Study of the Decay $D_s^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}\eta$ and Observation of the W-annihilation Decay $D_s^{+}\to a_0(980{)}^{+}{\rho }^0$

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 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 $$\mathcal{B}$(D^+s \to \pi^+ \pi^+ \pi^- \eta) = (3.12\pm0.13{\rm stat.}\pm0.09_{\rm syst.})$%. The first amplitude analysis of this decay reveals the sub-structures in $D_s^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}\eta$ and determines the relative fractions and the phases among these sub-structures. The dominant intermediate process is $D^{+}{s}\rightarrow a_1(1260)^+ \eta, a_1(1260)^+ \rightarrow \rho(770)^0\pi^+$withabranchingfractionof$(1.73 \pm 0.14{\rm stat.} \pm 0.08_{\rm syst.})$$D^{+}_{s}\rightarrow a_0(980)^+\rho(770)^0$,$a_0(980)^+ \to \pi^+ \eta$ with a branching fraction of $(0.21\pm {0.08}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.05}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}})$%, which is larger than the branching fractions of other measured pure W-annihilation decays by one order of magnitude.