diff --git "a/89FQT4oBgHgl3EQfIjX8/content/tmp_files/2301.13253v1.pdf.txt" "b/89FQT4oBgHgl3EQfIjX8/content/tmp_files/2301.13253v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/89FQT4oBgHgl3EQfIjX8/content/tmp_files/2301.13253v1.pdf.txt" @@ -0,0 +1,5775 @@ +Dense Nuclear Matter Equation of State from Heavy-Ion Collisions +Agnieszka Sorensen +Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA +Kshitij Agarwal +Physikalisches Institut, Eberhard Karls Universit¨at T¨ubingen, D-72076 T¨ubingen, Germany +Kyle W. Brown +Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI 48824, USA and +Department of Chemistry, Michigan State University, East Lansing, MI 48824, USA +Zbigniew Chajecki +Department of Physics, Western Michigan University, Kalamazoo, MI, 49008, USA +Pawe�l Danielewicz, William G. Lynch, Scott Pratt, and ManYee Betty Tsang +Department of Physics and Astronomy and Facility for Rare Isotope +Beams Michigan State University, East Lansing, MI 48824 USA +Christian Drischler +Department of Physics and Astronomy and Institute of Nuclear +and Particle Physics, Ohio University, Athens, OH 45701, USA +Stefano Gandolfi and Ingo Tews +Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA +Jeremy W. Holt and Che-Ming Ko +Department of Physics and Astronomy and Cyclotron Institute, +Texas A&M University, College Station, TX 77843, USA +Matthias Kaminski +Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA +Rohit Kumar +Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI 48824, USA +Bao-An Li and William Newton +Texas A& M University-Commerce, Commerce, TX 75429, USA +Alan B. McIntosh +Cyclotron Institute, Texas A&M University, College Station, Texas 77843, USA +Oleh Savchuk +Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI 48824, USA and +Bogolyubov Institute for Theoretical Physics, 03680 Kyiv, Ukraine +Maria Stefaniak +GSI Helmholtz Centre for Heavy-ion Research, Planckstr. 1, 64291 Darmstadt, Germany +arXiv:2301.13253v1 [nucl-th] 30 Jan 2023 + +2 +Ramona Vogt +Nuclear and Chemical Sciences Division, Lawrence Livermore +National Laboratory, Livermore, CA 94551, USA and +Department of Physics and Astronomy, University of California, Davis, CA 95616, USA +Hermann Wolter +Faculty of Physics, University of Munich, D-85748 Garching, Germany +Hanna Zbroszczyk +Faculty of Physics, Warsaw University of Technology, Koszykowa 75, Warsaw, Poland +Endorsing authors: +Anton Andronic +Westf¨alische Wilhelms-Universit¨at M¨unster, Institut f¨ur Kernphysik, 48149 M¨unster, Germany +Steffen A. Bass +Department of Physics, Duke University, Durham NC 27708 +Abdelouahad Chbihi +GANIL, CEA/DRF-CNRS/IN2P3, Boulevard Henri Becquerel, F-14076 Caen Cedex, France +Maria Colonna +INFN-LNS, Laboratori Nazionali del Sud, 95123 Catania, Italy +Mircea Dan Cozma +IFIN-HH, Reactorului 30, 077125 Mˇagurele-Bucharest, Romania +Veronica Dexheimer +Department of Physics, Kent State University, Kent OH 44242 USA +Xin Dong, Jørgen Randrup, and Nu Xu +Lawrence Berkeley National Laboratory, Berkeley, CA 94720 +Travis Dore +Fakult¨at f¨ur Physik, Universit¨at at Bielefeld, D-33615 Bielefeld, Germany +Lipei Du +Department of Physics, McGill University, Montreal, Quebec H3A 2T8, Canada +Steven P. Harris, Larry McLerran, and Sanjay Reddy +Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA +Huan Zhong Huang +University of California, Los Angeles, CA 90095 + +3 +Jos´e C. Jim´enez +Instituto de F´ısica, Universidade de S˜ao Paulo, Rua do Mat˜ao 1371, 05508–090 S˜ao Paulo-SP, Brazil +Joseph Kapusta +School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455 USA +Arnaud Le F`evre, Christian Sturm, and Wolfgang Trautmann +GSI Helmholtz Centre for Heavy-ion Research, Planckstr. 1, 64291 Darmstadt, Germany +Jacquelyn Noronha-Hostler +University of Illinois at Urbana-Champaign, Urbana, IL 61801 +Christopher Plumberg +Natural Science Division, Pepperdine University, Malibu, CA 90263, USA +Hans-Rudolf Schmidt +Physikalisches Institut, Eberhard Karls Universit¨at T¨ubingen, D-72076 T¨ubingen, Germany and +GSI Helmholtz Centre for Heavy-ion Research, Planckstr. 1, 64291 Darmstadt, Germany +Peter Senger +Facility for Antiproton and Ion Research, Planckstr. 1, Darmstadt, Germany +Richard Seto +University of California-Riverside, Riverside, California 92521, USA +Chun Shen +Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA and +RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA +Jan Steinheimer +Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany +Joachim Stroth +Institut f¨ur Kernphysik, Goethe-Universit¨at, 60438 Frankfurt, Germany and +GSI Helmholtz Centre for Heavy-ion Research, Planckstr. 1, 64291 Darmstadt, Germany +Kai-Jia Sun +Institute of Modern Physics, Fudan University, 200438,Shanghai,China +Giuseppe Verde +INFN Sezione di Catania, 64 Via Santa Sofia, I-95123 Catania, Italy +Volodymyr Vovchenko +Physics Department, University of Houston, Box 351550, Houston, TX 77204, USA +(Dated: February 1, 2023) + +4 +Executive Summary +The nuclear equation of state (EOS) is at the center of numerous theoretical and experimental +efforts in nuclear physics, motivated by its crucial role in our understanding of the properties of +nuclear matter found on Earth, in neutron stars, and in neutron-star mergers. With advances in +microscopic theories for nuclear interactions, the availability of experiments probing nuclear matter +under conditions not reached before, and the advent of multi-messenger astronomy, the next decade +will bring new opportunities for determining the nuclear matter EOS. +• Profound questions challenging our understanding of strong interactions remain +unanswered: It is still unknown whether the transition between a hadronic gas and a +quark-gluon plasma, which at zero baryon density is known to be consistent with a crossover +transition predicted by Lattice QCD, becomes of first order in the finite-density region of +the QCD phase diagram accessible in terrestrial experiments. The isospin-dependence of +the EOS, crucial to our understanding of both the structure of neutron-rich nuclei and the +properties of neutron stars, is poorly known above nuclear saturation density. Moreover, +recent observations of very heavy compact stars indicate that the EOS in neutron-rich mat- +ter becomes very stiff at densities of the order of a few times saturation density, leading to +values of the speed of sound exceeding 1/ +√ +3 of the speed of light (breaking the conformal +limit). Not only is the mechanism behind this striking behavior not known, but it is also +unknown whether a similar stiffening occurs in symmetric or nearly-symmetric nuclear mat- +ter. Resolving these and other questions about the properties of dense nuclear +matter is possible by taking advantage of the unique opportunities for studying +the nuclear matter EOS in heavy-ion collision experiments. +• Among controlled terrestrial experiments, collisions of heavy nuclei at interme- +diate beam energies (from a few tens of MeV/nucleon to about 25 GeV/nucleon +in the fixed-target frame) probe the widest ranges of baryon density and tem- +perature, enabling studies of nuclear matter from a few tenths to about 5 times the nuclear +saturation density and for temperatures from a few to well above a hundred MeV, respec- +tively. In the next decade, numerous efforts worldwide will be devoted to uncovering the +dense nuclear matter EOS through heavy-ion collisions, including studies at FRIB where +the isospin-dependence of the EOS can be probed in energetic collisions of rare isotopes. +Modern detectors and refined analysis techniques will yield measurements that +will elucidate the dependence of the EOS on density, temperature, and isospin +asymmetry. +• Hadronic transport simulations are currently the only means of interpreting +observables measured in heavy-ion collision experiments at intermediate beam +energies. This means that capitalizing on the enormous scientific effort aimed at uncovering +the dense nuclear matter EOS, both at RHIC and at FRIB, depends on the continued +development of state-of-the-art hadronic transport simulations. Support for the hadronic +transport community, and in particular for viable career pathways for early +career researchers, is imperative to maintain the health of and diversify the U.S. +hadronic transport community, and to fully realize the potential of U.S. efforts +leading the exploration of the dense nuclear matter EOS. + +5 +CONTENTS +I. Introduction +7 +A. Constraining the nuclear matter EOS using heavy-ion collisions +8 +B. Connections to fundamental questions in nuclear physics +9 +C. Upcoming opportunities +11 +D. Needs +12 +II. The equation of state from 0 to 5n0 +13 +A. Transport model simulations of heavy-ion collisions +13 +1. Transport theory +14 +2. Selected constraints on the EOS obtained from heavy-ion collisions +17 +3. Challenges and opportunities +19 +B. Microscopic calculations of the EOS +25 +1. Status +25 +2. Challenges and opportunities +27 +C. Neutron star theory +28 +1. Status +28 +2. Challenges and opportunities +31 +III. Heavy-ion collision experiments +33 +A. Experiments to extract the EOS of symmetric nuclear matter +35 +1. Measurements sensitive to the EOS +35 +2. Experiments probing densities between 1–2.5n0 +36 +3. Experiments probing densities above 2.5n0 +38 +4. Challenges and opportunities +39 +B. Experiments to extract the symmetry energy +42 +1. Experiments that probe low densities +42 +2. Measurements to extract symmetry energy up to 1.5n0 +42 +3. Selected constraints on the symmetry energy around 1.5n0 +44 +4. Challenges and opportunities +46 +IV. The equation of state from combined constraints +50 +A. Constraints +51 +B. EOS obtained by combining various constraint sets +53 +V. Connections to other areas of nuclear physics +54 +A. Applications of hadronic transport +54 +1. Detector design +55 +2. Space exploration, radiation therapy, and nuclear data +55 +B. Hydrodynamics +57 +1. Status +57 +2. Range of applicability +58 +3. Challenges and opportunities +60 +VI. Exploratory directions +60 +A. Dense nuclear matter EOS meeting extreme gravity and dark matter in supermassive +neutron stars +60 +B. Nuclear EOS with reduced spatial dimensions +61 + +6 +C. Interplay between nucleonic and partonic degrees of freedom: SRC effects on nuclear +EOS, heavy-ion reactions, and neutron stars +62 +D. High-density symmetry energy above 2n0 +63 +E. Density-dependence of neutron-proton effective mass splitting in neutron-rich matter +66 +Acknowledgments +68 +References +68 + +7 +I. +INTRODUCTION +The equation of state (EOS) is a fundamental property of nuclear matter, describing its emergent +macroscopic properties originating from the underlying strong interactions. Around the saturation +density of nuclear matter, the EOS controls the structure of nuclei through the binding energy and +the incompressibility. The EOS also determines, among other things, the neutron-skin thickness +in neutron-rich nuclei as well as the properties of nuclear matter at extreme densities and/or tem- +peratures, corresponding to conditions produced in experiments colliding heavy nuclei or observed +in neutron stars and neutron star mergers. Far beyond describing the properties of matter com- +posed of only protons and neutrons, the EOS can also reflect the appearance of new degrees of +freedom, e.g., strange particles in the cores of neutron stars or quarks and gluons in ultrarelativistic +heavy-ion collisions, or the emergence of new states of matter, e.g., chirally-restored matter, meson +condensates, or quarkyonic matter. +In heavy-ion collision experiments, the EOS is studied by detecting particles emerging from the +collision zone and measuring observables sensitive to the properties of nuclear matter. Crucially, +any interpretation of these observables, including quantitative constraints on the EOS, requires +comparisons of experimentally measured observables to results obtained in dynamic simulations. +This white paper highlights the essential role of hadronic transport simulations of +heavy-ion collisions in advancing our understanding of the EOS. It also elucidates the +many connections between inferences of the EOS from heavy-ion collision data and +other efforts aiming to describe and understand the properties of nuclear matter. +FIG. 1. Schematic depiction of the ranges of density and temperature probed in experiments and astronom- +ical observations sensitive to the EOS of nuclear matter (counterclockwise from bottom left): neutron star +crust physics, including nuclear pasta structures; properties of nuclei; structure of neutron stars; dynamics of +neutron star mergers; and outcomes of heavy-ion collisions which can probe both symmetric and asymmetric +matter. Figures adapted from [1–5]. + +100 +HIC (sym) +temperature [MeV] +10 +HIC (asym) +S +ROOKHAVEN +NS mergers +.NS crust +nuclear +properties +0.1 +Z +neutron stars (NS) +0 +1 +2 +3 +4 +5 +density np/no8 +A. +Constraining the nuclear matter EOS using heavy-ion collisions +FIG. 2. +Constraints on the zeroth (Sv) and +first (L) coefficient of the symmetry energy ex- +pansion. +Experimental constraints are derived +from heavy-ion collisions (HIC) [6], neutron-skin +thicknesses of Sn isotopes [7], giant dipole res- +onances (GDR) [8], the dipole polarizability of +208Pb [9, 10], nuclear masses [11], and isovector +skins (IAS+∆R) [12]. Also shown are constraints +from χEFT (GP-B) [13], microscopic neutron- +matter calculations (H, G) [14, 15], and from the +unitary gas limit (UG) [16]. Figure from [13]. +The last decade has brought tremendous progress +in extracting the EOS as a function of baryon den- +sity nB, temperature T, and the isospin asymme- +try δ (or, equivalently, the proton fraction) from +a variety of experimental and astronomical data as +well as theoretical calculations. Many-body theory, +based on sophisticated approaches with input from +nucleon scattering or nuclear structure data, can +now state the EOS below and near the saturation +density n0 with meaningful uncertainties (see Sec- +tion II B, “Microscopic calculations of the EOS”). +New classes of experiments have extracted the thick- +ness of neutron skins in nuclei, shedding light on the +isospin-dependence of the EOS (or, equivalently, the +symmetry energy) near or below n0. +High-energy +heavy-ion collisions have constrained the EOS of the +quark-gluon plasma at high temperatures and small +baryon densities, while ongoing experimental efforts +worldwide focus on the EOS of nearly-symmetric +dense baryonic matter, probed in collisions at in- +termediate energies. Meanwhile, collisions at lower +energies have led to experimental constraints on the +symmetry energy at sub- and suprasaturation den- +sities. +Most remarkably, a revolution in the qual- +ity and breadth of astronomical observations, high- +lighted by the first simultaneous detection of grav- +itational waves and electromagnetic signals from a +neutron-star merger, ushered in a new era of multi- +messenger astronomy (see Section II C, “Neutron +star theory”). Together with the newly available