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

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.