First Observation of \boldmath{$D^{+}\to a_0(980)\rho$ and $D^{+}\to a_0(980{)}^{+}{f}_0(500)$} in \boldmath{$D^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}\eta$ and $D^{+}\to {\pi }^{+}{\pi }^0{\pi }^0\eta$} 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 ${\mathrm{fb}}^{-1}$. 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 $(3.20\pm 0.06_{\text{stat.}}\pm 0.03_{\text{syst.}})\times 10^{-3}$ 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 $0.55\pm 0.08_{\text{stat.}}\pm 0.05_{\text{syst.}}$. These results offer a novel insight into our comprehension of the nature of the $a_0(980)$ and $f_0(500)$ states.