

 21 Jun 1994

Cavendish preprint HEP 94/4 DAMTP 94/50

The small x behaviour of g1

S. D. Bass1 and P. V. Landshoff2

1HEP Group, Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, England

2Department of Applied Mathematics and Theoretical Physics, University of Cambridge,

Silver Street, Cambridge CB3 9EW, England

Abstract We discuss the small x behaviour of the spin dependent structure function g1. We find a contribution from the exchange of two non-perturbative gluons which behaves as , (2 ln 1x \Gamma 1).

In recent years there has been much interest in polarised deep inelastic scattering as a result of the European Muon Collaboration (EMC) measurement of the polarised proton structure function gP1 (x; Q2) [1]. The naive parton model interpretation of the EMC data is that the quarks contribute a small fraction of the proton's spin.

The EMC experiment, which followed earlier polarised scattering experiments at SLAC [2], has inspired a new experimental programme. During 1993-1994 we have been presented with new data from the Spin Muon Collaboration (SMC) at CERN [3,4] and the E-142 experiment at SLAC [5]. New experiments are planned in the near future [6]. One of the key ingredients in using deep inelastic scattering experiments to test spin sum-rules (and hence learn about the spin structure of the nucleon) is to extrapolate the g1 structure function data to x = 0. This extrapolation introduces a theoretical error which must be included in the analysis.

There are two sum-rules which are important in polarised deep inelastic scattering. The first is the Bjorken sum-rule [7] for the iso-triplet part of g1Z

1 0 dx `g

p1 \Gamma gn

1 '(x; Q2) = 16 g3A C

q3(Q2) (1)

Here g3A is the iso-triplet axial charge and Cq3 is the non-singlet perturbative Wilson coefficient, which has been evaluated to O(ff3s) precision [8]. The Bjorken sum-rule was derived using current algebra before the advent of QCD and is a test of isospin. It is not expected to fail.

The Ellis-Jaffe sum-rule [9] is a test of Zweig's rule in the flavour singlet channel. If we assume that strange (and heavy) quarks do not play a significant role in polarised deep inelastic scattering, then we can derive sum-rules for each of the proton (R 10 dxgp1) and neutron (R 10 dxgn1 ) targets. The first moment of the flavour singlet part of g1 isZ

1 0 dx g1(x; Q

2)jS = 1

3 s

2 3 g

0 A C

q0(Q2) (2)

Here g0A is the flavour singlet axial charge and Cq0 is the singlet coefficient, which has been calculated to O(ff2s ) precision [10]. In the naive parton model \Sigma = p6g0A is interpreted as the quark spin content

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of the nucleon. The most recent determination of this quantity (which includes the SMC proton data [4]) is

\Sigma = 0:30 \Sigma 0:07(stat:) \Sigma 0:10(syst:) (3)

This number is two standard deviations below the naive quark model expectation (\Sigma = 0:68), which is based on the assumption that Zweig's rule is exact. In QCD one finds that the spin of the quarks is screened by the gauge symmetry of the background colour field in the nucleon (the physics of the axial anomaly). This result means that g1 does not have a simple explanation in terms of explicit quark and gluon spin degrees of freedom (for reviews see [11-13]).

The quantity which is measured in polarised experiments is the spin asymmetry A1. In order to test spin sum-rules for the first moment of g1 we first need to extract the structure function from the A1 data and then extrapolate g1 to x = 0 using Regge theory, which provides a good description of the small x behaviour of the unpolarised structure function in the NMC kinematic range.

