Isospin-breaking in $\varepsilon'/\varepsilon$: Impact of $η_0$ at the Dawn of the 2020s

Abstract

For direct CP-violation in $K\toππ$ decays, the usual isospin-breaking effects at the percent level are amplified by the dynamics behind the $ΔI=1/2$ rule and conventionally encoded in $Ω_{\rm IB}$ parameters. The updated prediction $Ω_{\rm IB}^{(8)}=(15.9\pm 8.2)\times 10^{-2}$ of the Chiral Perturbation Theory for the strong isospin-breaking due to $π_3-η_8$ mixing confirms such a tendency but is quite sensitive to the theoretical input value of the low-energy constant corresponding to the flavour-singlet $η_0$ exchange contribution in this truncated octet scheme. We rather exploit the phenomenological $η_8-η_0$ mixing as a probe for the non-negligible flavour-singlet component of the physical $η$ pole to find $Ω_{\rm IB}^{(9)}=(35\pm7)\times 10^{-2}$ 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 $π^+-π^0$ mass difference as the dominant electromagnetic isospin-breaking, we obtain for the effective parameter entering the ratio $ε'/ε$ an improved result $\hatΩ_{\rm eff}^{(9)}=(29\pm7)\times 10^{-2}$ to be compared with $\hatΩ_{\rm eff}^{(8)}=(17\pm9)\times 10^{-2}$ used in recent analyses of $ε'/ε$. Accordingly, we get a reduction from $(ε'/ε)_{\rm SM}^{(8)}=(17.4\pm 6.1)\times 10^{-4}$ to $(ε'/ε)_{\rm SM}^{(9)}=(13.9\pm 5.2)\times 10^{-4}$ and thereby an effective suppression of $(ε'/ε)_{\rm SM}$ by isospin-breaking corrections as large as $40\%$ relative to the recent RBC-UKQCD value.