Renormalized Polyakov Loop Research Articles

Finite-density QCD transition in a magnetic background field

Using numerical simulations of lattice QCD with physical quark masses, we reveal the influence of magnetic-field background on chiral and deconfinement crossovers in finite-temperature QCD at low baryonic density. In the absence of thermodynamic singularity, we identify these transitions with inflection points of the approximate order parameters: normalized light-quark condensate and renormalized Polyakov loop, respectively. We show that the quadratic curvature of the chiral transition temperature in the ``temperature--chemical potential'' plane depends rather weakly on the strength of the background magnetic field. At weak magnetic fields, the thermal width of the chiral crossover gets narrower as the density of the baryon matter increases, possibly indicating a proximity to a real thermodynamic phase transition. Remarkably, the curvature of the chiral thermal width flips its sign at $eB_{\mathrm{fl}} \simeq 0.6\,\mathrm{GeV}^2$, so that above the flipping point $B > B_{\mathrm{fl}}$, the chiral width gets wider as the baryon density increases. Approximately at the same strength of magnetic field, the chiral and deconfining crossovers merge together at $T \approx 140\,\mathrm{MeV}$. The phase diagram in the parameter space ``temperature-chemical potential-magnetic field'' is outlined, and single-quark entropy and single-quark magnetization are explored. The curvature of the chiral thermal width allows us to estimate an approximate position of the chiral critical endpoint at zero magnetic field: $(T_c^{\text{CEP}}, \mu_B^{\text{CEP}})= (100(25)\, \text{MeV},\ 800(140)\,\text{MeV})$.

Single quark entropy and the Polyakov loop

We study Quantum Chromodynamics (QCD) with 2 + 1 flavors with almost physical quark masses using the highly improved staggered quark action (HISQ). We calculate the Polyakov loop in a wide temperature range, obtain the free energy and the entropy of a single static quark and discuss the QCD crossover region in detail. We show that the entropy has a peak close to the chiral crossover and consider the consequences for the deconfinement aspects of the crossover phenomena. We study the renormalized Polyakov loop susceptibilities and place them into the context of the crossover. We also obtain a quantitative result for the onset of weak-coupling behavior at high temperatures.

Renormalization of the Polyakov loop with gradient flow

We use the gradient flow for the renormalization of the Polyakov loop in various representations. Using 2+1 flavor QCD with highly improved staggered quarks and lattices with temporal extents of $N_\tau=6$, $8$, $10$ and $12$ we calculate the renormalized Polyakov loop in many representations including fundamental, sextet, adjoint, decuplet, 15-plet, 24-plet and 27-plet. This approach allows for the calculations of the renormalized Polyakov loops over a large temperature range from $T=116$ MeV up to $T=815$ MeV, with small errors not only for the Polyakov loop in fundamental representation, but also for the Polyakov loops in higher representations. We compare our results with standard renormalization schemes and discuss the Casimir scaling of the Polyakov loops.

Thermodynamics of large-N gauge theories on a sphere: weak versus strong coupling

Recently lattice simulation in pure Yang-Mills theory exposes significant quadratic corrections for both the thermodynamic quantities and the renormalized Polyakov loop in the deconfined phase. These terms are previously found to appear naturally for ${\mathcal N}=4$ Super Yang-Mills~(SYM) on a sphere at strong coupling, through the gauge/gravity duality. Here we extend the investigation to the weak coupling regime, and for general large-$N$ gauge theories. Employing the matrix model description, we find some novel behavior in the deconfined phase, which is not noticed in the literature. Due to the non-uniform eigenvalue distribution of the holonomy around the time circle, the deviation of the Polyakov loop from one starts from $1/T^3$ instead of $1/T^2$. Such a power is fixed by the space dimension and do not change with different theories. This statement is also true when perturbative corrections to the single-particle partition functions are included. The corrections to the Polyakov loop and higher moments of the distribution function combine to give a universal term, $T/4$, in the free energy. These differences between the weak and strong coupling regime could be easily explained if a strong/weak coupling phase transition occurs in the deconfined phase of large-$N$ gauge theories on a compact manifold.

Quadratic thermal terms in the deconfined phase from holography

Recent lattice simulation has uncovered many interesting properties of SU(N) gauge theory at finite temperature. Especially, above the deconfinement phase transition all the thermodynamics quantities acquire significant quadratic contributions in inverse temperature. Such a term is also found to dominate the logarithmic of the renormalized Polyakov loop. Using the Hawking-Page transition in Anti-de Sitter space as an example, we show how such contributions can be naturally generated in the holographic approach.

