Gribov Copy Effects Research Articles

Toward background field independence within the Gribov horizon

We introduce a background gauge akin to the Landau-DeWitt gauge but deformed by the presence of a gauge parameter for the quantization of Euclidean Yang-Mills theories. In the limit where the background field vanishes, standard linear covariant gauges are recovered. This gauge allows for an explicit investigation of the effects of infinitesimal Gribov copies and their impact to background and gauge parameter dependence of physical correlators. Similarly to linear covariant gauges, the introduction of gauge-invariant dressed fields is essential to restore the Becchi-Rouet-Stora-Tyutin (BRST) symmetry. Hence, we construct a BRST symmetric action in linear covariant background gauges which eliminates regular infinitesimal Gribov copies in analogy to the recently introduced BRST invariant (refined) Gribov-Zwanziger action. The issue of background dependence and its relation to gauge parameter dependence is discussed in the light of nonperturbative effects driven by the elimination of Gribov copies.

Zero-momentum correlator in SU(2) Landau gauge gluodynamics under various boundary conditions

We make new simulations of S U (2) zero spatial momentum (ZM) gluon correlator in Landau gauge gluodynamics paying special attention to possible lattice artefacts and Gribov copy effect. In particular we started investigation of the correlator dependence on the choice of boundary conditions (BCs) by comparing results for periodic and zerofield (ZF) boundary conditions at various β values. Time behaviour of the ZM correlator for ZF BCs corresponds to approximately constant in time effective gluon mass thus providing additional evidence in favour of decoupling behaviour of momentum-dependent gluon propagator in the IR region. We have found that at fixed lattice sizes and β values the ZM gluon correlator and effective gluon mass for periodic and ZF boundary conditions can differ considerably.

Remarks on the effects of the Gribov copies on the infrared behavior of higher dimensional Yang-Mills theory

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-Zwanziger action from five to four dimensions, a non-perturbative coupling between the inverse of the Faddeev-Popov operator and the scalar field corresponding to the fifth component of the gauge field naturally arises, being in agreement with the recently proposed mechanism \cite{Capri:2014bsa} to generalize the Refined Gribov-Zwanziger construction to the matter sector.

Ghost sector and geometry in minimal Landau gauge: Further constraining the infinite-volume limit

We present improved upper and lower bounds for the momentum-space ghost propagator of Yang-Mills theories in terms of the two smallest nonzero eigenvalues (and their corresponding eigenvectors) of the Faddeev-Popov matrix. These results are verified using data from four-dimensional numerical simulations of SU(2) lattice gauge theory in minimal Landau gauge at beta = 2.2, for lattice sides N = 16, 32, 48 and 64. Gribov-copy effects are discussed by considering four different sets of numerical minima. We then present a lower bound for the smallest nonzero eigenvalue of the Faddeev-Popov matrix in terms of the smallest nonzero momentum on the lattice and of a parameter characterizing the geometry of the first Gribov region $\Omega$. This allows a simple and intuitive description of the infinite-volume limit in the ghost sector. In particular, we show how nonperturbative effects may be quantified by the rate at which typical thermalized and gauge-fixed configurations approach the boundary of Omega, known as the first Gribov horizon. As a result, a simple and concrete explanation emerges for why lattice studies do not observe an enhanced ghost propagator in the deep infrared limit. Most of the simulations have been performed on the Blue Gene/P--IBM supercomputer shared by Rice University and S\~ao Paulo University.

Lattice gluon propagators in 3DSU(2)theory and effects of Gribov copies

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.

Abelian monopoles in finite temperature latticeSU(2)gluodynamics: First study with improved action

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 earlier findings our results for these quantities are improved either with respect to effects of Gribov copies or with respect to lattice artifacts.

