Evaluating by Alpha and Feynman Parameters

Abstract

Feynman parameters¹ are very well known and often used in practical calculations. They are closely related to alpha parameters introduced in Chap. 2 so that we shall study both kinds of parametric representations of Feynman integrals in one chapter. The use of these parameters enables us to transform Feynman integrals over loop momenta into parametric integrals where Lorentz invariance becomes manifest. Using alpha parameters we shall first evaluate one and two-loop integrals with general complex powers of the propagators, within dimensional regularization, for which results can be written in terms of gamma functions for general values of the dimensional regularization parameter. We shall show then how these formulae, together with simple algebraic manipulations, enable us to evaluate some classes of Feynman integrals.