标题： 关于$d=2+ε$扩张的令人不安的消息 显示英文标题

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

摘要： $O(N)$非线性 sigma 模型 (NLSM) 在$d=2+\epsilon$中长期以来一直被认为描述了与从$\lambda(\phi^2)^2$模型在$d=4-\epsilon$中得到的威尔逊-费希尔 (WF)$O(N)$不动点相同的共形场论 (CFT)。在这项工作中，我们对该猜想提出质疑，基于最近的观察 [Jones (2024)]，NLSM CFT 拥有一个维度为$N-1$的保护算子，而该算子在 WF$O(N)$CFT 中不存在。 对于$N=3$，我们研究了通过多重态重组提升该算子的可能性——这是唯一已知可以解决这种不匹配的机制。 我们计算了可能参与重组的最轻算子的反常维度，并发现它仍然太重而无法允许这种情况。 这表明 NLSM 算子$O(3)$在$d=2+\epsilon$中的不动点与 WF 算子$O(3)$的共形场论并不连续连接，反而可能描述一个不同的普适类，例如在$3$维中对应于向列-柱状相变的刺猬抑制临界点。我们还讨论了如何将此分析推广到$N>3$。 显示英文摘要

摘要： 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 $N>3$.