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, leading to an apparent breaking of SU(N) to $U(1)^{N-1}$. This is turn yields a dual effective theory where magnetic monopoles give rise to string tensions for spatial Wilson loops. The spatial string tensions are calculable semiclassically from kink solutions of the dual system. We show that the spatial string tensions $σ^{(s)}_k$ associated with each $N$-ality $k$ obey a variant of Casimir scaling $σ^{(s)}_k /σ^{(s)}_1 \leq \sqrt{k(N-k)/(N-1)} $. Although lattice simulations indicate that the high temperature confining region is smoothly connected to the confining region of low-temperature pure SU(N) gauge theory, the electric and magnetic string tension scaling laws are different and readily distinguishable.
The reduced tension $σ_{cd}$ of the interface between the confined and the deconfined phase of $SU(3)$ pure gauge theory is related to the finite size effects of the first transfer matrix eigenvalues. A lattice simulation of the transfer matrix spectrum at the critical temperature $T_c = 1/L_t$ yields $σ_{cd} = 0.139(4) T_c^2$ for $L_t = 2$. We found numerical evidence that the deconfined-deconfined domain walls are completely wet by the confined phase, and that the confined-deconfined interfaces are rough.
We consider three-dimensional SU(N) gauge theories with massless Dirac or Majorana fermions in the adjoint representation. We use the Eguchi-Kawai volume reduction to two-dimensions to calculate the tension sigma_k of the k-string in such theories. Under some assumptions whose validity we discuss, we derive the previously conjectured sine formula, sigma_k ~ N sin pi k/N.
Continuum reduction and Monte Carlo simulation are used to calculate the heavy quark potential and the string tension in large N Yang-Mills theory in four dimensions. The potential is calculated out to a separation of nine lattice units on a lattice with extent six in each direction.
Using smeared Creutz ratios we extract the string tension for SU(N) pure gauge theory and $N$=3,4,5,6,8. We employ these results to extrapolate to large N. The same methodology is applied to the single-site Twisted Eguchi Kawai model. The corresponding string tension matches perfectly within errors with the extrapolated one, providing strong evidence in favour of the twisted reduction framework. Interesting results are also obtained on the behaviour of Creutz ratios for large sizes.
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…
Recently, we proposed a new method to extract the string tension from 4-dimensionally smeared Wilson loops. In this talk, we first show that the results obtained using this smearing method are identical to those obtained by Wilson flow, once the time step is sufficiently small. We then demonstrate the practical advantage of our method by applying it to the calculation of string tension in SU(3) Yang-Mills theory.
From continuum studies it is known that the Coulomb string tension $σ_C$ gives an upper bound for the physical (Wilson) string tension $σ_W$ [D. Zwanziger, Phys. Rev. Lett. 90, 102001 (2003)]. How does however such relationship translate to the lattice? In this paper we give evidence that there, while the two string tensions are related at zero temperature, they decouple at finite temperature. More precisely, we show that on the lattice the Coulomb gauge confinement scenario is always tied to the spatial string tension, which is known to survive the deconfinement phase transition and to cause …
We numerically compute the string tension in the large N limit of three dimensional Yang-Mills theory using Wilson loops. Space-time loops are formed as products of smeared space-like links and unsmeared time-like links. We use continuum reduction and both unfolded and folded Wilson loops in the analysis.
We calculate the k=2 string tensions in SU(4) and SU(5) gauge theories in 3+1 dimensions, and compare them to the k=1 fundamental string tensions. We find, from the continuum extrapolation of our lattice calculations, that K(k=2)/K(k=1) = 1.40(8) in the SU(4) gauge theory, and that K(k=2)/K(k=1) = 1.56(10) in SU(5). We remark upon the way this might constrain the dynamics of confinement and the intriguing implications it might have for the mass spectrum of SU(N) gauge theories. We also note that these results agree closely with the MQCD-inspired conjecture that the SU(N) string tensions satisf…
We exhibit the appropriate variables allowing the plus-minus (Z(2)) `reduction' of the Wilson loop operator which provides a direct measure of the thin, thick and `mixed' Z(2) topological, gauge-invariant vortices in SU(2) LGT. Simulations with the Wilson action, as well as a perfect action smoothing procedure, show the string tension to be reproduced from the contributions of these excitations.
