Observation of $D\to {a}_{0}\left(980\right)\pi $ in the decays ${D}^{0}\to {\pi }^{+}{\pi }^{-}\eta $ and ${D}^{+}\to {\pi }^{+}{\pi }^{0}\eta $

We report the first amplitude analyses of the decays ${D}^{0}\to {\pi }^{+}{\pi }^{-}\eta $ and ${D}^{+}\to {\pi }^{+}{\pi }^{0}\eta $ using a data sample taken with the BESIII detector at the center-of-mass energy of 3773 MeV, corresponding to an integrated luminosity of $7.9\text{}\text{}{\mathrm{fb}}^{-1}$. The contribution from the process ${D}^{0\left(+\right)}\to {a}_{0}\left(980{\right)}^{+}{\pi }^{-\left(0\right)}$ is significantly larger than the ${D}^{0\left(+\right)}\to {a}_{0}\left(980{\right)}^{-\left(0\right)}{\pi }^{+}$ contribution. The ratios $B\left({D}^{0}\to {a}_{0}\left(980{\right)}^{+}{\pi }^{-}\right)/B\left({D}^{0}\to {a}_{0}\left(980{\right)}^{-}{\pi }^{+}\right)$ and $B\left({D}^{+}\to {a}_{0}\left(980{\right)}^{+}{\pi }^{0}\right)/B\left({D}^{+}\to {a}_{0}\left(980{\right)}^{0}{\pi }^{+}\right)$ are measured to be ${7.5}_{-0.8\mathrm{stat}}^{+2.5}±{1.7}_{\mathrm{syst}}$ and $2.6±{0.6}_{\mathrm{stat}}±\phantom{\rule{0ex}{0ex}}{0.3}_{\mathrm{syst}}$, respectively. The measured ${D}^{0}$ ratio disagrees with the theoretical predictions by orders of magnitudes, thus implying a substantial contribution from final-state interactions.