QCD Critical End Point Research Articles

QCD critical end point from a realistic PNJL model

With parameters fixed by critical temperature and equation of state at zero baryon chemical potential, a realistic Polyakov-Nambu-Jona-Lasinio (rPNJL) model predicts a critical end point of chiral phase transition at (μEB = 720MeV; TE = 93MeV). The extracted freeze-out line from heavy ion collisions is close to the chiral phase transition boundary in the rPNJL model, and the kurtosis kσ2 of baryon number fluctuations from the rPNJL model along the experimental freeze-out line agrees well with the BES-I measurement. Our analysis shows that the dip structure of measured kσ2 is determined by the relationship between the freeze-out line and chiral phase transition line at low baryon density region, and the peak structure can be regarded as a clean signature for the existence of CEP.

Locating the QCD critical end point through peaked baryon number susceptibilities along the freeze-out line**Supported by NSFC (11275213, and 11261130311) (CRC 110 by DFG and NSFC), CAS key project KJCX2-EW-N01, and Youth Innovation Promotion Association of CAS

We investigate the baryon number susceptibilities up to fourth order along different freeze-out lines in a holographic QCD model with a critical end point (CEP), and we propose that the peaked baryon number susceptibilities along the freeze-out line can be used as a clean signature to locate the CEP in the QCD phase diagram. On the temperature and baryon chemical potential plane, the cumulant ratio of the baryon number susceptibilities (up to fourth order) forms a ridge along the phase boundary, and develops a sword-shaped “mountain” standing upright around the CEP in a narrow and oblate region. The measurement of baryon number susceptibilities from heavy-ion collision experiments is along the freeze-out line. If the freeze-out line crosses the foot of the CEP mountain, then one can observe the peaked baryon number susceptibilities along the freeze-out line, and the kurtosis of the baryon number distributions has the highest magnitude. The data from the first phase of the beam energy scan program at the Relativistic Heavy Ion Collider indicates that there should be a peak of the kurtosis of the baryon number distribution at a collision energy of around 5 GeV, which suggests that the freeze-out line crosses the foot of the CEP mountain and the summit of the CEP should be located nearby, around a collision energy of 3–7 GeV.

Multiplicity fluctuations near the QCD critical point

Statistical moments of particle multiplicities in heavy-ion collision experiments are an important probe in the exploration of the phase diagram of strongly interacting matter and, particularly, in the search for the QCD critical end point. In order to appropriately interpret experimental measures of these moments, however, it is necessary to understand the role of experimental limitations, as well as background contributions, providing expectations on how critical behavior should be affected by them. We here present a framework for calculating moments of particle multiplicities in the presence of correlations of both critical and spurious origins. We also include effects from resonance decay and a limited acceptance window, as well as detector efficiency. Although we focus on second-order moments, for simplicity, an extension to higher-order moments is straightforward.

Isentropic thermodynamics and scalar-mesons properties near the QCD critical end point

We investigate the QCD phase diagram and the location of the critical end point (CEP) in the SU(2) Polyakov$-$Nambu$-$Jona-Lasinio model with entanglement interaction giving special attention to the $\pi$ and $\sigma$-mesons properties, namely the decay widths $\sigma\rightarrow\pi\pi$, for several conditions around the CEP: we focus on the possible $\sigma\rightarrow\pi\pi$ decay along the isentropic trajectories close to the CEP since the hydrodynamical expansion of a heavy-ion collision fireball nearly follows trajectories of constant entropy. It is expected that the type of transition the dense medium goes through as it expands after the thermalization determines the behavior of this decay. It is shown that no pions are produced from the sigma decay in the chirally symmetric phase if the isentropic lines approach the first order line from chemical potentials above it. Near the CEP or above the $\sigma\rightarrow\pi\pi$ decay is possible with a high decay width.

