The Polyakov loop potential serves to distinguish between the confined hadronic and the deconfined quark-gluon plasma phases of QCD. For Nf=2+1 quark flavors with physical masses we determine the Polyakov loop potential at finite temperature and density and extract the location of the deconfinement transition. We find a cross-over at small values of the chemical potential running into a critical end-point at mu/T > 1.
We study the Polyakov loop and the correlator of two Polyakov loops at finite temperature in the weak-coupling regime. We calculate the Polyakov loop at order g^4. The calculation of the correlator of two Polyakov loops is performed at distances shorter than the inverse of the temperature and for electric screening masses larger than the Coulomb potential. In this regime, it is accurate up to order g^6. We also evaluate the Polyakov-loop correlator in an effective field theory framework that takes advantage of the hierarchy of energy scales in the problem and makes explicit the bound-state dyn…
A synthesis of Polyakov loop models of the deconfinement transition and quasiparticle models of gluon plasma thermodynamics leads to a class of models in which gluons move in a non-trivial Polyakov loop background. These models are successful candidates for explaining both critical behavior and the equation of state for the SU(3) gauge theory at finite temperature. Two sensitive lattice probes which can discriminate between models are the Polyakov loop P and e-3p.
We study the temperature dependence of the renormalized Polyakov loop in 2+1 flavor QCD for temperatures T<210 MeV. We extend previous calculations by the HotQCD collaboration using the highly improved staggered quark action and perform a continuum extrapolation of the renormalized Polyakov loop. We compare the lattice results with the prediction of non-interacting static-light hadron resonance gas, which describes the temperature dependence of the renormalized Polyakov loop up to T<140 MeV but fails above that temperature. Furthermore, we discuss the temperature dependence of the light and st…
In the heavy, static quark mass regime of QCD, the Polyakov loop is well known to be an order parameter of the deconfinement phase transition; however, the sensitivity of the Polyakov loop to the deconfinement of light, dynamical quarks is less clear. On the other hand, from the perspective of an effective Lagrangian written in the vicinity of the chiral transition, the Polyakov loop is an energy-like operator and should hence scale as any energy-like operator would. We show here that the Polyakov loop and heavy-quark free energy are sensitive to the chiral transition, i.e. their scaling is co…
We investigate correlations of the Polyakov loop fluctuations with eigenmodes of the lattice Dirac operator. Their analytic relations are derived on the temporally odd-number size lattice with the normal non-twisted periodic boundary condition for the link-variables. We find that the low-lying Dirac modes yield negligible contributions to the Polyakov loop fluctuations. This property is confirmed to be valid in confined and deconfined phase by numerical simulations in SU(3) quenched QCD. These results indicate that there is no direct, one-to-one correspondence between confinement and chiral sy…
Unquenching of the Polyakov-loop potential showed to be an important improvement for the description of the phase structure and thermodynamics of strongly-interacting matter at zero quark chemical potentials with Polyakov-loop extended chiral models. This work constitutes the first application of the quark backreaction on the Polyakov-loop potential at nonzero density. The observation is that it links the chiral and deconfinement phase transition also at small temperatures and large quark chemical potentials. The build up of the surface tension in the Polyakov-loop extended Quark-Meson model i…
We present the complete next-to-next-to-leading-order calculation of the correlation function of two Polyakov loops for temperatures smaller than the inverse distance between the loops and larger than the Coulomb potential. We discuss the relationship of this correlator with the singlet and octet potentials which we obtain in an Effective Field Theory framework based on finite-temperature potential Non-Relativistic QCD, showing that the Polyakov loop correlator can be re-expressed, at the leading order in a multipole expansion, as a sum of singlet and octet contributions. We also revisit the c…
We review theoretical aspects of quantum chromodynamics (QCD) at finite temperature. The most important physical variable to characterize hot QCD is the Polyakov loop, which is an approximate order parameter for quark deconfinement in a hot gluonic medium. Additionally to its role as an order parameter, the Polyakov loop has rich physical contents in both perturbative and non-perturbative sectors. This review covers a wide range of subjects associated with the Polyakov loop from topological defects in hot QCD to model building with coupling to the Polyakov loop.
