EM Decay of X(3872) as the 1{¹D₂}(2⁻⁺) charmonium

The recently BaBar results raise the possibility that X(3872) has negative parity. This makes people reconsider assigning X(3872) to the $1{^1D_2}(c\bar c)$ state. In this paper we give a general form of the wave function of $2^{-+}$ mesons. By solving the instantaneous Bethe-Salpeter equation, we get the mass spectrum and corresponding wave functions. We calculate electromagnetic decay widths of the first $2^{-+}$ state which we assume to be the X(3872) particle. The results are $\Gamma(2^{-+}(3872)\rightarrow J/\psi\gamma) = 1.59^{+0.53}_{-0.42}$ keV, $\Gamma(2^{-+}(3872)\rightarrow \psi(2S)\gamma) = 2.87^{+1.46}_{-0.97}$ eV and $\Gamma(2^{-+}(3872)\rightarrow \psi(3770)\gamma) = 0.135^{+0.066}_{-0.047}$ keV. The ratio of branch fractions of the second and first channel is about 0.002, which is inconsistent with the experimental value $3.4\pm 1.4$. So X(3872) is unlikely to be a $2^{-+}$ charmonium state. In addition, we obtain a relatively large decay width for $2^{-+}(3872)\rightarrow h_c\gamma$ channel which is $392^{+62}_{-111}$ keV.