Study of the quasi-two-body decays $$B^{0}_{s} \rightarrow \psi (3770) (\psi (3686))\pi ^+\pi ^-$$ Bs0→ψ(3770)(ψ(3686))π+π- with perturbative QCD approach

Abstract

Abstract In this note, we study the contributions from the S-wave resonances, $$f_{0}(980)$$ f0(980) and $$f_{0}(1500)$$ f0(1500) , to the $$B^{0}_{s}\rightarrow \psi (3770)\pi ^ {+}\pi ^{-}$$ Bs0→ψ(3770)π+π- decay by introducing the S-wave $$\pi \pi $$ 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 $$\Phi ^S_{\pi \pi }$$ ΦππS . Since the vector charmonium meson $$\psi (3770)$$ 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 $$f_{0}(980)$$ f0(980) is the main contribution of the considered decay, and the branching ratio of the $$\psi (2S)$$ ψ(2S) mode is in good agreement with the experimental data. We also take the S–D mixed effect into the $$B^{0}_{s}\rightarrow \psi (3686)\pi ^ {+}\pi ^{-}$$ Bs0→ψ(3686)π+π- decay. Our calculations show that the branching ratio of $$B^{0}_{s}\rightarrow \psi (3770)(\psi (3686))\pi ^ {+}\pi ^{-}$$ Bs0→ψ(3770)(ψ(3686))π+π- can be at the order of $$10^{-5}$$ 10-5 , which can be tested by the running LHCb experiments.