Coulomb Confinement from the Yang-Mills Vacuum State in 2+1 Dimensions
J. Greensite, S. Olejnik
备注：
8 pages, 5 figures (10 EPS files), RevTeX4.1. V2: original figs. 4 and 5 compressed into a new fig. 5; a new fig. 4; sec. IV.B slightly modified to reflect the changes. Version to appear in Phys. Rev. D. V3: a reference corrected.
The Coulomb-gauge ghost propagator, and the color-Coulomb potential, are computed in an ensemble of configurations derived from our recently proposed Yang-Mills vacuum wavefunctional in 2+1 dimensions. The results are compared to the corresponding values obtained by standard Monte Carlo simulations in three Euclidean dimensions. The agreement is quite striking for the Coulomb-gauge ghost propagator. The color-Coulomb potential rises linearly at large distances, but its determination suffers from rather large statistical fluctuations, due to configurations with very low values of $μ_0$, the lowest eigenvalue of the Coulomb-gauge Faddeev-Popov operator. However, if one imposes cuts on the data, effectively leaving out configurations with very low $μ_0$, the agreement of the potential in both sets of configurations is again satisfactory, although the errorbars grow systematically as the cutoff is eliminated.
We present recent results on quark confinement: in SU(3) Yang-Mills theory, confinement of fundamental quarks is obtained due to the dual Meissner effect originated from non-Abelian magnetic monopoles defined in a gauge-invariant way, which is distinct from the well-known Abelian projection scenario. This is achieved by using a non-Abelian Stokes theorem for the Wilson loop operator and a new reformulation of the Yang-Mills theory.
We derive a gauge-invariant low-energy effective model of the Yang-Mills theory. We find that the effective gluon propagator belongs to the Gribov-Stingl type and agrees with it when a mass term which breaks nilpotency of the BRST symmetry is included. We show that the effective model with gluon propagator of the Gribov-Stingl type exhibits both quark and gluon confinement: the Wilson loop average has the area law and the Schwinger function violates reflection positivity. However, we argue that both quark and gluon confinement can be obtained even in the absence of such a mass term.
The vortex picture of confinement is employed to explore the signals of confinement in Yang-Mills Green functions. By using SU(2) lattice gauge theory, it has been well established that the removal of the center vortices from the lattice configurations results in the loss of confinement. The running coupling constant, the gluon and the ghost form factors are studied in Landau gauge for both cases, the full and the vortex removed theory. In the latter case, a strong suppression of the running coupling constant and the gluon form factor at low momenta is observed. At the same time, the singulari…
We use 't Hooft loops of maximal size on finite lattices to calculate the free energy in the sectors of SU(2) Yang-Mills theory with fixed electric flux as a function of temperature and (spatial) volume. Our results provide evidence for the mass gap. The confinement of electric fluxes in the low temperature phase and their condensation in the high temperature phase are demonstrated. In a surprisingly large scaling window around criticality, the transition is quantitatively well described by universal exponents and amplitude ratios relating the properties of the two phases.
We investigate the connection between Color Confinement and thermal Abelian monopoles populating the deconfined phase of SU(2) Yang-Mills theory, by studying how the statistical properties of the monopole ensemble change as the confinement/deconfinement temperature is approached from above. In particular we study the distribution of monopole currents with multiple wrappings in the Euclidean time direction, corresponding to two or more particle permutations, and show that multiple wrappings increase as the deconfinement temperature is approached from above, in a way compatible with a condensati…
The dual superconductivity is a promising mechanism of quark confinement. In the preceding works, we have given a non-Abelian dual superconductivity picture for quark confinement, and demonstrated the numerical evidences on the lattice. In this talk, we focus on the the confinement and deconfinement phase transition at finite temperature in view of the dual superconductivity. By using our new formulation of lattice Yang-Mills theory and numerical simulations on the lattice, we extract the dominant mode for confinement by decomposing the Yang-Mills field, and we investigate the Polyakov loop av…
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…
The purpose of this paper is to review the recent progress in understanding quark confinement. The emphasis of this review is placed on how to obtain a manifestly gauge-independent picture for quark confinement supporting the dual superconductivity in the Yang-Mills theory, which should be compared with the Abelian projection proposed by 't Hooft. The basic tools are novel reformulations of the Yang-Mills theory based on change of variables extending the decomposition of the $SU(N)$ Yang-Mills field due to Cho, Duan-Ge and Faddeev-Niemi, together with the combined use of extended versions of t…
I review the main features of our model of the 4-dimensional Yang-Mills theory vacuum as a liquid of fractional instantons. The model provides a possible microscopic mechanism for Confinement in four dimensional Yang-Mills theory at $T=0$. It also connects this property to other non-perturbative properties of the theory which can be explained by the same model. This paper is a, somewhat enlarged, written up version of my recent talks on the subject given at ICTS-Bangalore, KITP-Santa Barbara and IMSC-Chennai.
