The elliptic double box and symbology beyond polylogarithms

Abstract

We study the elliptic double-box integral, which contributes to generic massless QFTs and is the only contribution to a particular 10-point scattering amplitude in N=4 SYM theory. Based on a Feynman parametrization, we express this integral in terms of elliptic polylogarithms. We then study its symbol, finding a rich structure and remarkable similarity with the non-elliptic case. In particular, the first entry of the symbol is expressible in terms of logarithms of dual-conformal cross-ratios, and elliptic letters only occur in the last two entries. Moreover, the symbol makes manifest a differential equation relating the double-box integral to a 6D hexagon integral, suggesting that it can be bootstrapped based on the latter integral alone.

5 pages + references and supplementary material, 2 figures, 4 ancillary files; v2: references added, identities now proven

References in corpus (14)

• Classical Polylogarithms for Amplitudes and Wilson Loops
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• The kite integral to all orders in terms of elliptic polylogarithms
• Elliptic polylogarithms and Feynman parameter integrals
• Cluster algebras for Feynman integrals
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Notes on cluster algebras and some all-loop Feynman integrals
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Traintrack Calabi-Yaus from Twistor Geometry
• Symbol Alphabets from Tensor Diagrams
• Modular transformations of elliptic Feynman integrals
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Prescriptive Unitarity with Elliptic Leading Singularities
• Double Box Motive

Cited by in corpus (14)

• Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
• Taming Calabi-Yau Feynman integrals: The four-loop equal-mass banana integral
• Yangian-invariant fishnet integrals in 2 dimensions as volumes of Calabi-Yau varieties
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• Bananas of equal mass: any loop, any order in the dimensional regularisation parameter
• Loop-by-loop Differential Equations for Dual (Elliptic) Feynman Integrals
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Schubert Problems, Positivity and Symbol Letters
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• An Infinite Family of Elliptic Ladder Integrals
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• A nice two-loop next-to-next-to-MHV amplitude in $\mathcal{N}=4$ super-Yang-Mills
• A Feynman integral depending on two elliptic curves