We improve upon recent calculations of the low-lying `glueball' spectra of SO(3) and SO(4) lattice gauge theories in 2+1 dimensions, and compare the resulting continuum extrapolations with SU(2). We find that these are reasonably consistent, as are the SU(2) and SO(4) string tensions when these are corrected for the differing representations of the flux. All this indicates that the different global properties of these groups do not play a significant role in the low-lying physics.
We numerically compute the spatial string tension in the deconfined phase of three dimensional QCD in the large N limit. Our results clearly show that the string tension grows linearly with temperature.
We address the question of whether the large-N expansion in pure SU(N) gauge theories requires that k-string tensions must have a power series expansion in 1/N^2, as in the sine law, or whether 1/N contributions are also allowable, as in Casimir scaling. We find that k-string tensions may, in fact, have 1/N corrections, and consistency with the large-N expansion in the open-string sector depends crucially on an exact cancellation, which we will prove, among terms involving odd powers of 1/N in particular combinations of Wilson loops. It is shown how these cancellations are fulfilled, and consi…
In the context of four-dimensional SU(N) gauge theories, we study the spectrum of the confining strings. We compute, for the SU(6) gauge theory formulated on a lattice, the three independent string tensions sigma_k related to sources with Z_N charge k=1,2,3, using Monte Carlo simulations. Our results, whose uncertainty is approximately 2% for k=2 and 4% for k=3, 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, and show deviations from the Casimir scaling. The sine formula is known to emerge in sup…
The reduced tension $σ_{cd}$ of the interface between the confined and the deconfined phase of $SU(3)$ pure gauge theory is determined from numerical simulations of the first transfer matrix eigenvalues. At $T_c = 1/L_t$ we find $σ_{cd} = 0.139(4) T_c^2$ for $L_t = 2$. The interfaces show universal behavior because the deconfined-deconfined interfaces are completely wet by the confined phase. The critical exponents of complete wetting follow from the analytic interface solutions of a $\Z(3)$-symmetric $Φ^4$ model in three dimensions. We find numerical evidence that the confined-deconfined inte…
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 present results for the confinement-deconfinement interface tension $σ_{cd}$ of quenched QCD. They were obtained by applying Binder's histogram method to lattices of size $L^2\times L_z\times L_t$ for $L_t=2$ and $L=8,10,12\mbox{ and }14$ and various $L_z\in [L,\, 4\, L]$. The use of a multicanonical algorithm and rectangular geometries have turned out to be crucial for the numerical studies. We also give an estimate for $σ_{cd}$ at $L_t=4$ using published data.
We present results of modeling the temperature dependence of the spatial string tension and thermal phase transition in a five-dimensional framework nowadays known as AdS/QCD. For temperatures close to the critical one we find a behaviour remarkably consistent with the lattice results.
We present an analytical continuum calculation, starting from first principles, of the vacuum wavefunction and string tension for pure Yang-Mills theories in $(2+1)$ dimensions, extending our previous analysis using gauge-invariant matrix variables. The vacuum wavefunction is consistent with what is expected at both high and low momentum regimes. The value of the string tension is in very good agreement with recent lattice Monte Carlo evaluations.
We compute, via numerical simulations, the non-perturbative Coulomb potential of pure SU(3) gauge theory in Coulomb gauge. We find that that the Coulomb potential scales nicely in accordance with asymptotic freedom, that the Coulomb potential is linear in the infrared, and that the Coulomb string tension is about four times larger than the asymptotic string tension. We explain how it is possible that the asymptotic string tension can be lower than the Coulomb string tension by a factor of four.
The spatial string tension for 3+1 dimensional QCD at finite temperature is measured. The gauge configurations we re generated with two light and one heavier strange quark on lattices of size $16^3 4$ and $24^3 6$. This spatial string tension is compared with the string tension of the 3 dimensional pure gauge theory together with the temperature dependent 2-loop running coupling. Further comparison is made with predictions from dimensionally reduced effective theories.
