Z. Phys. C - Particles and Fields 56, S 179-S 185 (1992) Zeitschrift P a r t i d e s ffir Physik C

and FE Is © Springer-Verlag 1992

A t t e m p t s t o u n d e r s t a n d g l ( x ) a n d g 2 ( x )

Andreas SchMer

Institut fCLr Theoretische Physik, Unlverslt~t Frankfurt, Germany

7.-10.5.1991

A b s t r a c t . The present theoretical understanding of the polarized nucleon structure functions is reviewed. The results of the European Muon Collaboration for g~(x) have generated an enormous theoretical activity, resulting in a large number of possible explanations. The presently available data are not sufficient to de- cide between them, but much improved experiments are planned for the next years. The possibility of an anoma- lous gluon contribution is most interesting on theoreti- cal grounds. Finally we discuss g2(x) which is extremely model dependent and thus could allow a clear decision between the various models proposed. Its measurement requires, however a much improved experimental preci- sion.

1 T h e d a t a a n d i ts i n t e r p r e t a t i o n

The EM collaboration measured the relative differ- ence in the polarised muon:proton scattering cross sec- tions between the case that the longitudional polarisa- tion of muon and proton are the same and the case that they are opposite. This quantity is called Ap(z) and is measured as a function of the Bjorken variable z. The results are shown in Fig.1.

The asymmetry is approximately given by the ra- tio of the polarised and unpolarised structure functions

d~'cr/dT2dE'( TT) - d"~/dY2dE'(T l) Ap(=) = d2q/dl)dE,(~T)-~ d2o./d~dE,(T ~)

D ( g l - 7292 gl-{- g2"~ :p7 ) (i)

%

D is the depolarization factor of the virtual photon and 7 = ~Q-~/v. The structure functions g1,~.(x) are defined as parametrizations of the nucleon scattering tensor.

1 f d4 x eiq. ~ W.~, = 2---~ d < N(P)IJ~'(z)J"(O)IN(p) >

• A =wL +

[ W~ A = %~x~,q;~ + q 'ps a - q. (2)

From Equ.(2) one can see why longitudional electron polarizations are needed to measure gi and g2- Their contribution must be antisymmetric in p and v. On the other hand it is proportional to )-~e~e~, where e u is the polarisation of the virtual photon. Obviously only transverse polarisation vectors like e~ ~ (0, 1, +i , 0) give rise to such ant isymmetric terms.

As FP(x), D, ~1 and 7 are all known one can extract g~(x) from Av in the measured z region. From ~ ( z ) in the measured range it is possible to extrapolate to all z values and thus to give an estimate for the integral

f i ~ ( z ) = 0.123 4- 0.013 4- 0.019 (3) dz

The validity of this extrapolation has been questioned [2] and it is indeed unclear whether non-perturbative effects could set in at very small values of x which are missed by any extrapolation. This is one of the reasons why it could be advantageous to concentrate on ~ ( z ) than on its first moment. (The other is that it con- tains more information.) From the very solid Bjorken sumrule (which relies only on the fact that isospin sym- metry holds on the nucleon level and that quarks have the charges 4-½, and Equ.(?) one gets the rst ment of the polarized neutron structure function. Finally adding some knowledge from weak decays one is able to derive values for the spin carried by the different quark flavours in a nucleon.

total spin carried by quarks and antiquarks

= 0.060 + 0.047 4- 0.069

spin carried by strange quarks and antiquarks

= -0.095 4- 0.016/=5.023

(4)

(5)

180

p

A1

1.0

08-

0.5

0.4-

0.2

0

• EM£ o SLAC oSLAC f

...,

............................. 5 ; +

I'" CorIitz ~, Kaur

l J , , i , , , 1 I .

