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Communications in Commun. Math. Phys. 114, 257-315 (1988)
Mathematical
Physics
©Springer-Verlagl988
Effective Action and Cluster Properties of the Abelian Higgs Model Tadeusz Balaban1'*, John Z. Imbrie2'**, and Arthur Jaffe2'*** 1 2
Department of Mathematics, Northeastern University, Boston, MA 02115, USA Harvard University, Cambridge, MA02138, USA
Abstract. We continue our program to establish the Higgs mechanism and mass gap for the abelian Higgs model in two and three dimensions. We develop a multiscale cluster expansion for the high frequency modes of the theory, within a framework of iterated renormalization group transformations. The expansions yield decoupling properties needed for a proof of exponential decay of correlations. The result of this analysis is a gauge invariant unit lattice theory with a deep Higgs potential of the shape required to exhibit the Higgs mechanism. Table of Contents 1. 2. 3. 4. 5.
Introduction Localized Kernels The First Renormalization Step The Inductive Hypothesis Renormalization and Decoupling in the General Step 5.1. Renormalization Transformation 5.2. Restrictions on the Fields 5.3. First Gauge Field Translation 5.4. Gauge Transformation 5.5. Second Gauge Field Translation 5.6. Expansion with Respect to the Fluctuation Field 5.7. The Gaussian Normalization Factors 5.8. Scalar Field Translation 5.9. Bounds on Fluctuation and Block Fields 5.10. The Interaction for the Fluctuation Fields 5.11. Mayer Expansion I 5.12. Conditional Integration 5.13. Decoupling of the Small Field Region 5.14. Resummation and Extraction of the Perturbation Expansion 5.15. Second Mayer Expansion and Scaling References
258 260 265 273 277 277 278 280 281 283 285 289 295 296 298 299 300 303 307 312 314
* Research partially supported by the National Science Foundation under Grant DMS8602207 and by the Air Force Office of Scientific Research under Grant AFOSR-86-0229 ** Alfred P. Sloan Research Fellow. Research partially supported by the National Science Foundation under Grants PHY-84-13285 and PHY-85-13554 *** Research partially supported by the National Science Foundation under Grant PHY-8513554
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T. Balaban, J. Z. Imbrie, and A. Jaffe
1. Introduction We wish to establish the existence of a mass gap for the abelian Higgs model on the subspace of gauge invariant observables. Earlier work on this problem has led to a method to establish these results and to a partial solution [1, 2]. Here we continue this study with the development of a multiscale expansion suitable for the problem. The basic formulation of the model is given in [2]. We consider an action function Sε which is defined for a gauge theory on a lattice with spacing ε. We use the Wilson form of lattice action, which is gauge invariant. Thus it is important to consider gauge invariant observables such as loop variables
(1.1) where y is a closed curve on the lattice, or string variables
, y, Γ) = φ(x) exp ieε £ A(b)\ φ(y) ,
(1.2)
where Γ is a lattice curve from x to y. These variables must be renormalized appropriately, by multiplying or subtracting ε-dependent terms. For gauge invariant operators (but not in general) we expect exponential clustering in the equilibrium state defined by Sε. This state is given by the limit of normalized finite volume expectations = ?-Se-ssB(u9φ)2u&φ. ^./
(1.3)
(We assume periodic boundary conditions, but this is not crucial since as a corollary we establish the existence of the infinite volume limit.) Thus for gauge invariant functions B, C we expect |-|^0(l)exp[-mdist(£,C)],
(1.4)
where 0 < m and dist(£, C) denotes the distance between the supports of B and C. For unit lattice models, (1.4) was established in [3] and here we investigate the corresponding estimates uniformly in the lattice spacing ε. The exponential decay or mass gap is intimately connected with the Higgs mechanism. We see the Higgs mechanism at work through the evolution of the effective action as we proceed lower in momentum. The action on the ε-lattice appears almost massless, but as we approach the unit lattice, the Higgs potential exhibits a pronounced ring of minima at |φ| = ρ0, which leads to a mass term for the gauge field. The apparently massless rotational degrees of freedom of φ can be gauged away. To obtain decay, we need a convergent expansion with a small parameter. Thus, we restrict the coupling constants (