ex- +perimental capabilities at the Facility for Rare Iso- +tope Beams (FRIB), there are unprecedented oppor- +tunities to probe the isospin-dependence of the EOS +through astronomical and terrestrial measurements. +Among the experimental efforts discussed above, heavy-ion collisions probe the widest range +of baryon densities and, moreover, represent the only means to address the EOS away from n0 +in controlled terrestrial experiments, see Fig. 1. +Indeed, heavy-ion reactions at beam energies +from a few tens of MeV/nucleon to about 25 GeV/nucleon in the fixed-target frame probe the +EOS of hadronic matter at baryon densities from a few tenths to about 5 times n0. Controlling the +properties of matter produced in these experiments is possible by varying the beam energy, collision +geometry, and isotopic composition of the target and projectile. Insights and constraints obtained +from transport model analyses of these experiments are relevant both for our understanding of +nuclear matter as found on Earth and for our understanding of neutron stars from crust to core. +Within ongoing efforts, the STAR experiment’s Beam Energy Scan (BES) fixed-target (FXT) +program at the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory +(BNL), which collided gold nuclei at intermediate beam energies and which completed data taking +in 2022, leads the US effort to constrain the EOS of nearly-symmetric nuclear matter at high + +100 +Constraints on S-L +HIC +80 +△R +X +AS +60 +GP-B +G +40 +H +Masses +pb +20 + Skin +UG +Analytic +UG +GDR +0 +26 +28 +30 +32 +34 +Symmetry Energy S [MeV9 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Prior +Astro + HIC +Pressure P (MeV fm–3) +100 +101 +102 +Number density n (nsat) +FIG. 3. +Pressure in neutron star matter as +a function of density from a Bayesian analysis +combining nuclear theory and data from multi- +messenger neutron-star observations and heavy- +ion collisions [17]: the dark blue and light blue +region corresponds to the 68% and 95% credible +interval, respectively, while the gray dashed line +shows the 95% bound obtained in χEFT calcu- +lations and used as a prior. Figure from [17]. +baryon densities up to around 5n0, corresponding +to densities present in the deep interiors of neu- +tron stars. +Among comparable efforts in Europe, +the HADES experiment at GSI, Germany, probes +matter at densities up to 2.5n0. Preliminary results +from these contemporary efforts, as well as measure- +ments from other heavy-ion collision experiments +in the past, have led to competitive constraints on +the EOS of symmetric nuclear matter, with future +measurements expected to shed more light on its +high-density behavior. Detailed constraints on the +isospin-dependence of the EOS can be obtained by +varying the isospin content of the target and pro- +jectile nuclei. +Here, the ability to use radioactive +isotopes, as in, e.g., intermediate-energy heavy-ion +collision experiments at RIKEN and FRIB, is cru- +cial to resolve the subtle effects arising from changes +in the isospin asymmetry of the colliding systems. +Above all, obtaining constraints on the +EOS from heavy-ion measurements would not +have been possible if not for advances in theory, and in particular for the collaborative +effort to test the robustness and quantify the uncertainties of hadronic transport sim- +ulations (see Section II A, “Transport model simulations of heavy-ion collisions”). At the same +time, much remains to be learned, as tight constraints on both the symmetric and asymmetric +EOS at higher densities have so far remained elusive. This is predominantly due to model uncer- +tainties, which themselves are rooted in the inherent complexity of nucleus-nucleus collisions and +the challenging task of describing all processes contributing to the final state observables. +B. +Connections to fundamental questions in nuclear physics +The wealth of data from efforts conducted in recent years not only helps to get a better grasp +on the nuclear matter EOS, but also has brought forward fascinating questions challenging our +understanding of strong interactions. +Following the successful BES-I campaign at RHIC, questions remain about the structure of the +QCD phase diagram at finite baryon densities, where the sign problem prevents obtaining predic- +tions with lattice QCD calculations. Surprisingly, the expected disappearance of the quark-gluon +plasma signatures has not been observed in BES-I, with some observables suggesting that the +QCD first-order phase transition may be located within the region probed by BES-II experiments, +including the region probed by the currently analyzed BES FXT data. If this is the case, then +constraining the EOS at lower densities and describing the approach to the transition from the +hadronic side, which would manifest as a softening of the EOS, will be crucial for a robust inter- +pretation BES-II measurements. Importantly, due to the largely out-of-equilibrium evolution of +collision systems probing that region of the QCD phase diagram, hadronic transport simulations +will play a dominant role in describing the dynamics of the collisions, and therefore in constraining +the EOS of nearly-symmetric dense nuclear matter. +Understanding the physics of neutron-rich matter across a range of densities is necessary not only +to explain the properties of rare neutron-rich isotopes and the structure of neutron stars, but also +to constrain microscopic interactions in isospin-asymmetric nuclear matter. At low densities, this + +10 +0.1 +− +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +'=0 +y'y +d +/ +1 +v +d +(AuAu) Protons (10-30%) +HADES +=0.25-0.45) +0 +(AuAu) Protons (b +FOPI +(AuAu) Z=1 (b=2-5.5fm) +FOPI +(AuAu) Z=1 +Plastic Ball +(AuAu) Z=1 (b=2-5.5fm) +INDRA +(AuAu) Protons (12-25%) +E895 +(AuAu) Protons +E877 +� +(AuAu) h +E877 +(AuAu) Protons (10-40%) +Star FXT +(AuAu) Protons (10-25%) +Star FXT +(AuAu) Protons (10-40%) +Star BES +(PbPb) Protons (12.5-33.5%) +NA49 +(PbPb) Protons (15-35%) +NA61/SHINE +1 +− +10 +1 +10 +2 +10 +(GeV) +N +-2m +NN +s +0.1 +− +0.05 +− +0 +0.05 +0.1 +2 +v +out-of-plane +in-plane +(AuAu) Protons (10-30%) +HADES +(AuAu) Protons (15-29%) +FOPI +(AuAu) Z=1 (20-30%) +FOPI +(AuAu) Z=1 (b=5.5-7.5fm) +INDRA +(AuAu) Protons +EOS +(AuAu) Protons (12-25%) +E895 +(AuAu) Protons +E877 +� +(AuAu) h +E877 +(AuAu) Protons (10-40%) +Star FXT +(AuAu) Protons (0-30%) +Star FXT +(AuAu) Protons (10-40%) +Star BES +(10-20%) +� +(AuAu) h +Star BES +(0-60%) +� +(AuAu) h +Star +(0-60%) +� +(AuAu) h +PHOBOS +(PbPb) Protons (12.5-33.5%) +NA49 +(10-30%) +� +(PbPb) h +WA98 +� +(PbAu) h +CERES +FIG. 4. Compilation of the world data on the slope of +the directed flow at mid-rapidity (dv1/dy|y′=0, top) and +the elliptic flow (v2, bottom) as functions of the reduced +center-of-mass energy √sNN − 2mN for protons, Z = 1 +nuclei, and inclusive charged particles. Figure from [18]. +challenge is addressed by experimental and +theoretical analyses of nuclear structure ob- +servables. An important objective of nuclear +many-body theorists is to accurately calcu- +late these observables and reliably deduce +the EOS using microscopic interactions de- +rived within the framework of chiral effective +field theory (χEFT). Probing the symmetry +energy over a range of densities wider than +found in nuclei is possible through heavy-ion +collisions and neutron star studies. Often, +knowledge of the isospin-asymmetric EOS is +encoded in terms of constraints on the Tay- +lor expansion coefficients of the symmetry +energy around n0. Numerous analyses yield +consistent constraints on the first few expan- +sion coefficients (see, e.g., Fig. 2), although +they rely on an assumption that the expan- +sion remains accurate away from n0. The re- +cent advent of Bayesian inference techniques +allows one to pursue a different approach, +within which the isospin-asymmetric EOS is +described in terms of the dependence of the +pressure on baryon density (see, e.g., Fig. 3). +Moreover, Bayesian analyses can shed more +light on densities at which measurements +constrain the symmetry energy and quan- +tify the uncertainties of the extracted EOS. +As a result, combining diverse measurements +and using advanced analysis techniques can +lead to significantly tighter constraints, es- +pecially on the high-density behavior of the +symmetry energy (or, equivalently, on the +higher-order symmetry energy expansion co- +efficients), which is so far poorly known. +Constraints on the EOS of neutron-rich +matter at high densities have been dramat- +ically affected by discoveries of heavy neutron stars. Combined with the properties of all known +compact stars, these observations indicate that while the EOS of neutron-rich matter is relatively +soft around (1–2)n0, the pressure steeply rises with density for nB >∼ 2n0. In fact, multiple analyses +show that describing the known population of neutron stars is only possible for EOSs in which the +speed of sound in neutron-star matter breaks the conformal limit at high densities, that is exceeds +1/ +√ +3 of the speed of light c for nB >∼ 2n0. This striking behavior remains to be understood. In par- +ticular, it is currently not known whether the speed of sound exceeds c/ +√ +3 above certain densities +at all isospin fractions of nuclear matter or, alternatively, only in neutron-rich matter. Importantly, +robust constraints on the symmetric matter EOS at nB >∼ 2n0, obtained from heavy-ion collisions +at intermediate to high beam energies, would also put constraints on the isospin-dependent part of +the EOS through comparisons with the EOS inferred from neutron star studies, thus uncovering +the magnitude of isospin-related effects at high baryon density. + +11 +C. +Upcoming opportunities +The next decade will be an era of high-luminosity heavy-ion collision experiments at high baryon +density with modern detector and analysis procedures, as well as detailed studies of the symmetry +energy with collisions of proton- and neutron-rich isotopes. +Many of the discoveries of the BES program in ultra-relativistic heavy-ion collisions at RHIC, +e.g., the discovery of the triangular flow and elliptic flow ���uctuations, illustrate that modern +analyses of heavy-ion collisions bring new quality to the understanding of the underlying processes. +Because of this, revisiting the intermediate to high beam energies, previously explored at the +AGS at BNL as well as at SIS18 at GSI and now explored by the STAR FXT program and +the HADES experiment, is imperative to enable putting tighter constraints on the EOS of dense +nuclear matter. Moreover, the future CBM experiment at the Facility for Antiproton and Ion +Research (FAIR), Germany, will be able to measure interaction rates exceeding those currently +used by several orders of magnitude, allowing for exploration of multiple high-statistics observables. +Furthermore, the explored beam energy range is where lower-order flow observables, reflecting the +collective motion of the colliding system due to the underlying hadronic EOS, are particularly +0.4 0.6 0.8 1.0 1.2 1.4 1.6 +1.8 +2.0150 +200 +250 +300 +350 +400 +450 +0.2 +0.3 +0.4 +0.5 +0.6 +Incompressibility (MeV) +dv1/dy'|y +'=0 +In-medium Xsection modification factor + free protons + free neutrons +Au+Au, Ebeam/A=1.23 GeV +b=6-9 fm +HADES data: 0.46+0.03 +−0.03 +0.4 0.6 0.8 1.0 1.2 1.4 +1.6 +1.8 +2.0150 +200 +250 +300 +350 +400 +450 +-0.08 +-0.06 +-0.04 +-0.02 +0.00 +Incompressibility (MeV) +v2 +In-medium Xsection modification factor + free protons + free neutrons +Au+Au, Ebeam/A=1.23 GeV +b=6-9 fm, |ycm|<0.05, pt>0.3 GeV/c +HADES data: -0.06+0.01 +−0.01 +FIG. 5. Predicted slope of the directed flow at mid- +rapidity (dv1/dy|y′=0, top) and elliptic flow (v2, bot- +tom) as functions of the incompressibility and the in- +medium nucleon-nucleon scattering cross section mod- +ification factor, generated in simulations of Au+Au +reactions using the isospin-dependent BUU (IBUU) +transport model [19, 20]. Figure from Ref. [21]. +prominent (see Fig. 4). +Therefore, the cor- +responding precision measurements carry with +them the opportunity to bring a richer perspec- +tive and a better understanding of the physics +underlying the complex dynamics of nuclear +matter at extreme conditions (see Section III A, +“Experiments to extract the EOS of symmetric +nuclear matter”). This advancement can only +occur provided a simultaneous development of +hadronic transport simulations, as only a de- +tailed understanding of various factors affect- +ing the dynamics of heavy-ion collisions can +lead to meaningful descriptions of the exper- +imental data, and, consequently, more robust +constraints on the EOS of nearly-symmetric +nuclear matter (see Section II A, “Transport +model simulations of heavy-ion collisions”). As +an example of the sensitivity of observables to +various details of the underlying physics, Fig. 5 +shows the dependence of the slope of the di- +rected flow (top panel) and of the elliptic flow +at midrapidity (bottom panel) on the stiffness +of the EOS, parametrized by the incompress- +ibility, and on the in-medium nucleon-nucleon +scattering cross-section modification factor. +Unprecedented possibilities are on the hori- +zon for studies of the isospin-dependence of the +EOS, which is critical for connecting heavy-ion +physics measurements to astrophysical obser- +vations. +The difficulties in using nuclei with +significant variations in the isospin asymme- +try, along with the paucity of neutron measure- +ments at midrapidity, have in the past greatly + +12 +restricted the capability to put tight constrains on the EOS of asymmetric nuclear matter. Fortu- +nately, at this time modern neutron detectors are available for heavy-ion measurements in many +facilities, including at accelerators performing collisions at high beam energies such as GSI, while +radioactive beam measurements are entering a new era at RIKEN and FRIB. FRIB will provide +proton- and neutron-rich beams of not only the highest-intensity worldwide, but also characterized +by the widest currently accessible range of the isospin asymmetry. Establishing a strong heavy-ion +program at FRIB will therefore enable previously inaccessible exploration of the symmetry en- +ergy (see Section III B, “Experiments to extract the symmetry energy”). Moreover, the proposed +FRIB400 beam energy upgrade would not only allow exploration of densities up to around 2n0, +but it would also provide increased resolution of the isospin-dependence of the EOS. In particular, +among observables sensitive to the symmetry energy, both charged pion yields and the absolute +magnitude of the elliptic flow (see Fig. 4) significantly increase between the current top FRIB +energy of 200 MeV/nucleon and the proposed 400 MeV/nucleon [22]. +The increase in available computing power and advances in statistical methods make it possible +to perform wide-ranging comparisons of heavy-ion collision simulations with experimental data +(e.g., using Bayesian analysis), allowing one to vary multiple model assumptions at the same +time