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We present the differential branching fractions for the $B\rightarrow \psi\pi\pi$ decays with the charmonia $\psi=J/\psi,\psi(2S)$ in the invariant massof the $P$-wave pion pairs in the perturbative QCD approach. The two-piondistribution amplitudes (DAs) corresponding to both longitudinal and transversepolarizations are constructed to capture important final state interactions inthe processes. The time-like form factors, normalizing the two-pion DAs,contains contributions from the $\rho$ resonance and radial excitations fittedto the BaBar data. Given the hadronic parameters for the two-pion DAsassociated with the longitudinal polarization which were determined in ourprevious study, and tuning those associated with the transverse polarization,we accommodate well the observed branching ratios and polarization fractions ofthe $B\rightarrow J/\psi \pi\pi$ decays. Our predictions for the $B\rightarrow\psi(2S) \pi\pi$ modes from the same set of parameters can be tested in futureLHCb and BelleII experiments. We also investigate the sources of theoreticaluncertainties in our calculation.

$P$-wave contributions to $B\to\psi\pi\pi$ decays in perturbative QCD approach