Title:First Observation of \boldmath{$D^+ \to a_0(980)ρ$ and $D^+ \to a_0(980)^+ f_0(500)$} in \boldmath{$D^+ \to π^+π^+π^-η$ and $D^+ \to π^+π^0π^0η$} Decays

Abstract:We perform the first amplitude analysis of the singly Cabibbo-suppressed decays $D^+ \to \pi^+ \pi^{+(0)} \pi^{-(0)} \eta$, using $e^+e^-$ collision data collected with the BESIII detector at the center-of-mass energy of 3.773\,GeV, corresponding to an integrated luminosity of 20.3 $\rm{fb}^{-1}$. The absolute branching fractions of the $D^+ \to \pi^+ \pi^+ \pi^- \eta$ and $D^+ \to \pi^+ \pi^0 \pi^0 \eta$ decays are measured to be $(3.20\pm0.06_{\text{stat.}}\pm0.03_{\text{syst.}})\times 10^{-3}$ and $(2.43 \pm 0.11_{\text{stat.}} \pm 0.04_{\text{syst.}}) \times 10^{-3}$, respectively. % , both achieving three times better precision than the current PDG values. The decay process $D^{+}\to a_0(980)^{+}f_0(500)$ is observed for the first time with an unexpectedly large branching fraction. Moreover, we observe the decays $D^+ \to a_0(980)^{+(0)} \rho(770)^{0(+)}$ and measure the ratio $r_{+/0} \equiv \frac{\mathcal{B}(D^+ \to a_0(980)^+ \rho(770)^0)}{\mathcal{B}(D^+ \to a_0(980)^0 \rho(770)^+)}$ for the first time to be $0.55\pm0.08_{\text{stat.}}\pm0.05_{\text{syst.}}$. These results offer a novel insight into our comprehension of the nature of the $a_0(980)$ and $f_0(500)$ states.