Little-Bang and Femto-Nova in Nucleus-Nucleus Collisions

We make a theoretical and experimental summary of the state-of-the-art status of hot and dense QCD matter studies on selected topics. We review the Beam Energy Scan program for the QCD phase diagram and present the current status of search for QCD Critical Point, particle production in high baryon density region, hypernuclei production, and global polarization effects in nucleus-nucleus collisions. The available experimental data in the strangeness sector suggests that a grand canonical approach in thermal model at high collision energy makes a transition to the canonical ensemble behavior at low energy. We further discuss future prospects of nuclear collisions to probe properties of baryon-rich matter. Creation of a quark-gluon plasma at high temperature and low baryon density has been called the"Little-Bang"and, analogously, a femtometer-scale explosion of baryon-rich matter at lower collision energy could be called the"Femto-Nova", which may possibly sustain substantial vorticity and magnetic field for non-head-on collisions.


INTRODUCTION
Nuclei are bound states of protons and neutrons via the Strong Interaction and the fundamental theory of the Strong Interaction is Quantum Chromodynamics (QCD). In QCD gluons are massless due to gauge symmetry and up (u) quarks are as light as m u = 3 − 4 MeV and down (d) quarks are heavier than u-quarks, i.e., m u /m d ∼ 0.5 (see Ref. [1] for a recent review of quark masses from the lattice-QCD). The strange (s) quark mass is comparable to the typical QCD energy scale; that is, m s = 80 − 90 MeV of the same order as Λ QCD = 100 − 200 MeV. Since charm (c) and bottom (b) quarks are much heavier than Λ QCD , they would make only small contributions to bulk thermodynamics and they serve as external probes. Here, we focus on two puzzling QCD features for the nucleons which are composed of N c valence quarks (where N c = 3 is the number of colors in QCD) and have a mass, m N 940 MeV ∼ N c Λ QCD . The first question is; how can almost massless particles form a bound state with a positive binding energy? The second question is; how can the nucleons become such extremely massive out of almost massless particles? The former question on the existence of bound states is referred to as confinement and the latter on the origin of the mass is via spontaneous chiral symmetry breaking.
The key to resolve these puzzles lies in the QCD vacuum structure. The vacuum in quantum field theory is not empty in general, but is full of quantum fluctuations dictated by fundamental interactions. Thus, the QCD vacuum is regarded as a "medium" in analogy to condensed matter physics. Just like spin systems for example, the QCD vacuum structure may be either an ordered/disordered state according to external environments such as the temperature T , the baryon density ρ B (or the baryon chemical potential µ B ), the magnetic field B, the vorticity ω, etc. The idea of the relativistic nucleus-nucleus collisions or heavy-ion collisions is to shake the QCD vacuum with high energy density to observe new states of matter out of quarks and gluons and to seek for traces of phase transitions associated with confinement and/or chiral symmetry breaking (see Ref. [2] for historical backgrounds).
The saturation density around the center of heavy nuclei is ρ 0 0.16 nucleons/fm 3 corresponding to the energy density 0 0.15 GeV/fm 3 . At the initial stage of the heavy-ion collisions the energy density can reach hundreds times larger than the saturation density depending on the collision energy √ s N N per nucleon. The Stefan-Boltzmann law converts the energy density = 0.8 − 1.0 GeV/fm 3 to T = 150 − 160 MeV that is comparable to LQCD, above which a quark-gluon plasma (QGP) is realized. In the history of the Universe such high-T matter should have existed shortly (of the order of µs) after the Big Bang. In this sense the QGP physics can be regarded as an emulation of the Big Bang in the laboratory on the Earth, which may well be called the Little Bang.
Now that we have learnt intriguing properties of the QGP in the regime at high T and low µ B , a next direction of the relativistic heavy-ion collision physics is expanding toward the regime at high baryon density. The phase structures could have much richer contents in these high-density regions, see Refs. [3][4][5] for reviews on the QCD phase diagrams. We have already known that symmetric nuclear matter has a first-order phase transition at µ B ≈ 923 MeV; that is, the nucleon mass minus the binding energy ∼ 16 MeV. Then, a terminal point of the first-order phase boundary, namely, a critical point of liquid-gas phase transition in nuclear matter ex- ists around T = (10 − 20) MeV (see Ref. [6] for a comprehensive review including experimental signatures). The question is what new state of matter is anticipated for nuclear matter at higher baryon density. The central cores of the neutron star would exhibit the most baryon-rich and equilibrated state of matter in the Universe, where the density could be as large as ∼ 5ρ 0 or even higher. The baryon density may become even larger in transient states such as from neutron star mergers. Analogously, by adjusting the collision energy in the heavy-ion collisions, the baryon density could transiently increase up to several times ρ 0 according to numerical simulations [7]. Such a femtometer-scale explosion of baryon rich matter may be called the Femto-Nova. This concept is illustrated in Fig. 1.
So far, most of the high-energy nuclear collisions have been designed at high energy region where the temperature of the formed matter is high in order to assure the formation of the QGP. Unfortunately, only little is known from theory about the phase structures in low-T and high-µ B (or high-ρ B ) regions. There are some speculations on the ground state structures in such regimes which will be partially reviewed in this article. The most important landmark is the QCD Critical Point. In the same way as the critical point of the nuclear matter liquid-gas phase transition, deconfined QCD matter may have a first-order phase boundary and the QCD Critical Point appears at the terminal point of the first-order line. Its exact location is not well constrained yet, and the experimental efforts to discover the QCD Critical Point are still continuing.

