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

Understand

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 $\alpha_{\rm s}$.