On $K\to \pi \pi$ Decays in Quenched and Unquenched Chiral Perturbation Theory

Abstract

We calculate the logarithmic corrections to the matrix elements for $K^{+}\to {\pi }^{+}$ and $K\to$vacuum (which are used on the lattice to determine $K\to \pi \pi$ amplitudes), in one-loop quenched and unquenched Chiral Perturbation Theory. We find that these corrections can be large. We also discuss, and present some results for, the direct determination of $K\to \pi \pi$ amplitudes. In particular, we address effects from choosing $m_s=m_d$ and vanishing external spatial momenta, finite volume and quenching. In the quenched octet case, we find enhanced finite-volume contributions which may make numerical estimates of this matrix element unreliable for large volumes.

Comments: LATTICE99(Matrix Elements), LaTeX(espcrc2.sty), 3p, 1 ps figure