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

Abstract

The three-body decays $B_s^0\to \psi (2S,3S){\pi }^{+}{\pi }^{-}$ 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 $B_s^0\to \psi (2S){\pi }^{+}{\pi }^{-}$ 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 $B_s^0\to \psi (2S){\pi }^{+}{\pi }^{-}$ is $6.0\times 10^{-5}$, which is in agreement with the LHCb data $\mathcal{B}(B_s^0\to \psi (2S){\pi }^{+}{\pi }^{-})=(7.2\pm 1.2)\times 10^{-5}$ within errors. The estimate of $\mathcal{B}(B_s^0\to \psi (3S){\pi }^{+}{\pi }^{-})$ can reach the order of $10^{-5}$, pending for the corresponding measurements.

7 pages, 3 figures, 2 tables

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