Chern-Simons diffusion rate across different phase transitions

We investigate how the dimensionless ratio given by the Chern-Simons diffusion rate ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{CS}}$ divided by the product of the entropy density $s$ and temperature $T$ behaves across different kinds of phase transitions in the class of bottom-up nonconformal Einstein-dilaton holographic models originally proposed by Gubser and Nellore. By tuning the dilaton potential, one is able to holographically mimic a first order, a second order, or a crossover transition. In a first order phase transition, ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{CS}}/sT$ jumps at the critical temperature (as previously found in the holographic literature), while in a second order phase transition it develops an infinite slope. On the other hand, in a crossover, ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{CS}}/sT$ behaves smoothly, although displaying a fast variation around the pseudo-critical temperature. In all the cases, ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{CS}}/sT$ increases with decreasing $T$. The behavior of the Chern-Simons diffusion rate across different phase transitions is expected to play a relevant role for the chiral magnetic effect around the QCD critical end point, which is a second order phase transition point connecting a crossover band to a line of first order phase transition. Our findings in the present work add to the literature the first predictions for the Chern-Simons diffusion rate across second order and crossover transitions in strongly coupled nonconformal, non-Abelian gauge theories.

Baryon effects on the location of QCD’s critical end point

The location of the critical end point of QCD has been determined in previous studies of ${N}_{f}=2+1$ and ${N}_{f}=2+1+1$ dynamical quark flavors using a (truncated) set of Dyson-Schwinger equations for the quark and gluon propagators of Landau-gauge QCD. A source for systematic errors in these calculations has been the omission of terms in the quark-gluon interaction that can be parametrized in terms of baryonic degrees of freedom. These have a potentially large dependence on chemical potential and therefore may affect the location of the critical end point. In this exploratory study we estimate the effects of these contributions, both in the vacuum and at finite temperature and chemical potential. We find only a small influence of baryonic contributions on the location of the critical end point. We estimate the robustness of this result by parameterizing further dependencies on chemical potential.

Susceptibilities and critical exponents within the Nambu–Jona-Lasinio model

In the mean field approximation of (2 + 1)-flavor Nambu–Jona-Lasinio model, we strictly derive several sets of coupled equations for the chiral susceptibility, the quark number susceptibility, etc. at finite temperature and quark chemical potential. The critical exponents of these susceptibilities in the vicinity of the QCD critical end point (CEP) are presented in SU(2) and SU(3) cases, respectively. It is found that these various susceptibilities share almost the same critical behavior near the CEP. The comparisons between the critical exponents for the order parameters and the theoretical predictions are also included.

Critical behaviors near the (tri-)critical end point of QCD within the NJL model

We investigate the dynamical chiral symmetry breaking and its restoration at finite density and temperature within the two-flavor Nambu-Jona-Lasinio model, and mainly focus on the critical behaviors near the critical end point (CEP) and tricritical point (TCP) of quantum chromodynamics. The multi-solution region of the Nambu and Wigner ones is determined in the phase diagram for the massive and massless current quark, respectively. We use the various susceptibilities to locate the CEP/TCP and then extract the critical exponents near them. Our calculations reveal that the various susceptibilities share the same critical behaviors for the physical current quark mass, while they show different features in the chiral limit.

Regularization dependence on phase diagram in Nambu–Jona-Lasinio model

We study the regularization dependence on meson properties and the phase diagram of quark matter by using the two flavor Nambu–Jona-Lasinio model. The model also has the parameter dependence in each regularization, so we explicitly give the model parameters for some sets of the input observables, then investigate its effect on the phase diagram. We find that the location or the existence of the critical end point highly depends on the regularization methods and the model parameters. Then we think that regularization and parameters are carefully considered when one investigates the QCD critical end point in the effective model studies.

Locating the critical end point of QCD

We summarize recent results for the phase structure of QCD at finite temperature and light-quark chemical potential for N_f=2+1 and N_f=2+1+1 dynamical quark flavors. We discuss order parameters for the chiral and deconfinement transitions obtained from solutions of a coupled set of (truncated) Dyson-Schwinger equations for the quark and gluon propagators of Landau gauge QCD. Based on excellent agreement with results from lattice-QCD at zero chemical potential we study the possible appearance of a critical end-point at large chemical potential.

Locate QCD critical end point in a continuum model study

With a modified chemical potential dependent effective model for the gluon propagator, we try to locate the critical end point (CEP) of strongly interacting matter in the framework of Dyson-Schwinger equations (DSE). Beyond the chiral limit, we find that Nambu solution and Wigner solution could coexist in some area. Using the CornwallJackiw-Tomboulis (CJT) effective action, we show that these two phases are connected by a first order phase transition. We then locate CEP as the end point of the first order phase transition line. Meanwhile, based on CJT effective action, we give a direct calculation for the chiral susceptibility and thereby study the crossover.

