An Infinite Family of Elliptic Ladder Integrals

Abstract

We identify two families of ten-point Feynman diagrams that generalize the elliptic double box, and show that they can be expressed in terms of the same class of elliptic multiple polylogarithms to all loop orders. Interestingly, one of these families can also be written as a dlog form. For both families of diagrams, we provide new 2l-fold integral representations that are linearly reducible in all but one variable and that make the above properties manifest. We illustrate the simplicity of this integral representation by directly integrating the three-loop representative of both families of diagrams. These families also satisfy a pair of second-order differential equations, making them ideal examples on which to develop bootstrap techniques involving elliptic symbol letters at high loop orders.

arXiv comment27 pages, 2 figures; v2: typo in ancillary file fixed, matches journal version

References in corpus (85)

• Harmonic Polylogarithms
• Gluon scattering amplitudes at strong coupling
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory
• Conformal properties of four-gluon planar amplitudes and Wilson loops
• Magic identities for conformal four-point integrals
• Numerical evaluation of multiple polylogarithms
• The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory
• Local Integrals for Planar Scattering Amplitudes
• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals
• From polygons and symbols to polylogarithmic functions
• Bootstrapping the three-loop hexagon
• Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
• Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
• The elliptic dilogarithm for the sunset graph
• Two-loop two-point functions with masses: asymptotic expansions and Taylor series, in any dimension
• Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
• Analytic treatment of the two loop equal mass sunrise graph
• Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
• Elliptic multiple zeta values and one-loop superstring amplitudes
• Bootstrapping an NMHV amplitude through three loops
• 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
• The massless higher-loop two-point function
• Heptagons from the Steinmann Cluster Bootstrap
• Jumpstarting the all-loop S-matrix of planar N=4 super Yang-Mills
• Hexagon functions and the three-loop remainder function
• Elliptic polylogarithms and iterated integrals on elliptic curves I: general formalism
• Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral
• The four-loop remainder function and multi-Regge behavior at NNLLA in planar N=4 super-Yang-Mills theory
• Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral
• Six-Gluon Amplitudes in Planar $\mathcal{N}=4$ Super-Yang-Mills Theory at Six and Seven Loops
• Cuts of Feynman Integrals in Baikov representation
• Analytic two-loop form factors in N=4 SYM
• The Elliptic Double-Box Integral: Massless Amplitudes Beyond Polylogarithms
• Elliptic Feynman integrals and pure functions
• Traintracks Through Calabi-Yaus: Amplitudes Beyond Elliptic Polylogarithms
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• Schouten identities for Feynman graph amplitudes; the Master Integrals for the two-loop massive sunrise graph
• Bootstrapping the QCD soft anomalous dimension
• Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series
• On hyperlogarithms and Feynman integrals with divergences and many scales
• The unequal mass sunrise integral expressed through iterated integrals on ${\overline{\mathcal{M}}}_{1,3}$
• Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
• Multiple polylogarithms with algebraic arguments and the two-loop EW-QCD Drell-Yan master integrals
• New differential equations for on-shell loop integrals
• Bootstrapping pentagon functions
• Analytic Structure of all Loop Banana Amplitudes
• An analytic solution for the equal-mass banana graph
• Cluster adjacency and the four-loop NMHV heptagon
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods
• The Double Pentaladder Integral to All Orders
• A Three-Point Form Factor Through Five Loops
• Rationalizing Loop Integration
• Mellin Amplitudes for Dual Conformal Integrals
• Taming Calabi-Yau Feynman integrals: The four-loop equal-mass banana integral
• Numerical evaluation of iterated integrals related to elliptic Feynman integrals
• Yangian-invariant fishnet integrals in 2 dimensions as volumes of Calabi-Yau varieties
• Multi-Loop Positivity of the Planar $\mathcal{N}=4$ SYM Six-Point Amplitude
• Bootstrapping a Stress-Tensor Form Factor through Eight Loops
• New Dual Conformally Invariant Off-Shell Integrals
• Algorithms and tools for iterated Eisenstein integrals
• Lifting Heptagon Symbols to Functions
• Star Integrals, Convolutions and Simplices
• Algorithms for minimal Picard-Fuchs operators of Feynman integrals
• The elliptic double box and symbology beyond polylogarithms
• All-Multiplicity Non-Planar MHV Amplitudes in sYM at Two Loops
• Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops
• Antipodal Self-Duality for a Four-Particle Form Factor
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Exponential suppression with four legs and an infinity of loops
• Traintrack Calabi-Yaus from Twistor Geometry
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Modular transformations of elliptic Feynman integrals
• Building Bases of Loop Integrands
• The Wilson-loop $d\log$ representation for Feynman integrals
• The Two-Loop Remainder Function for Eight and Nine Particles
• Cutting the traintracks: Cauchy, Schubert and Calabi-Yau
• The Three-loop MHV Octagon from $\bar{Q}$ equations
• Cuts and Isogenies
• Analytic results for two-loop planar master integrals for Bhabha scattering
• The Stratification of Rigidity

Cited by in corpus (6)

• Calabi-Yau Feynman integrals in gravity: $\epsilon$-factorized form for apparent singularities
• Classifying post-Minkowskian geometries for gravitational waves via loop-by-loop Baikov
• A double copy from twisted (co)homology at genus one
• A Calabi-Yau-to-Curve Correspondence for Feynman Integrals
• Les Houches 2023 -- Physics at TeV Colliders: Report on the Standard Model Precision Wishlist
• The spectrum of Feynman-integral geometries at two loops