A complete algebraic reduction of one-loop tensor Feynman integrals

Abstract

We set up a new, flexible approach for the tensor reduction of one-loop Feynman integrals. The 5-point tensor integrals up to rank R=5 are expressed by 4-point tensor integrals of rank R-1, such that the appearance of the inverse 5-point Gram determinant is avoided. The 4-point tensor coefficients are represented in terms of 4-point integrals, defined in $d$ dimensions, $4-2\epsilon \le d\le 4-2\epsilon +2(R-1)$, with higher powers of the propagators. They can be further reduced to expressions which stay free of the inverse 4-point Gram determinants but contain higher-dimensional 4-point integrals with only the first power of scalar propagators, plus 3-point tensor coefficients. A direct evaluation of the higher dimensional 4-point functions would avoid the appearance of inverse powers of the Gram determinants completely. The simplest approach, however, is to apply here dimensional recurrence relations in order to reduce them to the familiar 2- to 4-point functions in generic dimension $d = 4-2\eps$, introducing thereby coefficients with inverse 4-point Gram determinants up to power $R$ for tensors of rank $R$. For small or vanishing Gram determinants - where this reduction is not applicable - we use analytic expansions in positive powers of the Gram determinants. Improving the convergence of the expansions substantially with Padé approximants we close up to the evaluation of the 4-point tensor coefficients for larger Gram determinants. Finally, some relations are discussed which may be useful for analytic simplifications of Feynman diagrams.

57 pages, 3 figures, 2 tables; extended version; typo in equation (4.67) corrected, reference [20] added; reorganisation of text for journal publication

References in corpus (22)

• Automatized One-Loop Calculations in 4 and D dimensions
• One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits
• Fusing Gauge Theory Tree Amplitudes Into Loop Amplitudes
• Dimensionally Regulated Pentagon Integrals
• Reduction schemes for one-loop tensor integrals
• Connection between Feynman integrals having different values of the space-time dimension
• Scalar one-loop integrals for QCD
• Dimensionally Regulated One-Loop Integrals
• Reduction of one-loop tensor 5-point integrals
• An algebraic/numerical formalism for one-loop multi-leg amplitudes
• Algebraic reduction of one-loop Feynman graph amplitudes
• Reduction formalism for dimensionally regulated one-loop N-point integrals
• The SM and NLO multileg working group: Summary report
• A New Hypergeometric Representation of One-Loop Scalar Integrals in $d$ Dimensions
• One-Loop Tensor Integrals in Dimensional Regularisation
• A numerical evaluation of the scalar hexagon integral in the physical region
• Reduction of One-loop Tensor Form-Factors to Scalar Integrals: A General Scheme
• A complete reduction of one-loop tensor 5- and 6-point integrals
• Numerical Evaluation of One-Loop Diagrams Near Exceptional Momentum Configurations
• A recursive reduction of tensor Feynman integrals
• New results for loop integrals: AMBRE, CSectors, hexagon
• Some variations of the reduction of one-loop Feynman tensor integrals

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