Higher-order corrections to the splitting functions from differential equations in QCD

TL;DR

This paper advances the calculation of time-like splitting functions in QCD at NNLO by developing a differential equations method for master integrals, addressing gaps in previous indirect determinations.

Contribution

It introduces a first-principles approach to compute the missing parts of the three-loop time-like splitting functions in QCD.

Findings

Development of a differential equations method for master integrals.

Progress towards a complete NNLO calculation of time-like splitting functions.

Enhanced understanding of parton fragmentation in high-energy processes.

Abstract

We report on the status an ab initio computation of the time-like splitting functions at next-to-next-to-leading order in QCD. Time-like splitting functions govern the collinear kinematics of inclusive hadron production in $e^+e^-$ annihilation and the evolution of the parton fragmentation distributions. Current knowledge about them at three loops has been inferred by means of crossing symmetry from their related space-like counterparts, the deep-inelastic structure functions and parton densities. In this approach certain parts of the off-diagonal quark-gluon splitting function are left undetermined, which calls for an independent calculation from first principles. We outline the method for calculating master integrals from differential equations which are required to attack the problem.