Progress and bottlenecks in simulating lattice gauge theory with fermions

We present a progress report in lattice gauge theory computer simulations which includes the effects of light, dynamical fermions. Microcanonical and hybrid microcanonical-Langevin alogrithms are presented and discussed. A method for “accelerating” stochastic differential equations and defeating critical slowing down is reviewed. Physics applications such as the thermodynamics of quantum chromodynamics, hierarchal energy scales in unified gauge theories, and the phase diagram of theories with many fermion species are discussed. Prospects for future research are assessed.     

Scalar lattice gauge theory

Scalar lattice gauge theories are models for scalar fields with local gauge symmetries. No fundamental gauge fields, or link variables in a lattice regularization, are introduced. The latter rather emerge as collective excitations composed from scalars. For suitable parameters scalar lattice gauge theories lead to confinement, with all continuum observables identical to usual lattice gauge theories. These models or their fermionic counterpart may be helpful for a realization of gauge theories by ultracold atoms. We conclude that the gauge bosons of the standard model of particle physics can arise as collective fields within models formulated for other “fundamental” degrees of freedom.     

ZN Gauge theories on a lattice and quantum memory

In the present paper we shall study (2+1)-dimensional Z_N gauge theories on a lattice. It is shown that the gauge theories have two phases, one is a Higgs phase and the other is a confinement phase. We investigate low-energy excitation modes in the Higgs phase and clarify relationship between the Z_N gauge theories and Kitaev’s model for quantum memory and quantum computations. Then we study effects of random gauge couplings (RGC) which are identified with noise and errors in quantum computations by Kitaev’s model. By using a duality transformation, it is shown that time-independent RGC give no significant effects on the phase structure and the stability of quantum memory and computations. Then by using the replica methods, we study Z_N gauge theories with time-dependent RGC and show that nontrivial phase transitions occur by the RGC.     

Anomalous gauge theories as constrained Hamiltonian systems

Anomalous gauge theories considered as constrained systems are investigated. The effects of chiral anomaly on the canonical structure are examined first for nonlinear σ-model and later for fermionic theory. The breakdown of the Gauss law constraints and the anomalous commutators among them are studied in a systematic way. An intrinsic mass term for gauge fields makes it possible to solve the Gauss law relations as second class constraints. Dirac brackets between the time components of gauge fields are shown to involve anomalous terms. Based upon the Ward-Takahashi identities for gauge symmetry, we investigate anomalous fermionic theory within the framework of path integral approach.     

Theorems for asymptotic safety of gauge theories

We classify the weakly interacting fixed points of general gauge theories coupled to matter and explain how the competition between gauge and matter fluctuations gives rise to a rich spectrum of high- and low-energy fixed points. The pivotal role played by Yukawa couplings is emphasised. Necessary and sufficient conditions for asymptotic safety of gauge theories are also derived, in conjunction with strict no go theorems. Implications for phase diagrams of gauge theories and physics beyond the Standard Model are indicated.     

Fermion condensation in the field strength approach to non-abelian yang-mills theories

Within the field strength approach to Yang-Mills theories the fermionic sectors of gauge theories are bosonized for the SU(2) and SU(3) gauge group. The emerging effective meson theories are studied in the tree approximation. In this approximation the original minimal gauge coupling of the quarks to gluons is rendered into an effective local four-fermion interaction with non-trivial Lorentz and gauge structure. The Schwinger-Dyson equation is solved in the strong coupling limit and the quark condensates and constituent masses are evaluated.     

Dynamical supersymmetry breaking and gauge anomalies

Some aspects of supersymmetric gauge theories and discussed. It is shown that dynamical supersymmetry breaking does not occur in supersymmetric QED in higher dimensions. The cancellation of both local (perturbative) and global (non-perturbative) gauge anomalies are also discussed in supersymmetric gauge theories. We argue that there is no dynamical supersymmetry breaking in higher dimensions in any supersymmetric gauge theories free of gauge anomalies. It is also shown that for supersymmetric gauge theories in higher dimensions with a compact connected simple gauge group, when the local anomaly-free condition is satisfied, there can be at most a possibleZ ₂ global gauge anomaly in extended supersymmetricSO(10) (or spin (10)) gauge theories inD=10 dimensions containing additional Weyl fermions in a spinor representation ofSO(10) (or spin (10)). In four dimensions with local anomaly-free condition satisfied, the only possible global gauge anomalies in supersymmetric gauge theories areZ ₂ global gauge anomalies for extended supersymmetricSP(2N) (N=rank) gauge theories containing additional Weyl fermions in a representation ofSP(2N) with an odd 2nd-order Dynkin index.     

Chemical potentials in gauge theories

One-loop calculations of the thermodynamic potential Ω are presented for temperature gauge and non-gauge theories. Prototypical formulae are derived which give Ω as a function of both (i) boson and/or fermion chemical potential, and in the case of gauge theories (ii) the thermal vacuum parameter A₀=const (A_μ is the euclidean gauge potential). From these basic abelian gauge theory formulae, the one-loop contribution to Ω can readily be constructed for Yang-Mills theories, and also for non-gauge theories.     

Structure of the exact effective action and quark confinement in MSSM QCD

An expression for the exact (beyond perturbation theory) effective action in N=1 supersymmetric gauge theories where all particles, with the exception of gauge bosons, are massive is proposed. By analyzing the form of this expression, it is shown that, in supersymmetric theories, instanton effects can lead to quark confinement. On the basis of first principles, the characteristic scale of confinement is calculated within MSSM QCD, and the result is found to be consistent with experimental data. The proposed explanation differs drastically with the dual Higgs mechanism.     

