Polyakov Loop Research Articles

Thermodynamic Potential of the Polyakov Loop in SU(3) Quenched Lattice QCD

Thermodynamic Potential of the Polyakov Loop in SU(3) Quenched Lattice QCD

Pseudoscalar mesons from a PNJL model at zero temperature

We study pseudoscalar π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document}, K and η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta $$\\end{document} meson properties, such as masses and couplings, in dense matter at zero temperature. We use a recently proposed phenomenological quark model, known as the PNJL0, which takes into account the confinement/deconfinement phase transition by means of the traced Polyakov loop (Φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varPhi $$\\end{document}) which serves as an order parameter at zero temperature. We consider two different scenarios, namely, symmetric quark matter with equal chemical potentials for all the flavors, and the beta equilibrated matter. In the latter case the hadron-quark phase transition is implemented by a two model approach. For the hadron side we use a relativistic mean-field model with density dependent couplings. We show that Φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varPhi $$\\end{document} induces abrupt changes in the mesons properties with gap sizes regulated by the phenomenological gluonic sector of the model.

Thermal effects on sound velocity peak and conformality in isospin QCD

We study thermal effects on equations of state (EOS) in isospin QCD, utilizing a quark-meson model coupled to a Polyakov loop. The quark-meson model is analyzed at one-loop that is the minimal order to include quark substructure constraints on pions which condense at finite isospin density. In the previous study we showed that the quark-meson model at zero temperature produces the sound velocity peak and the negative trace anomaly in the domain between the chiral effective theory regime at low density and the perturbative QCD regime at high density, in reasonable agreement with lattice simulations. We now include thermal effects from quarks in the Polyakov loop background and examine EOS, especially the sound velocity and trace anomaly along isentropic trajectories. At large isospin density, there are three temperature windows; (i) the pion condensed region with almost vanishing Polyakov loops, (ii) the pion condensed region with finite Polyakov loops, and (iii) the quark gas without pion condensates. In the domain (i), the gap associated with the pion condensate strongly quenches thermal excitations. As the system approaches the domain (ii), thermal quarks, which behave as nonrelativistic particles, add energy density but little pressure, substantially reducing the sound velocity to the value less than the conformal value while increasing the trace anomaly toward the positive value. Approaching the domain (iii), thermal quarks become more relativistic as pion condensates melt, increasing sound velocity toward the conformal limit. Corrections from thermal pions are also briefly discussed. Published by the American Physical Society 2024

Split of the pseudocritical temperatures of chiral and confinement-deconfinement transitions by a temperature gradient

Searching of the critical endpoint (CEP) of the phase transition of quantum chromodynamics (QCD) matter in experiments is of great interest. The temperature in the fireball at a collider is location dependent, however, most theoretical studies address the scenario of uniform temperature. In this work, the effect of temperature gradients is investigated using lattice QCD approach. We find that the temperature gradient catalyzes chiral symmetry breaking, meanwhile the temperature gradient increases the Polyakov loop in the confined phase but suppresses the Polyakov loop in the deconfined phase. Furthermore, the temperature gradient decreases the pseudocritical temperature of chiral transition but increases the pseudocritical temperature of the confine-deconfine transition. It is also found that the temperature gradient can drive the system away from the singularity, implying that the CEP in the T−μ plane may move in the direction of the first-order phase transition in the phase diagram due to the temperature gradient. Published by the American Physical Society 2024

Reduction of pseudocritical temperatures of chiral restoration and deconfinement phase transitions in a magnetized PNJL model

We investigate the chiral restoration and deconfinement phase transitions under an external magnetic field in the frame of a Pauli-Villars regularized Polyakov loop extended Nambu-Jona-Lasinio model. A running Polyakov loop scale parameter T0(eB) is introduced to mimic the reaction of the gluon sector to the presence of magnetic fields. It is found that a decreasing T0(eB) with magnetic fields can realize the inverse magnetic catalysis phenomena of chiral condensates of u and d quarks, increase of Polyakov loop, and the reduction of pseudocritical temperatures of chiral restoration and deconfinement phase transitions. Published by the American Physical Society 2024

Medium induced mixing, spatial modulations, and critical modes in QCD

The mixing between the chiral condensate and the density in hot and dense quantum chromodynamics (QCD) matter is familiar. We show that the mixing relevant for the ground state is considerably more extensive, and in particular also involves gluonic degrees of freedom. As a result, the Hessian of the QCD effective action is non-Hermitian, but retains a symmetry under combined charge and complex conjugation. This can lead to complex-conjugate pairs of eigenvalues of this Hessian, signaling regimes with spatially modulated correlations. Furthermore, based on the analytic structure of the quark determinant at a chiral critical point, we demonstrate that the corresponding massless critical mode is composed of the chiral condensate, the density and the Polyakov loops. Due to an avoided crossing, the critical mode turns out to be disconnected from the chiral condensate in vacuum. We present general arguments for all these features and illustrate them through explicit model calculations. Published by the American Physical Society 2024

