A Feynman integral depending on two elliptic curves

Abstract

We study a two-loop four-point function with one internal mass. This Feynman integral is one of the simplest Feynman integrals depending on two elliptic curves. We transform the associated differential equation into an $\epsilon$-form. We study the entries of the differential equation, and in particular the entries which depend on both elliptic curves.

arXiv comment41 pages, v2: version to be published

References in corpus (65)

• Multiloop integrals in dimensional regularization made simple
• Algorithm FIRE -- Feynman Integral REduction
• Kira - A Feynman Integral Reduction Program
• Reduze - Feynman Integral Reduction in C++
• FIRE6: Feynman Integral REduction with Modular Arithmetic
• Integral Reduction with Kira 2.0 and Finite Field Methods
• The elliptic dilogarithm for the sunset graph
• Two-loop two-point functions with masses: asymptotic expansions and Taylor series, in any dimension
• Analytic treatment of the two loop equal mass sunrise graph
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• Feynman Integrals and Intersection Theory
• On the maximal cut of Feynman integrals and the solution of their differential equations
• 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
• Two-loop planar master integrals for Higgs$\to 3$ partons with full heavy-quark mass dependence
• Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral
• Elliptic integral evaluations of Bessel moments
• A Feynman integral via higher normal functions
• 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
• 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
• A K3 in phi4
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• The two-loop sunrise graph with arbitrary masses
• Schouten identities for Feynman graph amplitudes; the Master Integrals for the two-loop massive sunrise graph
• Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series
• An Elliptic Generalization of Multiple Polylogarithms
• The Two-Loop Master Integrals for $q\bar{q}\to VV$
• 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
• The unequal mass sunrise integral expressed through iterated integrals on ${\overline{\mathcal{M}}}_{1,3}$
• Elliptic polylogarithms and Feynman parameter integrals
• The iterated structure of the all-order result for the two-loop sunrise integral
• Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral
• Solving differential equations for Feynman integrals by expansions near singular points
• Analytic Structure of all Loop Banana Amplitudes
• An analytic solution for the equal-mass banana graph
• Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable
• Simplifying differential equations for multi-scale Feynman integrals beyond multiple polylogarithms
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods
• RationalizeRoots: Software Package for the Rationalization of Square Roots
• Rationalizing Loop Integration
• Elliptic Polylogarithms and Basic Hypergeometric Functions
• Numerical evaluation of iterated integrals related to elliptic Feynman integrals
• Algorithms and tools for iterated Eisenstein integrals
• Elliptic Functions and Maximal Unitarity
• Evaluating `elliptic' master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points
• The electron self-energy in QED at two loops revisited
• Integration by parts identities in integer numbers of dimensions. A criterion for decoupling systems of differential equations
• The elliptic double box and symbology beyond polylogarithms
• Meromorphic modular forms and the three-loop equal-mass banana integral
• Two-loop diagrams in non-relativistic QCD with elliptics
• On Epsilon Factorized Differential Equations for Elliptic Feynman Integrals
• Functional relations for elliptic polylogarithms
• Traintrack Calabi-Yaus from Twistor Geometry
• Three-loop contributions to the $\rho$ parameter and iterated integrals of modular forms
• Modular transformations of elliptic Feynman integrals
• Massive kite diagrams with elliptics
• Convolutions on the complex torus