Angular integrals with three denominators via IBP, mass reduction, dimensional shift, and differential equations

TL;DR

This paper develops a new method to evaluate complex angular integrals with three denominators in quantum field theory, using IBP, differential equations, and dimensional shifts, revealing novel geometric connections in the epsilon expansion.

Contribution

It introduces a systematic reduction and differential equation framework for three-denominator angular integrals, including the first identification of a Gram determinant term in the epsilon expansion.

Findings

Derived explicit recursion relations and reduction to master integrals.

Established epsilon-expansion results up to order epsilon for general cases.

Discovered a Gram determinant term expressed via Clausen functions, linking geometry and quantum field calculations.

Abstract

Angular integrals arise in a wide range of perturbative quantum field theory calculations. In this work we investigate angular integrals with three denominators in $d=4-2\varepsilon$ dimensions. We derive integration-by-parts relations for this class of integrals, leading to explicit recursion relations and a reduction to a small set of master integrals. Using a differential equation approach we establish results up to order $\varepsilon$ for general integer exponents and masses. Here, reduction identities for the number of masses, known results for two-denominator integrals, and a general dimensional-shift identity for angular integrals considerably reduce the required amount of work. For the first time we find for angular integrals a term contributing proportional to a Euclidean Gram determinant in the $\varepsilon$-expansion. This coefficient is expressed as a sum of Clausen functions with intriguing connections to Euclidean, spherical, and hyperbolic geometry. The results of this manuscript are applicable to phase-space calculations with multiple observed final-state particles.