Disturbing news about the $d=2+ε$ expansion

Résumé : The $O(N)$ Non-Linear Sigma Model (NLSM) in $d=2+\epsilon$ has long been conjectured to describe the same conformal field theory (CFT) as the Wilson-Fisher (WF) $O(N)$ fixed point obtained from the $\lambda(\phi^2)^2$ model in $d=4-\epsilon$. In this work, we put this conjecture into question, building on the recent observation [Jones (2024)] that the NLSM CFT possesses a protected operator with dimension $N-1$, which is instead absent in the WF $O(N)$ CFT. For $N=3$, we investigate the possibility of lifting this operator via multiplet recombination - the only known mechanism that could resolve this mismatch. We compute the anomalous dimension of the lightest operator that could participate in recombination, and find that it remains too heavy to allow for this scenario. This suggests that the NLSM $O(3)$ fixed point in $d=2+\epsilon$ is not continuously connected to the WF $O(3)$ CFT, and may instead describe an alternative universality class, such as the hedgehog-suppressed critical point, corresponding to the N\'eel-VBS phase transition in $3$D. We also discuss how to generalize this analysis to $N>3$.