Observation of the $W$-annihilation process $D_s^+ \to ωρ^+$ and measurement of $D_s^+ \to φρ^+$ in $D^+_s\to π^+π^+π^-π^0π^0$ decays

We present the first amplitude analysis and branching fraction measurement of the decay $D^+_s\to π^+π^+π^-π^0π^0$, using $e^+e^-$ collision data collected 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$^{-1}$, and report the first observation of the pure $W$-annihilation decay $D_s^+ \to ωρ^+$ with a branching fraction of $(0.99\pm0.08_{\rm stat}{\ ^{+0.05}_{-0.07}}_{\rm syst})\%$. %The absolute branching fraction is measured to be $(0.99\pm0.08_{\rm stat}\pm0.07_{\rm syst})\%$. In comparison to the low significance of the $\mathcal{D}$ wave in the decay $D_s^+ \to φρ^+$, the dominance of the $\mathcal{D}$ wave over the $\mathcal{S}$ and $\mathcal{P}$ waves, with a fraction of $(51.85\pm7.28_{\rm stat}{\ ^{+4.83}_{-7.90}}_{\rm syst})\%$ observed in the decay $D_s^+ \to ωρ^+$, provides crucial information for the``polarization puzzle", as well as for the understanding of charm meson decays. The branching fraction of $D^+_s\to π^+π^+π^-π^0π^0$ is measured to be ($4.41\pm0.15_{\rm stat}\pm0.13_{\rm syst}$)\%. Moreover, the branching fraction of $D_s^+ \to φρ^+$ is measured to be $(3.98\pm0.33_{\rm stat}{\ ^{+0.21}_{-0.19}}_{\rm syst})\%$, and the $R_φ= {\mathcal{B}(φ\toπ^+π^-π^0)}/{\mathcal{B}(φ\to K^+K^-)}$ is determined to be $(0.222\pm0.019_{\rm stat}{\ ^{+0.016}_{-0.016}}_{\rm syst}$), which is consistent with the previous measurement based on charm meson decays, but deviates from the results from $e^+e^-$ annihilation and $K$-$N$ scattering experiments by more than 3$σ$.