Observation of ${D}^{+}\to {K}_{S}^{0}{a}_{0}\left(980{\right)}^{+}$ in the Amplitude Analysis of ${D}^{+}\to {K}_{S}^{0}{\pi }^{+}\eta $

We perform for the first time an amplitude analysis of the decay ${D}^{+}\to {K}_{S}^{0}{\pi }^{+}\eta $ and report the observation of the decay ${D}^{+}\to {K}_{S}^{0}{a}_{0}\left(980{\right)}^{+}$ using $2.93\text{}\text{}{\mathrm{fb}}^{-1}$ of ${e}^{+}{e}^{-}$ collision data taken at a center-of-mass energy of 3.773 GeV with the BESIII detector. As the only $W$-annihilation-free decay among $D$ to ${a}_{0}\left(980\right)$ pseudoscalar, ${D}^{+}\to {K}_{S}^{0}{a}_{0}\left(980{\right)}^{+}$ is the ideal decay in extracting the contributions of the $W$-emission amplitudes involving ${a}_{0}\left(980\right)$ and to study the final-state interactions. The absolute branching fraction of ${D}^{+}\to {K}_{S}^{0}{\pi }^{+}\eta $ is measured to be $\left(1.27±0.0{4}_{\mathrm{stat}}±0.0{3}_{\mathrm{syst}}\right)%$. The branching fractions of intermediate processes ${D}^{+}\to {K}_{S}^{0}{a}_{0}\left(980{\right)}^{+}$ with ${a}_{0}\left(980{\right)}^{+}\to {\pi }^{+}\eta $ and ${D}^{+}\to {\pi }^{+}{\overline{K}}_{0}^{*}\left(1430{\right)}^{0}$ with ${\overline{K}}_{0}^{*}\left(1430{\right)}^{0}\to {K}_{S}^{0}\eta $ are measured to be $\left(1.33±0.0{5}_{\mathrm{stat}}±0.0{4}_{\mathrm{syst}}\right)%$ and $\left(0.14±0.0{3}_{\mathrm{stat}}±0.0{1}_{\mathrm{syst}}\right)%$, respectively.