We study spontaneous chiral-symmetry breaking in SU(3) QCD in terms of the dual superconductor picture for quark confinement in the maximally Abelian (MA) gauge, using lattice QCD Monte Carlo simulations with four different lattices of $16^4$, $24^4$, $24^3\times 6$ at $β=6.0$ (i.e., the spacing $a \simeq$ 0.1 fm), and $32^4$ at $β=6.2$ (i.e., $a \simeq$ 0.075 fm), at the quenched level. First, in the confinement phase, we find Abelian dominance and monopole dominance in the MA gauge for the chiral condensate in the chiral limit,using the two different methods of i) the Banks-Casher relation with the Dirac eigenvalue density and ii) finite quark-mass calculations with the quark propagator and its chiral extrapolation. In the high-temperature deconfined phase, the chiral restoration is observed also for the Abelian and the monopole sectors. Second, we investigate local correlation between the chiral condensate and monopoles, which topologically appear in the MA gauge. We find that the chiral condensate locally takes a quite large value near monopoles. As an interesting possibility, the strong magnetic field around monopoles is responsible to chiral symmetry breaking in QCD, similarly to the magnetic catalysis.
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 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.
Chiral monopoles are hedgehoglike structures in local chiral condensates in QCD. These monopoles are (i) made of quark and gluon fields; (ii) explicitly gauge-invariant; and (iii) they carry quantized and conserved chromomagnetic charge. We argue that the chiral condensate vanishes in a core of the chiral monopole while the density of these monopoles increases with temperature wiping out the quark condensate in quark-gluon plasma. We suggest that the dynamics of the chiral monopoles is responsible for the chiral symmetry restoration in high temperature phase of QCD. We also argue that the chir…
The general properties of the abelian projection are reviewed. We derive the explicit expression for the abelian functional integral for U(1) abelian theory which corresponds to the abelian projection of the SU(2) gluodynamics. The numerical results for the temperature dependence of the monopole condensate are presented.
Using the lattice gauge field theory, we study the relation among the local chiral condensate, monopoles, and color magnetic fields in quantum chromodynamics (QCD). First, we investigate idealized Abelian gauge systems of 1) a static monopole-antimonopole pair and 2) a magnetic flux without monopoles, on a four-dimensional Euclidean lattice. In these systems, we calculate the local chiral condensate on quasi-massless fermions coupled to the Abelian gauge field, and find that the chiral condensate is localized in the vicinity of the magnetic field. Second, using SU(3) lattice QCD Monte Carlo ca…
We observe the power law scaling behavior of the monopole mass and condensate in the pure compact U(1) gauge theory with the Villain action. In the Coulomb phase the monopole mass scales with the exponent ν_m=0.49(4). In the confinement phase the behavior of the monopole condensate is described with remarkable accuracy by the exponent β_{exp}=0.197(3). Possible implications of these phenomena for a construction of a strongly coupled continuum U(1) gauge theory are discussed.
We analyze the interplay of topological objects in four dimensional QCD. The distributions of color magnetic monopoles obtained in the maximum abelian gauge are computed around instantons in both pure and full QCD. We find an enhanced probability of encountering monopoles inside the core of an instanton on gauge field average. For specific gauge field configurations we visualize the situation graphically. Moreover we investigate how monopole loops and instantons are locally correlated with the chiral condensate.
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 study the embedded QCD monopoles (``quark monopoles'') using low-lying eigenmodes of the overlap Dirac operator in zero- and finite-temperature SU(2) Yang-Mills theory on the lattice. These monopoles correspond to the gauge-invariant hedgehogs in the quark-antiquark condensates. The monopoles were suggested to be agents of the chiral symmetry restoration since their cores should suppress the chiral condensate. We study numerically the scalar, axial and chirally invariant definitions of the embedded monopoles and show that the monopole densities are in fact globally anti-correlated with the …
To understand the relation between the chiral symmetry breaking and monopoles, the chiral condensate which is the order parameter of the chiral symmetry breaking is calculated in the $\overline{\mbox{MS}}$ scheme at 2 [GeV]. First, we add one pair of monopoles, varying the monopole charges $m_{c}$ from zero to four, to SU(3) quenched configurations by a monopole creation operator. The low-lying eigenvalues of the Overlap Dirac operator are computed from the gauge links of the normal configurations and the configurations with additional monopoles. Next, we compare the distributions of the neare…
The purpose of this study is to show that monopoles induce the chiral symmetry breaking. In order to indicate the evidence, we add one pair of monopoles with magnetic charges to the quenched SU(3) configurations by a monopole creation operator, and investigate the propaties of the chiral symmetry breaking using the Overlap fermion. We show that instantons are created by the monopoles. The pseudoscalar meson mass and decay constant are computed from the correlation functions, and the renormalization constant $Z_{S}$ is determined by the non perturbative method. The renormalization group invaria…
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 …
The purpose of this study is to show the relations between monopoles instantons and Chiral symmetry breaking. First, in order to show the relation between instantons and monopoles, we generate configurations, adding monopoles by a monopole creation operator. Then, we count the number of fermion zero modes in the configurations using Overlap fermions as a tool. As a result we find that one monopole with plus one charge and one anti-monopole with minus one charge make one instanton of charge plus or minus one. We have already reported these results elsewhere. In addition, in this report, the rel…
In the quenched approximation we use the abelian and monopole fields from abelian projection in SU(2) lattice gauge theory to numerically compute the value of the chiral condensate. The condensate calculated using abelian projection is observed to vanish at the same critical temperature as the full SU(2) theory predicts.
