  Using the QCD sum rules approach we study the mass and decay width of the
channel $J/\psi + \omega$ for the $Y(3940)$ state. We assume that it can be
described by a mixed charmonium-molecule scalar state, $(\chi_{c0})-(D^\ast
D^\ast)$ current, with $J^{PC} = 0^{++}$ quantum numbers. For the mixing angle
$(76.0 \pm 5.0)^0$, we obtain the value $M = (3.95 \pm 0.11)$ GeV for the mass,
which is in good agreement with the experimental mass of the $Y(3940)$ state.
For the decay width into the channel $Y \rightarrow J/\psi + \omega$, we find
the value $\Gamma_Y = (1.7 \pm 0.6)$ MeV, which is also compatible with the
experimental data. We thus conclude that the present description of the
$Y(3940)$ as a mixed charmonium-molecule state is a possible scenario to
explain the structure of this state.
