${ϵ}^{\mathrm{\prime }}\mathrm{/}ϵ$ at $O(p^4)$ in the Chiral Expansion

After updating the determination of the combination of Kobayashi-Maskawa elements $\mathrm{\Im }{V}_{td}{V}_{ts}^{\ast }$ according to our new estimate of the parameter ${\widehat{B}}_K$, we study the CP-violating ratio ${ϵ}^{\mathrm{\prime }}\mathrm{/}ϵ$ by means of hadronic matrix elements computed to $O(p^4)$ in the chiral expansion. It is the first time that this order in chiral perturbation theory is included. We also discuss the relevance of some $O(p^2)$ terms that are generally neglected in the calculation of the electroweak penguin matrix elements. The most important effect of this improved analysis is the substantial reduction (20%) of the leading electroweak penguin matrix element and accordingly a reduced cancelation between the electroweak and gluon penguin contributions. The ratio ${ϵ}^{\mathrm{\prime }}\mathrm{/}ϵ$ is thus larger than previously estimated and its predicted value enjoys a smaller uncertainty. We find $\epsilon’/\epsilon = 1.7 ^{+1.4}_{-1.0} \times 10^{-3}$\mathrm{.}Valuespositiveandoftheorderof$10^{-3}$ are therefore preferred.