Generating Tree Amplitudes in N=4 SYM and N=8 SG

TL;DR

The paper develops generating-function formalisms for n-point MHV and NMHV amplitudes in both N=4 SYM and N=8 supergravity, encoding external-state dependence via Grassmann variables and ensuring SUSY Ward identities and SU(8) symmetry are respected through a precise operator map. It shows how MHV generating functions yield efficient intermediate-state helicity sums and how NMHV amplitudes can be built from MHV data using MHV-vertex expansions anchored by 3-line shifts, while carefully analyzing large-z behavior to identify regimes where the approach is valid. The work demonstrates factorization properties linking gravity to gauge theory through KLT-like relations, tests the framework on explicit amplitudes, and discusses both the successes (many good shifts, clear NMHV structure) and the limitations (bad/very-bad amplitudes in N=8 SG, breakdown for n≥12) of the generating-function method. Overall, it provides a compact, symmetry-respecting bookkeeping tool for tree amplitudes with potential practical impact on loop calculations via unitarity, while highlighting deeper questions about gravity’s high-point behavior and hidden symmetries.

Abstract

We study n-point tree amplitudes of N=4 super Yang-Mills theory and N=8 supergravity for general configurations of external particles of the two theories. We construct generating functions for n-point MHV and NMHV amplitudes with general external states. Amplitudes derived from them obey SUSY Ward identities, and the generating functions characterize and count amplitudes in the MHV and NMHV sectors. The MHV generating function provides an efficient way to perform the intermediate state helicity sums required to obtain loop amplitudes from trees. The NMHV generating functions rely on the MHV-vertex expansion obtained from recursion relations associated with a 3-line shift of external momenta involving a reference spinor |X]. The recursion relations remain valid for a subset of N=8 supergravity amplitudes which do not vanish asymptotically for all |X]. The MHV-vertex expansion of the n-graviton NMHV amplitude for n=5,6,...,11 is independent of |X] and exhibits the asymptotic behavior z^{n-12}. This presages difficulties for n > 12. Generating functions show how the symmetries of supergravity can be implemented in the quadratic map between supergravity and gauge theory embodied in the KLT and other similar relations between amplitudes in the two theories.

Figures (6)

• Figure 1: Intermediate spin sums.
• Figure 2: Diagrams from the 2-line shift recursion relations (\ref{['recrel1']}) for MHV amplitudes in both gauge theory and gravity. The 3-vertex in the right hand diagram vanishes as a consequence of kinematics.
• Figure 3: 2-line shift recursion relations for NMHV amplitudes contain non-vanishing diagrams with NMHV vertices. It is an appealing feature of 3-line shifts that such diagrams vanish due to special kinematics and the recursion sum consequently contains MHV subdiagrams only.
• Figure 4: Generic MHV-vertex diagram from the 3-line shift recursion relations for NMHV amplitudes in both gauge theory and gravity.
• Figure 5: The six MHV-vertex diagrams needed for the 3-line recursion relation for the gluon NMHV amplitude $A_6(1^-,2^-,3^-,4^+,5^+,6^+)$. Amplitudes for other external particles of ${\cal N} =4$ theory are obtained by multiplying each gluon diagram by the appropriate spin factor.