The unequal mass sunrise integral expressed through iterated integrals on ${\overline{\mathcal{M}}}_{1,3}$

Abstract

We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter $\epsilon$. In order to do so, we transform the system of differential equations for the master integrals to an $\epsilon$-form. The sunrise integral with unequal masses depends on three kinematical variables. We perform a change of variables to standard coordinates on the moduli space ${\mathcal{M}}_{1,3}$ of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on ${\overline{\mathcal{M}}}_{1,3}$. On the hypersurface $τ=\mbox{const}$ our result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms.

arXiv comment34 pages, v2: version to be published

References in corpus (68)

• 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
• Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop 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
• Resolution of singularities for multi-loop integrals
• 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
• 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
• 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
• Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra
• Multiple Elliptic Polylogarithms
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• 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
• Symbolic Expansion of Transcendental Functions
• Transforming differential equations of multi-loop Feynman integrals into canonical form
• 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
• 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
• Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable
• 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
• Relations between elliptic multiple zeta values and a special derivation algebra
• Rationalizing Loop Integration
• 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
• Elliptic Functions and Maximal Unitarity
• Evaluating `elliptic' master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points
• A numerical test of differential equations for one- and two-loop sunrise diagrams using configuration space techniques
• The electron self-energy in QED at two loops revisited
• Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes
• Integration by parts identities in integer numbers of dimensions. A criterion for decoupling systems of differential equations
• Differential equations for loop integrals in Baikov representation
• Master Integrals for the two-loop, non-planar QCD corrections to top-quark pair production in the quark-annihilation channel
• Coordinate space calculation of two- and three-loop sunrise-type diagrams, elliptic functions and truncated Bessel integral identities
• Normalized Fuchsian form on Riemann sphere and differential equations for multiloop integrals

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• 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
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• Integral representation for three-loop banana graph
• The diagrammatic coaction beyond one loop
• Two-loop master integrals for pseudo-scalar quarkonium and leptonium production and decay