Long range plans to study the nuclear equation-of-state from sub-to supra-saturation densities with heavy-ion collisions.

. We cover here the present state-of-the-art in constraining the nuclear equation-of-state (EoS) and the symmetry energy using heavy-ion collisions (HIC), from sub-to supra-saturation densities, from Fermi to (ultra-) relativistic beam energies. We also discuss how HIC constraints on the EoS contribute to the knowledge of thermodynamical properties of neutron star matter. Necessary improvements and challenges are outlined, in particular in the perspective, for HICs, of staying competitive with future astrophysical multimessenger observations.


Introduction
Constraining the nuclear equation-of-state (EoS) of nuclear matter, from sub-to supra-saturation densities has given rise, since two decades, to a very intense experimental and theoretical activity worldwide.Important heavy-ion collision (HIC) projects are being carried-out at European heavy-ion accelerator facilities, from Fermi energies (GANIL, INFN-LNS) to (ultra-) relativistic energies (GSI, CERN).An early strong involvement in these studies has also been established in the USA, specifically at the Lawrence Berkeley National Laboratory (LBNL), at Brookhaven National Laboratory (BNL) and at the Michigan State University (MSU).Apart from a deeper knowledge of the EoS, which is a fundamental nuclear matter property, one of the main aims is to understand astrophysical processes like core-collapse supernovae explosions, neutron star properties like mass, radii and deformability, binary mergers and their associated phenomena such as gravitational wave emissions and kilonova signals.In a pioneering work [1], constraints on the pressure of symmetric nuclear matter (SNM) at large densities could be obtained from transport model calculations (pBUU) by analysing experimental data from LBNL and BNL on 197 Au+ 197 Au collisions at energies up to 10 GeV per nucleon.These collisions probe nuclear matter at 2-4.5 times saturation density (nsat»0.17fm -3 ).Similarly, around the same time and at later years, several experimental HIC campaigns at GSI Darmstadt have been performed to study the EoS of isospin symmetric matter at lower densities, ranging from 0.7 to 3×nsat.First, comparing the yields of kaons [2] produced in C+C and Au+Au collisions at sub-threshold incident energy, the KaoS collaboration, by using various transport models (IQMD [3] and RQMD [4]), has concluded that the EoS of SNM is soft (with a nuclear incompressibility K0»200 MeV) provided that the right momentum dependence of the nucleon-nucleon potential is taken into account, as demonstrated before in pA collision experiments.Thereafter, a complete excitation function campaign was performed by the FOPI collaboration with Au+Au collisions at energies between 90 AMeV and 1.5 AGeV.By studying the elliptic flow of light charged particles, the results of the KaoS collaboration could be confirmed.These new measurements with FOPI were performed with an unprecedented precision [5].Both IQMD and UrQMD, used to extract EoS contraints from the data, agreed on KaoS conclusions, leading to incompressibility values of K0=190 ± 30 MeV and 220 ± 40 MeV, respectively [6].In Fig. 1, the present constraints from HIC experiments on symmetric nuclear matter EoS are summarized from intermediate energies (GSI data, red band) to ultra-relativistic energies (AGS data, green band).As a comparison, recent predictions from chiral effective field theory are displayed as well (2-body N 2 LO (light blue band) and 3-body N 3 LO (dark blue band) interactions), showing a fair agreement with FOPI results.Another complementary valuable observable to constrain the EoS at high density is the continuum dilepton radiation, as addressed in White Papers by CBM and HADES, the authors believe that adding this observable will help reducing model uncertainties, which is a big issue in this field [7].

