Probing the soft rescattering parameters in $B$ decays involving a scalar meson with QCD factorization

Abstract

In this work, the soft rescattering parameters in the $B^{\pm }\to {\pi }^{\pm }{\pi }^{+}{\pi }^{-}$ and $B^{\pm }\to K^{\pm }{\pi }^{+}{\pi }^{-}$ decays with the light scalar meson $f_0(500)$ as the intermediate resonance are studied within the QCD factorization. Considering the interference effect between $\rho (770{)}^0$ and $f_0(500)$, we utilize the experimentally more direct event yields for fitting and get the soft rescattering parameters $|{\rho }_k^{SP}|=3.29\pm 1.01$ and $|{\rho }_k^{PS}|=2.33\pm 0.73$ in $B\to PS$ and $B\to SP$ decays ($P$ and $S$ denote pseudoscalar and scalar mesons, respectively), respectively. We also study the branching ratios and $CP$ asymmetries in the decay modes involving other scalar mesons, including $f_0(980)$, $a_0(980)$, $a_0(1450)$ and $K_0^{\ast }(1430)$, to test the rationality of the values of $|{\rho }_k^{SP}|$ and $|{\rho }_k^{PS}|$. Meanwhile, the wealth of experimental data facilitate the examination of the forward-backward asymmetry induced $CP$ asymmetries (FB-CPAs), and the localized $CP$ asymmetries (LACPs). We investigate these asymmetries resulting from the interference between $\rho (770{)}^0$ and $f_0(500)$ for $B^{\pm }\to {\pi }^{\pm }{\pi }^{+}{\pi }^{-}$ and $B^{\pm }\to K^{\pm }{\pi }^{+}{\pi }^{-}$ decays when the invariant mass of ${\pi }^{+}{\pi }^{-}$ locates in the low-energy region $0.445\mathrm{GeV}<m_{\pi \pi }<0.795\mathrm{GeV}$. Our theoretical results of FB-CPAs and LACPs align with the experimental findings. We propose that the interference between $\rho (770{)}^0$ and $f_0(500)$ can be extended to other beauty and charmed mesons decays.