THE SPIN STRUCTURE OF THE NUCLEON

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THE SPIN STRUCTURE OF THE NUCLEON

R. Jaffe

To cite this version:

R. Jaffe. THE SPIN STRUCTURE OF THE NUCLEON. Journal de Physique Colloques, 1990, 51

(C6), pp.C6-149-C6-161. �10.1051/jphyscol:1990612�. �jpa-00230875�

COLLOQUE DE PHYSIQUE

Colloque C6, supplkment au n022, Tome 51, 15 novembre 1990

R . L . JAFFE

Center for Theoretical Physics and Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge. Massachusetts 02139, U.S.A.

I have been asked to give an overview of the spin structure of the nucleon as probed by deep inelastic scattering of leptons. The modern era in this field dates from the surprising results reported by the European Muon Collaborationin 1987./1/ The subject has attracted lots of attention and my overview will, of necessity, be both skimpy and selective. I will only discuss deep inelastic scattering. I have resisted the urge to spend my entire allotted time on the "spin-puzzle" engendered by the EMC data. For a more complete review of this subject, see Ref. [2].

My talk wilI be organized as follows:

1. Introduction and Iiinematics.

2. gl(x, Q'): The Spin and Strangeness Content of the Nucleon.

Axial charges and the data.

Quick fixes.

The U(1)* Ward identity.

3. g2(x, 92): The Nucleon's Transverse Spin Structure Function g2 and twist-3.

The Burkhardt-Cottingham sum rule.

4. Higher Multipole Structure Functions.

bl (X, Q2 ).

A(x,

Q".

5. Conclusion

Since this conference is predominantly concerned with nuclear effects and low energies, I will try to emphasize some of those aspects of the subject.

1 - INTRODUCTION AND KINEMATICS

Inclusive deep inelastic lepton scattering has been an industry in particle physics for over two decades.

There are two good reasons for this: first, it provides clear, detailed and quantitative tests of QCD; second, it probes the quark and gluon substructure of hadrons with well-defined operators. Almost everything we know precisely about the quark and gluon content of the nucleon comes from these experiments.

"'

The kinematics of deep inelastic scattering are defined in Fig. 1. with q2 E q,qP E -Q2 < 0 and P . q ^{v .} Mostly we will be interested in the Bjorlien limit Q2, v -+ co with X Q2/2v and y

=

v/ME bounded by 0

<

X , y

5

1. Finally, its useful to define a measure,

of the approach to the Bjorken limit. For a spin-l/', target, its spin orientation is completely specified by the spin (Pauli-Lubanski) vector S' E ~ u y f l y s u obeying s

.

P = 0 and normalized to s2 = -M2. Later we will mention some additional features which arise for higher spin ( J

1

1) targets. For now we stick to J = 112.

his

work is supported in part by funds provided by the U.S Department of Energy (D.O.E.) under coyifact #DE-AC02-76ER03069.

There are a few notable exceptions: notably the nmgnetic moment, charge radius and axial charges measured by electroweak currents; the C-t,erm ineasurecl in low-energy nN-scattering.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990612

We also stick to electromagnetic scattering. Neutrino scattering may be similarly described, but my focus is spin physics and neutrino scattering from polarized targets is not (yet) feasible.

All of the information about the hadronic system is contained in the structure function, W,,, obtained by squaring the amplitude in Fig. l a ,

W,v(q,P,s) = - 1

C

^{(P, s}

^I

^{J,(O)I X ) ( X}

^I

JU(0)I p , S) (2r)4S4(p

+

^{q - X )}

4iT X

(1.2)

l

= - J

4~ d 4 t eiq'< (P, S

l

JP([) Jw(0)I P , S ) (1.3)

where, by the optical theorem, T,,(q, P, S ) is the amplitude for forward, virtual (q2 = -Q2

<

0) Compton scattering from the target. Current conservation, along with parity and time reversal invariance allows us to decompose W,, into four independent tensor structures for a spin-112 target like the nucleon

ie,,,pqa

+ ---

(P. *so

-

^S

.

qpP) gz(z, Q ~ )

.

u2

The Lorentz invariant structure functions Fl Fz, g1 and g2 are the objects of interest. Fl and Fz may be measured off unpolarized targets, whereas extraction of g1 and g2 requires polarization of both beam and target. Dimensional arguments based on the asymptotic (approximate) scale invariance of QCD predict that all these structure functions become functions of X , modulo logarithms of Q', in the Bjorken limit.