CURRENT STATUS -THEORY
Theoretical Background for the QCD Phase Diagram: It has been established that hadronic matter continuously changes into the QGP with increasing temperature T as long as µ B is sufficiently smaller than T . Although this continuous change of matter takes place around T = (156.5 ± 1.5) MeV according to the lattice-QCD simulation [8], there is no strict phase transition. In the QCD community this transitional but still continuous change of matter is commonly referred to as a crossover.
If masses of u and d quarks are zero and other quarks are massive, such 2-flavor QCD matter would go through a second-order phase transition for which various derivatives of thermodynamic quantities diverge with critical exponents belonging to the O(4) universality class (or a first-order transition if the axial anomaly is restored, see Ref. [9]). Physical values of m u and m d are much smaller than Λ QCD , and it is conceivable to observe some remnants of the second-order phase transition. Indeed, in the lattice-QCD simulation [10], it has been numerically confirmed that the magnetic scaling follows consistently with the O(4) universality class, from which the "pseudocritical" temperature, T pc , can be deduced.
The pseudo-critical temperature should be a function of the density. Generally T pc (µ B ) is a decreasing function with increasing µ B due to the Pauli blocking of quarks in phase space. Because of the notorious sign problem in the Monte-Carlo algorithm, expectation values of observables cannot be computed and the first-principles lattice-QCD simulation cannot access a region with a substantial value of baryon density, and so there is no reliable theoretical prediction for T pc (µ B ) at µ B much larger than T .
Instead, a phenomenologically determined boundary on the µ B -T plane is known, called the line of the Chemical Freeze-out, which has been identified from a hybrid approach of theory and experiment. The Chemical Freeze-out literally means that inelastic reactions stop and chemical compositions are fixed there. In QCD matter hadrons interact and particle species can change like chemical reactions. In the heavy-ion collision the physical system is expanding and the temperature rapidly drops down. Therefore, the average inter-particle distance would increase as the temperature gets lowered. In particular, around T pc , the entropy density significantly falls down, and correspondingly the degrees of freedom in thermal excitations decrease [11]. Since the average interparticle distance becomes large, the interaction among hadrons is considered to diminish. In this way the observed particle yields keep the footprint of hot and dense hadronic matter when the interaction was turned off, that is presumably when the matter underwent a crossover at T pc . If this scenario is the case, the particle yields should be sensitive to T pc (µ B ). As observed experimen-   [13], while the filled-squares are extracted from net-proton higher moments (up to third order) [14]. Representing the smoothcrossover region are the lattice-QCD results shown as greenband. The empirical thermal fit results to global hadron yield data are shown as yellow-line [15]. The coverage of the RHIC BES program, STAR fixed target program (FXT), and future (FAIR, JPARC-HI, and NICA) experimental facilities are also indicated at the top of the figure. The liquid-gas transition region that features a second order critical point is shown by the red-circle, and a first-order transition line is shown by the black dashed line, which connects the critical point to the ground state of nuclear matter.
tally, relative abundances of hadrons obey the thermal distribution at common T and µ B , so that the thermal fit can fix T and µ B , or a line of Chemical Freeze-out, T = T ch (µ B ) [12]. We note that the charge chemical potential µ Q is fixed from the proton/neutron ratio and the strangeness chemical potential µ S is fixed from the strangeness free condition. With various center-of-mass colliding energies, we can change accessible µ B to sample T ch (µ B ) from the thermal fit [13,14], see Fig. 2. Generally speaking, collisions at smaller √ s N N have larger baryon stopping, leading to larger values of µ B (and smaller values of T ) [15]. Thus, the line of Chemical Freeze-out on the µ B -T plane should be regarded as an experimentally determined QCD phase diagram which appears to be consistent with the lattice-QCD results at the vanishing µ B region [8], as also displayed in Fig. 2. This underlies the idea of the Beam Energy Scan (BES) program at RHIC. The thermal description is applicable for not only particle abundances but also thermodynamic quantities such as the pressure and the entropy density.
Observables for the QCD Critical Point: It is known from the theoretical analysis that the QCD crossover has a general tendency to become closer to a first-order transition at larger µ B (see discussions in a review [16]). It is thus a natural anticipation that the QCD crossover may turn to a first order phase transition in a high-density regime. If this is the case, as suggested by some effective model studies, there must be a "critical" value of µ B above which a first-order phase transition occurs and below which only a crossover is found. This separating point is the QCD Critical Point and critical fluctuations associated with the second-order phase transition should be expected at this point. Its exact location is still under dispute, and the lattice-QCD results [17] disfavors the existence of the QCD Critical Point for µ B /T 2.
Interestingly, the QCD Critical Point emerges with nonzero physical quark masses, so that it belongs to not the O(4) but the Z(2) universality class. Moreover, the dynamical universality class has been also identified as the model H (dynamics of the liquid-gas critical point of a fluid) [18] (see a review [19] for detailed classification). The dynamical critical exponents are important inputs for simulations including the critical slowing down effects [20].
