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QCD Phase Structure and Interactions at High Baryon Density: Continuation of BES Physics Program with CBM at FAIR

We advocate for an active US participation in the international collaboration of the CBM experiment that will allow the US nuclear physics program to build on its successful exploration of the QCD phase diagram, use the expertise gained at RHIC to make complementary measurements at FAIR, and contribute to achieving the scientific goals of the beam energy scan (BES) program.

content, which will enable a precise determination of properties of dense QCD matter not yet explored in the RHIC BES. In particular, studies of dense matter up to several times nuclear matter saturation density with varying isospin content could provide crucial constraints for both the nearly-symmetric and asymmetric equation of state, with the latter having particular significance for neutron stars. We propose high-quality and high-statistics measurements of spectra, flow, and femtoscopic correlations from a systematic scan of beam energies and target-projectile combinations at FAIR/CBM that will shed more light on the dynamics and the equation of state of dense nuclear matter, including potentially measuring the speed of sound in nuclear matter over a broad temperature and chemical potential domain and helping constrain the maximum mass of neutron stars.
These high-quality measurements in the FAIR/CBM energy region will build on the emerging success of studies pioneered at RHIC, help answer some of the key physics questions that originated the BES campaign, and allow reaching the full potential of the scientific program started at RHIC.
There is no doubt that the proposed CBM physics program will be essential to addressing questions concerning the phase structure of nuclear matter at the highest baryon densities achievable in a controlled experiment. The US participation in CBM will not only greatly enhance the CBM physics program, but will also strengthen US leadership in nuclear physics. The RHIC physics program has had defining contributions in the exploration of QCD matter, including the discovery of the Quark-Gluon Plasma (QGP) in very high-energy nuclear collisions [1][2][3][4]. For these high-energy collisions at high temperature and vanishing net-baryon density, the QGP-hadron transition is understood to be a smooth crossover based on lattice QCD calculations [5]. Since making these discoveries in the early 2000's, scientists have been asking: What is the structure of the QCD phase diagram in the high Model studies indicate that a first-order phase boundary is expected at large µ B , as illustrated by the white line in Fig. 1, showing the QCD phase diagram; see discussions in Refs. [6][7][8]. If this is the case, then there is a QCD critical point (strictly speaking, a critical region for finite systems) between the first-order phase boundary and the smooth crossover indicated by lattice QCD [5]. One of the most prominent goals of the BES program is to search for this critical point and the corresponding softening in the equation of state due to a first-order phase transition [9,10]. Nuclear collisions at BES energies, where in every collision a considerable number of hyperons are produced on a scale of several femtometers, present a unique opportunity to study hyperon-hyperon interactions, which are otherwise inaccessible due to their short lifetimes. Hyperon-nucleon and hyperon-hyperon interactions are fundamental ingredients to understand QCD and the equation of state (EoS) that governs the properties of nuclear matter and astrophysical objects such as neutron stars [11].

RHIC BES CBM
General arguments suggest that at nonzero quark density there is a quarkyonic phase, that is a phase of strongly-interacting matter which can be chirally symmetric yet confined [12].
Such a phase can contain both pion/kaon and quarkyonic condensates [13], as well as exhibit behavior of a quantum pion liquid [14]. Distinctive signatures of these exotic phenomena are predicted to be accessible in heavy-ion collisions through two-pion correlations [15] and dilepton spectra [16,17].
We anticipate the BES measurements to yield cutting-edge contributions toward resolving the physics questions described above, and more.  CBM energy range), the interplay between the expansion of the fireball and the influence of the spectators is highly dependent on baryonic interactions. Therefore, measurements of flow observables (in particular, the elliptic flow v 2 and the slope of the directed flow dv 1 /dy| y=0 ) can lead to significant constraints on the high-density EoS, with collisions at √ s NN ∼ 3-4 GeV and √ s NN ∼ 4-5 GeV being most sensitive to the EoS at densities 2-3 and 3-4 times the saturation density, respectively [35]. Recent studies [35,36] including the latest STAR measurements at √ s NN = 3.0 and 4.5 GeV [37,38] indicate that heavy-ion data are consistent with a relatively hard EoS at moderate densities (1-3 times saturation density) and a relatively soft EoS at high densities (3-5 times saturation density). This creates some tension with neutron star data, which for these density ranges indicate a steep rise in the speed of sound in asymmetric nuclear matter [39][40][41]. Moreover, while proton flow is relatively well described by models, pion flow (sensitive to the isospin dependence of the EoS, see, e.g., [42]) and Lambda baryon flow (sensitive to strange interactions, see, e.g., [43][44][45]) are not well described, see Fig. 2.
