Monopole Loops Research Articles

Phase transitions in Abelian lattice gauge theories

We study the phase transition in the U (1) lattice gauge theory using the Wilson-Polyakov line as the order parameter. The Wilson-Polyakov line remains very small at strong coupling and becomes non-zero at weak coupling, signalling a confinement-to-deconfinement phase transition. The decondensation of monopole loops is responsible for this phase transition. A finite size scaling analysis of the susceptibility of the Wilson line gives a ratio for / which is quite close to the corresponding value in the three-dimensional planar model. A scaling behaviour of the monopole loop distribution function is also established at the point of the second-order phase transition. A measurement of the plaquette susceptibility at the transition point shows that it does not scale with the four-dimensional volume as is expected of a first-order bulk transition.

Dual Ginzburg-Landau theory for confinement and chiral symmetry breaking

We introduce the dual Ginzburg-Landau (DGL) theory as a low energy effective theory of QCD. We study color confinement and dynamical chiral symmetry breaking of nonperturbative QCD by using the DGL theory, where color monopole and its condensation play an essential role on the nonperturbative dynamics in the infrared region. As a result of the dual Meissner effect, the linear static quark potential, which characterizes the quark confinement, is obtained in the long distance. We investigate then the dynamical chiral symmetry breaking by using the Schwinger-Dyson equation, where the gluon propagator includes the nonperturbative effect related to monopole condensation. We find a large enhancement of the chiral-symmetry breaking when the dual Meissner effect takes place. We study the recovery of the chiral symmetry and the deconfinement at finite temperature in the DGL theory. We discuss then the essential assumption of the DGL theory, which is the abelian dominance for the infrared physics, in the maximal abelian (MA) gauge in lattice QCD. The lattice QCD simulation demonstrates that the non-abelian gluons have a finite mass of order of 1 GeV in the MA gauge. We introduce further the instanton configuration as the source of the color monopole. In the MA gauge, a monopole circles around an instanton and with the increase of the instanton density, the monopole loop connects many instantons and a complicated monopole loop covers the whole 4 dimensional space. This study indicates that instantons may be playing an essential role even for color confinement.

Monopole clustering and color confinement in the multi-instanton system

We study color confinement properties of the multi-instanton system, which seems to carry an essence of the nonperturbative QCD vacuum. Here we assume that the multi-instanton system is characterized by the infrared suppression of instantons as $f(\rho)\sim \rho^{-5}$ for large size $\rho$. We first investigate a monopole-clustering appearing in the maximally abelian (MA) gauge by considering the correspondence between instantons and monopoles. In order to clarify the infrared monopole properties, we make the ``block-spin'' transformation for monopole currents. The feature of monopole trajectories changes drastically with the instanton density. At a high instanton density, there appears one very long and highly complicated monopole loop covering the entire physical vacuum. Such a global network of long-monopole loops resembles the lattice QCD result in the MA gauge. Second, we observe that the SU(2) Wilson loop obeys an area law and the static quark potential is approximately proportional to the distance $R$ between quark and anti-quark in the multi-instanton system using the SU(2) lattice with a total volume of $V=(10 fm)^4$ and a lattice spacing of $a=0.05 fm$. We extract the string tension from the $5 \times 10^{6}$ measurements of Wilson loops. With an instanton density of $(N/V)=(1/fm)^4$ and a average instanton size of $\bar{\rho}=0.4 fm$, the multi-instanton system provides the string tension of about $0.4 GeV/fm$.

Visible effect of a very heavy magnetic monopole at colliders

If a heavy Dirac monopole exists, the light-to-light scattering below the monopole production threshold is enhanced due to strong coupling of monopoles to photons. At the next Linear Collider with electron beam energy 250 GeV this photon pair production could be observable at monopole masses less than 2.5-6.4 TeV in the $e^+e^-$ mode or 3.7-10 TeV in the $\gamma\gamma$ mode, depending on the monopole spin. At the upgraded Tevatron such an effect is expected to be visible at monopole masses below 1-2.5 TeV. The strong dependence on the initial photon polarizations allows to find the monopole spin in experiments at $e^+e^-$ and $\gamma\gamma$ colliders. We consider the $Z\gamma$ production and the $3\gamma$ production at $e^+e^-$ and $pp$ or $p\bar{p}$ colliders via the same monopole loop. The possibility to discover these processes is significantly lower than that of the $\gamma\gamma$ case.

