Tree-level amplitudes in $\mathcal{N}=8$ supergravity

We present an algorithm for writing down explicit formulas for all tree amplitudes in $\mathcal{N}=8$ supergravity, obtained from solving the supersymmetric on-shell recursion relations. The formula is patterned after one recently obtained for all tree amplitudes in $\mathcal{N}=4$ super Yang-Mills theory 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 $\mathcal{N}=4$ to $\mathcal{N}=8$ superspace in the simplest manner possible—by squaring them—and certain additional nondual 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.

Figure 1

A diagrammatic representation of the relation (2.6) between a physical gravity amplitude ${\mathcal{M}}_n$ and the sum over its ordered subamplitudes $M(1,\dots ,n)$. 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 36 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. 2. 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 $u_1{v}_1;\dots u_r{v}_r;a_{p-1}{b}_{p-1};a_p{b}_p$ 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 $u_1{v}_1\dots u_r{v}_r{b}_{p-1}{a}_{p-1}$ for the left-most vertex. The next vertex to the right has the superscript $u_1{v}_1\dots u_r{v}_r{a}_{p-1}{b}_{p-1}$, i.e. the same as the first but with the final pair in alphabetical order. The next vertex has the superscript $u_1{v}_1\dots u_r{v}_r$ 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.