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The $c\bar c$ interaction above threshold and the radiative decay $X(3872)\rightarrow J/ψγ$

Radiative decays of $X(3872)$ are studied in single-channel approximation (SCA) and in the coupled-channel (CC) approach, where the decay channels $D\bar D^*$ are described with the string breaking mechanism. In SCA the transition rate $\tildeΓ_2=Γ(2\,{}^3P_1 \rightarrow ψγ)=71.8$~keV and large $\tildeΓ_1=Γ(2\,{}^3P_1\rightarrow J/ψγ)=85.4$~keV are obtained, giving for their ratio the value $\tilde{R_{ψγ}}=\frac{\tildeΓ_2}{\tildeΓ_1}=0.84$. In the CC approach three factors are shown to be equally important. First, the admixture of the $1\,{}^3P_1$ component in the normalized wave function of $X(3872)$ due to the CC effects. Its weight $c_{\rm X}(E_{\rm R})=0.200\pm 0.015$ is calculated. Secondly, the use of the multipole function $g(r)$ instead of $r$ in the overlap integrals, determining the partial widths. Thirdly, 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 the same as for low-lying charmonium states, then in the CC approach $Γ_1= Γ(X(3872)\rightarrow J/ψγ) \sim 3$~keV is very small, giving the large ratio $R_{ψγ}=\frac{\mathcal{B}(X(3872)\rightarrow ψ(2S)γ)}{\mathcal{B}(X(3872)\rightarrow J/ψγ)}\gg 1.0$. Arguments are presented why the gluon-exchange interaction may be suppressed for $X(3872)$ and in this case $Γ_1=42.7$~keV, $Γ_2= 70.5$~keV, and $R_{ψγ}=1.65$ are predicted for the minimal value $c_{\rm X}({\rm min})=0.185$, while for the maximal value $c_{\rm X}=0.215$ we obtained $Γ_1=30.8$~keV, $Γ_2=73.2$~keV, and $R_{ψγ}=2.38$, which agrees with the LHCb data.