Abstract
In the gauge-theoretic formulation of gravity the cubic vertex becomes simple enough for some graviton scattering amplitudes to be computed using Berends-Giele-type recursion relations. We present such a computation for the current with all same helicity on-shell gravitons. Once the recursion relation is set up and low graviton number cases are worked out, a natural guess for the solution in terms of a sum over trees presents itself readily. The solution can also be described either in terms of the half-soft function familiar from the 1998 paper by Bern, Dixon, Perelstein and Rozowsky or as a matrix determinant similar to one used by Hodges for MHV graviton amplitudes. This solution also immediate suggests the correct guess for the MHV graviton amplitude formula, as is contained in the already mentioned 1998 paper. We also obtain the recursion relation for the off-shell current with all but one same helicity gravitons.
Highlights
Of a matrix determinant was found in [9]
The computation of the all same helicity on-shell legs current presented in this paper can be viewed as the simplest available derivation of the MHV amplitude formula using Feynman diagrams
It is very hard to see that the solution to BCFW recursion organises itself into a sum over trees, while this directly follows from the Berends-Giele Feynman diagram derived recursion
Summary
We take the gauge-theoretic Feynman rules from [4], but describe them here in a language slightly more formal, and convenient for computations. The trace free part of the metric perturbation hμν, once the space-time indices are translated into the spinor ones, lives precisely in S+2 ⊗ S−2 Note that this space is parity-symmetric, and contains (in the case of Minkowski signature) real elements. The reference spinors pA′, pA are the same as used in the corresponding positive helicity spinor (2.2) It is this rule, together with the factors of M appearing in all formulas, that remember about the fact that this perturbation theory was originally set up in de Sitter space of cosmological constant Λ/3 = M 2. Note that to get the last relation we have used (2.8)
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