We study the string tension as a function of temperature, fitting the SU(3) lattice QCD finite temperature free energy potentials computed by the Bielefeld group. We compare the string tension points with order parameter curves of ferromagnets, superconductors or string models, all related to confinement. We also compare the SU(3) string tension with the one of SU(2) Lattice QCD. With the curve providing the best fit to the finite temperature string tensions, the spontaneous magnetization curve, we then show how to include finite temperature, in the state of the art confining and chiral invariant quark models.
We study the distribution of P-vortices near the confining string in the indirect Z(2) projection of SU(2) lattice gauge theory. It occurs that the density of vortices is constant at large distances and strongly suppressed near the line connecting the test quark-antiquark pair. This means that the condensate of P-vortices is broken inside the confining string. We also find that the width of the P-vortex density distribution is proportional to the logarithm of the distance between the quark and antiquark.
Using lattice Monte Carlo simulations of SU(3) pure gauge theory, we determine the spatial distribution of all components of the color fields created by a static quark and antiquark. We identify the components of the measured chromoelectric field transverse to the line connecting the quark-antiquark pair with the transverse components of an effective Coulomb-like field $\vec{E}^C $ associated with the quark sources. Subtracting $\vec{E}^C$ from the total simulated chromoelectric field $\vec{E}$ yields a non-perturbative, primarily longitudinal chromoelectric field $\vec{E}^{NP}$, which we iden…
The static gluon-quark-antiquark interaction is investigated using lattice QCD techniques. A Wilson loop adequate to the static hybrid three-body system is developed and, using a $24^3 \times 48$ periodic lattice with $β= 6.2$, the potential energy of the system is measured for different geometries. For the medium range behaviour, when the quarks are far apart, we find a string tension which is compatible with two fundamental strings. On the other hand, when the quark and antiquark are nearby, the string tension is larger than two fundamental strings and is compatible with the Casimir scaling.
We extend a previous numerical study of SU(3) Yang-Mills theory in which we measured the spatial distribution of all components of the color fields surrounding a static quark-antiquark pair for a wide range of quark-antiquark separations, and provided evidence that the simulated gauge invariant chromoelectric field can be separated into a Coulomb-like 'perturbative' field and a 'non-perturbative' field, identified as the confining part of the SU(3) flux tube field. In this paper we hypothesize that the fluctuating color fields not measured in our simulations do not contribute to the string ten…
In the low temperature confining phase of QCD or QCD-like theories it is challenging to capture the temperature dependence of observables through AdS/CFT. Using the blackfold approach we compute the quark-antiquark linear static potential in the low temperature confining phase, taking into account the thermal excitations of the string. We find the explicit temperature dependence of the string tension and notice that, as naturally expected, tension decreases as temperature increases. We have also generalized the blackfold approach for the computation of the Wilson loops, making it directly appl…
The colour fields, created by a static gluon-quark-antiquark system, are computed in quenched SU(3) lattice QCD, in a $24^3\times 48$ lattice at $β=6.2$ and $a=0.07261(85)\,fm$. We study two geometries, one with a U shape and another with an L shape. The particular cases of the two gluon glueball and quark-antiquark are also studied, and the Casimir scaling is investigated in a microscopic perspective. This also contributes to understand confinement with flux tubes and to discriminate between the models of fundamental versus adjoint confining strings, analogous to type-II and type-I supercondu…
We describe the Wuppertal university pilot project in applied parallel computing. We report on a comprehensive high statistics determination of the static quark-antiquark potential and related quantities from quenched quantum chromodynamics. New data for the string tension and the plaquette action for the region 5.5 < beta < 6.8 is presented.
