Integration-by-parts identities and differential equations for parametrised Feynman integrals

Daniele Artico; Lorenzo Magnea

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arXiv · 2310.03939

Integration-by-parts identities and differential equations for parametrised Feynman integrals

Daniele Artico·Lorenzo Magnea

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Abstract

Integration-by-parts (IBP) identities and differential equations are the primary modern tools for the evaluation of high-order Feynman integrals. They are commonly derived and implemented in the momentum-space representation. We provide a different viewpoint on these important tools by working in Feynman-parameter space, and using its projective geometry. Our work is based upon little-known results pre-dating the modern era of loop calculations: we adapt and generalise these results, deriving a very general expression for sets of IBP identities in parameter space, associated with a generic Feynman diagram, and valid to any loop order, relying on the characterisation of Feynman-parameter integrands as projective forms. We validate our method by deriving and solving systems of differential equations for several simple diagrams at one and two loops, providing a unified perspective on a number of existing results.

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Daniele Artico, Lorenzo Magnea. 2023-10-05. Integration-by-parts identities and differential equations for parametrised Feynman integrals. https://arxiv.org/abs/2310.03939

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