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

Understand

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.