Elliptic Feynman integrals and pure functions

Abstract

We propose a variant of elliptic multiple polylogarithms that have at most logarithmic singularities in all variables and satisfy a differential equation without homogeneous term. We investigate several non-trivial elliptic two-loop Feynman integrals with up to three external legs and express them in terms of our functions. We observe that in all cases they evaluate to pure combinations of elliptic multiple polylogarithms of uniform weight. This is the first time that a notion of uniform weight is observed in the context of Feynman integrals that evaluate to elliptic polylogarithms.

arXiv comment47 pages

References in corpus (123)

• One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits
• Harmonic Polylogarithms
• Fusing Gauge Theory Tree Amplitudes Into Loop Amplitudes
• Multiloop integrals in dimensional regularization made simple
• Differential Equations for Two-Loop Four-Point Functions
• Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond
• Transcendentality and Crossing
• Differential Equations for Feynman Graph Amplitudes
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Numerical evaluation of multiple polylogarithms
• Three-loop universal anomalous dimension of the Wilson operators in N=4 SUSY Yang-Mills model
• Local Integrals for Planar Scattering Amplitudes
• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory
• Multiple polylogarithms and mixed Tate motives
• Lectures on differential equations for Feynman integrals
• From polygons and symbols to polylogarithmic functions
• Two-Loop Four-Gluon Amplitudes in N=4 Super-Yang-Mills
• Bootstrapping the three-loop hexagon
• Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
• The elliptic dilogarithm for the sunset graph
• Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
• Analytic treatment of the two loop equal mass sunrise graph
• Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
• Higgs Boson Production at Hadron Colliders at N3LO in QCD
• The Two-Loop Hexagon Wilson Loop in N = 4 SYM
• Reducing differential equations for multiloop master integrals
• 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
• Two-Loop Quark and Gluon Form Factors in Dimensional Regularisation
• Bootstrapping an NMHV amplitude through three loops
• Soft triple-real radiation for Higgs production at N3LO
• Uniqueness of two-loop master contours
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• Three-loop corrections to the soft anomalous dimension in multi-leg scattering
• A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon
• Analytical and numerical methods for massive two-loop self-energy diagrams
• Heptagons from the Steinmann Cluster Bootstrap
• Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
• Hexagon functions and the three-loop remainder function
• Dressing and Wrapping
• An Analytic Result for the Two-Loop Hexagon Wilson Loop in N = 4 SYM
• Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD
• 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
• All Non-Maximally-Helicity-Violating One-Loop Seven-Gluon Amplitudes in N=4 Super-Yang-Mills Theory
• The four-loop remainder function and multi-Regge behavior at NNLLA in planar N=4 super-Yang-Mills theory
• Two-loop planar master integrals for Higgs$\to 3$ partons with full heavy-quark mass dependence
• Three-Loop Four-Point Correlator in N=4 SYM
• Pentagon functions for massless planar scattering amplitudes
• Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form
• Anomalous dimensions in N=4 SYM theory at order g^4
• Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral
• A Feynman integral via higher normal functions
• Leading singularities and off-shell conformal integrals
• Single-valued harmonic polylogarithms and the multi-Regge limit
• Analytic two-loop form factors in N=4 SYM
• Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph
• The Elliptic Double-Box Integral: Massless Amplitudes Beyond Polylogarithms
• The kite integral to all orders in terms of elliptic polylogarithms
• Iterative structure of finite loop integrals
• The $\epsilon$-form of the differential equations for Feynman integrals in the elliptic case
• A K3 in phi4
• Traintracks Through Calabi-Yaus: Amplitudes Beyond Elliptic Polylogarithms