Monopole loop distribution and confinement in SU(2) lattice gauge theory

The abelian-projected monopole loop distribution is extracted from maximal abelian gauge simulations. The number of loops of a given length falls as a power nearly independent of lattice size. This power increases with β=4/ g 2, reaching five around β=2.85, beyond which, it is shown, loops any finite fraction of the lattice size vanish in the infinite lattice limit.

Do large abelian monopole loops survive the continuum limit?

An analysis of the monopole loop length distribution is performed in Wilson-action SU(2) lattice gauge theory. A pure power law in the inverse length is found, at least for loops of length, l, less than the linear lattice size N. This power shows a definite β dependence, passing 5 around β = 2.9, and appears to have very little finite lattice size dependence. It is shown that when this power exceeds 5, no loops any finite fraction of the lattice size will survive the infinite lattice limit. This is true for any reasonable size distribution for loops larger than N. The apparent lack of finite size dependence in this quantity would seem to indicate that abelian monopole loops large enough to cause confinement do not survive the continuum limit. Indeed they are absent for all β > 2.9.

Magnetic monopole content of hot instantons

We study the Abelian projection of an instanton in R 3 × S 1 as a function of temperature (T) and non-trivial holonomic twist (ω) of the Polyakov loop at infinity. These parameters interpolate between the circular monopole loop solution at T = 0 and the static 't Hooft-Polyakov monopole/anti-monopole pair at high temperature.

Vortices and confinement at weak coupling

We discuss the physical picture of thick vortices as the mechanism responsible for confinement at arbitrarily weak coupling in SU(2) gauge theory. By introducing appropriate variables on the lattice we distinguish between thin, thick and `hybrid' vortices, the latter involving Z(2) monopole loop boundaries. We present numerical lattice simulation results that demonstrate that the full SU(2) string tension at weak coupling arises from the presence of vortices linked to the Wilson loop. Conversely, excluding linked vortices eliminates the confining potential. The numerical results are stable under alternate choice of lattice action as well as a smoothing procedure which removes short distance fluctuations while preserving long distance physics.

Monopole condensation and quark confinement at finite temperature QCD

We study the relation between the abelian monopole condensation and the deconfinement phase transition of the finite-temperature pure QCD, and find that the expectation value of the Polyakov loop becomes zero when a long monopole loop is distributed uniformly in the configuration of the confinement phase as suggested in the previous Monte-Carlo simulations, whereas this value becomes non-zero when the long monopole loop disappears in the deconfinement phase. We also discuss the relation between the monopole picture and the spontaneous breaking of Z(N) symmetry that is the usual interpretation of the deconfinement transition in the finite-temperature SU(N) gauge theory.

Instantons and monopoles are locally correlated with the chiral condensate

We perform a mutual analysis of the topological and chiral vacuum structure of four-dimensional QCD on the lattice at finite temperature. Concerning the topological sector, correlation functions between the distributions of color magnetic monopoles in the maximum Abelian gauge and the densities of topological charge are computed. An enhanced probability for monopoles inside the core of an instanton is found on the gauge field average and in specific configurations. In the chiral sector, clear evidence is found that monopole loops and instantons are locally correlated with the quark condensate. {copyright} {ital 1997} {ital The American Physical Society}

Magnetic monopole loop for the Yang-Mills instanton

We investigate 't Hooft--Mandelstam monopoles in QCD in the presence of a single classical instanton configuration. The solution to the maximal Abelian projection is found to be a circular monopole trajectory with radius $R$ centered on the instanton. At zero loop radius, there is a marginally stable (or flat) direction for loop formation to ${O(R}^{4}\mathrm{ln}R)$. We argue that loops will form, in the semiclassical limit, due to small perturbations such as the dipole interaction between instanton--anti-instanton pairs. As the instanton gas becomes a liquid, the percolation of the monopole loops may, therefore, provide a semiclassical precursor to the confinement mechanism.

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.