Fig. 1: a) Deep inelastic kinematics; b) Helicity amplitudes for forward, virtual and Compton scattering.

To get a feeling for the significance of the structure functions it's useful t o relate them to the helicity amplitudes for forward, virtual Compton scattering.131 Labeling helicities as shown in Fig. l b , I denote the helicity amplitudes by A h , , ~ ~ , h , ~ ~ . Since forward Compton scattering occurs along a line, the sum of initial and final helicities must be the same

h ~ + H i = h z + H z

.

(1.6)

Together with parity and time reversal invariance, Eq(1.6) leaves only four independent helicity amplitudes,

A + l , o ~

where

+,

^0,

-

labels the photon helicity and f,

1

labels the target helicity. -411 other helicity amplitudes are simply related by symmetries t o the set given in (1.7). The structure functions FI,z and gl,z are proportional to the imaginary parts of the helicity amplitudes. A short calculation gives

suppressing the "Im". So Fl measures the spin average coupling of transverse photons, g1 measures the spin asymmetry for transverse protons (remember n -+ 1 as

Q2

--+ cm);

$F2

-

Fl

measures the coupling longitudinal photons, which is small in the Bjorken limit. g2 is rather special: it cannot be measured (at large

Q2)

if the target is in a helicity eigenstate. The optimal way to measure g2 requires an equal admixture of

1

and

L,

i.e. transverse polarization for the target.

I

l

Renormalization

I

I +

group improved

1

perturbative QCD

I

'.

^I

Non- perturbative

Fig. 2: Separation of virtual forward Compton amplitude into vertex associated perturbative QCD eBects and targets associated non-pert~~rbative egects.

Loosely speaking, the

Q2

and X dependence of the deep inelastic structure functions test &CD and probe hadronic targets, respectively. This is illustrated in the cartoon of Fig. 2. Precise measurements of the

Q2

dependence of

Fz(x, Q ' )

has provided elegant and successful tests of QCD./4/ The X-dependence probes quark and gluon correlation functions in the target. These may be viewed either as "parton distributions"

for quarks and gluons in an infinite momentum frame, or equivalently/5/ (after taking moments in X ) as measures of local, gauge invariant operators of definite flavor, spin and "twist." Although these cannot yet be predicted from first principles, they are bench marlis for model builders, lattice enthusiasts - anyone who hopes t o understand non-perturbative aspects of hadron structure.

The structure function g l ( x ,

Q2)

was first measured over a linlited range of X and

Q2

in the 1970's in a heroic experiment by a SLAC-Yale collaboration./6/ More ~ r e c i s e data became available from the EMC in

1987./1/ The world's data on gl(x,

Q2)

is shown in Fig. 3.

0.8 0.6

CI

X

0.4

a. I-

0.2

0

0.01 0.02 0.05 0.1 0.2 0.5 1.0 X

Fig. 3: The X-dependence (integrated over

Q2)

of g 1 ( 5 ,

Q2)

from Ref. [l].

- + ^EMC

O SLAC

O SLAC ,

- -

-

-

I---:

- - - -

I I

-

I I l

!--:---~--l~l-\ ^.$,$#

- - - -

l

_{- - - - - - - - - - -} 'X.&-

The structure function gz(x, Q2) has never been measured. New experiments planned by the HERMES Collaboration a t HERA and the Spin Muon Collaboration (SMC) at CERN intend to measure g2(x, Q').

