud)s=i

\Fi

- -d t (uu)s=i

g-d 4- {uu)s=i ,

(57)

where the subscript S denotes the total spin of the two-quark component. In this limit, apart from charge and isospin, the u and d quarks in the proton would be identical, and the nucleon and A would, for example, be degenerate in mass. In deep-inelastic scattering, exact SU(6) symmetry would be manifested in equivalent shapes for the valence quark distributions of the proton, which would be related simply by uva\(x) = 2dva,\(x) for all x. From Eq.(32), for the neutron to proton structure function ratio this would imply: p

= |

[SU(6) symmetry].

(58)

In nature spin-flavor SU(6) symmetry is, of course, broken. The nucleon and A masses are split by some 300 MeV. Furthermore, it is known that the d

224

quark distribution in DIS is considerably softer than the u quark distribution, with the neutron/proton ratio deviating at large x from the SU(6) expectation. The correlation between the mass splitting in the 56 baryons and the large-x behavior of F^/F^ was observed some time ago 4 6 _ 4 8 . Based on phenomenological 47 and Regge 48 arguments, the breaking of the symmetry in Eq.(57) was argued to arise from a suppression of the 'diquark' configurations having 5 = 1 relative to the 5 = 0 configuration, namely: (qq)s=o > (qq)s=i,

x -> 1 .

(59)

Such a suppression is in fact quite natural 4 9 ' 5 0 if one observes that whatever mechanism leads to the observed N — A splitting (e.g. color-magnetic force, instanton-induced interaction, pion exchange), it necessarily acts to produce a mass splitting between the two possible spin states of the two quarks which act as spectators to the hard collision, (qq)s, with the 5 = 1 state heavier than the 5 = 0 state by some 200 MeV. From Eq.(57), a dominant scalar valence diquark component of the proton suggests that in the x —> 1 limit F$ is essentially given by a single quark distribution (i.e. the u), in which case: jpn

-I

-2p -¥ - ,

J

•0

[5 = 0 dominance] .

(60)

This expectation has, in fact, been built into most phenomenological fits to the parton distribution data 3 2 ~ 3 4 . An alternative suggestion, based on perturbative QCD, was originally formulated by Farrar and Jackson 51 . There it was argued that the exchange of longitudinal gluons, which are the only type permitted when the spins of the two quarks in (qq)s are aligned, would introduce a factor (1 - x)1/2 into the Compton amplitude — in comparison with the exchange of a transverse gluon between quarks with spins anti-aligned. In this approach the relevant component of the proton valence wave function at large x is that associated with states in which the total 'diquark' spin projection, Sz, is zero:

(qq)s,=o » (qq)s,=i,

x -> 1 .

(61)

Consequently, scattering from a quark polarized in the opposite direction to the proton polarization is suppressed by a factor (1 — x) relative to the helicityaligned configuration. This is related to the treatment based on counting rules where the large-x behavior of the parton distribution for a quark polarized parallel (ASZ = 1) or antiparallel (A Sz = 0) to the proton helicity is given by qn(x) = (1 x)2n~l+AS",

225

where n is the minimum number of non-interacting quarks (equal to 2 for the valence quark distributions). In the a; -+ 1 limit one therefore predicts: -j^p- -*• •= ,

> •=

[Sz = 0 dominance] .

(62)

Similar predictions can be made for the ratios of polarized quark distributions at large x (see Section 4.4). The biggest obstacle to an unambiguous determination of d/u at large x is the absence of free neutron targets. In practice essentially all information about the structure of the neutron is extracted from light nuclei, such as the deuteron. The deuteron cross sections must however be corrected for nuclear effects in the structure function, which can become quite significant52 at large x. In particular, whether one corrects for Fermi motion only, or in addition for binding and nucleon off-shell effects 44 , the extracted Fg/Fg ratio can differ by up to ~ 50% for beyond x ~ 0.5. A number of suggestions have been made how to avoid the nuclear contamination problem 5 3 ~ 5 7 . One of the more straightforward ones is to measure relative yields of TT+ and TT~ mesons in semi-inclusive scattering from protons in the current fragmentation region 58 . At large z (z being the energy of the pion relative to the photon) the u quark fragments primarily into a TT+ , while a d fragments into a TT~ , so that at large x and z one has a direct measure of d/u. From Eq.(52) the number of charged pions produced from a proton target per interval of x and z is at leading order in QCD given by 4 5 : Nf Nf

(a:, z, Q2) ~ 4u(x, Q2) D(z, Q2) + d(x, Q2) D(z, Q2) , 2

2

2

2

2

(x, z, Q ~ Au(x, Q ) D(z, Q ) + d{x, Q ) D(z, Q ) ,

(63) (64)

where D = D* = DJ is the leading fragmentation function (assuming isospin symmetry), and D(z) — D} = D£~ is the non-leading fragmentation function. Taking the ratio of these one finds:

V

' '*

;

Ng+

4 + d/u-D/D

V

;

In the limit z -> 1, the leading fragmentation function dominates, D > D, and the ratio Rn ->• (1/4)d/u. In the realistic case of smaller z, the D/D term in i?" contaminates the yield of fast pions originating from struck primary quarks, diluting the cross

226

section with pions produced from secondary fragmentation by picking up extra qq pairs from the vacuum. Nevertheless, one can estimate the yields of pions using the empirical fragmentation functions measured by the HERMES Collaboration 37 and the EMC 38 . Integrating the differential cross section over a range of z, as is more practical experimentally, the resulting ratios for cuts of z > 0.3 and z > 0.5 are shown in Fig. 4 at Q2 ~ 5 GeV2 for two different asymptotic x -> 1 behaviors 44 ' 50 ' 51 : d/u -¥ 0 (dashed) and d/u -> 1/5 (solid).

0.4

0.6

0.8

1

x Figure 4: Semi-inclusive pion ratio W as a function of x for fixed z fa 1. The dashed line represents the ratio constructed from the CTEQ parameterization 3 2 , while the solid includes the modified d distribution.

The HERMES Collaboration has previously extracted the d/u ratio from the 7r + -7r - difference using both proton and deuteron targets 37 to increase statistics. The advantage of using both p and d is that all dependence on fragmentation functions cancels, removing any uncertainty that might be introduced by incomplete knowledge of the hadronization process. On the other hand, at large x one still must take into account the nuclear binding and Fermi motion effects in the deuteron, and beyond x ~ 0.7 the difference between the ratios with corrected for nuclear effects and those which are not can be quite dramatic 5 8 . Consequently a d/u ratio obtained from such a measurement without nuclear corrections could potentially give misleading results. On the other hand, with the high luminosity electron beam available at CEBAF, one will be able to compensate for the falling production rates at large x and z, enabling good statistics to be obtained with protons alone. Such a measurement will become feasible with an upgraded 12 GeV electron beam, which will enable greater access to the region of large Q2 and W2. Since the

227

x -* 1 behavior is one of the very few predictions for the ^-dependence of quark distributions which can be drawn directly from QCD, the results of such measurements would clearly be of enormous interest. Quark-Hadron Duality Quark-hadron duality provides a beautiful illustration of the connection between structure functions and nucleon resonance form factors 5 9 - 6 3 . In particular, it allows the behavior of valence structure functions in the limit x -> 1 to be determined from the Q2 dependence of elastic form factors 59 ' 63 . The subject of quark-hadron duality, and the relation between exclusive and inclusive processes, is actually as old as the first deep-inelastic scattering experiments themselves. In the early 1970s the inclusive-exclusive connection was studied in the context of deep-inelastic scattering in the resonance region and the onset of scaling behavior. In their pioneering analysis, Bloom and Gilman B9 observed that the inclusive F2 structure function at low W generally follows a global scaling curve which describes high W data, to which the resonance structure function averages. Furthermore, the equivalence of the averaged resonance and scaling structure functions appears to hold for each resonance, over restricted regions in W, so that the resonance—scaling duality also exists locally 60 . Following Bloom and Gilman's empirical observations, de Rujula, Georgi and Politzer 64 pointed out that global duality can be understood from an operator product expansion of QCD moments of structure functions. Expanding the F2 moments in a power series in 1/Q2 (c.f. Eq.(39)), [

*• e~2F2

(£,Q 2 ) = f ] ( ( i V ~QfAy

A$

(aa(Q2))

,

(66)

where A is some mass scale, and the Nachtmann scaling variable £ = 2a;/(1 + y/1 + x2IT) takes into account target mass corrections, one can attribute the existence of global duality to the relative size of higher twists in deep-inelastic scattering. The Q2 dependence of the coefficients A^N' arises only through as(Q2) corrections, and the higher twist matrix elements A^ ' are expected to be of the same order of magnitude as the leading twist term Ayy. The weak Q2 dependence of the low F2 moments can then be interpreted as indicating that higher twist (1/Q2k suppressed) contributions are either small or cancel. Although global Bloom-Gilman duality of low structure function moments can be analyzed systematically within a perturbative operator product expansion, an elementary understanding of local duality's origins is more elusive.

228

This problem is closely related to the question of how to build up a scaling (as Q2 independent) structure function from resonance contributions 65 , each of which is described by a form factor GR{Q2) that falls off as some power of i/Q2. To illustrate the interplay between resonances and scaling functions, one can observe 59 ' 66 that (in the narrow resonance approximation) if the contribution of a resonance of mass MR to the F2 structure function at large Q2 is given by (c.f. Eqs.(19)-(20)) F 2 (fl) = IMv (GR(Q2))2 8{W2 - M%), then a form factor behavior GR(Q2) ~ (1/Q2)n translates into a structure function F 2 (fl) ~ (1 - xR)2n-\ where xR = Q2/(MR - M2 + Q2). On purely kinematical grounds, therefore, the resonance peak at XR does not disappear with increasing Q2, but rather moves towards x = 1. For elastic scattering, the connection between the l/Q2 power of the elastic form factors at large Q2 and the x -> 1 behavior of structure functions was first established by Drell and Yan 6 1 and West 6 2 . Although derived before the advent of QCD, the Drell-Yan—West form factor-structure function relation can be expressed in perturbative QCD language in terms of hard gluon exchange. The pertinent observation is that deep-inelastic scattering at x ~ 1 probes a highly asymmetric configuration in the nucleon in which one of the quarks goes far off-shell after exchange of at least two hard gluons in the initial state; elastic scattering, on the other hand, requires at least two gluons in the final state to redistribute the large Q2 absorbed by the recoiling quark 1 8 . If the inclusive-exclusive connection via local duality is taken seriously, one can use measured structure functions in the resonance region at large £ to directly extract elastic form factors 84 . Conversely, empirical electromagnetic form factors at large Q2 can be used to predict the x —>• 1 behavior of deepinelastic structure functions 5 9 . Integrating the elastic contributions to the structure functions in Eqs.(19)(22) over the Nachtmann variable £, where £ = 2a;/(1 + y/1 + X2/T), between the pion threshold £th and £ = 1, one finds 'localized' moments of the structure functions: d$ e~2 / Jin

f f

F^Q2)

= -£— G2M(Q2) , 4 - ^o

« {"-' F,((, C2) = j ^ -

G

^Q'\++Tf-(Q2) ,

(67)

(68)

229

where r = Q2/4M2 and £0 = 2/(1 + V^l + 1/r) is the value of f at a; = 1. Differentiating Eqs.(67)-(70) with respect to Q2 for N = 2 allows the inclusive structure functions near a; = 1 to be extracted from the elastic form factors and their Q 2 -derivatives 6 3 : Fi oc ^ M2 ,
(71) 1

GM(GM-GE) 51 K 92 a

2

4M (l + r)

(dG\

dG2M

(72)

+ rl + f "r\dQ ( ^2 f + ^ dQf 2 ) > . 2

GM(GM-GE)

4M2(l + r)2

+ +

1

.

2

+T

TT7V ,

T

~i2

fd(GsGM)

dQ

(d{GEGM)

TT^

dQ2

dGM

.

~WI

+

, dG 2 ,

dQ2J-

,

'

.

(?3) (?4)

Note that as r —• co each of the structure functions F\, F2 and g\ is determined by the slope of the square of the magnetic form factor, while g2 (which in deep-inelastic scattering is associated with higher twists) is determined by a combination of GE and GMEquations (71)-(74) allow the x ~ 1 behavior of structure functions to be predicted from empirical electromagnetic form factors. The ratios of the neutron to proton i*\7 F2 and gi structure functions are shown in Fig. 5 as a function of Q2, using typical parameterizations 67 of the global form factor data. While the F2 ratio varies quite rapidly at low Q2, beyond Q2 ~ 3 GeV2 it remains almost Q2 independent, approaching the asymptotic value (dG^/dQ2)/(dGp^/dQ2). This is consistent with the operator product expansion interpretation of de Rujula et al. 64 in which duality should be a better approximation with increasing Q2. Because the F"/F[ ratio depends only on GM, it remains flat over nearly the entire range of Q2. At asymptotic Q2 the model predictions for F\{x —> 1) coincide with those for F2; at finite Q2 the difference between F\ and F2 can be used to predict the x ->• 1 behavior of the longitudinal structure function, or the R = (JLI&T ratio. The pattern of SU(6) breaking for the spin-dependent structure function ratio <7"/<7i essentially follows that for F£/F%, namely 1/4 in the d quark suppression and 3/7 in the helicity flip suppression scenarios 44,50 . However, the pi structure function ratio approaches the asymptotic limit somewhat more slowly than F\ or F2, which may indicate a more important role played by higher twists in spin-dependent structure functions than in spin-averaged.

230 It appears to be an interesting coincidence that the helicity retention model prediction of 3/7 is very close to the empirical ratio of the squares of the neutron and proton magnetic form factors, fJ^/l^p w 4/9. Indeed, if one approximates the Q2 dependence of the proton and neutron form factors by dipoles, and takes G% « 0, then the structure function ratios are all given by simple analytic expressions, F£/F% « F?/Ff w sf /g[ -4 v?J\i\ as Q2 -> oo. On the other hand, for the gi structure function, which depends on both GE and GM at large Q2, one has a different asymptotic behavior, <72/fl2 ~~*• fin/(fJ,p(H~lJ,p)) w 0-345. 0.8 •2/3

^

t

0.6

•

F

2

«

•3/7

0.4 C

0.2

•1/4 1

/

I11/ //. 0

5

2

10

2 g (GeV )

15

20

Figure 5: Neutron to proton ratio for F% (dashed), F2 (solid) and 51 (dot-dashed) structure functions in the limit x —> I.

If the resonance structure functions at large £ are known, one can conversely extract the nucleon electromagnetic form factors from Eqs.(67)-(70). The GM form factor of the nucleon can be extracted directly from the measured Fi(€,Q2) structure function in Eq.(67). Unfortunately, only the F2(£,<22) structure function of the proton has so far been measured in the resonance region. Nevertheless, to a good approximation one can assume that the ratio of electric to magnetic form factors is reasonably well known (see, however, Ref. 68 ), and extract GM from the F 2 structure function in the resonance region via Eq.(68). Using the parameterization of the recent F 2 (£, Q2) data from Jefferson Lab 6 0 , in Fig. 6 we show the extracted GPM compared with a compilation of elastic data. The agreement with data is quite remarkable over the entire range of Q2 between 0 and 3 GeV 2 . The reliability of the duality predictions is of course only as good as the quality of the empirical data on the electromagnetic form factors and reso-

231 3 n—•

•L

1

1

duality extraction

2

1

"0

1

2 3 Q2 (GeV 2 )

4

5

Figure 6: Proton magnetic form factor extracted from the inclusive structure function via Eq.(68).

nance structure functions. While the duality relations 64 are expected to be progressively more accurate with increasing Q2, the difficulty in measuring form factors at large Q2 also increases. Experimentally, the proton magnetic form factor GPM is relatively well constrained to Q2 ~ 30 GeV 2 , and the proton electric GPE to Q2 ~ 10 GeV 2 . The neutron magnetic form factor G1^ has been measured to Q2 ~ 5 GeV 2 , although the neutron GE is not very well determined at large Q2 (fortunately, however, this plays only a minor role in the duality relations, with the exception of the neutron to proton g^ ratio, Eq.(74)). Obviously more data at larger Q2 would allow more accurate predictions for the x —» 1 structure functions, and new experiments at Jefferson Lab 68 will provide valuable constraints. Once data on the longitudinal and spindependent structure functions at large x become available, a more complete test of local duality between elastic form factors and x ~ 1 structure functions can be made. 4-2

Light Quark Sea

Over the past decade a number of high-energy experiments and refined data analyses have forced a re-evaluation of our view of the nucleon in terms of three valence quarks immersed in a sea of perturbatively generated qq pairs and gluons 69 . A classic example of this is the asymmetry of the light quark sea

232

of the proton, dramatically confirmed in recent deep-inelastic and Drell-Yan experiments at CERN 7 0 ' 7 1 and Fermilab 72 . Difference between quark or antiquark distributions in the proton sea almost universally signal the presence of phenomena which require understanding of strongly coupled QCD. Their existence testifies to the relevance of longdistance dynamics (which are responsible for confinement) even at large energy and momentum transfers. Because gluons in QCD are flavor-blind, the perturbative process g ->• qq gives rise to a sea component of the nucleon which is symmetric in the quark flavors. Although differences can arise due to different quark masses, because isospin symmetry is such a good symmetry in nature, one would expect that the sea of light quarks generated perturbatively would be almost identical, u(x) = d(x). It was therefore a surprise to many when measurements by the New Muon Collaboration (NMC) at CERN 70 of the proton and deuteron structure functions suggested a significant excess of J over u in the proton. Indeed, it heralded a renewed interest in the application of ideas from non-perturbative QCD to deep-inelastic scattering analyses. While the NMC experiment measured the integral of the antiquark difference, more recently the E866 experiment at Fermilab has for the first time mapped out the shape of the d/u ratio over a large range of x, 0.02 < x < 0.345. Specifically, the E866/NuSea Collaboration measured /z+fj,~ Drell-Yan pairs produced in pp and pd collisions. In the parton model the Drell-Yan cross section is proportional to: o* oc £ > 2 (qP(Xl) q\x2)

+
(75)

Q

where h = p or D, and x\ and x2 are the light-cone momentum fractions carried by partons in the projectile and target hadron, respectively. Using isospin symmetry to relate quark distributions in the neutron to those in the proton, in the limit x\ > x2 (in which q(xi)
^ 2aPP

=

I (i + 2\

fen u(x2)J

4 + d(si)/«(gi)

4 + d(x1)/u(x1)-d(x2)/u{x2)

. . '

{

'

Corrections for nuclear shadowing in the deuteron 7 3 ' 7 4 , which are important at x <£ 0.1, are small in the region covered by this experiment. The relatively large asymmetry found in these experiments, shown in Fig. 7 implies the presence of non-trivial dynamics in the proton sea which does not

233

have a perturbative QCD origin. The simplest and most obvious source of a non-perturbative asymmetry in the light quark sea is the chiral structure of QCD. Prom numerous studies in low energy physics, including chiral perturbation theory 1 3 , pions are known to play a crucial role in the structure and dynamics of the nucleon (see Section 2). However, there is no reason why the long-range tail of the nucleon should not also play a role at higher energies.

2

» i n| !

•

1.5

IS

l> 1

4$?-

""

-

'

E866 d a t a 0.5 0

0.1

0.2

0.3

x Figure 7: Flavor asymmetry of the light antiquark sea, including pion cloud (dashed) and Pauli blocking effects (dotted), and the total (solid).

As pointed out by Thomas 75 , if the proton's wave function contains an explicit 7r + n Fock state component, a deep-inelastic probe scattering from the virtual n+, which contains a valence d quark, will automatically lead to a J excess in the proton. To be specific, consider a model in which the nucleon core consists of valence quarks, interacting via gluon exchange for example, with sea quark effects introduced through the coupling of the core to qq states with pseudoscalar meson quantum numbers (many variants of such a model exist — see for example Refs. 76 ' 77 ). The physical nucleon state (momentum P) can then be expanded (in the one-meson approximation) as a series involving bare nucleon and two-particle meson-baryon states: \N(P))phys + ^

= VZ

{|iV(P)) bare

f dy d 2 k x gMNB

(t>BM{y,ki_) | B ( y , k x ) ; A f ( l - y , - k x ) > } , (77)

B,M

where M = ir, K, • • • and B = N, A, A, - • -. The function BM{y, kj_) is the probability amplitude for the physical nucleon N to be in a state consisting of

234

a baryon B and meson M, having transverse momenta kj_ and — kj_, and carrying longitudinal momentum fractions y = k+/P+ and 1-y = (P+ — k+)/P+, respectively. The bare nucleon probability is denoted by Z, and im splitting function) is given by 7 5 > 8 °- 8 2 : t U,\ - ^INN f d2kr J*N(**N) / - " ( » ) - -ifcHT J (l-y)y(M>-s„N)2

(k2T + y2M2\ [ 1-y ) •

(79)

For the nNN vertex a pseudoscalar 175 interaction has been used, although in the IMF the same results are obtained with a pseudovector coupling. The invariant mass squared of the irN system is given by: _ k2T + ml

SnN —

y

1

k2T + M2 '-.