A Scaling Relation, Zm-Type Deconfinement Phases, and Imaginary Chemical Potentials in Finite Temperature Large-N Gauge Theories

Abstract We show that the effective potentials for the Polyakov loops in finite temperature SU$(N)$ gauge theories obey a certain scaling relation with respect to temperature in the large-N limit. This scaling relation strongly constrains the possible terms in the Polyakov loop effective potentials. Moreover, by using the effective potentials in the presence of imaginary chemical potentials or imaginary angular velocities in several models, we find that phase transitions to $Z_m$-type deconfinement phases ($Z_m$ phase) occur, where the eigenvalues of the Polyakov loop are distributed $Z_m$ symmetrically. Physical quantities in the $Z_m$ phase obey the scaling properties of the effective potential. The models include Yang–Mills (YM) theories, the bosonic BFSS matrix model, and ${\mathcal {N}}=4$ supersymmetric YM theory on $S^3$. Thus, the phase diagrams of large-N gauge theories with imaginary chemical potentials are very rich and the stable $Z_m$ phase would be ubiquitous. Monte-Carlo calculations also support this. As a related topic, we discuss the phase diagrams of large-N YM theories with real angular velocities in finite volume spaces.

Fluctuations and Correlations of Conserved Charges Serving as Signals for QGP Production: An Overview from Polyakov Loop Enhanced Nambu–Jona-Lasinio Model

Quark–Gluon plasma driven by the strong force is subject to the conservativeness of the baryon number, net electric charge, strangeness, etc. However, the fluctuations around their mean values at specific temperatures and chemical potentials can provide viable signals for the production of Quark–Gluon plasma. These fluctuations can be captured theoretically as moments of different orders in the expansion of pressure or the thermodynamic potential of the system under concern. Here, we look for possible explanations in the methodologies used for capturing them by using the framework of the Polyakov–Nambu–Jona-Lasinio (PNJL) model under the 2 + 1 flavor consideration with mean-field approximation. The various quantities thus explored can act to signify meaningfully near the phase transitions. Justifications are also made for some of the quantities capable of serving necessarily under experimental scenarios. Additionally, variations in certain quantities are also made for the different collision energies explored in the high-energy experiments. Rectification of the quantitative accuracy, especially in the low-temperature hadronic sector, is of prime concern, and it is also addressed. It was found that most of the observables stay in close proximity with the existing lattice QCD results at the continuum limit, with some artifacts still remaining, especially in the strange sector, which needs further attention.

Study of the effects of external imaginary electric field and chiral chemical potential on quark matter

The behavior of quark matter with both external electric field and chiral chemical potential is theoretically and experimentally interesting to consider. In this paper, the case of simultaneous presence of imaginary electric field and chiral chemical potential is investigated using the lattice QCD approach with Nf=1+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_f=1+1$$\\end{document} dynamical staggered fermions. We find that overall both the imaginary electric field and the chiral chemical potential can exacerbate chiral symmetry breaking, which is consistent with theoretical predictions. However we also find a non-monotonic behavior of chiral condensation at specific electric field strengths and chiral chemical potentials. The behavior of Polyakov loop in the complex plane is not significantly affected by chiral chemical potential in the region of the parameters considered.

Wilson loops and random matrices

Linear confinement with Casimir scaling of the string tension in confining gauge theories is a consequence of a certain property of the Polyakov loop related to random matrices. This mechanism does not depend on the details of the theories (neither the gauge group nor dimensions) and explains approximate Casimir scaling below string-breaking length. In this paper, we study 3d SU(2) pure Yang-Mills theory numerically and find the same random-matrix behavior for rectangular Wilson loops. We conjecture that this is a universal feature of strongly coupled confining gauge theories.

Investigating two-dimensional adjoint QCD on the lattice

We present our investigations of SU(N) adjoint QCD in two dimensions with one Majorana fermion on the lattice. We determine the relevant parameter range for the simulations with Wilson fermions and present results for Polyakov loop, chiral condensate, and string tension. In the theory with massive fermions, all observables we checked show qualitative agreement between numerical lattice data and theory, while the massless limit is more subtle since chiral and non-invertible symmetry of the continuum theory are explicitly broken by lattice regularization. In thermal compactification, we observe N perturbative vacua for the holonomy potential at high-T with instanton events connecting them, and a unique vacuum at low-T. At finite-N, this is a cross-over and it turns to a phase transition at large-N thermodynamic limit. In circle compactification with periodic boundary conditions, we observe a unique center-symmetric minimum at any radius. In continuum, the instantons in the thermal case carry zero modes (for even N) and indeed, in the lattice simulations, we observe that chiral condensate is dominated by instanton centers, where zero modes are localized. We present lattice results on the issue of confinement vs. screening in the theory and comment on the roles of chiral symmetry and non-invertible symmetry.

