For direct CP-violation in $Ktopipi$ decays, the usual isospin-breaking effects at the percent level are amplified by the dynamics behind the $Delta I=1/2$ rule and conventionally encoded in $Omega_{rm IB}$ parameters. The updated prediction $Omega_{rm IB}^{(8)}=(15.9pm 8.2)times 10^{-2}$ of the Chiral Perturbation Theory for the strong isospin-breaking due to $pi_3-eta_8$ mixing confirms such a tendency but is quite sensitive to the theoretical input value of the low-energy constant corresponding to the flavour-singlet $eta_0$ exchange contribution in this truncated octet scheme. We rather exploit the phenomenological $eta_8-eta_0$ mixing as a probe for the non-negligible flavour-singlet component of the physical $eta$ pole to find $Omega_{rm IB}^{(9)}=(35pm7)times 10^{-2}$ in a complete nonet scheme. A large central value in the nonet scheme is thus substituted for a large uncertainty in the octet one. Including the experimental $pi^+-pi^0$ mass difference as the dominant electromagnetic isospin-breaking, we obtain for the effective parameter entering the ratio $epsilon/epsilon$ an improved result $hatOmega_{rm eff}^{(9)}=(29pm7)times 10^{-2}$ to be compared with $hatOmega_{rm eff}^{(8)}=(17pm9)times 10^{-2}$ used in recent analyses of $epsilon/epsilon$. Accordingly, we get a reduction from $(epsilon/epsilon)_{rm SM}^{(8)}=(17.4pm 6.1)times 10^{-4}$ to $(epsilon/epsilon)_{rm SM}^{(9)}=(13.9pm 5.2)times 10^{-4}$ and thereby an effective suppression of $(epsilon/epsilon)_{rm SM}$ by isospin-breaking corrections as large as $40%$ relative to the recent RBC-UKQCD value.
Download