The Landau gauge gluon propagator in 4D SU(2) lattice gauge theory revisited: Gribov copies and scaling properties
I. L. Bogolubsky, E. -M. Ilgenfritz, M. Müller-Preussker, A. Sternbeck
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6 pages, 3 figures, to appear in the Proceedings of the XXVII International Symposium on Lattice Field Theory, LATTICE 2009, July 26-31 2009, Peking University, Beijing, China
Lattice results for the gluon propagator in SU(2) pure gauge theory obtained on large lattices are presented. Simulated annealing is used throughout to fix the Landau gauge. We concentrate on checks for Gribov copy effects and for scaling properties. Our findings are similar to the ones in the SU(3) case, supporting the decoupling-type infrared behaviour of the gluon propagator.
We address the problem of Gribov copies in lattice QCD. The gluon propagator is computed, in the Landau gauge, using 302 ($β= 5.8$) $12^4$ configurations gauge fixed to different copies. The results of the simulation shows that: i) the effect of Gribov copies is small (less than 10%); ii) Gribov copies change essentially the lowest momenta components ($q < 2.6$ GeV); iii) within the statistical accuracy of our simulation, the effect of Gribov copies is resolved if statistical errors are multiplied by a factor of two or three. Moreover, when modelling the gluon propagator, different sets of Gri…
We have performed an exhaustive search for Gribov copies on the lattice and their possible dependence from finite temperature effects. We show that, for each value of lattice size, Gribov copies are dense in configuration space at low temperature but their density tend to lower when the temperature increases. We have investigated lattice sizes running from $16^3\times 8$ to $16^4$.
We compare two Landau gauge fixing methods, aiming to find the global maximum of the gauge fixing functional. Moreover, a systematic effect of Gribov copies in the gluon and ghost propagators computed in Landau gauge is presented and discussed.
We address the problem of the gauge fixing versus Gribov copies in lattice gauge theories. For the Landau gauge, results show that a suitable combination of evolutionary algorithms with traditional steepest descent methods identifies the global maximum of the optimisation function. We discuss the performance of the combined algorithm on small cubic lattices for SU(2) and SU(3).
Following a recent proposal by Cooper and Zwanziger we investigate via $SU(2)$ lattice simulations the effect on the Coulomb gauge propagators and on the Gribov-Zwanziger confinement mechanism of selecting the Gribov copy with the smallest non-trivial eigenvalue of the Faddeev-Popov operator, i.e.~the one closest to the Gribov horizon. Although such choice of gauge drives the ghost propagator towards the prediction of continuum calculations, we find that it actually overshoots the goal. With increasing computer time, we observe that Gribov copies with arbitrarily small eigenvalues can be found…
We fix $SU(2)$ lattice gauge fields to the Maximally Abelian gauge in both three and four dimensions. We extract the corresponding $U(1)$ fields and monopole current densities and calculate separately the confining string tensions arising from these $U(1)$ fields and monopole `condensates'. We generate multiple Gribov copies and study how the $U(1)$ fields and monopole distributions vary between these different copies. As expected, we find substantial variations in the number of monopoles, their locations and in the values of the $U(1)$ field strengths. The string tensions extracted from `extr…
Infrared behavior of the Landau gauge gluon propagators is studied numerically in the 3d SU(2) gauge theory on the lattice. A special accent is made on the study of Gribov copy effect. For this study we employ an efficient gauge-fixing algorithm and a large number of gauge copies (up to 280 copies per configuration). It is shown that, in the deep infrared region, the Gribov copy effects are very significant. Also we show that, in the infinite-volume limit, the zero-momentum value of the propagator does not vanish.
We study the influence of Gribov copies in the Coulomb gauge on the smeared hadronic correlation functions that are involved in the determination of the B meson decay constant. We find that the residual gauge freedom associated to Gribov copies induces observable noise effects, though at the level of numerical accuracy of our simulation these effects are not relevant to the final determination of f_B. Our results indicate that such effects may become important on bigger lattices.
