Amplitude analysis of $D_{s}^{+} \rightarrow π^{+}π^{0}η$ and first observation of the pure $W$-annihilation decays $D_{s}^{+} \rightarrow a_{0}(980)^{+}π^{0}$ and $D_{s}^{+} \rightarrow a_{0}(980)^{0}π^{+}$
We present the first amplitude analysis of the decay $D^{+}_{s} \rightarrow π^{+}π^{0}η$. We use an $e^{+}e^{-}$ collision data sample corresponding to an integrated luminosity of 3.19~${\mbox{\,fb}^{-1}}$ collected with the BESIII detector at a center-of-mass energy of $4.178$ GeV. We observe for the first time the pure $W$-annihilation decays $D_{s}^{+} \rightarrow a_{0}(980)^{+}π^{0}$ and $D_{s}^{+} \rightarrow a_{0}(980)^{0}π^{+}$. We measure the absolute branching fractions $\mathcal{B}(D_{s}^{+} \rightarrow a_{0}(980)^{+(0)}π^{0^(+)}, a_{0}(980)^{+(0)} \to π^{+(0)}η) = (1.46\pm0.15_{\rm stat.}\pm0.23_{\rm sys.})$\%, which is larger than the branching fractions of other measured pure $W$-annihilation decays by at least one order of magnitude. In addition, we measure the branching fraction of $D_{s}^{+} \rightarrow π^{+}π^{0}η$ with significantly improved precision.