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.

arXiv comment5 pages + references and supplementary material, 2 figures, 4 ancillary files; v2: references added, identities now proven

References in corpus (69)

• Harmonic Polylogarithms
• Classical Polylogarithms for Amplitudes and Wilson Loops
• The Amplituhedron
• Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals
• The elliptic dilogarithm for the sunset graph
• Motivic Amplitudes and Cluster Coordinates
• Analytic treatment of the two loop equal mass sunrise graph
• Uniqueness of two-loop master contours
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• Superconformal symmetry and two-loop amplitudes in planar N=4 super Yang-Mills
• Heptagons from the Steinmann Cluster Bootstrap
• Elliptic polylogarithms and iterated integrals on elliptic curves I: general formalism
• Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral
• On the periods of some Feynman integrals
• Pulling the straps of polygons
• Six-Gluon Amplitudes in Planar $\mathcal{N}=4$ Super-Yang-Mills Theory at Six and Seven Loops
• The Elliptic Double-Box Integral: Massless Amplitudes Beyond Polylogarithms
• Elliptic Feynman integrals and pure functions
• The kite integral to all orders in terms of elliptic polylogarithms
• The $\epsilon$-form of the differential equations for Feynman integrals in the elliptic case
• Multiple Elliptic Polylogarithms
• A K3 in phi4
• Traintracks Through Calabi-Yaus: Amplitudes Beyond Elliptic Polylogarithms
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• The two-loop sunrise graph with arbitrary masses
• Cluster adjacency properties of scattering amplitudes
• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series
• On hyperlogarithms and Feynman integrals with divergences and many scales
• The planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularisation parameter
• Analytic results for the planar double box integral relevant to top-pair production with a closed top loop
• The unequal mass sunrise integral expressed through iterated integrals on ${\overline{\mathcal{M}}}_{1,3}$
• Elliptic polylogarithms and Feynman parameter integrals
• Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral
• The iterated structure of the all-order result for the two-loop sunrise integral
• An analytic solution for the equal-mass banana graph
• Simplifying differential equations for multi-scale Feynman integrals beyond multiple polylogarithms
• Cluster adjacency and the four-loop NMHV heptagon
• Cluster algebras for Feynman integrals
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• A Three-Point Form Factor Through Five Loops
• Cluster adjacency beyond MHV
• Mellin Amplitudes for Dual Conformal Integrals
• Numerical evaluation of iterated integrals related to elliptic Feynman integrals
• Yangian Bootstrap for Conformal Feynman Integrals
• On Genera of Curves from High-loop Generalized Unitarity Cuts
• How tropical are seven- and eight-particle amplitudes?
• Landau Singularities and Symbology: One- and Two-loop MHV Amplitudes in SYM Theory
• Lifting Heptagon Symbols to Functions
• Algorithms and tools for iterated Eisenstein integrals
• Star Integrals, Convolutions and Simplices
• Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations
• The electron self-energy in QED at two loops revisited
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Notes on cluster algebras and some all-loop Feynman integrals
• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Traintrack Calabi-Yaus from Twistor Geometry
• Functional relations for elliptic polylogarithms
• Conformal anomaly of generalized form factors and finite loop integrals
• Symbol Alphabets from Tensor Diagrams
• The Double Box and Hexagon Conformal Feynman Integrals
• Modular transformations of elliptic Feynman integrals
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Feynman Integrals and Scattering Amplitudes from Wilson Loops
• Prescriptive Unitarity with Elliptic Leading Singularities
• An Elliptic Yangian-Invariant, `Leading Singularity'
• Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory
• 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