# Soft corrections to inclusive DIS at four loops and beyond

## TL;DR

This paper advances the understanding of threshold corrections in deep-inelastic scattering by calculating four-loop contributions to soft-gluon exponentiation, showing these corrections are small and improve the stability of perturbative QCD predictions.

## Contribution

It derives the fourth-order contribution to the coefficient of the quark form factor and the N$^4$LL exponentiation coefficient in DIS, extending soft-gluon resummation to five loops.

## Key findings

- N$^4$LL corrections are small.
- The results stabilize the perturbative expansion.
- An approximation scheme yields accurate N$^4$LL results for full QCD.

## Abstract

We study the threshold corrections to the structure functions in deep-inelastic scattering (DIS) at the fifth logarithmic (N$^4$LL) order of the soft-gluon exponentiation in massless perturbative QCD. Using recent results for the splitting functions and the quark form factor, we derive the fourth-order contribution to the coefficient $f^{\rm q}$ of the form factor and from it the N$^4$LL part of the exponentiation coefficient $B^{\rm DIS}$ in the limit of a large number of colours. An approximation scheme is shown that leads to sufficiently accurate N$^4$LL results for full QCD. The N$^4$LL corrections are small and lead to a further stabilization of the perturbative expansion for the soft-gluon exponent.