Fine structure of anomalous dimensions in N=4 super Yang-Mills theory

TL;DR

This work develops and applies an all-order Baxter-equation framework to the autonomous ${\rm SL}(2)$ sector of planar ${\cal N}=4$ SYM to resolve the fine structure of high-twist anomalous dimensions. By combining asymptotic Baxter methods for the lower part of the spectrum with semiclassical (WKB/finite-gap) techniques for the upper part, the authors derive explicit expressions and quantization conditions for ground and excited trajectories across the band, including up to three-loop corrections. They reveal an iterative, scaling-structure in the subleading terms governed by the cusp anomalous dimension and a hidden parameter $\xi=\frac{1}{L}\ln N$, and establish how the spectrum reorganizes as one moves from the low to the high ends of the band. The results illuminate the integrable underpinnings of the dilatation operator, provide concrete predictions for large-spin operators, and motivate further connections to AdS/CFT dual string configurations and the density of states at strong coupling.

Abstract

Anomalous dimensions of high-twist Wilson operators in generic gauge theories occupy a band of width growing logarithmically with their conformal spin. We perform a systematic study of its fine structure in the autonomous SL(2) subsector of the dilatation operator of planar N=4 super Yang-Mills theory which is believed to be integrable to all orders in 't Hooft coupling. We resort in our study on the framework of the Baxter equation to unravel the properties of the ground state trajectory and the excited trajectories in the spectrum. We use two complimentary approaches in our analysis based on the asymptotic solution of the Baxter equation and on the semiclassical expansion to work out the leading asymptotic expression for the trajectories in the upper and lower part of the band and to find how they are modified by the perturbative corrections.

Figures (4)

• Figure 1: The eigenvalues of the energy $\gamma_0$ and the conserved charge $\widehat{q}_{3, 0}={q}_{3, 0}/(N+{\frac{1}{2}} L)^3$ for three-site spin chain. For each value of the conformal spin $N$ there are $m = {\frac{1}{3}} (N-1) + {\frac{2}{3}} {\rm mod}(N-1, 3)$ eigenvalues of the anomalous dimensions with zero quasimomentum $\theta_ 0 = 0$.
• Figure 2: The exact eigenvalues of charge $\widehat{q}_1$ at two-loop order of perturbation theory.
• Figure 3: The exact eigenvalues of the conserved charge $\widehat{q}_3$ at one-, two- and three-loop order (left to right) of perturbation theory for twist-three operators shown together with a set of selected semiclassical trajectories computed from Eqs. (\ref{['q3-ch']}) and (\ref{['q3tot']}), respectively, for $n=0,1,4,7$.
• Figure 4: The exact eigenvalues of anomalous dimension matrix in the lowest three orders of perturbation theory for twist-three operators obtained from numerical solution of the Baxter equation and selected semiclassical trajectories for $n=0,1,4,7$ at corresponding orders in 't Hooft coupling determined by Eqs. (\ref{['LOadUPPER']}) and (\ref{['eps_fin']}), respectively.