Isospin-breaking in

For direct CP-violation in $$K\rightarrow \pi \pi $$ 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 $$\Omega _\mathrm{IB}$$ parameters. The updated prediction $$\Omega _\mathrm{IB}^{(8)}=(15.9\pm 8.2)\times 10^{-2}$$ 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 $$\Omega _\mathrm{IB}^{(9)}=(35\pm 7)\times 10^{-2}$$ 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 $$\varepsilon '/\varepsilon $$ an improved result $$\hat{\Omega }_\mathrm{eff}^{(9)}=(29\pm 7)\times 10^{-2}$$ to be compared with $$\hat{\Omega }_\mathrm{eff}^{(8)}=(17\pm 9)\times 10^{-2}$$ used in recent analyses of $$\varepsilon '/\varepsilon $$. Accordingly, we get a reduction from $$(\varepsilon '/\varepsilon )_\text {SM}^{(8)}=(17.4\pm 6.1)\times 10^{-4}$$ to $$(\varepsilon '/\varepsilon )_\text {SM}^{(9)}=(13.9\pm 5.2)\times 10^{-4}$$ and thereby an effective suppression of $$(\varepsilon '/\varepsilon )_\text {SM}$$ by isospin-breaking corrections as large as $$40\%$$ relative to the recent RBC-UKQCD value