On the Large Charge Sector in the Critical $O(N)$ Model at Large $N$

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 $d$.