2 - gl(x,Q2): THE SPIN AND STRANGENESS CONTENT OF THE NUCLEON

A. - Axial Charges and the EMC Data

The structure function gl(x, Q2) measures spin dependent quark distributions, qt(x, Q'), where a runs over flavors a = U, ii, d,

4

^{S, 3,}

. . .,

and S

=T

or

1.

qt(x, Q2) is the probability to find a quark of flavor a, helicity s and momentum fraction X in the target's infinite momentum frame when probed with a photon of virtuality -Q2. The analogous distributions for gluons gt(z, Q2) and gL(z, Q2) enter gl(x, Q2) indirectly as we will soon see Ref. [2].

Most attention has focussed on the 2-integral of gl(z,Q2). Long ago, Bjorken/7/ showed that

J:

dxgl(x, Q2) a t large Q2 is proportional to a sum of quark axial charges of different flavors weighted by the squared electric charge. The sum rule survives in QCD, with calculable radiative corrections. Other flavor combinations of axial charges are measured in hyperon and neutron P-decay. Using SU(3)f symmetry, the /?-decay data and the EMC values of gl(z, q2), one can separate the U-, d- and S-quark axial charges./8/

With the definition,

for U , d, S . .

.

^; is the renormalization point which is to be identified with the mean-Q2 (Q:) in the EMC experiment. The EMC obtain

EMC CANONICAL EXPECTATION

In the table above we present the values of Au, Ad and A s obtained from hyperon and neutron &decay plus the assumption A s = 0,/8/ which was the expectation before the experiment was done. The errors on the second column reflect the uncertainty in the SU(3)f rotation of /?-decay data as well as the errors on the data themselves. There has been a lot of discussion of the accuracy of both the experimental values of Au, Ad and As, and of the theoretical estimate. The reader should consult Ref. [g] for further discussion.

Theory and experiment disagree by several standard deviations. It is somewhat surprising to find polarized strange quarks in the nucleon, however the problem taltes on a new gravity when one constructs the sum:

EMC CANONICAL EXPECTATION

c(Q:) = 0.120 f 0.094 f 0.138 CO E 0.60 f 0.12 (2.3) where

C = A u + A d + A s

.

C can be shown to be the contribution of the U-, d- and S-quark's spin to the spin of the nucleon/2/ and the data show that this is compatible with zero! This result has raised a lot of eyebrows, but there are many caveats one should consider before worrying too deeply about it. For a more complete discussion of these (and other) issues, see Ref. [2]. Here is a short list:

Remember, quarks can have orbital angular lnomentun~ and gluons ca.11 contribute both spin and orbital angular momentum to the nucleon's spin.

After all, gluons carry 50% of the nucleon's "momentum" in an infinite illonlenturn frame.

Any quark model consistent with B-decay data and A s = 0 gives S ^g 0.60*0.12. In such models, there- fore, quark spin does not account colnpletely for the nucleon spin. There must be other contributions to the nucleon's spin - usually orbit.al angular lnomentuin found on the lower (Dirac) components of the quark wavefunctions./2,10/

(a) (b) (c)

Fig.

4:

Gluonic contribution to dxgl(x,Q". a) For two jet production; b ) In the Cornpton amplitude; c) After projecting out the first moment and the operator Gypy5q.

C is renormalization scale-dependent. This comes about in two loop order through the triangle anomaly./ll/ In principle, the value of C measured by EMC at large Q: can differ from the value estimated in quark models (at some unknown but small mass scale)./l2/ C depends only weakly on Q2 at large Q2 SO it would be awkward t o reconcile theory and experiment in this manner (see also Ref. [13]).

Higher twist effects (unknown 0 ( 1 / Q 2 ) corrections to the sum rule) could contaminate the EMC data,/l4/ though there are theoretical and experimental arguments against this./l5/

Last but not least, the data may be wrong, and should be confirmed. This is a major objective of the new CERN (SMC) and HERA (HERMES) experiments.

