Reduction with Degenerate Gram matrix for One-loop Integrals

Abstract

An improved PV-reduction method for one-loop integrals with auxiliary vector $R$ has been proposed in \cite{Feng:2021enk,Hu:2021nia}. It has also been shown that the new method is a self-completed method in \cite{Feng:2022uqp}. 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.

46 pages, no figures

References in corpus (14)

• Reducing full one-loop amplitudes to scalar integrals at the integrand level
• A Numerical Unitarity Formalism for Evaluating One-Loop Amplitudes
• D-dimensional unitarity cut method
• Techniques for the calculation of electroweak radiative corrections at the one-loop level and results for W-physics at LEP200
• Unitarity Cuts with Massive Propagators and Algebraic Expressions for Coefficients
• Numerical Evaluation of Six-Photon Amplitudes
• Integral Coefficients for One-Loop Amplitudes
• Closed-Form Decomposition of One-Loop Massive Amplitudes
• Polynomial Structures in One-Loop Amplitudes
• Single Cut Integration
• Analytic Tadpole Coefficients of One-loop Integrals
• PV-Reduction of Sunset Topology with Auxiliary Vector
• Reduction of General One-loop Integrals Using Auxiliary Vector
• Universal Treatment of Reduction for One-Loop Integrals in Projective Space

Cited by in corpus (5)

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• OPITeR: A program for tensor reduction of multi-loop Feynman Integrals
• Tensor Reduction for Feynman Integrals with Lorentz and Spinor Indices