Study of the decay ${D}_{s}^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}\eta $ and observation of the $W$-annihilation decay ${D}_{s}^{+}\to {a}_{0}\left(980{\right)}^{+}{\rho }^{0}$

The decay ${D}_{s}^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}\eta $ is observed for the first time, using ${e}^{+}{e}^{-}$ collision data corresponding to an integrated luminosity of $6.32\text{}\text{}{\mathrm{fb}}^{-1}$, 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 $B\left({D}_{s}^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}\eta \right)=\left(3.12±0.1{3}_{\mathrm{stat}}±0.0{9}_{\mathrm{syst}}\right)%$. The first amplitude analysis of this decay reveals the substructures in ${D}_{s}^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}\eta $ and determines the relative fractions and the phases among these substructures. The dominant intermediate process is ${D}_{s}^{+}\to {a}_{1}\left(1260{\right)}^{+}\eta ,{a}_{1}\left(1260{\right)}^{+}\to \rho \left(770{\right)}^{0}{\pi }^{+}$ with a branching fraction of $\left(1.73±0.1{4}_{\mathrm{stat}}±0.0{8}_{\mathrm{syst}}\right)%$. We also observe the $W$-annihilation process ${D}_{s}^{+}\to {a}_{0}\left(980{\right)}^{+}\rho \left(770{\right)}^{0}$, ${a}_{0}\left(980{\right)}^{+}\to {\pi }^{+}\eta $ with a branching fraction of $\left(0.21±0.0{8}_{\mathrm{stat}}±0.0{5}_{\mathrm{syst}}\right)%$, which is larger than the branching fractions of other measured pure $W$-annihilation decays by 1 order of magnitude.