Three-loop vacuum integrals with arbitrary masses

Abstract

Three-loop vacuum integrals are an important building block for the calculation of a wide range of three-loop corrections. Until now, only results for integrals with one and two independent mass scales are known, but in the electroweak Standard Model and many extensions thereof, one often encounters more mass scales of comparable magnitude. For this reason, a numerical approach for the evaluation of three-loop vacuum integrals with arbitrary mass pattern is proposed here. Concretely, one can identify a basic set of three master integral topologies. With the help of dispersion relations, each of these can be transformed into one-dimensional or, for the most complicated case, two-dimensional integrals in terms of elementary functions, which are suitable for efficient numerical integration.

17 pages; ancillary file with subtraction formulae for the U5 master integral in Mathematica format are supplied with this submission. v2: minor typos corrected and few references added, version to published in JHEP. v3: typos in eqs. (23), (31), (35) and (61) corrected

References in corpus (5)

• HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters
• Electroweak two-loop corrections to the effective weak mixing angle
• Gauss hypergeometric function: reduction, epsilon-expansion for integer/half-integer parameters and Feynman diagrams
• Evaluation of the general 3-loop vacuum Feynman integral
• Numerical multi-loop integrals and applications

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• Systematic study of one-loop realizations of $d=7$ long-range $0\nu \beta \beta$ decay operators
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• GKZ hypergeometric systems of the four-loop vacuum Feynman integrals