Isospin-breaking in $$\varepsilon '/\varepsilon $$: impact of $$\eta _0$$ at the dawn of the 2020s

Abstract

AbstractFor direct CP-violation in $$K\rightarrow \pi \pi $$K→ππ decays, the usual isospin-breaking effects at the percent level are amplified by the dynamics behind the $$\Delta I=1/2$$ΔI=1/2 rule and conventionally encoded in $$\Omega _\mathrm{IB}$$ΩIB parameters. The updated prediction $$\Omega _\mathrm{IB}^{(8)}=(15.9\pm 8.2)\times 10^{-2}$$ΩIB(8)=(15.9±8.2)×10-2 of the Chiral Perturbation Theory for the strong isospin-breaking due to $$\pi _3-\eta _8$$π3-η8 mixing confirms such a tendency but is quite sensitive to the theoretical input value of the low-energy constant corresponding to the flavour-singlet $$\eta _0$$η0 exchange contribution in this truncated octet scheme. We rather exploit the phenomenological $$\eta _8-\eta _0$$η8-η0 mixing as a probe for the non-negligible flavour-singlet component of the physical $$\eta $$η pole to find $$\Omega _\mathrm{IB}^{(9)}=(35\pm 7)\times 10^{-2}$$ΩIB(9)=(35±7)×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 $$\pi ^+-\pi ^0$$π+-π0 mass difference as the dominant electromagnetic isospin-breaking, we obtain for the effective parameter entering the ratio $$\varepsilon '/\varepsilon $$ε′/ε an improved result $$\hat{\Omega }_\mathrm{eff}^{(9)}=(29\pm 7)\times 10^{-2}$$Ω^eff(9)=(29±7)×10-2 to be compared with $$\hat{\Omega }_\mathrm{eff}^{(8)}=(17\pm 9)\times 10^{-2}$$Ω^eff(8)=(17±9)×10-2 used in recent analyses of $$\varepsilon '/\varepsilon $$ε′/ε. Accordingly, we get a reduction from $$(\varepsilon '/\varepsilon )_\text {SM}^{(8)}=(17.4\pm 6.1)\times 10^{-4}$$(ε′/ε)SM(8)=(17.4±6.1)×10-4 to $$(\varepsilon '/\varepsilon )_\text {SM}^{(9)}=(13.9\pm 5.2)\times 10^{-4}$$(ε′/ε)SM(9)=(13.9±5.2)×10-4 and thereby an effective suppression of $$(\varepsilon '/\varepsilon )_\text {SM}$$(ε′/ε)SM by isospin-breaking corrections as large as $$40\%$$40% relative to the recent RBC-UKQCD value.