Observation of D → a₀(980)π in the decays D⁰ → π⁺π⁻η and D⁺ → π⁺π⁰η

We report the first amplitude analyses of the decays 𝐷0 →𝜋+⁢𝜋−⁢𝜂 and 𝐷+ →𝜋+⁢𝜋0⁢𝜂 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  fb−1. The contribution from the process 𝐷0⁢(+)→𝑎0⁢(980)+⁢𝜋−(0) is significantly larger than the 𝐷0⁢(+)→𝑎0⁢(980)−(0)⁢𝜋+ contribution. The ratios ℬ⁡(𝐷0→𝑎0⁢(980)+⁢𝜋−)/ℬ⁡(𝐷0→𝑎0⁢(980)−⁢𝜋+) and ℬ⁡(𝐷+→𝑎0⁢(980)+⁢𝜋0)/ℬ⁡(𝐷+→𝑎0⁢(980)0⁢𝜋+) are measured to be 7.5+2.5−0.8⁢stat±1.7syst and 2.6±0.6stat±0.3syst, respectively. The measured 𝐷0 ratio disagrees with the theoretical predictions by orders of magnitudes, thus implying a substantial contribution from final-state interactions.