The Complete Four-Loop Four-Point Amplitude in N=4 Super-Yang-Mills Theory

TL;DR

This work delivers the complete four-loop four-point amplitude in ${\cal N}=4$ sYM, including non-planar contributions for a general gauge group, by applying the unitarity method with maximal cuts to a 50-integral ansatz. It analyzes ultraviolet behavior across dimensions, revealing cancellations in double-trace color structures and establishing patterns that feed into the N = 8 supergravity program via the double-copy. The paper also develops graphical and algebraic tools, including box cuts and color-kinematic duality, to efficiently construct non-planar contributions and to organize the UV structure in higher dimensions. These results provide a foundation for tests of infrared/ultraviolet structures in gauge theories and gravity, and offer a path toward higher-loop studies with non-planar dynamics.

Abstract

We present the complete four-loop four-point amplitude in N=4 super-Yang-Mills theory, for a general gauge group and general D-dimensional covariant kinematics, and including all non-planar contributions. We use the method of maximal cuts --- an efficient application of the unitarity method --- to construct the result in terms of 50 four-loop integrals. We give graphical rules, valid in D-dimensions, for obtaining various non-planar contributions from previously-determined terms. We examine the ultraviolet behavior of the amplitude near D=11/2. The non-planar terms are as well-behaved in the ultraviolet as the planar terms. However, in the color decomposition of the three- and four-loop amplitude for an SU(N_c) gauge group, the coefficients of the double-trace terms are better behaved in the ultraviolet than are the single-trace terms. The results from this paper were an important step toward obtaining the corresponding amplitude in N=8 supergravity, which confirmed the existence of cancellations beyond those needed for ultraviolet finiteness at four loops in four dimensions. Evaluation of the loop integrals near D=4 would permit tests of recent conjectures and results concerning the infrared behavior of four-dimensional massless gauge theory.

Figures (37)

• Figure 1: Rules for obtaining the color factors associated with cubic parent graphs. The roman letters $a,b,c$ are color indices. The sign of each structure constant $\tilde{f}^{abc}$ is fixed by the clockwise ordering of each vertex, as drawn.
• Figure 2: The box diagram is the only parent graph at one loop (up to permutations of the external legs).
• Figure 3: The planar and non-planar parent graphs contributing to the two-loop four-point amplitude. The prefactors $s_{12}$ are the integral numerators $N^{({\rm P})}$ and $N^{({\rm NP})}$, respectively.
• Figure 4: The nine parent graphs for the three-loop four-point ${\cal N}=4$ sYM amplitude. The prefactor of each diagram is the numerator polynomial $N^{(x)}$. The kinematic invariants are defined in eq. (\ref{['InvariantsDef']}).
• Figure 5: Planar contributions to the four-loop four-point amplitude.