0.01 002 0.05 0.1 02 0,5 1~

Fig. 1. The proton asyrra-aetry measured by the EM collaboration

While the derivation of these results seems to be straight forward it was argued [3] that there is no compelling reason to assume that isospin symmetry is valid for polarisation phenomena at small values of z, i.e. to assume (Au(x))p = (Ad(x))n etc•. Keeping these caveats in mind Eqs.(4) and (5) are still most surprising as they imply that the spin-distribution inside of a nucleon differs drastically from what simple valence quark models suggest. In fact the whole theoretical discussion of these results suggests that the polarized structure functions are very sensitive to the precise internal structure of the nucleon, causing a lot of theoretical controversy but also opening most promissing possibilities to understand this structure better.

2 Mode ls for ~ ( x )

The line in Fig.1 gives the prediction of a simple phenomenological model by Carlitz and Kaur published in 1977 [4]. It obviously is already pretty close to the data. Several authors proposed extensions of this model which allow for a satisfactory fit of. the new data [3,5]. All of these models use the information on the unpo- larized structure functions and add specific assumptions about the spin distribution among the quarks. They are basically valence quark models.

Another class of phenomenological models starts from the observation that in any relativistic bag model the valence quarks have angular momentum [6.7]. In various bag models this was used to explain the small amount of spin carried by quarks.

A subclass of such models are the Skyrme model, non-topological soliton model or chiral bag-model [8,9]. In these models additional scalar and pseudoscalar fields enter, which can carry angular momentum and thus explain why the spin carried by the quarks should be small. The extent to which this happens depends on the details of the models: Topological and non-topological soliton models give different results, it is not clear how the pion field should be treated, and the boosting into the infinite momentum frame of solitons seems to be a problem. Thus these models may point towards the cor- rect qualitative explanation but have difficulties to give reliable quantitative predictions. With all these caviats, such models can, however, give a good description not only of the first moment, but also of ~ (a:) as a function of • [9].

All these various nucleon bag models share the prob- lem that very similar results can be obtained with com- pletely different models• Also it should be noted that a spin-orbit force is usually needed in such models to get a good description of the hadron spectra. Such a force can, however, completely change the spin-distribution of the quarks.

Thus phenomenological models are capable of fitting the EMC spin data, but the physical interpretation of these data differes from model to model• Furthermore these models allow generally for a rather large number of modifications which alter their predictions for g~(z).

The most widely discussed possibility to interpret the EMC data is that 9~(z) is modified by an anomalous contribution clue to polarized gluons. It turned out that such a gluonic contribution is extremely sensitive to the precise understanding of QCD, Even slight conceptual

181

!.1. k+q V /

k k +

II II II I-

I I I II I

i k-P II

k III k+ O,-9

I

Fig . 2. The perturbative gluon contribution

differences can lead to completely different results. This fact generated rather fierce discussions and thus possibly helped a lot in understanding QCD better. Let me start the discussion with the perturbative hard contribution.

This contribution (Fig.2) turned out to be highly dependent on the kind and ratios of the infrared regu- lators (finite quark mass, minimal gluon.virtuality, min- imal transverse momentum etc.) one uses. As the use of infrared regulators is somewhat alien to the general concept of parton dynamics in the infinite momentum frame, several authors have put forward different ideas, leading to different results.

Adopting the usual procedure of the GLAP equations the anomalous contribution is written in the form

e 2 1 Q2

Here z2G ist the momentum distribution of polarised gluons and A is the splitting function to be calculated. The problems arising in its calculation are due to the fact tha t the dominating logarithmic contribution to A does not contribute to its first moment [10]. E.g. Carlitz. Collins and Mueller obtained

-

o~ r Q2 Nf (1-2z)[log-----~ - l o g x 2 - 21 (7)

