Conformal invariance for Wilson actions

H Sonoda

doi:10.1093/ptep/ptx114

Abstract

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.

Highlights

The study of conformally invariant field theories was initiated long ago byJ
We introduce the following the equation-of-motion composite operators: δ T
Note that the scale invariance, given by S = 0, is nothing but the exact renormalization group (ERG) differential equation for a fixed-point Wilson action, which is usually given in the form [4]

Summary

Introduction

The study of conformally invariant field theories (in dimensions D > 2) was initiated long ago by. Note that the scale invariance, given by S = 0, is nothing but the ERG differential equation for a fixed-point Wilson action, which is usually given in the form [4]. This rewriting is explained in Appendix B of Ref. We wish to rewrite the equation-of-motion composite operators (15) in terms of the generating functional W [J ] of connected correlations and the 1PI action [ ] associated with the Wilson action S[φ]. The invariance under the scale and special conformal transformations depends non-trivially on the cutoff function R. A similar expression was derived as (10) in Ref. [6]

Wilson–Fisher fixed point to order

Conclusion