From the amplitude analysis of the $D_s^{+}\to {\pi }^{+}{\pi }^0\eta$ decay, the BESIII Collaboration first 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 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 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 $D_s^{+}\to {\pi }^{+}{\pi }^0\eta$ 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 $K^{*}\overline{K}/K{\overline{K}}^{*}$ triangle diagrams. The intermediate $a_0(980)$ state can be dynamically generated from the final state interactions of coupled $K\overline{K}$ and $\pi \eta$ 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.

• Received 21 February 2021
• Accepted 12 April 2021

DOI:https://doi.org/10.1103/PhysRevD.103.116016

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Published by the American Physical Society