as well as to put robust uncertainties on the obtained constraints. Furthermore, given the +wealth of the upcoming independent data, e.g., from heavy-ion collision experiments, neutron star +observations, and microscopic nuclear theory calculations, global analyses of complementary efforts +have likewise a strong potential for putting tight constraints on the EOS (see Section IV, “The +EOS from combined constraints”). +Beyond the much-needed interpretation of intermediate energy heavy-ion collisions, advances +in transport theory can lead to significant contributions to other areas of nuclear physics. +In +particular, recently attention has been given to cross-cutting opportunities for employing state- +of-the-art hadronic transport codes in studies supporting space exploration and advanced medical +treatments (see Section V A, “Applications of hadronic transport”). Transport theories may also be +used in tests of extensions of hydrodynamic approaches supporting far-from-equilibrium evolution +(see Section V B, “Hydrodynamics”), which are a focus of intense studies due to their importance +for modeling heavy-ion collisions at high energies. Finally, constraining the dense nuclear matter +EOS through interpretations of heavy-ion collision measurements may have other profound conse- +quences, including helping to answer fundamental questions about the possible existence of dark +matter in the cores of neutron stars or providing the impetus for studies of nuclear systems in +fractional dimensions (see Section VI, “Exploratory directions”). +D. +Needs +The next-generation experimental measurements of observables sensitive to the nuclear matter +EOS are imminent, and further progress in resolving the nuclear matter EOS is contingent on +enhanced theory support. In particular, the development of transport theories based on +microscopic hadronic degrees of freedom, which are the only means of interpreting +measurements from heavy-ion collision experiments at intermediate to high beam +energies, must be strengthened and expanded to fully realize the potential of the +U.S. facilities leading the exploration of the dense nuclear matter EOS. Support for +both individual scientists and collaborations, and in particular for viable career pathways for early +career researchers, is imperative to maintain the health of and diversify the U.S. hadronic transport +community, and to fully capitalize on the U.S. efforts exploring the dense nuclear matter EOS. + +13 +II. +THE EQUATION OF STATE FROM 0 TO 5n0 +Efforts to determine the equation of state (EOS) of nuclear matter are at the forefront of nuclear +physics. An EOS contains fundamental information about the properties of a many-body system +(see, e.g., Section I B), and is, in essence, any nontrivial relation between the thermodynamic +properties of a given type of matter. In nuclear physics, the form of the EOS that is most often +pursued is the relation between energy per baryon or pressure and baryon density nB, isospin +excess δ, and temperature T. For symmetric matter, the isospin excess vanishes (δ = 0), and +for asymmetric matter the energy per baryon or pressure are commonly partitioned into a part +corresponding to symmetric matter and the remainder, which contains all information about the +isospin-dependence of the EOS. Due to the charge invariance of strong interactions, the latter +part is (to a very good accuracy) quadratic in the isospin excess δ at densities relevant to nuclear +experiments and astrophysical observations. The quadratic coefficients in the expansion around +δ = 0 are independent of δ, and are often referred to as the symmetry energy (denoted as S(nB) +at T = 0) or symmetry pressure, respectively. These, together with the EOS of symmetric matter, +are then sufficient to describe the EOS of nuclear matter at any isospin asymmetry. +While many approaches to constraining the nuclear matter EOS are pursued, here we describe +three research areas which have the capability to constrain the EOS over wide ranges of density: +inferences of the EOS from comparisons of experimental measurements to model simulations of +heavy-ion collisions (Section II A), microscopic calculations of the EOS using chiral effective field +theory (Section II B), and EOS inferences from neutron star studies (Section II C). +A. +Transport model simulations of heavy-ion collisions +soft EOS +hard EOS +temperature [MeV] +0 +50 +100 +150 +200 +250 +300 +density nB/n0 +0 +2 +4 +6 +8 +12.8 AGeV + 6.4 AGeV + 3.2 AGeV + 1.6 AGeV + 0.8 AGeV + 0.4 AGeV + 0.2 AGeV +FIG. 6. Phase diagram trajectories of the central re- +gion in Au+Au collisions at zero impact parameter, +obtained from UrQMD simulations with a soft or a hard +(characterized by K0 = 200 or K0 = 380 MeV, respec- +tively) EOS [23, 24]. The trajectories follow the evolu- +tion at times when temperature is fairly well-defined, +from the moment of the highest compression to densi- +ties around 0.5n0. +Heavy-ion collisions at very low to interme- +diate beam energies provide the means to probe +nuclear matter at different densities (from sub- +saturation to several times the saturation den- +sity), temperatures (from a few MeV to well +above one hundred), and neutron to proton +ratios (from near symmetric nuclear matter, +where Nn/Np ≈ 1, up to Nn/Np ≈ 2); see Fig. 6 +for an illustrative calculation of heavy-ion col- +lision trajectories in the T-nB phase diagram +from simulations using two schematic EOSs. +These wide ranges of system properties ac- +cessed in heavy-ion collisions position them as a +perfect tool to extract the nuclear matter EOS, +test predictions and extrapolations from regions +of the QCD phase diagram accessed by other +approaches, and provide a necessary input to +nuclear theory and nuclear astrophysics calcu- +lations. For example, the density-dependence +of both the symmetric and asymmetric EOS +can shed light on modeling effective nuclear in- +teractions in the medium [15, 25–27] or con- +strain approaches using the density functional +theory [28–30]. + +14 +However, systems created in heavy-ion collisions are short-lived, and their dynamic evolution +is out of equilibrium over significant fractions of the total collision time. +The evolution of a +colliding system depends on the energy and centrality of the collision, and progresses through initial +compression, growth of the compression zone, development of flows, and overall decompression +with a gradual local equilibration during the process, see Fig. 7. +The inherent complexity of +the evolution means that the corresponding transport equations cannot be solved directly due to +their high non-linearity, and therefore detailed inferences from heavy-ion collision experiments, +where the non-equilibrium evolution probes nuclear matter over substantial ranges of density, +require comparisons to results of collision simulations in transport models. Beyond modeling the +dynamics of the collisions, transport models provide a connection to the equilibrium limit allowing +for inferring the EOS [31], transport coefficients [32], as well as the in-medium properties and +cross-sections of hadrons [33–35]. +1. +Transport theory +At its core, transport theory aims to describe the time evolution of the one-body phase-space +distribution function in a semi-classical approximation for a dissipative system composed of a large +number of particles, here in particular for a system of two heavy nuclei colliding at an energy per +nucleon which is typically larger than the Fermi energy. The theoretical foundations of transport +theory include the BBGKY hierarchy of coupled equations for reduced density matrices [36] as well +as the equations of the nonequilibrium Green’s function theory [37, 38] such as obtained in Martin- +Schwinger (also known as Schwinger-Keldysh) formalism for non-equilibrium Green’s function (see +also Section V B). +To arrive at transport equations, one employs (among others) a Wigner transformation and +coarse-graining as well as a gradient expansion. The Wigner transformation and coarse-graining +nB +FIG. 7. Contour plots of the system-frame baryon density nB (top row) and local excitation energy E∗/A +(bottom row) at times t = 0, 5, 10, 15, and 20 fm/c (columns from left to right), obtained from a transport +simulation [39] of a 124Sn+124Sn reaction at beam energy Elab = 800 AMeV (√sNN = 2.24 GeV) and impact +parameter b = 5 fm. The contour lines for the density use increments of 0.4n0, starting from 0.1n0, while +the contour lines for the local excitation energy correspond to the values of E∗/A = {5, 20, 40, 80, 120} MeV; +for statistical reasons, contour plots for the energy have been suppressed for baryon densities nB < 0.1n0. + +15 +-0.5 +0.0 +0.5 +0 +100 +200 +300 +dN/dy +y + (GeV/c) +132Sn+ +124Sn +-0.5 +0.0 +0.5 +-0.1 +0.0 +0.1 +y + IBUU + pBUU + RVUU + SMF + IQMD + IQMD-BNU + IQMD-IMP + TuQMD +-0.5 +0.0 +0.5 +0 +100 +200 +300 +dN/dy +y + (GeV/c) +132Sn+ +124Sn +-0.5 +0.0 +0.5 +-0.1 +0.0 +0.1 +y + IBUU + pBUU + RVUU + SMF + IQMD + IQMD-BNU + IQMD-IMP + TuQMD +-0.5 +0.0 +0.5 +0 +100 +200 +300 +dN/dy +y + (GeV/c) +132Sn+ +124Sn +-0.5 +0.0 +0.5 +-0.1 +0.0 +0.1 +y + IBUU + pBUU + RVUU + SMF + IQMD + IQMD-BNU + IQMD-IMP + TuQMD +-0.5 +0.0 +0.5 +0 +100 +200 +300 +dN/dy +y + (GeV/c) +132Sn+ +124Sn +-0.5 +0.0 +0.5 +-0.1 +0.0 +0.1 +y + IBUU + pBUU + RVUU + SMF + IQMD + IQMD-BNU + IQMD-IMP + TuQMD +-0.5 +0.0 +0.5 +0 +100 +200 +300 +dN/dy +y + (GeV/c) +132Sn+ +124Sn +-0.5 +0.0 +0.5 +-0.1 +0.0 +0.1 +y + IBUU + pBUU + RVUU + SMF + IQMD + IQMD-BNU + IQMD-IMP + TuQMD +FIG. 8. +Comparison of results for rapidity distri- +butions (top) and transverse flow of nucleons (bot- +tom) as functions of the scaled rapidity, obtained +with different transport codes (identified in the leg- +end) within the TMEP initiative. The results shown +were obtained for 132Sn+124Sn collisions at Elab = +270 AMeV (√sNN = 2.01 GeV) and impact param- +eter b = 4 fm, using controlled input models for the +EOS and the cross sections as well as identically ini- +tialized nuclei [40]. +lead to positive-definite phase-space distribu- +tions [41] that can be efficiently sampled with +Monte-Carlo techniques, while the gradient ex- +pansion yields, for each particle species, the force +acting on a particle and the particle’s veloc- +ity as gradients of its total energy with respect +to the spatial position and momentum, respec- +tively. Knowledge of the kinematics of all parti- +cles, together with the elementary collision rates, +drives the evolution in the phase space. Finally, +to arrive at a set of Vlasov-Boltzmann–like equa- +tions, one employs the quasi-particle approxima- +tion, neglecting details of the spectral functions +and treating all particles as on-shell (we note +here that while there are some transport codes +with off-shell particle treatment, e.g., [42, 43], +this approach is still an outstanding challenge in +the transport theory, as will be discussed further +below). Alternative approaches to arriving at a +transport theory for heavy-ion collisions include +using the relativistic Landau quasiparticle the- +ory [44] or, in approaches starting from a molec- +ular picture, representing the global wavefunc- +tion as a product (sometimes antisymmetrized) +of single-particle Gaussian wavepackets [45]. +The particle species considered in transport +theory depend on the collision energy and may +range from nucleons, through pions and the delta +resonances, to higher resonances, kaons, and hy- +perons. Some transport formulations further in- +corporate light clusters (e.g., deuterons, tritons, +and 3He nuclei) as independent degrees of free- +dom, with recent extensions also including alpha +particles [46] which appear abundantly in exper- +iments and are of particular importance for colli- +sions at fixed-target beam energies on the order +of hundreds of MeV/nucleon. In some of these approaches, clusters are produced through multi- +particle reactions, as discussed further below. +For the lowest energy collisions, nonrelativistic +formulations of the transport theory may be employed, but the majority of the available codes are +relativistic, with many addressing collisions at energies from tens of MeV/nucleon to at least a few +GeV/nucleon (see [35, 47, 48] for reviews). +Transport approaches can be generally divided into those concentrating on a single-particle +characterization of the colliding system and those attempting to describe many-particle correla- +tions. Both types of approaches are highly complex and nonlinear, and the relevant equations are +solved by simulations. The single-particle approaches typically solve a set of Boltzmann-Vlasov– +type equations [47, 49] (also known as the Boltzmann-Uehling-Uhlenbeck, or BUU equations) in +which the evolution of the system is governed by a mean-field evolution of the phase space distribu- +tion (Vlasov equations) and a collision term which drives the dissipation (the Boltzmann collision +term). While, in principle, the Boltzmann-Vlasov equation is deterministic, numerical solutions + +16 +contain numerically-induced fluctuations due to the fact that the evolution is obtained using the +method of test particles, in which the continuous distribution function is represented by a large, +but finite, number of test particles sampling the phase space. To include fluctuations of a physi- +cal origin, one can add a fluctuation term to the two-particle collision term, thus arriving at the +Boltzmann-Langevin formulation [35, 50]. +In contrast, quantum molecular dynamics (QMD) approaches include classical many-body cor- +relations in the ansatz of the many-body wave function [47, 51], which is postulated as a product of +single-particle wave packets of a fixed width, with the width regulating the amount of fluctuations +and correlations in QMD. In Anti-Symmetrized Molecular Dynamics (AMD) [45], the product wave +function is anti-symmetrized and the formulation includes Pauli correlations in the propagation as +well as in, to a certain extent, the collision term. +The fact that hadronic transport approaches are built on firm theoretical foundations has been +crucial for the continued development of simulation frameworks. Reaching back to the roots of +the nuclear transport theory has made it possible to resolve ambiguities which would be otherwise +hard to tackle by purely phenomenological means, including descriptions of cluster production [52], +low relative-velocity correlations (Hanbury–Brown-Twiss correlations) [53], and off-shell transport +[42, 49, 54, 55]. +The strong theoretical foundation of transport theory has also been effective +in ensuring covariance of the theory and preserving conservation laws in case of interactions that +stray beyond outcomes of field-theoretic models, in particular interactions employing energy density +functionals [44, 56–58] which are often needed for realistic descriptions of bulk properties of nuclear +matter. +An important effort to validate conclusions reached from comparing transport model results +to data has been recently intensified by the formation