Renormalization group equations in lattice gauge theories

New recursion equations for renormalization group transformations of the Migdal-Kadanoff type are obtained for gauge systems including fermion variables on a d-dimensional Euclidean space-time lattice. It is shown that in the weak gauge coupling region these equations have β-functions similar to those of continuum field theories in the case of U(1), SU(2) gauge groups (QED, QCD). On the other hand in the strong-coupling limit there is an infrared attractive fixed point corresponding to a color-confining effective system in both groups. A possible entire trajectory of the non-Abelian system is briefly conjectured.     

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.     

Renormalization group approach to lattice gauge field theories

The fluctuation field integral, constructed in Part I, is represented by the exponentiated cluster expansion. It is proved that the terms of the expansion satisfy the inductive assumptions. This completes the construction of the sequence of effective actions in the small field approximation.Work supported in part by the Air Force under Grant AFOSR-86-0229 and by the National Science Foundation under Grant DMS-86-02207     

Renormalization group approach to lattice gauge field theories

We study four-dimensional pure gauge field theories by the renormalization group approach. The analysis is restricted to small field approximation. In this region we construct a sequence of localized effective actions by cluster expansions in one step renormalization transformations. We construct also -functions and we define a coupling constant renormalization by a recursive system of renormalization group equations.     

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O(N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g²-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly.The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P(N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P(N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a $\text{1}\text{N}$ expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.     

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.     

Renormalization and continuum limit of composite operators in lattice gauge theories

The gluon condensate of dimension 4 is extracted from different operators in a pureSU(2) lattice gauge theory. Multiplicative finite renormalization effects are observed, which are in qualitative agreement with one loop perturbative calculations. Asymptotic scaling is found in the range 2.45≦β≦2.85.     

Renormalization of an abelian gauge theory in stochastic quantization

The renormalization of an abelian gauge field coupled to a complex scalar field is disccused in the stochastic quantization method. The supper space formulation of the stochastic quantization method is used to derived the Ward Takahashi identities assocoated with supersymmetry. These Ward Takahashi identities together with previously derived Ward Takahshi identities associated with gauge invariance are shown to be sufficient to fix all the renormalization constant in temrs of scaling of the fields and of the parameters appearing in the stochastic theory.     

Adjoint sources in lattice gauge theory

Variance reduction techniques for the evaluation of Wilson loops in lattice gauge theory are analysed. The method is extended to Wilson loops in the adjoint representation. Variational methods are also applied to adjoint sources. The combination of these techniques allows the potential V(R) between two static adjoint sources to be determined in SU(2) gauge theory. One isolated static adjoint source is also studied and the energy and distribution of the gluon field of this “glue-lump” is obtained. This is relevant to the saturation of the adjoint potential V(R) at large R.     

Loop states in lattice gauge theory

We reformulate d-dimensional SU(2) lattice gauge theory in terms of gauge invariant loop state variables by solving the SU(2) Gauss law as well as the corresponding Mandelstam constraints. The loop states satisfying the Gauss law and the Mandelstam constraints in d dimension are explicitly constructed in terms of the SU(2) harmonic oscillator prepotential operators. We show that these mutually independent (orthonormal) loop states carry certain non-negative integer Abelian fluxes over the lattice links and are characterized by 3(d−1) gauge invariant angular momentum quantum numbers per lattice site. Thus, they provide a complete orthonormal loop basis in the physical Hilbert space of the gauge theory. Further, we derive the loop Hamiltonian and show that it counts, creates and annihilates the Abelian fluxes over the plaquettes. The generalization to SU(N) gauge group is discussed.     

Some calculations in lattice gauge theory

We look at the action proposed by Wilson on a lattice and calculate static constants like f_π and two-body decay amplitudes in a certain approximation. Results are good to factors of four to six. There is good agreement for some of the predicted meson masses.     

Renormalization of anisotropy and glueball masseson tadpole improved lattice gauge action

The Numerical calculations for tadpole-improved U(1) lattice gauge theory in three-dimensions on anisotropic lattices have been performed using standard path integral Monte Carlo techniques. Using average plaquette tadpole renormalization scheme, simulations were done with temporal lattice spacings much smaller than the spatial ones and results were obtained for the string tension, the renormalized anisotropy and scalar glueball masses. We find, by comparing the regular and sideways potentials, that tadpole improvement results in very little renormalization of the bare anisotropy and reduces the discretization errors in the static quark potential and in the glueball masses.Received: 19 March 2003, Revised: 26 August 2003, Published online: 24 October 2003     

Topological charge of (lattice) gauge fields

Using recently derived explicit formulae for the 2- and 3-cochains in SU(2) gauge theory, we are able to integrate the Chern-Simons density analytically. We arrive — in SU(2) — at a local algebraic expression for the topological charge, which is the sum of local winding numbers associated with the corners (lattice points) of the cells covering the manifold plus contributions from possible isolated gauge singularities which manifest themselves as “vortices” in the 1-, 2- or 3-cochains. Among others we consider hypercubic geometry — i.e. covering the manifold by hypercubes — which is of particular interest to lattice Monte Carlo applications. Finally, we extend our results to SU(3) gauge theory.     

Ising lattice gauge theory in three dimensions

By raising the transfer matrix to a finite power the partition function for a finite lattice Z(2) gauge model is obtained exactly. The zeros of the resultant polynomial are found and some plaquette-plaquette expectation values are extracted. An exponential fit for the inverse correlation length matches onto both strong- and weak-coupling results but breaks down close to the second-order phase transition point.Similar calculations for the three-dimensional Ising model are also discussed.     

Bosonic lattice gauge theory with noise

We apply the recently suggested linear updating algorithm of Kennedy and Kuti to four-dimensional SU(3) bosonic gauge theory with the Wilson action. The change in the action for each link update is estimated stochastically, and we find that the algorithm gives the mean plaquette correctly for reasonable parameter values. Our results indicate that this method should be efficient for Monte Carlo computations with complicated “improved” actions, and they also show the feasibility of using such “noisy” methods to include the dynamical effects of fermions.