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

Shuo-Ying Yu, Xian-Wei Kang

Published 2023-08-20Version 1

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 $J^{PC}=1^{++}$ ($J,\,P,\,C$ stand for spin, parity and charge conjugation, respectively) in accordance with the Particle Data Group assignment. 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 $\beta$ which 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 from the experimental values of their partial decay widths to electron-positron pair. As for $X(3872)$ we determine the value of $\beta$ by the decay width of $X(3872)\rightarrow \psi'\gamma$. Then we examine the width of $X(3872)\to J/\psi\gamma$ in a manner of parameter-free prediction and confront it with the experimental value. Both of the hypothesises that $X(3872)$ is the $1P$ state or $2P$ state are considered, although the predicted $1P$ charmonium mass is significantly lower than the mass of $X(3872)$. It is found that one will encounter the inconsistency or contradiction between the widths of $X(3872)\to J/\psi\gamma$ and $X(3872)\to \psi'\gamma$ if interpreting $X(3872)$ as a pure charmonium. We thus conclude that $X(3872)$ can not be a pure $c\overline c$ resonance and other components are necessary in its wave function.