Gravity MHV amplitudes via Berends-Giele currents

A bstract Berends and Giele derived the Parke-Taylor formula for Yang-Mills MHV amplitudes by computing Berends-Giele currents involving gluons of all-plus and all-but-one-plus helicities. Remarkably, the all-plus current already encodes much of the Parke-Taylor formula structure. The all-but-one-plus current satisfies a more intricate recursion relation than the all-plus case, but one that can still be solved explicitly. This current turns out to be proportional to the all-plus current, which explains why the essential features of the MHV formula are already present at the all-plus level. In this paper, we carry out an analogous program for gravity. The all-plus graviton Berends-Giele current satisfies a recursion relation that is more involved than in the Yang-Mills case, but whose explicit solution is known: a sum over spanning trees of the complete graph on n vertices. We derive and solve the recursion relation for the all-but-one-plus graviton current. The solution is again given by a sum over spanning trees, where each tree contributes a term proportional to the corresponding all-plus current, multiplied by a factor given by a sum over subtrees. Only a small subset of these terms contributes to the MHV amplitude, which we recover explicitly. This provides a direct derivation of the gravity MHV formula from the gravitational Feynman rules — achieving what Berends, Giele, and Kuijf in their 1987 paper regarded as “hard to obtain directly from quantum gravity”.