Excited Hexagon Wilson Loops for Strongly Coupled N=4 SYM

TL;DR

This paper analyzes the strong-coupling Regge limit of the planar six-gluon amplitude in ${\cal N}=4$ SYM using the Alday–Maldacena framework, recasting the area problem as the free energy of an auxiliary one-dimensional integrable system. By solving the associated Y-system/nonlinear integral equations and performing an analytic continuation to the mixed Regge region, it shows that excited-state contributions yield a computable, exponentiated correction to the BDS-like amplitude, with the leading exponent $E_2$ given by $E_2 = \frac{\sqrt{\lambda}}{2\pi}\left(\sqrt{2}+\frac{1}{2}\log(3+2\sqrt{2})\right)$. The Regge limit in the mixed region is obtained as a controlled exponential of a free-energy functional, with explicit cross-ratio dependence through quantities $\varepsilon$, $w$, and $c$ derived from the continuation path; this provides a concrete strong-coupling counterpart to the weak-coupling BFKL analysis. The work demonstrates the role of excited states in Regge physics at large $\lambda$ and outlines future directions for extending the approach to more external legs and alternative NLIE formulations.

Abstract

This work is devoted to the six-gluon scattering amplitude in strongly coupled N=4 supersymmetric Yang-Mills theory. At weak coupling, an appropriate high energy limit of the so-called remainder function, i.e. of the deviation from the BDS formula, may be understood in terms of the lowest eigenvalue of the BFKL hamiltonian. According to Alday et al., amplitudes in the strongly coupled theory can be constructed through an auxiliary 1-dimensional quantum system. We argue that certain excitations of this quantum system determine the Regge limit of the remainder function at strong coupling and we compute its precise value.

Figures (5)

• Figure 1: Kinematic configuration for the multi-Regge limit before (on the left) and after (on the right) the analytical continuation with the kinematic invariants from eq. \ref{['stinv']}. Momenta of are denoted by $p_i$ while dual coordinates of the Wilson loop by $x_i$.
• Figure 2: Upon analytic continuation along $\cal C$, the parameter $m$ is shifted from its initial value. The plot shows numerical results for $m=10$, $\cosh C= 3/5$ and $\phi=0$ at $\varphi =0$. Plots for different values of the parameter $C(\varphi=0)$ differ by terms of order $\exp(-m)$.
• Figure 3: The plot shows the dependence of $C$ on $\varphi$. The various curves correspond to the values $m=10$, $\phi=0$ and $\cosh C=c$ at the starting point $\varphi=0$. Here, we plot $C$ in its complex plane along the path given by \ref{['uthree']} while $\phi(\varphi)=0$.
• Figure 4: Positions of some central solutions to the equation $Y_3(\theta)=-1$ along the path depicted in figs. \ref{['fig:cpath']}, \ref{['fig:cpath2']}. Light dots correspond to positions in the first half of the path before any solution has crossed the real axis while the dark ones are associated with the second part.
• Figure 5: Positions of some central solutions to the equation $Y_2(\theta)=-1$ along the path depicted in figs. \ref{['fig:cpath']}, \ref{['fig:cpath2']}. We change from light to dark dots at the point where the first solutions of $Y_{3}= -1$ cross the real axis.