We investigate chiral and deconfinement transitions in the strong coupling lattice QCD for color SU(3). We combine the leading order Polyakov loop effective action of the strong coupling expansion and the next-to-next-to-leading order (1/g^4) fermionic effective action with one species of unrooted staggered fermion. Two approximation schemes are adopted to evaluate the Polyakov loop effects; a Haar measure method (no fluctuation from the mean field) and a Weiss mean-field method (with fluctuations). The Polyakov loop is found to suppress the chiral condensate and to reduce the chiral transitio…
In order to come closer to a realistic model of high-energy collisions, we simulate SU(2) lattice gauge theory under fluctuating temperature. The fluctuations are Euler-Gamma distributed, leading to a canonical state maximizing the Renyi and Tsallis entropy formulas. We test the random lattice spacing method numerically on the Polyakov Loop expectation value. The critical coupling and presumably also the critical deconfinement temperature shifts about 30 per cent to higher values with a realistic degree of fluctuations.
We use gauge/string duality to analytically evaluate the renormalized Polyakov loop in pure Yang-Mills theories. For SU(3), the result is in a quite good agreement with lattice simulations for a broad temperature range.
We investigate chiral and deconfinement transitions in the framework of the strong coupling lattice QCD for color SU(3) with one species of unrooted staggered fermion at finite temperature and quark chemical potential. We take account of the leading order Polyakov loop terms as well as the next-to-next-to-leading order (1/g^4) fermionic terms of the strong coupling expansion in the effective action. We investigate the Polyakov loop effects by comparing two approximation schemes, a Haar measure method (no fluctuation from the mean field) and a Weiss mean-field method (with fluctuations). The ef…
This is a progress report on the calculation of the effective potential for the Polyakov loop in $SU(N)$ pure gauge theory beyond two-loop order. We introduce a new approach using the Poisson resummation formula to compute sum-integrals with the holonomy. We show partial results for the free energy at order $g^3$ and $g^4$.
We investigate theta-vacuum effects on the QCD phase diagram for the realistic 2+1 flavor system, using the three-flavor Polyakov-extended Nambu-Jona-Lasinio (PNJL) model and the entanglement PNJL model as an extension of the PNJL model. The theta-vacuum effects make the chiral transition sharper. For large theta-vacuum angle the chiral transition becomes first order even if the quark number chemical potential is zero, when the entanglement coupling between the chiral condensate and the Polyakov loop is taken into account. We finally propose a way of circumventing the sign problem on lattice Q…
Based on first principle QCD arguments, it has been argued in arXiv:1204.2424[hep-ph] that the vacuum expectation value of the Polyakov loop can be represented in the hadron resonance gas model. We study this within the Polyakov-constituent quark model by implementing the quantum and local nature of the Polyakov loop hep-ph/0412308, hep-ph/0607338. The existence of exotic states in the spectrum is discussed.
We discuss a non-perturbative renormalization of n-point Polyakov loop correlation functions by explicitly introducing a renormalization constant for the Polyakov loop operator on a lattice deduced from the short distance properties of 2-point correlators. We calculate this constant for the SU(3)gauge theory.
We discuss the behavior of two point functions for Polyakov loops in a SU(3) gauge theory about the critical temperature, T_c. From a Z(3) model, in mean field theory we obtain a prediction for the ratio of masses at T_c, extracted from correlation functions for the imaginary and real parts of the Polyakov loop. This ratio is m_i/m_r = 3 if the potential only includes terms up to quartic order in the Polyakov loop; its value changes as pentic and hexatic interactions become important. The Polyakov Loop Model then predicts how m_i/m_r changes above T_c.
