Center dominance andZ2vortices in SU(2) lattice gauge theory

The European Physical Journal C, 2014

We highlight the progress, current status, and open challenges of QCD-driven physics, in theory and in experiment. We discuss how the strong interaction is intimately connected to a broad sweep of physical problems, in settings ranging from astrophysics and cosmology to strongly-coupled, complex systems in particle and condensed-matter physics, as well as to searches for physics beyond the Standard Model. We also discuss how success in describing the strong interaction impacts other fields, and, in turn, how such subjects can impact studies of the strong interaction. In the course of the work we offer a perspective on the many research streams which flow into and out of QCD, as well as a vision for future developments.

Physical Review D, 2008

Spectral sums of the Dirac-Wilson operator and their relation to the Polyakov loop are thoroughly investigated. The approach by Gattringer is generalized to mode sums which reconstruct the Polyakov loop locally. This opens the possibility to study the mode sum approximation to the Polyakov loop correlator. The approach is rederived for the ab initio continuum formulation of Yang-Mills theories, and the convergence of the mode sum is studied in detail. The mode sums are then explicitly calculated for the Schwinger model and SU (2) gauge theory in a homogeneous background field. Using SU (2) lattice gauge theory, the IR dominated mode sums are considered and the mode sum approximation to the static quark antiquark potential is obtained numerically. We find a good agreement between the mode sum approximation and the static potential at large distances for the confinement and the high temperature plasma phase.

Physical Review D, 1998

We discuss the implementation of the "direct" maximal center gauge (a gauge which maximizes the lattice average of the squared-modulus of the trace of link variables), and its use in identifying Z 2 center vortices in Yang-Mills vacuum configurations generated by lattice Monte Carlo. We find that center vortices identified in the vacuum state account for the full asymptotic string tension. Scaling of vortex densities with lattice coupling, change in vortex size with cooling, and sensitivity to Gribov copies is discussed. Preliminary evidence is presented, on small lattices, for center dominance in SU(3) lattice gauge theory.

Physical Review D, 2003

It is proposed that the cooling of a thermalized SU(N ) gauge theory can be formulated in terms of a cascade involving three effective theories with successively reduced (and spontaneously broken) gauge symmetries, SU(N ) → U(1) N −1 → Z N . The approach is based on the assumption that away from a phase transition the bulk of the quantum interaction inherent to the system is implicitly encoded in the (incomplete) classical dynamics of a collective part made of low-energy condensed degrees of freedom. The properties of (some of the) statistically fluctuating fields are determined by these condensate(s). This leads to a quasi-particle description at tree-level. It appears that radiative corrections, which are sizable at large gauge coupling, do not change the tree-level picture qualitatively. The thermodynamic self-consistency of the quasi-particle approach implies nonperturbative evolution equations for the associated masses. The temperature dependence of these masses, in turn, determine the evolution of the gauge coupling(s). The hot gauge system approaches the behavior of an ideal gas of massless gluons at asymptotically large temperature. A negative equation of state is possible at a stage where the system is about to settle into the phase of the (spontaneously broken) Z N symmetry.

Physical Review D

We propose a model for cosmic inflation which is based on an effective description of strongly interacting, nonsupersymmetric matter within the framework of dynamical abelian projection and centerization. The underlying gauge symmetry is assumed to be SU(N) with N≫1. Appealing to a thermodynamical treatment, the ground-state structure of the model is determined by a potential for the inflaton field (monopole condensate) which allows for nontrivially BPS saturated and thereby stable solutions. For T < MP this leads to an apparent decoupling of gravity from the inflaton dynamics. The ground state dynamics implies a heat capacity for the vacuum leading to inflation for temperatures comparable to the mass scale M of the potential. The dynamics has an attractor property. In contrast to the usual slow-roll paradigm we have m ≫ H during inflation. As a consequence, density perturbations generated from the inflaton are irrelevant for the formation of large-scale structure, and the model has to be supplemented with an inflaton independent mechanism for the generation of spatial curvature perturbations. Within a small fraction of the Hubble time inflation is terminated by a transition of the theory to its center symmetric phase. Due to the prevailing ZN symmetry relic vector bosons are stabilized and therefore potential originators of UHECR's beyond the GZK bound.

