Amplitudes at Weak Coupling as Polytopes in AdS_5

TL;DR

The paper reinterprets one-loop scalar box integrals in planar N=4 SYM as volumes of geodesic tetrahedra embedded in $AdS_5$ with boundary region momenta. It shows that 4-mass box functions equal twice the volume of ideal tetrahedra, with volumes expressed by the Bloch-Wigner dilogarithm of a boundary cross-ratio, and introduces horosphere regularisation to handle IR divergences, linking to dimensional regularisation. For one-loop MHV amplitudes, the sum of boxes forms a closed 3-polytope in $AdS_5$, providing a unified geometric picture of the amplitude as a single volumetric object. The work suggests natural extensions to higher MHV degrees and loops via higher-dimensional polytopes and Grassmannian/residue techniques, potentially offering new computational and conceptual tools for scattering amplitudes.

Abstract

We show that one-loop scalar box functions can be interpreted as volumes of geodesic tetrahedra embedded in a copy of AdS_5 that has dual conformal space-time as boundary. When the tetrahedron is space-like, it lies in a totally geodesic hyperbolic three-space inside AdS_5, with its four vertices on the boundary. It is a classical result that the volume of such a tetrahedron is given by the Bloch-Wigner dilogarithm and this agrees with the standard physics formulae for such box functions. The combinations of box functions that arise in the n-particle one-loop MHV amplitude in N=4 super Yang-Mills correspond to the volume of a three-dimensional polytope without boundary, all of whose vertices are attached to a null polygon (which in other formulations is interpreted as a Wilson loop) at infinity.

Figures (5)

• Figure 1: The 4-mass box function is defined by the integral \ref{['4mbdef']}. The region momenta are fixed up to overall translation by $x_i-x_{i+1} = p_i$, while the momentum running through a propagator is given by the difference of the region coordinates on either side of that propagator.
• Figure 2: An ideal tetrahedron in AdS, shown in both the Klein--Beltrami (l) and Poincaré (r) models. All vertices lie on the conformal boundary at infinity, and each edge is an AdS geodesic.
• Figure 3: The 3-mass box integral has an entire edge along the boundary at infinity, so its volume diverges. This edge lies along a null geodesic (shown in blue) and the quadric $X\cdot X=0$ is ruled by these null lines.
• Figure 4: The tetrahedra may be regularised by bringing the vertices into the interior of AdS along the geodesic (shown in blue) connecting their original locations to a chosen point $I$. For simplicity, we can keep all the vertices on the same horosphere, corresponding to giving the external states equal masses.
• Figure 5: One loop MHV amplitudes involve only '1-mass' and '2-mass easy' box functions.