The $\Delta S=1$ Weak Chiral Lagrangian as the Effective Theory of the Chiral Quark Model

Abstract

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 $K^0\to 2\pi$, 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.

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

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• CP violation in kaon decays
• Direct CP violation in $K^0\to \pi \pi$: Standard Model Status
• The Delta I = 1/2 Rule and B_K at O(p^4) in the Chiral Expansion
• Final State Interactions in Kaon Decays
• Obtaining K to pi pi from Off-Shell K to pi Amplitudes
• The $\Delta I=1/2$ Selection Rule
• Kaon CP violation and neutron EDM in the minimal left-right symmetric model
• Electromagnetism in nonleptonic weak interactions
• A New Estimate of ${\epsilon }^{\prime }/\epsilon$
• Isospin-Violating Contributions to ${\epsilon }^{\prime }/\epsilon$
• Hunting down massless dark photons in kaon physics
• New physics in ε' from chromomagnetic contributions and limits on Left-Right symmetry
• Nonfactorizable contributions to the decay mode D^0 -> K^0 \bar{K^0}
• The Large $N_c$ limit of QCD on the lattice
• A New Estimate of ${\epsilon }^{\prime }/\epsilon$
• A Heavy-Light Chiral Quark Model
• A gluonic mechanism for B --> D eta'
• The Strong Isospin-Breaking Correction for the Gluonic Penguin Contribution to epsilon'/epsilon at Next-to-Leading Order in the Chiral Expansion
• SU(3) analysis of four-quark operators: $K\to \pi \pi$ and vacuum matrix elements
• The Physics of $K^0-{\bar{K}}^0$ Mixing: ${\hat{B}}_K$ and $\Delta M_{LS}$ in the Chiral Quark Model
• K to ππ hadronic matrix elements of left-right current-current operators
• epsilon'/epsilon in the Standard Model
• On the Colour Suppressed Decay Modes B^0 --> D_s^+ D_s^- and B_s^0 --> D^+ D^-
• Electromagnetic penguin operators and direct CP violation in K --> pi l^+ l^-
• Effective Weak Chiral Lagrangian to $\mathcal{O}(p^4)$ in the Chiral Quark Model
• The Bosonization of the Electroweak Penguin Operators
• $\Delta S=1,2$ Effective Weak Chiral Lagrangian from the Instanton Vacuum
• The Isgur-Wise Function within a Modified Heavy-Light Chiral Quark Model
• Non-factorizable contribtion to $\overline{B_d^0}\to {\pi }^0{D}^0$
• On the Decay Modes $\overline{B_{d,s}^0}\to \gamma D^{\ast 0}$
• Theoretical overview of kaon decays
• On the color suppressed contribution to $\bar{B_{d}^0} \rightarrow \, π^0 π^{0}
• Final State Interactions and eps'/eps
• Chiral Logarithms in $\Delta S=1$ Kaon Decay Amplitudes in General Effective Flavor Theories
• Non-factorizable contributions to $\overline{B_d^0}\to D_s^{(\ast )}\overline{D_s^{(\ast )}}$
• CP conserved nonleptonic $K\to \pi \pi \pi$ decays in the Chiral Quark Model