Baryon form factors at high Q2 and the transition to perturbative QCD

PHYSICS REPORTS (Review Section of Physics Letters) 226, No. 3(1993)103—171. North-Holland

Baryon form factors at high

Q2

PHYSICS REPORTS

and the transition to perturbative QCD

Paul Stoler Physics Department, Rensselaer Polytechnic Institute, Tror, NY 12180, USA Received August 1992; editor: G.E. brown

Contents: I. Introduction 2. Form-factors and helicity amplitudes — notation 2.1. Electromagnetic helicity matrix elements 2.2. Elastic scattering 2.3. Resonance helicity form-factors 2.4. Resonance helicity amplitudes 3. Review of constituent quark models 3.1. Non-relativistic constituent quark models 3.2. Corrections to the independent quark shell model 3.3. Relativistic invariance of the constituent quark model 4. High-Q~form-factors 4.1. Light-front Fock state expansion 4.2. Helicity conservation 4.3. Form-factors for exclusive processes 4.4. Radiative corrections — Sudakov suppression

105 107 107 107 109 110 III Ill 114

115 117 118 118 118 128

5. Elastic scattering 5.1. Constituent quark model 5.2. Vector meson dominance and hybrid models 5.3. PQCD calculations 5.4. Radiative corrections — Sudakov suppression 6. Resonance form-factors 6.1. Analysis of inclusive data 6.2. Amplitude extraction 6.3. The P 33(1232) or A (1232) 6.4. The P11(1440) or Roper resonance 6.5. The S11 (1535) 6.6. The D13(1520) and F15 (1535) 6.7. Duality 7. Diquark models 8. Summary and outlook References

131 132 133 136 138 138 139 141 145 148 154 158 160 162 167 169

Abstract: The status of electromagnetic2 form-factors of the nucleons and non-strange nucleon resonances quark is reviewed. Of and special is the correspond to breakdown of the validity of constituent models, the interest region of Q2 question of what corresponding to the regions onset of in the Q validity ofperturbative QCD. Theoretical models based on perturbative QCD are reviewed and results of calculations compared to available experimental data. It is observed that relativistic constituent quark models become invalid for Q2 greater than about 2 GeV2/c2, compared with available experimental data. The Q2 dependence of experimental form-factors appears to begin to follow perturbative QCD scaling rules for Q2 as low as 5 or 6 GeV2/c2. A notable exception is the A(1232), whose form-factor decreases faster than other states. However the normalized magnitudes of form-factors at experimentally accessible values of Q2 obtained with perturbative QCD sum rule techniques is controversial and currently unsettled. The diquark model is discussed as a possible bridge between low and high-Q2 kinematic regions.

0370-l573/93/~24.00 © 1993

Elsevier Science Publishers B.V. All rights reserved

BARYON FORM FACTORS AT HIGH Q2 AND THE TRANSITION TO PERTURBATIVE QCD

Paul STOLER Physics Department, Rensselaer Polytechnic Institute, Troy, NY 12180, USA

NORTH-HOLLAND

1. Introduction The description of hadrons and their excitations in terms of elementary quark and gluon constituents is one of the fundamental problems in physics. For deep inelastic electron scattering it had been well established that perturbative QCD (PQCD) is the correct description for Q2 as low as a few GeV2/c2. These phenomena involve the virtual photon interacting with a single asymptotically free quark, followed by complicated hadronization processes. The situation for an exclusive reaction in which the baryon remains intact in its ground state or in a resonant excitation is quite selective, and becomes successively less probable with increasing Q2. This is because the momentum of the incident photon must be shared among the constituents in a manner such that the entire recoiling hadronic system remains in the ground or excited state of the original baryon, resulting in a reduction in the elastic and resonant cross sections relative to the deep inelastic cross sections with increasing Q2. This is seen in fig. 1, which shows some of the early inelastic electron scattering cross section measurements at SLAC. The relative decrease in cross sections for exclusive reactions is expressed in terms of formfactors. These form-factors are the electromagnetic matrix element for the exclusive transition, and thus contain the information about the charge and current structure of the baryon in the initial and final states, as well as the transition operator. The measurement and calculation of form-factors provide the meeting ground between theoretical conjecture and experimental reality. At low Q2 the incident photon has a long wavelength and acts coherently upon the entire nucleon, including the complex configurations of gluons and sea quarks. The virtualities k2 of the exchanged gluons are on the average very small, and the strong couplings ot 2) large, so that it is 5(k inappropriate to treat the problem perturbatively. Due to the complexity, non-relativistic constituent quark mean field models were developed to describe the properties of baryons. To account for spectroscopic splittings of the observed quantum states residual two-body “hyperfine” interactions were introduced. Quite remarkably, these models account rather well for many static (Q2 = 0) baryon properties, in terms of a limited number of phenomenological parameters. The situation was quite analogous to the earlier successes in the development of the nuclear shell model. The reasons for the early successes of the quark shell model are not well understood and considerable investigations into the extent of its validity are planned. As in the nuclear shell model, many excitations and phenomena were predicted which have yet to be experimentally studied, and there is considerable theoretical research to refine simple models. At high Q2 the situation becomes in one sense relatively simpler, and in another sense more complicated. At high Q2 the short wavelength photons interact with short range nucleon substructure. At asymptotically high Q2 it is widely believed that only the simplest nucleon substructure involving the minimum number of gluon exchanges is involved in exclusive transitions. This occurs when the photon happens to find the nucleon in a small-three-quark Fock substate, with dimensions comparable to the photon’s wavelength, such that only two hard gluon exchanges need to be considered. Such occurrences are rare (about a few percent at attainable momentum transfers). More complicated configurations are less probable to produce an exclusive reaction since the nucleon will more likely fragment. A difficulty in treating the exclusive reaction is how to extract the three-quark Fock state from the overall complexity of the nucleon ground and resonant state structure. 105

106

P. Stoler, Barvon j~rmfiwtorsat high

Q2

Q2

5x1:

~25OfI~~

I

(GeV2/c2)

2

: i~~ii i i i i i i i I I___~. deep inelastic

:~

reson~—~”

Excitation Energy W

15

(GeV)

Fig. I. Inclusive inelastic electron scattering cross sections (in nbGeV~sr~) at E = 13.5 GeV2/c2. The figure is from ref. [Fr-72].

A major advance in the calculation of exclusive form-factors at high Q2 was the development of the concept of factorization, which allows form-factors to be written in the form ~ 4~T4~.The three-quark distribution amplitudes 4~and 4~contain the non-perturbative parts of the formfactor, and the transition operator T is a sum of leading order purely perturbative terms. Further progress in determining the 4fs has come through the development of QCD sum rules, and currently via lattice calculations. The validity of calculations based on these techniques is very controversial. Closely related to these issues is the subject of “color transparency”, in which finite nuclei are used as filters to gain information about the size and structure of an elastic or resonance excitation in electron scattering from a nucleon within the nucleus at high Q2. This subject has been extensively reviewed and will not be treated here. Evidently there is a great deal of interest in the subject of exclusive reaction form-factors at high Q2, and particularly the questions of where in Q2 the limits of the usefulness of constituent quark models are, and where in Q2 PQCD models become justified. The purpose of this article is to review the literature pertaining to these questions, for elastic and resonant inelastic exclusive scattering.

P. Stoler, Baryon form factors at high

Q2

107

We find there is a paucity of existing experimental data, so that there is a lot of as yet untested theory in the literature. Therefore, significant progress in resolvingthe issues awaits the completion and construction of new high energy, high duty factor electron accelerators. This review is organized as follows. Section 2 reviews the concepts of form-factors and electromagnetic amplitudes, and their relationships to each other. Section 3 briefly reviews the low-Q2 constituent quark models, the SU(6) classification of baryon states, and the extent in Q2 of their validity. Section 4 summarizes the issues involving the calculations of form-factors at high Q2. Section 5 discusses the experimental status of elastic scattering and section 6 that of resonance transition form-factors, and the confrontation with theoretical calculations. This includes a recent analysis of available experimental inclusive data. Section 7 touches on diquark models. Finally, section 8 is a summary of the current status of the field, and the prognosis for future experimental programs involving new facilities. 2. Form-factors and helicity amplitudes

—

notation

The study of electron scattering from nucleons is one of the most important tools for studying the structure of baryons. The relatively weak nature of the coupling constant allows one to express the reaction in terms of a single (virtual) photon interacting weakly with a charge and current distribution in a precisely understood way, and the reaction can be interpreted unambiguously in terms of the charge and current structure of the baryon. All the information about the electromagnetic structure of the baryon is contained in structure functions and form-factors. The form-factors are the directly measured deviations of the reaction cross section from that which would be observed in scattering from a structureless point object. These form-factors are the charge and current transition matrix elements. In this section we review the notation and formulae which are referred to in subsequent sections. This section may be skipped without any loss of continuity. 2.1. Electromagnetic helicity matrix elements Electromagnetic transition helicity matrix elements correspond to transitions in which the initial state has helicity 2 and the final states have 2’. The concept of helicity is illustrated in fig. 2 for 2 = + 1/2. Transitions between a nucleon state N)’, and resonant state I R> can be expressed in terms of dimensionless helicity matrix elements ~

(2.1)

The notation follows that of ref. [Ca-86]. The polarization vectors s ±~0 correspond to right and left circularly polarized photons, and longitudinally polarized photons, respectively. 2.2. Elastic scattering Elastic electron scattering cross sections can be expressed as follows:

~7mfrec[IFiI2+ x,c2IF 2 + 2t(1F 2 tan2O/2]. 2I 1 + ,cF2I) In the above am is the Mott cross section for scattering from a point object, =

(2.2)

108

P. Stoler, Baryon form factors at high

A

Q2

+1/2

=

G~

k 112

A

A

+1/2

=

+1/2

=

K~Y\JThJ\J\JGO=GE

A

=

C513

+1/2

G..

A3~3

A = -3/2 Fig. 2. Graphical representation of helicity form-factors GA. and helicity amplitudes AA, as defined by eq. (2.1) in the text.

2(0/2)]2 [cccos(0/2)/2Esin where r Q2/4M~,and frec = E’/E is a recoil factor. The quantity ic is the value of the anomalous nucleon magnetic moment in nuclear magnetons: ~ = 1.79, and i~ = 1.91. F 1 is the helicity conserving Dirac form-factor, and F2 is the non-conserving Pauli 2 =helicity 0, F~’(0) = F~(0) = 1, and for form-factor. The convention adopted here is that for a proton at Q a neutron F?(0) = 0, F~(0)= 1. Often it is more convenient to express the cross section in terms of the Sachs form-factors =

—

GM

=

F 1 + KF2,

GE

=

F1

—

tKF2.

Substitution of eq. (2.3) into eq. (2.2) gives 2+XIGMI2 2 2 + 2rIGMI tan 0/2 j da = amfrec((1GE1 dQe \ 1+t / —

(23)

-

(24)

P. Stoler, Baryon form factors at high

Q2

109

In the static limit G~(0)= 1 + ic~F~(0) 2.79,

G~(0)= 1.91,

G~(0)= 1,

G~(0)= 0.

The elastic scattering helicity matrix elements (eq. 2.1) can be written as linear combinations of the form-factors. For elastic scattering, IR> (= IN>) has spin 1/2, and only 2’ = ±1/2 contributes. One then has = (Q/\/~MN)GM= (Q/~,,/~MN) (F 1 + KF2), G0 = GE = F1

(2.5)

2/4M~)KF —

(Q

2.

(2.6)

In terms of the helicity matrix elements the elastic cross section becomes

= amfrec(

01+2G+I + IG+I2tan2O/2).

2.3. Resonance helicity form-factors Inclusive inelastic sections are expressed in terms of structure functions 2, v) and W 2,scattering v) by the cross relation W1(Q 2(Q d2 dE’dQe = amfrec [W

2) + 2W

2(v, Q

2) tan20/2]

.

1(v, Q

(2.7)

The inelastic scattering cross section (eq. 2.7) for a resonance may be written in terms of longitudinal and transverse form-factors GE and GT,

dE’dQe =

amfrec(i±*

+ 2t*IGTI2 tan20/2)R(W).

(2.8)

In analogy with the Sachs form-factors for elastic scattering, the resonance longitudinal and transverse form-factors are written as 2 + IG_12) = T*1G12 0 = GE, ~(IG÷I A resonance line shape of the following form is introduced: G

R w

(

—

21r’WRMNFR (W2_W~)2+W~F2

(2.9)

21 (.0)

The analogous kinematic quantity is t~ = (Q2 + W~ M~)2/4M~Q2.Note that for elastic scattering .r* ~ t = Q2/4M~.The recoil factor is now —

frec = (E’/E)[l

(W~ M~)/2MNE]1.

110

P. Stoler, Baryon form factors at high

Q2

In the limit of a very narrow resonance, in which WR = MN and WE F 0, R(W) becomes a 5 function and the cross section reduces to that for elastic scattering. A distinction should be made between the form-factors used here, and one which is commonly found in the literature in describing the A(1232) resonance, namely G~.In terms of the form-factors used here one has (see ref. [Ca-86]) —~

G~I2=

l,

IGTI2 + 2~IGEI2).

2t

(2.11)

2+Q2(

At high Q2 G~c/cGT/Q.

Therefore, one must be careful in comparing form-factors used by various authors. 2.4. Resonance helicity amplitudes Equation (2.7) may be expressed as a product of a virtual photon absorption cross section and a virtual photon flux factor, d2 a dQdE’ = FV(aT +

£t7

(2.12)

1).

In eq. (2.12) aT and a1 are the cross sections for the absorption of transverse and longitudinal photons, respectively, and ~ is the virtual photon polarization parameter, 1 2/Q2)tan20/2~ = 1 + 2(1 + v

The virtual photon flux factor F~is given by ~ KE’

2

(2.13) where K (W2 M~)/2MNis the equivalent real photon energy. The virtual photon cross sections are related to the structure functions by equating eqs. (2.7) and (2.12), —

4~2~

aT=~(a+ +a4=_~k~_W1(v,Q2)~

a

2) 0 =

+

W2(v, Q

—

(2.14)

W

1(v, Q2)]

(2i5)

Electromagnetic transition matrix elements are commonly expressed as electromagnetic helicity amplitudes introduced in refs. [Co-69a, b]. The relationships between the helicity amplitudes and

P. Stoler, Baryon form factors at high

Q2

111

the helicity matrix elements introduced in eq. (2.1) are A 112 =~J~~G+1 A312 =

\J~~’

S112 = \/~~~Go.

(2.16)

In eq. (2.16) KR is the equivalent real photon energy at the resonance position. For transverse photons the total virtual photon cross section at the resonance position is given directly in terms of the electromagnetic helicity amplitudes aT(R) =

~M

(IA112 12 + IA312 2).

(2.17)

3. Review of constituent quark models This section reviews the main features of the constituent quark models (CQM) based on 2 over which such models areSU(6) valid flavor—spin symmetry. Of primary interest will be the range of Q in describing electromagnetic transition amplitudes. It is observed that relativistically invariant calculations in the framework of the CQM begin to deviate significantly from experimental data for Q2 greater than a few GeV2/c2. At higher Q2 it will be necessary to invoke new degrees of freedom, which at sufficiently high Q2 must merge into PQCD. 3.1. Non-relativistic constituent quark models Most of the early models of the structure of excited baryons in terms of quarks were founded directly on ideas and techniques developed in nuclear physics. Elaboration of these early models has led us to our current understanding of baryon st~uctureat large distance scales (low Q2). In nuclei the symmetry of nuclear forces with respect to spin, SU(2)~and isospin SU(2) 1 led to the ideas of SU(4) supermultiplet theory of nuclear excitations [Wi-37,41]. The concept was takencoupling over intowith particle 3)F symmetry spin physics SU(2) by inclusion of strangeness with isospin to give an SU( 5 symmetry resulting in the SU(6) spin—flavor supermultiplet theory of hadronic structure [Gu-64], which could account for the spectroscopic properties of baryon states. In the framework of SU(6) the spin—flavor part of the baryon wave functions may be classified into multiplets which have different symmetry properties. These combined with the spatial 0(3) wave function gives an overall even symmetry, since the color SU(3)~part of the wave function is odd. Physically the SU(6) multiplets correspond to three flavor quarks each having spin 1/2. These quark states can be combined to yield the following multiplet structure: 56S~7OMS$7OMA~2OA,

6®6®6 = where the subscripts denote the symmetry of the multiplets under interchange of two quarks. S and A denote symmetric and antisymmetric under the interchange of any two quarks. MA and MS denote mixed symmetry in which the wave function is antisymmetric and symmetric, respectively,

112

P. Stoler, Baryonform factors at high

Q2

with respect to interchange of the first two quarks. The states of SU(6) are frequently written as products of separate SU(3)F and SU(2) 5 states. Table3)F 1 gives the relationship between the SU(6) multiplets. Table 2 shows multiplets of multiplets and the products of the SU(2)~and SU( table 1 expressed in terms of products of flavor and spin states, where the SU(3)F and SU(2) 5 states 4~Fand Xs, respectively. areConsidering denoted only non-strange baryons, multiplets are composed of SU(2) 5 and SU(2)1. The overall SU(4) spin—isospin multiplets for three-quark states are sub-multiplets of SU(6), 2OS~2OMS~2OMA~4A.