Symmetry Energy
With the growing interest in properties of exotic nuclei and neutron star matter, a strong international activity aiming at constraining the nuclear symmetry energy has arisen.Numerous experimental on-earth methods based on heavy-ion collision studies have been developed.Sub-saturation densities have been probed on the one hand with measuring neutron-skin thicknesses of nuclei, collective resonances, short range correlations, isotope yields from the break-up of colliding heavy-ions, nuclear masses, and isobaric analog states, providing a more and more precise description of the density evolution of the symmetry energy as depicted on Figs. 2  and 5. On the other hand, at supra-saturation density, the symmetry energy has been constrained with the best so far achieved accuracy through the comparison of the elliptic flow of neutrons and light charged particles: this was obtained from FOPI-LAND re-analyzed data [8] and later improved by the ASY-EOS experiment at GSI [9].The successful S394 experiment held by the ASY-EOS Collaboration at the GSI [9] has pushed further the boundaries of our knowledge of the symmetry energy of the nuclear equation-of-state (EoS), up to two times the saturation density.The symmetry energy is found to be moderately soft with a nearly linear behaviour.After this achieved accuracy of the methods used by the ASY-EOS collaboration, the astrophysics community recently recognized (see the NEOS conferences, the Rußbach (Austria) School of Astrophysics and the Hirschegg 2020 Workshop) that laboratory-based HIC are also a powerful tool to constrain neutron star matter properties.Recently, an attempt has been done at RIKEN in Japan with the SpIRIT experiment [10], to contribute to this constraint at medium-high densities by measuring the yield ratio of p + over p -.This result is also shown in Fig. 2. It is consistent with the results obtained at GSI, though less accurate, mainly due to the omission of nonresonant pion production channels.Confirmation of this result using other transport models is required and foreseen for the near future.The most accurate experimental result on the neutronskin is presently provided by the PREX experiment [11], but it still remains in the present situation, with very large 1 sigma errors, far not as accurate as e.g. the flow method mentioned before or the chiral effective field theory to constrain the symmetry energy.In the future, developments of methods applicable to neutron-rich nuclei are needed, in order to provide precise and accurate data.At FAIR, proton elastic scattering of radioactive beams can be performed at the storage ring, after the collector ring will be implemented.In addition, an electron-heavy ion collider for radioactive beams at GSI/FAIR is proposed for the future as an outside experiment built at GSI.Then neutron density distributions and corresponding neutron skins can be extracted precisely.This also points to the need of bench-marking nuclear transport models, which is compulsory to draw robust constraints on the EoS.The new constraints coming from HIC have stimulated, since 2017, a renewed and intense activity on the theoretical side, including a world network initiative of code comparison TMEP (Transport Model Evaluation Project).This long-term international transport code comparison project has been pursued [12], and is still on-going, presently focusing on the pion production for better interpreting S π RIT and similar experimental data sets.A very close collaboration between theory and experiment is of utmost importance to obtain reliable information from HIC experiments.
In a pioneering attempt to extract a symmetry energy constraint at large density, the FOPI collaboration measured K + /K 0 yields from Ru+Ru and Zr+Zr collisions [13].The results remained inconclusive due to low experimental statistics, the difficulties of measuring kaon ratios for one collision system rather than double ratios, and uncertainties in describing kaon production.The advantage of using kaons relies on the fact that they are expected to be produced at the early stages of the collisions, when nuclear matter is in a high-density state due to compression, and do not interact significantly with the dynamically evolving system before being detected.The EoS is from the kaon multiplicity and kaon participant dependence [14].They may thus represent a more sensitive probe of the supra-saturation density dependence of the symmetry energy, as compared to the pion ratios [15].Recently, motivated by the increasing precision obtained by astrophysical multi-messenger constraints of the EoS of neutron star matter (with the measurement of gravitational waves, kilonovas, X-ray emissions of binary pulsars), an interdisciplinary research group gathering nuclear theorists, heavy-ion collision experimentalists, and astrophysicists has quantitatively demonstrated the impact of HICs on the knowledge of neutron star matter properties, that is remarkably consistent with astrophysical measurements, and therefore very complementary: "The simultaneous analysis of the HIC data and the full suite of astrophysical data on neutron stars within a selfconsistent Bayesian framework is a novel contribution to the literature » (S.Huth et al., Nature 606, 276 (2022)).Several conclusions have been drawn from this study (as it is illustrated by Fig. 3 and 4): 1. Constraints from HIC data favor higher pressures, similarly to what is deduced from NICER explorations, with an overall remarkable consistency with chiral EFT and astrophysical constraints 2. Up to densities of the order of 1.5r0, HICs constrain the EoS of neutron stars with an accuracy comparable to that of most recent astrophysical findings, favoring a somehow stiffer EoS (higher pressure).
3. Above 1.5r0, astrophysical measurements are still more accurate, and drive the neutron star EoS, but with lower statistics.
4. The most significant densities for constraining NS radii are as it follows: for 1.4 M⨀, ρ ≈ 1.6 nsat; for 2 M⨀ : ρ≈(2-2.5)nsat One of the main conclusions of this innovative work is to significantly contribute to constrain neutrons star radii and the density dependence of pressure in neutron stars, HIC should not only improve the accuracy of the symmetry energy constraints just above saturation density, but also probe it at larger densities, as compared to the so far limited reachable regime of about 1.5 nsat.
Various experimental HIC projects, focused especially on constraining the symmetry energy below and close to 3 nsat are planned.Among them, we mention: a) a second ASY-EOS experiment at GSI at higher incident energies, aiming at performing new measurements of elliptic flow with improved accuracy, probing higher densities with respect to the first ASY-EOS experiment, and profiting of the improved capabilities of the new NeuLAND detector; b) pion production measurements close to production threshold, to be performed at RIKEN.
Other experiments such as HADES at GSI, aiming at measuring kaon yields close to the production threshold, also promise to possess the potential of constraining the symmetry energy at larger densities.But the prerequisite would be to better bench-mark transport models to get more robust symmetry energy constraints, considering the complexity of mechanisms and dynamics ruling the kaon emission in the fireball.A robust description of pion production is a prerequisite for kaon studies [16].This concerns models like IQMD, dcQMD [17], UrQMD, PHQMD [18], SMASH [19].
At high densities, complementary information can be provided by investigating the isospin dependence of short-range correlations.A first experiment has been performed at R3B FAIR Phase-0.A full program with an upgraded setup will provide data on a wide isospin range along isotopes.In the farer future, short range correlations could be studied at a multi-GeV radioactive ion beam facility which is proposed as a later moderate upgrade of FAIR.Fig. 5. Compilation of constraints on symmetry energy obtained with heavy-ion studies.The blue hatched area are the most recent (preliminary) results obtained from the analysis of the spectator decay products of various projectiles as measured by the ALADiN@GSI S254 experiment [23].To guide the eye, density dependence functionals used in IQMD and FRIGA clustering algorithm [24] are shown as lines for various values of the power exponent of the symmetry potential used in IQMD-FRIGA (related to the stiffness of the symmetry energy dependence on density): 0.5 (green dasheddotted), 2/3 (black solid) and 1 (blue dashed).Various recent findings of the literature are displayed for comparison: AsyEOS (from Russotto et al. ( 2016) re-analysed with IQMD) as orange area, HIC (Sn+Sn) determined from isospin diffusion observables measured in mid-peripheral collisions of Sn isotopes [25] (grey area), IAS + Rnp from the analysis of isobaric analog states (IAS) supplemented with additional constraints from neutron-skin data [26] (magenta hatched area), along with results (as black markers) extracted from nuclear structure information by Brown [27], Zhang and Chen [28], Fan et al. [29], Roca-Maza et al. [30], Zhang and Chen [31] using dipole polarisability data by Tamii et al. [32].
As shown in the interdisciplinary study of S. Huth et al., Nature 606, 276 (2022), the combination of HIC constraints on the nuclear EoS with effective field theory of strong interactions is very successful in determining with an enhanced accuracy the thermodynamical properties of neutron star matter.This points out that powerful many-body theories must be further developed to support such joint efforts.In parallel to the chiral effective field theory which is commonly used in Bayesian analysis of neutrons-star multi-messenger studies, constraining strongly the subto near saturation region, one should note that HIC may also provide a valuable contribution, because they are increasingly accurate in constraining the symmetry energy at low densities.This is demonstrated by numerous recent experimental constraints such as the ones illustrated on Fig. 5, with a remarkable increasing accuracy and consistency between results using different observables and methods.A similar study has been recently published by Lynch et al. (Phys.Lett.B 830, 137098 (2022)), where also the attempt is made to determine the sensitive density of a particular observable more systematically.Similar efforts are ongoing using the isospin migration/distillation effect as a probe of the strength of the symmetry energy in neutron rich or isospin asymmetric colliding systems at the Fermi energy regime (see for instance Phys.Rev. C 78, 064618 (2008) and Nucl.Phys.A 730 (2004) 329), with e.g. the INDRA-VAMOS [20], INDRA-FAZIA [21] and CHIMERA-FARCOS [22] experiments.However, even at these energies there is a strong need for transport model benchmarking in order to provide robust probes, concerning in particular a more accurate description of the cluster formation in HIC's.Adding to the constraining power of probes, new opportunities with radioactive ion beams are presently emerging, with the need of neutron detection, at both INFN-LNS and GANIL.