A model for the infrared sector of Yang-Mills theory based on magnetic vortices represented by (closed) random surfaces is investigated using lattice Monte Carlo methods. The random surfaces are governed by a surface area action and a curvature action. The model generates a finite-temperature deconfinement transition; the coupling constants of the model can be chosen such as to reproduce the SU(2) Yang-Mills ratio of the deconfinement temperature to the square root of the zero-temperature string tension, T_c / sqrt{sigma_0} =0.69. This yields a physical trajectory in the space of coupling cons…
The study of Yang-Mills theories in three dimensions is an insightful playground to grasp important features for the four-dimensional case. Additionally, in three dimensions, the Chern-Simons term can be introduced with a mass parameter of topological nature. Quantizing such a theory in the continuum demands a gauge fixing which, in general, is plagued by Gribov copies. In this work, Yang-Mills-Chern-Simons theories are quantized in the maximal Abelian gauge and the existence of infinitesimal Gribov copies is taken into account. The elimination of copies modifies the (Abelian) gluon propagator…
Expanding the Landau gauge gluon and ghost two-point functions in a power series we investigate their infrared behavior. The corresponding powers are constrained through the ghost Dyson-Schwinger equation by exploiting multiplicative renormalizability. Without recourse to any specific truncation we demonstrate that the infrared powers of the gluon and ghost propagators are uniquely related to each other. Constraints for these powers are derived, and the resulting infrared enhancement of the ghost propagator signals that the Kugo-Ojima confinement criterion is fulfilled in Landau gauge Yang-Mil…
By making use of the background field method, we derive a novel reformulation of the Yang-Mills theory which was proposed recently by the author to derive quark confinement in QCD. This reformulation identifies the Yang-Mills theory with a deformation of a topological quantum field theory. The relevant background is given by the topologically non-trivial field configuration, especially, the topological soliton which can be identified with the magnetic monopole current in four dimensions. We argue that the gauge fixing term becomes dynamical and that the gluon mass generation takes place by a s…
We show that in the Landau gauge of the SU(2) Yang-Mills theory the residual global symmetry supports existence of the topological vortices which resemble disclination defects in the nematic liquid crystals and the Alice (half-quantum) vortices in the superfluid heluim 3 in the A-phase. The theory also possesses half-integer and integer charged monopoles which are analogous to the point-like defects in the nematic crystal and in the liquid helium. We argue that the deconfinement phase transition in the Yang-Mills theory in the Landau gauge is associated with the proliferation of these vortices…
Semi-classical configurations in Yang-Mills theory have been derived from lattice Monte Carlo configurations using a recently proposed constrained cooling technique which is designed to preserve every Polyakov line (at any point in space-time in any direction). Consequently, confinement was found sustained by the ensemble of semi-classical configurations. The existence of gluonic and fermionic near-to-zero modes was demonstrated as a precondition for a possible semi-classical expansion around the cooled configurations as well as providing the gapless spectrum of the Dirac operator necessary fo…
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…
We propose the reformulations of the $SU(N)$ Yang-Mills theory toward quark confinement and mass gap. In fact, we have given a new framework for reformulating the $SU(N)$ Yang-Mills theory using new field variables. This includes the preceding works given by Cho, Faddeev and Niemi, as a special case called the maximal option in our reformulations. The advantage of our reformulations is that the original non-Abelian gauge field variables can be changed into the new field variables such that one of them called the restricted field gives the dominant contribution to quark confinement in the gauge…
We study the spatial distribution of the stress tensor around static quark-anti-quark pair in SU(3) lattice gauge theory. In particular, we reveal the transverse structure of the stress tensor distribution in detail by taking the continuum limit. The Yang-Mills gradient flow plays a crucial role to make the stress tensor well-defined and derivable from the numerical simulations on the lattice.
