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Spin asymmetries for confined Dirac particles

We study the Bjorken `x' (or equivalently Nachtmann `xi') dependence of the virtual photon spin asymmetry in polarized deep inelastic scattering of electrons from hadrons. We use an exactly solved relativistic potential model of the hadron, treating the constituents as independent massless Dirac particles confined to an infinitely massive force center. The importance of including the p-wave components of the Dirac wave function is demonstrated. Comparisons are made to the observed data on the proton by taking into account the observed flavor dependence of the valence quark distribution functions.

We study the Bjorken x (or equivalently Nachtmann ξ) dependence of the virtual photon spin asymmetry in polarized deep inelastic scattering of electrons from hadrons. We use an exactly solved relativistic potential model of the hadron, treating the constituents as independent massless Dirac particles confined to an infinitely massive force center. The importance of including the p-wave components of the Dirac wave function is demonstrated. Comparisons are made to the observed data on the proton by taking into account the observed flavor dependence of the valence quark distribution functions. The structure functions of the nucleon as measured by deep inelastic scattering (DIS) of leptons from nucleons have, over the last 30 years, received much attention. In particular, the evolution with momentum transfer Q 2 has been quantitatively understood in terms of quantum chromodynamics (QCD).
During the last decade, the interest has been concentrated on the spin structure functions. Much of this interest was due to the fact that the integral over the experimental spin structure function g 1 (x) yielded values that were much lower than the ones expected in the naive quark model (for reviews see e.g. [1,2,3]). The presence of this "spin crisis" has led to many different ideas on how to account for the nucleon spin. The contribution of ss-components, the contribution of orbital angular momenta of the quarks and the contribution of gluons have been discussed. It also has been pointed out early on [4] that the non-relativistic quark model overestimates the quark contribution to the nucleon spin. Relativistic effects lead to a reduction which today, together with the gluon contribution via the triangle diagram, are believed to provide the main explanation for the low integral over the spin structure function g 1 ; in the present letter, we concentrate on the former.
Relativistic effects are expected to play an important role as the masses of quarks are small compared with their momenta. The non-relativistic quark models for instance also overestimate the axial vector weak coupling constants which experimentally amount to g A /g V = 1.26 rather than 5/3. Calculations with the MIT bag model [5] or using light-cone quantization [6] have indicated that the lower components of the wave function -present in a relativistic description using the Dirac equation -lead to an opposite contribution to the one of the upper com-ponents, which could generate, in the limit of massless quarks, a reduction factor of 0.65 [7].
In the present Letter we study a simple model for relativistic bound quarks. We want to address in particular the x-dependence of g 1 (x), for which otherwise little theoretical guidance is available.
We consider the calculation of the virtual photon spin asymmetry in DIS of a charged leptonic probe from a hadronic target within the model of Ref. [8]. The Hamiltonian in this model is chosen as where α and β are Dirac matrices in the standard representation [9]. It describes a massless Dirac particle in a linear confining well. The half-vector plus half-scalar structure of the confining potential is chosen for its spin symmetry [10] wherein spin-orbit doublets are degenerate. It is motivated by the relatively small spin-orbit splittings seen in meson spectra. Computations are simple with this choice since the lower components of the wave function are not coupled by the potential. The value of the string tension √ σ is assumed to be 1 GeV/fm as indicated by the slopes of baryon Regge trajectories. In Ref. [8] all the eigenstates of this model were obtained exactly for excitation energies up to ∼ 12 GeV. The ground state energy, E 0 for this string tension is 840 MeV. The model may be viewed as a heavy-light meson, such as tu, in the limit that the antiquark mass goes to infinity. However, it retains only the confining part of thetu interaction modeled by a flux tube.
The model neglects gluon and sea-quark contributions to DIS as well as the QCD evolution. However, the observed ratio of the g 1 (x) to F 1 (x), the unpolarized structure function is relatively independent of Q 2 [3], and our objective is to calculate the x-dependence of this ratio for the contribution of valence quarks to DIS. We hope that the model is useful in this context.
