⚛️ high-energy theory

Large $N$ conformal bootstrap and chirality

This paper utilizes the large $N$ critical point formalism to demonstrate that the anomalous dimension of the matter field in the bosonic sector of the $O(N)$ Wess-Zumino model is free of multiple zeta values up to order $O(1/N^3)$ .

Original authors: J. A. Gracey

Published 2026-07-14

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CC BY 4.0

Original authors: J. A. Gracey

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic video game where particles are the characters and the rules they follow are written in a complex code called "Quantum Field Theory." For decades, physicists have been trying to read this code at the highest levels of detail, looking for a specific pattern in the math: a set of numbers called "multiple zeta values." Think of these values as a secret, chaotic graffiti tag that usually appears when you zoom in really close on the game's physics, specifically when you calculate how particles interact over and over again (what scientists call "loops").

For a long time, it was a mystery why this graffiti tag showed up in some parts of the game but vanished in others. One theory suggested that Supersymmetry—a fancy rule where every particle has a "super-partner"—was the magic eraser that wiped the graffiti clean. Another theory thought it might be Chirality, a property where the interactions have a strict direction, like a one-way street where everything flows either all the way in or all the way out.

Enter J.A. Gracey, a physicist who decided to settle this debate using a powerful magnifying glass called the "large $N$ conformal bootstrap." Instead of trying to solve the whole game at once, this method looks at the system when there are a huge number of particle types (represented by the letter $N$ ), allowing for a clearer view of the underlying structure.

Gracey focused on a specific character in the game: the "bosonic sector" of the $O(N)$ Wess-Zumino model. You can think of this model as a special level in the game that has both Supersymmetry and Chirality. In previous studies, scientists found that the graffiti tag (multiple zetas) was missing in this level, but they couldn't tell if it was because of the super-partners or the one-way streets.

To find the answer, Gracey stripped away the super-partners and looked only at the scalar (bosonic) fields—the "matter" part of the Lagrangian. It's like removing the special power-ups to see if the level is still glitch-free. The result was a big reveal: even without the supersymmetry, the graffiti tag was still gone.

By calculating the "anomalous dimension" (a measure of how the particle's behavior changes) up to the $O(1/N^3)$ order, Gracey showed that the messy, complex numbers involving multiple zetas simply do not appear. Instead, the math is clean, involving only simpler functions called polygammas.

This finding explicitly rules out Supersymmetry as the sole reason for the missing graffiti. The paper argues that the true culprit is Chirality. Because the interactions in this theory are strictly directed (like a river flowing only downstream), the specific types of diagrams that usually generate the chaotic graffiti tags are forbidden from forming in the first place. It's as if the one-way street rules of the game prevent the messy loops from ever being drawn.

The paper is quite confident in this conclusion. The authors didn't just guess; they used a rigorous mathematical framework to compute the results and then cross-checked them by running a separate, four-loop calculation using a different method (renormalizing a six-dimensional Lagrangian down to four dimensions). The two methods matched perfectly. They also noted that in a related theory called the chiral Gross-Neveu model, the same "clean" result appeared, further supporting the idea that the directionality of the interactions is the key.

However, the authors are careful to note that this is a specific finding for the $O(1/N^3)$ order. They mention that while the current math is solid, future work at even higher orders (like $O(1/N^4)$ ) might need to address new complications, particularly regarding how certain mathematical tools handle a tricky concept called $\gamma_5$ in higher dimensions. But for now, the mystery of the missing graffiti in this specific sector has been solved: it's not the superpowers, it's the one-way streets.

Technical Summary: Large $N$ Conformal Bootstrap and Chirality

Problem Statement
The paper addresses the structural properties of perturbative renormalization group (RG) functions in quantum field theories at very high loop orders. Specifically, it investigates the appearance of multiple zeta values (MZVs) in these expansions. While MZVs are known to appear in scalar field theories (such as $\phi^4$ ) at high orders, previous work on the $O(N)$ Wess-Zumino model (a supersymmetric theory) revealed an absence of MZVs up to $O(1/N^3)$ in the matter field anomalous dimension ( $\eta$ ). The central question is whether this absence is driven by supersymmetry or chiral symmetry. To isolate the cause, the author examines the bosonic sector of the Wess-Zumino model independently, as the fermionic sector had already been studied in related chiral theories.