2~ L - - . - p ~ J

While the logarithmic term is uniqfle, the finite contri- butions depend on the regularization scheme adopted. Extreme care is needed when calculating A(x) to keep all terms contributing to the next to leading order. It was shown by Bodwin and Qiu [11] that if the quark mass and gluon virtuality are used as infrared cut-otis the resulting splitting function is identical zero. We showed [12] tha t introducing an additional cut-off in the trans- verse momentum leads to a finite contribution for small quark masses, which vanishes if the quark mass becomes

very large. (This is in agreement with the original obser- vation by Carlitz, Collins and Mueller.) We also found strong scaling violating effects. Qualitatively similar but quantitatively different results were obtained by Bass, Nikolaev and Thomas [13]. The main problems of this interpretation are that the gluon spin must be as large as 10 h, that strong scaling violation is predicted but not observed, and that it is hard to get an acceptable fit to g~(z) as a function of x [13,14].

It was argued by several authors [15,16] that a non- perturbative gluon contribution can exist. While this is possible in principle it is improbable that it could explain the data as it should occure only for very small z values. Its possible existence points, however, to the fact that the extrapolation of g~(x) to small z values might be problematic. So far nobody was able to actually calculate any non-perturbative gluon contribution to ~ (z ) .

A very severe objection to the whole concept of an anomalous gluon contribution was raised early on by Jaffe and Manohar [6], namely that there is no corre- sponding gauge-invariant local operator in the operator product expansion. This observation suggests that there cannot be any genuine point-like photon-gluon coupling and thus the gluon contribution could be rewritten in terms of a coupling to quark currents. To circumvent this argument it was suggested that the gauge non-invariance might be unimportant [16] and non-local operators were tried [17]. My personal understanding of this point is the following. For finite Q~ the contribution resulting from Fig.2 is not the anomaly. The loop moment is bound by Q2 and thus the contribution is not truely point- like. This could be the reason for the strong scaling violation, and is in accordance with the observation by Jaffe. Only for Q2 ---, oo does the graph collapse to the anomaly. Therefore it is in principle possible to resolve the gluon coupling into a polarised sea quark contribu- tion. However, as was pointed out by A1. Mueller, such a procedure is not reasonable. The quark loop containes

182

quarks with very large momentum which cannot be in- cluded naturally into the sea quark distribution. It is much more sensible to speak of this contribution as due to gtuons and to make model assumptions for the dis- tr ibution function of polarized gluons, i•e. to interpret it as anomaly plus infra-red corrections than as due to a crazy sea-quark component.

At tempt were also made to tie the divergence of the isoscalar a.xialvector current to the r/ in the same sense as PCAC ties the isovector current to the pion. Different authors have derived completely differnt results along these lines and it is probably fair to say that PCAC is just not good enough a symmetry in this case to be of much use.

3 T h e p h y s i c a l m e a n i n g o f g2(x)

The primary aim of future experiments will be to measure with high precission the polarized nucleon struc- ture functions gl(x). To achieve this the second polarized s t ructure function g2(x) has also to be known with some accuracy (see Equ.(1)). It can be determined from a comparison of the results for longitudionally polarized electrons scattering off either longitudionally or trans- versely polarized nuclei• This perspective of a rough experimental determination challenged theoreticians to analyse the properties of g~.(¢) and to make model-based predictions.

The most crucial y of g2(x) is that it contains im- por tant contributions from higher twist. This fact makes any predictions of the exact form of g2 as a function of x very difficult• On the other hand 92 offers the only chance known so far to measure isolated higher twist ef- fects directly. Because of this complication g2(z) can be analysed only in a rather formal manner, using the lan- guage of operator product expansion. Before we review this analyses let us, however, give a intuitive, physical argument for the physical content of g2(x).

g~ (a:) parametrizes the difference in the cross-sections for a longitudionally respectively transversely polarized proton. If a probabilistic interpretation would be appli- cable (as it is for the leading twist contribution) this difference would be proportional to the difference be- tween the probability to find a transversely polarized quark in a transversely polarized proton and the prob- ability to find a longitudionally polarized quark in a longitudionally polarized proton. For free particles this difference should obviously vanish and indeed g~(x) = 0 for free quarks. Thus g2(x) is exclusively determined by interaction effects. One such effect is the following. The s tructure functions are defined in the infinite momen- tum frame and describe e.g. the boosted proton. For an interacting field theory a boost changes the particle content. If this change depends on the polarization of the proton relative to the boost direction this leads to a contribution to g~(a:). From this it should already be clear tha t g.x.(z) is much more sensitive to subtle effects than 91(x) or the unpolarized structure functions. It will

even become clear that nuclear factorisation might not apply at all.