The virtual photon asymmetry is defined as [3] with σ 1 2 and σ 3 2 the helicity cross sections for the target angular momentum antiparallel and parallel to the photon helicity, respectively. We may calculate the inclusive virtual photon helicity cross sections in the rest frame of the target as where |q| and ν are the momentum and energy transferred to the target, σ M is the Mott cross section and we have assumed that the virtual photon is in theẑ direction. The ground states |0, j z = ± 1 2 have the total angular momentum projection j z = ± 1 2 . The operator α + corresponds to a virtual photon with positive helicity, and |I are eigenstates of the Hamiltonian H [Eq.(1)] with energies E I .
The calculation of the virtual photon helicity cross sections proceeds in this model, without approximations, exactly as the calculation of the unpolarized structure functions described in Ref. [8]. When |q| is large the σ/σ M depend only onỹ = |q| − ν. the scaling variableỹ, and their sum the unpolarized structure function. The conventionally used Bjorken and Nachtmann scaling variables are related toỹ by [11]: where M T is the target mass. Thus small (large) negativeỹ correspond to small (large) x. We note that the σ 1 2 (ỹ) and σ 3 2 (ỹ) are not proportional, which implies that the A q 1 of a confined relativistic quark has a largeỹ or equivalently x dependence.
The ground state |0, j z of the confined quark is given by: where, f 0 (r) and g 0 (r) are the radial functions for the sand p-waves, respectively, and Y ℓ j,jz are the spin-angle functions obtained by coupling spin and orbital angular momentum to j = 1 2 . The interference in the DIS between the s-and pwaves contributes significantly to theỹ dependence of the σ 1 2 helicity cross-section, A q 1 and F q 1 . The effect of interference is shown in Fig.(2) where we compare the polarized cross section σ 1 2 including interference terms (solid curve, labeled 'full') with the polarized cross section neglecting interference terms (dotted curve). Also shown are the polarized cross sections obtained after keeping only the s-or p-waves in the j z = − 1 2 target. We note that the interference shifts σ 1 2 to more negativeỹ corresponding to larger values of ξ. Only the p-waves contribute to σ 3 2 shown in Fig.(1). The virtual photon asymmetry is given in terms of the spin-dependent structure functions g 1 and g 2 [3] by where γ 2 = 4M 2 T x 2 /Q 2 , in the scaling regime, Q 2 → ∞. As mentioned earlier, the observed A 1 of the proton, A p 1 is largely independent of Q 2 , and is used to extract values of g p 1 /F p 1 [3]. Using the structure functions given in Fig.(1) we can easily calculate the virtual photon asymmetry A q 1 or equivalently the ratio g q 1 /F q 1 for a single confined quark, as a function ofỹ. In order to compare it with the data on protons we have to convert it to a function of ξ by providing a mass scale M T (see Eq. (6)). Our model target has infinite mass associated with the center of the confining potential. However, that mass is not relevant since only the confined quark contributes to DIS. We use M T = 2.5 GeV ∼ 3E 0 , where E 0 is the energy of a single confined quark in the ground state. With this choice the F q 1 (ξ) becomes small at ξ ∼ 0.8 as in the proton. The solid curve in Fig.(3) shows the A q 1 (ξ) or equivalently g q 1 (ξ)/F q 1 (ξ) of a confined quark. The calculated ratio goes to zero at small ξ, and this behavior is independent of the chosen value of M T . The dip at ξ = 0 is due to the shift of σ 1 2 to larger values of ξ, produced by the interference effect shown in Fig.(2). When the interference terms are omitted we obtain the dashed curve in Fig.(3) which has g q 1 /F q 1 ∼ 0.6 at ξ = 0. Alternatively we could have chosen the string tension √ σ such that 3E 0 = M N , the nucleon mass. However, since √ σ provides the only mass scale in the Hamiltonian (1)], this choice gives exactly the same A q 1 (ξ) as the previous.
