Multi-Regge kinematics and the moduli space of Riemann spheres with marked points

TL;DR

This work reframes MRK amplitudes in planar N=4 SYM as single-valued iterated integrals on the moduli space M_{0,N-2}, connecting kinematics to the geometry of genus-zero curves with marked points. It establishes a robust factorisation framework for MHV and non-MHV amplitudes at leading-log accuracy, enabling all-order constructions via a limited set of building blocks and Fourier–Mellin convolutions. The authors classify leading singularities, prove key factorisation results, and provide explicit analytic results up to five loops for MHV and four loops for NMHV with many legs, supported by a comprehensive single-valued polylogarithm structure. The approach leverages cluster algebras, KZ equations, Drinfeld associators, and single-valued variants of hyperlogarithms to deliver an efficient, unified procedure for MRK amplitudes with broad applicability beyond N=4 SYM. This framework opens avenues for extending MRK analyses to higher orders and other theories, with potential impacts on resummations, integrability, and the analytic understanding of scattering amplitudes.

Abstract

We show that scattering amplitudes in planar N = 4 Super Yang-Mills in multi-Regge kinematics can naturally be expressed in terms of single-valued iterated integrals on the moduli space of Riemann spheres with marked points. As a consequence, scattering amplitudes in this limit can be expressed as convolutions that can easily be computed using Stokes' theorem. We apply this framework to MHV amplitudes to leading-logarithmic accuracy (LLA), and we prove that at L loops all MHV amplitudes are determined by amplitudes with up to L + 4 external legs. We also investigate non-MHV amplitudes, and we show that they can be obtained by convoluting the MHV results with a certain helicity flip kernel. We classify all leading singularities that appear at LLA in the Regge limit for arbitrary helicity configurations and any number of external legs. Finally, we use our new framework to obtain explicit analytic results at LLA for all MHV amplitudes up to five loops and all non-MHV amplitudes with up to eight external legs and four loops.

Figures (6)

• Figure 1: The algebraically independent set of $3N-15$ cross-ratios.
• Figure 2: The $\mathcal{A}$-coordinates for the initial quiver for ${\rm Gr}(4,N)$ with frozen nodes in boxes.
• Figure 3: The $\mathcal{X}$-coordinates for the initial quiver for ${\rm Gr}(4,N)$.
• Figure 4: The three cross ratios associated to the reggeized propagator $|{\bf q}_{i}|^2$: $u_{1i}$ (left), $u_{2i}$ (center) and $u_{3i}$ (right). Solid lines denote square distances in the numerator, and dashed lines in the denominator, respectively.
• Figure 5: The dual coordinates in the transverse space. Dashed lines indicate the forward momenta with zero transverse momentum, which are strictly speaking absent in the transverse momentum space because they are orthogonal to it.

Theorems & Definitions (2)

• Claim 1
• Claim 2