Vector model in various dimensions
Mikhail Goykhman, Michael Smolkin
Abstract
We study behavior of the critical $O(N)$ vector model with quartic interaction in $2\ensuremath{\le}d\ensuremath{\le}6$ 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 nontrivial fixed point and explore the corresponding conformal field theory (CFT). In particular, we use conformal techniques to calculate the multiloop 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 $6\ensuremath{-}\ensuremath{\epsilon}$ 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 operator product expansion coefficient vanishes up to $\mathcal{O}(1/{N}^{3/2})$ order.