1-2/

,

(oU)

and for the functional form of the nNN vertex function ^ ^ ( s w j v ) we can take a simple dipole parameterization, . T V J V ^ J V ) = ((A2 + M 2 )/(A 2 + s^jv)) , normalized so that the coupling constant Q^NN has its standard value (= 13.07) at the pole (J-(M2) = 1). Note that the contribution from the pion cloud in Eq.(78) is a leading twist effect, which scales with Q2 (at leading order the Q2 dependence of d — u in Eq.(78) enters through the leading-twist quark distribution in the pion, • A + + 7r~, the A will actually cancel some of the J excess generated through the

235

nN component, although this will be somewhat smaller due to the larger mass of the A. The relative contributions are partly determined by the irNN and irNA vertex form factor. The form factor cut-offs A can be determined phenomenologically by comparing against various inclusive and semi-inclusive data 83 , although the most direct way to fix these parameters is through a comparison of the axial form factors for the nucleon and for the iV-A transition. Within the framework of PCAC these form factors are directly related to the corresponding form factors for pion emission or absorption. The data on the axial form factor are best fit, in a dipole parameterization, by a 1.3 (1-02) GeV dipole for the axial AT (N-A transition) form factor 84 , which gives a pion probability in the proton of PS 13%(10%). With these parameters Fig .6 shows the d/u ratio in the proton due to -KN and 7r A components of the nucleon wave function (dashed line) 8 5 . Data 32 on the sum of the u and d (which is dominated by perturbative contributions) has been used to convert the calculated d—u difference to the d/u ratio. The results suggest that with pions alone one can account for about half of the observed asymmetry, leaving room for possible contributions from other mechanisms. Another mechanism which could also contribute to the d — u asymmetry is associated with the effects of antisymmetrization of qq pairs created inside the core 2 6 , 8 6 . As pointed out originally by Field and Feynman 87 , because the valence quark flavors are unequally represented in the proton, the Pauli exclusion principle will affect the likelihood with which qq pairs can be created in different flavor channels. Since the proton contains 2 valence u quarks compared with only one valence d quark, uu pair creation will be suppressed relative to dd creation. In the ground state of the proton the suppression will be in the ratio d : u = 5 : 4. Phenomenological analyses in terms of low energy models (specifically, the MIT bag model 2 5 ) suggest that the contribution from Pauli blocking can be parameterized as {d — u ) P a u h = r P a u h ( a + 1)(1 - x)a, where a is some large power, with normalization, T P a u h , less than « 25%. Phenomenologically, one finds a good fit with a sa 14 and a normalization r P a u l i « 7%, which is at the lower end of the expected scale but consistent with the bag model predictions 25 . Together with the integrated asymmetry from pions, rn ~ 0.05, the combined value r = T* + r P a u h « 0.12 is in quite reasonable agreement with the experimental result, 0.100 ± 0.018 from E866. Although the combined pion cloud and Pauli blocking mechanisms are able to fit the E866 data reasonable well at small and intermediate x (x < 0.2), it is difficult to reproduce the apparent trend in the data at large x towards zero asymmetry, and possibly even an excess of u for x > 0.3. Unfortunately, the

236

error bars are quite large beyond x ~ 0.25, and it is not clear whether any new Drell-Yan data will be forthcoming in the near future to clarify this. A solution might be available, however, through semi-inclusive scattering, tagging charged pions produced off protons and neutrons. Taking the ratio of the isovector combination of cross sections for n+ and ir~ production 88 : N% + "

p

3 fu-d-d

=

+ u\

(D + D\

m )

NS the difference d — u can be directly measured provided the u and d quark distributions and fragmentation functions are known. The HERMES Collaboration has in fact recently measured this ratio 8 9 , although there the rapidly falling cross sections at large x make measurements beyond x ~ 0.3 challenging. On the other hand, a high luminosity electron beam such as that available at Jefferson Lab, could, with higher energy, allow the asymmetry to be measured well beyond x ~ 0.3 with relatively small errors. This would parallel the semi-inclusive measurement of the d/u ratio through Eq.(65) at somewhat larger x. 0.4

0.2

pn pion cloud


«*-T

•""-.*--

s'''n~p

symmetric core

-0.2

-0.4

0.5

1

1.5

2

Q (GeV ) Figure 8: Neutron electric form factor in the pion cloud model 90 . The direct ir~ coupling contribution is labeled "7r~p", and the recoil proton "p-K~".

If a pseudoscalar cloud of qq states plays an important role in the d/u asymmetry, its effects should also be visible in other flavor-sensitive observables, such as electromagnetic form factors 90 . An excellent example is the electric form factor of the neutron 9 1 , a non-zero value for which can arise from a pion cloud, n —> piv~. Although in practice other effects 9 2 _ 9 4 such

237

as spin-dependent interactions due to one gluon exchange between quarks in the core will certainly contribute at some level, it is important nevertheless to test the consistency of the above model by evaluating its consequences for all observables that may carry its signature. To illustrate the sole effect of the pion cloud, all residual interactions between quarks in the core can be switched off, so that the form factors have only two contributions: one in which the photon couples to the virtual IT and one where the photon couples to the recoil nucleon:

FiAQ2) = f dy (/$(
•

(82)

The recoil nucleon contribution is described by the functions: f{N)(,i 71

(12\ W V

'

= ^&NN

'

rf2k

f

-L

167T3 J 2/2(1 _

^(gJVTT.t) -^(SJVTT,/)

_ M*)(8N„,f

y) {sNni

- M^)

x(fci + M 2 ( l - j / ) 2 - ( l - y ) 2 « r ) , # ) ,

12

Q2)

^

W

_ 2>9lNN

'

f

^k-L

F(sNn,i)

16TT3 J y 2 ( i _ y) x(-2M2)(l-j/)2,

(sjV]ri .

(83)

F(sNnj)

_ M*)(sN*,f

- M*) (84)

where the squared center of mass energies are: SN7r,i(f) =S*N

+ —

• ((1 - y)~

± k±j

,

(85)

with S„N defined in Eq.(80). The contribution from coupling directly to the pion is: f(*)( Jl

rf2k Ql\ = Z&NN f -L F(s„N,i) F(s*Nj) "' ' 167T3 7 l / 2 ( 1 _ 2 / ) ( S 7 r J v . _ M 2 ) ( s ^ / _ M 2 )

K V

x(*i + M2(l-y)2-y2^) J2 u/, v ;

j

167r3

y2{1_

y) {s^Ni

,

(86)

__ M2){SnNJ

2

x {2M y(l - y)) ,

_

M2)

(87)

where the nN squared center of mass energies are:

s^N,i(f) = S*N + YZ~ • {y x

± k±

) •

( 88 )

238

The N -> TTN splitting functions are related to the distribution functions in Eqs.(83) and (86) by:

fl*Hv,Q2 = o) = Mv), N)

fi (v,

2

Q = 0) = fN*(y)

(89) = UN{\

- y) .

(90)

Using the same pion cloud parameters as in the calculation of the d/u asymmetry in Fig. 7, the relative contributions to G% from the n~ and recoil proton are shown in Fig. 8. Both are large in magnitude but opposite in sign, so that the combined effects cancel to give a small positive G%, consistent with the data. Note, however, that the Pauli blocking effect plays no role in form factors, since any suppression of u relative to J here would be accompanied by an equal and opposite suppression of u s e a relative to d sea , and form factors always contain charge conjugation odd (valence) combinations of flavors. The fact that the model prediction underestimates the strength of the observed G% suggests that other mechanisms, such as the color hyperfine interaction generated by one-gluon exchange between quarks in the core 9 2 ' 9 3 , are likely to be responsible for some of the difference. The lowest order Hamiltonian for the color-magnetic hyperfine interaction 92 between two quarks is proportional to (as/mimj)Si • Sj. Because this force is repulsive if the spins of the quarks are parallel and attractive if they're antiparallel, from the SU(6) wave function in Eq.(57) it naturally leads to an increase in the mass of the A and a lowering of the mass of the nucleon. The same force also leads to the softening 50 of the d quark distribution relative to the u (see Eq.(59)). Furthermore, it leads to a distortion of the spatial (and hence charge) distributions of quarks in the neutron, pushing the two (negatively charged) d to the periphery of the neutron, while forcing the (positively charged) u in the center, giving rise to a negative charge radius 9 3 , (J2ieiTV) • In the harmonic oscillator model at leading order in as, the hyperfine interaction gives rise to a neutron electric form factor 93 :

G£(Q2) = - g ( I > r i ) 2 2 ex P H2 2 /6a 2 ) , \

i

(91)

In

where a can be related to decay amplitudes and charge radii 9 2 . Taking the value a = 0.243 from the ratio of neutron to proton charge radii, the resulting form factor in Fig. 9 agress quite well with the available G^ data. More accurate data, which will soon be available from Jefferson Lab and elsewhere over a range of Q2 will allow more systematic comparison of the various mechanisms which contribute to SU(6) symmetry breaking.

239 0.1 0.08 «g0.06

Hyperfine interaction

°0.04 0.02 0

0

0.5

1 2 2 (GeV 2 )

1.5

2

Figure 9: Electric neutron form factor in the hyperfine perturbed quark model with a harmonic oscillator potential.

4-3

Strange Quarks in the Nucleon

A complication in studying the light quark sea is the fact that non-perturbative features associated with u and d quarks are intrinsically correlated with the valence core of the proton, so that effects of qq pairs can be difficult to distinguish from those of antisymmetrization or residual interactions of quarks in the core. The strange sector, on the other hand, where antisymmetrization between sea and valence quarks plays no role, is therefore more likely to provide direct information about the non-perturbative origin of the nucleon sea 9 5 . Evidence for non-perturbative strangeness is currently being sought in a number of processes, ranging from semi-inclusive neutrino induced deepinelastic scattering to parity violating electron-proton scattering. As for the d - u asymmetry, perturbative QCD alone generates identical s and s distributions, so that any asymmetry would have to be non-perturbative in origin. In deep-inelastic scattering, the CCFR collaboration 96 analyzed charm production cross sections in v and v reactions, which probe the s and s distributions in the nucleon, respectively. The resulting difference s-s, indicated in Fig. 10 by the shaded area, has been extracted from the s/s ratio and absolute values of s + s from global data parameterizations. The curve in Fig. 10 corresponds to the chiral cloud model prediction for the asymmetry (in analogy with the pion cloud in Section 4.2), in which the strangeness in the nucleon is carried by kaons and hyperons, so that the s and s quarks have quite different origins 77 ' 97 . Taking the A hyperon as an

240

0.1

0.05

0 -0.05

•

0

0.2

0.4

•

0.6

x Figure 10: Strange quark asymmetry in the proton arising from a kaon cloud of the nucleon. The shaded region indicates current experimental limits from the C C F R Collaboration 9 6 .

illustration (the results generalize straightforwardly to other hyperons such as the E), the difference between the s and s can be written 9 8 : s(x) - s(x) = f J

^ x

{fAK(y)

sA(x/y)

- fKA(y)

sK(x/y))

,

(92)

y

where the K distribution function /KA is the analog of the TTN splitting function in Eq.(79), and the corresponding A distribution f\K(y) = fK\(l - y)In the IMF parameterization of the KNY (Y = A, S) vertex function, because the s distribution in a kaon is much harder than the s distribution in a hyperon, the resulting s — s difference is negative at large x, despite the kaon distribution in the nucleon being slightly softer than the hyperon distribution 9 8 . With a dipole cut-off mass of A ~ 1 GeV, the kaon probability in the nucleon is w 3%. On the other hand, the exact shape and even sign of the s - s difference as a function of x is quite sensitive to the shape of the KNY vertex 9 8 . Overall, while the current experimental s — s difference is consistent with zero, it does also consistent with a small amount of non-perturbative strangeness, which would be generated from a kaon cloud around the nucleon 98 . Of course other, heavier strange mesons and hyperons can be added to the analysis, although in the context of chiral symmetry " the justification for inclusion of heavier pseudoscalar as well as vector mesons is less clear. The addition of the towers of heavier mesons and baryons has also been shown in a quark model 77

241 to lead to significant cancellations, leaving the net strangeness in the nucleon quite small.

0.08 • HAPPEX

CO

0.04

total

+ 0

Eq

ti

M

•

E

-0.04 0.25

0.5 0.75 Q2 (GeV 2 )

Figure 11: Strange electromagnetic form factors of the proton compared with a kaon cloud prediction, with the magnetic (M), electric (E) and total contributions indicated. For the HAPPEX data 1 0 1 , r m 0.4.

Within the same formalism as used to discuss strange quark distributions one can also calculate the strangeness form factors of the nucleon, which are being measured in parity-violating electron scattering experiments at MITBates 10° and Jefferson Lab 101 . The HAPPEX Collaboration at Jefferson Lab 1 0 1 has recently measured the left-right asymmetry ALR in ep —> ep elastic scattering, which measures the 7*Z interference term: A =

OR - VL PR +CTL

eGEGpE{z)+r

-GF •KCLe. rp

1

lV2

G%+TG%

/ - ( p(Z)

(1

4 sin2 6W) e' GPM GpA^z))

, (93)

with e = (l + 2(l + r)tan 2 (0/2)) \ and e' = 0 - ( i + T ) ( 1 - e 2 ) (the Q2 dependence in all form factors is implicit). Using isospin symmetry, one can relate the electric and magnetic form factors for photon and Z-boson exchange via: rp(Z)

^E,M

_ 1^(7=1) — 4UB,M

e V sin' w G EM

~lpE,M J

(94)

242

where GE M' is the isovector form factor. At forward angles, the asymmetry is sensitive to a combination of strange electric and magnetic form factors shown in Fig. 11 for Q 2 = 0.48 GeV 2 . With a soft KNY form factor the contributions to both GE and GSM are small and slightly positive 98 , in agreement with the trend of the data. This result is consistent with the earlier experiment by the SAMPLE Collaboration at MIT-Bates 1 0 °, G8M = +0.23 ± 0.44 at Q 2 = 0.1 GeV2 in a similar experiment but at backward angles. Although the experimental results on non-perturbative strangeness in both structure functions and form factors are still consistent with zero, they are nevertheless compatible with a soft kaon cloud around the nucleon. Future data on GSE ,M from the HAPPEX-II and GO experiments at Jefferson Lab with smaller error bars and over a large range of Q2, as well as the remaining data on the proton and deuteron from SAMPLE, will hopefully provide conclusive evidence for the presence or otherwise of a tangible non-perturbative strange component in the nucleon. 4-4

Polarized Quarks

Most of the discussion thus far has dealt with the flavor dependence of quarks in the nucleon. On the other hand, there has been considerable interest over the past decade in how the spin of the nucleon is distributed amongst its constituents 1 0 2 - 1 0 4 . Spin degrees of freedom allow access to information about the structure and interactions of hadrons which would otherwise not be available through unpolarized processes. Indeed, experiments involving spin-polarized beams and targets have often yielded surprising results and presented severe challenges to existing theories. A fundamental sum rule for the spin in the nucleon states that: \ = JQ + J9 ,

(95)

where Jq and Jg are the total quark and gluon angular momenta, which can be decomposed into their helicity and orbital contributions: Jq = ^ A E + Lq ,

(96)

Jg = AG +Lg .

(97)

In particular, A S , which is defined as the forward matrix element of the axial current, AE = (7V|^7375^|JV), measures the total helicity of the nucleon carried by quarks, which for three flavors is: AE = Au + Ad + As .

(98)

243

In non-relativistic quark models the spin of the nucleon is carried entirely by valence quarks, so that A E N R Q M = 1The gluon helicity, AG, can be measured in high-energy polarized protonproton collisions, via charm production through quark-gluon fusion, or in production of jets with high transverse momentum 105 . The orbital contributions, Lq and Lg, can in principle be extracted from measurements of off-forward parton distributions in DVCS, or deeply-virtual meson production experiments 39 . Currently there is little empirical information on the gluon polarization, and the angular momentum distributions are totally unknown. Note that each term in Eqs.(96) and (97) is renormalization scheme and scale dependent, and only the quark helicity can be defined in a gauge-invariant manner 3 9 . Experimentally, AE (which is also referred to as the singlet axial charge) can be determined from a combination of triplet and octet axial charges, = Au - Ad = gA , gs = Au + Ad -As ,

(99) (100)

ff3

which are determined from /3-decays of the nucleon and hyperons, and the spin-dependent g\ structure function of the nucleon. In the MS scheme, the lowest moment of the g\ structure function (at lowest order in perturbative QCD) is given by 2 3 ' 1 0 4 ' 1 0 6 :

jf*^%.«J*)-[i-(^)](4» + £» + JAE), (ioi) where the ± refers to the proton or neutron. The first spin structure function experiments at CERN 107 suggested a rather small value for AE, in fact consistent with zero, which prompted the so-called 'proton spin-crisis'. A decade of subsequent measurements of inclusive spin structure functions using proton, deuteron and 3 He targets have 1 0 8 determined AE much more accurately, with the current world average value 104 (at a scale of Q2 - 10 GeV2 in the MS scheme) being AE « 0.3. While the spin fractions carried by quarks in the nucleon require only the first moment of the inclusive spin-dependent structure functions, to determine the x dependence of the polarized distributions requires independent linear combinations of Aq which at present can only be obtained from semi-inclusive scattering. Generalizing Eq.(52) to the production of hadrons with a polarized beam and target, eN ->• e'hX, the difference between the number of hadrons produced at a given x, z and Q2 with electron and nucleon spins parallel and antiparallel is (at leading order) given by: ANh

= N^-utix,z,Q2)

~ £e q

2

Aq(x,Q2)

ADhq{z,Q2)

,

(102)

244

where the polarized fragmentation function ADg gives the probability for a polarized quark q to hadronize into a hadron h. In analogy with the large-x behavior of the unpolarized u and d distributions in Section 4.1, the x -)• 1 limit of polarized quark distributions provides a sensitive test of various mechanisms of spin-flavor symmetry breaking. For SU(6) symmetry, the ratio of the polarized to unpolarized quark distributions is:

v

=

I'

ir

=

4

[su(6) s m m e t r

y

y] •

(103)

If the symmetry is broken through the suppression of the 5 = 1 diquark contributions in the nucleon, then in the limit x —• 1: — —> 1 , u

^r

—> -I d o

[5 = 0 dominance] .

(104)

The perturbative QCD prediction (where the dominant configurations of the proton wave function are those in which the spins of the interacting quark and proton are aligned) on the other hand, is: — —-* 1 ,

-j

—*• 1

[St = 0 dominance] .

(105)

Note that the predictions for the Ad quark in particular are quite different in the perturbative and non-perturbative models, even differing by sign. The spin-flavor distributions can be directly measured via polarization asymmetries for the difference between ?r+ and n~ production cross sections on the proton 1 0 9 ' 1 1 0 : + — A M*~ A AT* _ AN£ ANf + ~~ AN£ + ANf +

7T"*"— 7T

l

P

AA.. . _ AJ . 4Au va l - Ad va l

4u v a l - dvai

(106)

where the dependence on fragmentation functions and sea quarks cancels. A combination of inclusive and semi-inclusive asymmetries using protons and deuterons at an energy upgraded Jefferson Lab will allow the spin-dependent Au and Ad distributions to be determined up to x ~ 0.8 with good statistics 1 1 1 , which should be able to discriminate between the various model scenarios in Eqs.(103)-(105). At smaller x, similar combinations of asymmetries could also be able to measure the polarized antiquark distributions, Ad and Au. These are particularly interesting in view of the qualitatively different predictions in nonperturbative models. While chiral (pion) cloud models do not allow any polarization in the antiquark sea, the Pauli exclusion principle on the other hand

245

predicts quite a large asymmetry, (Au — Ad)/(d—u) = 5/3, even bigger than in the unpolarized sea 2 6 ' 1 1 2 . Measurement of the polarized asymmetry in semiinclusive scattering would then enable the relative sizes of the pion and Pauli blocking contributions to d — u to be disentangled. While data on the polarized quark distributions has slowly been accumulating from various experiments, and plans are under way to systematically measure the polarization of the gluons, until recently there has been very little discussion about the fraction of the nucleon spin residing in angular momentum 1 1 3 . This changed somewhat when it was demonstrated 39 that the orbital angular momentum contributions could be determined from off-forward parton distributions measured in deeply-virtual Compton scattering (see Section 3.4). In particular, it was shown 39 that a sum rule can be derived relating moments of the OFPDs to the total angular momentum carried by quarks and gluons: I

dxx(H(x,£,t)

+ E{x,Z,t)}

=A(t)+B{t)

,

(107)

where A and B are form factors of the energy-momentum tensor in QCD 1 1 4 : T£" = \ (rh^W^^ + WiD^tp) ,

(108)

T»" = ^g^F2

(109)

- F'taFva

.

for quarks and gluons, respectively, where the braces {• • •} represent symmetrization of indices. The matrix elements of T^v can be expanded as 3 9 : (P'\T^\P)

= u(P') [i;4(t) 7 < " ( P + P , )">

+ ^B{t)

(P + P')^ia^a(P'

-P)a

+ - • -]u(P) .

(110)

One can then show that the total angular momentum carried by quarks is given by: Jq = \(A(0)

+ B(0J).

(Ill)

Combining the extracted Jq with AE measured in inclusive DIS, one can then determine the orbital angular momentum of the quarks in the nucleon. An analogous sum rule can also be written for the total gluon angular momentum, Jg, which can be obtained from OFPDs measured in deeply-virtual meson

246

production. From this the gluon orbital angular momentum can be extracted once the gluon helicity AG is known. The program to measure the off-forward parton distributions H,E,- • • in deeply-virtual Compton scattering and meson production experiments is difficult, requiring a large coverage of kinematics and knowledge of background such as the Bethe-Heitler process 115 for DVCS. The first steps along the road to mapping out these fundamental quantities are already being taken at Jefferson Lab and HERMES. 5

Conclusion

Thanks to recent advances in accelerator technology that have enabled precise data to be collected at the world's particle accelerators, we have been able to probe the fascinating inner structure of the nucleon with unprecedented clarity. Though much has been learned from inclusive DIS experiments, future analyses of nucleon structure will focus more on semi-inclusive reactions, which will enable the spin and flavor composition of protons and neutrons to be resolved with greater precision. Furthermore, there is a growing appreciation of the need to understand the common underlying physics revealed through a range of observables, from elastic form factors to deep-inelastic structure functions. Some of the most exciting recent developments have been in the study of the non-perturbative structure of the proton sea through asymmetries in sea quark distributions, which illustrate the relevance of chiral symmetry breaking in QCD even at high energies. Important breakthroughs in our understanding of the proton spin have opened the way to accessing for the first time information about the full helicity and orbital momentum distributions in the nucleon. In addition, perhaps longest overdue is the need to determine the valence quark distributions in the region of large Bjorken-a;, which should settle the long-standing puzzle of the precise x -> 1 behavior of structure functions and shed light on the mechanisms of spin-flavor symmetry breaking in the nucleon. To make the inroads necessary to achieve a deeper understanding of these issues will require full utilization of the high luminosities and machine duty factors at modern accelerator facilities such as Jefferson Lab. We can anticipate the new generations of experiments to reveal much more of the intriguing world of subnucleon dynamics. Acknowledgments I would like to thank Jose Goity for organizing an excellent HUGS summer school. This work was supported by the Australian Research Council and DOE

247

contract DE-AC05-84ER40150.