Physical Review D, 2000

By fixing lattice Yang-Mills configurations to the maximal center gauge and subsequently applying the technique of center projection, one can identify center vortices in these configurations. Recently, center vortices have been shown to determine the string tension between static quarks at finite temperatures (center dominance); also, they correctly reproduce the deconfining transition to a phase with vanishing string tension. After verifying center dominance also for the socalled spatial string tension, the present analysis focuses on the global topology of vortex networks. General arguments are given supporting the notion that the deconfinement transition in the center vortex picture takes the guise of a percolation transition. This transition is detected in Monte Carlo experiments by concentrating on various slices through the closed vortex surfaces; these slices, representing loops in lattice universes reduced by one dimension, clearly exhibit the expected transition from a percolating to a non-percolating, deconfined, phase. The latter phase contains a large proportion of vortex loops winding around the lattice in the Euclidean time direction. At the same time, an intuitive picture clarifying the persistence of the spatial string tension in the deconfined phase emerges.

Physical Review D, 2003

The de-confinement phase transition in SU(2) Yang-Mills theory is revisited in the vortex picture. Defining the world sheets of the confining vortices by maximal center projection, the percolation properties of the vortex lines in the hypercube consisting of the time axis and two spatial axis are studied. Using the percolation cumulant, the temperature for the percolation transition is seen to be in good agreement with the critical temperature of the thermal transition. The finite size scaling function for the cumulant is obtained. The critical index of the finite size scaling function is consistent with the index of the 3D Ising model. PACS numbers: 11.15.Ha, 12.38.Aw, 12.38.Gc Understanding de-confinement at finite temperatures is one of the major challenges of QCD, the theory of strong interactions. Large scale collision experiments (RHIC) are currently operating in order to detect new states of matter at high temperatures . The general believe is that the confinement of quarks is induced by the gluonic state, and that the mechanism can be anticipated within the pure SU (N ) gauge theory. Using dimensional reduction in combination with lattice simulations of the emerging effective 3D theory, it was recently shown [7] that the non-perturbative effects of SU (N ) gauge theory are strong for temperatures which are accessible in collider experiments. The universality class of the transition is of particular importance because the corresponding degrees of freedom are weakly interacting close to the transition, thus dictating the dynamics of the theory.

Physical Review D, 2004

The structures of confining vortices which underlie pure SU (3) Yang-Mills theory are studied by means of lattice gauge theory. Vortices and Z 3 monopoles are defined as dynamical degrees of freedom of the Z 3 gauge theory which emerges by center gauge fixing and by subsequent center projection. It is observed for the first time for the case of SU (3) that these degrees of freedom are sensible in the continuum limit: the planar vortex density and the monopole density properly scales with the lattice spacing. By contrast to earlier findings concerning the gauge group SU (2), the effective vortex theory only reproduces 62% of the full string tension. On the other hand, however, the removal of the vortices from the lattice configurations yields ensembles with vanishing string tension. SU (3) vortex matter which originates from Laplacian center gauge fixing is also discussed. Although these vortices recover the full string tension, they lack a direct interpretation as physical degrees of freedom in the continuum limit.

Physical Review D, 2005

Physical Review D, 2008

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) monopoles without vortices and -to a less extent -the remaining vortices without monopoles are unstable under smearing whereas smearing of the unmodified Monte Carlo ensemble effects the monopoles and vortices only by smoothing, reducing the density only slightly.

Physical Review D, 2013

The spherical vortex as introduced in [Phys. Rev. D77, 014515 (2008)] is generalized. A continuum form of the spherical vortex is derived and investigated in detail. The discrepancy between the gluonic lattice topological charge and the index of the lattice Dirac operator described in previous papers is identified as a discretization effect. The importance of the investigations for Monte Carlo configurations is discussed.

Particles

In search for an effective model of quark confinement we study the vacuum of SU(2) quantum chromodynamic with lattice simulations using Wilson action. Assuming that center vortices are the relevant excitations causing confinement, we analyzed their physical size and their color structure. We present confirmations for a vanishing thickness of center vortices in the continuum limit and hints at their color structure. This is the first time that algorithms for the detection of thick center vortices based on non-trivial center regions has been used.

Physical Review D, 2003

We calculate the one loop gluon vacuum energy in the background of a color magnetic vortex for SU(2) and SU(3). We use zeta functional regularization to obtain analytic expressions suitable for numerical treatment. The momentum integration is turned to the imaginary axis and fast converging sums/integrals are obtained. We investigate numerically a number of profiles of the background. In each case the vacuum energy turns out to be positive increasing in this way the complete energy and making the vortex configuration less stable. In this problem bound states (tachyonic modes) are present for all investigated profiles making them intrinsically unstable.

