Kaon B parameter with the Wilson quark action using chiral Ward identities

A lattice QCD calculation of the kaon B parameter BK is carried out with the Wilson quark action in the quenched approximation at β=6/g2=5.9–6.5. The mixing problem of the Δs=2 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 BK with the nonperturbative mixing coefficients show small scaling violation beyond the lattice cutoff a-1∼2.5 GeV. Our estimate concludes BK(NDR,2 GeV)=0.69(7) at a-1=2.7–4.3 GeV, which agrees with the value obtained with the Kogut-Susskind quark action. For comparison we also calculate BK 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