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.

• Received 3 July 2022
• Accepted 2 August 2022

DOI:https://doi.org/10.1103/PhysRevD.106.034010

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Published by the American Physical Society