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

This follow-up paper assesses the multiplet recombination scenario proposed to reconcile the $d=2+\epsilon$ and $d=4$ fixed points of the $O(N)$ Non-Linear Sigma Model for $N=3$ and $N=4$, concluding that one-loop calculations showing increasing scaling dimensions for candidate operators make this recombination mechanism unlikely.

The Gist

The Big Picture: Two Families That Look Alike But Aren't

Imagine the universe of physics as a vast landscape of "theories" (rules that describe how particles behave). For a long time, physicists believed there were two specific families of theories that were actually the same family, just viewed from different angles.

1. The "Wilson-Fisher" Family: These theories are well-understood. They describe how things behave in 4 dimensions (like our familiar space plus time, but slightly tweaked).
2. The "NLSM" Family: These theories describe how things behave on a curved surface (like a sphere) in 2 dimensions.

For decades, scientists thought: "If we stretch the 2D theory into 3D, it should smoothly turn into the 4D theory. They are just the same family."

The Problem:
In a previous paper, the authors found a "fingerprint" on the 2D family that the 4D family didn't have.

• The Fingerprint: A special, protected object (let's call it a "Magic Shield") that exists in the 2D world. Its properties are locked in place; they cannot change no matter how much you stretch or squeeze the dimensions.
• The Conflict: The 4D family has no such shield. If the two families were truly connected, the shield should either appear or disappear smoothly as you move between them. But because the shield is "protected," it can't just vanish.

The Proposed Solution: The "Recombination" Scenario

To save the idea that these two families are connected, the authors proposed a clever escape hatch called "Multiplet Recombination."

The Analogy: The Lego Tower
Imagine the "Magic Shield" is a small, sturdy Lego tower (a Short Multiplet). In physics, these towers are special because they are "protected"—they can't grow taller or shorter.

However, if this tower meets a larger, wobbly tower of the same shape but made of different blocks (a Long Multiplet), they might snap together.

• The Merge: The small tower eats the big tower.
• The Result: They form one giant, unstable tower (a Long Multiplet).
• The Consequence: Once merged, the "protection" is gone! The new giant tower is free to change its size. As you move from 2D to 3D, this giant tower could shrink down until it matches the size of the 4D family, effectively "hiding" the fact that the shield existed.

If this happens, the two families are connected, just not in a simple, smooth way.

The Investigation: Did the Towers Merge?

The authors of this paper decided to test this "Recombination" theory. They asked: "Is there a wobbly tower (Long Multiplet) nearby that is the right size to get eaten by the Magic Shield?"

To do this, they looked at two specific cases:

• Case N=3: Think of this as a 3D sphere.
• Case N=4: Think of this as a 4D sphere.

They had to:

1. Find the Candidates: Search through a massive library of possible "wobbly towers" (mathematical operators) to find the lightest ones that could possibly merge with the shield.
2. Check the Weight: Calculate how heavy these towers are (their "scaling dimension").
3. Watch the Growth: See what happens to their weight as you move from 2D toward 3D.

The Rule for Success:
For the recombination to work, the "wobbly tower" must get lighter as you move toward 3D, eventually becoming light enough to merge with the shield.

The Results: The Towers Got Heavier

The authors did the heavy mathematical lifting (using complex equations and computer assistance) and found a disappointing result for the "connection" theory:

• For N=3: The candidate tower didn't get lighter. It got heavier.
• For N=4: The candidate tower also got heavier.

The Metaphor:
Imagine you are trying to merge a small boat (the shield) with a large ship (the candidate) to cross a river. For them to merge, the ship needs to shrink down to the size of the boat.
Instead, as the authors watched, the ship started inflating like a balloon. It grew bigger and bigger. It became impossible for the small boat to eat it.

The Conclusion: They Are Different Families

Because the "wobbly towers" grew heavier instead of lighter, the Recombination Scenario failed.

• The "Magic Shield" in the 2D family cannot be lifted or hidden.
• Therefore, the 2D family (NLSM) and the 4D family (Wilson-Fisher) cannot be connected, even continuously.

The Final Verdict:
The authors conclude that these are two distinct, separate families of theories. They look similar in some ways, but they are fundamentally different. The "Magic Shield" proves they live in different universes.

