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Valence Quark Spin Distribution Functions

The hyperfine interactions of the constituent quark model provide a natural explanation for many nucleon properties, including the Delta-N splitting, the charge radius of the neutron, and the observation that the proton's quark distribution function ratio d(x)/u(x)->0 as x->1. The hyperfine-perturbed quark model also makes predictions for the nucleon spin-dependent distribution functions. Precision measurements of the resulting asymmetries A_1^p(x) and A_1^n(x) in the valence region can test this model and thereby the hypothesis that the valence quark spin distributions are"normal".


I. INTRODUCTION
The quark model has enjoyed so much success as a qualitative guide to hadronic structure that the discovery that only about 30% of the proton's spin could be attributed to quark spin came as a surprise. Since the quark model remains unjustified within QCD, it is a misnomer to call this "proton spin surprise" the "proton spin crisis". However, whatever we call it, this result has generated much very productive experimental and theoretical activity.
While in general the spin of the proton could reside on any mixture of its quark and gluon constituents or in their orbital angular momenta, a conservative interpretation [1] of the current situation is that the valence quarks carry the spin expected by the quark model but that the low x sea of qq pairs is negatively polarized. In this case Σ (defined to be twice the expectation value of the quark plus antiquark spin along the spin direction of a polarized proton, so that Σ = 1 would saturate the proton spin), when decomposed into its valence and sea components, would be where Σ v = dxΣ v (x) is twice the spin on the valence quarks and ∆q sea = dx∆q sea (x) and ∆q sea = dx∆q sea (x) are, respectively, twice the spin on the sea quarks and antiquarks of flavor q. If the valence quarks were in nonrelativistic S-waves as in the naive quark model, then Σ v would be unity. However, as has been appreciated for nearly thirty years [2], in realistic valence quark models lower components of quarks spinors convert about 25% of the quark spin into orbital angular momentum so that Σ v ≃ 0.75. If in addition each of the three light quark flavors carries ∆(q +q) sea ≃ −0.15, a very modest per flavor effect, Σ ≃ 0.30 would follow. Sea quark polarizations of just this sign and magnitude have recently been obtained in a realistic model of qq pair creation [1]. (In a more general context, such small ∆(q +q) sea values are perfectly consistent with a 1/N c expansion of QCD (where sea quarks appear at order 1/N c via quark-antiquark loops). Note that the condition ∆(q +q) sea ≪ 1, not how accurately Σ v approximates unity, determines the applicability of the 1/N c expansion: any nonzero ∆(q +q) sea would lead to a "spin crisis" as N f (the number of light flavors) tends to infinity. ) In the conservative scenario just described, both the 25% relativistic quenching of spin from Σ v and the negative polarization of ∆(q +q) sea are compensated by orbital angular momentum. In general, however, we are only guaranteed that (where L q is the quark and antiquark orbital angular momentum and 1 2 Σ g is the total angular momentum residing in the gluonic fields), so major experimental efforts are planned to measure the component parts of Eq. (2) in an effort to disentangle the "spin crisis". These efforts begin with planned extensions of deep inelastic lepton scattering measurements of the proton and neutron spin structure functions down to very small x to complete the integrals required to calculate Σ, and studies of the Q 2 -dependence of spin structure functions to make inferences about ∆g(x), the gluon helicity contribution to Σ g (x). Major efforts are also planned to directly measure ∆g(x) based on helicity-dependent gluon-parton cross sections. In addition to these classical inclusive measurements, flavor-tagging semi-inclusive experiments are planned to measure separately ∆s sea (x), ∆s sea (x), ∆ū sea (x), ∆d sea (x), and also the quark contributions ∆u(x) ≡ ∆u v (x) + ∆u sea (x) and ∆d(x) ≡ ∆d v (x) + ∆d sea (x).
(Note that it is not possible to experimentally separate the quark contributions ∆u sea (x) and ∆d sea (x) from ∆u v (x) and ∆d v (x): this separation is conceptual only.) Additional complementary information on the ss content of the proton is expected from planned measurements of the electric and magnetic form factors G s E and G s M of thesγ µ s current using parity-violating electron-nucleon elastic scattering.
