Experimental evidences at the relativistic heavy ion collisions (RHIC) and large hadron collider (LHC) have demonstrated the formation of quark gluon plasma (QGP) in ultra-relativistic heavy-ion collisions at a small baryon chemical potential, where the phase transition from hadronic matter to QGP is suggested to be a crossover from state-of-the-art lattice quantum chromodynamics (QCD) calculations. It has been conjectured that there is a first-order phase transition and a critical point at a finite μB region in the QCD phase diagram. This study reviewed recent progress in searching for the QCD critical point from RHIC-STAR experiments.

Recent progress in studies on quantum chromodynamics (QCD) phase transition and related critical phenomena within the functional renormalization group (fRG) approach were reviewed, including the nonperturbative critical exponents and baryon number fluctuations, which are pertinent to the critical end point (CEP) in the QCD phase diagram. The fRG is a nonperturbative continuum field approach, in which quantum thermal fluctuations are successively integrated with the evolution of the renormalization group (RG) scale. Different methods of finding solutions to the flow or fixed-point equations of a nonperturbative effective potential have been discussed, for example, the Taylor expansion, expansion of the spatial dimension ε=4-d, and the recently proposed direct solution of the global potential. Furthermore, the baryon number of fluctuations is relevant to the critical phenomena of the CEP. Both have been discussed, and one explores the underlying reasons for the observed non-monotonic dependence of the kurtosis of the net proton number of distributions on collision energy in experiments.

Several experiments are being conducted at heavy-ion colliders around the world to determine the location of the proposed critical end point of quantum chromodynamics (QCD) in the T-μB phase diagram. As the presence of a very strong magnetic field is relevant to peripheral heavy-ion collisions, magnetars, and the early Universe, it is important to investigate the effect of a high magnetic field strength on QCD phase diagrams. We summarize the recent status and new developments in studies investigating QCD phase transitions under an extremely strong magnetic field. By doing so, we believe that this work will promote both theoretical and experimental research in this field. TheT-B phase diagrams are produced by Lattice QCD simulations. Other phase diagrams (E-B, μB-B,μI-B, andΩ-B) are mainly studied by using the chiral effective Nambu Jona-Lasinio model. A rotating magnetic field is adopted for the study of color superconductivity. The Ginzburg-Landau approximation is used to studyπ-superfluidity andρ-superconductivity in a very strong magnetic field. Physical effects, besides a magnetic fieldB, can also be measured when sketching a QCD phase diagram, such as temperatureT, strong electric fieldE, chemical potentialsμ, and rotational angular velocityΩ. We present five QCD phase diagrams: T-B,E-B, μB-B,μI-B, andΩ-B. The following phases are present in many (if not all) of the five QCD phase diagrams: chiral symmetry breaking, chiral symmetry restoration, inhomogeneous chiral phase, π0-condensation,π-superfluidity,ρ-superconductivity, and color superconductivity. The running of the coupling constant with magnetic field is consistent with the decrease of the pseudo-critical deconfinement temperature, providing a natural explanation for the inverse magnetic catalysis effect. We also found that a chiral anomaly induces pseudoscalar condensation in a parallel electromagnetic field, and that there appears to be a chiral-symmetry restoration phase in theE-B phase diagram. Without consideration of confinement, color superconductivity is typically favored for large baryon chemical potential; however, chiral density wave is also possible in the largeB and relatively smallμB region of the phase diagram. In an external magnetic field, theπ-superfluid with finite isospin chemical potential acts similarly to a Type-II superconductor with finite electric chemical potential. Bothπ-superfluidity andρ-superconductivity are possible in a parallel magnetic field and rotation, but the latter is more favored for largerΩ particles.

The exploration of the critical point on the QCD (Quantum Chromodynamics) phase diagram is one of the most important goals of the beam energy scan program in relativistic heavy-ion collisions (RHIC-BES). Preliminary experimental measurement observed the non-monotonic behavior of net-proton fluctuations as a function of collision energy, which qualitatively agrees with the prediction of the static theoretical models and this hints the existence of the QCD critical point. The system created in heavy-ion collision is highly expanding system with which the dynamical effects dramatically modify the critical fluctuations near the QCD critical point. To confirm the existence of QCD critical point and study the phase structure of QCD system at finite temperature and finite density region, a series of dynamical models near the QCD critical point has been developed. This paper reviews the recent developments related to the exploration of the QCD critical point from experimental and theoretical viewpoints. In particular, we emphasize on the developments and challenges of the dynamical model near the QCD critical point and the first-order phase transition.

