The role of the short-distance interaction in e⁺e⁻→ γ X(3872)

In this study, we analyze the cross-section data from the e⁺e⁻→ γ J/ψω process to explore both short-distance and long-distance interactions for the radiative transition Y(4200)→ γ X(3872). We investigate the short-distance effects through the E1 transition among the cc̄ components, and the long-distance effects via hadronic loop diagrams. Our numerical analysis reveals that short-distance interactions play a significantly larger role in the radiative transition than that of the long-distance interactions. This finding underscores the importance of the compact cc̄ components in both the initial and final states for accurately understanding the cross-section σ[e⁺e⁻→ γ J/ψω]. Furthermore, with the help of relative branch ratio R we estimate the Γ[Y(4200)→γ X(3872)] implied in the the experimental study. Finally, we also discuss the possible existence of ψ(4040) signal within the cross-section data.