Nuclear Physics A527 (1991) 559~~564~ North-Holland, Amsterdam

559c

THEORETICAL ATTEMPTS TO UNDERSTAND g;(t): AN OVERVIEW

Andreas SCHAFER*

Max-Planck-Institut fiir Keruphysik, Postfach 10 39 90, D-6900 Heidelberg

The EMC data on the polarized function of the proton have led to intense theoretical discussion on the internal spin structure of the nucleons. Theoretical attempts to understand these data can be grouped into different classes. The major ones are reviewed in some detail. It is argued that, while many different models are in principle capable of describing the data, none of them is really compelling.

The improved measurement of the polarized proton structure function g:(z) by the EMC

group l-2 has generated great excitement in the high-energy physics community and conse- quently a multitude of propositions have been made to int.erpret and explain these data. I shall try to give a short overview of these theoretical activities. To do so I shall group them in several categories. In section 1.) experimental data are presented and it is discussed whether they really imply the astonishing consequences which have been claimed. In sec- tion 2.) we shall review a number of simple phenomenological models which are all able to describe the data but differ completely in their respective assumptions on the internal nucleon structure. In section 3.) the anomalous gluon contribution to g;(z) is discussed. The ambiguities and conceptual problems associated with it are mentioned. It is also argued that this contribution is very unlikely to explain the data.

1.) THE EXPERIMENTAL DATA The EMCcollaboration measured the relative difference in the polarised muon-proton

scattering cross sections between the case that the longitudinal polarisation of muon and proton are the same and the case that they are opposite. This quantity is called Ap(r) 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 ratio of the polarised and unpolarised

structure functions 2zgf(s)/F,P(z). Thus knowing F:(s) one gets g;(z) in the measured x

region. However the results for F,P(z) have been controversial and were only settledrecently3. As a consequence the numbers for g:(x) have changed slightly between the first’ and second’ publication of EMC and the theoretical error was substantially reduced. From g!(z) in the measured range it is possible to extrapolate to all I values and thus to give an estimate for

the integral

J 1

g;(z) dx = 0.123 + 0.013 & 0.019 (1) 0

The validity of this extrapolation has been questioned4 and it is indeed unclear whether non-perturbative effects could set in at very small values of I which are missed by any extrapolation. This is one of the reasons why it could be advantageous to concentrate on

*supported by DFG

0375-9474/91/$03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)

560~ A. Schbfer / g{ (x): An overview

08 l EMC 0 SLAC

06 oSLAC

C0rbt2

L Kour

I 1 1 , I I

0.01 002 a05 01 02 OS lo

X

Fig.1 : The EMC data

g:(z) than on its first moment. (The other is that it contains more information.) From the very solid Bjorken sumrule (which relies only on the fact that isospin symmetry holds on the nucleon level and that quarks have the charges *If, *$) and Equ.(l) one gets the first moment 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 quurks and antiquarks = 0.060 * 0.047 f 0.069 (2)

spin carried by strange quarks and untiqtmrks = -0.095 f 0.016 f 0.023 (3)

While the derivation of these results seems to be straightforward, it was argued5 that there is no compelling reason to assume that isospin symmetry is valid for polarisation phenomena at small values of 2, i.e. to assume (Au(z))~ = (Ad(z)), etc..

2.) PHENOMENOLOGICAL MODELS The line in Fig.1 gives the prediction of a simple phenomenological model by Carlitz and

Kaur published in 19776. 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 data5,7. All of these models use the information on the unpolarized 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 rel- ativistic bag model the valence quarks have angular momentums~g. 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-modellOJ1. 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 correct qualitative explanation but

A. S&lifer / g f @): An oven&w 56lC

have difficulties to give reliable quantitative predictions. With all these caveats, such models can, however, give a good description not only of the first moment, but also of g;(z) as a function of +,

All these various nucleon bag models share the problem that very similar results can be obtained with completely 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 phenomenolog~~al models are capable of fitting the EMC spin data, but the physical interpretation of these data differs from model to model. Furthermore these models allow generally for a rather large number of modifications which alter their predictions for g:(z).