It should be kept in mind that the properties of g~(x) have some important experimental consequences. If one would conclude e.g. that the nuclear factorisation (i.e. the usual treatment of the nuclear binding effects on the distribution functions) does not apply for g~(x) one would have to measure g2(z) separately for the p,d and aHe target. Otherwise a measurement for e.g. p and d would be sufficient.

To obtain the OPE one investigates the virtual pho- ton forward Compton amplitude [19,20]

T,. = i f < N(p)IT(J~,(z)J,,(o))IN(p) >

• A

[ s° q 'vs" - (8) T[, A = ieu.~,q ~ ~7~al (x) + (q -y6, 5 =

The time ordered product of currents gives the propa- gator which is expanded in the usual manner.

{ 2~ '+I T/A=ieu"~'q~ E \ - - ~ ] qu'"•qu'~

n=0,2,4,...

< PSIO ~'{m""} IPS >

OO{Ul--.u-} = i '~¢Ta@D {~1Du~ •..DU"}¢

- traces + O(twist 4) (9)

O has a mixed symmetry, it is symmetrized in all but the first indices, and contains twist 2 and twist 3 contribu- tions. The twist 2 contribution is completely symmetric in all indices and can be extracted according to

O~{,1...~,} = O{~ul..."-} + O[~'{i'~] '--"-} (10)

< PSIO Mml ' ' ' ""} ]PS >

d, - n + 1 [(s Pl - + ' - t r a c e s ]

< ps]o{ ¢'~'~ ...u,~} tps >

= a,, t,'accs] (11) n + l

Here O [~{~j ' '~ '} is symmetric in all p's and antisym- metric under exchange of c~ with one # index. As the total deep inelastic cross section is proportional to the imaginary part of the forward Compton scattering am- plitude a l , o2 and gI, g-, are connected by a dispersion relation

1 [ 1 d , ,

X n = 0 , 2 , 4 . . . .

n = 0 , 2 , 4 . . . .

183

jfO l

/01 rl ") a n &jy g~(y, Q ' ) = _

4 n = 0 , 2 , 4 .... (13)

n 2 d y y g~(y, Q ) = - - - 1 n

4 n + 1 (d. - a.)

n = 2, 4, ... (14)

Thus the moments of gl are determined exclusively by the twist 2 matr ixelements an, while g2 has a leading twist 3 contribution. This is the reason why it is so difficult to make reliable predictions for g2. As the twist 3 contribution is hard to be calculated one might hope that it is negligable. Under this assumption all the moments of g-" are determined by the moments of gl and both functions are related [21]

ggw (~, @,) = -g~ (~, O ~) + gl(v, O") (15)

However, this approximation is not justified. Jaffe and Ji have shown that e.g. for the bag model 92 calculated f rom

0.6

0.4

0.2

0.0

- 0 . 2

/ 1 / " " , . ~

l "% ,-" I " " . . . . " "

l , . ! . . . . I . . . . I . . . . .

0 0.2 0.4 0.6 0.8 1

X

1/ < PS*I¢(g - ) Q~h71-rs¢(0) + . . .[PS': > (16)

differs substant ia l ly from the Wandzura-Wilczek predic- tion, see Fig. 3. Other models give different predictions (Fig.4,5). However, none of these predictions is reliable. E.g. in the bag model all correlators of two quark and one gluon opera tor (see Eq.(19)) are substi tuted by the bag- boundary condition. This crude approximation works quite well for leading twist contributions but it can- not describe higher twist contributions correctly. On the other hand light-cone wavefunctions as used by Dziem- bowski [27] (Fig.4) contain in principle all higher twist te rms correctly, but up to now their precise form is still not sufficiently well known.