Monopoles, instantons and chiral condensate in lattice QCD

We analyze the interplay of topological objects in four-dimensional QCD at finite temperature on the lattice. The distributions of color magnetic monopoles in the maximum abelian gauge are computed around instantons. Studies are performed in both pure and full QCD and in both the confinement and deconfinement phase. We find an enhanced probability for monopoles inside the core of an instanton on gauge field average. This is independent of the topological charge definition used. For specific gauge field configurations we visualize the situation graphically. Moreover the correlation of monopole loops and instantons with the chiral condensate is investigated. Strong evidence is found that clusters of the quark condensate and topological objects coexist locally on individual configurations.

Instantons, monopoles and chiral symmetry breaking

We analyze the interplay of topological objects in four dimensional QCD. The distributions of color magnetic monopoles obtained in the maximum abelian gauge are computed around instantons in both pure and full QCD. We find an enhanced probability of encountering monopoles inside the core of an instanton. We show this by means of local correlation functions of the topological variables. For specific gauge field configurations we visualize the situation graphically. Motivated by the fact that a fermion in the field of a static monopole has an energy zero mode we investigate how monopole loops and instantons are locally correlated with the chiral condensate. The observed correlations suggest that monopoles are involved in the mechanism of breaking of chiral symmetry.

Coexistence of monopoles and instantons for different topological charge definitions and lattice actions

We compute instanton sizes and study correlation functions between instantons and monopoles in maximum abelian projection within .SU(2) lattice QCD at finite temperature. We compare several definitions of the topological charge, different lattice actions and methods of reducing quantum fluctuations. The average instanton size turns out to be σ ≈ 0.2 fm. The correlation length between monopoles and instantons is ξ ≈ 0.25 fm and hardly affected by lattice artifacts as dislocations. We visualize several specific gauge field configurations and show directly that there is an enhanced probability for finding monopole loops in the vicinity of instantons. This feature is independent of the topological charge definition used.

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.

Instantons in the Maximally Abelian gauge

We investigate the Maximally Abelian (MA) Projection for a single SU(2) instanton in continuum gauge theory. We find that there is a class of solutions to the differential MA gauge condition with circular monopole loops of radius R centered on the instanton of width ϱ. However, the MA gauge fixing functional G decreases monotonically as R/ ϱ → 0. Its global minimum is the instanton in the singular gauge. We point out that interactions with nearby anti-instantons are likely to excite these monopole loops.

Monopole condensation and Polyakov loop in finite-temperature pure QCD

We study the relation between the abelian monopole condensation and the deconfinement phase transition of the finite-temperature pure QCD. The expectation value of the monopole contribution to the Polyakov loop becomes zero when a long monopole loop is distributed uniformly in the configuration of the confinement phase. On the other hand, it becomes non-zero when the long monopole loop disappears in the deconfinement phase. We also discuss the relation between the monopole behaviors and the usual interpretation of the spontaneous breaking of Z(N) symmetry in finite-temperature SU(N) QCD. It is found that the boundary condition of the space direction is important to understand the Z(N) symmetry in terms of the monopoles.

Visualization of topological objects in QCD

Recently evidence appeared that instantons and monopoles have a certain local correlation in four-dimensional pure SU (2) and SU (3) gauge theory. We visualize several specific gauge field configurations and show directly that there is an enhanced probability for finding maximally projected abelian monopole loops in the vicinity of instantons. This feature is independent of the topological charge definition used.

Non-Gaussian Fixed Point in Four-Dimensional Pure Compact U(1) Gauge Theory on the Lattice

The line of phase transitions, separating the confinement and the Coulomb phases in the four-dimensional pure compact U(1) gauge theory with extended Wilson action, is reconsidered. We present new numerical evidence that a part of this line, including the original Wilson action, is of second order. By means of a high precision simulation on homogeneous lattices on a sphere we find that along this line the scaling behavior is determined by one fixed point with distinctly non-Gaussian critical exponent nu = 0.365(8). This makes the existence of a nontrivial and nonasymptotically free four-dimensional pure U(1) gauge theory in the continuum very probable. The universality and duality arguments suggest that this conclusion holds also for the monopole loop gas, for the noncompact abelian Higgs model at large negative squared bare mass, and for the corresponding effective string theory.