For experimental signatures, we can in principle seek for enhanced fluctuations coupled to the critical modes. Since the critical modes appear in a mixed channel of scalar (i.e., chiral condensate) and vector (i.e., baryon density) at the QCD Critical Point [21], the baryon number fluctuations are sensitive to the criticality. Let us denote the baryon number fluctuation by δN = N − N where N is the number of net baryons at each collision event and · · · stands for the ensemble average taken over collision events. At the critical point, generally, the correlation length ξ diverges, and it was pointed out in Ref. [22] that the non-Gaussian fluctuations behave as Here, the subscript c represents a part of the correlation function corresponding to the connected diagrams (to extract non-Gaussian fluctuations) and η is the anomalous dimension (which is usually η 1). Higher-order fluctuations are more sensitive to the criticality, but they need more statistics especially to construct connected contributions. Now, the third order (k = 3) and the fourth order (k = 4) in Eq. (1) are common measures for the QCD Critical Point search, normalized by the k = 2 fluctuation (variance), σ 2 = (δN ) 2 ; namely, S and κ are called the skewness and the kurtosis and characterize how skewed and how sharp the distribution of δN appears, respectively. It is noted that, in the QCDphysics context, κ (or the fourth-order cumulant) was first considered in the lattice-QCD simulation to diagnose whether quarks are confined or deconfined [23]. This idea can be easily generalized to other observables coupled to the critical modes. Because observables in the heavy-ion collisions are integrated quantities over the whole dynamical evolutions, we should look at fluctuations of conserved charges; otherwise, critical enhancement would be wiped off through the dynamical evolution. There are three representative candidates available in the heavy-ion collisions, i.e., the baryon number (B), the electric charge (Q), and the strangeness (S). In thermodynamics those fluctuations are defined by the derivatives of the pressure with respect to the chemical potentials corresponding to conserved charges, i.e. [24] where q = B, Q, S. In terms of these fluctuations the skewness and the kurtosis are represented as S q σ q = χ q , respectively. Susceptibilities in mixed channels can also be defined in a similar fashion.
The baseline to be compared for the critical enhancement is estimated by an approximation of non-interacting and dilute hadronic gases described by the Boltzmann distribution. Then, in this Boltzmann gas approximation, the chemical potential dependence is factored out, yielding, Since only the net charge is conserved, calculated as a difference between the particle and the anti-particle contributions, the above estimate is often referred to as the baseline by the Skellam distribution that is the probability distribution of two statistically independent variables. It is also possible to apply the Hadron Resonance Gas (HRG) model to estimate the baselines, and then, S q σ q and κ q σ 2 q are generally suppressed by quantum statistical effects, that reflects deviations of the Bose and the Fermi distribution functions from the Boltzmann distribution. The major strategy for the QCD Critical Point search is to measure S and κ at various √ s N N and look for enhancement as compared to the baseline (4).
Baryon-rich Matter, an Approximate Triple Point, and Strangeness: From the HRG model estimate, the baryon number density along the Chemical Freeze-out line is maximized around T 150 MeV and µ B 400 MeV between √ s N N = 3 − 19.6 GeV. This has been experimentally confirmed through the K + /π + ratio peaked around √ s N N 8 GeV. It is intuitively easy to understand that K + /π + is such sensitive to the baryon density, while K − /π − is not. In the heavy-ion collisions the time scale is much shorter than the weak interaction, so the net strangeness should be vanishing. This means that the net chemical potential coupled to s quarks must be zero. Since s quarks have S = −1 and B = 1/3, the strangeness free condition leads to which means that the dense baryonic matter should contain strange baryons or hyperons which must be cancelled by mesons involvings quarks such as K + . Therefore, K + is enhanced at high density, while K − is not. In this sense, this point reached around √ s N N 8 GeV plays a special role to tell us about a regime transition; at smaller µ B 400 MeV physics is dominated by mesonic degrees of freedom, and at larger µ B 400 MeV more baryons dominate. Roughly speaking, the QGP transition is understood from the Hagedorn transition with the exponentially rising mesonmass spectrum, while the the transition at dense region arises from the Hagedorn transition with the baryon-mass spectrum, and two Hagedorn transition lines cross just around √ s N N 8 GeV. In this way, the most baryonic point around √ s N N 8 GeV could be regarded as a QCD Triple Point approximately facing baryon-less deconfined matter, baryon-rich deconfined matter, and confined hadronic matter [25].
From the correlations between baryon number and strangeness, the QCD Triple Point can be a landmark for the realization of the most strange matter which contains hyperons. In particular the interactions between nucleons (N ) and hyperons (Y ), i.e., Y -N and Y -Y interactions are important parameters for the neutron star structures. The most strange matter would provide us with chances to constrain those interactions. We note that N in Eq. (2) is the baryon number, but neutrons are not electrically charged, and the proton number is used experimentally as a proxy for the baryon number. Results are shown for both central (0 − 5%, small impact parameter) and peripheral (70 − 80%, large impact parameter) collisions. Also shown are the expectations from the HRG model and a transport based model called UrQMD, namely, theories for central Au+Au collisions without including critical fluctuations.
The following conclusions can be drawn: (a) As we go from lower order moments (Sσ) to higher order moments (κσ 2 ) deviations between central and peripheral collisions for the measured values increases. (b) Central κσ 2 data show a non-monotonic variation with collision energy at a significance of ∼ 3σ [26]. (c) Experimental data show deviation from heavy-ion collision models without a critical point. Although a non-monotonic variation of the experimental data with collision energy looks promising from the point of view of the QCD Critical Point search, a more robust conclusion can be de- rived when the uncertainties get reduced and significance above 5σ is reached. The goal of the second phase of the BES program (BES-II) at RHIC and the fixed traget (FXT) programs is to have high precision measurements below The data presented in Fig. 3 provides the most relevant measurements over the widest range in µ B (20−450 MeV) to date for the critical point search, and for comparison with the baryon number susceptibilities computed from QCD to understand the various features of the QCD phase structure. The deviations of κσ 2 below the baseline (4) are qualitatively consistent with theoretical considerations including a critical point [27]. However, the conclusions on the experimental confirmation of the QCD Critical Point might be made only after improving the precision of the measurements at lower collision energies and by comparing to the QCD calculations with critical point behavior which includes the dynamics associated with heavy-ion collisions. See Ref. [28] for the latest report.
