Generic multiloop methods and application to N=4 super-Yang-Mills

TL;DR

The review addresses the problem of efficiently constructing multiloop, multileg amplitudes in ${\cal N}=4$ sYM across dimensions, including non-planar contributions. It presents a cohesive framework based on cubic-graph representations, generalized unitarity, and the maximal-cut method, complemented by D-dimensional recursive cuts and the BCJ color–kinematics duality. A central contribution is showing how a small set of master graphs and kinematic Jacobi relations can fix large classes of numerators, with gravity amplitudes arising from double-copy constructions. The discussion includes concrete two-loop and three-loop examples, emphasizes dimensional regularization, and highlights the broader significance for less-supersymmetric theories and gravity beyond gauge theory.

Abstract

We review some recent additions to the tool-chest of techniques for finding compact integrand representations of multiloop gauge-theory amplitudes - including non-planar contributions - applicable for N=4 super-Yang-Mills in four and higher dimensions, as well as for theories with less supersymmetry. We discuss a general organization of amplitudes in terms of purely cubic graphs, review the method of maximal cuts, as well as some special D-dimensional recursive cuts, and conclude by describing the efficient organization of amplitudes resulting from the conjectured duality between color and kinematic structures on constituent graphs.

Figures (18)

• Figure 1: In four dimensions, the kinematics of the on-shell three-point vertex comes in two chiral classes: the $(+)$ vertex is supported on $\lambda$s (thus all $\widetilde{\lambda}$s are parallel and their $\left[i\,j\right]$ vanish), and the $(-)$ vertex is supported on $\widetilde{\lambda}$s (thus all $\lambda$s are parallel and their $\left\langle i\,j\right\rangle$ vanish).
• Figure 2: In a $D=4$ cut, one can encounter two three-vertices of the same chirality joined by a common line ( e.g. see cuts in figure \ref{['PentaBoxFigure']}(b) and \ref{['non-planarFigure']}). For the purpose of solving the kinematics, one can treat this configuration as an effective four-point vertex of definite chirality (note that a corresponding amplitude is not defined). All intermediate channels of this effective kinematic vertex are on-shell, as all legs are proportional to the same spinor.
• Figure 3: A quadruple cut of a four-point amplitude in $D=4$, and a penta-cut of a five-point amplitude in $D>4$. In $D=4$, we use $(+)$ and $(-)$ labels on three-point vertices to specify the chirality, but in $D>4$ the chirality is not a well-defined notion.
• Figure 4: A hepta-cut of a four-point amplitude, and an octa-cut of a five-point amplitude, both in $D=4$.
• Figure 5: A non-planar hepta-cut of a two-loop four-point amplitude, and a non-planar deca-cut of a three-loop amplitude, both in $D=4$.