The dependence of the Landau gauge gluon and ghost propagators on the choice of Gribov copies is studied in pure SU(3) lattice gauge theory. Whereas the influence on the gluon propagator is small, the ghost propagator becomes clearly affected by the copies in the infrared region. We compare our data with the infrared exponents predicted by the Dyson-Schwinger equation approach
The dependence of the gluon and ghost propagator in pure SU(3) gauge theory on the choice of Gribov copies in Landau gauge is studied. Simulations were performed on several lattice sizes at $β$=5.8, 6.0 and 6.2. In the infrared region the ghost propagator turns out to depend on the choice, while the impact on the gluon propagator is not resolvable. Also the eigenvalue distribution of the Faddeev-Popov operator is sensitive to Gribov copies.
Perturbative QCD uses the Faddeev-Popov gauge-fixing procedure, which leads to ghosts and the local BRST invariance of the gauge-fixed perturbative QCD action. In the asymptotic regime, where perturbative QCD is relevant, Gribov copies can be neglected. In the nonperturbative regime, one must adopt either a nonlocal Gribov-copy free gauge (e.g., Laplacian gauge) or attempt to maintain local BRST invariance at the expense of admitting Gribov copies. These issues are explored. In addition, we discuss the relationship between recent Dyson-Schwinger based model calculations of the infrared behavio…
We present lattice results for the gluon propagator for SU(2) and SU(3) in the Laplacian gauge which avoids lattice Gribov copies. In SU(3) we compare with the most recent lattice calculation in Landau gauge and with various approximate solutions of the Dyson Schwinger equations (DSE).
We perform the careful study of the Gribov copies problem in SU(2) lattice gauge theory for maximal direct and maximal indirect center projections. We find that this problem is much more severe than it was thought before. The projected string tension is not in agreement with the physical string tension. We also show that the particle--like objects, nexuses, might be important for the confinement dynamics.
Write-up for Lattice'92, held in Amsterdam. preprint LTH 291. Comes with 6 PostScript figures and 1 sty-file. We study the nature of gauge fixing ambiguities in two dimensional gauge theories. We find that these ambiguities can be related to the asociated spin model. They can be eliminated by means of a (multigrid) annealing algorithm.
We reconsider the lattice measurement of Green functions in Coulomb gauge, both in 2+1 and 3+1 dimensions, using an improved gauge fixing scheme. The influence of Gribov copies is examined and we find clear indications that Green functions are more strongly affected than previously assumed, in particular for low momenta. Qualitatively, our improved lattice results in the infra-red compare more favourably with recent continuum calculations in the Hamiltonian approach.
We study the influence of Gribov copies on gluon and ghost propagators in lattice Landau gauge. For the gluon propagator, Gribov noise seems to be of the order of magnitude of the numerical accuracy. On the contrary, for the ghost propagator, Gribov noise is clearly observable, at least in the strong- coupling regime. We also observe, in the limit of large lattice volume, a gluon propagator decreasing as the momentum goes to zero. Finally, we introduce an implementation of the method of Fourier Acceleration which avoids the use of the fast Fourier transform, being well suited for parallel and …
The covariant gauges are known to suffer from the Gribov problem: even after fixing a gauge non-perturbatively, there may still exist residual copies which are physically equivalent to each other, called Gribov copies. While the influence of Gribov copies in the relevant quantities such as gluon propagators has been heavily debated in recent studies, the significance of the role they play in the Faddeev--Popov procedure is hardly doubted. We concentrate on Gribov copies in the first Gribov region, i.e., the space of Gribov copies at which the Faddeev--Popov operator is strictly positive (semi)…
We study the influence of Gribov copies on gluon and ghost propagators, evaluated numerically in pure SU(2) lattice gauge theory in the minimal Landau gauge. Simulations are done at four different values of $β$ (namely $β$ = 0, 0.8, 1.6 and 2.7) and for volumes up to $16^4$ (up to $24^4$ at $β$ = 1.6). For the gluon propagator, Gribov noise seems to be of the order of magnitude of the numerical accuracy, even at very small values of the coupling $β$. On the contrary, for the ghost propagator, Gribov noise is clearly observable for the three values of $β$ in the strong-coupling regime. In parti…
We discuss the influence of Dirac sheets and zero-momentum modes on the gauge variant photon correlators $Γ(τ;\vp)$ with $\vp\ne 0$ and $\vp =0$ in the pure gauge U(1) theory. A special attention has been paid to the $β$- and volume-dependence of this influence. Numerical simulations are performed on $12\times 6^3$ and $24\times 12^3$ lattices at different $yβ$'s in the Coulomb phase.