4®4®4 = For example, the 20s is the SU(4) non-strange component of the S6~multiplet of SU(6). The nucleons and deltas are members of the 56~SU(6) multiplet, or the 2Os of SU(4). The nucleons comprise a multiplicity of 4 (S = 1/2, I = 1/2), and the deltas a multiplicity of 16 (S = 3/2, I = 3/2). Table 3 exhibits the known low-excitation spectrum of non-strange baryons, and fig. 3 shows their ordering. The notation is L 21, 2J( WR), where L is the orbital angular momentum of the single pion decay channel, I and J are the isospin and total angular momentum, respectively, and WR is the mass. Determining the ordering and energy spacing of the states requires a dynamical physical model. In 1964 it was suggested [Gr-64] that the observed baryon spectra might be described by colored quarks (obeying “para-Fermi statistics”) moving in single particle orbits such that the total space [0(3)] and spin—flavor [SU(6)] wave function must be symmetric under permutations, i.e. a quark shell model. The idea of the shell model was elaborated in ref. [Da-65]. To address the specific energy spectrum of the multiplets requires assumptions about the specific nature of the interactions among the constituents, as well as the dynamical properties of the constituents themselves. In fig. 3 it is seen that the level bands alternate in parity and are very Table 1 3)F and SU(6) multiplets SU(2)expressed in terms of the constituent SU( 5. The notation is (SU(3)F, SU(2)5) S M A

(10,4) (lO,2)+(8,4)

+(8,2) +(8,2)+(l,2) (8,2)

=

+(l,4)

Table 2 The multiplets oftable 1 expressed in terms ofthe spin x and flavor is from ref. [C1-78] S

4

56

=70 =20

three-body wave functions. The table

4~y~m(10,4) 4’M,AXMA) (8,2)

M 5

(1/V/~)(~MsxMs + 4’SXM.S (10,2) 4’M.SXS (8,4) (1/~~/~)(—t~M5xMS + 4~M,AXM.A) (8,2) 4’AXM.A (1,2)

A

MA

4’SZM.A 4’M.AXS

(l/~/~)(~MsxMA + 4JM,AXMS) ~AXM.S 4’AXS

(1/\/~)(~MSxMA —

4’M,AXM.S)

~(1,4) (8,2)

(10,2) (8,4) (8,2) (1.2)

P. Stoler, Baryon form factors at high

Q2

113

Table 3 The assignment of non-strange resonances in terms of a harmonic oscillator 0(3) basis, and SU(6) spin—flavor multiplets. In the first column N is the oscillator shell quantum number. In the second column dim gives the SU(6) multiplet. Columns 3 through 6 show the contributing SU(3)F and SU(2) 5 sub-multiplets of tables 1 and 2. The sum of the multiplicities in columns 3 through 6 equals dim in column 2. The numbers in parentheses are the masses, standard font numerals denoting four-star, and italic denoting three- or two-star states according to ref. [PDG-92]. The table is from ref. [Gi-91] N

(dim,L~)

28

0

(56,0k)

P11(939)

1

(70,1—)

S~~(1535)

~8

210

~iO P33(1232)

S11(1650) D13(1530)

S31(1620) Di3(1700)

D33(1700)

D15(1675) 2

(56,0’) (70,O’) (56,2k)

P11(1400) P11(1710) P13(1720) F15(1680)

P33(1600) P13

P31(1910) p31 P33 (1920)

F35 (1905) F37 (1950) (70,2k)

P13

P11

P33

F15 (2000)

P13

F35 (2000)

F15 F17 (1990)

roughly evenly spaced. This suggested the use of simple harmonic two-body forces [Fa-69] to try and calculate the level spectrum. The use of harmonic two-body forces results in very attractive calculational simplifications. (1) The solutions are simple analytic functions. (2) It is easily shown that the problem of three bodies moving with harmonic two-body forces is mathematically equivalent to three bodies moving independently of each other under the influence of a simple harmonic central potential, i.e. a single particle shell model. The energy scheme for the harmonic oscillator states is simply (1 + 2n + 3/2)hw. In the harmonic oscillator model the nucleon N11(938) i~(1232)[or have three 3 with total LR =and 0~,and belong P33(1232)] to the 56s spin—flavor quarks in the lowest n and 1 orbitals (is) multiplet, or in terms of spin—isospin SU(4) they are members of the 2Os multiplet. The band of negative parity levels in fig. 3 corresponds to the E~= 1 hw shell configuration (ls)2(lp)1 combining to give L~= 1. The overall spatial symmetry with respect to interchange of any two quarks is mixed M 5 and MA70MA withand S and A referring interchange of quarks 1 and These 70M5 states oftoSU(6), respectively, to give an2.overall states must be combined with the symmetric spin—flavor—space state. The SU (6) 70 multiplet may be expressed in terms of flavor and spin multiplets (F, S). From table 1 70 = (10,2) + (8,4) + (8,2) + (1,2). The states in fig. 3 and table 3 belonging to the negative parity multiplets are identified as follows: (10,2): S 31(1620), D33(1700) (8,2):

S~~(i535), D13(1520)

(8,4):

S11(1650), D13(1700), D15(1675).

114

P. Stoler, Baryon form factors at high

_____ l56,2~i

Q2

W(GeV) F 37 P31 p33

F35 P13

1.8 SD D’13_____

F15

_____

1.6 D13 1.4

P33

1.2

[56,o~) 1.0 P

~11

poaitive panty

negative parity

Fig. 3. The spectrum of non-strange nucleon resonances up to an excitation of 2 GeV.

Higher energy level assignments become more uncertain because of the increased multiplicity of predicted states. Likewise, as seen in fig. 3, there are many experimentally observed broadly overlapping states. The positive parity states shown in fig. 3 with energies greater 56s than about states of 1650 MeV are candidates for the LR = 2 + even symmetry spatial states coupled with SU(6). Some of these are (10,2):

F 35(1905), P31(1910), P33(1920)

(8,2):

F15(1680), P13(1720)

Among the states at E~= 2hw are those with radial excitation involving projection one 56s multiplet with the L~= 0~.The of Roper quark to the 2s level, Pwhich would be a member of a resonance, designated 11 (1440) in fig. 3, is a candidate for this state. (This resonance is also a candidate for a state having hybrid quark—gluon characteristics.) 3.2. Corrections to the independent quark shell model The SU(6) shell model has seen many important improvements and refinements over the past two decades. A notable example is the model of Isgur and Karl (I-K) [Is-78, 79 et seq.], which adds

P. Stoler, Baryon form factors at high

Q2

115

a quark—quark color hyperfine interaction composed of contact and tensor terms having the form 2

=

j
[~7rSi.Sit~(rii) + 1(3(Sj.rjj)(S,.r

13)

m,m3

= Hcontact

r1~

—

nj

+ Htensor

The contact term causes splitting of degenerate states with different spin combinations, e.g. N and A. The tensor term mixes states with the same overall J, but having and 312Sdifferent L12S S combinations. For example, the two negative parity L = 1 SU(6) states 112 and ‘ 1~2 become two states that are linear combinations of the two pure SU(6) states, thereby breaking the pure SU(6) symmetry. An example is the structure of the A(1232), in which the tensor interaction introduces a small (few percent) contribution of L = 2 into the predominantly L = 0 structure. The simple I—K model, although marking an important turning point, is too naive. Its shortcomings and improvements are summarized in refs. [Ca-89, Gi-90, 91]. However, there has been, 2 models and continues to be [CL-90]. considerable work withadding the goal of making theoflow-Q more sophisticated Thesetheoretical include explicitly gluonic degrees freedom, and relativizing the basic Hamiltonian. 3.3. Relativistic invariance of the constituent quark model To investigate the range of validity of CQM baryon wave functions Dziembowski [Dz-88a, b] calculated electromagnetic form-factors as a function of Q2 in a relativistically covariant light-front (or light-cone) Fock state basis. The light-front formalism is a natural framework for relativistically treating systems in which there are fixed numbers of constituents. The formalism was introduced in ref. [Di-49]. Applications in quark model problems have been treated in detail in, e.g., refs. [Le-80, Na-85]. The light-front variables are defined in terms of Cartesian coordinates, z+ =t=zo+z

3 P~mPo+P3,

z

~z0—z3,

P=P0P3,

z1=(z1,z2), PiP1+P2.

The variable p~plays the role of the light-front energy. One of the principal simplicities is that it is linear in p3 so that there is only one solution, in contrast to the quadratic relationship between ordinary energy and momentum, which leads to the creation of positive and negative energy states. Note, in this reference frame events which are connected by the speed of light all occur at the same light-front time ‘r. The light-front momentum fraction variables are x,

pj+/p+

k1, =p~

—

xk1

where the subscript i refers to a single parton, and ~ = ~ ~

and p~.=

116

P. Stoler, Baryon form factors at high

Q2

The important advantages of the light-front formalism may be summarized as follows: (1) The basis state can be factorized into a part which is dependent only on the baryon’s internal structure, and an external part dependent only upon the baryon’s overall momentum, =

2~)

~externai I/I (xi, k

11, The internal wave function ~Ji is independent of the external center of mass motion of the baryon. (2) The wave function t/, is invariant under all kinematical Poincaré transformations (translations, three-boosts, and rotations about the three-axis). The meaning of i~1’ is then the frame independent probability amplitude for finding the constituents with momenta ~ and x 1p1 + k1. (3) A Fock state having a given number of constituents in one frame will continue to have the same constituents in all frames. Thus, a specific three-quark Fock2.state at rest is invariant under all Lorentz boosts, enabling one form-factors at any The CQM wave function in to thecalculate rest frame in the notation of Qref. [Dz-88a, b] is expressed in the form

cl’A(k~,2~)=

4~cm(Xi,k±~)x~(2

1)

in which 4cm is a CQM momentum space state, and X1 is a spinor with helicity 2 which is built from products of three quark spinors with helicities 2~.This can be expressed as a Lorentz invariant light-front wave function, 1/2

(n~~) —

~A(xt, k±~, 2~)=

~LC(Xi,

kjt)XA(xt, k1~,2~)

Note that the spinors now become functions of x, and ~ The ground state model wave function used in ref. [Dz-88a, b] (and earlier in ref. [Br-81b]) is a solution of a SHO Hamiltonian, 4~cm(1~i) = ANeXP(

—

~

k~/2~2) = ANexP(

—

i~1

k~~+ m~/62)

(3.1)

The theory has two parameters, the hadronic parameter and the used constituent quark mass 3 the size model wave cs, functions by Dziembowski quantities of nucleons such as magnetic moments and radii very well. Elastic and transition form-factors as functions of Q2 are obtained by evaluating the matrix elements of the electromagnetic current operator ~ according to the relationships given in ref. [Br-80]. *)For elastic scattering the helicity conserving (Dirac) and non-conserving (Pauli) form-factors are related to the matrix elements as follows: m,. With b]~ reproduce —~ 320 MeV/c m, —~MN/ [Dz-88a, staticand electromagnetic

eFj(Q2)=~4~, p

~

MN

P

J~ip1>.

*hln a Drell—Yan coordinate system the matrix elements are diagonal, and ignoring quark anomalous magnetic moments one can restrict oneself to the component y ~. For the N —~N* transition Konen and Weber [Ko-90] point out that the non-orthogonality of CQM wave functions leads to a violation of gauge invariance, which can be rectified by redefining the current operator J~. A more general discussion on gauge invariance in quantum mechanics is given in ref. [Bo-87].

Q2

P. Stoler, Baryonform factors at high

117

In the above the longitudinal momentum transfer is q~ q 1 F1

>~

—

iq2. Note that when q~~ MN,

F2, and the matrix elements are related to the wave functions by
2’~J~/p~p2> = J[dx][d2k±]~~(J+/p+)~,

with the light-front current J + = uy + u3. 2 Calculated elastic form-factors reproduce the empirical dipole form-factors very well at low Q but begin to fall off too rapidly near Q2 2 GeV2/c2. Figure 4, from ref. [Dz-88b], shows the result for the proton magnetic form-factor. The divergence from experiment above 2 GeV2/c2 is attributed to the invalidity of the “soft” mechanism inherent in the CQM, and the onset of “hard” mechanisms which culminate in lowest order PQCD in the high-Q2 limit. Thus Dziembowski [Dz-88b] concludes that PQCD effects must already be considered at values of Q2 exceeding a few GeV2/c2. 4. High-Q~form-factors In the previous section it was seen that constituent quark models (CQM) are useful in describing electromagnetic form-factors at low Q2. However, even when relativistic effects are taken into account [Dz-88a, b] CQM5 become inadequate at Q2 greater than a few GeV2/c2, where the calculated form-factors become significantly smaller than experimental values. Already at Q2 = 5 GeV2/c~the CQM form-factors are more than a factor of two too small, and above = 10 GeV2/c2 they essentially vanish in comparison with experiment. In effect the Gaussian factors in the constituent quark wave functions are responsible for this strong decrease in the form-factors with increasing Q2. (Other potentials may be constructed [De-93] which give slower falloff of the form-factor with Q2, but do not change the conclusions about the inadequacy of the CQM with increasing Q2.) In the extreme high-Q2 limit it is widely believed that exclusive reactions are appropriately treated using the techniques of perturbative QCD (PQCD), involving current quarks interacting

PROTON MAGNETIC FORM FACTOR ~‘G~(Q2)/G~(O) 0.4.

,

• S

S

S

0 0

0. ~

20

a2 I GeV

/c

Fig. 4. The magnetic form-factor ofthe proton, GM, as a function of Q2. The experimental points are due to Arnold et al. [Ar-86]. The solid curve is the result of a calculation employing a relativistically invariant constituent quark model. The figure is from ref. [Dz-88b]. The curve denoted C-Z is the result of a PQCD calculation in ref. [Ch-84a], as discussed in section 4.

118

P. Stoler, Baryonform factors at high

Q2

via the exchange of high-momentum or hard gluons. At high Q2 an exclusive reaction is most likely to involve the exchange of the minimum number of hard gluons among the minimum number of quarks, and becomes increasingly less probable with greater numbers of active constituents. For elastic scattering or resonant inelastic scattering the leading order processes involve three quarks and two gluons. Each exchanged gluon carries a sizeable fraction of Q2, so that quark—gluon couplings are relatively small. The question is then what is the lower limit in Q2 where leading order PQCD is a valid description of exclusive reactions? This is a very controversial question which cannot yet be unequivocally answered. However, some of the most important issues can be identified. In this section we briefly review some of the issues involved in calculating high-Q2 electromagnetic form-factors using PQCD. Detailed discussions of the theoretical developments may be found in other reviews, e.g. ref. [Br-89a]. 4.1. Light-front Fock state expansion

In order to calculate form-factors it is necessary to know the wave functions of the involved baryon states in terms of their quark and gluon contents. If one wishes to limit this to a fixed number of constituents then in general one immediately runs into calculational difficulties, since the interactions create new quanta from the vacuum. Thus, the calculation of a properly normalized wave function requires the calculation of contributions from all orders of particle production from the vacuum. These difficulties are minimized by working with light-front variables (see section 3.3). The baryon wave function is written in terms of its projections on the free (on-shell) quark and gluon momentum space basis to define a Fock state expansion, i.e., the baryon state is expanded as a series of succeedingly more complex components, which is symbolically written as

Ii’>

=

qqq> + IqqqG> + qqq~q>+ qqqciqG> +

In the above, each component describes a state of a fixed number of quarks and gluons, evaluated at equal light-front time t = t + v/c. 4.2. Helicity conservation Helicity is the projection ofa particle’s spin along its direction of motion. Point spin 1/2 particles of zero mass conserve helicity in interactions with vector quanta such as photons, gluons or weakons. To the extent that quarks can be treated as massless on-shell objects, their helicities are conserved in a hadronic interaction with a photon. In cases where the interactions involve only very short distances, or a spherically symmetric baryon distribution amplitude, then L~= 0 and at high Q2 hadron helicities are the sum of individual quark helicities [Ra-92]. Therefore, at high Q2 the initial nucleon has the same helicity as the final nucleon or resonance. This is schematically illustrated in fig. 5. Helicity conservation implies that only F 1 in eq. (2.2) survives and that F1 = GM 2. at high Q 4.3. Form-factors for exclusive processes It was shown [Br-80, Le-79, 80] that at high Q2, where one can ignore the transverse momentum k 1 of the valence quark entering the hard scattering, and invoking hadron helicity

P. Stoler, Baryonform factors at high

A

:

Q2

119

+1/2

—

__

G~

A 1~3

4—

A = +1/2

A=

+1/2

-+

G0 = GE

4-

A

-1/2

A=

+1/2

CI,2

—p

G... -p —4

A,

A3~2

1

A = -3/2 Fig. 5. Schematic representation of helicity form-factors for a three-quark baryon state.

conservation, baryon form-factors may be written as 11

2) = JJ[dx] [dy] ~f(x)*TH~j(y). F1(Q

(4.1)

00

The two main ingredients are the hard transition operator TH, which contains the short distance perturbative physics, and the distribution amplitudes cb, and ~i 1, which are determined by the long range non-perturbative physics. Both TH and ~ are functions of the initial and final longitudinal momentum 2. fractions x (~x1, x2,amplitudes x3) and y ( ~ are 1, Y~ y3), respectively,as well asmomentum of the moThe distribution related to the three-quark mentum distribution transfer Q function by eq. (4.7). The integral in eq. (4.1) is understood to imply fraction [dx] dx 1 dx2 dx3 5(1 ~x1) and similarly for [dy]. —

120

P. Stoler. Barvon form factors at high

Q2

Later Li and Sterman [Li-92b, c] have extended the concept of factorization to include the effects of k±.Their procedure, considered as a higher twist effect, allows factorization to be useful for calculating form-factors at lower Q2 than in eq. (4.1). This is discussed in more detail in section 4.4. Using eq. (4.1) and the concepts of QCD sum rules to obtain the distribution amplitudes, it has been possible to obtain reasonable fits to the elastic form-factor F 2). However, the procedures have been the subject of strong criticism which has not yet been 1(Q resolved. The remainder of this section briefly reviews these procedures, and some of the unresolved difficulties. 4.3.1. Operator The transition operator is constructed perturbatively by considering successive orders of Feynman diagrams involving gluon exchanges amongst the three quarks, as well as higher processes. In leading order two gluons are exchanged among three quarks. Some of the diagrams which must then be evaluated and their corresponding mathematical structures are shown in fig. 6. All the leading order diagrams result in the following form of TH: TH = ;(k

1);(k2)~~

(4.2)

Q~x~Y)f(XY)

(a)

2)cv,(x,y 3)

________________

—yi)Q 2x3y3 3Q Q4lcs.((1 —xj)(1 (1—xI)2(1—y 1)

(b)

1

1 a.((1—x 2)a,(x2y 2) (1—x 2(1—yl)2x2y2 2Q 1)(1—y5)Q 1)

Q4

(c)

__________

_____

L Q4

a,(x

2)cs,(xsyaQ2)

2y2Q (1 — x3)(1

—

y 2)x3y2xays

Fig. 6. Some leading order contributions to the scattering amplitude TH.

P. Stoler, Baryon fisemfactors at high

Q2

121

where k

1 and k2 are the four-momentum transfers of the two exchanged gluons. This results in the well-known dimensional F1 = GMc decrease Q~. Combining and (2.16) onerunning obtains 3. There scaling is also rule a logarithmic due to theeqs. Q2 (2.5) dependence of the G± cc A112 cc Q coupling constant ;, as well as a Q2 dependence due to higher-twist terms. Additionally, the evolution of the cIi(x) with Q2 exhibits a small variation with Q2 as a logarithm raised to a fractional power. The transition amplitudes of fig. 6 immediately present a difficulty. For example, consider the amplitude corresponding to fig. 6c, 1 ;(x Q4(1 —x2

2) (x Y2 Q

(4 3

2) 3 j~Q

3)(1 —y2)x2y2x3y3 2, then at Since the virtualities k~ and k~ofconstants the exchanged are directly proportional to Qtreatment sufficiently high Q2 the coupling will begluons small enough so that a perturbative will be appropriate. However, this is strictly true only for frozen quarks, each having x, = 1/3. When the momentum distributions of the ~‘s are taken into account the k’s also depend on the x’s and y’s, and for small enough values of the x’s and y’s the gluon virtualities k2 are small and PQCD is inappropriate. Another difficulty is that the perturbative logarithmic dependence of the coupling constants,

=

4ir/[f3ln(k2/A2)] ,

(4.4)

causes the integral in eq. (4.1) to blow up at k2 = A2. One way of dealing with this problem is to treat ; as a constant equal to its “average” value, and factor it outside the integral in eq. (4.1). Another way is to insert a low-k2 cutoff in; in terms of an effective gluon mass. Thus, eq. (4.4) becomes = 4ir/flln[(k2

(4.5)

+ m~)/A2] .

