We study the confinement between vortex strings in the dual lattice gauge theory of the abelian Higgs model. The dual lattice gauge theory is described by a 2-form gauge field. We calculate the string-antistring potential from the surface operator of the 2-form gauge field. The linear confining potential appears in a confinement phase and it disappears in a deconfinement phase. The phase diagram of the theory is also obtained.
It is shown, in D=2+1 dimensions, that by merely imposing non-abelian gauge invariance on the temporal gauge ground state wavefunctional of an abelian gauge theory, a confining state is obtained.
Using a new type of simulation algorithm for the standard SU(3) lattice gauge theory that yields results with unprecedented precision, we confirm the presence of a $γ/r$ correction to the static quark potential at large distances $r$, with a coefficient $γ$ as predicted by the bosonic string theory. In both three and four dimensions, the transition from perturbative to string behaviour is evident from the data and takes place at surprisingly small distances.
For a particular lattice gauge theory with ${\mathbb Z}_2$ gauge invariance there is confinement for all couplings. The gauge fields, on lattice links, lie in the closed interval $[-1,1]$. It is proved that the expectation value of a gauge-invariant loop operator decays as the exponential of minus the area.
In this talk, first, we show that the color $N$-dependent area law falloffs of the double-winding Wilson loop averages for the $SU(N)$ lattice gauge model are reproduced from the $Z_N$ lattice Abelian gauge model due to the center group dominance in quark confinement. Next, we discuss lattice gauge-scalar models which allow analytic continuation for gauge invariant operators between confinement region and Higgs region. Applying the cluster expansion, we try to understand non-trivial contribution from scalar field in quark confinement mechanism. In order to understand quark confinement further,…
We review some recent developments on BPS string solutions and monopole confinement in the Higgs or (color) superconducting phase of deformed N=2 and N=4 super Yang-Mills theories. In particular, the monopole magnetic fluxes are shown to be always integer linear combinations of string fluxes. Moreover, a bound for the threshold length of the string breaking is obtained. When the gauge group SU(N) is broken to Z_N, the BPS string tension satisfies the Casimir scaling law. Furthermore in the SU(3) case the string solutions are such that they allow the formation of a confining system with three m…
We propose to use the different kinds of vacua of the gauge theories coupled to matter as a laboratory to test confinement ideas of pure Yang-Mills theories. In particular, the very poor overlap of the Wilson loop with the broken string states supports the 't Hooft and Mandelstam confinement criteria. However in the Z(2) gauge-Higgs model we use as a guide we find that the condensation of monopoles and center vortices is a necessary, but not sufficient condition for confinement.
The property of color confinement ("C confinement"), meaning that all asymptotic particle states are color neutral, holds not only in QCD, but also in gauge-Higgs theories deep in the Higgs regime. In this talk we describe a new and stronger confinement criterion, separation-of-charge confinement or "S$_c$ confinement," which is an extension of the Wilson area-law criterion to gauge + matter theories. We will show that there is a transition between S$_c$ and C confinement in the phase plane of gauge-Higgs theories, and we will also explain what symmetry is actually broken in the Higgs phase of…
In this talk we review some generalizations of 't Hooft and Mandelstam ideas on confinement for theories with non-Abelian unbroken gauge groups. In order to do that, we consider N=2 super Yang-Mills with one flavor and a mass breaking term. One of the spontaneous symmetry breaking is accomplished by a scalar that can be in particular in the representation of the diquark condensate and therefore it can be thought as being itself the condensate. We analyze the phases of the theory. In the superconducting phase, we show the existence of BPS Z_k-strings and calculate exactly their string tension i…
Quark confinement is proposed to be a dual Meissner effect of nonAbelian kind. Important hints come from physics of strongly-coupled infrared-fixed-point theories in N=2 supersymmetric QCD, which turn into confining vacua under a small relevant perturbation. The quest for the semiclassical origin of the nonAbelian monopoles, ubiquitous as the infrared degrees of freedom in supersymmetric gauge theories, motivates us to study the quantum dynamics of 2D CP(N-1)model defined on a finite-width worldstrip, with various boundary conditions. The model is found to possess a unique phase ("confinement …
All colored particles including dynamical quarks and gluons are confined if the color confinement criterion proposed by Kugo and Ojima is satisfied. The criterion was obtained under the gauge fixing of the Lorenz type. However, it was pointed out that the Kugo-Ojima criterion breaks down for the Maximal Abelian gauge, which is quite strange in view of the fact that quark confinement has been verified according to the dual superconductivity caused by magnetic monopole condensations. In order to make a bridge between color confinement due to Kugo and Ojima and the dual superconductor picture for…
I review investigations of the quark confinement mechanism that have been carried out in the framework of SU(N) lattice gauge theory. The special role of Z(N) center symmetry is emphasized.