Methodology
The study employs the large $N$ critical point formalism (conformal bootstrap) to compute critical exponents to high orders in the $1/N$ expansion.

Theoretical Framework: The author utilizes the universal Lagrangian for the scalar sector of the $O(N)$ Wess-Zumino model, which involves two doublets of fields $\{\phi_i, \bar{\phi}_i\}$ and $\{\sigma, \bar{\sigma}\}$ . This sector is characterized by chiral vertices where edges are directed entirely toward or away from interaction points.
Calculation of $\eta_3$ : The anomalous dimension of the matter field, $\eta$ $η$ , is expanded in powers of $1/N$ $1/ N$ . The paper focuses on the third-order term, $\eta_3$ $η_{3}$ . Using the conformal bootstrap method, the calculation involves:
- Analytic regularization where the vertex anomalous dimension $\chi$ is shifted ( $\chi \to \chi + \Delta$ ).
- Replacing vertices with conformal triangles to eliminate vertex subgraphs.
- Evaluating specific Feynman graphs (two graphs contribute to $\eta_3$ in this sector) to determine the coefficient.
Verification: To ensure the credibility of the large $N$ results, the author performed a standard perturbative renormalization of the corresponding six-dimensional Lagrangian to four loops using the Forcer package. This allowed for the calculation of $\beta$ -functions and field anomalous dimensions, which were then matched against the large $N$ expansion at the Wilson-Fisher fixed point ( $\mu = 3 - \epsilon$ ).

Key Contributions and Results

Computation of the Bosonic Sector: The paper successfully determines the $O(1/N^3)$ correction to the anomalous dimension ( $\eta_3$ ) for the scalar fields in the $O(N)$ Wess-Zumino bosonic sector.
Absence of Multiple Zetas: The explicit calculation reveals that $\eta_3$ in this scalar chiral theory does not contain the integral function $I(\mu)$ (associated with two-loop self-energy graphs that generate MZVs) nor the function $\hat{\Omega}(\mu)$ . Consequently, the result is free of multiple zeta values to this order.
Comparison with Fermionic Sector: The author contrasts this with the chiral Gross-Neveu model (fermionic sector), where $\eta_3$ also lacks MZVs despite containing $I(\mu)$ in intermediate steps, due to cancellations involving Lie group Casimirs.
Verification of Chirality: By demonstrating that the scalar sector (which lacks supersymmetry but retains chiral symmetry) shares the MZV-free property with the full supersymmetric model and the fermionic sector, the paper concludes that chirality is the driving factor for the absence of multiple zetas, rather than supersymmetry.
Consistency Check: The perturbative four-loop results for the six-dimensional theory were found to be in full agreement with the large $N$ expansion when mapped to the critical dimension, validating the bootstrap results.

Significance and Claims
The paper claims that the structural absence of multiple zeta values in the renormalization group functions of these theories is fundamentally due to chiral symmetry.

Mechanism: Chirality restricts the contributing Feynman graphs to those with even-edge subgraphs, effectively excluding the topologies that generate the specific integrals (like $I(\mu)$ ) responsible for MZVs.
Supersymmetry vs. Chirality: The results clarify that while supersymmetry is often linked to maximal transcendentality, the specific absence of MZVs in the Wess-Zumino model at $O(1/N^3)$ is a consequence of the chiral nature of the vertices, which persists even when supersymmetry is broken (i.e., in the scalar sector alone).
Future Implications: The author notes that this chiral property allows for the circumvention of $\gamma_5$ treatment issues in $d$ -dimensions for Weyl fermions up to the orders studied. However, the paper modestly notes that recent four-loop computations suggest care is needed with $\gamma_5$ at six loops, implying that the large $N$ approach may face similar challenges at $O(1/N^4)$ . The work suggests that using the universal theory at the Wilson-Fisher fixed point with Weyl fermions could offer an alternative route for handling $\gamma_5$ or supersymmetry in dimensional regularization.

Technical Summary: Large NNN Conformal Bootstrap and Chirality

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Technical Summary: Large $N$ Conformal Bootstrap and Chirality