Observation of $D \to a_{0}(980)π$ in the decays $D^{0} \rightarrow π^{+}π^{-}η$ and $D^{+} \rightarrow π^{+}π^{0}η$

We report the first amplitude analysis of the decays $D^{0} \to π^{+} π^{-} η$ and $D^{+} \rightarrow π^{+}π^{0}η$ 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 ${\rm fb}^{-1}$. The contribution from the process $D^{0(+)} \to a_{0}(980)^{+} π^{-(0)}$ is significantly larger than the $D^{0(+)} \to a_{0}(980)^{-(0)} π^{+}$ contribution. The ratios $\mathcal{B}(D^{0} \rightarrow a_{0}(980)^{+}π^{-})/\mathcal{B}(D^{0} \rightarrow a_{0}(980)^{-}π^{+})$ and $\mathcal{B}(D^{+} \rightarrow a_{0}(980)^{+}π^{0})/\mathcal{B}(D^{+} \rightarrow a_{0}(980)^{0}π^{+})$ are measured to be $7.5^{+2.5}_{-0.8\,\mathrm{stat.}}\pm1.7_{\mathrm{syst.}}$ and $2.6\pm0.6_{\mathrm{stat.}}\pm0.3_{\mathrm{syst.}}$, respectively. The measured $D^{0}$ ratio disagrees with the theoretical predictions by orders of magnitudes, thus implying a substantial contribution from final-state interactions.