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

AbstractBased on the fitting results of the LHCb collaboration on the contributions of various intermediate resonances to the$$B^{+}\rightarrow \pi ^{+}\pi ^{+}\pi ^{-}$$B+→π+π+π-decay, we calculate the branching fractions and localizedCPasymmetries of the quasi-two-body$$B^{+}\rightarrow \pi ^{+} \left( \rho (770), \omega (782), \rho (1450), f_{2}\left( 1270\right) \rightarrow \right) \pi ^{+}\pi ^{-}$$B+→π+ρ(770),ω(782),ρ(1450),f21270→π+π-decays within the perturbative QCD (PQCD) approach systematically. 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-bodyBmeson decays in PQCD and the wave functions of$$\pi \pi $$ππpair, we also calculate the branching fractions of the corresponding two-body$$B^{+}\rightarrow \pi ^{+}\rho (770)$$B+→π+ρ(770),$$B^{+}\rightarrow \pi ^{+}\omega (782)$$B+→π+ω(782)and$$B^{+}\rightarrow \pi ^{+}f_{2}\left( 1270\right) $$B+→π+f21270decays under the narrow-width-approximation, which are in consistence with the experimental data. The directCPasymmetries of$$B^{+}\rightarrow \pi ^{+}(\rho (770/1450)\rightarrow ) \pi ^{+}\pi ^{-}$$B+→π+(ρ(770/1450)→)π+π-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,$$B^{+}\rightarrow \pi ^{+} (f_{2}\left( 1270\right) \rightarrow )\pi ^{+}\pi ^{-}$$B+→π+(f21270→)π+π-decay has a large directCPasymmetry, which is also discovered by LHCb collaboration. A relatively largeCPasymmetry is also expected in the$$B^+\rightarrow \pi ^+ (\omega \rightarrow )\pi ^+\pi ^-$$B+→π+(ω→)π+π-decay occurring via$$\rho -\omega $$ρ-ωmixing, which would be measured LHCb and Belle-II experiments in future.