The Singlet Contribution to the Structure Function $g_1(x,Q^2)$ at Small $x$

TL;DR

This paper investigates the small-x behavior of the polarized singlet structure function g1 by resumming the leading logarithmic terms that arise in the singlet evolution. Using a renormalization-group approach, it derives the dominant small-x contributions to the splitting-function matrix and predicts the leading terms of the three-loop ($P^{(2)}$) matrix in the MSbar scheme. The authors provide explicit expressions for the small-x limit of the evolution kernels, demonstrate supersymmetric constraints, and perform numerical studies with GRSV input, revealing large potential corrections at small x that are highly sensitive to subleading terms and to the polarized gluon distribution. The work highlights the need to compute subleading small-x contributions for robust polarized DIS predictions and underscores the significant impact of input uncertainties on small-x evolution.

Abstract

The resummation of $O(α_s^{l+1} \ln^{2l} x)$ terms in the evolution equation of the singlet part of $g_1(x,Q^2)$ is carried out. The corresponding singlet evolution kernels are calculated explicitely. The leading small-$x$ contribution to the three-loop splitting function matrix is determined in the $\overline{\rm MS}$ scheme. Relations are derived for the case of ${\cal N} = 1$ supersymmetric Yang--Mills field theory. Numerical results are presented for the polarized singlet and gluon densities, and the structure functions $g_1^{\, p}(x,Q^2)$ and $g_1^{\, n}(x,Q^2)$. They are compared for different assumptions on the non--perturbative input distributions, and the stability of the results against presently unknown subleading contributions is investigated.