Can the nature of a 0 ( 980 ) be tested in the D s + → π + π 0 η decay?

From the amplitude analysis of the ${D}_{s}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{0}\ensuremath{\eta}$ decay, the BESIII Collaboration first observed the ${D}_{s}^{+}\ensuremath{\rightarrow}{a}_{0}(980{)}^{+}{\ensuremath{\pi}}^{0}$ and ${D}_{s}^{+}\ensuremath{\rightarrow}{a}_{0}(980{)}^{0}{\ensuremath{\pi}}^{+}$ decay modes, which are expected to occur through the pure $W$-annihilation processes. The measured branching fraction $\mathcal{B}[{D}_{s}^{+}\ensuremath{\rightarrow}{a}_{0}(980{)}^{+(0)}{\ensuremath{\pi}}^{0(+)},{a}_{0}(980{)}^{+(0)}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+(0)}\ensuremath{\eta}]$ is, however, found to be larger than those of known $W$-annihilation decays by 1 order of magnitude. This apparent contradiction can be reconciled if the two decays are induced by internal $W$-conversion or external $W$-emission mechanisms instead of a $W$-annihilation mechanism. In this work, we propose that the ${D}_{s}^{+}$ decay proceeds via both the external and internal $W$-emission instead of $W$-annihilation mechanisms. In such a scenario, we perform a study of the ${D}_{s}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{0}\ensuremath{\eta}$ decay by taking into account the contributions from the tree diagram ${D}_{s}^{+}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{+}\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{0}\ensuremath{\eta}$ and the intermediate ${\ensuremath{\rho}}^{+}\ensuremath{\eta}$ and ${K}^{*}\overline{K}/K{\overline{K}}^{*}$ triangle diagrams. The intermediate ${a}_{0}(980)$ state can be dynamically generated from the final state interactions of coupled $K\overline{K}$ and $\ensuremath{\pi}\ensuremath{\eta}$ channels, and it is shown that the experimental data can be described fairly well, which supports the interpretation of ${a}_{0}(980)$ as a molecular state.