Four-loop two-mass tadpoles and the $\rho$ parameter

TL;DR

This paper discusses the calculation of four-loop quantum chromodynamics corrections to the electroweak rho parameter considering a non-zero bottom quark mass, highlighting the complex mathematical functions involved.

Contribution

It presents the status of four-loop calculations of the rho parameter, emphasizing the mathematical structures like elliptic integrals that appear at this order.

Findings

Elliptic integrals contribute at three loops.

Mathematical structures at four loops are being analyzed.

Progress on four-loop QCD corrections to the rho parameter.

Abstract

We calculate four-loop QCD corrections to the electroweak $\rho$ parameter with a non-vanishing $b$ quark mass. At three loops, it was observed that elliptic integrals contribute to this observable. This prompts the question of which classes of functions appear at the next order. We report on the status of our calculation with a focus on the mathematical structures that emerge at four loops.