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 3×3 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.
Highlights
The study of the mathematical structures that characterize multiloop Feynman integrals has played a crucial role for the most recent developments in the computation of higher order radia-A
We showed in [40] that an integral representation for the homogeneous solutions of the differential equation satisfied by any Feynman integral can be obtained by computing its maximal cut
The first, which has recently received a lot of attention, has to do with the explicit construction of a full set of homogeneous solutions of the system of couple differential equations satisfied by a family of Feynman integrals
Summary
We notice here that there is a clear correspondence between this simple idea and the methods described in [48] to count the number of independent master integrals.2 While this can be seen very already in the case of the two-loop massive sunrise graph, here we move one step forward and consider the first example of a Feynman graph that fulfills an irreducible third-order differential equation: the three-loop massive banana graph. This calculation requires finding three independent homogeneous solutions, for which we show how to derive integral representations from the study of its maximal cut only. From this point of view, we believe our results are complementary to the ones presented in [52]
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