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

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.

• The paper finds that multiplet recombination is strongly disfavored as scaling dimensions increase with ε for N=3 and N=4.
• The methodology systematically constructs composite operators using unconstrained π fields and analyzes their one-loop anomalous dimensions via both momentum-space Feynman diagrams and the OPE.
• The results affirm that NLSM and Wilson-Fisher fixed points are distinct, clarifying universality classes in critical phenomena and phase transitions.

Assessment of the Recombination Scenario in the $d=2+\epsilon$ Expansion of the $O(N)$ Non-Linear Sigma Model

Context and Motivation

The $d=2+\epsilon$ expansion for the $O(N)$ Non-Linear Sigma Model (NLSM) has been historically viewed as providing a conformal field theory (CFT) fixed point that coincides with the Wilson-Fisher (WF) fixed point family upon analytic continuation from near $d=4$ dimensions. Recent works, notably De Cesare and Rychkov’s prior study (Cesare et al., 27 May 2025), identified a protected operator in the $d=2+\epsilon$ NLSM—a closed form with $N-1$ indices and scaling dimension exactly $N-1$, independent of $\epsilon$. This operator is absent in the WF family at $d=4-\epsilon$, thus challenging the presumed analytic connection between the two families of fixed points.

A plausible resolution involves multiplet recombination: the protected operator may recombine with a long multiplet, thereby lifting its protected scaling dimension at a critical $\epsilon_c$, and forging a continuous (though not analytic) path between NLSM and WF fixed points. This paper ("Disturbing news about the $d=2+\epsilon$ expansion II. Assessing the recombination scenario" (2602.10194)) rigorously investigates the likelihood of this scenario for $N=3$ and $N=4$.

Operator Identification and Methodology

The first technical challenge is identifying the lowest-dimension candidates for the long multiplet suitable for recombination. These must be $O(N)$ pseudoscalar primaries in the Lorentz $N$-form representation, with a scaling dimension that can reach $N$ in $d=2+\epsilon$, thereby fulfilling the recombination criterion.

The analysis proceeds by constructing composite operators using products of $n^a(x)$ and their derivatives, subject to the spherical constraint $n^a n^a = 1$. Operators are built via Kronecker and Levi-Civita tensors, and systematic reduction eliminates redundancies both from the constraints and the equations of motion. The approach leverages expansion into unconstrained field variables ($\pi^i$), transforming operator counting into a linear algebra problem.

It is established that the lightest candidate primary operators for recombination possess $N+4$ derivatives, with two pairs of contracted indices. The basis of independent operators is determined, and total derivatives are classified to extract the subset of primaries—these contribute to the scaling spectrum under renormalization.

Mixing and Renormalization Analysis

Multiplet recombination hinges on mixing between the short (protected) and long (candidate) multiplets under renormalization. The scaling dimension shift is governed by the one-loop anomalous dimension of the relevant primary operators. The methodology utilizes both momentum-space Feynman diagrams and the operator product expansion (OPE) in position space. Operator mixing matrices are block-diagonal, and the focus is placed on extracting the anomalous dimensions of the primaries (i.e., non-descendants).

The computation is carried out for $N=3$ and $N=4$, with explicit bases constructed and primaries isolated. Renormalization constants and anomalous dimensions are obtained using the expansion in the unconstrained $\pi$ fields and analysis of divergent parts of relevant correlation functions.

Numerical Results and Interpretation

For $N=3$, the primary candidate operator has classical scaling dimension $7+2\epsilon$, and its one-loop anomalous dimension $\gamma=-\frac{2t}{3\pi}$ evaluated at the fixed point yields a full scaling dimension of $7+\frac{2}{3}\epsilon$. Analogous calculations for $N=4$ yield scaling dimension $8+\frac{5}{12}\epsilon$.

Crucially, for recombination to occur, these scaling dimensions would have to decrease with $\epsilon$ and cross $N$ within the interval $2$\epsilon\in(0,1)$). Instead, results demonstrate unequivocally that the scaling dimensions increase with $\epsilon$—multiplet recombination is strongly disfavored.

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 $2$\Delta = N-1$.

The analysis is further corroborated for $N=3$ by multi-loop data from WF theory [Henriksson et al., (Henriksson et al., 16 Jul 2025, Henriksson et al., 20 Nov 2025)], demonstrating that the scaling dimension of the operator $\widetilde{B}$ remains strictly above the protected dimension across the relevant range, supporting the absence of a continuous connection.

Figure 2: Scaling dimension of the operator $\widetilde{B}$ as a function of $d$, obtained via Padé approximation of multi-loop results, safely above that of the protected operator for all shown cases, further disfavoring the recombination scenario.

Implications and Future Directions

Theoretical Consequences

The finding that multiplet recombination does not occur for $N=3,4$ at one loop implies that the NLSM and WF fixed points constitute genuinely distinct CFTs throughout $2$d=2+\epsilon$ expansion, as conventionally applied to the $O(N)$ NLSM, is not equivalent to the WF expansion, even in the physically relevant $d=3$ limit.

Practical and Phenomenological Impact

This result clarifies ambiguities regarding critical phenomena, phase transitions, and universality classes in $d=3$ systems described by NLSMs, particularly those exhibiting $O(N)$ symmetry. The presence of a protected operator with dimension $N-1$ in NLSM CFTs is a sharp distinguishing feature. Theoretical studies and numerical bootstrap analyses should henceforth regard NLSM and WF fixed points as separate entities.

Prospective Research Directions

The analysis should be extended to larger $N$, where computational complexity escalates. It is essential to investigate whether the protected operator vanishes for $N>4$ at $d=3$, opening the door to special cases where the theories possibly coincide only at integer dimensions.

Nonperturbative approaches, such as conformal bootstrap techniques, could probe the existence and characteristics of NLSM CFTs—especially their correlation functions involving protected operators, conserved currents, and pseudoscalars. Further study of marginal operator dimensions, and the behavior near the merger point $d = d_*$ where NLSM CFT ceases to exist, will help chart the phase structure more comprehensively.

Conclusion

The rigorous operator mixing and scaling analysis performed in this paper decisively demonstrates the unlikelihood of multiplet recombination in the $d=2+\epsilon$ expansion of the $O(N)$ NLSM for $N=3$ and $N=4$. The protected operator persists and scaling dimensions rise rather than fall, confirming the distinct nature of NLSM and Wilson-Fisher fixed points across the physically relevant dimension interval. This result provides strong guidance for the identification, classification, and study of fixed points and universality classes in quantum field theory and statistical physics models exhibiting $O(N)$ symmetry. Future work will be required to generalize to higher $N$ and leverage non-perturbative tools to further elucidate the structure and existence of NLSM CFTs in both integer and non-integer dimensions.