Precise measurements of differential flow observables will be necessary for more accurate model comparisons. The CBM experiment, with its excellent acceptance, will enable detailed π + SMASH π -SMASH π + UrQMD π + JAM 3.0 GeV Au+Au collisions at 10-40% centrality from STAR measurements [38] and UrQMD, JAM, and SMASH hadronic transport simulations [35,38]. All simulations used a relatively hard EoS at moderate densities, with the EoS used in SMASH becoming significantly softer at high densities; see [35,38] and references therein for more details.
studies of the rapidity-dependence of flow, leading to more meaningful constraints on the symmetric nuclear matter EoS. The unprecedented interaction rate at CBM will also allow to put tighter constraints on hyperon interactions (see Section II C for more details). Moreover, while the RHIC BES program was largely based on a single collision system, Au+Au, at FAIR collisions of a variety of nuclei of different size and isospin content can be achieved.
Crucially, a comparison of isobar species with the same mass numbers but different nuclear shape and/or isospin content [46][47][48] provides a lever arm to vary the initial condition and observe its importance for the final state [49][50][51][52][53], as well as allows one to the extract information about the isospin dependence of the EoS [54,55]. A scan of nuclei with different mass numbers could also provide insights into the onset of the hadron-quark transition while minimizing the impact of volume fluctuations [56,57] and baryon conservation [8], complementing corresponding upcoming studies within the RHIC BES program. and their dependence on the transverse momentum p T , rapidity y, and particle species, will provide stringent tests on the modeling approaches and may help determine the correct dynamical description in terms of the microscopic degrees of freedom.
Altogether, the FAIR system scan program will provide an unprecedented diverse set of flow data necessary for understanding the EoS of symmetric and asymmetric dense nuclear matter, the initial state of heavy-ion collisions, and QCD dynamics, complementing the flow measurements performed at RHIC.  [62][63][64]. For a comparison with a noncritical baseline, model results from HRG, based on both Canonical Ensemble (CE) and Grand-Canonical Ensemble (GCE) [57], and transport model UrQMD (cascade mode) [65,66] are also shown. The energy range covered by the CBM experiment is shown as the yellow hatched area. The nonmonotonic curve indicates the qualitative shape expected due to the contribution of critical fluctuations. correlations as compared to a non-critical baseline including effects due to, e.g., baryon number conservation [70], with the largest deviations seen at √ s NN = 7.7 GeV. In lowenergy collisions, net-proton cumulants have been also recently proposed as a means for extracting the speed of sound and its logarithmic derivative [71].
Significantly improved statistical precision (and likely reduced systematic uncertainties) are expected from RHIC BES II, as indicated by the green band in Figure 3; In the right panel of Fig. 4, the experimentally measured ratio S 3 ≡ 3 Λ H/ 3 He Λ/p , in which trivial factors (such as the chemical potential and canonical effects) cancel to reveal the strength of the hyperon-nucleon interaction, is shown along with model calculations [81][82][83][84].
New precision data from STAR show a gradual increase of S 3 as a function of √ s NN , with the value approaching the equilibrium limit at the LHC [85]. It is interesting to note that the limiting value in the canonical calculations of S 3 is about 2/3, which is commonly used in modeling of Λ-N interactions [86]. Conversely, in collisions at √ s NN ≤ 10 GeV, S 3 is further away from the thermal limit, suggesting an effect due to probing a high net-baryon density region. In addition, yields of light nuclei and hypernuclei are potentially sensitive to multiple-baryon correlations, which constitute an exquisite test of QCD. For example, the interaction Λ-N -N may be important for the EoS relevant to physics inside neutron stars [11].
