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<p>Angular integrals arise in a wide range of perturbative quantum field theory calculations. In this work we investigate angular integrals with three denominators in <i>d</i> = 4 – 2<i>ε</i> 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 <i>ε</i> 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 <i>ε</i>-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.</p>

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

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  • 发表日期：

    2025年3月19日

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  • Juliane Haug,
  • Fabian Wunder

摘要

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 – 2ε 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 ε 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 ε-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.