High Baryon Density Matter: Figure 4 (1) in the upper panel shows the energy dependence of K/π particle yield ratio. The results are from AGS [29][30][31], SPS [32,33], and RHIC [34]. These ratios reflect the strangeness content relative to entropy of the system formed in heavy-ion collisions. The thermal model calculation is shown as yellow band for K + /π + and green band for K − /π − . The dot-dashed line represents the net-baryon density at the Chemical Freeze-out as a function of collision energy, calculated from the thermal model [35].
The following observations can be made. (a) The collision energy dependence of both the ratios is fairly well described by a thermal model calculation. (b) A peak position in energy dependence of K + /π + is observed and has been suggested to be a signature of a change in degrees of freedom (baryon to meson [36] or hadrons to QGP [37]) while going from lower to higher energies. (c) The calculated net baryon density exhibits a maximum as the collision energy is scanned, with a value of about three-fourth of the normal nuclear saturation density (i.e., ρ 0 0.16 nucleons/fm 3 ). The collision energy where the maximum net-baryon density occurs is very close to the peak position of the K + /π + ratio. This way of representation of the results from experimental measurement and theory calculation serves to clearly demonstrate that the freeze-out density and K + /π + ratio could be related. (d) The K − /π − ratio seems unaffected by the changes in the net-baryon density with collision energy and shows a smooth increasing trend.
Through these measurements we have the knowledge of regions in collision energy where the maximal net-baryon density is reached. This is an important aspect in the context of planning of experiments that seek to explore compressed baryonic matter.
Tests of Thermal Model -GCE vs. CE: Relativistic statistical thermodynamics has been applied to systems ranging from cosmology to heavy-ion collisions in laboratory. The cosmological applications usually deal with systems having large volumes and matter or radiation, hence the Grand Canonical Ensemble (GCE) is a suitable description. For heavy-ion collisions recorded in labora- tory, the situation is complicated due to the femtometerscale nature of the systems. Often one assumes (approximate) local thermal equilibrium for such processes. Further, such thermal models based on the GCE employ chemical potentials to account for conservation of quantum numbers on average. These GCE models have been able to explain the particle production successfully for a wide range of collision energies [12]. However, conservation laws do impose restriction on particle production if the available phase space is reduced. Hence, the relativistic statistical thermodynamics provides two choices of the formalisms: a GCE and a canonical ensemble (CE) approachs [38]. In the thermodynamic (large volume) limit, the GCE and the CE formalisms are equivalent, but it is an interesting question to ask where and when the transition from a GCE picture to a CE one occurs for finite volume systems produced in collisions at laboratory, where the collision energy spans from a few GeV to a few TeV (three orders in magnitude). Figure 4 (2) in the lower panel shows the energy dependence of φ/K − yield ratio. For most collision energies the ratio remains constant. Similar to K − /π − ratio, the φ/K − ratios seem not affected by the net-baryon density. Below the collision energy where the freeze-out netbaryon density peaks [shown by the dot-dashed line in Fig. 4 (1)] the φ/K − ratio starts to increase. Thermal model calculations, adopting the GCE, which has been quite successful in accounting for the observed yields of the hadrons in heavy-ion collisions, explains the measurements up to collision energy of 5 GeV. Then the GCE model values decrease, while the increase in φ/K − at lower energies is explained by using a thermal model within the CE framework for strangeness. We note that a control parameter, r sc , is introduced for strangeness CE results in Fig. 4 (2). The physical meaning of r sc is a typical spatial size of ss correlations. For smaller r sc , pairs of s ands stick together and the strangeness free condition is satisfied locally, which suppresses the yield of K − and thus enhances φ/K − . This makes a quantitative difference from the GCE results. For a given volume of the whole system, r sc determines how close to the GCE/CE situation the strangeness sector in the system should be. Since r sc reflects the intrinsic properties of matter, the shifting from the GCE to the CE in strangeness signals a considerable change of the medium properties. Future measurements of φ/K − at lower collision energies can be used as an observable to estimate the volume in which the open strangeness is produced (reflected by the value of r sc ).

Lifetime of Hypernuclei:
Hypernuclei are bound states of nucleons and hyperons, hence they are natural hyperon-nucleon correlated systems [41]. They can be used as an experimental probe to study the hyperonnucleon (Y -N ) interaction. Studying hypernuclei properties is one of the best ways to investigate the strengths of Y -N interactions. Theoretically, the lifetime of a hypernucleus depends on the strength of the Y -N interactions. Therefore, a precise determination of the lifetime of hypernuclei provides direct information on the Y -N interaction strength. The high energy heavy-ion collisions at RHIC and LHC create favorable conditions to produce hypernuclei in significant quantities. At the moment, the experiments have measured the production of the lightest hypernuclei, i.e., the hypertriton, 3 Λ H, which is a bound state of a proton, a neutron and a Λ. Figure 5 shows a compilation of the measurement of the hypertriton lifetime from various experiments and theory calculations [39,40]. The lifetime of the (anti-) hypertriton is determined by reconstructing the mesonic decay channels. A statistical combination of all the experimental results yields a global average lifetime of 206 +15

−13
picoseconds. The lifetime is about 22% shorter than the lifetime of a free Λ of 263.2 ± 2.0 picoseconds, indicating a possibility of a reasonable hyperon-nucleon interaction in the hypernucleus system. Most calculations predict the hypertriton lifetime to be in the range of 213 − 256 picoseconds. The Y -N interaction is of fundamental interest, for it controls the onset of strange degrees of freedom in high density nuclear matter, such as matter in the neutron star. The lifetime measurements of hypernuclei thus provides a crucial input for models attempting to understand physics of the neutron star. One should be aware of discrepancies in the measured lifetime of 3 Λ H from RHIC and LHC. High statistics data are called for in order to resolve these discrepancies.
