Problems of the Strategy of Regions
V. A. Smirnov
TL;DR
This work examines limitations and ambiguities of the generalized strategy of regions for the asymptotic expansion of Feynman integrals in Minkowski-space limits. By analyzing characteristic two-loop vertex diagrams, Smirnov demonstrates that naive region sets can miss contributions unless nonstandard regions (e.g., ultracollinear) are included, and he reformulates the approach in the alpha-parameter representation to clarify bookkeeping and avoid double counting. The alpha-language provides a Lorentz-invariant framework in which region contributions correspond to relative orders among alpha parameters and offers practical recipes for identifying regions, including sector-based analyses of singularities. Overall, the paper highlights both the utility and the caveats of the region strategy, especially in on-shell limits, and provides guidance on testing expansions via pole cancellations and analytic checks while illustrating the emergence of nontrivial regions.
Abstract
Problems that arise in the application of general prescriptions of the so-called strategy of regions for asymptotic expansions of Feynman integrals in various limits of momenta and masses are discussed with the help of characteristic examples of two-loop diagrams. The strategy is also reformulated in the language of alpha parameters.