Study of the $B_{s}$ ${\to}$ $ϕf_{0}(980)$ ${\to}$ $ϕ\,π^{+}π^{-}$ decay with perturbative QCD approach

Abstract

The rare cascade $B_{s}$ ${\to}$ $ϕf_{0}(980)$ ${\to}$ $ϕ\,π^{+}π^{-}$ 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 $<$ $m(π^{+}π^{-})$ $<$ 1600 MeV is ${\cal B}_{\rm theo}(B_{s}^{0}\, {\to}\,ϕ\,f_{0}(980)\,{\to}\,ϕ\,π^{+}\,π^{-})$ $=$ $[1.31_{-0.31}^{+0.40}(a_{ππ}){}_{-0.16}^{+0.19}(m_{b}) {}_{-0.09}^{+0.10}(\text{CKM})]{\times}10^{-6}$, where the uncertainties come from the parameter $a_{ππ}$ 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, ${\cal B}_{\rm exp}(B_{s}^{0}\,{\to}\,ϕ\, f_{0}(980)\,{\to}\,ϕ\,π^{+}\,π^{-})$ $=$ $(1.12{\pm}0.16^{+0.09}_{-0.08}{\pm}0.11){\times}10^{-6}$, where the errors are statistical, systematic and from the normalization, respectively.