KaonBparameter in quenched QCD

I calculate the kaon B parameter $B_K,$ defined via ${8(m}_K{f}_K{)}^2{B}_K/3=〈\overline{K}|\overline{s}{\gamma }_{\mu }\left(1\mathrm{}-{\gamma }_5\right)d\overline{s}{\gamma }_{\mu }\left(1\mathrm{}-{\gamma }_5\right)d|K〉,$ with a lattice simulation in the quenched approximation. The lattice simulation uses an action possessing exact lattice chiral symmetry, an overlap action. Computations are performed at two lattice spacings, about 0.13 and 0.09 fm (parametrized by Wilson gauge action couplings $\beta =5.9\mathrm{}$ and 6.1) with nearly the same physical volumes and quark masses. I describe particular potential difficulties which arise due to the use of such a lattice action in finite volume. The continuum-extrapolated result for quenched $B_K^{\mathrm{NDR}}(\mu =2\mathrm{}\mathrm{GeV})=0.55\mathrm{}\left(7\right)\mathrm{}$ where the uncertainty includes statistics, extrapolation, and an estimate of uncertainty from the choice of matching factor and strange quark mass.