Thermal power terms in the Einstein-dilaton system

We employ the gauge/string duality to study the thermal power terms of various thermodynamic quantities in gauge theories and the renormalized Polyakov loop above the deconfinement phase transition. We restrict ourselves to the five-dimensional Einstein gravity coupled to a single scalar, the dilaton. The asymptotic solutions of the system for a general dilaton potential are employed to study the power contributions of various quantities. If the dilaton is dual to the dimension-4 operator ${\rm Tr} F_{\mu\nu}^2$, no power corrections would be generated. Then the thermal quantities approach their asymptotic values much more quickly than those observed in lattice simulation. When the dimension of the dual operator is different from 4, various power terms are generated. The lowest power contributions to the thermal quantities are always quadratic in the dilaton, while that of the Polyakov loop is linear. As a result, the quadratic terms in inverse temperature for both the trace anomaly and the Polyakov loop, observed in lattice simulation, cannot be implemented consistently in the system. This is in accordance with the field theory expectation, where no gauge-invariant operator can accommodate such contributions. Two simple models, where the dilaton is dual to operators with different dimensions, are studied in detail to clarify the conclusion.

Polyakov loop in2+1flavor QCD

We study the temperature dependence of the renormalized Polyakov loop in $2+1$ flavor QCD for temperatures $T<210\text{ }\text{ }\mathrm{MeV}$. 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 noninteracting static-light hadron resonance gas, which describes the temperature dependence of the renormalized Polyakov loop up to $T<140\text{ }\text{ }\mathrm{MeV}$ but fails above that temperature. Furthermore, we discuss the temperature dependence of the light and strange quark condensates.

Towards the phase diagram of dense two-color matter

We study two-color QCD with two flavors of Wilson fermion as a function of quark chemical potential $\ensuremath{\mu}$ and temperature $T$. We find evidence of a superfluid phase at intermediate $\ensuremath{\mu}$ and low $T$ where the quark number density and diquark condensate are both very well described by a Fermi sphere of nearly free quarks disrupted by a BCS condensate. Our results suggest that the quark contribution to the energy density is negative (and balanced by a positive gauge contribution), although this result is highly sensitive to details of the energy renormalization. We also find evidence that the chiral condensate in this region vanishes in the massless limit. This region gives way to a region of deconfined quark matter at higher $T$ and $\ensuremath{\mu}$, with the deconfinement temperature, determined from the renormalized Polyakov loop, decreasing only very slowly with increasing chemical potential. The quark number susceptibility ${\ensuremath{\chi}}_{q}$ does not exhibit any qualitative change at the deconfinement transition. We argue that this is because ${\ensuremath{\chi}}_{q}$ is not an appropriate measure of deconfinement for two-color QCD at high density.

Casimir scaling and renormalization of Polyakov loops in large-N gauge theories

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 qualitatively and quantitatively very similar for the various gauge groups. In particular, close to the deconfinement transition, the renormalized Polyakov loop increases with the temperature, and its logarithm reveals a characteristic dependence on the inverse of the square of the temperature. At higher temperatures, the renormalized Polyakov loop overshoots one, reaches a maximum, and then starts decreasing, in agreement with weak-coupling predictions. The implications of these findings are discussed.

Chiral transition and deconfinement transition in QCD with the highly improved staggered quark (HISQ) action

We report preliminary results on the chiral and deconfinement aspects of the QCD transition at finite temperature using the Highly Improved Staggered Quark (HISQ) action on lattices with temporal extent of N τ = 6 and 8. The chiral aspects of the transition are studied in terms of quark condensates and the disconnected chiral susceptibility. We study the deconfinement transition in terms of the strange quark number susceptibility and the renormalized Polyakov loop. We find reasonably good agreement between our results and the recent continuum extrapolated results obtained with the stout staggered quark action.

Trace anomaly and dimension two gluon condensate above the phase transition

The dimension two gluon condensate has been used previously within a simple phenomenological model to describe power corrections from available lattice data for the renormalized Polyakov loop and the heavy quark-antiquark free energy in the deconfined phase of QCD [1,2]. The QCD trace anomaly of gluodynamics also shows unequivocal inverse temperature power corrections which may be encoded as dimension two gluon condensate. We analyze lattice data of the trace anomaly and compare with other determinations of the condensate from previous references, yielding roughly similar numerical values.

How wide is the transition to deconfinement?

Pure SU(3) glue theories exhibit a deconfining phase transition at a nonzero temperature, Tc. Using lattice measurements of the pressure, we develop a simple matrix model to describe the transition region, when T > Tc. This model, which involves three parameters, is used to compute the behavior of the 't Hooft loop. There is a Higgs phase in this region, where off diagonal color modes are heavy, and diagonal modes are light. Lattice measurements of the latter suggests that the transition region is narrow, extending only to about 1.2 Tc. This is in stark contrast to lattice measurements of the renormalized Polyakov loop, which indicates a much wider width. The possible implications for the differences in heavy ion collisions between RHIC and the LHC are discussed.