We show that a modification of the BRST lattice quantization allows to circumvent an old paradox, formulated by Neuberger, related to lattice Gribov copies and non-perturbative BRST invariance. In the continuum limit the usual BRST formulation is recovered.
The standard techniques of gauge-fixing, such as covariant gauge fixing, are entirely adequate for the purposes of studies of perturbative QCD. However, they fail in the nonperturbative regime due to the presence of Gribov copies. These copies arise because standard local gauge fixing methods do not completely fix the gauge. Known Gribov-copy-free gauges, such as Laplacian gauge, are manifestly non-local. These issues are examined and the implications of non-local gauge-fixing for ghost fields, BRST invariance, and the proof of renormalizability of QCD are considered.
Beyond perturbation theory the number of gauge copies drastically increases due to the Gribov-Singer ambiguity. Any way of treating them defines, in principle, a new, non-perturbative gauge, and the gauge-dependent correlation functions can vary between them. Herein various such gauges will be constructed as completions of the Landau gauge inside the first Gribov region. The dependence of the propagators and the running coupling on these gauges will be studied for SU(2) Yang-Mills theory in two, three, and four dimensions using lattice gauge theory, and for a wide range of lattice parameters. …
We review the equivalence of maximal center gauge fixing to the problem of finding the best fit, to a given lattice gauge field, by a thin vortex configuration. This fit is necessarily worst at the location of P-plaquettes. We then compare the fits achieved in Gribov copies generated by (i) over-relaxation; (ii) over-relaxation after Landau gauge preconditioning; and (iii) simulated annealing. Simulated annealing yields the best fit if all links on the lattice are included, but the situation changes if we consider only the lattice volume exterior to P-plaquettes. In this exterior region, the f…
We present results of the careful study of the Gribov copies problem in SU(2) lattice gauge theory for the direct maximal center projection widely used in confinement studies. Applying simulated annealing algorithm we demonstrate that this problem is more severe than it was thought before. The projected (gauge noninvariant) string tension is not in the agreement with the physical string tension. We do not find any indications that P-vortices reproduce the full SU(2) string tension neither in the infinite volume limit nor in the continuum limit.
We study the influence of Gribov copies, in the Landau gauge, on the lattice renormalisation constant of the axial current, obtained from a Ward identity on quark state correlation functions, with the Clover action, in quenched SU(3) gauge theory. A comparison between the gauge invariant determination of the renorma= lization constant and the gauge dependent one is discussed. We find that the residual gauge freedom associated to Gribov copies induces observable effects, which, at the level of numerical accuracy of our simulation, are included in the statistical uncertainty inherent in a Monte …
We calculate the complex phase of chiral determinant by the vacuum overlap formula with configurations of two-dimensional U(1) gauge field fixed in Landau and Laplacian gauge. The complex phase fluctuates over the Gribov copies, which appear in the process of Landau gauge fixing and contain vortex-like singularities. In the Laplacian gauge, the fluctuation can be reduced and the phase can be determined uniquely. If it is used as a preconditioning for Landau gauge fixing, the most smooth configuration is obtained among the copies generated.
The infrared behavior of the gluon and ghost propagators is studied in SU(2) Euclidean Yang-Mills theory in the maximal Abelian gauge within the Gribov-Zwanziger framework. The nonperturbative effects associated with the Gribov copies and with the dimension two condensates are simultaneously encoded into a local and renormalizable Lagrangian. The resulting behavior turns out to be in good agreement with the lattice data.
The continuum Yang-Mills functional integral within the first order formalism and in Coulomb gauge is studied. In particular, the temporal zero-modes of the Faddeev-Popov operator are explicitly accounted for. It is shown that the treatment of these zero-modes results in the constraint that the total color charge of the system vanishes at all times. Further, it is argued that the functional integral is effectively fully gauge-fixed once Gauss' law has been resolved in Coulomb gauge.
