Finite-size behavior near the critical point of QCD phase-transition

It is pointed out that the finite-size effect is not negligible in locating the critical point of quantum colordynamics （QCD） phase transitions at current relativistic heavy ion collisions. The finite-size scaling form of the critical related observable is suggested. Its fixed point behavior at critical incident energy can be served as a reliable identification of a critical point and nearby boundary of QCD phase transition. How to experimentally find the fixed point behavior is demonstrated by using 3D-Ising model as an example. The validity of the method at finite detector acceptances at RHIC is also discussed.

Correlations of conserved charges and QCD phase structure

Correlations of conserved charges,i.e.,the baryon number,electric charge,and strangeness,are calculated at finite temperature and chemical potentials up to the fourth order.The calculations are done ina 2+1 flavor low energy effective theory,in which the quantum and thermal fluctuations are encoded through the evolution of flow equations within the functional renormalization group approach.Strangeness neutrality and a fixed ratio of the electric charge to the baryon number density are implemented throughout the computation.We find that higher-order correlations incorporate more sensitive critical dynamics than the quadratic ones.In addition,a non-monotonic dependence of the fourth-order correlations between the baryon number and strangeness,i.e.,-χ_(31)^(BS)/χ_(2)^(S)and χ_(22)^(BS)/χ_(2)^(S),on the collision energy is also observed.

QCD phase diagram at finite isospin and baryon chemical potentials with the self-consistent mean field approximation

The self-consistent mean field approximation of the two-flavor NJL model,with a free parameter a to reflect the competition between the"direct"channel and the"exchange"channel,is employed to study the QCD phase structure at finite isospin chemical potentialμr,finite baryon chemical potentialμB and finite temperature T,and especially to study the location of the QCD critical point.Our results show that in order to match the corres-ponding lttice results of isospin density and energy density,the contributions of the"exchange"channel need to be considered in the framework of the NJL model,and a weighting factorα=0.5 should be taken.It is also found that for fixed isospin chemical potentials,the lower temperature of the phase transition is obtained with increasing a in the T-μplane,and the largest difference of the phase transition temperature with differentα's appears atμ1~1.5mm.Atμ=0 the temperature of the QCD critical end point(CEP)decreases with increasing r,while the critical baryon chemical potential increases.At high isospin chemical potential(μr=500 MeV),the temperature of the QCD tricritical point(TCP)increases with increasing a,and in the low temperature regions the system will.transition from the pion superfluidity phase to the normal phase asμB increases.At low density,the critical temperature of the QCD phase transition with differentα's rapidly increases withμ1 at the beginning,and then increases smoothly aroundμ1>300 MeV.In the high baryon density region,the increase of the isospin chemical potential will raise the critical baryon chemical potential of the phase transition.

QCD at finite temperature and density within the fRG approach:an overview

In this paper,we present an overview on recent progress in studies of QCD at finite temperature and densities within the functional renormalization group(fRG)approach.The f RG is a nonperturbative continuum field approach,in which quantum,thermal and density fluctuations are integrated successively with the evolution of the renormalization group(RG)scale.The f RG results for the QCD phase structure and the location of the critical end point(CEP),the QCD equation of state(EoS),the magnetic EoS,baryon number fluctuations confronted with recent experimental measurements,various critical exponents,spectral functions in the critical region,the dynamical critical exponent,etc,are presented.Recent estimates of the location of the CEP from first-principle QCD calculations within f RG and Dyson-Schwinger equations,which pass through lattice benchmark tests at small baryon chemical potentials,converge in a rather small region at baryon chemical potentials of about 600 MeV.A region of inhomogeneous instability indicated by a negative wave function renormalization is found withμ_(B)■420 MeV.It is found that the non-monotonic dependence of the kurtosis of the net-proton number distributions on the beam collision energy observed in experiments could arise from the increasingly sharp crossover in the regime of low collision energy.

Locating the QCD critical end point through peaked baryon number susceptibilities along the freeze-out line

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 Ge V, 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.

