Helicity Evolution at Small $x$: the Single-Logarithmic Contribution

TL;DR

This paper derives single-logarithmic corrections to small-$x$ helicity evolution equations, improving the understanding of polarized parton distributions at high energies by including important subleading effects and running coupling corrections.

Contribution

It introduces the first calculation of single-logarithmic contributions to small-$x$ helicity evolution, extending the double-logarithmic approximation with new correction terms and running coupling effects.

Findings

Derived single-logarithmic correction terms for helicity evolution.

Unified small-$x$ evolution kernel with polarized DGLAP splitting functions.

Included running coupling corrections to the evolution kernel.

Abstract

We calculate single-logarithmic corrections to the small-$x$ flavor-singlet helicity evolution equations derived recently in the double-logarithmic approximation. The new single-logarithmic part of the evolution kernel sums up powers of $\alpha_s \ln(1/x)$, which are an important correction to the dominant powers of $\alpha_s \ln^2(1/x)$ summed up by the double-logarithmic kernel at small values of Bjorken $x$ and with $\alpha_s$ the strong coupling constant. The single-logarithmic terms arise separately from either the longitudinal or transverse momentum integrals. Consequently, the evolution equations we derive employing the light-cone perturbation theory simultaneously include the small-$x$ evolution kernel and the leading-order polarized DGLAP splitting functions. We further enhance the equations by calculating the running coupling corrections to the kernel.