Recent developments in Quantum Chromodynamics (QCD) are reviewed on three major topics where nonperturbative gluon excitations of the QCD vacuum and the physical properties of the confining flux play a central role: (1) New lattice results on the spectrum and wave functions of heavy quark-antiquark molecules, known as heavy hybrids, will be discussed. (2) Recent advances on the glueball spectrum in lattice QCD will be presented with some theoretical observations. (3) Progress in our understanding of the nonperturbative internal structure of the confining flux and its excitation spectrum will b…
We represent the Polyakov loop correlator as a spectral sum of correlators of eigenvectors of the lattice Dirac operator. This spectral representation is studied numerically using quenched SU(3) configurations below and above the deconfinement temperature. We analyze whether the individual Dirac eigenvector correlators differ in the confined and deconfined phases. The decay properties of the normalized Dirac eigenvector correlators turn out to be essentially identical in the two phases, but the amplitudes change. This change of the amplitudes shifts the relative contributions of the individual…
SU(N) gauge theories, extended with adjoint fermions having periodic boundary conditions, are confining at high temperature for sufficiently light fermion mass m. Lattice simulations indicate that this confining region is smoothly connected to the confining region of low-temperature pure SU(N) gauge theory. In the high temperature confining region, the one-loop effective potential for Polyakov loops has a Z(N)-symmetric confining minimum. String tensions associated with Polyakov loops are smooth functions of m/T. In the magnetic sector, the Polyakov loop plays a role similar to a Higgs field, …
The chromoelectric and chromomagnetic fields, created by a static gluon-quark-antiquark system, are computed in quenched SU(3) lattice QCD, in a $24^3\times 48$ lattice at $β=6.2$ and $a=0.07261(85) fm$. We compute the hybrid Wilson Loop with two spatial geometries, one with a U shape and another with an L shape. The particular cases of the two gluon glueball and quark-antiquark are also studied, and the Casimir scaling is investigated in a microscopic perspective. This microscopic study of the colour fields is relevant to understand the structure of hadrons, in particular of the hybrid excita…
We study the distribution of the color fields due to a static quark-antiquark pair in SU(2) lattice gauge theory. We find evidence of dual Meissner effect. We put out a simple relation between the penetration length and the string tension.
We study the long distance interaction for hybrid hadrons, with a static gluon, a quark and an antiquark with lattice QCD techniques. A Wilson loop adequate to the static hybrid three-body system is developed and, using a 24^3 x 48 periodic lattice with beta=6.2 and a ~ 0.075 fm, two different geometries for the gluon-quark segment and the gluon-antiquark segment are investigated. When these segments are perpendicular, the static potential is compatible with confinement realized with a pair of fundamental strings, one linking the gluon to the quark and another linking the same gluon to the ant…
SU(N) gauge theories, extended with adjoint fermions having periodic boundary conditions, are confining at high temperature for sufficiently light fermion mass $m$. In the high temperature confining region, the one-loop effective potential for Polyakov loops has a Z(N)-symmetric confining minimum. String tensions associated with Polyakov loops are calculable in perturbation theory, and display a novel scaling behavior in which higher representations have smaller string tensions than the fundamental representation. In the magnetic sector, the Polyakov loop plays a role similar to a Higgs field,…
We study the ``effective string picture'' of confinement, deriving theoretical predictions for the interquark potential at finite temperature. At low temperatures, the leading string correction to the linear confining potential between a heavy quark-antiquark pair is the "Lüscher term''. Assuming a Nambu-Goto effective string action, subleading contributions can be worked out in an analytical way. We also discuss the contribution given by a possible ``boundary term'' in the effective action, and compare these predictions with high precision results from simulations of lattice Z(2) gauge theory…
The separation of a heavy quark and antiquark pair leads to the formation of a tube of flux, or string, which should break in the presence of light quark-antiquark pairs. This expected zero temperature phenomenon has proven elusive in simulations of lattice QCD. We present simulation results that show that the string does break in the confining phase at nonzero temperature.
The separation of a heavy quark and antiquark pair leads to the formation of a tube of flux, or string, which should break in the presence of light quark-antiquark pairs. This expected zero temperature phenomenon has proven elusive in simulations of lattice QCD. We present simulation results that show that the string does break in the confining phase at nonzero temperature.
