Tree-Level Amplitudes in N=8 Supergravity

TL;DR

The paper builds a practical framework to derive explicit tree-level amplitudes in N=8 supergravity by combining a supersymmetric on-shell recursion with the squared dual conformal invariants from N=4 SYM. It expresses gravity amplitudes as sums over the same R_invariants used in SYM, but squared, and augmented by gravity-specific dressing factors G_α, enabling closed-form MHV, NMHV, and NNMHV formulas and a path toward higher N^pMHV amplitudes. The authors provide explicit constructions for MHV, NMHV, and NNMHV sectors, including detailed definitions of the auxiliary f, G^L, G^R, and Z/P structures, and outline proofs via on-shell shifts and factorization. This unified approach paves the way for efficient tree-level gravity calculations and informs potential extensions to loop-level unitarity methods in N=8 supergravity.

Abstract

We present an algorithm for writing down explicit formulas for all tree amplitudes in N=8 supergravity, obtained from solving the supersymmetric on-shell recursion relations. The formula is patterned after one recently obtained for all tree amplitudes in N=4 super Yang-Mills which involves nested sums of dual superconformal invariants. We find that all graviton amplitudes can be written in terms of exactly the same structure of nested sums with two modifications: the dual superconformal invariants are promoted from N=4 to N=8 superspace in the simplest manner possible--by squaring them--and certain additional non-dual conformal gravity dressing factors (independent of the superspace coordinates) are inserted into the nested sums. To illustrate the procedure we give explicit closed-form formulas for all NMHV, NNMHV and NNNMV gravity superamplitudes.

Figures (5)

• Figure 1: A diagrammatic representation of the relation (\ref{['main']}) between a physical gravity amplitude ${\mathcal{M}}_n$ and the sum over its ordered subamplitudes $M(1,\ldots,n)$. We draw an arrow indicating the cyclic order of the indices between the special legs $n$ and $1$.
• Figure 2: An alternative rooted tree diagram for tree-level SYM amplitudes. The figure is the same as the tree diagram presented in Drummond:2008cr except that the labels in the vertices appear in a different order, meaning that the $R$-invariants appearing in the amplitude are slightly different. Also the limits, written to the left and right of each line, are treated differently.
• Figure 3: The rule for going from line $p-1$ to line $p$ (for $p>1$) in Fig. \ref{['newYMtree']}. For every vertex in line $p-1$ of the form given at the top of the diagram, there are $r+2$ vertices in the lower line (line $p$). The labels in these vertices start with $u_{1}v_{1};\ldots u_{r}v_{r};a_{p-1}b_{p-1};a_{p}b_{p}$ and they get sequentially shorter, with each step to the right removing the pair of labels adjacent to the last pair $a_p,b_p$ until only the last pair is left. The summation limits between each line are also derived from the labels of the vertex above. The left superscripts which appear on the associated $R$-invariants start with $u_1v_1\ldots u_rv_r b_{p-1} a_{p-1}$ for the left-most vertex. The next vertex to the right has the superscript $u_1v_1 \ldots u_r v_r a_{p-1} b_{p-1}$, i.e. the same as the first but with the final pair in alphabetical order. The next vertex has the superscript $u_1v_1 \dots u_rv_r$ and thereafter the pairs are sequentially deleted from the right.
• Figure 4: The recursion for MHV amplitudes.
• Figure 5: The two kinds of diagrams contributing to the recursion of NMHV amplitudes.