Physical Review D, 2005

The confining and topological properties of the compact Abelian Higgs model with doubly-charged Higgs field in three space-time dimensions are studied. We consider the London limit of the model. We show that the monopoles are forming chainlike structures (kept together by ANO vortices) the presence of which is essential for getting simultaneously permanent confinement of singly-charged particles and breaking of the string spanned between doubly-charged particles. In the confinement phase the chains are forming percolating clusters while in the deconfinement (Higgs) phase the chains are of finite size. The described picture is in close analogy with the synthesis of the Abelian monopole and the center vortex pictures in confining non-Abelian gauge models. The screening properties of the vacuum are studied by means of the photon propagator in the Landau gauge.

Physical Review D, 2001

We consider full non-Abelian, Abelian and center projected lattice field configurations built up from random instanton gas configurations in the continuum. We study the instanton contribution to theQQ force with respect to (i) instanton density dependence, (ii) Casimir scaling and (iii) whether various versions of Abelian dominance hold. We check that the dilute gas formulation for the interaction potential gives an reliable approximation only for densities small compared to the phenomenological value. We find that Casimir scaling does not hold, confirming earlier statements in the literature. We show that the lattice used to discretize the instanton gas configurations has to be sufficiently coarse (a ≈ 2ρ compared with the instanton sizeρ) such that maximal Abelian gauge projection and center projection as well as the monopole gas contribution to theQQ force reproduce the non-Abelian instantonmediated force in the intermediate range of linear quasi-confinement. We demonstrate that monopole clustering also depends critically on the discretization scale confirming earlier findings based on monopole blocking.

Physical Review D, 2010

We reveal the center vortex content of SU (2) calorons and ensembles of them. We use Laplacian Center Gauge as well as Maximal Center Gauges to show that the vortex in a single caloron consists of two parts. The first one connects the constituent dyons of the caloron (which are monopoles in Laplacian Abelian Gauge) and extends in time. The second part is predominantly spatial, encloses one of the dyons and can be related to the twist in the caloron gauge field. This part depends strongly on the caloron holonomy and degenerates to a plane when the holonomy is maximally nontrivial, i.e. when the asymptotic Polyakov loop is traceless. Correspondingly, we find the spatial vortices in caloron ensembles to percolate in this case. This finding fits perfectly in the confinement scenario of vortices and shows that calorons are suitable to facilitate the vortex confinement mechanism.

Physical Review Letters, 2005

By using the method of center projection the center vortex part of the gauge field is isolated and its propagator is evaluated in the center Landau gauge, which minimizes the open 3-dimensional Dirac volumes of non-trivial center links bounded by the closed 2-dimensional center vortex surfaces. The center field propagator is found to dominate the gluon propagator (in Landau gauge) in the low momentum regime and to give rise to an OPE correction to the latter of √ σ /p 3 .The * supported in part by DFG Re 856/4-3 † Speaker.

Journal of High Energy Physics, 2002

We prove that magnetic charge does not exist as a physical observable on the physical Hilbert space of the pure SU (2) gauge theory. The abelian magnetic monopoles seen in lattice simulations are then interpreted as artifacts of gauge fixing. The apparent physical scaling properties of the monopole density in the continuum limit observed on the lattice are attributed to the correct scaling properties of physical objects-magnetic vortices, as first argued by Greensite et. al. We can show that a local gauge transformation of a certain type can "create" abelian monopole-antimonopole pairs along magnetic vortices. This gauge transformation exists in pure SU (N) gauge theory at any N .

Physical Review D, 2008

The vacuum of Yang-Mills theory contains singular stringlike objects identified with center (magnetic) vortices. Percolation of magnetic vortices is known to be responsible for the color confinement in the low-temperature phase of the theory. In our work we study properties of the vortices at finite temperature using lattice simulations of SU (2) gauge theory. We show that magnetic vortices provide a numerically large contribution to thermodynamic quantities of the gluon plasma in Yang-Mills theory. In particular, we observe that in the deconfinement phase at temperatures Tc < T 3Tc the magnetic component of the gluon plasma produces a negative (ghostlike) contribution to the anomaly of the energy-momentum tensor. In the confinement phase the vortex contribution is positive. The thermodynamical significance of the magnetic objects allows us to suggest that the quark-gluon plasma may contain a developed network of magnetic flux tubes. The existence of the vortex network may lead to observable effects in the quark-gluon plasma because the chromomagnetic field of the vortices should scatter and drag quarks.

