Study of $D_{s}^{+} \rightarrow f_{0}(980)ρ^+$ and $ϕπ^+$ decays through $D_{s}^{+}\rightarrow π^{+}π^{+}π^{-}π^{0}$
We perform the first amplitude analysis of $D^+_s \to π^+π^+π^-π^0$ 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 $D_{s}^{+} \to f_0(980)ρ(770)^{+}$ with a statistical significance greater than 10$σ$ and determine the branching fractions $\mathcal{B}(D_s^+\toπ^+π^+π^-π^0|_{{\rm non}-η})=(2.04\pm0.08_{\rm stat.}\pm0.05_{\rm syst.})\%$ and $\mathcal{B}(D_s^+\toηπ^+)=(1.56\pm0.09_{\rm stat.}\pm0.04_{\rm syst.})\%$. Moreover, we measure the relative branching fraction between $ϕ\toπ^+π^-π^0$ and $ϕ\to K^+K^-$ to be $\frac{\mathcal{B}(ϕ(1020) \to π^+π^-π^0)}{\mathcal{B}(ϕ(1020) \to K^+K^-)}=0.230 \pm 0.014_{\rm stat.} \pm 0.010_{\rm syst.}$, which deviates from the world average value by more than $4σ$.