New recursion relations for tree amplitudes of gluons

Abstract

We present new recursion relations for tree amplitudes in gauge theory that give very compact formulas. Our relations give any tree amplitude as a sum over terms constructed from products of two amplitudes of fewer particles multiplied by a Feynman propagator.The two amplitudes in each term are physical, in the sense that all particles are on-shell and momentum conservation is preserved. This is striking, since it is just like adding certain factorization limits of the original amplitude to build up the full answer. As examples, we recompute all known tree-level amplitudes of up to seven gluons and show that our recursion relations naturally give their most compact forms. We give a new result for an eight-gluon amplitude, A(1 + , 2 − , 3 + , 4 − , 5 + , 6 − , 7 + , 8 − ). We show how to build any amplitude in terms of three-gluon amplitudes only.

Taming tree amplitudes in general relativity

“…More recently, a new set of recursion relations for tree level amplitudes of gluons was introduced by Britto, Feng and the third author [7]. These recursion relations were inspired by [8,9] and reproduced very compact results obtained in [10] by studying the IR behavior of N = 4 one-loop amplitudes.…”

“…One in which an amplitude is given as a sum of terms containing the product of two physical on-shell amplitudes where the momenta of only two gravitons have been complexified. These recursion relations, originally discovered in [7] in gauge theory, were proven using the power of complex analysis in [11]. The BCFW construction opened up the possibility for using complex analysis in many other situations.…”

MHV, CSW and BCFW: field theory structures in string theory amplitudes

“…In particular, the supersymmetric Ward identities determine in these four dimensions [19] A(+ + · · · +) = 0 , 12) with an exception only for the three particle A(+ + −) and A(− − +) amplitude which vanish generically only for all momenta in R (1,3) . The input in this argument is nothing more than simple representation theory of on-shell N = 1 space-time supersymmetry and the absence of helicity-violating fermion amplitudes.…”

MHV-vertices for gravity amplitudes

“…In an alternate approach to computing tree level amplitudes, Britto, Cachazo, Feng and Witten [17] obtained a recursion relation based on analytically shifting a pair of external legs, λ i −→ λ i + zλ j ,λ j −→λ j − zλ i , (1.4) and determining the physical amplitude, A n (0), from the poles in the shifted amplitude, A n (z). This leads to a recursion relation in the number of external legs, n, of the form, 5) where the factorisation is only on these poles, z α , where legs i and j are connected to different sub-amplitudes.…”