We study whether broken dual gauge symmetry, as detected by a monopole order parameter introduced by the Pisa group, is necessarily associated with the confinement phase of a lattice gauge theory. We find a number of examples, including SU(2) gauge-Higgs theory, mixed fundamental-adjoint SU(2) gauge theory, and pure SU(5) gauge theory, which appear to indicate a dual gauge symmetry transition in the absence of a transition to or from a confined phase. While these results are not necessarily fatal to the dual superconductor hypothesis, they may pose some problems of interpretation for the prese…
The vortex theory which emerges from SU(2) lattice gauge theory by center projection is briefly reviewed. In this vortex picture, quark confinement is due to percolating (closed) vortices which are randomly linked to the Wilson loop. The deconfinement phase transition appears as a de-percolation phase transition.
We study aspects of confinement in the M theory fivebrane version of QCD (MQCD). We show heavy quarks are confined in hadrons (which take the form of membrane-fivebrane bound states) for N=1 and softly broken N=2 SU(Nc) MQCD. We explore and clarify the transition from the exotic physics of the latter to the standard physics of the former. In particular, the many strings and quark-antiquark mesons found in N=2 field theory by Douglas and Shenker are reproduced. It is seen that in the N=1 limit all but one such meson disappears while all of the strings survive. The strings of softly broken N=2, …
We review the confinement scenario in Coulomb gauge. We show that when thin center vortex configurations are gauge transformed to Coulomb gauge, they lie on the common boundary of the fundamental modular region and the Gribov region. This unifies elements of the Gribov confinement scenario in Coulomb gauge and the center-vortex confinement scenario. We report on recent numerical studies which support both of these scenarios.
We study the spatial volume dependence of electric flux energies for SU(2) Yang-Mills fields on the torus with twisted boundary conditions. The results approach smoothly the rotational invariant Confinement regime. The would-be string tension is very close to the infinite volume result already for volumes of $(1.2 \ {\rm fm.})^3$. We speculate on the consequences of our result for the Confinement mechanism.
We study confining strings in ${\cal{N}}=1$ supersymmetric $SU(N_c)$ Yang-Mills theory in the semiclassical regime on $\mathbb{R}^{1,2} \times \mathbb{S}^1$. Static quarks are expected to be confined by double strings composed of two domain walls - which are lines in $\mathbb{R}^2$ - rather than by a single flux tube. Each domain wall carries part of the quarks' chromoelectric flux. We numerically study this mechanism and find that double-string confinement holds for strings of all $N$-alities, except for those between fundamental quarks. We show that, for $N_c \ge 5$, the two domain walls con…
I will review essential features of the Hamiltonian approach to QCD in Coulomb gauge showing that Gribov's confinement scenario is realized in this gauge. For this purpose I will discuss in detail the emergence of the horizon condition and the Coulomb string tension. I will show that both are induced by center vortex gauge field configurations, which establish the connection between Gribov's confinement scenario and the center vortex picture of confinement. I will then extend the Hamiltonian approach to QCD in Coulomb gauge to finite temperatures, first by the usual grand canonical ensemble an…
We review certain properties of confinement with added focus on the ones we study with holography. Then we discuss observables whose unique behavior can indicate the presence of confinement. Using mainly the Wilson loop in the gauge/gravity formalism, we study two main features of the QCD string: the string tension dependence on the temperature while in the confining phase, and the logarithmic broadening of the flux tube between the heavy static charges that turns out to be a generic property of all confining theories. Finally, we review the k-string bound state and we show that for a wide cla…
We give an outline of a recent proof that the low-energy effective gauge theory exhibiting quark confinement due to magnetic monopole condensation can be derived from QCD without any specific assumption. We emphasize that the low-energy effective abelian gauge theories obtained here give the dual description of the same physics in the low-energy region. They show that the QCD vacuum is nothing but the dual (type II) superconductor.