Equilibrium statistical–thermal models in high-energy physics

We review some recent highlights from the applications of statistical–thermal models to different experimental measurements and lattice QCD thermodynamics that have been made during the last decade. We start with a short review of the historical milestones on the path of constructing statistical–thermal models for heavy-ion physics. We discovered that Heinz Koppe formulated in 1948, an almost complete recipe for the statistical–thermal models. In 1950, Enrico Fermi generalized this statistical approach, in which he started with a general cross-section formula and inserted into it, the simplifying assumptions about the matrix element of the interaction process that likely reflects many features of the high-energy reactions dominated by density in the phase space of final states. In 1964, Hagedorn systematically analyzed the high-energy phenomena using all tools of statistical physics and introduced the concept of limiting temperature based on the statistical bootstrap model. It turns to be quite often that many-particle systems can be studied with the help of statistical–thermal methods. The analysis of yield multiplicities in high-energy collisions gives an overwhelming evidence for the chemical equilibrium in the final state. The strange particles might be an exception, as they are suppressed at lower beam energies. However, their relative yields fulfill statistical equilibrium, as well. We review the equilibrium statistical–thermal models for particle production, fluctuations and collective flow in heavy-ion experiments. We also review their reproduction of the lattice QCD thermodynamics at vanishing and finite chemical potential. During the last decade, five conditions have been suggested to describe the universal behavior of the chemical freeze-out parameters. The higher order moments of multiplicity have been discussed. They offer deep insights about particle production and to critical fluctuations. Therefore, we use them to describe the freeze-out parameters and suggest the location of the QCD critical endpoint. Various extensions have been proposed in order to take into consideration the possible deviations of the ideal hadron gas. We highlight various types of interactions, dissipative properties and location-dependences (spatial rapidity). Furthermore, we review three models combining hadronic with partonic phases; quasi-particle model, linear sigma model with Polyakov potentials and compressible bag model.

Exploring a possible origin of the QCD critical endpoint

We discuss a new way to develop the exactly solvable model of the QCD critical endpoint by matching the deconfinement phase transition line for the system of quark-gluon bags with the line of their vanishing surface tension coefficient. In contrast to all previous findings in such models the deconfined phase is defined not by an essential singularity of the isobaric partition function, but by its simple pole. As a result we find out that the first order deconfinement phase transition which is defined by a discontinuity of the first derivative of system pressure is generated by a discontinuity of the derivative of surface tension coefficient.

On the Higher Moments of Particle Multiplicity, Chemical Freeze-Out, and QCD Critical Endpoint

We calculate the first six nonnormalized moments of particle multiplicity within the framework of the hadron resonance gas model. In terms of the lower order moments and corresponding correlation functions, general expressions of higher order moments are derived. Thermal evolution of the first four normalized moments and their products (ratios) are studied at different chemical potentials, so that it is possible to evaluate them at chemical freeze-out curve. It is found that a nonmonotonic behaviour reflecting the dynamical fluctuation and strong correlation of particles starts to appear from the normalized third order moment. We introduce novel conditions for describing the chemical freeze-out curve. Although the hadron resonance gas model does not contain any information on the criticality related to the chiral dynamics and singularity in the physical observables, we are able to find out the location of the QCD critical endpoint atμ~350 MeV and temperatureT~162 MeV.

Finite-size scaling as a tool for the search of the critical endpoint of QCD in heavy-ion data

We briefly discuss the role played by the finiteness of the system created in high-energy heavyion collisions (HIC’s) in the experimental search of the QCD critical endpoint and, in particular, the applicability of the predicting power of finite-size scaling plots in data analysis of current HIC’s.

