Intercepts of the non-singlet structure functions

TL;DR

The paper develops two-dimensional infrared evolution equations for the non-singlet DIS structure functions at small $x$, incorporating running $oldsymbol{\alpha_s}$ to all orders and distinguishing the $f_1^{NS}$ and $g_1^{NS}$ cases via signature effects. By solving the IREE, the authors obtain square-root type intercepts $\\omega_0^{(+)}$ and $\\omega_0^{(-)}$ that govern the power-like small-$x$ growth, and they provide asymptotic forms $f_1^{NS}\sim x^{-\\omega_0^{(+)}}(Q^2/\\mu^2)^{\\omega_0^{(+)}/2}$ and $g_1^{NS}\sim x^{-\\omega_0^{(-)}}(Q^2/\\mu^2)^{\\omega_0^{(-)}/2}$, with numerical estimates $\\omega_0^{(+)}\approx0.37$ and $\\omega_0^{(-)}\approx0.40$ for $\\mu$ in the few-GeV range. The work compares these results to LLA and to DGLAP, highlighting the role of running coupling and $\\pi^2$ terms, the limited reliability of LLA for the $Q^2$-dependence at HERA, and the connection to DGLAP through a consistent resummation of the leading small-$x$ contributions. Overall, it provides a more realistic small-$x$ framework for non-singlet DIS and clarifies the regimes where LLA and DGLAP approximations are applicable.

Abstract

Infrared evolution equations for small-$x$ behaviour of the non-singlet structure functions $f_1^{NS}$ and $g_1^{NS}$ are obtained and solved in the next-to-leading approximation, to all orders in $α_s$, and including running $α_s$ effects. The intercepts of these structure functions, i.e. the exponents of the power-like small-$x$ behaviour, are calculated. A detailed comparison with the leading logarithmic approximation (LLA) and DGLAP is made. We explain why the LLA predictions for the small-$x$ dependence of the structure functions may be more reliable than the prediction for the $Q^2$ dependence in the range of $Q^2$ explored at HERA.

Figures (5)

• Figure 1: The Born graphs for the DIS amplitude $M_{\mu\nu}$.
• Figure 2: Two-loop graph obtained from Fig. \ref{['Born']}b, contributing to $\Im_s M_{\mu\nu}^{(-)}$ (a), and the corresponding physical process (b).
• Figure 3: The evolution equation for the DIS amplitude $M_{\mu\nu}$.
• Figure 4: The evolution equation for the quark scattering amplitude $M_0$.
• Figure 5: Dependence of the intercept $\omega_0$ on infrared cut-off $\eta=\ln(\mu^2/\Lambda_{QCD})$. 1: for $f_1^{NS}$, 2: for $g_1^{NS}$, 3 and 4: for $f_1^{NS}$ and $g_1^{NS}$, respectively, without accounting for $\pi^2$-terms.