Vector model in various dimensions

Abstract: We study behaviour of the critical $O(N)$ vector model with quartic interaction in $2 \leq d \leq 6$ dimensions to the next-to-leading order in the large-$N$ expansion. We derive and perform consistency checks that provide an evidence for the existence of a non-trivial fixed point and explore the corresponding CFT. In particular, we use conformal techniques to calculate the multi-loop diagrams up to and including 4 loops in general dimension. These results are used to calculate a new CFT data associated with the three-point function of the Hubbard- Stratonovich field. In $6-\epsilon$ dimensions our results match their counterparts obtained within a proposed alternative description of the model in terms of $N+1$ massless scalars with cubic interactions. In $d=3$ we find that the OPE coefficient vanishes up to $\mathcal{O}(1/N^{3/2})$ order.