Planar Double Box Integral for Top Pair Production with a Closed Top Loop to all orders in the Dimensional Regularization Parameter.

We compute systematically for the planar double box Feynman integral relevant to top pair production with a closed top loop the Laurent expansion in the dimensional regularization parameter ϵ. This is done by transforming the system of differential equations for this integral and all its sub-topologies to a form linear in ϵ, where the ϵ^{0} part is strictly lower triangular. This system is easily solved order by order in the dimensional regularization parameter ϵ. This is an example of an elliptic multiscale integral involving several elliptic subtopologies. Our methods are applicable to similar problems.

Simplifying Differential Equations for Multiscale Feynman Integrals beyond Multiple Polylogarithms.

In this Letter we exploit factorization properties of Picard-Fuchs operators to decouple differential equations for multiscale Feynman integrals. The algorithm reduces the differential equations to blocks of the size of the order of the irreducible factors of the Picard-Fuchs operator. As a side product, our method can be used to easily convert the differential equations for Feynman integrals which evaluate to multiple polylogarithms to an ϵ form.