The S-wave resonance contributions in the $B^0_s$ decays into $ \psi(2S,3S)$ plus pion pair

The three-body decays $B^0_s \rightarrow \psi(2S,3S) \pi^+ \pi^-$ are studied based on the perturbative QCD approach. With the help of the nonperturbative two-pion distribution amplitudes, the analysis is simplified into the quasi-two-body processes. Besides the traditional factorizable and nonfactorizable diagrams at the leading order, the next-to-leading order vertex corrections are also included to cancel the scale dependence. The $f_0(980)$, $f_0(1500)$ resonance contributions as well as the nonresonant contributions are taken into account using the presently known $\pi\pi$ time-like scalar form factor for the $s\bar{s}$ component. It is found that the predicted $B^0_s \rightarrow \psi(2S) \pi^+ \pi^-$ decay spectra in the pion pair invariant mass shows a similar behavior as the experiment. The calculated S-wave contributions to the branching ratio of $B^0_s \rightarrow \psi(2S) \pi^+ \pi^-$ is $6.0\times 10^{-5}$, which is in agreement with the LHCb data $\mathcal {B}(B^0_s \rightarrow \psi(2S) \pi^+ \pi^-)=(7.2\pm 1.2)\times 10^{-5} $ within errors. The estimate of $\mathcal {B}(B^0_s \rightarrow \psi(3S) \pi^+ \pi^-)$ can reach the order of $10^{-5}$, pending for the corresponding measurements.