Hard-soft renormalization and the exact renormalization group

Abstract

The Wilsonian exact renormalization group gives a natural framework in which ultraviolet and infrared divergences can be treated separately. In massless QED we introduce, as the only mass parameter, a renormalization scale Λ R > 0. We prove, using the flow equation technique, that infrared convergence is a necessary consequence of any zero-momentum renormalization condition at Λ R compatible with the effective Ward identities and axial symmetry. The same formalism is applied to renormalize gauge-invariant composite operators and to prove their infrared finiteness; in particular we consider the case of the axial current operator and its anomaly. * Work supported in part by M.U.R.S.T.

The nonperturbative functional renormalization group and its applications

“…This rarely happens, the only known examples are perturbation theory and self-consistent nPI schemes, [67,304,598,605,[743][744][745][746][747][748][749]. Moreover, even in the known examples, perturbation theory and nPI schemes, a comparison is only possible if taking into account the generically different RG scheme in the functional RG, for related work see [67,80,[82][83][84][85][86][87][88][89].…”

“…The set of composite operators O k [Φ] with the flow (85) includes general correlation functions with their connected and disconnected parts as well as functions of the source J[Φ]. For the latter case further terms enter (85), see [67]. An educative example for the former case of general correlation functions and the necessity of including the disconnected terms is the full two-point function…”

A generalised manifestly gauge invariant exact renormalisation group for SU(N) Yang–Mills

“…The answer is that we have discarded these terms already, on the basis that they vanish at O(p 2 ). 26 There are several strategies to demonstrate that the B ′ (0) terms vanish. In reference [16], it was demonstrated algebraically that the B ′ (0) terms cancel, at O(p 2 ): Taylor expanding the standard set sub-diagrams to zeroth order in p, we can algebraically substitute for all constituent structures.…”

Wilsonian flow and mass-independent renormalization

“…In the following we will call for short HS (hard-soft) scheme any renormalization scheme in which zero-momentum renormalization conditions at a Wilsonian scale Λ R > 0 are used. For a discussion of these schemes see [6,15]. For the beta and gamma functions one finds, using (14,16)…”

“…A recent discussion of the HS schemes in the Wilsonian approach can be found in [6]. The problem of deducing the renormalization group equation within these Wilsonian HS schemes has not yet been solved for a generic HS cut-off function.…”