The role of the $\omega(4S)$ and $\omega(3D)$ states in mediating the $e^+e^-\to \omega\eta$ and $\omega\pi^0\pi^0$ processes

The $e^+e^-\to \omega\eta$ and $e^+e^-\to\omega\pi^0\pi^0$ processes are ideal platforms to search for higher $\omega$ states. Focusing on the observations of two enhancement structures around 2.2 GeV existing in $e^+e^-\to \omega\eta$ and $e^+e^-\to\omega\pi^0\pi^0$ at BESIII, we analyze how the $\omega(4S)$ and $\omega(3D)$ states play the role in the $e^+e^-\to \omega\eta$ and $e^+e^-\to\omega\pi^0\pi^0$ processes. The present study is supported by theoretical $\omega$ mesonic spectroscopy. For reproducing the data of the cross sections of $e^+e^-\to\omega\eta$ and $\omega\pi^0\pi^0$, the intermediate $\omega(4S)$ and $\omega(3D)$ should be introduced, which indicates that the enhancement structures around 2.2 GeV existing in $e^+e^-\to\omega\eta$ and $\omega\pi^0\pi^0$ contain the $\omega(4S)$ and $\omega(3D)$ signals. Nonetheless, in the process $e^+e^-\to\omega\eta$, the $\omega(4S)$ plays a dominant role, while the $\omega(4S)$ and $\omega(3D)$ have similar sizable contributions in the process of $e^+e^-\to\omega\pi^0\pi^0$, 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 $e^+e^-\to \omega\eta$ and $e^+e^-\to\omega\pi^0\pi^0$ at BESIII. The present study provides valuable information to construct $\omega$ meson family, which can be accessible at future experiment like BESIII.