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From topological amplitude to rescattering dynamics

We proposed a theoretical framework to correlate the topological diagram at quark level and rescattering dynamics at hadron level. In the framework, both the hadronic triangle diagram, and the topological-scattering diagram, which is the intermediate structure between topological diagram and triangle diagram, are expressed in the tensor form. The completeness of topological-scattering diagram is confirmed by the quark substructure of meson-meson scattering. The coefficient of each triangle diagram can be derived from the topological-scattering diagram and the total rescattering amplitudes are consistent with the ones derived from the chiral Lagrangian. If only the short-distance $T$ diagram is considered as the weak vertex in triangle diagram, the rescattering contributions in the $C$, $E$ and $P$ diagrams have definite proportional relation of $L(C):L(E):L(P)=-2:1:1$ under the $SU(3)_F$ symmetry, and the rescattering contributions in the $T$ and $A$ diagrams only arise from the $SU(3)_F$ breaking effects. Taking $D\to Kπ$ and $D\to ππ$ modes as examples, we present our framework in detail. We find the Isospin relations in these decays are still valid in terms of triangle diagrams. Besides, the conclusions in the $D$ meson decays under the $SU(3)_F$ symmetry can be generalized to the $B$ meson decays under the $SU(4)_F$ symmetry.