Wilson Effective Action Research Articles

THE BV MASTER EQUATION FOR THE WILSON ACTION IN GENERAL YANG-MILLS GAUGE THEORY

We study the four dimensional gauge theory within Wilsonian Renormalization Group (WRG) method. The Wilson effective action for general Yang-Mills gauge theory is shown to satisfy the usual form of Batalin-Vilkovisky (BV) master equation, despite that a momentum cutoff apparently breaks the gauge invariance. In the case of Abelian gauge theory, in particular, it actually deduces the Ward-Takahashi identity for Wilson action recently derived by Sonoda. We elucidated the relation of our Wilson master action with that derived by Ref. 2 and, in particular, showed that our BV Master equation really reproduced the Sonoda's WT identity for the Wilson action in QED. (This is a proceeding to the conference based on the poster given by E.I.).

The BV Master Equation for the Wilson Action in General Yang-Mills Gauge Theory

The Wilson effective action for general Yang-Mills gauge theory is shown to satisfy the usual form of Batalin-Vilkovisky (BV) master equation, despite that a momentum cutoff apparently breaks the gauge invariance. In the case of Abelian gauge theory, in particular, it actually deduces the Ward-Takahashi identity for Wilson action recently derived by Sonoda.

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

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.

Weak coupling instabilities of two-dimensional lattice electrons

We present a consistent and systematic study of a two-dimensional extended Hubbard model in the weak coupling limit. Quite generally, the electron gas is unstable towards a superconducting state even in the absence of phonons, since high-energy spin fluctuations create an effective attraction between the quasi-particles. However in the special case of a half-filled nearest-neighbor tight-binding band, the Fermi surface is nested and the system is at a Van Hove singularity. In this situation, there are six competing instabilities: s- and d-wave superconductivity, spin- and charge-density waves and two phases with circulating charge and spin currents, respectively. The required renormalization group formalism can be presented on a most elementary level, connecting the idea of the “parquet summation” to the more modern concept of Wilson's effective action. As a result, a rich phase diagram is obtained as a function of the model interaction. We also argue that the one-loop approximation is insufficient in a case where the Fermi surface is at a Van Hove singularity but not nested.

Weakly interacting electrons and the renormalization group

AbstractWe present a general method to study weak‐coupling instabilities of a large class of interacting electron models in a controlled and unbiased way. Quite generally, the electron gas is unstable towards a superconducting state even in the absence of phonons, since high‐energy spin fluctuations create an effective attraction between the quasi‐particles. As an example, we show the occurrence of d‐wave pairing in the repulsive Hubbard model in two dimensions. In one dimension or if the Fermi surface is nested, there are several competing instabilities. The required renormalization group formalism for this case is presented to lowest (one‐loop) order on a most elementary level, connecting the idea of the “parquet summation” to the more modern concept of Wilson's effective action. The validity and restrictions of the one‐loop approximation are discussed in detail. As a result, three different renormalization group approaches known in the literature are shown to be equivalent within the regime of applicability. We also briefly discuss the open problem of a two‐dimensional Fermi system at Van Hove filling without nesting.

Weakly interacting electrons and the renormalization group

We present a general method to study weak-coupling instabilities of a large class of interacting electron models in a controlled and unbiased way. Quite generally, the electron gas is unstable towards a superconducting state even in the absence of phonons, since high-energy spin fluctuations create an effective attraction between the quasi-particles. As an example, we show the occurrence of d-wave pairing in the repulsive Hubbard model in two dimensions. In one dimension or if the Fermi surface is nested, there are several competing instabilities. The required renormalization group formalism for this case is presented to lowest (one-loop) order on a most elementary level, connecting the idea of the “parquet summation” to the more modern concept of Wilson's effective action. The validity and restrictions of the one-loop approximation are discussed in detail. As a result, three different renormalization group approaches known in the literature are shown to be equivalent within the regime of applicability. We also briefly discuss the open problem of a two-dimensional Fermi system at Van Hove filling without nesting.

Analytic calculation of the 1-loop effective action for the O( [formula omitted])-symmetric 2-dimensional nonlinear σ-model

Starting from the 2-dimensional nonlinear σ-model living on a lattice Λ of lattice spacing a with action S[φ]=− 1 2 β∫ zφΔφ , φ(z)∈S N we compute a manifestly covariant closed form expression for the Wilson effective action S eff[ Φ] on a lattice of lattice spacing ã in a 1-loop approximation for a Gaussian choice of blockspin, where Cφ(x)≡ Cφ(x)/| Cφ(x)| fluctuates around Φ( x). C is averaging of φ( z) over a block x. The limiting case of a δ-function is also considered. The result extends Polyakov which had furnished those contributions to the effective action which are of order ln a ̃ /a . The additional terms which remain finite as a↦0 include corrections other than coupling constant renormalization: a current–current interaction and a contribution from an augmented Jacobian which has a field dependence of a different kind than S has. Particular attention is paid to S eff's domain of validity in field space. It turns out that Hasenfratz and Niedermayer's choice of a low value of the parameter κ which governs the width of the Gaussian is optimal also in this respect.

