We report on a calculation of $B_K$ with domain-wall fermion action in quenched QCD. Simulations are made with a renormalization group improved gauge action at $\beta =2.6\mathrm{}$ and $2.9$ corresponding to $a^{-1}\approx 2$ and 3 GeV. Effects due to finite fifth dimensional size $N_5$ and finite spatial size $N_{\sigma }$ are examined in detail. Matching to the continuum operator is made perturbatively at one loop order. We obtain $B_K(\mu =2\mathrm{}\hspace{0.5em}\mathrm{GeV})=0.5746\mathrm{}\left(61\right)\mathrm{}\left(191\right),$ where the first error is statistical and the second error represents an estimate of scaling violation and $\mathcal{O}\left({\alpha }^2\right)$ errors in the renormalization factor added in quadrature, as an estimate of the continuum value in the modified minimal subtraction $\left(\overline{\mathrm{MS}}\right)$ scheme with naive dimensional regularization. This value is consistent, albeit somewhat small, with $B_K(\mu =2\mathrm{}\hspace{0.5em}\mathrm{GeV})=0.628\mathrm{}\left(42\right)\mathrm{}$ obtained by the JLQCD Collaboration using the Kogut-Susskind quark action. Results for light quark masses are also reported.

• Received 21 May 2001

DOI:https://doi.org/10.1103/PhysRevD.64.114506