of the Transport Model Evaluation Project +(TMEP) [47]. Within this endeavor, predictions from different models are compared in controlled +settings (e.g., ensuring the same physical input such as the EOS, initial densities, and cross sec- +tions), oftentimes with comparisons to known results that can be achieved analytically or by other +methods. Similar controlled comparisons of complex simulations have been done in other fields +of physics: from atomic traps, through ultra-relativistic heavy-ion collisions, to core-collapse su- +pernova calculations [59–62], and they are known to be very fruitful for their respective fields. +The TMEP analyses not only enable identifying models that produce outlier predictions, but also +determine details of implementation or physical assumptions behind the diverging results. An ex- +ample of such a comparison of codes for simulations of heavy-ion collisions at lower energies, with +controlled input, can be seen in Fig. 8, showing results for rapidity distributions (left) and the +transverse flow (right) [40]. In general, the codes agree with each other reasonably well, however, +differences between the codes are visible and, moreover, can be traced to specific model choices +in the simulations. For example, the generally lower values of the transverse flow in the case of +QMD codes are a result of an approximation used in the evaluation of a non-linear term in the +mean-fields, which becomes relevant when density fluctuations become large, as often occurs in +QMD. Beyond identifying this and similar problems, the Project has yielded recommendations +for optimal algorithms used in transport codes, e.g., for ensuring obeying the Pauli principle in +elementary two-body collisions [63] or for integration of equations of motion with mean-fields [64]. +Moreover, the project has identified a set of tests for transport codes that ensure their credibility +when addressing different heavy-ion collision observables. Stringent tests of hadronic transport +codes are especially important for studies aimed at constraining the nuclear symmetry energy, +which, compared to other model parameters, has a comparatively weak effect on heavy-ion observ- +ables and which therefore demands maximal precision from transport simulations. Below, we will +also discuss the role that such comparisons can play in determining the uncertainty of transport +model investigations. + +17 +2. +Selected constraints on the EOS obtained from heavy-ion collisions +A selection of important constraints on the EOS obtained from heavy-ion collisions can be found +in Fig. 9 for both symmetric matter (pressure as a function of density, left panel) and asymmetric +matter (symmetry energy as a function of density, right panel). +We note here that while many results are reported in terms of constraints on the incompress- +ibility K0, in the context of heavy-ion collision studies of the EOS, K0 should be understood as +a parameter which specifies the behavior of the EOS in the range of densities probed by a given +study. For example, in the case of experiments probing mostly densities above 2n0, constraints on +K0 are only indicative of the behavior of the EOS above 2n0, and in particular do not constrain +the behavior of the EOS around n0. This subtle, and often confusing, point is a consequence of +simple parametrizations of the EOS used in many transport codes, where the only parameter con- +trolling the behavior of the EOS both around n0 and at higher densities is K0. Recently, flexible +parametrizations of the EOS have been developed (see, e.g., [57, 58]) and implemented (e.g., in +hadronic transport code SMASH [75, 76]) which allow one to vary the incompressibility K0 and the +high-density behavior of the EOS independently. +The collective behavior of matter created in the collisions, especially the directed and elliptic +flow, has been shown to be a very sensitive probe of the EOS [31, 67, 77–79]. +In contrast to +collisions at the Fermi energies, where all nucleons within nuclei participate in the collisions, and +unlike in collisions at ultrarelativistic energies, where the evolution of the colliding nuclei can be +understood in terms of participant nucleons, at intermediate energies the interplay between the +expanding collision zone and the dynamics of the spectators are key ingredients to understanding +Le Fèvre et al. +Lynch et al. from Fuchs et al. +Oliinychenko et al. +Danielewicz et al. +Walecka model +Fermi gas +pressure [MeV/fm3] +1 +10 +100 +baryon density nB/n0 +1 +2 +3 +4 +5 +HIC(isodiff) +HIC(n/p) +mass(Skyrme) +IAS +mass(DFT) +PREX II +HIC(π) +Tsang et al. +ASY-EOS +FOPI-LAND +symmetry energy S(nB) [MeV] +0 +20 +40 +60 +80 +baryon density nB/n0 +0 +0.5 +1 +1.5 +2 +FIG. 9. Left: Selected constraints on the symmetric EOS obtained from comparisons of experimental data to +hadronic transport simulations in [31] (region with black horizontal stripes), [65, 66] (region with red forward +stripes), [67] (region with blue backward stripes), and [58] (region with green vertical stripes); see text for +more details. Also shown are results of analytical calculations for the free Fermi gas (green dotted line) +and in the linear Walecka model (pink dashed line). Right: Selected constraints on the symmetry energy +obtained from comparisons of hadronic transport simulations to experimental data in [6] (region with purple +forward stripes), [68] (region with green backward stripes), [69] (the solid orange region), and [70] (the red +circle, square, and triangle symbols). Also shown are symmetry energy constraints obtained in [70] based on +a novel interpretation of analyses of nuclear masses in DFTs [11, 71] (cyan diamond symbol) and in Skyrme +models [72] (cyan star symbol), of Isobaric Analog States (IAS) energies [73] (magenta plus symbol), and of +PREX-II experiment [74] (blue inverted triangle symbol). + +18 +experimental results. +A seminal constraint on the symmetric nuclear matter EOS [31] in the +density range (2–4.5)n0 was obtained by comparing measurements of collective flow from heavy- +ion collisions [80–83] at beam energies Elab = 0.15–10 AGeV (corresponding to nucleon-nucleon +center-of-mass energies √sNN = 1.95–4.72 GeV) with results from hadronic transport simulations +using EOSs with different values of the incompressibility at saturation density K0. The outcome of +this study suggests a symmetric-matter EOS to lie between those labeled with K0 = 210 MeV and +K0 = 300 MeV (see the region with black horizontal stripes in the left panel of Fig. 9). For densities +in the range (1.0–2.5) n0, probed in collisions below Elab <∼ 1.5 AGeV (√sNN <∼ 2.5 GeV), the EOS +may be inferred from meson yields [84–86]. Indeed, subthreshold production of strange mesons +(specifically, K+ and K0), which interact weakly with nuclear matter, depends on the highest +densities sampled in the collision, which in turn depend on the stiffness of the EOS [87]. In [65], +ratios of experimentally measured kaon yields in Au+Au and C+C collisions have been reproduced +in hadronic transport simulations with soft mean-field interactions yielding K0 = 200 MeV and +an EOS [66] consistent with the constraint from [31] (see the region with red forward stripes +in the left panel of Fig. 9). +In [67], the elliptic flow data measured at Elab = 0.4–1.5 AGeV +(√sNN = 2.07–2.52 GeV) by the FOPI collaboration [88] were used together with simulations +from Isospin Quantum Molecular Dynamics (IQMD) [23, 89] to constrain the incompressibility at +K0 = 190 ± 30MeV, again indicating a rather soft EOS (see the region with blue backward stripes +in the left panel of Fig. 9). Recently, new measurements by the STAR collaboration from the fixed +target (FXT) program at RHIC have become available, providing an opportunity to expand the +set of world data utilized to deduce the baryonic EOS. A Bayesian analysis study [58], in which the +speed of sound was independently varied in specified intervals of baryon density (thus providing +a more flexible EOS at higher densities), suggests a tension between the E895 [83, 90–92] and +STAR [93, 94] data. Using only the STAR measurements, the study [58] further found that EOSs +which simultaneously describe the slope of the directed flow and the elliptic flow, in the considered +energy range of Elab = 2.9–9 AGeV (√sNN = 3.0–4.5 GeV), are relatively stiff at lower densities +and relatively soft at higher densities (see the region with green vertical stripes in the left panel of +Fig. 9). However, the model used in that work did not include the momentum dependence of the +EOS, which likely results in a spuriously stiff EOS at intermediate densities. As such, the study +should be treated as a proof of principle that a tight constraint on the EOS at high densities can +be achieved by using a combination of precise data, flexible forms of the EOS used in simulations, +state-of-the-art models, and advances in analysis techniques. +The symmetry energy contribution to the EOS can be studied at low collision energies Elab <∼ +1.0 AGeV (√sNN <∼ 2.32 GeV), where in particular observables such as charged pion yields [95] or +neutron and proton flow [96, 97] have been proposed as sensitive to the asymmetric contribution +to the EOS. Some of the constraints derived from such studies are shown in the right panel of +Fig. 9, where, in addition to the usual EOS constraint bands, symbols with uncertainty bars +represent results from analyses in which the symmetry energy has been determined for the most +sensitive density of a given measurement. At incident energies below Elab = 100 AMeV (√sNN = +1.93 GeV), low densities are probed after the initial impact and compression of the projectile and +target [6, 98]. Since the symmetry potentials for neutrons and protons have opposite signs, emission +of a particular nucleon type is enhanced or suppressed depending on the asymmetry. A comparison +of the experimental measurements of isospin diffusion and the ratio of neutron and proton spectra in +collisions of 112Sn+124Sn at Elab = 50 AMeV (√sNN = 1.90 GeV) to results from ImQMD simulations +produced a constraint on the symmetry energy for densities (0.3–1) n0 [6] (see the region with purple +forward stripes in the right panel of Fig. 9). Collisions at higher energies (Elab > 200 AMeV, or +√sNN > 1.97 GeV) probe the EOS at n > n0. In the FOPI-LAND experiment, constraints on +the symmetry energy were obtained from studies of the ratio of the elliptic flow of neutrons and +hydrogen nuclei in Au+Au collisions at Elab = 0.4 AGeV (√sNN = 2.07GeV) [68], while the ASY- + +19 +EOS experiment used neutron to charged fragments ratios measured in Au+Au collisions [69] (see +the region with green backward stripes and the solid orange region, respectively, in the right panel +of Fig. 9). In [70], a comprehensive analysis was performed with the goal of identifying the values of +the symmetry energy at densities to which given experiments are most sensitive. Using the isospin +diffusion in collision systems with different proton to neutron ratios [99], neutron to proton energy +spectra in Sn+Sn systems [100], and spectral pion ratios measured by the SπRIT collaboration in +Sn+Sn collisions at Elab = 270 AMeV (√sNN = 2.01 GeV) [101, 102], that work [70] put constraints +on the values of the symmetry energy at about 0.2n0, 0.4n0, and 1.5n0, respectively (see the red +circle, square, and triangle symbols in the right panel of Fig. 9). Also shown in the right panel +of Fig. 9 are symmetry energy constraints obtained in [70] based on a novel interpretation of the +analyses of nuclear masses in DFTs [11, 71] (cyan diamond symbol) and in Skyrme models [72] +(cyan star symbol), of the Isobaric Analog State (IAS) energies [73] (magenta plus symbol), and +of the PREX-II experiment result [74] (blue inverted triangle symbol). +3. +Challenges and opportunities +Selected results presented in Fig. 9 showcase significant achievements in determining the EOS +and, simultaneously, the need to develop improved transport models to obtain tighter and more +reliable constraints. Answering this need will require support for a sustained collaborative effort +within the community to address remaining challenges in modeling collisions, in particular in the +intermediate energy range (Elab ≈ 0.1–25 AGeV, or √sNN ≈ 1.9–7.1 GeV). In the following, we +will address selected areas where we see the need for such developments: (1) comprehensive treat- +ment of both mean-field potentials and the collision term in transport codes, (2) use of microscopic +information on mean fields and in-medium cross sections, such as discussed in Section II B, in trans- +port, (3) better description of the initial state of heavy-ion collisions in hadronic transport codes, +(4) deeper understanding of fluctuations in transport approaches, which affect many aspects of +simulations, (5) inclusion of correlations beyond the mean field into transport, which is crucial for +a realistic description of light-cluster production, (6) treatment of short-range-correlations (SRCs) +in transport, which are tightly connected to multi-particle collisions as well as off-shell transport, +(7) sub-threshold particle production, (8) the study of new observables, e.g., azimuthally resolved +spectra, to obtain tighter constraints on the EOS, (9) the question of quantifying the uncertainty of +results obtained in transport simulations, and (10) the use of emulators and flexible parametriza- +tions for wide-ranging explorations of all possible EOSs. Fortunately, advances in transport theory +as well as the greater availability of high-performance computing make many of these improvements +possible. Support for these developments will lead to a firm control and greater understanding of +multiple complex aspects of the collision dynamics, allowing comparisons of transport model cal- +culations and heavy-ion experiment measurements to provide an important contribution to the +determination of the EOS of dense nuclear matter, which, in particular, cannot be determined by +any other method at intermediate densities (1–5)n0. +Comprehensive treatment of mean-field potentials and the collision term +Notably, driven by specific experimental needs over the last two decades, the refinement of +hadronic transport codes has diverged into two complementary branches: Codes which were ap- +plied to describing experiments at very low to low energies (Elab <∼ 1.5 AGeV, or √sNN <∼ 2.5 GeV), +such as IQMD, AMD and pBUU, have become progressively better at describing the momentum- and +isospin-dependence of the interaction, while codes which were primarily used as afterburners for +simulations of ultra-relativistic heavy-ion collisions (Elab >∼ 25 AGeV, or √sNN >∼ 7 GeV), such +as SMASH [75] or UrQMD, were developed to offer a fully relativistic evolution as well as scattering + +20 +and decay modes taking into account all established particle and resonance species. As heavy-ion +collisions are entering an era of precision data on symmetric nuclear matter at higher densities +(e.g., in experiments at HADES, BES