Quark Model Description of Polarised Deep Inelastic Scattering and the prediction of $g_2$

TL;DR

This paper examines polarized deep inelastic scattering to predict the structure function $g_2$ from $g_1$ within a covariant quark parton model that neglects gluons. By preserving quark transverse momentum and analyzing the operator product expansion, the authors show that $g_2$ is fixed by $g_1$ through the WW relation in the massless limit, while addressing apparent contradictions in earlier formulations. They extend the analysis to finite quark masses, where WW is violated but the Burkhardt-Cottingham sum rule remains intact, and propose a plausible mechanism linking parton polarization components to derive a mass-dependent relation between $g_1$ and $g_2$. Phenomenological fits indicate WW is consistent with data for small quark masses, with mass effects introducing measurable, though typically small, deviations that could, with improved data, constrain light-quark masses. The work thus provides a coherent framework tying g1 and g2, clarifies the role of quark masses, and offers a pathway to test the validity of a pure quark-level description of polarized DIS against experimental measurements.

Abstract

We show how the operator product expansion evaluated in the approximation of ignoring gluons leads to the covariant formulation of the quark parton model. We discuss the connection with other formulations and show how the free quark model prediction, $g_2=0$, changes smoothly into the Wandzura-Wilczek (WW) relation for quark masses small relative to the nucleon mass. Previous contradictory parton model predictions are shown to follow from an inconsistent treatment of the mass shell conditions. The description is extended to include quark mass corrections.

Figures (4)

• Figure 1: a) Form of the operator matrix elements in the absence of gluons. b) Form of the structure function using the OPE and neglecting the gluons both in the coefficient functions and the operator matrix elements.
• Figure 2: Fit to the data on $g_1(x)$ from refs(4,5,6) using eq.(6) and the comparison of the resulting prediction for $g_2(x)$ from eq.(7) with the preliminary data of ref(1). This is the $m=0$ case.
• Figure 3: Fits to the data on $g_1(x)$ from refs(4,5,6) using eqs.(25,31) for $m/M$ up to 0.2 and the comparison of the resulting predictions for $g_2(x)$ from eqs.(25,31) with the preliminary data of ref(1).
• Figure 4: The two components $g_i^{(1)}(x)$ and $g_i^{(3)}(x)$ for the fits shown in Fig.3