Study of the $B^{+}\to\pi^{+}\left(\pi^{+}\pi^{-}\right)$ decay in PQCD Approach

Based on the fitting results of the LHCb collaboration on the contributions of various intermediate resonances to the $B^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decay, we make systematically calculate the branching fractions and localized $CP$ asymmetries of the quasi-two-body $B^{+}\to \pi^{+} \left( \rho(770), \omega(782), \rho(1450), f_{2}\left(1270\right)\to\right)\pi^{+}\pi^{-}$ decays within the perturbative QCD (PQCD) approach. Our theoretical predictions for the branching fractions agree well with the data within the errors. In order to further test the framework of the three-body $B$ decays in PQCD and the wave functions of $\pi\pi$ pair, we also calculate the branching fractions of the corresponding two-body $B^{+}\to\pi^{+}\rho(770)$, $B^{+}\to\pi^{+}\omega(782)$ and $B^{+}\to\pi^{+}f_{2}\left(1270\right)$ decays under the narrow-width-approximation, which are in consistence with the experimental data. The direct $CP$ asymmetries of $B^{+}\to\pi^{+}(\rho(770/1450)\to) \pi^{+}\pi^{-}$ decays are found to be very small because these decay modes are tree-dominated. However, due to the large penguin contributions from the chiral enhanced annihilation diagrams, $B^{+}\to\pi^{+} (f_{2}\left(1270\right)\to )\pi^{+}\pi^{-}$ decay has a large direct $CP$ asymmetry, which is also discovered by LHCb collaboration. A relatively large $CP$ asymmetry is also expected in the $B^+\to\pi^+ (\omega\to)\pi^+\pi^-$ decay occurring via $\rho-\omega$ mixing, which would be measured LHCb and Belle-II experiments in future.