KaonBparameter with the Wilson quark action using chiral Ward identities

A lattice QCD calculation of the kaon $B$ parameter $B_K$ is carried out with the Wilson quark action in the quenched approximation at $\beta {=6/g}^2=5.9\mathrm{}–\mathrm{6.5.}$ The mixing problem of the $\Delta s=2\mathrm{}$ four-quark operators is solved nonperturbatively with full use of chiral Ward identities employing four external quarks with an equal off-shell momentum in the Landau gauge. This method, without invoking any effective theory, enables us to construct the weak four-quark operators exhibiting good chiral behavior. Our results for $B_K$ with the nonperturbative mixing coefficients show small scaling violation beyond the lattice cutoff $a^{-1}\sim 2.5\mathrm{GeV}.$ Our estimate concludes $B_K(\mathrm{N}\mathrm{D}\mathrm{R},2\mathrm{}\mathrm{}\mathrm{GeV})=0.69\mathrm{}\left(7\right)\mathrm{}$ at $a^{-1}=2.7\mathrm{}–4.3\mathrm{GeV},$ which agrees with the value obtained with the Kogut-Susskind quark action. For comparison we also calculate $B_K$ with one-loop perturbative mixing coefficients. While this yields incorrect values at finite lattice spacing, a linear extrapolation to the continuum limit as a function of $a$ leads to a result consistent with those obtained with the Ward identity method.