On level crossing in conformal field theories

TL;DR

This work analyzes level-crossing phenomena in unitary CFTs where planar scaling dimensions collide as parameters vary but must respect the non-crossing rule when $N$ is finite. It develops a universal resummation framework for the two-level dilatation operator, yielding Δ_± = {Δ_1+Δ_2 over 2} ± sqrt{ε^2/4 + γ^2/N^2}, with ε the planar-level separation and γ set by leading nonplanar corrections; OPE data are resummed consistently to give finite C_{φφ O_±}^2 that smoothly interpolate between the planar limit and the crossing region. The approach is validated against known level-crossing instances in four-dimensional N=4 SYM (Konishi vs double-trace), three-dimensional CFTs near the Ising point, and QCD baryonic operators, showing agreement in both scaling dimensions and OPE coefficients. The results highlight a universal mechanism by which nonplanar effects restore non-crossing, and they have potential implications for dualities, bootstrap analyses, and nonperturbative operator mixing in gauge theories. The paper thus provides a coherent, cross-model description of level-crossing phenomena across high-energy and conformal systems.

Abstract

We study the properties of operators in a unitary conformal field theory whose scaling dimensions approach each other for some values of the parameters and satisfy von Neumann-Wigner non-crossing rule. We argue that the scaling dimensions of such operators and their OPE coefficients have a universal scaling behavior in the vicinity of the crossing point. We demonstrate that the obtained relations are in a good agreement with the known examples of the level-crossing phenomenon in maximally supersymmetric $\mathcal N=4$ Yang-Mills theory, three-dimensional conformal field theories and QCD.

Figures (4)

• Figure 1: The scaling dimensions $\Delta_\pm$ (left panel) and the OPE coefficients $(C_{\phi\phi O_\pm} / C_{\phi\phi O_1} )^2$ (right panel) in the transition region (\ref{['crossing']}) as a function of the coupling constant $g$ for $\gamma/N=0.1$. The dashed lines represent the same quantities in the planar limit, $\Delta_i(g) = \alpha_i g + O(g^2)$.
• Figure 2: The scaling dimensions of the three lowest spin$-2$ operators and their OPE coefficients (multiplied by $4^{\Delta_i}$) as found in Ref. El-Showk:2014dwa.
• Figure 3: Dependence of the scaling dimensions of the spin$-2$ operators and their OPE coefficients $\widehat{C}_{\sigma\sigma \pm}^2 = C_{\sigma\sigma \pm}^2 /C_{\sigma\sigma 1}^2$ on the level splitting $\epsilon$. Dots stand for the exact values found in El-Showk:2014dwa, solid lines are described by (\ref{['ad-3d']}) and (\ref{['C-3d']}).
• Figure 4: The flow of energies of parity-even eigenstates of the Hamiltonian (\ref{['H-bar']}) for $S=20$. The vertical dotted line indicates the spectrum of $H_1$. The two highest levels are shown by red and blue lines. The flow of energies close to the crossing point is zoomed in on the right panel.