Study of the f₀(980) through the decay $Dₛ⁺→ π⁺π⁺π⁻π⁰$

We perform the first amplitude analysis of D⁺ₛ → π⁺π⁺π⁻π⁰ decays, based on data samples of electron-positron collisions recorded with the BESIII detector at center-of-mass energies between 4.128 and 4.226 GeV, corresponding to an integrated luminosity of 7.33~fb⁻¹. We report the observation of Dₛ⁺ → f₀(980)ρ(770)⁺ with a statistical significance greater than 10σ and determine the branching fractions B(Dₛ⁺→π⁺π⁺π⁻π⁰|_ non-η)=(2.04±0.08stat.±0.05syst.)% and B(Dₛ⁺→ηπ⁺)=(1.56±0.09stat.±0.04syst.)%. Moreover, we measure the relative branching fraction between φ→π⁺π⁻π⁰ and φ→ K⁺K⁻ to be B(φ(1020) → π⁺π⁻π⁰)/B(φ(1020) → K⁺K⁻)=0.230 ± 0.014stat. ± 0.010syst., which deviates from the world average value by more than 4σ.