Reconstruction of baryon number distributions

The maximum entropy method(MEM)and Gaussian process(GP)regression,which are both well-suited for the treatment of inverse problems,are used to reconstruct net-baryon number distributions based on a finite number of cumulants of the distribution.Baryon number distributions across the chiral phase transition are reconstructed.It is deduced that with the increase of the order of cumulants,distribution in the long tails,i.e.,far away from the central number,would become increasingly important.We also reconstruct the distribution function based on the experimentally measured cumulants at the collision energy sNN−−−√=7.77 GeV.Given the sizable error of the fourth-order cumulant measured in the experiments,the calculation of MEM shows that with the increasing fourth-order cumulant,there is another peak in the distribution function developed in the region of the large baryon number.This unnaturalness observed in the reconstructed distribution function could in turn be used to constrain the cumulants measured in the experiments.

Yield ratio of hypertriton to light nuclei in heavy-ion collisions from √sNN=4.9 GeV to 2.76 TeV

We argue that the difference in the yield ratio S3=NΛ3H/NΛ/N3He/Np measured in Au+Au collisions at √sNN=200 GeV and in Pb-Pb collisions at √sNN=2.76 TeV is mainly owing to the different treatment of the weak decay contribution to the proton yield in the Au+Au collisions at √sNN=200 GeV.We then use the coalescence model to extract from measured S3 the information about the Λ and nucleon density fluctuations at the kinetic freeze-out of heavy-ion collisions.We also show,using available experimental data,that the yield ratio S2=NΛ3H/NΛNd is a more promising observable than S3 for probing the local baryon-strangeness correlation in the produced medium.

High cumulants of conserved charges and their statistical uncertainties

We study the influence of measured high cumulants of conserved charges on their associated statistical uncertainties in relativistic heavy-ion collisions. With a given number of events, the measured cumulants randomly fluctuate with an approximately normal distribution, while the estimated statistical uncertainties are found to be correlated with corresponding values of the obtained cumulants. Generally, with a given number of events, the larger the cumulants we measure, the larger the statistical uncertainties that are estimated. The error-weighted averaged cumulants are dependent on statistics. Despite this effect, however, it is found that the three sigma rule of thumb is still applicable when the statistics are above one million.

Effects of mean-field and softening of equation of state on elliptic flow in Au+Au collisions at (~SNN)^(1/2)=5 GeV from the JAM model

We perform a systematic study of elliptic flow（v2） in Au＋Au collisions at √^SNN = 5 GeV by using a microscopic transport model, JAM. The centrality, pseudorapidity, transverse momentum and beam energy dependence of v2 for charged as well as identified hadrons are studied. We investigate the effects of both the hadronic mean-field and the softening of equation of state（EoS） on elliptic flow. The softening of the EoS is realized by imposing attractive orbits in two body scattering, which can reduce the pressure of the system. We found that the softening of the EoS leads to the enhancement of v2, while the hadronic mean-field suppresses v2 relative to the cascade mode. It indicates that elliptic flow at high baryon density regions is highly sensitive to the EoS and the enhancement of v2 may probe the signature of a first-order phase transition in heavy-ion collisions at beam energies of a strong baryon stopping region.

Do we need to use regularization for the thermal part in the NJL model?

The Nambu–Jona-Lasinio(NJL)model is one of the most useful tools for studying non-perturbative strong interactions in matter.Because it is a nonrenormalizable model,the choice of regularization is a subtle issue.In this paper,we discuss one of the general issues regarding regularization in the NJL model,which is whether we need to use regularization for the thermal part by evaluating the quark chiral condensate and thermal properties in the two-flavor NJL model.The calculations in this work include three regularization schemes that contain both gauge covariant and invariant schemes.We found that,regardless of the regularization scheme we choose,it is necessary to use regularization for the thermal part when calculating physical quantities related to the chiral condensate and to not use regularization for the thermal part when calculating physical quantities related to the grand potential.

Critical behavior of higher cumulants of order parameter in the 3D-Ising universality class

QCD deconfinement phase transition is supposed to be the same universality class as the 3D-Ising model. According to the universality of critical behavior, the Binder-like ratios and ratios of higher cumulants of order parameter near the critical temperature in the 3D-Ising model are studied. The Binder-like ratio is shown to be a step function of temperature. The critical point is the intersection of the ratios of different system sizes between two platforms. The normalized cumulant ratios, like the Skewness and Kurtosis, do not diverge with correlation length, contrary to the corresponding cumulants. Possible applications of these characters in locating critical point in relativistic heavy ion collisions are discussed.