Exact results for ${Z}_m^{\mathrm{OS}}$ and ${Z}_2^{\mathrm{OS}}$ with two mass scales and up to three loops

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 $\mathcal{O}$(ϵ2) and $\mathcal{O}$(ϵ) at two- and three-loop order which are crucial ingredients 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.