Observation of an Altered $a_{0}(980)$ Line-shape in $D^{+} \rightarrow π^{+}ηη$ due to the Triangle Loop Rescattering Effect
Using 20.3~${\rm fb}^{-1}$ of $e^{+}e^{-}$ collision data taken with the BESIII detector at the center-of-mass energy 3.773~GeV, we report the first amplitude analysis of the hadronic decay $D^{+} \rightarrow π^{+}ηη$. The intermediate process $D^{+} \to a_{0}(980)^{+}η, a_{0}(980)^{+} \to π^{+}η$ is observed and is found to be the only component and its branching fraction is measured to be $(3.67\pm0.12_{\mathrm{stat.}}\pm 0.06_{\mathrm{syst.}})\times 10^{-3}$. Unlike the $a_{0}(980)$ line-shape observed in the decays of charmed mesons to $a_{0}(980)π$ and in the decay $D^{0} \to a_{0}(980)^{-}e^{+}ν_{e}$, where the low-mass side of the $a_0(980)$ is wider than the high-mass side, the $a_{0}(980)$ line-shape in $D^{+} \to a_{0}(980)^{+}η$ is found to be significantly altered, with the high-mass side being wider than the low-mass side. We establish that the $a_0(980)$ line-shape arises from the triangle loop rescattering of $D^+ \to \bar{K}_0^*(1430)^0K^+ \to a_0(980)^+ η$ and $D^+ \to K_0^*(1430)^+\bar{K}^0 \to a_0(980)^+ η$ with a significance of 5.8$σ$. This is the first experimental confirmation of the triangle loop rescattering effect.