Two-loop splitting functions in QCD

TL;DR

The paper advances the understanding of perturbative QCD by deriving universal two-loop splitting functions that govern the collinear limits of massless partons. It achieves this by extracting the limits from explicit two-loop four-point helicity amplitudes in the 't Hooft–Veltman scheme and cross-checking with unitarity-based results for gluon splittings. The work furnishes tree-, one-, and two-loop results for quark-gluon, gluon-gluon, and quark-antiquark splittings, and provides a complete two-loop soft splitting function, with the pole structure consistent with Catani’s infrared factorization. These results enable rigorous NNLO/NNNLO computations of jet observables and offer robust cross-checks for multi-loop QCD amplitudes.

Abstract

We present the universal two-loop splitting functions that describe the limits of two-loop $n$-point amplitudes of massless particles when two of the momenta are collinear. To derive the splitting amplitudes, we take the collinear limits of explicit two-loop four-point helicity amplitudes computed in the 't Hooft-Veltman scheme. The $g \to gg$ splitting amplitude has recently been computed using the unitarity sewing method and we find complete agreement with the results of Ref. \cite{Bern:2lsplit}. The two-loop $q \to qg$ and $g \to q\bar q$ splitting functions are new results. We also provide an expression for the two-loop soft splitting function.