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$ε$-Expansion in the Gross-Neveu CFT

Avinash Raju

TL;DR

This paper extends the conformal bootstrap approach of Rychkov and Tan to the fermionic $O(N)$ Gross-Neveu model in $d=2+ε$, providing a non-perturbative framework to compute anomalous dimensions near the Wilson-Fisher fixed point. By developing a fermionic 'cow-pie' contraction method and exploiting conformal multiplet structure and operator maps between free and interacting theories, the authors derive explicit free-theory OPE coefficients and feed them into matching conditions that fix the interacting dimensions. The main results are the anomalous dimensions $γ_1$ and $γ_2$ for the lowest operators, which agree with known perturbative calculations, along with a prescription to extract OPE coefficients $q_i$ from three-point functions. The work broadens the reach of CFT-based ε-expansion to fermionic models and provides a scalable, diagram-free route to operator data in Gross-Neveu CFTs, with potential applications to other fermionic fixed points.

Abstract

We use the recently developed CFT techniques of Rychkov and Tan to compute anomalous dimensions in the $O(N)$ Gross-Neveu model in $d=2+ε$ dimensions. To do this, we extend the "cowpie contraction" algorithm of arXiv:1506.06616 to theories with fermions. Our results match perfectly with Feynman diagram computations.

$ε$-Expansion in the Gross-Neveu CFT

TL;DR

This paper extends the conformal bootstrap approach of Rychkov and Tan to the fermionic Gross-Neveu model in , providing a non-perturbative framework to compute anomalous dimensions near the Wilson-Fisher fixed point. By developing a fermionic 'cow-pie' contraction method and exploiting conformal multiplet structure and operator maps between free and interacting theories, the authors derive explicit free-theory OPE coefficients and feed them into matching conditions that fix the interacting dimensions. The main results are the anomalous dimensions and for the lowest operators, which agree with known perturbative calculations, along with a prescription to extract OPE coefficients from three-point functions. The work broadens the reach of CFT-based ε-expansion to fermionic models and provides a scalable, diagram-free route to operator data in Gross-Neveu CFTs, with potential applications to other fermionic fixed points.

Abstract

We use the recently developed CFT techniques of Rychkov and Tan to compute anomalous dimensions in the Gross-Neveu model in dimensions. To do this, we extend the "cowpie contraction" algorithm of arXiv:1506.06616 to theories with fermions. Our results match perfectly with Feynman diagram computations.

Paper Structure

This paper contains 12 sections, 53 equations, 9 figures.

Table of Contents

  1. Introduction
  2. $O(N)$ Gross-Neveu model in $2+\epsilon$ dimensions
  3. Counting contractions
  4. $\mathbf{f_{2p}}$
  5. $\mathbf{f_{2p+1}}$
  6. $\mathbf{f_{2p}\rho_{2p}}$
  7. $\mathbf{f_{2p+1}\rho_{2p+1}}$
  8. Matching with the Free Theory
  9. OPE coefficients from 3-point function
  10. Computing $f_{2p}\rho_{2p}$ and $f_{2p+1}\rho_{2p+1}$ from cow-pies
  11. $\mathbf{f_{2p}\rho_{2p}}$
  12. $\mathbf{f_{2p+1}\rho_{2p+1}}$

Figures (9)

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  • ...and 4 more figures