Constructing Gravity Amplitudes from Real Soft and Collinear Factorisation

TL;DR

The paper develops soft-lifting functions, generalising half-soft functions, to construct gravity amplitudes from real soft and collinear factorisation. It shows how to build $n$-point MHV tree amplitudes and the rational terms of one-loop $ ext{N}=4$ supergravity using soft-lifting seeds (3- or 4-point) and a one-loop link-diagram interpretation. It provides analytic, diagrammatic proofs of the correct soft and various collinear limits, including $a^+b^+$ and mixed helicity collinear cases, demonstrating the robustness of the construction. The work suggests a unifying framework for gravity amplitudes via soft and collinear structures and opens avenues for extending to higher-MHV amplitudes and linking to twistor-space formulations.

Abstract

Soft and collinear factorisations can be used to construct expressions for amplitudes in theories of gravity. We generalise the "half-soft" functions used previously to "soft-lifting" functions and use these to generate tree and one-loop amplitudes. In particular we construct expressions for MHV tree amplitudes and the rational terms in one-loop amplitudes in the specific context of N=4 supergravity. To completely determine the rational terms collinear factorisation must also be used. The rational terms for N=4 have a remarkable diagrammatic interpretation as arising from algebraic link diagrams.

Figures (9)

• Figure 1: The topologies of the link diagrams for the seven point MHV amplitude. The vertices are labelled by the five positive helicity legs.
• Figure 2: Diagrammatic representation of the soft-lifting functions. The particular figure contributes to $\hat{S}^8[P^4;Q^8]$
• Figure 3: The box and bubble functions appearing in the $\mathcal{N} = 4$ MHV one-loop amplitude
• Figure 4: The loop part of a link diagram producing a term in $R_n^{r}$. The $r$ positive helicity legs of $P^r$ lie in the loop.
• Figure 5: A link diagram corresponding to a term in $C_{n-10} \times \hat{S}^{8}[P^{n-10};Q^8]$