Isospin-breaking in ε^′/ ε: impact of η at the dawn of the 2020s

For direct CP-violation in K→ ππ decays, the usual isospin-breaking effects at the percent level are amplified by the dynamics behind the Δ I= 1 / 2 rule and conventionally encoded in Ω _IB parameters. The updated prediction ΩIB(8)=(15.9±8.2)×10-2 of the Chiral Perturbation Theory for the strong isospin-breaking due to 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 η exchange contribution in this truncated octet scheme. We rather exploit the phenomenological η₈- η mixing as a probe for the non-negligible flavour-singlet component of the physical η pole to find 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 π⁺- π mass difference as the dominant electromagnetic isospin-breaking, we obtain for the effective parameter entering the ratio ε^′/ ε an improved result Ω^eff(9)=(29±7)×10-2 to be compared with Ω^eff(8)=(17±9)×10-2 used in recent analyses of ε^′/ ε. Accordingly, we get a reduction from (ε′/ε)SM(8)=(17.4±6.1)×10-4 to (ε′/ε)SM(9)=(13.9±5.2)×10-4 and thereby an effective suppression of (ε′/ε)SM by isospin-breaking corrections as large as 40 % relative to the recent RBC-UKQCD value.