Isospin-breaking in ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$: Impact of ${\eta }_0$ at the Dawn of the 2020s

For direct CP-violation in $K\to \pi \pi$ decays, the usual isospin-breaking effects at the percent level are amplified by the dynamics behind the $\Delta I=1/2$ruleandconventionallyencodedin$\Omega_{\rm IB}$ parameters. The updated prediction ${\mathrm{\Omega }}_{\mathrm{I}\mathrm{B}}^{(8)}=(15.9\pm 8.2)\times {10}^{-2}$ 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 ${\mathrm{\Omega }}_{\mathrm{I}\mathrm{B}}^{(9)}=(35\pm 7)\times {10}^{-2}$ 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 ${ϵ}^{\mathrm{\prime }}\mathrm{/}ϵ$ an improved result ${\widehat{\mathrm{\Omega }}}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{(9)}=(29\pm 7)\times {10}^{-2}$ to be compared with ${\widehat{\mathrm{\Omega }}}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{(8)}=(17\pm 9)\times {10}^{-2}$ used in recent analyses of ${ϵ}^{\mathrm{\prime }}\mathrm{/}ϵ$. Accordingly, we get a reduction from $({ϵ}^{\mathrm{\prime }}\mathrm{/}ϵ{)}_{\mathrm{S}\mathrm{M}}^{(8)}=(17.4\pm 6.1)\times {10}^{-4}$ to $({ϵ}^{\mathrm{\prime }}\mathrm{/}ϵ{)}_{\mathrm{S}\mathrm{M}}^{(9)}=(13.9\pm 5.2)\times {10}^{-4}$ and thereby an effective suppression of $({ϵ}^{\mathrm{\prime }}\mathrm{/}ϵ{)}_{\mathrm{S}\mathrm{M}}$ by isospin-breaking corrections as large as $40\mathrm{\%}$ relative to the recent RBC-UKQCD value.