Limit Of Lattice Gauge Theory Research Articles

Bloch Waves in Minimal Landau Gauge and the Infinite-Volume Limit of Lattice Gauge Theory.

By exploiting the similarity between Bloch's theorem for electrons in crystalline solids and the problem of Landau gauge fixing in Yang-Mills theory on a "replicated" lattice, we show that large-volume results can be reproduced by simulations performed on much smaller lattices. This approach, proposed by Zwanziger [Nucl. Phys. B412, 657 (1994)NUPBBO0550-321310.1016/0550-3213(94)90396-4], corresponds to taking the infinite-volume limit for Landau-gauge field configurations in two steps: first for the gauge transformation alone, while keeping the lattice volume finite, and second for the gauge-field configuration itself. The solutions to the gauge-fixing condition are then given in terms of Bloch waves. Applying the method to data from MonteCarlo simulations of pure SU(2) gauge theory in two and three space-time dimensions, we are able to evaluate the Landau-gauge gluon propagator for lattices of linear extent up to 16 times larger than that of the simulated lattice. This approach is reminiscent of the Fisher-Ruelle construction of the thermodynamic limit in classical statistical mechanics.

Some exact properties of the gluon propagator

Recent numerical studies of the gluon propagator in the minimal Landau and Coulomb gauges in space-time dimension 2, 3, and 4 pose a challenge to the Gribov confinement scenario. We prove, without approximation, that for these gauges, the continuum gluon propagator $D(k)$ in SU(N) gauge theory satisfies the bound ${d-1 \over d} {1 \over (2 \pi)^d} \int d^dk {D(k) \over k^2} \leq N$. This holds for Landau gauge, in which case $d$ is the dimension of space-time, and for Coulomb gauge, in which case $d$ is the dimension of ordinary space and $D(k)$ is the instantaneous spatial gluon propagator. This bound implies that $\lim_{k \to 0}k^{d-2} D(k) = 0$, where $D(k)$ is the gluon propagator at momentum $k$, and consequently $D(0) = 0$ in Landau gauge in space-time $d = 2$, and in Coulomb gauge in space dimension $d = 2$, but D(0) may be finite in higher dimension. These results are compatible with numerical studies of the Landau-and Coulomb-gauge propagator. In 4-dimensional space-time a regularization is required, and we also prove an analogous bound on the lattice gluon propagator, ${1 \over d (2 \pi)^d} \int_{- \pi}^{\pi} d^dk {\sum_\mu \cos^2(k_\mu/2) D_{\mu \mu}(k) \over 4 \sum_\lambda \sin^2(k_\lambda/2)} \leq N$. Here we have taken the infinite-volume limit of lattice gauge theory at fixed lattice spacing, and the lattice momentum componant $k_\mu$ is a continuous angle $- \pi \leq k_\mu \leq \pi$. Unexpectedly, this implies a bound on the {\it high-momentum} behavior of the continuum propagator in minimum Landau and Coulomb gauge in 4 space-time dimensions which, moreover, is compatible with the perturbative renormalization group when the theory is asymptotically free.

Microscopic Spectrum of the Wilson Dirac Operator

We calculate the leading contribution to the spectral density of the Wilson Dirac operator using chiral perturbation theory where volume and lattice spacing corrections are given by universal scaling functions. We find analytical expressions for the spectral density on the scale of the average level spacing, and introduce a chiral random matrix theory that reproduces these results. Our work opens up a novel approach to the infinite-volume limit of lattice gauge theory at finite lattice spacing and new ways to extract coefficients of Wilson chiral perturbation theory.

Spin foam model for pure gauge theory coupled to quantum gravity

We propose a spin foam model for pure gauge fields coupled to Riemannian quantum gravity in four dimensions. The model is formulated for the triangulation of a four-manifold which is given merely combinatorially. The Riemannian Barrett-Crane model provides the gravity sector of our model and dynamically assigns geometric data to the given combinatorial triangulation. The gauge theory sector is a lattice gauge theory living on the same triangulation and obtains from the gravity sector the geometric information which is required to calculate the Yang-Mills action. The model is designed so that one obtains a continuum approximation of the gauge theory sector at an effective level, similarly to the continuum limit of lattice gauge theory, when the typical length scale of gravity is much smaller than the Yang-Mills scale.

D-PARTICLE RECOIL SPACE–TIMES AND "GLUEBALL" MASSES

We discuss the properties of matter in a D-dimensional anti-de Sitter-type space–time induced dynamically by the recoil of a very heavy D(irichlet)-particle defect embedded in it. The particular form of the recoil geometry, which from a world sheet view point follows from logarithmic conformal field theory deformations of the pertinent sigma-models, results in the presence of both infrared and ultraviolet (spatial) cutoffs. These are crucial in ensuring the presence of mass gaps in scalar matter propagating in the D-particle recoil space–time. The analogy of this problem with the Liouville-string approach to QCD, suggested earlier by John Ellis and one of the present authors, prompts us to identify the resulting scalar masses with those obtained in the supergravity approach based on the Maldacena's conjecture, but without the imposition of any supersymmetry in our case. Within reasonable numerical uncertainties, we observe that agreement is obtained between the two approaches for a particular value of the ratio of the two cutoffs of the recoil geometry. Notably, our approach does not suffer from the ambiguities of the supergravity approach as regards the validity of the comparison of the glueball masses computed there with those obtained in the continuum limit of lattice gauge theories.

