Testing two versions of lattice gauge theory: Creutz ratios inU(1)3

In our simplicial version of lattice gauge theory, we approximate Euclidean path integrals by tiling space-time with simplexes and by linearly interpolating the fields throughout each simplex from their values at the vertices. We compare this method with Wilson's lattice gauge theory forU(1) in three dimensions. As a standard of comparison, we compute the exact values of Creutz ratios of Wilson loops in the continuum theory. Monte Carlo computations using the simplicial method give Creutz ratios within a few percent of the exact values for reasonably sized loop atβ=1, 2, and 10. Similar computations using Wilson's method give ratios that typically differ from the exact values by factors of 2 or more for 1⩽β⩽3.5 and that have the wrongβ dependence. The better accuracy of the simplicial method is due to its use of the action and domain of integration of the exact theory, unaltered apart from the granularity of the simplicial lattice. We also present data on the action density and the mass gap. Research supported by the U.S. epartment of Energy under grant DE-FG04-84ER40166.     

Scaling behaviour of Creutz ratios inSU(2) lattice gauge theory

An analysis of the scaling behaviour of Creutz ratios on large lattices is given forSU(2) gauge theory. The β-interval is 2.5≦β≦2.8. Under a factor 2 scaling test, after multiplicative corrections for lattice artifacts, the Monte Carlo data show deviations from scaling, which are similar for all values of β. The ratios can be fitted successfully by a sum of three perturbative terms and an exponentially decreasing nonperturbative term. For many ratios the latter turns out to be very small, and its size dependence at fixed β is consistent with that of an area term in the Wilson loops. The deviation of the corresponding exponents from the ones expected for an area term gives a coherent cxplanation of the observed departures from scaling. It is well possible that for fixed spatial extension (in lattice units) nonperturbative contributions vanish so fast that they cannot be interpreted as physical effects.     

Simplicial gauge theory and quantum gauge theory simulation

We propose a general formulation of simplicial lattice gauge theory inspired by the finite element method. Numerical tests of convergence towards continuum results are performed for several SU(2) gauge fields. Additionally, we perform simplicial Monte Carlo quantum gauge field simulations involving measurements of the action as well as differently sized Wilson loops as functions of β.     

Scaling and string tension in pureSU(2) lattice gauge theory

Wilson loops inSU(2) lattice gauge theory without fermions are determined on lattices of size 12⁴, 16⁴ and 24⁴ at β=2.4, 2.5 and 2.6. At β=2.6 the static quark-antiquark potential is extracted for distances up to 8 lattice units. A string tension smaller by a factor 2 than in previous investigations is found. Deviations from asymptotic scaling for multiplicatively improved Creutz ratios are certain, and their magnitude depends on the geometrical size of the ratios. This implies deviations from scaling.     

Large-N universality of variant actions

Corrections to large-N universality in mixed action lattice gauge theories imply constraints on the cut-off dependence. We state the form of these constraints and show that they are satisfied up to and including two loops (weak coupling). This implies that the second coefficient of the β function on the lattice is indeed universal as expected. We then evaluate the complete form of the corrections to large-N universality up to and including two loops. They give a sizeable contribution and determine qualitative aspects of equal string tension data like the bending seen for SU(2).     

U(2) four-dimensional simplicial lattice gauge theory

Monte Carlo simulations for pureU (2) gauge theory on a four-dimensional simplicial lattice with six sites in each direction are reported. Wilson loops and the string tensions for squares and triangles are presented. A first-order phase transition, similar to that found for the hypercubical lattice, is observed and found to confineSU (2) colour and deconfineU (1) charge.     

Monte Carlo simulation of non-compact QCD with stochastic gauge fixing

A non-compact lattice model of quantum chromodynamics is studied numerically. Whereas in Wilson's lattice theory the basic variables are the elements of a compact Lie group, the present lattice model resembles the continuum theory in that the basic variables A are elements of the corresponding Lie algebra, a non-compact space. The lattice gauge invariance of Wilson's theory is lost. As in the continuum, the action is a quartic polynomial in A, and a stochastic gauge fixing mechanism - which is covariant in the continuum and avoids Faddeev-Popov ghosts and the Gribov ambiguity — is also transcribed to the lattice. It is shown that the model is self-compactifying, in the sense that the probability distribution is concentrated around a compact region of the hyperplane div A = 0 which is bounded by the Gribov horizon. The model is simulated numerically by a Monte Carlo method based on the random walk process. Measurements of Wilson loops, Polyakov loops and correlations of Polyakov loops are reported and analyzed. No evidence of confinement is found for the values of the parameters studied, even in the strong coupling regime.     

Monte Carlo simulation of SU(2) gauge theory with fermions on a four-dimensional lattice

After integration over the fermions in an SU(2) lattice gauge theory, the effective fermionic action may be expressed as a sum over all possible closed gauge field loops with corresponding weight factors. We approximate this sum and perform a Monte Carlo simulation of a coupled fermion-gauge system on a 4⁴ lattice. We compare our results for 〈S_eff〉 and $\text{〈}\text{ψ}\text{ψ〉}$ for different values of the gauge field coupling β and fermion coupling κ with the free fermion theory on a lattice. 〈S_eff〉 turns out to be quite small for $\text{κ?}\text{1}\text{8}$.     

