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On the maximal cut of Feynman integrals and the solution of their differential equations. (English) Zbl 1356.81136
Summary: The standard procedure for computing scalar multi-loop Feynman integrals consists in reducing them to a basis of so-called master integrals, derive differential equations in the external invariants satisfied by the latter and, finally, try to solve them as a Laurent series in \(\epsilon = (4 - d) / 2\), where \(d\) are the space-time dimensions. The differential equations are, in general, coupled and can be solved using Euler’s variation of constants, provided that a set of homogeneous solutions is known. Given an arbitrary differential equation of order higher than one, there exists no general method for finding its homogeneous solutions. In this paper we show that the maximal cut of the integrals under consideration provides one set of homogeneous solutions, simplifying substantially the solution of the differential equations.

MSC:
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
35F35 Systems of linear first-order PDEs
35G35 Systems of linear higher-order PDEs
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