The experiments which have been carried out to date use a longitudinally polarised beam and target [14]. The spin structure function is related to the spin asymmetry A1 via the equation:

g1(x; Q2) = A1(x; Q

2)F2(x; Q2)

2x(1 + R(x; Q2)) (4)

In order to extract g1 from A1, the experimenters use the NMC parametrisation of F2 [15] and the SLAC parametrisation of R(x; Q2), which was determined by Whitlow et al. [16]. This parametrisation of R(x; Q2) was obtained from a fit to data on the longitudinal structure function FL in the region x ? 0:1. It includes the effects of leading order QCD evolution and twist four terms. At the present time, the smallest x data point for gp1 (which comes from SMC) has been measured at ! x ?= 0:005 and ! Q2 ?= 1:3 GeV2 [4]. This measurement is obtained from A1(x; Q2) via an extrapolation of R(x; Q2) from x = 0:1 down to x = 0:005. It seems reasonable to us that one should question the size of the error that results from using this extrapolation of R(x; Q2) to such small x and Q2. One should also bear in mind possible shadowing corrections when one extracts gn1 from measurements on either a deuteron target (SMC) or a 3He target (E-142). Shadowing in F2 in the deuteron is typically 3-6% at ! x ?= 0:005 [17] and may be a factor of two greater in g1 [18].

Given that we have extracted the structure function g1 from A1 how should we extrapolate it to x = 0 ? It is well known [19, 20] that the iso-triplet piece of g1, which appears in the Bjorken sum-rule, behaves as

g(3)1 , x

\Gamma ffa1 ; x ! 0 (5)

where ffa1 is the intercept of the a1 Regge trajectory. Heimann quotes this intercept as ffa1 = \Gamma 0:14 [19], which is within the phenomenological range \Gamma 0:5 ^ ffa1 ^ 0 discussed by Ellis and Karliner [21]. If one assumes that the slope of the a1 trajectory is the same as the slope of the ae; ! trajectory, then one finds that ffa1 ss \Gamma 0:4. As a model-independent statement, one can say that the iso-triplet part of g1 converges as x ! 0.

Usually g1 is extrapolated to small x as g1 , constant, which is consistent with assuming that g1 is almost pure iso-triplet at small x if we take Heimann's value of ffa1. On the other hand, the latest gp1 data from SMC show evidence of a possible rise in g1 in the smallest x bins [4]. In the rest of this paper we discuss the small x behaviour of the flavour singlet part of g1 and report on a gluonic exchange contribution to spin-dependent processes which has hitherto been missed in the Regge spin literature.

In the flavour singlet channel we have to consider gluonic exchanges. The small x piece of the unpolarised structure function F2 is governed by pomeron exchange. The NMC data at small x (in the range 0:005 ! x ! 0:1) are described by the physics of the non-perturbative pomeron [22]. Strictly speaking, the non-perturbative pomeron is relevant to deep inelastic scattering in that it describes the total cross section for a real or almost-real high-energy photon scattering from the nucleon. At very small x (eg. at HERA) the structure function F2 rises faster than the prediction of simple nonperturbative pomeron exchange [23]. This tells us that off-shell (finite Q2) effects are important in the range of the HERA kinematics and may be evidence of a perturbative pomeron [24].

At the present time, we have data on the spin dependent structure function g1 for x ? 0:005; that is, in the same x range as the NMC unpolarised structure function data. Whilst the usual pomeron (with

2

Regge intercept +1) does not contribute to g1 [19,20], it seems reasonable that the physics which leads to pomeron exchange may also be important in understanding the small x behaviour of g1. Indeed, we find that nonperturbative two-gluon exchange, which seems to be an important component of the nonperturbative pomeron [26-29], contributes to g1 a small x behaviour that is roughly constant.

The perturbative parton model suggests that there is more to the small x behaviour of g1 than the iso-triplet a1 exchange in equ.(5). In the parton model the spin dependent gluon distribution \Delta g(x; Q2) contributes to the flavour singlet part of g1 as \Delta g \Omega Cg, where Cg is the gluonic Wilson coefficient. This gluonic contribution is clearly dependent on the shape of \Delta g(x; Q2) but can easily lead to a contribution which diverges as x ! 0 [25]. At the present time, there is no experimental measurement of \Delta g(x; Q2) or even the sign of \Delta g = R 10 dx\Delta g(x; Q2).

