From Partons to Strings: Scattering on the Coulomb Branch of $\mathcal{N}=4$ SYM

TL;DR

This paper explores the scattering amplitudes on the Coulomb branch of planar ${ m N}=4$ SYM at finite coupling, revealing rich structures and connecting weak and strong coupling regimes through integrability and bootstrap techniques.

Contribution

It introduces a finite-coupling model for scattering amplitudes that unifies weak and strong coupling behaviors using integrability, bootstrap, and Padé approximations.

Findings

The four-point amplitude exhibits nonlinear Regge trajectories and dual conformal invariance.

The spectrum of bound states has an accumulation point and relates to the cusp Hamiltonian.

The amplitude nearly saturates bootstrap bounds at strong coupling.

Abstract

We study scattering on the Coulomb branch of planar ${\mathcal{N}}=4$ SYM at finite 't Hooft coupling. This setup defines a family of classical open-string S-matrices that smoothly interpolates between perturbative parton scattering at weak coupling and flat-space string scattering at strong coupling. We focus on the four-point amplitude, which exhibits a remarkably rich structure: nonlinear Regge trajectories, dual conformal invariance, an intricate spectrum of bound states with an accumulation point, and a two-particle cut. Dual conformal invariance relates the spectrum of Regge trajectories to the energy spectrum of the Maldacena-Wilson cusp Hamiltonian. This connection allows us to use integrability to compute the leading and subleading Regge trajectories at finite coupling, which we then input in the bootstrap analysis. At strong coupling, we use the worldsheet bootstrap to construct the first $AdS$-curvature correction to the Veneziano amplitude. We apply dispersion relations and S-matrix bootstrap techniques to derive bounds on Wilson coefficients, couplings to bound states, and the overall shape of the amplitude. We find that the $\mathcal{N}=4$ amplitude saturates the bootstrap bounds at weak coupling and nearly saturates them at strong coupling. At intermediate coupling, the amplitude traces a nontrivial path through the allowed space of observables. To characterize this path, we combine the weak- and strong-coupling information about the amplitude to construct a finite-coupling model for Wilson coefficients using a Pad\'{e} approximation. The resulting model satisfies bootstrap constraints and yields sharp predictions for the finite-coupling behavior of the amplitude. We provide evidence that complete monotonicity of the scattering amplitude, previously observed perturbatively, persists at finite coupling.