Logarithmic scaling in gauge/string correspondence

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.