Infrared exponents and the running coupling of Landau gauge QCD and their relation to confinement

The infrared behaviour of the gluon and ghost propagators in Landau gauge QCD is reviewed. The Kugo-Ojima confinement criterion and the Gribov-Zwanziger horizon condition result from quite general properties of the ghost Dyson-Schwinger equation. The numerical solutions for the gluon and ghost propagators obtained from a truncated set of Dyson-Schwinger equations provide an explicit example for the anticipated infrared behaviour. The results are in good agreement with corresponding lattice data obtained recently. The resulting running-coupling approaches a fix point in the infrared, . Two different fits for the scale dependence of the running coupling are given and discussed.Received: 30 September 2002, Published online: 22 October 2003PACS: 12.38.Aw General properties of QCD (dynamics, confinement, etc.) - 14.70.Dj Gluons - 12.38.Lg Other nonperturbative calculations - 11.15.Tk Other nonperturbative techniques - 02.30.Rz Integral equations     

The Landau gauge gluon propagator in lattice QCD

A Monte Carlo calculation of the gluon propagator in the Landau gauge in SU(3) lattice gauge theory is described. The results of calculations at β = 5.6 (200 4³ × 8 lattices), β = 5.8 (400 4³ × 10 lattices and 100 6³ × 12 lattices), and β = 6.0 (100 4³ × 8 lattices) indicate that the gluon propagator resembles a massive particle propagator in which the mass grows with separation. At the largest distances accessible with these lattices, the mass is about 600 MeV.     

Convergence of strong-coupling expansion and linear potential in lattice gauge theories

We give a numerical lower bound on the radius of convergence of the strong coupling expansion in lattice Yang-Mills theories. From this we infer that the static potential rises linearly starting at ≈1.4 fm.     

Strong-coupling study of the Gribov ambiguity in lattice Landau gauge

We study the strong-coupling limit β=0 of lattice SU(2) Landau gauge Yang–Mills theory. In this limit the lattice spacing is infinite, and thus all momenta in physical units are infinitesimally small. Hence, the infrared behavior can be assessed at sufficiently large lattice momenta. Our results show that at the lattice volumes used here, the Gribov ambiguity has an enormous effect on the ghost propagator in all dimensions. This underlines the severity of the Gribov problem and calls for refined studies also at finite β. In turn, the gluon propagator only mildly depends on the Gribov ambiguity.     

Infrared gluon and ghost propagator exponents from lattice QCD

The compatibility of the pure power law infrared solution of QCD and lattice data for the gluon and ghost propagators in Landau gauge is discussed. For the gluon propagator, the lattice data are well described by a pure power law with an infrared exponent κ∼0.53, in the Dyson–Schwinger notation. κ is measured using a technique that suppresses finite volume effects. This value is consistent with a vanishing zero momentum gluon propagator, in agreement with the Gribov–Zwanziger confinement scenario. For the ghost propagator, the lattice data seem not to follow a pure power law, at least for the range of momenta accessed in our simulation.     

Universality and the approach to the continuum limit in lattice gauge theory

The universality of the continuum limit and the applicability of renormalized perturbation theory are tested in the SU(2) lattice gauge theory by computing two different non-perturbatively defined running couplings over a large range of energies. The lattice data (which were generated on the powerful APE computers at Rome II and DESY) are extrapolated to the continuum limit by simulating sequences of lattices with decreasing spacings. Our results confirm the expected universality at all energies to a precision of a few percent. We find, however, that perturbation theory must be used with care when matching different renormalized couplings at high energies.     

Factorisation in large-N limit of lattice gauge theories revisited

We prove using the Schwinger-Dyson equations that the factorisation property holds for all gauge-invariant Green’s function in the large-N limit of a Wilson-Polyakov lattice gauge theory.     