In the remaining parts of this letter we attempt to estimate the ξ dependence of g p 1 /F p 1 ratio for the proton in the "naive" quark model in which the total PDF (parton distribution functions) are approximated by the sum of the three valence quark contributions. The unpolarized PDF of valence quarks in j z = ± 1 2 states are denoted by u(± 1 2 , ξ) and d(± 1 2 , ξ). We extract these from the MRST [12] PDF fitted to the experimental data, and use only the calculated asymmetry A q 1 (ξ). In order to obtain the asymmetry of the proton we must account for the flavor dependence of the PDF's, as extracted from experiment. It is well known that the flavor dependence of the valence PDF's can be understood in terms of the state of the residual system after removal of the struck quark [11,13]. The residual diquark in DIS from protons can be in a spin-0 or spin-1 state. Removal of a d quark results in a spin-1 diquark, while removal of a u quark results in a spin-0 diquark 3/4 of the time and spin-1 for the remainder.  Let q 0 (ξ) and q 1 (ξ) be the unpolarized PDF when the diquark is in spin-0 and 1 state respectively. From the proton spin-flavor wave function we obtain The empirically known, total unpolarized u(ξ) and d(ξ) are given by Solving the above equations for q 0 (ξ) and q 1 (ξ) using the MRST [12] u(ξ) and d(ξ) we find that q 0 is shifted to larger values of ξ with respect to q 1 as expected from the smaller mass of the spin-0 diquark. Eliminating q 0 and q 1 from Eqs. (9) and (10) we obtain: Due to the presence of p-waves both the j z = ± 1 2 PDF's have spin ↑ and ↓ contributions. For example the u(j z , ξ) are given by: is the spin asymmetry of a quark in the j = 1 2 ground state. In the following we will use A q 1 = g q 1 /F q calculated above using linear confinement, and shown in Fig.(3).
The total spin ↑, ↓ PDF's, summed over j z , are obtained from the above equations. They are given by: We may now compute the spin asymmetry in the proton. Using (u ↑ (ξ) − u ↓ (ξ)) and obtain This ratio is plotted in Fig.(4) as the solid curve and is in fair agreement with the data from Ref. [3] at all ξ. The PDF's evolved to Q 2 = 5 GeV 2 at next-to leading order are used to obtain the results shown in Fig.(4), but the ratio, (4u(ξ) − 3d(ξ))/(4u(ξ) − d(ξ)) is fairly insensitive to Q 2 in the range 2 < Q 2 < 20 GeV 2 . It should be pointed out that at ξ < 0.2 the sea quark contributions to the F p 1 (ξ) are large, particularly at large Q 2 . However, they are neglected in the present calculation.
If we neglect p-waves in the valence quark orbitals, then A q 1 (ξ) = 1. The resulting g p 1 /F p 1 is significantly above the experimental data as shown in Fig.(4). In the SU (6) limit of the naive quark model we have u(ξ) = 2d(ξ) and Eq.(18) reduces to g p 1 (ξ) F p 1 (ξ) Results obtained with this approximation are shown by the dot-dashed curve in Fig.(4). It lies below the data at large ξ. Approximating the A q 1 by unity in the above equation gives the ξ independent result, g p 1 /F p 1 = 5/9, of Kuti and Weisskopf [14].
In conclusion the present work suggests that at least two different effects shape the ξ dependence of A p 1 (ξ). The p-waves in bound quark wave functions interfere with the dominant s-waves to suppress A q 1 (ξ) at small ξ; and the difference in the shapes of the u(ξ) and d(ξ) enhances the A p 1 (ξ) at large ξ. Our model is certainly too simple; it approximates the problem of three interacting quarks by a relativistic onequark problem. Nevertheless p-waves occur very naturally in the wave functions of spin-half relativistic particles, and their effect will presumably exist in more refined treatments of spin asymmetries.