More precise measurements are needed in order to understand the Y -N and Y -Y interactions at high baryon density, which have substantial implications for the inner structure of compact stars. The RHIC BES data sets for collisions in the fixed-target mode contain on the order of 100M events at each collision energy, which is not enough for these analyses Real and virtual photons radiated from the hot QCD medium carry sensitive information on the thermal and chiral properties of the medium. A combination of blue-shift-free invariant mass spectra of virtual photons (detected via dileptons) with the blue-shifted p T spectra for different masses provides particular constraints of the medium temperature as well as the collective flow throughout different phases of the evolution [91,92]. In consequence, they offer unique insights into the EoS of the medium in the finite temperature and density regime [89,93,94]. Figure 5 shows the integrated dilepton excess yields in the mass region of 0.3-0.7 GeV/c 2 as a function of µ B from STAR, NA60, and HADES [87][88][89].
A recent theoretical study has shown that a softening of the EoS due to a first-order phase transition can result in an increase of the low-mass dilepton yield relative to a cross-over scenario [95]; for an example, see the dashed line in Fig. 5. At the same time, the slope parameter of the dielectron excess mass spectra, as a measure of the medium temperature, may exhibit a distinct sharp change due to a first-order phase transition [90]. Therefore, if the phase transition between hadrons and QGP occurs in the energy range √ s NN = 2.4-10 GeV (corresponding to 800 ≥ µ B ≥ 400 MeV), a systematic measurement of the excitation function of dielectron production will provide an excellent opportunity to probe the firstorder phase transition line, as well as to conduct studies of the relationship between chiral symmetry restoration and the onset of deconfinement in this regime, including the appearance of quarkyonic matter [17]. The CBM experiment at FAIR has dedicated apparatus for both di-electron and di-muon measurements. Together with three order of magnitude higher interaction rates compared to current experiments, the CBM experiment is expected to offer unprecedented precision on dilepton observables. Moreover, the conditions achieved in heavy-ion collisions at HADES, BES-II FXT, and the future CBM experiment at FAIR are comparable to those encountered in the simulations of neutron star mergers at early times [89,96], and thus of fundamental significance for nuclear astrophysics. Is the strong φ spin alignment a consequence of the proposed φ mean field in the nuclear medium, as put forward in Ref. [103]? Could the vector meson mean field be connected to high baryon density and a strong vorticity field, as suggested by recent calculations [104]? Answering these questions requires precision measurements at large µ B . The CBM experiment at FAIR, due to its unprecedented interaction rates and the capability of performing a clean φ reconstruction via both hadronic (K + K − ) and dilepton channels, will provide a unique opportunity to address these questions, offering fresh insights into the structure and spin transport in dense nuclear matter in the presence of strong vorticity fields.

F. Connections to Nuclear Astrophysics
Understanding high-energy nuclear collisions and astrophysical phenomena, such as supernova explosions or neutron-star collisions, requires a robust knowledge of matter at high densities [105]. While an EoS that is soft at low densities is needed to fit within the LIGO gravitational wave constraints, a steep rise in the speed of sound is needed above saturation density to produce recently observed M ∼ 2M neutron stars. In fact, if mysterious heavy and compact objects as massive as GW190814 [106] are confirmed to be neutron stars, then the speed of sound may reach nearly the causal limit [41].
New methods are being developed [35] to connect the neutron star EoS to that extracted II D), then the consequences of this first-order phase transition could be seen in neutron star mergers [107,108], in core collapse supernovae [109], or in the existence of mass twins [110].
Moreover, while microscopic models of the neutron star EoS often predict strange baryons to appear at large densities [111,112], they struggle to reconcile the softening hyperons cause with observational data (which is known as the hyperon puzzle). Measurements of strange baryon interactions as envisaged in Sec. II C, could provide crucial insight into this puzzle [113].
Heavy-ion collisions have given rise to an experimentally-driven field that has been crucial to the theoretical formulation of far-from-equilibrium relativistic fluid dynamics [114][115][116][117][118][119][120][121] and simulations that incorporate shear and bulk viscosity [122][123][124][125] as well as charge diffusion [126,127]. Furthermore, the mathematical formalism and theoretical tools developed to explore heavy-ion collisions have recently been applied to neutron star mergers, where weak interactions lead to an effective bulk viscosity [128], opening up entirely new fields of research to explore [129,130]. Following this parallel, one wonders whether the development of critical fluctuations in a relativistic system [131] could have consequences for neutron star mergers, especially given theoretical predictions for a low-temperature critical point induced by the axial anomaly in dense QCD matter [132]. New accelerator facilities are being designed to have the highest possible luminosity.
