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

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.