Gauge symmetry as symmetry of matrix coordinates

We propose a new point of view to gauge theories based on taking the action of symmetry transformations directly on the coordinates of space. 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 noncommutativity of coordinates appears as the lattice spacing parameter. Through the statements concerning the continuum limit of lattice gauge theories, this suggestion arises that noncommutative 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 manifested.

String effects in the Wilson loop: a high precision numerical test

We test numerically the effective string description of the infrared limit of lattice gauge theories in the confining regime. We consider the 3D Z 2 lattice gauge theory, and we define ratios of Wilson loops such that the predictions of the effective string theory do not contain any adjustable parameters. In this way we are able to obtain a degree of accuracy high enough to show unambiguously that the flux-tube fluctuations are described, in the infrared limit, by an effective bosonic string theory.

COLLECTIVE FIELD APPROACH TO GAUGED PRINCIPAL CHIRAL FIELD AT LARGE-N

The lattice model of principal chiral field interacting with the gauge fields, which have no kinetic term, is considered. This model can be regarded as a strong coupling limit of lattice gauge theory at finite temperature. The complete set of equations for collective field variables is derived in the large-N limit and the phase structure of the model is studied.

Fundamental modular region, Boltzmann factor, and area law in Lattice gauge theory

The thermodynamic limit of lattice gauge theory is derived in a gauge which is optimized to make all link variables as close to unity as possible. The derivation rests upon (1) a precise bound on the fundamental modular region and (2) a direct evaluation of the functional integral of the Wilson lattice by the saddle-point method that is valid in the thermodynamic limit. The result confirms the calculational scheme obtained previously, which differs from the Faddeev-Popov scheme by the incorporation of non-perturbative effects, but which remains perturbatively renormalizable. The lattce Faddeev-Popov propagator, which appears in the modified action, acquires a dipole singularity at zero momentum, characteristic of long-range correlations. This is sufficient to produce an area law for Wilson loops, provided that unevaluated terms to not cancel the effect found.

Fundamental modular region, Boltzmann factor and area law in lattice theory

The thermodynamic limit of lattice gauge theory is derived in a gauge which is optimized to make all link variables as close to unity as possible. The derivation rests upon (1) a precise bound on the fundamental modular region and (2) a direct evaluation of the functional integral of the Wilson lattice by the saddle-point method that is valid in the thermodynamic limit. The result confirms the calculational scheme obtained previously, which differs from the Faddeev-Popov scheme by the incorporation of non-perturbative effects, but which remains perturbatively renormalizable. The lattice Faddeev-Popov propagator, which appears in the modified action, acquires a dipole singularity at zero momentum, characteristic of long range correlation. This is sufficient to produce an area law for Wilson loops, provided that unevaluated terms do not cancel the effect found.

Renormalizability of the critical limit of lattice gauge theory by BRS invariance

The critical limit of lattice gauge theory obtained previously, and which contains a parameter with dimension of (mass) 4, reflecting the boundary of the fundamental modular region, is shown to be renormalizable. The proof relies on BRS symmetry. It is also proven that the exact propagator of the Fermi ghost possesses a 1/( q 2) 2 singularity at q = 0. The relation of these results to confinement, the gluon condensate, and the fundamental modular region of gauge theory is discussed briefly.

Renormalizability of the critical limit of lattice gauge theory by BRS invariance

The critical limit of lattice gauge theory, which contains a parameter with dimension of ( mass) 4, reflecting the boundary of the fundamental modular region (no Gribov copies), is shown to be renormalizable.

Strong coupling gauge theory, quantum spin systems and the spontaneous breaking of chiral symmetry

We demonstrate equivalence between a large class of generalized quantum antiferromagnets and the strong coupling limit of lattice gauge theories with dynamical fermions. We prove that an SU(N c) lattice gauge theory with N L lattice flavors (=2 [ d/2]−1 N L continuum flavors in d spacetime dimensions) of staggered fermions is equivalent to a quantum U( N L) antiferromagnet for N c ⩾3 and to a Sp(2 N L) antiferromagnet for N c=2 (for N L=1 the latter is identical with a spin 1 2 Heisenberg antiferromagnet). We also show that U( N c ) gauge theory with N L lattice flavors and N L even is equivalent to an SU( N L) antiferromagnet. We use this correspondence and results about Heisenberg antiferromagnets to obtain a rigorous proof that, when N L=2, the strong coupling limit of U( N c ) gauge theory breaks chiral symmetry when d⩾4 and for all N c and in d=3 for N c ⩾2, and similarly for SU(2) gauge theories with N L=1 in d⩾3. (The corresponding result is trivial for SU( N c ) gauge theories with N L=1 and N c ⩾3.) The extension of this proof to other gauge theories would require a better understanding of SU( N L), U( N L>1) and Sp(2 N L>2) antiferromagnets.