Polarization and Spin Alignment: Recently, it was realized that the initial condition of the QGP in relativistic heavy-ion collisions is subjected to two extraordinary parameters: the angular momentum and the magnetic field. The angular momentum of the order of 10 7 is theorized to be imparted to the system through the torque generated when two nuclei collide at non-zero impact parameter with center-of-mass energies per nucleon of a few 100 GeV [42]. This leads to a thermal vorticity of the order of 10 21 per second for QCD matter formed in the collisions [43]. Further, when the two nuclei collide in the LHC, an extremely strong magnetic field of the order of 10 15 T is generated by the spectator protons, which pass by the collision zone without breaking apart in inelastic collisions. The effect of the angular momentum (which is a conserved quantity) is expected to be felt throughout the evolution of the system. In contrast to that, the magnetic field is transient in nature and stays for a short time of the order of ∼ 0.1 fm/c unless the electric conductivity is large (but this is disfavored, see discussions in Ref. [44]). Just to give an idea of the magnitude of these values, the highest angular momentum measured for nuclei (near the Yrast line) is ∼ 70 and the strongest magnetic field we have managed to produce in the laboratory is ∼ 10 3 T using the electromagnetic flux-compression technique. Getting experimental signatures of these phenomena is not easy due to the femtoscopic nature of the system (both in space and time) formed in the heavy-ion collisions. Nevertheless, the experiments at RHIC and LHC have been able to address this challenging problem.
It is known that the spin-orbit LS coupling causes the fine structure in atomic physics and the shell structure in nuclear physics, and, is a key ingredient in the field of spintronics in materials sciences. It is also expected to affect the development of the rotating QGP created in collisions of nuclei at high energies. The extremely large initial value of the orbital angular momentum is expected to lead to the polarization of quark spin along the direction of the angular momentum of the plasma's rotation due to the LS coupling [45]. This should in turn cause the spins of vector (spin = 1) mesons (K * 0 and φ) to align [46] and hyperons like Λ baryons to be polarized [43]. Both the hyperon polarization and the spin alignment can be studied by measuring the angular distribution of the decay products of Λ and vector mesons. The hyperon polarization is found to increase with decrease in heavy-ion collision energy. The thermal vorticity values thus show that the QGP formed in the collisions, along with exhibiting the emergent properties of relativistic fluid, is also the most vortical fluid found in nature [43]. Meanwhile, the observed spin alignment of vector mesons (with J = 1) was quantified by obtaining the probability of finding a vector meson in a J z = 0 state along the z direction that is the direction of the orbital angular momentum of the rotating QGP. The momentum dependence of these probability values indicated polarization of quarks in the presence of large initial angular momentum in heavy-ion collisions and a subsequent hadronization by the process of recombination [46].

FUTURE DIRECTIONS -THEORY
There remain a lot of theoretical challenges in understanding physics of dense baryonic/quark matter with magnetic field and rotation. We give brief discussions on some topics in order.

QCD Phase Structures and Quark Matter at High
Baryon Density: Theoretically, it is highly nontrivial how quarks can melt from hadrons in cold and dense matter. Unlike hot QCD even an approximate measure for quark deconfinement is still unknown or such an order parameter simply may not exist.
The QCD Critical Point is a landmark on the QCD phase diagram and the next intriguing question is where we can find quark matter. One might naively think that the asymptotic freedom with a large quark chemical potential, µ q Λ QCD , makes quarks unbound from hadrons, but this is not necessarily true. When µ q is large, quarks form a Fermi sphere, and the typical energy scale of quarks near the Fermi surface is ∼ µ q . However, gluons can still carry soft momenta, mediating confining forces. Therefore, excitations on top of the Fermi surface are still confined, while the Fermi sphere itself is dominated by quarks, and this refined picture of a dense baryonic state is called Quarkyonic Matter [47].
One can develop a more precise definition of Quarkyonic Matter by deforming the fundamental theory; in reality N c = 3 where N c is the number of colors, and one can significantly simplify theoretical treatments by taking the N c → ∞ limit. In this special limit the ground state could have an inhomogeneous crystalline shape [48] (see also a review [4] for comprehensive studies of inhomogeneous phases). In reality mesonic fluctuations would destroy inhomogeneity, but some remnant correlations can still remain. Those remnants of enhanced spatial correlations would increase the density fluctuation. For experimental detections to discriminate it from bubble formation associated with a first-order transition beyond the QCD Critical Point, more theoretical works are needed.
Neutron Star Phenomenology: We specifically pick two problems here in neutron star phenomenology. One is a question of whether quark matter is found in cores of the neutron star, which is a continued subject from the above problem of the phase diagram, and the other is what is called the hyperon puzzle.
It is the experimental fact that massive neutron stars whose masses are greater than 2M exist, where M represents the solar mass. This observations is strong enough to constrain the stiffness of the equation of state (EoS) and a strong first-order phase transition has already been excluded. Thus, even if quark matter existed in cores of the neutron star, it is likely that there is only a smooth crossover or a weak first-order transition from nuclear to quark matter.
Matter created in the heavy-ion collision is regarded better as hot and dense baryonic matter. Since the physical observables are sensitive to the EoS, the global analysis of experimental data would quantify the most likely regions of EoS parameters. Such a program of the global Bayesian analysis has already been successful at large √ s N N where the experimentally inferred EoS is found to be consistent with the lattice-QCD results [49]. The same machinery would in principle constrain the EoS of hot and dense baryonic matter. One of the most interesting EoS parameters is the speed of sound c 2 s , which hints the presence of quark matter as discussed in Ref. [50]. We also mention that the global analysis of the flow measurements could constrain the viscosities of dense matter (apart from leptonic contributions), which should be useful for considerations of the r-mode evolutions of the neutron star [51]. See, for example, Ref. [52] for a theoretical estimate of viscosities of dense nuclear matter.
