We address the issue why the number and the location of magnetic monopoles detected on lattice configurations are gauge dependent, in contrast with the physical expectation that monopoles have a gauge invariant status. By use of the Non-Abelian Bianchi Identities we show that monopoles are gauge invariant, but the efficiency of the technique usually adopted to detect them depends on the choice of the gauge in a well understood way. In particular we have studied a class of gauges which interpolates between the Maximal Abelian gauge, where all monopoles are observed, and the Landau gauge, where all monopoles escape detection.
We study the action of the lattice monopoles in quenched SU(2) QCD in the Maximal Abelian projection. We relate the lattice action of the monopole currents to the monopole degrees of freedom of the continuum dual superconductor model and obtain the value of the monopole condensate.
We review briefly current interpretation of the lattice monopoles, defined within the Maximal Abelian Projection, in terms of the continuum theory. We emphasize, in particular, that the lattice data, at the presently available lattices, indicate that the monopoles are associated with singular fields. This note is prompted by a recent analysis hep-lat/0211005 which is based on an implicit assumption that the fields are regular.
We consider lattice implementation of the recently proposed gauge invariant definition of the monopole charge. Because of the lattice discretization the algorithm gives rise to specific lattice artifacts and an effective Ising model. The Ising-model problem might in principle be solved and we discuss the role of the Maximal Abelian gauge in this respect. The lattice artifacts are much more difficult to deal with since they are mixed up with monopoles thus obscuring the physical observables. Nevertheless, it is possible to extract the density of physical monopoles which seems to scale correctly…
It is shown that the creation of a monopole is a gauge invariant statement, based on topology. Creating a monopole is independent on the abelian projection in which it is created. This is fundamental in defining an order parameter for detecting dual superconductivity of the QCD vacuum.
We study the temperature dependence of the monopole condensate in different Abelian projections of the SU(2) lattice gauge theory. Using the Frohlich-Marchetti monopole creation operator we show numerically that the monopole condensate depends on the choice of the Abelian projection. Contrary to the claims in the current literature we observe that in the Abelian Polyakov gauge and in the field strength gauge the monopole condensate does not vanish at the critical temperature and thus is not an order parameter.
By numerical calculations we show that the abelian monopole currents are locally correlated with the density of SU(2) lattice action. This fact is established for the maximal abelian projection. Thus, in the maximal abelian projection the monopoles are physical objects, they carry SU(2) action. Calculations on the asymmetric lattice show that the correlation between monopole currents and the density of SU(2) lattice action also exists in the deconfinement phase of gluodynamics.
We discuss some properties of abelian monopoles in the Maximal Abelian projection of the SU(2) lattice gluodynamics. We show that in the maximal abelian projection abelian monopoles carry fluctuating electric charge and that the monopole currents are correlated with the magnetic and the electric parts of the SU(2) action density.
Within the context of the Abelian Projection of QCD monopole-like quantum excitations of gauge fields are studied. We start with certain classical solutions, of the SU(2) Yang-Mills field equations, which are not monopole-like and whose energy density diverges as $r \to \infty$. These divergent classical solutions are then quantized using a modified version of Heisenberg's quantization technique for strongly interacting, nonlinear fields. The modified Heisenberg quantization technique leads to a system of equations with mixed quantum and classical degrees of freedom. By applying a Feynman path…
The long standing problem is solved why the number and the location of monopoles observed in Lattice configurations depend on the choice of the gauge used to detect them, in contrast to the obvious requirement that monopoles, as physical objects, must have a gauge-invariant status. It is proved, by use of non-abelian Bianchi identities, that monopoles are indeed gauge-invariant: the technique used to detect them has instead an efficiency which depends on the choice of the abelian projection, in a known and controllable way.
