Taming Tree Amplitudes In General Relativity

TL;DR

This paper provides a rigorous proof of BCFW recursion relations for all tree-level graviton amplitudes in General Relativity, confirming their validity and enabling many prior results to be established rigorously.

Contribution

It extends the proof of BCFW recursion relations to all tree-level graviton amplitudes in General Relativity using Feynman diagram analysis.

Findings

BCFW recursion relations are valid for all tree-level graviton amplitudes.

The proof confirms the vanishing of amplitudes at infinity under complex deformations.

Many previous results assuming BCFW validity are now rigorously justified.

Abstract

We give a proof of BCFW recursion relations for all tree-level amplitudes of gravitons in General Relativity. The proof follows the same basic steps as in the BCFW construction and it is an extension of the one given for next-to-MHV amplitudes by one of the authors and P. Svr\v{c}ek in hep-th/0502160. The main obstacle to overcome is to prove that deformed graviton amplitudes vanish as the complex variable parameterizing the deformation is taken to infinity. This step is done by first proving an auxiliary recursion relation where the vanishing at infinity follows directly from a Feynman diagram analysis. The auxiliary recursion relation gives rise to a representation of gravity amplitudes where the vanishing under the BCFW deformation can be directly proven. Since all our steps are based only on Feynman diagrams, our proof completely establishes the validity of BCFW recursion relations. This means that many results in the literature that were derived assuming their validity become true statements.