Abelian Gauge Fixing Research Articles

Cho decomposition, Abelian gauge fixing, and monopoles in G(2) Yang-Mills theory

By extending the Cho decomposition method to G(2) gauge group, monopoles of this group are studied. Since SU(2) and SU(3) are subgroups of G(2), discussions are done mostly based on these subgroups of G(2). A direct relation between root vectors of G(2) and the associated magnetic charges is presented by group theoretical issues. In addition, G(2) monopoles are obtained by an Abelian gauge fixing method, and it is shown that the results agree with the ones we obtain by the Cho decomposition method.

Coulomb, Landau and maximally Abelian gauge fixing in lattice QCD with multi-GPUs

A lattice gauge theory framework for simulations on graphic processing units (GPUs) using NVIDIA’s CUDA is presented. The code comprises template classes that take care of an optimal data pattern to ensure coalesced reading from device memory to achieve maximum performance. In this work we concentrate on applications for lattice gauge fixing in 3+1 dimensional SU(3) lattice gauge field theories. We employ the overrelaxation, stochastic relaxation and simulated annealing algorithms which are perfectly suited to be accelerated by highly parallel architectures like GPUs. The applications support the Coulomb, Landau and maximally Abelian gauges. Moreover, we explore the evolution of the numerical accuracy of the SU(3) valued degrees of freedom over the runtime of the algorithms in single- (SP) and double-precision (DP). Therefrom we draw conclusions on the reliability of SP and DP simulations and suggest a mixed-precision scheme that performs the critical parts of the algorithm in full DP while retaining 80%–90% of the SP performance. Finally, multi-GPUs are adopted to overcome the memory constraint of single GPUs. A communicator class which hides the MPI data exchange at the boundaries of the lattice domains, via the low bandwidth PCI-Bus, effectively behind calculations in the inner part of the domain is presented. Linear scaling using 16 NVIDIA Tesla C2070 devices and a maximum performance of 3.5 Teraflops on lattices of size down to 643×256 is demonstrated.

Gauge Fixing in Lattice QCD with Multi-GPUs

Here we present the cuLGT code for gauge fixing in lattice gauge field theories with graphic processing units (GPUs). Implementations for SU(3) Coulomb, Landau and maximally Abelian gauge fixing are available and the overrelaxation, stochastic relaxation and simulated annealing algorithms are supported. Performance results for single and multi-GPUs are given.

On ghost condensation, mass generation and Abelian dominance in the maximal Abelian gauge

Recent work claimed that the off-diagonal gluons (and ghosts) in pure Yang–Mills theories, with maximal Abelian gauge (MAG) fixing, attain a dynamical mass through an off-diagonal ghost condensate. This condensation takes place due to a quartic ghost interaction, unavoidably present in MAG for renormalizability purposes. The off-diagonal mass can be seen as evidence for Abelian dominance. We discuss why ghost condensation of the type discussed in those works cannot be the reason for the off-diagonal mass and Abelian dominance, since it results in a tachyonic mass. We also point out what the full mechanism behind the generation of a real mass might look like.

A fresh look at the flux tube in Abelian-projected SU(2) gluodynamics

We reconsider the properties of the QQ¯ flux tube within Abelian-projected SU(2) lattice gauge theory in terms of electric field and monopole current. In maximal Abelian gauge fixing, we assess the influence of Gribov copies on the apparent flux-tube profile. For the profile corresponding to optimal gauge fixing, we study the independence of the lattice spacing for β = 2.3, 2.4, and 2.5115 on a 32 4 lattice. We apply the decomposition of the Abelian Wilson loop into monopole and photon parts and compare the electric and monopole profile emerging from these three different sources with the field strength and monopole current within the DGL theory.

On the Landau–Ginzburg theory of MAG projected SU(2) lattice gauge theory

Maximal Abelian gauge fixing and subsequent Abelian projection of SU(2) lattice gauge theory defines closed trajectories of magnetic monopoles. These trajectories can be interpreted in terms of an effective scalar field theory of the MAG monopoles using the world line representation of the functional determinants. Employing the monopole world lines detected in the numerical simulation, we show that a scalar bound state exists. The screening mass m of this state properly scales towards the continuum limit. We find m≈1.4 GeV when the string tension σ=440 MeV is used as reference scale.

Abelianization of first class constraints

We show that a given set of first class constraints becomes Abelian if one maps each constraint to the surface of other constraints. There is no assumption that first class constraints satisfy a closed algebra. The explicit form of the projection map is obtained at least for irreducible first class constraints. Using this map we give a method to obtain gauge fixing conditions such that the set of Abelian first class constraints and gauge fixing conditions satisfy the symplectic algebra.

Reflections on topology - viewpoints on abelian projections

This talk discusses two topological features in non-abelian gauge theories, related by the notion of abelian projection and the Hopf invariant. Minimising the energy of the non-linear sigma model with a Skyrme-like term (the Faddeev-Niemi model), can be identified with a non-linear maximal abelian gauge fixing of the SU(2) gauge vacua with a winding number equal to the Hopf invariant. In the context of abelian projection the Hopf invariant can also be associated to a monopole world line, through the Taubes winding, measuring its contribution to topological charge. Calorons with non-trivial holonomy provide an explicit realisation. We discuss the identification of its constituent monopoles through degenerate eigenvalues of the Polyakov loop (the singularities or defects of the abelian projection). It allows us to study the correlation between the defect locations and the explicit constituent monopole structure, through a specific SU(3) example.

