Analytic Continuation and Reciprocity Relation for Collinear Splitting in QCD

It is well-known that direct analytic continuation of the 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 due to splitting functions not being analytic functions of external momenta. However, splitting functions can be constructed from the squares of (generalized) splitting amplitudes. We establish the rules 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.