Exact results for $Z_m^{\rm OS}$ and $Z_2^{\rm OS}$ with two mass scales and up to three loops

TL;DR

This paper derives exact analytic expressions for on-shell mass and wave function renormalization constants with two mass scales up to three loops, extending previous numerical results and aiding future high-order calculations.

Contribution

It provides new analytic formulas involving harmonic polylogarithms for $Z_m^{ m OS}$ and $Z_2^{ m OS}$ with two masses, including higher-order epsilon terms for four-loop computations.

Findings

Analytic expressions in harmonic polylogarithms for two-mass renormalization constants.

Explicit epsilon expansion terms for three-loop order.

Results facilitate high-precision numerical evaluation in various mass regimes.

Abstract

We consider the on-shell mass and wave function renormalization constants $Z_m^{\rm OS}$ and $Z_2^{\rm OS}$ up to three-loop order allowing for a second non-zero quark mass. We obtain analytic results in terms of harmonic polylogarithms and iterated integrals with the additional letters $\sqrt{1-\tau^2}$ and $\sqrt{1-\tau^2}/\tau$ which extends the findings from Ref. [1] where only numerical expressions are presented. Furthermore, we provide terms of order ${\cal O}(\epsilon^2)$ and ${\cal O}(\epsilon)$ at two- and three-loop order which are crucial ingrediants for a future four-loop calculation. Compact results for the expansions around the zero-mass, equal-mass and large-mass cases allow for a fast high-precision numerical evaluation.