Exact Renormalization Group Formalism Research Articles

Gradient Flow Exact Renormalization Group for Scalar Quantum Electrodynamics

Gradient Flow Exact Renormalization Group (GF-ERG) is a framework to define the renormalization group flow of Wilsonian effective action utilizing coarse-graining along the diffusion equations. We apply it for Scalar Quantum Electrodynamics and derive flow equations for the Wilsonian effective action with the perturbative expansion in the gauge coupling. We focus on the quantum corrections to the correlation functions up to the second order of the gauge coupling and discuss the gauge invariance of the GF-ERG flow. We demonstrate that the anomalous dimension of the gauge field agrees with the standard perturbative computation and that the mass of the photon keeps vanishing in general spacetime dimensions. The latter is a noteworthy fact that contrasts with the conventional Exact Renormalization Group formalism in which an artificial photon mass proportional to a cutoff scale is induced. Our results imply that the GF-ERG can give a gauge-invariant renormalization group flow in a non-perturbative way.

Correction to: Chiral anomaly as a composite operator in the gradient flow exact renormalization group formalism

Correction to: Chiral anomaly as a composite operator in the gradient flow exact renormalization group formalism

Chiral anomaly as a composite operator in the gradient flow exact renormalization group formalism

Abstract The gradient flow exact renormalization group (GFERG) is an idea that incorporates gauge-invariant gradient flows into the formalism of the exact renormalization group (ERG). GFERG introduces a Wilson action with a cutoff while keeping the vector gauge invariance manifestly. The details of the formalism are still to be worked out. In this paper, we apply GFERG to construct the Wilson action of massless Dirac fermions under the background chiral gauge fields. By formulating the chiral anomaly as a “composite operator”, we make the scale invariance of the anomaly manifest. We argue that the same result extends to QCD.

Products of current operators in the exact renormalization group formalism

Abstract Given a Wilson action invariant under global chiral transformations, we can construct current composite operators in terms of the Wilson action. The short-distance singularities in the multiple products of the current operators are taken care of by the exact renormalization group. The Ward–Takahashi identity is compatible with the finite momentum cutoff of the Wilson action. The exact renormalization group and the Ward–Takahashi identity together determine the products. As a concrete example, we study the Gaussian fixed-point Wilson action of the chiral fermions to construct the products of current operators.

Operator product expansion coefficients in the exact renormalization group formalism

We study how to compute the operator product expansion coefficients in the exact renormalization group formalism. After discussing possible strategies, we consider some examples explicitly, within the $\epsilon$-expansions, for the Wilson-Fisher fixed points of the real scalar theory in $d=4-\epsilon$ dimensions and the Lee-Yang model in $d=6-\epsilon$ dimensions. Finally we discuss how our formalism may be extended beyond perturbation theory.

Derivation of a gradient flow from the exact renormalization group

We establish a concrete correspondence between a gradient flow and the renormalization group flow for a generic scalar field theory. We use the exact renormalization group formalism with a particular choice of the cutoff function.

Products of composite operators in the exact renormalization group formalism

We discuss a general method of constructing the products of composite operators using the exact renormalization group formalism. Considering mainly the Wilson action at a generic fixed point of the renormalization group, we give an argument for the validity of short distance expansions of operator products. We show how to compute the expansion coefficients by solving differential equations, and test our method with some simple examples.

Geometry of the theory space in the exact renormalization group formalism

We consider the theory space as a manifold whose coordinates are given by the couplings appearing in the Wilson action. We discuss how to introduce connections on this theory space. A particularly intriguing connection can be defined directly from the solution of the exact renormalization group (ERG) equation. We advocate a geometric viewpoint that lets us define straightforwardly physically relevant quantities invariant under the changes of a renormalization scheme.

A holographic form for Wilson's RG

An attempt is made to make precise the connection between Wilson's RG and “Holographic RG” by writing Wilson's RG in a holographic form. A functional formulation is given for the exact RG evolution of a scalar field in d (flat) dimensions. It is shown that a change of variables maps the action to that for a scalar field in AdSd+1. This provides a holographic form for Wilson's RG that can be called “Holographic RG”. This mapping can only be done for a specific form of the cutoff function in the Exact Renormalization Group formalism. The notion of scale and conformal invariance in the presence of a finite UV cutoff is emphasized. The discussion is primarily about the two-point function and the Gaussian fixed point. Some remarks are made about nontrivial fixed points.

Conformal invariance for Wilson actions

We discuss the realization of conformal invariance for Wilson actions using the formalism of the exact renormalization group. This subject has been studied extensively in the recent works of O. J. Rosten. The main purpose of this paper is to reformulate Rosten's formulas for conformal transformations using a method developed earlier for the realization of any continuous symmetry in the exact renormalization group formalism. The merit of the reformulation is simplicity and transparency via the consistent use of equation-of-motion operators. We derive equations that imply the invariance of the Wilson action under infinitesimal conformal transformations which are non-linearly realized but form a closed conformal algebra. The best effort has been made to make the paper self-contained; ample background on the formalism is provided.

Gauge invariant composite operators of QED in the exact renormalization group formalism

Using the exact renormalization group (ERG) formalism, we study the gauge invariant composite operators in QED. Gauge invariant composite operators are introduced as infinitesimal changes of the gauge invariant Wilson action. We examine the dependence on the gauge fixing parameter of both the Wilson action and gauge invariant composite operators. After defining ‘gauge fixing parameter independence,’ we show that any gauge independent composite operators can be made ‘gauge fixing parameter independent’ by appropriate normalization. As an application, we give a concise but careful proof of the Adler–Bardeen non-renormalization theorem for the axial anomaly in an arbitrary covariant gauge by extending the original proof by A Zee.