# Two-loop triangle integrals with 4 scales for the $HZV$ vertex

## TL;DR

This paper analytically computes two-loop triangle integrals with four scales for the $HZV$ vertex, enabling faster numerical evaluations and deeper understanding of multi-loop multi-scale Feynman integrals, with applications to Higgs decay and production.

## Contribution

It provides explicit analytic results for complex two-loop integrals with multiple scales, improving computational efficiency and understanding of multi-scale Feynman diagrams.

## Key findings

- Analytic expressions for two-loop triangle integrals with four scales.
- Efficient evaluation of bottom quark loop contributions.
- Application to Higgs decay and ZH production processes.

## Abstract

We calculate analytically the two-loop triangle integrals entering the $\mathcal{O}(\alpha\alpha_s)$ corrections to the $HZV$ vertex with $V=Z^*,\gamma^*$ using the method of differential equations. Our result provides a prototype to study the analytic properties of multi-loop multi-scale Feynman integrals, and also allows fast numeric evaluation for phenomenological studies. We apply our results to the leptonic decay of the Higgs boson and to $ZH$ production at electron-positron colliders. Besides the top quark loop, we include also the bottom quark loop contributions, whose evaluation takes a lot of time using purely numeric methods, but is very efficient with our analytic results.