Two-loop scalar functions with $I$ internal lines, $I \le 5$

TL;DR

This paper demonstrates a mixed analytical and numerical approach to compute two-loop scalar functions with up to five internal lines, applicable to Higgs sector amplitudes, including divergent cases, with results validated against existing codes.

Contribution

It introduces a proof of concept for a mixed method to evaluate complex two-loop scalar functions with up to five propagators, expanding computational tools for Higgs sector amplitudes.

Findings

Method successfully computes divergent and non-divergent amplitudes.

All amplitudes can be generated by four master topologies.

Numerical results agree with existing public code within error margins.

Abstract

This article displays a proof of concept of the mixed analytical/numerical method, presented in previous publications, to compute two-loop functions with up to five massive propagators in a scalar theory having three- and four-leg vertices as the Higgs sector of the Standard Model. Several amplitudes are considered with two, three and four external legs. Some of them diverge in the UV region and we demonstrate that the method works in that case. It is shown that all these classes of amplitudes can be generated by four master topologies. Results of a numerical evaluation for some kinematics are presented, they are compared to a public code and agree well within the error bars quoted by the different programs.