Quark Mass Renormalization in the $\MSbar$ and RI schemes up to the NNLO order

We compute the relation between the quark mass defined in the minimal modified $\MSbar$ scheme and the mass defined in the ``Regularization Invariant" scheme (RI), up to the NNLO order. The RI scheme is conveniently adopted in lattice QCD calculations of the quark mass, since the relevant renormalization constants in this scheme can be evaluated in a non-perturbative way. The NNLO contribution to the conversion factor between the quark mass in the two schemes is found to be large, typically of the same order of the NLO correction at a scale $\mu \sim 2$ GeV. We also give the NNLO relation between the quark mass in the RI scheme and the renormalization group-invariant mass.