From topological amplitudes to rescattering dynamics in charmed baryon decays

Abstract

Charmed baryon decays play an important role in studying the weak and strong interactions, which have been studied in the rescattering dynamics and topological diagram approach. In this work, we establish a theoretical framework to correlate the topological diagram at quark level and rescattering dynamics at hadron level. Note that the chiral Lagrangian involving octet baryons is constructed via (1,1)-rank octet tensors, while topological diagrams are constructed by 3-rank octet tensors. We propose that, the (1,1)-rank amplitudes, which are linear combinations of topological diagrams, will be a bridge between topological amplitudes and rescattering dynamics. The possible meson-meson or meson-baryon coupling configurations are constructed via tensor contractions. The rescattering amplitudes derived from topological amplitudes are consistent with those derived directly from the chiral Lagrangian. The $u$-, $t$-, and $s$-channel rescattering amplitudes contributing to each (1,1)-rank amplitudes in the $SU(3)_F$ limit are derived. Isospin sum rules for all isospin systems in $ B_{c\overline{3}}\to B_8P$ decays are checked in terms of rescattering amplitudes. The rescattering amplitudes contributing to penguin diagrams are found to be comparable to those contributing to tree diagrams, indicating potential observable $CP$ violation in charmed baryon decays. Furthermore, it is found that the K\"orner-Pati-Woo theorem is not consistent with the rescattering dynamics. The proof of the K\"orner-Pati-Woo theorem is questionable when the color changes of quarks arising from gluons are considered. We suggest precisely measuring the branching fraction of the $\Lambda^+_c\to \Sigma^+K^0_S$ mode on Belle (II) to test the K\"orner-Pati-Woo theorem.