The simple and successful four parameter model of non-perturbative pomeron exchange involves the exchange of two non-perturbative gluons [26-29]. This is shown for deep inelastic scattering in Fig. 1. Here p is the nucleon target momentum, q is the photon momentum and we use xBj to denote the Bjorken variable. (The crossed-quark-box graph gives only non-leading terms as * = p:q ! 1.) We use the Sudakov variables:

k = xp + yq + kT

l = ,p + j q2* + lT (6) to denote the quark and gluon momenta respectively. The box graph contribution in Fig. 1 carries the same quantum numbers as the perturbative photon gluon fusion process, which has been discussed in the g1 literature in connection with the contributions to g1 from polarised glue and the axial anomaly [11-13,30,31]. We now calculate the contribution to g1 from this process involving the exchange of two non-perturbative gluons, which provides a successful description of the physics of the non-perturbative pomeron. We do not attempt to interpret our calculation in terms of separate contributions from either the polarised glue or sea and the effect of the axial anomaly.

The non-perturbative gluon and quark propagators which appear in the vertical lines of Fig.1 are taken from the model of refs.[26,27]:

G_* (l) = g_* D(l2) (7)

(where we work in Feynman gauge) and

Q(k) = (fl:k + m)S(k2) (8) (where m is the constituent quark mass) respectively. Confinement is built into the model by requiring that D(l2) and S(k2) do not have any poles at time-like momentum on the physical sheet [26]. This requirement ensures that the moments of D(l2) and S(k2) are finite. The leading term in the hadronic cross section comes when the struck quark travels only a very short distance between interactions with the gluons [27]. This means that the horizontal quark lines in Fig.1 should be treated as perturbative; these quarks are placed on-shell when we take the imaginary part of the diagram.

The calculation of the two non-perturbative gluon exchange contribution to hadronic cross sections involves taking the first and second moments of D2(l2) and S2(k2). These moments are determined in terms of four parameters. The physics of the gluon propagator involves the coupling fi0 of the pomeron to a quark in the target and the mass parameter _0 in the quark pomeron form-factor [28], which are determined from experiment to be fi0 ss 2:0 GeV

\Gamma 1 and _0 ss 1 GeV [29]. (These quantities are

related to the values of the gluonic condensate G_* G_* (0) in the vacuum and its correlation length [26].) The moments of the quark propagator S(k2) are determined by the vacuum quark condensate ! vacjqqjvac ?= \Gamma m30 where m0 ss 225 MeV and the constituent quark mass m ss 330 MeV [27].

The spin-dependent part of the hadronic tensor in deep inelastic scattering enters as the antisymmetric term in the proton line flae(fl:p \Gamma fl:l)floe. This is

ifflaeoefffi(p \Gamma l)fffl5flfi ! ifflaeoefffi(p \Gamma l)ffsfi (9) The anti-symmetric part of W_* is linear in the nucleon spin vector s. In order to pick out the part proportional to g1 we choose to evaluate it with s set equal to p -- even though this is unphysical

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it removes the contribution from the other structure function g2 [14]. The leading term at * ! 1 is independent of the nucleon mass, which we set to zero. The anti-symmetric part of the term that results from contracting equ.(9) with the trace of the quark loop is:

\Gamma 4(m2 + k2)l2(, \Gamma 2) (10) The delta function constraints on the horizontal lines evaluate as:Z

d4l ffi((p \Gamma l)2) = ss2 Z d, Z dl2 (11)

and Z

d4k ffi((k \Gamma l)2)ffi((k + q)2) = ss2 12* 1, Z d(\Gamma k2) (12)

Inserting numerical factors, we find a contribution to the flavour singlet part of g1:

\Gamma g4` ss2 '

22 Z 1

xBj d,(1 \Gamma

2 , ) Z dl

2l2D2(l2) Z

1

0 d(\Gamma k

2)(m2 + k2)S2(k2) (13)

where g is the quark-gluon coupling. The l2 integral is [28]:

g4 Z dl2l2D2(l2) = 18ssfi20 _20 (14) The integrand in the k2 integral is the sum of two terms with opposite signs, which makes the value of this integral particularly model dependent. If we use the same model (non-perturbative) quark propagator

S(k2) = 4ss

2

3

m30 m5 exp

k2 m2 (15) as that used in [27], then the two non-perturbative gluon exchange process gives a contribution to the flavour singlet part of g1, which is equal to

N `2 ln 1x \Gamma 1' (16a) with

N = ss18 fi20_20`m0m '

6 ' 0

:09 (16b)

The best fit of the form (16a) to the small x SMC data yields N = 0:085 \Sigma 0:01, which corresponds to \Sigma = 25 \Sigma 11% [32].

