On the roughening transition in abelian lattice gauge theories

We study the width of the confining string between static quarks in abelian lattice gauge theories using strong coupling expansions. We consider gauge groups Z_n and U(1) in 3 and 4 dimensions. This extends previous work with Lüscher, where SU(2) and Z₂ were studied. In ν = 3 dimensions we find evidence for a roughening transition. It is characterized by a divergence of the string width for an infinitely far separated quark-antiquark pair, while the string tension remains non-zero. In ν = 4 dimensions for the abelian groups we do not have evidence for a roughening transition away from a phase transition.     

Roughening transition in lattice gauge theories in arbitrary dimension: (II). The groups Z3, U(1), SU(2), SU(3)

Strong coupling expansions for the string tension and other quantities in lattice gauge theories are computed in arbitrary dimension for the groups Z₃, U(1), SU(2), SU(3). This enables us to determine the location of the roughening transition, which seems to be group independent when measured in an appropriate variable. In four dimensions, the strong coupling expansion of string tension calculated up to fourteenth order for SU(2), and twelfth order for SU(3) agrees nicely with Monte Carlo data up to the roughening point.     

High-temperature expansions for the free energy of vortices and the string tension in lattice gauge theories

We derive high-temperature cluster expansions for the free energy of vortices in SU(2) and Z₂ lattice gauge theories in 3 and 4 dimensions. The expected behaviour of the vortex free energy is verified. It obeys an area law behaviour. The coefficient of the area is shown to be equal to the string tension between static quarks. We calculate its expansion up to 12th order. For SU(2) in 4 dimensions the result is compared with Monte Carlo calculations of Creutz and is in good agreement at strong and intermediate coupling.     

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.     

Linked cluster expansions for U(1) lattice gauge theory in 2 + 1 and 3 + 1 dimensions

Compact U(1) lattice gauge theory is studied in 2 + 1 and 3 + 1 dimensions using strong coupling series expansions and the recently proposed exact linled cluster expansion alborithm Results for the vacuum energy, specific heat and axial string tension in 2 + 1 dimensions are in agreement with previous finite lattice estimates. In 3 + 1 dimensions, we present new strong coupling series results (order g^?40) which together with the ELCE estimates show evidence of a continuous phase transition at x = 1/g⁴ = 0.72 ± 0.08. The associated critical index for the vanishing string tension is μ = 0.65 ± 0.12. The axial string tension in D = 3 + 1 appears to undergo a non-deconfining roughening transition at smaller x (0.56 ± 0.07).     

Generalized roughening transition and its effect on the string tension

The well-understood roughening transition of an interface in the d = 3 Ising model implies an essential singularity in the string tension of the dual Z₂ gauge model. The roughening transition corresponds to the delocalization of the string due to strong long-wavelength fluctuations, and this reformulation can be generalized to other gauge groups and to d = 4 also. It is not a deconfining transition - it is expected to occur deep in the confining region - but its presence would raise serious questions about the continuation of strong coupling expansions of the tension beyond this point. In this paper predictions on the roughening transition are confronted with the available information on the string tension for different gauge groups in three dimensions.     

How well do Monte Carlo simulations distinguish between U(1) and SU(2)?: A study of the string tension in U(1) lattice gauge theory

To investigate how a system with a known deconfining phase transition behaves when studied on finite lattices via Monte Carlo simulations, we have made such studies of compact U(1) lattice gauge theory for 8⁴, 10⁴, and 12⁴ lattices. We have concentrated on the mean plaquette energy and the string tension. The string tension does not vanish on a finite lattice, but using finite size scaling arguments the indications are that it does vanish on an infinite lattice, where we predict the critical coupling β_c = 1.008 and the correlation length exponent $\text{ν =}\text{1}\text{3}$. We compare our results to those for SU(2) and find that although there are differences, they are not yet definitive.     

SU(2) string tension,glueball mass and interquark potential by Monte Carlo computations

Monte Carlo simulations of the SU(2) gauge system on a large (16⁴) lattice and with high statistics are performed to determine several quantities of physical interest. Previous evaluations of the ratio between string tension, σ, and scale constant, Λ₀, are confirmed. The mass of the glueball is found to be approximately 3√σ and the potential between static charges at very small separation is measured.     

SU(2) Lattice gauge theory in (2+1)D

Cluster expansion methods are applied to theSU(2) lattice gauge model in (2+1) dimensions. Strong-coupling series are calculated for the vacuum energy per site, the axial string tension, and the scalar mass gap; while ELCE approximants are used to estimate the string tension beyond its roughening transition. The simple scaling behaviour expected of this super-renormalizable theory is clearly seen, and we estimate that in the continuum limit the string tension σ～(0.14±0.01)g ⁴, while the mass gapM _s ～(2.2±0.25)g ². More accurate Monte Carlo simulations are needed to check the universality between the Hamiltonian and Euclidean versions of this model.     

