Abstract
For direct CP-violation in Krightarrow 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)}=(15.9pm 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)}=(35pm 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 varepsilon '/varepsilon an improved result hat{Omega }_mathrm{eff}^{(9)}=(29pm 7)times 10^{-2} to be compared with hat{Omega }_mathrm{eff}^{(8)}=(17pm 9)times 10^{-2} used in recent analyses of varepsilon '/varepsilon . Accordingly, we get a reduction from (varepsilon '/varepsilon )_text {SM}^{(8)}=(17.4pm 6.1)times 10^{-4} to (varepsilon '/varepsilon )_text {SM}^{(9)}=(13.9pm 5.2)times 10^{-4} and thereby an effective suppression of (varepsilon '/varepsilon )_text {SM} by isospin-breaking corrections as large as 40% relative to the recent RBC-UKQCD value.
Highlights
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 η0 exchange contribution in this truncated octet scheme
A large central value in the nonet scheme is substituted for a large uncertainty in the octet one
We note that the excited-state contamination seen at 1.5 GeV in LQCD data might be residual effects of the scalar resonance at O( p2, 0) and meson evolution at O( p0, 1/N ), those being essential for the estimate of the Q6 hadronic matrix elements around 1 GeV in DQCD
Summary
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 η0 exchange contribution in this truncated octet scheme. The new RBC-UKCD result has been obtained by raising the matching scale from 1.5 GeV to 4 GeV in order to reduce the systematic error. As a consequence, this makes it difficult if not impossible for us to confront the numerical result (2) with the analytical ones displayed in (3) and (4). The LQCD approach is purely based on the quark and gluon dynamics with no reference to the meson degrees of freedom at work in both the ChPT and DQCD ones Comments on both hadronic matrix elements and final state interactions will be given later on. Still uncertainties are not in full theoretical control: discrepancies of O(10−3) between the experimental value and the SM prediction could be either positive, as expected from DQCD, or negative as allowed by the new RBC-UKQCD result
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