Quasi-two-body decays $B_{(s)}\to P{\rho }^{\prime }(1450),P{\rho }^{\prime \prime }(1700)\to P\pi \pi$ in the perturbative QCD approach

Abstract

In this work, we calculate the $CP$-averaged branching ratios and direct $CP$-violating asymmetries of the quasi-two-body decays $B_{(s)}\to P{\rho }^{\prime }(1450),P{\rho }^{\prime \prime }(1700)\to P\pi \pi$ by employing the perturbative QCD (PQCD) factorization approach, where $P$ is a light pseudoscalar meson $K,\pi ,\eta$ and ${\eta }^{\prime }$. The considered decay modes are studied in the quasi-two-body framework by parameterizing the two-pion distribution amplitude ${\Phi }_{\pi \pi }^{\mathrm{P}}$. The $P$-wave time-like form factor $F_{\pi }$ in the resonant regions associated with the ${\rho }^{\prime }(1450)$ and ${\rho }^{\prime \prime }(1700)$ is estimated based on available experimental data. The PQCD predictions for the $CP$-averaged branching ratios of the decays $B_{(s)}\to P{\rho }^{\prime }(1450),P{\rho }^{\prime \prime }(1700)\to P\pi \pi$ are in the order of $10^{-7}\sim 10^{-9}$. The branching ratios of the two-body decays $B_{(s)}\to P{\rho }^{\prime }(1450),P{\rho }^{\prime \prime }(1700)$ are extracted from the corresponding quasi-two-body decay modes. The whole pattern of the pion form factor-squared $|F_{\pi }{|}^2$ measured by {\it BABAR} Collaboration could also be understood based on our studies. Our PQCD predictions will be tested by the precise data from the LHCb and the future Belle II experiments.

Comments: 8 pages, 1 figure