FXT, and future CBM) and on asymmetric nuclear mat- +ter at normal and supranormal densities (e.g., at FRIB and future FRIB400), where features of +both diverging branches of hadronic transport codes are important, a vigorous development of +transport models is needed. In particular, numerous studies show the importance of including +the momentum-dependence of the interactions, which is observed in elastic scattering of hadrons +off nuclei. +Moreover, momentum-dependence naturally occurs in microscopic effective interac- +tions [38, 103] where it contributes to the calculated mean fields, whether near or away from sat- +uration density. Incorporating single-particle energies with momentum dependence different than +that in free space, which is often quantified with effective masses, is crucial in hadronic transport +both for studies of symmetric nuclear matter [31, 79, 104, 105] as well as studies of the symmetry +energy and its relation to effects such as the neutron-proton effective mass splitting [106–108] (see +also Section VI E for more discussion on effective masses and the nuclear symmetry energy). Some +of the theoretical and implementation solutions have already been established, while others will +require devising new approaches. When possible, the best practices need to be carried over across +the domains, as has been exemplified in, e.g., the development of the SMASH code, which uses many +implementation solutions from pBUU. +Microscopic input to transport +One of the most prominent opportunities for improvement in transport models concerns imple- +mentations of the EOS informed by state-of-the-art many-body studies. Such efforts are especially +timely given that sophisticated microscopic calculations of the properties of nuclear matter are +currently becoming available for large ranges of baryon density, temperature, and isospin fraction +(see Section II B for more details). To incorporate the effects of the resulting EOSs in hadronic +transport calculations, the corresponding Lorentz-covariant single-particle potentials as well as the +in-medium interactions (both as functions of density, asymmetry, and momentum) are needed. A +particular challenge is to determine the connection between the EOS inferred from a transport +calculation and the zero-temperature EOS obtained from microscopic calculations [109], or even +the finite-temperature EOSs that are becoming increasingly available [110, 111]. In a heavy-ion +collision, the medium progresses through a set of non-equilibrium states that relax toward a local +equilibrium, however, the nature of the local equilibrium also evolves during the collision due to the +system expansion, so that even if the system approaches a local equilibrium at any given moment +of the evolution, that agreement is only temporary. Errors incurred due to differences between +non-equilibrium and equilibrium states of high-density matter contribute to the systematic error in +inferring the EOS when comparing transport to experimental data (see Fig. 9 and [31]). Here, the +availability of state-of-the-art microscopic calculations at finite temperature could reduce system- +atic errors in connecting the finite- and zero-temperature EOSs. Moreover, the use of microscopic +input would provide a consistency between the effective in-medium cross sections in the collision +term and the mean fields used in the propagation of the phase space distribution. It could also +help address the question of the extent to which nonlocalities in the microscopic theory should +be reflected in the propagation and the collision term [112, 113] (where, in particular, departures +from standard approaches modify the entropy to take a form different than that obtained in the +Landau quasiparticle theory [44, 114]). To accelerate progress at the interface of the transport +description of heavy-ion collisions and microscopic nuclear matter theory, direct collaboration of +practitioners in the two research areas is required to assess how the needs of transport simulations +can be answered by what can be currently calculated in microscopic theories. Conversely, the use +of microscopic interactions in transport could validate the many-body theory results in regions of +density and temperature which are only accessible by heavy-ion collisions [115]. + +21 +Initial state +Numerous studies point toward the dependence of outcomes of heavy-ion collision experiments +on details of the initial conditions. In ultrarelativistic heavy-ion collisions, understanding these +effects have led to the discovery of higher order flow harmonics [116, 117] and flow fluctuations [118]. +(Interestingly, the importance of the initial state for experimental outcomes also positions heavy-ion +collisions at high energies as an unusual, but complementary probe of nuclear structure, see, e.g., a +white paper on Imaging the initial condition of heavy-ion collisions and nuclear structure across the +nuclide chart [119].) Given the high sensitivity of flow observables to both the EOS and the initial +state of collisions, the impact of the initial conditions on outcomes of heavy-ion collisions needs to +be thoroughly understood in order to narrow the constraints on the EOS of both symmetric and +asymmetric matter. Aspects of initial conditions that need to be considered include event-by-event +fluctuations of the initial state [116–118], relative distributions of neutrons and protons and shell +effects [120], and correlations tied to deformation [121] or short-range correlations [122]. Some of +these elements will be further discussed below in the context of the dynamics of heavy-ion collisions. +Fluctuations +Fluctuations of the phase space distribution are an important ingredient of transport simula- +tions. In particular, fluctuations of the one-body density are important for including the conse- +quences of the dissipation-fluctuation theorem in the reaction dynamics as well as for describing +effects due to the largely unknown, neglected many-body correlations, thus going beyond the mean- +field description. The question of how to include them properly and of their consistency with the +nucleon-nucleon correlations explicitly implemented in transport theories, however, has not been +completely clarified. As discussed above, fluctuations are included in a different manner in the two +families of transport approaches. While in the BUU transport fluctuations can be introduced by +the Langevin extension of the Boltzmann-Vlasov equation, which adds a fluctuation term to the +collision term (and which is still rarely implemented), in the molecular dynamics approach fluctu- +ations are introduced in a classical way by using finite-size particles, the width of which regulates +the amount of fluctuations. Fluctuations then affect the outcome of simulations in many ways, in- +cluding by regulating the formation of intermediate-mass fragments (IMFs) which appear through +the growth of fluctuations in regions of spinodal instability. It was also shown in box calculations +that fluctuations have a strong influence on the efficiency of Pauli-blocking [63] and even on the +calculation of the force in the Vlasov term for QMD codes in which non-linear parametrizations of +the fields are used [64]. +Correlations +Correlations in transport simulations strive to address intermediate-range correlations beyond +the mean-field picture. Physically, such correlations are also a source of fluctuations, but at the +same time have other additional impacts, including, e.g., influencing the production of light clusters +(LCs), that is light nuclei up to the alpha particle which are copiously produced in heavy-ion +collisions. The mean-field models used in transport calculations are usually not detailed enough to +realistically describe very light nuclei with their particular spin-isospin structure reflecting strong +quantum effects. An additional complication results from the fact that in a collision, clusters often +appear in the nuclear medium where their properties are drastically changed (e.g., the binding +energy of clusters is reduced with increasing density until the Mott point, at which they dissolve). +Currently, most codes describe the production of clusters by using a cluster-finding algorithm, +based on particle proximity in coordinate and/or momentum space (coalescence) toward the end +of the evolution, which in more advanced versions also takes into account criteria related to the +binding energy of the produced clusters [123]. However, these late-stage algorithms do not take into +account the dynamic role played by both correlations and LCs in the evolution of the collision. One + +22 +of the known approaches to this problem has been to consider LCs as separate degrees of freedom, +with their own distribution functions and corresponding transport equations, where the collision +terms can lead to creation or destruction of clusters (pBUU, SMASH) and which in particular can +also take into account the in-medium modifications of clusters. However, this approach becomes +increasingly complex as heavier clusters are characterized by more and more production channels, +and consequently it is significantly challenging to include, e.g., alpha particles. Another approach +is to modify the phase space of the correlated nucleons according to the Wigner function of the +cluster, but then to propagate them after the collision again as nucleons (as is done in, e.g., AMD +[41]), which still requires using a cluster-finding step at the end. In both cases, the production and +destruction of clusters necessarily requires multi-particle collisions to ensure energy-momentum +conservation. Finally, at lower incident energies the LC production can also be described in terms +of the catalyzing effect of spectator nucleons in few-particle collisions [46, 124]. To explain LC +production in high-energy collisions, where LCs are produced in numbers that cannot be obtained +through nucleon catalysis due to the relatively few nucleons present in the final stages of these +collisions, a similar mechanism of catalysis by pions [52, 125, 126] can be invoked. +Short-range correlations +A particular aspect of describing correlations in transport simulations is the treatment of short- +range-correlations (SRCs), which have been measured in nucleon knock-out experiments [127–130]. +Along with the experiments, microscopic many-body calculations show that SRCs introduce a +high-momentum tail (HMT) into the nucleon momentum distribution and, moreover, reduce the +kinetic symmetry energy relative to the Fermi gas kinetic energy, which is a consequence of the fact +that SRCs are more pronounced in symmetric relative to asymmetric matter [131–138] (see also +Section VI C). Phenomenological methods have been used to include SRCs in transport models, +e.g., by initializing nuclei with a HMT, but such a procedure does not take into account the dynamic +role of SRCs in the initial state, which in the case of the on-shell semiclassical equations of motion +results in obtaining nonstationary, excited states of nuclei. In on-shell transport approaches, three- +and many-body collisions, incorporated into transport codes within varying approximations, have +been suggested as a way of treating SRCs. In particular, in an investigation [139] of three-body +collisions for pion production processes (e.g., NNN → NN∆), it was found that SRCs between +two of the incident nucleons give a noticeable contribution to pion yields. Another approach [140], +based on a mean-free-path approximation to the collision integral, observed large effects also on +bulk observables. The incorporation of n-body collisions in transport equations within a schematic +cluster approximation was also studied [141], however, there the effects were found to be rather +small. +So far, none of these methods have been widely exploited in the description of heavy- +ion reactions. +Since HMTs are tied to the tails of the nucleon spectral functions (away from +the quasiparticle peaks), a consistent description of SRCs should involve an off-shell transport +formulation. Dynamical spectral functions of all considered particles, including those which are +stable in free space like nucleons, have been accounted for in the off-shell transport approaches +implemented, with some differences in detail, in the codes GiBUU [42] and PHSD [43]. A subsequent +study [142] demonstrated that the momentum distribution automatically develops a HMT within +the approach used in GiBUU. Differences in the results from the two approaches have yet to be +investigated systematically, including the impact on symmetry energy inferences from heavy-ion +collision data based on, e.g., charged pion subthreshold production yields. Fully quantum transport +approaches with SRCs (or equivalent content), without any semi-classical expansions as are present +in current off-shell transport approaches, remain a long-term goal, and progress in this area has +not ventured yet beyond schematic models [143, 144]. However, increasing computational power +combined with emulation techniques may make such efforts more realistic and enable, e.g., a +seamless integration of the treatment of shell effects in the initial state and collision dynamics. + +23 +Threshold effects +An important influence of mean-field potentials in heavy-ion transport appears in the form +of threshold shifts and the related subthreshold production of particles. Thresholds of particle +production are modified in a medium since the mean-field potentials have to be taken into account in +the energy-momentum balance of a two-body collision. Specifically, when the mean-field potentials +are momentum-dependent and/or as a consequence of other model assumptions for the mean-field +potentials of the produced particles, the thresholds are shifted away from their free-space values. +This may strongly change the production rates of particles. Moreover, the threshold shifts make +it necessary to involve other nucleons, besides the two collision partners in the process, to ensure +the energy-momentum conservation. Various schemes to achieve this locally or globally have been +in use [115, 145]. Indeed, explaining recent heavy-ion collision subthreshold pion yields, measured +by the SπRIT Collaboration [102], required invoking many-body elementary effects in the form of +mean-field effects on thresholds in two-particle collisions [86, 101]. However, because the physics +invoked in describing the threshold effects is similar to that invoked for other multi-particle effects, +alternative multi-particle options remain to be investigated, including producing pion degrees of +freedom in multi-particle collisions or in the aftermath of an off-shell propagation between binary +collisions. (We note here that there is a physics overlap between these mechanisms and the impact +of SRCs on pion production [42, 122, 139].) Notably, theoretical explorations find sequences of +on-shell binary processes to dominate the production at higher beam energies [43, 55, 139], and +no comparable difficulties