We investigate the relationship between colour confinement and the gauge independent Cho-Duan-Ge Abelian decomposition.
The decomposition is defined in terms of a colour field $n$; the principle novelty of our study is that we have defined this field in terms of the eigenvectors of the Wilson Loop. This establishes an equivalence between the path ordered integral of the non-Abelian gauge fields with an integral over an Abelian restricted gauge field which is tractable both theoretically and numerically in lattice QCD. We circumvent path ordering without needing an additional path integral. By using Stokes' theorem, we can compute the Wilson Loop in terms of a surface integral over a restricted field strength, and show that the restricted field strength may be dominated by certain structures, which occur when one of the quantities parametrising the colour field $n$ winds itself around a non-analyticity in the colour field. If they exist, these structures will lead to a area law scaling for the Wilson Loop and provide a mechanism for quark confinement.
We search for these structures in quenched lattice QCD. We perform the Abelian decomposition, and compare the electric and magnetic fields with the patterns expected theoretically. We find that the restricted field strength is dominated by objects which may be peaks a single lattice spacing in size or extended string-like lines of electromagnetic flux. The objects are not isolated monopoles, as they generate electric fields in addition to magnetic fields, and the fields are not spherically symmetric, but may be either caused by a monopole/anti-monopole condensate, some other types of topological objects or a combination of these. Removing these peaks removes the area law scaling of the string tension, suggesting that they are responsible for confinement.
Decomposition of SU(2) gauge field into the monopole and monopoleless components is studied in the Maximal Abelian gauge using Monte-Carlo simulations in lattice SU(2) gluodynamics as well as in two-color QCD with both zero and nonzero quark chemical potential. The interaction potential between static charges is calculated for each component and their sum is compared with the non-Abelian static potential. A good agreement is found in the confinement phase. Implications of this result are discussed.
We calculate the static quark potential for an SU(N) gauge theory in the Maximal Abelian gauge as well as its Abelian projection up to two loops in perturbation theory. We discuss its renormalization properties. The result is compared with a recent lattice result at $r \lesssim 0.5$ fm.
In this report, I discuss lattice Monte Carlo evidence of excitations of isolated static charges in q=2 Abelian Higgs theory. The localized excitations are excited states of the interacting fields surrounding the static charges.
This paper complements a new lattice formulation of SU(2) Yang-Mills theory written in terms of new variables in a compact form proposed in the previous paper. The new variables used in the formulation were once called the Cho--Faddeev--Niemi or Cho--Faddeev--Niemi--Shabanov decomposition. Our formulation enables us to explain the infrared ``Abelian'' dominance, in addition to magnetic monopole dominance shown in the previous paper, in the gauge invariant way without relying on the specific gauge fixing called the maximal Abelian gauge used in the conventional investigations. In this paper, es…
We investigate the relationship between colour confinement and topological structures derived from the gauge invariant Abelian (Cho-Duan-Ge) decomposition. This Abelian decomposition is made imposing an isometry on a colour field $n$ which selects the Abelian direction; the principle novelty of our study is that we have defined this field in terms of the eigenvectors of the Wilson Loop. This allows us to establish an equivalence between the path ordered integral of the non-Abelian gauge fields with an integral over an Abelian restricted gauge field which is tractable both theoretically and num…
We study decomposition of $SU(2)$ gauge field into monopole and monopoleless components. After fixing the Maximal Abelian gauge in $SU(2)$ lattice gauge theory we decompose the nonabelian gauge field into the Abelian field created by monopoles and the modified nonabelian field with monopoles removed. We then calculate respective static potentialis and show that the potential due to the modified nonabelian field is nonconfining while, as is well known, the Abelian field produces linear potential. We further find that the sum of these potentials approximates the nonabelian static potential with …
A common approach while considering confinement is to study the dominance of an Abelian subgroup of the SU(3) gauge Links. A good way to find the Abelian component of the field is through the Cho-Guan-De gauge invariant Abelian Decomposition, which uses a carefully chosen direction vector $n$ to split the gauge field into an Abelian restricted field and a remnant coloured field. The restricted field can be further subdivided into topological and non-topological terms. We show that there is a choice of $n$ which allows us to exactly represent the Wilson Loop of full QCD as a function of only th…
Dual formulations of Abelian U(1) and Z(N) LGT with a static fermion determinant are constructed at finite temperatures and non-zero chemical potential. The dual form is valid for a broad class of lattice gauge actions, for arbitrary number of fermion flavors and in any dimension. The distinguished feature of the dual formulation is that the dual Boltzmann weight is strictly positive. This allows to gain reliable results at finite density via the Monte Carlo simulations. As a byproduct of the dual representation we outline an exact solution for the partition function of the (1 + 1)-dimensional…
