Feynman Integrals and Scattering Amplitudes from Wilson Loops

We study Feynman integrals and scattering amplitudes in ${\cal N}=4$ super-Yang-Mills by exploiting the duality with null polygonal Wilson loops. Certain Feynman integrals, including one-loop and two-loop chiral pentagons, are given by Feynman diagrams of a supersymmetric Wilson loop, where one can perform loop integrations and be left with simple integrals along edges. As the main application, we compute analytically for the first time, the symbol of the generic ($n\geq 12$) double pentagon, which gives two-loop MHV amplitudes and components of NMHV amplitudes to all multiplicities. We represent the double pentagon as a two-fold $\mathrm{d} \log$ integral of a one-loop hexagon, and the non-trivial part of the integration lies at rationalizing square roots contained in the latter. We obtain a remarkably compact "algebraic words" which contain $6$ algebraic letters for each of the $16$ square roots, and they all nicely cancel in combinations for MHV amplitudes and NMHV components which are free of square roots. In addition to $96$ algebraic letters, the alphabet consists of $152$ dual conformal invariant combinations of rational letters.