Recent approaches to quark confinement are reviewed, with an emphasis on their connection to renormalization group methods. Basic concepts related to confinement are introduced: the string tension, Wilson loops and Polyakov lines, string breaking, string tension scaling laws, center symmetry breaking, and the deconfinement transition at non-zero temperature. Current topics discussed include confinement on $R^3\times S^1$, the real-space renormalization group, the functional renormalization group, and the Schwinger-Dyson equation approach to confinement.
We propose a novel description for the Higgs mechanism by which a gauge boson acquires the mass. We do not assume spontaneous breakdown of gauge symmetry signaled by a non-vanishing vacuum expectation value of the scalar field. In fact, we give a manifestly gauge-invariant description of the Higgs mechanism in the operator level, which does not rely on spontaneous symmetry breaking. This enables us to discuss the confinement-Higgs complementarity from a new perspective. The "Abelian" dominance in quark confinement of the Yang-Mills theory is understood as a consequence of the gauge-invariant H…
We identify the artifact that causes confinement at strong coupling in compact U(1) lattice gauge theory and show that most of the string tension in compact SU(2) lattice gauge theory is due to plaquettes of negative trace.
We discuss the effect of the choice of an Abelian projection on the dual superconductor mechanism of confinement in SU(2) gluodynamics. Using qualitative arguments we show that the dual superconductor Lagrangian corresponding to the Abelian Polyakov gauge has a different structure compared to the dual Lagrangian in the Maximal Abelian gauge. A difference between these Lagrangians reflects the fact that in continuum limit the monopoles should be static in the Abelian Polyakov gauge and therefore these monopoles can not give rise to the confinement of static quarks. Using the Abelian Polyakov ga…
We show that the Higgs and confinement phases of a gauge Higgs theory, with the Higgs field in the fundamental representation of the gauge group, are distinguished both by a broken or unbroken realization of the global center subgroup of the gauge group, and by the type of confinement in each phase. This is color confinement in the Higgs phase, and a stronger property, which we call "separation-of-charge" confinement, in the confining phase.
We discuss theories with the exceptional centerless gauge group G(2), paying attention to confinement and the pattern of chiral symmetry breaking. Exploiting the Higgs mechanism to break the symmetry down to SU(3), we also present how the familiar features of confinement and chiral symmetry breaking of SU(3) gauge theories reemerge. G(2) gauge theories show up as an unusual theoretical framework to study SU(3) gauge theories without the ``luxury'' of a center.
Starting from the bosonic part of N=2 Super QCD with a 'Seiberg-Witten' N=2 breaking mass term, we obtain string BPS conditions for arbitrary semi-simple gauge groups. We show that the vacuum structure is compatible with a symmetry breaking scheme which allows the existence of Z_k-strings and which has Spin(10) -> SU(5) x Z_2 as a particular case. We obtain BPS Z_k-string solutions and show that they satisfy the same first order differential equations as the BPS string for the U(1) case. We also show that the string tension is constant, which may cause a confining potential between monopoles i…
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…
The effective string picture of confinement is used to derive theoretical predictions for the interquark potential at finite temperature. At short distances, the leading string correction to the linear confining potential between a heavy quark-antiquark pair is the "Lüscher term''. We assume a Nambu--Goto effective string action, and work out subleading contributions in an analytical way. We discuss the contribution given by a possible ``boundary term'' in the effective action, comparing these predictions with results from simulations of lattice $Z_2$ gauge theory in three dimensions, obtained…
We report on connections between the confining color Coulomb potential, center vortices, and the unbroken realization of remnant gauge symmetry in Coulomb gauge.
Based on our exact effective spin model, the zero temperature $SU(N)$ gauge field confinement has been studied by the deconfinement-confinement phase transition order parameter, i.e. the Wilson line, rather than the quark antiquark potential. The procedure would allow one to study the confinement in the true continuum limit. For the $(1+1)$ $SU(2)$ gauge field, the confinement and a linear zero temperature Polyakov quark antiquark potential were shown. It is also shown, for the first time, that the Polyakov and Wilson potentials are identical.
