The $\rho$ parameter at three loops and elliptic integrals

TL;DR

This paper analytically computes complex master integrals for three-loop corrections to the $ ho$ parameter, involving elliptic integrals and hypergeometric functions, advancing precision in quantum field theory calculations.

Contribution

It introduces a novel method to evaluate non-factorizable differential equations using hypergeometric and elliptic integrals for the $ ho$ parameter calculation.

Findings

Master integrals expressed in terms of elliptic integrals.

Differential equations solved with hypergeometric functions.

Enhanced analytical techniques for three-loop quantum corrections.

Abstract

We describe the analytic calculation of the master integrals required to compute the two-mass three-loop corrections to the $\rho$ parameter. In particular, we present the calculation of the master integrals for which the corresponding differential equations do not factorize to first order. The homogeneous solutions to these differential equations are obtained in terms of hypergeometric functions at rational argument. These hypergeometric functions can further be mapped to complete elliptic integrals, and the inhomogeneous solutions are expressed in terms of a new class of integrals of combined iterative non-iterative nature.