Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semiclassical Canovaccio for Composite Operators

Oleg Antipin, Jahmall Bersini, Jacob Hafjall, Giulia Muco, Francesco Sannino

TL;DR

The paper develops a global semiclassical framework that maps scaling dimensions of heavy neutral composite operators in CFTs to the energy spectrum of time-periodic homogeneous solutions on the cylinder, via the state-operator correspondence. It systematizes the calculation through a double-scaling limit, with a classical term $C_0$ and a quantum correction $C_1$ obtained from fluctuations encoded by Lamé-type operators and stability angles, using a Gutzwiller-like trace formula and Bohr–Sommerfeld quantization. The method is concretely tested in the $O(N)$ $\phi^4$ theory near $d=4-\epsilon$ and the $O(N)$ $\phi^6$ theory near $d=3-\epsilon$, yielding the full spectra of neutral operators in traceless symmetric Lorentz representations and reproducing known perturbative results while predicting higher-order terms. It also analyzes large-$n$ behavior and instabilities at strong coupling, offering a path toward extending to gauge, Yukawa, and higher-spin sectors with potential cross-checks against bootstrap and Monte Carlo techniques. Overall, the work provides a pedagogical, scalable route to a wide class of CFT data beyond traditional perturbative or bootstrap approaches, including detailed operator towers and their degeneracy lifting at NLO.

Abstract

We present a novel semiclassical framework tailored to determine the scaling dimensions of heavy neutral composite operators in conformal field theories (CFTs) which are inaccessible with other current methodologies. It utilizes the state-operator correspondence to map the desired scaling dimensions to the semiclassical energy spectrum of periodic homogeneous field configurations on a cylinder. As concrete applications, we provide detailed analyses for the $φ^4$ theory near four dimensions and $φ^6$ near three dimensions, semiclassically determining the full spectrum of neutral operators in the traceless symmetric Lorentz representations. Our methodology is presented pedagogically and is readily applicable to a vast class of CFTs.

Semiclassical Canovaccio for Composite Operators

TL;DR

The paper develops a global semiclassical framework that maps scaling dimensions of heavy neutral composite operators in CFTs to the energy spectrum of time-periodic homogeneous solutions on the cylinder, via the state-operator correspondence. It systematizes the calculation through a double-scaling limit, with a classical term and a quantum correction obtained from fluctuations encoded by Lamé-type operators and stability angles, using a Gutzwiller-like trace formula and Bohr–Sommerfeld quantization. The method is concretely tested in the theory near and the theory near , yielding the full spectra of neutral operators in traceless symmetric Lorentz representations and reproducing known perturbative results while predicting higher-order terms. It also analyzes large- behavior and instabilities at strong coupling, offering a path toward extending to gauge, Yukawa, and higher-spin sectors with potential cross-checks against bootstrap and Monte Carlo techniques. Overall, the work provides a pedagogical, scalable route to a wide class of CFT data beyond traditional perturbative or bootstrap approaches, including detailed operator towers and their degeneracy lifting at NLO.

Abstract

We present a novel semiclassical framework tailored to determine the scaling dimensions of heavy neutral composite operators in conformal field theories (CFTs) which are inaccessible with other current methodologies. It utilizes the state-operator correspondence to map the desired scaling dimensions to the semiclassical energy spectrum of periodic homogeneous field configurations on a cylinder. As concrete applications, we provide detailed analyses for the theory near four dimensions and near three dimensions, semiclassically determining the full spectrum of neutral operators in the traceless symmetric Lorentz representations. Our methodology is presented pedagogically and is readily applicable to a vast class of CFTs.
Paper Structure (23 sections, 206 equations, 2 figures, 2 tables)

This paper contains 23 sections, 206 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Free field theory
  3. The canovaccio as semiclassical blueprint
  4. Semiclassical quantization of periodic solutions in QFT
  5. The $O(N)$ $\phi^4$ theory in $d=4-\epsilon$
  6. Classical solution and leading order: $C_0$
  7. Renormalization of the action
  8. Fluctuation operators
  9. Leading quantum correction: $C_1$
  10. Perturbative semiclassics
  11. Ising CFT
  12. $O(N)$ CFT
  13. On the instabilities at large $\lambda n$
  14. The $O(N)$ $\phi^6$ theory in $d=3-\epsilon$
  15. Classical solution and leading order: $C_0$
  16. ...and 8 more sections

Figures (2)

  • Figure 1: The band structure for the $\kappa =1$ (left) and $\kappa =2$ (right) Lamé operator $L_\kappa$ as a function of $m$. The allowed energy regions are filled in light blue. The solid lines denote the band edges whose values are given in the main text.
  • Figure 2: Values of $m$ for which complex stability angles occur. The numbers above the red intervals denote the corresponding value of $\ell$. For graphical reasons we do not display the numbers for the obvious $\ell=10, 11, 12$ modes.