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We present non-perturbative renormalization factors for $\Delta S=2$four-quark operators in quenched domain-wall QCD using the Schroedingerfunctional method. Non-perturbative renormalization factor for $B_K$ isevaluated at hadronic scale. Combined with the non-perturbative RG runningobtained by the Alpha collaboration, our result yields renormalization factorwhich converts lattice bare $B_K$ to the renormalization group invariant one.We apply the renormalization factor to bare $B_K$ previously obtained by theCP-PACS collaboration with the quenched domain-wall QCD(DWQCD). We compare ourresult with previous ones obtained by perturbative renormalization factors,different renormalization schemes or different quark actions. We also show thatchiral symmetry breaking effects in the renormalization factor are numericallysmall.

Non-perturbative renormalization of four-quark operators and B_K with Schroedinger functional scheme in quenched domain-wall QCD