Abstract
In this paper we develop and demonstrate a method to obtain epsilon factorized differential equations for elliptic Feynman integrals. This method works by choosing an integral basis with the property that the period matrix obtained by integrating the basis over a complete set of integration cycles is diagonal. The method is a generalization of a similar method known to work for polylogarithmic Feynman integrals. We demonstrate the method explicitly for a number of Feynman integral families with an elliptic highest sector.
Highlights
In this paper we develop and demonstrate a method to obtain epsilon factorized differential equations for elliptic Feynman integrals
For Feynman integrals in the traditional canonical form it is often straight forward to integrate them up order by order in the epsilon expansion, giving results in the function class of generalized polylogarithms [103]
Many attempts have been made to extend this function class to elliptic cases and beyond [22, 28, 40, 46], yet none of those seem directly applicable to the form of the differential equations derived in this paper
Summary
We will in this work encounter the complete elliptic integrals of respectively first, second and third kind. We will in this paper make significant use of generalized unitarity cuts, in particular maximal cuts which refers to cuts of all genuine propagators (and differs from the notation in the N =4 literature, in which a maximal is defined to fix all degrees of freedom). This is due to the fact that differential equations [36], as well as IBP relations [78], remain the same after such a cutting procedure, and the Baikov parametrization is suited for generalized unitarity cuts at the integral level [67, 80].
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