How to Integrate Divergent Integrals: a Pure Numerical Approach to Complex Loop Calculations

Abstract

Loop calculations involve the evaluation of divergent integrals. Usually [1] one computes them in a number of dimensions different than four where the integral is convergent and then one performs the analytical continuation and considers the Laurent expansion in powers of epsilon =n-4. In this paper we discuss a method to extract directly all coefficients of this expansion by means of concrete and well defined integrals in a five dimensional space. We by-pass the formal and symbolic procedure of analytic continuation; instead we can numerically compute the integrals to extract directly both the coefficient of the pole 1/epsilon and the finite part.