Given the substantial effort being devoted to this problem, it is surprising that we still do not know whether our original simple picture of the spin structure of the valence quarks is right! To some degree this is because this question is not well-defined: in contrast to other methods (e.g., QCD sum rules [3]), the quark model is not normally embedded in a fieldtheoretic framework. As a result, there are many difficulties in making comparisons between the "predictions" of the quark model and the precisely defined quantities measured in deep inelastic scattering. As two illustrations of such difficulties, I note that: 1) the separation of Eq. (2) is Q 2 -dependent (e.g., ∆g might be small at low Q 2 but very important at large Q 2 ) and, as mentioned above, 2) the u and d contributions to ∆u v (x) and ∆d v (x) cannot be disentangled from those to ∆u sea (x) and ∆d sea (x). However, beyond x ≃ 0.3, sea quarks and antiquarks are scarce and, since gluons are too, such intrinsically field-theoretic issues as the factorization scheme dependence of Σ v (x) associated with the gluon anomaly [5]  What are the valence quark model predictions for the resulting polarization asymmetries A p 1 (x) and A n 1 (x) in the valence region? Ignoring Q 2 evolution, they are where u v (x) and d v (x) are the unpolarized valence distribution functions which integrate to 2 and 1, respectively. From these formulas it is clear that the predictions depend on knowing the interplay between the valence quark spin and momentum wavefunctions so that there can be no unique prediction of the valence quark model for these asymmetries. However, I will argue here that its predictions are sufficiently well-determined that they can be used to answer the simple question of whether the valence spin structure is "normal" or not.
Aside from this observation, there is little in this paper that could not be extracted from earlier work on this subject to which I will refer below. However, the results of this earlier work vary widely since they are based on diverse methods of dealing with relativistic internal quark motion, various prescriptions for boosting to the infinite momentum frame, ad hoc versus dynamical origins for the assumed SU(6)-breaking, potential versus bag models, and choices of quark masses. Here I will assume that the hyperfine interaction is responsible for the d(x)/u(x) ratio as x → 1, and then normalize predictions for the valence quark spin distribution functions to the data on this ratio. In doing so, I will not only avoid much model dependence, but also most of the pitfalls discussed above associated with not knowing precisely how to embed the quark model in field theory.  The body of this paper builds up to these predictions in steps. In the next Section I will review the naive SU(6) predictions and then modify them within the context of SU (6) by allowing the quarks to have relativistic internal motions. I then describe the breaking of SU(6)-symmetric quark spin distributions in the hyperfine-perturbed quark model and close with a brief historical overview.
II. THE SU (6) AND "RELATIVIZED SU (6)" DISTRIBUTION FUNCTIONS I begin by recalling that in SU(6) one may simply write where since C A and ψ S are the antisymmetric color and symmetric L = 0 spatial wavefunctions, is the unique spin-1 2 wavefunction which is symmetric in the first two quarks as required by the Pauli principle [6]. In the nonrelativistic SU(6) quark model one therefore expects where v SU (6) (x) is the universal SU(6) distribution function associated with ψ S . These distributions lead to the standard SU(6) predictions d(x)/u(x) = 1/2, A p 1 (x) = 5/9 and A n 1 (x) = 0, and G A = 5/3. When the relativistic quenching mentioned above [2] is turned on, it creates an x-dependent probability which we denote by 1 2 c A (x) for a spin up (down) quark to be flipped to down (up). This reshuffling of probability leads to the "relativistic SU(6)" spin distributions d where with the nucleon expectation value 1−c A (x) N ≃ 3 5 G A ≃ 0.75, the integrated valence spins become so that the "relativistic SU(6)" spin distributions satisfy the Bjorken sum rule. However, among other problems, the "relativistic SU(6)" model still makes the incorrect prediction d(x)/u(x) = 1/2. Note that the model also predicts that A p 1 (x) = [1 − c A (x)]5/9 and A n 1 (x) = 0 as x → 1, which, since c A (x) → 0 as x → 1, is not obviously wrong (see Fig. 1).

