Precise determination of $B_K$ and light quark masses in quenched domain-wall QCD

TL;DR

This paper non-perturbatively determines the $B_K$ parameter and light quark masses in quenched domain-wall QCD, achieving high precision and reducing systematic errors through the Schr"{o}dinger functional method and continuum extrapolation.

Contribution

It provides the first fully non-perturbative renormalization of $B_K$ and light quark masses in quenched domain-wall QCD with minimal systematic errors.

Findings

$B_K^{ m RGI}=0.782(5)(7)$ with less than 2% total error

Light quark masses $m_{u,d}^{ m RGI}=5.613(66)$ MeV and $m_s^{ m RGI}=147.1(17)$ MeV

Results are consistent with previous estimates but with significantly reduced uncertainties.

Abstract

We calculate non-perturbative renormalization factors at hadronic scale for $\Delta S=2$ four-quark operators in quenched domain-wall QCD using the Schr\"{o}dinger functional method. Combining them with the non-perturbative renormalization group running by the Alpha collaboration, our result yields the fully non-perturbative renormalization factor, which converts the lattice bare $B_K$ to the renormalization group invariant (RGI) $\hat{B}_K$. Applying this to the bare $B_K$ previously obtained by the CP-PACS collaboration at $a^{-1}\simeq 2, 3, 4$ GeV, we obtain $\hat{B}_K=0.782(5)(7)$ (equivalent to $B_K^{\bar{\rm MS}}({\rm NDR}, 2 {\rm GeV}) = 0.565(4)(5)$ by 2-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 $m_{u,d}^{\rm RGI}=5.613(66)$ MeV and $m_s^{\rm RGI}=147.1(17)$ MeV (equivalent to $m_{u,d}^{\bar{\rm MS}}(2 {\rm GeV})=4.026(48)$ MeV and $m_{s}^{\bar{\rm MS}}(2 {\rm GeV})=105.6(12)$ MeV by 4-loop running) in the continuum limit.