Solving Recurrence Relations for Multi-Loop Feynman Integrals

Abstract

We study the problem of solving integration-by-parts recurrence relations for a given class of Feynman integrals which is characterized by an arbitrary polynomial in the numerator and arbitrary integer powers of propagators, {\it i.e.}, the problem of expressing any Feynman integral from this class as a linear combination of master integrals. We show how the parametric representation invented by Baikov can be used to characterize the master integrals and to construct an algorithm for evaluating the corresponding coefficient functions. To illustrate this procedure we use simple one-loop examples as well as the class of diagrams appearing in the calculation of the two-loop heavy quark potential.

24 pages, 5 ps figures, references added, minor modifications, published version

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• Feynman integral relations from parametric annihilators
• ep-Finite Basis of Master Integrals for the Integration-By-Parts Method
• Numerical evaluation of loop integrals
• Integration by parts: An introduction
• The Pentabox Master Integrals with the Simplified Differential Equations approach
• Applying Groebner Bases to Solve Reduction Problems for Feynman Integrals
• $M(B_c^{\ast })-M(B_c)$ Splitting from Nonrelativistic Renormalization Group
• Two-Loop Static QCD Potential for General Colour State
• Analytic Computing Methods for Precision Calculations in Quantum Field Theory
• Two-Loop Vertices in Quantum Field Theory: Infrared Convergent Scalar Configurations
• Two-Loop Matching Coefficients for Heavy Quark Currents
• Fermionic Corrections to the Three-Loop Matching Coefficient of the Vector Current
• Two-Loop Tensor Integrals in Quantum Field Theory
• An Algorithm to Construct Groebner Bases for Solving Integration by Parts Relations
• Evaluating multiloop Feynman integrals by Mellin-Barnes representation
• Hypergeometric representation of a four-loop vacuum bubble
• S-bases as a tool to solve reduction problems for Feynman integrals
• On the reduction of Feynman integrals to master integrals
• Two-Loop Vacuum Diagrams in Background Field and Heisenberg-Euler Effective Action
• Some recent results on evaluating Feynman integrals
• Heavy quark and gluino potentials to two loops
• Two-Loop Diagrammatics in a Self-Dual Background
• "Background Field Integration-by-Parts" and the Connection Between One-Loop and Two-Loop Heisenberg-Euler Effective Actions
• A new reduction strategy for special negative sectors of planar two-loop integrals without Laporta algorithm
• Groebner Bases Applied to Systems of Linear Difference Equations