• Four-point functions in N=4 supersymmetric Yang-Mills theory at two loops
• The two-loop sunrise graph with arbitrary masses
• Schouten identities for Feynman graph amplitudes; the Master Integrals for the two-loop massive sunrise graph
• Simple one-dimensional integral representations for two-loop self-energies: the master diagram
• Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams
• All Next-to-Maximally-Helicity-Violating One-Loop Gluon Amplitudes in N=4 Super-Yang-Mills Theory
• The three-loop cusp anomalous dimension in QCD
• Evaluating single-scale and/or non-planar diagrams by differential equations
• Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series
• The three-loop cusp anomalous dimension in QCD and its supersymmetric extensions
• Sharpening The Leading Singularity
• An Elliptic Generalization of Multiple Polylogarithms
• Inclusive Heavy Flavor Hadroproduction in NLO QCD: the Exact Analytic Result
• Soft-virtual corrections to Higgs production at N${}^3$LO
• The Two-Loop Master Integrals for $q\bar{q}\to VV$
• Transforming differential equations of multi-loop Feynman integrals into canonical form
• A non-planar two-loop three-point function beyond multiple polylogarithms
• 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
• Four-loop photon quark form factor and cusp anomalous dimension in the large-$N_c$ limit of QCD
• N${}^3$LO Higgs and Drell-Yan production at threshold: the one-loop two-emission contribution
• Multi-Regge kinematics and the moduli space of Riemann spheres with marked points
• The three-loop form factor in N=4 super Yang-Mills
• Iterated Binomial Sums and their Associated Iterated Integrals
• Four-gluon scattering at three loops, infrared structure and Regge limit
• The iterated structure of the all-order result for the two-loop sunrise integral
• Master integrals for the NNLO virtual corrections to $\mu e$ scattering in QED: the non-planar graphs
• A note on four-point functions of conformal operators in N=4 Super-Yang Mills
• Analytic result for the nonplanar hexa-box integrals
• Two-loop sunset diagrams with three massive lines
• Master integrals for the NNLO virtual corrections to $\mu e$ scattering in QED: the planar graphs
• Analytic results for two-loop master integrals for Bhabha scattering I
• The last of the simple remainders
• Multiple Modular Values and the relative completion of the fundamental group of $M_{1,1}$
• Relations between elliptic multiple zeta values and a special derivation algebra
• DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory
• The Two Loop Crossed Ladder Vertex Diagram with Two Massive Exchanges
• Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons
• Three-loop master integrals for ladder-box diagrams with one massive leg
• Two-Loop Master Integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering
• Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation
• On the reduction of generalized polylogarithms to ${\text{Li}}_n$ and ${\text{Li}}_{2,2}$ and on the evaluation thereof
• Next-to-Leading Order QCD Corrections to the Decay Width $H\to Z\gamma$
• Elliptic Polylogarithms and Basic Hypergeometric Functions
• On Genera of Curves from High-loop Generalized Unitarity Cuts
• Two-Loop Master Integrals for the Planar QCD Massive Corrections to Di-photon and Di-jet Hadro-production
• Star Integrals, Convolutions and Simplices
• Periods and Feynman integrals
• Elliptic Functions and Maximal Unitarity
• Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes
• Two-Loop Master Integrals for $q\bar{q}\to VV$: the Planar Topologies
• The Principle of Maximal Transcendentality and the Four-Loop Collinear Anomalous Dimension
• The seven-gluon amplitude in multi-Regge kinematics beyond leading logarithmic accuracy
• Analytic results for planar three-loop integrals for massive form factors
• NNLO QCD Corrections to $\gamma +{\eta }_c({\eta }_b)$ Exclusive Production in Electron-Positron Collision
• Two-loop master integrals for the leading QCD corrections to the Higgs coupling to a $W$ pair and to the triple gauge couplings $ZWW$ and ${\gamma }^{\ast }WW$
• Two-Loop integrals for CP-even heavy quarkonium production and decays: Elliptic Sectors
• Global Structure of Curves from Generalized Unitarity Cut of Three-loop Diagrams
• A Direct Evaluation of the Periods of the Weierstrass Zeta Function

