A self-consistent framework of topological amplitude and its SU(N) decomposition

We propose a systematic theoretical framework for the topological amplitudes of the heavy meson decays and their SU(N) decomposition. In the framework, the topologies are expressed in invariant tensors and classified into tree- and penguin-operator-induced diagrams according to which four-quark operators, tree or penguin, being inserted into their effective weak vertexes. The number of possible topologies contributing to one type of decay can be counted by permutations and combinations. The Wigner-Eckhart theorem ensures the topological amplitudes under flavor symmetry are the same for different decay channels. By decomposing the four-quark operators into irreducible representations of SU(N) group, one can get the SU(N) irreducible amplitudes. Taking the D → PP decay (P denoting a pseudoscalar meson) with SU(3)F symmetry as an example, we present our framework in detail. The linear correlation of topologies in the SU(3)F limit is clarified in group theory. It is found there are only nine independent topologies in all tree- and penguin-operator-induced diagrams contributing to the D → PP decays in the Standard Model. If a large quark-loop diagram, named TLP, is assumed, the large ∆ACP and the very different D0→ K+K− and D0→ π+π− branching fractions can be explained with a normal U-spin breaking. Moreover, our framework provides a simple way to analyze the SU(N) breaking effects. The linear SU(3)F breaking and the high order U-spin breaking in charm decays are re-investigated in our framework, which are consistent with literature. Analogous to the degeneracy and splitting of energy levels, we propose the concepts of degeneracy and splitting of topologies to describe the flavor symmetry breaking effects in decay. As applications, we analyze the strange-less D decays in SU(3)F symmetry breaking into Isospin symmetry and the charm-less B decays in SU(4)F symmetry breaking into SU(3)F symmetry.

Two-loop QCD penguin contribution to the width difference in Bs − $$ {\overline{B}}_s $$ mixing

We consider two-loop QCD corrections to the element $$ {\Gamma}_{12}^q $$ Γ 12 q of the decay matrix in Bq−$$ {\overline{B}}_q $$ B ¯ q mixing, q = d, s, in the leading power of the Heavy Quark Expansion. The calculated contributions involve one current-current and one penguin operator and constitute the next step towards a theory prediction for the width difference ∆Γs matching the precise experimental data. We present compact analytic results for all matching coefficients in an expansion in mc/mb up to second order. Our new corrections are comparable in size to the current experimental error and slightly increase ∆Γs.

EXTENDED ANALYSIS OF THE “PENGUIN” PART OF THE K→ππ AMPLITUDE

We make an attempt to clarify the role of the annihilation or “penguin” mode in the description of the K→ππ decay within the Standard Model. Attention is centered on new operators in the effective ΔS=1 Hamiltonian and the violation of factorization for mesonic matrix elements of the local four-quark operators. We propose a regular method to evaluate the mesonic matrix elements of K→ππ transitions based on the study of three-point correlators via QCD sum rules using the chiral effective theory as an underlying low energy model for strong interaction. Matrix elements of the QCD penguin operator are calculated within this approach. The total “penguin” contribution is found to be relatively large, which improves the theoretical description of the ΔI=1/2 rule in nonleptonic kaon decays.

THE PROBLEM OF BOSONIZATION OF LOCAL FOUR-QUARK OPERATORS

The problem of bosonization of standard local four-quark operators, by involving the penguin operator, with ΔI = 1/2 and ΔS = 1 selection rules, is studied. We show that the well-known Cronin's operator determined within the effective chiral Lagrangian approach with nonlinear realization of chiral SU (3) × SU (3) symmetry (so-called chiral perturbation theory at the hadronic level) can realize the hadronic level description of local four-quark operators with (V − A) × (V − A) quark structure only. Also, it is shown that any attempts to apply Cronin's operator for bosonization of the penguin operator, possessing (V − A) × (V + A) quark structure, are doomed to failure. The latter is explained by the inability of Cronin's operator to describe correctly the s-wave final-state interaction, which is very important for transitions governed by the penguin operator. The asymmetry of quark condensate is calculated. The sign of this asymmetry is opposite to that one assumed in the framework of QCD sum rules.

IS THE CRONIN’S REPRESENTATION CORRECT FOR THE PENGUIN OPERATOR?

It is shown that the Cronin’s operator, being used for a phenomenological description of non-loptonic K-meson decays within the effective chiral Lagrangian approach with non-linear realization of chiral SU (3)× SU (3) symmetry (so-called, chiral perturbation theory at the hadronic level), cannot be applied to the determination of the Penguin-operator (O5-operator) at the hadronic level. The affirmation is based on the proof of the violation of the low-energy theorem [Formula: see text] which is valid within the framework of the Cronin’s representation. The violation of this low-energy theorem is obtained at the quark level and connected with the account of the low-energy interactions of pions caused by the overlap of wave functions of pions in the |π+π−> state in the low-energy limit p+→0. The chiral transformation properties of the normal-ordered Penguin operator are studied.