Evaluating `elliptic' master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points

Abstract

This is a sequel of our previous paper where we described an algorithm to find a solution of differential equations for master integrals in the form of an $\epsilon$-expansion series with numerical coefficients. The algorithm is based on using generalized power series expansions near singular points of the differential system, solving difference equations for the corresponding coefficients in these expansions and using matching to connect series expansions at two neighboring points. Here we use our algorithm and the corresponding code for our example of four-loop generalized sunset diagrams with three massive and two massless propagators, in order to obtain new analytical results. We analytically evaluate the master integrals at threshold, $p^2=9m^2$, in an expansion in $\epsilon$ up to ${\epsilon }^1$. With the help of our code, we obtain numerical results for the threshold master integrals in an $\epsilon$-expansion with the accuracy of 6000 digits and then use the PSLQ algorithm to arrive at analytical values. Our basis of constants is build from bases of multiple polylogarithm values at sixth roots of unity.

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References in corpus (48)

• Harmonic Polylogarithms
• Multiloop integrals in dimensional regularization made simple
• Differential Equations for Two-Loop Four-Point Functions
• Differential Equations for Feynman Graph Amplitudes
• LiteRed 1.4: a powerful tool for the reduction of the multiloop integrals
• The Four-Loop QCD Beta-Function and Anomalous Dimensions
• Algorithm FIRE -- Feynman Integral REduction
• Numerical evaluation of multiple polylogarithms
• FIRE5: a C++ implementation of Feynman Integral REduction
• HPL, a Mathematica implementation of the harmonic polylogarithms
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The planar topologies
• FIESTA 4: optimized Feynman integral calculations with GPU support
• Quark mass relations to four-loop order
• Presenting LiteRed: a tool for the Loop InTEgrals REDuction
• Reducing differential equations for multiloop master integrals
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The non-planar topologies
• New results for the epsilon-expansion of certain one-, two- and three-loop Feynman diagrams
• FIRE4, LiteRed and accompanying tools to solve integration by parts relations
• Tops from Light Quarks: Full Mass Dependence at Two-Loops in QCD
• 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
• Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA
• Two-loop planar master integrals for Higgs$\to 3$ partons with full heavy-quark mass dependence
• 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
• Critical points and number of master integrals
• 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
• Master Integrals for Four-Loop Massless Propagators up to Transcendentality Weight Twelve
• The $\epsilon$-form of the differential equations for Feynman integrals in the elliptic case
• $\overline{\mathrm{MS}}$-on-shell quark mass relation up to four loops in QCD and a general SU$(N)$ gauge group
• epsilon: A tool to find a canonical basis of master integrals
• Virtual amplitudes and threshold behaviour of hadronic top-quark pair-production cross sections
• Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams
• High-precision epsilon expansions of single-mass-scale four-loop vacuum bubbles
• An Elliptic Generalization of Multiple Polylogarithms
• Transforming differential equations of multi-loop Feynman integrals into canonical form
• Solving differential equations for Feynman integrals by expansions near singular points
• The Two Loop Crossed Ladder Vertex Diagram with Two Massive Exchanges
• Precise numerical evaluation of the two loop sunrise graph Master Integrals in the equal mass case
• Threshold and pseudothreshold values of the sunset diagram
• Single mass scale diagrams: construction of a basis for the $\epsilon$-expansion
• Threshold expansion of the sunset diagram
• Two-loop $gg\to Hg$ amplitude mediated by a nearly massless quark
• Three-loop massive tadpoles and polylogarithms through weight six
• On the computation of finite bottom-quark mass effects in Higgs boson production
• "Sixth root of unity" and Feynman diagrams: hypergeometric function approach point of view
• Fuchsia and master integrals for splitting functions from differential equations in QCD

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