  We perform the first lattice QCD study on the radiative decay of the scalar
glueball to the vector meson $\phi$ in the quenched approximation. The
calculations are carried out on three gauge ensembles with different lattice
spacings, which enable us to do the continuum extrapolation. We first revisit
the radiative $J/\psi$ decay into the scalar glueball $G$ and obtain the
partial decay width $\Gamma(J/\psi \to \gamma G)=0.578(86)~\text{keV}$ and the
branching fraction $\text{Br}(J/\psi \to \gamma G) = 6.2(9)\times 10^{-3}$. We
then extend the similar calculation to the process $G\to \gamma \phi$ and get
the partial decay width $\Gamma(G \to \gamma \phi)= 0.074(47)~\text{keV}$,
which implies that the combined branching fraction of $J/\psi\to\gamma G\to
\gamma\gamma\phi$ is as small as $\mathcal{O}(10^{-9})$ such that this process
is hardly detected by the BESIII experiment even with the large $J/\psi$ sample
of $\mathcal{O}(10^{10})$. With the vector meson dominance model, the
two-photon decay width of the scalar glueball is estimated to be
$\Gamma(G\to\gamma\gamma)=0.53(46)~\text{eV}$, which results in a large
stickiness $S(G)\sim \mathcal{O}(10^4)$ of the scalar glueball by assuming the
stickiness of $f_2(1270)$ to be one.