We attempt to describe the interplay of confinement and chiral symmetry breaking in QCD by using the string model. We argue that in the quasi-abelian picture of confinement based on the condensation of magnetic monopoles and the dual Meissner effect, the worldsheet dynamics of the confining string can be effectively described by the $1+1$ dimensional massless electrodynamics, which is exactly soluble. The transverse plane distribution of the chromoelectric field stretched between the quark and antiquark sources can then be attributed to the fluctuations in the position of the string. The depen…
The flux tube between a quark and an antiquark in Coulomb gauge is imagined in the gluon-chain model as a sequence of constituent gluons bound together by Coulombic nearest-neighbor interactions. We diagonalize the transfer matrix in SU(2) lattice gauge theory in a finite basis of states containing a static quark-antiquark pair together with zero, one, and two gluons in Coulomb gauge. We show that while the string tension of the color-Coulomb potential (obtained from the zero-gluon to zero-gluon element of the transfer matrix) overshoots the true asymptotic string tension by a factor of about …
We study the QCD phase diagram, in particular we study the critical points of the two main QCD phase transitions, confinement and chiral symmetry breaking. Confinement drives chiral symmetry breaking, and, due to the finite quark mass, at small density both transitions are a crossover, while they are a first or second order phase transition in large density. We study the QCD phase diagram with a quark potential model including both confinement and chiral symmetry. We present the confinement of quark fields into a flux tube, obtained in SU(3) quenched lattice QCD, to illustrate the importance o…
We study the action and the energy densities of the confining string in the indirect Z(2) projection of SU(2) lattice gauge theory. We find that the width of the confining string is proportional to the logarithm of the distance between the quark and antiquark. In $2q2\bar{q}$ system we observe the effect of the reconstruction of the flux tube when we change the distance between two $q\bar{q}$ pairs.
In lattice QCD, a confining potential for a static quark-antiquark pair can be computed with the Wilson loop technique. This potential, dominated by a linear potential at moderate distances, is consistent with the confinement with a flux tube, an extended and scalar system also directly observable in lattice QCD. Quantized flux tubes have also been observed in another class of confinement, the magnetic confinement in type II superconductors. On the other hand the solution of Schwinger Dyson Equations, say with the Landau gauge fixing and the truncation of the series of Feynman diagrams, alread…
The distribution of gluon fields in hadrons is of fundamental interest in QCD. Using lattice QCD we have observed the formation of gluon flux tubes within 3 quark (baryon) and quark plus antiquark (meson) systems for a wide variety of spatial distributions of the color sources. In particular we have investigated three quark configurations where two of the quarks are close together and the third quark is some distance away, which approximates a quark plus diquark string. We find that the string tension of the quark-diquark string is the same as that of the quark-antiquark string on the same lat…
We investigate the dual superconductivity hypothesis in pure SU(2) lattice gauge theory. We find evidence of the dual Meissner effect both in the maximally Abelian gauge and without gauge fixing. We also obtain a rather good extimation of the string tension using the value of the London penetration length.
We study the approach, initiated by Marinari et al., to the static inter-quark potential based on Polyakov lines of finite temporal extent, evaluated in Coulomb gauge. We show that, at small spatial separations, the potential can be understood as being between two separately gauge invariant colour charges. At larger separations Gribov copies obstruct the non-perturbative identification of individually gauge invariant colour states. We demonstrate, for the first time, how gauge invariance can be maintained quite generally by averaging over Gribov copies. This allows us to extend the analysis of…
We consider the four point connected correlator representing a static quark-antiquark pair separated by a spatial distance R, propagating for a Euclidean time T. This function is computed by lattice Monte Carlo in SU(2) pure gauge theory at lattice couplings $β=2.2$ and $β=2.5$ in both Coulomb and Landau gauges. The Coulomb gauge correlator is well behaved, and is dominated at large T by a state whose energy grows linearly as $σR$, with $σ$ the known asymptotic string tension. The connected correlator in Landau gauge behaves differently. At intermediate R there is clear evidence of a linear po…
Motivated by the gluon-chain model of flux tube formation, we compute and diagonalize the transfer matrix in lattice SU(2) gauge theory for states containing heavy static quark-antiquark sources, with separations up to one fermi. The elements of the transfer matrix are calculated by variational Monte Carlo methods, in a basis of states obtained by acting on the vacuum state with zero, one, and two-gluon operators in Coulomb gauge. The color Coulomb potential is obtained from the zero gluon to zero gluon element of the transfer matrix, and it is well-known that while this potential is asymptoti…
We apply smoothing to SU(2) lattice field configurations in 3+1 dimensions before fixing to the maximally Abelian gauge. The Abelian projected string tension is shown to be stable under this, whilst the monopole string tension declines by O(30%). Blocking of the SU(2) fields reduces this effect, but the use of extended monopole definitions does not. We discuss these results in the context of additional confining excitations in the U(1) vacuum.