One of the main goals of relativistic heavy-ion collision (HIC) is to search for the critical end point (CEP) of quantum chromodynamics (QCD), and distribution of the net-proton number from experimental measurements shows non-monotonic behavior, which indicates the existence of a CEP. The purpose of this work is to investigate the relationship between the net-proton number fluctuation and collision energy, and to explain the experimentally measured behavior. This study investigates the three-flavor Polyakov-loop Nambu-Jona-Lasinio (PNJL) model, which contains quark degrees from the NJL (Nambu-Jona-Lasinio) model and effective gluon contributions from Polyakov-loop, based on the equilibrium assumption and mean-field approximation. In addition, we study the phase diagram and C4/C2 of baryon number fluctuation as a function of collision energy along the freeze-out lines fitted from experimental data. With an appropriate form of freeze-out line, the collision energy decreases in the region of 7.7~200 GeV, and C4/C2 decreases slightly then increases, which is in agreement with the experimental data. Additionally, these results indicate that the equilibrium assumption is appropriate for the exploration of the system evolution after HIC, and the relationship between the freeze-out and phase transition lines is highly sensitive for observables.

The goal of relativistic heavy-ion collisions is to determine the phase boundary of quantum chromodynamics (QCD) phase transitions. Critically sensitive observables are suggested to be higher-order cumulants of conserved charges. The non-monotonous behavior of higher cumulants was observed at the relativistic heavy-ion collider (RHIC). However, it remains unclear whether these non-monotonous behaviors are critically related. We studied the influences of non-critical fluctuations, finite system size, and limited evolution time to determine if they cause non-monotonous behavior. First, we examined the minimum statistics required for measuring the fourth cumulant. The minimum statistic obtained using the centrality bin width correction (CBWC) method was 25 M. We suggest using a 0.1% centrality bin in the CBWC method instead of each Nch. With a 0.1 centrality bin width, 1 M statistics are sufficient. We then pointed out the statistical fluctuations from the limited number of final particles. By assuming the independent emission of each positive (or negative) charged particle, the statistical fluctuations of positive (or negative) charged particles were presented by a Poisson distribution, and the statistical fluctuations of net-charged particles were their evolution. The obtained statistical fluctuations for net protons, net electronic charges, and net baryons were consistent with those from the Hadron Resonance Gas model. In addition, the measured cumulants at RHIC/STAR are dominated by these Poisson-like statistical fluctuations. At the end of this section, we suggest the pooling method of mixed events and demonstrate that the sample of mixed events accurately presents the contributions of the background. Dynamic cumulants were defined as the cumulant of the original sample minus that of the mixed sample. Dynamical cumulants were shown to simultaneously reduce the influence of the statistical fluctuations, centrality bin width effects, and detector efficiency. Second, because the system is finite, the correlation length at the critical point is not developed to infinity in contrast to the system at thermal limits. Using a Monte Carlo simulation of the three-dimensional three-state Potts model, we demonstrated the fluctuations of the second- and fourth-order generalized susceptibilities near the temperatures of the external fields of the first-, second-, and crossover regions. Both the second- and fourth-order susceptibilities showed similar peak-like and oscillation-like fluctuations in the three regions. Therefore, non-monotonic fluctuations are associated with the second-order phase transition and the first-order phase and crossover in a finite-size system. The exponent of finite-size scaling (FSS) characterizes the order of transitions or crossover. To determine the parameters of the phase transition using the FSS, we studied the behavior of a fixed point in the FSS. To quantify the behavior of the fixed point, we define the width of the scaled observables of different sizes at a given temperature and scaling exponent ratio. The minimum width reveals the position of the fixed point in the plane of the temperature and scaling exponent ratio. The value of this ratio indicates the nature of the fixed point, which can be a critical, first-order phase transition line point, or crossover region point. To demonstrate the effectiveness of this method, we applied it to three typical samples produced by a three-dimensional three-state Potts model. The results show that the method is more precise and effective than conventional methods. Possible applications of the proposed method are also discussed. Finally, because of the limited evolution time, some processes in relativistic heavy-ion collisions may not reach thermal equilibrium. To estimate the influence of the nonequilibrium evolution, we used the three-dimensional Ising model with the Metropolis algorithm to study the evolution from nonequilibrium to equilibrium on the phase boundary. The order parameter exponentially approaches its equilibrium value, as suggested by the Langevin equation. The average relaxation time is defined. The relaxation time is well represented by the average relaxation time, which diverges as the zth power of the system size at a critical temperature, similar to the relaxation time in dynamical equations. During nonequilibrium evolution, the third and fourth cumulants of the order parameter could be positive or negative depending on the observation time, which is consistent with the calculations of dynamical models at the crossover side. The nonequilibrium evolution at the crossover side lasts briefly, and its influence is weaker than that at the first-order phase transition line. These qualitative features are instructive for experimentally determining the critical point and phase boundary in quantum chromodynamics.

