Maximally Abelian Research Articles

Monopoles of the Dirac type and color confinement in QCD: Study of the continuum limit

Non-Abelian gauge fields having a line-singularity of the Dirac type lead us to violation of the non-Abelian Bianchi identity. The violation as an operator is equivalent to violation of Abelian-like Bianchi identities corresponding to eight Abelian-like conserved magnetic monopole currents of the Dirac type in $SU(3)$ QCD. It is very interesting to study if these new Abelian-like monopoles are responsible for color confinement in the continuum $SU(3)$ QCD, since any reliable candidate of color magnetic monopoles is not known yet. If these new Abelian-like monopoles exist in the continuum limit, the Abelian dual Meissner effect occurs, so that the linear part of the static potential between a quark-antiquark pair is reproduced fully by those of Abelian and monopole static potentials. These phenomena are called here as perfect Abelian and monopole dominances. It is shown that the perfect Abelian dominance is reproduced fairly well, whereas the perfect monopole dominance seems to be realized for large $\beta$ when use is made of the smooth lattice configurations in the maximally Abelian (MA) gauge. Making use of a block spin transformation with respect to monopoles, the scaling behaviors of the monopole density and the effective monopole action are studied. Both monopole density and the effective monopole action which are usually a two-point function of $\beta$ and the number of times $n$ of the block spin transformation are a function of $b=na(\beta)$ alone for $n=1,2,3,4,6,8,12$. If the scaling behavior is seen for up to larger $n$, it shows the existence of the continuum limit, since $a(\beta)\to 0$ when $n\to\infty$ for fixed $b=na(\beta)$. Along with the previous results without any gauge fixing, these new results obtained in MA gauge suggest that the new Abelian-like monopoles play the role of color confinement in $SU(3)$ QCD.

Clear correlation between monopoles and the chiral condensate in SU(3) QCD

We study spontaneous chiral-symmetry breaking in SU(3) QCD in terms of the dual superconductor picture for quark confinement in the maximally Abelian (MA) gauge, using lattice QCD Monte Carlo simulations with four different lattices of $16^4$, $24^4$, $24^3\times 6$ at $\beta=6.0$ (i.e., the spacing $a \simeq$ 0.1 fm), and $32^4$ at $\beta=6.2$ (i.e., $a \simeq$ 0.075 fm), at the quenched level. First, in the confinement phase, we find Abelian dominance and monopole dominance in the MA gauge for the chiral condensate in the chiral limit,using the two different methods of i) the Banks-Casher relation with the Dirac eigenvalue density and ii) finite quark-mass calculations with the quark propagator and its chiral extrapolation. In the high-temperature deconfined phase, the chiral restoration is observed also for the Abelian and the monopole sectors. Second, we investigate local correlation between the chiral condensate and monopoles, which topologically appear in the MA gauge. We find that the chiral condensate locally takes a quite large value near monopoles. As an interesting possibility, the strong magnetic field around monopoles is responsible to chiral symmetry breaking in QCD, similarly to the magnetic catalysis.

Perfect Abelian dominance of confinement in mesons and baryons in SU(3) lattice QCD

For a long time, the quark confinement mechanism has been one of the most difficult problems in theoretical physics. In particular, there is no clear correspondence between the confinement and non-Abelian nature of QCD. We study the static interquark potential and its Abelian projection in both mesons and baryons in the maximally Abelian (MA) gauge in SU(3) quenched lattice QCD. Remarkably, we find that the quark confining force in QCD can be perfectly described only with Abelian variables in theMAgauge, which we call “perfect Abelian dominance” of the quark confinement.

Gribov-Zwanziger action in SU(2) maximally Abelian gauge withU(1)3Landau gauge

We construct the local Gribov-Zwanziger action for SU(2) Euclidean Yang-Mills theories in the maximally Abelian (MA) gauge with $\mathrm{U}(1{)}_{3}$ Landau gauge fixing based on Zwanziger's work in the Landau gauge. By restricting the functional integral region to the Gribov region in the MA gauge, we give the nonlocal action. We localize the action with new fields and obtain the action with the shift of the new scalar fields, which has the terms corresponding to the localized action of the horizon function in the MA gauge. The diagonal gluon propagator in the MA gauge at tree level behaves like the propagator from the Gribov-Zwanziger action in the Landau gauge, and shows the violation of the Kallen-Lehmann representation. From our result, the validity of the Gribov-Zwanziger scenario in the MA gauge can be studied by comparing with the lattice result in two dimensions, which is qualitatively consistent with the result of the original Gribov-Zwanziger action in the Landau gauge.

