Evaluation by Differential Equations

Abstract

In contrast to the method of alpha parametric representation, the method of MB representation and many other methods of evaluating individual Feynman integrals, the two methods presented in this and subsequent chapter are oriented at the evaluation of master integrals. This means that we have a solution of the IBP relations [21] for a given family of Feynman integrals, using some technique described in the previous chapter. The method of differential equation (DE) was suggested in [31–35] and developed in [46] and later papers (see references below). The idea of the method is to take some derivatives of a given master integral with respect to kinematical invariants and masses. Then the result of this differentiation is written in terms of Feynman integrals of the given family and, according to the known reduction, in terms of the master integrals. Therefore, one obtains a system of differential equations for the master integrals which can be solved with appropriate boundary conditions.