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arXiv · 2210.09898

Loop-by-loop Differential Equations for Dual (Elliptic) Feynman Integrals

Mathieu Giroux·Andrzej Pokraka
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Abstract

We present a loop-by-loop method for computing the differential equations of Feynman integrals using the recently developed dual form formalism. We give explicit prescriptions for the loop-by-loop fibration of multi-loop dual forms. Then, we test our formalism on a simple, but non-trivial, example: the two-loop three-mass elliptic sunrise family of integrals. We obtain an epsilon-form differential equation within the correct function space in a sequence of relatively simple algebraic steps. In particular, none of these steps relies on the analysis of $q$-series. Then, we discuss interesting properties satisfied by our dual basis as well as its simple relation to the known epsilon-form basis of Feynman integrands. The underlying K3-geometry of the three-loop four-mass sunrise integral is also discussed. Finally, we speculate on how to construct a "good" loop-by-loop basis at three-loop.

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Mathieu Giroux, Andrzej Pokraka. 2022-10-18. Loop-by-loop Differential Equations for Dual (Elliptic) Feynman Integrals. https://doi.org/10.1007/jhep03(2023)155

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