Analyticity of gluon and Faddeev--Popov ghost propagators and their form factors on the complex momentum-squared plane is exploited to continue analytically the ultraviolet asymptotic form calculable by perturbation theory into the infrared non-perturbative solution. We require the non-perturbative multiplicative renormalizability to write down the renormalization group equation. These requirements enable one to settle the value of the exponent characterizing the infrared asymptotic solution with power behavior which was originally predicted by Gribov and has recently been found as approximate…
We study the Z_N flux tubes and monopole confinement in deformed N=2* super Yang-Mills theories. In order to do that we consider an N=4 super Yang-Mills theory with an arbitrary gauge group G and add some N=2, N=1 and N=0 deformation terms. We analyze some possible vacuum solutions and phases of the theory, depending on the deformation terms which are added. In the Coulomb phase for the N=2* theory, G is broken to U(1)^r and the theory has monopole solutions. Then, by adding some deformation terms, the theory passes to the Higgs or color superconducting phase, in which G is broken to its cente…
We show that, starting from known exact classical solutions of the Yang-Mills theory in three dimensions, the string tension is obtained and the potential is consistent with a marginally confining theory. The potential we obtain agrees fairly well with preceding findings in literature but here we derive it analytically from the theory without further assumptions. The string tension is in strict agreement with lattice results and the well-known theoretical result by Karabali-Kim-Nair analysis. Classical solutions depend on a dimensionless numerical factor arising from integration. This factor e…
This work is a step towards merging the ideas that arise from semi-classical methods in continuum QFT with analytic/numerical lattice field theory. In this context, we consider Yang-Mills theories coupled to fermions in the adjoint representation. These theories have the remarkable property that confinement and discrete chiral symmetry breaking can persist at weak coupling on small (non-thermal) $\mathbb R^3 \times S^1$. This work presents a lattice investigation of Yang-Mills with one adjoint Majorana fermion, $\mathcal N=1$ super Yang-Mills, and opens the prospect to understand a number of n…
We examine a double trace deformation of SU(N) Yang-Mills theory which, for large $N$ and large volume, is equivalent to unmodified Yang-Mills theory up to $O(1/N^2)$ corrections. In contrast to the unmodified theory, large $N$ volume independence is valid in the deformed theory down to arbitrarily small volumes. The double trace deformation prevents the spontaneous breaking of center symmetry which would otherwise disrupt large $N$ volume independence in small volumes. For small values of $N$, if the theory is formulated on $\R^3 \times S^1$ with a sufficiently small compactification size $L$…
The center vortex model for the infrared sector of Yang-Mills theory, previously studied for the SU(2) gauge group, is extended to SU(3). This model is based on the assumption that vortex world-surfaces can be viewed as random surfaces in Euclidean space-time. The confining properties are investigated, with a particular emphasis on the finite-temperature deconfining phase transition. The model predicts a very weak first order transition, in agreement with SU(3) lattice Yang-Mills theory, and also reproduces a consistent behavior of the spatial string tension in the deconfined phase. The geomet…
We study the volume dependence of electric flux energies for SU(2) gauge theory with twisted boundary conditions. The curves interpolate smoothly between the perturbative semiclassicalresults and the Confinement regime. On the basis of our results, we propose that the Confinement property might be caused by a class of non-dilute multi-instanton configurations.
