Space-time S-matrix and Flux-tube S-matrix at Finite Coupling

TL;DR

The paper proposes a non-perturbative, OPE-based framework for planar N=4 SYM scattering amplitudes, via a decomposition into Pentagon transitions connected to the GKP flux tube S-matrix. It provides a finite-coupling solution for gluonic transitions, with perturbative and strong-coupling checks that recover known results and link to Y-system/TBA descriptions. The approach unifies weak and strong coupling descriptions and suggests a path toward a Thermodynamic Bethe Ansatz-like resummation for finite coupling, with extensions to non-gluonic sectors and non-planar theories.

Abstract

We propose a non-perturbative formulation of planar scattering amplitudes in N=4 SYM or, equivalently, polygonal Wilson loops. The construction is based on the OPE approach and introduces a new decomposition of the Wilson loop in terms of fundamental building blocks named Pentagon transitions. These transitions satisfy a simple relation to the worldsheet S-matrix on top of the so called Gubser-Klebanov-Polyakov vacuum which allows us to bootstrap them at any value of the coupling. In this letter we present a subsector of the full solution to scattering amplitudes which we call the gluonic part. We match our results with both weak and strong coupling data available in the literature. For example, the strong coupling Y-system can be understood in this approach.

Figures (6)

• Figure 1: Decomposition of $n$-sided Null Polygons into sequences of $n-3$ null squares. Any two adjacent squares form a pentagon and any middle square is shared by two pentagons. There are $n-4$ pentagons and $n-5$ middle squares. Every middle square in the decomposition shares two of its opposite cusps with the big polygon; the positions of the other two cusps (which are not cusps of the big polygon) are fixed by the condition that they are null separated from their neighbours. For example, in (a) we have an hexagon. It has a single middle square whose symmetries $\tau,\sigma$ and $\phi$ parametrize its three conformal cross-ratios OPEpaper.
• Figure 2: For any middle square in the framing we associate a GKP time $\tau$, space $\sigma$, and angle $\phi$ for rotations in the two dimensional space transverse to it. They are the three conformal cross ratios associated with an hexagon that is formed by the two pentagons overlapping on that square. Note that cusps $\bf 2$ and $\bf 5$ are the only cusps of the hexagon that are not shared with the big polygon. Cusps $\it 1$ and $\it 4$ are the only cusps of the hexagon that are shared with the middle square. An hexagon is symmetric under $\phi \leftrightarrow -\phi$. The relative sign between $\phi_i$ and $\phi_{i+1}$ is physical and fixed by demanding that in the measure limit, $\sigma_i,\sigma_{i+1}\to-\infty$, they only appear in the combination $\phi_i+\phi_{i+1}$.
• Figure 3: We construct a conformal invariant finite ratio by dividing the expectation value of the WL by all the pentagons in the decomposition and multiplying it by all the middle squares, $\mathcal{W}\equiv \langle W\rangle \times \frac{\langle W_{1^\text{st} \text{middle sq.}}\rangle\langle W_{2^\text{nd} \text{middle sq.}}\rangle\dots }{ \langle W_{1^\text{st} \text{pent.}}\rangle \langle W_{2^\text{nd} \text{pent.}}\rangle \dots }$. This is a generalization of the ratios considered in OPEpaperheptagonPaper.
• Figure 4: Two fundamental building blocks: the expectation value of the square ( a) and pentagon WL ( b) with GKP excitations inserted on their bottom and top. One natural way to insert these excitations is to start from the hexagon or heptagon (regulated as in fig.\ref{['calW']}) and take the collinear limit $\tau_i\to\infty$. In this way we can extract the transitions from known Amplitudes/WL in perturbation theory and match them with the integrability predictions; more details in toappear.
• Figure 5: Flipping the sign of both momenta is equivalent to a reflection of the pentagon.