We aim to study the effects of chemical potential and angular velocity on the critical endpoint of quantum chromodynamics (QCD). We used several probes (drag force, jet quenching parameter, heavy vector meson spectral function) to characterize the phase transition and studied gravitational waves from the holographic QCD phase transition in the early universe. We used different holographic QCD models to discuss the QCD phase transition, energy loss, spectral function, and gravitational waves. We found that the chemical potential and angular velocity changed the location of the critical endpoint, and the drag force and jet quenching parameter were temperature dependent and enhanced near the phase transition temperature. The magnetic field had a nontrivial effect on the spectral function. We conclude that the chemical potential decreases ωc, and the angular velocity decreases μc and the phase transition temperature. The jet quenching parameter and drag force can characterize the phase transition, and the magnetic field promotes the dissociation of heavy vector mesons. Moreover, the energy density of gravitational waves decreases as the gluon condensate increases, and the peak frequency shifts downward with increasing gluon condensate.Exploring the phase structure of QCD is an important task in high-energy heavy ion collision physics, and recently, there has been considerable interest in the QCD phase transition for rotating backgrounds.

We review the current status of quantum chromodynamics (QCD) properties in strong magnetic fields from lattice QCD. After a general introduction, we briefly present the implementation of a background magnetic field onto a lattice and discuss the recent progress on QCD properties at zero temperature, QCD transition temperature and inverse magnetic catalysis, and QCD phase structure in strong magnetic fields. Finally, we summarize this study.

The quantum chromodynamics (QCD) phase diagram is of great interest to researchers in the field of high energy nuclear physics. We review the present research status of several aspects of this topic. This review includes the search for the phase transition mechanism resulting in high-order baryon number fluctuations, how chiral imbalance, finite volume, and under rotations affect the QCD diagram, and the applications of the equation of states of dense QCD matter in the study of compact stars. The Nambu-Jona-Lasinio model and Dyson-Schwinger equations approach are the most commonly used methods described in this review. It is found that the theoretical results of high-order baryon number fluctuations are in good agreement with the experimental data. The chiral imbalance, finite volume, and rotation of quark-gluon plasma (QGP) have a quantitative impact on the chiral condensate and the QCD phase structure. In the study of compact stars, the theoretical results from equation of states of dense QCD matter agree well with pulsar observations. Further research will be required to form a complete understanding of the QCD phase diagram, particularly given the abundance of QGP.

Exploring the quantum chromodynamics (QCD) phase diagram at finite bayron density regime through the beam energy scan (BES) program at the relativistic heavy-ion collider (RHIC) is one of the key frontiers in high energy nuclear physics. The high precision data anticipated from the second phase of the BES program would potentially enable the discovery of the conjectured QCD critical point, a landmark point on the phase diagram. In this paper, the progress made by the beam energy scan theory (BEST) collaboration, which was formed with the goal of providing a theoretical framework for analyzing data from BESII, is reviewed. In addition, the challenge of investigating the QCD phase diagram with future facilities is discussed.