We study the Polyakov loop correlator in the weak coupling expansion and show how the perturbative series re-exponentiates into singlet and adjoint contributions. We calculate the order $g^7$ correction to the Polyakov loop correlator in the short distance limit. We show how the singlet and adjoint free energies arising from the re-exponentiation formula of the Polyakov loop correlator are related to the gauge invariant singlet and octet free energies that can be defined in pNRQCD, namely we find that the two definitions agree at leading order in the multipole expansion, but differ at first or…
The dual quark condensate $Σ^{(n)}$ proposed recently as a new order parameter of the spontaneous breaking of the ${\mathbb Z}_3$ symmetry are evaluated by the Polyakov-loop extended Nambu--Jona-Lasinio model, where $n$ are winding numbers. The Polyakov-loop extended Nambu--Jona-Lasinio model well reproduces lattice QCD data on $Σ^{(1)}$ measured very lately. The dual quark condensate $Σ^{(n)}$ at higher temperatures is sensitive to the strength of the vector-type four-quark interaction in the Polyakov-loop extended Nambu--Jona-Lasinio model and hence a good quantity to determine the strength.
Chiral quark models with Polyakov loop at finite temperature have been often used to describe the phase transition. We show how the transition to a hadron resonance gas is realized based on the quantum and local nature of the Polyakov loop.
We explore the NJL model with Polyakov loops for a system of three colors and two flavors within the mean-field approximation, where both chiral symmetry and confinement are taken into account. We focus on the phase structure of the model and study the chiral and Polyakov loop susceptibilities.
While the Polyakov loop is an order parameter of the deconfinement transition in the heavy quark mass regime of QCD, its sensitivity to the deconfinement of light, dynamical quarks in QCD is not apparent. On the other hand, the quark mass dependence of the Polyakov loop is sensitive to the appearance of a chiral phase transition. Using lattice QCD calculations in the staggered fermion discretization scheme at finite values of the lattice spacing, $aT = 1/8$, we show here, for the first time, that the Polyakov loop expectation value, and the heavy quark free energy extracted from it, behave lik…
The phase structure of the two-flavor Polyakov-loop extended Nambu-Jona-Lashinio model is explored at finite temperature and imaginary chemical potential with a particular emphasis on the confinement-deconfinement transition. We point out that the confined phase is characterized by a $\cos3μ_I/T$ dependence of the chiral condensate on the imaginary chemical potential while in the deconfined phase this dependence is given by $\cosμ_I/T$ and accompanied by a cusp structure induced by the Z(3) transition. We demonstrate that the phase structure of the model strongly depends on the choice of the P…
We simulate SU(2) gauge theory at temperatures ranging from slightly below $T_c$ to roughly $2T_c$ for two different values of the gauge coupling. Using a histogram method, we extract the effective potential for the Polyakov loop and for the phases of the eigenvalues of the thermal Wilson loop, in both the fundamental and adjoint representations. We show that the classical potential of the fundamental loop can be parametrized within a simple model which includes a Vandermonde potential and terms linear and quadratic in the Polyakov loop. We discuss how parametrizations for the other cases can …
A synthesis of Polyakov loop models of the deconfinement transition and quasiparticle models of gluon plasma thermodynamics leads to a class of models in which gluon quasiparticles move in a non-trivial Polyakov loop background. These models are successful candidates for explaining both critical behavior and the equation of state for the SU(3) gauge theory at temperatures above the deconfinement temperature T_c. Polyakov loops effects are most important at intermediate temperatures from T_c up to roughly 2.5 T_c, while quasiparticle mass effects provide the dominant correction to blackbody beh…
The Polyakov loop extended Nambu--Jona-Lasinio (PNJL) model with imaginary chemical potential is studied. The model possesses the extended ${\mathbb Z}_{3}$ symmetry that QCD does. Quantities invariant under the extended ${\mathbb Z}_{3}$ symmetry, such as the partition function, the chiral condensate and the modified Polyakov loop, have the Roberge-Weiss (RW) periodicity. The phase diagram of confinement/deconfinement transition derived with the PNJL model is consistent with the RW prediction on it and the results of lattice QCD. The phase diagram of chiral transition is also presented by the…
We calculate the Polyakov loop susceptibilities in the SU(3) lattice gauge theory using the Symanzik improved gauge action on different-sized lattices. The longitudinal and transverse fluctu- ations of the Polyakov loop, as well as, that of its absolute value are considered. We analyze their properties in relation to the confinement-deconfinement phase transition. We also present results based on simulations of (2+1)-flavor QCD on 32^3 x 8 lattice using Highly Improved Staggered Quark (HISQ) action by the HotQCD collaboration. The influences of fermions on the Polyakov loop fluctuations are di…
We study the spatial inhomogeneity of the Polyakov loop induced by inhomogeneous chiral condensates. We formulate an effective model of gluons on the background fields of chiral condensates, and perform its lattice simulation. On the background of inhomogeneous chiral condensates, the Polyakov loop exhibits an in-phase spatial oscillation with the chiral condensates. We also analyze the heavy quark potential and show that the inhomogeneous Polyakov loop indicates the inhomogeneous confinement of heavy quarks.
