Reduction with degenerate Gram matrix for one-loop integrals

• Published in: The journal of high energy physics Vol. 2022; no. 8; pp. 110 - 46
• Main Authors: Feng, Bo, Hu, Chang, Li, Tingfei, Song, Yuekai
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 09.08.2022; Springer Nature B.V; SpringerOpen
• Subjects: Algebra; Algorithms; Classical and Quantum Gravitation; Coefficients; Decomposition; Divergence; Elementary Particles; High energy physics; Integrals; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Reduction; Regular Article - Theoretical Physics; Relativity Theory; Renormalization and Regularization; Scattering Amplitudes; Singularity (mathematics); String Theory; Taylor series; Topology; Scattering Amplitudes; Renormalization and Regularization
• ISSN: 1029-8479; 1126-6708; 1127-2236; 1029-8479
• DOI: 10.1007/JHEP08(2022)110

A bstract An improved PV-reduction (Passarino-Veltman) method for one-loop integrals with auxiliary vector R has been proposed in [ 1 , 2 ]. It has also been shown that the new method is a self-completed method in [ 3 ]. Analytic reduction coefficients can be easily produced by recursion relations in this method, where the Gram determinant appears in denominators. The singularity caused by Gram determinant is a well-known fact and it is important to address these divergences in a given frame. In this paper, we propose a systematical algorithm to deal with this problem in our method. The key idea is that now the master integral of the highest topology will be decomposed into combinations of master integrals of lower topologies. By demanding the cancellation of divergence for obtained general reduction coefficients, we solve decomposition coefficients as a Taylor series of the Gram determinant. Moreover, the same idea can be applied to other kinds of divergences.