We derive an all-loop recursion relation for the rational integrands of scattering amplitudes in planar N=4 super Yang-Mills theory. The recursion relation is generated by an all-line shift of momentum twistors. We solve the recursion relation in detail for 1-loop and 2-loop integrands of MHV and NMHV amplitudes, and show that the general solution of the recursion relation is the MHV diagram formalism, thus proving its validity for all rational loop integrands.
Bullimore, M. (2011). MHV diagrams from an all-loop recursion relation. Journal of High Energy Physics, 2011(08), Article 107. https://doi.org/10.1007/jhep08%282011%29107