I compute the deconfinement order parameter for the SU(2) lattice gauge theory, the Polyakov loop, using the fixed scale approach for two different scales and show how one can obtain a physical, renormalized, order parameter. The generalization to other gauge theories, including quenched or full QCD, is straightforward.
We study the interpretation of Lattice data about the thermodynamics of the deconfinement phase of SU(3) Yang-Mills theory, in terms of gluon quasiparticles propagating in a background of a Polyakov loop. A potential for the Polyakov loop, inspired by the strong coupling expansion of the QCD action, is introduced; the Polyakov loop is coupled to tranverse gluon quasiparticles by means of a gas-like effective potential. This study is useful to identify the effective degrees of freedom propagating in the gluon medium above the critical temperature. A main general finding is that a dominant part …
The relation between the deconfinement and chiral phase transition is explored in the framework of an Polyakov-loop-extended two-flavor quark-meson (PQM) model. In this model the Polyakov loop dynamics is represented by a background temporal gauge field which also couples to the quarks. As a novelty an explicit quark chemical potential and N_f-dependence in the Polyakov loop potential is proposed by using renormalization group arguments. The behavior of the Polyakov loop as well as the chiral condensate as function of temperature and quark chemical potential is obtained by minimizing the grand…
We investigate the rapid rise of the dressed Polyakov loop in the Nambu--Jona-Lasinio (NJL) model as a function of temperature. In QCD such a behaviour is interpreted as a confinement- deconfinement phase transition. However, we demonstrate that in the NJL model this is simply a remnant of the chiral transition.
A model for particle production at the deconfining phase transition in QCD is developed, as the semi-classical decay of a condensate for the Polyakov loop. In such a model, generically particle production, as measured on an event-by-event basis, exhibits significant deviations from statistical behavior.
We summarize recent results on the hadron resonance gas description of QCD. In particular, we apply this approach to describe the equation of state and the vacuum expectation value of the Polyakov loop in several representations. Ambiguities related to exactly which states should be included are discussed.
We study Casimir scaling and renormalization properties of Polyakov loops in different irreducible representations in SU(N) gauge theories; in particular, we investigate the approach to the large-N limit, by performing lattice simulations of Yang-Mills theories with an increasing number of colors, from 2 to 6. We consider the twelve lowest irreducible representations for each gauge group, and find strong numerical evidence for nearly perfect Casimir scaling of the bare Polyakov loops in the deconfined phase. Then we discuss the temperature dependence of renormalized loops, which is found to be…
The susceptibilities of the real and imaginary parts, as well as of the modulus of the Polyakov loop, are computed in SU(3) lattice gauge theory. We show that the ratios of these susceptibilities are excellent probes of the deconfinement transition, independent of the renormalization of the Polyakov loop and only weakly dependent on the system size. The ratios are almost temperature independent above and below the transition and exhibit a discontinuity at the transition temperature. This characteristic behavior can be understood in terms of the global Z(3) symmetry of the Yang-Mills Lagrangian…
A non-perturbative method of Field Correlators is applied to calculate the Polyakov loop dependence on temperature, $L(T)$, in the 2+1 flavor QCD with small quark masses, so the only relevant scale is the color-electric string tension $σ(T)$. Polyakov loop in the temperature range 100 MeV $\lesssim T\lesssim$ 400 MeV is expressed via the heavy-light meson mass that decreases with $T$ as $\sqrt{σ(T)}$. The latter is deduced from a gradually vanishing chiral condensate. The resulting $L(T)$ is in good agreement with recent lattice data.