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.
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 review the method of blocking of topological defects from continuum used as a non--perturbative tool to construct effective actions for these defects. The actions are formulated in the continuum limit while the couplings of these actions can be derived from simple observables calculated numerically on lattices with a finite lattice spacing. We demonstrate the success of the method in deriving the effective actions for Abelian monopoles in the pure SU(2) gauge models in an Abelian gauge. In particular, we discuss the gluodynamics in three and four space--time dimensions at zero and non--zero…
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…
We discuss dynamics of N=2 supersymmetric SU(n_c) gauge theories with n_f quark hypermultiplets. Upon N=1 perturbation of introducing a finite mass for the adjoint chiral multiplet, we show that the flavor U(n_f) symmetry is dynamically broken to U(r) \times U(n_f-r), where r\leq [n_f/2] is an integer. This flavor symmetry breaking occurs due to the condensates of magnetic degrees of freedom which acquire flavor quantum numbers due to the quark zero modes. We briefly comment on the USp(2n_c) gauge theories. This talk is based on works with Giuseppe Carlino and Ken Konishi, hep-th/0001036 and h…
We classify the lattice by elementary 3-cubes which are associated to dual links occupied by, or free of monopoles. We then compute the quark condensate, the quark charge and the chiral density on those cubes. By looking at distributions we demonstrate that monopole trajectories carry considerably more chirality with respect to the free vacuum.
Recently (hep-th/0104171) we considered N=2 super Yang-Mills with a N=2 mass breakingn term and showed the existence of BPS Z_{k}-string solutions for arbitrary simple gauge groups which are spontaneously broken to non-Abelian residual gauge groups. We also calculated their string tensions exactly. In doing so, we have considered in particular the hypermultiplet in the same representation as the one of a diquark condensate. In the present work we analyze some of the different phases of the theory and find that the magnetic fluxes of the monopoles are multiple of the fundamental Z_{k}-string fl…
We simulate four flavor noncompact lattice QED using the Hybrid Monte Carlo algorithm on $10^4$ and $16^4$ lattices. Measurements of the monopole susceptibility and the percolation order parameter indicate a transition at $β= {1/e^2} = .205(5)$ with critical behavior in the universality class of four dimensional percolation. We present accurate chiral condensate measurements and monitor finite size effects carefully. The chiral condensate data supports the existence of a power-law transition at $β= .205$ in the same universality class as the chiral transition in the two flavor model. The resul…
We show that there exists a special compactification of QCD on $\mathbb{R}^3 \times S^1$ in which the theory has a domain where continuous chiral symmetry breaking is analytically calculable. We give a microscopic derivation of the chiral lagrangian, the chiral condensate, and the Gell-Mann-Oakes-Renner relation $m_π^2 f_π^2 = m_q \langle \bar{q} q \rangle$. Abelian duality, monopole operators, and flavor-twisted boundary conditions, or a background flavor holonomy, play the main roles. The flavor twisting leads to the new effect of fractional jumping of fermion zero modes among monopole-insta…
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…
The abelian projection of lattice gluodynamics is reviewed. The main topics are: abelian and monopole dominance, monopole condensate as the disorder parameter, effective abelian Lagrangian, monopoles in the instanton field, Aharonov -- Bohm effect on the lattice.
In the pure U(1) lattice gauge theory with the Villain action we find that the monopole mass in the Coulomb phase and the monopole condensate in the confinement phase scale according to simple power laws. This holds outside the coupling region in which on finite toroidal lattices the metastability phenomena occur. A natural explanation of the observed accuracy of the scaling behaviour would be the second order of the phase transition between both phases in the general space of couplings not far away from the Villain action.
