Determination of the running coupling constant $\alpha_s$ for Nf=2+1 QCD with the Schroedinger functional scheme

We present an evaluation of the running coupling constant and the quark mass renormalization factor for $N_f=2+1$ QCD. The Schr\"odinger functional scheme is used as the intermediate scheme to carry out non-perturbative running from the low energy region, where physical input is introduced, to deep in the high energy perturbative region, where conversion to the ${\ovl{\rm MS}}$ scheme is safely performed. For numerical simulations we adopted Iwasaki gauge action and non-perturbatively improved Wilson fermion action with the clover term. Seven renormalization scales are used to cover from low to high energy region and three lattice spacings to take the continuum limit at each scale. Physical inputs are introduced from the previous $N_f=2+1$ simulation of the CP-PACS/JL-QCD collaboration, which covered the up-down quark mass range heavier than $m_\pi\sim 500$ MeV, and that of PACS-CS collaboration for much lighter quark masses down to $m_\pi=155$ MeV.