Walking Sudakov: From Cusp to Octagon

TL;DR

This paper investigates the behavior of the Sudakov form factor and four-point scattering amplitude in planar N=4 SYM on the Coulomb branch, revealing a novel scaling limit with a walking anomalous dimension that interpolates between known regimes.

Contribution

It introduces a new scaling limit with a walking anomalous dimension and proposes an all-loop form for it based on two-loop results and expected all-order structures.

Findings

Identified a double-logarithmic scaling limit with a walking anomalous dimension.

Interpolates between cusp and octagon anomalous dimensions as mass scales vary.

Proposed an all-loop expression involving unknown functions of the coupling.

Abstract

We study the Sudakov form factor and the four-point scattering amplitude on the Coulomb branch of planar $\mathcal{N}=4$ SYM as functions of the Coulomb-branch parameters and kinematic invariants. This setup provides a controlled probe of the interpolation between on- and off-shell regimes of infrared-sensitive quantities in gauge theories. We identify a novel scaling limit in which both observables exhibit double-logarithmic behavior governed by a walking anomalous dimension. As the mass scales are varied, this walking anomalous dimension interpolates between the cusp anomalous dimension of the on-shell regime and the octagon anomalous dimension of the off-shell regime. Based on the explicit two-loop result and the expected all-order structure, we propose an all-loop form for the walking anomalous dimension both for the form factor and for the four-point scattering amplitude. These all-loop expressions depend on new, presently unknown functions of the 't Hooft coupling.