Beyond densities of nsat
In the next decade, astrophysical multi-messenger observations are foreseen to deliver new constraints on the neutron matter EoS, with unprecedented accuracy, thanks to more precise apparata and larger statistics, being able to detect events coming from larger distances in the universe.In the context of HIC explorations, in order to maintain a competitive contribution to our understanding of neutron star physics, it is compulsory to probe densities that are higher than 3×nsat.Such task will concern both the symmetric part of the EoS, which is still not accurately constrained (see Fig. 1, green area) at these densities, and the symmetry energy.Such a program could be performed using the already existing HADES set-up, as well as the future CBM [33] and R3B detector ensembles at GSI/FAIR, with HIC's at bombarding energies above 2 A×GeV.In particular, the precise measurement of the collective flow of protons and light fragments in HIC at energies up to 10A GeV by the CBM experiment will significantly improve our knowledge on the high-density EOS of symmetric nuclear matter.On the theoretical side, a strong effort of the community working on transport models for relativistic energy collisions needs to be sustained in order to deliver observables that are both robust and sensitive to the nuclear EoS (for both isospin symmetric and asymmetric matter).Indeed, elliptic flow phenomena, that are very successful to constrain the EoS at lower energies, will be significantly reduced above 2-3 A×GeV incident energies.The reason is the very fast escape of spectators with respect to the expanding evolution of the fireball, which eliminates the spectator shadowing effect that is needed to generate a negative elliptic flow of particles emitted from the participant region, whose amplitude informs on the expansion velocity -hence on the compressibility -of this initially compressed region [34].Due to the highly relativistic nature of the dynamics at such bombarding energies, a new generation of relativistic transport models needs to be developed (within the scientific context of SMASH, PHQMD and other approaches), using the state-of-the-art description of in-medium potentials, momentum/effective mass dependencies, short-range correlations, off-shell transport with spectral functions for all particles, and possibly sub-nuclear degrees-of-freedom, benchmarked with e.g.data taken at SIS18 at the highest available beam energies (FOPI, ASY-EOS, HADES).