3-dimensionalSU(2) lattice gauge theory in terms of gauge invariant variables

As a first step towards a duality transformation for theSU(2) lattice gauge theory in 3 dimensions, the integration over all gauge variant variables is performed explicitly after introducing gauge invariant auxiliary variables. The resulting new Hamiltonian is complex and involves a sum over closed loops. Each of these loops is confined to an elementary cube of a dual lattice. Like in a previous investigation for theO(4) symmetric Heisenberg ferromagnet Rühl's boson representation is used to derive the result.     

MONTE CARLO STUDY OF MANTON'S ACTION FOR FOUR-DIMENSIONAL SU(5) LATTICE GAUGE THEORY

The feature of Manton's lattice action in the SU(5) four-dimensional lattice gauge theory is investigated by means of Monte Carlo simulation. A rather smooth curve of the internal energy is obtained, no discontinuity in the average action perplaquette is found in opposition to the resultes of usually adopted Wilson's action. An evidence of a higher order phase transition with Manton's action is also found.     

Finite size Wilson loops

A study of Wilson loop averages for finite size loops is initiated. Within the framework of euclidean four-dimensional lattice SU(2) gauge theory with elementary Wilson action we compute the expectation values of all rectangular loops to 12th order in the strong coupling expansion. The leading term for weak coupling is evaluated for loops up to 4 × 4. A comparison to Monte Carlo data is presented. Other related issues are also discussed.     

The relation between the waveguide and overlap implementations of Kaplan's domain wall fermions

Recently, Narayanan and Neuberger proposed that the fermion determinant for a lattice chiral gauge theory be defined by an overlap formula. The motivation for that formula comes from Kaplan's five-dimensional lattice domain wall fermions. In the case that the target continuum theory contains 4n chiral families, we show that the effective action defined by overlap formula is identical to the effective action of a modified waveguide model that has extra bosonic ghost fields. This raises serious questions about the viability of the overlap formula for defining chiral gauge theories on the lattice.     

On the quark-antiquark potential at short distances

We investigate the static quark-antiquark potential up to distances of 8 lattice units for pure SU(2) gauge theory on lattices with anisotropic couplings. The action is the Wilson action with a coupling for time-like plaquettes which differs from those for space-like ones. Numerical simulations are performed in a large range of β The potential is obtained by fitting “cooled” Wilson loops with up to four exponential terms. An interpolation of the potentials by a sum of a perturbative term, a linear term and by lattice artifacts shows poor scaling in comparison with he isotropic case. If the coupling in the time-like region is reduced, the linear term is much smaller than in the isotropic case, and vice versa. Consequences for the bag picture for hadrons are discussed.     

Does one see gluon condensation after subtraction of mean field perturbation theory from monte carlo data?

We do a linearised mean field calculation in axial gauge for the four dimensional mixed fundamental adjointSU (2) lattice gauge theory and extract the gluon condensate parameter from the expectation values of the plaquette and the action by subtracting mean field perturbation theory from monte carlo data.     

The Λ parameter for improved lattice gauge actions

The ratios of the scale parameters are calculated for lattice gauge theories with action including up to six-link loops. Comparison with the results of Monte Carlo calculations are also made.     

SU(2) Lattice Gauge Theory Formulation of the (2+1)-Dimensional Antiferromagnetic Heisenberg Model

The equivalence of 2+1 antiferromagnetic Heisenberg model and the SU(2) Kogut Susskind lattice gauge theory is recapitulated and the naive Euclidean lattice action of the threedimensional an tiferromagnetic Heisen berg model is derived. The three-dimensional lattice gauge fermion theory is formulated to give the consistent lattice gauge theory of antiferromagnetic Heisenberg model. In continu um limit the two copies of two flavor fermions are resulted, which give the negative results of the microscopic derivation of the Chern-Simons terms. The Chern-Simons terms, the gauge invariant problem of effective action and the '%hiralityn are discussed.     

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.     

The free energy of lattice gauge theories at large β

An upper and a lower bound for the free energy density of a lattice gauge teory with compact gauge group are derived, valid for all values of β. For large β the first two terms of the asymptotic expansions of these bounds coincide, thus determining the corresponding terms of the free energy density. Moreover the gauge groups U(N) and SU(N) are treated explicitly.     

Critical thermodynamics of two-dimensional systems in the five-loop renormalization-group approximation

The paper is devoted to the calculation of renormalization-group (RG) functions in the O(n)-symmetry two-dimensional model of the λ? ⁴ type in the five-loop approximation and to an analysis of the critical behavior of systems described by this model. Five-loop expansions for the β function and the critical indices are determined in bulk theory. They are summed up using the Padé-Borel and Padé-Borel-Le Roy methods, making it possible to optimize the summation procedure and to estimate the accuracy of the obtained numerical values. It is shown that in the Ising (n=1) case, as well as in other cases, the inclusion of the five-loop contribution to the β function displaces the coordinate of the Wilson fixed point only insignificantly, leaving it outside the interval formed by the results of computations on lattices; even “spreads” of the error in the renormalization group and lattice estimates do not overlap. This discrepancy is attributed to the effect of the nonanalytic com-ponent of the β function, which cannot be determined in perturbation theory. A computation of critical indices proves that, although the inclusion of the five-loop terms in the corresponding RG expansion slightly improves the concordance with the exact results, the nonanalytic contributions are apparently also significant in this case.