Let us turn into the hyperon puzzle that has twofold manifestations. If the baryon density reaches several times ρ 0 , inside the neutron star to balance the gravitational force, it is energetically more favorable to activate the strangeness degrees of freedom. One problem is that the introduction of strangeness generally softens the EoS and it would become more difficult to support the neutron stars with the mass 2M . Another problem is, once hyperons are favored, the direct Urca process would shorten the time scale of the neutron star cooling, which makes the neutron star too cold. Thus, the threshold of the neutron star mass to open the direct Urca process is an important parameter, and this is dictated by the Y -N and Y -Y interactions as well as three-body forces involving hyperons.
Theoretically speaking, the most promising approach is the first-principles calculation of the baryon interactions including nontrivial strangeness from the lattice-QCD simulation [53]. In the HAL QCD method the Nambu-Bethe-Salpeter wave functions are computed on the lattice, from which the potential is extracted, see Ref. [54] for a review on the HAL QCD method including hyperon results. For example, pΞ − correlation has been theoretically predicted to have attractive interaction [55]; this is an interesting system since Ξ − ∼ dss is a multi-strange baryon, and the experimental signature is reported [56]. Also, the correlations of ΩΩ and N Ω have also been estimated in Ref. [57] based on the lattice-QCD determined potential. For more comprehensive discussions to quantify the potential from the correlations in heavy-ion collisions, see a recent review [58].
Dibaryons and Diquarks: ΩΩ is an interesting candidate for one of possible dibaryons [59] which are six-quark objects. There is a long history of the dibaryon hunting (see Ref. [60] for a review); the idea is traced back to the conjecture on the H-dibaryon [61]. One might think that the deuteron is also a six-quark bound state, but what is special about the H-dibaryon is that the diquark correlation plays an essential role. From the one-gluon exchange interaction the color-triplet diquarks are favored, and the low-energy reduction leads to the Breit interaction involving the color and the spin degrees of freedom. It is an established notion that the energetically most favored channel is the spin-singlet and the flavor-triplet, and diquarks in this channel are called "good diquarks", while the second stable diquarks, i.e., "bad diquark" are found in the spin-triplet and the flavor-sextet channel. The structure of the H-dibaryon is considered to be dominated by three good diquarks, i.e., H ∼ (ud)(ds)(su).
It is still challenging to find a direct signature of the strong diquark correlation. From the theoretical point of view, the difficulty lies in the fact that diquarks are not gauge invariant. Nevertheless, the density-density correlation in baryon wave-functions could quantify the diquark correlation in a gauge-invariant way [62]. Interestingly, the diquark correlations would be more prominent at higher baryon density. Actually, it is a solid theoretical prediction that QCD matter at asymptotically high density should be a color superconductor in which the diquarks form condensates. If there is no sharp transition separating baryonic matter from color-superconducting quark matter, as is conjectured in the quark-hadron continuity scenario, one can expect some remnants of the diquark correlations in density regions accessible by the heavy-ion collision. The interesting question is whether diquarks are treated as active thermal degrees of freedom, participating in the thermal model in dense matter, see Ref. [63] for a model with colored thermal excitations like diquarks. Since the lattice-QCD calculation is not functional at finite density, the test can be made only in comparison to experimental data.
Femto-Nova Rotating with Magnetic Fields: The profound feature of matter created in the heavy-ion collision is that non-central collisions are accompanied by vorticity and magnetic fields as illustrated in Fig. 1. Such a hot, dense, and rotating object exposed under the magnetic field can be found as an emulator of the proto-neutron star after the supernova, and we may well call this heavyion system the Femto-Nova.
The Femto-Nova investigations have a lot of research potentials. Relativistic rotation and magnetic fields would change the properties of matter, or even the phase diagram should be affected [64]. In numerical simulations of the supernovae and the neutron-star-merger, such effects of rotation and magnetic fields are not taken into account yet. Thus, the heavy-ion collision experiments can constrain uncertainties in the interplay of rotation and magnetic field in strongly interacting matter. In particular, relativistic formulations of spin-and magnetohydrodynamics are still in the middle way of developments.
Rotation and magnetic fields are of paramount importance also from the point of view of the topological effects. Once the density (finite µ B ), the rotation (finite angular velocity ω), and the magnetic field B are coupled together, the theory tells us that the chiral seperation effect (CSE) and the chiral vortical effect (CVE) should appear (see Ref. [65] for a review): where σ s ∝ µ B if the particle masses are negligible. The coefficient σ v has two components; one is ∝ T 2 and the other ∝ µ 2 B . Here, J 5 is the axial current, and its physical meaning is the spin expectation value of matter. Therefore, the first in Eq. (6) physically represents the spin polarization, and what is nontrivial in relativistic systems is that the spin and the momentum of massless fermions are tightly correlated with a certain handedness. Therefore, the global spin polarization results from the CSE leading to the chirality separation associated with a chirality flow along the polarization.
The second equation, i.e., the CVE, looks like a counterpart of the CSE with B replaced by ω, but physical interpretations are rather nontrivial. In this context the physical meaning of the CVE is the relativistic realization of the Barnett effect (see Ref. [66] for a review by Barnett himself); a mechanical rotation yields nonzero magnetization [67]. One might then wonder if the chiral anomaly mechanism could be an independent origin from the conventional LS coupling. Actually, in the nonrelativistic Barnett effect, the magnetization is inversely proportional to the gyromagnetic ratio, and so it is proportional to the mass; this has motivated the nuclear Barnett effect experiment [68]. The CVE is more prominent, however, for massless fermions, and their mass dependences look competing. The theoretical framework is not yet complete to incorporate all those effects consistently for arbitrary masses of fermions. Establishing a firm bridge between nonrelativistic and relativistic (as seen in the heavy-ion collision) Barnett effects is a challenging subject in theory.

