Taming tree amplitudes in general relativity

Abstract

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. Svrček 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 results in the literature that were derived assuming their validity become true statements.

KLT and new relations for $ \mathcal{N} = 8 $ SUGRA and $ \mathcal{N} = 4 $ SYM

“…Based on the BCFW recursion relations, the third beautiful work along the line of S-matrix program is given by Benincasa and Cachazo [7]. In the paper, by assuming the applicability of BCFW recursion relations for four-particle amplitudes, they have easily re-derived many well-known (but difficult to prove) fundamental facts in S-matrix program, such as the non-Abelian structure for gauge theory and all matters couple to gravity with same coupling constant.…”

“…One important observation in [7] is that three point amplitude (on-shell) of any theory is uniquely fixed by the Lorentz invariance and the spin of external particles. For example, the three gluon amplitudes are given by…”

“…The idea of [7] has intrigued several important works in last few months. Staring form the antisymmetry of (1) plus the validity of BCFW recursion relations, four important properties of color-ordered gluon amplitudes have been proved in the frame of S-matrix program in [9].…”

Constructing amplitudes from their soft limits

“…The requirements will be discussed in later sections and will be related to the BCFW on-shell recursion formula, which we now discuss. The BCFW on-shell recursion relations are an efficient method for computing amplitudes at tree level in gauge theories [10,11] and gravity [30][31][32]. Two external particles, i and j, are singled out and their helicity spinors are deformed as:…”

“…A BCFW recursion exists in gauge theory and gravity for specific helicity choices of the deformed particles i and j [30][31][32][33]. In this paper, we will explicitly develop a relationship between BCFW and the soft limits of amplitudes.…”