Physical Review D, 2008

We give a definition of gauge-invariant magnetic monopoles in Yang-Mills theory without using the Abelian projection due to 't Hooft. They automatically appear from the Wilson loop operator. This is shown by rewriting the Wilson loop operator using a non-Abelian Stokes theorem. The magnetic monopole defined in this way is a topological object of co-dimension 3, i.e., a loop in four-dimensions. We show that such magnetic loops indeed exist in four-dimensional Yang-Mills theory. In fact, we give an analytical solution representing circular magnetic monopole loops joining a pair of merons in the four-dimensional Euclidean SU(2) Yang-Mills theory. This is achieved by solving the differential equation for the adjoint color (magnetic monopole) field in the two-meron background field within the recently developed reformulation of the Yang-Mills theory. Our analytical solution corresponds to the numerical solution found by Montero and Negele on a lattice. This result strongly suggests that a meron pair is the most relevant quark confiner in the original Yang-Mills theory, as Callan, Dashen and Gross suggested long ago.

Physical Review D, 2016

We give a gauge-covariant decomposition of the Yang-Mills field with an exceptional gauge group G(2), which extends the field decomposition invented by Cho, Duan-Ge, and Faddeev-Niemi for the SU (N) Yang-Mills field. As an application of the decomposition, we derive a new expression of the non-Abelian Stokes theorem for the Wilson loop operator in an arbitrary representation of G(2). The resulting new form is used to define gauge-invariant magnetic monopoles in the G(2) Yang-Mills theory. Moreover, we obtain the quantization condition to be satisfied by the resulting magnetic charge. The method given in this paper is general enough to be applicable to any semi-simple Lie group other than SU (N) and G(2).

Physical review, 2022

Inspired by direct and indirect maximal center gauge methods which confirm the existence of vortices in lattice calculations and by using the connection formalism, we show that under some appropriate gauge transformations vortices and chains appear in the QCD vacuum of the continuum limit. In the direct method, by applying center gauge transformation and "center projection," QCD is reduced to a gauge theory including vortices, which corresponds to the nontrivial first homotopy group Π 1 ðSOð3ÞÞ ¼ Z 2. On the other hand, using the indirect method, in addition to the center gauge transformation and center projection, an initial step called Abelian gauge transformation and then Abelian projection are applied. Therefore, instead of single vortices, chains that contain monopoles and vortices appear in the theory.

Universe

Maximal center gauge (MCG) aims to detect some of the most important vacuum configurations, suggesting thick magnetic flux tubes quantized to non-trivial center elements of the gauge group being responsible for confinement. Due to the NP-hardness of a global maximization of the gauge functional only numeric procedures aiming for local maxima are possible. We observe a linear decrease in the string tension with increasing gauge functional value of the local maxima. This implies that the request to get as close as possible to the absolute maximum is unattainable. We compare global properties of the ensemble of local maxima with other methods to detect center vortices and with determinations of the string tension from full configurations. This comparison indicates that the information about the number and positions of center vortices is contained in the structure of the ensemble of local maxima. This may pave the way for a future more successful formulation of the gauge condition.

Physical Review Letters, 2004

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 singularity of the ghost form factor at vanishing momentum disappears. This observation establishes an intimate correlation between the ghost singularity and confinement. The result also shows that a removal of the vortices generates a theory for which Zwanziger's horizon condition for confinement is no longer satisfied.

Journal of Physics A: Mathematical and Theoretical, 2009

In this paper, we present an exact procedure to deal with Dirac strings or worldsheets in gauge theories containing ensembles of monopoles interacting with charged fields. For SU (2) Yang-Mills theory, initially we construct the appropriate change of variables of the charged fields (including charged ghosts and auxiliary fields) so that the only change in the integrand of the partition function, in the Maximal Abelian gauge, is the addition of given closed Dirac worldsheets. Next, we derive our main result, namely, we show that it is always possible to choose them in such a manner that the total (open plus closed) Dirac worldsheets explicitly decouple from the charged sector, leaving only the effect of their associated gauge invariant borders (where the monopoles are placed), without missing any information about the center vortex sector. This procedure serves as a simplifying basis to deal with ensembles of monopoles and center vortices in the framework of the Cho-Faddeev-Niemi gauge field decomposition, by writing the partition function only in terms of the physical part of the defects to be integrated.

Physical Review D, 2012

In this work, we derive a recently proposed Abelian model to describe the interaction of correlated monopoles, center vortices, and dual fields in three dimensional SU (2) Yang-Mills theory. Following recent polymer techniques, special care is taken to obtain the end-toend probability for a single interacting center vortex, which constitutes a key ingredient to represent the ensemble integration.

Universe, 2022

The wealth of theoretical and phenomenological information about Quantum Chromodynamics at short and long distances collected so far in major collider measurements has profound implications in cosmology. We provide a brief discussion on the major implications of the strongly coupled dynamics of quarks and gluons as well as on effects due to their collective motion on the physics of the early universe and in astrophysics.