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 invari…
Spatial string tension is computed in finite temperature gluodynamics on asymmetric lattices in a spherical model approximation. Conditions of scaling behavior are specified. Discrepancies with a standard renormalisation procedure are discussed.
A review article on string tension concept and their relevance as non-perturbative quantity on the study of quark confinement in lattice gauge theories. A detailed description of a variety of methods to measure the string tension on the lattice and an indication of the most promising developments is proposed.
In earlier papers we established quark confinement analytically in anisotropic $(2+1)$-dimensional Yang-Mills theory with two gauge coupling constants. Here we point out a few features of the confining phase. These are: 1) the string tension in the $x^{2}$-direction as a function of representation obeys a sine law, and 2) static adjoint sources are not confined.
The spatial string tension of a SU(N) gauge theory without quarks is calculated using gauge/string duality for 1.2T_c<T<3T_c. The result is remarkably consistent with the available lattice data for N=2,3. Some evidence is found that to leading order the large distance physics of a string can be described by a five-dimensional geometry without a need for an internal space.
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…
Monopole and photon contributions to abelian Wilson loops are calculated using Monte-Carlo simulations of finite-temperature $SU(2)$ QCD in the maximally abelian gauge. The string tension is reproduced by monopole contribution alone also in finite temperature SU(2) QCD. The spatial string tension scales as $\sqrtσ \propto g^{2}(T)T$ and is reproduced almost by monopole contribution alone. Each configuration has one long monopole loop, and the long monopole loops alone are responsible for the string tension in the confinement phase. On the other hand, the spatial string tension in the deconfine…
Monopole and photon contributions to abelian Wilson loops are calculated using Monte-Carlo simulations of finite-temperature $SU(2)$ QCD in the maximally abelian gauge. Long monopole loops alone are responsible for the behavior of the string tension in the confinement phase up to the critical $β_c$. Short monopole loops and photons do not contribute to the string tension. The abelian and the monopole spacial string tensions (both of which agree with the normal ones for $β< β_c$) show a $g^{4}(T) T^2$ scaling behavior in the deconfinement phase. The abelian spacial string tension is in agreemen…
We investigate Abelian and monopole contributions to spatial string tension in the deconfined phase of finite temperature SU(2) gauge theory without imposing any gauge fixing conditions. Lattice calculations of non-Abelian and Abelian spatial string tensions from the Wilson action at gauge coupling $β=2.74$ and lattice volume $24^3\times{N_t}$ $(N_{t}=\left\{ 24,8,6,4,2)\right\}$ show that these string tensions agree with each other within error bars at any adopted value of $N_t$, which implies Abelian dominance. From measurements of non-Abelian, Abelian, and monopole forces that arise from th…
In response to recently renewed interests in SU(N) pure-gauge dynamics with large N, both from M/string duality and from finite-temperature QCD phase structure, we calculate string tensions acting between the fundamental 4, diquark 6 and other color charges in SU(4) pure-gauge theory at temperatures below the deconfining phase change and above the bulk phase transition. Our results suggest 4 and 6 representations have different string tensions, with a ratio of about 1.3. We also found the deconfining phase change is not strong.
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.
The critical curve of string tension as a function of the temperature is computed in SU(3) Lattice QCD. We present the results for the string tension utilizing a pair of Polyakov loop and antiloop, with two different techniques. We compare the colour averaged loop-antiloop which is gauge invariant but is only adequate to study the string tension, and the colour singlet loop-antiloop using the Landau gauge fixing which also enables to compute the coulomb part of the free energy.
Wilson loops in the adjoint representation are evaluated on cooled lattices in SU(2) lattice gauge theory. It is found that the string tension of an adjoint Wilson loop vanishes, if the loop is evaluated in a sub-ensemble of configurations in which no center vortex links the loop. This result supports our recent proposal that the adjoint string tension, in the Casimir-scaling regime, can be attributed to a center vortex mechanism.
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.
In pure SU(N) gauge theory in four dimensions, we determine the string tension at large N from smeared rectangular Wilson loops on the lattice. We learn how well loops of sizes barely on the strong-coupling side of the large-N transition in their eigenvalue distribution can be described by effective string theory.