Why Does This Matter?

This is a big deal for physics because:

1. It closes a door: It tells us we can't just assume these theories are the same. We have to treat them as separate entities.
2. It opens a window: It suggests there might be entirely new, undiscovered theories in 3D (our physical world) that are different from the ones we already know.
3. It solves a mystery: It explains why the math didn't add up when people tried to connect them. The "fingerprint" (the shield) was real, and it meant the families were never the same to begin with.

In short: The authors checked if two distant cousins were actually the same person by looking for a shared birthmark. They found the birthmark was permanent and unique to one side, proving they are definitely different people.

Technical Summary

1. Problem Statement and Motivation

The paper addresses a fundamental conflict in the study of the $O(N)$ Non-Linear Sigma Model (NLSM) in $d = 2 + \epsilon$ dimensions.

• The Conflict: It was previously believed that the $O(N)$ NLSM fixed point in $d = 2 + \epsilon$ is analytically connected to the Wilson-Fisher (WF) fixed point family obtained by continuing from $d = 4 - \epsilon$. However, the authors' previous work [1] identified a protected operator in the $d = 2 + \epsilon$ NLSM: a closed differential form with $N-1$ indices. Its scaling dimension is exactly $\Delta = N-1$, independent of $\epsilon$.
• The Obstacle: The Wilson-Fisher family in $d = 4 - \epsilon$ lacks such a protected operator. Therefore, the two families cannot be connected via analytic continuation.
• The Hypothesis: A potential loophole is multiplet recombination. In Conformal Field Theory (CFT), a protected short multiplet $S$ (with dimension $\Delta_S = N-1$) can become a long multiplet $L$ if it "eats" a long multiplet $L'$ of higher dimension. For this to happen, the dimension of the candidate operator in $L'$ must decrease as $\epsilon$ increases, eventually reaching $\Delta = N$ (the dimension required for the shortening condition) at some critical $\epsilon_c$.
• The Goal: This paper quantitatively tests the recombination scenario for $N=3$ and $N=4$. Specifically, it asks: Do the scaling dimensions of the lightest candidate operators (which could be eaten) decrease with $\epsilon$ to allow recombination, or do they increase?

2. Methodology

The authors employ a perturbative approach within the $d = 2 + \epsilon$ expansion, focusing on one-loop anomalous dimensions.

A. Identification of Candidate Operators

To recombine, a candidate operator $O'$ must satisfy specific quantum numbers:

• Lorentz: An $N$-index antisymmetric form.
• Global Symmetry: An $O(N)$ pseudoscalar.
• Scaling Dimension: Must be $\Delta = N$ at the point of recombination.
• Derivative Count: The lightest operators with these quantum numbers have $N+2$ derivatives but are descendants. The authors focus on the second-lightest class, which have $N+4$ derivatives ($N$ antisymmetrized, 4 contracted).

Construction Strategy:

1. Basis Construction: They construct $O(N)$ pseudoscalar operators using building blocks involving the Levi-Civita tensor ($E$-block) and Kronecker deltas ($D$-blocks).
2. Constraint Handling: They impose the sphere constraint ($n^a n^a = 1$) and eliminate operators proportional to equations of motion.
3. Linear Independence: Due to the large number of potential operators (40+), they resolve the constraint by expressing fields in terms of unconstrained components $\pi^i$ ($i=1 \dots N-1$). They treat the problem as a linear algebra task, factoring out denominators to determine the rank of the coefficient matrix and isolate a basis of independent operators.
4. Primary vs. Descendant: They distinguish between total derivatives (descendants) and primary operators. They construct a basis where the first $r_D$ elements are total derivatives and the remaining $r_P$ are primaries.

B. Solving the Mixing Problem

The core calculation involves determining the anomalous dimensions of the primary operators.