We study the effective monopole action obtained in the Maximal Abelian projection of quenched SU(2) lattice QCD. We determine the quadratic part of the lattice action using analytical blocking from continuum dual superconductor model to the lattice model. The leading contribution to the quadratic action depends explicitly on value of the monopole condensate. We show that the analytical monopole action matches the numerically obtained action in quenched SU(2) QCD with a good accuracy. The comparison of numerical and analytical results gives us the value of the monopole condensate in quenched SU…
We study the deconfinement phase transition and monopole properties in the finite temperature 3D compact Abelian gauge model on the lattice. We predict the critical coupling as function of the lattice size in a simplified model to describe monopole binding. We demonstrate numerically that the monopoles are sensitive to the transition. In the deconfinement phase the monopoles appear in the form of a dilute gas of magnetic dipoles. In the confinement phase both monopole density and string tension differ from semiclassical estimates if monopole binding is neglected. However, the analysis of the m…
It is shown that the SU(2) Higgs model on a lattice is equivalent to the Georgi--Glashow model in the limit of a small coupling constant between the Higgs and gauge fields. It can therefore be concluded that the transition between the confinement and symmetric phases in the 3+1 dimensional SU(2) Higgs model at finite temperature is accompanied by condensation of Nambu monopoles.
Magnetic monopoles are known to emerge as leading non-perturbative fluctuations in the lattice version of non-Abelian gauge theories in some gauges. In terms of the Dirac quantization condition, these monopoles have magnetic charge |Q_M|=2. Also, magnetic monopoles with |Q_M|=1 can be introduced on the lattice via the 't Hooft loop operator. We consider the |Q_M|=1,2 monopoles in the continuum limit of the lattice gauge theories. To substitute for the Dirac strings which cost no action on the lattice, we allow for singular gauge potentials which are absent in the standard continuum version. On…
We show that the extended Abelian magnetic monopoles in the Maximal Abelian projection of lattice SU(2) gluodynamics are locally correlated with the magnetic and the electric parts of the SU(2) action density. These correlations are observed in the confined and in the deconfined phases.
The monopole confinement mechanism in the abelian projection of lattice gluodynamics is reviewed. The main topics are: the abelian projection on the lattice and in the continuum, a numerical study of the abelian monopoles in the lattice gauge theory. Additionally, we briefly review the notation of differential forms, duality, and the BKT transformation in the lattice gauge theories.
By numerical calculations we show that the abelian monopole currents are locally correlated with the density of SU(2) lattice action. The correlations are larger by the order of magnitude in the maximal abelian projection than in the projections which correspond to the diagonalization of Polyakov line and to the diagonalization of the plaquette. These facts show that (at least) in the maximal abelian projection the monopoles are the physical objects, they carry the SU(2) action. The larger value of β, the larger the relative action carried by monopole. Calculations on the asymmetric lattice sh…
In this paper we begin on a new lattice formulation of the non-linear change of variables called the Cho--Faddeev--Niemi decomposition in SU(2) Yang-Mills theory. This is a compact lattice formulation improving the non-compact lattice formulation proposed in our previous paper. Based on this formulation, we propose a new gauge-invariant definition of the magnetic monopole current which guarantees the magnetic charge quantization and reproduces the conventional magnetic-current density obtained in the Abelian projection based on the DeGrand--Toussaint method. Finally, we demonstrate the magneti…
We investigate the dual superconductor hypothesis in finite-temperature SU(2) lattice gluodynamics in the Spatial Maximal Abelian gauge. This gauge is more physical than the ordinary Maximal Abelian gauge due to absence of non-localities in temporal direction. We show numerically that in the Spatial Maximal Abelian gauge the probability distribution of the abelian monopole field is consistent with the dual superconductor mechanism of confinement: the abelian condensate vanishes in the deconfinement phase and is not zero in the confinement phase.
We study the Abelian and non-Abelian action densitynear the monopole in the maximal Abelian gauge of SU(2) lattice gauge theory. We find that the non-Abelian action density near the monopoles belonging to the percolating cluster decreases when we approach the monopole center. Our estimate of the monopole radius is R_mon ~ 0.04 fm.