References 1. H. Yukawa, Proc. Phys. Math. Soc. Japan 17, 48 (1935). 2. G.C. Wick, Nature 142, 993 (1938). 3. M. Gell-Mann, Phys. Lett. 8, 214 (1964); G. Zweig, CERN preprint 8419/TH 412 (1964). 4. O.W. Greenberg, Phys. Rev. Lett. 13, 598 (1964); M.Y. Han and Y. Nambu, Phys. Rev. 139, 1006 (1965). 5. W.K.H. Panofsky, in Proceedings of the 14th International Conference on High Energy Physics, Vienna, 1968 (CERN Scientific Information Service, Geneva); E.D. Bloom et al., Phys. Rev. Lett. 23, 930 (1969); J.I. Friedman and H.W. Kendall, Ann. Rev. Nucl. Part. Sci. 22, 203 (1972). 6. H. Fritzsch and M. Gell-Mann, in Proceedings of the 16th. International Conference on High Energy Physics, ed. J.P. Jackson and A. Roberts (National Accelerator Lab., Batavia, Illinois, 1972); H. Fritzsch, M. GellMann and H. Leutwyler, Phys. Lett. B 47, 365 (1973); D.J. Gross and F. Wilczek, Phys. Rev. D 8, 3633 (1973). 7. N. Isgur, in Proceedings of the 26th International Conference on High Energy Physics (ICHEP 92), Dallas, Texas, August 1992. 8. A.J. Buras, Rev. Mod. Phys. 52, 199 (1980). 9. F.J. Yndurain, Quantum Chromodynamics (Springer-Verlag, Berlin, 1983). 10. T. Muta, Foundations of Quantum Chromodynamics (World Scientific, Singapore, 1987). 11. G. Sterman et al., Rev. Mod. Phys. 67, 157 (1995). 12. K.G. Wilson, Phys. Rev. D 10, 2445 (1974); D.G. Richards, these proceedings, nucl-th/0006020. 13. S. Weinberg, Physica (Amsterdam) 96 A, 327 (1979); J. Gasser and H. Leutwyler, Ann. Phys. 158, 142 (1984); B.R. Holstein, hepph/9911449, and these proceedings. 14. A.J.G. Hey and J.E. Mandula, Phys. Rev. D 5, 2610 (1972). 15. E. Leader and E. Predazzi, An Introduction to Gauge Theories and Modern Particle Physics (Cambridge Univ. Press, 1996). 16. I. Niculescu, private communication. 17. R. Hofstadter and R.W. McAllister, Phys. Rev. 98, 183 (1955); R.W. Mcallister and R. Hofstadter, Phys. Rev. 102, 851 (1956). R. Hofstadter et al., Phys. Rev. Lett. 5, 263 (1960).

18. G.P. Lepage and S.J. Brodsky, Phys. Rev. Lett. 43, 545 (1979); Phys. Rev. D 22, 2157 (1980). 19. R.P. Feynman, in Proceedings of the 3rd Topical Conference on High Energy Collisions of Hadrons, Stony Brook, NY, ed. C.N. Yang et al. (Gordon & Breach, New York, 1969); Phys. Rev. Lett. 23, 1415 (1969). 20. J.D. Bjorken and E.A. Paschos, Phys. Rev. 185, 1975 (1969). 21. K. Wilson, Phys. Rev. 179, 1499 (1969); R.A. Brandt and G. Preparata, Nucl. Phys. B 27, 541 (1971); O. Nachtmann, Nucl. Phys. B 63, 237 (1973). 22. R.L. Jaffe, Nucl. Phys. B 229, 205 (1983). 23. T. Weigl and W. Melnitchouk, Nucl. Phys. B 465, 267 (1996). 24. A. Le Yaouanc, L. Oliver, O. Pene and J.C. Reynard, Phys. Rev. D 9, 2636 (1974); R.L. Jaffe, Phys. Rev. D 11, 1953 (1975); N. Cabbibo and R. Petronzio, Nucl. Phys. B 137, 395 (1978); C.J. Benesh and G.A. Miller, Phys. Rev. D 36, 1344 (1987). 25. A.I. Signal and A.W. Thomas, Phys. Rev. D 40, 2832 (1989); A.W. Schreiber, A.W. Thomas and J.T. Londergan, Phys. Lett. B 237, 120 (1989). 26. A.W. Schreiber, A.I. Signal and A.W. Thomas, Phys. Rev. D 44, 2653 (1991). 27. R.L. Jaffe and G.G. Ross, Phys. Lett. 93 B , 313 (1980). 28. G. Parisi and R. Petronzio Phys. Lett. 62 B , 331 (1976). 29. H. Georgi and H.D. Politzer, Phys. Rev. D 9, 416 (1974). 30. D.J. Gross and F. Wilczek, Phys. Rev. D 9, 920 (1974). 31. Yu.L. Dokshitzer, Sov. Phys. JETP 46, 641 (1977); V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15, 439 (1972); L.N. Lipatov, Sov. J. Nucl. Phys. 20, 181 (1974); G. Altarelli and G. Parisi, Nucl. Phys. B 126, 278 (1977). 32. H.L. Lai et al., Eur. Phys. J. C 12, 375 (2000). 33. A.D. Martin, R.G. Roberts, W.J. Stirling and R.S. Thorne, Eur. Phys. J. C 4, 463 (1998). 34. M. Gluck, E. Reya and A. Vogt, Eur. Phys. J. C 5, 461 (1998). 35. P.V. Landshoff and J.C. Polkinghorne, Nucl. Phys. B 33, 221 (1971); J. Ellis, Phys. Lett. B 35, 537 (1971); J.D. Stack, Phys. Rev. Lett. 28, 57 (1972); T. Sloan, G. Smadja and R. Voss, Phys. Rep. 162, 45 (1980); R. Renton and W.S.C. Williams, Ann. Rev. Nucl. Part. Sci. 31, 193 (1981). 36. W. Melnitchouk, in Proceedings of the Workshop on Physics with a High-Luminosity Polarized Electron-Ion Collider, IUCF, April 1999, hepph/9906488.

249

37. K. Ackerstaff, First Results from the HERMES Experiment using Unpolarized Targets, PhD thesis, Univ. Hamburg, 1996. 38. J.J. Aubert et al., Phys. Lett. B 110, 73 (1982); Nucl. Phys. B 213, 213 (1983). 39. X. Ji, Phys. Rev. Lett. 78, 610 (1997); J. Phys. G 24, 1181 (1998). 40. A.V. Radyushkin, Phys. Rev. D 56, 5524 (1997). 41. R.L. Jaffe and X. Ji, Phys. Rev. Lett. 67, 552 (1991); A.V. Manohar, Phys. Rev. Lett. 66, 289 (1991). 42. X. Ji, W. Melnitchouk and X. Song, Phys. Rev. D 56, 5511 (1997). 43. S. Kuhlmann et al., Phys. Lett. B 476, 291 (2000). 44. W. Melnitchouk and A.W. Thomas, Phys. Lett. B 377, 11 (1996). 45. F.E. Close, An Introduction to Quarks and Partons (Academic Press, 1979). 46. R.P. Feynman, Photon Hadron Interactions (Benjamin, Reading, Massachusetts, 1972). 47. F.E. Close, Phys. Lett. B 43, 422 (1973). 48. R. Carlitz, Phys. Lett. B 58, 345 (1975). 49. F.E. Close and A.W. Thomas, Phys. Lett. B 212, 227 (1988). 50. N. Isgur, Phys. Rev. D 59, 034013 (1999). 51. G.R. Farrar and D.R. Jackson, Phys. Rev. Lett. 35, 1416 (1975). 52. W. Melnitchouk, A.W. Schreiber and A.W. Thomas, Phys. Lett. B 335, 11 (1994); Phys. Rev. D 49, 1183 (1994). 53. W. Melnitchouk and J.C. Peng, Phys. Lett. B 400, 220 (1997). 54. W. Melnitchouk, M. Sargsian and M.I. Strikman, Z. Phys. A 359, 99 (1997). 55. P. Souder, in Proceedings of Workshop on CEBAF at Higher Energies, CEBAF, Newport News, 1994; R. Michaels, in Physics and Instrumentation with 6-12 GeV Beams, Jefferson Lab, Newport News, June 1998. 56. HI Collaboration, C. Adloff et al., DESY-99-107, hep-ex/9908059; ZEUS Collaboration, J. Breitweg et al., DESY-99-059, hep-ex/9907010. 57. G.G. Petratos et al., in Proceedings of Workshop on Experiments with Tritium at JLab, Jefferson Lab, Newport News, Virginia, September 1999; I.R. Afnan et al., nucl-th/0006003. 58. W. Melnitchouk, J. Speth and A.W. Thomas, Phys. Lett. B 435, 420 (1998). 59. E.D. Bloom and F.J. Gilman, Phys. Rev. Lett. 16, 1140 (1970). 60. I. Niculescu, Ph.D. thesis, Hampton University, 1999; C. Keppel, talk presented at 7th International Workshop on Deep Inelastic Scattering and QCD (DIS 99), Zeuthen, Germany, Apr. 1999. 61. S.D. Drell and T.-M. Yan, Phys. Rev. Lett. 24, 181 (1970).

250

62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76.

77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90.

G.B. West, Phys. Rev. Lett. 24, 1206 (1970). W. Melnitchouk, preprint ADP-00-13/T397, JLAB-THY-00-08. A. De Riijula, H. Georgi and H.D. Politzer, Ann. Phys. 103, 315 (1975). N. Isgur, S. Jeschonnek, W. Melnitchouk and J.W. Van Orden, in preparation. C.E. Carlson and N.C. Mukhopadhyay, Phys. Rev. D 58, 094029 (1998); Phys. Rev. D 4 1 , 2343 (1989). P. Mergell, U.-G. Meifiner and D. Drechsel, Nucl. Phys. A 596, 367 (1996). M.K. Jones et al., Phys. Rev. Lett. 84, 1398 (2000). A. Watson, Science 283, 472 (1999). P. Amaudraz et al., Phys. Rev. Lett. 66, 2712 (1991). A. Baldit et al., Phys. Lett. B 332, 244 (1994). E.A. Hawker et al., Phys. Rev. Lett. 80, 3715 (1998). W. Melnitchouk and A.W. Thomas, Phys. Rev. D 47, 3783 (1993). A.W. Thomas and W. Melnitchouk, Nucl. Phys. A 631, 296 (1998). A.W. Thomas, Phys. Lett. B 126, 97 (1983). S. Theberge, G.A. Miller and A.W. Thomas, Phys. Rev. D 22, 2838 (1980); A.W. Thomas, Adv. Nucl. Phys. 13, 1 (1984); D.H. Lu, A.W. Thomas and A.G. Williams, Phys. Rev. C 57, 2628 (1998). P. Geiger and N. Isgur, Phys. Rev. D 55, 299 (1997). S. Weinberg, Phys. Rev. 150, 1313 (1966); S.D. Drell, D.J. Levy, and T.M. Yan, Phys. Rev. D 1, 1035 (1970). J. Speth and A.W. Thomas, Adv. Nucl. Phys. 24, 83 (1998). J.D. Sullivan, Phys. Rev. D 5, 1732 (1972). V.R. Zoller, Z. Phys. C 54, 425 (1992); Z. Phys. C 60, 141 (1993). W. Melnitchouk and A.W. Thomas, Phys. Rev. D 47, 3794 (1993). W. Melnitchouk and A.W. Thomas, Phys. Rev. D 47, 3794 (1993); H. Holtmann, A. Szczurek and J. Speth, Nucl. Phys. A 596, 631 (1996). T. Kitagaki et a l , Phys. Rev. D 42, 1331 (1990). W. Melnitchouk, J. Speth and A.W. Thomas, Phys. Rev. D 59, 014033 (1999). F.M. Steffens and A.W. Thomas, Phys. Rev. C 55, 900 (1997). R.D. Field and R.P. Feynman, Phys. Rev. D 15, 2590 (1977). J. Levelt, P.J. Mulders and A.W. Schreiber, Phys. Lett. B 263, 498 (1991). K. Ackerstaffet al., Phys. Rev. Lett. 81, 5519 (1998). W. Melnitchouk, in Proceedings of the Workshop on Exclusive & SemiExclusive Processes at High Momentum Transfer, Jefferson Lab, June 1999, hep-ph/9909463.

251

91. S. Platchkov et al., Nucl. Phys. A 510, 740 (1990); P. Stoler, Phys. Rep. 226, 103 (1993). 92. N. Isgur, G. Karl and R. Koniuk, Phys. Rev. Lett. 41, 1269 (1978). 93. N. Isgur, G. Karl and D.W.L. Sprung, Phys. Rev. D 23, 163 (1981). 94. N. Isgur, Phys. Rev. Lett. 83, 272 (1999). 95. X. Ji and J. Tang, Phys. Lett. B 362, 182 (1995). 96. A.O. Bazarko et al., Z. Phys. C 65, 189 (1995). 97. A.I. Signal and A.W. Thomas, Phys. Lett. B 191, 206 (1987). 98. W. Melnitchouk and M. Malheiro, Phys. Rev. C 55, 431 (1997); Phys. Lett. B 451, 224 (1999); Phys. Rev. C 56, 2373 (1997). 99. A.W. Thomas, W. Melnitchouk and F.M. Steffens, hep-ph/0005043. 100. B. Mueller et al., Phys. Rev. Lett. 78, 3824 (1997). 101. K.A. Aniol et al., Phys. Rev. Lett. 82, 1096 (1999). 102. F.E. Close and R.G. Roberts, Phys. Lett. B 316, 165 (1993). 103. S.D. Bass and A.W. Thomas, Prog. Part. Nucl. Phys. 33, 449 (1994). 104. B. Lampe and E. Reya, hep-ph/9810270. 105. G.K. Mallot, J. Phys. G 25, 1539 (1999). 106. X. Ji and P. Unrau, Phys. Lett. B 333, 228 (1994); X. Ji and W. Melnitchouk, Phys. Rev. D 56, 1 (1997). 107. J. Ashman et al., Phys. Lett. B 206, 364 (1988). 108. K. Abe et al., Phys. Rev. Lett. 79, 26 (1997); B. Adeva et a l , Phys. Rev. D 58, 112001 (1998); K. Abe et al., ibid D 58 112003 (1998). 109. L.L. Frankfurt, M.I. Strikman, L. Mankiewicz, A. Schafer, E Rondio, A. Sandacz and V. Papavassiliou, Phys. Lett. B 230, 141 (1989). 110. K. Ackerstaff, Phys. Lett. B 464, 123 (1999). 111. T.A. Forest, private communication. 112. B. Dressier, K. Goeke, M.V. Polyakov and C. Weiss, Eur. Phys. J. C 14, 147 (2000); M. Gliick and E. Reya, hep-ph/0002182; R.S. Bhalerao, hep-ph/0003075. 113. L. Sehgal, Phys. Rev. D 10, 1663 (1974); X. Ji, J. Tang and P. Hoodbhoy, Phys. Rev. Lett. 76, 740 (1996); S.V. Bashinsky and R.L. Jaffe, Nucl. Phys. B 536, 303 (1998); N. Mathur, S.J. Dong, K.F. Liu, L. Mankiewicz and N.C. Mukhopadhyay, hep-ph/9912289. 114. R.L. Jaffe and A. Manohar, Nucl. Phys. B 337, 509 (1990). 115. M. Vanderhaeghen, P.A.M. Guichon and M. Guidal, Phys. Rev. Lett. 80, 5064 (1998); Phys. Rev. D 60, 094017 (1999).

252

HIGH E N E R G Y ELECTRON N U C L E U S SCATTERING B. W. Filippone Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, CA 91125 The use of high energy electrons to probe the short-range structure of nuclei is discussed. The basic concepts are introduced by reviewing the theoretical and experimental study of nucleon structure as probed with electrons. Then, electron scattering from nuclei is presented with emphasis on several experimental studies including the EMC effect, j/-scaling and Color Transparency. An overview of previous experiments at SLAC, CERN and Jefferson Laboratory (JLAB) is presented, and prospects for future high energy measurements at JLAB are discussed.

1

Overview

These lectures present an introduction to the physics of high energy electron scattering from nuclei. The term "high energy" will be crudely denned in terms of the momentum transfered from the electron to the proton or neutron (nucleon) with values > 1 GeV. This corresponds to a wavelength for the virtual photon exchanged between the electron and nucleus that is comparable to the size of a nucleon; ~ 1 x 1 0 - 1 3 cm or 1 fm. We will first discuss "inclusive" scattering where only the scattered electron is detected. This is the simplest electron scattering experiment and, at least for scattering from a nucleon, it is fairly well understood. We understand this e~nucleon process in terms of scattering from charged point-like quarks within the nucleon. The concepts of the Parton Model and the subsequent "scaling" laws that result are an important introduction to the physics of electron-nucleus scattering. Inclusive e~-nucleus scattering, or A(e,e'), is certainly more complex than e~-nucleon scattering. But here also there is some evidence for "scaling" behavior in terms of the constituents of the nucleus, namely nucleons. Both the existence and violation of the scaling provide important clues into the structure of nuclei at very short distance scales. There is also the potential to probe the quark structure of nuclei and determine if the nuclear medium modifies the quark distributions. It is important to note that the density inside a heavy nucleus corresponds to an average distance between nucleons that is close to the size of the nucleon itself. We will also discuss the relation between inclusive scattering and a type of semi-inclusive scattering A(e, e'p) where the scattered electron and a scattered

253

proton are detected. The A(e, e'p) process can provide important information on the distribution of nucleons in nuclei and may provide access to a novel prediction of Quantum ChromoDynamics (QCD) called Color Transparency. Throughout the discussion, the role of Jefferson Laboratory (JLAB) in addressing this physics will be explored, both in terms of existing data and data that could be obtained with an upgraded, higher energy accelerator. Before beginning the detailed discussion, I would like to acknowledge that a considerable amount of the physics issues and results in these lectures are based on the PhD thesis of two of my students: Tom O'Neill* and John Arrington 2 . 2

Inclusive Electron Scattering from Nucleons

In this section we will develop the formalism of inclusive electron-nucleon, e~ — N, scattering. We will focus on unpolarized scattering and assume that parity is conserved. The physics of high energy polarized scattering will be discussed in a later series of lectures 3 . Parity conservation is a reasonable assumption for momentum transfers that are much less than the mass of the weak vector bosons (W± and Z°, with M ~ 80 - 90 GeV), which is the energy regime of the present lectures. Much of the physics associated with e~—N scattering will be paralleled in the discussion of electron-nucleus, e~-A, scattering. 2.1

Elastic Electron Scattering

First consider unpolarized electron scattering from a structureless spin \ object (eg. a muon). We can write the cross section as

da = -7r^r-

\V!-V2\

( J L ) ( - M |M|2dLips \2EkJ

\2E2J

(1)

where dLips represents the Lorentz Invariant Phase Space factor and the 1 and 2 refer to the initial particles before scattering. Quantum Electrodynamics 4 gives the formula for the squared amplitude with e4 |M| 2 = pL^W*"

(2)

where £*„ is the electron tensor and Wj£ is the muon tensor. The product of these tensors can be evaluated via Trace Theorems 4 , giving for the laboratory frame: da (E' 1 + Q tan 2 (0 — — (TMott i ,-, (3) d£l V E

fe

2

254

(k,E) e" Virtual 4 ^[/l Photon •— y~^ Target Figure 1: Schematic diagram of inclusive electron scattering.

where E, E' and 9 are the measured laboratory variables as denned in Fig. 1 and Q2 is the invariant 4-momentum transfer given by Q2 = 4EE' sin2(9/2). The first term in the above bracket is due to scattering from the muon's charge, while the second term is due to the muon's magnetic moment. The prefactor QMott is the Mott cross section _ 4q2(E')W(f) Mott -QI •

a

(4)

For elastic scattering from a spin | structured object (eg. a proton), the structure must be contained in the hadron tensor - Wp". Defining the 4-vectors as shown in Fig. 2, PI=P'Z

= M2P

fi = ~Q2

(5)

Pn + In = P'n

(p» + q»)2 = MZ=pl + 2Pfiq»-Q2 the structure (eg. the hadron tensor) can only depend on Q2 as there is no other independent scalar quantity. The hadron tensor is determined from the electromagnetic current 4 , which for the muon is J£ = u^^u. The most general form for a proton is Jl = u\Fx (Q2)7li + ^ ^ l i a ^

+ F3(Q2)q,]u

(6)

255

P^ (0, M) Figure 2: Definition of the four-vectors for elastic scattering.

u,u are Dirac spinors and K is the anomalous magnetic moment with K = A*-A*Dirac (« = 1-79 for the proton and K = —1.91 for the neutron). The above simplifies using the conservation of current (sometimes called conservation of probability) which requires d^J1* = q^J*1 = 0 giving F3 = 0. Now if we define (~i

771

771

KQ

UE = -fl — 4 p ^ 2 GM = FX+ then we have for the cross section: da E' jr>

dil

~~ a M o t t i

,-,

(7)

KF2

* I ^

+2.GW(*

(8)

\ E

where r = y*^-. This gives the e~ — p or e~ — n elastic scattering cross section in terms of GE{Q2) and GM(Q2) which are the electric and magnetic elastic form factors respectively. Experimentally GE(Q2), GM(Q2) fall rapidly with Q 2 for Q 2 £ l(GeV/c)2. This is important for A(e,e') experiments. In particular GE = (1+.71C?V2/<=2)

GM = iipGE " M — ^nGE

(9)

256

Pu (0, M) Figure 3: Definition of four-vectors for inelastic scattering.