III. PREDICTIONS OF THE HYPERFINE-PERTURBED QUARK MODEL
Since the zeroth-order nucleons are pure S-waves, in the hyperfine-perturbed quark model [7], only the Fermi contact part of the hyperfine interaction (the S i · S j δ 3 ( r ij ) force responsible for the ∆ − N mass splitting) is operative in perturbing the nucleon's energy in first order.
What does this perturbation do? In the nucleon rest frame, quark pairs with spin 1 have their energies raised (as in the ∆) while pairs with spin zero have their energies lowered.
Since χ λ has the two u quarks in a pure spin one state, while each ud pair is in a mixture of spin one and spin zero (with spin zero dominant so that the net perturbation in a nucleon decreases its energy), up quarks acquire higher average energy than down quarks. This physics then immediately suggests that the neutron will have a negative charge radius and where (ψ ρ , ψ λ ) are mixed symmetry wavefunctions of the permutation group S 3 which are antisymmetric (ρ) and symmetric (λ) under 1 ↔ 2 interchange, where and where θ m is a small mixing angle induced by SU(6)-breaking interactions [9]. (Since, as explained above, the L = 0 ground state energies are perturbed in first order only by the S i · S j interaction, one can ignore L = 2 and totally antisymmetric L = 0 admixtures.) It follows that the rest frame probabilities of spin up and spin down d quarks are, to first order in θ m , Similarly one finds where now the wavefunction ψ ρ does play a role. I have now suppressed coordinate labels which indicate that the probability P (u ↑) (P (u ↓)) is that for finding a spin up (spin down) u quark at a point r u while the other up quark is at position α and the d quark is at position β.
Note that, as advertized, the net leading-order effect of the SU(6)-breaking in the spinaveraged probabilities is to create distributions of mixed symmetry that allow the d quark to have a different probability distribution from the two u quarks. With the calculated quark model value [9] sinθ m ≃ −0.23, the distortion of the SU(6)-symmetric probabilities is substantial. I now make the natural assumption that this distortion translates into the observation that d(x)/u(x) → 0 as x → 1, and associate the measured u(x) and d(x) with functions u v (x) and d v (x) associated with the spin-averaged probability ψ S − 1 2 θ m ψ λ 2 .
This remarkably simple picture then leads to the "standard" prediction F n 2 /F p 2 → 1 4 as x → 1. is due to u ↑ (x). This means that The resulting predictions for A p 1 (x) and A n 1 (x) in the valence region, shown in Fig. 1, can be obtained without engaging in an elaborate parameterization of structure functions. Using the rough parameterizations d(x)/u(x) ≃ κ(1 − x) as x → 1 (with 0.5 < κ < 0.6) and c A (x) = nx(1 − x) n (which builds in c A (x) → 0 as x → 1 and x → 0 and for 2 < n < 4 gives the required quenching of G A ), produces the narrow bands shown in the figure. As x → 1 both A p 1 and A n 1 tend to 1, but I show the predictions only in the region where the valence quark wavefunction is large since very small effects might become important at the endpoint [11].

IV. SOME HISTORY
The history of the prediction of the effects of SU(6)-breaking on the quark distribution functions in the valence region is somewhat convoluted. It perhaps begins with the parton model discussion by Feynman [12] who argues that as a u or d quark approaches x = 1, it must leave behind "wee" partons with either I = 0 or I = 1, and that these two configurations are unlikely to have the same x-dependence. He then notes that if the I = 0 configuration dominates as x → 1, the observed ratio F n 2 /F p 2 = 1/4 would follow. If we take the modern view that this high x behaviour will be controlled by the valence quarks, and note the quark model correlation between isospin and spin in the valence quark sector, this argument would also naively lead to the conclusion that u v ↑ (x) will dominate as x → 1.
While correct, since Feynman's argument relies on the "wee" partons being uncorrelated with the leading quark, and so does not take into account the required antisymmetrization between the leading u quark and the "wee" u quark, its predictions for the full valence region are unclear.
