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

We identify constraints in the energy spectra of quantum theories that have a global O(N) O(N) symmetry, where N N is treated as a continuous parameter. We point out that a class of evanescent states fall out of the spectrum at integer values of N N in pairs, via an annihilation mechanism. This forces the energies of the states in such a pair to approach equality as N N approaches a certain integer, with both states disappearing at precisely integer N N 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.