Logarithmic scaling in gauge/string correspondence

TL;DR

Belitsky, Gorsky, and Korchemsky investigate the logarithmic scaling of anomalous dimensions for high-twist Wilson operators with large Lorentz spin N in N=4 SYM, comparing weak- and strong-coupling regimes. They exploit integrability, via the Baxter Q-operator in gauge theory and classical spectral curves in the AdS5×S5 string, to identify scaling regimes controlled by the parameter ξ = ln(N/L)/L and ξ_str = λ ln^2(N/L)/L^2. The authors show that the conventional semiclassical Bethe Ansatz breaks down for large ξ due to collision of Bethe root cuts, and they develop a beyond-semiclassical approach that yields a uniform one-loop expression for the minimal anomalous dimension across the entire regime. On the string side, logarithmic scaling arises from two-spike configurations near the AdS boundary, with the spectral curve degenerating to an elliptic curve whose branching points approach √λ′, which leads to γ(λ) ∼ (√λ/π) ln(N/√λ) in the strong-coupling limit. The results illuminate the universal role of soft-gluon dynamics and connect BMN-like regimes with true cusp-like scaling through all-coupling resummations.

Abstract

We study anomalous dimensions of (super)conformal Wilson operators at weak and strong coupling making use of the integrability symmetry on both sides of the gauge/string correspondence and elucidate the origin of their single-logarithmic behavior for long operators/strings in the limit of large Lorentz spin. On the gauge theory side, we apply the method of the Baxter Q-operator to identify different scaling regimes in the anomalous dimensions in integrable sectors of (supersymmetric) Yang-Mills theory to one-loop order and determine the values of the Lorentz spin at which the logarithmic scaling sets in. We demonstrate that the conventional semiclassical approach based on the analysis of the distribution of Bethe roots breaks down in this domain. We work out an asymptotic expression for the anomalous dimensions which is valid throughout the entire region of variation of the Lorentz spin. On the string theory side, the logarithmic scaling occurs when two most distant points of the folded spinning string approach the boundary of the AdS space. In terms of the spectral curve for the classical string sigma model, the same configuration is described by an elliptic curve with two branching points approaching values determined by the square root of the 't Hooft coupling constant. As a result, the anomalous dimensions cease to obey the BMN scaling and scale logarithmically with the Lorentz spin.

Figures (4)

• Figure 1: The definition of the $\alpha-$cycles and $\gamma-$contours on the Riemann surface $\Gamma_L$. The dashed lines represent the part of the path on the lower sheet of the surface.
• Figure 2: Symmetric two-cut configuration (a) resulting in the BMN scaling of the anomalous dimension in gauge theory for $\xi < 1$. For $\xi\gg 1$ the two cuts collide at the origin (b) yielding the logarithmic scaling. The same configuration in string sigma model (c) for $\xi_{\rm str}\gg 1$ -- the minimal value for the inner end of the cut is given by the BMN coupling $\sqrt{\lambda^\prime}$ which prevents the cuts to collide.
• Figure 3: The transfer matrix $\cos p(x)=\tau_0(x)/2$ as a function of $1/x$ for the symmetric two-cut solution for $\beta=3/4$ and $m=1$. The cuts $[-a,-b] \cup [b,a]$ with $a=0.45$ and $b=0.15$ correspond to $\cos^2 p(x)> 1$. The double points $x_{2j}$ are denoted by crosses, $\cos p(x_{2j})=\pm 1$ and $1/x_{2j}^2 > 1/b^2$. The "large" and "small" roots of the transfer matrix, $\cos p (\delta_n) = 0$, are shown by full and light blobs, respectively.
• Figure 4: Left panel: "small" roots of the transfer matrix $\delta_n$ (with $n=2,\ldots,L-1$) for $s={\frac{1}{2}}$, $L=10$ and two values of the spin $N = 10^2$ and $N=10^{10}$. Crosses stand for the solutions to the quantization condition (\ref{['system-arg']}) and the lines correspond to (\ref{['delta-fin']}) for $k_n=L+1-2n$. The roots tend to approach the line as $\xi=\ln(N/L)/L$ increases from $\xi = 0.23$ to $\xi=2.07$. Right panel: the minimal energy for $s={\frac{1}{2}}$, $L=10$ and the total spin $0 \le N \le 100$. Crosses denote the exact values, Eq. (\ref{['Energy-ABA']}), while the solid line stands for the semiclassical expression (\ref{['Energy-2-cut']}). We do not display the data for the asymptotic energy (\ref{['energy-as']}) with roots deduced from the quantization conditions (\ref{['system-arg']}), since they are not distinguishable from the exact spectrum.