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Study of the $D_s^+\to a_0(980) ρ$ and $a_0(980) ω$ decays

We study $D_{s}^{+}\toρ^{0(+)}a^{+(0)}_{0}$, $D_{s}^{+}\toωa^{+}_{0}$, and the resonant $D_{s}^{+}\toρa_0$,$a_{0}\to ηπ(KK)$ decays. In the final state interaction, where $D_s^+\to (η^{(\prime)}π^+,K^+\bar K^0)$ are followed by the $(η^{(\prime)}π^+,K^+\bar K^0)$ to $ρ^{0(+)}a^{+(0)}_{0}$ rescatterings, we predict ${\cal B}(D_{s}^{+}\toρ^{0(+)}a^{+(0)}_{0})=(3.0\pm 0.3\pm 1.0)\times 10^{-3}$. Due to the cancellation of the rescattering effects and the suppressed short-distance $W$ annihilation contribution, we expect that ${\cal B}(D_{s}^{+}\toωa^{+}_{0}) \simeq {\cal B}(D_s^+\toπ^+π^0)<3.4\times 10^{-4}$. In our calculation, ${\cal B}(D_{s}^{+}\toρ^{0}(a^{+}_{0}\to)ηπ^{+}) =(1.6^{+0.2}_{-0.3}\pm 0.6)\times 10^{-3}$ agrees with the data, whereas ${\cal B}(D_{s}^{+}\toρ^{+}(a^{0}_{0}\to)K^+K^-)$ is 10 times smaller than the observation, which requires a careful examination.