Precise determination ofBKand light quark masses in quenched domain-wall QCD

Abstract

We calculate nonperturbative renormalization factors at hadronic scale for $\ensuremath{\Delta}S=2$ four-quark operators in quenched domain-wall QCD using the Schr\"odinger functional method. Combining them with the nonperturbative renormalization group running by the Alpha Collaboration, our result yields the fully nonperturbative renormalization factor, which converts the lattice bare ${B}_{K}$ to the renormalization group invariant (RGI) ${\stackrel{^}{B}}_{K}$. Applying this to the bare ${B}_{K}$ previously obtained by the CP-PACS Collaboration at ${a}^{\ensuremath{-}1}\ensuremath{\simeq}2,3,4\text{ }\text{ }\mathrm{GeV}$, we obtain ${\stackrel{^}{B}}_{K}=0.782(5)(7)$ [equivalent to ${B}_{K}^{\overline{\mathrm{MS}}}(\mathrm{NDR},2\text{ }\text{ }\mathrm{GeV})=0.565(4)(5)$ by two-loop running] in the continuum limit, where the first error is statistical and the second is systematic due to the continuum extrapolation. Except the quenching error, the total error we have achieved is less than 2%, which is much smaller than the previous ones. Taking the same procedure, we obtain ${m}_{u,d}^{\mathrm{RGI}}=5.613(66)\text{ }\text{ }\mathrm{MeV}$ and ${m}_{s}^{\mathrm{RGI}}=147.1(17)\text{ }\text{ }\mathrm{MeV}$ [equivalent to ${m}_{u,d}^{\overline{\mathrm{MS}}}(2\text{ }\text{ }\mathrm{GeV})=4.026(48)\text{ }\text{ }\mathrm{MeV}$ and ${m}_{s}^{\overline{\mathrm{MS}}}(2\text{ }\text{ }\mathrm{GeV})=105.6(12)\text{ }\text{ }\mathrm{MeV}$ by four-loop running] in the continuum limit.