Three-loop QCD+QED corrections to on-shell quark renormalization

We present the three-loop QCD+QED mixed corrections to the on-shell quark mass and wave-function renormalization constants through orders $\mathcal{O}(\alpha_s^m\alpha^n)$ with $m+n=3$. We further derive the three-loop relation between the pole mass and the $\overline{\text{MS}}$ mass of a heavy quark, including the complete mixed QCD+QED contributions. The corresponding quark-mass anomalous dimension in the presence of both interactions is also extracted. Moreover, we provide the explicit conversion formulae, up to the same perturbative order, between the pole mass and the trace-anomaly subtracted $\sigma$-mass of heavy quark.