Feynman Integrals and Scattering Amplitudes from Wilson Loops

Abstract

We study Feynman integrals and scattering amplitudes in $\mathcal{N}=4$ 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 ($n\ge 12$) 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.

arXiv comment8 pages, 4 figures, 1 ancillary file; v3: important updates, a compact form for the symbol of double pentagon integral added; typos corrected

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• Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations
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Cited by in corpus (15)

• Implications of the Landau Equations for Iterated Integrals
• The elliptic double box and symbology beyond polylogarithms
• Antipodal Self-Duality for a Four-Particle Form Factor
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Schubert Problems, Positivity and Symbol Letters
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• A study of Feynman integrals with uniform transcendental weights and the symbology from dual conformal symmetry
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• The Wilson-loop $d\log$ representation for Feynman integrals
• The Two-Loop Remainder Function for Eight and Nine Particles
• The Three-loop MHV Octagon from $\bar{Q}$ equations
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• 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