In the modern formulation of lattice gauge-fixing, the gauge fixing condition is written in terms of the minima or stationary points (collectively called solutions) of a gauge-fixing functional. Due to the non-linearity of this functional, it usually has many solutions called Gribov copies. The dependence of the number of Gribov copies, n[U] on the different gauge orbits plays an important role in constructing the Faddeev-Popov procedure and hence in realising the BRST symmetry on the lattice. Here, we initiate a study of counting n[U] for different orbits using three complimentary methods: 1.…
We study the approach, initiated by Marinari et al., to the static inter-quark potential based on Polyakov lines of finite temporal extent, evaluated in Coulomb gauge. We show that, at small spatial separations, the potential can be understood as being between two separately gauge invariant colour charges. At larger separations Gribov copies obstruct the non-perturbative identification of individually gauge invariant colour states. We demonstrate, for the first time, how gauge invariance can be maintained quite generally by averaging over Gribov copies. This allows us to extend the analysis of…
The gluon and ghost propagators are computed using both quenched and dynamical configurations for the same lattice spacings. The Wilson fermions are simulated at several quark masses. Furthermore, the effect of the Gribov copies is evaluated for all sets of configurations.
The effects of the Gribov copies on the gluon and ghost propagators are investigated in SU(2) Euclidean Yang-Mills theory quantized in the maximal Abelian gauge. The diagonal component of the gluon propagator displays the characteristic Gribov type behavior. The off-diagonal component of the gluon propagator is found to be of the Yukawa type, with a dynamical mass originating from the dimension two condensate <A^2>, which is also taken into account. Finally, the off-diagonal ghost propagator exhibits infrared enhancement.
We construct a smooth gauge for the adjoint field which is free of ambiguities on the lattice. In this Laplacian Center Gauge, center vortices and monopoles appear together as local gauge defects. A numerical study of center vortices in SU(2) and SU(3) supports equality of the $Z_N$ and SU(N) string tensions in the continuum limit, and only then.
In this paper, a proposal for the restriction of the Euclidean functional integral to a region free of infinitesimal Gribov copies in linear covariant gauges is discussed. An effective action, akin to the Gribov-Zwanziger action of the Landau gauge, is obtained which implements the aforementioned restriction. Although originally non-local, this action can be cast in local form by introducing auxiliary fields. As in the case of the Landau gauge, dimension two condensates are generated at the quantum level, giving rise to a refinement of the action which is employed to obtain the tree-level gluo…
We calculate the Faddeev-Popov operator corresponding to the maximally Abelian gauge for gauge group SU(N). Specializing to SU(2) we look for explicit zero modes of this operator. Within an illuminating toy model (Yang-Mills mechanics) the problem can be completely solved and understood. In the field theory case we are able to find an analytic expression for a normalizable zero mode in the background of a single `t Hooft instanton. Accordingly, such an instanton corresponds to a horizon configuration in the maximally Abelian gauge. Possible physical implications are discussed.
In this paper, we discuss non-perturbative infrared features of Yang-Mills theory in Euclidean space-time dimensions greater than four in the Landau gauge and within the Refined Gribov-Zwanziger framework, which enables us to take into account the existence of gauge copies by restricting the domain of integration in the path integral to the Gribov region. Evidences for a decoupling/massive solution for the gluon propagator in higher dimensions are provided. This behavior is strengthened the bigger the dimension is. Further, we show that, by a dimensional reduction of the Refined Gribov-Zwanzig…
We study the problem of the Landau gauge fixing in the case of the SU(2) lattice gauge theory. We show that the probability to find a lattice Gribov copy increases considerably when the physical size of the lattice exceeds some critical value $\approx2.75/\sqrtσ$, almost independent of the lattice spacing. The impact of the choice of the copy on Green functions is presented. We confirm that the ghost propagator depends on the choice of the copy, this dependence decreasing for increasing volumes above the critical one. The gluon propagator as well as the gluonic three-point functions are insens…
We study the gluon propagator in quenched lattice QCD using the Laplacian gauge which is free of lattice Gribov copies. We compare our results with those obtained in the Landau gauge on the lattice, as well as with various approximate solutions of the Dyson Schwinger equations. We find a finite value $\sim (445 \rm{MeV})^{-2}$ for the renormalized zero-momentum propagator (taking our renormalization point at 1.943 GeV), and a pole mass $\sim 640 \pm 140$ MeV.