We present the results of the numerical calculation of the probability distribution of the value of the monopole creation operator in $SU(2)$ lattice gluodynamics. We work in the maximal abelian projection. It occurs that at the low temperature, below the deconfinement phase transition the maximum of the distribution is shifted from zero, which means that the effective constraint potential is of the Higgs type. Above the phase transition the minimum of the potential (the maximum of the monopole field distribution) is at the zero value of the monopole field. This is the direct proof of the exis…
3D electroweak sphalerons on the lattice are used as test configurations for definitions of various topological defects. In the maximally Abelian gauge they are shown to contain a symmetric array of Abelian monopoles and anti-monopoles connected by two kinds of Abelian vortex strings. Gauge-invariant lattice definitions of the Nambu monopole and the Z-vortex string are formulated which correspond to Abelian projection from the unitary gauge. The sphalerons contain in their core just one (non-Abelian) Nambu monopole-anti-monopole pair (connected by a Z-string) in an unstable saddle point bound …
The Abelian monopoles in lattice gluodynamics are associated with continuum monopoles blocked to the lattice. This association allows to predict the lattice monopole action and density of the (squared) monopole charges from a continuum monopole model. The method is applied to the static monopoles in high temperature gluodynamics. We show that the numerical data both for the density and the action of the lattice monopoles can be described in terms of a Coulomb gas of Abelian monopoles in the continuum.
We study properties of the Abelian monopoles in the Maximal Abelian projection of the three dimensional pure SU(2) gauge model. We match the lattice monopole dynamics with the continuum Coulomb gas model using a method of blocking from continuum. We obtain the Debye screening length and the monopole density in continuum using numerical results for the lattice density of the (squared) monopole charges and for the monopole action. The monopoles treated within our blocking method provide about 75% contribution to the non--Abelian Debye screening length. We also find that monopoles form a Coulomb …
SU(2) gluodynamics is investigated numerically and analytically in the (Indirect) Maximal Center gauge at finite temperature. The center vortices are shown to be condensed in the confinement phase and dilute in the deconfinement phase. A new physical object, center monopole, is constructed. We show that the center monopole condensate is the order parameter of deconfinement phase transition. The linking of the vortex worldsheets and quark trajectories is identified with the Aharonov-Bohm interaction in an effective Abelian Higgs theory. We conclude that the confinement in the Maximal Center gau…
We considered the lattice electroweak theory at realistic values of $α$ and $θ_W$ and for large values of the Higgs mass. We investigated numerically the properties of topological objects that are identified with quantum Nambu monopoles. We have found that the action density near the Nambu monopole worldlines exceeds the density averaged over the lattice in the physical region of the phase diagram. Moreover, their percolation probability is found to be an order parameter for the transition between the symmetric and the broken phases. Therefore, these monopoles indeed appear as real physical ob…
To check the dual superconductor picture for the quark-confinement mechanism, we evaluate monopole dominance as well as Abelian dominance of quark confinement for both quark-antiquark and three-quark systems in SU(3) quenched lattice QCD in the maximally Abelian (MA) gauge. First, we examine Abelian dominance for the static $Q\bar Q$ system in lattice QCD with various spacing $a$ at $β$=5.8-6.4 and various size $L^3$x$L_t$. For large physical-volume lattices with $La \ge$ 2fm, we find perfect Abelian dominance of the string tension for the $Q\bar Q$ systems: $σ_{Abel} \simeq σ$. Second, we acc…
We analyze the zero-modes of the Dirac operator in quenched SU(3) gauge configurations at non-zero temperature and compare periodic and anti-periodic temporal boundary conditions for the fermions. It is demonstrated that for the different boundary conditions often the modes are localized at different space-time points and have different sizes. Our observations are consistent with patterns expected for Kraan - van Baal solutions of the classical Yang-Mills equations. These solutions consist of constituent monopoles and the zero-modes are localized on different constituents for different boundar…
The site-reduction of U(1) lattice gauge theory along the spatial directions is used to model the monopole dynamics. The reduced theory is that of the angle-valued coordinates on the discrete worldline. Below the critical coupling $g_{c}=1.125$ and temperature $T_c=0.335$ the model exhibits a first order phase transition. It is argued that the phase structure matches with the proposed role for magnetic monopoles in the confinement mechanism based on the dual Meissner effect.
