Meromorphic modular forms and the three-loop equal-mass banana integral

Abstract

We consider a class of differential equations for multi-loop Feynman integrals which can be solved to all orders in dimensional regularisation in terms of iterated integrals of meromorphic modular forms. We show that the subgroup under which the modular forms transform can naturally be identified with the monodromy group of a certain second-order differential operator. We provide an explicit decomposition of the spaces of modular forms into a direct sum of total derivatives and a basis of modular forms that cannot be written as derivatives of other functions, thereby generalising a result by one of the authors form the full modular group to arbitrary finite-index subgroups of genus zero. Finally, we apply our results to the two- and three-loop equal-mass banana integrals, and we obtain in particular for the first time complete analytic results for the higher orders in dimensional regularisation for the three-loop case, which involves iterated integrals of meromorphic modular forms.

arXiv comment60 pages, 3 figures

References in corpus (66)

• Harmonic Polylogarithms
• Multiloop integrals in dimensional regularization made simple
• Differential Equations for Two-Loop Four-Point Functions
• Numerical evaluation of multiple polylogarithms
• Local Integrals for Planar Scattering Amplitudes
• Multiple polylogarithms and mixed Tate motives
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The planar topologies
• The elliptic dilogarithm for the sunset graph
• Numerical Evaluation of Harmonic Polylogarithms
• Analytic treatment of the two loop equal mass sunrise graph
• Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
• Elliptic multiple zeta values and one-loop superstring amplitudes
• Uniqueness of two-loop master contours
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• Numerical Evaluation of Two-Dimensional Harmonic Polylogarithms
• Analytical and numerical methods for massive two-loop self-energy diagrams
• 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
• Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral
• A Feynman integral via higher normal functions
• Cuts of Feynman Integrals in Baikov representation
• Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph
• 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
• Simple one-dimensional integral representations for two-loop self-energies: the master diagram
• Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams
• Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series
• Inclusive Heavy Flavor Hadroproduction in NLO QCD: the Exact Analytic Result
• Master integrals with 2 and 3 massive propagators for the 2-loop electroweak form factor - planar case
• Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
• The iterated structure of the all-order result for the two-loop sunrise integral
• Analytic Structure of all Loop Banana Amplitudes
• An analytic solution for the equal-mass banana graph
• HandyG -- rapid numerical evaluation of generalised polylogarithms in Fortran
• Two-loop sunset diagrams with three massive lines
• The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods
• On Rationally Parametrized Modular Equations
• 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
• ep-Finite Basis of Master Integrals for the Integration-By-Parts Method
• The Two Loop Crossed Ladder Vertex Diagram with Two Massive Exchanges
• Maximal Cuts in Arbitrary Dimension
• Baikov-Lee Representations Of Cut Feynman Integrals
• On the reduction of generalized polylogarithms to ${\text{Li}}_n$ and ${\text{Li}}_{2,2}$ and on the evaluation thereof
• Towards multiple elliptic polylogarithms
• Numerical evaluation of iterated integrals related to elliptic Feynman integrals
• On Genera of Curves from High-loop Generalized Unitarity Cuts
• Algorithms and tools for iterated Eisenstein integrals
• Numerical Implementation of Harmonic Polylogarithms to Weight w = 8
• Star Integrals, Convolutions and Simplices
• Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes
• On the algebraic structure of iterated integrals of quasimodular forms
• Three-loop contributions to the $\rho$ parameter and iterated integrals of modular forms
• Modular transformations of elliptic Feynman integrals
• $\epsilon$-regular basis for non-polylogarithmic multiloop integrals and total cross section of the process $e^{+}{e}^{-}\to 2(Q\bar{Q})$
• Cuts and Isogenies
• From the Deligne-Ihara conjecture to Multiple Modular Values
• Picard-Fuchs Uniformization: Modularity of the Mirror Map and Mirror-Moonshine
• Towards algebraic iterated integrals on elliptic curves via the universal vectorial extension

Cited by in corpus (21)

• Taming Calabi-Yau Feynman integrals: The four-loop equal-mass banana integral
• Duals of Feynman Integrals, 2: Generalized Unitarity
• Bananas of equal mass: any loop, any order in the dimensional regularisation parameter
• 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
• The ice cone family and iterated integrals for Calabi-Yau varieties
• Loop-by-loop Differential Equations for Dual (Elliptic) Feynman Integrals
• Calabi-Yau meets Gravity: A Calabi-Yau three-fold at fifth post-Minkowskian order
• Canonical Differential Equations Beyond Genus One
• Constraints on Sequential Discontinuities from the Geometry of On-shell Spaces
• Calabi-Yau Feynman integrals in gravity: $\epsilon$-factorized form for apparent singularities
• Classifying post-Minkowskian geometries for gravitational waves via loop-by-loop Baikov
• Bananas: multi-edge graphs and their Feynman integrals
• Schottky-Kronecker forms and hyperelliptic polylogarithms
• Three-loop banana integrals with four unequal masses
• Canonical differential equations and intersection matrices
• Electroweak double-box integrals for Moller scattering
• An Elliptic One-Loop Amplitude in Anti-de-Sitter Space
• Analytic and numerical approaches for depictive 3-loop integrals using sector decomposition
• The spectrum of Feynman-integral geometries at two loops
• A Feynman integral depending on two elliptic curves