On the structure of scattering amplitudes in N=4 super Yang-Mills and N=8 supergravity

TL;DR

This paper develops a coherent framework linking singularities of Feynman integrals to the structure of multi-loop scattering amplitudes in ${\cal N}=4$ SYM and ${\cal N}=8$ supergravity. It introduces the rung-rule for constructing a scalar integral basis, analyzes hidden (Jacobian) singularities that enable iterative reductions to lower loops, and connects these ideas to infrared behavior and dual conformal invariance to constrain both bases and coefficients. The authors demonstrate how one-loop amplitudes are fixed by maximal cuts and extend the approach to higher loops, showing how spurious singularities necessitate corrections and how gravity modifies the resummation with $u$-type factors. They propose a principled, IR-informed procedure to generate the finite dual-conformal basis to all loops, discuss non-planar and gravity extensions, and outline the remaining challenges and consistency requirements for higher-loop amplitudes. Overall, the work provides a predictive, diagrammatic, and symmetry-guided route to assembling four-pparticle amplitudes at high loops, with implications for both planarity and non-planarity in gauge/gravity theories.

Abstract

Exploiting singularities in Feynman integrals to get information about scattering amplitudes has been particularly useful at one-loop in theories where no triangles or bubbles appear. At higher loops the integrals possess subtle singularities. In this paper we give these singularities a physical interpretation and show how they turn tedious computations into purely pictorial manipulations. We illustrate our methods with various examples from the computation of four-particle amplitudes in N=4 super Yang-Mills and N=8 supergravity. Along the way we find clues towards an understanding i) of the rung-rule as a consequence of infra-red singularities, ii) of the non rung-rule integrals included in the basis as corrections to the rung-rule and iii) of the coefficients - including signs - of these two types of contribution. The role of corrections is to cancel unphysical singularities generically present in rung-rule integrals. A further byproduct, coming from the fact such unphysical singularities are located where conformal cross-ratios become unity, is the possibility of understanding the dual conformal invariance ansatz for constructing the basis of four-particle amplitudes in N=4 super Yang-Mills.

Figures (16)

• Figure 1: Computation of a coefficient using the leading singularity of a box. The lines circling the propagators represent the $T^4$ contour of integration. The left hand side of the figure represents the sum of all 1-loop Feynman diagrams - note that only those Feynman diagrams that contain the displayed propagators actually contribute to this particular contour integral.
• Figure 2: Main resummation formula in ${\cal N}=4$ SYM. The left of the diagram represents the sum of all 1-loop Feynman diagrams, integrated over the displayed contour. The external states may be any members of the ${\cal N}=4$ multiplet.
• Figure 3: Main resummation formula in ${\cal N}=8$ supergravity. The external states may be any members of the ${\cal N}=8$ supermultiplet.
• Figure 4: The $L$-loop ladder integral, shown for both the scalar integrals and as a factorization channel of the $L$-loop Feynman diagrams.
• Figure 5: The Jacobian provides factorization channels of the $(L-1)$-loop amplitude. In a ladder integral, one of these factorization channels does not contribute because for generic, fixed external momenta, $s$ is always non-zero.