Kaon Matrix Elements and CP-violation from Quenched Lattice QCD: (I) the 3-flavor case

Abstract

We report the results of a calculation of the K --> pi pi matrix elements relevant for the $\text{DIhalf}$ rule and $\text{epe}$ in quenched lattice QCD using domain wall fermions at a fixed lattice spacing $a^{-1}\sim 2$ GeV. Working in the three-quark effective theory, where only the u, d and s quarks enter and which is known perturbatively to next-to-leading order, we calculate the lattice K --> pi and K --> |0> matrix elements of dimension six, four-fermion operators. Through lowest order chiral perturbation theory these yield K --> pi pi matrix elements, which we then normalize to continuum values through a non-perturbative renormalization technique. For the ratio of isospin amplitudes |A_0|/|A_2| we find a value of $25.3\pm 1.8$ (statistical error only) compared to the experimental value of 22.2, with individual isospin amplitudes 10-20% below the experimental values. For $\text{epe}$, using known central values for standard model parameters, we calculate $(-4.0\pm 2.3)\times 10^{-4}$ (statistical error only) compared to the current experimental average of $(17.2\pm 1.8)\times 10^{-4}$. Because we find a large cancellation between the I = 0 and I = 2 contributions to $\text{epe}$, the result may be very sensitive to the approximations employed. Among these are the use of: quenched QCD, lowest order chiral perturbation theory and continuum perturbation theory below 1.3 GeV. We have also calculated the kaon B parameter, B_K and find $B_{K,\overline{MS}}(2\mathrm{GeV})=0.532(11)$. Although currently unable to give a reliable systematic error, we have control over statistical errors and more simulations will yield information about the effects of the approximations on this first-principles determination of these important quantities.

Comments: 158 pages, 51 tables, 41 figures. No numerical data or results changed in this revision, except for an improved analysis of B_K. Many improvements and modifications to the text and explanations. Table XLVII changed to include improved values for B_K. Submitted to Physical Review D