Decomposition of Triple Collinear Splitting Functions

TL;DR

This paper analyzes the structure of triple collinear splitting functions in multi-parton scattering amplitudes, explicitly decomposing them into products of two-particle functions and finite remainders, clarifying unresolved singularities.

Contribution

It introduces a decomposition of triple collinear splitting functions into products of two-particle functions and finite remainders, enhancing understanding of unresolved singularities.

Findings

Explicit decomposition of triple collinear splitting functions.

Identification of finite remainders when two particles are collinear.

Analysis of single unresolved singularities in the remainders.

Abstract

In the kinematic region where three particles $i$, $j$, $k$ are collinear, the multi-parton scattering amplitudes factorise into a product of a triple collinear splitting function and a multi-parton scattering amplitude with two fewer particles. These triple collinear splitting functions contain both iterated single unresolved contributions, and genuine double unresolved contributions. We make this explicit by rewriting the known triple collinear splitting functions in terms of products of two-particle splitting functions, and a remainder that is explicitly finite when any two of $\{i,j,k\}$ are collinear. We analyse all of the single unresolved singularities present in the remainder.