A recursion relation for gravity amplitudes

TL;DR

This paper extends the success of BCFW-like recursions from Yang–Mills to gravitons by formulating a gravity-specific recursion that accounts for gravity’s richer multi-particle pole structure. It derives the recursion, shows the MHV gravity amplitudes have favorable large-z behavior so boundary terms vanish, and obtains explicit 4-, 5-, and 6-point results that agree with known BGK/KLT formulas. Crucially, it proposes a new general closed-form for the n-point MHV gravity amplitude and validates it numerically up to n=11, highlighting the method’s potential to simplify tree-level gravity calculations. The authors also discuss the broader implications for recursion relations in other massless field theories, suggesting a general and practical framework beyond gauge theories.

Abstract

Britto, Cachazo and Feng have recently derived a recursion relation for tree-level scattering amplitudes in Yang-Mills. This relation has a bilinear structure inherited from factorisation on multi-particle poles of the scattering amplitudes - a rather generic feature of field theory. Motivated by this, we propose a new recursion relation for scattering amplitudes of gravitons at tree level. Using this recursion relation, we derive a new general formula for the MHV tree-level scattering amplitude for n gravitons. Finally, we comment on the existence of recursion relations in general field theories.

Figures (3)

• Figure 1: One of the terms contributing to the recursion relation for the MHV amplitude ${\cal M} (1^- , 2^- , 3^+ , \ldots , n^+)$. The gravity scattering amplitude on the right is symmetric under the exchange of gravitons of the same helicity. In the recursion relation, we sum over all possible values of $k$, i.e. $k=3, \ldots , n$. This amounts to summing over cyclical permutations of $(3, \ldots , n)$.
• Figure 2: This class of diagrams also contributes to the recursion relation for the MHV amplitude ${\cal M} (1^- , 2^- , 3^+ , \ldots , n^+)$; however, each of these diagrams vanishes if the shifts \ref{['shiftsonceagain']} are performed.
• Figure 3: One of the two diagrams contributing to the recursion relation for the MHV amplitude ${\cal M} (1^- , 2^- , 3^+ , 4^+)$. The other is obtained from this by cyclically permuting the labels $(3,4)$ -- i.e. swapping $3$ with $4$.