The infrared behavior of Landau gauge gluon and ghost propagators are investigated in Yang-Mills theory at non-vanishing temperatures. Self-consistent solutions are presented for temperatures below the presumed phase transition and in the infinite temperature limit. Gluon confinement is manifest in the infrared behavior of these propagators. As expected confinement prevails below the phase transition. In the infinite-temperature limit a qualitative change is observed: the chromoelectric sector exhibits a near-perturbative behavior while long-range chromomagnetic interactions, mediated by soft …
Yang-Mills gauge field with gauge group SU(2) decomposes into a single charge neutral complex vector, and two spinless charged scalar fields. At high energies these constituents are tightly confined into each other by a compact U(1) interaction, and the Yang-Mills Lagrangian describes the dynamics of asymptotically free massless gauge vectors. But in a low energy and finite density environment the interaction between the constituents can become weak, and a spin-charge separation may occur. We suggest that the separation between the spin and charge with the ensuing condensation of the charged s…
We give new descriptions of lattice SU(N) Yang-Mills theory in terms of new lattice variables. The validity of such descriptions has already been demonstrated in the SU(2) Yang-Mills theory by our previous works from the viewpoint of defining and extracting topological degrees of freedom such as gauge-invariant magnetic monopoles and vortices which play the dominant role in quark confinement. In particular, we have found that the SU(3) lattice Yang-Mills theory has two possible options, maximal and minimal: The existence of the minimal option has been overlooked so far, while the maximal optio…
In the preceeding works, we have given a non-Abelian dual superconductivity picture for quark confinement, and demonstrated the numerical evidences on the lattice. In this talk, we discuss the confinement and deconfinement phase transition at finite temperature in view of the dual superconductivity. We investigate chromomagnetic monopole currents induced by chromoelectric flux in both confinement and deconfinement phase by the numerical simulations on a lattice at finite temperature, and discuss the role of the chromomagnetic monopole in the confinement/deconfinement phase transition.
We have proposed the non-Abelian dual superconductivity in SU(3) Yang-Mills theory for the mechanism of quark confinement,and we presented the numerical evidences in preceding lattice conferences by using the proposed gauge link decomposition to extract magnetic monopole in the gauge invariant way. In this talk, we focus on the dual Meissner effects in view of the magnetic monopole in SU(3) Yang-Mills theory. We measure the chromoelectric and chromomagnetic flux due to a pair of quark and antiquark source at finite temperature. Then, we measure the correlation function of Polyakov loops and Po…
The infrared behavior of Yang-Mills theory at finite temperature provides access to the role of confinement. In this review recent results on this topic from lattice calculations and especially Dyson-Schwinger studies are discussed. These indicate persistence of a residual confinement even in the high-temperature phase. The confinement mechanism is very similar to the one in the vacuum for the chromomagnetic sector. In the chromoelectric sector screening occurs at the soft scale g^2T, although not leading to a perturbative behavior.
The dual superconductivity is a promising mechanism for quark confinement. We have proposed the non-Abelian dual superconductivity picture for SU(3) Yang-Mills theory, and showed the restricted field dominance (called conventionally Abelian dominance), and non-Abelian magnetic monopole dominance in the string tension. We have further demonstrated by measuring the chromoelectric flux that the non-Abelian dual Meissner effect exists and determined that the dual superconductivity for SU(3) case is of type I, which is in sharp contrast to the SU(2) case: the border of type I and type II. In this t…
The dual superconductivity is a promising mechanism for quark confinement. We have presented a new formulation of the Yang-Mills theory on the lattice that enables us to change the original non-Abelian gauge field into the new field variables such that one of them called the restricted field gives the dominant contribution to quark confinement in the gauge independent way. We have pointed out that the SU(3) Yang-Mills theory has another reformulation using new field variables (minimal option), in addition to the way adopted by Cho, Faddeev and Niemi (maximal option). In the past lattice confer…
