Abstract
The known isospin-breaking contributions to the K → ππ amplitudes are reanalyzed, taking into account our current understanding of the quark masses and the relevant non-perturbative inputs. We present a complete numerical reappraisal of the direct CP-violating ratio ∈′/∈, where these corrections play a quite significant role. We obtain the Standard Model prediction Re (∈′/∈) = (14 ± 5) · 10−4, which is in very good agreement with the measured ratio. The uncertainty, which has been estimated conservatively, is dominated by our current ignorance about 1/NC-suppressed contributions to some relevant chiral-perturbation-theory low-energy constants.
Highlights
Dynamics are, key for understanding the decay amplitudes, even at the qualitative level, since gluonic interactions are responsible for the empirical ∆I = 1/2 rule that governs the measured non-leptonic decay rates, i.e., a huge enhancement of the isoscalar K → ππ amplitude over the isotensor one, 16 times larger than the naive expectation without QCD
While isospin symmetry is an excellent approximation for most phenomenological applications, the isospin violations induced by the quark mass difference mu − md and the electromagnetic interaction can get strongly enhanced in some observables [2, 3], owing to the ∆I = 1/2 rule, when a tiny isospin-violating correction to the dominant amplitude feeds into the suppressed one
Where the three complex quantities A∆I are generated by the ∆I = 1/2, 3/2, 5/2 components of the electroweak effective Hamiltonian, in the limit of isospin conservation
Summary
Knowing the symmetry properties of the relevant QCD amplitudes, one can build their effective χPT realization in terms of the pseudoscalar meson fields as systematic expansions in powers of momenta, p2, quark masses, mq, and electric charges, e2q. The relevant ∆S = 1 electroweak Lagrangian contains O(e2GF p0) and O(e2GF p2) terms: L∆EWS=1 = e2 G8 gewk F 6 λU †QU + e2 G8 F 4 Zi OiEW + O(GF e2p4) This Lagrangian transforms as (8L, 8R) under chiral transformations and provides the needed low-energy realization of the electromagnetic penguin operators in eq (2.1). We outline below the relevant sources of isospin breaking up to NLO in χPT
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