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Nonperturbative renormalization of scalar quantum electrodynamics in d=3

### Abstract

For scalar quantum electrodynamics on a three-dimensional toroidal lattice with a fine lattice spacing, we consider the renormalization problem of choosing counter terms depending on the lattice spacing, so that the theory stays finite as the spacing goes to zero. We employ a renormalization group method which analyzes the flow of the mass and the vacuum energy as a problem in discrete dynamical systems. The main result is that counter terms can be chosen so that at the end of the iteration these quantities take preassigned values. No use is made of perturbation theory. The renormalization group transformations are defined with bounded fields, an approximation which can be justified in Balaban’s approach to the renormalization group.

© 2015 AIP Publishing LLC

Received 13 May 2015
Accepted 01 October 2015
Published online 30 October 2015

Article outline:

I. INTRODUCTION
A. Overview
B. The model
C. The scaled model
II. RG TRANSFORMATION FOR SCALARS
A. Block averages
B. The transformation
C. Compositions of averaging operators
D. Free flow
E. The next step
III. GREEN’S FUNCTIONS
A. Basic properties
B. Changes in background field
C. Local estimates
D. Random walk expansion
E. More random walk expansions
IV. RG TRANSFORMATIONS FOR GAUGE FIELDS
A. Axial gauge
B. Free flow
C. The next step
D. Other gauges
E. Parametrization of the fluctuation integral
F. Representation for *C*_{k}
,
G. Random walk expansions
V. POLYMER FUNCTIONS
A. A preliminary lemma
B. A regularity result
C. Bounded fields
D. Definition of polymer functions
E. Symmetries
F. Normalization
G. Arranging normalization
H. Localized Green’s functions
VI. THE MAIN THEOREM
A. The theorem
B. Proof of the theorem
1. Preliminaries
2. Gauge field translation
3. First localization
4. Restoration of dressed fields
5. Scalar field translation
6. Estimates
7. Adjustments
8. Second localization
9. Cluster expansion
10. Scaling
11. Completion of the proof
VII. NORMALIZATION FACTOR
A. Single scale
B. Improved single scale
C. Resummation
D. Photon self-energy
1. Estimates
2. Removal of averaging operators from interaction
3. An explicit representation
4. Removal of averaging operators from propagators
5. Proof of Lemma 33
VIII. THE FLOW

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