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 $λφ^4$ theory in $d=4-ε$, 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 $λ\rightarrow 0$ with $λn$ 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 $ε$-expansion for the family of $φ^n$ operators. Finally, in three dimensions the next-to-leading order semiclassical results supersede any other existing methodology for $n \gtrsim \mathcal{O}(10)$.

Figures (2)

• Figure 1: Comparison of the band structure of the Lamé operator $L_2$ (allowed bands colored in light blue) and $\Lambda_\ell$ for $\ell=0,1,2,5,10$ (dashed lines) as a function of $m$. Complex stability angles arise when $\Lambda_\ell$ takes values inside the forbidden zones i.e., when dashed lines are inside the uncolored zones.
• Figure 2: Solid lines: Relative difference for $\Delta_n$ (in $\%$) between the bootstrap Simmons-Duffin:2016wlqHenriksson:2022gpa and the $\epsilon$-expansion truncated to various loop orders. Dashed lines: same quantity evaluated for the corresponding Padé approximants.