We propose to consider lattice monopoles in gluodynamics as continuum monopoles blocked to the lattice. In this approach the lattice is associated with a measuring device consisting of finite-sized detectors of monopoles (lattice cells). Thus a continuum monopole theory defines the dynamics of the lattice monopoles. We apply this idea to the static monopoles in high temperature gluodynamics. We show that our suggestion allows to describe the numerical data both for the density of the lattice monopoles and for the lattice monopole action in terms of a continuum Coulomb gas model.
In this series of three papers, we generalize the derivation of dual photons and monopoles by Polyakov, and Banks, Myerson and Kogut, to obtain approximative models of SU(2) lattice gauge theory. Our approach is based on stationary phase approximations. In this third article, we start from the plaquette representation of 3-dimensional SU(2) lattice gauge theory. By extending a work of Borisenko, Voloshin and Faber, we transform the expectation value of a Wilson loop into a path integral over a dual gluon field and monopole variables. The action contains the tree-level Coulomb interaction and a…
This review is devoted to the Multiple Point Principle (MPP), according to which several vacuum states with the same energy density exist in Nature. The MPP is implemented to the Standard Model (SM), Family replicated gauge group model (FRGGM) and phase transitions in gauge theories with/without monopoles. Lattice gauge theories are reviewed. The lattice results for critical coupling constants are compared with those of the Higgs Monopole Model (HMM), in which the lattice artifact monopoles are replaced by the point-like Higgs scalar particles with a magnetic charge. Considering our (3+1)-dime…
A monopole creation operator is constructed: its vacuum expectation value is an order parameter for dual superconductivity in that, if different from zero it signals spontaneous breaking of the U(1) symmetry corresponding to monopole charge conservation. The operator is tested on compact U(1) gauge theory on lattice. For SU(2) gauge theory it clearly demonstrates that confinement is produced by dual superconductivity.
We review recent developments in understanding the physics of the magnetic monopoles in unbroken non-Abelian gauge theories. Since numerical data on the monopoles are accumulated in lattice simulations, the continuum theory is understood as the limiting case of the lattice formulation. In this review, written for a memorial volume dedicated to the memory of Academician A.B. Migdal, we emphasize physical effects related to the monopoles. In particular, we discuss the monopole-antimonopole potential at short and larger distances as well as a dual formulation of the gluodynamics, relevant to the …
Using a plaquette formulation for lattice gauge models we describe monopoles of the three dimensional SU(2) theory which appear as configurations in the complete axial gauge and violate the continuum Bianchi identity. Furthemore we derive a dual formulation for the Wilson loop in arbitrary representation and calculate the form of the interaction between generated electric flux and monopoles in the region of a weak coupling relevant for the continuum limit. The effective theory which controls the interaction is of the sine-Gordon type model. The string tension is calculated within the semiclass…
We analyze the interplay of topological objects in four-dimensional QCD on the lattice. The distributions of color magnetic monopoles in the maximum abelian gauge are computed around instantons in both pure and full QCD. We find an enhanced probability for monopoles inside the core of an instanton on gauge field average. This feature is independent of the topological charge definition used. For specific gauge field configurations we visualize the situation graphically. Moreover we investigate how monopole loops and instantons are correlated with the chiral condensate. Strong evidence is found …
We compute instanton sizes and study correlation functions between instantons and monopoles in maximum abelian projection within SU(2) lattice QCD at finite temperature. We compare several definitions of the topological charge, different lattice actions and methods of reducing quantum fluctuations. The average instanton size turns out to be $σ\approx 0.2$ fm. The correlation length between monopoles and instantons is $ζ\approx 0.25$ fm and hardly affected by lattice artifacts as dislocations. We visualize several specific gauge field configurations and show directly that there is an enhanced p…
We study topological defects constructed from diagonal and non-diagonal gluons in Abelian projection of zero temperature lattice QCD. We compare results obtained in the quenched and non-quenched vacuum field configurations and show that the density of hybrids is higher than the density of monopoles. The density of some hybrids is sensitive to the presence of the virtual fermions.