LATTICE QCD GLUON PROPAGATORS NEAR TRANSITION TEMPERATURE

Landau gauge gluon propagators are studied numerically in the SU (3) gluodynamics as well as in the full QCD with the number of flavors nF = 2 using efficient gauge fixing technique. We compare these propagators at temperatures very close to the transition point in two phases: confinement and deconfinement. The electric mass mE has been determined from the momentum space longitudinal gluon propagator. Gribov copy effects are found to be rather strong in the gluodynamics, while in the full QCD case they are weak ("Gribov noise"). Also we analyze finite volume dependence of the transverse and longitudinal propagators.

Landau gauge gluon and ghost propagators at finite temperature from quenched lattice QCD

The behavior of the Landau gauge gluon and ghost propagators is studied in pure SU(3) gauge theory at non-zero temperature on the lattice. We concentrate on the momentum range [0.6, 2.0] GeV. For the longitudinal as well as for the transverse component of the gluon propagator we extract the continuum limit. We demonstrate the smallness of finite-size and Gribov-copy effects at temperatures close to the deconfinement phase transition at T=T_c and within the restricted range of momenta. Since the longitudinal component D_L(q) turns out to be most sensitive with respect to the phase transition we propose some combinations of D_L(q) signalling the transition much like "order parameters".

Study of the thermal Abelian monopoles with proper gauge fixing

The properties of the thermal abelian monopoles are studied in the deconfinement phase of the SU(2) gluodynamics. To remove effects of Gribov copies the simulated annealing algorithm is applied to fix the maximally abelian gauge. Computing the density of the thermal abelian monopoles in the temperature range between 1.5T_c and 6.9T_c we show, by comparison with earlier results, that the Gribov copies effects might be as high as 20% making proper gauge fixing mandatory. We find that in the infinite temperature limit the monopole density converges to its value in 3-dimensional theory. To study the interaction between monopoles we calculate the monopole-monopole and monopole-antimonopole correlators at different temperatures in the region (1.5T_c, 6.9T_c). Using the result of this study we determine the screening mass, monopole-monopole coupling constant, monopole size and monopole mass. In addition we check the continuum limit of our results.

SU(2)lattice gluon propagators at finite temperatures in the deep infrared region and Gribov copy effects

We study numerically the SU(2) Landau gauge transverse and longitudinal gluon propagators at non-zero temperatures T both in confinement and deconfinement phases. The special attention is paid to the Gribov copy effects in the IR-region. Applying powerful gauge fixing algorithm we find that the Gribov copy effects for the transverse propagator D_T(p) are very strong in the infrared, while the longitudinal propagator D_L(p) shows very weak (if any) Gribov copy dependence. The value D_T(0) tends to decrease with growing lattice size; however, D_T(0) is non-zero in the infinite volume limit, in disagreement with the suggestion made in [1]. We show that in the infrared region D_T(p) is not consistent with the pole-type formula not only in the deconfinement phase but also for T < T_c. We introduce new definition of the magnetic infrared mass scale ('magnetic screening mass') m_M. The electric mass m_E has been determined from the momentum space longitudinal gluon propagator. We study also the (finite) volume and temperature dependence of the propagators as well as discretization errors.

Thermal Abelian monopoles as self-dual dyons

The properties of the thermal Abelian monopoles are studied in the deconfinement phase of the SU(2) gluodynamics. To remove effects of Gribov copies the simulated annealing algorithm is applied to fix the maximally Abelian gauge. To study monopole profile we complete the first computations of excess of the nonabelian action density as a function of the distance from the center of the thermal Abelian monopole. We have found that starting from the distances $\approx$ 2 lattice spacings the chromoelectric and chromomagnetic action densities created by monopole are equal to each other, from what we draw a conclusion that monopole is a dyon. Furthermore, we find that the chromoelectric and chromomagnetic fields decrease exponentially with increasing distance. These findings were confirmed for different temperatures in the range $T/T_c \in (1.5, 4.8)$.