1. Renormalization: They analyze the mixing matrix $\delta Z$ relating bare and renormalized operators. Crucially, total derivatives only mix among themselves. The mixing matrix is block-diagonal, allowing the isolation of the anomalous dimension for the primary operator $O$.
2. One-Loop Calculation:
   • They expand the action and operators in terms of the unconstrained fields $\pi$.
   • They compute the divergent part of correlation functions $\langle \pi \dots \pi O \rangle$ in dimensional regularization ($d = 2 + \epsilon$).
   • The divergence arises from two sources:
     • 1PR (One-Particle Reducible): Wave-function renormalization of the $\pi$ fields.
     • 1PI (One-Particle Irreducible): Diagrams involving the interaction vertex $V = (\pi \partial \pi)^2$.
3. OPE Technique: Instead of standard momentum-space Feynman diagrams, they utilize the Position-Space Operator Product Expansion (OPE) method. This involves contracting fields in the interaction vertex with the external operator, Taylor expanding the result, and integrating over the interaction point to extract the $1/\epsilon$ pole.
4. Fixed Point Evaluation: The anomalous dimension $\gamma$ is calculated at the NLSM fixed point coupling $t^* \approx \frac{2\pi}{N-2}\epsilon$. The full scaling dimension is $\Delta = \Delta_{classical} + \gamma$.

3. Key Results

Case $N=3$

• Operator Count: There are 8 independent operators with the required quantum numbers. 7 are total derivatives; 1 is a primary.
• Scaling Dimension:
  • Classical dimension: $\Delta_0 = 7 + 2\epsilon$ (7 derivatives + 4 $\pi$ fields).
  • Anomalous dimension at one-loop: $\gamma = -\frac{2t}{3\pi}$.
  • At the fixed point ($t^* \propto \epsilon$): $\Delta = 7 + \frac{2}{3}\epsilon$.
• Conclusion: The dimension increases with $\epsilon$ (slope $+2/3$). For recombination to occur, the dimension would need to decrease from 7 down to $N=3$. The result strongly disfavors recombination.

Case $N=4$

• Operator Count: There are 10 independent operators. 9 are total derivatives; 1 is a primary.
• Scaling Dimension:
  • Classical dimension: $\Delta_0 = 8 + \frac{5}{2}\epsilon$.
  • Anomalous dimension at one-loop: $\gamma = -\frac{25t}{12\pi}$.
  • At the fixed point: $\Delta = 8 + \frac{5}{12}\epsilon$.
• Conclusion: The dimension increases with $\epsilon$ (slope $+5/12$). For recombination, it would need to drop to $N=4$. Again, the result strongly disfavors recombination.

Cross-Check from $d=4-\epsilon$

For $N=3$, the authors also analyzed the Wilson-Fisher side using five-loop results from literature. They tracked the dimension of the candidate operator $\tilde{B}$ (the $d=4$ analogue of the protected operator). The dimension remains above 2 for $d > 2.1$ and only approaches 2 very close to $d=2$. This supports the conclusion that the two families do not merge continuously in the interval $2 < d < 3$.

4. Significance and Conclusions

• Ruling Out Recombination: The paper provides strong quantitative evidence that the multiplet recombination scenario is unlikely for $N=3$ and $N=4$. Since the candidate operators' dimensions grow with $\epsilon$ rather than shrinking, the protected operator in the NLSM cannot be "lifted" to merge with the Wilson-Fisher family.
• Distinct CFTs: Consequently, the $O(N)$ NLSM in $d=2+\epsilon$ and the Wilson-Fisher fixed point are likely distinct Conformal Field Theories throughout the interval $2 < d < 4$ for these values of $N$. They do not coincide even continuously.
• Implications for $d=3$: Since recombination is ruled out, the NLSM fixed point in $d=3$ (physical dimension) is likely a distinct theory from the standard Wilson-Fisher fixed point. This suggests the existence of a new universality class for $O(N)$ symmetric systems in 3D, potentially detectable via the presence of the protected operator (which becomes a conserved current for $N=3$ or a pseudoscalar for $N=4$).
• Future Directions: The authors suggest that for $N > 4$, the protected operator vanishes at $d=3$, leaving open the possibility of a merger at that specific dimension. They propose using the conformal bootstrap to search for these distinct 3D CFTs, specifically looking for correlation functions involving the protected operator as a unique signature.

In summary, this work delivers "disturbing news" for the unification of the $d=2+\epsilon$ and $d=4-\epsilon$ expansions, effectively closing the door on the recombination hypothesis for low $N$ and reinforcing the existence of distinct $O(N)$ CFTs in physical dimensions.