On the collinear limit of scattering amplitudes at strong coupling

TL;DR

This work analyzes the collinear limit of gluon scattering amplitudes at strong coupling in planar ${\cal N}=4$ SYM, showing that the problem maps to correlators of twist operators in the $O(6)$ non-linear sigma model and is efficiently captured by a pentagon form-factor framework. By combining the worldsheet minimal-area description with fluctuations in the $S^5$, the authors derive a non-perturbative expression for the amplitude and reveal a new exponential competition between AdS$_5$ area and sphere dynamics, leading to a refined strong-coupling formula $\mathcal{W}_n = C\,\lambda^B\, e^{A\sqrt{\lambda}}$. They obtain detailed UV/IR analyses, a short-distance OPE relating pentagon and hexagon operators, and a cross-over regime where the $O(6)$ dynamics govern the leading corrections to the classical result, with a proposed relation $B=-\tfrac{3}{2}A$ linking scaling dimensions to the prefactor. The results illuminate non-perturbative IR effects on the string worldsheet and lay groundwork for a systematic inclusion of $\alpha'$ and massive-mode corrections beyond the collinear limit, potentially enabling a bootstrap-like determination of the overall prefactor $C$.

Abstract

In this letter we consider the collinear limit of gluon scattering amplitudes in planar N=4 SYM theory at strong coupling. We argue that in this limit scattering amplitudes map into correlators of twist fields in the two dimensional non-linear O(6) sigma model, similar to those appearing in recent studies of entanglement entropy. We provide evidence for this assertion by combining the intuition springing from the string worldsheet picture and the predictions coming from the OPE series. One of the main implications of these considerations is that scattering amplitudes receive equally important contributions at strong coupling from both the minimal string area and its fluctuations in the sphere.

Figures (8)

• Figure 1: A null polygon Wilson loop sources a colour flux in the gauge theory whose dual description is that of an open string ending on it. The Wilson loop expectation value can be mapped to a correlator in the flux tube theory or, equivalently, into an open string partition function.
• Figure 2: Under a mirror transformation $\theta\to\theta+i{\pi\over2}$ an excitation is sent from one edge to its neighbour. For a pentagon, we need five such transformations to move the excitation all the way around.
• Figure 3: (a) The world-sheet of the string ending on a pentagon can be viewed as made out of five quadrants. (b) Equivalently, we can engineer these five quadrants starting from the square by inserting the twist operator $\phi_{\pentagon}$. This one generates an excess angle of $2\pi/4$.
• Figure 4: At strong coupling, the hexagon Wilson loop in the collinear limit is given by a correlator of two twist operators in the $O(6)$ sigma model (on the left), corresponding to the two pentagons in its decomposition (on the right).
• Figure 5: The pattern of auxiliary rapidities arising in the construction of the matrix part has a group theoretical interpretation. The three sets of rapidities can be identified with the three nodes of the $O(6)$ dynkin diagram. The occupation numbers are fixed such that the overall state with $n$ particles belongs to the $O(6)$ singlet representation. The cartoon depicted here is related to (\ref{['mpart']}) by identifying the solid lines with $f$'s and the dashed lines with $g$'s.