In QCD with adjoint fermions (aQCD) the deconfining transition takes place at a lower temperature than the chiral transition. We study the two transitions by use of the Polyakov Loop, the monopole order parameter and the chiral condensate. The deconfining transition is first order, the chiral is a crossover. The order parameters for confinement are not affected by the chiral transition. We conclude that the degrees of freedom relevant to confinement are different from those describing chiral symmetry.
We compute, in SU(3) pure gauge theory, the vacuum expectation value (vev) of the operator which creates a $Z_3$ vortex wrapping the lattice through periodic boundary conditions (dual Polyakov line). The technique used is the same already tested in the SU(2) case. The dual Polyakov line proves to be a good disorder parameter for confinement, and has a similar behaviour to the monopole condensate. The new features which characterise the construction of the disorder operator in SU(3) are emphasised.
We describe numerical calculation results for the probability distribution of the value of the monopole creation operator in the SU(2) lattice gluodynamics. We work in the maximal abelian projection. It occurs that at low temperatures, 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 fact confirms the existence of the abeli…
Calculations of the chiral condensate on the lattice using staggered fermions and the Lanczos algorithm are presented. Four gauge fields are considered: the quenched non-Abelian field, an Abelian projected field, and monopole and photon fields further decomposed from the Abelian field. Abelian projection is performed in maximal Abelian gauge and in Polyakov gauge. The results show that monopoles in maximal Abelian gauge largely reproduce the chiral condensate values of the full non-Abelian theory, in both SU(2) and SU(3) color.
Non-compact lattice QED with two flavors of light dynamical quarks is simulated on $16^4$ lattices, and the chiral condensate, monopole density and susceptibility and the meson masses are measured. Data from relatively high statistics runs at relatively small bare fermion masses of 0.005, 0.01, 0.02 and 0.03 (lattice units) are presented. Three independent methods of data analysis indicate that the critical point occurs at $β=0.225(5)$ and that the monopole condensation and chiral symmetry breaking transitions are coincident. The monopole condensation data satisfies finite size scaling hypothe…
We consider the lattice realization of the Standard Model with an additional $Z_6$ symmetry. Numerical simulations were performed on the asymmetric lattice, which corresponds to the finite temperature theory. Our choice of parameters corresponds to large Higgs masses ($M_H > 90$ Gev). The phase diagram was investigated and has been found to be different from that of the usual lattice realization of the Standard Model. It has been found, that the confinement-deconfinement phase transition lines for the SU(2) and SU(3) fields coincide. The transition line between Higgs and symmetric deconfinemen…
We aim to show the effects of the magnetic monopoles and instantons in quantum chromodynamics (QCD) on observables; therefore, we introduce a monopole and anti-monopole pair in the QCD vacuum of a quenched SU(3) by applying the monopole creation operator to the vacuum. We calculate the eigenvalues and eigenvectors of the overlap Dirac operator that preserves the exact chiral symmetry in lattice gauge theory using these QCD vacua. We then investigate the effects of magnetic monopoles and instantons. First, we confirm the monopole effects as follows: (i) The monopole creation operator makes the …
Calculations of the chiral condensate $\langle \barψ ψ\rangle$ on the lattice using staggered fermions and the Lanczos algorithm are presented. Three gauge fields are considered: the quenched non-Abelian field, the Abelian field projected in the maximal Abelian gauge, and the monopole field further decomposed from the Abelian field. The results show that the Abelian monopoles largely reproduce the chiral condensate values of the full non-Abelian theory, both in SU(2) and in SU(3).
We discuss phenomenology of the vacuum condensate <(A_μ)^2> in pure gauge theories, where A_μis the gauge potential. Both Abelian and non-Abelian cases are considered. In case of the compact U(1) the non-perturbative part of the condensate <(A_μ)^2> is saturated by monopoles. In the non-Abelian case, a two-component picture for the condensate is presented according to which finite values of order Λ_{QCD}^2 are associated both with large and short distances. We obtain a lower bound on the <(A_μ)^2> by considering its change at the phase transition. Numerically, it produces an estimate similar t…
A comparative study of Abelian and Monopole dominance in the Laplacian and Maximally Abelian projected gauges is carried out. Clear evidence for both types of dominance is obtained for the Laplacian projection. Surprisingly, the evidence is much more ambiguous in the Maximally Abelian gauge. This is attributed to possible ``long-distance imperfections'' in the maximally abelian gauge fixing.
