Large N, large j: Calculating scaling dimensions of large-charge operators in the O(N) vector model

Abstract

We employ the saddle point approximation, a standard technique for evaluating functional integrals in quantum field theory, to calculate scaling dimensions for quantum operators in the O(N) vector model. We do this in a relatively unstudied set of limits, taking both the number of degrees of freedom in the model (N) and the charge (quantum number) of the relevant operator (j) to infinity simultaneously, while leaving the dimension of the space completely general. We begin with a toy model (d = 0) and examine the correspondence between results obtained with the saddle point approximation in large N and results from standard Feynman diagram perturbation theory. We then expand the analysis to general dimension d. First, we explain how to calculate scaling dimensions for operators of fixed charge j using Feynman diagrams, and we extend that approach to find a perturbative expansion for scaling dimensions in the large-charge limit. Finally, we employ the saddle point approximation to confirm this perturbative expansion, and we explain how this approach could be extended to generate more exact results.