Can the nature of $a_0(980)$ be tested in the $D_s^{+}to pi^{+}pi^0 eta$ decay?

From the amplitude analysis of the $D^+_s to pi^+ pi^0 eta$ decay, the BESIII Collaboration firstly observed the $D^+_s to a_0(980)^+pi^0$ and $D^+_s to a_0(980)^0pi^+$ decay modes, which are expected to occur through the pure $W$-annihilation processes. The measured branching fraction $mathcal{B}[D_{s}^{+}to a_{0}(980)^{+(0)}pi^{0(+)},a_{0}(980)^{+(0)}to pi^{+(0)}eta]$ is, however, found to be larger than those of known $W$-annihilation decays by one 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 $W$-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 $D^+_s to pi^+pi^0eta$ decay by taking into account the contributions from the tree diagram $D^+_s to rho^+ eta to pi^+ pi^0 eta$ and the intermediate $rho^+ eta$ and $K^*bar{K}/Kbar{K}^*$ triangle diagrams. The intermediate $a_0(980)$ state can be dynamically generated from the final state interactions of coupled $K bar{K}$ and $pi eta$ 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.