Soft corrections to inclusive deep-inelastic scattering at four loops and beyond

TL;DR

This paper derives four-loop soft and collinear corrections for inclusive deep-inelastic scattering, extends threshold resummation to N$^4$LL order, and analyzes the numerical impact of these high-order corrections in QCD.

Contribution

It provides the complete four-loop soft and collinear contributions to DIS Wilson coefficients and extends threshold resummation to N$^4$LL order, including precise approximations for QCD.

Findings

Derived four-loop soft and collinear contributions to DIS Wilson coefficients.

Extended threshold resummation exponent to N$^4$LL order.

Numerical analysis of N$^4$LL corrections' impact on DIS.

Abstract

We study the threshold corrections for inclusive deep-inelastic scattering (DIS) and their all-order resummation. Using recent results for the QCD form factor, related anomalous dimensions and Mellin moments of DIS structure functions at four loops we derive the complete soft and collinear contributions to the DIS Wilson coefficients at four loops. For a general $SU(n_c)$ gauge group the results are exact in the large-$n_c$ approximation and for QCD with $n_c=3$ we present precise approximations. We extend the threshold resummation exponent $G^N$ in Mellin-$N$ space to the fifth logarithmic (N$^4$LL) order collecting the terms $\alpha_{\rm s}^{\,3} (\alpha_{\rm s} \ln N)^n$ to all orders in the strong coupling constant $\alpha_{\rm s}$. We study the numerical effect of the N$^4$LL corrections using both the fully exponentiated form and the expansion of the coefficient function in towers of logarithms. As a byproduct, we derive a numerical result for the complete pole structure of the QCD form factor in the parameter of dimensional regularization $\varepsilon$ at four loops.