Wilsonian Flow and Mass-Independent Renormalization

Abstract

We derive the Gell-Mann and Low renormalization group equation in the Wilsonian approach to renormalization of massless $g{\varphi }^4$ in four dimensions, as a particular case of a non-linear equation satisfied at any scale by the Wilsonian effective action. We give an exact expression for the $\beta$ and ${\gamma }_{\varphi }$ functions in terms of the Wilsonian effective action at the Wilsonian renormalization scale ${Ł}_R$; at the first two loops they are simply related to the gradient of the flow of the relevant couplings and have the standard values; beyond two loops this relation is spoilt by corrections due to irrelevant couplings. We generalize this analysis to the case of massive $g{\varphi }^4$, introducing a mass-independent Wilsonian renormalization scheme; using the flow equation technique we prove renormalizability and we show that the limit of vanishing mass parameter exists. We derive the corresponding renormalization group equation, in which $\beta$ and ${\gamma }_{\varphi }$ are the same as in the massless case; ${\gamma }_m$ is also mass-independent; at one loop it is the gradient of a relevant coupling and it has the expected value.

29 pages, Revtex

References in corpus (1)

• Hard-Soft Renormalization and the Exact Renormalization Group

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