Conclusions for a long-range plan
According to the above-mentioned physical motivations, it will be important to support the following actions: -High precision measurements of cluster production at low density, of isospin diffusion and migration effects using HICs, and exotic systems at GANIL, INFN-LNS and GSI; -High precision measurements of elliptic flow of neutrons and protons at relativistic energies to probe supra-saturation densities up to around 3×nsat (i.e. up to about 2 A×GeV) at GSI (R3B); -Perform high precision measurements of collective flow of protons and light fragments in HICs up to 10 A×GeV (up to 5×nsat) at FAIR; -Stimulate significant efforts in developing new neutron detectors to be used in various energy regimes, characterized by good angular and energy resolutions with a high granularity and efficiency; -A strong collaboration to develop and use transport models in Europe (IQMD, UrQMD, dcQMD, SMASH, BUU, PHQMD, and others) aiming at providing new and robust observables sensitive to the nuclear EoS and to the symmetry energy.
-The opportunity/challenge to apply Bayesian statistics just for the model-to-data comparison to extract the nuclear equation of state from heavy-ion collisions with transport codes.
-Develop alternative methods to PREX for neutron-rich nuclei, in order to provide precise and accurate data

Fig. 1 .
Fig. 1.Constraints from HIC experiments on symmetric nuclear matter EoS.See text for details.

Fig. 2 .
Fig. 2. Compilation of nuclear symmetry energy constraints obtained with heavy ion experiments, compared with recent astrophysical constraints.Taken from Ref. [9] and complemented with results from Horowitz et al., J. Phys.G (2014).Recent PREX [11] and SpRIT data have been added.

Fig. 4 .
Fig. 4. Posterior constraints on the EoS of neutron-star matter, obtained from the Bayesian analysis on distributions for the pressure at 1.5 nsat (left) and 2.5nsat (right) at different stages of the analysis, with respective constraints coming from HIC's (orange curve), astrophysical multi-messenger observations (green curve), and with combining both (light blue shaded region) respectively.The dashed curve is the prior distribution.From Ref. (S. Huth at al., Nature 606, 276 (2022)).