We present a calculation for the $K^{+}\to {\pi }^{+}{\pi }^0$ decay amplitude using a quenched simulation of lattice QCD with the Wilson quark action at $\beta {=6/g}^2=\mathrm{6.1.}\mathrm{}$ The decay amplitude is extracted from the ratio, the $\overrightarrow{K}\pi \pi$ three-point function divided by either $K$ and $\pi$ meson two-point functions or $K$ meson two-point function and $I=2\mathrm{}\hspace{0.5em}\pi \pi$ four-point function; the two different methods yield consistent results. Finite size effects are examined with calculations made on ${24}^3\times 64$ and ${32}^3\times 64$ lattices, and are shown that they are explained by one-loop effects of chiral perturbation theory. The lattice amplitude is converted to the continuum value by employing a one-loop calculation of chiral perturbation theory, yielding a value in agreement with experiment if extrapolated to the chiral limit. We also report on the $K$ meson $B$ parameter $B_K$ obtained from the $K^{+}\to {\pi }^{+}{\pi }^0$ amplitude using chiral perturbation theory.

• Received 25 November 1997

DOI:https://doi.org/10.1103/PhysRevD.58.054503