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

**Authors:** Fabiana De Cesare, Slava Rychkov  
**arXiv:** [2505.21611](https://arxiv.org/abs/2505.21611)  
**Date:** 2025-05-27  
**Summary page:** https://www.summarizepaper.com/en/paper/2505.21611v1/

## Key points

- Authors Fabiana De Cesare and Slava Rychkov challenge the conjecture that the $O(N)$ Non-Linear Sigma Model (NLSM) in $d=2+\epsilon$ is equivalent to the Wilson-Fisher (WF) $O(N)$ fixed point from the $\lambda(\phi^2)^2$ model in $d=4-\epsilon.
- Recent observation by Jones (2024) highlights a key difference between these two theories: NLSM CFT has a protected operator with dimension $N-1, absent in WF O(N) CFT.
- Investigation for N=3 shows discrepancy may not be resolved through multiplet recombination due to heavy anomalous dimension of potential lightest operator.
- Suggestion that NLSM O(3) fixed point may not be continuously connected to WF O(3) CFT, possibly belonging to an alternative universality class related to hedgehog-suppressed critical points associated with N\'eel-VBS phase transition in 3D systems.
- Discussion on extending analysis to cases where N>3, hinting at broader implications for understanding critical phenomena in diverse physical systems.
- Work sheds new light on quantum field theory fundamentals and opens avenues for exploring novel universality classes beyond established paradigms.

## Summary

In their paper titled "Disturbing news about the $d=2+\epsilon$ expansion," authors Fabiana De Cesare and Slava Rychkov challenge the long-standing conjecture that the $O(N)$ Non-Linear Sigma Model (NLSM) in $d=2+\epsilon$ describes the same conformal field theory (CFT) as the Wilson-Fisher (WF) $O(N)$ fixed point obtained from the $\lambda(\phi^2)^2$ model in $d=4-\epsilon$. They base their argument on a recent observation by Jones (2024), which highlights a key difference between these two theories. Specifically, they point out that the NLSM CFT features a protected operator with dimension $N-1$ that is absent in the WF $O(N)$ CFT. Focusing on the case of $N=3$, De Cesare and Rychkov investigate whether this discrepancy can be resolved through multiplet recombination, a known mechanism for addressing such mismatches. By computing the anomalous dimension of the lightest operator that could participate in recombination, they determine that it remains too heavy to facilitate this scenario. This finding leads them to suggest that the NLSM $O(3)$ fixed point in $d=2+\epsilon$ may not be continuously connected to the WF $O(3)$ CFT as previously thought. Instead, they propose that it could belong to an alternative universality class, potentially related to phenomena such as the hedgehog-suppressed critical point associated with the N\'eel-VBS phase transition in 3D systems. Furthermore, De Cesare and Rychkov discuss how their analysis can be extended to cases where $N>3$, hinting at broader implications for understanding critical phenomena in diverse physical systems. Their work sheds new light on fundamental aspects of quantum field theory and opens up avenues for exploring novel universality classes beyond established paradigms. The paper spans 33 pages and includes 6 figures, inviting further insights and discussions from interested readers within the scientific community.