Investigating $B_s$ three-body decays to scalar mesons in perturbative QCD approach

In this work, we study the branching ratios of B^{0}_{s}\rightarrow a_{0}(980)[ \rightarrow K\overline{K}, \pi\eta]a_{0}(980), B^{0}_{s} \rightarrow f_{0}(980)[ \rightarrow \pi^{+}\pi^{-}, K^{+}K^{-}]f_{0}(980) and B^{0}_{s} \rightarrow f_{0}(500)[ \rightarrow \pi^{+}\pi^{-}]f_{0}(500) decays in the pQCD approach, in which the scalar mesons a_{0}(980), f_{0}(980) and f_{0}(500) are regarded as the lowest-lying q\overline{q} state. In the SU(3) nonet, there exists a mixing between the scalars f_{0}(980) and f_{0}(500), so we have considered the mixing effect in our calculations to obtain more reliable data, and the mixing angle \theta is set at [15^{\circ}, 82^{\circ}] and [105^{\circ}, 171^{\circ}]. Based on the isospin symmetry, we also estimate the branching ratios of the B^{0}_{s} \rightarrow f_{0}(980)[ \rightarrow \pi^{0}\pi^{0}]f_{0}(980) and B^{0}_{s} \rightarrow f_{0}(500)[ \rightarrow \pi^{0}\pi^{0}]f_{0}(500) decays. The branching ratios of the B^{0}_{s}\rightarrow a_{0}(980)[ \rightarrow K\overline{K}, \pi\eta]a_{0}(980) decays are much small, while the results of the B^{0}_{s} \rightarrow f_{0}(980)[ \rightarrow \pi\pi, K\overline{K}]f_{0}(980) and B^{0}_{s} \rightarrow f_{0}(500)[ \rightarrow \pi\pi]f_{0}(500) decays are at the order of 10^{-6}\sim10^{-5}, which can be tested in the LHCb and Belle II experiments, hopefully.