have been encountered in describing the data [146, 147] by transport +models without multi-particle effects. +The contrasting struggles of transport models which do +not include threshold or other multi-particle effects of this type [102] , together with expected +further theoretical explorations and future measurements of the subthreshold production in heavy- +ion collisions, offer exciting possibilities for gaining understanding of the more exotic in-medium +processes. +New observables +Upcoming precision data will further bring unprecedented observables that could be previously +considered only in theory, such as triple-differential spectra tied to a fixed orientation of the reaction +plane [18, 148–150] not only for protons and most abundant mesons, but also for deuterons,tritons, +light nuclei, and hypernuclei. The potential of such spectra for the determination of the EOS +is still to be fully explored, but a preliminary investigation [149] indicates a rich structure with +spectra which exhibit a maximum away from the beam direction, characterized by slopes dependent +on azimuthal angle and slope discontinuities. Models that might have agreed with each other in +describing low-order Fourier coefficients of flow will likely find describing such detailed observables +difficult. Challenges remain even at the level of the low-order coefficients, as many models now +reproduce proton flow, but not Lambda or pion flow (see, e.g., Fig. 14). Understanding the relations +between observables for various particle species will lead to constraints on the physics driving +the evolution of heavy-ion collisions in simulations and, through that, to understanding cluster +formation, hyperon yields, in-medium interactions with of strange hadrons, and more (see also the +white paper on QCD Phase Structure and Interactions at High Baryon Density: Continuation of +BES Physics Program with CBM at FAIR [151]). +Quantifying uncertainties of transport predictions +In the era of multi-messenger physics, where information on the EOS is derived from different +areas of physics such as nuclear structure, nuclear reactions, and astrophysics, the ability to assess +the uncertainty of a particular result is of crucial importance. This problem is especially relevant for +evaluations of constraints on the EOS from transport simulations of heavy-ion reactions, since it has +been found that using different transport models to describe the same data can lead to very different + +24 +conclusions. As found in the TMEP comparisons (see [47] for a review), even with controlled input +the results from different models may vary considerably due to different implementation strategies +which in themselves are not dictated by the underlying physics. In such a situation, calculating +the mean and variance of different model predictions is not a reliable way of determining the +uncertainties. An approach currently considered for ensuring a robust quality control in combining +inferences from different models is to weigh the models with a Bayesian weight which could be +based, e.g., on the performance of a given model in benchmark tests and/or its ability to reproduce +all key observables of a given reaction (for example, flow observables, particle multiplicities, and +spectra). Bayesian analysis can be also used for model selection through a comparison of results +from a list of available models with data, during which one assigns to each model a probability +of being correct based on the quality of the fit. However, this approach implicitly assumes that +among the considered models there is at least one “true” model (also known as the M-closed +assumption), which is often not fulfilled. Efforts have been taken to analyze data with an M-open +assumption, where the existence of a perfect model is not assumed. For nuclear physics efforts, +this is being attempted within the Bayesian Analysis of Nuclear Dynamics (BAND) group [152] by +using Bayesian model mixing, where information from different models is combined for inference. +Emulators and flexible EOS parametrizations +Robust explorations of the possible physics underlying various observables often necessitate +repeating the calculations many times for different combinations of physics parameters. When high +event statistics is needed, the computational task can easily overwhelm the available computational +resources. +An additional computational strain often arises from assessing Bayesian probability +distributions for any conclusions. Increasingly, emulators are going to be used for this task, with +some steps having been already made [58, 102, 153]. Notably, similar issues emerge in the area of +applications of hadronic transport [154] (see also Section V A). +For explorations focused on the EOS, it may be of advantage to fit various possible EOSs with +flexible relativistic density functionals as suggested in [57, 76]. This approach, given the complete +freedom in varying both the functional form of the EOS as well as the EOS parameters, is particu- +larly amenable to Bayesian analyses (see, e.g., [58] for a Bayesian analysis with a parametrization +of the EOS in terms of the functional dependence of the speed of sound on density). +The above list of issues facing the application of transport theory to heavy-ion collisions high- +lights the fact that this approach to putting tighter constraints on the EOS rests on overcoming +certain challenges. In simple terms, one attempts here to use a very dynamic and complex non- +equilibrium process to obtain information describing a relatively simple and well-defined system, +namely the equilibrated EOS of nuclear matter for different densities, temperatures, and isospin +asymmetries. To achieve this in a reliable way, multiple complex issues of many-body physics have +to be well controlled. On the other hand, several of the needed improvements are relatively well- +understood, and tackling some of the unresolved problems poses an exciting intellectual challenge. +As a reward for undertaking this effort, one gains the opportunity to obtain information on the EOS +in a region which cannot be accessed through any other means: For densities below saturation, +there is strongly constraining information from nuclear structure, with significant contributions +coming also from low-energy heavy-ion collisions. Astrophysical observations on neutron stars and +neutron star mergers are mainly sensitive to densities above about 3n0. The gap between these +domains can only be filled with intermediate energy heavy-ion collisions, and transport studies are +the essential tool to extract the information on the EOS from experimental data. + +25 +B. +Microscopic calculations of the EOS +Over the past decade, many-body nuclear theory has made significant progress in deriving +microscopic constraints on the nuclear EOS at low densities from chiral effective field theory +(χEFT) [155–158]. The progress has been driven by improved two-nucleon (NN) and three-nucleon +(3N) interactions, rigorous uncertainty quantification, and algorithmic and computational advances +in the frameworks used to solve the many-body Schr¨odinger equation with these interactions (see +also the recent white paper on Dense matter theory for heavy-ion collisions and neutron stars [159]). +1. +Status +Chiral EFT [160–164] provides a systematic way to construct nuclear interactions consistent with +the low-energy symmetries of QCD, using nucleons (N’s), pions (π’s), and (in the case of delta-full +χEFT), ∆-resonances (∆’s) as the relevant effective degrees of freedoms. Nuclear interactions in +χEFT are expanded in powers of momenta or the pion mass over a hard scale at which χEFT breaks +down; this breakdown scale is expected to be of the order of the ρ-meson mass, Λb ≈ 600 MeV. +At each order in the EFT expansion, only a finite number of diagrams enter the description of the +interaction according to a chosen power counting scheme, of which the Weinberg power counting +has been predominant. +For example, at the leading-order (LO) in Weinberg’s power counting +one includes contribution from the one-π exchange between two nucleons as well as momentum- +independent contact interactions, which allow one to describe key features of the nuclear interaction +already at the lowest order. At next-to-leading-order (NLO), two-π exchanges are included as well +as momentum-dependent contact interactions, and similarly, more involved terms appear at higher +orders. The various low-energy coupling constants are determined from fits to experimental data, +e.g., the π-N couplings are fit to π-N scattering, while those describing NN short-range interactions +are fit to NN scattering data. The advantage of χEFT over phenomenological approaches is that +multi-nucleon interactions, such as the important 3N interactions, naturally emerge in the EFT +expansion and, moreover, are consistent with the NN sector. Forces involving increasingly more +nucleons are correspondingly more suppressed, e.g., the leading contribution to 3N forces (four- +nucleon (4N) forces) appears at N2LO (at N3LO) in Weinberg’s power counting. Furthermore, there +are only two new low-energy couplings appearing in the three- and four-body forces to N3LO, which +govern the strengths of the intermediate- and short-range contribution to the leading 3N forces, +respectively. Consequently, χEFT 3N and 4N interactions at N3LO are completely determined by +constraints on the coupling constants obtained from NN and π-N scattering”, usually resulting in +tight constraints on very neutron-rich matter from χEFT. +Another key feature of χEFT is that order-by-order calculations in the χEFT expansion have +enabled estimation of theoretical uncertainties due to truncating the chiral expansion at a fi- +nite order [13, 158, 165, 166]. Quantifying and propagating these EFT truncation errors enables +meaningful comparisons between competing nuclear theory predictions, see Fig. 10, and/or con- +straints from nuclear experiments and neutron-star observations in the multi-messenger astron- +omy era [167]. Such comparisons are facilitated by Bayesian methods in a statistically rigorous +way [158, 167, 168] to take full advantage of the great variety of empirical EOS constraints we +anticipate in the next decade. +Chiral EFT also provides nuclear Hamiltonians governing the interactions in nuclear systems. +However, to calculate properties of a many-body system, computational methods able to solve the +Schroedinger equation for this system are necessary. Among various frameworks used to solve the +nuclear many-body problem in dense matter, quantum Monte Carlo (QMC) methods and many- +body perturbation theory (MBPT) have been the main tools employed to study the physics of + +26 +FIG. 10. Comparison of the energy per particle E/N (left) and the pressure P (right) as functions of density +for pure neutron matter in different many-body calculations using interactions from χEFT. The left panel +also shows low-density QMC results of Ref. [169] and the conjectured unitary-gas lower bound on the energy +per particle of pure neutron matter from Ref. [16]. Figure from Ref. [170]. +neutron-star matter in recent years. Both methods have recently made tremendous advances in +predicting properties of nuclei and calculating the nuclear matter EOS [156, 158, 171–175]. +QMC frameworks, such as the auxiliary field Diffusion Monte Carlo (AFDMC) method, are +based on imaginary-time propagation of a many-body wave function and enable us to extract +ground-state properties of a nuclear many-body system with high statistical precision [156, 171]. +Their nonperturbative nature also allows for the treatment of nuclear interactions at high mo- +mentum cutoffs, providing important insights into nuclear interactions at relatively short distances +that may help to improve the modelling of χEFT interactions. QMC calculations of binding en- +ergies, radii, and electroweak transitions of nuclei up to A = 16 [176–182] using χEFT NN and +3N interactions are in very good agreement with experimental data [183–186]. +QMC methods +were also used to calculate the EOSs of matter up to about twice the nuclear saturation density +n ≈ 2 n0 [187–191]. The calculated EOSs include estimates of systematic truncation uncertainties, +and are commonly used to constrain properties of neutron stars [188, 192, 193]. +The past decade has also seen a renaissance for many-body perturbation theory (MBPT) calcu- +lations in nuclear physics [158, 175]. Key to this development has been the discovery that nuclear +potentials with momentum-space cutoffs in the range 400 MeV <∼ Λ <∼ 500 MeV (not to be confused +with the breakdown scale of χEFT, Λb) are sufficiently soft to justify the use of perturbation theory +methods [194] (see [195] for a Weinberg eigenvalue analysis). Such low-momentum potentials can +be obtained from renormalization group methods [196] or by directly constructing chiral effective +field theory potentials at a coarse resolution scale. Furthermore, recent advances in automatic +diagram generation [197] combined with automatic code generation [198] and high-performance +computing have led to a fully automated approach to MBPT calculations in nuclear physics [158], +in which chiral two- and multi-nucleon forces can be included to high orders in the chiral and +MBPT expansions. MBPT has been demonstrated to be a computationally efficient and versatile +tool for studying the nuclear EOS as a function of baryon number density nB, isospin asymmetry +δ = (nn −np)/(nn +np), and temperature T [110, 111, 199, 200] with implications for neutron star +structure [158] and astrophysical simulations [201]; here, nn and np correspond to the neutron and + +25 +5 +Hebeler et al.,ApJ (2013) +Hebeler et al.,ApJ(2013) +Tews et al.,PRL(2013) +Tewsetal.,PRL(2013) +Lynn et al.,PRL (2016) +Lynn et al., PRL (2016) +20 +4 +Drischler et al.,PRL (2019) +Drischler et al.,PRL (2019) +Drischler et al.,GP-B (2020) +Drischler et al.,GP-B (2020) +Gezerlis, Carlson, PRC (2010) +Unitary gas (s = 0.376) +3 +fm +[MeV +10 +P2 +5 +1 +0 +0. +0 +0.05 +0.1 +0.15 +0.2 +0 +0.05 +0.1 +0.15 +0.2 +n [fm-3] +n [fm-3]27 +proton densities, respectively. In particular, MBPT allows us to compute the EOS of neutron-star +(i.e., β-equilibrated) matter explicitly, which can help improve isospin asymmetry expansions of +the low-density nuclear EOS such as the standard quadratic expansion [199, 202–206]. MBPT also +allows us to study nuclear properties other than the nuclear EOS, including the linear response +and transport coefficients that could be used to inform more accurate numerical simulations of +supernovae and neutron-star mergers [207]. Furthermore, MBPT for (infinite) nuclear matter has +been used to construct a microscopic global optical potential with quantified uncertainties based +on χEFT NN and 3N interactions [208, 209]. +Altogether, MBPT calculations of nuclear matter +properties can provide important constraints that enable microscopic interpretations of future nu- +clear reaction experiments [210] (e.g., at the Facility