Note that at low Q2, GE and GM are simply the Fourier transform of the charge and magnetization densities, respectively. 2.2

Inelastic Electron Scattering

Inelastic scattering corresponds to scattering where the mass of the final state system is greater than the mass of the initial particle. This process is depicted in Fig. 3. Again via QED we have

ass-£(!)»-."*•

«

and the four vectors are related via

4 = -Q1 <

= (PM+
(11)

(p*) 2 = M\ = W2 = Pi + 2p„q» + ql = M2 + IMv -Q2 ^ constant The structure is again contained in the hadron tensor, but there are now two independent scalars (eg. any two of Q2, W2 or v = E — E'). The most general form for W£„ is obtained via QED assuming parity conservation. In this case only symmetric terms in n, v are allowed and explicit parity violating terms are absent. Then

257 W-y

Wd

W5

where g^u is metric tensor (with gn = 1,922 = 933 = 9a = _ 1 and all other components = 0). If we now apply current conservation {q^W^v — 0) while noting that q^g^ = +qv, q^p^ - vM, q^q^ = -Q2 we get:

-W1 + ffi(-Q*) + m(Mv)=0

(13)

leading to only two independent structure functions W\{u, Q2) and W2(v, Q2). With this we can rewrite:

^^)[(**-^)(*-^4-

(14) Note that in the lab frame we can choose the z-axis along the direction of the virtual photon 7*, eg. % = (y, 0,0, q3) p„ = (M, 0,0,0).

(15)

Note also that we can write Wu = W22 = Wi

W33 = f 1 - i | W + (&) (f^) 2 q2W2

(16)

= {i-$)w1 + (£) {$) w2. We now have finally for cross section (fa dQ.dE'

CMott

W2(u,Q2) +2W1(v,Q2)t&n2

(|

(17)

which is the inelastic e~ — N cross section. The above cross section is sometimes expressed in terms of a helicity decomposition of the virtual photon, leading to d?
- = Y{(jT + eaL) dCtdE' with:

(18)

258 r 1

<*

_

a

(ML\

(W2-M2\

(_2_\

a

47r Q \E J \1-EJ

\

2M

e = [ l + 2(l + ^ ) t a n 2 ( | ) ] _ 1 PT,L

=

4TT

a

K

)

(19)

W:T,L

where <JT,L are the transverse and longitudinal "photo-production" cross sections (corresponding to transverse and longitudinal polarization of the virtual photon) and K = W 2 M ^ is the "flux" of virtual photons. Note also that WT = Wn = Wi WL = ( § ) W33 = ( l + £ ) W2 -

(20)

WL

The quantity R = ^i. c a n provide information on the substructure of the nucleon since for spin 0 constituents R —> oo, while for spin 1/2 constituents R-¥0. 2.3

The Parton Model

In order to interpret inclusive electron scattering data it is useful to have a simple model to discuss nucleon structure. The Parton Model, due to R. P. Feynman, assumes that the nucleon is composed of structureless, spin | charged constituents (eg. "partons" or quarks). Within this model we can calculate how W\ and W? should behave. Starting with the invariant form of the e~ — p, cross section d2a dQ2dv

4na2 Q4

(E' \E

Q2

cos

1 +

(!)

2^

2 (9 t a n

2

Slv

2M

(21)

then if "quarks" have charge ef, and mass m»;

dV 2

dQ dv

Ana2

(E

~ Q*

\E

= e,-l

cos

1 + - ^ > tan 2 2m?

2) J

V

2m

i

• (22)

Now if we recall that the e~ — N inelastic cross section is d2a 2

47ra2

(E'

m

cos "Ujr2+^itan"u dQ dV- Q 4 - u j we see that we can define a W\ and W2 for quark scattering as

(23)

259

To complete the derivation we need the second assumption of the Parton Model, namely that the quark carries a fraction x of the nucleon's fourmomentum. In the lab frame then we have m* = xM, giving

Wi = & (»-£:)= e? (*) S(x-Ss)

(25)

and Because the quarks are confined within the proton they cannot have a well-defined momentum (from the Heisenberg Uncertainty Principle), rather they must have a momentum distribution fi(x), leading to

»W2=Xi£dxfi{x)eix6(x-$i;) = Eie2xfi(x)=F2(x)^f(Q2) and MW1 =

(26)

±Eie*ifi(x)=F1(x)?f{Q*).

The final result of the Parton Model is that a particular form of the structure function (yWi in one case and MW\ in the other) is independent of Q2. This phenomenon is called "scaling" of the structure functions. It arises naturally in the Parton Model because in this model the inelastic scattering is actually due to elastically scattering from the nucleon's constituents. Scaling is present throughout physics whenever an apparently complicated behavior can be simplified with an assumption about the underlying physics. The appearance of scaling generally signals some simplifying physics or symmetry principle. 3

Phenomenology of Inclusive e~ — N Scattering Data

In this section we will discuss the evidence for scaling and the associated scaling violations seen in e~~ - TV inclusive inelastic scattering. Before addressing the data we will discuss interpretation of the data in terms of the Parton Model. 3.1

Kinematics of e~ — q Scattering

We begin by discussing the kinematics of electron-quark scattering as shown in Fig. 4. We assume that the quark's momentum four-vector is directly related

260

% 05. v) Ej-P„2=m5~0 quark with 4-momentum = c P u Nucleon

P^ (0, M) Figure 4: Definition of four-vectors for inelastic scattering from a "free" quark.

to the nucleon's four-vector via P£ = £PM, where £ = 0 -> 1. Then using energy and momentum conservation:

El-P*q={ZM

+

vY-\ctf

0 = vv2 - |
(27)

'

-Q2 2

then at high energies (Q , v ->• oo), M 2 £ 2 is small compared to Q2 and 2Mv£ and we can approximate £ via

**&VSX

(28)

where x is the Bjorken scaling variable and is related the longitudinal momentum fraction of the struck quark (see next section). Note that if we do not take the high energy limit, we can solve for £ giving *=

i2X

(29)

1 + V /T7W which is called the Nachtmann scaling variable. This variable will be discussed in more detail in later sections. 3.2

The Interpretation of fi{x)

A simple interpretation of fi(x) can be obtained by moving to a special reference frame called the "Infinite Momentum Frame" or IMF. This frame is

F2

1V±N

lTiA

A f~^ \7^ T VJV^ T

Figure 5: The structure function F^ vs. invariant mass of the final state system - W2.

obtained by a boost along the direction of the virtual photon 7* such that all the partons are effectively moving in the longitudinal direction. Also in this frame, because of time dilation, the quark-7* interaction takes place effectively free of complicating quark-quark interactions. In this IMF x is the parton's (quark's) longitudinal momentum fraction and fi(x) is the parton's longitudinal momentum distribution. Because W\ and W2 are part of an invariant amplitude, they can be measured in the laboratory frame. 3.3

The Scaling of Fi,F2

Within the Parton Model, we might expect scaling to occur when Q2 is the only mass scale in the process (i.e. Q2 » M"ff,M2x ~ 1 GeV 2 ). In addition, the approximation in the Parton Model that the quarks are free requires minimal interactions of the struck quark with the other quarks in the nucleon. If the final state has the mass of a nucleon (MJV) this approximation is certainly wrong, as there must be additional interactions to allow the other quarks to acquire sufficient momentum to keep the nucleon "intact". Thus we expect scaling only if W2 = M\ is larger than the mass of the nucleon and its excited states. Fig. 5 shows the structure function F2 vs. W2, where elastic scattering and the nucleon resonances are clearly visible. Beyond about 4 GeV2 the resonance structure disappears. Experimentally for Q2 > 1 GeV 2 and W2 > 4 GeV 2 the data shows evidence for scaling (i.e. F2 and Pi are independent of Q 2 ). This will be discussed in the next section. In this kinematic regime the inclusive scattering is called Deep Inelastic Scattering (DIS) and the structure functions F\ = MW\ and F2 = uW2 are called the

262 Table 1: Summary of Deep Inelastic Scattering Experiments. Experiment Many HI ZEUS BCDMS NMC E665 CDHSW CCFR

Laboratory SLAC DESY DESY CERN CERN Fermilab CERN Fermilab

Beam/Target e~ - N e± - N e± - N H-N fi-N fi-N

u-Fe v - Fe

DIS structure functions. 3.4

Overview of DIS Experimental Data

A considerable amount of experimental data has been compiled over the last 30 years on DIS structure functions. This data has come from a variety of experiments at several accelerator laboratories. Electron, muon and neutrino beams can all be used to measure the structure functions. The neutrino beams provide access to the quark distributions through the weak charged-current interaction which couples to the quarks with a different charge than electromagnetic probes (eg. e ± ,ju ± ). A summary of the different experiments is given in Table I. An example of the quality of the scaling of the proton structure function is shown in Fig. 6. Here F% is plotted vs. Q2 for different values of the Bjorken x variable. Certainly for the higher x points shown in this figure (x > .02) the structure function is nearly independent of Q2 over three orders of magnitude in Q2. Useful compilations of the structure function data can be found in Refs. 5 and 6. Another way to present the data is to look at .Ff plotted vs. x for many different values of Q2. This is shown in Fig. 7 for a subset of the experiments (BCDMS and NMC). The shape of the structure function is clearly evident, but at each value of x there is a smearing of F 2 due to so-called scaling violations. These are effects that introduce a Q2 dependence to the structure functions. This is the topic of the next section.

263

14

— I

1—I

J I I 111

1—I

1

l I I I l|

I I 1 I 11

T "

x = 0.000032 • x = 0.00005 B« D x = 0.00008

1

1—TT

.

12

a

• fl ° * •„ * = 0.00013 " ^ . D. * p a = 0.0002 „ •D°

• •" • **

-°»

. •• * *

„ * = 0.0005

.8°"

. q , " f l 'I D B

8

• E665 o NMC * BCDMS

_D

.m°

3

D ZEUS

x = 0.00032

gS

10

Proton • HI

„ x = 0.0008 x = 0.0013

6»

f

an 0

*

adb

,o«° '

* = 0.002 9

.anfl8

$

4

1 * = 0.0032

»5"D'

fiB p fl* * ^..a-"8*"*

• °B • • B«

6

. . A <"A * = 0 - 0 0 8 T, f

1 . . » • • p* . . . 9 * J ?

. • • • " • * ^

* = 0.013 .oOo» K I a; = 0.02

* ' «

B

.

•0«0»l*

ot»»« ««>•>" "

i=0.032 o o o « » » "

..•••a

a; = 0.05 o o o o o« «<»o«5 • •

X = 0.13

t

0.1

III

o»*4 *

AA

•

•

#

•

* ' , *

OOODOOOOtfCtl

x = 0.32 M

»

I°OJO!HI*IJ

o o o o o o »*#*^****] V *

X = 0.2

)

B

. • • • • • • • • • • •

X = 0.08 o o o O o o CUKMV^ B

I

a; = 0.005

I

oo o oa
. t i n !

l

10

i

I

100

« •

•

a

•

J_

1000

10000

Q2 (GeV/c) 2

Figure 6: Proton structure function F2 vs. Q2 for several values of x. Experimental points are from a number of different experiments, see Ref. 5.

264

.6

BCDMS and NMC d a t a Q2 = 0.75 - 230 GeV2

.4 x

.8

Figure 7: The structure function i<2 plotted vs. x for many different values of Q2.

3.5

Scaling Violations

Often following the observation of some intriguing scaling phenomenon in physics, there is intense interest in the origins and structure of the violations of this scaling. This is certainly true for the nucleon structure functions. In the limit that the quarks behave as free particles at high momentum transfer (which is the result of the decrease in the value of the strong interaction coupling constant with increasing energy and is called asymptotic freedom) there should be perfect scaling in the Parton Model. However as the momentum transfer is increased, the Parton Model does not account for radiative corrections due to the strong interaction. These corrections include the creation of a gluon by the struck quark and can be calculated with perturbative QCD. Some examples of these effects are shown in Fig. 8. In fact, the successful calculation

Figure 8: Perturbative QCD radiative corrections corresponding to radiation of a gluon.

265

of this Q2 dependence was seen as strong confirmation of the correctness of the theory of QCD. These corrections introduce a logarithmic dependence on Q2 F

2

~ l n ( ^ — J

; AQCD ~ 0.3 GeV.

(30)

The other dominant form of scaling violation also involves the creation of a gluon, but in this case the gluon is exchanged with another quark. These corrections are called higher twist. The interaction between the quarks represents the non-applicability of asymptotic freedom at low energies, as the quark in the final state is not a free quark. An example of this type of correction is shown in Fig. 9. These corrections represent correlations between the quarks,

Figure 9: Higher Twist correction corresponding to the exchange of a gluon between quarks.

and must vanish at very high momentum transfer. This can be seen qualitatively since the exchanged gluon must have a Q on the order of the virtual photon's Q in order to influence the interaction. This requires that the quarks be separated by a transverse distance on the order ofl/Q, and the probability that this will occur will drop as ~ 1/Q2- In fact the higher twist corrections fall as powers of M2/Q2. Both of these scaling violations can be seen in the data as shown in Fig. 10, where at low Q2 a power law dependence on Q2 is visible, while at higher Q2 a logarithmic dependence is evident. Other scaling violations are also present including so-called "Target Mass" effects. These violations occur at low Q2 where in our original derivation of the kinematics of the Parton Model (see Eqs. 27 and 28) we ignored the mass of the nucleon. These violations can be removed if the Nachtmann scaling variable (introduced in Eq. 29) is used instead of the Bjorken scaling variable. As can be seen in Eq. 29 as Q2 -> oo the two variables become equal.

266 .40

.35

.30 -

.25

Figure 10: The structure function F2 plotted vs. Q2 for two values of x. At both of these x points there are two types of Q2 dependence visible: a ln(Q2) dependence at high Q2 representative of Perturbative QCD scaling violations and a power law dependence at lower Q2 representative of "higher twist" effects.

4

Inclusive Electron Scattering From Nuclei

For high energy electron-nucleus ( e - —A) scattering two obvious reaction mechanisms are expected to dominate: deep inelastic scattering (DIS) where quarks are the relevant degrees-of-freedom and quasi-elastic scattering (QE) where nucleons are the relevant degrees-of-freedom. These two reaction mechanisms are shown schematically in Fig. 11. While there is no direct way to identify the reaction mechanism in an inclusive scattering experiment, the two processes appear to be dominant in different kinematic regimes. For the Bjorken variable x < 1, the DIS mechanism is more important (at least for the standard DIS region of Q2 > 1 GeV 2 and W2 > 4 GeV 2 ). When x > 1 the quasi-elastic mechanism dominates (at least for the presently measured Q2 values). In the DIS regime the data shows the so-called "EMC effect", where the nuclear structure function is clearly different from that of the free nucleon. For the QE regime the data exhibits another type of scaling behavior known as "y-scaling". Both of these phenomena will be discussed below. 4-1

The EMC effect in e —A scattering

The general form for the e

- nucleus inclusive cross section is

267

ys

Figure 11: Schematic diagrams of DIS and QE scattering for e

d?c dftdv

CM

Wf + 2Wf tan2

— A scattering.

(31)

where W^(x,Q2) and W^(x,Q2) are the nuclear structure functions. In the simplest approximation, the scaling structure function F* — vW£ is directly related to the free proton and neutron structure functions F^Fg via F^ = ZF$ + NFg, where Z and iV are the numbers of protons and neutrons in the nucleus. Or since Fg ~ F£, we have F£ — AF^, where A is the total number of nucleons A = Z + N. The experimental data however do not agree with this simple picture as shown in Fig. 12. Understanding this difference between the free nucleon structure function and the nuclear structure function (the "EMC effect" named after the first experiment to observe the effect) remains a

268

Figure 12: Ratio of nuclear structure function to the free nucleon structure function. The deviation from unity is known as the EMC effect.

theoretical challenge. That such a difference might exist (at least for x > 0.2) because of the structure of the nucleus can be understood with some simple kinematic arguments. The increase above unity seen in Fig. 12 at high x > 0.8 can be understood because of the Fermi motion of the nucleons in a nucleus. Thus while the structure function for a free nucleon must kinematically vanish at x = 1, the Fermi motion allows the nuclear structure function to be nonzero all the way up to x = A. We will discuss this region in more detail in the next section. For the significant drop below 1 in the region x ~ 0.6 we can understand part of the effect by recalling that nucleons are bound in the nucleus with an average binding energy of ~ 10-50 MeV. We will show that even with a binding energy that is small compared to the nucleon mass or Q2, the effects of the binding persist. Consider a stationary bound nucleon with four-vector momentum P^ = (Eo, 0,0,0) with (32) E0 = Mn- E,sep where E3ep is the separation energy or binding energy mentioned above. Following the arguments associated with the derivation of Eq. 11 from Fig. 3,

269

the invariant mass of the residual nucleon from which the quark is extracted is then given by M2x = W2 = = = s

(Pli + q„)2 El + 2E0u - Q2 M2 + El - 2MEsep M 2 - Q 2 [1 - i ]

- Q2 [1 - I (%)]

, > {66 >

where x' = x ( j ^ ) > x. The only difference between this calculation and the one done in connection with Eq. 11 is that here the nucleon has energy E0 ^ M. Thus in a nucleus for a given value of x we are probing the nucleon structure function (F^) at an effectively larger x value (= a;'). Recalling the shape of F^{x) as shown in Fig. 7, where F^ drops rapidly with x, we see that Fj 4 < F£*, at least for the region x ~ 0.6 where Fermi motion effects are not too large. A number of models for the EMC effect have been discussed, most of them relying on shifting F^ with respect to F™. In the Dynamical Rescaling model it is hypothesized that QCD at nuclear densities may cause a nucleon to "swell" such that in a nucleus RJ$ > Rft. This larger confinement size "softens" the quark momentum distributions (via the Uncertainty Principle) pushing the distribution to lower x. But the required increase in the nucleon size Rfj ~ 1.15Rtf - would lead to a significant violation of y-scaling (discussed in the next section) which is not observed. In the Multiquark Cluster Model it is hypothesized that a fraction of the nucleons in a nucleus may exist as 6 or 9 quark "Bags". These bags would have a larger size and therefore softer quark momentum distributions - fi(x). Up to now there is no evidence from other experiments that indicates any appreciable 6 or 9 quark components in the nuclear wavefunction. The "Convolution" + Pion Model attempts to account for the full x dependence of the effect in a somewhat self-consistent manner. This model uses nuclear binding to reduce the contribution to F£ from quarks in nucleons, thus producing the drop in the ratio F^/F^ at medium x ~ 0.6. The source of the nuclear binding (pions) then produces an enhanced contribution in the nucleus, leading to an increase of F^/F^ above unity for x ~ 0.1 — 0.2. The drop below one for x < 0.1 is attributed to a phenomenon known as shadowing whereby real photons fluctuate into vector mesons with increased hadronic interactions with the nucleons in a nucleus. This effect should become important at very low x, since real photons have Q2 = 0 and therefore x — 0.

270

In the Convolution Model we can calculate F^ via MA

F2A(x) = J"" dyf(y)F2N ( j )

(34)

where f(y) is the probability to find a nucleon with a given value of longitudinal momentum y. This f(y) is then given by:

f(y) = J dEd3P5 (y - ^ Q (l + g ) S(p, E)

(35)

where k+ = ko + k3 is the longitudinal momentum, S(p, E) is nucleon spectral function which gives the probability to find a nucleon with a given value of E,p and (1 + f£) is the "flux factor" for a moving nucleon 4 . Despite its apparent success there are serious problems with the Convolution + 7r Model. First, the pions and nucleons are treated incoherently in this model, by separately adding fn(y) and fN(y), without accounting for the obvious correlations between the two. Second, the nucleons are treated as being asymptotically free, allowing application of perturbative QCD. While this is sensible for the struck quark in the case of a free nucleon it is questionable for a nucleon in a nucleus where the nucleon remnant is likely moving slowly with respect to the residual nucleus and can easily interact with the residual nucleus. Thus we see that there is no complete, satisfactory explanation of the EMC effect. In fact it is likely that all three of the models discussed above may be active at some level. Ideally any complete explanation for the effect should also be able to account for the region x > 1 which we will discuss next. 4-2

Quasi-Elastic Scattering

What does a nucleus look like at different length scales? We can answer this question by performing electron scattering from nuclei using virtual photons of different wavelengths. This is depicted graphically in Fig. 13, where the structure function is shown vs. x for three values of Q2. At low Q2, elastic scattering from the nucleus and excitation of discrete states will be important. At moderate Q 2 , quasi-elastic nucleon scattering (Fermi-broadened) will dominate, while at high Q2, DIS scattering from individual quarks will be the most important contribution to the structure function. If the electron scattering process is dominated by quasi-elastic scattering (see lower diagram in Fig. 11), elastic e~ — N scattering takes place with

271

Low Q*; X ~ 10 fm L

2

-+-X

A

If

Moderate Cf ; A, ~ 1 fm

-•X

High Q2;X~ 0.1 fm

-•X

A Figure 13: The nuclear structure function vs. x for three values of Q2. The wavelength of the virtual photon corresponding to each Q2 is also given.

