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The role of the $ω(4S)$ and $ω(3D)$ states in mediating the $e^+e^-\to ωη$ and $ωπ^0π^0$ processes

The $e^+e^-\to ωη$ and $e^+e^-\toωπ^0π^0$ processes are ideal platforms to search for higher $ω$ states. Focusing on the observations of two enhancement structures around 2.2 GeV existing in $e^+e^-\to ωη$ and $e^+e^-\toωπ^0π^0$ at BESIII, we analyze how the $ω(4S)$ and $ω(3D)$ states play the role in the $e^+e^-\to ωη$ and $e^+e^-\toωπ^0π^0$ processes. The present study is supported by theoretical $ω$ mesonic spectroscopy. For reproducing the data of the cross sections of $e^+e^-\toωη$ and $ωπ^0π^0$, the intermediate $ω(4S)$ and $ω(3D)$ should be introduced, which indicates that the enhancement structures around 2.2 GeV existing in $e^+e^-\toωη$ and $ωπ^0π^0$ contain the $ω(4S)$ and $ω(3D)$ signals. Nonetheless, in the process $e^+e^-\toωη$, the $ω(4S)$ plays a dominant role, while the $ω(4S)$ and $ω(3D)$ have similar sizable contributions in the process of $e^+e^-\toωπ^0π^0$, which leads to a difference in the line shape of enhancement structure in the cross sections under the interference effect. Thus, we find a solution to alleviate the puzzling difference of resonance parameter of two reported enhancement structures around 2.2 GeV existing in $e^+e^-\to ωη$ and $e^+e^-\toωπ^0π^0$ at BESIII. The present study provides valuable information to construct $ω$ meson family, which can be accessible at future experiment like BESIII.