Study of D_{s}^{+}→f_{0}(980)ρ^{+} and ϕπ^{+} Decays through D_{s}^{+}→π^{+}π^{+}π^{-}π^{0}

We perform the first amplitude analysis of D_{s}^{+}→π^{+}π^{+}π^{-}π^{0} 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 D_{s}^{+}→f_{0}(980)ρ(770)^{+} with a statistical significance greater than 10σ and determine the branching fractions B(D_{s}^{+}→π^{+}π^{+}π^{-}π^{0}|_{non-η})=(2.04±0.08_{stat}±0.05_{syst})% and B(D_{s}^{+}→ηπ^{+})=(1.56±0.09_{stat}±0.04_{syst})%. Moreover, we measure the relative branching fraction between ϕ→π^{+}π^{-}π^{0} and ϕ→K^{+}K^{-} to be [B(ϕ(1020)→π^{+}π^{-}π^{0})/B(ϕ(1020)→K^{+}K^{-})]=0.230±0.014_{stat}±0.010_{syst}., which deviates from the world average value by more than 4σ.