Two nearby states in the $X(3872)$ region: Resolving the radiative-decay ratio tension with $\eta_{c2}$

Abstract

Recently, LHCb reported the radiative-decay ratio ${\cal R}^{\psi\gamma}\equiv {\cal B}[X(3872)\to \psi'\gamma]/{\cal B}[X(3872)\to J/\psi\gamma]=1.67\pm 0.25$ extracted from $B^+\to K^+(J/\psi\gamma, \psi'\gamma)$. This result differs markedly ($\sim4.6\sigma$) from the BESIII value obtained from $e^+e^-\to \gamma(J/\psi\gamma, \psi'\gamma)$, ${\cal R}^{\psi\gamma}=-0.04\pm 0.28$. Such a significant tension suggests that more than one state in the $X(3872)$ region contributes to the processes. We therefore propose a two-state scenario: a shallow $D^{*0}\bar{D}^0$ bound state with $J^{PC}=1^{++}$ and a $2^{-+}$ charmonium candidate, $\eta_{c2}$, slightly above the $D^{*0}\bar{D}^0$ threshold. We show that this hypothesis consistently describes these ratios along with other branching fractions and lineshapes across multiple processes. By contrast, fits without the $\eta_{c2}$ component fail to reproduce the radiative-ratio data. We also predict helicity-angle distributions that motivates the future experiments to test the two-state hypothesis and search for the so-far missing $\eta_{c2}$.