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Integrands for colour ordered scattering amplitudes in planar N=4 SYM aredual to those of correlation functions of the energy-momentum multiplet of thetheory. The construction can relate amplitudes with different numbers of legs.  By graph theory methods the integrand of the four-point function ofenergy-momentum multiplets has been constructed up to six loops in previouswork. In this article we extend this analysis to seven loops and use it toconstruct the full integrand of the five-point amplitude up to five loops, andin the parity even sector to six loops.  All results, both parity even and parity odd, are obtained in a concise localform in dual momentum space and can be displayed efficiently through graphs. Wehave verified agreement with other local formulae both in terms ofsupertwistors and scalar momentum integrals as well as BCJ forms where thoseexist in the literature, i.e. up to three loops.  Finally we note that the four-point correlation function can be extracteddirectly from the four-point amplitude and so this uncovers a direct link fromfour- to five-point amplitudes.

Local integrands for the five-point amplitude in planar N=4 SYM up to five loops