B. - Quick Fixes

After the publication of the EMC data several conceptually simple and relatively painless explanations of the EMC result have been proposed. Although they maybe part of the solution, I don't believe any of these "quick fixes" do the job alone./2/ Here I will briefly discuss the two attempts which have attracted the most attention.

eSkyrme model:

Brodsky, Ellis and Karliner/lG/ showed that C l/Nc in a11 SU(3)f-symmetric Skyrme model. Later Ryzak/l7/ estimated the O(l/N,) correction and found C 2 0.2 f 0.1. Recently, hoivever, Icaplan and Klebanov/l8/ showed that the model also predicts (iiu

+

^&l

+

^3s)

-

l / N c disagreement with data on the a N

-

C term and baryon masses./l9/ Also, one must be suspicious of predictions of the Skyrme model in the axial singlet sector since it requires 1/NC. To measure the success of this prediction compare &Iq, with m p , which is O(Nc) in the model, so M V , / m , 1/N: but in Nature hf,,~/nz, E 1.02 (!)

*Anomalous gluon contribution:

Efremov and Teyaev/20/ pointed out that there is an anomalous gluon contribution to

So]

gl(x, Q2)dr which is associated with the triangle graph. Their work was corrected and expancled by Altarelli and Ross/21/ and by Carlitz, Collins and Mueller./22/ Since then, it has been the subject of many papers. After considerable controversy, there now appears to be some consensus emerging regarding this subject (although I am far from a disinterested observer). Perhaps the best way to present the issue is via Fig. 3a, which shows the subprocesses y*

+

^{g -+} 2 jets which is one contribution to gl(n., Q'). Reference [22] define the jets by a kl cut-off with respect to the axis defined by the initial virtual photon and gluon. With this definition, the contribution of this graph to

S,

1 ^dz gl(x, Q2) can be computed using standard parton model methods. The result is a contribution

A ~ ( Q ~ ) =

--

N ~ A ~ ( Q Z )

Zn (2.5)

where AqO (Q2) includes all the rest of the contributions to C. h (2.5), Nf is the number of flavors which can participate in deep inelastic scattering at this Q? Ag(Q2) is the gluon spin asymmetry,

Most workers agree that gluons make a contribution of the form of (2.5) to C. The disagreement centers around the interpretation of Aqo which enters (2.6). Originally the authors of Refs. [20-221 sought to identify Aq0(Q2) as the "canonical quark" expectation for C, i.e. Ay0

r

0.60 0.12. Then, they argued, experiment (E 0) calls for a large value of AI' and Ag to cancel AqO. This led them to predict large spin asymmetries in other hard scattering processes. There is now convincing evidence that this interpretation is not warranted.

Looking back, perhaps the strongest argument in favor of it is that the class of contributions to Aqo generates a conserved piece of the flavor singlet axial current analogous to the conserved non-singlet currents which can be unambiguously identified with the quark flavor differences Au

-

Ad and Au

+

^Ad

-

2As. On the other hand there are many arguments against this interpretation:

The decomposition of C into Aqo and AI' is infrared regularization scheme dependent.1231 AI' can be defined by a specific two jet algorithm but other gluon contributions land in Ago.

For non-zero quark mass there are gluonic contributions arising from other regions of the box graph (Fig. 3b) which end up in Aq0./2/ For heavy quarks (Q2

>>

nz2

>>

A2) these contributions cancel AT to order 1 / r n ~ / ~ , / 2 , 2 4 / so heavy quarks contribute very little to C. If one defines AI' a l6 Refs. [20 -221 and if it's large, then the intrinsic heavy quark contributions, AcO and Abo, must be large, too. This is not a very useful definition of an intrinsic quark contribution.

The X-dependence of the contribution of the box graph to gl(x, Q2) does not look like the EMC data,/25/

so the data bound A r to be too small to account for C 0.

