Beyond cusp anomalous dimension from integrability

TL;DR

The paper addresses the subleading constant in the high-spin expansion of finite-twist operators in ${\cal N}=4$ SYM by deriving and solving a linear integral equation with the BES kernel to compute the subleading function $f_{sl}(g,L)$. It establishes a clear separation between universal BES-like contributions and twist-dependent parts, enabling detailed weak-coupling expansions that reproduce known four-loop results and strong-coupling analyses that reveal a leading $-f(g)\\ln g$ behavior with string-theory-consistent subleading terms. Numerical and analytic techniques are used to extract constants and verify that twist dependence cancels at $O(s^0)$, supporting a universal structure tied to the collinear anomalous dimension $f_{sl'}(g)$. The results strengthen the gauge/string duality picture, provide high-precision data for wrapping effects up to several loops, and lay groundwork for the simultaneous study of high-spin and large-twist regimes (as in FIR).

Abstract

We study the first sub-leading correction $O((\ln s)^0)$ to the cusp anomalous dimension in the high spin expansion of finite twist operators in ${\cal N}=4$ SYM theory. Since this approximation is still governed by a linear integral equation (derived already from the Bethe Ansatz equations in a previous paper), we finalise it better in order to study the weak and strong coupling regimes. In fact, we emphasise how easily the weak coupling expansion can be obtained, confirms the known four loop result and predicts the higher orders. Eventually, we pay particular attention to the strong coupling regime showing agreement and predictions in comparison with string expansion; speculations on the 'universal' part (upon subtracting the collinear anomalous dimension) are brought forward.

Figures (1)

• Figure 1: Plot of $f_{sl}(g)$ with magnification for $g\in (0,1)$.