for Rare Isotope Beams) and neutron star +observations. +To date, theoretical predictions for the nuclear EOS, optical potentials, and in-medium NN +scattering cross sections have been computed at finite temperature at various levels of approxi- +mation starting from fundamental two- and multi-nucleon forces. These quantities are inputs to +transport model simulations [89, 211] of heavy-ion collisions used to extract constraints on the +properties of hot and dense nuclear matter (see Section II A for more details). In transport simu- +lations, the EOS, single-particle potentials, and in-medium NN cross sections are usually obtained +from effective phenomenological interactions [212, 213] that are fitted to the properties of finite +nuclei and cold nuclear matter, and then extrapolated into the finite-temperature regime. Recently, +some effort has been devoted to benchmarking [109] the temperature dependence of these effective +interactions against predictions from χEFT or directly using EFT constraints in fitting effective +interactions [207, 214, 215]. To enable such comparisons, the free energy of homogeneous nuclear +matter as a function of temperature, baryon number density, and isospin asymmetry has been cal- +culated using χEFT interactions up to second order in many-body perturbation theory [110] and +within the Self-Consistent Green’s Function (SCGF) approach [216], which resums particle-particle +and hole-hole ladder diagrams to all orders. The resulting EOS has been shown to be consistent +with the critical endpoint of the symmetric nuclear matter liquid-gas phase transition [110, 216] as +well as the low-density/high-temperature pure neutron matter EOS from the virial expansion [204]. +Furthermore, single-particle potentials have been computed at finite temperature at the Hartree- +Fock level [217], from G-matrix effective interactions [218], and in SCGF theory [201, 219]. Of +particular importance is the associated nucleon effective mass, which is obtained from a momen- +tum derivative of the single-particle energy. The nucleon effective mass is directly related to the +density of states and hence governs entropy generation at finite temperature, with consequences for +the dynamical evolution of core-collapse supernovae and neutron star mergers. Finally, in-medium +NN scattering cross sections have been computed at finite density and zero [220] as well as at +finite [218] temperature using high-precision nuclear forces. In the next decade, the use of effective +field theory methods will enable a consistent framework for describing all of these quantities with +uncertainty estimates for input into transport simulations of heavy-ion collisions and astrophysical +simulations. +2. +Challenges and opportunities +To fully capitalize on experimental and observational data and extract key information on fun- +damental questions in nuclear physics, continued progress in nuclear theory is crucial. The combi- +nation of χEFT with modern computational approaches like machine learning, artifical intelligence, +emulators, and Bayesian inference have provided EOS results for a wide range of densities, and at +various proton-to-neutron asymmetries and temperatures, with quantified uncertainties [111, 166]. +Future progress in the development of fundamental interactions, combined with these tools, will + +28 +increase the precision of the results and enable us to answer open problems in chiral EFT. Among +these, the most pressing is at which densities and how χEFT breaks down [166, 188]. In particular, +for studies of neutron-star mergers it is of great importance to describe dense matter at finite +temperatures [200, 201, 204], however, these might influence the breakdown of the theory in dense +matter. In the next decade, it will be crucial to reliably determine how far one can push the χEFT +approach in nucleonic matter. +While microscopic calculations have been very successful in calculating properties of nuclei and +homogeneous matter at densities up to 1-2 times the nuclear saturation density, we need improved +microscopic descriptions of neutron-rich dense matter beyond that regime, at a few times nuclear +saturation density and finite temperatures, with quantified uncertainties. This can be achieved +by employing models derived within relativistic mean-field or density functional theory that are +firmly rooted in microscopic theory at lower densities. Such models will be very important to +connect theoretical calculations within the framework of χEFT to heavy-ion collision experiments +at accelerator facilities around the world. Heavy-ion collision experiments at intermediate beam +energies bridge the low- and high-density regimes of the EOS and provide complimentary informa- +tion to that obtained from nuclear structure or neutron-star studies [17] (see Section II A). Robust +inferences from the experimental data will require more accurate predictions from transport the- +ory, which strongly depend on, among others, mean-field or density functional models. It will be +imperative to test and constrain such models for the EOS with more rigorous microscopic calcu- +lations. Beyond their use in hadronic transport simulations, these models are also a crucial input +for calculations of properties of neutron star crusts (see Section II C). +Additional theoretical constraints might be provided by high-density calculations within the +framework of perturbative QCD (pQCD) [221], which can be applied at very high densities of +the order of 40 times the nuclear saturation density, where the strong interactions among quarks +become perturbative. Constraints on the EOS based on pQCD, together with assumptions on +causality and stability, have been used to constrain the EOS at lower densities probed in the core +of neutron stars [222–225]. +However, it has been found that the constraining power of pQCD +calculations is strongly dependent on the way in which they are implemented [225, 226]. Future +studies have to establish to what extent pQCD constraints are robust at densities of the order of +several times nuclear saturation density, and how constraining future higher-order calculations may +become. In this regard, improved microscopic calculations of the nuclear EOS using the functional +renormalization group [227, 228] will provide important insights. +C. +Neutron star theory +1. +Status +Measurements of the EOS, masses of neutron-rich isotopes far from the band of stability, and +experimental constraints on nucleon effective masses provide essential input into neutron star mod- +els, progressing our understanding of the structure and dynamics of these astronomically important +objects. Several properties of neutron stars, including the mass-radius relation and their tidal de- +formabilities, can be calculated once the EOS is provided. This, in turn, enables us to constrain +the EOS once those properties are observed [229]. +Nuclear EOSs for neutron stars can be constructed from, for example, ab initio calculations and +density functionals [230–233] or, more schematically, from meta-models [234–236] parameterized +by nuclear matter parameters, which can be used to make contact with heavy-ion collisions [17]. +Ab initio calculations take into account more fundamental properties of the nuclear force (see Sec- +tion II B), but prohibit the calculation of large ensembles of EOSs spanning the nuclear parameter + +29 +FIG. 11. Impact of nuclear physics theory and experiment, and +different astrophysical measurements on constraining the cold +neutron-star EOS. Blue lines show a family of EOS that are con- +strained by chiral EFT at low densities. At higher densities, the +EOS can then be constrained using GWs from inspirals of neutron +star mergers, data from radio and X-ray observations of pulsars, +and electromagnetic signals associated with neutron star mergers. +The indicated boundaries between regions affected by these mea- +surements are not strict and depend on the EOS and properties +of the astrophysical system. Figure from [237]. +space. +Meta-models allow rapid +computation of such large ensem- +bles, but encode mainly bulk prop- +erties of nuclear matter, which ex- +cludes them from being used to +model finite nuclei. +Density func- +tionals represent a compromise, al- +lowing both rapid computation of +EOSs and use in finite nuclear mod- +els, and thus are more suited to +combining nuclear experimental and +astrophysical information. +Many +of these models can be smoothly +extrapolated from the saturation- +density to arbitrarily high density, +in which case astronomical obser- +vations can be used to constrain +the saturation-density nuclear mat- +ter parameters and their density de- +pendence [236, 238]. This extrapo- +lation, however, is model-dependent, +as different density functionals have +different dependence on density. Ad- +ditionally, this extrapolation might +not be physically well-founded. +As densities inside neutron stars +can reach up to several times nuclear saturation density, at some (as-yet not determined) density a +description in terms of purely nucleonic degrees of freedom is expected to break down. Heavy-ion +collisions can help us constrain that point, and the nature of any phase transitions that occur above +saturation density. The nuclear EOSs can be then combined with models describing the EOS at +higher densities. Models that explicitly include a range of possible high-density degrees of freedom, +such as hyperons and quarks, can be constructed; the predicted neutron star compositions are then +dependent on the particular model used. Another approach is to use more general models that give +up the explicit dependence on the underlying degrees of freedom, thus losing information on, e.g., +appearance of exotic particles at high densities, in favor of spanning the full space of physically con- +sistent EOSs, reducing the model dependence of inferences from astrophysical observations [239]. +These schemes include piecewise polytropes [14, 240–242], line segments [192, 243], speed-of-sound +models [188, 189, 244–246], spectral models [247] and non-parametric models generated from Gaus- +sian processes (GPs) [168, 248–251] or machine learning techniques [252]. If these more general +approaches are used down to the nuclear saturation density, extra modeling is required to connect +them to the microscopic nuclear EOS and nuclear observables [253]. +Once the EOS is specified, the solution of the Tolman-Oppenheimer-Volkov equations and their +extensions including rotation, determining the structure of a neutron star through balancing the +attractive force of gravity and the repulsion coming from the EOS, provide predictions for bulk +properties of the neutron star such as radii, tidal deformabilities, moments of inertia, and break-up +frequencies of neutron stars as a function of their mass. All of these properties can be compared +with multi-messenger observations, including gravitational waves and electromagnetic signals from +neutron-star mergers and isolated neutron stars [193]. +The systematic construction of neutron star EOS models and statistical inference of EOS pa- + +103 +8 +102 +[Mev fm +EM: Kilonovae / GRB +GWs (post-merger) +Pressure +101 +GWs (inspiral) +Radio and X-ray pulsars +100 +Nuclear Physics +Experiment and Theory +2 +4 +6 +8 +Number density [nsat30 +rameters from data is an endeavor that is just over a decade old [14, 240–242]. This effort has +matured in the current era of multi-messenger astronomy with a large push to explore the model- +dependence of EOS inferences [244, 254] and ways of connecting the EOS with astrophysical and +nuclear data [17, 167, 193, 245, 246, 255]. Different choices of which observables to include or infer +can be made. For example, astrophysical observations can be used to infer the EOS, which can +then be connected to nuclear models to inform their parameters and predict nuclear observables. +Conversely, nuclear observables can be used to infer nuclear parameters, which can then inform +the neutron star models and predict astrophysical observables. The future lies in combining more +and more sets of data of both types to understand nuclear and neutron star models better. +Exciting progress has been made in gathering astrophysical data to constrain our dense matter +theories (see Fig. 11 for an illustration of density regions affected by different observables). Neutron- +star data from the last 5 years identified the heaviest neutron star known to date with a mass of +2.08(7)M⊙ [256, 257] (where M⊙ is the solar mass), while the kilonova AT2017gfo, associated with +GW170817, has placed an upper limit on the maximum mass to be on the order of 2.3M⊙ [258, +259]. +The detection of GW170817 by the LIGO-Virgo Collaboration has enabled us to place +constraints on the tidal deformability of this system, ˜ΛGW170817 ≤ 720 [260, 261]. Neutron Star +Interior Composition Explorer Mission (NICER) has provided two mass-radius measurements by +observing X-ray emission from several hot spots on the neutron star surface, finding a radius of +13.02+1.24 +−1.06km for a star with mass 1.44+0.15 +−0.14M⊙ (PSR J0030+0451) and 13.7+2.6 +−1.5km for a star with +mass 2.08(7)M⊙ (PSR J0740+6620) in the analyses of Refs. [255, 262–264]. X-ray observations of +the temperature of the neutron star in the Cas A supernova remnant have revealed core cooling +on the timescale of years, hinting at the possible superfluid properties of the core [265]. These +observations have enabled meaningful constraints on the EOS to be set and have already allowed us +fascinating glimpses into the possible properties of high-density matter. For example, perturbative +QCD predicts that the speed of sound squared approaches the conformal limit of 1/3 from below +as the density becomes arbitrarily high. +Meanwhile, inferences of the neutron star EOS from +observational data indicate that the speed of sound rises in the core to significantly above c2 +s = +1/3 [188, 266–269]. Consequently, this suggests that the speed of sound has a non-trivial behavior +with increasing density [188, 221, 270]. At the same time, tentative evidence for quark matter in +neutron star cores, which in turn indicates a softening of the EOS, has likewise been suggested [246]. +If we want to leverage the substantial data we have on neutron star cooling and dynamical +evolution, additional EOS quantities need to be supplied consistently for each EOS model, such +as the effective masses (see also Section VI E) and superfluid neutron and proton gaps, essential +for modeling thermal and dynamical properties of neutron stars. For example, the mutual friction +of the core – the strength of the coupling between the charged particles (electrons, protons) and +superfluid neutrons – depends on the effective neutron mass and the proton fraction [271], which +both also correlate with the symmetry energy [108]. A consistent extraction of both symmetry +energy parameters and effective masses from heavy-ion collision data is therefore required. +In contrast to efforts devoted to systematic, statistically