Confinement without symmetry breaking in chiral gauge theories
School of Physics, Sun Yat-Sen University（中山大学物理学院）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所跨学科理论与数学科学中心（iTHEMS））；Centre for Cosmology, Particle Physics and Phenomenology (CP3), Université catholique de Louvain（天主教鲁汶大学宇宙学、粒子物理与现象学中心（CP3））
The lattice regularized pure gauge compact U(1) theory is an ideal laboratory to explore how confinement is realized as its phase diagram has a confined and a deconfined phase that depends on the value of the coupling constant, i.e. on $β$. Herein, the connection between confinement and positivity violation through the Schwinger function associated with the Landau gauge photon propagator is investigated. The simulations reported show a very clear link between the realization of confinement and positivity violation of the photon Schwinger function and, therefore, of the photon Källén-Lehmann sp…
We give another derivation of quark confinement in QCD from the viewpoint of the low-energy effective Abelian gauge theory of QCD obtained via Abelian projection. It is based on the recently discovered Berezinskii-Kosterlitz-Thouless phase transition in the four-dimensional Abelian gauge theory. Moreover, we show that there exists a critical gauge coupling constant in QCD, above which confinement set in and below which there is no confinement. In a SU(N) gauge theory with $N_f$ flavored fermions, we argue that this leads to a critical value of fermion flavors $N_f^c$ for the confinement as wel…
We calculate in the SU(6) gauge theory the mass of the lightest flux loop that winds around a spatial torus, as a function of the torus size, taking care to achieve control of the main systematic errors. For comparison we perform a similar calculation in SU(4). We demonstrate approximate linear confinement and show that the leading correction is consistent with what one expects if the flux tube behaves like a simple bosonic string at long distances. We obtain similar but less accurate results for stable (k-)strings in higher representations. We find some evidence that for k>1 the length scale …
Adopting the cooling technique to smooth the discontinuous Z(2) lattice gauge field, we found that on SU(2) gauge configurations obtained by this smoothing there exists clear discontinuity of the topological property at almost the same point as the confinement-deconfinement phase transition of the original Z(2) gauge theory. This observation suggests the possibility that Z(2) gauge model might be a testbed for analyzing the relation between the confinement and the chiral phase transition in which the topological objects are believed to play crucial roles.
The colour confinement criterion proposed by Kugo and Ojima is tested in the lattice Landau gauge QCD simulation. The renormalization effects are studied by measuring the gluon propagator, ghost propagator, three gluon vertex and the ghost-antighost-gluon vertex.
We show that the color $N$ dependent area law falloffs of the double-winding Wilson loop averages for the $SU(N)$ lattice gauge theory obtained in the preceding works are reproduced from the corresponding lattice Abelian gauge theory with the center gauge group $Z_N$. This result indicates the center group dominance in quark confinement.
I analyze the IR fixed point structure of Landau gauge QCD. Precisely the fixed point with a strong kinematic singularity of the quark-gluon vertex that proved crucial for the recently proposed confinement mechanism in the quenched approximation is absent in dynamical QCD. Therefore, the IR singularities do not induce asymptotic quark confinement but the long-range interaction is screened by unquenching loops at scales of the order of the quark mass. This provides the prerequisite for a microscopic description of deconfinement and string breaking. The fixed points determine the qualitative for…
There has been substantial progress in understanding a class of SU(N) gauge theories that are confining at high temperatures. This class includes theories with center-symmetric Polyakov loop deformations or with periodic adjoint fermions. The crucial role of monopoles in lattice gauge theories of this type can be understood analytically. The basic mechanisms occur in the two-dimensional O(3) spin model, deformed by appropriate mass term to give an XY model. Vortices of the XY model are constituents of O(3) instantons just as SU(N) magnetic monopoles are constituents of KvBLL instantons. Simila…
We report on further progress in our programme of understanding confinement in 3d and 4d SU(2) gauge theory in terms of Z(2) monopoles. A sufficient condition for confinement was previously translated into Z(2) monopole correlation inequalities in a related SO(3) gauge theory. We shall discuss the physical picture underlying this scenario and present some Monte Carlo evidence concerning the monopole correlation inequalities.
In axial gauge, the (2+1)-dimensional SU($N$) Yang-Mills theory is equivalent to a set of (1+1)-dimensional integrable models with a non-local coupling between charge densities. This fact makes it possible to determine the static potential between charges at weak coupling in an anisotropic version of the theory, and understand features of the spectrum.