W2 = Mjy. As discussed in Sec. 2.1, elastic e~ — N scattering depends on the elastic electric and magnetic form factors GE(Q2), GM(Q2)Furthermore, elastic scattering from a free stationary nucleon occurs at x = 1. Thus we might expect to see a peak in the cross section near x = 1. However since the nucleon is not at rest inside a nucleus, quasi-elastic scattering can occur at other values of x leading to a finite width for the quasi-elastic peak that is determined by the nucleon momentum distribution. Experimental cross section measurements are shown in Fig. 14, where a quasi-elastic peak is clearly observed, at least at the lower beam energies. Thus the measured cross section contains information on the momentum distribution of the nucleons in a nucleus. This distribution is expected to behave approximately as an interacting degenerate Fermi gas. The behavior of this momentum distribution

272

SLAC E121 3He, 9 300

5

.1

.2 .3 .4 v (GeV)

0.5 1.0 v (GeV)

1

2 3 v (GeV)

Figure 14: Measured cross section vs. energy transfer v for inclusive electron scattering from 3 He from SLAC experiment E121 7 . The quasi-elastic peak is clearly visible at the lower beam energies. The kinematic point x = 1 is indicated by an arrow for each beam energy.

at large values of momentum is very sensitive to the short-range correlations between nucleons. These correlations represent the the short-distance (or high momentum) behavior of the nucleon-nucleon interaction. Theoretically this involves incorporating the nucleon-nucleon interaction into a finite many-body system. While there have been some successes in calculating this for 2- and 3-body systems and for nuclear matter (A -» oo), there has been no successful treatment for finite nuclei. Experimental information is needed for both few-

273

body nuclei as well as heavy nuclei in order to compare with the existing calculations, and provide input to future theoretical efforts. Inclusive electron scattering is one piece of experimental input that may be able to provide information on these short-range correlations. We now discuss the relation of y-scaling to the momentum distribution of nucleons in a nucleus. 4-3

Simple Derivation of y-scaling

Recall that in the Parton Model, the charge-weighted sum of the momentum distributions of the quarks (see Eq. 26) is contained in the scaling structure functions F\ and F% which are determined from the inclusive cross section via

+^ / n

W ^

(36)

For quasi-elastic scattering from a nucleus we can try to construct an analogous "structure function" that depends on the momentum distribution of the nucleons in the nucleus. We can do this by trying to remove most of the Q2 dependence of the cross section via d,v

dadE

'

=KF(y)

(37)

dil>e~ —N elastic

where K is a kinematic factor, F(y) is the new nuclear scaling function and y is the appropriate scaling variable. West 8 first derived the nuclear scaling variable by considering the simple reaction mechanism shown in Fig. 15. With the four-vectors as denned in the figure, we have for the final mass of the outgoing nucleon (which must be M): M2 = E'2 - P12 = (E0 + v)2 -(P + q)2 = E2 + 2E0v + v2 - q2 - 2 P • qNoting that v2 - q2 = —Q2 and assuming that v, q » EQ ~ M we can rewrite the previous equation as 2P-q = 2\P\\q\cos9pq = 2Mu-Q2 or with y — P cos 6pq

2Mv-Q2 y=

—2W~ =

(38) P2. P and EQ and that (39)

, yw

x

m

274

N

< p P' E p)

(0, M.) Figure 15: Kinematics of quasi-elastic scattering.

where yw is the West scaling variable. Thus we see that yw represents the component of P along the virtual photon direction. Note that if P is parallel to q, cos#pg = 1 and yw > 0, while if P is anti-parallel to q, cos6pq = - 1 and Vw < 0. Note also that we can rewrite yw as 2Mi/(l - x) Vw

(41)

such that if yw = 0 then x = 1 if yw > 0 then x < 1 if yw < 0 then x > 1.

(42)

Thus in the above simple model j2„e_ A a (7

dUdE' •^

(43)

dil) e—N elastic

should "scale" for QE scattering with y as the scaling variable and F(y) related to the nucleon momentum distribution in the nucleus. Of course, in analogy with the Parton Model, there are also effects that violate the scaling presented above. We will discuss these scaling violations after we have discussed the assumptions and approximations that are involved in a more complete derivation of y-scaling.

275

4-4

Assumptions and Approximations for y-scaling in the Plane Wave Impulse Approximation

y-scaling can be derived within the Plane Wave Impulse Approximation. In this approximation the scattering takes place as shown in Fig. 15, where the final nucleon is "free" and does not interact with the other nucleons in the final state. We begin by conserving energy and momentum at the *y*NN vertex giving for the scattered nucleon's mass: Wl = M2 = {y + E0)2 - (q + Pf

(44)

or V + EQ =

^M* + {q + P)2.

(45)

Also from energy conservation at the A — (A — 1) - N vertex we have E0 = MA-

yjM*2_x + P 2 = M - E8

(46)

where M* is the mass of the final A — 1 system (which may be in an excited state) and Es is the nucleon's separation or removal energy (a.k.a. binding energy). Note that EQ ^ M2 + P2 because the initial state nucleon is off its mass shell (due to the binding energy). Combining the above two equations gives the condition for energy conservation: Arg = v + M A - ^Mf^

+ P 2 - ^ M 2 + (q + P) 2 = 0.

(47)

Now we can construct the cross section within the impulse approximation:

d^7 = E / d 3 p J'dEsa°JJSi(P,E8)8{k,g)

(48)

where 5, is the spectral function for nucleon i (treating n and p separately), a°JJ is the elastic electron nucleon cross section for an off-mass-shell nucleon, and the delta function ensures energy conservation. To determine a°% requires some assumptions for the off-mass-shell nucleon current. There are several prescriptions to do this; see for example Ref. 9. Noting that Arg = v + M A - ^Mf^

+ P 2 - \ / M 2 + q2 + P 2 + 2Pqcos(9

(49)

276

we can now integrate over d9 using the 6 function. But because — 1 < cos 6 < 1, the integration limits for P and Es are restricted: Pmax

cPa dQ.dE1

fEi

P2dP / JO o

JPmin

f dE ^ saf-°NfS{P,E s)

SArg dcos#

(50)

where ef°^ = ^ / cr°%d<j) represents the integration over d<j> and the factor comes from the S function integration with dArg 9cos0

Pq

a 9

(51)

2

0W + {q + P)2

Now

^M2 + (q + P)2

d2 d£ldE>

dE,

S(P,Ea)a°N. (52)

In the limit that q -> oo, we have PTOax and i£ ma x -^ °o giving J2

(•OO *«C

TOO

PdP / 7 S 27TCT2S- / dUdE' Jpmin Jo

S(P,Es)dEs.

(53)

Here the a°% has been removed from the integration by assuming that there is only a small variation of a over the range in Es, P where the spectral function S(P, Es) is large. This usually results in a 5-10% error, but can be checked explicitly for any given S(P, Ea). Now letting V = -Pmin when — • +£-

(54)

V = Pmin when -2->- - A and noting that y can be calculated from the requirement that v + MA- sjM\I + y^ _ ^ M 2 + (q + j/) 2 = 0,

(55)

we have dUdE1 =.off "eN

'OO

2TT

/

/"O

S(P,Es)dE3=F(y)

(56)

PdP / where P(j/) is the scaling function. ,vl Then •'o since J0°° S(P, Es)dEg = n(P), where n(P) is the nucleon momentum distribution, we can in principle extract n(P)

^^

277

from F(y) via ,„, -1 fldF\ n{P) = 7T- - ^ ~ • (57) v ; 2TT \ydy)y=P This exercise was meant to show what approximations and assumptions are used in the standard derivation of y-scaling. As one example of this, if q < co then the upper integration limits in Eq. 53 are not oo and the scaling function approaches a scaling limit from below; eg. F(y,qi)
if q2 > qx.

(58)

This is one example of a mechanism that breaks the scaling and gives a Q2 dependence to F(y). We will discuss other mechanisms in the next two sections. 4-5

Evidence for y-Scaling

The first evidence for y-scaling was seen in SLAC experiment E121 7 using a 3 He target. Heavier targets were investigated in SLAC experiment NE3 1 0 and more recently in JLAB experiment 89-008 2 . Cross section data from E121 was shown in Fig. 14. Cross sections for an Fe target from 89-008 are shown in Fig. 16. In this experiment the beam energy was fixed at 4.045 GeV and the spectrometer angle was varied to access different Q2. Here we see cross sections that vary by nearly seven orders-of-magnitude for the Q2 range of 0.5 - 7 GeV 2 for the experiment. In addition a quasi-elastic peak is barely visible only at the smallest angles (the expected location of the QE peak - x — 1 - is indicated by an arrow in the figure). From these cross section data the scaling function F(y) can then be extracted. This is shown in Fig. 17. While the data for y < 0 show evidence for scaling, the data at y > 0 clearly do not. This scaling violation is chiefly due to the small contribution from QE scattering at y > 0 (recall that y > 0 corresponds to x < 1). At the measured values of Q2, the region of y > 0 is dominated by resonance production and DIS. Neither of these processes will have a Q2 dependence that matches e~ — N elastic scattering leading to a significant scaling violation. For the region y < 0 there appears to be an approach to a scaling limit. But as shown in Fig. 18, the approach to scaling is from above with a decreasing F(y) with increasing Q2. This is contrary to the scaling violations expected in the PWIA (see sec. 4.4) or from contaminating processes (eg. resonance production and DIS scattering which will increase as Q2 increases). Rather this new type of scaling violation is suggestive of the power law violations (~ M2/Q2) seen in e" - N DIS scattering. This is the topic of the next section.

278 l5J. dl'-kW ' ' I ' ' ' ' I 23", Q"=1.94

m

30°, Q*^8.7B

>

^mmii»^m0mmm

37°, Q»=3.53

55° Q«=4.92

CI

a X)

b

0.0

0.5

1.0

1.5 2.0 v (GeV)

2.5

3.0

3.5

Figure 16: Inclusive cross section for e scattering from an Fe target from JLAB experiment 89-008. The kinematic point x = 1 is shown by an arrow for each spectrometer angle.

4-6

Scaling Violations in y-Scaling: Final State Interactions

In analogy with the "higher twist" scaling violations discussed in Sec. 3.5, similar "nuclear higher twist" effects can contribute to the QE cross section. Two examples of these processes are illustrated in Fig. 19. The process shown in Fig. 19(a) represents a final-state interaction with the mean field of the residual nucleus. This process adds a significant contribution to the cross section because the virtual photon can strike a low momentum off-mass-shell nucleon (in a region where the spectral function is large) and the exchanged pion can bring the final nucleon onto the mass-shell and contribute to the large y region. In analogy with the qualitative discussion of higher twist in Sec. 3.5, this process must decrease rapidly with increasing Q2 as the exchanged pion must be produced at shorter distance scales as Q2 increases. As the probability of finding a nucleon at this shorter distance scale is reduced, this process is suppressed at higher Q2. Calculations 11 indicate that the fall-off should go as M/q, where q is the virtual photon's three-momentum. The second process shown in Fig. 19(b) represents a final-state interaction with a high-momentum correlated nucleon. As this second nucleon is strongly correlated with the struck nucleon, there is a significant probability

279

io2

—i—i—r~ r

101

io"4 io"5

i

i

i

r

[

i

i

i

i

|

i

1

' Loc£&°' ' =

^gS**"**1****10^

1 1 lllllll

[GeV

io"

|

fr

x D

\

+

1 lllllll

llllll

F(y)

1

3

i

1

10°

io"8

i

\

rr1

io"

i

0

Q 2 =0.97 Q 2 =1.94 Q 2 =2.78

! i :

Q2=3.53

]

2

r

ill

/

r J , , l . i

-1.0

0 i

-0.8

i

1

-0.6

l

l

l

l

l

-0.4 -0.2 7 [GeV/c]

Figure 17: Nuclear scaling function vs. y for e experiment 89-008.

l

l

0.0

Q =4.24 Q 2 =4.92 Q 2 =5.75 —1—1

1

1

0.2

1

1

-, : ] 1

1

0.4

scattering from an Fe target from JLAB

that this nucleon will be close to the struck nucleon and can exchange a soft (low-momentum) pion. Thus this process has a much weaker dependence on Q2. Some calculations 12 suggest that this process may be Q2 independent leading to an additional contribution (on top of the PWIA contribution) to F(y) at all Q2. As this process relies on significant high-momentum components in the nuclear wavefunction, the nuclear scaling function at large y may still provide direct access to these components once this process is better understood theoretically. 4-7

Other Scaling Phenomena in e~ — A Scattering

At sufficiently high energies, the strong Q2 dependence of the the nucleon elastic form factors (see Eqs. 9) will eventually reduce the QE component of the cross section even for x > 1. At these energies we would expect to see scaling of ^ ( a ; ) and the cross section dominated by DIS from quarks for x > 1. The present data for e~ — A scattering are certainly much too low in Q2 to observe this scaling as shown in Fig. 20. An interesting speculation regarding scaling of F£ is that a large part of the scaling violation could be due to so-called target mass effects as discussed

280

.35 .30 .25

_l

1 1 1 1 1 1

1

|

1

!

|

!

1

I

I

|

1

1

1

1

|

1

1

1

1

[

I

I

I

l _

y = - 0 . 3 (xl)

'-

•

y = - 0 . 4 (x2)

:

r £

•

y = - 0 . 5 (x4)

-

-

* 1i

tit] $

:

I

J*

*

.10

<3>

.00

1

•

.15

.05

1

]J

% .20 -J>5 fcT

1

1

".

[

it

*If i

i

i

i

i

i

1

i

.. :

i

j J I- -

1, . . , 1 , . , , 1

i

2

3 4 5 Q2 [ G e V / c ] 2

, I I , "

6

7

Figure 18: Nuclear scaling function vs. Q2 for several values of y. The open symbols are from the NE3 experiment and the filled symbols are from JLAB 89-008.

Figure 19: Examples of final state interaction processes that contribute to quasi-elastic scattering: (a) nucleon interacting with the mean field of the residual nucleus, (b) nucleon interacting with a high momentum correlated nucleon.

in Sec. 3.5. There we discussed that using the Nachtmann variable largely eliminates this source of scaling violation. Thus it might be interesting to look at i ^ ( £ ) with the existing data. This was first done for the SLAC NE3 13 data where an intriguing scaling behavior was observed. This effect is shown in Fig. 21 where i/WbCO = F*(Q is shown for e~ - Fe scattering. Clearly there is a suggestion of scaling for £ > 1 even at Q2 ~ 2 — 4 GeV 2 . One possibility that has been suggested 13 is that this behavior is a reflection of a

281

10° r Fe

10 * lO"2 If)"3 >

10"*

*

lO"5 lO"6 lO"7

* n * * * * «

Q a =0.97 Q 2 =1.94 Q 8 =2.78 Q 2 =3.53 Q z =4.24 Q 2 =4.92 Q 2 =5.75

lh *I

8

lO"

1 Figure 20: Nuclear structure function per nucleon vs. x for an Fe target from JLAB experiment 89-008.

x o * • ! . » ! o — I

Q 2 =0.97 Q*=1.94 Q z =2.78 Q 2 =3.53 Q z =4.24 Q 2 =4.92 Q 2 =5.75 1

1

1

1

1

1-

I

I

I

I

I

l _

1.6 Figure 21: Nuclear structure function per nucleon vs. £ for an Fe target from JLAB experiment 89-008.

phenomena known as local duality. This behavior has been observed 14 for the free nucleon structure function first with a slightly different scaling variable u' = 1 + W2/Q2 ~ 1/f. This is shown in Fig. 22, where data from Ref. 15 at several Q2 are compared to the DIS scaling limit structure function from

282 0.4

0.35

0.3 0.25

£0.2 > 0.15 0.1 0.05 0 1

2

3 , 4 , co' = 1 + W 2 /Q 2

5 6 7 8 9

10

Figure 22: Nucleon structure function vs. modified scaling variable a/. The data are from Ref. 15 and the scaling limit curve is from Ref. 16.

Ref. 16. In the modified scaling variable (meant to account for target mass effects) even the resonances appear to average to the scaling limit curve. This has been explained in terms of QCD 17 , where it is shown that the higher twist effects of the resonances only cause a redistribution of strength but do not modify the scaling limit structure function. The Fermi motion of the nucleons in the nucleus (which accounts for the finite width of the quasi-elastic peak) may provide the averaging needed to smooth the quasi-elastic peak and resonances to the scaling limit curve as suggested by local duality. However it has also been suggested 18 that this scaling is accidental because of the different variation with Q2 of the quasi-elastic vs. the DIS component. Measurements at still higher Q2 should help to resolve this issue.

283

4-8

Summary and Open Questions for y-scaling

A fairly substantial data set now exists for high energy A(e, e') scattering cross sections. The recent JLAB data confirms the presence of y-scaling for heavy nuclei and, for the first time, extends the range where y-scaling is observed to y ~ —0.3 to - 0 . 5 GeV/c. In addition, the JLAB experiment has shown the importance of DIS in producing y-scaling violations for Q2 ~ 3 GeV 2 and y ~ 0 to -0.2 GeV/c. The question of the role of final-state interactions (FSI) remains a theoretical challenge. While a significant part of the FSI appears to fall rapidly with Q2, a part of the FSI may remain that is nearly independent of Q2. These theoretical issues need to be resolved before reliable information on the nuclear spectral function can be extracted from inclusive electron scattering data. The experimental observation of the scaling of F£ with £ remains intriguing. The suggestions of local duality may allow the extraction of the nuclear quark distributions in nuclei at x > 1 from existing and future data. However the possibility of an accidental scaling must be resolved through further measurements at higher Q2. An approved experiment at JLAB - 99-015 - would extend the previous data up to a beam energy of 6 GeV, and extend the Q2 reach up to 12 GeV 2 . Future measurements with beam energies up to 12 GeV, promise extension of these data to Q2 ~ 20 GeV 2 . 5

Semi-Inclusive A(e, e'p) Scattering

High energy semi-inclusive scattering, where a proton is detected in coincidence with the scattered electron, can provide powerful, additional information on the short-range structure of nuclei. We will discuss two aspects of this process: first, at low to medium energies direct information on the nuclear spectral function S(P, Eg) can be obtained and second, at higher energies a novel prediction of QCD known as Color Transparency can be explored. 5.1

Access to Nuclear Structure

At low energies A(e, e'p) data can provide information on the wavefunction of nucleons in a nucleus. Recalling the shell model for eg. 1 2 C, the ground state has nucleons in the two lowest shell model states (see Fig. 23). Because of the coincident detection of both the scattered electron and recoiling proton and the measurement of both particles four-momentum the energy and momentum of the initial nucleon can be reconstructed. This allows reconstruction of the ground state spectral function once corrections are made for absorption of the

284

IP3/2

lsl/2

Q

-

6

(f)
4—f—

—$—$—$—&-

—4—f-

P Figure 23: Shell model states and occupation for a

n 12

C nucleus in its ground state.

outgoing proton. These effects are much larger than in the inclusive A(e, e') case because interactions of the struck proton with any other nucleons in the nucleus can modify the four-momentum of the outgoing proton. In contrast, only nucleons that are within a short distance (~ 1/Q) of the struck proton can contribute to the A(e, e') cross section. This larger final-state interaction actually provides the needed sensitivity to search for the Color Transparency effects discussed in the next section. Measurements of A(e, e'p) have been carried out at lower energy electron accelerator laboratories at Saclay, NIKHEF, Mainz and Bates to provide a relatively consistent picture of the nuclear spectral function in the region where the spectral function is large. Data is sparse in the high momentum region where short-range correlations dominate because the smallness of the spectral function limits the cross section. Experiments are under way at JLAB to extend these measurements to large values of P and Es for the initial nucleons using the high current and high duty factor of CEBAF. 5.2

Color Transparency

In 1982 Brodsky 19 and Mueller 20 predicted a novel nuclear QCD effect. Called Color Transparency, this phenomenon leads to a vanishing of the N — N interaction cross section for a nucleon produced in a high Q2 scattering process. For free N — N scattering, the interaction cross section is approximately a constant 40 mb for EN > 2 GeV. Color transparency predicts that this cross section should vanish (for a very short time-scale or propagation distance) if the nucleon is produced via high Q2 processes. Qualitatively, the high Q2 process selects a particular basis state of the proton wavefunction with small inter-quark separation. As shown in Fig. 24, in order for the nucleon to remain intact after interaction with the photon, the struck quark must exchange gluons with the remaining valence quarks to

285

yield a final state nucleon. As Q2 is increased the gluon must also be higher

N N Figure 24: Diagram of high Q2 elastic nucleon scattering.

energy (to appropriately share the momentum among the quarks). But a high energy virtual gluon can only be produced for a short time (t ~ h/AE). Thus there must be a small inter-quark separation in the initial state. This "small proton" then has a reduced interaction with other nucleons for a time-scale comparable to the time it takes the proton to fluctuate back to its characteristic size (t ~ Tpjc in the proton rest frame). Thus the strong interactions are suppressed for a distance (in the lab frame) of

'-(ft)©-

«

For proton energies of a 5 - 10 GeV this corresponds to I ~ 4 — 8 fm which is comparable to the size of a heavy nucleus. Now a nucleon interacting with a nucleus with the "full" 40 mb cross section will be strongly absorbed by the nucleus. Averaging over the size of the nucleus the absorption probability -PA- for a gold nucleus is ~ 80%. The signature for Color Transparency is then the observation of a nearly transparent nucleus. Defining the nuclear transparency as T^ = 1 - PA we would expect to have Tjv ->• 1 as Q2 increases. (60) Thus in the absence of Color Transparency effects we would expect to see approximately constant nuclear transparencies on the order of TV ~ 50% for a carbon nucleus and T ~ 20% for a Au nucleus. 5.3

Evidence for Color Transparency from A(p, 2p)

The first suggestive evidence for Color Transparency (CT) came from a proton scattering experiment, where two final-state protons were detected in a kine-

286

matics corresponding to high momentum transfer from the incident proton. This A(p, 2p) experiment has potentially even more sensitivity to CT effects as both final state protons can experience the transparency effects. Thus for a carbon target, the expected transparency in the absence of CT effects is only 20-30%. The experiment - E834 21 - was performed at Brookhaven National Laboratory on a variety of nuclear targets. Results from the experiment are shown in Fig. 25. The momentum transfer range of this experiment corresponds to 1.00

j i i i 11 11 i I i i i 11 i i i i 11 11 i I i

o 0.50

(a) Various Nuclei + -Li

a

o -d

CD U

x -Al

cfi 0.20 ft w ti 0.10

• -du

ctf

u EH

%

.-)(

x -Pb 0.05

1111111111111111111 i 11 i 11 i 11 I I 111M

i 11 i 11 11 i I I i i 11 11 i

(b) Aluminum x - 6 GeV/c 0 - 10 GeV/: n - 12 GeV/:

0.5 0

C

XI L+JI


a

X

0.2 -

l

[]°, I]

a u

[]

EH

0.1 I I I I I I I I I I I I I I I I I I I I 0-1 I I

0

2.5

5

7.5

10

12.5

15

Incident Momentum Gev/c Figure 25: Nuclear transparency from E834 using the A(p, 2p) reaction. The dashed line in the lower panel is the expected transparency assuming no Color Transparency effects.

