Adjoint BFKL at finite coupling: a short-cut from the collinear limit

TL;DR

The paper derives the color-adjoint BFKL data at finite coupling in planar ${\cal N}=4$ SYM by analytically continuing the collinear OPE from the six-gluon hexagon amplitude. It introduces a sister dispersion relation that governs the Mandelstam-region discontinuity and expresses the BFKL eigenvalue $-\omega(\nu,m)$ and the measure $\hat{\mu}_{\mathrm{BFKL}}(\nu,m)$ in terms of the BES kernel $K(t)$, valid to all loops. At strong coupling, the trajectories become universal and connect to a semiclassical string description with a giant-hole interpretation, including a folded AdS$_3$ worldsheet ending on a boundary null polygon, and a link to dipole-dipole scattering via Wilson lines. The work provides a coherent, nonperturbative bridge between the collinear OPE and high-energy Regge limits, offering exact finite-coupling predictions and a path toward understanding wrapping and higher-point amplitudes in this integrable setting.

Abstract

In the high energy Regge limit, the six gluons scattering amplitude is controlled by the adjoint BFKL eigenvalue and impact factor. In this paper we determine these two building blocks at any value of the 't Hooft coupling in planar $\cal{N}$=4 SYM theory. This is achieved by means of analytic continuations from the collinear limit, where similar all loops expressions were recently established. We check our predictions against all available data at weak and strong coupling.

Figures (12)

• Figure 1: (a) The most familiar continuation, away from integer spin, of the spectrum of the superstring in flat space. (b) The familiar and sister trajectories, which both enter the partial wave decomposition of the six-particle amplitude.
• Figure 2: (a) Sketch of the string worldsheet exchanged between two sets of highly boosted particles (2, 3 and 4 vs. 5, 6 and 1) in the classical limit. Here, particles 1 and 2 are in-coming while the rest are out-going. (b) In the Mandelstam region where $3$ and $6$ are in-coming while the rest are out-going, the exchanged state is obtained by folding the picture in (a). These states lie on the "sister" trajectories of fig. \ref{['fig:stringspectrum']}.
• Figure 3: (a) Gluon Reggeization in the planar limit. The multiple gluons exchanged between the target and projectile are sourced by a single color source on each side (heavy blue line), whose location in the transverse plane is the dynamical variable in the BFKL evolution equation. This equation originates from (factorized) high-energy sub-loops as shown for example on the left. (b) The amplitude in the Mandelstam region (compare with fig. \ref{['fig:foldedstring']}(b)). At high energies it factorizes into dipole-dipole scattering.
• Figure 4: a). The hexagon representing the six-gluon amplitude in Euclidean kinematic, where all non-adjacent cusps are space-like separated. Cusps 5-6 and 2-3 in this figure are extended in the transverse space while all other cusps stand in the plotted ${\mathbb R}^{1,1}$ plane. In the collinear limit, these two cusps are flattened along the red square. After the analytic continuation $\{\sigma,\tau\}\to\{\sigma+i\pi/2,\tau-i\pi/2\}$ we arrive at the $3\to3$ kinematical configuration in (b). To be more precise, for the momentum transfers $s_{345}$ and $s_{456}$ to become time-like (while all other kept space-like), we also have to drag the cusp 5-6 to its antipodal point (at the top in (b)). This flip is not seen at the level of the conformal cross ratios (\ref{['crossratios']}). In the multi-Regge limit, cusp 5-6 (2-3) approaches the antipodal point of cusp 1-6 (3-4).
• Figure 5: (a) The function $F^{\downarrow}$ is originally defined on a real line below the cut. (b) Contour for $(\sigma-t)$ in the convolution (\ref{['Kaction2']}) after $\sigma$ has been continued above the cut, passing to the left of the branch point. Only the range of integration between the two dashed lines contributes to the discontinuity [(b)-(a)].