Constraints on the $O(n)$ model from a negative number of flavors
Treating the number of flavors $n$ as a variable in the $O(n)$ model leads to non-perturbative constraints on the spectrum of operators. Using the duality between $O(n)$ and $Sp(-n)$, we extend these relations to negative even values of $n$ and we make them explicit by decomposing operators with general flavor structure into irreducible representations. In perturbation theory, we exploit this structure to reveal novel degeneracies in the scaling dimensions of different operators, which persist for arbitrary $n$. This allows us to derive the two-loop anomalous dimension of any $ϕ^k$-type operator from existing results without additional loop calculations. The same mechanism dictates patterns in renormalization group mixing matrices, yielding new non-renormalization results in a specific operator basis. We comment on extending this framework to relations between operator product expansion coefficients and to analogous constraints arising from evanescence under continuation in the number of spacetime dimensions.