The $ΔS = 1$ Weak Chiral Lagrangian as the Effective Theory of the Chiral Quark Model

Details The $ΔS = 1$ Weak Chiral Lagrangian as the Effective Theory of the Chiral Quark Model The $ΔS = 1$ Weak Chiral Lagrangian as the Effective Theory of the Chiral Quark Model

1995-11-07

arxiv.org/abs/hep-ph/9511255v4

Nucl.Phys. B469 (1996) 143-180

Physics

High Energy Physics

High Energy Physics - Phenomenology

Revised version to appear in Nucl. Phys. B 48 pages, uuencoded LATEX file including 4 eps figures

Scientific paper

10.1016/0550-3213(96)00144-7

We use the chiral quark model to estimate the coefficients of the weak chiral lagrangian as obtained from the bosonization of the ten relevant operators of the $\Delta S = 1$ effective quark lagrangian. All contributions of order $N_c^2$ as well as $N_c$ and $\alpha_s N_c$ are included. The chiral coefficients are given in terms of $f_{\pi}$, the quark and gluon condensates 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. The $\gamma_5$-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 result to the evaluation of the hadronic matrix elements for the decay $K^0 \rightarrow 2 \pi$, 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.

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