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From the amplitude analysis of the $D^+_s \to \pi^+ \pi^0 \eta$ decay, theBESIII Collaboration firstly observed the $D^+_s \to a_0(980)^+\pi^0$ and$D^+_s \to a_0(980)^0\pi^+$ decay modes, which are expected to occur throughthe 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 contradictioncan be reconciled if the two decays are induced by internal $W$-conversion orexternal $W$-emission mechanisms instead of $W$-annihilation mechanism. In thiswork, we propose that the $D^+_s$ decay proceeds via both the external andinternal $W$-emission instead of $W$-annihilation mechanisms. In such ascenario, we perform a study of the $D^+_s \to \pi^+\pi^0\eta$ decay by takinginto account the contributions from the tree diagram $D^+_s \to \rho^+ \eta \to\pi^+ \pi^0 \eta$ and the intermediate $\rho^+ \eta$ and$K^*\bar{K}/K\bar{K}^*$ triangle diagrams. The intermediate $a_0(980)$ statecan be dynamically generated from the final state interactions of coupled $K\bar{K}$ and $\pi \eta$ channels, and it is shown that the experimental datacan be described fairly well, which supports the interpretation of $a_0(980)$as a molecular state.

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