Isospin-breaking in $\varepsilon'/\varepsilon$: Impact of $\eta_0$ at the Dawn of the 2020s

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 $$\\varepsilon \u0027/\\varepsilon $$ 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 $$\\varepsilon \u0027/\\varepsilon $$. Accordingly, we get a reduction from $$(\\varepsilon \u0027/\\varepsilon )_\\text {SM}^{(8)}=(17.4\\pm 6.1)\\times 10^{-4}$$ to $$(\\varepsilon \u0027/\\varepsilon )_\\text {SM}^{(9)}=(13.9\\pm 5.2)\\times 10^{-4}$$ and thereby an effective suppression of $$(\\varepsilon \u0027/\\varepsilon )_\\text {SM}$$ by isospin-breaking corrections as large as $$40\\%$$ relative to the recent RBC-UKQCD value.