Cited by in corpus (86)

• PolyLogTools - Polylogs for the masses
• Collider Physics at the Precision Frontier
• A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops
• The unequal mass sunrise integral expressed through iterated integrals on ${\overline{\mathcal{M}}}_{1,3}$
• Elliptic polylogarithms and Feynman parameter integrals
• Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
• An analytic solution for the equal-mass banana graph
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• Deriving canonical differential equations for Feynman integrals from a single uniform weight integral
• Energy-energy correlations at next-to-next-to-leading order
• Cluster adjacency beyond MHV
• On a procedure to derive $\epsilon$-factorised differential equations beyond polylogarithms
• Numerical evaluation of iterated integrals related to elliptic Feynman integrals
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• Implications of the Landau Equations for Iterated Integrals
• Tree-level amplitudes from the pure spinor superstring
• An algorithmic approach to finding canonical differential equations for elliptic Feynman integrals
• Algorithms and tools for iterated Eisenstein integrals
• The ice cone family and iterated integrals for Calabi-Yau varieties
• Four-graviton scattering to three loops in $\mathcal{N}=8$ supergravity
• Evaluating two-loop non-planar master integrals for Higgs + jet production with full heavy-quark mass dependence
• The electron self-energy in QED at two loops revisited
• Loop-by-loop Differential Equations for Dual (Elliptic) Feynman Integrals
• Algorithms for minimal Picard-Fuchs operators of Feynman integrals
• One-loop open-string integrals from differential equations: all-order alpha'-expansions at n points
• The elliptic double box and symbology beyond polylogarithms
• The complete set of two-loop master integrals for Higgs + jet production in QCD
• Meromorphic modular forms and the three-loop equal-mass banana integral
• Monodromy relations from twisted homology
• Two-loop diagrams in non-relativistic QCD with elliptics
• On Epsilon Factorized Differential Equations for Elliptic Feynman Integrals
• A Numerical Routine for the Crossed Vertex Diagram with a Massive-Particle Loop
• Linear power corrections to $e^{+}{e}^{-}$ shape variables in the three-jet region
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Canonical Differential Equations Beyond Genus One
• Properties of infinite product amplitudes: Veneziano, Virasoro, and Coon
• Three-loop contributions to the $\rho$ parameter and iterated integrals of modular forms
• Modular transformations of elliptic Feynman integrals
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• Coaction and double-copy properties of configuration-space integrals at genus zero
• Prescriptive Unitarity with Elliptic Leading Singularities
• Cutting the traintracks: Cauchy, Schubert and Calabi-Yau
• Intersection theory rules symbology
• An Elliptic Yangian-Invariant, `Leading Singularity'
• On $\epsilon$-factorised bases and pure Feynman integrals
• Massive kite diagrams with elliptics
• $\epsilon$-regular basis for non-polylogarithmic multiloop integrals and total cross section of the process $e^{+}{e}^{-}\to 2(Q\bar{Q})$
• Bananas: multi-edge graphs and their Feynman integrals
• Iterated integrals over letters induced by quadratic forms
• An Infinite Family of Elliptic Ladder Integrals
• Constructing multi-loop scattering amplitudes with manifest singularity structure
• Gravity loop integrands from the ultraviolet
• Towards Analytic Structure of Feynman Parameter Integrals with Rational Curves
• Three-loop banana integrals with four unequal masses
• Cuts and Isogenies
• Canonical differential equations for the elliptic two-loop five-point integral family relevant to $t\bar{t}+$jet production at leading colour
• A Calabi-Yau-to-Curve Correspondence for Feynman Integrals
• Two dialects for KZB equations: generating one-loop open-string integrals
• On series and integral representations of some NRQCD master integrals
• Non-planar elliptic vertex
• Master integrals for $\mathcal{O}(\alpha {\alpha }_s)$ corrections to $H\to Z{Z}^{\ast }$
• Integrating three-loop modular graph functions and transcendentality of string amplitudes
• A first computation of three-loop master integrals for the production of two off-shell vector bosons with different masses
• The analytic leading color contribution to the Higgs-gluon form factor in QCD at NNLO
• Canonical differential equations and intersection matrices
• Illustrations of Integrand-Basis Building at Two Loops
• All Two-Loop, Color-Dressed, Six-Point Amplitude Integrands in sYM
• Discontinuities of banana integrals in dispersion relation representation
• Electroweak double-box integrals for Moller scattering
• Sunrise integral with two internal masses and pseudo-threshold kinematics in terms of elliptic polylogarithms
• An Elliptic One-Loop Amplitude in Anti-de-Sitter Space
• Calculation of master integrals in terms of elliptic multiple polylogarithms
• Massive Two-Loop Heavy Particle Diagrams
• The spectrum of Feynman-integral geometries at two loops
• Integrands of Less-Supersymmetric Yang-Mills at One Loop
• One-loop Feynman integrals for $2\to 3$ scattering involving many scales including internal masses
• Analytic results for two-loop planar master integrals for Bhabha scattering
• Co-homology of Differential Forms and Feynman diagrams
• Integral representation for three-loop banana graph
• Recent Developments in Gluon Fusion Higgs Calculations
• A Feynman integral depending on two elliptic curves
• A construction of single-valued elliptic polylogarithms