Q2 = 4 - 1 1 GeV 2 . While there is evidence for an increase in transparency

287

with increasing Q2, the transparency then decreases as Q2 is further increased. Thus this is certainly not a straightforward CT signal. Most of theoretical interpretations of these data suggest that it is the more complicated nature of the (p, 2p) reaction that is responsible for the ambiguity of the observed signal. Thus it was proposed that a process with a better understood reaction mechanism [eg. A(e, e'p)] might be provide a better signature for the effects. 5-4 A(e, e'p) at high Q2 as a Probe of Color Transparency The first experiment to look for Color Transparency effects using the A(e, e'p) process was SLAC experiment NE18 2 2 ' 1 . At low momentum transfer (Q2 = 1 GeV 2 ) the experiment was able to observe the nuclear shell structure discussed in Sec. 5.1. This is shown in Fig. 26 where the separation energy (called missing energy in the experiment) of the initial-state proton is shown for a carbon target. The narrow, less tightly bound lp shell and the broader Is

1 1 • • 11

50 E

' > • 1 1'

100 (MeV)

150

200

Figure 26: Missing energy spectrum for C(e, e'p) for experiment NE18.

shell are both visible in this data. The extracted nuclear transparencies for all of the targets from that experiment are shown in Fig. 27. Here we see a transparency for H and D that is nearly unity (D is expected to have T ~ 95%). The nuclear targets show a

288

~r 2

4 Q2 ( G e V / c ) 2

Figure 27: Measured nuclear transparency vs. Q2 for a variety of nuclear targets. The open symbols are from an experiment23 at Bates laboratory and the filled symbols are from SLAC experiment NE18.

transparency that is approximately independent of Q2, with no clear evidence of Color Transparency. The apparent increase in T at low Q2 is expected as the N — N cross section is smaller at these energies and several nuclear effects also become important (eg. Pauli blocking). A comparison of the measured transparencies with several theoretical calculations is shown in Fig. 28. While the data rule out a simple parton model prediction a definitive statement about CT cannot be made. Part of the difficulty is that the key signature is a Q2 dependence for T. Because the standard nuclear physics calculations presently have ~ 20% uncertainty, the value for the lower Q2 transparency (which "anchors" the theoretical predictions) can move up or down by this amount. Improved calculations are in progress and should help to clarify the situation. New data have also recently been obtained at JLAB in experiment 91013 25 . These measurements overlap the Q2 of the NE18 experiment but have much higher statistical precision. Results from 91-013 are shown in Fig. 29, in comparison with the previous Bates and NE18 experiments. A new experiment at JLAB - 94-139 - has also recently taken data at a momentum transfer of 8

289 1.0

JL,

j _

Naive Parton n Quantum Diffusion • No Color TraJisparency

C Data Au Data

0.8

- - ' - * -

Q2 (GeV/c) 2 Figure 28: Comparison of nuclear transparency for C and Au targets with several calculations from Ref. 24.

GeV2 with statistical precision comparable to JLAB 91-013. It is clear that improved calculations of the nuclear transparency in the absence of CT effects, may allow increased sensitivity to CT in this new experiment. 5.5

Future JLAB Color Transparency Experiments

Measurements of A(e, e'p) at higher Q2 are possible at JLAB with higher beam energies. Estimates have been made for possible future experiments 26 at JLAB to extend the search for Color Transparency to Q2 ~ 16 GeV 2 . Using standard nuclear physics calculations for the nuclear transparency (eg. no Color Transparency) and reasonable estimates for spectrometer acceptance and beam currents, measurements with several % uncertainty would be possible for several nuclear targets with a running time of ~ 30 days. These estimates are shown in Table 2. 5.6

Summary

The present status of the study of nuclei with unpolarized high energy electron scattering has been presented. The EMC effect, y-scaling and Color Trans-

290

0.9 0.S 6.7 0.6 [• O.0.S J? 0.4 h

l

05 r

a* I , .t

.j,!*..,,,.*

J...

,j..l

1

10

Qa (GeV8)

Figure 29: Nuclear transparency vs. Q2 for JLAB experiment 91-013 (solid symbols). The open symbols are from Ref. 1 and Ref. 23.

Table 2: Kinematics for a future A(e,e'p) experiment with higher energy beams at JLAB. Q'2

E

PP

(GeV^)

(GeV)

(GeV/c)

6 8 10 12 16

4.7 5.8 6.8 7.9 10

4.0 5.1 6.2 7.3 94.

Op

Coincidence Rate (1/hr)

18° 14° 12° 10° 8°

320 100 40 10 1

parency have all been discussed. While there are a variety of intriguing signals present in the data, each of these processes remains a challenge to understand in detail theoretically. Additional experimental data can provide key insights into several open questions associated with these processes.

291

Bibliography Deep Inelastic Scattering: F. E. Close: An Introduction to Quarks and Partons (Academic Press, San Diego, 1979). I. J. R. Aitchison and A. J. G. Hey: Gauge Theories in Particle Physics (Adam Hilger, Bristol, 1982). S. Bom, C. Giusti, F. D. Pacati, Phys. Rep. 226, 1 (1993). A(e, e'): EMC effect: D. Geesaman, et al, Ann. Rev. Nucl. Part. Sci 45, 337 (1995). y-scaling: E. Pace and G. Salme, Phys. Lett. B 110, 411 (1982). «/-scaling: D. Day, J. McCarthy, W. Donnelly and I. Sick: Ann. Rev. Nucl. Part. 5«"40, 357 (1990). T. deForest, Nucl. Phys. A 392, 232 (1983). O. Benhar et al, Phys. Rev. C C44, 2324 (1991). C.C.d.Atti and S. Simula, Phys. Lett. B 325, 276 (1994). X. Ji and B. W. Filippone, Phys. Rev. C 42, R2279 (1990)). References 1. T. O'Neill et al, Phys. Lett. B 351, 87 (1995), and PhD thesis (Caltech, 1995) unpublished. 2. J. Arrington, et al, Phys. Rev. Lett. 82, 2056 (1999), and PhD thesis (Caltech, 1998) unpublished. 3. S. Pate, included in these proceedings (HUGS99). 4. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGrawHill, New York, 1964). 5. Particle Data Group, Eur. Phys. J. 3, 1 (1998), and http://pdg.lbl.gov/. 6. http://durpdg.dur.ac.uk/HEPDATA. 7. D. B. Day et al, Phys. Rev. Lett. 43, 1143 (1979). 8. G. B. West, Phys. Rev. Lett. 24, 1206 (1970). 9. T. deForest, Nucl. Phys. A 392, 232 (1983). 10. D. B. Day et al, Phys. Rev. Lett. 59, 427 (1987). 11. M. Butler and S. E. Koonin, Phys. Lett. B 205, 123 (1988). 12. O. Benhar, Phys. Rev. Lett. 83, 3130 (1999). 13. B. W. Filippone et al, Phys. Rev. C 45, 1582 (1992). 14. E. Bloom and F. Gilman, Phys. Rev. D 4, 2901 (1971). 15. S. Rock, et al, Phys. Rev. D 46, 24 (1992). 16. L. W. Whitlow, et al, Phys. Lett. B 282, 475 (1992). 17. A. DeRujula, H. Georgi, and H. D. Politzer, Ann. Phys. 103, 315 (1977).

292

18. Benhar and Liuti, Phys. Lett. B 358, 173 (1995). 19. S. J. Brodsky in Proceedings of the 13th International Symposium on Multiparticle Dynamics, eds. W. Kittel, W. Metzger, and A. Stergiou (World Scientific, Singapore, 1982), p. 963. 20. A. H. Mueller in Proceedings of the XVII Rencontre de Moriond, ed. J. I r a n Thanh Van (Editions Prontieres, Gif-sur-Yvette, France 1982), p.13. 21. A. S. Carroll et al, Phys. Rev. Lett. 6 1 , 1698 (1988). 22. N. C. R. Makins, et al, Phys. Rev. Lett. 72, 1986 (1994). 23. G. Garino, et al, Phys. Rev. C 45, 780 (1992). 24. G. Farrar, et al, Phys. Rev. Lett. 6 1 , 686 (1988). 25. D. Abbott, et al, Phys. Rev. Lett. 80, 5072 (1998). 26. B. W. Filippone, Proceedings of the Workshop on CEBAF at Higher Energies (N. Isgur, P. Stoler, eds.), p. 405 4/14-16, 1994.

293 The H E R M E S Experiment

Stephen F. Pate Physics Department New Mexico State University Las Cruces NM 88003 An introduction to deep inelastic scattering is given, in the context of results from the HERMES experiment being carried out at the HERA storage ring at DESY. The overall goal of HERMES is the investigation of the polarized and unpolarized quark structure of hadrons. I will discuss in particular the measurement of the flavor asymmetry of the light-quark sea, the polarized quark distribution functions, and the polarization of lambdas produced in deep inelastic scattering.

1 1.1

Introduction: Overview of Deep Inelastic Scattering and the Spin Structure of the Nucleon Deep Inelastic Scattering

Deep inelastic scattering occurs when a high energy lepton interacts with an individual quark inside of a nucleon or nucleus, with sufncient momentum transfer (more than 1 GeV/c) such that the interaction can be treated in an impulse approximation. Deep inelastic scattering has been the fundamental experimental probe of the internal structure of the nucleon for the past three decades. The incident lepton, of energy E, interacts with a quark of flavor / (up, down, strange...) via a virtual photon of energy v and momentum q, so that the scattered lepton has remaining energy E' = E — v. The positive square of the four-momentum transfer q of the virtual photon, denoted by Q2 = -q2, and the Bjorken momentum fraction of the struck quark, x = Q2/2Mv, are used to characterize the kinematics of the deep inelastic scattering process. (M is the mass of the nucleon.) Deep inelastic scattering of leptons from nucleons probes the quark distribution functions: • qt(x,Q2) probability of finding a quark of flavor / and Bjorken momentum fraction x with helicity parallel (+) to the nucleon spin, at a given momentum transfer Q2 • qf(x,Q2) tions

= q~j(x,Q2) + qJ(x,Q2)

unpolarized distribution func-

• Aqf(x, Q2) = q~j(x, Q2) — qJ(x, Q2) polarized distribution functions

294

k = (E, k)

lepton four mometum

q 2 = (k — k') 2

mometum transfer

2

= 2m + 2(kk' cos 6 -

EE')

Q2 = - q 2

=

2EE'(cosQ-l)

Relativisic limit : m «

E,

2

= 4EE' sin (0/2) v = E — E'

virtual photon energy

W2 = (P + q) 2

photon — nucleon invariant mass

x = Q2/2Mv

parton momentum fraction; ^Bjorken

y = v/E z = Ehjv

virtual photon energy fraction individual hadron energy fraction

E « k

295

1.2

Inclusive Deep Inelastic Scattering

The cross section for inclusive deep inelastic scattering, in which only the scattered lepton is observed, d2a _ Ana2 dxdQ2 ~ 1 ^

^Fl(l,^) +

ftfec!)(1-„-^)

measures the unpolarized structure functions Fi and i*2. In the quarkparton model, these are f and F2(x) = 2xF1(x)

(Callan - Gross relation)

These are now well measured. If the lepton and nucleon are longitudinally polarized, then the polarized structure function

9i (x) =

-^2e}Aqf(x) 1

f is obtained from the asymmetry in the yield of scattered leptons d2
_ ~

d2
N~(x)-N+(x) N-(x)+N+(x)'

where iV is the normalized yield of leptons for beam and target helicities parallel (+) or antiparallel (—). This polarized structure function has only recently been well measured for the proton and neutron. The first good measurement, by the European Muon Collaboration, brought about the "nucleon spin crisis" which continues to be of interest today and is the motivation for the HERMES experiment. 1.3

Naive Quark Model for the Spin Structure of the Proton

The total spin of the nucleon is a sum of contributions from its constituents (quarks and gluons), and they can contribute though their own intrinsic spin and/or though orbital angular momentum:

296

where Auv and Adv are the polarized valence u and d quark distribution functions, Aqs represents the polarized quark distribution functions for all sea quarks, Ag is the polarized gluon distribution function, and Lq and Lg are the orbital angular momentum contributions from the quarks and gluons. The terms Ag, Lq, and Lg are not separately gauge invariant, but the sum Ag + Lg + Lg is. In the naive quark model we assume • s-wave quark states • sea-quarks unpolarized • gluons unpolarized and under those assumptions you can write down a fully antisymmetrized proton spin wavefunction; ——\2ututdl+2dl.u J \-u't+2u J tdlu^-uld'tu-f-u J \-uld J \v 18 -u^u^d^— d^u^u^— w t d t " | — dtw|w4->

|$p > =

Prom this one can easily compute the moments Aw, Ad and As for the proton: Aw = <

u^-ul>

= i [+4 -2 + 4 - 2 + 4 - 2 - 0 - 0 - 0 - 0 - 0 - 0 ] 18 = +4/3 Ad= < d^-di> = i [ - 4 - 4 - 4 + 1 + 1 + 1 + 1 + 1 + 1] 18 = -1/3 As = 0 1.4

The Ellis-Jaffe Sum Rule

The moments Aw, Ad, and As can be related to the measurement of the polarized structure function gi(x). As 9i(x) =

2

-^e}Aqf{x) /

297

then 9{{x) = \ \^Au{x)

+ ^Ad(x)

+ ^As(x)

+ ^Au(x)

+ ^Ad(x)

+ ^Au(x)

+ ^As(x)

+ ^Ad(x) + ^Au(x)

+

^As(x)j

and 9i(x) = \ \^Ad{x)

+ ^As(x]

where we have assumed isospin symmetry; thus un(x) — dp(x) = d(x), and up(x) = dn(x) = u(x). Then the moments are related to the integral of gi(x):

I* = J* 9{{x)dx = \ (^Au + \ Ad + IA I? = fQ fiix)dx = \ (±Ad + \AU + 1 As) where we adopt the convention that Aw (for example) sums over quark and anti-quark contributions, which inclusive deep-inelastic scattering cannot distinguish. The integral of g\ (x) can also be given in terms of the matrix elements of the quark SU(3) axial vector currents; ao, a3, and as'. 1

ri =

(

1

4

12 r + T T 8 + 3 a °

where a0 = Aw + Ad + As 0,3 = Au — Ad a8 - ~j= {Au + Adv3

2As).

Values for a3 and ag can be gotten from data on neutron and hyperon /?-decay, as the same axial currents are involved: o3 = D + F a8 =

^=(3F-D).

where F and D are F = 0.477 ±0.012 D = 0.756 ± 0.011

298

Combining this information with a measurement of gi (x) (for one of the proton or neutron) gives a0 and thus the moments of the quark polarizations: Au = -(2a0 + 3a3 + VSas) Ad = -(2ao - 3a3 + VSa8) As - - ( o 0 - %/3a8) If one assumes As = 0 (as in the naive quark model), then one may calculate the first moments of the polarized structure function g\ for the proton and the neutron from these basic QCD considerations. This result for the integrals J0 g^(x)dx and J0 g"(x)dx is called the Ellis-Jaffe sum rule:1 T\=

f g{{x)dx = 0.185 Jo

r ? = J g?(x)dx = -0.024 Jo 1.5

A Crisis is Born

In the late 1980's, the European Muon Collaboration (EMC) measured g\{x) via an inclusive deep inelastic scattering of polarized muons from a polarized hydrogen target. In 1987 they reported 2 a value for the g\(x) integral of r ? = 0.126 ±0.010 ±0.015 considerably shy of the value of T^ = JQ g\{x)dx = 0.185 from the Ellis-Jaffe result which assumed As = 0. However, this result indicates not only that perhaps As ^ 0, but also (and more importantly!) that the contribution of the quarks alone is insufficient to account for the spin of the proton. Using more recent data for T^, and combining with the data from /3-decay, one may calculate the moments of the quark polarizations from experiment:

Au = +0.79 ± 0.04 Ad = -0.45 ± 0.04 As = -0.16 ±0.04

299 Thus the quark contribution to the proton spin is A S = Au + Ad + As « 0.18 a far cry from 1.0! This result is the source of the "nucleon spin crisis" which was the motivation for the HERMES experiment. 1.6

Semi-inclusive Deep Inelastic Scattering

The quark which is struck in a deeply inelastic process will fragment into one or more hadrons. The number of hadrons of type h (pion, kaon, lambda...) produced is Nh(x,z)<x^e2fqf(x)Dhf(z), f where D'f(z) is the fragmentation function, expressing the probability that a quark of flavor / will produce a hadron of type h with energy fraction z — Eh/u, the fraction of the virtual photon energy carried away by the hadron. In the expression for the hadron multiplicity Nh(x,z), the factor associated with the hard scattering process (qf(x)) and the factor associated with the subsequent fragmentation^^(z)) are independent of each other. This expresses the underlying assumption of the impulse approximation, namely that the fragmentation process is entirely separate from the hard scattering process. We call this separation "factorization." The hadron produced with the largest energy fraction z, the so-called "leading hadron," will most likely carry the struck quark. More detailed information about the quark distribution functions is made available by detecting the leading hadron in coincidence with the scattered lepton. In particular, by examining the spin-dependent asymmetry in the yields of hadrons produced in deep-inelastic processes, the individual polarized quark distribution functions can be extracted. A detailed knowledge of these functions is vital to a resolution of the "nucleon spin crisis." This is the main thrust of the HERMES physics program! 2

Description of the H E R M E S Experiment

HERMES is an internal target experiment located in the East Hall of the HERA storage ring complex at DESY. A beam of longitudinally polarized 27.5 GeV positrons in the HERA storage ring passes through the HERMES internal gas target. The target gases are polarized 3 He (1995), polarized 1 H (1996 and 1997), and various unpolarized gases (hydrogen, deuterium, 3 He,

300

nitrogen) in all years. In 1998 and 1999 HERMES employed a polarized 2 H target. The HERMES spectrometer 3 is constructed with an open geometry to allow detection of scattered leptons in coincidence with quark fragmentation products. Identification of detected particles is achieved through tracking, momentum reconstruction, and the differential responses of the various detector elements. • typical HERA positron beam properties: energy = 27.5 GeV, current = 40 mA, lifetime = 10 h, polarization = 60% • typical HERMES target properties: 3 He: thickness = 10 15 nucleons/cm 2 , polarization = 45% X H: thickness = 10 14 nucleons/cm 2 , polarization = 90% unpolarized gases: thickness in excess of 10 15 nucleons/cm 2 • detector acceptance properties: angular range: ± (40 - 140) mrad vertical, ±170 mrad horizontal Bjorken x: 0.021 < x < 0.85 momentum transfer: 0.8 GeV 2 < Q2 < 20 GeV 2 , with (Q 2 ) « 2.5 GeV 2 dipole field integral: 1.3 Tm tracking resolution: Ap/p = 0.7-1.3%, A0 = 0.3-0.6 mrad 2.1

Internal Gas Targets

The use of internal gas targets (10 14 - 10 15 nucleons/cm 2 , a very low density compared to solid targets), in combination with a high intensity stored beam (10-40 mA), permits experiments to be done at useful luminosities of 10 31 10 32 c m - 2 s _ 1 in a relatively background-free environment and a minimum of material between the detector and the event interaction point. The storage cell is an elliptical cylinder, 400mm long, 29mm wide, and 9.8mm high, made from 125/xm thick aluminum sheet (see fig. 1). The collimator does not actually intercept the beam, but is in place to block the flux of synchrotron photons generated by the beam upstream of the target area. The dimensions of the cell and collimator are chosen so that the cell is not illuminated by any direct synchrotron radiation - only radiation which has scattered at least twice can hit the cell. Particles scattered out of the beam can pass through a very thin-walled (0.3mm) stainless-steel exit window on their way to the detector. The "wakefield suppressors" provide smooth electrical continuity of the beam pipe around the beam, so that the target chamber does

301 gas injection target vacuum chamber spectrometer wake field suppressors

thin walled beam pipe \ f^^

fixed collimator

e-beam

x pumps

Figure 1: Schematic diagram of the target region of HERMES.

not act as an inadvertant resonant cavity and absorb energy from t h e b e a m . In 1995 H E R M E S used a polarized 3 H e t a r g e t , 4 and in subsequent years has used polarized hydrogen and deuterium atomic gas t a r g e t s . 5 2.2

The HERMES

Spectrometer

T h e H E R M E S Spectrometer is built as a large acceptance, open geometry system, t o achieve the goal of observing semi-inclusive deep inelastic scattering events (see fig. 2). It is composed of forward tracking devices, a magnet for m o m e n t u m measurement, back tracking devices, and a system for particle identification. These devices and their performance are summarized here and in t h e following subsections. • Forward Tracking: Microstrip Vertex Chambers, Drift Vertex C h a m b e r s , Front Chambers, Magnet Chambers • Back Tracking: Back Chambers • Particle Identification: Cherenkov Detector, Transition Radiation Detector, Preshower (H2), Calorimeter; also Time-of-Flight (hodoscopes) • Luminosity Monitor

302 . FIELD CLAMPS

Hi

H2

FEED TUBE B

LUMINOSITY

CALORIMETER

1

I 7

m

8

Figure 2: Layout of the HERMES Spectrometer. For details, see below, or in K. AckerstafF et al. 3

• Fast Trigger: Scintillator Hodoscopes HO, HI, and H2; Calorimeter; Luminosity Monitor Tracking Chambers • Microstrip Vertex Chambers: high resolution measurement of the scattering angle and vertex position aluminum strips etched on a high-resistivity substrate form alternating cathode (85 /mi) and anode (7 fxm) lines; these face a cathode plane on the other gas of a 3 mm thick gas layer 6 planes - VUXXVU - 65 /im resolution per plane • Drift Vertex Chambers: increased acceptance for the muon physics program 6 mm drift cell 6 planes - XX'UU'VV - 220 ^m resolution per plane

303

• Front Drift Chambers: good spatial resolution immediately in front of the magnet 7 mm drift cell 12 planes - 2(UU'XX'VV) - 225 /an resolution per plane • Magnet Chambers: standard proportional chamber 0.5 mm wire separation 9 planes - 3(UXV) - 700 /j,m resolution per plane • Back Drift Chambers: tracking for momentum measurement 15 mm drift cell 24 planes - 4 ( U U ' X X ' W ) - 280 //m resolution per plane Particle Identification Detectors • Threshold Cherenkov Detector: 1995: pure nitrogen gas was used, for which the n/K/p momentum thresholds are 5.6/19.8/37.6 GeV respectively; therefore only detected pions and positrons in 1995, for detector commissioning purposes 1996-1997: 70% N 2 + 30% C 4 Fi 0 gas was used, for which the ir/K/p momentum thresholds are 3.8/13.6/25.8 GeV respectively; this gives better pion acceptance - a very few kaons with 13.6 GeV momentum or more are there as a small background present: Ring Imaging Cherenkov (RICH): much greater acceptance and identification for pions, kaons and protons • Transition Radiation Detector (TRD): transition radiation occurs when a charged particle of passes through an interface between two differing indices of refraction; only electrons and positrons will produce such radiation at HERMES energies

304

6 planes - each plane is composed of a radiator followed by a 90% Xe + 10% CH4 gas filled proportional counter which acts to detect the transition radiation X-rays • Hodoscopes (HO, HI, H2): fast scintillator detectors, primarily for triggering 11 mm thick Pb sheet ("preshower") in front of H2 enlarges the pulse height in H2 for electrons and positrons; can identify protons and deuterons below 2 GeV momentum via TOF • Lead-Glass Calorimeter (CALO): designed to respond significantly only to electrons and positrons, to aid in triggering and particle identification 420 blocks of F101 lead-glass in 42x10 stacks, above and below the beam each block is 9x9x50 cm 3 ; 18 radiation lengths deep; n = 1.65 Particle Identification Performance No single detector in the HERMES spectrometer can perform the full particle identification job alone - some scheme to let the detectors "vote" is needed. By comparing the responses of the various detectors, and by carrying out various calibrations, one can learn to calculate two numbers for each event and for each detector: the probability that a track corresponds to an electron or positron, P e , and the probability that it is a hadron, P h . These values can be combined into a single number, for example P I D 3 = l o g 1 0 [ ( P 5 a l ^ P r e ^ h e ) / (^al^Pre^Che)l

which can be used to make a single cut to distinguish electrons and positrons from hadrons. (PID3<0 implies a hadron.) In 1995, for about 97% of our events, we are able to make a two-dimensional cut on the correlation TRD vs. PID3. This provides an electron or positron identification with 1% or less hadron contamination. Luminosity Monitor (LUMI) HERMES uses many target gases, and we need to have a monitor of the beamtarget luminosity which is somewhat independent of the target in use. Since all target gases contain electrons (!), we can observe elastic scattering of the beam

305 (DIS event with K, in hadronic final state):

transverse vertex distribution:

Ka mass peak:

no background from wall scattering!

