Tree Expansion of the Wilson Effective Action and Reduction of the Polchinski Flow Equations

Abstract

In a tree expansion of the Wilson effective action, reduction of the Polchinski flow equations to their 1PI part is performed. We extract a certain subset of the connected cutoff Green functions that determines the renormalization group flow of the 1PI part of the Wilson effective action. 1. Introduction In the exact renormalization group (RG) approach to continuum field theories, there are two types of flow equations for the effective actions: the Polchinski flow equations (PFE) 1) for the Wilson effective action Seff ,a nd the fl ow equations 2), 3) for the Legendre effective actions Γeff . We call the latter “Legendre flow equations” (LFE). Although these two types of the flow equations are formally equivalent for exact effective actions Seff and Γeff, there is a clear distinction in their structure when used with some approximation method, such as truncation in a field polynomial expansion of the effective actions. Consider, for example, a NambuJona-Lasinio type model. In LFE, 4-fermi flow equations are not affected by higherdimensional multi-fermi operators. By contrast, the PFE requires the contribution of 6-fermi operators in order to determine the RG flow of 4-fermi operators. This difference originates from the fact that the action is 1PI in one case but not in the other. Furthermore, the Wilson effective action is known to have a strong cutoff scheme dependence, as pointed out in Ref. 4). In spite of these disadvantages, the Wilson effective action plays an important role in symmetry considerations. In particular, even if the standard realization of a symmetry is incompatible with the RG regularization, an effective but exact symmetry may be realized along the RG flow. The regularization-dependent symmetry which has attracted interest in recent years is characterized by the quantum master equation for the Wilson effective action. 5) It is not at all obvious if the realization of this symmetry is preserved under the Legendre transformation. Therefore, it is worth considering the RG approach based upon the Wilson effective action to which the PFE applies. The purpose of this note is to investigate the structure of Seff and the PFE in detail. Using a tree expansion, we extract a certain subset of the connected cutoff Green functions that determines the RG flow of the 1PI part of Seff , precisely in the same manner as the LFE. This is the 1PI reduction of the PFE. The entire set of PFE is shown to be decomposed into infinitely many sets of equations generated by the reduced PFE we have obtained. As an application of our formalism, we obtain the 6-fermi operators needed to describe the 4-fermi flow equations.