Is the $J^P=2^-$ assignment for the $X(3872)$ compatible with the radiative transition data?

The very recent analysis by BaBar Collaboration indicates that the $X(3872)$ may favor the quantum number $J^{PC}=2^{-+}$ rather than the previously assumed $1^{++}$. By pretending the $η_{c2}(1D)$ charmonium to be the $X(3872)$, we study the parity-even radiative transition processes $η_{c2}(1D)\to J/ψ(ψ')+γ$ within several phenomenological potential models. We take the ${}^3D_1$ admixture in $ψ'$ into account, and consider the contributions from the magnetic dipole ($M1$), electric quadrupole ($E2$), and magnetic octupole ($M3$) amplitudes. It is found that the ratio of the branching fractions of these two channels, as well as the absolute branching fraction of $η_{c2}\to ψ'γ$, are in stark contradiction with the existing \textsc{BaBar} measurements. This may indicate that the $2^{-+}$ assignment for the $X(3872)$ is highly problematic.

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