Figure 3: Exemplary HERMES event, in which a A"-short is produced by a deep inelastic process, decays to two pions, and is reconstructed from the two pion tracks. T h e distribution of vertices in the target-beam interaction region (summed over many events) shows no evidence of scattering from the target cell walls.

particles from these electrons and use that as our luminosity monitor. Elastic scattering of positrons and electrons (Bhabha scattering), or of electrons and electrons (Moller scattering), is observed by placing two small NaBi(W0 4 ) 2 calorimeters on either side of the beam and requiring a coincidence between two signals of approximately equal energy, about half the beam energy each this corresponds to elastic scattering at about #CM = 90°. In order that the luminosity yield not have a spin-dependent component (fatal to any spin-asymmetry measurement), it is important that the electrons in the target not be polarized. For the 3 He target, this is not an issue. For the *H target, the RF system must produce equal populations of atoms with

306

parallel and anti-parallel electron spins, but during normal operations this is not hard to achieve. Conversely, it is possible to produce electron polarization in the target on purpose and then use the resulting asymmetry in the luminosity yield to check on the operation of the 1 H target. Physics Triggers • Deep Inelastic Trigger: This is the primary trigger for HERMES, and all the results shown in this set of talks are based on it. We look for a scattered beam particle (positron beam during 1995-1997, electron beam in 1998-1999) with an energy of at least 3.5 GeV. (The beam energy is 27.5 GeV). The detector requirement is then: All four of HO, HI, H2, and CALO must fire in the correct time relationship, with the additional requirement that the CALO must record at least 3.5 GeV of energy (at least in 1995 - this was later reduced to 1.4 GeV). Only an electron or positron can produce any significant light output in the calorimeter. • Photoproduction Trigger: We also seek to study some physics processes in which the scattered beam particle is not observed. This usually means that the beam particle was deflected at such a small angle that it "went down the beam pipe" and missed the spectrometer altogether. At such small angles, the virtual photon produced is almost a real photon, hence the name for this type of trigger. Lacking the scattered beam particle, we look for one or more secondary particles (pions and the like) using some combination of hodoscopes and prompt wire chamber information. • Luminosity Trigger: A simple coincidence between the left and right luminosity detector clusters. • Other triggers: A variety of triggers under test and development are always in use. For example, triggering on a coincidence between one lumi cluster and a "hadron" in the main detector increases our acceptance for low-<22 events. 3

The Gottfried Sum Rule and the Light-Quark Sea Flavor Asymmetry

The Gottfried Sum Rule is an integral over the difference in the F2 structure functions of the proton and neutron: 6

SG= Jof-(*?(*)--*?(*)) x

307

= j\x(J£e}qpf(x)-J2eh1fW = jf1 dx (^UP(x) + l-dP{x) + ±8»(X) + ^UP(X) + l-dP{X) + ±P>(X)^ - (^un(x)

+ \
+ ±un(x) + \dn{x)

+ !*"(*))

We will assume isospin symmetry is valid - switching the n and p labels just means switching u and d quarks: up(x) = dn(x) = u{x)

dP{x) = un(x) = d{x)

sp(x) = sn(x)

u»(x) = dn(x) = u(x)

dP{x) = un{x) = d(x)

sp(x) = sn(x)

Now we define the valence quark distributions: uv(x) = u(x) — u(x)

dv(x) = d(x) — d(x)

Then the Gottfried sum takes this form:

1 r1 SG = dx{Au(x) + d(x) + Au(x) + d{x) 9 Jo — Ad{x) — u(x) — Ad(x) — u(x)) = — I dx (u(x) — d(x)) + — I dx (u(x) — d(x)) 3 Jo 3 J0 SG = q / dx (uv(x) - dv(x)) + dx (u(x) - d(x)) 3 Jo o J0 If we now further assume that the light quark sea is flavor symmetric u(x) = d(x) then the Gottfried sum is reduced to SG

=

If1 3 /

dx

^Uv^

~

d v

^

=

1 3 (2 ~ ^

=

1 3

In the early 1990's, the New Muon Collaboration (NMC) measured F£(x) and F^ix) in inclusive deep inelastic scattering of unpolarized muons from hydrogen and deuterium targets. 7 From F%(x) and F£(x) they are able to extract F£(x) (by assuming that F${x) = (F%{x) + F£(x))/2), and hence the Gottfried sum. In 1994 they published a value of

308

SG = 0.235 ± 0.026. Assuming that isospin symmetry is still valid, then this measurement suggests that the light quark sea is not flavor symmetric after all, i.e. u(x) ^ d(x). Two methods are available to study the ^-dependence of these unpolarized anti-quark distributions: • the Drell-Yan process, as employed in CERN Experiment NA51 8 and Fermilab Experiment E866 9 • semi-inclusive deep inelastic scattering, as in HERMES In semi-inclusive deep inelastic scattering, we must consider the fragmentation functions. 3.1

Fragmentation

Functions

Recall that the number of hadrons of type h (pion, kaon, lambda...) produced is Nh(x,z)cx1£e2fqf(x)Dhf(Z), f where D'l(z) is the fragmentation function. Given that there are six types of quarks commonly found in the nucleon (it, u, d, d, s, s) and two kinds of charged pions (n+, 7r~), then in principle there are 12 fragmentation functions for the production of charged pions from the nucleon. Application of charge and isospin symmetries leaves only three independent fragmentation functions: D+ (z) = Df

(z) = Df

(z) = Df

(z) = Df

(z)

"favoured"

D~ (z) = Df

(z) = Df

(z) = Df

(z) = Df

(z)

"unfavoured"

Ds(z) = Df(z)

= Df(z)

= Df(z)

= Df(z)

"strange"

These functions are hard enough to measure - you have to build a big accelerator like HERA and then a detector apparatus like HERMES. But calculating them is nearly impossible. Deep inelastic scattering is (by definition) done at a high enough energy to be treated perturbatively, but the subsequent fragmentation of the quarks into hadrons is strictly non-perturbative, and therefore a struggle to calculate. As a result, if your experimental program depends on a knowledge of the fragmentation functions, you are going to have to measure them first - no theory can give them to you.

309 3.2

The Light Quark Sea Flavor Asymmetry from Semi-Inclusive Deep Inelastic Scattering

The flavor asymmetry is extracted from the observed ratio in the difference between pion yields from the proton and neutron data, r{x,z) -

TV"" - N*~ _ +

where the normalized yield of pions from a proton or neutron target is N^(x,z)<xY/e2fqf(x)Df(z) f as before. Using isospin symmetry and the definitions of the fragmentation functions listed previously, you can show that 1 4- r(x, z) _ u(x) — d(x) + u(x) — d(x) l — r(x,z) uv(x) — dv(x) where (

_ 3 fl + D-/D+(z)\ >- 5 \l-D-/D+(z)J-

The equation giving r(x, z) is a product of two factors, one only a function of x, the other only a function of z. Those two factors can be extracted separately by fitting the data for r(x,z) along x and z independently, if factorization is valid. In this way the fragmentation function ratio D~ /D+(z) and the light quark sea asymmetry are determined together (see fig. 4). The result for the fragmentation function ratio D~ /D+(z) is consistent with a previous EMC measurement. 3.3

A Check on the Validity of Factorization

The quantity of interest for the light quark sea asymmetry is d(x) - u(x) _ J{z)[l - r(x, z)] - [1 + r(x, z)} u{x) - d(x) ~ J(z)[l - r(x, z)] + [1 + r(x, z)]' As a check on factorization, we extract this quantity as a function of z for different values of x (see fig. 5). Clearly, there is no dependence on z for any value of x, so factorization is valid.

310

D+/(z)

j

+

o o o o o

Q "O

-

o

O

o

m

Hermes (preliminary)

0

EMC Nucl.Phys. B321 (1989)541

•

o

° •0 £>

• o

•

•

0

o

•
10

* 1.

; -

10 0.4

"

*

•\

,

l

I

I

1

,

i

1

,

-

,

,

I

,

,

i

•

I

I

,

,

,

I

,

,

,

Rel. Systematic E•rror

0.2

\ "

•

-o-

S 10

I

I

I

I

.

1

0.2

i

0.4

i

» i

i

•i

i

0.6

i

0.8

Figure 4: Favoured and unfavoured fragmentation functions vs. z measured at HERMES, in comparison to results from the EMC experiment.

3.4

HERMES Results for the Light Quark Sea Flavor

Asymmetry

The final HERMES results for d(x) - u(x) _ J(z)[l - r(x, z)] - [1 + r(x, z)] u(x) - d(x) ~~ J(z)[l-r(x,z)] + [l + r(x,z)] are published in K. AckerstafF et al. 10 The results confirm the asymmetric nature of the light quark sea. HERMES measures ^X<~^X( directly, while the Drell-Yan experiment E866 9 at Fermilab measures d/u. To calculate the Gottfried sum, d(x) —u(x) is needed, and so some transformations of both data sets are required.

311

~ 1 •a i

s •

to

0.02<x<0.05

-.

I

I

I

,

!

.

•i

r-

1

0.05<x<0.075

0.075<x<0.115 i

i

I

i

.

i

i

.

i

0.115<x<0.15 i

i

I

i

i

i

I

0.15<x<0.3

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 5: Check on factorization: there is no correlation between x and z in the extraction — u(x) of d(x) u(x) — d(x) , so the factorization assumption is valid.

• HERMES calculates d(x) - u(x) from ^ I ^ ] by using the "GRV 94 LO" parametrizations of the u(x) and d(x) distributions of M. Gluck et

al.n • E866 calculates d(x)-u(x) from d/u by using the "CTEQ4M" parametrization of d(x) + u(x) of H.L. Lai et al.12 Using d(x) — u(x) as a common quantity for comparison, we see that the HERMES and E866 data are in agreement (see fig. 6). This is significant because: • The two experiments use very different experimental techniques.

o) . GRV 94 LO C i l O
0. 0.

10 is 1. 4

2 1 0. 8 0, 6 0,4 0, 2 0

-2 X

;

! T :

\

^ - ~ - GRV 94 LO, G/=54.0 Ge\T

\ J^vN

GRV 9 4 LO

b)

2

• HERMES =2.3GeV2 ° E866 Q2=54.0 GeV"

| "

"

^ < Q 3 > ~ 2 . 3 GeV*

:

-0.2 -2

10

< '

X

I] i

!

10

-1

Figure 6: HERMES 1 0 and E866 9 results for the light quark sea flavor asymmetry.

The momentum transfer, Q2, of each experiment is very different as well.

5

Possible explanations for d(x) > u(x) • Pauli Blocking? The original assumption that d(x) = u(x) comes from the idea that all sea quarks arise from the process g —> qq, and since the u and d masses are equally small you don't expect a large asymmetry in their production. Perhaps, however, since u{x) > d(x), then g -> uu might be suppressed relative to g ->• dd. But Ross and Sachrajda 13 showed this is a small effect, on the order of 1%.

313

• Meson Cloud? One may write an expansion of the proton wave function as a "bare" proton plus contributions from various "dressed" configurations of the form |JV7r > and |A7r >. The pions which "dress" a proton will produce additional anti-quarks in the environment. • Chiral Perturbation Theory? Couplings between resident quarks and Goldstone bosons, in processes like u -¥ d-K+ and d -¥ uir~, produce additional anti-quarks. Hence, since u(x) > d(x), then there will be an excesss of d over u (excess of 7r+ over 7T~). The origin of the nucleon sea is still under investigation! Additional experimental and theoretical work is needed. 4

Spin—Dependent D e e p Inelastic Scattering

Two types of inclusive asymmetries will be measured at HERMES: one for beam and target both polarized longitudinally (A\\) An =

an

_ art

(jU + o-tt

and one for longitudinal beam and transverse target polarization {A±_)

A, =

,rt-> _ ,rt
+

CTt<-

•

However, the physically more meaningful quantities are the asymmetries in virtual photon absorption, which are directly related to the structure functions: A\

=

01/2 - 03/2

—

Si - 7 2 52

^1/2 + C3/2

A2 =

Fi

&TL _ 7(gl + 92) CJT

FI

where cr

aT =

(CTJ/2

i/2
314

Note that the kinematic factor 7 = y/Q^/v suppresses g2 relative to gi in A\. Also, it is anticipated theoretically, and found experimentally, that #2 is small, nearly zero:

The virtual photon absorption asymmetries are related to the experiment asymmetries: All=D-(A1+r]-

A2)

A±=d-(A2-li-A1)

where the kinematical factors D, d, rj, and £ are D = V =

2/(2 - y) y* + 2(l-y)(l + R) 27(1-2/) (2-y)

d = 2V

2£

1+ e 1+ e Z = T) 2e and 1-2/ e =

R —

1

2T0

(JLJCTT

(long./trans, cross section ratio)

y= —

(virtual photon polarization)

(fractional energy transfer to the nucleon).

Note that A2 is suppressed in A\\, and Ax is suppressed in A±: =>

i4|| « £> • Ai

=>•

Aj_ « d • .42

We noted already that A\ « 51 / F i :

This is a useful guide to thought.

315 e 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4

A

+ *+ 4 \*\ *|* { i

s 0.1 0 -0.1

ft. , . *

*<>

-0.2

'*'+'

-0.3 -0.4 -0.5

U

HERMES

A

SLAC-E154

-0.6 -0.7

=r=F 10

10

X Figure 7: The structure function ratio A" =
4-1

HERMES Results from Inclusive Spin-Dependent Deep Inelastic Scattering

In 1995, a longitudinally polarized 3 He target was used at HERMES to study the spin structure of the neutron. The result for the polarized structure function g"(x) from this data (see fig. 7) has been published in K. Ackerstaff et al.14 Our value for the integral of g\ (x) over the range of our measurement is

/ 0.023 Jo

g?(x)dx = -0.034 ± 0.013(stat.) ± 0.005(syst.)

316

•

HERMES ( 1 9 9 7 )

A SLACE-143

0.6

a SMC (1993+1996)

0.4 0.2 i-jtl*6***

0 0.01

O) 0.7

i*& 0.1

• H E M E S (1997 )

• HEBMES (1997)

0.6

A SLAC E-143

D SMC ( 1993+1996 )

0.5

Q2=2GeV2

0.4 0.3

Qz=10GeV2

if

wm fm&

0.2

«*

0.1 0.02

0.1

H 0.5

0.02

0.1

0.5

Figure 8: T O P : Comparison of HERMES polarized proton target data with recent results obtained at SLAC (E-143) 1 7 and CERN (SMC) 1 8 . The inner error bars represent the statistical uncertainties and the outer ones the quadratic sum of the statistical and systematic uncertainties. B O T T O M : The spin structure function g\(x). The HERMES data are evolved to Ql = 2 (GeV/c) 2 (left) and Q 2 = 10 (GeV/c) 2 (right) assuming gi/Fi to be independent of Q . Also shown are results for Q2 > 1 ((GeV/c) 2 from E-143 and from SMC, the latter shown for x > 0.01 only.

at (Q2) = 2.5 (GeV/c) 2 . Extrapolating our data to x = 0 and x = 1, our value for the Ellis-Jaffe sum rule integral is

17 = / g?(x)dx = -0.037 ± 0.013(stat.) ± 0.005(syst.) ± 0.006(extrapol.). Jo But beware: such extrapolations are fraught with assumptions about the behaviour of gi(x) as x —> 0 and x —> 1. The most precise measure of g"(x) to date is that of the SLAC E154 Experiment, published in K. Abe et al.15 Their value for the first moment of

317

gi(x) over the range of their measurement is /•0.7

/

g?(x)dx = -0.036 ± 0.004(stat.) ± 0.005(syst.)

at ( < 2 2 ) = 5 ( G e V / c ) 2 . In 1996 and 1997 HERMES used a polarized *H target to study the spin structure of the proton. Our 1997 data for the polarized structure function g\(x) (see fig. 8) have been published in A. Airapetian et al.16 4.2

Why is \tf{x)\ »

\g?{x)\ ?

A large cancelation occurs in the neutron - even the naive quark model gets this right. Recall

f\l(x)dx=1-(lAu+1-Ad+1-As) f1

1/4 1 ]o9n*)dx=-(-Ad+-Au+-As)

1

\

In the naive quark model, we saw that AM

= 4/3

Ad = - 1 / 3

As = 0

Then, jf 1 9{{x)dx = \ ^ ( 4 / 3 ) + | ( - l / 3 ) + 1(0)) = 5/18

jf g?(x)dx = \ ( | ( - l / 3 ) + 1(4/3) + 1(0)) = 0

318

0.8 0.6

(c)

(b)

(a)

A?" (p)

A, (p)

• HERMES

• HERMES A

E143

• SMC

0.4

^

p*

ul_

0.3 0.2 0.1

•h

il\ •'•

3

A; + ( He)

A, f He) • HERMES A

E154

Miit

0 -0.1 -0.2 -0.3

0.04 0.1

1 0.04 0.1

1 0.04 0.1

Figure 9: Asymmetries from which the quark polarizations are extracted. Inclusive (a) and semi-inclusive asymmetries for positive (b) and negative (c) hadrons are shown. The inclusive asymmetries are compared to the SLAC 1 7 ' 1 5 results for g\/F\ (open triangles). The semi-inclusive asymmetries are compared to the SMC 1 8 results for the proton (open squares), truncated to the HERMES x-range. Error bars for HERMES are statistical only, while those of SLAC and SMC are total uncertainties. The error bands are the systematic uncertainties for HERMES. In these plots, no effort has been made to correct any data for the considerable differences in Q2 in the various experiments.

4-3

Semi-Inclusive Hadron

Asymmetries

We consider the asymmetry in the yield of hadrons produced in deep-inelastic scattering:

"

N-(x)+N+(x)

where Nh is the normalized yield of hadrons for a given beam-target helicity state (parallel or anti-parallel):

Nh{x,z)<xY,efatWD)W f

319 We can relate A1?, to a more fundamental asymmetry Ah similarly as in the case of inclusive scattering: A\ w A\/D where in terms of the quark-parton model, Ah,

n2

o2 \ n . i „ k n2\nh , _ Zfe2f*q f(x,Q*)D (x,Q*)

Al(x,Q,z)-

(l + R(x,Q*))

Efe2qf{XjQ2)Dhf(XyQ2)

( 1 + 7

2)

"

For the purposes of the HERMES analysis, this expression is re-written 19 thus: f

y

if(x)

(1 + 7 2 )

where PH(X)=

*,qf{x)JlaD}{z)dz_ 2

Zf<e f,qf'(x)J*2Di},(z>)dz> are the integrated purities. We similarly express the inclusive asymmetry Ax as A l ( l )

~^

P / ( l )

( 1l + 77 2 )

qf(x) Qf(x)

where p x

f( )

4-4

2 e f1f(x) = v ^ 2 / v

Extraction of the Quark Polarizations

^-(x)

Altogether we can summarize these relations in a matrix form: A(x) = P(x) • Q(x) • the vector A(x) = (Alp, A^, A\~, A1He, A^e, A^]je) contains the measured inclusive and semi-inclusive asymmetries for the polarized hydrogen and helium targets • the vector Q(x) contains the quark and anti-quark polarizations A.q(x)/q(x) • the matrix operator P(x) contains the purities, indexed by type of measurement and quark flavor / , as well as the factor (1 + R(x))/(1 + 7 2 )

320 0.6

(Au+Au)/(u+u)

0.4

• +

0.2

0 0.5

(Ad+Ad)/(d+d)

0

I |

-0.5 -1 A(

0.5

?A

0 -0.5

-1 0.03

0.1

0.5

Figure 10: The flavor decomposition of the quark polarization as a function of x, derived from the HERMES inclusive and semi-inclusive asymmetries, assuming a flavor symmetric sea polarization. Error bars are statistical; error bands are systematic.

