Master integrals for splitting functions from differential equations in QCD

TL;DR

This paper introduces a method using differential equations and basis selection to efficiently compute phase-space master integrals in QCD, enabling higher-order precision calculations for splitting functions.

Contribution

It presents a novel algorithm for constructing a basis that simplifies differential equations, facilitating recursive solutions in terms of (G)HPLs for QCD master integrals.

Findings

Successfully calculated NLO time-like splitting function integrals.

Demonstrated the method's power for recursive solutions in epsilon expansion.

Discussed potential applications at NNLO precision.

Abstract

A method for calculating phase-space master integrals for the decay process $1 \to n$ massless partons in QCD using integration-by-parts and differential equations techniques is discussed. The method is based on the appropriate choice of the basis for master integrals which leads to significant simplification of differential equations. We describe an algorithm how to construct the desirable basis, so that the resulting system of differential equations can be recursively solved in terms of (G)HPLs as a series in the dimensional regulator $\epsilon$ to any order. We demonstrate its power by calculating master integrals for the NLO time-like splitting functions and discuss future applications of the proposed method at the NNLO precision.