The spatial string tension across a crossover from the low temperature phase to the high temperature phase is computed in QCD with two flavors of non-perturbatively improved Wilson fermions at small lattice spacing a \sim 0.12fm. We find that in the low temperature phase spatial string tension agrees well with zero temperature string tension. Furthermore, it does not show increasing for temperatures up to T = 1.36 T_{pc}, the highest temperature considered. Our results agree with some theoretical predictions.
A lower bound on the string tension for large beta in SU(2) LGT is derived. The derivation is from first principles and bounds the string tension from below by the expectation for the excitation of a single `tagged' thick vortex winding around the lattice. Thus confinement follows if this expectation remains nonvanishing at large beta. Numerical simulations are presented to show that this is indeed the case.
We report a method to select optimal smearing parameters before production runs and discuss the advantages of this selection for the determination of the string tension.
Heavy quark transport coefficients in a strongly coupled Quark-Gluon Plasma can be evaluated using a gauge/string duality and lattice QCD. Via this duality, one can argue that for low momenta the drag coefficient for heavy quarks is proportional to the spatial string tension. Such a tension is well studied on the lattice that allows one to straightforwardly make non-perturbative estimates of the heavy quark diffusion coefficients near the critical point. The obtained results are consistent with those in the literature.
The spatial string tension, a classic non-perturbative probe for the convergence of the weak-coupling expansion at high temperatures, can be determined in full QCD as well as in a dimensionally reduced effective theory. Comparing both approaches, we find surprisingly good agreement almost down to the critical temperature of the deconfinement phase transition.
We study the perfect monopole action as an infrared effective theory of SU(2) QCD. It is transformed exactly into a lattice string model. Since the monopole interactions are weak in the infrared SU(2) QCD, the string interactions become strong. The strong coupling expansion of string model shows the quantum fluctuation is small. The classical string tension is estimated analytically, and we see it is very close to the quantum one in the SU(2) QCD. We also discuss how to calculate the glueball mass in our model.
We discuss the k dependence of the k-string tension sigma_k in SU(N) supersymmetric gluodynamics. As well known, at large N the k-string consists, to leading order, of k noninteracting fundamental strings, so that sigma_k=k sigma_1. We argue, both from field-theory and string-theory side, that subleading corrections to this formula run in powers of 1/N^2 rather than 1/N, thus excluding the Casimir scaling. We suggest a heuristic model allowing one to relate the k-string tension in four-dimensional gluodynamics with the tension of the BPS domain walls (k-walls). In this model the domain walls a…
We compute, for the four-dimensional 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 compatible with sigma_k \propto sin (k pi/N), and show sizeable deviations from Casimir scaling.
We study the properties of the color-Coulomb string tension obtained from the instantaneous part of gluon propagators in Coulomb gauge using quenched SU(3) lattice simulation. In the confinement phase, the dependence of the color-Coulomb string tension on the QCD coupling constant is smaller than that of the Wilson loop string tension. On the other hand, in the deconfinement phase, the color-Coulomb string tension does not vanish even for $T/T_c = 1 \sim 5$, the temperature dependence of which is comparable with the magnetic scaling, dominating the high temperature QCD. Thus, the color-Coulomb…
We present a method for locating center vortices (``fluxons'') in thermalized lattice gauge field configurations. We find evidence, in lattice Monte Carlo simulations, that the asymptotic string tension of fundamental-representation Wilson loops is due to fluctuations in the number of center vortices linking those loops.
We consider whether the 1/N corrections to k-string tensions must begin at order 1/N^2, as in the Sine Law, or whether odd powers of 1/N, as in Casimir Scaling, are also acceptable. The issue is important because different models of confinement differ in their predictions for the representation-dependence of k-string tensions, and corrections involving odd powers of 1/N would seem to be ruled out by the large-N expansion. We show, however, that k-string tensions may, in fact, have leading 1/N corrections, and consistency with the large-N expansion, in the open string sector, is achieved by an …
Global string networks may be relevant in axion production in the early Universe, as well as other cosmological scenarios. Such networks contain a large hierarchy of scales between the string core scale and the Hubble scale, log(f/H) around 70, which influences the network dynamics by giving the strings large tensions T = pi f^2 log(f/H). We present a new numerical approach to simulate such global string networks, capturing the tension without an exponentially large lattice.