Such an effective gluon mass procedure had already been proposed in ref. [Co-82] in calculations involving the possible existence of glueballs. With this ansatz Ji et al. [Ji-87], for example, were able to achieve a good fit to the experimental elastic form-factor with several available Ws, with an effective gluon mass of 300 MeV. Still, it has been pointed out in refs. [Is-84, 89, Ra-91] that due to the structure of terms in TH, such as in eq. (4.3), a disproportionately high contribution to the form-factor comes from the region near the kinematic limits of the x’s and y’s, which is just where a PQCD approach is not valid, and only at very high values of Q2 does the perturbative region dominate the form-factor. At such high Q2 it is speculated that form-factors would be too small to be practically measured. In addition, one may very well argue that, to be consistent, m~should also appear in the gluon propagators. In fact, it is shown in ref. [Ha-92] that the effect of a gluon mass can be obtained in the gluon propagators by explicitly keeping kI contributions in the expression for the amplitude. Equation (4.3) then becomes 1 f54 11 ~.1 ~

;(x —

2 + m~);(x

2+ m~)

2y2Q i 3y3Q 2~ X 5(1 3J~I — Y2,’.X2Y2 -1- mg,~x3y3-r

2

mg

(

.

)

122

P. Stoler, Baryon form factors at high

Q2

By introducing an effective gluon mass as in eq. (4.6), it has been pointed out [Is-84, 89, Ra-91] that the contributions near the kinematic end points are no longer dominant. The obtained form-factors are reduced by a very large factor, and strongly underestimate the experimental values for most quark distribution amplitudes [Ha-92]. In any case, it may be argued that the introduction of an effective gluon mass is really a way of parameterizing non-perturbative effects and therefore hybridizing the theory. A discussion of the effects of these procedures on the resulting elastic form-factors is given in section 5. 4.3.2. Distribution amplitudes The distribution amplitudes P(x), in eq. (4.1) [Br-79, Le-79], contain the physics of the baryon structure. They are determined primarily by the dominant long range non-perturbative interactions of the quarks among themselves and with the physical vacuum. A significant advance in determining these distributions has been made possible by the successful application of PQCD sum rules [Sh-79, Ch-84a] Suppose ~P(x,k 1) gives the distribution of the valence quarks as a function of x ( x1, x2, x3), k1(m k11, k12, k13), and the spin, flavor and color. Then

2).

2k ~(x)

[d

(4.7)

1] ~(x, k1) 0(k~
qi(x) is the probability amplitude for the baryon to exist in a three-quark Fock state with quark longitudinal momentum fractions x~,collinear up to a momentum scale min~(x 1Q),with characteristic transverse size parameter b —~ 1/Q. The overall probability that the baryon exists in the leading 2 = P3q. three-quark Fock state is given by J[dx]J~(x1)I Since the baryon color state is an antisymmetric singlet, the distribution amplitude is composed of products of longitudinal momentum fraction distributions 4~N and spin and flavor functions with overall even symmetry. For example, the distribution amplitude for a nucleon whose spin is polarized in the + z direction may be written in the symmetric form [Ji-87] ~(x,

Q2) =f;~) ><

{u1u~d~ 4N(x,Q2) + u1u1d14N(x, Q2)—

Ut utd1[~(x, Q2)

+ 4~.

2)]}

1(x,Q

(4.8)

The dimensional constant fN(Q2) is fixed by the value of the distribution amplitude at the origin and the condition J[dx] 4~N= 1. It will be useful to rewrite eq. (4.8) in terms of symmetric and antisymmetric distribution amplitudes and spin—flavor functions (see refs. [Ca-87a, b]) with respect to interchange of quarks 1 and 3, =

~

/~x~ +perm.,

(4.9)

where ~s=(1/~~/~)IUdu_duu_uud>tit, ~A(1/~/~)Iuudduu> 111

P. Stoler, Baryonform factors at high

Q2

123

The distribution amplitudes in eqs. (4.8) and (4.9) are related as follows: 4N(x)

=

(4\/2/fN) [4~pS(x)

—

In order to obtain 4Ns or more generally ~ for a baryon state, it is necessary to evaluate the gauge invariant matrix element of a tn-local operator [Ch-84a, Ki-87] which connects the vacuum with a baryon composed of three quarks and their interactions with each other and with the physical vacuum, via the strong gauge field, <01 [eA13u 5(zi

)]i

~

,

(4.10)

in which

Aim =

j’A~(a)da~.

(4.11)

For light-like separations and light-front gauge [A + (0) = 0] eq. (4.11) vanishes, and the leading order three-quark light-cone Fock state reduces to <0Iu~(zi)uj(z2)dk(z3)IB>F.~u!c.

(4.12)

In eq. (4.12) ~, /3, y are spinor indices, i,j, k are color indices, and z1, z2, z3 are light-front coordinates. Upon coupling the quark spinors to the appropriate baryon spinors in eq. (4.12) one obtains the following expression for the distribution amplitude: (412) = *fB[aV V(z~.p)+ aAA(z1.p) + aTT(Z1p)] ,

(4.13)

where p is the proton light-front momentum (see section 3.2). The coefficients ay, aA and aT denote

vector, axial vector and tensor-type couplings of the quark spinors, respectively, and V, A, T are invariant functions corresponding to the respective couplings. The Fourier transforms of these invariant functions are V(x) =

J

V(z~.p)fl (d(zrP)exp[iXi(zi.p)])

(4.14)

etc. V(x), T(x), A(x) are the longitudinal light-front momentum fraction amplitudes. The functions V(x) and T(x) are symmetric, and the function A (x) is antisymmetric with respect to interchange of arguments 1 and 2. The three functions V(x), A(x) and T(x) are not independent. The identity of the two up quarks and the requirement that the total wave function of the three quarks have the isospin of the baryon lead to two relationships between the functions so that finally there is only one independent

124

P. Stoler. Bar von form factors at high

Q2

function which determines the distribution amplitude. For example, for the nucleon Chernyak and Zhitnitsky [Ch-84a] take 4N(x)

=

VN(x)

—

AN(x).

The problem then is to determine VN(x), AN(x), and TN(x). In the extreme non-relativistic constituent quark model the nucleondistribution amplitude takes the trivial form =

ö(x 1

—

1/3)ö(x2

—

1/3),

which may be immediately disregarded since it yields proton and neutron elastic form-factors which are both of the wrong sign, and are much smaller than experiment [Az-80, Ch-84a]. 2 One needsfrom a more quark distribution amplitude. The first step to separateequation the Q dependence the realistic x dependence. The Q2 dependence is described by anis evolution [Le-80]. The solutions to the characteristic equation of the evolution equation are Appell polynomials t/(x), which form a complete set of basis states. There is also a logarithmic dependence on Q2, which reflects the Q2 dependence of ;(Q2). The distribution amplitude may then be expanded in terms of Appell polynomials as follows:

~N(X,

Q2) =

120x

2/AQCD)]m n(Q 1x2x3~B~~(x) [ln(Q 0/

-

(4.15)

QCD)

The exponents are positive calculated anomalous dimensions. The actual nucleon 4N(X, Q2) y~ is contained then in the expansion coefficients B~.The factor physical 120 ensures that state B 0 1. 4~as(’~)= 120x In the2)asymptotic limit the x dependence has the symmetric form However, evolves very slowly with Q2 so that one does not expect &s to be valid1x2x3. at values of Q2 cbN(x, Q which are experimentally accessible. The general form of the nucleon distribution amplitude is then expressed in terms of &s,

~NCX,

Q2) =

~as(X)~Bn4~1(X)[lfl(Q2)/lfl(AQcD)]~”.

(4.16)

The distribution amplitude 4N(x~,Q2) cannot be evaluated analytically. On the other hand, approximate information has been obtained by a number of authors [Ce-84a, b, Ga-86a, b, Ca-87a, b, 88, Ne-83, Ra-91, St-89] by evaluating the first few derivatives of the distribution amplitude at the origin z . p 0 (a knowledge of all derivatives completely determines the distribution amplitude) using QCD sum rule techniques developed in ref. [Sh-79]. A knowledge of the moments of the distribution amplitude is equivalent to a knowledge of the derivatives at the origin. This is seen as follows. The moments are defined as —~

~,ç(fl1fl2fl3)

—

=

~,4 1 fli 112 111 ,J~ ( LUXJX1 X 2 X3 ~~NkX1,)~2,X3

.

P. Stoler, Baryon form factors at high

Q2

125

Re-inverting eq. (4.14) one obtains

~

Thus, moments of the distribution amplitude can be expressed as =

fN ~1fl2113)

(p. Z)~1

+112+113)

(iz - i3~)fll (iz- ~)fl2(j~

ô 3)~4N(Z . P)Io.

(4.17)

The derivatives in eq. (4.17) are with respect to z . p. The normalization for the moments is 4~N(0)= fN ~00), with 4~00) = 1. Since the distribution amplitudes are determined primarily by long range non-perturbative interactions they cannot be calculated analytically. There are two ways in which they have been obtained. One way is by means of lattice calculations, in which some progress has been made in the past few years [Ma-89]. The second and most common approach has been viaoperator the technique of 4N in terms of an product QCD sum rules. This involves expressing the moments of expansion (OPE) consisting of a perturbative part involving three mutually interacting quarks, and non-perturbative parts involving vacuum expectation values of sea quarks and gluons. Though the non-perturbative parts cannot be calculated from first principles, they also appear in the calculations of other reactions, such as the decay rates of mesons involving heavy quarks, from which they can be parametrized. The basis of the QCD sum rule calculations is the relationship between the moments of 4N and the OPE. This relationship can be seen as follows. One can define an operator which projects moments of each of the distribution amplitudes VN, AN and TN, or ~ = VN AN. For example, following ref. [Ch-84a] the moment operator of VN ~S —

[(iz”D,,)111u(0)]iC(zPy)[(izI~D,

~(n1n2n3)(o)

1)112u(0)] 3 [(iz’~D,~)~~y5d(0)]”c1Jk ,

(4.18)

in which a quantity with a hat denotes an operator, and C is the charge conjugation operator. In eq. (4.18) the effects of QCD are included by replacing the derivatives in (3.15) with covariant derivatives =

ô.,~—igt5A~

Similarly, the distribution amplitude moments are then <0I~~52113)(x)IN> cc fN~1fl2fl3) ,

(4.19)

where ~/.‘N VN AN. Similar expressions are obtained for AN and TN. In order to project the moments of a state at the origin with the correct isospin and parity an auxiliary operator J(0) is defined, u(0)]iCz~y~u(0)i(y m j(mOO) = ~a[(iz~’D,~) 5dIc(0)) + b [(iz~D~)mu(0)]tCz5~d3(0)(y5uIc(0)) —

+ c[(iz~D,~)mu(0)]lCz15y,,u3(0)(y5ul~(0)) + } ~,,

.

(4.20)

126

Q2

P. Stoler, Baryon form factors at high

The matrix elements of eq. (4.20) are ccfN(

+ T~°°~).

~°°~

(4.21)

In eq. (421) the order of the derivative m = 0 or 1. For a nucleon, a = b = 1 and c = 0, and j( 00) projects a state with positive parity, isospin 1/2. Alternatively, for a baryon such as a A the —

1m

constants a = b = c = 1/4, and ,j~mOO)projects a state with positive parity, isospin 3/2. The distribution amplitude moments then are obtained using eqs. (4.17) and (4.19) by evaluating (4.4) f2

4~(fllfl2fl3)(i4~(mOO)

+ T~’°°~) = <01 T{

1112fl3)(x)JN(0)(lOo)}I0>

(4.22)

.

Matrix elements such as in eq. (4.22) are evaluated using the operator product expansion (OPE), in which time ordered products of operators are expanded in a sum of progressively higher order products of the quark and gluon operators contained in the operators, i.e., T{~m112113)(zj)JN(0)(lOO)} =

2(0)

C 1(z1) + C~q(zj)~(0)q(0) + C02(z1)G + C~Dq(zj)~(0)Dq(0) +

...

(4.23)

.

The C’s in eq. (4.23) are the Wilson coefficients [Wi-69]. In the limit of small distances (z~ 0) it is necessary to retain only a few leading order terms, since this corresponds to large momenta. Taking the expectation value of eq. (4.23), and converting to q space one finally obtains the correlatons of the type —~

In2113)(q2)(z.

q)fl

+fl2+fl3+4

=

i Jd4x e~ x
(x)JN(0)~°°~} 0> (z . y).

(4.24)

Expressing eq. (4.24) in terms of the OPE one obtains I~hm112113)(q2)= I~fh112113)(q2)

+J

52113)

(q2)

+ I~?~~~(q2) .

(4.25)

In eq. (4.25) only three terms have been retained. (1) Purely perturbative effects involving three quarks interacting with each other via gluonic fields. This is done exactly in lowest order perturbation theory yielding I~m112113) = (i/1~4)c~n1n2n3) q2

ln(— q2/A2)

(4.26)

.

The next two terms involve non-perturbative corrections due to the presence of the nonperturbative vacuum, which is a dense collection of strongly interacting quark and gluon fields. (2) Absorption and emission of gluons by the valence quarks from the physical vacuum. This is proportional to the gluon energy density of the gluon fields, I~.~I253) (q2) =

(1/~2)c~nln2n3)q_

2 <(;/it)

G2>

-

(4.27)

2

127

P. Stoler, Baryon form factors at high Q

(3) Absorption and emission of quarks from and to the physical vacuum, J~J~2113)(q2) =

(4.28)

(1/ir)c~~15253)q_4<~,/~~q>2.

The expectation values of the physical vacuum condensates of gluons and quarks cannot be calculated, but are phenomenologically fixed from calculations compared with the known rates of decay of, e.g., charmonium. The standard values which are used in the literature are =

1.2

x

10_2 GeV4,

<,.Jq>

=

1.8

x

104GeV6.

(4.29)

The coefficients c~fh112113,n) are calculable. The moments, eq. (4.22), are obtained by evaluating the correlators with the parameterizations of the non-perturbative matrix elements of eqs. (4.29). The evaluation involves technical mathematical procedures beyond the scope of this review. These are described in detail in, e.g., refs. [Sh-79, Ch-84a, Ca-88]. Using sum rule constraints Chernyak and Zhitnitskii (C-Z) [Ch-84a] had obtained the lowest six independent moments of the quark distribution amplitudes at a Q2 normalization ~2 = 1 GeV2/c2, expanded in the six lowest order Appell polynomials. Later Zhitnitskii et al. [Zh-88] have extended the calculation of ref. [Ch-84] to include all moments with order n 3. Their result was later verified with small corrections by King and Sachrajda (K-S) [Ki-87]. The distribution amplitude denoted C-O-Z from ref. [Zh-88] is shown graphically in fig. 7 plotted on the x 1, x2, x3 Mandelstam plane. features are and noteworthy. 4N(X, j~) takes Two on both positive negative values. Thus, it is not appropriate to interpret (1) ~N(X, it) as a quark probability distribution. Stefanis [St-89] suggests instead that I 4’N(X, iz)12 is more appropriate for considering quark relative longitudinal momentum fraction probabilities. However, even this is strictly speaking not correct since the quark probability distribution is given by q(x,~) J[dk±]I~(x,ki)I2.

(4.30)

~‘coz

X21 Xi

Z3 =

1

1

Fig. 7. The C-O-Z quark distribution amplitude obtained by means of QCD sum rule calculation [Ch-84a, Zh-88].

128

P. Stoler, Baryon form factors at high

Q2

From distribution amplitudes such as C-O-Z the probability of the nucleon consisting of three quarks has been estimated [Sh-89] to be as much as 5% at Q2 = 10 GeV2/c2. (2) The C-Z (or C-O-Z) distribution amplitude exhibits a strong asymmetry, with a large maximum near x 1 = 1. In terms of the probability distribution as defined above, the large asymmetry in the C-Z distribution amplitude implies that the struck u quark with helicity the same as the overall proton carries about 60% of the proton’s longitudinal momentum, with the other two quarks roughly equally sharing the remainder. (King and Sachrajda [Ki-87] and Carlson and Mukhopadhyay [Ca-92] have found small errors in the C-Z distribution amplitudes, which, however, do not appreciably change the basic conclusions.) Apparently, the sum rule constraints do not completely pin down the distribution, due to uncertainties entering the calculation of higher moments, and the degree of truncation of the Appell polynomial series. An alternative treatment (G-S) has been presented by Gari and Stefanis [Ga-87], in which all Appell polynomials are included which incorporate all second order polynomials, with the additional restriction that the ratio of neutron to proton Dirac form-factors is minimized. Unlike the C-Z distribution amplitude this yields a large magnitude neutron Dirac form-factor. (This is discussed further in section 5). The two leading polynomials in C-Z and G-S are still about the same, but the higher order coefficients are quite different. In particular, while the large perturbative matrix elements yield similar results, the non-perturbative parts of the calculation yield very different results. The G-S distribution amplitude is less asymmetric than the C-Z amplitude leading to important interpretational differences. Even though the G-S distribution amplitude is somewhat asymmetric, it t(x yieldst(x a flatter probability t(x t(x distribution, such that pairs of quarks having the same helicity, e.g. u 1)u 2) or u 1)d 3), equally share most (80%) of the proton’s longitudinal momentum fractions. This led to the concept of di-quark clustering, and the possibility of a cluster expansion of the distribution functions. Diquark models are discussed in section 8. In the recent study of Hansper et al.from [Ha-92] wasmoments pointed out is considerable 4~Nstrictly sum itrule sincethat onlythere a finite number of uncertainty in determining moments are known. Furthermore these moments themselves have rather large uncertainties (—~20%). Hansper et al. [Ha-92] consider several different model distribution amplitudes, including C-Z, C-O-Z, G-S, and K-S. They also have proposed several alternative distribution amplitudes which are within the uncertainties of the sum rule moments. They expanded in a larger number of basis polynomials, up to order three (ten polynomials), than in previous work, which used up to order two (six polynomials). They also include transverse momentum components in TH, and find that resulting physical observables, i.e. the elastic form-factors F 1~and the ratio 4~N~ F1~/F1~, extremely to the choice of Figure are 8 shows one ofsensitive their distribution amplitudes, çb~.It is quite apparent that it is qualitatively very different from that of C-O-Z (fig. 7). 4.4. Radiative corrections

—

Sudakov suppression

The PQCD expression for the form-factors of exclusive reactions, eq. (4.1), is based on the approximation that at asymptotically high Q2 the transverse momenta of the valence quarks entering the hard scattering process can be ignored. The valence quarks in 4,(x) are all considered to be on or near their mass shell, whereas all lines in TH are considered to be very far off-shell. When this formalism is applied to the regime of experimentally accessible Q2 important difficulties arise as discussed in section 4.3.1. In particular, (1) the logarithmic dependence on Q2 of; leads to divergences in the integral in eq. (4.1); (2) near the kinematic limits x 1, y~ 0 the gluon virtualities —+

P. Stoler, Baryon form factors at high

~

Q2

129

(4)

‘PN

=1 Si = 1

Fig. 8. The quark distribution amplitude from ref. [Ha-92].

are small so that PQCD becomes inappropriate. Regarding the first point, the divergences due to; have been circumvented by inserting in ; an effective gluon mass parameter [see, e.g., ref. [Ji-87] and eq. (4.5)], which is varied to obtain the correct magnitude of the form-factor. Regarding the second point, Hansper et a!. [Ha-92] have shown that the retention of transverse momentum components in TH has the effect of inserting an effective gluon mass term in the denominator of TH [see eq. (4.6)], thereby avoiding the kinematic region of small gluon virtuality. Recently, Li and Sterman [Li-92b, c] have entirely rederived eq. (4.1), retaining transverse momentum components k 1 throughout. The retention of transverse components is equivalent to the inclusion of transverse spatial dimensions b 1/k1, where b b1 b2 b3. This allows one to take into account radiative corrections of the quark lines in the leading order diagrams such as in fig. 6. When the quarks have a large transverse separation, they experience large color interactions and undergo color radiative corrections. When they are close together (small b), color fields are small and color radiative corrections are small. Examples of radiative corrections considered by Li and Sterman [Li-92b, c] are schematically illustrated in fig. 9. These corrections, known as Sudakov corrections, depend strongly on the separations b, and are found to suppress the form-factor at large b (or small k1) regardless of x. Thus Sudakov corrections have the effect of suppressing the non-perturbative contributions to the form-factors, while preferentially retaining the perturbative contributions. The inclusion of k1 also modifies the structure of the hard scattering amplitude in a way which is mathematically equivalent to the insertion of an effective gluon mass. ‘~

(a)

(b)

(d)

(C)

Fig. 9. Radiative correction diagrams considered in the calculations by Li and Sterman [Li-92b, ci Sudakov suppressions.

of the elastic

form-factors with

130

P. Stoler, Baryon form factors at high

Q2

The resulting elastic form-factors are then written as

F

2)=

[dx][dy][dk±][dkl]

~(x,k~)TH(x,y,k±,

k~)~~(y,k~).

