On multi-propagator angular integrals

TL;DR

This paper develops new methods for evaluating complex multi-propagator angular integrals in phase space, providing explicit solutions and recursive relations that advance computational techniques in particle physics.

Contribution

It introduces an Euler integral representation, recursive IBP reduction, and explicit solutions for integrals with multiple denominators, extending the computational toolkit for phase space integrals.

Findings

Explicit calculation of four-denominator angular integrals for any mass configuration.

Development of recursive IBP reduction and dimensional shift relations.

All-order epsilon expansion of the massless three-denominator integral with soft logarithm resummation.

Abstract

We study multi-propagator angular integrals, a class of phase-space integrals relevant to processes with multiple observed final states and a test-bed for transferring loop-integral technology to phase space integrals without reversed unitarity. We present an Euler integral representation similar to Lee-Pomeransky representation and explicitly describe a recursive IBP reduction and dimensional shift relations for the general case of $n$ denominators. On the level of master integrals, applying a differential equation approach, we explicitly calculate the previously unknown angular integrals with four denominators for any number of masses to finite order in $\varepsilon$. Extending the idea of dimensional recurrence, we explore the decomposition of angular integrals into branch integrals reducing the number of scales in the master integrals from $(n+1)n/2$ to $n+1$. To showcase the potential of this method, we calculate the massless three denominator integral to establish all-order results in $\varepsilon$ including a resummation of soft logarithms.