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Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph. (English) Zbl 1370.81073

Summary: We consider the calculation of the master integrals of the three-loop massive banana graph. In the case of equal internal masses, the graph is reduced to three master integrals which satisfy an irreducible system of three coupled linear differential equations. The solution of the system requires finding a \(3 \times 3\) matrix of homogeneous solutions. We show how the maximal cut can be used to determine all entries of this matrix in terms of products of elliptic integrals of first and second kind of suitable arguments. All independent solutions are found by performing the integration which defines the maximal cut on different contours. Once the homogeneous solution is known, the inhomogeneous solution can be obtained by use of Euler’s variation of constants.

MSC:

81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
81T18 Feynman diagrams
11G55 Polylogarithms and relations with \(K\)-theory
33E05 Elliptic functions and integrals

Software:

Reduze; SecDec
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References:

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