On the Resummation of $α\ln^2 x$ Terms for Non-Singlet Structure Functions in QED and QCD

TL;DR

The paper addresses the resummation of leading small-$x$ contributions to non-singlet evolution kernels in both QED and QCD, focusing on terms of order $O(\alpha^{l+1} \ln^{2l} x)$. It derives all-order resummed kernels from positive/negative signature amplitudes and presents closed-form solutions for $\Gamma_{x\rightarrow 0}^{\pm}$, translating them into $K_{x\rightarrow 0}^{\pm}(x,a)$ and confirming consistency with known NLO results in the MS-bar scheme. The authors predict the $ obreak{a^3\ln^4 x}$ contributions to the non-singlet splitting functions and perform a detailed numerical study for unpolarized and polarized nucleon structure functions, finding small resummation effects in the unpolarized case (about 1% even at $x$ as small as $10^{-15}$) but potentially larger, input-dependent effects in the polarized case (up to ~15% near $x\sim10^{-5}$). Overall, the results support fixed-order perturbation theory as the reliable framework for non-singlet evolution at small $x$ within current experimental reach, while highlighting scheme-dependent uncertainties and the role of unknown subleading terms.

Abstract

The resummation of $O(α^{l+1} \ln^{2l} x)$ terms in the evolution kernels of non--singlet combinations of structure functions is investigated for both QED abd QCD. Numerical results are presented for unpolarized and polarized structure functions.