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

TL;DR

This paper refines the theoretical prediction of isospin-breaking effects in direct CP-violation in kaon decays, highlighting the significant impact of the $ ext{eta}_0$ meson component on the $rac{ ext{epsilon}'}{ ext{epsilon}}$ ratio.

Contribution

It introduces a complete nonet scheme analysis of isospin-breaking effects, emphasizing the role of the $ ext{eta}_0$ meson, and updates the Standard Model prediction for $rac{ ext{epsilon}'}{ ext{epsilon}}$ with reduced uncertainties.

Findings

Updated isospin-breaking parameter $ ilde{ m Omega}_{ m eff}^{(9)}$ is approximately 29% with 7% uncertainty.

Predicted $rac{ ext{epsilon}'}{ ext{epsilon}}$ in the Standard Model is reduced to about 13.9×10^{-4}.

Isospin-breaking effects can suppress $rac{ ext{epsilon}'}{ ext{epsilon}}$ by up to 40% compared to previous estimates.

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 $\Omega_{\rm IB}$ parameters. The updated prediction $\Omega_{\rm IB}^{(8)}=(15.9\pm 8.2)\times 10^{-2}$ 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 $\Omega_{\rm IB}^{(9)}=(35\pm7)\times 10^{-2}$ 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 $\pi^+-\pi^0$ mass difference as the dominant electromagnetic isospin-breaking, we obtain for the effective parameter entering the ratio $\epsilon'/\epsilon$ an improved result $\hat\Omega_{\rm eff}^{(9)}=(29\pm7)\times 10^{-2}$ to be compared with $\hat\Omega_{\rm eff}^{(8)}=(17\pm9)\times 10^{-2}$ used in recent analyses of $\epsilon'/\epsilon$. Accordingly, we get a reduction from $(\epsilon'/\epsilon)_{\rm SM}^{(8)}=(17.4\pm 6.1)\times 10^{-4}$ to $(\epsilon'/\epsilon)_{\rm SM}^{(9)}=(13.9\pm 5.2)\times 10^{-4}$ and thereby an effective suppression of $(\epsilon'/\epsilon)_{\rm SM}$ by isospin-breaking corrections as large as $40\%$ relative to the recent RBC-UKQCD value.