In this series of three papers, we generalize the derivation of dual photons and monopoles by Polyakov, and Banks, Myerson and Kogut, to obtain approximative models of SU(2) lattice gauge theory. The papers take three different representations as their starting points: the representation as a BF Yang-Mills theory, the spin foam representation and the plaquette representation. The derivations are based on stationary phase approximations. In this first article, we cast 3- and 4-dimensional SU(2) lattice gauge theory in the form of a lattice BF Yang-Mills theory. In several steps, the expectation…
We study Abelian monopole and vortex condensation in lattice pure gauge theories. Condensation is detected by means of a disorder parameter defined in terms of a gauge-invariant effective action introduced using the lattice Schrödinger functional. Dirac monopoles condense in the confined phase of U(1) lattice gauge theory. Abelian monopoles and Abelian vortices condense in the confined phase of SU(2) and SU(3) lattice gauge theories.
We investigate the dynamics of lattice gauge theories in an Abelian monopole background field. By means of the gauge-invariant lattice Schrodinger functional we study the Abelian monopole condensation in U(1) lattice gauge theory at zero temperature and in SU(3) lattice gauge theory at finite temperature.
Non-compact lattice QED is simulated for various numbers of fermion species $N_f$ ranging from 8 through 40 by the exact Hybrid Monte Carlo algorithm. Over this range of $N_f$, chiral symmetry breaking is found to be strongly correlated with the effective monopoles in the theory. For $N_f$ between 8 and 16 the chiral symmetry breaking and monopole percolation transitions are second order and coincident. Assuming powerlaw critical behavior, the correlation length exponent for the chiral transition is identical to that of monopole percolation. This result supports the conjecture that monopole pe…
We explicitly construct a monopole creation operator: its vacuum expectation value is an order parameter for dual superconductivity, in that, if different from zero, it signals a spontaneous breaking of the $U(1)$ symmetry corresponding to monopole charge conservation. This operator is tested by numerical simulations in compact $U(1)$ gauge theory. Our construction provides a general recipe for detection of the condensation of any topological soliton. In particular our operator can be used to detect dual superconductivity of the QCD vacuum.
We investigated numerically properties of Nambu monopoles in lattice Electroweak theory at realistic values of $α$ and $θ_W$. Our choice of parameters of lattice Lagrangian corresponds to large values of the Higgs boson mass $M_H > 2 M_W$. We find that the density of Nambu monopoles cannot be predicted by the choice of the initial parameters of Electroweak theory and should be considered as the new external parameter of the theory. We also investigate the difference between the versions of Electroweak theory with the gauge groups $SU(2)\otimes U(1)$ and $SU(2)\otimes U(1)/Z_2$. We do not detec…
Effective monopole actions for various extended monopoles are derived from vacuum configurations after abelian projection in the maximally abelian gauge in $T=0$ and $T\ne 0$ $SU(3)$ lattice QCD. The actions obtained appear to be independent of the lattice volume adopted. At zero temperature, monopole condensation is seen to occur from energy-entropy balance in the strong coupling region. Larger $β$ is included in the monopole condensed phase as more extended monopoles are considered. The scaling seen in the $SU(2)$ case is not yet observed. The renormalization flow diagram suggests the existe…
We consider fermions in the field of static monopole-like configurations in the Euclidean space-time. In all the cases considered there exists an infinite number of zero modes, labeled by frequency iω. The existence of such modes is a manifestation of instability of the vacuum in the presence of the monopoles and massless fermions. In the Minkowski space the corresponding phenomenon is well known and is a cornerstone of the theory of the magnetic catalysis. Moreover, the well known zero mode of Jackiw and Rebbi corresponds to the limiting case, ω= 0. We provide arguments why the chiral condens…
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.
U(1) gauge fields are decomposed into a monopole and photon part across the phase transition from the confinement to the Coulomb phase. We analyze the leading Lyapunov exponents of such gauge field configurations on the lattice which are initialized by quantum Monte Carlo simulations. We observe a strong relation between the sizes of the monopole density and the Lyapunov exponent. Evidence is found that monopole fields stay chaotic in the continuum whereas the photon fields are regular.