The non-perturbative two gluon exchange process is a simple model of non-perturbative pomeron exchange in unpolarised hadronic interactions. In that case, it is necessary to correct the model result xq , 0:2 by a factor x

\Gamma 0:08 [27] to account for the fact that the pomeron is really more complicated

than just two gluons, and reggeises. We do not know whether the two-gluon exchange contribution we have calculated for deep inelastic scattering also reggeises, or whether it is a fixed pole.

The Regge theory successfully describes the small x part of the unpolarised structure function as measured by the NMC [15]. It seems reasonable that the same should be true of the polarised structure function in the same kinematic range of x and Q2. The gluonic exchange contribution to the flavour singlet part of g1 in equ.(16) should be present in the SMC data and may be responsible for the rise in gp1 which is seen in the smallest x bins. We show this in Fig. 2 where we plot our gluonic exchange contribution to gp1 together with the SMC data.

It is interesting to include the perturbative photon gluon fusion process into the model. Here the quarks carry large transverse momentum. We re-calculate the diagram in Fig. 1 using purely perturbative quark propagators and keep the quark transverse momentum k2T * *2, where *2 * 2 GeV2. In this case, equ.(13) is replaced by

1 96ss2 g

2ffs(*) Z dl2l2D2(l2) Z 1

0 d,`ln

2,*

*2 \Gamma 1' ' 0:0008 (17)

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which makes a negligible contribution to g1 at small x compared to the non-perturbative contribution in equ.(16).

Finally, we discuss how our results may be manifest in the SMC and E-142 low x data on the spin dependent nucleon structure function. The EMC and SMC data tells us that gp1 is positive in the x range shown in Fig.2, whereas gn1 (which has been extracted modulo shadowing corrections) is negative over this range of x. This suggests that there is a large iso-triplet piece to g1 at small x, which is described by equ.(5). The rise in gp1 which is seen in the smallest x bins of the SMC may be due to the non-perturbative gluon exchange process which we have described here (see Fig. 2). This flavour singlet contribution will come to dominate the convergent iso-triplet contribution at very small x. Close and Roberts [32, 33] have also discussed a possible rising contribution to g1 at small x associated with the pomeron-pomeron cut [34]. At the present time the size of the experimental error and the theoretical uncertainty in the normalisation of equ.(16) mean that it is not possible to determine the relevant weights of the a1 and gluon exchange contributions. This uncertainty means that the error which is quoted on tests of the Ellis-Jaffe sum-rule is probably too small (see also [32]). Of course, tests of the iso-triplet Bjorken sum-rule are independent of any theoretical discussion about the small x behaviour of the flavour singlet part of g1. It is sufficient within the present experimental error to use the same small x extrapolation given in equ.(5) for both gp1 and gn1 in order to test the Bjorken sum-rule. At first sight, our result may appear disappointing in that it is harder than hitherto expected to make a precise test of the Ellis-Jaffe sum-rule. On the other hand, the real test for theoretical models is whether they can predict the structure function g1 over a complete range of x rather than just one moment! [11]

Acknowledgements: We thank F. E. Close, N. N. Nikolaev, R. G. Roberts and A. W. Thomas for helpful discussions. This research is supported in part by the EU Programme "Human Capital and Mobility", Network "Physics at High Energy Colliders", contract CHRX-CT93-0537 (DG 12 COMA).

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l q

p

k

Fig. 1: The gluon exchange contribution to g1.

Fig. 2: The gluonic exchange contribution to gp1 at small x (equ.(16)) together with the SMC data.

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