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

We report the first amplitude analysis of the decays D⁰ → π⁺ π⁻ η and D⁺ → π⁺π⁰η 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 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)⁰π⁺) 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.