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

Abstract

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 $\Delta I=1/2$ 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 ${ϵ}^{\prime }/ϵ$ 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 ${ϵ}^{\prime }/ϵ$. Accordingly, we get a reduction from $({ϵ}^{\prime }/ϵ{)}_{\mathrm{SM}}^{(8)}=(17.4\pm 6.1)\times 10^{-4}$ to $({ϵ}^{\prime }/ϵ{)}_{\mathrm{SM}}^{(9)}=(13.9\pm 5.2)\times 10^{-4}$ and thereby an effective suppression of $({ϵ}^{\prime }/ϵ{)}_{\mathrm{SM}}$ by isospin-breaking corrections as large as $40\%$ relative to the recent RBC-UKQCD value.

Comments: 18 pages, no figures, typos removed, additional clarifying comments and one reference added, results unchanged. Version to appear in EPJC