Despite these objections, the work of Ref. [20-221 has stimulated much thought about the consequences of a polarized gluon distribution in the nucleon. Studies of Drell-Yan and single photon production in polarized ppcollisions may lead to experimental determinations of Ag which would settle the theoretical debate once and for a11./26/

C. - U ( l ) A Ward Identity

Some insight into the subtlety of C can be found by studying the Ward identity for the flavor axial current, A:./2,27/ The matrix elements of

are related to the gluonic operators by the triangle anomaly/28/

An analysis exactly like the one that leads to the Goldberger-Treiman relation for massless quarks, leads, for the singlet axial current, to/2/

E = n(0) , (2.10)

where ~ ( t ) is the form factor for an insertion of GG at nlomentum transfer L,, with k,L' r t:

At first sight (2.10) looks very peculiar: C, which measures the quark spin contribution to the nucleon's spin, is identified with the matrix element of a gluon operat,or. However, a similar relation has been known for many years in QCD:

2M2 = (P IT",lP) = (2.12)

(2.12) follows from the "trace anomaly" in the QCD stress tensor T"". At the classical level, massless QCD is scale invariant so the scale current (d" is conserved,

a#cd"

⁼ 0, and consequently T'" is traceless./29/

Renormalization breaks scale invariance and gives T"" a trace proportional to the QCD p-function. (2.12) says that the nucleon's mass comes from glue. Well, that's right, in the sense that if we turn off the coupling to glue the "proton" would consist of three free, massless quarks. Y G 2 is a measure of the gluonic effects which confine the quarks and give mass to the nucleon. Likewise, (2.10) expresses the fact that the nucleon's axial vector charge would be zero if we turned off the QCD coupling constant, since three free, massless quarks with J~ = 1/2+ have vanishing axial charge.

Fig. 5: An insertion of GG carrying momentum .'k

Equation (2.12) makes it clear that it is not easy to distinguish in a precise way between quark and gluon contributions to the axial charge. TheJwo are related by the QCD equations of motion. If we try to evaluate the nucleon matrix element of GG in the spirit of the parton model we run into a conceptual problem which as far as I know, has not been fully resolved. Take the nucleon to be a superposition of quarks and gluons (in Fock space) at infinite momentum. A simple calculation based on Fig. 5 yields

This is the same sort of calculation that expresses the flavor charges as dx (qa(x)

-

?(X)) or the quark

"momentum" as

J:

dx X (p(x)

+ q ( x ) ) .

Combining (2.10) and (2.13) we end up with the surprising result that C is to be identified with -asNfAg/2n. Fritzch/SO/ has proposed that this is correct, and that C is small because a,NfAg/2r is a small quantity. This may be correct, but Forte,/31/ for example, has argued that there are non-perturbative contributions to n(0) associated with topologically non-trivial gauge field configurations which are missed in the parton model forillula (2.13). This is an area where further work is necessary.

3. - gz(x,Q2): THE NUCLEON'S TRANSVERSE SPIN STRUCTURE FUNCTION

Both the HERMES collaboration/32/ at HERA and the SMC/33/ a t CERN have proposed to measure g2(x, Q2). Although g2(x, Q') has not received as much attention as the more accessible structure functions, it has unique properties which make it a particularly interesting object for further study by both theorists and experimenters. Here I'd like to introduce two issues regarding gz: first its unique sensitivity to twist-3 - that is, to quark gluon correlations in hadronic targets; and second, the Burkhardt-Cottingham sum rule which may (or may not) satisfied by gz. More detailed treatment of these subjects can be found in Refs. [34]

and [35].

A. - g2 and twist-3

One of the most important aspects of g2 is how it is measured. As mentioned in the Introduction, a transverse component of the target polarization is required. To be more specific, consider the kinematics defined in Fig. 6. a! is the polar angle made by the target spin relative to the bean1 axis, 6 is the polar scattering angle, cp is the azimuthal angle between the scattering plane and the plane defined by the beam and the target spin. Define cross sections fox right- and left-handed incident leptons:

then

dAu(a) e4 Y y2 Y

-

dx dy dq5 - - -{cosa 4n2Q2 ^[(l

-

: 2 - ⁴⁽⁸ - 1) g1(x,Qz) -

-

l)g2(x,Qz)

l

Fig. 6: Kinematics for polarized deep inelastic lepton scattering in the target rest frame.