As g=(x) is zero in the limit of infinitely heavy quark masses all higher twist corrections which are due to finite quark masses should cancel part of the twist two contr ibution, i.e. should have the oposite sign for most x values. For the models leading to Fig.4 and 5 this is obviously the case. In these models all higher twist effects depend on to the finite quark masses and would vanish for zero quark masses. For the MIT bag model a similar cancelation occures, however, in this case it is not clear whether the bag boundary condition can be related to any effetive mass term. Most people in the field think tha t twist 2 and twist 3 parts should allways have different signs, but we are not aware of any formal proof for this assumption.

Thus not much more is known about g2(x) than that its absolut value should be comparable and probably somewhat smaller than the pure twist 2 contribution from gq(9) . A precise measurement of g, would be very helpfull in determining the nucleon wave function at least in the long run but even a rough est imate which is all one can hope for with the limited precission of

Fig. 3. g2(x) in the MIT bag model [22]. The dotted line is the Wandzura.Wilczek contribution of twist 2 (Eq.(9)). The solid line was calculated from Eq.(10). The dashed line is the difference to be identified with the twist 3 contribution.

g~

1.0

0.5

O0

-0.5

-l,O

I ' ' I ' I I

I !

I / l I

/ J. -1.5 I ,, i ,, J,,, I , , ,

0.0 0.2 0.4 0.6 0.8 1.0

X

Fig. 4. Same as figure 1 for the llght-cone quark model [27]. The predictions are obviously completely different for small x.

experiments feasible in the near future could rule out some model predictions. As the value of g~ for very small x is very sensitive to the specific model, hotoproduction experiments (x = 0), if feasible, could give valuable additional information.

184

05

0.~

gz0 z 0.I , ~

.21 t .... 0.1 0.2

f

J , 1 I i i i i

0.3 0,~. 0.5 0.6 0.7 0.8 0.9 1.0 X

F i g . 5. g2(x) in a s imple phenomenological model with only small h igher twist cont r ibut ions [28].

The predictions for g.~(x) in Figures 3,4 and 5 fullfill the so called Burkhardt-Cott ingham sumrule (BCSR) [22].

j~01 g 2 ( x ) : 0 (17) dx

This sumrule is also suggested by Eq.(8). While this equation is only valid for even values of n larger than zero one can think of continuing it analytically to n = 0. Although this continuation is not unique the ambiguity could be irrelevant. Two functions with the same higher moments can only differ by a contribution proportional to 5(x), which is unobservable in deep inelastic scattering anyway (though it would be seen in photoproduction). Higher twist contribution could give a small finite con- tr ibution to the BCSP~ for finite Q2, corresponding to washing out this singular contribution to some extend, but still it seems somewhat unphysical and thus is as- sumed by most people to be absent. In the following we assume that the naive extrapolation to n = 0 is valid. Then the BCSR should hold if d,, stays finite for n ---* 0 (a0 is finite).

If the matrixelement do is analysed in in the light- cone formalism [23] one finds not surprisingly that it can only diverge for x ---+ e% i.e. if the hit polarized quark has vanishing momentum fraction. More precisely the two relevant contributions can be written in the form

lim n < PS , x'~(StT - 5jA)IPS, > (18) r~--* 0

and

lira < PSi[ g- ( + Sa Ol°g /VIPS > (ZO)

The first term depends on the transverse momentum and is thus related to the angular momentum of the struck quark. The second term contains the coupling to a gluon. It can be shown, that soft gluons (y - x ~ 0) nor soft outgoing quarks (y ---+ 0) can lead to divergences. Thus only the limit x --~ 0 can lead to any divergence in any of the two relevant expressions. Thus whenever there is no substantial sea-polarization for x ~ 0 BCSR must hold, as is the case for e.g the bag model. The fact that model calculations respect the BCSR is thus usually a direct consequence of the model assumptions and could be misleading.