We construct the integration measure over the moduli space of an arbitrary number of N kinds of dyons of the pure SU(N) gauge theory at finite temperatures. The ensemble of dyons governed by the measure is mathematically described by a (supersymmetric) quantum field theory that is exactly solvable and is remarkable for a number of striking features: 1) The free energy has the minimum corresponding to the zero average Polyakov line, as expected in the confining phase; 2)The correlation function of two Polyakov lines exhibits a linear potential between static quarks in any N-ality non-zero repre…
We discuss some recent results obtained in the study of strong quark-antiquark interactions in the presence of intense external magnetic fields by means of lattice QCD simulations. We confirm previous findings and show that both at zero and finite temperature the external field induces anisotropies in the static quark potential. An in-depth study suggests that the effects are essentially due to the variation of the string tension whose angular dependence can be nicely parametrized by the first allowed term in a Fourier expansion. In the confined phase at high temperature, we observe that the s…
We study the distribution of the color fields due to a static quark-antiquark pair in SU(2) lattice gauge theory. We find that the London penetration length measured after Abelian projection in the Abelian Covariant gauge (Maximal Abelian gauge) agrees with the one obtained without gauge fixing. Moreover the penetration length scales according to asymptotic freedom. We put out a simple relation between the penetration length and the string tension.
We investigate, by numerical lattice simulations, the static quark-antiquark potential, the flux tube properties and the chiral condensate for $N_f = 2+1$ QCD with physical quark masses in the presence of strong magnetic fields, going up to $eB = 9$ GeV$^2$, with continuum extrapolated results. The string tension for quark-antiquark separations longitudinal to the magnetic field is suppressed by one order of magnitude at the largest explored magnetic field with respect to its value at zero magnetic background, but is still non-vanishing; in the transverse direction, instead, the string tension…
It has recently been pointed out that simple scaling properties of Polyakov correlation functions of gauge systems in the confining phase suggest that the ratios of k-string tensions in the low temperature region is constant up to terms of order T^3. Here we argue that, at least in a three-dimensional Z_4 gauge model, the above ratios are constant in the whole confining phase. This result is obtained by combining numerical experiments with known exact results on the mass spectrum of an integrable two-dimensional spin model describing the infrared behaviour of the gauge system near the deconfin…
It was recently pointed out that simple scaling properties of Polyakov correlation functions of gauge systems in the confining phase suggest that the ratios of k-string tensions in the low temperature region is constant up to terms of order T^3. Here we argue that, at least in a three-dimensional Z_4 gauge model, the above ratios are constant in the whole confining phase. This result is obtained by combining numerical experiments with known exact results on the mass spectrum of an integrable two-dimensional spin model describing the infrared behaviour of the gauge system near the deconfining t…
We simulate five different systems belonging to the universality class of the gauge dual of three-dimensional random percolation to study the underlying effective string theory at finite temperature. All the data for the finite temperature string tension, when expressed by means of adimensional variables, are nicely described by a unique scaling function. We calculate the first few terms of the string tension up to order $T^6$ and compare to different theoretical predictions. We obtain unambiguous evidence that the coefficients of $T^2$ and $T^4$ terms coincide with those of the Nambu-Goto str…
Thermodynamics of a heavy quark-antiquark pair in SU(3)-QCD is studied below the deconfinement critical temperature, $T_c$. In the quenched case, a model of the string passing through heavy valence gluons yields a correct estimate of $T_c$ and a behavior of the string tension near $T_c$. For two light flavors, entropy and internal energy can be obtained from the partition function of heavy-light mesons and baryons. They are in a good qualitative agreement with the lattice results.
It is widely thought that the early onset of the asymptotic Casimir energy with unit conformal charge signals bosonic string formation of the confining flux connecting a static quark-antiquark pair in QCD. This is observed on a scale where most of the string eigenmodes do not exist and the few stable modes above the ground state are displaced. Hints for the resolution of this paradox are suggested.
The emergence of flux-tubes as the distance is increased between a quark and an antiquark is explored in a three-dimensional confining gauge theory using the gauge/gravity duality. We delineate the shape of the flux-tube corresponding to the classical open string in the bulk and further explore the fluctuation in its thickness induced by the fluctuation of the open string along the radial direction in the bulk. The relationship between the intrinsic shape of a flux-tube and its effect on the heavy quark potential is also discussed.
