Symmetries of Feynman integrals and the Integration By Parts method

Abstract

Integration By Parts (IBP) is an important method for computing Feynman integrals. This work describes a formulation of the theory involving a set of differential equations in parameter space, and especially the definition and study of an associated Lie group G. The group acts on parameter space and foliates it into G-orbits. The differential equations essentially provide the gradient of the integral within the orbit in terms of integrals associated with degenerate diagrams. In this way the computation of a Feynman integral at a general point in parameter space is reduced to the evaluation of the Feynman integral at some freely chosen base point on the same orbit, together with a line integral inside the G-orbit and the degenerate integrals along the path. This paper restricts to vacuum diagrams and integrals without numerators, but the method is expected to generalize. The method is demonstrated by application to the two-loop vacuum diagram. A relation between the reducibility of a diagram though IBP and the reducibility of the associated electric circuit though the Y-Delta transform is speculated.

arXiv comment16 pages

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Cited by in corpus (7)

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• The vacuum seagull: evaluating a 3-loop Feynman diagram with 3 mass scales
• Bubble diagram through the Symmetries of Feynman Integrals method
• The algebraic locus of Feynman integrals
• Two-loop vacuum diagram through the Symmetries of Feynman Integrals method
• Algebraic aspects of when and how a Feynman diagram reduces to simpler ones
• Kite diagram through Symmetries of Feynman Integrals