Renormalized Polyakov loop in the fixed scale approach

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.

Deconfinement and chiral transition with the highly improved staggered quark (HISQ) action

We report on investigations of the chiral and deconfinement aspects of the finite temperature transition in 2+1 flavor QCD using the Highly Improved Staggered Quark (HISQ) action on lattices with temporal extent Nτ = 6 and Nτ = 8. We have performed the calculations for physical values of the strange quark mass ms and the light quark masses ml = 0.2ms and 0.05ms. Several finite temperature observables, including the renormalized Polyakov loop, the renormalized chiral condensate and the chiral susceptibility have been calculated. We also study the fluctuations and correlations of different conserved charges as well as the trace anomaly at finite temperature. We compare our findings with previous calculations that use different improved staggered fermion formulations: asqtad, p4 and stout.

Equation of state for physical quark masses

We calculate the QCD equation of state for temperatures corresponding to the transition region with physical mass values for two degenerate light quark flavors and a strange quark using an improved staggered fermion action (p4-action) on lattices with temporal extent ${N}_{\ensuremath{\tau}}=8$. We compare our results with previous calculations performed at twice larger values of the light quark masses as well as with results obtained from a resonance gas model calculation. We also discuss the deconfining and chiral aspects of the QCD transition in terms of renormalized Polyakov loop, strangeness fluctuations, and subtracted chiral condensate. We show that compared to the calculations performed at twice larger value of the light quark mass the transition region shifts by about 5 MeV toward smaller temperatures.

Renormalized Polyakov Loop in the Deconfined Phase of SU(N) Gauge Theory and Gauge-String Duality

We use gauge-string duality to analytically evaluate the renormalized Polyakov loop in pure Yang-Mills theories. For SU(3), the result is in quite good agreement with lattice simulations for a broad temperature range.

Suppression of the shear viscosity near the critical temperature in hot QCD

We consider QCD near but above critical temperature T c . The pressure, susceptibilities and the renormalized Polyakov loop — which is an order parameter for the deconfining phase transition — dramatically change up to temperatures a few times T c . We refer to this region as a “semi”-QGP, where partial confinement plays important role. We show that the shear viscosity η is suppressed by two powers of the Polyakov loop. This suggests that η / T 3 decreases markedly as QCD cools down to temperatures near T c . We also show a ratio of the viscosity to the entropy becomes small near T c [Y. Hidaka and R.D. Pisarski, Phase," arXiv:0803.0453 [Phys. Rev. D (to be published)]].

The powers of deconfinement

The trace anomaly of gluodynamics encodes the breakdown of classical scale invariance due to interactions around the deconfinement phase transition. While it is expected that at high temperatures perturbation theory becomes applicable we show that current lattice calculations are far from the perturbative regime and are dominated instead by inverse even power corrections in the temperature, while the total perturbative contribution is estimated to be extremely small and compatible with zero within error bars. We provide an interpretation in terms of dimension-two gluon condensate of the dimensionally reduced theory which value agrees with a similar analysis of power corrections from available lattice data for the renormalized Polyakov loop and the heavy quark-antiquark free energy in the deconfined phase of QCD [E. Megias, E. Ruiz Arriola and L. L. Salcedo, JHEP 0601 (2006) 073; E. Megias, E. Ruiz Arriola and L. L. Salcedo, Phys. Rev. D75 (2007) 105019].

Suppression of the shear viscosity in a “semi”-quark-gluon plasma

We consider QCD at temperatures $T$ near ${T}_{c}$, where the theory deconfines. We distinguish between a ``complete'' quark-gluon plasma (QGP), where the vacuum expectation value of the renormalized Polyakov loop is near unity, essentially constant with $T$, and the ``semi''-QGP, where the loop changes strongly with $T$. Lattice simulations indicate that in QCD, there is a semi-QGP from below ${T}_{c}$ to a few times that. Using a semiclassical model, we compute the shear viscosity, $\ensuremath{\eta}$, to leading order in perturbation theory. We find that near ${T}_{c}$, where the expectation value of the Polyakov loop is small, that $\ensuremath{\eta}/{T}^{3}$ is suppressed by two powers of the loop. For heavy ions, this suggests that during the initial stages of the collision, hydrodynamic behavior at the LHC will be characterized by a shear viscosity which is significantly larger than that at RHIC.

Renormalized Polyakov loops in many representations

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 up to the 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. We present the first results for the nonvanishing renormalized octet loop in the thermodynamic limit below the SU(3) phase transition, and estimate the associated string breaking distance and the gluelump binding energy. By studying the connection of the direct renormalization procedure with a generalization of an earlier suggestion which goes by the name of the QQ renormalization procedure, we find that they are functionally equivalent.