Double Parton Scattering Singularity in One-Loop Integrals

TL;DR

This paper analyzes the double parton scattering singularity in one-loop integrals, deriving a simple formula to identify divergences and explaining why some amplitudes are finite while others diverge.

Contribution

It introduces a general formula for DPS singularities in one-loop diagrams with arbitrary external particles, clarifying their occurrence in specific photon amplitudes.

Findings

Derived a simple formula for DPS singular parts of four-point diagrams.

Explained why certain six-photon amplitudes are finite or divergent at DPS points.

Established consistency with the two-parton GPD framework, but not with the double PDF framework.

Abstract

We present a detailed study of the double parton scattering (DPS) singularity, which is a specific type of Landau singularity that can occur in certain one-loop graphs in theories with massless particles. A simple formula for the DPS singular part of a four-point diagram with arbitrary internal/external particles is derived in terms of the transverse momentum integral of a product of light cone wavefunctions with tree-level matrix elements. This is used to reproduce and explain some results for DPS singularities in box integrals that have been obtained using traditional loop integration techniques. The formula can be straightforwardly generalised to calculate the DPS singularity in loops with an arbitrary number of external particles. We use the generalised version to explain why the specific MHV and NMHV six-photon amplitudes often studied by the NLO multileg community are not divergent at the DPS singular point, and point out that whilst all NMHV amplitudes are always finite, certain MHV amplitudes do contain a DPS divergence. It is shown that our framework for calculating DPS divergences in loop diagrams is entirely consistent with the `two-parton GPD' framework of Diehl and Schafer for calculating proton-proton DPS cross sections, but is inconsistent with the `double PDF' framework of Snigirev.