A New Estimate of $\varepsilon '/\varepsilon$

Abstract

We discuss direct $CP$ violation in the standard model by giving a new estimate of $\varepsilon'/\varepsilon$ in kaon decays. Our analysis is based on the evaluation of the hadronic matrix elements of the \mbox{$\Delta S =1$} effective quark lagrangian by means of the chiral quark model, with the inclusion of meson one-loop renormalization and NLO Wilson coefficients. Our estimate is fully consistent with the $\Delta I =1/2$ selection rule in $K\to \pi\pi$ decays which is well reproduced within the same framework. By varying all parameters in the allowed ranges and, in particular, taking the quark condensate---which is the major source of uncertainty---between $(-200\ {\rm MeV})^3$ and $(-280\ {\rm MeV})^3$ we find $$ -5.0 \times 10^{-3}\ <\varepsilon'/\varepsilon <\ 1.4 \times 10^{-3}\ .$$ Assuming for the quark condensate the improved PCAC result \mbox{$\vev{\bar qq} = -(221\: \pm 17\ {\rm MeV})^3$} and fixing $\Lambda_{\rm QCD}^{(4)}$ to its central value, we find the more restrictive prediction $$\varepsilon '/\varepsilon = ( 4 \pm 5 ) \,\times \,10^{-4}\ , $$ where the central value is defined as the average over the allowed values of Im $\lambda_t$ in the first and second quadrants. In these estimates the relevant mixing parameter Im $\lambda_t$ is self-consistently obtained from $\varepsilon$ and we take $m_t^{\rm pole} = 180 \pm 12$ GeV. Our result is, to a very good approximation, renormalization-scale and $\gamma_5$-scheme independent.