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

Comments: 5 pages + appendices, 4 tables; v3: title change and clarifying text in intro added to better describe the nature of the derived constraints ('evanescent-degeneracy'), emphasized distinct mathematics from Racah-Speiser algorithm, other minor changes