Threshold resummation at N3LL accuracy and approximate N3LO corrections to semi-inclusive DIS

We advance the threshold resummation formalism for semi-inclusive deep-inelastic scattering (SIDIS) to next-to-next-to-next-to-leading logarithmic (N3LL) order, including the three-loop hard factor. We expand the results in the strong coupling to obtain approximate next-to-next-to-next-to-leading order (N3LO) corrections for the SIDIS cross section. In Mellin moment space, these corrections include all terms that are logarithmically enhanced at threshold, or that are constant. We also consider a set of corrections that are suppressed near threshold. Our numerical estimates show modest changes of the cross section by the approximate N3LO terms, suggesting a very good perturbative stability of the SIDIS process.