(4.31)

1(Q The integral in eq. (4.31) is understood to imply [dx]

dx

1dx2dx35(1 >x~)and similarly for [dy], [dk1], and [dk’1]. The distribution amplitude W is obtained by evaluating the matrix element of a tn-local operator, as in eq. (4.12) et seq. The result is that the distribution function is expressed in terms of three functions VN(x, k1), AN(x, k1) and TN(x, k1). Isospin constraints reduce the number of independent functions to one, which, e.g., may be taken as —

k1) = VN(x, k1)

~N(X,

—

AN(x,

k.1j,

in terms of which the nucleon elastic form-factor may then be written as 2) = JJ[dx] [dy] [dk

1] [dkl] ~(x, k1) TH(X, y, k1, k~~(y, k~).

F1(Q

(4.32)

Since the effect of the Sudakov suppressions depend upon transverse separations, Li and Sterman [Li-92b, c] find it convenient to transform eq. (4.32) to b space. The transformed distribution amplitude takes the form J[dk±]e~ b~N(xk1) cc exp[—s(x, b, Q)] ~N(X),

(4.33)

4’N(X) is the where ~ lb12 I~jb23 I without and 1b311 are theofdistances between the pairs of quarks. distribution amplitude inclusion the transverse momentum effects, such as C-Z, etc. The factor exp[—s(x, b, Q)] is the result of the radiative corrections. The scattering amplitude TH is also expressed in b space. The net result for the formfactor is

F 1 cc J[dx] [dy] [db] ~(x,

cc

b) TH(X, y, ~, Q)~N(~,b)

JdxdY~N(Y)~N(x) JdbTH(x~y, b, Q) exp[—S(x, b, Q)].

(4.34)

(4.35)

The exponent S(x, b, Q) cc ln(Q) ln(ln(Q)/ln(b)) suppresses the form-factor at large separations b, thus limiting the soft-pion non-perturbative contributions. The effect of the Sudakov suppression on the elastic form-factor calculations using QCD sum rule distribution amplitudes is discussed further in section 5.4.

P. Stoler, Baryon form factors at high

Q2

131

5. Elastic scattering The seminal early works involving elastic electron scattering from the proton were performed by Hofstadter et a!. [Ho-55, 56, Ch-56]. The form-factors were observed to follow a dipole shape 1/(q2 + rn2)2 as seen in fig. 10, which shows some of their early data. The dipole dependence of the charge form-factor would be obtained if vector mesons propagate between the virtual photons and the nucleon (fig. ha). This led to the prediction of the existence of vector mesons [Na-57, Fr-59a, b] with approximately the correct mass even before their discovery [Fr-59a, b, 60], and the idea of the vector meson dominance (VMD) model which has historically been the most successful theory of low-Q2 nucleon and meson form-factors.

PROTON

~XPONENTI4L MOOEL Q8i

~

iii iii iii iii El —

liii ~ •-200MEV

o.~o

~

a•—300MEV - 400MEV

O.I5-~00~--—

0.100

I

2

4

8

10

12

14

Fig. 10. Form-factor for elastic scattering from a proton measured in the 1950s [Ch-56, Ho-56]. The solid curve is obtained using an exponential charge distribution Po exp( — r/r~), with r, = 0.8 x 10-13 cm. The abscissa is in units of 10 26 cm2. The figure is from ref. [Ch-56].

0~0000 00000~,

Fig. 11.

Representation of elastic electron scattering, (a) with the intermediary of a vector meson (VMD model), and photon—proton coupling. (c) A schematic quark model representation of the VMD model.

(b) by direct

P. Stoler, Baryon form factors at high

132

Q2

5.1. Constituent quark model Elastic form-factors at low Q2 have also been calculated in the basis of the non-relativistic CQM. In terms of constituent quarks the proton form-factors are written as GE(Q2) = (P~etexp(_iQ.ni)~P)~

GM(Q2) = (P~itiaizexP(_iQ.ri)~P).

In the SHO basis, the nucleon is in a 12N(56, 0k)> state of SU(6), with spatial wave function ~ ccexP(_1/2~r?). For the proton this leads to Gaussian shaped form-factors, GM(Q2) cc GE(Q2) cc exp(—Q2/6c2)

(5.1)

.

Equation (5.1) decreases much too fast with increasing Q2 in comparison with the experimental dipole shape, as seen in fig. 12. The situation is somewhat improved by including hyperfine interactions such as in the I-K model (see section 3.2), which mixes higher states of SU(6) into the nucleon wave function so that higher Q2 components are added to the wave function [Gi-83,

Is-87]. One then obtains GE(Q2) = [AE

GM(Q2)

=

+ BE(Q2/6o~)+ CE(Q2/6o~)2]exp(—Q2/6c~2),

/2P[AM

+ BM(Q2/6c~)+ CM(Q2/6ot)2]exp(—Q2/6c~2).

::~

~ N

I

0

I

I

I

I 1

I

I

I

I 2

~2

I

~

I 3

(CeV2)

Fig. 12. The proton magnetic form-factor compared with three calculations employing the constituent quark model. The figure is from ref. [Gi-9 1]. The long dashed curve is due to a non-relativistic model with hyperfine interactions [Is-78]. The solid curve is the result of a constituent quark model with relativistic corrections [Wa-90a, ci. The short dashed curve is a dipole fit.

P. Stoler, Baryon form factors at high

Q2

133

Figure 12 shows that the result of including hyperfine interactions improves the agreement with data, but the calculated form-factors are still too small at high Q2. The range of Q2 for which the CQM is valid is extended to a few GeV2/c2 by treating the wave functions in a relativistically invariant framework (see section 3.3). 5.2. Vector meson dominance and hybrid models The VMD model, whose origins predate the quark model, accounts rather well for the shape of the elastic form-factor in the transition region between the very low-Q2 region and the high-Q2 region where PQCD effects should become manifest. The VMD model explicitly recognized the crucial role played by strongly interacting virtual vector mesons as intermediaries in the coupling between a photon and nucleon. The VMD model also fails to reproduce the data at higher Q2, which can be partially rectified by including an additional form-factor due to the internal structure of the nucleon [Ga-86a]. The elements of the VMD model (e.g. refs. [Ia-73, Ho-76, Ga-86a, Ca-87a, Br-89a]) are illustrated in fig. 1 la, b. The propagator represents the intermediary vector meson, either isovector such as the p, or isoscalar such as the w. The open circle schematically represents the internal quark—gluon structure of the interacting hadrons. This allows a more general expression for the nucleon form-factors F 1 and F2 in terms of isovector and isoscalar components. In the notation of refs. [Ga-86a,b] we have 2)

~[F~(Q2) + 2I

=

F1(Q

2)]

,

F

3Fr’(Q

2) = ~[,c~,F~(Q2)+ 2I 2(Q

2)] ,

(5.2)

3K~F~(Q

where 13 is the third component of isospin, 13 = ±1/2 for the proton and neutron, respectively, and the superscripts IS and IV denote isoscalar and isovector mesons, respectively, i.e.,

~‘IV,ISj 1

~‘

2\ )

— —

2

~

mp0,

2

2

i

fl2 Cp

t

j

0) J 1k

2~ 1

t’IV,ISi

lCp0)

‘

~1~

1 2

2\

—

)

—

k

mpe, 2

mp0,

~‘

,-~2~

~

i

2

J2 ~

~1~ ~

In eqs. (5.3) cp~,,= g~/f~, and g~~/2 are effective couplings at the meson—nucleon vertices, and m~/f~~ is the photon—meson coupling. 2 dependence of the form-factor is then obtained by assuming an intrinsic The overall dipole Q form-factor

f

2) ~f

1(Q

2(Q~)

2/(A2 + A

Q2),

A

0.8 GeV/c.

Although this procedure yields the requisite low-Q2 form-factor, it does not account for the high-Q2 behavior, where the VMD model becomes inadequate. The VMD model can be retained by adding additional terms in eqs. (5.2) and (5.3) to account for the direct interaction of a virtual photon with the nucleon, as schematically illustrated in fig. 1 lb. The additional term requires the form-factors to join smoothly with PQCD expectations at high Q2 while retaining their VMD model characteristics at low Q2.

P. Stoler, Baryon form factors at high

134

Q2

In the hybrid model of refs. [Ga-86a, b] this is accomplished in two ways: (1) Redefining f

t

and f2,

42 “1

2

7

42 “2

\

42 “1

2

A~+Q2k~A~+Q2)’

I

7

42 “2

\2

A1+Q2~A~+Q2

2

(2) Adding an intrinsic term to the form-factor, 7 rlv,iSi ~1 k

KjSiV

2~— ~

~

2

m~0, 2 ~fl2cP,O~k1

\

F2iv

is

(Q 2 )

7 =

I

\

~Ii ~ 2’~ ,, /

cP(O,,Jlk

\m~,fflT~

2 ____________

c~()1c~,)2

,i2

+ (K15

iv

—

~

~

m~,~-T-~

;f2(Q2 )

.

/

(5.4)

2, while F In eqs. (5.4) both F1 and F2 approach the dipole shape for small Q 1 and F2 approach the 4and Q~6dependence, respectively, in the high Q2 limit. requisite Q~ Finally, effects due to the running coupling constant are taken into account by the substitution 2)/A~cD]

2

Q

2log[(A~ + Q log(A~/A~CD)

A “satisfactory” description of GE, and GM is obtained with the following values for the various constants: =

0.776 GeV/c2

=

3.706,

‘~~s = —0.12

=

0.377

C(U

=

0.795 GeV/c,

,

,

=

m,,

,

i~

=

6.62

,

tc,,

=

0.163

0.411

A 1

0.784 GeV/c2

=

A2

=

2.27 GeV/c,

2 AQCD = 0.29 GCV/c

The proton form-factors are well represented at lower Q2. However, the situation with the neutron is controversial. Near Q2 = 0 the G~is large due to the sizeable neutron anomalous magnetic moment [G~(0) = 1.9], while G~is very small [G~(0) = 0] due to the net zero charge. On the other hand, Gari and Krupelmann [Ga-86a] suggest that surprisingly at high Q2 it may be that G~>>G~.This can be seen as follows. The isoscalar and isovector coupling are nearly equal, c, c( 5.Since 13 = 1/2, eq. (5.2) implies G~and G~are dominated by Fr. so that Fr”one obtains F~” G~ rG~,,with G~>G~for Q2 > 4M~.The result of the analysis Thus, 0, from eq. (2.3) of Gari and Krupelmann [Ga-86a] compared with existing data is shown in fig. 13. The recent experimental results bearing on this are as follows. Experimental elastic neutron cross sections up to Q2 = 10 GeV2/c2 were extracted from quasi-elastic scattering data on the deuteron, utilizing a model deuteron wave function [Ro-92]. The ratio of neutron to proton cross sections is shown in fig. 14, along with the predictions of various models, including the hybrid model -~

—

-~

P. Stoler, Baryon form factors at high

Q2

135

10 •G~

oG~e~io

g 1.0

Rocketal

~ £

~ ~

1

2

3

4

5

6

G~’e~48° 8 ~48° Albrecht et at,

7 8 9 02 [1GeV/c 2]

10

11

12

13

14

Fig. 13. Neutron electric and magnetic form-factors, G~and G~,calculated using the hybrid model of ref. [Ga-86a]. The quoted

data

are from refs. [Al-68, Ro-82]. The figure is from ref. [Ga-86a].

0.6

I

I

I

I

4 6 8 Q2 [(GeV/c)2]

10

!~‘‘

I

• This Expi. ~ NE4 D Bartei

0

0

2

0 Albrecht U

Budnitz

12

Fig. 14. The ratio of the elastic scattering cross section for the neutron and proton versus Q2 at an electron scattering angle of 100. The black dots are data of ref. [Ro-92]. The inner error bars include statistical and systematic uncertainties. The outer error bars include estimates of the error in the neutron cross section due to the assumed deuterium wave function used in extracting the neutron cross section from deuterium data. The data points represented by open symbols are due to earlier measurements [Al-68, Bu-68, Ba-73] referenced in ref. [Ro-92]. The thick dashed curve is due to the hybrid model of ref. [Ga-86a]. The dash—dotted and solid curves are due to vector dominance models of refs. [Ho-76, Ko-77]. The figure is from ref. [Ro-92].

discussed here. The data are consistent with models for which F 1 is small and is larger than 2. In other words, if the neutron form-factor is dominated at high Q2 G~ by F G~at high Q 6 behavior, and the proton form-factor is dominated at high Q2 by F2, which has an asymptotic Q4 behavior, then the ratio G~/GL~ 1, whichwith has should approach Q2, which is consistent an asymptotic Q the data in fig. 14. However, preliminary data from SLAC experiment NE-li [St-92, Bo-92] give G~consistent with zero for Q2 up to 4 GeV2/c2, which contradicts the above result, as.well as the PQCD form

136

P. Stoler, Baryon form factors at high

Q2

factor corresponding to the G-S distribution amplitude, see sections 4.3.2 and 5.3. Clearly, a measurement of the neutron elastic form-factors at higher Q2 should contribute greatly to assessing distribution functions. 5.3. PQCD calculations The elastic proton form-factor has been measured [Ar-86] up to a maximum Q2 of 31 GeV2/c2, and has served as the best available exclusive test of PQCD theory. The experimental form-factors are shown in fig. 15. Ji et al. [Ji-87] have calculated this form- factor using the QCD sum rule distribution functions C-Z, G-S, and K-S (see section 4.3.2). In the calculations a gluon effective mass was inserted into the strong coupling constant ; as in eq. (4.5) in order to avoid blowup at k2 = A2. The results are shown in fig. 15. Good fits to the data were obtained for all three distribution functions, with an effective gluon mass of 0.3 GeV. The curves in fig. 15 are not applicable at lower Q2 since they are based on leading order PQCD. Zhitnitskii et al. [Zh-88] have improved the distribution function of Chernyak and Zhitnitskii [Ch-84a] by including more moments. This is denoted C-O-Z in the literature and in fig. 7. The results are qualitatively similar. The slow decrease with Q2 of Q4F 1 is identified by Ji et al. [Ji-87] as due to the monotonic decrease of;. Alternatively Hansper et al. [Ha-92] point out that what is really measured is GM, and the falloff can be explained by the decrease in the F2 contribution to GM. The procedures used to obtain form-factors such as those shown in fig. 15 have been strongly criticized [Is-84, 89, Ra-91]. In section 4.3.2 it was pointed out that the sum rule calculations of Chernyak and Zhitnitskii [Ch-84a] yield asymmetric nucleon distribution amplitudes (see fig. 7). Also, the transition amplitude becomes large near the kinematic limits of x and y [see, e.g., eq. (4.3)]. Thus, most of the form-factor comes from contributions near the kinematic limits of

o

Previt~~~

cz 0.5

~,

a8 Inside Integral 2 r4:03

~2

(G:V1c2)

[(Gev/c)2]

Fig. 15. Results ofthe leading order calculation of ii eta!. [Ji-87] ofthe proton Dirac form-factor with three different models for the distribution amplitude as discussed in the text.

P. Stoler, Baryon form factors at high

Q2

137

x and y. However, it is just in this kinematic region where the gluon momentum is not necessarily large, and PQCD would not be expected to be operative. The effective gluon mass is introduced in order to keep; finite at low k2, and thus enable the evaluation of the integral in eq. (4.1). To be consistent it is argued that one should also include an effective mass for the gluon appearing in the denominator of the terms in the transition amplitude, eqs. (4.3) and (4.6). The transition amplitude is now no longer very large near the kinematic limits of x and y, which significantly reduces the value of the calculated form-factor. Hansper et al. [Ha-92] have shown that an effective mass in the denominators of the terms in the transition amplitude occurs quite naturally if one explicitly retains the k 1 components flowing through the transition. In fact when the effect of these contributions is approximated by a constant they obtain an effective mass of roughly the same magnitude as used by Ji et al. [Ji-87] above. The net result is that the form-factor is reduced by an order of magnitude when calculated with any of the distribution functions used above, and is therefore an order of magnitude smaller than the experimental values. Hansper et al. [Ha-92] argue that this does not eliminate the possibility that PQCD is still valid. Their point is that the expansion of the distribution function in Appell polynomials (eq. 4.15) needs a larger number of polynomials than used for example in ref. [Ch-84a] to obtain the C-Z distribution. They increase the order of the expansion from 2 to 3, thus increasing the number of polynomials from 6 to 10. In addition they introduce an arbitrary exponential function which suppresses the contributions from the troublesome region near the kinematic limits. They find indeed that they still obtain the moments of the distribution function consistent with C-Z and C-O-Z. The form-factors 2 dependence is theycorrect, obtain they are comparable in magnitude to the experimental data. Although the Q2, and when it is not point out that the Pauli form-factor F 2 isofquite atas lowseen Q in fig. 16. added to their PQCD calculated F 2 dependence GM issignificant improved, 1 the Q

1.5

D~poIe

f a51

‘

Q2

:2~2~3b35 (GeV2/c2)

Fig. 16. The proton magnetic form-factor Q’GM. The experimental points represented by full dots are due to Arnold et al. [Ar-86]. The results ofthe hybrid model of ref. [Ga-86a] are given by the heavy solid curve. The helicity conserving part Q4F 1 calculated by Hansper et al. [Ha-92] using4FPQCD is given by the thick dashed curve. The light solid curve is the helicity non-conserving 4GM = Q4F 4cF Pauli form-factor Q 2 obtained from the hybrid model. The net magnetic form-factor is Q 1 +Q 2

P. Stoler, Baryon form factors at high

138

Q2

Although the shape of the distribution function (fig. 8) obtained in their analysis looks rather odd, one must conclude that in fact one can find acceptable distribution functions which reproduce the magnitude of the experimental Dirac form-factor. It appears that one really needs more stringent constraints on the distribution function than are currently available from QCD sum rule calculations before one can make strong statements about whether the Q2 dependence of the proton form-factor is evidence relating to the validity of the PQCD calculations. As mentioned in section 5.2 above, preliminary results from a recent measurement [Lu-92, Bo-92] of G~are consistent with zero for Q2 less than 4 GeV2/c2. This would seem to rule out the G-S or [Ha-92] distribution functions, which predict a rather large neutron Dirac form factor Q4F~ 0.57 at Q2 = 20 GeV2/c2, and supports C-Z. —

5.4. Radiative corrections

—

Sudakov suppression

In section 4.4 it was pointed out that by retaining transverse components of the parton momentum in deriving the factorization formula for the nucleon elastic form-factor, Li and Sterman [Li-92b, c] were able to account for radiative corrections to the leading order scattering process. The radiation corrections depend on quark spatial separations and are most important at the largest separations b, corresponding to the smallest k 1. Since the net effect of the radiative corrections is to suppress the nucleon distribution amplitude, the form-factor contributions from the non-perturbative kinematic region are suppressed, leaving the dominant contribution from the perturbative region. According to Li and Sterman [Li-92b, c] the Sudakov extend the validitybyofaPQCD 2. suppressions This argument is supported good based calculations down to accessible values of Q agreement with the experimental elastic proton form-factor achieved using, e.g., the C-Z distribution amplitude, without the arbitrary insertion of a phenomenological gluon effective mass. The result of the elastic form-factor calculation using the C-Z distribution amplitude is shown in fig. 17. Also shown is the contribution from the different intervals of quark relative transverse separations. The various curves are the result of cutting off the form-factor integral over b, eq. (4.34), at different values of b~,in units of AQCD. Nearly all of the form-factor is accounted for by transverse separations less than l/AQCD. The curves in fig. 17 are calculated with AQCD = 0.1 GeV/c. Also shown is the result of the calculation with AQCD = 0.2 GeV/c. The relative insensitivity of the magnitude of the form-factor to different values of AQCD is another indication that most of the form-factor comes from the perturbative region corresponding to rather small values of b, and the non-perturbative regions are strongly suppressed. 6. Resonance form-factors A large body of exclusive data at high Q2 including both inelastic resonant and elastic form-factors is needed to further constrain wave functions and test other aspects of theory. However, except for the ~(1232) the resonances are largely overlapping even at relatively low excitation. The separation of the contributing electromagnetic multipoles will require measurements of exclusive reactions such as (e, e’ it) and (e, e’ r~)to as high Q2 as possible, a very difficult task made more difficult since the form-factors become small at high Q2, and there are significant contributions from non-resonant processes. The current experimental situation is that exclusive (e, e’ it) and (e, e’ 11) data exist only for Q2 less than 3 GeV2/c2. The status of the existing resonance data, especially relating to high-Q2 formfactors will be discussed in this section.

