Study of the decay D⁺_s → π⁺π⁺π⁻η and observation of the W-annihilation decay D⁺_s → a₀(980)⁺ρ⁰

The decay D+s→π+π+π−η is observed for the first time, using e+e− collision data corresponding to an integrated luminosity of 6.32  fb−1, collected by the BESIII detector at center-of-mass energies between 4.178 and 4.226 GeV. The absolute branching fraction for this decay is measured to be B(D+s→π+π+π−η)=(3.12±0.13stat±0.09syst)%. The first amplitude analysis of this decay reveals the substructures in D+s→π+π+π−η and determines the relative fractions and the phases among these substructures. The dominant intermediate process is D+s→a1(1260)+η,a1(1260)+→ρ(770)0π+ with a branching fraction of (1.73±0.14stat±0.08syst)%. We also observe the W-annihilation process D+s→a0(980)+ρ(770)0, a0(980)+→π+η with a branching fraction of (0.21±0.08stat±0.05syst)%, which is larger than the branching fractions of other measured pure W-annihilation decays by 1 order of magnitude.