Additional Isospin-Breaking Effects in ǫ′/ǫ

Isospin-breaking effects, in particular those associated with electroweakpenguin contributions and π-η, η mixing, have long been known to affect the Standard Model prediction of ǫ/ǫ in a significant manner. We have found an heretofore unconsidered isospin-violating effect of importance; namely, the u-d quark mass difference can spawn |∆I| = 3/2 components in the matrix elements of the gluonic penguin operators. Using chiral perturbation theory and the factorization approximation for the hadronic matrix elements, we find within a specific model for the low-energy constants that we can readily accommodate an increase in ǫ/ǫ by a factor of two. INTRODUCTION The recent measurement of a non-zero value of Re (ǫ′/ǫ) [1] establishes the existence of CP violation in direct decay and thus provides an important first check of the mechanism of CP violation in the Standard Model (SM). Nevertheless, the world average which emerges is Re (ǫ′/ǫ) = (19.3±2.4)·10−4 [2], which is larger than the “central” SM prediction of 7.0 ·10−4 [3,4] by nearly a factor of three. This compels us to scrutinize the SM prediction in further detail: we study isospin-violating effects arising from the u-d quark mass difference. Isospin violation plays an important role in the analysis of ǫ′/ǫ, for the latter is predicated by the difference of the imaginary to real part ratios in the |∆I| = 1/2 and |∆I| = 3/2 K → ππ amplitudes. The differing charges of the u and d quarks engender |∆I| = 3/2 electroweak penguin contributions, whereas π-η, η′ mixing, driven by the u-d quark mass difference, modifies the relative contribution of the |∆I| = 1/2 and |∆I| = 3/2 amplitudes in a significant way. Here we describe isospin-breaking effects in the matrix elements of the gluonic penguin operators [5], such as Q6. These operators have always been thought to 1) Talk at the 3rd Int. Conf. on Symmetries in Subatomic Physics, Adelaide, Mar. 13-17, 2000 — based on work performed in collaboration with G. Valencia. induce exclusively |∆I| = 1/2 transitions, but this is true only in the limit of isospin symmetry. The difference in the up and down quark masses effectively distinguishes the interaction of gluons with up and down quarks, so that the 〈ππ|Q6|K〉 matrix element possesses a |∆I| = 3/2 component as well [6]. Let us begin by showing why the numerical prediction of ǫ′/ǫ is sensitive to the presence of isospin violation. The value of ǫ′/ǫ is inferred from a ratio of ratios, namely