SU(3)Landau gauge gluon and ghost propagators using the logarithmic lattice gluon field definition

We study the Landau gauge gluon and ghost propagators of $SU(3)$ gauge theory, employing the logarithmic definition for the lattice gluon fields and implementing the corresponding form of the Faddeev-Popov matrix. This is necessary in order to consistently compare lattice data for the bare propagators with that of higher-loop numerical stochastic perturbation theory. In this paper we provide such a comparison, and introduce what is needed for an efficient lattice study. When comparing our data for the logarithmic definition to that of the standard lattice Landau gauge we clearly see the propagators to be multiplicatively related. The data of the associated ghost-gluon coupling matches up almost completely. For the explored lattice spacings and sizes discretization artifacts, finite size, and Gribov-copy effects are small. At weak coupling and large momentum, the bare propagators and the ghost-gluon coupling are seen to be approached by those of higher-order numerical stochastic perturbation theory.

Spectral sum for the color-Coulomb potential inSU(3)Coulomb gauge lattice Yang-Mills theory

We discuss the essential role of the low-lying eigenmodes of the Faddeev-Popov (FP) ghost operator on the confining color-Coulomb potential using $SU(3)$ quenched lattice simulations in the Coulomb gauge. The color-Coulomb potential is expressed as a spectral sum of the FP ghost operator and has been explored by partially summing the FP eigenmodes. We take into account the Gribov copy effects that have a great impact on the FP eigenvalues and the color-Coulomb potential. We observe that the lowest eigenvalue vanishes in the thermodynamic limit much faster than that in the Landau gauge. The color-Coulomb potential at large distances is governed by the near-zero FP eigenmodes; in particular, the lowest one accounts for a substantial portion of the color-Coulomb string tension comparable to the Wilson string tension.

Mass gap without vacuum energy

We consider soft nonlocal deformations of massless theories that introduce a mass gap. By use of a renormalization scheme that preserves the ultraviolet softness of the deformation, renormalized quantities of low mass dimension, such as normal mass terms, vanish via finite counterterms. The same applies to the renormalized cosmological constant. We connect this discussion to gauge theories, since they are also subject to a soft nonlocal deformation due to the effects of Gribov copies. These effects are softer than usually portrayed.

The effects of Gribov copies in 2D gauge theories

In previous works, we have motivated that the Gribov–Zwanziger action, which implements the restriction of the domain of integration in the path integral to the Gribov region, generates extra dynamical effects which influence the infrared behaviour of the gluon and ghost propagator in SU(N) Yang–Mills gauge theories. The latter are in good agreement with the most recent lattice data obtained at large volumes, both in 4D and in 3D. More precisely, the gluon propagator is suppressed and does not vanish at zero momentum, while the ghost propagator keeps a 1/p2 behaviour for p2≈0. Instead, in 2D, the lattice data revealed a vanishing zero momentum gluon propagator and an infrared enhanced ghost, in support of the usual Gribov–Zwanziger scenario. We will now discuss that the 2D version of the Gribov–Zwanziger action still gives results in qualitative agreement with these lattice data, as the peculiar infrared nature of 2D gauge theories precludes the analogue of the dynamical effect otherwise present in 4D and 3D. Simultaneously, we also observe that the Gribov–Zwanziger restriction serves as an infrared regulating mechanism.

Infrared behavior and Gribov ambiguity inSU(2)lattice gauge theory

For SU(2) lattice gauge theory, we study numerically the infrared behavior of the Landau-gauge ghost and gluon propagators with a 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 continues to grow. Our best copy zero-momentum gluon propagator shows a tendency to decrease with growing lattice size, which excludes singular solutions. Our results seem compatible with the so-called decoupling solution with a nonsingular gluon propagator. However, we do not yet consider the Gribov copy problem to be resolved.

