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

We present the differential branching fractions for the $B\to \psi \pi \pi$ decays with the charmonia $\psi =J/\psi ,\psi (2S)$ in the invariant mass of the $P$-wave pion pairs in the perturbative QCD approach. The two-pion distribution amplitudes (DAs) corresponding to both longitudinal and transverse polarizations are constructed to capture important final state interactions in the processes. The timelike form factors, normalizing the two-pion DAs, contain contributions from the $\rho$ resonance and radial excitations fitted to the BABAR $e^{+}{e}^{-}$ annihilation data. Given the hadronic parameters for the two-pion DAs associated with the longitudinal polarization which were determined in our previous study, and tuning those associated with the transverse polarization, we accommodate well the observed branching ratios and polarization fractions of the $B\to J/\psi \pi \pi$ decays. Our predictions for the $B\to \psi (2S)\pi \pi$ modes from the same set of parameters can be tested in future LHCb and BelleII experiments. We also investigate the sources of theoretical uncertainties in our calculation.

Figure 1

Leading-order Feynman diagrams for the quasi-two-body decays $B\to \psi \rho (\to \pi \pi )$, where $\rho$ represents a $P$-wave $\pi \pi$ intermediate state, with (a) and (b) the factorizable amplitudes, and (c) and (d) the nonfactorizable amplitudes.

Figure 2

$P$-wave contribution to the differential branching fractions of the modes (a) $B^0\to J/\psi {\pi }^{+}{\pi }^{-}$ and (b) $B^0\to \psi (2S){\pi }^{+}{\pi }^{-}$.

Figure 3

$\rho (770)$, $\rho (1450)$, and $\rho (1700)$ resonance contributions to the differential branching fractions of (a) $B^0\to J/\psi {\pi }^{+}{\pi }^{-}$ and (b) $B^0\to \psi (2S){\pi }^{+}{\pi }^{-}$, which are displayed by the solid red, dashed green, and dotted blue curves, respectively.