Angular integrals with three denominators via IBP, mass reduction, dimensional shift, and differential equations
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 – 2ε dimensions. We derive integration-by-parts relations for this class of integrals, leading to explicit recursion rela...
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| Format: | Article |
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SpringerOpen
2025-03-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP03(2025)141 |
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| author | Juliane Haug Fabian Wunder |
| author_facet | Juliane Haug Fabian Wunder |
| author_sort | Juliane Haug |
| collection | DOAJ |
| description | 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 – 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. |
| format | Article |
| id | doaj-art-db4a8d5b059e4cf1a4d2dc087cb00f77 |
| institution | DOAJ |
| issn | 1029-8479 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj-art-db4a8d5b059e4cf1a4d2dc087cb00f772025-08-20T03:06:48ZengSpringerOpenJournal of High Energy Physics1029-84792025-03-012025313610.1007/JHEP03(2025)141Angular integrals with three denominators via IBP, mass reduction, dimensional shift, and differential equationsJuliane Haug0Fabian Wunder1Institute for Theoretical Physics, University of TübingenInstitute for Theoretical Physics, University of TübingenAbstract 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.https://doi.org/10.1007/JHEP03(2025)141Higher-Order Perturbative CalculationsRenormalization and RegularizationDeep Inelastic Scattering or Small-x Physics |
| spellingShingle | Juliane Haug Fabian Wunder Angular integrals with three denominators via IBP, mass reduction, dimensional shift, and differential equations Journal of High Energy Physics Higher-Order Perturbative Calculations Renormalization and Regularization Deep Inelastic Scattering or Small-x Physics |
| title | Angular integrals with three denominators via IBP, mass reduction, dimensional shift, and differential equations |
| title_full | Angular integrals with three denominators via IBP, mass reduction, dimensional shift, and differential equations |
| title_fullStr | Angular integrals with three denominators via IBP, mass reduction, dimensional shift, and differential equations |
| title_full_unstemmed | Angular integrals with three denominators via IBP, mass reduction, dimensional shift, and differential equations |
| title_short | Angular integrals with three denominators via IBP, mass reduction, dimensional shift, and differential equations |
| title_sort | angular integrals with three denominators via ibp mass reduction dimensional shift and differential equations |
| topic | Higher-Order Perturbative Calculations Renormalization and Regularization Deep Inelastic Scattering or Small-x Physics |
| url | https://doi.org/10.1007/JHEP03(2025)141 |
| work_keys_str_mv | AT julianehaug angularintegralswiththreedenominatorsviaibpmassreductiondimensionalshiftanddifferentialequations AT fabianwunder angularintegralswiththreedenominatorsviaibpmassreductiondimensionalshiftanddifferentialequations |