Phenomenological studies on decay * *Supported by National Natural Science Foundation of China (11575023, 11775024, 11947001, 11605150, 11805153), Natural Science Foundation of Zhejiang pvovince (LQ21A050005), Achievements of the Basic Scientific Research Business Foundation Project of Universities in Zhejiang Province and Natural Science Foundation of Ningbo (2019A610067)

Abstract

Within the quasi-two-body decay model, we study the localized CP violation and branching fraction of the four-body decay when the and pair invariant masses are and , with the pairs being dominated by the , , , and , and , , and resonances, respectively. When dealing with the dynamical functions of these resonances, , , and are modeled with the Bugg model, Gounaris-Sakurai function, Flatté formalism and LASS lineshape, respectively, while the others are described by the relativistic Breit-Wigner function. Adopting the end point divergence parameters and , our predicted results are and , based on the hypothetical structures for the scalar mesons in the QCD factorization approach. Meanwhile, we calculate the CP violating asymmetries and branching fractions of the two-body decays and all the individual four-body decays , respectively. Our theoretical results for the two-body decays , , , , and are consistent with the available experimental data, with the remaining predictions await testing in future high precision experiments.