Weakly correlated electrons on a square lattice: Renormalization-group theory

We formulate the exact Wilsonian renormalization group for a system of interacting fermions on a lattice. The flow equations for all vertices of the Wilson effective action are expressed in the form of the Polchinski equation. The advantage of this renormalization scheme is that the flow itself has a physical interpretation, i.e., the cutoff has the meaning of the temperature. We apply this method to the Hubbard model on a square lattice using both zero- and finite-temperature methods. Truncating the effective action at the sixth term in fermionic variables and neglecting self-energy renormalization, we obtain the one-loop functional renormalization equations for the effective interaction. We find the temperature of the instability ${T}_{c}^{\mathrm{RG}}$ as a function of doping. Furthermore we calculate the renormalization of the angle-resolved correlation functions for the superconductivity (SC) and for the antiferromagnetism (AF). The dominant component of the SC correlations is of the type ${d}_{{x}^{2}\ensuremath{-}{y}^{2}},$ while the AF fluctuations are of the type s. Following the strength of both SC and AF fluctuations along the instability line, we obtain the phase diagram. The temperature ${T}_{c}^{\mathrm{RG}}$ can be identified with the crossover temperature ${T}_{\mathrm{co}}$ found in the underdoped regime of the high-temperature superconductors, while in the overdoped regime ${T}_{c}^{\mathrm{RG}}$ corresponds to the superconducting critical temperature.

1-Loop improved lattice action for the nonlinear σ-model

In this paper we show the Wilson effective action for the 2-dimensional O( N + 1)-symmetric lattice nonlinear σ-model computed in the 1-loop approximation for the nonlinear choice of blockspin /gf( x ), /gf( x ) = C /gf( x )/| C /gf( x ) where C is averaging of the fundamental field /gf( z ) over a square x of side a . The result for S eff is composed of the classical perfect action with a renormalized coupling constant β eff , an augmented contribution from a Jacobian, and further genuine 1-loop correction terms. Our result extends Polyakov's calculation which had furnished those contributions to the effective action which are of order 1n a a , where a is the lattice spacing of the fundamental lattice. An analytic approximation for the background field which enters the classical perfect action will be presented elsewhere [1].

Wilson effective Kähler potential for supersymmetric nonlinearσmodels

Renormalization group methods are used to determine the evolution of the low energy Wilson effective action for supersymmetric nonlinear $\ensuremath{\sigma}$ models in four dimensions. For the case of supersymmetric ${\mathrm{CP}}^{N\ensuremath{-}1}$ models, the K\"ahler potential is determined exactly and is shown to exhibit a nontrivial ultraviolet fixed point in addition to a trivial infrared fixed point. The strong coupling behavior of the theory suggests the possible existence of additional relevant operators or nonperturbative degrees of freedom.

On the holomorphic structure of a low energy supersymmetric Wilson effective action

The Wilson (exact) renormalization group equations are used to determine the evolution of a general low energy N = 1 supersymmetric action containing a U(1) gauge vector multiplet and a neutral chiral multiplet. The effective theory evolves towards satisfying a fixed relation where the Kähler potential and effective gauge coupling are obtained from a N = 2 supersymmetric holomorphic prepotential.

THE EXACT RENORMALIZATION GROUP AND APPROXIMATE SOLUTIONS

We investigate the structure of Polchinski’s formulation of the flow equations for the continuum Wilson effective action. Reinterpretations in terms of I.R. cutoff Green’s functions are given. A promising nonperturbative approximation scheme is derived by carefully taking the sharp cutoff limit and expanding in “irrelevancy” of operators. We illustrate with two simple models of four-dimensional λφ4 theory: the cactus approximation, and a model incorporating the first irrelevant correction to the renormalized coupling. The qualitative and quantitative behaviour give confidence in a fuller use of this method for obtaining accurate results.

Consistency of Anomalies in SUSY Gauge Theories

We consider two problems in supersymmetric gauge theories - the anomaly supermultiplet and the calculation of the exact β functions in perturbation theory - which turn out to be interrelated. The main findings are:(i) generalization of the non-renormalization theorems for gauge F terms ∫ d2θ W2;(ii) solution of the anomaly puzzle and superextension of the Adler-Bardeen theorem;(iii) a constructive algorithm for computation of all-order terms in β functions;(iv) a new bosonic anomaly counting the number of the bosonic zero modes. We also show that Z factors of the matter superfields affect the magnitude of the observable gauge coupling. The basic technical element which ensured the breakthrough is the use of the Wilson operator expansion and distinction between the Wilson effective action Sw and the generator of 1PI vertices Γ.

Solution of the anomaly puzzle in SUSY gauge theories and the Wilson operator expansion

The present paper completes a series of works on β functions and the anomaly problem in supersymmetric theories. Exact expressions for the β functions are obtained within the framework of standard perturbation theory. The key observation is that the Wilson effective action S W( μ) does not coincide with the sum of vacuum loops in the external field Γ(μ). The difference is due to infrared effects. The coefficient 1/ g 2 in front of the operator W 2 in S W is renormalized only at one-loop level (extension of the non-renormalization theorem for F-terms). This fact results in the one-loop form of the anomalous operator equation for the supercurrent (generalization of the Adler-Bardeen theorem). The full Gell-Mann-Low function emerges after passing to matrix elements of the operators. The quantity entering observable amplitudes differs from 1/ g 2 by Σ i ln Z i where the factors Z i describe renormalization of the fields. (In this sense the Z factors of the matter fields become observable.) We discuss the relation with calculations of the instanton type.