Critical O(N) model to order $\epsilon^4$ from analytic bootstrap

We compute, using the method of large spin perturbation theory, the anomalous dimensions and OPE coefficients of all leading twist operators in the critical $O(N)$ model, to fourth order in the $\epsilon$-expansion. This is done fully within a bootstrap framework, and generalizes a recent result for the CFT-data of the Wilson-Fisher model. The anomalous dimensions we obtain for the $O(N)$ singlet operators agree with the literature values, obtained by diagrammatic techniques, while the anomalous dimensions for operators in other representations, as well as all OPE coefficients, are new. From the results for the OPE coefficients, we derive the $\epsilon^4$ corrections to the central charges $C_T$ and $C_J$, which are found to be compatible with the known large $N$ expansions. Predictions for the central charge in the strongly coupled 3d model, including the 3d Ising model, are made for various values of $N$, which compare favourably with numerical results and previous predictions.