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

Abstract

We introduce and study the Wilson-loop $\mathrm{d}\log$ representation of certain Feynman integrals for scattering amplitudes in $\mathcal{N}=4$ 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 $\mathrm{d}\log$ 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 $\mathrm{d}\log$ 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 $\mathrm{d}\log$ 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 $\mathrm{d}\log$ 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.

arXiv comment34 pages, many figures. v3: alphabet of double-pentagon corrected

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• Notes on cluster algebras and some all-loop Feynman integrals
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Schubert Problems, Positivity and Symbol Letters
• A study of Feynman integrals with uniform transcendental weights and the symbology from dual conformal symmetry
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Feynman Integrals and Scattering Amplitudes from Wilson Loops
• The Two-Loop Remainder Function for Eight and Nine Particles
• An Infinite Family of Elliptic Ladder Integrals
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• Emergent unitarity, all-loop cuts and integrations from the ABJM amplituhedron
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• Bootstrapping octagons in reduced kinematics from $A_2$ cluster algebras
• A nice two-loop next-to-next-to-MHV amplitude in $\mathcal{N}=4$ super-Yang-Mills