Classical gauge theories as dynamical systems—Regularity and chaos

In this review we present the salient features of dynamical chaos in classical gauge theories with spatially homogeneous fields. The chaotic behaviour displayed by both abelian and non-abelian gauge theories and the effect of the Higgs term in both cases are discussed. The role of the Chern-Simons term in these theories is examined in detail. Whereas, in the abelian case, the pure Chern-Simons-Higgs system is integrable, addition of the Maxwell term renders the system chaotic. In contrast, the non-abelian Chern-Simons-Higgs system is chaotic both in the presence and the absence of the Yang-Mills term. We support our conclusions with numerical studies on plots of phase trajectories and Lyapunov exponents. Analytical tests of integrability such as the Painlevé criterion are carried out for these theories. The role of the various terms in the Hamiltonians for the abelian Higgs, Yang-Mills-Higgs and Yang-Mills-Chern-Simons-Higgs systems with spatially homogeneous fields, in determining the nature of order-disorder transitions is highlighted, and the effects are shown to be counter-intuitive in the last-named system.     

Gauge dependence of ultraviolet behavior of gauge theories

The gauge dependence of ultraviolet behavior of gauge theories is examined on the basis of renormalization-group equation. Non-Abelian gauge theories are confirmed to be asymptotically free in an arbitrary gauge. It is also shown that the effective gauge parameter approaches a finite value in the ultraviolet limit in contrast with the case of QED.     

Solution of the gauge hierarchy problem

We propose a novel solution to the gauge hierarchy problem in theories with softly broken supersymmetry. Quantum effects can resuscitate classically sick theories, producing the large scale from the small supersymmetry breaking scale. We use this mechanism to construct realistic SU(6) and SU(5) GUTs which do not suffer from gauge hierarchy or fine tuning problems.     

Gauge symmetry as symmetry of matrix coordinates

We propose a new point of view on gauge theories, based on taking the action of symmetry transformations directly on the space coordinates. Via this approach the gauge fields are not introduced at the first step, and they can be interpreted as fluctuations around some classical solutions of the model. The new point of view is connected to the lattice formulation of gauge theories, and the parameter of the non-commutativity of the coordinates appears as the lattice spacing parameter. Through the statements concerning the continuum limit of lattice gauge theories, the suggestion arises that the non-commutative spaces are the natural ones to formulate gauge theories at strong coupling. Via this point of view, a close relation between the large-N limit of gauge theories and string theory can be made manifest. Received: 16 June 2000 / Published online: 8 September 2000     

Monte Carlo computations in lattice gauge theories

The formulation of gauge theories on Euclidean space-time lattices and the application of the Monte Carlo computational technique to the ensuing systems are reviewed. A variety of numerical results obtained for lattice gauge theories are presented and discussed.     

On a Mathematical Definition of a Renormalization Transformation for Lattice Gauge Theories

In lattice gauge theories, the renormalization transformation and its properties are formally defined and formally proved by making use of Dirac's function and its properties. In this Letter, we shall give a mathematically rigorous definition of a renormalization transformation for lattice pure gauge field theories and show the required properties, which are use to show ultraviolet stability of lattice gauge theories.     

Dispersion relation approach to the anomalous magnetic moment in gauge theories

The reason why the dispersion relations for the anomalous magnetic moment are not valid in gauge theories is explained. New sum rules are derived based on unitary bounds of scattering amplitudes. In gauge theories these sum rules give the correct values for the anomalous magnetic moment, while, in the case of conventional renormalizable theories which contain no massive vector bosons, they are identical with the usual dispersion relations for the anomalous magnetic moment.     

Geometry and topology of chiral anomalies in gauge theories

Geometrical and topological aspects of chiral anomalies in gauge theories are reviewed. Geometrical and topological concepts and results for chiral anomalies in gauge theories are considered, including differential forms, Lie groups, homotopy, homology, cohomology, Riemannian manifolds, fibre bundles, characteristic classes, index theorems and spectral flow. Gauge theories and their formulation in terms of differential forms and fibre bundles are described. The quantisation of gauge theories is performed using path integrals, and the orbit space, BRST symmetries and ? vacuum are discussed. Gauge theories with fermions are formulated, including realistic models of strong and weak interactions. Chiral anomalies and related issues such as the existence of Schwinger terms, their origins in terms of differential forms, cohomology, the orbit space, BRST identities, Hamiltonian systems and relations to index theorems are analysed. Constraints on models for particle physics from chiral anomalies and theories involving spontaneously broken chiral symmetry described by effective Lagrangians are also mentioned.     

Gauge bosons at zero and finite temperature

Gauge theories of the Yang–Mills type are the single most important building block of the standard model of particle physics and beyond. They are an integral part of the strong and weak interactions, and in their Abelian version of electromagnetism. Since Yang–Mills theories are gauge theories their elementary particles, the gauge bosons, cannot be described without fixing a gauge. Therefore, to obtain their properties a quantized and gauge-fixed setting is necessary.     

The ’t Hooft vertex revisited

In 1976 ’t Hooft introduced an elegant approach towards understanding the physical consequences of the topological structures that appear in non-Abelian gauge theories. These effects are concisely summarized in terms of an effective multi-fermion interaction. These old arguments provide a link between a variety of recent and sometimes controversial ideas including discrete chiral symmetries appearing in some models for unification, ambiguities in the definition of quark masses, and flaws with some simulation algorithms in lattice gauge theory.     

Crossover and scaling in SU(3) lattice gauge theories with Susskind fermions

The phase diagram of SU(3) lattice gauge theories with Susskind fermions is investigated by Monte Carlo methods. For three flavors in the continuum a significant shift in the location of the peak in the specific heat is found, as compared to the pure gauge case. These results suggest that the crossover region moves to smaller β when fermion polarization effects are included.