Computing NMHV Gravity Amplitudes at Infinity

In this note we show how the solutions to the scattering equations in the NMHV sector fully decompose into subsectors in the $z\to \infty$ limit of a Risager deformation. Each subsector is characterized by the punctures that coalesce in the limit. This naturally decomposes the $E(n-3,1)$ solutions into sets characterized by partitions of $n-3$ elements so that exactly one subset has more than one element. We present analytic expressions for the leading order of the solutions in an expansion around infinite $z$ for any $n$. We also give a simple algorithm for numerically computing arbitrarily high orders in the same expansion. As a consequence, one has the ability to compute Yang-Mills and gravity amplitudes purely from this expansion around infinity. Moreover, we present a new analytic computation of the residue at infinity of the $n=12$ NMHV tree-level gravity amplitude which agrees with the results of Conde and Rajabi. In fact, we present the analytic form of the leading order in $1/z$ of the Cachazo-Skinner-Mason/CHY formula for graviton amplitudes for each subsector and to all multiplicity. As a byproduct of the all-order algorithm, one has access to the numerical value of the residue at infinity for any $n$ and hence to the corrected CSW (or MHV) expansion for NMHV gravity amplitudes.