Integrable spin chains and scattering amplitudes

TL;DR

This work analyzes high-energy multi-particle scattering in planar $N=4$ SYM, showing that Mandelstam-cut contributions from adjoint reggeized-gluon composites govern the Regge-limit behavior beyond Regge poles. It establishes that the BKP Hamiltonian for these composite states is equivalent to the local Hamiltonian of an integrable open spin chain and constructs their wavefunctions using integrals of motion via the Baxter–Sklyanin separation of variables. The analysis connects these integrable structures to known results for the two-loop remainder function and extends to general $n$-gluon states, highlighting a deep link between scattering amplitudes and integrable systems in the planar limit. This framework provides a path toward exact results for Regge-limit amplitudes and enhances understanding of the interplay between high-energy QCD dynamics and integrable models in $\,N=4$ SYM.

Abstract

In this review we show that the multi-particle scattering amplitudes in N=4 SYM at large Nc and in the multi-Regge kinematics for some physical regions have the high energy behavior appearing from the contribution of the Mandelstam cuts in the complex angular momentum plane of the corresponding t-channel partial waves. These Mandelstam cuts or Regge cuts are resulting from gluon composite states in the adjoint representation of the gauge group SU(Nc). In the leading logarithmic approximation (LLA) their contribution to the six point amplitude is in full agreement with the known two-loop result. The Hamiltonian for the Mandelstam states constructed from n gluons in LLA coincides with the local Hamiltonian of an integrable open spin chain. We construct the corresponding wave functions using the integrals of motion and the Baxter-Sklyanin approach.

Figures (13)

• Figure 1: The analytic representation of the $2 \to 4$ scattering amplitude. The dashed lines denote possible energy discontinuities.
• Figure 2: Contributions of $\mathbf{a)}$ the Regge poles and $\mathbf{b)}$ the Mandelstam cut to the $2 \to 4$ scattering amplitude in the $t_2$-channel. The wavy lines represent reggeized gluons. The Mandelstam cut appears as a bound state of two reggeized gluons.
• Figure 3: The diagrammatic structure of the Mandelstam cut.
• Figure 4: Twisting a planar $2 \to 4$ amplitude.
• Figure 5: The $2 \to 6$ scattering amplitude.