$P$-wave contributions to $B_{(s)}\to\psi K\pi$ decays in perturbative QCD approach

In this work, we studied the quasi-two-body decays $B_{(s)} \to \psi [K^*(892), K^*(1410),$ $K^*(1680)] \to \psi K\pi$ by employing the perturbative QCD (PQCD) factorization approach, where the charmonia $\psi$ represents $J/\psi$ and $\psi(2S)$. The corresponding decay channels are studied by constructing the kaon-pion distribution amplitude (DA) $\Phi_{K \pi}^{\rm P}$, which contained the important final state interactions between the kaon and pion in the resonant region. The relativistic Breit-Wigner formulas are adopted to parameterize the time-like form factor $F_{K\pi}$ appeared in the kaon-pion DAs. The SU(3) flavor symmetry breaking effect resulting from the mass difference between kaon and pion is taken into account, which makes significant contributions to the longitudinal polarizations. We accommodate well the observed branching ratios and the polarization fractions of the $B_{(s)} \to \psi K^*(892) \to \psi K\pi$ by tuning the hadronic parameters for the kaon-pion DAs. The PQCD predictions for the $B_{(s)} \to \psi [K^*(1410), K^*(1680)] \to \psi K\pi$ modes from the same set of parameters can be tested by the future precise data from the LHCb and the Belle II experiments.