The Twelve-Graviton Next-to-MHV Amplitude from Risager's Construction

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Abstract
The MHV or CSW expansion of tree-level Yang-Mills amplitudes provides an elegant and simple way of obtaining analytic formulas for S-matrix elements. Inspired by the BCFW technique, a systematic approach to obtain the MHV expansion was introduced by Risager, using a particular complex deformation. Although it works for Yang-Mills amplitudes, Risager's technique fails to provide an MHV expansion already for Next-to-MHV gravity amplitudes with more than eleven particles, as shown by Bianchi, Elvang and Freedman in 2008. This fact implies that in this sector there is a contribution at infinity starting at n = 12, which we call the "Anomaly". In this note we determine the explicit analytic form of the anomaly for n = 12. Together with the terms of the Risager MHV expansion, the anomaly completes the first full CSW-like analytic expression of the twelve-graviton NMHV amplitude. Our technique can also be used to compute the anomaly for higher points.

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