Q2

P. Stoler, Baryon form factors at high

139

1.2

O.9~

•!:~:4:!~i~0;A

~

b~=0.4/A

0.3 bc = 0 . 2/A 0.0

I

I 10

I

I

I

20

30

I

Q2 (Gev2) Fig. 17. The proton magnetic form-factor Q4GM versus Q2. The data are due to Arnold et al. [Ar-86]. The curves are the results of calculations of Li and Sterman [Li-92b, c] including Sudakov suppressions due to radiative corrections, using a C-Z proton distribution amplitude as discussed in sections 4.4 and 5.3.2. The curves correspond to successively larger cutoffs b~in the spatial integrals (eq. 4.34) in units of l/A~ 0.The solid curves are for AQCD = 0.1 GeV/c, and the dashed curve is for AQCD = 0.2 Gev/c.

6.1. Analysis of inclusive data

2 = 3 GeV2/c~,there is a considerable Although there is adata total in absence of exclusive data above Q mostly at SLAC [Po-74, Br-76, 84, amount of inclusive the resonance region obtained Bo-79, Ri-73, Ro-91,92]. Figure 18 shows examples of previously obtained spectra, vW 2 as 2/Q2), at various values of Q2 covering the range Q2 = ito 21 GeV2/c2. a function ofw’(=w + W Consider the spectra in fig. 18a at Q2 = 1 GeV2/c2. The most significant feature above the elastic scattering peak is the existence of three maxima: the first, second and third resonance regions. In this interval there are many resonances (see fig. 3), but only a few are dominant. The first maximum is due to the isolated A(1232) resonance. The second resonance region is dominated by two strong negative-parity states, the D 2 (< i GeV2/c2) the D 13(1520) the S11(1535). At low Q 2 (-~3 and GeV2/c2) there is some evidence that the S~1(1535) 13(1520) becomes dominates, whereas at higher Q dominant [Br-84, Ha-79]. In the third resonance region, the strongest excitation at low Q2 is the F 15(1680) state. 2. The relative strength of the other states is not well determined, especially at increasing Q Available inclusive data were fit with resonance curves [St-91a,b] to extract information about the Q2 dependence of resonance cross sections at higher Q2 than is available from exclusive measurements. The purpose was to see whether any of the features predicted by PQCD are observable. Although the statistical accuracy of the data diminishes with increasing Q2, the second and third resonances remain prominent. In the fits for Q2 > 4 GeV2/c2 it was assumed that only the S 1 1(1535)the contributes to the second 2 4 GeV2/c2 corrections were made to subtract contribution from the resonance. For Q D 2. It was assumed 13(1520). The third resonance region dominated the F15(i680) at low Q that this remains one state, although it isispossible thatby several states contribute strongly at high Q2.

140

P. Stoler, Baryon form factors at high 0.4

Q2

0025

Q2

iCe 0.020

I—

—

0.3

02

Q2=l2Gev2/c2 0.0 0015 10

0.1

00

0.005

2

3

4

5

6

7

0.000

1

1.1

1.2

1.3

1.4

0.30

Q3 0.25

—

0.15 0.20

—

=

3 Gev3/c2

0.008 —

0006 0 004

~

0.10 0.002

1

1.25

1.5

1.75

2

2.25

2.5

0.05

1

1.05

1.1

1.15

1.2

1.25

0.005 /

—

0.004 0.03 0.003

o.o4

,

0.02 0 002 0.01 0.00

0.001

1

1.1

1.2

1.3

1.4

1.5

1.6

0.000

1

1.05

1.1

1.15

1.2

1.25

Fig. 18. The structure functions vW 2, where ci’ 1 + W2/Q2. The experimental 2 versusdata w’ for were inclusive obtained inelastic from scattering refs. [Bo-79, in the Br-76, resonance 84, Ri-73, region Po-74, for various Ro-91,92]. values The ofnominal solid curves Q are resonant fits to the data as discussed in the text. The dashed curves are fits to the data in the scaling region extrapolated down to the resonance region as a test of duality (see section 6.6). The figure is from ref. [St-91b].

All the fits included a state at W = 1440 MeV, corresponding to the possible location of the P 11 (1440), but none of the fits exhibited any positive contribution from this state. Thus, the resonance in the fits included the peaks at W 1232, 1535 and 1680 MeV.

Q2

P. Stoler, Baryon form factors at high

141

6.2. Amplitude extraction Transverse resonance amplitudes are usually expressed in terms of the transverse electromagnetic helicity amplitudes [Co-69a, b] A 112 and A312 [see eqs. (2.16) and (2.17)]. These helicity amplitudes are matrix elements for the absorption of photons in which the resonance emerges with helicity 1/2 and 3/2, respectively, as illustrated in fig. 2. The A112 amplitude is helicity conserving, while A312 is helicity non-conserving. The transverse helicity amplitudes are related to the virtual photon transverse cross section at the resonance position by

2) = FRWR IAH(Q2)12,

aT(WR,

(6.1)

Q

where IAH(Q2)12 = 1A 2)12 + A 2)12. The virtual photon cross section a~is related to the 112(Q 312(Q inclusive electron scattering cross section by eq. (2.12). For single meson production the electromagnetic helicity amplitudes are obtained from the Walker [Wa-69] helicity amplitudes by factoring out the strong interaction part. For the A(1232) and F 15(1680) both A112 and A312 contribute. 2 limit PQCD predicts helicity is conserved for photons interacting with spin 1/2 In the high-Q quarks, leading to the ansatz of hadron helicity conservation. Helicity conservation implies that as Q2 becomes large A 2A 2 only A 112 x Q 312, so that at asymptotic 112 contributes. In the 2. values of Q case of the S~~(1535) onlyare A112 contributes at all Q functions v W Data in the literature presented as structure 2 (eq. 2.7) or as cross sections do]dQdE. For the three dominant resonances there is no evidence of a significant longitudinal contribution to the resonant cross section, unlike the situation for the non-resonant cross section, which is known to have a significant longitudinal component [Bo-79]. In the conversion between vW2 and da/dQdE and the subsequent fits in ref. [St-91a,b] the longitudinal resonant component was taken as zero. 2 dependence of the proton elastic form-factors, analogous In order to transverse compare with the Q GT(Q2) have been defined in terms of the helicity amplitudes dimensionless form-factors according to eqs. (2.8), (2.9), and (2.16), ~

,-‘

2

‘-‘Tk’z~

I

—

1

2A’fN~,

—

~j

~YkVVR

s,i’2~

1,,2 —

IVIN)

A jç~2~ 2 ~Hk’~

I

Helicity amplitudes and associated transition form-factors for the three dominant resonances 2 by fitting resonance cross near W =with 1232, and 1680 MeV were a function of Q sections the1535 following functional formextracted for each asresonance: 2 /IIWRTTI’, —

~.~(W2

—

W~)2+ (WRF)2

A

( HI,

6

2~2 ~

.

In eq. (6.3) the subscript i denotes the decay channel, it—N, i

~(~k~)’ 2

=

2M

K

1—N, p—N, or

it—is,

and (6.4)

142

P. Stoler, Baryon form factors at high

Q2

where x

1 is the branching ratio for channel i, and KR is the equivalent real photon energy needed to excite the baryon to a mass WR. The subscript R denotes the quantity is taken at the resonance energy WR. The partial widths are given by —

~

2I+l)(p*2 ~p*2

iR

+ + X2\’ X2)’

—

/K*’\2” (K~2+ R~k~) ~K*2 +

x2\~’ x2)

/p*\(

The meson momenta P7 are in the center of mass. X is a damping parameter whose value is not well determined, but 1/X is on the order of the strong interaction radius. The values of X used were taken from ref. [Wa-69]. It was found that the extracted amplitudes are relatively insensitive to the precise value of X. The total width F is the sum of the partial widths, weighted by their branching ratios. For the A(1232) resonance the only decay channel considered was single pion emission so that the total width (neglecting the photon width) is F = F~.The best fit width for the A(1232) was FR = 120 MeV. For the S~~(1535)decay there is a large ii branching ratio. With similar expressions as for the it, the total cross section is then given by aT = aT~ + 0Ti 1~ The branching ratios in the fit were taken as 50% for each channel [PDG-92] so that F~R= F~R= O.SFR, and the total width in the resonance denominator in eq. (6.3) was taken as F

=

FR (0.5 P~/P~R + 0.SP$/P$R).

A small two-pion decay contribution observed width the S11 is somewhat 2. was For ignored. Q2> 10 The GeV2/c2 a width FR for = 95 MeV was used. narrower than of forthe theFA at higher Q In the case 15(1680) there is a significant p decay channel and a smaller it—A decay channel, which were taken into account. The non-resonant background contribution is always large, and was phenomenologically included as in ref. [Ha-79] in the fit by the form

=

~W—

Wth ~ C~(W—Wth)~,

where C, are fitting parameters, and Wth = MN + m~.This procedure in principle is not valid since it ignores any interferences between background and resonances of the same multipoles, and also lumps other resonances into the background. However, this appears to be the best one can do with the available inclusive data. The resulting fits are shown for the spectra in fig. 18. Also shown in fig. 18 is the background extrapolated from the scaling region using an empirical expression obtained in ref. [Bl-70]. The scaling of the background together with the resonances, which is known as “duality”, is discussed in section 6.6. The extracted form-factors are shown in figs. 19 and 20 relative to a dipole shape of the form 2)dipoie = 3/(1 + Q2/0.71)2 versus Q2. The quantity F(Q2)/F(Q2)dIPOI~is plotted rather than F(Q Q4F(Q2) in order to better display the low-Q2 behavior. For high Q2 the quantity Q4F(Q2)dt~ 0,~ constant. The proton elastic form fator is also shown in figs. 19 and 20. The errors are due to the statistics of fitting. -+

P. Stoler, Baryon form factors at high 1.50

Q2

I

1.25

-

1.00

oo~ •

0.75

-

0.50

-

0.25

—

Proton Elastk. ~ 0 -

(a) 1 I I I

-

I I

I I I I

I I

1.00

I

F

~(i232)

0.75

—

0.50

—

~ 0.25

—.

III~IIIl

IIIIj)Il~f~

N(1535)

(c) :1111

—

—

III

III

III

I

N(1680

2.0

-

f

(d) ~

0

—

—

—

i!E

-

—

(b)

2.0

0.0

143

liii

5

10 QS

III

15

I

I

20

(GeV’/c1)

Fig. 19. (a) The absolute value of the proton elastic magnetic form-factor divided by the dipole shape, GM/Gdj~Je,versus Q2, where GdI~j~ = p,.(l + Q2/0.71)2. The data are from ref. [Ar-86]. The curves at the right of the figure are the results of calculations [Ji-87] at Q2 = 20 GeV2/c2 using the proton distribution functions c-z (solid), G-S (dot—dashed), and K-S (dashed). (b)—(d) The quantity GT/GdI~,OIOversus Q2 for transitions to the first, second and third resonances, respectively, with GdI 2/0.71y2. The first resonance (b) is the ~(1232).The second resonance (c) at Q2 = 0 consists of contributions from the D11~,j.= 3(1 + Q 2 3 GeV2/c2 it is mainly due to the S 2 is dominated by the13(1520) F and S12(1535), but at Q 12(1535). The third resonance (d) at low Q 15(1680). The resonance form-factor GT is defined in the text (eq. 9). The fits for GT were based 2 denoted on inclusive by (x)data are form-factors reconstructed derived from refs. from [Br-76] amplitudes ( + ), [Ro-92] (0), obtained from [Po-74, exclusiveBo-79, (e, e’p)ir°and Ri-73] (LI, (e, 0). e’p) ~i Also datashown obtained at lower from Q refs. [Br-84, PDG-92, Bu-91]. The errors shown are statistical. Estimated systematic errors are discussed in the text.

For the A(1232), uncertainties in the radiative corrections, the width and background lead to estimated systematic errors in the amplitudes of between 10 and 15%, depending on Q2. For the A(1232) the uncertainty in background shape at high Q2 adds a significant uncertainty to the lower bound, and it is not clear whether Q4F(Q2) continues to decrease past Q2 = 10 GeV2/c2. For the second and third resonances, uncertainties in the branching ratios for r~,p, A and 2it decay channels, resonance widths, background shape, and the possibility of small longitudinal resonant contributions add about another 10 to 15% depending upon Q2, so that the overall systematic uncertainty is estimated at about 15 to 20%. Also shown at lower Q2 are form-factors extracted from data obtained from exclusive (e, e’, p) it°and (e, e’, p)r~experiments [Br-76, Ha-79]. Figure 20 shows the data of fig. 19 for Q2 less than 5 GeV2/c2. The extracted form-factors from all data sets agree within their quoted errors.

144

P. Stoler, Baryon form factors at high 1.50 1.25

-

1.00

-

0.75

—

0.50

-

I

0.25

~

I

I

I

I

I

I

I

I

I

I

I

I

I

Proton Elastie o

0

~

-

— —

(a) I 1*1 I

1.00

~

I

Q2

—

I I I I

I I I I

I I I I

I I

I

~(1232)

—

—

0.75—

-

0.50

—

—

0.25

—

—

(b) —I——I——lI

0.00

Ill

III

—

liii

N(1535)

~

1.0

Ill)

Iii

—

0.5

—

(c) I I 1.25 1.00

—

0.75

—

0.50

—

I II

I

(III)! +N(1680)

T

-k~

0.25 0.00

I I

-

— —

(d) 0

-

JA.JI

—

1

I

2

lillilIll

3

4

III

5

Q’ (GV’/e’) Fig. 20. The same as in fig. 19, but for

Q2 <5

GeV2/c2.

Table 4 Elastic scattering form-factors obtained in ref. [Ar-86]. GM,dP p~.(l+ Q2/0.71)2. The subscripts ö,~,and il~denote statistical and systematic errors, respectively

Q2

Q4GM

2.0 2.9 3.7 5.0 5.0 5.0 7.3 9.7 12.0 15.8 19.5 23.3 27.1 31.3

0.80 0.938 1.013 1.080 1.096 1.079 1.119 1.102 1.094 1.046

0.971 0.979 0.904 0.945

~SI~I

0.0 0.006

0.011 0.003 0.008 0.008 0.011 0.014 0.014 0.025 0.033 0.039 0.053 0.086

~

~

F/GM

0.0 0.017 0.017 0.017 0.017 0.017 0.020 0.020 0.017 0.017 0.017 0.017 0.014 0.017

0.0 0.017 0.020 0.017 0.020 0.020 0.022 0.025 0.008 0.022 0.036 0.042 0.056 0.089

0.97 1 0.963 0.956 0.930 0.944 0.922 0.887 0.839 0.811 0.755 0.690 0.682 0.597 0.653

dip

0.0 0.017 0.018 0.014 0.017 0.017 0.017 0.019 0.006 0.016 0.026 0.029 0.037 0.061

P. Stoler, Baryonform factors at high

Q2

145

Figure 19 shows that the obtained form-factors for the second and third resonance regions are consistent with a Q ~ dependence, although with large statistical uncertainty. On the other hand, the A(1232) form-factor is falling relative to both the elastic as well as the second and third resonances. However, the A(1232) data stops at Q2 = 10 GeV2/c2, and it would be very interesting to see if in fact it does begin to follow a Q4 dependence at higher Q2. The resonant form-factors are discussed in the next sections in the context of the overall known structure of the individual resonances. The amplitudes and form-factors of all the data shown in figs. 19 and 20 are tabulated in tables 4, 5 (in section 6.3), 6 (in section 6.5) and 7 (in section 6.6). 6.3. The P 33 (1232) or A (1232) 2 models 6.3.1. The Low-Q P 33(1232) or A(1232) has been the most studied excited baryon since its discovery in it° photoproduction [St-SO, Br-51] and56sthen in charged-pion scattering [An-52]. In the pure SU(6) multiplet (see section 3). At low Q2 in the non-relativistic model the A(1232) is in the lowest CQM the N A(1232) excitation is characterized as a single quark spin-flip transition within the same 56s multiplet as the nucleon and its mass is degenerate with the nucleon mass. Although the transition 1/2k to 3/2k admits M 1 + and E1 + (in alternative notation Ml and E2) multipole contributions, it was pointed out in ref. [Be-69] that the E1 + identically vanishes in the SU(6) limit, and the transition is pure M1 ÷. The reason the E1 + contribution vanishes is simply that a spatial operator with L > 0 cannot connect two wave functions with L = 0. In the notation of helicity amplitudes the vanishing of E1 + implies that A312/A112 = —*

Table 5 Helicity amplitudes and form-factors for the transition P —~~(1232) obtained in refs. [St-91a,b]. The data in columns 6 and 7 are plotted in figs. 19(b) and 20(b). Column 8 gives the data source reference from which the amplitudes and form-factors were fit. Data denoted [Ha-79] were derived from exclusive experiments. The prefix SL denotes SLAC data. See sections 6.2 and 6.3 for further details 2 A~ hA GT 8GT GT/GdIP h(GT/GdIP) source Q 2) (GeV”2) (GeV”2) 1 (GeV 0.9 0.170 0.0013 0.647 0.0051 1.109 0.0087 [Br-76] 1.9 0.0813 0.0018 0.213 0.0047 0.959 0.021 [Br-76] 2.4 0.0606 0.0017 0.141 0.0041 0.904 0.026 [Br-76] 2.4 0.0610 0.0011 0.142 0.0026 0.909 0.017 SL E133 2.9 0.0464 0.0016 0.098 0.0034 0.847 0.029 [Br-76] 2.9 0.0482 0.0006 0.102 0.0012 0.880 0.011 SL E89 2.9 0.0457 0.0017 0.097 0.0036 0.835 0.031 [Br-76] 3.9 0.0288 0.0012 0.053 0.0022 0.740 0.031 SL E133 3.9 0.0283 0.0015 0.052 0.0028 0.727 0.040 [Br-76] 3.9 0.0273 0.0016 0.050 0.0029 0.701 0.041 [Br-76] 5.9 0.0127 0.0008 0.019 0.00120 0.543 0.035 SL E133 7.8 0.0052 0.0013 0.0067 0.0016 0.321 0.079 SL E133 7.9 0.0055 0.0011 0.0070 0.0014 0.344 0.067 SL E89 8.6 0.0069 0.0007 0.0085 0.0009 0.488 0.052 SL E89 9.6 0.0071 0.0017 0.0083 0.0020 0.582 0.14 SL E19 9.8 0.0026 0.0014 0.0030 0.0016 0.221 0.12 SL E133 1.0 2.0 3.0

0.163 0.072

0.012 0.005

0.55 0.186 0.089

0.034 0.012 0.005

1.062 0.903 0.805

0.067 0.057 0.0430

[Ha-79] [Ha-79] [Ha-79]

146

P. Sw/er, Barvon form factors at high

Q2

The SU(6) symmetry is broken by the color hyperfine interaction so that the A(1232) wave function is a mixture of L = 0 and L 2 configurations, and can be written as P

4S~>+ b~IA4S~> + bDIA4DS> + b~IA4DM> (6.5) 33(1232)> = b5IA The notation (see section 3) is 1A2s+ 1 Li>, where S, L and t denote total spin, orbital angular momentum and symmetry. Typically [Gi-90a] the values of the coefficients are .

b 5

=

0.963

b~= 0.231

,

bD

,

=

—0.119

b~= 0.075.

