Expansion by regions meets angular integrals

TL;DR

This paper develops a method to analyze the small-mass asymptotic behavior of angular integrals in quantum field theory using expansion by regions, converting integrals into parametric forms and applying Mellin-Barnes techniques.

Contribution

It introduces a novel approach combining parametric representations and automation tools to evaluate asymptotics of angular integrals with multiple denominators.

Findings

Explicit calculation of asymptotics for three and four denominators.

Connection established between contributing regions and algebraic decompositions.

Formulation of a conjecture for general cases of denominators and masses.

Abstract

We study the small-mass asymptotic behavior of so-called angular integrals, appearing in phase-space calculations in perturbative quantum field theory. For this purpose we utilize the strategy of expansion by regions, which is a universal method both for multiloop Feynman integrals and various parametric integrals. To apply the technique to angular integrals, we convert them into suitable parametric integral representations, which are accessible to existing automation tools. We use the the code \texttt{asy.m} to reveal regions contributing to the asymptotic expansion of angular integrals. To evaluate the contributions of these regions in an epsilon expansion we apply the method of Mellin-Barnes representation. Our approach is checked against existing results on angular integrals revealing a connection between contributing regions and angular integrals constructed from an algebraic decomposition. We explicitly calculate the previously unknown asymptotics for angular integrals with three and four denominators and formulate a conjecture for the leading asymptotics and the pole part for a general number of denominators and masses.