Remember that K - 1 = 4x2M2/Q2 is a measure of the approach to the scaling limit and will be small in both HERA and CERN experiments. The dominant term in (3.2) is therefore cosa(1

-

y/2)gl(x, Q2). In general, extracting g2(x, Q2) requires searching for a subdominant Q2 variation. However a = 7r/2 is special:

the "dominant" term vanishes identically; the cross-section falls faster by a power of than at any other a ; but g2(z, Q2) can be extracted from the measured Aa merely by subtracting iygl(x, Q2). As we will soon see, g2(x, Q2) has a twist-3 component which is sensitive to quark-gluon correlations. Similar effects occur at 0(1/Q2) in Fl, Fz and gl, but g2 is the only case where these effects can be measured without first measuring data over a wide range of Q2, fitting it a t large Q2 to the dominant Q2-dependence predicted by a &CD and then subtracting it away to expose the 0(1/Q2) piece of interest.

This feature of gz(x, Q2) was apparently first recognized by Shuryak and Vainshteyn./36/ They showed that in general in QCD, g2 can be written as

The first two terms in (3.3) ( G gYW(x, Q 2 ) ) were originally found by SiVandzura and Wilczek/37/ (who incorrectly argued that gz = 0). The last term g2(x, Q2) contains quark mass effects, and more importantly, quark gluon correlations. This is most clear from the perspective of the operator product expansion, which identifies moments of g2 with specific, local quark gluon operators, for example

(3.4) is only intended symbolically: the Lorentz indices don't match, flavor indices have been ignored, etc.

The details can be found in Refs. [35], [3G] and 1381. The i~nportant point is that g2 directly measures quark gluon correlations (note the factor of the color gauge coupling, g, in (3.4)) in a precise enough way that phenomenologists, lattice &CD people and others should be able to analyze them.

The parton model for g2 is still obscure. It is l i n o ~ m that g2 vanishes only in free field theory1351 and that parton off-shellness and transverse momenta are sources of contributions to g2./39/ However, it is not yet clear whether parton model "intuition" will yield any insight into the magnitude or shape of gz. To learn something about 32, Ji and I calculated it in the old MIT bag mode1./35/ There, quark interactions at the bag boundary mock up the effect of gluons and give rise to both confinement and a non-vanishing 82. The bag model predictions for gl(x) and gz(x) are shown in Fig. 7. The model has some limitations: it violates translation invariance so g2(x) fails to vanish above X = 1;/41/ it lacks Regge behavior near X = 0 because the bag has strictly finite spatial extent;/41/ it presumably applies a t some small renormalization scale and must be evolved to a large Q2 where the experiment will be done./42/ To measure the limitations we compare gl(x) with the data in Fig. 7a. Some of these diseases could perhaps be cured or a t least treated by methods like those applied to Fl, F2 and g1 by Tllolnas and his collaborators./43/

B. - The Burkhardt-Cottingham Sum R.ule

A glance at the definition of g?W, (3.3), shows that

S,'

dz g?"(x, Q ~ ) = 0. A careful look at Fig. 7 shows that dxgZ(x) = 0 in the bag model. These are examples of a sum rule derived many years ago by Burkhardt and Cottingham,/44/

4 1 d x , ( r , a 2 , = o

.

(3.5)

Fig. 7: a ) &(X) in the bag model compared with the EMC data (statistical errors only); b) g [ ( x ) in the bag model, the dotted line is the twist-& contribrction, the dashed line is twist-$, the solid line is the s u m .

The sum rule is of central importance to experimentalists since it const,rains the area under 9 2 . When

~ Z ( X , Q') is measured, the sum rule may well attain nearly the same prominence as the spin sum rule for g1 has enjoyed over the past few years. It is one of a class of suin rules which follow from technical assumptions of Regge theorjr rather than from asymptotic perturbative &CD. Another such sum rule, the Drell, Hearn, Gerasimov/45/ Sum Rule for polarized photoproduction, seems to be obeyed by resonance photoproduction amplitudes./46/

The original derivation of Ref. [44] started from a dispersion relation for the helicity flip Compton amplitude, A2(9', v), whose imaginary part is proportional to gz(x, Q'),

CO clv'

A 2 ( q z , ~ ) = g /

-

Im ,42 (q2, v')

.

