Five-Point Amplitudes in N=4 Super-Yang-Mills Theory and N=8 Supergravity

TL;DR

This work constructs and verifies explicit duality-satisfying five-point integrands for ${\cal N}=4$ sYM and ${\cal N}=8$ supergravity at one and two loops, valid for general D-dimensional internal kinematics and any gauge group. By organizing the external-state dependence with kinematic prefactors $\beta_{abcde}$ and $\gamma_{ij}$ and employing a master-graph approach, the authors obtain compact numerators that satisfy color-kinematics Jacobi relations and enable gravity amplitudes via the BCJ double-copy construction. They systematically fix all free parameters using unitarity cuts, presenting complete one-loop and two-loop five-point MHV amplitudes and computing their ultraviolet divergences in the critical dimensions $D=8$ and $D=7$, respectively. The results provide strong evidence for the color-kinematics duality at multipoint and multiloop levels, demonstrate the double-copy structure of ${\cal N}=8$ supergravity, and establish a framework for extending these constructions to higher points and loops, with implications for understanding the UV behavior of maximal supergravity.

Abstract

We present the complete integrands of five-point superamplitudes in N=4 super-Yang-Mills theory and N=8 supergravity, at one and two loops, for four-dimensional external states and D-dimensional internal kinematics. For N=4 super-Yang-Mills theory we give the amplitudes for general gauge group -- including all non-planar contributions. The results are constructed using integral diagrams that manifestly satisfy the conjectured duality between color and kinematics, providing additional nontrivial evidence in favor of the duality for multipoint and multiloop amplitudes. We determine the ultraviolet poles by integrating the amplitudes in the dimensions where logarithmic divergences first occur. We introduce new kinematic prefactors which offer a convenient decomposition of the external state structure of the nonplanar five-point amplitudes in the maximally supersymmetric theories to all loop orders.

Figures (5)

• Figure 1: The two diagrams that appear in the five-point one-loop amplitudes.
• Figure 2: The six diagrams that appear in the five-point two-loop amplitudes.
• Figure 3: A boxcut for the two-loop pentabox graph excises the box (dropping the propagators) and writes the result in terms of two one-loop diagrams times a kinematic factor.
• Figure 4: The four-dimensional cuts used to verify the duality-satisfying form of the two-loop five-point ${{\cal N}=4}$ sYM amplitude. All cyclically distinct leg orderings on each blob are counted. Cuts (a) and (d) are also evaluated through a $D$-dimensional boxcut.
• Figure 5: The two vacuum integrals that capture the $D=7$ ultraviolet divergence for planar and nonplanar diagrams, respectively.