Disturbing news about the $d=2+ε$ expansion II. Assessing the recombination scenario

TL;DR

The paper investigates whether the $d=2+\epsilon$ fixed point of the $O(N)$ NLSM can be continuously connected to the Wilson-Fisher fixed point through multiplet recombination, focusing on a protected operator of dimension $\Delta = N-1$ absent in WF. The authors identify candidates for the recombination partner $\mathcal{O}'$ in the $N$-form pseudoscalar sector with $p=N+4$ derivatives and analyze their one-loop anomalous dimensions by diagonalizing a reduced mixing problem (one primary, $r_P=1$). For $N=3$ and $N=4$ they compute $\gamma$ at the WF fixed point and find $\Delta = 7 + \frac{2}{3}\epsilon$ and $\Delta = 8 + \frac{5}{12}\epsilon$, respectively, both of which increase with $\epsilon$ rather than decrease to $N$, thereby ruling out recombination at one loop. The results support the conclusion that the two CFTs remain distinct for $2<d<4$, with implications for the existence and character of possible mergers or annihilations at some $d_*>2$ and for future bootstrap or higher-loop analyses. The work narrows the landscape of possible reconciliations between the $d=2+\epsilon$ NLSM and WF theories and guides targeted tests at higher orders and larger $N$.

Abstract

In [De Cesare, Rychkov (2025)], we revisited the $d=2+ε$ expansion in the $O(N)$ Non-Linear Sigma Model (NLSM), emphasizing the existence of a protected operator which is a closed form with $N-1$ indices. The scaling dimension of this operator stays exactly equal to $N-1$, independently of $ε$. Its existence is problematic for the identification of the NLSM fixed point in $d=2+ε$ with the Wilson-Fisher fixed point family obtained by analytically continuing from near $d=4$, which does not possess such a protected operator. Multiplet recombination is one scenario discussed in [De Cesare, Rychkov (2025)], which could allow to connect the two families continuously (although not analytically). In this scenario, the protected dimension is lifted at some critical value of $ε$, thanks to the short conformal multiplet of scaling dimension $N-1$ eating a long conformal multiplet of higher scaling dimension. In this followup work, we assess this scenario for the cases $N=3$ and $N=4$. We identify the lowest candidates for the long multiplet which could be eaten, and compute their one-loop anomalous dimensions. We find that at one loop, scaling dimensions of these candidates grow with $ε$, while it should decrease down to $N$ for the recombination to occur. We conclude that multiplet recombination is unlikely.

Figures (2)

• Figure 1: Scaling dimension of the lightest primary operator ($P$) in the $N$-form Lorentz representation in $d=2+\epsilon$ as a function of $d$ for $N=3$ (left) and $N=4$ (right). For recombination to occur, these dimensions must reach $\Delta=N$ (dashed line) in the interval $2<d<3$, which is strongly disfavored by our results. The scaling dimension of the protected operator ($B$) is also shown; it remains protected at $\Delta = N-1$.
• Figure 2: Scaling dimension of the operator $\widetilde{B}$ as a function of $d$ (dashed lines), obtained via a Padé approximation of the results of Henriksson:2025hwiHenriksson:2025vyi, with the use of Padé$_{m,n}$ approximants. For this test, we show all approximants with $m+n=4$ and 5 ($m,n\ge 1$), discarding those which have poles in the range $2<d<4$. The Pade-approximated scaling dimension remains safely above that of the protected operator $B$ (solid line) for all the shown cases. We therefore conclude that the continuous connection between $d=2+\epsilon$ and $d=4-\epsilon$ expansions is unlikely.

Theorems & Definitions (2)

• Remark 4.1
• Remark 4.2