Asymptotic expansions with the strategy of regions
Dr. Bernd Jantzen (RWTH Aachen)

When loop integrals involve many different scales from masses and kinematical parameters, it can be hard or even impossible to evaluate them exactly. The integrand may be simplified before integration by exploiting hierarchies of parameters and expanding in powers of small parameter ratios. Naive expansions of the integrand often generate new singularities, but there are sophisticated methods of asymptotic expansions to solve this problem. One of them is the so-called "strategy of regions" or "expansion by regions" 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 the correctness of its prescriptions has not been available.
This talk 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.
The talk illustrates the application of the expansion by regions using a variety of simple, pedagogical one-loop examples.