3.) ‘IKE ANOMALOUS GLUON CONTRIBUTION The gluon contribution to g:(s) is the most widely discussed possibility to explain the

EMC data. The main reason is that one would assume to be on safe grounds in calculating it from QCD. It turned out, however, that the gluonic contribution is extremely sensitive to the precise understanding of QCD. Even slight conceptual 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.

+

Fig.2 : The perturbative gluon contribution

This contribution (Fig.2) turned out to be highly dependent on the kind and ratios of the infrared regulators (finite quark mass, minimal gluon virtuality, minimal 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

Here AG is 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 that the dominating logarithmic contribution to A does not contribute to its first momenP. E.g, Carlitz, Collins and Mueller obtained

562~ A. Schtifer / g f (x): An overview

A(z) = -g Nf * (1 - 22) [log $ - log 5s _ 2] .

While the logarithmic term is unique, the finite contributions depend on the regularization scheme adopted. Extreme care is needed when calculating A(z) to keep all terms contribut- ing to the next to le:ding order. It was shown by Bodwin and Qiu13 that if the quark mass and gluon virtuality are used as infrared cut-offs the resulting splitting function is identical

zero. We showed14 that introducing an additional cut-off in the transverse 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 observation by Carlitz, Collins and Mueller.) We also found strong scaling violating effects. Qualitatively similar but quantitatively dif- ferent results were obtained recently by Bass, Nikolaev and Thomasis. 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 Equ(4) d oes not give an acceptable fit to g:(r) as a function of Z, unless rather strange assumptions are made’5~‘6. Thus it seems at present unlikely that the perturbative gluon ,contribution could explain the data.

It was argued by several authors 16,17 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 occur only for very small z values. Its possible existence points, however, to the fact that the extrapolation of g:(x) to small 2 values might be problematic. So far nobody was able to actually calculate any non-perturbative gluon contribution to g;(z).

A very severe objection to the whole concept of an anomalous gluon contribution was raised early on by Jaffe and Manoha?, namely that there is no corresponding 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” and non-local operators were triedlg. 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 Q” 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 + 0;) does the graph collapse to the anomaly. Therefore it is in principle possible to resolve the gluon coupling into a polarised sea quark contribution. However, as was pointed out so clearly by Dr. Mueller in his talk, such a procedure is not reasonable. The quark loop contains quarks with very large momentum which cannot be included naturally into the sea quark distribution. It is much more sensible to speak of this contribution as due to gluons and to make model assumptions for the distribution function of polarized gluons, i.e. to interpret it as anomaly plus infra-red corrections than as due to a crazy sea-quark component.

Attemptswerealso made to tie the divergence of the isoscalar axialvector current to the 7’ in the same sense as PCAC ties the isovector current to the pion. Different authors have derived completely different results along these linesz2 and it is probably fair to say that PCAC is just not good enough a symmetry in this case to be of much use.

4.)CONCLUSIONS The small values of g:(x) measured by EMC suggest that the spin structure of the

nucleons is different from naive expectations. Whether they imply that the total spin carried by the quarks is close to zero and that the strange quark contribution is large is questionable.

Simple phenomenological models give good fits to the data but differ widely in their interpretations.

A. Schdfer / g 1 (x): An overview 563c

The anomalous gluon contribution is present in principle, but has problems to explain the data. Possibly it is just an additional small correction. Its precise contribution to g:(l) is still very controversial.

There have also been put forward a rather Iarge number of unconventional ideas. While such ideas might perhaps prove successful if future experiments give additional puzzling results they have not been discussed here. The long-standing question about the role of the strange quark has also not been reviewed.

One has to conclude that new complementary experiments are needed to decide which explanation of the EMC data is correct.

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