We investigate the confinement-deconfinement phase transition at finite temperature of the SU(3) Yang-Mills(YM) theory on the lattice from a viewpoint of the dual superconductor picture based on the novel reformulation of the YM theory. In particular, we compare the conventional Abelian dual superconductor picture with the non-Abelian dual superconductor picture proposed in our previous works as the mechanism of quark confinement in the SU(3) YM theory. For the SU(3) YM theory, the reformulation allows two possible options called maximal and minimal. The maximal option corresponds to the manif…
We generalize the (2+1)-dimensional Yang-Mills theory to an anisotropic form with two gauge coupling constants $e$ and $e^{\prime}$. In an axial gauge, a regularized version of the Hamiltonian of this gauge theory is $H_{0}+{e^{\prime}}^{2}H_{1}$, where $H_{0}$ is the Hamiltonian of a set of (1+1)-dimensional principal chiral nonlinear sigma models. We treat $H_{1}$ as the interaction Hamiltonian. For gauge group SU(2), we use form factors of the currents of the principal chiral sigma models to compute the string tension for small $e^{\prime}$, after reviewing exact S-matrix and form-factor me…
The dual superconductivity is a promising mechanism for quark confinement. We proposed the non-Abelian dual superconductivity picture for SU(3) Yang-Mills theory, and demonstrated the restricted field dominance (called conventionally "Abelian" dominance), and non-Abelian magnetic monopole dominance in the string tension. In the last conference, we have demonstrated by measuring the chromoelectric flux that the non-Abelian dual Meissner effect exists and determined that the dual superconductivity for SU(3) case is of type I, which is in sharp contrast to the SU(2) case: the border of type I and…
In order to clarify the mechanism of quark confinement in the Yang-Mills theory with mass gap, we propose to investigate the massive Yang-Mills model, namely, Yang-Mills theory with ``a gauge-invariant gluon mass term'', which is to be deduced from a specific gauge-scalar model with a single radially-fixed scalar field under a suitable constraint called the reduction condition. The gluon mass term simulates the dynamically generated mass to be extracted in the low-energy effective theory of the Yang-Mills theory and plays the role of a new probe to study the phase structure and confinement mec…
The question of the role of the center of the gauge group in the phenomenon of confinement in Yang-Mills theory is addressed. The investigation is performed from the most general perspective of considering all possible choices for the gauge symmetry group. In this context, an interesting role is played by G(2) Yang-Mills theory: the simplest pure gauge theory with a trivial center and without 't Hooft flux vortices. Numerical simulations show the presence of a first order finite temperature deconfinement phase transition in G(2) Yang-Mills theory in (3+1) dimensions. Interestingly, the G(2) ga…
Dual superconductivity is believed to be a promising mechanism for quark confinement and has been investigated on a lattice effectively by a particular gauge called the maximal Abelian (MA) gauge. We propose a new formulation of SU(3) Yang-Mills theory on a lattice based on a non-linear change of variables where the new field variables are expected to reduce to those of the Cho-Faddeev-Niemi- Shabanov decomposition in the continuum limit. By introducing a new variable, say color field, carrying the color direction with it, this formulation enables us to restore and maintain color symmetry that…
The SU(2) Yang-Mills gluon and ghost Green-functions are studied in Landau gauge by means of lattice gauge simulations. A focal point is their low energy behavior since in particular the Kugo-Ojima confinement criterion relates a diverging ghost form factor at vanishing momentum transfer to the confining capabilities of the theory. This divergence is verified by numerical simulations. Removing the confining vortices from the lattice ensembles converts SU(2) Yang-Mills theory in a non-confining theory. It is shown that in this modified theory the divergence of the ghost propagator disappears.
The solution of symmetry equation of Yang-Mills self dual system is found in explicit form of its raising Hamiltonian operator. Thus explicit form of equations of self dual Yang Mills hierarchy is constructed.
We have previously found analytically a very unusual and unexpected form of confinement in SU(3) Yang-Mills theory. This confinement occurs in the deconfined phase of the theory. The free energy of a single static test quark diverges, even though it is contained in deconfined bulk phase and there is no QCD string present. This phenomenon occurs in cylindrical volumes with a certain choice of spatial boundary conditions. We examine numerically an effective model for the Yang-Mills theory and, using a cluster algorithm, we observe this unusual confinement. We also find a new way to determine the…
