We compute the symbol of the first two-loop amplitudes in planar $\mathcal{N}=4$ SYM with algebraic letters, the eight-point Next to Maximally Helicity Violating (NMHV) amplitude (or the dual octagon Wilson loops). We show how to apply $\overline{Q}$ equations of [S. Caron-Huot and S. He, J. High Energy Phys. 07 (2012) 174] for computing the differential of two-loop $n$-point NMHV amplitudes and present the result for $n=8$ explicitly. The symbol alphabet for octagon consists of 180 independent rational letters and 18 algebraic ones involving Gram-determinant square roots. We comment on all-loop predictions for final entries and aspects of the result valid for all multiplicities.

• Received 5 December 2019
• Accepted 19 February 2020

DOI:https://doi.org/10.1103/PhysRevD.101.061701

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Published by the American Physical Society