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.
We present non-perturbative renormalization factors for $Delta S=2$ four-quark operators in quenched domain-wall QCD using the Schroedinger functional method. Non-perturbative renormalization factor for $B_K$ is evaluated at hadronic scale. Combined
with the non-perturbative RG running obtained by the Alpha collaboration, our result yields renormalization factor which converts lattice bare $B_K$ to the renormalization group invariant one. We apply the renormalization factor to bare $B_K$ previously obtained by the CP-PACS collaboration with the quenched domain-wall QCD(DWQCD). We compare our result with previous ones obtained by perturbative renormalization factors, different renormalization schemes or different quark actions. We also show that chiral symmetry breaking effects in the renormalization factor are numerically small.