A Bridge between Trace Anomalies and Deconfinement Phase Transitions

Inspired by the fact that both the dilaton potential encoding the trace anomalies of QCD and the Polyakov loop potential measuring the deconfinement phase transition can be expressed in the logarithmic forms, as well as the fact that the scale symmetry is expected to be restoring and colors are deconfined in extreme conditions such as high temperatures and/or densities, we conjecture a relation between the dilaton potential and the Polyakov loop potential. Explicitly, we start from the Coleman–Weinberg type potential of a real scalar field—a dilaton or conformal compensator—and make an ansatz of the relation between this scalar field and the Polyakov loop to obtain the Polyakov loop potential, which can be parameterized in Lattice QCD (LQCD) in the pure glue sector. We find that the coefficients of Polyakov potential fitted from Lattice data are automatically satisfied in this ansatz, the locations of deconfinement and scale restoration are locked to each other, and the first-order phase transition can be realized. Extensions to the low-energy effective quark models are also discussed. The conjectured relation may deepen our understanding of the evolution of the universe, the mechanism of electroweak symmetry breaking, the phase diagram of QCD matter, and the properties of neutron stars.

Order parameters for gauge invariant condensation far from equilibrium

Nuclear collisions at sufficiently high energies are expected to produce far-from-equilibrium matter with a high density of gluons at early times. We show gauge condensation, which occurs as a consequence of the large density of gluons. To identify this condensation phenomenon, we construct two local gauge invariant observables that carry the macroscopic zero mode of the gauge condensate. The first order parameter for gauge condensation investigated here is the correlator of the spatial Polyakov loop. We also consider, for the first time, the correlator of the gauge invariant scalar field, associated with the exponent of the Polyakov loop. Using real-time lattice simulations of classical-statistical SU(2) gauge theory, we find gauge condensation on a system-size-dependent timescale tcond∼L1/ζ with a universal scaling exponent ζ. Furthermore, we suggest an effective theory formulation describing the dynamics using one of the order parameters identified. The formation of a condensate at early times may have intriguing implications for the early stages in heavy-ion collisions. Published by the American Physical Society 2024

Effects of tensor spin polarization on the chiral restoration and deconfinement phase transitions

Effects of tensor spin polarization (TSP) on the chiral restoration and deconfinement phase transitions are studied in Polyakov loop extended Nambu–Jona-Lasinio (PNJL) model. For chiral phase transition, the higher the polarized degree of quark-antiquark pairs under the strong magnetic field, the higher the phase transition temperature. The TSP corrects the position of the critical end point. The small impact of TSP on the phase transition temperature is found for the deconfinement phase transition. On the other hand, we divide the phase space into three ranges based on the phase diagram obtained from the PNJL model: the confinement phase with chiral symmetry broken, the deconfinement phase with restored chiral symmetry, and the confinement phase with restored chiral symmetry (quarkyonic phase). It is found that TSP has only a very small effect on the anisotropic pressure in the deconfined phase with chiral symmetry restored and the quarkyonic phase, but it has a very strong effect on the anisotropic pressure in the confined phase with chiral symmetry broken. This is because TSP is closely related to chiral symmetry. The restoration of chiral symmetry means the dissociation of spin polarization condensate. Published by the American Physical Society 2024

Deconfinement, center symmetry and the ghost propagator in Landau gauge pure SU(3) Yang-Mills theory

The temperature dependence of the Landau gauge ghost propagator is investigated in pure SU(3) Yang-Mills theory with lattice QCD simulations. Its behavior around the confined-deconfined phase transition temperature, Tc ∼ 270 MeV, is investigated. The simulations show that in the deconfined phase, the ghost propagator is enhanced for small momenta, ≲ 1 GeV. Furthermore, the analysis of the spontaneous breaking of center symmetry on the ghost propagator is studied. Similarly as observed for the gluon propagator, the simulations result in a decoupling of the sectors where the phase of the Polyakov loop is either 0 or ±2π/3 sectors, with the latter remaining indistinguishable. The results point to the possible use of the ghost propagator as an “order parameter” for the confined-deconfined phase transition.

