Study of $ {D}_{s}^{+}\to {f}_{0}\left(980\right){\rho }^{+}$ and $ \varphi {\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\text{}\text{}{\mathrm{fb}}^{-1}$. We report the observation of $ {D}_{s}^{+}\to {f}_{0}\left(980\right)\rho (770{)}^{+}$ with a statistical significance greater than $ 10\sigma $ and determine the branching fractions $ B({D}_{s}^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}{\pi }^{0}{|}_{\mathrm{non}-\eta })=(2.04\pm 0.0{8}_{\mathrm{stat}}\pm 0.0{5}_{\mathrm{syst}})\%$ and $ B({D}_{s}^{+}\to \eta {\pi }^{+})=(1.56\pm 0.0{9}_{\mathrm{stat}}\pm \phantom{\rule{0ex}{0ex}}0.0{4}_{\mathrm{syst}})\%$. Moreover, we measure the relative branching fraction between $ \varphi \to {\pi }^{+}{\pi }^{-}{\pi }^{0}$ and $ \varphi \to {K}^{+}{K}^{-}$ to be $ \left[B\right(\varphi \left(1020\right)\to {\pi }^{+}{\pi }^{-}{\pi }^{0})/B(\varphi \left(1020\right)\to {K}^{+}{K}^{-}\left)\right]=0.230\pm 0.01{4}_{\mathrm{stat}}\pm 0.01{0}_{\mathrm{syst}}$., which deviates from the world average value by more than $ 4\sigma $.