Amplitude analysis of $D_s^{+}\to {\pi }^{+}{\pi }^0\eta$ and first observation of the pure $W$-annihilation decays $D_s^{+}\to a_0(980{)}^{+}{\pi }^0$ and $D_s^{+}\to a_0(980{)}^0{\pi }^{+}$

We present the first amplitude analysis of the decay $D^{+}{s} \rightarrow \pi^{+}\pi^{0}\eta$\mathrm{.}Weusean$e^{+}e^{-}$ collision data sample corresponding to an integrated luminosity of 3.19~$f{b}^{-1}$ 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 $D{s}^{+} \rightarrow a_{0}(980)^{+}\pi^{0}$and$D_{s}^{+} \rightarrow a_{0}(980)^{0}\pi^{+}$. We measure the absolute branching fractions $$\mathcal{B}$(D_{s}^{+} \rightarrow a_{0}(980)^{+(0)}\pi^{0^(+)}, a_{0}(980)^{+(0)} \to \pi^{+(0)}\eta) = (1.46\pm0.15_{{\rm stat.}}\pm0.23_{{\rm sys.}})$%, 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 $D_s^{+}\to {\pi }^{+}{\pi }^0\eta$ with significantly improved precision.