T -q2/z v12-v'

Regge asymptotics applied to Compton amplitudes suggest v) -+ 0 as v -+ ca. [Absence of "J = 0 fixed poles."] If the limit is taken under the integral in (3.6) we obtain a "super-convergence" relation

which is the same as (3.5). The sum rule would fail if (a) there is a. J = 0 fixed pole in Az, or (h) Im Az(q2, v) falls too slowly with v to allow the limit under the integral. In case (b) the sum rule fails to converge. In case (a) it converges, but not to zero.

Ioffe, Khoze and Lipatov/47/ have argued that the sum rule does converge when Q2 is large. However, it may not converge to zero. A "modern" derivation of the sum rule illustrates the problem. The operator product expansion leads to a tower of sum rules for g2 of the general form,

(3.4) is an example of one of them. Cn(Q2) is the nucleon matrix element of some operator. If one carelessly set n = 0 in (3.8), that would give the sum rule. Instead one has to argue that it is possible to analytically continue in n to n = 0. The problem is that yz could conta.in something like &(X) which would ruin the analytic continuation [&(X) would not contribute for n = 2,4,6,. . .]. A simple example is given in Ref. [35],

While formally dz g2(x) = 0, the 6(x) is inaccessible to experiment so the integral over the data fails to vanish. Just such a problem is known to ruin the "Schwinger term sum rule" a t lowest non-trivial order in perturbative QCD,/48/ (which is why you're unlikely to have heard of it). On the other hand, Antonaides and I<ounnas/49/ have studied the Burkhardt-Cottingham suill rule to the same order in or, in QCD and find no problem. This proves little, however, because its the small-X behavior of the non-perturbative parton distributions which is the issue here. In summary, we aren't yet sure whether the Burkhardt-Cottingham sum rule is valid in QCD. It is an important subject requiring further work.

IV. - HIGHER MULTIPOLE STRUCTURE FUNCTIONS

During the early days of deep inelastic lepton scattering little specific attention was paid to nuclei.

Apparently no one noticed that nuclei of spin greater or equal to unity have additional deep inelastic structure functions beyond F l , F2, g1 and gz. Frankfurt and Strickman analyzed the first such structure function in the course of their study of the deuteron./50/ Independently, Hoodbhoy, Manohar and I recognized the existence of such structure functions and their generality./3,51/ The case of spin-l is treated in Refs. [3] and [53] and t o some extent in Ref. [52]. The arbitrary spin case is treated in the Bjorken limit in Ref. [51]. In this talk I will restrict myself to two novel structure functions which occur at spin-l and may be measured in the HERMES experiment.

Unfortunately there are no fundamental hadrons with spin greater than 1/2,* so the interest in these structure functions comes entirely from their possible role in nuclear chromodynamics. Since they vanish for spin-112 (and spin-0) targets these structure functions are probes of non-nucleonic (and non-pionic) components of nuclei. They receive small, calculable contributions from nucleon Fermi motion, but any excess beyond this would be a signature of exotic components in the nuclear wavefunction.

A. - bl(x,Q2): The Quadrupole Structure Function

For a spin-l target the list of helicity amplitudes given in (1.7) must be enlarged. The quadrupole combination

A++,++

+

^-4+-,+- - 2A+o,+o K bi(z, Q2) (4.1) scales in QCD and can be identified with helicity-dependent quarlc distributions. [The notation in (4.1) is the same as (1.7), though now the target helicity takes on the values +l, 0 and -l]. The quark distributions in a spin-l target can be denoted q:(x, Q2), where H is the target helicity (we suppress the quarlc flavor indices a). There are three independent distributions (as opposed to two for spin-1/2), q!(x, Q2), qt(x, Q2)

aJld

$(X, Q ~ ) (q; = q l l , qt = q t l and = Q: by parity) and together they determine F l , g1 and bl for a spin- l target:

F1 =

3

1 ^(Q;

+

^4t

+ Q;)

Apparently, bl measures the correlation of the quark's momentum with the target hadron's helicity. bl is not quark-spin dependent (q? and q? appear with equal weight) and not surprisingly, it can be measured with an unpolarized lepton beam by taking appropriate coinbinations of target helicities. bl is interesting in its own right. Moreover, as pointed out in the HERMES proposal,/32/ g1 for the neutron cannot be measured from a deuterium target unless b1 is also measured or is negligibly small.