We study quark confinement in a system of two parallel domain walls interpolating different color dielectric media. We use the phenomenological approach in which the confinement of quarks appears considering the QCD vacuum as a color dielectric medium. We explore this phenomenon in QCD_2, where the confinement of the color flux between the domain walls manifests, in a scenario where two 0-branes (representing external quark and antiquark) are connected by a QCD string. We obtain solutions of the equations of motion via first-order differential equations. We find a new color confining potential…
In $\mathrm{U(1)}$ lattice gauge theory in three spacetime dimensions, confinement can be analytically shown to persist at all values of the coupling. Furthermore, the explicit predictions for the dependence of string tension $σ$ and mass gap $m_0$ on the coupling allow one to tune their ratio at will. These features, and the possibility of obtaining high-precision numerical results via an exact duality map to a spin model, make this theory an ideal laboratory to test the effective string description of confining flux tubes. In this contribution, we discuss our investigation of next-to-leading…
Topological configurations, monopoles and vortices, successfully describe quark confinement and the spontaneous breakdown of chiral symmetry. Despite their infinite action, these configurations are relevant due to a subtle cancellation between action and entropy. A natural explanation for this intrinsic fine-tuning is that smooth low action configurations exist which confine and which appear as singular topological objects in certain gauges. To reveal these confining semi-classical configurations, a new cooling method is proposed which largely reduces the action while preserving the asymptotic…
We study the spectrum of the confining strings in four-dimensional SU(N) gauge theories. We compute, for the SU(4) and SU(6) gauge theories formulated on a lattice, the string tensions sigma_k related to sources with Z_N charge k, using Monte Carlo simulations. Our results are consistent with the sine formula sigma_k/sigma = sin k pi/N / sin pi/N for the ratio between sigma_k and the standard string tension sigma. For the SU(4) and SU(6) cases the accuracy is approximately 1% and 2%, respectively. The sine formula is known to emerge in various realizations of supersymmetric SU(N) gauge theorie…
The gluon excitations of the QCD vacuum are investigated in the presence of a static quark-antiquark source. It is shown that the ground state potential and the excitation spectrum of dynamical gluon degrees of freedom, as determined in our lattice simulations, agree remarkably well with model predictions based on the diaelectric properties of the confining vacuum described as a dual superconductor. The strong chromoelectric field of the static Q-Qbar source creates a bubble (bag) in the condensed phase where weakly interacting gluon modes can be excited. Some features and predictions of the b…
A recently developed model for the QCD analytic invariant charge is compared with quenched lattice simulation data on the static quark-antiquark potential. By employing this strong running coupling one is able to obtain the confining quark-antiquark potential in the framework of the one-gluon exchange model. To achieve this objective a technique for evaluating the integrals of a required form is developed. Special attention is paid here to removing the divergences encountered the calculations. All this enables one to examine the asymptotic behavior of the potential at both small and large dist…
Phases of SU(N) gauge theories in which the global Z(N) symmetry breaks spontaneously to a subgroup Z(L) can be realized by adding appropriate Wilson line terms to the gauge action. These phases are partially confining, in the sense that quarks are confined but bound states of L quarks are not. At temperatures large compared to the normal deconfinement temperature, the phase diagram, pressure, string tensions, and 't Hooft loop surface tensions can be calculated analytically. Approximate scaling laws emerge naturally for both string tensions and surface tensions.
We study the free energy of a quark-antiquark pair near the deconfinement temperature by particle mesh Ewald's method for non-interacting dyon ensemble. We show that the free energy of the quark-antiquark pair increases linearly by increasing the distance between them. The string tension decreases by increasing the temperature, as expected.