Hamiltonian approach to 1 + 1 dimensional Yang–Mills theory in Coulomb gauge

We study the Hamiltonian approach to 1 + 1 dimensional Yang–Mills theory in Coulomb gauge, considering both the pure Coulomb gauge and the gauge where in addition the remaining constant gauge field is restricted to the Cartan algebra. We evaluate the corresponding Faddeev–Popov determinants, resolve Gauss’ law and derive the Hamiltonians, which differ in both gauges due to additional zero modes of the Faddeev–Popov kernel in the pure Coulomb gauge. By Gauss’ law the zero modes of the Faddeev–Popov kernel constrain the physical wave functionals to zero colour charge states. We solve the Schrödinger equation in the pure Coulomb gauge and determine the vacuum wave functional. The gluon and ghost propagators and the static colour Coulomb potential are calculated in the first Gribov region as well as in the fundamental modular region, and Gribov copy effects are studied. We explicitly demonstrate that the Dyson–Schwinger equations do not specify the Gribov region while the propagators and vertices do depend on the Gribov region chosen. In this sense, the Dyson–Schwinger equations alone do not provide the full non-abelian quantum gauge theory, but subsidiary conditions must be required. Implications of Gribov copy effects for lattice calculations of the infrared behaviour of gauge-fixed propagators are discussed. We compute the ghost–gluon vertex and provide a sensible truncation of Dyson–Schwinger equations. Approximations of the variational approach to the 3 + 1 dimensional theory are checked by comparison to the 1 + 1 dimensional case.

Magnetic monopoles in the high temperature phase of Yang–Mills theories

We investigate the properties of thermal Abelian magnetic monopoles in the high temperature phase of Yang–Mills theories, following a recent proposal for their identification on lattice configurations. The study is done for SU(2) pure gauge theory, for temperatures going up to about 10 times the deconfining temperature and using the Maximal Abelian gauge to perform the Abelian projection. We find that the monopole density has a well defined continuum limit. Its temperature dependence disagrees with a free particle gas prediction and is instead well described by a T 3 / ( log ( T / Λ ) ) α behaviour in all the explored range, with α ∼ 2 and Λ ∼ 100 MeV . Also the study of spatial correlations of thermal monopoles shows the presence of non-trivial interactions among them. Finally, we discuss the gauge dependence of our results, showing that it is significant and that, even within the Maximal Abelian gauge, Gribov copy effects are important.

Numerical Study of the Ghost-Gluon Vertex in Landau gauge

We present a numerical study of the ghost-gluon vertex and of the corresponding renormalization function 1(p 2) in minimal Landau gauge for SU(2) lattice gauge theory. Data were obtained for three different lattice volumes V( = 44,84,164 ) and for three lattice couplings ? = 2.2,2.3,2.4. Gribov-copy effects have been analyzed using the so-called smeared gauge fixing. We also consider two different sets of momenta (orbits) in order to check for possible effects due to the breaking of rotational symmetry. The vertex has been evaluated at the asymmetric point (0;p,?p) in momentum-subtraction scheme. We find that 1(p 2) is approximately constant and equal to 1, at least for momenta p1 GeV. This constitutes a nonperturbative verification of the so-called nonrenormalization of the Landau ghost-gluon vertex. Finally, we use our data to evaluate the running coupling constant ? s (p 2).

Propagators and running coupling from SU(2) lattice gauge theory

We perform numerical studies of the running coupling constant α R ( p 2) and of the gluon and ghost propagators for pure SU(2) lattice gauge theory in the minimal Landau gauge. Different definitions of the gauge fields and different gauge-fixing procedures are used, respectively, for gaining better control over the approach to the continuum limit and for a better understanding of Gribov-copy effects. We find that the ghost–ghost–gluon vertex renormalization constant is finite in the continuum limit, confirming earlier results by all-order perturbation theory. In the low momentum regime, the gluon form factor is suppressed while the ghost form factor is divergent. Correspondingly, the ghost propagator diverges faster than 1/ p 2 and the gluon propagator appears to be finite. Precision data for the running coupling α R ( p 2) are obtained. These data are consistent with an IR fixed point given by lim p→0 α R ( p 2)=5(1).