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

Abstract: 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. We investigate the possibility of lifting this operator via multiplet recombination - the only known mechanism that could resolve this mismatch while preserving a connection between the two theories. We also explore an alternative scenario in which the NLSM $O(N)$ fixed point in $d=2+\epsilon$ is not continuously connected to the WF $O(N)$ CFT, and instead corresponds to a different universality class. For $N=3$, this could be related to the hedgehog-suppressed critical point, which describes the N\'eel-VBS phase transition in 3D.