We study the properties of configurations from which P-vortices on one hand or Abelian monopoles on the other hand have been removed. We find that the zero modes and the band of non-zero modes close to zero disappear from the spectrum of the overlap Dirac operator, confirming the absence of topological charge and quark condensate. The different behavior of the modified ensembles under smearing compared to the unmodified Monte Carlo ensemble corroborates these findings. The gluonic topological susceptibility rapidly approaches zero in accordance with Q_{index}=0. The remaining (ultraviolet) mon…
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 compare U(1) lattice gauge theory to an effective model of Maxwell and London equations. In the effective model there is only one free parameter, the London penetration depth lambda. It turns out that one can get good agreement between both models if one modifies the usual definition of magnetic monopole currents in U(1) lattice gauge theory. This comparison also shows that already at small distances fluctuations of the occuring string are important. Further, we investigate the beta-dependence of the penetration depth and determine the suppression of the monopole condensate in flux tubes.
We study the relation between the instanton distribution and the monopole loop length in the SU(2) gauge theory with the abelian gauge fixing. We measure the monopole current from the multi-instanton ensemble on the $16^4$ lattice using the maximally abelian gauge. When the instanton density is dilute, there appear only small monopole loops. On the other hand, in the dense case, there appears one very long monopole loop, which is responsible for the confinement property, in each gauge configuration. We find a clear monopole clustering in the histogram of the monopole loop length from 240 gauge…
In any Abelian gauge theory with an action periodic in the link variables one can perform a duality transformation not only in the partition function, but also in correlation functions including Polyakov loops. The calculation of expectation values in the confinement phase, like electric field strength or monopole currents in the presence of external charges, becomes significantly more efficient simulating the dual theory. We demonstrate this using the ordinary Wilson action. This approach also allows a quantitative analysis of the dual superconductor model, because the dual transformed U(1) t…
Magnetic monopoles are suggested to play an important role in strongly coupled quark-gluon plasma (sQGP) near the deconfinement temperature. So far, their many-body treatment has only been done classically, with just binary scattering solved in quantum mechanics. In this paper we start quantum many-body studies of the monopole ensembles. Specifically, we carry out numerical simulations of the path integral for one- and two-component Coulomb Bose systems. We determine the relation between the critical temperature for the Bose-Einstein condensation phase transition $T_\text{c}$ and the Coulomb c…
We extend our previous study of magnetic monopole currents in the maximally Abelian gauge [hep-lat/9712003] to larger lattices at small lattice spacings (20^4 at beta = 2.5 and 32^4 at beta = 2.5115). We confirm that at these weak couplings there continues to be one monopole cluster that is very much longer than the rest and that the string tension, K, is entirely due to it. The remaining clusters are compact objects whose population as a function of radius follows a power law that deviates from the scale invariant form, but much too weakly to suggest a link with the analytically calculable si…
The confining mechanisms of 't Hooft and Mandelstam have a simple microscopic realization in 3D Z2 gauge theory: the center vortex and the magnetic monopole condensation are associated, in the set of configurations contributing to the confining phase, to the presence of two kinds of infinite clusters. These generate the area law of the large Wilson loops and the universal finite size effects produced by the quantum fluctuations of the bosonic string describing the infrared behavior of the flux tube.
The normal confining phase of gauge theories is characterised by the condensation of magnetic monopoles and center vortices. Sometimes in coupled gauge system one finds another phase with simultaneous condensation of electric and magnetic charges. In both phases the confining string breaks down at a given scale because of pair creation, however the mechanism is different. In the former case the string breaking is a mixing phenomenon which is invisible in the Wilson loop. On the contrary, in presence of both electric and magnetic condensates the string breaking can be observed even in the Wilso…
Using a renormalization group motivated smoothing technique, we investigate the large scale structure of lattice configurations at finite temperature, concentrating on Abelian monopoles identified in the maximally Abelian, the Laplacian Abelian, and the Polyakov gauge. Monopoles are mostly found in regions of large action and topological charge, rather independent of the gauge chosen to detect them. Gauge invariant properties around Abelian monopoles, the local non-Abelian action and topological density, are studied. We show that the local averages of these densities along the monopole traject…
Three dimensional SU(2) lattice gauge theory is studied after eliminating thin monopoles and the smallest thick monopoles. Kinematically this constraint allows the formation of thick vortex loops which produce Z(2) fluctuations at longer length scales. The thick vortex loops are identified in a three dimensional simulation. A condensate of thick vortices persists even after the thin vortices have all disappeared. The thick vortices decouple at a slightly lower temperature (higher beta) than the thin vortices and drive a phase transition.
SU(2) lattice gauge theory is studied after eliminating thin monopoles and the smallest thick monopoles. Kinematically this constraint allows thick vortex loops which produce long range Z(2) fluctuations. The thick vortex loops are identified in a three dimensional simulation. A condensate of thick vortices persists even after the thin vortices have all disappeared. They decouple at a slightly lower temperature (higher $β$) than the thin vortices and drive a Z(2) like phase transition.
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…
