ALBERT

Description: For direct CP-violation in $$K ightarrow pi pi $$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 $$Omega _mathrm{IB}^{(8)}=(15.9pm 8.2)imes 10^{-2}$$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 $$Omega _mathrm{IB}^{(9)}=(35pm 7)imes 10^{-2}$$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 $$hat{Omega }_mathrm{eff}^{(9)}=(29pm 7)imes 10^{-2}$$Ω^eff(9)=(29±7)×10-2 to be compared with $$hat{Omega }_mathrm{eff}^{(8)}=(17pm 9)imes 10^{-2}$$Ω^eff(8)=(17±9)×10-2 used in recent analyses of $$varepsilon '/varepsilon $$ε′/ε. Accordingly, we get a reduction from $$(varepsilon '/varepsilon )_ext {SM}^{(8)}=(17.4pm 6.1)imes 10^{-4}$$(ε′/ε)SM(8)=(17.4±6.1)×10-4 to $$(varepsilon '/varepsilon )_ext {SM}^{(9)}=(13.9pm 5.2)imes 10^{-4}$$(ε′/ε)SM(9)=(13.9±5.2)×10-4 and thereby an effective suppression of $$(varepsilon '/varepsilon )_ext {SM}$$(ε′/ε)SM by isospin-breaking corrections as large as $$40\%$$40% relative to the recent RBC-UKQCD value.