We study gauge fixing via the standard local extremization algorithm for 2-dimensional $U(1)$. On a lattice with spherical topology $S^2$ where all copies are lattice artifacts, we find that the number of these 'Gribov' copies diverges in the continuum limit. On a torus, we show that lattice artifacts can lead to the wrong evaluation of the gauge-invariant correlation length, when measured via a gauge-fixed procedure; this bias does not disappear in the continuum limit. We then present a new global approach, based on Hodge decomposition of the gauge field, which produces a unique smooth field …
We present results for the fundamental string tension in SU(2) lattice gauge theory after projection to maximal abelian and direct maximal center gauges. We generate 20 Gribov copies/configuration. Abelian and center projected string tensions slowly decrease as higher values of the gauge functionals are reached.
%In order to understand how gauge fixing can be affected on the %lattice, we first study a simple model of pure Yang-mills theory on a %cylindrical spacetime [$SU(N)$ on $S^1 \times$ {\bf R}] where the %gauge fixed subspace is explicitly displayed. On the way, we find that %different gauge fixing procedures lead to different Hamiltonians and %spectra, which however coincide under a shift of states. The lattice %version of the model is compared and lattice gauge fixing issues are %discussed. (---TALK GIVEN AT LATTICE 92---AMSTERDAM, 15 SEPT. 92)
For the Lorentz gauge the influence of Gribov copies on the fermion propagator is investigated in quenched lattice compact QED. In the Coulomb phase zero-momentum modes of the gauge fields are shown to be the main reason for a significant deviation from ordinary perturbation theory.
We extend our study of the gluon propagator in quenched lattice QCD using the Laplacian gauge to a finer lattice. We verify the existence of a pole mass as we take the continuum limit and deduce a value of $\sim 600^{+150}_{-30}$ MeV for this pole mass. We find a finite value of $(454(5){\rm MeV})^{-2}$ for the renormalized zero-momentum propagator, in agreement with results on coarser lattices.
We reinvestigate the problem of Gribov ambiguities within the Landau (or Lorentz) gauge for the ghost and gluon propagators in pure SU(2) lattice gauge theory. We make use of the full symmetry group of the action taking into account {\it large}, i.e. non-periodic $\mathbb{Z}(2)$ gauge transformations leaving lattice plaquettes invariant. Enlarging in this way the gauge orbits for any given gauge field configuration the Landau gauge can be fixed at higher local extrema of the gauge functional in comparison with standard (overrelaxation) techniques. This has a clearly visible effect not only for…
The Gribov copies and their consequences on the infrared behavior of the gluon propagator are investigated in Euclidean Yang-Mills theories quantized in linear covariant gauges. Considering small values of the gauge parameter, it turns out that the transverse component of the gluon propagator is suppressed, while its longitudinal part is left unchanged. A Green function, G_{tr}, which displays infrared enhancement and which reduces to the ghost propagator in the Landau gauge is identified. The inclusion of the dimension two gluon condensate <A^2> is also considered. In this case, the transvers…
The Gribov ambiguity problem is studied for compact U(1) lattice theory within the Lorentz gauge. In the Coulomb phase, it is shown that apart from double Dirac sheets all gauge (i.e. Gribov) copies originate mainly from the zero-momentum modes of the gauge fields. The removal of the zero-momentum modes turns out to be necessary for reaching the absolute maximum of the gauge functional $F(θ)$. A new gauge fixing procedure - zero-momentum Lorentz gauge - is proposed.
