Three-loop banana integrals with four unequal masses

Abstract

We present a system of canonical differential equations satisfied by the three-loop banana integrals with four distinct non-zero masses in $D = 2-2\eps$ dimensions. Together with the initial condition in the small-mass limit, this provides all the ingredients to find analytic results for three-loop banana integrals in terms of iterated integrals to any desired order in the dimensional regulator. To obtain this result, we rely on recent advances in understanding the K3 geometry underlying these integrals and in how to construct rotations to an $\eps$-factorized basis. This rotation typically involves the introduction of objects defined as integrals of (derivatives of) K3 periods and rational functions. We apply and extend a method based on results from twisted cohomology to identify relations among these functions, which allows us to reduce their number considerably. We expect that the methods that we have applied here will prove useful to compute further multiloop multiscale Feynman integrals attached to non-trivial geometries.

25 pages plus 9 pages in the appendices

References in corpus (69)

• Multiloop integrals in dimensional regularization made simple
• Local Integrals for Planar Scattering Amplitudes
• The elliptic dilogarithm for the sunset graph
• Analytic treatment of the two loop equal mass sunrise graph
• 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
• 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
• Vector Space of Feynman Integrals and Multivariate Intersection Numbers
• Expansion by regions: revealing potential and Glauber regions automatically
• Geometric approach to asymptotic expansion of Feynman integrals
• 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
• Foundation and generalization of the expansion by regions
• 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
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• Decomposition of Feynman Integrals by Multivariate Intersection Numbers
• 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
• Analytic Structure of all Loop Banana Amplitudes
• An analytic solution for the equal-mass banana graph
• Feynman integral relations from parametric annihilators
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods
• Picard-Fuchs equations for Feynman integrals
• Emergence of Calabi-Yau manifolds in high-precision black hole scattering
• Duals of Feynman Integrals, I: Differential Equations
• Duals of Feynman Integrals, 2: Generalized Unitarity
• Taming Calabi-Yau Feynman integrals: The four-loop equal-mass banana integral
• On a procedure to derive $\epsilon$-factorised differential equations beyond polylogarithms
• Yangian-invariant fishnet integrals in 2 dimensions as volumes of Calabi-Yau varieties
• Baikov representations, intersection theory, and canonical Feynman integrals
• On the computation of intersection numbers for twisted cocycles
• 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
• On Genera of Curves from High-loop Generalized Unitarity Cuts
• An algorithmic approach to finding canonical differential equations for elliptic Feynman integrals
• Periods and 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
• Calabi-Yau meets Gravity: A Calabi-Yau three-fold at fifth post-Minkowskian order
• Meromorphic modular forms and the three-loop equal-mass banana integral
• Differential equations for loop integrals in Baikov representation
• Analytical expressions of 3 and 4-loop sunrise Feynman integrals and 4-dimensional lattice integrals
• Intersection Numbers from Higher-order Partial Differential Equations
• Reduction to master integrals via intersection numbers and polynomial expansions
• Canonical Differential Equations Beyond Genus One
• Topological polylogarithms and $p$-adic interpolation of $L$-values of totally real fields
• Calabi-Yau Feynman integrals in gravity: $\epsilon$-factorized form for apparent singularities
• Classifying post-Minkowskian geometries for gravitational waves via loop-by-loop Baikov
• On $\epsilon$-factorised bases and pure Feynman integrals
• Bananas: multi-edge graphs and their Feynman integrals
• Position Space Equations for Banana Feynman Diagrams
• Global Structure of Curves from Generalized Unitarity Cut of Three-loop Diagrams
• Identifying regions in wide-angle scattering via graph-theoretical approaches
• Elliptic hyperlogarithms
• Real time lattice correlation functions from differential equations
• Self-duality from twisted cohomology
• On canonical differential equations for Calabi-Yau multi-scale Feynman integrals
• Intersection Numbers from Companion Tensor Algebra
• Twisted Riemann bilinear relations and Feynman integrals
• From equations in coordinate space to Picard-Fuchs and back
• Asymptotics of the banana Feynman amplitudes at the large complex structure limit

Cited by in corpus (3)

• Canonical differential equations and intersection matrices
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