The validity of the BCSR can also be analysed in terms of Regge theory. It turns out that in this lan- guage BCSR holds unless there is a J=0 fixed pole in the Compton amplitude. Whether this can be the case or not seems to be a question of debate among special- ists. As said before the crucial question is whether the relevant matr ix elements diverge as 1/:c or not. One can show that in perturbat ion theory this does not happen for QCD. (It does e.g. for a scalar analog of QCD.) 1/x singularities can, however, also appear on the constituent quark level and it does not seem to be possible to de- cide unambiguously whether it does or not. A general problem of this approach is also that many arguments apply strictly only in the Q~ ---* co limit such that it is not clear whether finite Q2 corrections are important for realistic values.

This brings us to the more general question of Q2 evolution for g2(x). Experiments at HERA like HER- MES can measure the polarized structure functions only in a rather limited Q2 range, depending on x. Thus for a very high precission comparison the precise Q~ evolution of g~. becomes important . In practice the experimental precission necessairy to observe this evolution will cer- tainly not be reached in the near future. Still it is a very interesting theoretical problem. All, Braun and Hiller have recently presented a comprehensive analyses of it [24,25].

It was noted already some time ago that the Q2 evolution of higher twist contributions can in general not be described by Gribov-Lipatov-Altarelli-Parisi type equations. Instead various contributions are mixed [24]. 2~his holds also for the twist 3 part of g,. Higher and higher moments get contributions from more and more different matrixelements.

The fact that g2 contains a leading higher twist con- tribution, which has led to all the problems discussed so far, can still lead to another qualitative difference to the situation for gl- For g2 simple factorization of nuclear effects does not apply and thus it is not clear how large nuclear effects will be, i.e. it is possible that g~ deter- mined for 3He is quite different from that obtained from experiments with deuterium. Such nuclear effects would be analogous to the first EMC effect, namely the nuclear dependence of the unpolarised structure functions. How- ever, as g~(x) is so strongly dependent on higher twist effects its nuclear dependence could be much more pro- nounced than that of F2(x). This fact implies also, that to constrain g, sufficiently to reduce the error on gz it

should be measured for each nucleus separately. The size of these nuclear effects has not yet been de termined the- ore t ical ly bu t at least a formal i sm was developped [26]. Unless the nuclear effects are surpr is ingly large HER- MES will no t reach the precission to ac tual ly measure them. One reason why these effects could be large is the following: One par t of the twist 3 con t r ibu t ion depends on the pe rpend icu la r m o m e n t u m of the struck quark, which in t u r n is related to its angular m o m e n t u m , see Eq.(12). As fac tor iza t ion does not apply g, could depend on the to ta l hadronic angu la r m o m e n t u m . For 3He one has a s t rong D state a d m i x t u r e and thus one could hope for s u b s t a n t i a l nuclear effects.

4 C o n c l u s i o n s

Measurements of the spin structure functions play an impor tan t role in improving our understanding of the internal s t ructure of the nucleons. The theoretical discus- sion of the last years has already lead to important new results. Still, without new, more precise experiments a decission between the various interpretations is not pos- sible. Thus one of the major tasks at the moment is to find additional ways of extracting important information (e.g. Drell-Yan experiments, semi-inclusive experiments etc.). If such high precission data will become available they could become one of the most sensitive ways to investigate thge internal structure of the nucleons.

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references therein. The calculations for g2 (z) are unpublished.

This article was processed using Springer-Verlag ~ Z.Physik C macro package 1991 and the AMS fonts, developed by the American Mathematical So- ciety.