EM Decay of X(3872) as the 1¹D₂(2⁻⁺) charmonium

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The recently BaBar results raise the possibility that X(3872) has negative parity. This makes people reconsider assigning X(3872) to the 1¹D₂(c 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 Γ(2⁻⁺(3872)→ J/ψγ) = 1.59⁺⁰.⁵³₋₀.₄₂ keV, Γ(2⁻⁺(3872)→ ψ(2S)γ) = 2.87⁺¹.⁴⁶₋₀.₉₇ eV and Γ(2⁻⁺(3872)→ ψ(3770)γ) = 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± 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)→ hcγ channel which is 392⁺⁶²₋₁₁₁ keV.