A combined analysis of $Y(2240)$ in the $e^+ e^- \to K^+ K^-$ and $e^+ e^- \to K_S^0 K_L^0$ processes

In the present work, we perform a combined analysis of the cross sections for $e^+ e^- \to K^+ K^-$ and $e^+ e^- \to K_S^0 K_L^0$ by including the interference between direct coupling process and resonance intermediate process. When setting the isospin factor as a free parameter, we can well reproduce the structures in the cross sections for $e^+ e^- \to K^+ K^-$ and $e^+ e^- \to K_S^0 K_L^0$, however, the isospin factor deviates with the isospin symmetry conservation predicted value of one. By fixing the isospin factor to be one, we find the cross sections for $e^+ e^-\to K^+ K^-$ can be accurately fitted, while the same model parameters fail to adequately reproduce the observed cross section for the process $e^+ e^- \to K_S^0 K_L^0$, which indicate that there should be contributions from other vector meson states, such as $\rho(2150)$ and $\omega(2220)$. More precise experimental measurement below 2.0 GeV and between $2.3\sim 2.5$ GeV may provide more information about the vector resonances in this energy range, which should be accessible for the BESIII collaboration.