The Wilson-loop $d \log$ representation for Feynman integrals

Abstract

We introduce and study the Wilson-loop ${\rm d}\log$ representation of certain Feynman integrals for scattering amplitudes in ${\cal N}=4$ SYM and beyond, which makes their evaluation completely straightforward. Such a representation was motivated by the dual Wilson loop picture, and it can also be derived by partial Feynman parametrization of loop integrals. We first introduce it for the simplest one-loop examples, the chiral pentagon in four dimensions and the three-mass-easy hexagon in six dimensions, which are represented by two- and three-fold ${\rm d}\log$ integrals that are nicely related to each other. For multi-loop examples, we write the $L$-loop generalized penta-ladders as $2(L{-}1)$-fold ${\rm d}\log$ integrals of some one-loop integral, so that once the latter is known, the integration can be performed in a systematic way. In particular, we write the eight-point penta-ladder as a $2L$-fold ${\rm d}\log$ integral whose symbol can be computed without performing the integration; we also obtain the last entries and the symbol alphabet of these integrals. Similarly we compute and study the symbol of the seven-point double-penta-ladder, which is represented by a $2(L{-}1)$-fold integral of a hexagon; the latter can be written as a two-fold ${\rm d}\log$ integral plus a boundary term. We comment on the relation of our representation to differential equations and resum the ladders by solving certain integral equations.