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

TL;DR

The paper develops a comprehensive program to understand scattering on the Coulomb branch of planar ${\mathcal{N}}=4$ SYM at finite coupling. It combines integrability (via the Quantum Spectral Curve) to obtain leading and subleading Regge trajectories from cusp energies, a worldsheet bootstrap to compute AdS-curvature corrections to flat-space amplitudes, and S-matrix bootstrap with dispersion relations and enhanced unitarity to bound Wilson coefficients and the full finite-coupling amplitude. A Padé-based finite-coupling model ties weak- and strong-coupling data into a coherent prediction, which is checked against bootstrap constraints and evidence for complete monotonicity of the amplitude beyond perturbation theory. The work reveals a rich interpolation between hydrogen-like bound states at weak coupling and open-string dynamics at strong coupling, with a Regge/cusp correspondence extending to higher trajectories and a robust framework for exploring finite-coupling observables in a highly constrained S-matrix. The results provide sharp predictions for finite-coupling Wilson coefficients, three-point couplings, and the full amplitude, and they establish a blueprint for applying integrability and bootstrap techniques to other nonconformal, Coulomb-branch settings.

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 Padé 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.

Figures (37)

• Figure 1: Scattering of open strings $\chi$ attached to distinct probe D3-branes at positions $z_i = 1/m_i$. The strings stretched between the probe branes and the stack of $N$ coincident D3-branes correspond to the massive W-bosons.
• Figure 2: A characteristic shape of the leading Regge trajectory at finite coupling. We also depict the relevant limits of the Wilson line with a cusp. The Regge trajectory is curved, it has a logarithmic asymptotic at large $s$, and it exhibits an accumulation point at $s=4m^2$. For $s>4 m^2$ it develops a non-zero imaginary part.
• Figure 3: Roadmap illustrating the topics discussed in the paper and the connections among them.
• Figure 4: Planar diagrams with external $\chi$ fields (wavy lines) attached to a frame made of W-bosons (thick lines), filled with massless fields from the unbroken gauge group $U(N)$ (dashed lines).
• Figure 5: One-loop box diagram in double-line notation.