The Kugo-Ojima confinement criterion is verified in the unquenched Landau gauge QCD simulation. The valence quark propagator of the Kogut-Susskind fermion with use of the fermion action including the Naik term and the staple contribution is calculated on MILC Asqtad unquenched gauge configurations, and it shows infrared suppression of the quark propagator.
The confinement problem is studied using the thick center vortex model. It is shown that the $SU(3)$ Cartan sub algebra of the decomposed $G(2)$ gauge theory can play an important role in the confinement. The Casimir eigenvalues and ratios of the $G(2)$ representations are obtained using its decomposition to the $SU(3)$ subgroups. This leads to the conjecture that the $SU(3)$ subgroups also can explain the $G(2)$ properties of the confinement. The thick center vortex model for the $SU(3)$ subgroups of the $G(2)$ gauge theory is applied without the domain modification. Instead, the presence of …
Gauge-independent transition separating confinement and Higgs phases in lattice SU(2) gauge theory with a scalar field in the fundamental representation
The magnetic vortices which arise in SU(2) lattice gauge theory in center projection are visualized for a given time slice. We establish that the number of vortices piercing a given 2-dimensional sheet is a renormalization group invariant and therefore physical quantity. We find that roughly 2 vortices pierce an area of 1 fm^2.
We demonstrate that confinement in $SU(2)$ gauge theory is produced by dual superconductivity of the vacuum. We show that for $T < T_c$ (temperature of deconfining phase transition) the $U(1)$ symmetry related to monopole charge conservation is spontaneously broken; for $T > T_c$ the symmetry is restored.
A gauge invariant procedure for extracting combined SO(3)-Z2 monopoles in positive-plaquette SU(2) lattice gauge theory is shown. When these monopoles are eliminated through a constraint, the theory deconfines for all $β$ on $12^4$ and $20^4$ lattices even in the strong coupling limit, despite a rather strong average plaquette of around 0.64. This corresponds to an effective Wilson $β$ of 2.45, at which Wilson-action lattices would be far into the confining region. This suggests that Wilson-action confinement may be a strong-coupling lattice artifact; the continuum limit may not confine.
We apply to the case of gauge group G = SU(2) in three dimensions a recently proposed gauge-independent mechanism for confinement that is based on a particular form of the dual spin foam framework for lattice gauge theory. Explicit formulae for interaction factors and their asymptotics are introduced and their behavior in different sectors of the theory are identified and analyzed. We arrive at several elementary properties of the dual theory that represent one scenario by which confinement may be realized at weak coupling. We conclude with an outlook for further development of this approach.
A generalization of the Wilson loop area-law criterion is proposed, which is applicable to gauge theories with matter in the fundamental representation of the gauge group. This new criterion, like the area law, is stronger than the statement that asymptotic particle states are massive color singlets, which holds even for theories described by the Brout-Englert-Higgs mechanism.
Abelian mechanism of non-Abelian color confinement is observed in a gauge-independent way by high precision lattice Monte Carlo simulations in gluodynamics. An Abelian gauge field is extracted with no gauge-fixing. A static quark-antiquark potential derived from Abelian Polyakov loop correlators gives us the same string tension as the non-Abelian one. The Hodge decomposition of the Abelian Polyakov loop correlator to the regular photon and the singular monopole parts also reveals that only the monopole part is responsible for the string tension. The investigation of the flux-tube profile then …
We review the physics of confinement based on non-Abelian dual superconductor picture, relying on exact solutions in N=2 supersymmetric QCD and based on the recent developments in our understanding of non-Abelian vortices and monopoles. The non-Abelian monopoles, though they are basically just the 't Hooft-Polyakov SU(2) monopoles embedded in various corners of the larger gauge group, require flavor symmetry in an essential way for their very existence. The phenomenon of flavor-color-flavor separation characterizes the multiple roles flavor symmetry plays in producing quantum-mechanical non-Ab…
As a color confinement mechanism, a dual Meissner effect due to Abelian monopoles involved in QCD has been discussed so far in various ways. But still there is an important problem unsolved. It is gauge invariance of the schemes or, in other words, the reason why non-Abelian color confinement is explained by means of the Abelian dual Meissner effect. Here it is shown that a random Abelian method based on the violation of non-Abelian Bianchi identity (VNABI) could prove SU(3) invariance and explain why color-singlets alone can survive in the confinement phase of QCD in the framework of the Abel…