A more complete quark model argument is given in the papers of Close [13] and Carlitz and Kaur [14]. They argued that SU(6)-breaking changes Eq. (5) into where χ 0 ud = 1 2 (↑↓ − ↓↑), χ 11 ud =↑↑, and χ 10 ud = 1 2 (↑↓ + ↓↑) are the S = 0 and two S = 1 ud spin wavefunctions. For ψ 0 = ψ 1 = ψ S , this wavefunction collapses to (5). Referring to hyperfine forces as driving the physics (which is equivalent to Feynman's assumption in this case), these papers posit that SU(6)-breaking leads to ψ 0 = ψ 1 , which would in turn 3 v 1 (x) in terms of distribution functions v 0 and v 1 associated with ψ 0 and ψ 1 , respectively, with v 1 /v 0 → 0 as x → 1. This model thus also leads to F n 2 /F p 2 → 1 4 as x → 1 and it predicts that both A p 1 (x) and A n 1 (x) → 1 as x → 1. While assuming ψ 0 = ψ 1 is very natural, since these diquark spin states are eigenstates of the hyperfine interaction, this assumption is not consistent with the Pauli principle unless ψ 0 and ψ 1 have very special properties under the permutation group S 3 or the wavefunction (28) is antisymmetrized. Thus, as with Feynman's argument, it is unclear what these models predict for the full valence region. The closely related model of Close and Thomas [15] is based on examining the energy of the spectator diquark after the deep inelastic scattering in a rest frame calculation of deep inelastic structure functions. Although, as pointed out by the authors, their calculation suffers from the fact that the diquark is a colored object which cannot have a well-defined energy, this calculation emphasizes the same physics and reaches the same conclusions as Refs. [13,14]. Given that the impact of the hyperfine interaction is implemented somewhat intuitively in this work, it is once again unclear whether the results presented are reliable for anything other than the x → 1 behaviour.
Although they do not use the hyperfine-perturbed quark model, the formalism required to deal explicitly with the fully antisymmetrized nucleon wavefunction seems to have first been applied to the valence quark spin distribution functions by Le Yaouanc et al. [16].
They introduce an SU(6) intraband mixing between the ground state [56, 0 + ] and the mixed symmetry [70, 0 + ] in an attempt to account for the observed behaviour F n 2 /F p 2 → 1 4 as x → 1, i.e., d(x)/u(x) → 0 as x → 1. This is precisely the kind of mixing introduced in (18) as required by color hyperfine interactions. (In fact, using this formalism makes calculations much simpler than in the uds basis, though perhaps less physically transparent.) They then make a prescription to boost this mixed wavefunction into the infinite momentum frame, fit the mixing angle to the data, and discuss the implications of such mixing to a wide range of phenomena.
More recently, a number of authors [17][18][19][20] have addressed the connection between the hyperfine-perturbed quark model (either potential-based or bag-like) and the quark distribution functions. Most of these papers find the same two key effects I have emphasized here: axial current quenching by internal quark motion and u quark dominance as x → 1.
Despite this extensive body of work [12][13][14][15][16][17][18][19][20], it does not seem to be widely appreciated that the hyperfine-perturbed valence quark model makes quite clear predictions for the asymmetries A p 1 (x) and A n 1 (x) in the valence region. I attribute this state of affairs to the fact that this work has been very ambitious: most authors have attempted "absolute" calculations of structure functions. In doing so they encountered many obstacles, which forced them to a variety of assumptions, approximations, and "procedures". The result is a wide range of predictions for the structure functions with apparent agreement only on their qualitative features.

V. CONCLUSIONS
In this paper I have shown that once it is assumed that the hyperfine perturbations of the quark model are responsible for the SU(6)-breaking observed in the structure functions, a very narrow band of predictions follows. In a broader context, I have argued that the extensive measurements and theoretical studies engendered by the "spin crisis" should be anchored in knowledge of whether the valence quark spin distributions are in fact anomalous. Thus whether the distributions described here prove to be correct when confronted with the data will be interesting, but not as important as the fact that such data will indicate whether the valence spin structure functions are in fact anomalous, and thus guide the search for where the resolution of the "spin crisis" is to be found.