Exact Results for the Spectrum of the Ising Conformal Field Theory

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 $n \gtrsim \mathcal{O}(10)$.