FUTURE DIRECTIONS -EXPERIMENT
More on the QCD Critical Point: As discussed above, the search for the QCD Critical Point has been led by the RHIC BES program, where the collision energy has been dialed down from 200 GeV (see Fig. 3). It spans a µ Brange from 20 to 400 MeV of the phase diagram. The fluctuations near the QCD Critical Point are predicted to make κσ 2 swing below its baseline value (= 1.0) as the critical point is approached, then going well above, with both the dip and the rise being significant for headon nuclear collisions [27]. The data show a substantial drop and intriguing hints of a rise for the lowest energy collisions, although the uncertainties at present are too large to draw definitive conclusions [see panel (2) of Fig. 3]. The ongoing phase-II of the BES program and the fixed target program at RHIC aim to gather high statistics data to look for this important landmark in the QCD phase diagram. The lattice-QCD calculations suggest that the QCD Critical Point, if exists, lies for µ B /T 2 [17]. Thus, the role of upcoming high baryon density experiments as listed below becomes important in the critical point search program. Not only they extend the search to high µ B regions (≈ 750 MeV) of the phase diagram, they also provide a reverse approach of studying the critical point observable by dialing up the beam energy. This approach is complimentary to the current searches and the observable studied from both directions of collision energy (i.e., from both above and below the QCD Critical Point) is expected to meet at a common point. This will complete test of the theoretical prediction of non-monotonic variation of κσ 2 with √ s N N .
Light Hypernuclei Production: Figure 6 shows the mid-rapidity yields of light-nuclei and (multi- )hypernuclei, from thermal (HRG) model calculations, shown as a function of colliding energy. All of the data points are from Refs. [69,70] (see also Ref. [71] for a theoretical analysis). As one can see in the figure, all of the light hypernuclei yields are peaked around 3 − 8 GeV range fully covered by both STAR fixed target (FXT) program [72] (hatched region) and future CBM experiment at FAIR [73].
Data of K + over pion ratios show a peak at the centerof-mass energy of 8 GeV implying that the baryon density at Chemical Freeze-out reaches maximum around this colliding energy, see Fig. 4 (1) at top panel. Due to the relatively low production threshold, the production of the Λ hyperon becomes abundant. The coalescence process [74] combines these advantages and leads to the copious production of the hypernuclei in this energy region. The strangeness degrees of freedom is therefore introduced into the dense nuclear matter. The cross sections for hypernuclei productions in high energy nuclear collisions are much higher than that in either elementary collisions or Kaon induced interactions, making the heavy-ion collision as a hypernuclei factory (HNF). The HNF offers a great opportunity for studying fundamental interactions of hyperon-nucleon (Y -N ), hyperon-hyperon (Y -Y ) within the many-body baryonic system and the spectroscopy of nuclear structure with strangeness [75]. In addition, these nuclear collisions provide the means to study the inner dynamics of compact stars in the laboratory. We should note that most of the studies on hypernuclei so far utilized the "light system" with electron or pion or Kaon beams. In such cases the hypernuclei were produced in vacuum. Data on hyperon produc- Interaction rates (in Hz) for high-energy nuclear collision facilities. Collider mode: the second phase RHIC beam energy scan (BES-II) [72] for 7.7 GeV < √ sNN < 19.6 GeV (filled-red-circles) and NICA (filled-redsquares) [79]. Fixed target mode: STAR fixed target (FXT) program for 3.0 GeV < √ sNN < 7.2 GeV (filled-black-circles), FAIR (CBM, SIS) [73], HADES [80], J-PARC [81], and HIAF [82]. Also shown for reference the rates of ALICE at LHC [83] and sPHENIX at RHIC [84]. tion in nuclear collisions is scarce [76]. Measurements of hypernuclei collectivity in the truly heavy-ion, Au+Au, collisions, for example, allow one to extract information on the transport properties (crucial for the neutron star stability, see Ref. [52]) as well as the Y -N interaction driven EoS, with the strangeness degrees of freedom, in the hot and dense environment where the baryon density could be very high. Simulations for neutron star inner properties crucially depend on the EoS, see Ref. [77] for the effect of Y -Y interactions, and also Ref. [78] for the hyperon effects including a possibility of quark mixture. Furthermore, an additional benefit of the unique high baryon density environment is the enhanced production of multi-Λ hypernuclei as already suggested in Fig. 6.
The future accelerator based experiments, as introduced below and aimed for high baryon density matter, compresses the baryonic matter in heavy-ion collisions around 2 − 8 GeV, which takes place in an ideal location for serving as the HNF as shown in Fig. 6. Hence, these future experiments can make tremendous contributions towards detecting and measuring the yields of hypernuclei and their life-time. This will then provide valuable inputs to understanding the hyperon-nucleon (Y -N ) interactions in heavy-ion collisions and inner dynamics of the compact stars.
Fluid Vorticity of High Baryon Density Matter: Experiments at RHIC and LHC have observed that the polarization of hyperons and vector mesons have a distinct energy dependence. Their values increase with decrease in collision energy. The physics reasons attributed for the observed energy dependence are twofolds. Firstly, the baryon stopping is enhanced and shear flow patterns in the beam direction emerge, and secondly, a shorter lifetime of the fluid phase thereby allows perseverance of the initial vorticity in the system from getting diluted [85]. The possibility of high interaction rate experiments in high baryon density matter at the upcoming facilities, opens up an unique facility to study relativistic interplays of the spin, the orbital angular momentum, and the magnetic field in QCD matter. This will guide theoretical developments in the field of relativistic spin-and magneto-hydrodynamics.  Fig. 7. Note that all the new facilities under construction are focused in the energy region where the baryon density is high. Below we discuss briefly the salient features of these four experiments.
