Observation of $D\to a_0(980)\pi$ in the decays $D^0\to {\pi }^{+}{\pi }^{-}\eta$ and $D^{+}\to {\pi }^{+}{\pi }^0\eta$

We report the first amplitude analysis 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 3.773 GeV, corresponding to an integrated luminosity of 7.9 ${\mathrm{f}\mathrm{b}}^{-1}$. The contribution from the process $D^{0(+)}\to a_0(980{)}^{+}{\pi }^{-(0)}$ is significantly larger than the $D^{0(+)}\to a_0(980{)}^{-(0)}{\pi }^{+}$ contribution. The ratios $\mathcal{B}(D^0\to a_0(980{)}^{+}{\pi }^{-})\mathrm{/}\mathcal{B}(D^0\to a_0(980{)}^{-}{\pi }^{+})$ and $\mathcal{B}(D^{+}\to a_0(980{)}^{+}{\pi }^0)\mathrm{/}\mathcal{B}(D^{+}\to a_0(980{)}^0{\pi }^{+})$ are measured to be $7.5_{-0.8\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}^{+2.5}\pm {1.7}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}}$ and $2.6\pm {0.6}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.3}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}}$, respectively. The measured $D^0$ ratio disagrees with the theoretical predictions by orders of magnitudes, thus implying a substantial contribution from final-state interactions.