Renormalization group analysis of differential equations subject to slowly modulated perturbations

The application of renormalization group (RG) theory to the asymptotic analysis of differential equations is considered. It is found that there is a class of small structural perturbations whose effects cannot be systematically treated using the Gell-Mann–Low RG approach applied in this context. Guided by a reinterpretation of the calculational procedure employed in this approach, whose motivation is provided by the unnecessarily complicated nature of its ‘standard’ interpretation, we formulate a generalized perturbative analysis and an RG approach which naturally and systematically treat equations subject to perturbations of this class. This formulation of RG theory is demonstrated with a number of examples for which the Gell-Mann–Low formulation fails to provide a systematic theoretical framework. For one representative example, it is found that the Wilson RG formulation also fails. The implications of this failure are discussed.