Strong coupling expansions for the mass gap in lattice gauge theories

We derive strong coupling expansions for the mass gap in euclidean lattice gauge theories in any space-time dimension. For gauge groups SU(2), SU(3), Z₂ and Z₃ the series are calculated up to order g^?16. They are used to get rough estimates for the lowest glueball mass in continuum SU(2) and SU(3) gauge theories, assuming a sudden crossover from strong to weak coupling behaviour in the lattice theory.     

The string tension and the scaling behavior of SU(2) gauge theory on a random lattice

The SU(2) gauge theory on an 8⁴ random lattice has been studied by the Monte Carlo method. The string tensions have been evaluated. They display the expected scaling behavior for β=1.2−1.3. The scale parameter Λ_RAN has been determined approximately.     

The string tension of compact U(1) four-dimensional lattice gauge theory

Monte Carlo simulations of the string tension are calculated for four-dimensional U(1) gauge theory on a 6⁴ lattice. The string tension follows the result -1n(β/2) in the high temperature region and is zero for β > β_c, where β_c is the critical inverse temperature.     

Thermodynamics of SU(3) lattice gauge theory

The pressure and the energy density of the SU(3) gauge theory are calculated on lattices with temporal extent N_τ = 4, 6 and 8 and spatial extent N_σ = 16 and 32. The results are then extrapolated to the continuum limit. In the investigated temperature range up to five times T_c we observe a 15% deviation from the ideal gas limit. We also present new results for the critical temperature on lattices with temporal extent N_τ = 8 and 12. At the corresponding critical couplings the string tension is calculated on 32⁴ lattices to fix the temperature scale. An extrapolation to the continuum limit yields T_c/√σ = 0.629(3). We furthermore present results on the electric and magnetic condensates as well as the temperature dependence of the spatial string tension. These observables suggest that the temperature dependent running coupling remains large even at T ≅ 5T_c. For the spatial string tension we find √σ_s/T=0.566(13)g²2(T) with g² (5T_c) ≅ 1.5.     

String tensions for lattice gauge theories in 2 + 1 dimensions

Compact U(1) and SU(2) lattice gauge theories in 3 euclidean dimensions are studied by standard Monte Carlo techniques. The question of extracting reliable string tensions from these theories is examined in detail, including a comparison of the Monte Carlo Wilson loop data with weak coupling predictions and a careful error analysis: our conclusions are rather different from those of previous investigations of these theories. In the case of U(1) theory, we find that only a tiny range of β values can possibly be relevant for extracting a string tension and we are unable to convincingly demonstrate the expected exponential dependence of the string tension on β. For the SU(2) theory we are able to determine, albeit with rather large errors, a string tension from a study of Wilson loops.     

Wilson loops for U(4) and SU(4) lattice gauge theories in four dimensions

Monte Carlo simulations are used to calculate Wilson loops for pure U(4) and SU(4) gauge theories on a 6⁴ lattice. The first-order phase transitions previously observed in the average action per plaquette for U(4) and SU(4) is also seen in the string tension. U(4) and SU(4) color seem to be confined while U(4) charge in U(4) appears to be deconfined.     

String tensions of SU(3) gauge theory on a random lattice

The SU(3) gauge theory on a random lattice is studied using the Monte Carlo method. The string tensions are measured. A scaling window around g₀⁻² = 1.4 is detected and the scale parameter is estimated.     

One-loop threshold effects in string unification

Like grand unification of old, string unification predicts simple tree-level relations between the couplings of all unbroken gauge groups such as SU(3)_C or SU(2)_W. I show here how to compute one-loop corrections to these relations for any four-dimensional model based on a classical vacuum of the heterotic string. The result can be used to calculate both sin²θ_Wand Λ_QCD in terms of α_QCD and M_Planck.     

Adjoint Wilson lines and the effective gluon mass

The effective mass of the gluon is defined in terms of the energy stored in the string between adjoint sources. For SU(2) lattice gauge theory, I present Monte Carlo evidence that this mass scales correctly in the weak coupling regime and hence is non-zero and finite in the continuum limit. Various systematic errors (chief among them are temperature effects) still make it difficult to give a precise numerical value for this mass at zero temperature, but a range of 500 to 800 MeV seems reasonable (using a value for the string tension of (420 MeV)²). Some attempt is made to relate this quantity to a true “constituent gluon mass.”     

A visualization of the SU(2) vacuum on the lattice

Configurations of pure SU(2) gauge field theory on lattice are transformed to Landau gauge. After Fourier transformation, large momentum amplitudes are suppressed (filtered) by a variable amount, and the configurations are transformed back to x-space. Spectacular peaks in electric and magnetic field strengths are found, which share many properties with either almost pointlike instantons or with extended anti-instantons. The environment around those peaks are visualized with respect to the action, to the topological charge density, to the gauge fields and to electric and magnetic field strengths. The density of the peaks is of order 1 fm^–4, and it scales according to the string tension under a variation of the coupling constant. The evolution of the peaks under the amount of Fourier filtering is visualy compared to the evolution under cooling, and the gauge dependence of the peaks is discussed. Advantages and shortcomings of this method are discussed, with emphasis on possible strong distortions of the action and topological charge density, which become gauge dependent. Finally, I compare the character of the SU(2)-configurations to those of noncompact abelian theory.