Gluon propagators in maximally Abelian gauge in SU(3) lattice QCD

In SU(3) lattice QCD, we study diagonal and off-diagonal gluon propagators in the maximally Abelian (MA) gauge with U(1)$_3 \timesU(1)_8$ Landau gauge fixing. These propagators are studied both in the coordinate space and in the momentum space. The Monte Carlo simulation is performed on $16^4$ at $\beta$=6.0 and $32^4$ at $\beta$=5.8 and 6.0 at the quenched level. In the four-dimensional Euclidean space-time, the effective mass of diagonal gluons is estimated as $M_{\mathrm{diag}} \simeq 0.3\mathrm{GeV}$ and that of off-diagonal gluons as $M_{\mathrm{off}} \simeq 1\mathrm{GeV}$ in the region of $r =0.4-1.0$fm. In the momentum space, the effective mass of diagonal gluons is estimated as $M_{\mathrm{diag}} \simeq 0.3\mathrm{GeV}$ and that of off-diagonal gluons as $M_{\mathrm{off}} \simeq 1\mathrm{GeV}$ in the region of $p <1.1$GeV. The off-diagonal gluon propagator is relatively suppressed in the infrared region and seems to be finite at zero momentum, while the diagonal gluon propagator is enhanced. Furthermore, we also study the functional form of these propagators in momentum space. These propagators are well fitted by $Z/(p^2+m^2)^\nu$ with fit parameters, $Z, m$ and $\nu$ in the region of $p <3.0$GeV. From the fit results and lattice calculations, all of the spectral functions of diagonal and off-diagonal gluons would have negative regions.

Vacuum type of SU(2) gluodynamics in maximally Abelian and Landau gauges

The vacuum type of SU(2) gluodynamics is studied using Monte-Carlo simulations in maximally Abelian (MA) gauge and in Landau (LA) gauge, where the dual Meissner effect is observed to work. The dual Meissner effect is characterized by the coherence and the penetration lengths. Correlations between Wilson loops and electric fields are evaluated in order to measure the penetration length in both gauges. The coherence length is shown to be fixed in the MA gauge from measurements of the monopole density around the static quark-antiquark pair. It is also shown numerically that a dimension 2 gluon operator A^+A^-(s) and the monopole density has a strong correlation as suggested theoretically. Such a correlation is observed also between the monopole density and A^2(s)= A^+A^-(s) + A^3A^3(s) condensate if the remaining U(1) gauge degree of freedom is fixed to U(1) Landau gauge (U1LA). The coherence length is determined numerically also from correlations between Wilson loops and A^+A^-(s) and A^2(s) in MA + U1LA gauge. Assuming that the same physics works in the LA gauge, we determine the coherence length from correlations between Wilson loops and A^2(s). Penetration lengths and coherence lengths in the two gauges are almost the same. The vacuum type of the confinement phase in both gauges is near to the border between the type 1 and the type 2 dual superconductors.

Abelianization of QCD in the maximally Abelian gauge and the Nambu-'t Hooft picture for color confinement

We study the Nambu-'t Hooft picture for color confinement in terms of the abelianization of QCD and monopole condensation in the maximally abelian (MA) gauge. In the MA gauge in the Euclidean metric, the off-diagonal gluon amplitude is strongly suppressed, and then the off-diagonal gluon phase shows strong randomness, which leads to rapid reduction of the off-diagonal gluon correlation. In SU(2) and SU(3) lattice QCD in the MA gauge with the abelian Landau gauge, the Euclidean gluon propagator indicates a large effective mass of the off-diagonal gluon as M off≅ 1 GeV in the intermediate distance as 0.2fm < r <_ 0.8fm. Due to the infrared inactiveness of off- diagonal gluons, infrared QCD is well abelianized like nonabelian Higgs theories in the MA gauge. We investigate the inter-monopole potential and the dual gluon field B μ in the MA gauge, and find longitudinal magnetic screening with m B = 0.5 GeV in the infrared region, which indicates the dual Higgs mechanism by monopole condensation. We define the “gluonic Higgs scalar field” providing the MA projection, and find the correspondence between its hedgehog singularity and the monopole location in lattice QCD.

