Renormalization and asymptotics

After a gentle introduction to the Stuckelberg-Petermann style (i.e. field-theoretical) renormalization group (RG) theory, its application to the study of asymptotic behaviors of differential equations is explained through simple examples. The essence of singular perturbation methods to study asymptotic behaviors of differential equations is to reduce it to equations governing long time scale behaviors (i.e. the so-called reductive perturbation). The RG approach gives the reduced equation as an RG equation (this is called the reductive renormalization group approach). Once the RG equation is written down, the asymptotic behavior can be obtained by solving it. The RG equation also facilitates the error analysis of the asymptotic solutions. A new approach via 'proto-RG equation' explained in this article further simplifies the reductive use of RG. For example, to the lowest nontrivial order the approach does not require any explicit calculation of perturbative results.