Three-loop contributions to the $\rho$ parameter and iterated integrals of modular forms

TL;DR

This paper analytically computes three-loop contributions to the $ ho$ parameter in the Standard Model, revealing the involvement of elliptic polylogarithms and modular forms, and provides methods for their analytic continuation and series expansion.

Contribution

It introduces a fully analytic calculation of three-loop diagrams involving different quark flavors, utilizing advanced functions like elliptic polylogarithms and modular forms.

Findings

Results involve elliptic polylogarithms and modular forms.

Analytic continuation of these functions to all parameter regions.

Fast-converging series expansions for the functions.

Abstract

We compute fully analytic results for the three-loop diagrams involving two different massive quark flavours contributing to the $\rho$ parameter in the Standard Model. We find that the results involve exactly the same class of functions that appears in the well-known sunrise and banana graphs, namely elliptic polylogarithms and iterated integrals of modular forms. Using recent developments in the understanding of these functions, we analytically continue all the iterated integrals of modular forms to all regions of the parameter space, and in each region we obtain manifestly real and fast-converging series expansions for these functions.