Gluonic higgs scalar, abelianization and monopoles in QCD - similarity and difference between QCD in the MA gauge and the NAH theory

We study the similarity and the difference between QCD in the maximally abelian (MA) gauge and the nonabelian Higgs (NAH) theory by introducing the ``gluonic Higgs scalar field'' $\vec \phi(x)$ corresponding to the ``color-direction'' of the nonabelian gauge connection. The infrared-relevant gluonic mode in QCD can be extracted by the projection along the color-direction $\vec \phi(x)$ like the NAH theory. This projection is manifestly gauge-invariant, and is mathematically equivalent to the ordinary MA projection. Since $\vec \phi(x)$ obeys the adjoint gauge transformation and is diagonalized in the MA gauge, $\vec \phi(x)$ behaves as the Higgs scalar in the NAH theory, and its hedgehog singularity provides the magnetic monopole in the MA gauge like the NAH theory. We observe this direct correspondence between the monopole appearing in the MA gauge and the hedgehog singularity of $\vec \phi(x)$ in lattice QCD, when the gluon field is continuous as in the SU($N_c$) Landau gauge. In spite of several similarities, QCD in the MA gauge largely differs from the NAH theory in the two points: one is infrared monopole condensation, and the other is infrared enhancement of the abelian correlation due to monopole condensation.

Gauge problem of monopole dynamics inSU(2)lattice gauge theory

Gauge problem of monopole dynamics is studied in SU(2) lattice gauge theory. We study first abelian and monopole contributions to the static potential in four smooth gauges, i.e., Laplacian Abelian (LA), Maximally Abelian Wilson Loop (MAWL) and L-type gauges in comparison with Maximally Abelian (MA) gauge. They all reproduce the string tension in good agreement with the SU(2) string tension. MA gauge is not the only choice of the good gauge which is suitable for the color confinement mechanism. Using an inverse Monte-Carlo method and the blockspin transformation, we determine effective monopole actions and the renormalization group (RG) flows of its coupling constants in various abelian projection schemes. Every RG flow looks to converge to a unique curve which suggests gauge independence in the infrared region.

Strong randomness of off-diagonal gluon phases and off-diagonal gluon mass in the maximally abelian gauge in QCD

We study abelianization of QCD in the maximally abelian (MA) gauge. In the MA gauge, the off-diagonal gluon amplitude is strongly suppressed, and then the off-diagonal gluon phase shows strong randomness, which leads to a large off-diagonal gluon mass. Using lattice QCD, we find a large effective off-diagonal gluon mass in the MA gauge: M off ⋍ 1.2GeV in SU(2) QCD, M off ⋍ 1.1GeV in SU(3) QCD. Due to the large off-diagonal gluon mass in the MA gauge, infrared QCD is well abelianized like nonabelian Higgs theories. We investigate the inter-monopole potential and the dual gluon field B μ in the MA gauge, and find longitudinal magnetic screening with m B ⋍ 0.5 GeV in the infrared region, which indicates the dual Higgs mechanism by monopole condensation. We propose a gauge invariant description of the MA projection by introducing the “gluonic Higgs scalar field”.