*

This isn't quite true: the photon's structure function can be nleasured and include novel spin-l effects.

See Ref. [53] and further references therein.

B. - A(x,Q2): The Double Helicity Flip Structure Function

One of the helicity amplitudes for Compton scattering from a spin-l target is particularly exotic,

A(x, Q2) might be expected to vanish in the Bjorken limit because the dominant quasi-elastic quark scattering graph (Fig. 8a) can only transfer one unit of helicity from the virtual photon to the target. QCD is, however, more subtle. Gluons carry spin-l and therefore the box graph of Fig. 8b can in principle transfer two units of helicity. But, one might complain, the quarks at the upper vertex of Fig. 8b can only transfer one unit of helicity. The subtlety is that the loop integral in Fig. 8b is too singular to allow us t o take Q2 + W

underneath and treat the quarks as partons. Instead we must view the gluons a partons and the box graph as a local probe of the gluon distribution (Fig. 8). The result is that A(x, Q') scales (modulo a factor of a,(Q2)) in the Bjorken limit and measures a gluon distribution in a spin-l target with n o quark contamination. A straightforward calculation yields

where Q is the quark charge matrix (Q = diag (213, -113, -113.. .)) and a(x, Q ~ ) is a rather unusual gluon distribution:

a(x, Q2) af (X, Q')

-

ap(x, Q ~ ) (4.5)

where ai: (ag) is the probability t o find a gluon linearly polarized along f

(6)

in a target circularly polarized along f moving at high momentum in the 2-direction (!).

Fig. 8

A(x, Q2) can be measured with an unpolarized lepton beam and a target polarized transverse to the beam. Then

2% --- du

dxdydd~ = 1

- -

1 ^{4 1}

-

Y ) A ( ~ , Q2)

d o cos 24

2 x y V l ( z , Q2)

+

^{(1 -} y)F2($, Q2) (4.6) dx dy

A(x, Q2) may be sma11/54/ in the Bjorken limit but it is particularly interesting because it probes gluons which cannot be assigned to individual nucleons within the nucleus.

V. - CONCLUSIONS

We know considerably more about the spin structure of the nucleon than we did before the EMC data appeared. There are excellent prospects for learning much more in the not too distant future. In addition to the HERMES and SMC experiments, E142 at SLAC will measure g1 for the neutron using a 3He target.

Lower energy machines like CEBAF and a possible European electron machine in the 10 - 20 GeV range can complement high-energy measurements with low-Q2 data that may be essential in isolating g2 and other higher twist effects.

So far, assuming EMC data are correct, we have learned that strange quarks carry a significant fraction of the nucleon's spin or equivalently that the fraction of the nucleon's spin to be found on the spin of the quarks is small. As yet, there is no simple explanation of this fact: The Skyrme model does not seem reliable here; there is a "gluonic" contribution, but there's no reason to expect it to be particularly large. An

outstanding issue is the role of the U(l)A Ward Idei~tity and the non-perturbative as well as perturbative role of gluons.

The measurement of various fiavor axial vector inatrix elements in the nucleon complements the mea- surement of other local quark operator matrix elements. The overall picture (reviewed in Ref. [2]) is puzzling.

Some S-quark matrix elements are large, some are small. No theoretical study has succeeded in explaining the ensemble of measurements, although theorists are beginning to address the issue./55/

As for the near-term future, we may expect to see attempts to measure the transverse structure function g2(x, Q 2 ) . g2 can provide significant data on quark-gluon correlations within hadrons and a test of the Burkhardt-Cottingham sum rule. At the same time it may be possible to measure higher spin structure functions which are sensitive to unusual components in nuclear wavefunctions.

Even within the narrow perspective of inclusive deep inelastic lepton scattering, the prospects for progress are encouraging. There will be plenty to challenge spin physics conferences for years to come.

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