Perturbative QCD uses the Faddeev-Popov gauge-fixing procedure, which leads to ghosts and the local BRST invariance of the gauge-fixed perturbative QCD action. In the asymptotic regime, where perturbative QCD is relevant, Gribov copies can be neglected. In the nonperturbative regime, one must adopt either a nonlocal Gribov-copy free gauge (e.g., Laplacian gauge) or attempt to maintain local BRST invariance at the expense of admitting Gribov copies. These issues are explored and discussed. In addition, the relationship between recent Dyson-Schwinger based calculations of the infrared behavior o…
The center vortex model, capable of explaining confinement and chiral symmetry breaking, has been plagued by the lattice equivalent of Gribov copies: different maxima of the gauge functional lead to different predictions of the string tension. It is possible to resolve this problem using center regions, loops evaluating to center elements, as guide for the gauge fixing procedure. The success of this approach was already shown, but the algorithms came with an arbitrary free parameter. In recent development this parameter has been fixed, even improving the results.
We propose a modified lattice Landau gauge based on stereographically projecting the link variables on the circle S^1 -> R for compact U(1) or the 3-sphere S^3 -> R^3 for SU(2) before imposing the Landau gauge condition. This can reduce the number of Gribov copies exponentially and solves the Gribov problem in compact U(1) where it is a lattice artifact. Applied to the maximal Abelian subgroup this might be just enough to avoid the perfect cancellation amongst the Gribov copies in a lattice BRST formulation for SU(N), and thus to avoid the Neuberger 0/0 problem. The continuum limit of the Land…
We suggest a method of introducing the Gribov--Zwanziger horizon functional, $H$, for Yang--Mills theories by using the composite fields technique: $σ(ϕ)=H$. A different form of the same horizon functional in gauges $χ$ and $χ^{\prime }$ is taken into account via (gauged) field-dependent BRST transformations connecting quantum Yang--Mills actions in these gauges. We introduce generating functionals of Green's functions with composite fields and derive the corresponding Ward identities. A study of gauge dependence shows that the effective action in Yang--Mills theories with the composite field …
The properties of the thermal Abelian color-magnetic monopoles in the maximally Abelian gauge are studied in the deconfinement phase of the lattice SU(2) gluodynamics. To check universality of the monopole properties we employ the tadpole improved Symanzik action. The simulated annealing algorithm combined with multiple gauge copies is applied for fixing the maximally Abelian gauge to avoid effects of Gribov copies. We compute the density, interaction parameters, thermal mass and chemical potential of the thermal Abelian monopoles in the temperature range between Tc and 3Tc. In comparison with…
For SU(2) lattice gauge theory we study numerically the infrared behavior of the Landau gauge ghost and gluon propagators with the special accent on the Gribov copy dependence. Applying a very efficient gauge fixing procedure and generating up to 80 gauge copies we find that the Gribov copy effect for both propagators is essential in the infrared. In particular, our best copy dressing function of the ghost propagator approaches a plateau in the infrared, while for the random first copy it still grows. Our best copy zero-momentum gluon propagator shows a tendency to decrease with growing lattic…
We perform a study of the effects from maximal abelian gauge Gribov copies in the context of the dual superconductor scenario of confinement, on the basis of a novel approach for estimation of systematic uncertainties from incomplete gauge fixing. We present numerical results, in SU(2) lattice gauge theory, using the overrelaxed simulated annealing gauge fixing algorithm. We find abelian and non-abelian string tensions to differ significantly, their ratio being 0.92(4) at BETA = 2.5115. An approximate factorization of the abelian potential into monopole and photon contributions has been confir…
We study numerically the so-called fundamental modular region Lambda, a region free of Gribov copies, in the minimal Landau gauge for pure SU(2) lattice gauge theory. To this end we evaluate the influence of Gribov copies on several quantities --- such as the smallest eigenvalue of the Faddeev-Popov matrix, the third and the fourth derivatives of the minimizing function, and the so-called horizon function --- which are used to characterize the region Lambda. Simulations are done at four different values of the coupling: beta = 0, 0.8, 1.6, 2.7, and for volumes up to 16^4. We find that typical …
The Landau background gauge, also known as the Landau-DeWitt gauge, has found renewed interest during the past decade given its usefulness in accessing the confinement-deconfinement transition via the vacuum expectation value of the Polyakov loop, describable via an appropriate background. In this Letter, we revisit this gauge from the viewpoint of it displaying gauge (Gribov) copies. We generalize the Gribov-Zwanziger effective action in a BRST and background invariant way; this action leads to a restriction on the allowed gauge fluctuations, thereby eliminating the infinitesimal background g…
