Study of the quasi-two-body decays B${}_s^0$→ψ(3770)(ψ(3686))π${}^{+}$π${}^{-}$ with perturbative QCD approach

In this note, we study the contributions from the S-wave resonances, f${}_0$(980) and f${}_0$(1500), to the B${}_s^0$→ψ(3770)π${}^{+}$π${}^{-}$ decay by introducing the S-wave ππ distribution amplitudes within the framework of perturbative QCD approach. Both the resonant and the non-resonant contributions are contained in the scalar form factor in the S-wave distribution amplitude Φ${}_{\pi \pi }^S$. Since the vector charmonium meson ψ(3770) is a S–D wave mixed state, we calculate the branching ratios of S-wave and D-wave, respectively, and the results indicate that f${}_0$(980) is the main contribution of the considered decay, and the branching ratio of the ψ(2S) mode is in good agreement with the experimental data. We also take the S–D mixed effect into the B${}_s^0$→ψ(3686)π${}^{+}$π${}^{-}$ decay. Our calculations show that the branching ratio of B${}_s^0$→ψ(3770)(ψ(3686))π${}^{+}$π${}^{-}$ can be at the order of 10${}^{-5}$, which can be tested by the running LHCb experiments.