Evaluating Master Integrals by Dimensional Recurrence and Analyticity

Abstract

In this chapter, the method called ‘dimensional recurrence and analyticity’ (DRA) is described following recent papers by Lee [24–26] and his papers with coauthors [27–32, 34].It is based on so-called dimensional recurrence relations (DRR) which express a given master integral considered in dimension \(d-2\) or \(d+2\) as a linear combination of Feynman integrals in dimension \(d\) with shifted indices. In the next section, two kinds of such relations are described. As in the case of the method of differential equations it is assumed that one can perform an IBP reduction [12] for a given family of Feynman integrals. Using a solution of IBP relations with the help of the algorithms described in Chap. 6, the linear combinations on the right-hand sides of the DRR can be represented as linear combinations of master integrals so that we obtain a difference equation (or, a system of difference equations) with respect to the variable \(d\). Then this equation is solved by finding its solution in the form of a series and fixing then the arbitrariness encoded in the solution of the corresponding homogeneous equation with the help of information about properties of the given Feynman integral as an analytic function of \(d\).