Study of the Decay $D^{+}_{s}rightarrow pi^{+}pi^{+}pi^{-}eta$ and Observation of the W-annihilation Decay $D^{+}_{s}rightarrow a_0(980)^+rho^0$

The decay $D^{+}_{s}rightarrow pi^{+}pi^{+}pi^{-}eta$ 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 $mathcal{B}(D^+_s to pi^+ pi^+ pi^- eta) = (3.12pm0.13_{rm stat.}pm0.09_{rm syst.})$%. The first amplitude analysis of this decay reveals the sub-structures in $D^{+}_{s}rightarrow pi^{+}pi^{+}pi^{-}eta$ and determines the relative fractions and the phases among these sub-structures. The dominant intermediate process is $D^{+}_{s}rightarrow a_1(1260)^+ eta, a_1(1260)^+ rightarrow rho(770)^0pi^+$ with a branching fraction of $(1.73 pm 0.14_{rm stat.} pm 0.08_{rm syst.})$%. We also observe the W-annihilation process $D^{+}_{s}rightarrow a_0(980)^+rho(770)^0$, $a_0(980)^+ to pi^+ eta$ with a branching fraction of $(0.21pm0.08_{rm stat.}pm0.05_{rm syst.})$%, which is larger than the branching fractions of other measured pure W-annihilation decays by one order of magnitude.