Analytic Structure of all Loop Banana Amplitudes

Abstract

Using the Gelfand-Kapranov-Zelevinsk\uı system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana amplitudes with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel $\hat{\Gamma }$-class evaluation in the ambient spaces of the mirror, while the imaginary part of the amplitude in this regime is determined by the $\hat{\Gamma }$-class of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius $\kappa$-constants, which determine the behaviour of the amplitudes, when the momentum square equals the sum of the masses squared, in terms of zeta values. We extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogenous) Picard-Fuchs differential equations for arbitrary masses. This allows to evaluate the amplitude as well as other master integrals with raised powers of the propagators in very short time to very high numerical precision for all values of the physical parameters. Using a recent $p$-adic analysis of the periods we determine the value of the maximal cut equal mass four-loop amplitude at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins.

arXiv comment40 pages, 1 figure, new appendix concerning a pari program to compute equal mass amplitude, minor corrections the paper is now also published in JHEP but under the name "Analytic structure of all loop banana integrals", some changes are done in the published version

References in corpus (42)

• K3--Fibrations and Heterotic-Type II String Duality
• Calabi-Yau fourfolds for M- and F-Theory compactifications
• The elliptic dilogarithm for the sunset graph
• Mirror Symmetry, Mirror Map and Applications to Complete Intersection Calabi-Yau Spaces
• High-precision calculation of the 4-loop contribution to the electron g-2 in QED
• On the maximal cut of Feynman integrals and the solution of their differential equations
• Direct Integration of the Topological String
• Feynman graph polynomials
• Counting higher genus curves in a Calabi-Yau manifold
• Elliptic integral evaluations of Bessel moments
• A Feynman integral via higher normal functions
• Numbers and Functions in Quantum Field Theory
• Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph
• The kite integral to all orders in terms of elliptic polylogarithms
• A K3 in phi4
• Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams
• An analytic solution for the equal-mass banana graph
• Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties
• Feynman integral relations from parametric annihilators
• The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods
• 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
• The Galois coaction on ${\varphi }^4$ periods
• Feynman integrals as A-hypergeometric functions
• Integration by parts: An introduction
• Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures
• Quantum geometry of del Pezzo surfaces in the Nekrasov-Shatashvili limit
• Calculation of two-loop self-energies in the electroweak Standard Model
• Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems
• Periods and Feynman integrals
• Evaluating two-loop non-planar master integrals for Higgs + jet production with full heavy-quark mass dependence
• Counting BPS states on the Enriques Calabi-Yau
• A One Parameter Family of Calabi-Yau Manifolds with Attractor Points of Rank Two
• GKZ-hypergeometric systems for Feynman integrals
• Counting the number of master integrals for sunrise-type Feynman diagrams via Mellin-Barnes representation
• A Proof of a Recursion for Bessel Moments
• Analytical expressions of 3 and 4-loop sunrise Feynman integrals and 4-dimensional lattice integrals
• Three-loop contributions to the $\rho$ parameter and iterated integrals of modular forms
• WKB Method and Quantum Periods beyond Genus One
• Wrońskian factorizations and Broadhurst-Mellit determinant formulae
• Wick rotations, Eichler integrals, and multi-loop Feynman diagrams
• Some algebraic and arithmetic properties of Feynman diagrams

Cited by in corpus (37)

• Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
• Emergence of Calabi-Yau manifolds in high-precision black hole scattering
• Taming Calabi-Yau Feynman integrals: The four-loop equal-mass banana integral
• Yangian-invariant fishnet integrals in 2 dimensions as volumes of Calabi-Yau varieties
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• Bananas of equal mass: any loop, any order in the dimensional regularisation parameter
• 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
• Algorithms for minimal Picard-Fuchs operators of Feynman integrals
• Meromorphic modular forms and the three-loop equal-mass banana integral
• Kinematic singularities of Feynman integrals and principal A-determinants
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Canonical Differential Equations Beyond Genus One
• Bananas: multi-edge graphs and their Feynman integrals
• Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges
• Recursive computation of Feynman periods
• An Infinite Family of Elliptic Ladder Integrals
• Cohen-Macaulay Property of Feynman Integrals
• Flux Vacua and Modularity for ${\mathbb{Z}}_2$ Symmetric Calabi-Yau Manifolds
• Three-loop banana integrals with four unequal masses
• Dyson-Schwinger Equations in Minimal Subtraction
• A Calabi-Yau-to-Curve Correspondence for Feynman Integrals
• Feynman Integrals from Positivity Constraints
• Mirror Symmetry for Five-Parameter Hulek-Verrill Manifolds
• Twisted Riemann bilinear relations and Feynman integrals
• Feynman Integrals of Grassmannians
• GKZ-system of the 2-loop self energy with 4 propagators
• GKZ hypergeometric systems of the three-loop vacuum Feynman integrals
• Canonical differential equations and intersection matrices
• Discontinuities of banana integrals in dispersion relation representation
• Quantum Periods and TBA-like Equations for a Class of Calabi-Yau Geometries
• Symmetries of Calabi-Yau Prepotentials with Isomorphic Flops
• Asymptotics of the banana Feynman amplitudes at the large complex structure limit
• Vanishing cycles and analysis of singularities of Feynman diagrams
• A Feynman integral depending on two elliptic curves
• GKZ hypergeometric systems of the four-loop vacuum Feynman integrals
• Integral representation for three-loop banana graph