,

Davidson et al. [Da-90, 91] have made a thorough review of the available photoproduction data and theoretical approaches, and conclude for the process ‘y + N A —+

M1±= 285 ±37, A112

=

E1±= —4.60 ±2.58,

—135 ±16,

A312

=

E1+/M1+

—(1.57 ±0.72)x 102,

=

—251 ±33

in units of iO~GeV 1/2 The helicity non-conserving amplitude A312 is dominant over the helicity conserving 2 >amplitude 0 the E A112, and the cross section is much larger than for any other resonance. For Q 1 ±/M1+ ratio remains Figure shows the by 2 ~ small. 3 GeV2/c2. At213 GeV2/c2 theresult valueof of an the analysis ratio is still Burkert [Bu-92] available datathe up errors to Q indicate statistical and systematic errors respectively. small, 0.08 ±0.03of±0.03, where -

6.3.2. High-Q2 PQCD limit Using PQCD sum rule techniques Carlson and Poor [Ca-88] and Farrar et a!. [Fa-89] have calculated the distribution amplitude for the A(1232) excitation. The helicity amplitudes for the transition form-factors between the A(1232) and the proton were calculated in ref. [Ca-88] with three different proton distribution functions, the C-Z, K-Z and G-S distributions. 0.2

I

I

I

I

I

I

I

I

Re(E~M

I

I

I

I

I

2 1+)/lU1+1

iT -.o.1~-

—0.2

0

-

I

I

I

I

1

I

I

I

I

2

I

I

I

I

I

3

I

Qe (GeV’) Fig. 21. Evaluation by Burkert [Bu-92] of existing data for the ratio E 1+

+

P. Stoler, Baryonform factors at high

Q2

147

The proton three-quark Fock state, eq. (4.8), can be expressed in terms of symmetric and antisymmetric distribution functions and spin—flavor functions with respect to interchange of quarks 1 and 3. For a proton with helicity + 1/2 this may be written as 1/2)

=

4s x ~s + 4~Ax

=

(1/~/~udu duu

=

—

—

+ perm.,

~A

uud>~~ 7,

For the A(2

=

=

~A

=

(1/~/~)Iuud duu>,11. —

+ 1/2) Carlson and Poor [Ca-88] take the purely symmetric form 1/2) = 4~sx ~ + perm.

(1/~)~udu duu

=

(6.6)

—

—

(6.7)

,

uud>111,

~~(Xj)

=

J[d2kT]~(xi,kT).

The normalization is such that 2kT]/~(xt, ~

kT)12

P

=

J[dxj] [d

3q.

(6.8)

P3q is the probability of finding the three-quark Fock component in the A.*) The isospin 3/2 requirement constrains the three distribution functions V~,A~,and T~such that there is only one independent function, = A~.The calculation of the first few moments of the distribution function proceeds with the evaluation of the correlators as in the case of the nucleon ground states [see eq. (4.24)], ~

i Jd4x e’~
—

t100~}I 0>(z. y).

JA(0)

projects out the moments of order n

1~~2uI3)

1n2n3 and the auxiliary operator J~(0) projects a state with isospin 3/2. The moments of the A distribution function are related to the correlator as in eq. (4.24). As in the case of the nucleon the correlator can be evaluated in terms of Feynman diagrams involving quarks and gluons. Three types of processes are considered: (1) purely perturbative effects involving three quarks interacting with each other via gluonic fields, (2) absorption and emission of gluons by the valence quarks interacting with the physical vacuum, and (3) absorption and emission of quarks from and to the physical vacuum. The results have the same form as eqs. (4.26), (4.27), and (4.28), except that now the physical constants c~mn2ul3)are different since the auxilliary operator JA projects an isospin 3/2 state rather than one with isospin 1/2. The moments of the distribution function are then obtained as with eq. (4.24). Carlson and POor [Ca-88] note that, unlike the case of the nucleon distribution function, only three stable moments can be obtained for the A. The resulting distribution function in terms of the Appell polynomials is =

x1x2x3(0.3440

—

0.12452 + 0.14453 + 0.06455).

S

2kT] I~(x~, kT)12

[Ch-84a} the *)Chernyak definition inand ref. Zhitnitskii [Br-81a] by a factordefine ~/~Ij the normalization of eq. (6.8) such that [dx1] [d 5.

=

1. This is related to

148

P. Stoler, Barvon form factors at high

Q2

Transverse transition form-factors have been obtained using eq. (4.1) with a constant average value of the strong coupling constant; = 0.3, for the three available proton distribution functions C-Z, K-S, and G-S. The C-Z and K-S proton distributions are rather similar, and both yield small transition form-factors Q4F1A(Q2) Q4GT(Q2) 0.07 and 0.11 GeV4, respectively. [The transverse form-factor G 1 is defined in eq. (2.9).] This can be traced to a cancellation of the matrix elements connecting the symmetric A(1232) distribution function eq. (6.7), with the symmetric and antisymmetric proton distribution functions, respectively (eq. 6.6), i.e., —+

<45AITHI4~> and

<45AITHI45~>

(6.9)

.

On the other hand,2)the0,G-S which the constraint that the neutron doesdistribution, not yield this largehas cancellation and gives Q4F 2) Fermi 0.61 elastic GeV4 form-factor F~(Q 1A(Q for large Q2. [In section 5 it was pointed out that the available data up to Q2 = 4 GeV2/c2 are not compatible with Q4F~(Q2)= 0.] The results of the calculation of the P A transition form-factors, as well as the experimental situation are shown in fig. 19. The experimental transition form-factor is falling faster than the Q ~ dependence predicted by the leading order PQCD amplitude. If the leading amplitude of the P A transition is indeed small, the shape of the transition form-factor might be explained as follows. At high Q2, the helicity conserving amplitude (analogous to the Dirac form-factor F 1 in elastic scattering) dominates over 2.theThat helicity is, A non-conserving 2A amplitude (analogous to the Pauli form factor F2) in proportion to Q 112 cc Q 312. In terms of multipoles, this implies the asymptotic 2equality + and M1 +. of the symmetric and due to of theE1cancellation If the A112 amplitude is suppressed at all Q antisymmetric matrix elements (eq. 6.9), then one might expect the A 312G~(Q2)would amplitude would remain 2 than otherwise expected, and Q4 decrease as dominant over a larger range of Q a function of Q2 over a larger range in Q2 than for other resonances. In fact, as noted above, the E 2 up to 3 GeV2/c2 (see fig. 21), which is consistent with the 1 + /M1 + ratio is still small for Q If that were in fact the explanation, then this would favour the dominance of non-leading processes. C-Z and K-Z proton distribution amplitudes over the G-S amplitude, since the G-S amplitude does not predict a small helicity conserving form-factor. The data in fig. 19 end at Q2 10 GeV2/c2. It will be interesting in the future to find whether Q4GT(Q2) does indeed level off above this Q2, and where the E 1 + amplitude becomes comparable to the M1 +. This would confirm the PQCD description. —+

—+

6.4. The P11 (1440) or Roper resonance This is one of the more controversial baryon states, whose structure is currently not understood. According to the non-relativistic potential model an N* radial excitation is expected with N = 2, L = 0 having energy 2hw, which is a member of the second (56,0k), and otherwise having the same quantum numbers as the nucleon. Soon after the CQM was proposed it was speculated that the P11 (1440) is a candidate for this state. Likewise, the analogous A excitation was speculated to be the P33(1700). The P1 1(1440) is a four-star resonance which is observed quite distinctly in it—N scattering (fig. 22 shows the P11 phase shift analysis from ref. [PDG-88]), and is clearly observed in real photon absorption and exclusive photoproduction experiments. However, there were several problems with this hypothesis. The P11(1440) energy lies considerably below the excitation energy characteristic of 2hw (see fig. 3). Even when the SU(6) configuration mixing of the I-K model is invoked, the state is still 150 to 200 MeV below where it is expected.

P. Stoler, Baryon form factors at high

Q2

149

N( 1440)

0

—.25 El)

I

.25

1100

.50 1100

1400

1700

2000

2300

ENERGY (I~eV)

I

1400

irN ELASTIC P11 AMPLITUDE 1700

I

41

2300

ENERGY

Fig. 22. The

~l

(Mev)

amplitude versus excitation energy for elastic pion scattering [PDG-88].

The first attempts to calculate the helicity amplitudes [Co-69a, b] using the non-relativistic CQM gave the wrong sign for A 112. However, this difficulty was subsequently eliminated. It was shown in ref. [Ku-76] that when relativistic corrections are added to the non-relativistic CQM a spin—orbit term is generated in the Hamiltonian whose effect is to reverse the sign of A112 and A312 to give the correct sign. The effect of SU(6) mixing in the I-K non-relativistic model [Ko-80] also reverses the signs thereby giving the correct sign. Recently there have been several calculations which are relativized and take into account the hyperfine interaction. The calculation of Li and Close [Li-90] appears to account rather well for the experimental amplitudes on both the proton and neutron. That of Warns et al. [Wa-90b, c] gives results for the proton only, which are rather far from the experimental result. There are two important remaining difficulties with the hypothesis that the P11 (1440) is a N = 2 radial excitation. 2 = 0~then the non-relativistic (1) Ifpredicts the P11(i440) is indeed a N = should 2 radialincrease excitation with L CQM that the cross section relative to the N = 1 resonances having L~= 1, which are in the second resonance region [Ab-72, Wa-90b, c, Li-92a]. This can be simply seen by noting that the harmonic oscillator spatial state (ls)2(2s)1 has one more node in the radial wave function than the (ls)2(ls)1 state, leading to the transition matrix element ratio (~fr

2

.

(6.10)

00 çfr~o)/(çfroo çfr1o,~fr1i) k ‘~

The notation in eq. (6.10) is that ç1i~is the nucleon ground state, i/’~the monopole excitation and ~ 10 and 4,11 correspond to states for which N = 1 and L = 1 with orbital and spin-flip, respectively. However, there is no experimental evidence that the P 11 (1440) increases relative to the 2 = 1 states as a function of Q2. Rather, the existing data suggest otherwise. Figure 23, N = 1, L —

150

P. Stoler, Baryon form factors at high

Q2

from ref. [Li-92a], summarizes the existing experimental situation for the A 112 amplitude. The dashed curve in fig. 23 shows their calculated cross section in the framework of nonelativistic CQM. It can be seen that there is a complete discrepancy between the model and the data. (2) If the P11 (1440) is a radial excitation, then the transition should have a large longitudinal amplitude S112. In fact an evaluation of existing inclusive data by Drees et al. [Dr-81]indicates 2/c2, awhich possible in the L/T the position of the P11the (1440) near 1 GeV may enhancement be taken as evidence for anratio S112 at component. In any case, S 112 amplitude is still not in accord with the theoretical prediction in ref. [Li-92a] as seen in fig. 23. An alternative interpretation of the P11 (1440) is that it is a hybrid state. A hybrid state is defined as 3one constituent waveP function can be described primarily as three quarks and a gluon G). whose The possibility of the (q 11 (1440) being a hybrid state had earlier been ruled out [Go-82, Ba-83a, b] based on a rigorous group theoretical SU(6) selection rule, the Barnes—Close selection rule, which is a generalization of the earlier Moorhouse selection rule [Mo-66]. The coupling of one spin 1 gluon plus three quarks results in a 70 SU(6) multiplet. Transitions between this multiplet and the proton, which is in a pure 56 SU(6) multiplet, forbidden due to orthogonality. 2 = are 0 the electromagnetic proton to Since the observed experimental situation is that at Q P 11 (1440) amplitude is quite large, this would appear to rule out the P11 (1440) as a hybrid partner of the proton. Li [Li-91] has pointed out that Barnes and Close [Ba-83a] did not take into account relativistic effects, in particular the function radiative-like involvingand quark—gluon couplings. 3 wave to getcorrections a q3G component, the pure hybrid q3GThis statecauses gets athe q3 proton SU(6) q component, destroying the orthogonality, so that the Barnes—Close selection rule does not apply. The diagrams which they considered are illustrated in fig. 24. In particular, the orthogonal nucleon

Q2(GeV/c)2 Fig. 23. P 1 1(1440) transverse 3 I-K non-relativistic A~12helicity amplitudes. CQM as in The ref.curves [Ko-80]. are The calculations solid curve ofLiis and due Burkert to a hybrid [Li-92a]. q3 G model The long as in dashed ref. [Li-92a]. curve is Thea data points referenced in ref. [Li-92a] are from refs. [Ge-80] (diamond) and [Bo-86] (crosses). the result assuming q

P. Stoler, Baryon form factors at high

N

151

N

(a)

N

Q2

_________________

9

(b)

+

_________________

(d)

(c)

Fig. 24. Electromagnetic couplings to the constituents of the Roper resonance where the resonance is assumed to be a hybrid (from ref. [Li-92a]).

and hybrid states may be written as

IN>

1 +

=

IN*>

=

2~2[b0>

+

±2~2[b0>

~(I2Ng> + I4Ng>)], + +(I2Ng> +

I4Ng>)].

(6.11)

In eqs. (6.11) the state IN 3 state, and 12N 4N 3G states with total is a pure qThe mixing parameter 5> and 15 is 5> are pure qby the quark—gluon quark spin 1/2 and 3/2, 0> respectively. determined radiative correction interaction, and found in ref. [Li-92a] to be 5 -~ 0.35. With this value of ö Li et al. find a substantial transverse helicity amplitude at Q2 = 0, which decreases as a function of Q2 in comparison with that from a q3 excitation with N = 2. They obtain —

A

3G; N = O)/A 3 N = 2) 1/Q2 , 51/2 = 0. 112(q 112(q Li et al. [Li-92a] also predict a longitudinal amplitude S 3G state, while an N = 2 112 = 0 for a q radial excitation would have a large value of S 112. Their results, shown in figs. 23 and 25, are not inconsistent with the data. 2 PQCD The possibility that P11 (1440) a hybridand stateMukhopadhyay has important implications in the26high-Q regime, as recently observed by isCarison [Ca-91]. Figure illustrates the leading order PQCD diagrams for a hybrid electroproduction from a nucleon. The extra quark and gluon propagators modify the powers of 1/Q to the amplitude. In particular, each new internal quark propagator adds a factor 1/Q to the amplitude. Furthermore, a Fock state description requires gluon lines such as in fig. 26c to be on-shell, and hence transverse. In order to conserve helicity this requires the photon to be longitudinal. A transverse photon will not conserve helicity,

152

P. Stoler, Baryon form factors at high 20

IIIIIIIIIE

IlIp

0

Q2 1111

p

Q2(GeV/c)2 Fig. 25. The longitudinal amplitude S~’

3 configuration (solid curve). For the hybrid state S~, 12for the P1 ~(1440)assuming a q 2vanishes. The dot-dashed curve is due to an analysis ofdata by Gerhardt [Ge.80]. The data points are from refs. [Ge-80] (squares) and [Bo-86] (crosses). The figure is from ref. [Li-92a].

(a)

2

_

(c) Fig. 26. Leading order PQCD diagrams from ref. [Ca-9l] contributing to the Roper resonance form-factors assuming the resonance has a hybrid structure.

