Pole mass of the W boson at two-loop order in the pure MS-bar scheme

I provide a calculation at full two-loop order of the complex pole squared mass of the W boson in the Standard Model in the pure MS-bar renormalization scheme, with Goldstone boson mass effects resummed. This approach is an alternative to earlier ones that use on-shell or hybrid renormalization schemes. The renormalization scale dependence of the real and imaginary parts of the resulting pole mass are studied. Both deviate by about $\pm 4$ MeV from their median values as the renormalization scale is varied from 50 GeV to 200 GeV, but the theory error is likely larger. A surprising feature of this scheme is that the 2-loop QCD correction has a larger scale-dependence, but a smaller magnitude, than the 2-loop non-QCD correction, unless the renormalization scale is chosen very far from the top-quark mass.