Polyakov loop eigenvalues and their N-dependence are studied in 2 and 4 dimensional SU(N) YM theory. The connected correlation function of the single eigenvalue distributions of two separated Polyakov loops in 2D YM is calculated and is found to have a structure differing from the one of corresponding hermitian random matrix ensembles. No large $N$ non-analyticities are found for two point functions in the confining regime. Suggestions are made for situations in which large-N phase transitions involving Polyakov loops might occur.
We investigate the quark backreaction on the Polyakov loop and its impact on the thermodynamics of quantum chromodynamics. The dynamics of the gluons generating the Polyakov-loop potential is altered by the presence of dynamical quarks. However, this backreaction of the quarks has not yet been taken into account in Polyakov-loop extended model studies. In the present work, we show within a 2+1 flavour Polyakov-quark-meson model that a quark-improved Polyakov-loop potential leads to a smoother transition between the low-temperature hadronic phase and the high-temperature quark-gluon plasma phas…
The deconfinement transition in SU(2) gauge theory and the magnetization transition in the Ising model belong to the same universality class. The critical behaviour of the Ising model can be characterized either as spontaneous breaking of the Z_2 symmetry of spin states or as percolation of appropriately defined spin clusters. We show that deconfinement in SU(2) gauge theory can be specified as percolation of Polyakov loop clusters with Fortuin-Kasteleyn bond weights, leading to the same (Onsager) critical exponents as the conventional order-disorder description based on the Polyakov loop expe…
We present a study of the free energy of parameterized Polyakov loops P in SU(2) and SU(3) lattice gauge theory as a function of the parameters that characterize P. We explore temperatures below and above the deconfinement transition, and for our highest temperatures T > 5 T_c we compare the free energy to perturbative results.
We calculate the next-to-next-to-leading correction to the expectation value of the Polyakov loop or equivalently to the free energy of a static charge. This correction is of order $g^5$. We show that up to this order the free energy of the static charge is proportional to the quadratic Casimir operator of the corresponding representation. We also compare our perturbative result with the most recent lattice results in SU(3) gauge theory.
The order parameter for the pure Yang-Mills phase transition is the Polyakov loop which encodes the symmetries of the Z_N center of the SU(N) gauge group. On the other side the physical degrees of freedom of any asymptotically free gauge theory are hadronic states. Using the Yang-Mills trace anomaly and the exact Z_N symmetry we construct a model able to communicate to the hadrons the information carried by the order parameter.
We compare two renormalization procedures, one based on the short distance behavior of heavy quark-antiquark free energies and the other by using bare Polyakov loops at different temporal extent of the lattice and find that both prescriptions are equivalent, resulting in renormalization constants that depend on the bare coupling. Furthermore these renormalization constants show Casimir scaling for higher representations of the Polyakov loops. The analysis of Polyakov loops in different representations of the color SU(3) group indicates that a simple perturbative inspired relation in terms of t…
We construct a new order parameter for finite temperature QCD by considering the quark condensate for U(1)-valued temporal boundary conditions for the fermions. Fourier transformation with respect to the boundary condition defines the dual condensate. This quantity corresponds to an equivalence class of Polyakov loops, thereby being an order parameter for the center symmetry. We explore the duality relation between the quark condensate and these dressed Polyakov loops numerically, using quenched lattice QCD configurations below and above the QCD phase transition. It is demonstrated that the Di…
In an extension of the Nambu-Jona-Lasinio model where the quarks interact with the temporal gluon field, represented by the Polyakov loop, we explore the relation between the deconfinement and chiral phase transitions. The effect of Polyakov loop dynamics on thermodynamic quantities, on the phase structure at finite temperature and baryon density and on various susceptibilities is presented. Particular emphasis is put on the behavior and properties of the fluctuations of the (approximate) order parameters and their dependence on temperature and net--quark number density. We also discuss how th…
We study the hadron resonance gas model and describe the equation of state of QCD and the vacuum expectation value of the Polyakov loop in the confined phase, in terms of hadronic states with light quarks in the first case, and with exactly one heavy quark in the second case. Comparison with lattice simulations is made.