P. Stoler, Baryonform factors at high

Q2

153

and would add another factor of 1/Qto the amplitude. This reverses the relative Q2 dependence of G + and G 0 from the case of a normal baryon. Thus, applying the PQCD2scaling rules finds baryon, that for than for an one ordinary a hybrid the transverse amplitude decreases faster by a factor of 1/Q and the longitudinal amplitude decreases at the same rate as that for the normal baryon. In summary G+(N-+q3)ccl/Q3,

G+(N—+q3G)ccl/Q5,

GO(N

G

3G) cc 1/Q4. 0(N q For a hybrid q3G state, the longitudinal amplitude would asymptotically be much greater than that for an ordinary q3 resonance. However, due to the arguments of duality (see, e.g., refs. [Bl-70, 71, Ca-90] and section 6.6) it would still be expected to follow the same Q2 dependence as the longitudinal background. On the other hand, since transverse amplitudes for q3 resonances typically have the same Q2 dependence as transverse background, then Gq3 resonances should fall faster, and not be observable in the transverse cross sections. An estimate of how large the longitudinal amplitudes are expected to be will be needed in order to assess the viability of such an experiment. —+

q3) cc 1/Q4,

—+

Table 6 Helicity amplitudes and form-factors for the transition P —~N5(1535) obtained in refs. [St-91a,b]. The data in columns 6 and 7 are plotted in figs. 19(c) and 20(c). Column 8 gives the data source reference from which the amplitudes and form-factors were fit. Data denoted [Ha-79] were derived from exclusive experiments. The prefix SL denotes SLAC data. Near Q2 = 0 the resonance is dominated by the D 2 = 3 GeV2/c2 it is dominated by the S and the 1(1535). Q (1535), 2 its13(1520) composition hasS1not beenAt determined. See sections 6.2 and 6.5 in the11text for and at higher Q further details

(GeV2)

AT (GeV112)

bAT (GeV112)

G

8GT

GT/GdiP

b(GT/G 4IP)

1

source

1.8 2.3 2.3 2.7 2.8 3.8 5.7 7.7 7.7 8.4 9.3 9.6 10.5 11.7 16.5 20.6

0.0643 0.0533 0.0478 0.0436 0.0409 0.0288 0.0204 0.0124 0.0144 0.0117 0.0083 0.0100 0.0077 0.0089 0.0053 0.0036

0.0012 0.0013 0.0017 0.0009 0.0013 0.0015 0.0008 0.0006 0.0004 0.0004 0.0013 0.0005 0.0006 0.0013 0.0011 0.0006

0.264 0.193 0.173 0.146 0.134 0.081 0.047 0.025 0.029 0.022 0.015 0.0178 0.0131 0.0144 0.0071 0.0043

0.0049 0.0047 0.0063 0.0031 0.0043 0.0042 0.0018 0.0012 0.0007 0.0007 0.0023 0.0008 0.0010 0.0020 0.0015 0.000k

1.098 1.157 1.037 1.120 1.094 1.094 1.277 1.151 1.338 1.221 0.992 1.248 1.088 1.463 1.399 1.29 1

0.020 0.028 0.039 0.024 0.035 0.056 0.049 0.054 0.034 0.040 0.153 0.057 0.082 0.205 0.295 0.227

[Br-76] [Br-76] SL E133 SL E89 [Br-76] [Br-76] SL E133 SL E133 SL E89 SL E89 SL E19 SL E133 SL E89 SL E26 SL E26 SL E89

0.2 0.4 0.6 0.6 1.0 1.0 2.0 3.0

0.089 0.143 0.085 0.087 0.078 0.076 0.042 0.039

0.054 0.019 0.007 0.008 0.004 0.007 0.008 0.003

1.002 1.025 0.600 0.6 17 0.429 0.417 0.162 0.124

0.062 0.113 0.053 0.058 0.048 0.085 0.030 0.031

0.585 0.835 0.680 0.700 0.829 0.806 0.784 1.130

0.036 0.091 0.059 0.066 0.092 0.164 0.144 0.278

[Ha-79] [Ha-79] [Ha-79] [Ha-79] [Ha-79] [Ha-79] [Ha-79] [Ha-79]

154

P. Stoler, Baryon jirm factors at high

Q2

6.5. The S 11(1535) The S1 1(1535), in the second resonance region, is one of the strongest excitations and therefore has been the subject of considerable investigation. In the non-relativistic CQM the S11(1535) spatial wave function corresponds a mixed 1 hw harmonic 2(lp)1, andtooverall J’~=symmetry 1/2~,or IN 2P1/2>M. As oscillator such it is aexcitation member ofwith the orbital configuration (ls) (70,1-) spin—flavor SU(6) multiplet (see fig. 3). The S~~(1535) is uniquely interesting for two reasons.

(1) The S~~(1535) and the D are the two most strongly excited states in the second 2 = 0 the 13(1520) D 2 the resonance region. At Q 13(1520) is dominant, but, as seen in fig. 27, at increasing Q S~~(1535) grows relative to the D 2 = 3 GeV2/c2 it dominates the inclusive 13(1520) so that at structure Q cross section. Figure 18 shows that the peak in the function near the position of the S 2. 1 (2) ~(1535)remains the most striking feature of the inclusive data at all higher values of Q The S 11(1535) is the only low lying nucleon resonance which has a large branching ratio to the Ni decay channel ~ 50%), so that it may be cleanly separated from other strong resonances in the ii electroproduction reaction. In fact, this resonance has been most commonly studied by measuring the recoil proton in the reaction N(e, e’ P)r~,and reconstructing2 the ~‘sGeV2/c2. missing mass = 0.3 The [Br-84, 78]. Figure 28 shows the 1) electroproduction cross section at Q asymmetry in the energy dependence is due to the fact that the threshold for r~production occurs at

=

0.3(GeV/c)2

100.0

[pb]

D

15

13

-

1

30.0

1~OO~~2I.O~

6.O~

2[GeV2]

G

Fig. 27. Total virtual photon cross section for excitation of the S 2. The data are from [Ha-79, Br-84]. The belowof 3 GeV2/c2 11(1535) and refs. D13(1520) resonances as data a function Q are the result of fits to exclusive 11 and it~’angular distribution data. The data above 3 GeV2/c2 are the results of fitting inclusive data assuming the second resonance is due only to the S 11(1535).

__ w(6eV) Fig. 28. The total virtual photon cross section for ~lproduction from a proton at Q2 = 0.3 GeV2/c2 from ref. [Br-84]. The curve is a fit using a relativistic Breit—Wigner distribution with s-wave threshold behavior. The asymmetry in the shape is due to the i threshold at W= 1487 Mev.

P. Stoler, Baryon form factors at high

Q2

155

W = 1.48 7 GeV, which is close to the resonance peak. This also makes the resonance peak appear very narrow in the inclusive cross sections. In the 11 decay channel the non-resonant background contribution is relatively small in the region of the S 1 1(1535), which would further help to isolate the resonance. However, the question of the relative contribution between resonance and background is not settled [Be-9 1a,b]. Figure 29 shows the angular distribution in the 11 decay channel from ref. [Br-84]. The constant distribution is consistent with an S-wave resonance with at most a small P-wave contribution. 2 requires a relativistically covariant treatment. This has Thecarried extension CQM and to higher Q [Ko-90], employing the light-front formulation as in been out ofbytheKonen Weber refs. [Dz-88a, b] (see section 3.3). The ground state nucleon model wave function used in refs. [Dz-88a, b] is a solution of a SHO Hamiltonian,

45cm(ki)

=

ANexP(_~ k~/2~2).

The theory has two parameters, the hadronic size parameter ~, and the constituent quark mass (—~MN/3). With ~ 320 MeV/c the model wave functions used by Dziembowski [Dz-88a, b] reproduce static electromagnetic quantities of nucleons such as magnetic moments and radii very well, as well as low-Q2 elastic form-factors (Q2 <2 GeV2/c2). For the S~~(1535) state, as in the case of the nucleon Konen and Weber [Ko-90] arbitrarily also choose a symmetric Gaussian. Since the S 11(1535) state has negative parity, the spin part must also be a negative parity invariant. This is accomplished by substituting u -y5u for one of the quark spinors in eqs. (4.8). The non-orthogonality of the nucleon and S1 1(1535) wave functions leads to a violation of gauge invariance, which is restored by adding a three-body convection current term to the overall current operator. The resulting transverse and scalar helicityless amplitudes are shown in figs.non-relativistic 30 and 31. NoteCQM that 2 dependence falls much rapidly than the purely the calculated Q prediction. Given the ad hoc choice of the distribution function for the S 11(1535) the calculation -+

•~I1~Of1 f I I

0. -1

0 cos

+1.

eqs

Fig. 29. Angular distribution from ref. [Br-84] of TI production at W -~1535 MeV. The horizontal line is a constant fit, which is indicative of an s-wave behavior.

156

P. Stoler, Baryon form factors at high

Q2

02(GeV2] Fig. 30. Helicity transition amplitude A 112 ofthe S11 (1535) resonance from the proton and neutron. The solid and dashed curves are the results of the relativized quark model calculation of Konen and Weber [Ko-90] and Warns et al. [Wa-90b,c], respectively. The dot—dashed curves are the non-relativistic reduction of Konen and Weber [Ko-90]. References to theoretical curves and data in ref. [Ko-90J are [Fo-83, Bo-87, Cr.83].

60\

-

~2 (GeV9 Fig. 31. Same as figure 30, but for the longitudinal amplitudes.

P. Stoler, Baryon form factors at high

Q2

157

does a reasonable job of reproducing the helicity amplitudes up to Q2 = 3 GeV2/c2, which is the limit of existing experimental data. This demonstrates that the “anomalously” small Q2 falloff of the S 11(1535) helicity amplitude might be explained by incorporating a relativistically invariant formulation. Presumably a better quantitative fit would be obtained with a wave function which is a solution to the model 2, justHamiltonian. as in the case encountered in the elastic form-factor (fig. 4), one expects At even higher Q the helicity amplitudes to fail to reproduce the data since the fast falloff of the Gaussian wave function cannot account for “hard” processes, which produce the slow falloff of the Q2 scaling laws inherent in PQCD. Figure 19 shows that at high Q2 the form-factor for the resonance at W 1.53 5 GeV approaches the Q ~ dependence predicted by PQCD. The likelihood that this peak continues to be dominated by the S 2 11(1535) is good because the apparent the peak inratio, Wis in narrow entire Q range covered, which would be consistent with awidth largeof ~t1 branching whichover the itheproduction threshold is close to the resonant energy. Carlson and Poor [Ca-88] have calculated the P transition form-factor in the PQCD limit. It is expected to behave similarly to the elastic form-factor because the distribution amplitude ~ obtained is similar to that of a proton, with the antisymmetric part about half as large. The results for Q4G~.h1(Q2) obtained for the three proton distribution functions are C-Z 0.60, K-S 0.77 and G-S 0.66 GeV4, and are also plotted in fig. 19. Although all these results are several times lower than the data, theoretical uncertainties in the distribution functions, and higher order contributions to ; are probably great enough to encompass these discrepancies. —~

‘~

Table 7 Helicity amplitudes and form-factors for the transition P —* N*(1680) obtained in refs. [St-9 la, b]. The data in columns 6 and 7 are plotted in figs. 19(d) and 20(d). Column 8 gives the data source reference from which the amplitudes and form-factors were fit. Data denoted [Bu-91] were derived from exclusive experiments. The prefix SL denotes SLAC data. Near Q2 = 0 the resonance is dominated by the F 2 its composition has not been determined. See sections 6.2 and 6.6Qin the text for further details 15(1880). At high

Q2 (GeV2)

AT (GeV112)

bAT

(GeV112)

G 1

0.8 2.6 2.7 3.7 7.4 8.3 9.0 10.3 11.4 16.1 20.1 0.5 0.5 0.6 0.6 1.0 1.0 2.0 3.0

bGT

GT/Gd~~ b(GT/GdjP)

source

0.0827 0.0425 0.0393 0.0295 0.0137 0.0110 0.0080 0.0079 0.0060 0.0034 0.0034

0.0023 0.0011 0.0012 0.0013 0.0004 0.0004 0.0013 0.0005 0.0014 0.0011 0.0008

0.583 0.166 0.151 0.097 0.032 0.024 0.0169 0.0156 0.01 12 0.0054 0.0047

0.0159 0.0043 0.0048 0.0041 0.0009 0.0010 0.0026 0.0010 0.0027 0.0017 0.0011

0.88 1.20 1.16 1.24 1.38 1.31 1.05 1.25 1.09 1.01 1.35

0.024 0.031 0.037 0.053 0.040 0.051 0.17 0.08 0.26 0.32 0.30

[Br-76] SL E89 [Br-76] [Br-76] SL E89 SL E89 SL E89 SL E89 SL E26 SL E26 SL E89

0.090 1.077 0.080 0.080 0.064 0.078 0.035 0.026

0.014 0.021 0.022 0.010 0.010 0.010 0.010 0.009

0.808 0.947 0.651 0.645 0.408 0.488 0.156 0.105

0.12 0.18 0.18 0.081 0.065 0.063 0.045 0.033

0.78 0.95 0.72 0.75 0.77 0.97 0.76 0.95

0.12 0.18 0.19 0.094 0.12 0.12 0.22 0.30

[Bu-91] [Bu-91] [Bu.91] [Bu-91] [Bu-91] [Bu-91] [Bu-91] [Bu-91]

158

P. Stoler, Baryon form factors at high

Q2

6.6. The D 13(1520) and F15(1535)

2. At These states the third second and third resonance respectively, at low Q with increasing Q2 thedominate second and resonance peaks persist. regions, In the second resonance region increasing Q2 the S 11(1535) form-factor increases relative to that of the D13(1520) so that 2 = 3 GeV2/c2 the S at Q 11(1535) form-factor is already twice as great as for the D13(1520), and the virtual photon cross section is dominated by the S~~(1535). third resonance 2, at least outInto the 3 GeV2/c2. Currentlyregion, there the F15(1680) remains very strong with increasing Q exist separated helicity amplitude data for both the D 13(1520) andQ2 F15(1680) out to a maximum 2 = 3 GeV2/c2, although the statistical errors are quite large for > 1.5 GeV2/c2. Q Figure 32 shows the cross section for the reaction y + p it~+ nat O~= 00, at various values of Q2. At Q2 = 0, i.e. for real photons, the second and third resonances are absent, whereas they become quite prominent at higher Q2. Since A 312 cannot at 6,,important = 00 this with showsincreasing that A112Q2. is very small for 2 = 0,contribute but becomes The situation both the D13 (1520) and F15 (1680) at Q is summarized in fig. 33, which shows the helicity asymmetry (A~ 12 A~12)/(A?12+ A~12)for the 2 = 0, A 2 the A112 amplitude D13(1520) and F15(1680) resonances. Near Q 112 0, whereas at larger Q dominates over the A 312. The transverse form-factors due to the individual helicities are shown in 2, figs. 34 and 35. The helicity 1/2 amplitudes appear to level off to a dipole shape at a low value of Q whereas the helicity 3/2 amplitudes diminish rapidly in the same Q2 region. -+

—

20.0

02 (6eV2) 0 F, ~

0.4 ~ I

06

I

-

0.0

I

1.3

I

~•5

.

I

I

1.7 W 1GeV)

.

1.9

Fig. 32. Differential cross sections for it + production by virtual photons for O~= 0~as a function of invariant mass W for different values of Q2. The figure is from ref. [Fo-83]. The data are from NINA and DESY.

P. Stoler, Baryon form factors at high

I’d Cd

I’d OII’1

9~p ~~ /~ / ~

Q2

159

_______

,0

£

:~

BOflfl 83

~

/

(1520)

. DESY79 o02_o 1

2

3

C_________

~2 (GeV/c) Fig. 33. Helicity asymmetry as a function of Q2 for the D 13(1520) and the F15(1680). The figure is from ref. [Bu-89]. The curves correspond to non-relativistic CQM calculations referenced in ref. [Bu-89].

It is incorrect to ascribe the crossovers in fig. 33, and the subsequent dominance of the 2 this behavior is

A112 amplitudeofastheannon-relativistic indication of CQM the onset of PQCD processes. At low a consequence [Co-69a,b]. In the SHO model theQD

13(1520) is in the 70 SU(6) multiplet with lhw excitation and L = 1, while the F15(1680) is in the 56~multiplet with N = 2hw excitation and L = 2. Taking a simple non-relativistic interaction Hamiltonian containing spin and orbital current contributions, u• B and p A, for both the D13(1520) and F15(1680), the transitions can be written as a sum of spin-flip and orbital-flip matrix elements. The matrix elements can be evaluated analytically using SHO wave functions [Co-69b]. For the proton target this yields for the D13(1520) 2/~2). (6.12) (A112/A312) = (1/%,,/~)(1 gq The gq2 term comes from the spin-flip matrix element, where q2 is the three-momentum squared. The quantity g is the constituent quark gyromagnetic ratio defined by j~= eg/2mq. For a constituent quark with mq 0.330 GeV, and magnetic moment equal to that of the proton, 0.13 GeV1, one obtains g 1. The ratio A 2 = ~2/g. For the F 112/A312 vanishes if q 15(1680) one obtains —

—

2/2x2) (A112/A312) =

—

(1/~.,/~)(1 gq —

.

(6.13)

160

Q2

P. Stoler, Baryon form factors at high

2

I

T~

I

I

I

D 13(1520) 1

L

—

3/2

=

•

—1.25 —1.00

—

—0.75

H

—0.50

I

I

I

I

I

I

[I

—~ I

I

I

I

I

•I

,I~.

~

1.5

i—I:

—

D13(1520)

QQo

.

j

I

I

—0.25

0

I

~

I

L’

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

—

I

I

I

X.,.9=1/2’ —

F16(1680)

s11(1535) -

1

I

0

I

1111

—1

0

è

F:

.

I

III II 1 2 Q’ (GeV/c)’ I

0.0

~1

~I_ .

—1.00

I I I I

I

I I

I I

I I

I I

I I

I I

I

I

I

I I

I I

I I

i0:50i~l —0.75-

Fig. 34. Helicity 3/2 form-factors, as defined in eqs. (2.9) and (2.16), for the D 13(1 520) and F1, (1680), derived from data cornpiled by Burkert [Bu-91]. The figure is from ref. [St-91b].

I I.

—

F15(1680)

III

3

I

-~

025 II

—

2 2 (GeV2/c2)

0.00

I

I

I

I

I

I

I

I

I I

I

Q

Fig. 35. Helicity 1/2 form-factors, as defined in section 2, for the D13(1520), S1 1(1535) and F1 ~(1680), derived from data compiled by Burkert [Bu-91]. The figure is from ref. {St-9lb].

2 = 2cs2/g. At Q2 = 0 both eqs. (6.12) and (6.13) can be satisfied if This q2(F also vanishes 2(D~ if q 2/c2 and 3),2/c2. whichWith experimentally is approximately i.e., q(F15) = 1.035 an GeVoscillator q(D 15) 2q these experimental values true, one obtains [Co-69b] 13) = 0.762 parameter ~2 = GeY 0.17 GeV2, a value consistent with that obtained from fits to other quantities such as photoproduction couplings. For electroproduction the cancellation is quickly destroyed [Cl-72], and the ratio A 112/A312 2 since the magnitude of the three-momentum q2 increases with varies dramatically with Q increasing Q2. The spin-flip contribution, and hence A 112 is predicted in the SHO model to 2 0.5 GeV2/c2. dominate already around Q In summary, the rate of decrease of the helicity 3/2 form-factors for the D 13 (1520) and ~ (1680) resonances, as shown in fig. 34, is already a manifestation of the non-relativistic CQM, and should not be construed as generallyfor indicative a transition to GeV PQCD. 2 the form-factor the peakofnear W = 1.680 shown in fig. 19 is consistent with At high Q the predicted Q behavior. However, the errors are large, and it is not clear how many resonances contribute strongly to this peak, since there are at least five with masses in this region [PDG-92]. —

6.7. Duality Figures 2 and 18 show that the resonance cross sections decrease at a greater rate with Q2 than the cross sections in the non-resonant deep inelastic scattering region, but in the resonance region they decrease at about the same rate. This was noted by Bloom and Gilman [Bl-70, 71] soon after

P. Stoler, Baryon form factors at high

Q2

161

the first SLAC inclusive electron scattering data were obtained. Another observation they made was, that if the structure functions were plotted as a function of variable w’ 1 + W2/Q2, rather than the usual scaling variable w, scaling appears to occur down into the resonance region if one averages over the resonance bumps. The similar behavior of the resonances and background with Q2, known as the Bloom—Gilman duality, was shown by Carlson and Mukhopadhyay [Ca-91] to follow naturally from PQCD scaling rules. Their development is as follows. According to QCD in lowest order perturbation theory near w’ 1, i.e. W2 4 Q2, the behavior of electron scattering from a proton was determined [Br-73, 75, Fa-75] to follow —p

‘—‘(1— w’)3

(6.14)

.

