Phenomenological studies on the ${\bar{B}}^0\to [K^{-}{\pi }^{+}{]}_{S/V}[{\pi }^{+}{\pi }^{-}{]}_{V/S}\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-}$ decay

Abstract

Within the quasi-two-body decay model, we study the localized $CP$ violation and branching fraction of the four-body decay ${\bar{B}}^0\to [K^{-}{\pi }^{+}{]}_{S/V}[{\pi }^{+}{\pi }^{-}{]}_{V/S}\to K^{-}{\pi }^{+}{\pi }^{-}{\pi }^{+}$ when $K^{-}{\pi }^{+}$ and ${\pi }^{-}{\pi }^{+}$ pair invariant masses are $0.35<m_{K^{-}{\pi }^{+}}<2.04\mathrm{GeV}$ and $0<m_{{\pi }^{-}{\pi }^{+}}<1.06\mathrm{GeV}$, with the pairs being dominated by the ${\bar{K}}_0^{\ast }(700{)}^0$, ${\bar{K}}^{\ast }(892{)}^0$, ${\bar{K}}^{\ast }(1410{)}^0$, ${\bar{K}}_0^{\ast }(1430)$ and ${\bar{K}}^{\ast }(1680{)}^0$, and $f_0(500)$, ${\rho }^0(770)$ , $\omega (782)$ and $f_0(980)$ resonances, respectively. When dealing with the dynamical functions of these resonances, $f_0(500)$, ${\rho }^0(770)$, $f_0(980)$ and ${\bar{K}}_0^{\ast }(1430)$ are modeled with the Bugg model, Gounaris-Sakurai function, Flatt$\acute{\mathrm{e}}$ formalism and LASS lineshape, respectively, while others are described by the relativistic Breit-Wigner function. Adopting the end point divergence parameters ${\rho }_A\in [0,0.5]$ and ${\varphi }_A\in [0,2\pi ]$, our predicted results are ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}({\bar{B}}^0\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-})\in [-0.383,0.421]$ and $\mathcal{B}({\bar{B}}^0\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-})\in [7.36,199.69]\times 10^{-8}$ based on the hypothetical $q\bar{q}$ 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 ${\bar{B}}^0\to SV(VS)$ and all the individual four-body decays ${\bar{B}}^0\to SV(VS)\to K^{-}{\pi }^{+}{\pi }^{-}{\pi }^{+}$, respectively. Our theoretical results for the two-body decays ${\bar{B}}^0\to {\bar{K}}^{\ast }(892{)}^0$$f_0(980)$, ${\bar{B}}^0\to {\bar{K}}_0^{\ast }(1430{)}^0$$\omega (782)$, ${\bar{B}}^0\to {\bar{K}}^{\ast }(892{)}^0{f}_0(980)$, ${\bar{B}}^0\to {\bar{K}}_0^{\ast }(1430{)}^0\rho$,

Comments: 20 pages, 4 tables