We study the interplay between topological observables and chiral and Higgs transitions in lattice scalar QED with quenched fermions. Emphasis is put on the chiral transition line and magnetic monopole percolation at strong gauge coupling. We confirm that at infinite gauge coupling the chiral transition is described by mean field exponents. We find a rich and complicated behaviour at the endpoint of the Higgs transition line which hampers a satisfactory analysis of the chiral transition. We study in detail an intermediate coupling, where the data are consistent both with a trivial chiral trans…
We investigate the recent conjecture that the chiral phase transition in non-compact lattice QED is driven by monopole condensation. The comparison of analytic and numerical results shows that we have a quantitative understanding of monopoles in both the quenched and dynamical cases. We can rule out monopole condensation.
We present the first implementation of the Cho--Faddeev--Niemi decomposition of the SU(2) Yang-Mills field on a lattice. Our construction retains the color symmetry (global SU(2) gauge invariance) even after a new type of Maximally Abelian gauge, as explicitly demonstrated by numerical simulations. Moreover, we propose a gauge-invariant definition of the magnetic monopole current using this formulation and compare the new definition with the conventional one by DeGrand and Toussaint to exhibit its validity.
We study the structure of the isolated static monopoles in the maximal Abelian projection of SU(2) lattice gluodynamics. Our estimation of the monopole radius is $ \approx 0.06 fm$.
In this series of three papers, we generalize the derivation of dual photons and monopoles by Polyakov, and Banks, Myerson and Kogut, to obtain approximative models of SU(2) lattice gauge theory. Our approach is based on stationary phase approximations. In this second article, we start from the spin foam representation of 3-dimensional SU(2) lattice gauge theory. By extending an earlier work of Diakonov and Petrov, we approximate the expectation value of a Wilson loop by a path integral over a dual gluon field and monopole-like degrees of freedom. The action contains the tree-level Coulomb int…
To find a perfect lattice action in terms of monopole action on the dual lattice, we performed simulations of a monopole effective action obtained numerically from vacuum configurations in SU(2) QCD. Although the Polyakov loop behavior near $T_c$ is well reproduced by the action, a small but repulsive term is needed in addition to get the string tension correctly. It is reported also a monopole effective action in $SU(3)$ QCD which is expressed by one kind of monopole currents.
Classical solutions corresponding to monopole-antimonopole pairs are found in 3d and 4d SU(2) and U(1) lattice gauge theories. The stability of these solutions in different theories is studied.
In 4D compact U(1) lattice gauge theory with a monopole term added to the Wilson action we first reveal some properties of a third phase region at negative $β$. Then at some larger values of the monopole coupling $λ$ by a finite-size analysis we find values of the critical exponent $ν$ close to, however, different from the Gaussian value.
This is the short review of Monte-Carlo studies of quark confinement in lattice QCD. After abelian projections both in the maximally abelian and Polyakov gauges, it is seen that the monopole part alone is responsible for confinement. A block spin transformation on the dual lattice suggests that lattice $SU(2)$ QCD is always ( for all $β$) in the monopole condensed phase and so in the confinement phase in the infinite volume limit.
We obtain an almost perfect monopole action numerically after abelian projection in pure SU(3) lattice QCD. Performing block-spin transformations on the dual lattice, the action fixed depends only on a physical scale b. Monopole condensation occurs for large b region. The numerical results show that two-point monopole interactions are dominant for large b. We next perform the block-spin transformation analytically in a simplified case of two-point monopole interactions with a Wilson loop on the fine lattice. The perfect operator evaluating the static quark potential on the coarse b-lattice are…
We find that Polyakov lines, computed in abelian-projected SU(2) lattice gauge theory in the confined phase, have finite expectation values for lines corresponding to two units of the abelian electric charge. This means that the abelian-projected lattice has at most Z(2), rather than U(1), global symmetry. We also find a severe breakdown of the monopole dominance approximation, as well as positivity, in this charge-2 case. These results imply that the abelian-projected lattice is not adequately represented by a monopole Coulomb gas; the data is, however, consistent with a center vortex structu…