Confinement, chiral symmetry breaking, and the axial anomaly from domain formation at intermediate resolution

Based on general renormalization group arguments, Polyakov's loop-space formalism, and recent analytical lattice arguments, suggesting, after Abelian gauge fixing, a description of pure gluodynamics by means of a Georgi-Glashow like model, the corresponding vacuum fields are defined in a nonlocal way. Using lattice information on the gauge invariant field strength correlator in full QCD, the resolution scale ${\ensuremath{\Lambda}}_{b},$ at which these fields become relevant in the vacuum, is determined. For SU(3) gauge theory it is found that ${\ensuremath{\Lambda}}_{b}\ensuremath{\sim}2.4$ GeV, 3.1 GeV, and 4.2 GeV for ${(N}_{F}=4,$ ${m}_{q}=18$ MeV), ${(N}_{F}=4,$ ${m}_{q}=36$ MeV), and pure gluodynamics, repectively. Implications for the operator product expansion of physical correlators are discussed. It is argued that the emergence of magnetic (anti)monopoles in the vacuum at resolution ${\ensuremath{\Lambda}}_{b}$ is a direct consequence of the randomness in the formation of a low entropy Higgs condensate. This implies a breaking of chiral symmetry and a proliferation of the axial U(1) anomaly at this scale already. Justifying Abelian projection, a decoupling of non-Abelian gauge field fluctuations from the dynamics occurs. The condensation of (anti)monopoles at ${\ensuremath{\Lambda}}_{c}<{\ensuremath{\Lambda}}_{b}$ follows from the demand that vacuum fields ought to have vanishing action at any resolution. As monopoles condense they are reduced to their cores, and hence they become massless. Apparently broken gauge symmetries at resolutions ${\ensuremath{\Lambda}}_{c}<\ensuremath{\Lambda}<~{\ensuremath{\Lambda}}_{b}$ are restored in this process.

Abelian and center gauge fixing in continuum Yang–Mills theory for general gauge groups

A prescription for center gauge fixing for pure Yang–Mills theory in the continuum with general gauge groups is presented. The emergence of various types of singularities (magnetic monopoles and center vortices) appearing in the course of the gauge fixing procedure are discussed.

Laplacian Abelian projection: Abelian dominance and Monopole dominance

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.

Yang-Mills theory as a deformation of topological field theory, dimensional reduction, and quark confinement

We propose a reformulation of Yang-Mills theory as a perturbative deformation of a novel topological (quantum) field theory. We prove that this reformulation of four-dimensional QCD leads to quark confinement in the sense of an area law of the Wilson loop. First, Yang-Mills theory with a non-Abelian gauge group G is reformulated as a deformation of a novel topological field theory. Next, a special class of topological field theories is defined by both Becchi-Rouet-Stora-Tyupin (BRST) and anti-BRST exact actions corresponding to the maximal Abelian gauge leaving the maximal torus group Hof G invariant. Then we find topological field theory $(D>2)$ has a hidden supersymmetry for a choice of maximal Abelian gauge. As a result, the D-dimensional topological field theory is equivalent to the $(D\ensuremath{-}2)$-dimensional coset $G/H$ nonlinear sigma model in the sense of the Parisi-Sourlas dimensional reduction. After maximal Abelian gauge fixing, the topological property of the magnetic monopole and antimonopole of four-dimensional Yang-Mills theory is translated into that of an instanton and anti-instanton in a two-dimensional equivalent model. It is shown that the linear static potential in four dimensions follows from the instanton--anti-instanton gas in the equivalent two-dimensional nonlinear sigma model obtained from the four-dimensional topological field theory by dimensional reduction, while the remaining Coulomb potential comes from the perturbative part in four-dimensional Yang-Mills theory. The dimensional reduction opens a path for applying various exact methods developed in two-dimensional quantum field theory to study the nonperturbative problem in low-energy physics of four-dimensional quantum field theories.

Existence of chiral-asymmetric zero modes in the background of QCD-monopoles

We study topological aspects of the QCD vacuum structure in SU(2) lattice gauge theory with the abelian gauge fixing. The index of the Dirac operator is measured by using the Wilson fermion in the quenched approximation. We find chiral-asymmetric zero modes in background fields dominated by QCD-monopoles without any cooling.

Clustering of monopoles in the instanton vacuum

We generate a random instanton vacuum with various densities and size distributions. We perform numerically the maximally abelian gauge fixing of these configurations in order to find monopole trajectories induced by instantons. We find that instanton-induced monopole loops form enormous clusters occupying the whole physical volume, provided instantons are sufficiently dense. It indicates that confinement might be caused by instantons.

Distribution of instanton and monopole clustering

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, these 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 configurations.

New algorithm for gauge fixing in SU(2) lattice gauge theory

An overrelaxed variant of simulated annealing is applied to the problem of maximally abelian gauge fixing. The superiority of this algorithm over the commonly used relaxation procedure is demonstrated. Biases on non gauge invariant quantities due to gauge fixing ambiguities are discussed.

Hot gluon propagator

A general expression of the hard thermal loop gluon propagator is given for a variety of abelian gauge fixing conditions (Coulomb gauge, axial gauge, covariant gauge, temporal gauge ...) The current --- current interaction energy, in particular the Coulomb energy, is shown to be independent of the gauge fixing condition (not only of the parameter).

Fractal dimension of gauge-fixing defects

The fractal dimension D ƒ of sites resisting Landau or maximal Abelian (MA) gauge fixing in lattice SU(3) gluodynamics is defined and computed. In Landau gauge such sites clump into D ƒ ∼ 1 clusters in the confining phase. In the finite temperature phase their dimensionality drops to D ƒ < 1 , that is, clustering seems to dissipate. In contrast, MA gauge resistant sites fail to exhibit a notable tendency to cluster at any temperature.