Three-loop vacuum integrals with arbitrary masses

TL;DR

This paper introduces a numerical method to evaluate three-loop vacuum integrals with multiple arbitrary masses, extending previous results limited to fewer mass scales, thus aiding complex calculations in particle physics models.

Contribution

It develops a numerical approach for three-loop vacuum integrals with arbitrary mass patterns, using dispersion relations to transform integrals into forms suitable for efficient computation.

Findings

Numerical evaluation method for complex three-loop integrals.

Transformation of integrals into elementary functions for efficiency.

Applicable to Standard Model and extensions with multiple mass scales.

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.