NICA a JINR: At this new accelerator complex under construction the plan is to provide accelerated particle beams both in collider (Multi Purpose Detector) and fixed target (Baryonic Matter at Nuclotron) modes [79]. The gold nuclei collision energies will be in the range, √ s N N = 4 − 11 GeV. The physics goals are dominantly to explore the QCD phase diagram through measurements of particle yields, collective flow, femtoscopy etc. In addition, it also emphasizes studying polarization of hyperons and investigating hyperon-nucleon (Y -N ) interactions through hypernuclei production. As one sees in Fig. 7, NICA connects the high-energy collider experiments with the FXT experiments nicely.
CBM a FAIR: This facility currently under construction will offer the opportunity to study nuclear collisions at extreme interaction rates. It will initially comprise of the SIS100 ring which provides energies for gold beams of √ s N N = 2.7 − 4.9 GeV and µ B > 500 MeV. The CBM detector at FAIR has been designed as a multi-purpose device which will be capable to measure hadrons, electrons, and muons in heavy-ion collisions over the above beam energy range at interaction rates up to 10 MHz for selected observables. The physics goals include studying the phase structure of the QCD phase diagram (i.e., the order of the transition, the QCD Critical Point, and chiral symmetry), possible modification of properties of hadrons in dense baryonic matter, and the EoS at high density as is expected to be relevant to the cores of neutron stars through measurements of hypernuclei and heavy multi-strange objects.
JPARC-HI a KEK/JAEA: The idea of this facility is under discussions for several years and the planned J-PARC-HI will provide heavy-ion beams up to uranium for center-of-mass energies of 2 − 6.2 GeV. This corresponds to exploring the QCD phase diagram in very high baryon densities [81]. It excepts to carry out important measurements including dileptons to understand QCD transitions, in-medium modifications of ρ, ω, and φ mesons decaying into dileptons, rare particles such as multi-strangeness hadrons, exotic hadrons, and hypernuclei utilizing high rate beams. According to the plan, this will be the experiment with the highest beam rate capability up to 100MHz allowing precision measurements for rare processes in heavy-ion collisions.

CEE a HIAF:
The complex of the HIAF is under construction and it is expected to be in operation in 2025. The machine is designed to deliver bright ion beams of protons and heavy nuclei such as uranium with the center-of-mass energy up to 10 GeV and 4 GeV, respectively. A superconducting dipole magnet spectrometer experiment (CEE) [87] is also under construction. In many respects, this is a simple hadron spectrometer with the main physics focused on the measurements of (i) proton, light nuclei including hypernuclear production and correlation for understanding the QCD phase structure and (ii) meson ratios for extracting the EoS at the high baryon density region.
Future new experiments are all designed with high rates, large acceptance, and the-state-of-the-art particle identification, at the energy region where baryon density is high, i.e., 500 MeV < µ B < 800 MeV, see Fig. 7.

CONCLUSIONS
We reviewed what we have understood so far and what we are trying to understand in the future using the relativistic heavy-ion collisions. There are three important physics targets: (1) Scanning the QCD phase diagram and seeking for the QCD Critical Point.
(2) Constraining the Y -N and Y -Y interactions and the EoS in dense baryonic matter including strangeness degrees of freedom.
(3) Exploring the effects of large angular momentum and strong magnetic fields.
For (1) "Criticality" is essential to detect the critical point, and the extraction of the EoS as (2) needs the global analysis including "Collectivity", and the physics of (3) exhibits topologically nontrivial effects once nonzero "Chirality" is involved. These three C's abbreviate the future directions of the heavy-ion collision physics. The first-principles calculations from the lattice-QCD simulation have shown tremendous progresses with the cutting-edge computing technologies also toward the high-density region. Now understanding the QCD phase structure is in need of experimental guides together with theoretical approaches. The QCD Critical Point is a landmark, and the next question is what awaits beyond it. If the first-order phase transition is reached, the spinodal decomposition and the nucleation processes would lead to characteristic patterns of baryon fluctuations.
In the heavy-ion collisions of center-of-mass energy below 15 GeV, one of the important features is that the baryon density is high enough to be above the threshold for the strangeness production. The interesting observation is transitional behavior from the grand canonical to the canonical ensembles in the strangeness sector with different collision energies. Also, with abundant strangeness, N -N , Y -N , and Y -Y interactions can be investigated not only in the vacuum but in an environment with surrounding baryonic mean field. Under such a situation, measurements of baryon correlations and collectivity involving multi-strange hadrons such as φ-meson, Λ, Ξ, and Ω-baryons, and hypernuclei will provide us with the information on the EoS relevant to the neutron star structures and simulations of the supernovae and the neutron star merger. Although it is not covered in this article, as long as the penetrating observables are concerned, dilepton mass distributions for example give us informaion on the initial thermal properties for matter created in the heavy-ion collision (see a recent review [88]).
The unique property of matter in the heavy-ion collision is the presence of rotation (or the angular momentum) and the external magnetic fields. Such extreme environments at high baryon density, rapid rotation, and strong magnetic fields can be found not only in the heavy-ion collisions but also in astrophysical phenomena. Therefore, revealing those effects in the controlled laboratory experiments shall bring conner stones for understanding the nature of visible matter, through the Femto-Nova, in the Universe.