  We study operators in the rank-$j$ totally symmetric representation of $O(N)$
in the critical $O(N)$ model in arbitrary dimension $d$, in the limit of large
$N$ and large charge $j$ with $j/N\equiv \hat{j}$ fixed. The scaling dimensions
of the operators in this limit may be obtained by a semiclassical saddle point
calculation. Using the standard Hubbard-Stratonovich description of the
critical $O(N)$ model at large $N$, we solve the relevant saddle point equation
and determine the scaling dimensions as a function of $d$ and $\hat{j}$,
finding agreement with all existing results in various limits. In $4<d<6$, we
observe that the scaling dimension of the large charge operators becomes
complex above a critical value of the ratio $j/N$, signaling an instability of
the theory in that range of $d$. Finally, we also derive results for the
correlation functions involving two "heavy" and one or two "light" operators.
In particular, we determine the form of the "heavy-heavy-light" OPE
coefficients as a function of the charges and $d$.