Monopoles and gluon fields in QCD in the maximally abelian gauge

We study monopoles and gluon fields in QCD in the maximally abelian (MA) gauge in the context of the dual superconductor picture for confinement. In the abelian gauge, unit charge magnetic monopoles appear, but multi-charge monopoles do not in general cases. The appearance of the monopole is studied in relation to the SU(Nc) singular gauge transformation. The relevant role of off-diagonal gluons is found for the appearance of monopoles in the abelian gauge in QCD. We study the gluon-field properties around the monopole in the MA gauge in terms of the action density using lattice QCD. The monopole provides infinitely large field fluctuations in the abelian sector. In the MA gauge, off-diagonal gluons are strongly suppressed but largely remain around the monopole, which indicates the effective size and the structure of monopoles. We find large cancellation between the abelian part and the off-diagonal part of the action density around the monopole in the MA gauge. Owing to this cancellation, the monopole can appear in QCD without large cost to the QCD action. Finally, we generalize the framework of the abelian projection, i.e. the extraction of the abelian gauge manifold from QCD, by introducing the `gluonic Higgs field' φD[Aμ(x)] defined from the SU(Nc) covariant derivative D̂μ. By way of φD[Aμ(x)], the maximally abelian projection can be performed in a gauge-covariant manner without the notion of gauge fixing in principle.

Quark confinement physics from quantum chromodynamics

We show the construction of the dual superconducting theory for the confinement mechanism from QCD in the maximally abelian (MA) gauge using the lattice QCD Monte Carlo simulation. We find that essence of infrared abelian dominance is naturally understood with the off-diagonal gluon mass m off ≈- 1.2GeV induced by the MA gauge fixing. In the MA gauge, the off-diagonal gluon amplitude is forced to be small, and the off-diagonal gluon phase tends to be random. As the mathematical origin of abelian dominance for confinement, we demonstrate that the strong randomness of the off-diagonal gluon phase leads to abelian dominance for the string tension. In the MA gauge, there appears the macroscopic network of the monopole world-line covering the whole system. We investigate the monopole-current system in the MA gauge by analyzing the dual gluon field B μ . We evaluate the dual gluon mas as m B = 0.4 ∼ 0.5GeV in the infrared region, which is the lattice-QCD evidence of the dual Higgs mechanism by monopole condensation. Owing to infrared abelian dominance and infrared monopole condensation, QCD in the MA gauge is describable with the dual Ginzburg-Landau theory.

Recent topics of infrared effective lattice QCD

Three topics concerning infrared effective lattice QCD are discussed. (1) Perfect lattice action of infrared SU(3) QCD and perfect operators for the static potential are analytically given when we assume two-point monopole interactions alone. The assumption seems to be justified from numerical analyses of pure SU(3) QCD in maximally abelian gauge. (2) Gauge invariance of monopole dominance can be proved theoretically if the gauge invariance of abelian dominance is proved. The gauge invariance of monopole condensation leads us to confinement of abelian neutral but color octet states after abelian projection. (3) A stochastic gauge fixing method is developed to study the gauge dependence of the Abelian projection, which interpolates between the maximally abelian (MA) gauge and no gauge fixing. Abelian dominance for the heavy quark potential holds even in the gauge which is far from Maximally Abelian one.

Quark confinement and extended monopole clustering in the multi-instanton system

We study the SU (2) string tension using the 36 4 lattice with a = 0.05 fm in the random multi-instanton configurations with an entire volume of V = (10 fm) 4. We take the standard instanton-size distribution as ϱ −5 in the infrared region. From about 5 × 10 6 measurements of Wilson loops, we find the string tension σ ⋍ 0.4 GeV/fm in the intermediate region up to 1 fm, when the average instanton size is taken to be ϱ = 0.4 fm and the instanton density ( N V ) 1 4 = 1/ fm . Quantitatively, the string tension strongly depends on the average instanton size as well as the instanton density. We investigate also the clustering of extended monopoles in the maximally abelian (MA) gauge in the multi-instanton ensemble.

Infrared abelian dominance and dual Higgs mechanisms in MA gauge

We study infrared abelian dominance and the dual Higgs mechanism in the maximally abelian (MA) gauge using the lattice QCD Monte Carlo simulation. In the MA gauge, the off-diagonal gluon phase tends to be random, and the off-diagonal gluon A μ ± acquires the effective mass as M off ≅ 1.2 GeV. From the monopole current in the MA gauge, we extract the dual gluon field B μ and estimate the dual gluon mass as m B ≅ 0.5 GeV. The QCD-monopole structure is also investigated in terms of off-diagonal gluons. From the lattice QCD in the MA gauge, the dual Ginzburg-Landau (DGL) theory can be constructed as a realistic infrared effective theory based on QCD

