Analytic Continuation and Reciprocity Relation for Collinear Splitting in QCD

TL;DR

This paper addresses the breakdown of analytic continuation of splitting functions in QCD at three loops, establishing a new rule for continuation of splitting amplitudes and deriving time-like splitting functions without ambiguity.

Contribution

It introduces a novel analytic continuation rule for splitting amplitudes and derives three-loop time-like splitting functions, also proposing a reciprocity relation in QCD.

Findings

Derived time-like splitting functions at three loops without ambiguity.

Established a new rule for analytic continuation of splitting amplitudes.

Provided evidence supporting a reciprocity relation in QCD at three loops.

Abstract

It is well-known that direct analytic continuation of DGLAP evolution kernel (splitting functions) from space-like to time-like kinematics breaks down at three loops. We identify the origin of this breakdown as splitting functions are not analytic function of external momenta. However, splitting functions can be constructed from square of (generalized) splitting amplitudes. We establish the rule of analytic continuation for splitting amplitudes, and use them to determine the analytic continuation of certain holomorphic and anti-holomorphic part of splitting functions and transverse-momentum dependent distributions. In this way we derive the time-like splitting functions at three loops without ambiguity. We also propose a reciprocity relation for singlet splitting functions, and provide non-trivial evidence that it holds in QCD at least through three loops.