Study of the $B_s$ $\to$ $\ddot{I}•f_0(980)$ $\to$ $\ddot{I}•\ddot{I}{€}^{+}\ddot{I}{€}^{-}$ decay with perturbative QCD approach

Abstract

The rare cascade $B_s$ $\to$ $\ddot{I}•f_0(980)$ $\to$ $\ddot{I}•\ddot{I}{€}^{+}\ddot{I}{€}^{-}$ decay is studied with the perturbative QCD approach based on the formula for the quasi two-body decay, where the two-pion pair originates from the $S$-wave resonant $f_0(980)$ state. It is found that with the introduction of the nonperturbative two-pion distribution amplitudes and the Flatté parameterization of the scalar form factor for the $f_0(980)$ resonance, the branching ratio in the mass range 400 MeV $<$ $m(\ddot{I}{€}^{+}\ddot{I}{€}^{-})$ $<$ 1600 MeV is ${\mathcal{B}}_{\mathrm{theo}}(B_s^0\to \ddot{I}•f_0(980)\to \ddot{I}•\ddot{I}{€}^{+}\ddot{I}{€}^{-})$ $=$ $[1.31_{-0.31}^{+0.40}(a_{\ddot{I}€\ddot{I}€}){}_{-0.16}^{+0.19}(m_b){}_{-0.09}^{+0.10}(\text{CKM})]\times 10^{-6}$, where the uncertainties come from the parameter $a_{\ddot{I}€\ddot{I}€}$ of the two-pion distribution amplitudes, the $b$ quark mass $m_b$, the Cabibbo-Kobayashi-Maskawa (CKM) factors, respectively. This result agrees with the recent LHCb measurement within uncertainties, ${\mathcal{B}}_{\exp }(B_s^0\to \ddot{I}•f_0(980)\to \ddot{I}•\ddot{I}{€}^{+}\ddot{I}{€}^{-})$ $=$ $(1.12\pm 0.16_{-0.08}^{+0.09}\pm 0.11)\times 10^{-6}$, where the errors are statistical, systematic and from the normalization, respectively.

27 pages, 2 figures