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

Abstract

We consider the on-shell mass and wave function renormalization constants $Z_m^{\mathrm{OS}}$ and $Z_2^{\mathrm{OS}}$ 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 $\mathcal{O}({ϵ}^2)$ and $\mathcal{O}(ϵ)$ 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.

Comments: 14 pages, 3 figures