Physical mechanism of the (tri)critical point generation

We discuss some ideas resulting from a phenomenological relation recently declared between the tension of string connecting the static quark-antiquark pair and surface tension of corresponding cylindrical bag. This relation analysis leads to the temperature of vanishing surface tension coefficient of the QGP bags at zero baryonic charge density as T_\sigma = 152.9 +- 4.5 MeV. We develop the view point that this temperature value is not a fortuitous coincidence with the temperature of (partial) chiral symmetry restoration as seen in the lattice QCD simulations. Besides, we argue that T_\sigma defines the QCD (tri)critical endpoint temperature and claim that a negative value of surface tension coefficient recently discovered is not a sole result, but should also exist in ordinary liquids at the supercritical temperatures.

Physics around the QCD (tri)critical endpoint and new challenges for femtoscopy

On the basis of exactly solvable models with the tricritical and critical endpoints I discuss the physical mechanism of endpoints formation which is similar to the usual liquids. It is demonstrated that the necessary condition for the transformation of the 1-st order deconfinement phase transition into the 2-nd order phase transition at the (tri)critical endpoint is the vanishing of surface tension coefficient of large/heavy QGP bags. Using the novel model of the confinement phenomenon I argue that the physical reason for the cross-over appearance at low baryonic densities is the negative value of QGP bag surface tension coefficient. This implies the existence of highly non-spherical or, probably, even fractal surfaces of large and heavy bags at and above the cross-over, which, perhaps, can be observed via some correlations. The model with the tricritical endpoint predicts that at the deconfinement transition line the volume (mass) distribution of large (heavy) QGP bags acquires the power law form at the endpoint only, while in the model with the critical endpoint such a power law exists inside the mixed phase. The role of finite width of QGP bags is also discussed.

Finite-size effects and signatures of the QCD critical endpoint

We discuss finite-size effects on the phase diagram of strong interactions and consequences for the experimental search of the critical endpoint in high-energy heavy-ion collisions. We point out that a finite-size scaling analysis may be crucial in the process of data analysis in the Beam Energy Scan program at RHIC and in future experiments at FAIR-GSI.

UA(1)anomaly in hot and dense QCD and the critical surface

We discuss the chiral phase transition in hot and dense QCD with three light flavors. Inspired by the well-known fact that the ${U}_{\mathrm{A}}(1)$ anomaly could induce first order phase transitions, we study the effect of the possible restoration of the ${U}_{\mathrm{A}}(1)$ symmetry at finite density. In particular, we explore the link between the ${U}_{\mathrm{A}}(1)$ restoration and the recent lattice QCD results of de Forcrand and Philipsen, in which the first order phase transition region near zero chemical potential ($\ensuremath{\mu}$) shrinks in the quark mass and $\ensuremath{\mu}$ space when $\ensuremath{\mu}$ is increased. Starting from the Ginzburg-Landau theory for general discussions, we then use the Nambu--Jona-Lasinio model for quantitative studies. With the partial ${U}_{\mathrm{A}}(1)$ restoration modeled by the density dependent 't Hooft interaction, we fit the shrinking of the first order region found in de Forcrand and Philipsen's lattice calculation at low $\ensuremath{\mu}$. At higher $\ensuremath{\mu}$, the first order region might shrink or expand, depending on the scenarios. This raises the possibility that despite the shrinking of the first order region at lower $\ensuremath{\mu}$, the QCD critical end point might still exist due to the expansion at higher $\ensuremath{\mu}$. In this case, very high precision lattice data will be needed to detect the recently observed backbending of the critical surface with the currently available analytic continuation or Taylor expansion approaches. Lattice computations could, however, test whether the ${U}_{\mathrm{A}}(1)$ restoration is responsible for the shrinking of the first order region by computing the ${\ensuremath{\eta}}^{\ensuremath{'}}$ mass or the topological susceptibility at small $\ensuremath{\mu}$.

Compressed baryonic matter—experiments at GSI and FAIR

Experiments on strangeness production in nucleus-nucleus collisions at SIS energies address fundamental aspects of modern nuclear physics: the determination of the nuclear equation-of-state at high baryon densities and the properties of hadrons in dense nuclear matter. Experimental data and theoretical results will be reviewed. The planned “Compressed Baryonic Matter” (CBM) experiment at the future Facility for Antiproton and Ion Research (FAIR) aims at the exploration of the QCD phase diagram at the highest baryon densities. This includes the search for the deconfinement and chiral phase transition and for the QCD critical endpoint. Possible observables and the experimental setup will be discussed.