Exact Results for the Spectrum of the Ising Conformal Field Theory

TL;DR

This paper develops a semiclassical approach to exactly determine the spectrum of composite operators in the critical Ising conformal field theory, providing high-order results in various dimensions and regimes.

Contribution

It introduces a semiclassical framework that resums infinite Feynman diagrams to compute scaling dimensions of composite operators in the Ising CFT, surpassing existing methods for large operator numbers.

Findings

Full spectrum of composite operators in 4D near criticality obtained.

Complete five-loop scaling dimensions for $^n$ operators derived.

Semiclassical results in 3D outperform previous methodologies for large $n$.

Abstract

We develop a semiclassical framework to determine scaling dimensions of neutral composite operators in scalar conformal field theories. For the critical Ising $\lambda\phi^4$ theory in $d=4-\epsilon$, we obtain the full spectrum of composite operators built out of $n$ fields transforming in the traceless-symmetric Lorentz representations to next-to-leading order in the double-scaling limit $n\rightarrow \infty$ and $\lambda \rightarrow 0$ with $\lambda n$ fixed. At any given order the semiclassical expansion resums an infinite number of Feynman diagrams. Combining our results with existing perturbative computations further yields the complete five-loop scaling dimensions in the $\epsilon$-expansion for the family of $\phi^n$ operators. Finally, in three dimensions the next-to-leading order semiclassical results supersede any other existing methodology for $n \gtrsim \mathcal{O}(10)$.