We calculate the spatial string tension in (2+1) flavor QCD with physical strange quark mass and almost physical light quark masses using lattices with temporal extent N_tau=4,6 and 8. We compare our results on the spatial string tension with predictions of dimensionally reduced QCD. This suggests that also in the presence of light dynamical quarks dimensional reduction works well down to temperatures 1.5T_c.
We numerically compute the string tension in the large N limit of three dimensional Yang-Mills theory using Wilson loops. Space-time Wilson loops are formed using smeared space-like links and unsmeared time-like links. We use partial reduction and both unfolded and folded Wilson loops in the analysis.
We calculate the low-lying glueball spectrum and various string tensions in SU(N) lattice gauge theories in 2+1 dimensions, and extrapolate the results to the continuum limit. We do so for for the range N=2 to N=16 so as to control the N-dependence with a useful precision. We observe a number of striking near-degeneracies in the various J^PC sectors of the glueball spectrum, in particular between C=+ and C=- states. We calculate the string tensions of flux tubes in a number of representations, and provide evidence that the leading correction to the N-dependence of the k-string tensions varies …
I report on the status of a computation of fundamental and some higher representation string tensions in pure gauge SU(3). An order a^2 tadpole improved action and an anisotropic lattice are used. At present, the static quark potentials and the string tensions are calculated by measuring Wilson loops on an 8^3 x 24 lattice. Wilson loops for higher representations are measured in terms of Wilson loops of the fundamental representation. At the small and intermediate distances available, rough agreement with Casimir scaling is observed, and no color screening for the 8 representation is seen.
We propose a simple mean-field ansatz to study phase transitions from a topological phase to a trivial phase. We probe the efficiency of this approach by considering the string-net model in the presence of a string tension for any anyon theory. Such a perturbation is known to be responsible for a deconfinement-confinement phase transition which is well described by the present variational setup. We argue that mean-field results become exact in the limit of large total quantum dimension.
We investigate the features of ${\rm CP}^{N-1}$ models concerning confinement and topology. In order to study the approach to the large-$N$ asymptotic regime, we determine the topological susceptibility and the string tension for a wide range of values of $N$, in particular $N=4,10,21,41$. Quantitative agreement with the large-$N$ predictions is found for the ${\rm CP}^{20}$ and the ${\rm CP}^{40}$ models. Problems related to the measure of the topological susceptibility and the string tension on the lattice are discussed.
We discuss a new method for testing confinement and measuring the string tension (in the Coulomb gauge). Our numerical simulations demonstrate that the problems related to Gribov copies are not harmful and that the method is effective in the case of pure gauge Q.C.D.. We discuss the relevance of the method for studying gauge theories coupled to fermionic matter.
A center vortex model for the infrared sector of SU(4) Yang-Mills theory is constructed such as to reproduce both the ratio between the zero-temperature quark and diquark string tensions known from lattice Yang-Mills theory, as well as the properties of the deconfinement transition. On this basis, the temperature dependence of the spatial quark and diquark string tensions is predicted. Though still phenomenologically viable, details of the construction of the SU(4) center vortex model corroborate previous arguments that modeling infrared Yang-Mills dynamics purely in terms of vortex world-surf…
Compact U(1) lattice gauge theory in (2+1) dimensions is studied on anisotropic lattices using Standard Path Integral Monte Carlo techniques. We extract the static quark potential and the string tension from 1.0 <= Dtau <= 0.333 simulations at 1.0 <= beta <= 3.0. Estimating the actual value of the renormalization constant, (c = 44), we observe the evidence of scaling in the string tension for 1.4142 <= beta <= 2.5; with the asymptotic behaviour in the large-beta limit given by K sqrt(beta) = e^(-2.494 beta +2.29). Extrapolations are made to the extreme anisotropic or "Hamiltonian" limit, and c…
