Study of the $B_s$ $\to$ $\varphi f_0(980)$ $\to$ $\varphi {\pi }^{+}{\pi }^{-}$ decay with perturbative QCD approach

The rare cascade $B_s$ $\to$ $\varphi f_0(980)$ $\to$ $\varphi {\pi }^{+}{\pi }^{-}$ 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 Flatt'{e} parameterization of the scalar form factor for the $f_0(980)$ resonance, the branching ratio in the mass range 400 MeV $<$ $m({\pi }^{+}{\pi }^{-})$ $<$ 1600 MeV is ${\cal B}{\rm theo}(B{s}^{0}, {\to},{\phi},f_{0}(980),{\to},{\phi},{\pi}^{+},{\pi}^{-})$ $=$ $[1.31_{-0.31}^{+0.40}(a_{{\pi}{\pi}}){}{-0.16}^{+0.19}(m{b}) {}{-0.09}^{+0.10}($\text{CKM}$)]{\times}10^{-6}$, where the uncertainties come from the parameter $a_{\pi \pi }$ 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, ${\cal B}{\rm exp}(B_{s}^{0},{\to},{\phi}, f_{0}(980),{\to},{\phi},{\pi}^{+},{\pi}^{-})$ $=$ $(1.12\pm 0.16_{-0.08}^{+0.09}\pm 0.11)\times {10}^{-6}$, where the errors are statistical, systematic and from the normalization, respectively.