The heavy-ion collision rates for some of the current and future facilities and upgrades are summarized in Fig. 7. The NA61/SHINE experiment is an ongoing experiment at the Super Proton Synchrotron (SPS) at CERN, investigating hadron production in collisions of beam particles (pions, protons, and beryllium, argon, and xenon ions) with a variety of fixed nuclear targets. The current program will continue until the end of 2024, and a program beyond that is being discussed. The NA60 + experiment is planned as an upgrade to the NA60 experiment at the CERN SPS, and it will study dilepton and heavy-quark production in nucleus-nucleus and proton-nucleus collisions with collision energies of √ s NN = nuclear matter at and around saturation density, focusing in particular on the properties of exotic nuclei and the density-dependence of the symmetry energy.
The CBM experiment at FAIR will provide an unprecedented interaction rate at high net-baryon density, with the capability of taking data at event rates up to 10 MHz [72,133].
This is more than four orders of magnitude higher than what was previously possible over a similar energy range, and it will allow unique precision measurements of crucial observables.
FAIR also has the ability to collide a variety of nuclei, including isobaric species with varying isospin content, allowing a precise determination of various properties of dense QCD matter.
The first phase of the CBM experiment will use the heavy-ion synchrotron SIS100 to cover the center-of-mass energy range 2.9 ≤ √ s NN ≤ 4.9 GeV, corresponding to baryon chemical potentials 800 ≥ µ B ≥ 500 MeV (see Table I for a comparison with the BES energies). The STAR BES-II data analyses are ongoing and expected to yield further insights on the nuclear phase diagram, particularly on the critical point search. However, even if the critical point is discovered in BES-II, the lower energy range covered by FAIR will be essential for the understanding of the critical point (or rather a critical region because of inevitable smearing effects) and the associated first-order phase boundary. If, on the other hand, the critical point is not evident from BES-II, the FAIR energy range will be critical to complete the search. With respect to both the rapidity coverage and event statistics, the CBM experiment will allow to build on and improve a number of measurements in the baryon-rich region, pioneered by the RHIC BES program.
Cost-wise, the US-CBM team would require a modest support, which would include support for student participation and for an upgrade of the Si-pixel-based vertex detector in CBM. Because the culmination of the RHIC-BES physics program, especially in the fixedtarget mode, will likely take place around the start of the CBM program, members of the RHIC community currently involved in the BES program will be able to seamlessly transition to the CBM program. The continued US participation in experiments probing high-density nuclear matter will not only make US-led measurements of key properties of dense nuclear matter possible, but will also support and preserve the US expertise in the field. The latter, in particular, may be of significant strategic value given the proposed 400 MeV/u upgrade of the FRIB linear accelerator (FRIB400, corresponding to √ s NN ≈ 2.06 GeV).
The discovery potential of the CBM experiment includes, but is not limited to, probing the first-order phase transition boundary, locating the QCD critical point, measuring multistrange hypernuclei, and constraining the nuclear matter EoS at high baryon density. Among other observables, the following key measurements will be carried out by CBM: 1. differential collective flow of protons, pions, and deuterons to study the symmetric and asymmetric dense nuclear matter equation of state; 2. higher-order proton cumulants and cumulant ratios to search for the QCD critical point and study medium properties at high baryon densities; 3. hyperon correlations and (multi)hypernuclei to study hyperon-nucleon and hyperonhyperon interactions, with impacts for the QCD phase diagram and our understanding of the inner structure of compact stars; 4. dilepton spectra and collective flow to characterize the medium temperature, study the chiral symmetry restoration and search for the first-order phase boundary in the high baryon density region; and 5. global polarization and spin alignment at extreme baryon densities to study vorticity and spin transport.
These measurements reflect some of the most fundamental questions, and are key to our understanding of the phases of nuclear matter, including the inner structure of astronomical objects. There is no doubt that the proposed CBM physics program, building on the success of the RHIC BES program, will be essential to the understanding of QCD matter at the highest baryon densities achievable in laboratory. CBM with its high rate capabilities permits the measurement of extremely rare probes that are essential for the understanding of strongly interacting matter at high densities [135]."