Reduction of Feynman integrals in the parametric representation II: reduction of tensor integrals

• Published in: The European physical journal. C, Particles and fields Vol. 81; no. 3; pp. 244 - 10
• Main Author: Chen, Wen
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2021; Springer; Springer Nature B.V; SpringerOpen
• Subjects: Astronomy; Astrophysics and Cosmology; Elementary Particles; Hadrons; Heavy Ions; Integrals; Mathematical analysis; Measurement Science and Instrumentation; Nuclear Energy; Nuclear Physics; Parameterization; Physics; Physics and Astronomy; Polynomials; Quantum Field Theories; Quantum Field Theory; Reduction; Regular Article – Theoretical Physics; Regular – Theoretical Physics; Representations; String Theory; Tensors
• ISSN: 1434-6044; 1434-6052
• DOI: 10.1140/epjc/s10052-021-09036-5

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.