  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 $\tilde{\Gamma}_2=\Gamma(2\,{}^3P_1 \rightarrow \psi\gamma)=71.8$~keV and
large $\tilde{\Gamma}_1=\Gamma(2\,{}^3P_1\rightarrow J/\psi\gamma)=85.4$~keV
are obtained, giving for their ratio the value
$\tilde{R_{\psi\gamma}}=\frac{\tilde{\Gamma}_2}{\tilde{\Gamma}_1}=0.84$. 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
$\Gamma_1= \Gamma(X(3872)\rightarrow J/\psi\gamma) \sim 3$~keV is very small,
giving the large ratio $R_{\psi\gamma}=\frac{\mathcal{B}(X(3872)\rightarrow
\psi(2S)\gamma)}{\mathcal{B}(X(3872)\rightarrow J/\psi\gamma)}\gg 1.0$.
Arguments are presented why the gluon-exchange interaction may be suppressed
for $X(3872)$ and in this case $\Gamma_1=42.7$~keV, $\Gamma_2= 70.5$~keV, and
$R_{\psi\gamma}=1.65$ 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
$\Gamma_1=30.8$~keV, $\Gamma_2=73.2$~keV, and $R_{\psi\gamma}=2.38$, which
agrees with the LHCb data.
