Reduction of Feynman Integrals in the Parametric Representation II: Reduction of Tensor Integrals

Abstract

In a recent paper by the author (Chen in JHEP 02:115, 2020), the reduction of Feynman integrals in the parametric representation was considered. Tensor integrals were directly parametrized by using a generator method. The resulting parametric integrals were reduced by constructing and solving parametric integration-by-parts (IBP) identities. In this paper, we furthermore show that polynomial equations for the operators that generate tensor integrals can be derived. Based on these equations, two methods to reduce tensor integrals are developed. In the first method, by introducing some auxiliary parameters, tensor integrals are parametrized without shifting the spacetime dimension. The resulting parametric integrals can be reduced by using the standard IBP method. In the second method, tensor integrals are (partially) reduced by using the technique of Gröbner basis combined with the application of symbolic rules. The unreduced integrals can further be reduced by solving parametric IBP identities.

arXiv comment16 pages, major revision, matches published version

References in corpus (16)

• Algorithm FIRE -- Feynman Integral REduction
• Connection between Feynman integrals having different values of the space-time dimension
• FIRE6: Feynman Integral REduction with Modular Arithmetic
• Presenting LiteRed: a tool for the Loop InTEgrals REDuction
• Integration-by-parts reductions from unitarity cuts and algebraic geometry
• Sector Decomposition
• Explicit solutions of the multi-loop integral recurrence relations and its application
• Two-Loop QCD Helicity Amplitudes for $e^{+}{e}^{-}\to 3$~Jets
• Critical points and number of master integrals
• Feynman integral relations from parametric annihilators
• Physical projectors for multi-leg helicity amplitudes
• Applying Groebner Bases to Solve Reduction Problems for Feynman Integrals
• A prescription for projectors to compute helicity amplitudes in D dimensions
• Lecture Notes on Multi-loop Integral Reduction and Applied Algebraic Geometry
• An Algorithm to Construct Groebner Bases for Solving Integration by Parts Relations
• Modern techniques of multiloop calculations

Cited by in corpus (5)

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• Soft Theorem to Three Loops in QCD and $\mathcal{N}=4$ Super Yang-Mills Theory
• Reduction of Feynman Integrals in the Parametric Representation III: Integrals with Cuts
• Double-Real-Virtual and Double-Virtual-Real Corrections to the Three-Loop Thrust Soft Function
• Unregulated Divergences of Feynman Integrals