Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization

TL;DR

The paper introduces a spectral-parameter deformation of amplitudes in planar $N=4$ SYM by solving Yang-Baxter equations in a Graßmannian framework, producing an R-matrix structure for both four-point and three-point building blocks. The spectral parameter is interpreted as a central charge and a local helicity deformation, enabling a symmetry-preserving regulator for loop amplitudes—demonstrated at one loop by a regulated four-point box integral in four dimensions. Gluing three-point R-matrices via plabic diagrams yields higher-point deformed amplitudes, establishing a link between scattering amplitudes and integrability. If extended to all loops, the method could provide a non-perturbative, symmetry-respecting approach using 2D quantum inverse scattering techniques, with potential applicability beyond integrable theories by relaxing helicity quantization.

Abstract

Planar N=4 super Yang-Mills appears to be integrable. While this allows to find this theory's exact spectrum, integrability has hitherto been of no direct use for scattering amplitudes. To remedy this, we deform all scattering amplitudes by a spectral parameter. The deformed tree-level four-point function turns out to be essentially the one-loop R-matrix of the integrable N=4 spin chain satisfying the Yang-Baxter equation. Deformed on-shell three-point functions yield novel three-leg R-matrices satisfying bootstrap equations. Finally, we supply initial evidence that the spectral parameter might find its use as a novel symmetry-respecting regulator replacing dimensional regularization. Its physical meaning is a local deformation of particle helicity, a fact which might be useful for a much larger class of non-integrable four-dimensional field theories.

Figures (5)

• Figure 1: Bootstrap equations for the three-point R-matrices.
• Figure 2: Four-point R-matrix from three-point R-matrices.
• Figure 3: Lattice encoding $\mathcal{R}_{n,k}$
• Figure 4: Dictionary for plabic diagrams.
• Figure 5: Plabic diagram for the one-loop four-point MHV case. A regulating assignment of spectral parameters is added. Note that the spectral parameters of the external and internal lines are the difference of the numbers assigned to the faces. For instance, the parameter associated to the line connecting particles 1 and 4 is $z = 4\bar{\epsilon} - 3 \bar{\epsilon} = \bar{\epsilon}$, with the sign being determined by the choice of helicity flowing upward.