Renormalization of a gauge theory in a nonlinear gauge

We discuss renormalization of an O(3) gauge model with the gauge fixing term given by ℒ_g.f.=-1/ζ|(∂_μ-igA ₃ ^μ )W ^+μ|²-(1/2α)(∂A ³)². We utilize earlier results on the general theory of renormalization of gauge theories in quadratic gauges to prove multiplicative renormalizability of the theory together with a subtractive renormalization of gauge fixing and ghost terms. We show that this model has a double BRS invariance and that it is preserved under renormalization.     

Optical Abelian lattice gauge theories

We discuss a general framework for the realization of a family of Abelian lattice gauge theories, i.e., link models or gauge magnets, in optical lattices. We analyze the properties of these models that make them suitable for quantum simulations. Within this class, we study in detail the phases of a U(1)$U\left(1\right)$-invariant lattice gauge theory in 2+1$2+1$ dimensions, originally proposed by P. Orland. By using exact diagonalization, we extract the low-energy states for small lattices, up to 4×4$4\times 4$. We confirm that the model has two phases, with the confined entangled one characterized by strings wrapping around the whole lattice. We explain how to study larger lattices by using either tensor network techniques or digital quantum simulations with Rydberg atoms loaded in optical lattices, where we discuss in detail a protocol for the preparation of the ground-state. We propose two key experimental tests that can be used as smoking gun of the proper implementation of a gauge theory in optical lattices. These tests consist in verifying the absence of spontaneous (gauge) symmetry breaking of the ground-state and the presence of charge confinement. We also comment on the relation between standard compact U(1)$U\left(1\right)$ lattice gauge theory and the model considered in this paper.     

Renormalization of gauge theories in nonlinear gauges

The usual rules for constructing the Lagrangian of fictitious particles in gauge theories is generalized. The gauge invariant and multiplicative renormalizability of the theory with generalized action in nonlinear gauges is proved.Translated from Izvestiya Vysshikh, Uchebnykh Zavedenii, Fizika, No. 6, pp. 11–15, June, 1981.The author thanks I. A. Batalin and B. L. Voronov for discussing the problems considered here.     

Superspace Ward identities in supersymmetric gauge theories

In superspace formulation of supersymmetric gauge theories, gauge invariance requires an infinite set of identities between the infinite set of renormalization constants. Using Ward identities in superspace, the same is derived. These identities at one loop level are also demonstrated.     

Gribov ambiguity in gauge theories

Gribov ambiguity in gauge field theories is discussed and it is shown that such an ambiguity exists even for Abelian theories in covariant gauge at finite temperature. Both geometric and algebraic proofs are presented. In view of the importance of non-perturbative methods, some special gauges are given in which such ambiguities do not exist or are not relevant. The significance of these in the study of confinement in QCD is pointed out.     

General theory of renormalization of gauge invariant operators

We study the question of renormalization of gauge invariant operators in the gauge theories. Our discussion applies to gauge invariant operators of arbitrary dimensions and tensor structure. We show that the gauge noninvariant (and ghost) operators that mix with a given set of gauge invariant operators form a complete set of local solutions of a functional differential equation. We show that this set of gauge noninvariant operators together with the gauge invariant operators close under renormalization to all orders. We obtain a complete set of local solutions of the differential equation. The form of these solutions has recently been conjectured by Kluberg Stern and Zuber. With the help of our solutions, we show that there exists a basis of operators in which the gauge noninvariant operators “decouple” from the gauge invariant operators to all orders in the sense that eigenvalues corresponding to the eigenstates containing gauge invariant operators can be computed without having to compute the full renormalization metrix. We further discuss the substructure of the renormalization matrix.     

Infinite-dimensional gauge groups and special nonlinear gravitons

A gauge theory in flat space—time, in which the gauge algebra is the (infinite-dimensional) algebra of vector fields on a surface, determines a curved space—time metric. This note deals with some completely integrable examples, concentrating on the N → ∞ limit of the Euler—Arnol'd equations [geodesics on SO(N)]. In this case, the metric turns out to be flat, which points the way to a coordinate transformation that solves the original equations.     

Charges in gauge theories

In this article we investigate charged particles in gauge theories. After reviewing the physical and theoretical problems, a method to construct charged particles is presented. Explicit solutions are found in the abelian theory and a physical interpretation is given. These solutions and our interpretation of these variables as the true degrees of freedom for charged particles, are then tested in the perturbative domain and are demonstrated to yield infra-red finite, on-shell Green’s functions at all orders of perturbation theory. The extension to collinear divergences is studied and it is shown that this method applies to the case of massless charged particles. The application of these constructions to the charged sectors of the standard model is reviewed and we conclude with a discussion of the successes achieved so far in this programme and a list of open questions. An erratum to this article is available at .     