The purities describe the probability that a virtual photon hit a quark of flavor / if a hadron of type h has been detected in the experiment. They are calculated in a Monte Carlo simulation which includes • the acceptance of the spectrometer • a LUND string model for the fragmentation process • parameterizations of the unpolarized quark distributions and for R(x). The fragmentation functions for the sea quarks are much smaller than for the valence quarks, producing an unstable numerical situation if you try to extract all six u, d, and s polarized distribution functions — some additional constaints must be placed on these functions in order to extract a result. Several choices for these constraints are possible! In the system presented here, we

321

make a distinction between the valence and sea contributions of the u quarks u{x) = uv(x) + u$(x)

us(x) = u(x)

and similarly for the d quarks. Then, we seek to obtain a solution for the vector

\

(u + u)

(d + a)

(s + s)

J

with the requirement that all sea quarks have the same polarization:

qs 4-5

us

as

s

u

a

s

HERMES Results for the Quark Polarizations —2(x)

Using the techniques described above, the experimental inclusive and semiinclusive asymmetries (see fig 9) were used to extract the quark polarizations, 20 shown in fig. 10. The qualitative features of the quark polarizations are not surprising: The u-quark polarizations are positive and grow with increasing x, the d-quark polarizations are much smaller, negative, and grow in magnitude with increasing x as well. But several quantitative features are also deserving of discussion. • Strange quark polarization — conflict with previous results? Recall that by assuming SU(3)-flavor symmetry, we were able to use the F and D coefficients from /3-decay and an inclusive measurement of g\{x) to get the moments Au, Ad, and As, and we found a significant negative As. Our new results, which assumed a asymmetric sea polarization, give an unpolarized sea. But these two results are each bound by their own (and different) assumptions, and neither is a direct measure of the strange quark polarization - this will have to wait (see later).

• Non-singlet contribution — In the quark-parton model, the combination AqNS(x) = Au(i) + Au(ar) - Ad(sc) - Ad(x) is related directly to the proton and neutron spin structure functions by AqNS(x)=6(gp1(x)-g?(x)).

322

S 0.5

De Florian/(1+R) GI0ck/(1+R) Gehrmann

0.4

-0.1

0.03

0.1

0.5

Figure 11: The non-singlet contribution xAqNs(x) = x{Au{x) + Au{x) - Ad(x) - Ad(x)) at Q 2 = 2.5 GeV 2 . The result is compared to parametrizations 2 1 ' 2 2 ' 2 3 derived from inclusive data from other experiments. Error bars are statistical, error bands are systematic.

Our results for this combination from our data can be compared to parametrizations of other world inclusive data, with which they are in agreement (see fig. 11). The integral of the non-singlet contribution is related to the fundamental Bjorken sum-rule

/' Jo

(S?(x) - g"(x))dx = \ga/gv\ • (QCD corrections)

and our value is consistent with the theoretical one. • Try a n o t h e r a s s u m p t i o n a b o u t t h e sea? Instead of a sea polarization independent of flavor, we can instead assume a singlet spin distribution for the sea: Aqs(x) = Aus(x)

= Ads(x)

= As(x) = Au(x) = Ad(x) -

As(x)

The results for the quark polarizations are essentially the same (see fig. 12). This result also implies that we are not yet sensitive enough

323 to test the validity of flavor-SU(3) symmetry. This again will require a direct measurement of the strange quark polarization. oo 0.6

< x 0.5 0.4

• Equal Aqs/qs o Equal Aqs DeFlorian/(1+R) Gluck/(1+R) — Gehrmann

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 0.03

0.1

0.5 x

Figure 12: The SU(3)-flavor octet combination xA(j8(x) = x(Au(x) + Au(x) + Ad(x) + Ad(x) — 2(As(x) + AS(x))) at Q2 = 2.5 GeV 2 . The result is compared to parametrizat i o n s 2 1 ' 2 2 ' 2 3 derived from inclusive data from other experiments. Error bars are statistical, error bands are systematic. The closed circles assume the flavor-independent sea polarization, while the open circles assume a singlet sea spin distribution - the results are essentially identical.

4-6

The RICH Detector allows a measurment of the Strange Quark Polarization ^f(x)

Until 1998, HERMES was restricted to the observation of leptons, pions, and heavier unidentified hadrons. [In certain limited kinematic regimes, heavier particles such as kaons and protons could be identified.] With the introduction of a Ring Imaging Cherenkov (RICH) detector detector, now charged kaons and also protons can be identified in each event over a broad kinematic range. The observation of the asymmetry in the semi-inclusive production of K+ and K~ gives us a stronger handle on the strange quark polarization in the nucleon (see fig. 13).

324

>tZ

0.02-Xbj-0.08

"

£

l n t U = 100 pb"'

(0

<

1"

+

-

0

-0.1

f-

:

•02

-0.3 •

•

Sea Unpolarized

•

Strange Sea Polarized

HERMES 9 9 - 2 0 0 0

-0.4 ,

>

,

1 02

,

,

,

1 0.4

,

i 0.6

,

,

,

i 0.8

,

,

,

i 1 Z-KAON

Figure 13: Results of a Monte Carlo simulation of the asymmetry in the production of negative kaons, AK (z), for an unpolarized sea (Aq = 0), and for a maximally negatively polarized strange sea (As = —s), with 0.02 < XBJ < 0.08. T h e error bars indicate the statistical precision with which Ap (z) would be measured in HERMES using a polarized deuterium target in two years of normal running.

5

The Polarization of As produced at H E R M E S

The A is the lightest strange baryon: a uds quark state. In hadronic interactions, it is always produced in associated with a strange meson, typically a kaon. The A has a spin of 1/2, while the kaon is spinless. The A primarily decays via the weak neutral current channel: A —> pn~ As this decay is parity-violating, it will display an angular distribution with respect to the polarization axis of the A: dN oc (1 + /3PA cos @pA)dn

325

where (3 = 0.642 ± 0.013

decay asymmetry parameter

P\ = A polarization 0 p A = p — A angle in A c m . system Hence, the detection of the A through the A —> pn~ channel includes information about its polarization. In the naive quark model, the spin of the A is carried by the s-quark: |$A > =

—^=\u^dis't-uidJ\-st-d'tuls't+diuJtsJt v 12 + dis^u^-d^s^u].— uls'fd't+u'fs^dl + stut^4--stwJ-rft— stdt^4-+st^4-"t>

so that Au = 0

Ad = 0

As = 1.

However... some facts to consider. • The spin of the A is not entirely due to its s-quark: The same naive quark model which is wrong for the nucleon is also wrong for the A. • HERMES has significant acceptance for the production of A's through current fragmentation only: Within the present detector configuration, those produced in target fragmentation tend not to decay into the detector acceptance. • It is tempting to consider the production of As as a source of information about the polarization of s-quarks in the A, but in fact u-quarks are much more probably found in the target nucleon than are s-quarks and the interaction rate in DIS is higher by the ratio (e„/e s ) 2 = 4. So the dominant production mechanism for As in HERMES will be via interactions with u-quarks (see fig. 14). • Not only the A but also the S° can be produced in DIS, and the S° -> A7 decay will contribute to the production of A's. Since this decay can have a significant spin-flip amplitude, any measurement of A polarization must be corrected for E° production.

326

• To the extent that Flavor-SU(3) is a good symmetry, then values of moments like Au, etc... in the proton will be related to those same moments in the A.

- ,

5 0.25 •

'

- * •

'A

O Data •

.'

u

'

Monte Carlo

!

+ \ • .... d

L^,y •

s 0.05

j 0.1

0.15

.hYjliTSIi4»>t*u. ,j ....i . 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Figure 14: Taking into account not only the ratio of quark charges but also the distribution functions, a Monte Carlo simulation of the production and detection of A's from a proton target in HERMES shows the dominance of the it-quark in this process even down to low XBJ-

5.1

Polarized Fragmentation

In the case of a longitudinally polarized electron beam and an unpolarized hydrogen target, the longitudinal polarization of the A is 2 4 ' 2 5 y(2-y) Zfe2fq?(x,Q2)G$(z,Q*) PA=Pe 1 + (1 - y) 2 £ / e 2 g f (x, Q2)D$(z, Q 2 ) ' or

PA =

PeD(y)D£L,

where Pe is the beam polarization, D(y) = j f f e ^ r , GJ(z,Q2) is the longitudinal spin-dependent fragmentation function for the production of a A from a

327

quark of flavor / , and the longitudinal spin transfer coefficient D^L, contains all the information concerning the transfer of the quark polarization to the A during fragmentation: _J2fe2fq?(x,Q2)GJ(z,Qi) U

T.T.i LL

—

'~Efe}q^(x,Q2)DJ(z,Q^)

Taking the notion that the u-quarks dominate the process to the limit, this reduces to

DlALV

_Gt(z,Q22) Dt{z,Q y

so that PA = PeD(y)

Gt(z,Q2)

For typical HERMES conditions this gives ,G*(z,Q2) PA ~ 0.3

DZ(z,Q*y

If a significant A polarization is observed, then it most likely arises from the u-quarks and confirms the notion of significant u-quark polarization in the A. 350 300

*

250

•

+

200

•

+

150 100 50 1.08 1.1 1.12 1.14 1.16 1.18

1.2

2

Invariant mass (GeV/c ) Figure 15: (a) Invariant filled and hatched areas analysis, (b) Spectrum asymmetric appearance HERMES spectrometer

0

b)

•

+

* +

i t

" ^ ^ = J ^=4= 1 -0.75-0.5-0.25 0 0.25 0.5 0.75 1

cos(0pA)

mass spectrum from the reconstruction of candidate A events. The respectively indicate the signal and background regions used in the of C O S 0 P A for the two beam helicities (circles and squares). The of these spectra is almost entirely due to the acceptance of the for reconstructing A decays.

328

5.2

Extraction of A Polarization and D^L,

The parity-violating decay A —• pn~ produces an angular distribution of the decay products according to the polarization of the A: NA OC (1 4- 0.642 • PA cos(0 pA )) Analysis of this angular distribution (see fig. 15) under reversal of the beam helicity permits extraction of FA a n d hence the spin-transfer factor D^L,. Our present data set includes runs with unpolarized hydrogen and deuterium gases during 1996 and 1997. The beam helicity was positive in 1996 and negative in 1997 — not an example of rapid helicity reversal! Selecting a data sample for which the luminosity-weighted average beam polarization is zero {i.e., using equal amounts of data from each beam helicity state), and assuming that the detector acceptance is constant, the spin transfer to the A can be related to the asymmetry in A production by r>A

—

( YHA pB,i cos QpA,i) \

/

(/TOI££Ai0(vOcos2eMii)

where the sum is over the A events in the dataset, and \\Pg\\ is the luminosityweighted average square of the beam polarization. This expression maximizes the statistical significance of the measurement and minimizes any false asymmetry due to the detector acceptance. The preliminary HERMES result for longitudinal A polarization in the current fragmentation region is DlL, = 0.10 ± 0.17 ± 0.09(syst) or P A = 0.03 ± 0.06 ± 0.03(syst) 2

at < z > = 0.5 and < Q > = 2.5 GeV 2 . 5.3

Remember SV'(3)?

Recall that by assuming fiavor-SU(3) symmetry, we can calculate the moments Au, Ad, and As for the proton by using the values of the F and D coefficients from /3-decay and a measurement of g\{x) from inclusive DIS: a0 - Au + Ad + As = AE (quark contribution to proton spin)

329 a% = Au — Ad a8 = —= (Au + Ad - 2As) V3 combined with

19*{x)dx = h (° 3 + T I 0 8 + 1 0 0

and a 3 = £> + F a8 = - ^ ( 3 F - I > ) (where F and D are F = 0.477 ± 0.012 and £> = 0.756 ± 0.011), finally yields (as it is usually written): Au" = (AS + D + 3 F ) / 3 Ad p = (AS - 2D)/3 As p = (AS + D- 3 F ) / 3 where I have attached an explicit "proton" label to these values. Then, as we have already seen, Aup = +0.79 ± 0.04 Ad? = -0.45 ± 0.04 Asp = -0.16 ± 0.04. 5.4

The Proton, the A, and Flavor SU(3)

Flavor-SU(3) symmetry additionally allows you to relate these moments among all the members of the SU(3) baryon octet. 2 6 These relations are summarized in Table 1, using the values of A S , D, and F to parametrize the moments. Then using the current values of F , D, and A S , we can predict the values of the moments for the entire baryon octet (table 2). So, in particular for the A, we see that the current data (assuming flavor-SU(3) is valid) predicts a very significant —0.20 contribution from the u and d quarks to the overall spin. Significantly, the moments of the polarized fragmentation functions, i.e.

Gj = / GUz)dz Jo

330 Table 1: Parametrization of the moments of the polarized quark distributions of the members of the SU(3) baryon octet, in terms of A S , D, and F.

p n E+

E° E" A H°

Au ±(AE + L> + 3F) |(AE-2£>) | ( A E + D + 3F) §(A£ + I>) i(AE + Z3-3F) |(AS-U) |(AE-2Z?) ±(A£ + £ > - 3 F )

Ad ±(A£-2D) ±(A£ + £> + 3F) |(AS + D - 3 F ) §(AE + £>) i ( A S + D + 3F) |(AE-D) |(AS + £>-3F) i(AE-2D)

As i(AE + D-3F) |(AE + I>-3F) | ( A E - 2D) |(AE-2£)) |(A£-2£>) | ( A S + 2JD) i ( A E + Z) + 3F) | ( A E + £> + 3F)

Table 2: Predicted values for the moments of the polarized quark distributions of the members of the SU(3) baryon octet, using the known values of A E , D, and F.

p n E+

E° EA =0

E~

Au 0.79 ± 0.04 -0.45 ± 0.04 0.79 ± 0.04 0.32 ± 0.04 -0.16 ±0.05 -0.20 ±0.04 -0.45 ± 0.04 -0.16 ±0.05

Ad -0.45 ± 0.04 0.79 ± 0.04 -0.16 ±0.05 0.32 ±0.04 0.79 ± 0.04 -0.20 ± 0.04 -0.16 ±0.05 -0.45 ± 0.04

As -0.16 ±0.05 -0.16 ±0.05 -0.45 ± 0.04 -0.45 ± 0.04 -0.45 ± 0.04 0.58 ± 0.04 0.79 ± 0.04 0.79 ± 0.04

obey the same algebra as the moments of the polarized quark distribution functions. In this context, we place a "carat" over the (unknown) parameters of that algebra: AE F D. So, for example, Gt = ( A S - D) / 3 , just as Au A = (AE - D) / 3 . We can learn about these unknown parameters AE, F, and D by comparing the results of the HERMES deep-inelastic polarized A production and that done via the Z°-pole at ALEPH 27 and OPAL. 2 8

331 A L E P H , OPAL The ALEPH and OPAL experiments at CERN study (among other things) e+e~ annihilation at the Z°-pole. The e + e~ —> A + X production process can be used to learn about q —> A fragmentation. The parity-violating e~Z° and qZ° couplings produces a "polarized quark beam:" UL

-

UR

+ dh - d,R :

SL

-

SR

= 61% : 39%

At the same time, the parity-violating decay of the A allows a study of the polarized q —> A fragmentation functions, just as in HERMES. As the "polarized quark beam" is nearly in a flavor singlet state, it probes A S most strongly (actually A S + .161?). The experimental result is that the A-polarization is about 32%. That means unlike DIS, where A S is small, A S is large. This could mean that s-fragmentation is PS 100% polarized and that u, ^-fragmentation is completely unpolarized. That's what we would expect in a naive quark model picture of the fragmentation. HERMES As already described above, HERMES uses the semi-inclusive deep-inelastic process ep —> e'AX and extracts the A polarization from the parity-violating decay A —> pn~. And, as we already saw in Figure 14, the nucleon target is largely a u and d target. From isospin symmetry, we have that G^ — G§, so that we can treat it as pure u target, and thus we are largely sensitive to A S — D. The HERMES experimental result is that the observed A polarization PA W 0 but probably positive. That implies, to a large extent, that u, d fragmentation into As is nearly unpolarized, and that the SU(S)fiavor parameters D and A S are approximately equal. Unforunately, that also means

A£>f = i(AS + p) is large and positive, which has the consequence that we can expect significant contamination in our A polarization signal from S° in G^A" = G^ — | G ^ • (Note that u —> S° —> A7 is indistinguishable from u —> A if the 7 is not detected, as happens in HERMES.) The factor — | comes from spin-flip and depolarization in the S° ->• A7 decay. The bottom line is that As from S° decay will have a negative polarization. That negative contribution from the secondary As partially cancels an apparently positive polarization from the primary As! It is possible to correct the result for the production and subsequent decay of the S° in two ways:

332

• Fit existing data on the fragmentation of produced A's, which will implicitly correct for the S°. • Monte Carlo the production process, including explicit relative production rates, a model for the fragmentation process, and the unpolarized and polarized quark distributions.

t c

Scenario 3 _;-- Scenario 2 .^— Scenario 1

_„.-••"

'5. 0.6

_ * HERMES

0.4

0.2 -

:: . -0.2

-0.4. , 0.2

,

.

I

0.3

.

,

,

I

0.4

,

0.5

0.6

0.7

0.8

0.9

Figure 16: Comparison to Models - Spin transfer D^,, as a function of z. The curves [deFlorian et al. 2 9 ] are Q 2 -evolved fits to the existing data on A production. Scenarios 1 and 2 use an Ansatz for the polarized fragmentation which is inspired by the spin structure of the A in the naive Q P M (scenario 1) and the Burkardt-Jaffe SU(3) model (scenario 2 ) . 3 0 Scenario 3 assumes an flavor symmetric contribution to the A polarization. T h e data point is the preliminary HERMES result at the average z of 0.5.

Figure 16 shows the experimental result for the spin transfer in comparison with the predictions made in deFlorian et al. 29 They are based on fits to A production data in e+e~ collider experiments. The different scenarios in Figure 16, which all describe the recent results on A polarization in ALEPH 2 7 and OPAL 2 8 quite well, are:

333

1 The naive QPM, where only the s quark is polarized in the A and contributes to polarized A production 2 The Burkardt-Jaffe 30 model, in which u and d quarks also contribute with a negative sign 3 A rather extreme scenario, in which all the light quark flavors contribute equally to the A polarization. The trend implied by the results of these scenarios has been confirmed by a Monte Carlo simulation (Schnell 26 ) including spin transfers from decaying hyperons, i.e. the S°. The resulting spin transfers are based on the spin structure of the various produced hyperons using either the naive QPM or the Burkardt-Jaffe model. At this level of measurement, polarized fragmentation looks like the naive quark parton model! Thus, no "spin crisis" in polarized fragmentation. But we have not enough data yet to make definitive statements about models of polarization in the A, particularly in regards to a possible u-quark polarization in the A. This will have to await the analysis of additional A production data in HERMES collected in the years 1998-2000. Acknowledgements I would like to thank G. Schnell and M. Burkardt for extremely useful discussions. This work was supported by the U.S. Department of Energy and the Thomas Jefferson National Accelerator Facility. References 1. J. Ellis and R.L. Jaffe, Phys. Rev. D 9, 1444 (1974); Phys. Rev. D 10, 1669 (1974). 2. J. Ashman et al. (The EMC Collaboration), Nucl. Phys. B 328, 1 (1989). 3. K. Ackerstaff et al. (The HERMES Collaboration), Nucl. Instrum. Methods A 417, 230 (1998). 4. D. DeSchepper, L.H. Kramer, S.F. Pate, et al, Nucl. Instrum. Methods A 419, 16 (1998). 5. K. Ackerstaff et al. (The HERMES Collaboration), Phys. Rev. Lett. 82, 1164 (1999). 6. K. Gottfried, Phys. Rev. Lett. 18, 1174 (1967). 7. M. Arneodo et al. (The NMC Collaboration), Phys. Rev. D 5 0 , 1 (1994).

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8. A. Baldit et al. (The NA51 Collaboration), Phys. Lett. B 332, 244 (1994). 9. E. A. Hawker et al, Phys. Rev. Lett. 80, 3715 (1998); J.C. Peng et al, Phys. Rev. D 58, 092004 (1998), and references therein. 10. K. Ackerstaff et ak (The HERMES Collaboration), Phys. Rev. Lett. 81, 5519 (1998). 11. M. Gliick et al, Z. Phys. C 67, 433 (1995). 12. H.L. Lai et al, Phys. Rev. D 55, 1280 (1997). 13. D.A. Ross and C.T. Sachrajda, Nucl. Phys. B 149, 497 (1979). 14. K. Ackerstaff et al. (The HERMES Collaboration), Phys. Lett. B 404, 383 (1997). 15. K. Abe et al. (The SLAC E154 Collaboration), Phys. Rev. Lett. 79, 26 (1997). 16. A. Airapetian et al. (The HERMES Collaboration), Phys. Lett. B 442, 484 (1998). 17. K. Abe et al. (The SLAC E143 Collaboration), Phys. Rev. Lett. 74, 346 (1995); Phys. Lett. B 364, 61 (1995); Phys. Rev. D 58, 112003 (1998). 18. D. Adams et al (The SMC Collaboration), Phys. Lett. B 329, 399 (1994); Phys. Rev. D 56, 5330 (1997); B. Adeva et al, Phys. Lett. B 412, 414 (1997); Phys. Rev. D 58, 112001 (1998) and references therein. 19. J.M. Niczyporuk, E.E.W. Bruins, Phys. Rev. D 58, 091501 (1998). 20. K. Ackerstaff et al. (The HERMES Collaboration), Phys. Lett. B 464, 123 (1999). 21. D. de Florian, O. Sampayo, R. Sassot, Phys. Rev. D 57, 5803 (1998). 22. M. Gliick et al, Phys. Rev. D 53, 4775 (1996). 23. T. Gehrmann and W.J. Stirling, Phys. Rev. D 53, 6100 (1996). 24. R.L. Jaffe, Phys. Rev. D 54, R6581 (1996), and references therein. 25. P.J. Mulders and R.D. Tangerman, Nucl. Phys. B 461, 197 (1996), and references therein. 26. G. Schnell, Ph.D. thesis, New Mexico State University, May 1999 (unpublished). 27. D. Buskulic et al. (The ALEPH Collaboration), Phys. Lett. B 374, 319 (1996). 28. K. Ackerstaff et al. (The OPAL Collaboration), Eur. Phys. J. C 2, 49 (1998). 29. D. de Florian, M. Stratman, and W. Vogelsang, Phys. Rev. D 57, R5811 (1997). 30. M. Burkardt and R.L. Jaffe, Phys. Rev. Lett. 70, 2537 (1993).