Non-perturbative renormalization of quark mass in N_f = 2 + 1 QCD with the Schrödinger functional scheme

We present an evaluation of the quark mass renormalization factor for N f = 2 + 1 QCD. The Schrödinger functional scheme is employed as the intermediate scheme to carry out non-perturbative running from the low energy region, where renormalization of bare mass is performed on the lattice, to deep in the high energy perturbative region, where the conversion to the renormalization group invariant mass or the \( \overline {\text{MS}} \) scheme is safely carried out. For numerical simulations we adopted the Iwasaki gauge action and nonperturbatively improved Wilson fermion action with the clover term. Seven renormalization scales are used to cover from low to high energy regions and three lattice spacings to take the continuum limit at each scale. The regularization independent step scaling function of the quark mass for the N f = 2 + 1 QCD is obtained in the continuum limit. Renormalization factors for the pseudo scalar density and the axial vector current are also evaluated for the same action and the bare couplings as two recent large scale N f = 2 + 1 simulations; previous work of the CP -PACS/JLQCD collaboration, which covered the up-down quark mass range heavier than m π ∼ 500 MeV and that of PACS-CS collaboration for much lighter quark masses down to m π = 155MeV. The quark mass renormalization factor is used to renormalize bare PCAC masses in these simulations.