A New Estimate of ${\epsilon }^{\prime }/\epsilon$

Abstract

We discuss direct $CP$ violation in the standard model by giving a new estimate of ${\epsilon }^{\prime }/\epsilon$ 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 $(-200\mathrm{MeV}{)}^3$ and $(-280\mathrm{MeV}{)}^3$ we find $$-5.0\times 10^{-3}<{\epsilon }^{\prime }/\epsilon <1.4\times 10^{-3}.$$ Assuming for the quark condensate the improved PCAC result \mbox{$\text{vev}\bar{q}q=-(221\pm 17\mathrm{MeV}{)}^3$} and fixing ${\Lambda }_{\mathrm{QCD}}^{(4)}$ to its central value, we find the more restrictive prediction $${\epsilon }^{\prime }/\epsilon =(4\pm 5)\times 10^{-4},$$ 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 $\epsilon$ and we take $m_t^{\mathrm{pole}}=180\pm 12$ GeV. Our result is, to a very good approximation, renormalization-scale and ${\gamma }_5$-scheme independent.

Comments: 40 pages, uuencoded LATEX2e file including 13 eps figures, revised version to appear in Nucl. Phys. B