We study the implications of the spontaneous and explicit Z(3) center symmetry breaking for the Polyakov loop susceptibilities. To this end, ratios of the susceptibilities of the real and imaginary parts, as well as of the modulus of the Polyakov loop are computed within an effective model using a color group integration scheme. We show that the essential features of the lattice QCD results of these ratios can be successfully captured by the effective approach. Furthermore we discuss a novel scaling relation in one of these ratios involving the explicit breaking field, volume, and temperature.
In the Nambu--Jona-Lasinio model with Polyakov loops, we explore the relation between the deconfinement and chiral phase transitions within the mean-field approximation. We focus on the phase structure of the model and study the susceptibilities associated with corresponding order parameters.
We study the thermodynamics of the Nambu--Jona-Lasinio model with Polyakov loops, where spontaneous chiral symmetry breaking and confinement are taken into account. We focus on the phase structure of the model and explore the susceptibilities associated with corresponding order parameters under the mean-field approximation.
The asymptotic behavior of the correlator for Polyakov loop operators separated by a large distance $R$ is determined for high temperature QCD. It is dominated by nonperturbative effects related to the exchange of magnetostatic gluons. To analyze the asymptotic behavior, the problem is formulated in terms of the effective field theory of QCD in 3 space dimensions. The Polyakov loop operator is expanded in terms of local gauge-invariant operators constructed out of the magnetostatic gauge field, with coefficients that can be calculated using resummed perturbation theory. The asymptotic behavior…
We present a renormalization procedure for Polyakov loops which explicitly implements the fact that the renormalization constant depends only on the ultraviolet cutoff. Using this we study the renormalized Polyakov loops in all representations upto the {\bf 27} of the gauge group SU(3). We find good evidence for Casimir scaling of the Polyakov loops and for approximate large-N factorization. By studying many loops together, we are able to show that there is a matrix model with a single coupling which can describe the high temperature phase of QCD, although it is hard to construct explicitly. W…
I give a brief review of the Polyakov Loops Model and tests thereof. I concentrate especially on how in a pure SU(N) gauge theory, Polyakov loops with Z(N) charges two and three affect the effective potential for charge-one loops.
We calculate the colour averaged and colour singlet free energies of static quark anti-quark sources placed in a thermal gluonic heat bath. We discuss the renormalization of these free energies using the short distance properties of the zero temperature heavy quark potential. This leads to the definition of the renormalized Polyakov loop as an order parameter for the deconfinement phase transition of the SU(3) gauge theory which is well behaved in the continuum limit.
We study the free energy of a static quark anti-quark pair at finite temperature in three flavor QCD with degenerate quark masses using N_tau=4 and 6 lattices with Asqtad staggered fermion action. The static free energy was calculated for different values of the quark mass and the entropy contribution at large distances has been extracted. We also calculate the renormalized Polyakov loop following the approach by Kaczmarek et al.
In this work we have re-explored our earlier study on the vector meson spectral function and its spectral property in the form of dilepton rate in a two-flavour Polyakov loop extended Nambu-Jona-Lasinio (PNJL) model in presence of a strong entanglement between the chiral and Polyakov loop dynamics. The entanglement considered here is generated through the four-quark scalar type interaction in which the coupling strength depends on the Polyakov loop and runs with temperature and chemical potential. The entanglement effect is also considered for the four-quark vector type interaction in the same…
We show that in four-dimensional gauge theories dual to five-dimensional Einstein gravity coupled to a single scalar field in the bulk the derivative of the single heavy quark free energy in the deconfined phase is $dF_{Q}(T)/dT \sim -1/c_s^2(T)$, where $c_s(T)$ is the speed of sound. This general result provides a direct link between the softest point in the equation of state of strongly-coupled plasmas and the deconfinement phase transition described by the expectation value of the Polyakov loop. We give an explicit example of a gravity dual with black hole solutions that can reproduce the l…
