Nature of $X(3872)$ from its radiative decay

We study the radiative decay of $X(3872)$ based on the assumption that $X(3872)$ is regarded as a $c\overline{c}$ charmonium with quantum number $J^{PC}=1^{++}$ ($J,\,P,\,C$ represent the spin, parity and charge conjugation, respectively). The form factors of $X(3872)$ transitions to $J/\psi\gamma$ and $\psi'\gamma$ ($\psi'$ denotes $\psi(2S)$ throughout the paper) are calculated in the framework of the covariant light-front quark model. The phenomenological wave function of a meson depends on the parameter $\beta$, whose inverse essentially describes the confinement scale. In the present work, the parameters $\beta$ for the vector $J/\psi$ and $\psi'$ mesons will be determined through their decay constants, which are obtained from the experimental values of their partial decay widths to the electron-positron pair. For $X(3872)$, we determined the value of $\beta$ by the decay width of $X(3872)\rightarrow \psi'\gamma$. Then, we examined the width of $X(3872)\to J/\psi\gamma$ in a manner of parameter-free prediction and compared it with the experimental value. As a result, an inconsistency or contradiction occurs between the widths of $X(3872)\to J/\psi\gamma$ and $X(3872)\to \psi'\gamma$. We thus conclude that $X(3872)$ cannot be a pure $c\overline c$ resonance and that other components are necessary in its wave function.