Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph

Abstract

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.

39 pages, 3 figures; Fixed a typo in eq. (6.16)

References in corpus (10)

• Classical Polylogarithms for Amplitudes and Wilson Loops
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• Two-loop planar master integrals for Higgs$\to 3$ partons with full heavy-quark mass dependence
• Elliptic integral evaluations of Bessel moments
• The kite integral to all orders in terms of elliptic polylogarithms
• Simplifying differential equations for multi-scale Feynman integrals beyond multiple polylogarithms
• The algebraic structure of cut Feynman integrals and the diagrammatic coaction
• The Two Loop Crossed Ladder Vertex Diagram with Two Massive Exchanges
• Differential equations on unitarity cut surfaces
• Cuts and coproducts of massive triangle diagrams

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