Foundation and generalization of the expansion by regions

The expansion by regions is a method of asymptotic expansion developed by Beneke and Smirnov in 1997. It expands the integrand according to the scaling prescriptions of a set of regions and integrates all expanded terms over the whole integration domain. This method has been applied successfully to many complicated loop integrals, but a general proof for its correctness has still been missing. This paper shows how the expansion by regions manages to reproduce the exact result correctly in an expanded form and clarifies the conditions on the choice and completeness of the considered regions. A generalized expression for the full result is presented that involves additional overlap contributions. These extra pieces normally yield scaleless integrals which are consistently set to zero, but they may be needed depending on the choice of the regularization scheme. While the main proofs and formulae are presented in a general and concise form, a large portion of the paper is filled with simple, pedagogical one-loop examples which illustrate the peculiarities of the expansion by regions, explain its application and show how to evaluate contributions within this method.