Splitting amplitudes at N$^3$LO in QCD

TL;DR

This paper calculates universal splitting amplitudes in QCD up to third loop order for both time-like and space-like processes, enhancing understanding of parton collinear limits at high precision.

Contribution

It provides the first complete third-loop splitting amplitudes for two collinear partons in QCD, including space-like cases with factorization violation, using a novel expansion-by-subgraph method.

Findings

Third-loop splitting amplitudes for time-like processes computed.

Space-like splitting amplitudes at third loop order derived.

New expansion-by-subgraph technique developed for these calculations.

Abstract

In the limit where partons become collinear to each other, scattering amplitudes factorize into a product of universal, process-independent building blocks and scattering amplitudes involving fewer partons. We compute these universal building blocks -- known as splitting amplitudes -- for two collinear QCD partons up to third loop order in QCD. Our results describe arbitrary time-like splitting processes. Due to the violation of strict collinear factorization in space-like splitting processes, we specifically present space-like splitting amplitudes for three-parton QCD scattering amplitudes at third loop order. To achieve our results, we perform a collinear expansion of three-loop scattering amplitudes using a new expansion-by-subgraph technology, which is based on the method of regions.