c(c)over-bar interaction above threshold and the radiative decay X(3872) -> J/psi gamma

Radiative decays of X(3872) are studied in single-channel approximation (SCA) and in the coupled-channel (CC) approach, where the decay channels DD¯∗ are described with the string-breaking mechanism. In SCA the transition rate Γ2=Γ(2P31→ψγ)=71.8keV and large Γ1=Γ(2P31→J/ψγ)=85.4keV are obtained, giving for their ratio the value Rψγ=Γ2Γ1=0.84. In the CC approach, three factors are shown to be equally important: First, the admixture of the 1P31 component in the normalized wave function of X(3872) due to the CC effects. Its weight cX(ER)=0.200±0.015 is calculated. Second, the use of the multipole function g(r) instead of r in the overlap integrals, determining the partial widths. Third, the choice of the gluon-exchange interaction for X(3872), as well as for other states above threshold. If for X(3872) the gluon-exchange potential is taken to be the same as for low-lying charmonium states, then in the CC approach Γ1=Γ(X(3872)→J/ψγ)∼3keV is very small, giving the large ratio Rψγ=B(X(3872)→ψ(2S)γ)B(X(3872)→J/ψγ) 1.0. Arguments are presented why the gluon-exchange interaction may be suppressed for X(3872) and in this case Γ1=42.7keV, Γ2=70.5keV, and Rψγ=1.65 are predicted for the minimal value cX(min)=0.185, while for the maximal value cX=0.215 we obtain Γ1=30.8keV, Γ2=73.2keV, and Rψγ=2.38, which agrees with the LHCb data.