Graphs, determinants and gravity amplitudes

TL;DR

The paper develops a unified framework linking graph-theoretic and determinant expressions in gravity amplitudes via the matrix-tree theorem. It shows that Hodges' determinant for tree-level MHV gravity amplitudes and the NSVW tree-diagram expansion are two faces of the same underlying graph-determinant structure, and it extends this viewpoint to half-soft and soft-lifting functions, proving key recursion and square identities. The approach also yields a determinant-based formulation for the one-loop rational part of $ ext{N}=4$ supergravity, introducing loop-counting matrices and expanded determinant terms that reproduce known results. Overall, the work clarifies the structural simplicity of gravity amplitudes and opens avenues for higher-loop and non-MHV generalizations, potentially connecting to twistor-string/Grassmannian perspectives.

Abstract

We apply the matrix-tree theorem to establish a link between various diagrammatic and determinant expressions, which naturally appear in scattering amplitudes of gravity theories. Using this link we are able to give a general graph-theoretical formulation for the tree-level maximally-helicity-violated gravity amplitudes. Furthermore, we use the link to prove two identities for half-soft functions of gravity amplitudes. Finally we recast the diagrammatic formulation of one-loop rational part of $\mathcal{N}=4$ supergravity into a matrix form.

Figures (3)

• Figure 1: There are 3 labeled spanning trees of the complete graph with 3 vertices, $K_3$ (top). An example of labeled forests with 10 vertices, which has 4 trees; if one chooses e.g. vertices 1,5,7,10 to be the roots, it becomes a rooted forest (bottom).
• Figure 2: Diagrammatic expansions of Hodges' determinant formula for gravity MHV tree amplitudes. A special choice (reference points $1,2$) gives the NSVW formula as the sum of weighted spanning tree (left). The most general diagrammatic expansion of the formula is the sum of weighted forests with 3 trees, which contain $\{i,r\},\{j,s\},\{k,t\}$ (or $S_3$ permutations) respectively (right).
• Figure 3: Diagrammatic formula for the rational part of one-loop MHV amplitudes. A general connected, one-loop diagram (left) is a loop with $r$ nodes, attached with forests, which have $r$ trees containing the $r$ nodes as their roots. The 7-pt case (right): we pick two reference labels, and sum over all possible assignments of the remaining $n{-}2$ ones to the vertices.