Constraints on the spectrum of field theories with non-integer $O(N)$ symmetry from quantum evanescence

Abstract

We identify constraints in the energy spectra of quantum theories that have a global O(N)O(N) symmetry, where NN is treated as a continuous parameter. We point out that a class of evanescent states fall out of the spectrum at integer values of NN in pairs, via an annihilation mechanism. This forces the energies of the states in such a pair to approach equality as NN approaches a certain integer, with both states disappearing at precisely integer NN and the point of would-be degeneracy. These constraints occur between different irreducible representations of the analytic continuation of O(N)O(N) and hold non-perturbatively. We give examples in the spectra of the critical O(N)O(N) model.