Optimized Negative Dimensional Integration Method (NDIM) and multiloop Feynman diagram calculation

Abstract

We present an improved form of the integration technique known as NDIM (Negative Dimensional Integration Method), which is a powerful tool in the analytical evaluation of Feynman diagrams. Using this technique we study a theory in $D=4-2\epsilon$ dimensions, considering generic topologies of $L$ loops and $E$ independent external momenta, and where the propagator powers are arbitrary. The method transforms the Schwinger parametric integral associated to the diagram into a multiple series expansion, whose main characteristic is that the argument contains several Kronecker deltas which appear naturally in the application of the method, and which we call diagram presolution. The optimization we present here consists in a procedure that minimizes the series multiplicity, through appropriate factorizations in the multinomials that appear in the parametric integral, and which maximizes the number of Kronecker deltas that are generated in the process. The solutions are presented in terms of generalized hypergeometric functions, obtained once the Kronecker deltas have been used in the series. Although the technique is general, we apply it to cases in which there are 2or3 different energy scales (masses or kinematic variables associated to the external momenta), obtaining solutions in terms of a finite sum of generalized hypergeometric series de 1 and 2 variables respectively, each of them expressible as ratios between the different energy scales that characterize the topology. The main result is a method capable of solving Feynman integrals, expressing the solutions as hypergeometric series of multiplicity $(n-1)$, where $n$ is the number of energy scales present in the diagram.

49 pages, 14 figures

References in corpus (14)

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• Scalar One-Loop Integrals using the Negative-Dimension Approach
• Gauss hypergeometric function: reduction, epsilon-expansion for integer/half-integer parameters and Feynman diagrams
• The two-loop scalar and tensor pentabox graph with light-like legs
• Application of the negative-dimension approach to massless scalar box integrals
• Negative Dimensional Integration: "Lab Testing" at Two Loops
• Massless and massive one-loop three-point functions in negative dimensional approach
• General massive one-loop off-shell three-point functions
• An easy way to solve two-loop vertex integrals
• Recursive method to obtain the parametric representation of a generic Feynman diagram
• Non-planar double-box, massive and massless pentabox Feynman integrals in negative dimensional approach
• Loop integrals in three outstanding gauges: Feynman, Light-cone and Coulomb
• Solutions for a massless off-shell two-loop three-point vertex

Cited by in corpus (14)

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• Mellin-Barnes meets Method of Brackets: A novel approach to Mellin-Barnes representations of Feynman integrals
• Multiloop calculations with Implicit Regularization in massless theories
• Modular application of an Integration by Fractional Expansion (IBFE) method to multiloop Feynman diagrams II
• Analytic calculation of doubly heavy hadron spectral density in coordinate space
• The Moments of the Hydrogen Atom by the Method of Brackets
• Feynman Diagrams and a Combination of the Integration by Parts (IBP) and the Integration by Fractional Expansion (IBFE) Techniques
• A Generalized Ramanujan Master Theorem and Integral Representation of Meromorphic Functions
• On the Method of Brackets: Rules, Examples, Interpretations and Modifications