Hamiltonian structure and gauge symmetries of Poincaré gauge theory

This is a review of the constrained dynamical structure of Poincaré gauge theory which concentrates on the basic canonical and gauge properties of the theory, including the identification of constraints, gauge symmetries and conservation laws. As an interesting example of the general approach, we discuss the teleparallel formulation of general relativity.     

Recent advances in lattice gauge theories

Recent progress in the field of lattice gauge theories is briefly reviewed for a nonspecialist audience. While the emphasis is on the latest and more definitive results that have emerged prior to this symposium, an effort has been made to provide them with minimal technicalities.     

2PI functional techniques for gauge theories: QED

We discuss the formulation of the prototype gauge field theory, QED, in the context of two-particle-irreducible (2PI) functional techniques with particular emphasis on the issues of renormalization and gauge symmetry. We show how to renormalize all n-point vertex functions of the (gauge-fixed) theory at any approximation order in the 2PI loop-expansion by properly adjusting a finite set of local counterterms consistent with the underlying gauge symmetry. The paper is divided in three parts: a self-contained presentation of the main results and their possible implementation for practical applications; a detailed analysis of ultraviolet divergences and their removal; a number of appendices collecting technical details.     

Topological gravity from a transgression gauge field theory

It is shown that a topological action for gravity in even dimensions can be obtained from a gravity theory whose Lagrangian is given by a transgression form invariant under the Poincaré group.     

Weighted power counting and Lorentz violating gauge theories. I: General properties

We construct local, unitary gauge theories that violate Lorentz symmetry explicitly at high energies and are renormalizable by weighted power counting. They contain higher space derivatives, which improve the behavior of propagators at large momenta, but no higher time derivatives. We show that the regularity of the gauge-field propagator privileges a particular spacetime breaking, the one into space and time. We then concentrate on the simplest class of models, study four dimensional examples and discuss a number of issues that arise in our approach, such as the low-energy recovery of Lorentz invariance.     

Quark masses,the Dashen phase,and gauge field topology

The CP violating Dashen phase in QCD is predicted by chiral perturbation theory to occur when the up–down quark mass difference becomes sufficiently large at fixed down-quark mass. Before reaching this phase, all physical hadronic masses and scattering amplitudes are expected to behave smoothly with the up-quark mass, even as this mass passes through zero. In Euclidean space, the topological susceptibility of the gauge fields is positive at positive quark masses but diverges to negative infinity as the Dashen phase is approached. A zero in this susceptibility provides a tentative signal for the point where the mass of the up quark vanishes. I discuss potential ambiguities with this determination.     

Renormalization of gauge theories: Non-linear gauges

The renormalization action for gauge theories quantized with non-linear gauge conditions is written down in its most general form. A natural interpretation of the well-known quartic ghost term generated by renormalization is given in terms of a generalized Faddeev-Popov determinant.     

Dynamics of N = 2 supersymmetric gauge theories in three dimensions

We study the structure of the moduli spaces of vacua and superpotentials of N = 2 supersymmetric gauge theories in three dimensions. By analyzing the instanton corrections, we compute the exact superpotentials and determine the quantum Coulomb and Higgs branches of the theories in the weak coupling regions. We find candidates for non-trivial N = 2 superconformal field theories at the singularities of the moduli spaces. The analysis is carried out explicitly for gauge groups U(N_c) and SU(N_c) with N _f flavors. We show that the field theory results are in complete agreement with the intersecting branes picture. We also compute the exact superpotentials for arbitrary gauge groups and arbitrary matter content.     

Gauge/string duality in confining theories

This is the content of a set of lectures given at the “XIII Jorge André Swieca Summer School on Particles and Fields”, Campos do Jordão, Brazil in January 2005. They intend to be a basic introduction to the topic of gauge/gravity duality in confining theories. We start by reviewing some key aspects of the low energy physics of non‐Abelian gauge theories. Then, we present the basics of the AdS/CFT correspondence and its extension both to gauge theories in different spacetime dimensions with sixteen supercharges and to more realistic situations with less supersymmetry. We discuss the different options of interest: placing D–branes at singularities and wrapping D–branes in calibrated cycles of special holonomy manifolds. We finally present an outline of a number of non‐perturbative phenomena in non‐Abelian gauge theories as seen from supergravity.     

Generally covariant vs. gauge structure for conformal field theories

We introduce the natural lift of spacetime diffeomorphisms for conformal gravity and discuss the physical equivalence between the natural and gauge natural structure of the theory. Accordingly, we argue that conformal transformations must be introduced as gauge transformations (affecting fields but not spacetime point) and then discuss special structures implied by the splitting of the conformal group.