Abstract
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.
Highlights
The topological diagram amplitude (TDA) and irreducible representation amplitude (IRA) approaches seem to be equivalent in the SU(3)F limit
The topologies are expressed in invariant tensors and classified into tree- and penguin-operatorinduced diagrams according to which four-quark operators, tree or penguin, being inserted into their effective weak vertexes
In order to match the tensor form of topology, we suggest to classify the topologies in the Standard Model into tree- and penguin-operator-induced diagrams according to which operators, tree or penguin, being inserted into the effective vertexes, no matter whether the topologies involving quark loop or not
Summary
We study the topological amplitudes and the SU(3) irreducible amplitudes model-independently, taking the D → P P decay as an example. 2.1 Topological amplitude The weak Hamiltonian of charm decay in a general effective theory can be written as. A(Dγ → PαPβ) = PαPβ|Heff |Dγ (Dγ )i(H(p))jl k(Pα)nm(Pβ)sr × PnmPsr|Oj(pk)l|Di , p Per. in which Per. present summing over all the possible full contractions of PnmPsr|Oj(pk)l|Di. Under the flavor symmetry, decay amplitude is a complex number without flavor indices, i.e., a SU(N ) invariant. While Ol(jp)l presents the quark loop induced in one effective vertex in the topological diagram. If the operator with three same indices (for instance (uu)(uc)) is inserted, all the 14 diagrams in figure 3 contribute to the D → P P decays. One can find the sole difference between the TDA and IRA approaches is whether the four-quark operators (or equivalent, (H)ikj) are decomposed into the SU(3) irreducible representations or not. The decomposition of b decay is discussed in appendix B, in which some mistakes in refs. [93, 94] are cleared
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