Effective mass generation of off-diagonal gluons and abelian dominance in the maximally abelian gauge in QCD

We study the properties of gluons in QCD in the maximally abelian (MA) gauge. In the MA gauge, the off-diagonal gluon behaves as the massive vector boson with the mass M off ≅ 1.2 GeV, and therefore the off-diagonal gluon cannot carry the long-range interaction for r ⪢ M off −1 ≅ 0.2 fm. The essence of the infrared abelian dominance in the MA gauge is physically explained with the generation of the off-diagonal gluon mass M off ≅ 1.2 GeV induced by the MA gauge fixing, and the off-diagonal gluon mass generation would predict general infrared abelian dominance in QCD in the MA gauge. We report also the off-diagonal gluon propagator at finite temperature.

Off-diagonal gluon mass generation and infrared Abelian dominance in the maximally Abelian gauge in lattice QCD

We study effective mass generation of off-diagonal gluons and infrared Abelian dominance in the maximally Abelian (MA) gauge. Using SU(2) lattice QCD, we investigate the propagator and the effective mass of the gluon field in the MA gauge with $\mathrm{U}{(1)}_{3}$ Landau gauge fixing. The Monte Carlo simulation is performed on the ${12}^{3}\ifmmode\times\else\texttimes\fi{}24$ lattice with $2.2&lt;~\ensuremath{\beta}&lt;~2.4,$ and also on the ${16}^{4}$ and ${20}^{4}$ lattices with $2.3&lt;~\ensuremath{\beta}&lt;~2.4.$ In the MA gauge, the diagonal gluon component ${A}_{\ensuremath{\mu}}^{3}$ shows long-range propagation, and infrared Abelian dominance is found for the gluon propagator. In the MA gauge, the off-diagonal gluon component ${A}_{\ensuremath{\mu}}^{\ifmmode\pm\else\textpm\fi{}}$ behaves as a massive vector boson with the effective mass ${M}_{\mathrm{off}}\ensuremath{\simeq}1.2\mathrm{GeV}$ in the region of $r\ensuremath{\gtrsim}0.2\mathrm{fm},$ and its propagation is limited within a short range. We conjecture that infrared Abelian dominance can be interpreted as infrared inactivity of the off-diagonal gluon due to its large mass generation induced by the MA gauge fixing.

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 $0.4 GeV/fm$.

Maximally Abelian gauge and the gauge-invariance condition

We reformulate the maximally Abelian (MA) gauge in terms of the gauge-connection formalism with the covariant derivative ${D}_{\ensuremath{\mu}}.$ In this expression, the residual symmetry and the local MA-gauge-fixing condition are naturally derived. In addition to the local $\mathrm{U}{(1)}^{{N}_{c}\ensuremath{-}1}$ symmetry, the global Weyl symmetry persists as the relic of $\mathrm{SU}{(N}_{c})$ in the MA gauge, and causes an ambiguity in the global definition of the electric and magnetic charges. We show the criterion on $\mathrm{SU}{(N}_{c})$ gauge invariance for the variable defined in the MA gauge in terms of the residual symmetry. Finally, we widen this criterion on gauge invariance to general partial gauge fixing.

Abelian dominance for confinement and random phase property of off-diagonal gluons in the maximally abelian gauge

We study abelian dominance for confinement in terms of the local gluon properties in the maximally abelian (MA) gauge both in the analytical consideration and in the lattice QCD. In the MA gauge, the off-diagonal gluon amplitude | A μ ±| is strongly suppressed, which is observed as “microscopic abelian tendency” on the link variable in lattice QCD. According to the suppression of the off-diagonal gluon amplitude, the phase variable of the off-diagonal gluon tends to be random in the MA gauge, besides the residual gauge degrees of freedom. Within the random-variable approximation for the off-diagonal phase variable, we demonstrate that the off-diagonal gluon contribution to the Wilson loop obeys the perimeter law in the MA gauge, and show exact abelian dominance for the string tension, although a small deviation is brought about by the finite size effect of the Wilson loop. We numerically confirm these results using the lattice QCD simulations.