Polyakov loop models in the large N limit: Correlation function and screening masses

We explore the ’t Hooft-Veneziano limit of the Polyakov loop models at finite baryon chemical potential. Using methods developed by us earlier we calculate the two- and N-point correlation functions of the Polyakov loops. This gives a possibility to compute the various potentials in the confinement phase and to derive the screening masses outside the confinement region. In particular, we establish the existence of complex masses and an oscillating decay of correlations in a certain range of parameters. Furthermore, it is shown that the calculation of the N-point correlation function in the confinement phase reduces to the geometric median problem. This leads to a large N analog of the Y law for the baryon potential. Published by the American Physical Society 2024

A New Perspective on Thermal Transition in QCD

Abstract Motivated by the picture of partial deconfinement developed in recent years for large-N gauge theories, we propose a new way of analyzing and understanding thermal phase tsuppransition in QCD. We find nontrivial support for our proposal by analyzing the WHOT-QCD collaboration’s lattice configurations for SU(3) QCD in 3 + 1 spacetime dimensions with up, down, and strange quarks. We find that the Polyakov line (the holonomy matrix around a thermal time circle) is governed by the Haar-random distribution at low temperatures. The deviation from the Haar-random distribution at higher temperatures can be measured via the character expansion, or equivalently, via the expectation values of the Polyakov loop defined by the various nontrivial representations of SU(3). We find that the Polyakov loop corresponding to the fundamental representation and loops in the higher representation condense at different temperatures. This suggests that there are three phases, one intermediate phase existing in between the completely-confined and the completely-deconfined phases. Our identification of the intermediate phase is supported also by the condensation of instantons: by studying the instanton numbers of the WHOT-QCD configurations, we find that the instanton condensation occurs for temperature regimes corresponding to what we identify as the completely-confined and intermediate phases, whereas the instantons do not condense in the completely-deconfined phase. Our characterization of confinement based on the Haar-randomness explains why the Polyakov loop is a good observable to distinguish the confinement and the deconfinement phases in QCD despite the absence of the $\mathbb {Z}_3$ center symmetry.

Color confinement and random matrices. A random walk down group manifold toward Casimir scaling

We explain the microscopic origin of linear confinement potential with the Casimir scaling in generic confining gauge theories. In the low-temperature regime of confining gauge theories such as QCD, Polyakov lines are slowly varying Haar random modulo exponentially small corrections with respect to the inverse temperature, as shown by one of the authors (M. H.) and Watanabe. With exact Haar randomness, computation of the two-point correlator of Polyakov loops reduces to the problem of random walk on group manifold. Linear confinement potential with approximate Casimir scaling except at short distances follows naturally from slowly varying Haar randomness. With exponentially small corrections to Haar randomness, string breaking and loss of Casimir scaling at long distance follow. Hence we obtain the Casimir scaling which is only approximate and holds only at intermediate distance, which is precisely needed to explain the results of lattice simulations. For (1 + 1)-dimensional theories, there is a simplification that admits the Casimir scaling at short distances as well.

On Thermal Transition in QCD

Abstract We describe how the general mechanism of partial deconfinement applies to large-N QCD and a partially deconfined phase inevitably appears between completely confined and completely deconfined phases. Furthermore, we propose how partial deconfinement can be observed in real-world QCD with the SU(3) gauge group. For this purpose, we employ lattice configurations obtained by the WHOT-QCD Collaboration and examine our proposal numerically. In the discussion, the Polyakov loop plays a crucial role in characterizing the phases, without relying on center symmetry, and hence we clarify the meaning of the Polyakov loop in QCD at large N and finite N. At both large N and finite N, the complete confinement is characterized by the Haar-random distribution of the Polyakov line phases. Haar-randomness, which is stronger than unbroken center symmetry, indicates that Polyakov loops in any nontrivial representations have vanishing expectation values, and deviation from the Haar-random distribution at higher temperatures is quantified with the loops. We discuss that the transitions separating the partially deconfined phase are characterized by the behaviors of Polyakov loops in various representations. The lattice QCD data provide us with the signals exhibiting two different characteristic temperatures: deconfinement of the fundamental representation and deconfinement of higher representations. As a nontrivial test for our proposal, we also investigate the relation between partial deconfinement and instanton condensation and confirm the consistency with the lattice data. To make the presentation more easily accessible, we provide a detailed review of the previously known aspects of partial deconfinement.

Gravitational waves from composite dark sectors

We study under which conditions a first-order phase transition in a composite dark sector can yield an observable stochastic gravitational-wave signal. To this end, we employ the Linear-Sigma model featuring Nf = 3, 4, 5 flavours and perform a Cornwall-Jackiw-Tomboulis computation also accounting for the effects of the Polyakov loop. The model allows us to investigate the chiral phase transition in regimes that can mimic QCD-like theories incorporating in addition composite dynamics associated with the effects of confinement-deconfinement phase transition. A further benefit of this approach is that it allows to study the limit in which the effective interactions are weak. We show that strong first-order phase transitions occur for weak effective couplings of the composite sector leading to gravitational-wave signals potentially detectable at future experimental facilities.