Study of $D_s^{+}\to f_0(980){\rho }^{+}$ and $\phi {\pi }^{+}$ decays through $D_s^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}{\pi }^0$

We perform the first amplitude analysis of $D_s^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}{\pi }^0$ 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 $D_s^{+}\to f_0(980)\rho (770{)}^{+}$ with a statistical significance greater than 10$\sigma$ and determine the branching fractions $\mathcal{B}(D_s^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}{\pi }^0{\mathrm{\mid }}_{\mathrm{n}\mathrm{o}\mathrm{n}-\eta })=(2.04\pm {0.08}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.05}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}})\mathrm{\%}$ and $\mathcal{B}(D_s^{+}\to \eta {\pi }^{+})=(1.56\pm {0.09}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.04}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}})\mathrm{\%}$. Moreover, we measure the relative branching fraction between $\phi \to {\pi }^{+}{\pi }^{-}{\pi }^0$ and $\phi \to K^{+}{K}^{-}$ to be $\frac{\mathcal{B}(\phi (1020)\to {\pi }^{+}{\pi }^{-}{\pi }^0)}{\mathcal{B}(\phi (1020)\to K^{+}{K}^{-})}=0.230\pm {0.014}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.010}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}}$, which deviates from the world average value by more than $4\sigma$.