On the chiral limit in lattice gauge theories with Wilson fermions

The chiral limit κ ? κ _c (β) in lattice gauge theories with Wilson fermions and problems related to near-to-zero (’exceptional’) eigenvalues of the fermionic matrix are studied. For this purpose we employ compact lattice QED in the confinement phase. A new estimator $\tilde m_\pi$ for the calculation of the pseudoscalar mass m _π is proposed which does not suffer from ’divergent’ contributions at κ ? κ _c (β)We conclude that the main contribution to the pion mass comes from larger modes, and ’exceptional’ eigenvalues play no physical role. The behaviour of the subtracted chiral condensate $\left\langle {\bar \psi \psi } \right\rangle _{subt}$ near κ _c (β) is determined. We observe a comparatively large value of $\left\langle {\bar \psi \psi } \right\rangle _{subt} \cdot Z_P^{ - 1}$ , which could be interpreted as a possible effect of the quenched approximation.     

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.     

Attraction of like-charged macroions in the strong-coupling limit

Like-charged macroions attract each other as a result of strong electrostatic correlations in the presence of multivalent counterions or at low temperatures. We investigate the effective electrostatic interaction between i) two like-charged rods and ii) two like-charged spheres using the recently introduced strong-coupling theory, which becomes asymptotically exact in the limit of large coupling parameter (i.e. for large counterion valency, low temperature, or high surface charge density on macroions). In contrast to previous applications of the strong-coupling theory, we deal with curved surfaces and an additional parameter, referred to as Manning parameter, is introduced, which measures the ratio between the radius of curvature of macroions to the Gouy-Chapman length. This parameter, together with the size of the confining box enclosing the two macroions and their neutralizing counterions, controls the counterion-condensation process that directly affects the effective interactions. For sufficiently large Manning parameters (weakly-curved surfaces), we find a strong long-ranged attraction between two macroions that form a closely-packed bound state with small surface-to-surface separation of the order of the counterion diameter in agreement with recent simulations results. For small Manning parameters (highly-curved surfaces), on the other hand, the equilibrium separation increases and the macroions unbind from each other as the confinement volume increases to infinity. This occurs via a continuous universal unbinding transition for two charged rods at a threshold Manning parameter of , while the transition is strongly discontinuous for spheres because of a pronounced potential barrier at intermediate distances. Unlike the cylindrical case, the attractive forces between spheres disappear slowly for increasing confinement volume due to the complete de-condensation of counterions. Scaling arguments suggest that for moderate values of coupling parameter, strong-coupling predictions remain valid for sufficiently small surface-to-surface separations.Received: 20 August 2003, Published online: 2 March 2004PACS: 87.15.-v Biomolecules: structure and physical properties - 82.70.Dd Colloids - 87.15.Nn Properties of solutions; aggregation and crystallization of macromolecules     

Infrared finiteness of normalization constants of currents in lattice gauge theories

A simple proof of the infrared finiteness of the renormalization of the strength of the flavour currents induced by lattice artifacts is presented.     

On the realization of Landau gauge

Starting with the determination of the two-points function in the Landau gauge given by Wightman and Strocchi, we build a more economic realization of this gauge, following the basic principles they have proposed.     

Definition and general properties of the transfer matrix in continuum limit improved lattice gauge theories

When operators of dimension 6 are added to the standard Wilson action in lattice gauge theories, physical positivity is lost in general. We show that a transfer matrix can nevertheless be defined. Its properties are, however, unusual: complex eigenvalues may occur (leading to damped oscillatory behaviour of correlation functions), and there are always contributions in the spectral decomposition of two-point functions that come with a negative weight.     

A path-functional field theory of lattice gauge models and the large-N limit

We transform lattice gauge models to a theory of functional fields defined on a set of closed paths. Some relevant properties of the formalism are discussed in detail, with emphasis on symmetry and topological structure. We then investigate the large-N limit of the U(N) lattice gauge model in arbitrary dimensions using this formalism. Assuming the existence of the limit, we show, to arbitrary order of the strong coupling expansion parameter (g²N)^?, which is kept fixed, that for the leading contribution in the limit: (i) the flow of indices in color space can be represented by planar diagrams; (ii) when the diagrams are immersed in space-time they are random surfaces without handles; (iii) there are interactions of the surfaces which can be depicted as the formation of multisheet bubblesw in the surfaces. This formalism also makes it possible to set up a gauge-invariant mean-field approximation.     

Mean field approximation is exact in the many-component limit of potts lattice gauge model

The statement in the title concerning free energy is proved for a version of mean field approximation with previously fixed temporal gauge.