The ΔS = 1 weak chiral lagrangian as the effective theory of the chiral quark model

Abstract

We use the chiral quark model to construct the complete O( p 2) weak ΔS = 1 chiral lagrangian via the bosonization of the ten relevant operators of the effective quark lagrangian. The chiral coefficients are given in terms of f π the quark and gluon condesates and the scale-dependent NLO Wilson coefficients of the corresponding operators; in addition, they depend on the constituent quark mass M, a parameter characteristic of the model. All contributions of order N c 2 as well as N c and α s N c are included. The γ 5-scheme dependence of the chiral coefficients, computed via dimensional regularization, and the Fierz transformation properties of the operator basis are discussed in detail. We apply our results to the evaluation of the hadronic matrix elements for the decays K → 2 π, consistently including the renormalization induced by the meson loops. The effect of this renormalization is sizable and introduces a long-distance scale dependence that matches in the physical amplitudes the short-distance scale dependence of the Wilson coefficients.