The $\epsilon$-form of the differential equations for Feynman integrals in the elliptic case

Abstract

Feynman integrals are easily solved if their system of differential equations is in $\epsilon$-form. In this letter we show by the explicit example of the kite integral family that an $\epsilon$-form can even be achieved, if the Feynman integrals do not evaluate to multiple polylogarithms. The $\epsilon$-form is obtained by a (non-algebraic) change of basis for the master integrals.

arXiv comment15 pages, version to be published

References in corpus (55)

• Multiloop integrals in dimensional regularization made simple
• Differential Equations for Two-Loop Four-Point Functions
• Differential Equations for Feynman Graph Amplitudes
• Connection between Feynman integrals having different values of the space-time dimension
• Lectures on differential equations for Feynman integrals
• 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
• Generalized Recurrence Relations for Two-loop Propagator Integrals with Arbitrary Masses
• Feynman Diagrams and Differential Equations
• Elliptic multiple zeta values and one-loop superstring amplitudes
• Space-time dimensionality D as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D
• Explicit solutions of the multi-loop integral recurrence relations and its application
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• Analytic two-loop results for selfenergy- and vertex-type diagrams with one non-zero mass
• Analytical and numerical methods for massive two-loop self-energy diagrams
• Magnus and Dyson Series for Master Integrals
• On the maximal cut of Feynman integrals and the solution of their differential equations
• 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
• Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA
• Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form
• 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
• The kite integral to all orders in terms of elliptic polylogarithms
• Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra
• Multiple Elliptic Polylogarithms
• Maximal Unitarity at Two Loops
• epsilon: A tool to find a canonical basis of master integrals
• 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
• An Elliptic Generalization of Multiple Polylogarithms
• On the Generalized Harmonic Polylogarithms of One Complex Variable
• Transforming differential equations of multi-loop Feynman integrals into canonical form
• Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral
• Simplifying differential equations for multi-scale Feynman integrals beyond multiple polylogarithms
• Two-loop sunset diagrams with three massive lines
• Picard-Fuchs equations for Feynman integrals
• On the evaluation of a certain class of Feynman diagrams in x-space: Sunrise-type topologies at any loop order
• Monodromy Relations in Higher-Loop String Amplitudes
• 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
• Two-Loop Formfactors in Theories with Mass Gap and Z-Boson Production
• Towards multiple elliptic polylogarithms
• Elliptic Polylogarithms and Basic Hypergeometric Functions
• Two-Loop Master Integrals for the Planar QCD Massive Corrections to Di-photon and Di-jet Hadro-production
• Calculation of two-loop self-energies in the electroweak Standard Model
• A numerical test of differential equations for one- and two-loop sunrise diagrams using configuration space techniques
• Differential equations for loop integrals in Baikov representation
• On the algebraic structure of iterated integrals of quasimodular forms
• Normalized Fuchsian form on Riemann sphere and differential equations for multiloop integrals

Cited by in corpus (73)

• Conservative Dynamics of Binary Systems at Fourth Post-Minkowskian Order in the Large-eccentricity Expansion
• Radiation Reaction and Gravitational Waves at Fourth Post-Minkowskian Order
• Collider Physics at the Precision Frontier
• A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec
• Elliptic Feynman integrals and pure functions
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• Radiation-Reaction in the Effective Field Theory Approach to Post-Minkowskian Dynamics
• Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops
• Analytic results for the planar double box integral relevant to top-pair production with a closed top loop
• The planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularisation parameter
• 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
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• Deriving canonical differential equations for Feynman integrals from a single uniform weight integral
• RationalizeRoots: Software Package for the Rationalization of Square Roots
• Numerical evaluation of iterated integrals related to elliptic Feynman integrals
• Taming Calabi-Yau Feynman integrals: The four-loop equal-mass banana integral
• On a procedure to derive $\epsilon$-factorised differential equations beyond polylogarithms
• Bananas of equal mass: any loop, any order in the dimensional regularisation parameter
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• Tree-level amplitudes from the pure spinor superstring
• An algorithmic approach to finding canonical differential equations for elliptic Feynman integrals
• The three-loop equal-mass banana integral in $\epsilon$-factorised form with meromorphic modular forms
• Four-graviton scattering to three loops in $\mathcal{N}=8$ supergravity
• Evaluating `elliptic' master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points
• Analytic Computing Methods for Precision Calculations in Quantum Field Theory
• The electron self-energy in QED at two loops revisited
• Loop-by-loop Differential Equations for Dual (Elliptic) 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
• Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings
• The complete set of two-loop master integrals for Higgs + jet production in QCD
• Status of Intersection Theory and Feynman Integrals
• Meromorphic modular forms and the three-loop equal-mass banana integral
• Two-loop leading colour QCD helicity amplitudes for top quark pair production in the gluon fusion channel
• On Epsilon Factorized Differential Equations for Elliptic Feynman Integrals
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Canonical Differential Equations Beyond Genus One
• All-order alpha'-expansion of one-loop open-string integrals
• Modular transformations of elliptic Feynman integrals
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• Coaction and double-copy properties of configuration-space integrals at genus zero
• Calabi-Yau Feynman integrals in gravity: $\epsilon$-factorized form for apparent singularities
• Cutting the traintracks: Cauchy, Schubert and Calabi-Yau
• On $\epsilon$-factorised bases and pure Feynman integrals
• Iterated integrals over letters induced by quadratic forms
• Symmetric $\epsilon$- and $(\epsilon +1/2)$-forms and quadratic constraints in "elliptic" sectors
• Constructing multi-loop scattering amplitudes with manifest singularity structure
• Two-loop Feynman integrals for leading colour $t\bar{t}W$ production at hadron colliders
• Self-duality from twisted cohomology
• Three-loop banana integrals with four unequal masses
• Canonical differential equations for the elliptic two-loop five-point integral family relevant to $t\bar{t}+$jet production at leading colour
• Multivariate hypergeometric solutions of cosmological (dS) correlators by $\text{d}\log$-form differential equations
• A Calabi-Yau-to-Curve Correspondence for Feynman Integrals
• Two dialects for KZB equations: generating one-loop open-string integrals
• Systematic approximation of multi-scale Feynman integrals
• On series and integral representations of some NRQCD master integrals
• Non-planar elliptic vertex
• Classification of Feynman integral geometries for black-hole scattering at 5PM order
• Master integrals for $\mathcal{O}(\alpha {\alpha }_s)$ corrections to $H\to Z{Z}^{\ast }$
• The analytic leading color contribution to the Higgs-gluon form factor in QCD at NNLO
• Generalized Polylogarithms in Maple
• Canonical differential equations and intersection matrices
• Electroweak double-box integrals for Moller scattering
• Discontinuities of banana integrals in dispersion relation representation
• Sunrise integral with two internal masses and pseudo-threshold kinematics in terms of elliptic polylogarithms
• Rationalizing roots: an algorithmic approach
• The spectrum of Feynman-integral geometries at two loops
• Integral of the double-emission eikonal function for a massive and a massless emitter at an arbitrary angle
• From elliptic curves to Feynman integrals
• Massive Two-Loop Heavy Particle Diagrams
• A Feynman integral depending on two elliptic curves
• Higher-Order Corrections to Higgs Boson Amplitudes with Full Quark Mass Dependence in Quantum Chromodynamics