At the peak of a resonance the structure functions can be written in terms of the leading helicity conserving matrix element G±(eq. 2.1) as follows: vW2

=

+

MN(

(6.15)

2/Q2)G±.

2 PQCD scaling Equation (6.15) obtained by combining eqs. (2.14) (2.17).where The high-Q behavior of the isleading helicity matrix elements is Gthrough + = g + /Q3, g + is a constant up to a log(Q2). Using the relationship 1/Q~= (w’ 1)/ WR, eq. (6.15) becomes —

2M~

vW

2=(w

,

—1)

which has the same w’ dependence as eq. (6.14) above. Bloom and Gilman [Bl-71] obtained a good global fit to the then available deep inelastic scattering data with the following polynomial: 3 + 2.1978(1 1/w’)4 2.5954(1 1/co’)~ (6.16) vW2(w’) = 0.557(1 1/w’) In fig. 18 the dashed curves are plots of eq. (6.16) extended down into the resonance region. It is seen that eq. (6.16) works rather well in the resonance region and at higher Q2 than was originally fit in ref. [Bl-71]. The relationship between eq. (6.16) and the actual structure functions in the resonance region is formally expressed by a finite energy sum rule [Bl-70, 71], —

—

—

.

—

Vm

—~

j’dvvW

2) =

2(v~Q

J

dw’ vW 2(w’),

(6.17)

where v W2 on the l.h.s. of eq. (6.17) is the actually observed structure function, and that on the r.h.s. is the extrapolation of the structure functions fit from the scaling region in the region of integration, eq. (6.16). Figure 36, from ref. [Bl-7l], shows the result of applying eq. (6.16) to inelastic electron scattering data, a function the limits of the integration invariant 2/c2.asThe sum ruleofappears to be remarkably wellinsatisfied formass Q2 asWM, lowfor as = 1, 2, and 3 GeV 2 GeV2/c2. e.g.

162

P. Stoler, Baryonform factors at high 0.6 I

0.4

I

I

Q2 I

q2’I.O

.

-

~ ~ —0.4

H

-

-0.6 -0.8

-

0.6

H

I

I

q2~2O

0.4 0.2 0

-

0.4

-

0.2

-

q2’3.0

0.8.0

I

1.2

1.4

1.6

1.8

a.o

2.2

Wm (0eV) Fig. 36. The difference in the right and left hand sides of eq. (6.17), divided by the right hand side, as a function ofthe cutoff, Wm, for = 1, 2, and 3 GeV2/c2. The figure is from ref. [Bl.71].

The sum rule may also be expressed in a local version, with the limits of integration in eq. (6.17) taken over finite intervals covering the region of particular resonances. A test of finite interval duality has been carried out recently by Carlson and Mukhopadhyay [Ca-92] for Q2 up to 20 GeV2/c2, from the same data used by Stoler [St-91a, b] in constructing figs. 18, 19 and 20. They have considered the three intervals corresponding to the first, second and third resonance regions. Their result, shown in fig. 37, indicates that indeed the sum of resonance plus background exhibits the same Q2 behavior as the resonances themselves, and that in the regions of the second and third resonances both appear to obey the PQCD scaling rules above Q2 = 5 GeV2/c2.

7. Diquark models The CQM cannot account for observed form-factors for Q2 above about 2 GeV2/c2, even within a relativistically invariant framework (see section 3). This is due to the exponential falloff with Q2 of the single particle wave functions in a confining potential. The PQCD formalism discussed in

P. Stoler, Baryon form factors at high

10

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I I

I

I

Q2 I

163

I

II

I

I

+ 1~(1232)

8

—

—

DN’(1535)

°N’(1680) 6

—

4

—

—

a. ,/

P—t3------B

—

~.—B

4 24

0

0

I

—-~

I

I

I

I

5

I

I

I

I

I

10 Q2 (0eV2)

I

I

I

15

I

I

I

I

I

20

-

I

I

I

I

25

Fig. 37. The integral of vW 2 data in the regions of the first, second and third resonances, divided by the integral of the scaling region curve extrapolated down to the resonance region, over the same values ofthe scaling variable x. The dashed lines connecting the data points 2, andarethat merely the non-resonant to guide the eye. background The figurewhich suggests underlies the validity the second of the Bloom—Gilman and third resonances sum rule follows up tothe the PQCD highest scaling measured rules values above of Q about 5 GeV2/c2. From ref. [Ca-92].

sections 4 to 6 is expected to be valid only at high momentum transfer since it requires all internal quark—gluon couplings ;(k2) to be small. At such high Q2 the basic assumption is that only the lowest order Feynman diagrams contribute. There is considerable controversy as to how high a is required to insure these conditions. In addition, a basic ingredient of factorization, i.e. the assumption of helicity conservation on the quark level and helicity conservation on the baryon level, precludes the calculation of helicity non-conserving amplitudes, which are known to be important and even dominant at low and intermediate Q2. The diquark model [An-87, Dz-90, Kr-91a, b, 92] is a semi-phenomenological approach motivated by the large asymmetries obtained in sum rule nucleon distribution function calculations. Since helicity conservation is not an intrinsic part of the model, it can be used in a non-perturbative domain and as a bridge between the CQM and PQCD regimes. The model has been applied to the calculation of elastic and resonance form-factors. The diquark model assumes that the baryon distribution function can be expressed in terms of two constituents, a quark and a diquark which consists of a correlated quark pair. The correlated pair supplies the momentum components necessary to maintain a large form-factor at high Q, and provides a large asymmetry to the momentum fraction distribution function. The quark and diquark are given constituent masses Mq and MD, respectively. The two-body system is treated perturbatively by calculation of the Feynman diagrams in fig. 38. Form-factors are introduced into the couplings between the photon or gluon with the diquark to account for the internal nonperturbative structure of the diquark. The form-factors are chosen so that in the high-Q2 limit the diquark model becomes equivalent to the three-quark PQCD model. The advantages of the diquark model are (1) the perturbative part of the calculation involves only two bodies; (2) non-perturbative effects are phenomenologically taken into account; (3) the quarks constituting the diquarks can couple to both spin 1 and 0 allowing for helicity flips to occur. Thus the calculation also includes helicity non-conserving as well as helicity conserving amplitudes, and the theory therefore is applicable in the intermediate-Q2 range.

164

P. Stoler, Baryonformfactors at high

1)1

____________

Q2

1~’

F

Fig. 38. Leading order diagrams in the diquark model [Kr-91a].

Its major disadvantage is that it is highly phenomenological, and much of the interesting physics is parametrized into constituent quark masses and diquark form-factors. In the diquark model the transition amplitude is obtained from leading order Feynman diagrams in terms of the relative momentum fraction x between the quark and diquark. In addition there are form-factors inserted at the ‘yDD and GDD vertices such that asymptotically in Q2 the overall form-factor evolves into the pure three-valence-quark PQCD result. Denoting the S = 1, I = 1 coupling as vector and the S = 0, I = 0 coupling as scalar, these form-factors are written F~31(Q2)= ~(Q

2)=

2)(Qs)

(7.1)

~(Q2)(Q~1)

F~(Q

For the ‘yGDD vertices one has

F~4~(Q2) = asFs(Q2),

F~(Q2)=

(7.2)

av~(Q2~Q2).

In eqs. (7.1) and (7.2) Q~and Q~are scalar and vector scale parameters and as and ay are strength parameters which take into account that the diquarks may be far off-shell and may undergo excitation and breakup. 5(Q) is a parameter which accounts for the Q2 dependence of as. The baryon distribution function is written as a sum of two terms, corresponding to scalar (5) and vector (V) diquarks. In the notation of Kroll et al. [Kr-91a, b, 92] IB> =fv~(x)>C~’Iq~, V~>+fs~s8(x)~C~’Iq 1,S~>

(7.3)

.

t~are Clebsch—Gordan In eq. (7.3) the indices i andj denote flavor, color and spin helicity. The C coefficients. The coefficients f~and f~are normalization factors determined by the probability that the baryon is in a Fock state consisting of a qD 5 or qD~pair. The distribution amplitudes for a nucleon 4~(x)describing the q—D relative momentum fraction is phenomenologically written in a light-cone harmonic oscillator basis, 3 exp{—b2[(m~/x) + m?~/(1 x)]}. (7.4) ~x) = ~x) = 4~~(x) = Ax(1 x) For a resonance —

—

2) 4~B(x)=

.

B4H0(l + c1x + c2x

The parameters c 1 and c2 are phenomenological.

(7.5)

P. Stoler, Baryon form factors at high

Q2

165

Figure 39 shows the fit to the proton magnetic form-factor GM. The following parameters were used: Q~= 3.22 GeV2

Q~=

,

as=av=O.286,

1.58 GeV2

,

=

66.1 MeV,

f~= 120.2 MeV,

lCvl.16,

where ic~is an anomalous chromomagnetic moment of V diquarks. Note that at low and moderate Q2(~7GeV2/c2) the helicity non-conserving form-factor F 2 makes an important contribution to GM, although its ~ 6 behavior reduces its contribution asymptotically. 4G~(Q2),which is For the transition, fig. 40Q4GT(Q2) shows theby results of the calculation Q related to Pthe A(1232) transverse form-factor eq. (2.11), under threeofdifferent theoretical assumptions for the distribution amplitudes. The dashed curve assumes = = 4HO• Although the magnitude is about right, the decrease with Q2 is not reproduced. It was proposed in refs. [St-91a, b] that the decrease in the form-factor with Q2 might be due to a significant helicity flip component which remains dominant over a larger Q2 range for the A than for the other large resonances. This would be consistent with the calculation by Carlson and Poor [Ca-88], which gives for the A an abnormally small helicity conserving form-factor at high Q2 [see section 6.2 and eq. (6.9)]. If that were in fact the explanation, then this would be additional evidence favoring the C-Z and K-Z proton distribution amplitudes over the G-S amplitude, since the G-S amplitude does not predict a small helicity conserving form-factor. With this in mind Kroll et al. [Kr-92] have been able to reproduce the Q2 dependence of the data using 4~as in eq. (7.5). The solid curve in fig. 40 is the result. The distribution amplitude which gives the best fit to the data is —+

çb~(x)= BA4HO(l + 5.15x + 5.45x2) 1.4

I

(7.6)

.

I

I

I

I

I

I

I

25

30

çO.8 *

0.4

0

* Hoehier A Arnold

0.2 0

0

5

I

I

10

15 20 Q2 [GeV2]

35

Fig. 39. The proton elastic magnetic form-factor Q4 GM(Q2) versus Q2. The black triangles are data from ref. [Ar-86], the opentriangles from ref. [Ho-76]. The solid and dashed curves are the result of the diquark calculation with two different model distribution amplitudes. The figure is from ret [Kr.91b].

166

P. Stoler, Baryon form factors at high

i.e

Q2

I

Q2 [GeV] Fig. 40. The transverse form-factor for the electroproduction of the A(1232) obtained in the diquark model [Kr-92] using distribution amplitudes corresponding to eqs. (7.5) and (7.6). The ordinate G~is related to the transverse form-factor GT as in eq. (2.11). The curves are the results of using different model A distribution amplitudes. The solid curve is obtained using the distribution amplitude of eq. (7.6), which has a large longitudinal contribution. The dashed curve is obtained using a harmonic oscillator distribution amplitude similar to eq. (7.4), which gives a small longitudinal component. The dot-dashed curve is obtained using a model distribution amplitude which gives a small longitudinal contribution to the form-factor. The values due to the three-quark PQCD calculations using distribution amplitudes G-S, K-S, and C-Z are also shown.

In eq. (7.6) the quantity B~is an overall normalizing constant. The distribution amplitude given by eq. (7.6) has a large helicity flip component, and contains a rather large asymmetry, in contradiction to the sum rule result of ref. [Ca-88]. However, Kroll et al. [Kr-92] point out that this distribution amplitude is still within the uncertainty of the sum rule moments. A more serious problem is that the j.i’~(x)of eq. (7.6) yields a large longitudinal component to the form-factor (also shown in fig. 40), which is in strong disagreement with all available data [Ba-76, Dr-81]. A recent analysis of available data by Burkert [Bu-92] yields S 2 = + 0.05 0.03 ±0.05 (statistical and inclusive systematic errors, Q 3 1+/M1+ GeV2/c2.= Also, the±analysis of the experimental cross sectionrespectively) agrees with at exclusive measurements at low Q2 under the assumption of zero longitudinal contribution to the resonance. One may conclude that the anomalous behavior of the A form-factor remains unresolved. Kroll et al. [Kr-92] have also calculated the S~1(1535) transverse form-factor, the results of which are shown in fig. 41, compared with the analysis of the experimental inclusive cross sections. A reasonable fit is obtained assuming 4~S11 4~,although the best fit is obtained for the following distribution amplitude: 4~S11(x)

=

Bs~Ho(1 6.68x + 6.53x2) —

.

(7.7)

This distribution amplitude is not inconsistent with the sum rule amplitude obtained in ref. [Ca-88], and it appears that the situation for the S~~(1535) is in better shape than that of the A(1232).

P. Stoler, Baryon form factors at high I

I

I

Q2

167 I

~2 [Gev21 Fig. 41. As in fig. 40, but for the S 11(l535). The dashed curve is the result of using a proton-like harmonic oscillator distribution function similar to eq. (7.4). The solid curve is the result of using the S~~(1535) as in eq. (7.6).

8. Summary and outlook The constituent an appropriate forstrong treatingquark—gluon electromagnetic of 2 lessquark than model a few is GeV2/c2. At low basis Q2 the and properties gluon—gluon baryons at Q couplings render baryon wave functions too complex to treat in terms of the elementary QCD constituents, although progress is being made in lattice calculations. But somehow the great complexity leads to a relatively simple parameterization of nucleon structure in terms of massive constituent quarks moving in one-body effective potentials, with additional perturbations. There has been considerable success of the CQM in describing static and low-Q2 phenomena of nucleons. However, due to the complicated overlay of nucleon resonances, the data necessary to characterize individual excitations has been limited to the few strongest resonances. The advent of new powerful accelerator and detection facilities, especially CEBAF, will make possible a careful and systematic study of resonances. A large experimental program is already planned, which has been paralleled by new theoretical activity, and an overall renewal of interest in the field, as exemplified by recent conferences in Troy (1988) [Ad-89] and New Haven (1992) [Ga-92]. At Q2 greater than a few GeV2/c2 the constituent quark model becomes less effective in describing baryon form-factors, in that the simplifications of the CQM begin to break down. Although the structure involved in these transitions becomes on one hand simpler, the problem becomes much more difficult since it is no longer possible to parametrize the complexity in terms of a simple constituent quark basis, and this region of Q2 is difficult to treat theoretically. Up to now there has been very little experimental investigation of this kinematic regime. However, a major CEBAF program is planned for the region Q2 > 3 GeV2/c2. At high Q2 the reaction mechanism continues to become simpler such that in the asymptotic limit the reaction is predicted to select the simplest Fock state, which consists of three current

168

P. Stoler, Baryonforinfi~ctorsat high

Q2

quarks exchanging two gluons of high enough momentum so that the elementary QCD interactions are small, and the problem is treatable perturbatively. An important difficulty then lies in obtaining the quark distribution amplitudes, which are determined mainly by low-k2 non-perturbative physics. A useful technique has been the QCD sum rules, in which the moments of the non-perturbative interaction of the three-quark Fock state with the physical vacuum are determined phenomenologically from other experimental data. This has led to various theoretical predictions about elastic and inelastic resonance form-factors. The results of the sum rule calculations are controversial as well as contradictory, depending on specific ad-hoc assumptions. In principle the predictions are experimentally testable. In practice nearly no data exist at high Q2. An exception is the elastic G~,which has been measured up to a Q2 of about 30 GeV2/c2. In addition, there has been an analysis of the Q2 dependence of form-factors from inclusive data in the resonance region. The indication is that these form-factors begin to approach the Q ~dependence predicted by PQCD at Q2 as low as about 6 GeV2/c2. An exception is the A(1232), whose form factor decreases at a greater rate up to about 10 GeV2/c2, beyond which there are no data. It must be kept in mind though, that, except for the A(1232), the resonance form-factors have been extracted from inclusive (e,e’) data, and therefore do not unambiguously separate individual states. The analysis of the inclusive data also appear to confirm the concept of duality for resonances plus non-resonant structure functions out to a Q2 greater than 10 GeV2/c2 [Ca-92]. The magnitudes of the form-factors are very controversial. For example, the magnitude of G~can only be reproduced with rather extreme properties of the proton’s distribution amplitude, such that most of the form-factor comes from a region of momentum fraction very near the kinematic limits, where coupling constants are large, and it is difficult to justify PQCD. A central question is how low in Q2 do perturbative contributions dominate the formfactors for exclusive reactions? Recent calculations [Li-92b, c] indicate that radiative corrections strongly suppress the non-perturbative contributions (Sudakov suppression) so that perturbative contributions dominate the nucleon form-factor at experimentally accessible values of Q2, providing optimism that the transition from the non-perturbative to perturbative regimes can be studied with currently planned accelerator facilities. Another positive note is that the concept of duality between resonances and background also appears to be valid at modest values of Q2. Experimentally, some primary obstacles against measuring form-factors at high Q2 are: (1) due to the difficulty in separating individual resonances which are highly overlapping, a large amount of coincidence data will be required; (2) the cross sections are falling at least as Q so that very high luminosities will have to be attainable as well as measurable; (3) measurements will require accelerator facilities with high energy, and high duty factor, as well as a new generation of particle detectors, which will be very expensive. The prognosis is that the region of Q2 accessible to CEBAF will continuously increase as the energy of the accelerator is upgraded. This will enable us to examine the electromagnetic structure of baryons in the transition region between constituent quark models and PQCD. However, to investigate the validity of PQCD at Q2 near and beyond about 15 GeV2/c2 it will be necessary to utilize a new generation, higher energy facility, similar to the proposed European Electron Facility (EEF). Given these possibilities, the future looks bright for investigating the transition from the realm of constituent quarks to PQCD in exclusive reactions. ~,

P. Stoler, Baryon form factors at high

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169

Acknowledgements The author would like to thank Volker Burkert, Carl E. Carlson, and Nimai C. Mukhopadhyay for reading portions of the manuscript and making valuable comments and suggestions. This work was supported in part by the National Science Foundation under Grant No. PHY8905162.

References [Ab-72] [Ad-89] [Al.68] [An-52] [An-87] [Ar-86] [Az.80] [Ba-73] [Ba-76] [Ba-83a] [Ba-83b] [Be-69] [Be-91a] [Be-9lb] [Bl-70] [Bl-71] [Bo-79] [Bo-86] [Bo.87] [Bo-93] [Br-Si] [Br-73] [Br-75] [Br-76] [Br-78] [Br-79] [Br-80] [Br-81a] [Br-81b] [Br-84] [Br-89a] [Br-89b] [Bu-68] [Bu-89] [Bu-91] [Bu-92] [Ca-86] [Ca-87a] [Ca-87b] [Ca-88]

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