We study the QCD phase diagram, in particular we study the critical points of the two main QCD phase transitions, confinement and chiral symmetry breaking. Confinement drives chiral symmetry breaking, and, due to the finite quark mass, at small density both transitions are a crossover, while they are a first or second order phase transition in large density. We study the QCD phase diagram with a quark potential model including both confinement and chiral symmetry. This formalism, in the Coulomb gauge hamiltonian formalism of QCD, is presently the only one able to microscopically include both a…
Thermodynamics of a heavy quark-antiquark pair in SU(3)-QCD is studied both below and above the deconfinement critical temperature $T_c$. In the quenched case, a model of the string passing through heavy valence gluons yields a correct estimate of $T_c$ and the critical behavior of the string tension below $T_c$. For two light flavors, entropy and internal energy below $T_c$ can be obtained from the partition function of heavy-light mesons and baryons. To calculate the free energy of the system above $T_c$, we apply second-order perturbation theory in the interaction of the quark-gluon plasma …
As an approach to describe the long-range properties of non-Abelian gauge theories at non-zero temperature T < T_c, we consider a non-interacting ensemble of dyons (magnetic monopoles) with non-trivial holonomy. We show analytically, that the quark-antiquark free energy from the Polyakov loop correlator grows linearly with the distance, and how the string tension scales with the dyon density. In numerical treatments, the long-range tails of the dyon fields cause severe finite-volume effects. Therefore, we demonstrate the application of Ewald's summation method to this system. Finite-volume eff…
We study the QCD phase diagram, in particular we study the critical points of the two main QCD phase transitions, confinement and chiral symmetry breaking. Confinement drives chiral symmetry breaking, and, due to the finite quark mass, at small density both transitions are a crossover, while they are a first or second order phase transition in large density. We study the QCD phase diagram with a quark potential model including both confinement and chiral symmetry. This formalism, in the Coulomb gauge hamiltonian formalism of QCD, is presently the only one able to microscopically include both a…
The masses of the lowest-lying states in the meson and in the gluonic sector of an SU(2) gauge theory with two Dirac flavors in the adjoint representation are measured on the lattice at a fixed value of the lattice coupling $β= 4/g_0^2 = 2.25$ for values of the bare fermion mass $m_0$ that span a range between the quenched regime and the massless limit, and for various lattice volumes. Even for light constituent fermions the lightest glueballs are found to be lighter than the lightest mesons. Moreover, the string tension between two static fundamental sources strongly depends on the mass of th…
We investigate the static $\overline{Q}Q$-potential for $N_f = 2+1$ QCD at the physical point in the presence of a constant and uniform external magnetic field. The potential is found to be anisotropic and steeper in the directions transverse to the magnetic field than in the longitudinal one. In particular, when compared to the standard case with zero background field, the string tension increases (decreases) in the transverse (longitudinal) direction, while the absolute value of the Coulomb coupling and the Sommer parameter show an opposite behavior.
We introduce a D-dimensional Hamiltonian formalism for the study of Polyakov loop models of finite temperature gauge theories in D+1 dimensions. Polyakov loop string tensions are obtained from energy eigenstates of the Hamiltonian. For D=1, the gauge theory reduces to quantum mechanics on the gauge group; for D>1, the Hamiltonian includes hopping terms that link sites on the transverse lattice. The deconfined phase is associated with a ground state which breaks Z(N) symmetry, and Svetitsky-Yaffe critical universality emerges naturally for D>1. A minimal model is proposed which naturally reprod…
We calculate the string tensions of $k$-strings in SU($N$) gauge theories in both 3 and 4 dimensions. In D=3+1, we find that the ratio of the $k=2$ string tension to the $k = 1$ fundamental string tension is consistent, at the $2 σ$ level, with both the M(-theory)QCD-inspired conjecture and with `Casimir scaling'. In D=2+1 we see a definite deviation from the MQCD formula, as well as a much smaller but still significant deviation from Casimir scaling. We find that in both D=2+1 and D=3+1 the high temperature spatial $k$-string tensions also satisfy approximate Casimir scaling. We point out tha…
The complete expression of the heavy quark-antiquark potential up to order $1/m^2$ is known from QCD in terms of Wilson loop expectation values. We use that expression and a mapping, assumed to be valid at large distances, between Wilson loop expectation values and correlators evaluated in the effective string theory, to compute the potential. We obtain previously unknown results for the spin and momentum-independent parts of the potential. These are linearly rising with the distance and may be interpreted as relativistic corrections to the string tension. We confirm known results for the othe…
Numerical data for the SU(2) confining string in the maximal abelian projection are analysed. The distribution of the electric flux and monopole currents are perfectly described by the classical equations of motion for the dual Abelian Higgs model. The mass of the vector boson is equal to the mass of the monopole (Higgs particle) within numerical errors. The classical energy per unit length of the Abrikosov vortex reproduces 94% of the full non-Abelian string tension.