We present lattice Monte Carlo evidence of stable excitations of isolated static charges in the Higgs phase of the charge $q=2$ abelian Higgs model. These localized excitations are excited states of the interacting fields surrounding the static charges. Since the $q=2$ abelian Higgs model is a relativistic version of the Landau-Ginzburg effective action of a superconductor, we conjecture that excited states of this kind might be relevant in a condensed matter context. Taken together with recent related work in SU(3) gauge Higgs theory, our result suggests that a massive fermion excitation spec…
Various field strength correlators are investigated in the maximal Abelian projection of pure SU(2) lattice gauge theory. High precision measurements of the colour fields, monopole currents, their curls and divergences allow for detailed checks of the dual superconductor scenario. On this basis, we perform a Ginzburg-Landau type analysis of the flux tube profile from which we derive the size of the penetration length, LAMBDA = 0.16(2) fm, and coherence length of the monopole condensate wave function, XI = 0.27(3) fm. The ratio of these numbers is KAPPA = LAMBDA/XI = 0.59(13) which is below the…
Faddeev and Niemi (FN) have introduced an abelian gauge theory which simulates dynamical abelianization in Yang-Mills theory (YM). It contains both YM instantons and Wu-Yang monopoles and appears to be able to describe the confining phase. Motivated by the meson degeneracy problem in dynamical abelianization models, in this note we present a generalization of the FN theory. We first generalize the Cho connection to dynamical symmetry breaking pattern SU(N+1) -> U(N), and subsequently try to complete the Faddeev-Niemi decomposition by keeping the missing degrees of freedom. While it is not poss…
In the past few years, we have presented a new way of considering quark confinement. Through a careful choice of a Cho-Duan-Ge Abelian Decomposition, we can construct the QCD Wilson Loop in terms of an Abelian restricted field. The relationship between the QCD and restricted string tensions is exact; and we do not need to gauge fix, apply any path ordering of gauge links, or additional path integrals. This hints at why mesons are colour neutral. Furthermore, the Abelian restricted field contains two parts: a Maxwell term, and a topological term. The topological term can describe magnetic monop…
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…
In earlier work, we used a gauge independent Abelian Decomposition to show that Abelian degrees of freedom are wholly responsible for the static quark potential. The restricted Abelian field can be split into two terms, a Maxwell term and a $θ$ (Dirac) term. The $θ$ term's contribution to the string tension can be analysed theoretically and numerically, and arises because of the existence of a certain type of monopole. While the Abelian field can be constructed without gauge fixing, its two component parts are gauge-dependent, with a gauge transformation moving the topological features from on…
We present results of $SU(3)$ Monte-Carlo studies of a new color confinement scheme due to Abelian-like monopoles of the Dirac type without any gauge-fixing. We get (1) perfect Abelian dominance with respect to the static potentials on $12^4\sim 16^4$ lattice at $β=5.6-5.8$ using the multilevel method, (2) monopole as well as Abelian dominances with respect to the static potentials by evaluating Polyakov loop correlators on $24^3\times4$ lattice at $β=5.6$. (3) Abelian dual Meissner effects are studied directly by measuring Abelian color fields and monopole currents around the static source. T…
We revisit the issue of the geometrical separability of the Hilbert space of physical states on lattice Abelian theories in the context of entanglement entropy. We discuss the conditions under which vectors in the Hilbert space, as well as the gauge invariant algebra, admit a tensor product decomposition with a geometrical interpretation. With the exception of pure gauge lattices with periodic boundary conditions which contain topological degrees of freedom, we show that the Hilbert space is geometrically separable.
After fixing the Maximal Abelian gauge in SU(2) lattice gauge theory we decompose the nonabelian gauge field into the so called monopole field and the modified nonabelian field with monopoles removed. We then calculate respective static potentials and find that the potential due to the modified nonabelian field is nonconfining while, as is well known, the monopole field potential is linear. Furthermore, we show that the sum of these potentials approximates the nonabelian static potential with 5% or higher precision at all distances considered. We conclude that at large distances the monopole f…
We study the structure of the isolated static monopoles in the maximal Abelian projection of SU(2) lattice gluodynamics. Our estimation of the monopole radius is $ \approx 0.06 fm$.
Using matrix product states, we explore numerically the phenomenology of string breaking in a non-Abelian lattice gauge theory, namely 1+1 dimensional SU(2). The technique allows us to study the static potential between external heavy charges, as traditionally explored by Monte Carlo simulations, but also to simulate the real-time dynamics of both static and dynamical fermions, as the latter are fully included in the formalism. We propose a number of observables that are sensitive to the presence or breaking of the flux string, and use them to detect and characterize the phenomenon in each of …
Various field strength correlators are investigated in the maximal abelian projection of pure SU(2) lattice gauge theory. High precision measurements of the colour fields, monopole currents, their curl and divergence allow for detailed checks of the dual superconductor scenario. A further decomposition of abelian observables into monopole and photon parts reveals that the flux tube is built up from the monopole part alone.