Critical limit of lattice gauge theory

The critical limit of lattice gauge theory is studied in a gauge which is optimized to make all link variables as close to unity as possible. Hadronic mass arises as the thermodynamic parameter conjugate to the constraint imposed by the restriction to the fundamental modular region (no Gribov copies). This parameter, of dimension (mass) 4, multiplies a new term in the action which is sufficiently soft that renormalizability is preserved. The new term is made local by integration over auxiliary fields, and the resulting perturbation theory differs from the Faddeev-Popov theory by terms which are finite in every order. The renormalized theory contains a single dimensionful parameter Λ h, which sets the hadronic mass scale. To all orders in renormalized perturbation theory, the gluon propagator vanishes like k 2 at k = 0, and has a pole at imaginary ( k 2) = const. × Λ h 2 that describes confined gluons. Various predictions may be verified by Monte Carlo simulation and numerical gauge fixing.

Proof of chiral symmetry breaking in strongly coupled lattice gauge theory

We study chiral symmetry in the strong coupling limit of lattice gauge theory with staggered fermions and show rigorously that chiral symmetry is broken spontaneously in massless QED and the gauge-invariant Nambu-Jona-Lasinio model if the dimension of spacetime is at least four. The results for the chiral condensate as a function of the mass imply that the mean-field approximation is an upper bound for this observable which becomes exact as the dimension goes to infinity. For the model with gauge groupU(N),N=2,3,4, we prove that chiral long-range order exists at zero mass in four or more dimensions.

CAUSTICS IN A CUBIC SU(2) LATTICE MODEL WITH ANTIPERIODIC BOUNDARY CONDITIONS

In order to investigate the weak coupling limit of lattice gauge theories, it has been suggested recently to apply the semiclassical approximation to the Schrödinger equation in the Hamiltonian formalism. This method is used to study pure SU(2) gauge theory on a cube with sides of length one lattice constant and with antiperiodic boundary conditions. We show the existence of caustics, i.e. envelopes of families of classical trajectories where the ground state wave function peaks, and describe their shape.

Semiclassical analysis of the weak-coupling limit of SU(2) lattice gauge theory: The subspace of constant fields.

This paper contains the first part of a systematic semiclassical analysis of the weak-coupling limit of lattice gauge theories, using the Hamiltonian formulation. The model consists of an N/sup 3/ cubic lattice of pure SU(2) Yang-Mills theory, and in this first part we limit ourselves to the subspace of constant field configurations. We investigate the flow of classical trajectories, with a particular emphasis on the existence and location of caustics. There the ground-state wave function is expected to peak. It is found that regions densely filled with caustics are very close to the origin, i.e., in the domain of weak field configurations. This strongly supports the expectation that caustics are essential for quantities of physical interest.

Caustics in a simpleSU(2) lattice gauge theory model

This paper describes the beginning of an attempt to study the weak coupling limit of lattice gauge theories, using the Hamiltonian formulation and the semiclassical approximation. The topics are caustics and the behavior of the ground state wave function in the vicinity of caustics. For two very simpleSU(2) models (one plaquette, two plaquettes) we demonstrate the existence of caustics, determine their locations and study the peaking behavior of the ground state wave function on them in the limitg2→0.

Continuum limit of lattice gauge theory: A perturbative study.

In a weak-coupling expansion, the Creutz ratio is calculated on relatively large lattices up to O(${g}^{4}$) for improved actions. A universality in size dependence of the Creutz ratio is found among lattice actions. By extrapolating the results on finite lattices to an infinite lattice, the artifacts are studied, and the minimum size of Creutz ratios for which the artifact becomes less than 10% is determined. The scale transformation and universality between lattice actions on finite lattices are studied.

Towards the Continuum Limit of Lattice Gauge Theory with Dynamical Fermions

Simulations of SU(3) lattice gauge theory with four species of light quarks show a large peak in the bulk specific heat near ..beta.. = 6/g-italic/sup 2/roughly-equal5.10 separating strong and weak coupling. satisfies asymptotic-freedom scaling for ..beta.. = 6/g-italic/sup 2/approx. >5.10 and /sup 1/3//..lambda../sub M//sub S-bar/ = (2.6 +- 0.1)(4..pi../sup 2/..beta../25)/sup 4/25/ (MS-bar denotes the modified minimal-subtraction scheme). Finite-temperature simulations on a 6 x 10/sup 3/ lattice show a smooth but rapid crossover at T-italic/..lambda../sub M//sub S-bar/ = 2.14 +- 0.10. We find no evidence for metastable states.