meaningful inferences of the EOS in the +cores of neutron stars, modeling the neutron star crust is still in its infancy: The first calculations +of large ensembles of systematically parameterized crust models and their use in statistical analysis +have only been carried out recently [272–276]. However, much more nuclear experimental data can +be brought to directly bear on crust physics, and we have entered an era where we can access +information about the crust with unprecedented fidelity. For example, we have now observed the +same neutron-star crust as it first cooled, then became heated by accreted matter, and then cooled +again [277–281]. We have followed a pulsar through a glitch – a sudden change in the rotation period +of the pulsar – and glitch recovery with a resolution of a few seconds [282]. These observations +have provided very strong evidence that the crust is solid, that there exist superfluid neutrons in +the inner crust which can be decoupled from the nuclei in the crustal lattice, and that nuclear + +31 +reactions from accreted material sinking into the crust provide deep crustal heating [279, 283, 284]. +Additionally, models of the neutron star crust predict that, prior to the transition to homoge- +neous matter, isolated nuclei in the crust fuse to form cylindrical, planar, and more exotic shapes, +termed “nuclear pasta”, that can affect neutron-star observations [285–287]. This crust-core bound- +ary region, often referred to as the mantle, is likely a complex fluid. Density functional theory and +molecular dynamics calculations of these structures reveal a complex energy landscape with many +coexisting shapes, and correspondingly complex mechanical and transport properties [288–294], +which are strongly influenced by the EOS at around 0.5n0 through the pressure, proton fraction, +and surface energy of the structures. +These properties can also be studied in multifragmenta- +tion reactions, which probe, among others, the competition between nuclear surface energy and +Coulomb energy at sub-saturation density [295–297]. +Inhomogeneous matter in the crust of a neutron star, including the dripped neutrons expected in +the inner crust, can be modeled using a variety of nuclear theory techniques. These usually involve +calculations within a single, repeating unit (Wigner-Seitz cell) of matter, typically containing a sin- +gle nucleus [298–300]. The compressible liquid drop model (CLDM) treats the nuclear matter inside +and outside of nuclei as homogeneous and described by the bulk matter EOS, while the surface +energy is specified by a separate function with additional parameters [288, 300–303]. The surface +parameters and those that define the dimensions of the cell and nucleus are minimized to obtain +the ground state. The Thomas-Fermi model employs the local density approximation, modeling +matter with a specified form of the inhomogeneous nuclear matter density in the unit cell; here, +the parameters of the density distribution are varied to obtain the ground state configuration [304]. +Microscopic approaches to describing inhomogeneous nuclear matter, in which individual neutrons +and protons are the degrees of freedom, include quantum Hartree-Fock or Relativistic Mean Field +models [305–310], and semi-classical molecular dynamics approaches [292, 311]. +There is a great need for nuclear physics input into models of the neutron star crust, which +analyses of heavy-ion collision data can provide. For example, the thickness, mass and moment +of inertia of the crust depend on the higher-order symmetry energy parameters L, Ksym, and +Qsym [272, 274, 312]. Thus measurements of the symmetry energy parameters up to third order +in heavy-ion collision experiments are essential to understand the properties of the crust. The +symmetry energy, effective masses, and surface energies of nuclear clusters strongly affect the +proton fraction on either side of the crust-core transition density, the extent of nuclear pasta near +the crust-core boundary, the mechanical and transport properties, the thermal conductivity and +specific heat, the electrical conductivity, and the shear modulus of the crust [298, 304, 309, 313]. +Nuclear experiment can thus constrain neutron star crust models, and astrophysical observables +associated with the crust can measure nuclear observables as well as measurements of neutron star +bulk properties. For example, the symmetry energy can be constrained by combining nuclear data +with crust and core observables, e.g., through a potential multi-messenger measurement of the +resonant frequency of crust-core interface oscillations [276]. +2. +Challenges and opportunities +The next decade will provide a wealth of new data on neutron stars, as the LIGO-VIRGO- +KAGRA detectors are expected to observe many new binary neutron-star mergers, some of them +with electromagnetic counterparts [314–316]. As NICER continues to measure more neutron star +masses and radii, next-generation X-ray timing missions such as Strobe-X [317] and radio tele- +scopes such as the Square-Kilometer Array will increase the number of pulsars we see and are able +to measure by an order of magnitude. Long-timescale observations of individual pulsars (using +radio timing) and persistent gravitational waves from deformations of neutron stars will lead to + +32 +measurements of their moments of inertia. These new data points might enable us to pin down +the nuclear matter EOS, to discover or rule out the existence of phase transitions to exotic forms +of matter in the cores of neutron stars, and to reliably constrain microscopic interactions between +fundamental particles. +Although model-agnostic extrapolations to higher densities such as through the use of poly- +tropes [194, 240], speed of sound schemes [188, 244, 318], Gaussian processes [249, 250] and spectral +methods [247], combined with robust data analysis, will eventually allow us to pin down the dense- +matter EOS, they cannot answer the question about the relevant microscopic degrees of freedom at +high densities. Hence, it is crucial to develop improved microscopic models with well-quantified un- +certainties in this regime. At the same time, creating ensembles of outer core and crust models that +allow for inclusion of astrophysical and nuclear data requires underlying nuclear models to have +enough freedom to explore a large region of parameter space, and allow fast computation of relevant +quantities that also capture the essential physics. Currently, it is energy density functionals like +Skyrme, Gogny, and Relativistic Mean Field models that provide these properties. Consequently, +progress could be made by making a stronger connection between these models and microscopic +approaches, e.g., connecting energy-density functionals to ab initio calculations allowing a more +direct link to χEFT [299, 319, 320]. In the same spirit, EFT calculations of the EOS can be used as +a “low-density limit” to calibrate higher-density models for neutron stars and heavy-ion collisions. +The crust can be modeled consistently with nucleonic matter in the core using density functional +theory to model both. When choosing a model, a compromise must be made between accurate +modeling of microscopic quantum effects, such as shell effects in the nucleus and surrounding +neutron gas, and the computational expediency required to construct large ensembles of crust +models needed for statistical inference. For example, quantum shell effects strongly determine the +evolution of the mass and charge number of nuclei with density, alter the effective mass of dripped +neutrons, and drive the complex energy landscape of nuclear pasta. Fully microscopic quantum +calculations include shell effects self-consistently, but are computationally expensive. The CLDM +approach can be used to construct large numbers of crust models, but requires shell effects to be +added by hand. Future work needs to develop schemes of incorporating such microscopic effects in +large ensembles of crust models. The method that may allow that is the Extended Thomas-Fermi +method, incorporating shell effects through the Strutinsky Integral: see, e.g., [321]. +Models should also incorporate nuclear pasta, as its extended structures may contribute to the +mechanical and thermal properties of matter at the crust-core boundary. It is computationally +demanding to model transport and mechanical properties of the crust microscopically or in simula- +tions [322], particularly in the nuclear pasta phases, and it is unrealistic to include these quantities +in large ensembles of crust models. Simpler schemes that extrapolate the mechanical and trans- +port properties across the parameters space based on microscopic models could be developed. Also, +representative crust models inferred from data can be used to calculate these crust properties. +There is also a need for a balance between accuracy and precision. A model can be accurate +but not precise (predicting the correct value of a physical quantity but having large error bars), +or precise but not accurate predicting very small error bars, but not predicting the correct value +of some physical observable). +Individual crust models can be created from mass models that +are precisely fit to data and which predict precise values for, for example, the symmetry energy +parameters. However, to make accurate inferences of nuclear matter parameters from astrophysical +observables, and to include their experimentally measured ranges, ensembles of models spanning +the parameter space should be employed. +Both strategies are important, and the precision-fit +models can act as benchmarks against which we assess the outcomes of statistical inferences. +When older neutron stars accrete matter in the crust the matter gets gradually pushed down +into the core and replaced by the accreted matter. The temperatures in the crust are well below +the nuclear potential energies, so the replacement crust cannot easily attain nuclear statistical + +33 +equilibrium. Ensembles of accreted crust models are yet to be constructed, but are necessary to +correctly account for deep crustal heating and therefore to fully utilize the observations of cooling +of accreted crusts in low mass X-ray binaries. +In all this work, effort must be made to calculate the different observables consistently as well +as to combine different data sets in a well-controlled way. This is expanded upon in Section IV. +III. +HEAVY-ION COLLISION EXPERIMENTS +Establishing the equation of state (EOS) of nuclear matter has been a major focus of heavy-ion +collision experiments. While very low energy collisions can probe nuclear matter at densities smaller +than the saturation density n0, highly-compressed nuclear matter is produced in the laboratory +by colliding heavy nuclei at relativistic velocities. At even higher energies, in the ultra-relativistic +regime, quarks in the colliding nuclei become almost transparent to each other and therefore escape +the collision region, which means that matter measured at midrapidity is characterized by a nearly- +zero net baryon number. Heavy-ion collision experiments at top beam energies at the Relativistic +Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) provided convincing evidence that +at high temperatures and near-zero baryon density, nuclear matter becomes a quark-gluon plasma +(QGP) [323–329], a deconfined but strongly-interacting state composed of color charges, confirming +Lattice QCD (LQCD) calculations of the EOS at zero density [330–332]. +While the region of the QCD phase diagram explored in ultra-relativistic heavy-ion collisions +is relatively well understood, the EOS of dense nuclear matter at moderate-to-high temperatures +and moderate-to-high baryon densities is not known well due to the break-down of first-principle +approaches in this regime. Answering pressing questions about the QCD EOS in this region, such as +whether the quark-hadron transition becomes of first-order at high densities or what is the minimal +energy required to produce the QGP, is the driving force behind Phase II of the Beam Energy Scan +(BES) program at RHIC, the HADES experiment at GSI, and the future Compressed Baryonic +Matter (CBM) experiment at the Facility for Antiproton and Ion Research (FAIR), Germany. +This renewed interest in the nuclear matter EOS at high densities, accessible in heavy-ion +collisions at intermediate energies, coincides with an increased effort to constrain the EOS of +neutron-rich matter, probed in studies of neutron stars and neutron star mergers (see Section II C +as well as recent white papers on QCD Phase Structure and Interactions at High Baryon Den- +sity: Continuation of BES Physics Program with CBM at FAIR [151] and Dense matter theory for +heavy-ion collisions and neutron stars [159]). Furthermore, studies show that heavy-ion collisions +in this regime and neutron star mergers probe similar temperatures and baryon densities [333, 334]. +However, while matter created in collisions of heavy-ions has comparable numbers of protons and +neutrons, matter inside neutron stars is neutron-rich. Establishing the much needed connection +between the studies of the nuclear EOS as probed in heavy-ion collisions and as inferred from +neutron star observations is possible by leveraging the experimental capabilities of the newly com- +missioned Facility for Rare Ion Beams (FRIB), where energetic beams of proton- and neutron-rich +nuclei can be produced. Heavy-ion collision experiments at FRIB can put tight constraints on the +dependence of the nuclear matter EOS on the relative proton and neutron abundances [22], and +thus enable a description of both dense nuclear and dense neutron-rich matter within a unified +framework. +Indeed, if we assume that the core of a neutron star is composed of mostly uniform nucleonic +matter, then nuclear matter and neutron stars should be described by a common EOS, specifying +the relationship between the pressure and the temperature, density, and isospin content. +The +theoretical construct of symmetric nuclear matter consisting of equal amounts of neutrons and + +34 +Number density +Astro +HIC(asym) +Nuclei properties +Theory +Crust +HIC(SNM) +FIG. 12. This schematic plot illustrates the approximate density +ranges that are explored in the studies of chiral effective field +theory, nuclei properties, heavy-ion collision experiments, and +observations of neutron stars and their crusts in astronomy. +protons has been successful to derive +properties of symmetric matter such +as the saturation density and bind- +ing energy, however, an additional +term in the EOS is needed to de- +scribe nuclear matter with unequal +neutron-proton composition. +This +second term depends on the asymme- +try δ, defined as δ = (nn − np)/nB, +where nn, np, and nB are the neu- +tron, proton, and total baryon densi- +ties, respectively. Consequently, one +can view the asymmetry as the neu- +tron excess fraction. Mathematically, +the energy per nucleon can be then +expressed as a sum of two terms: +ϵ(nn, np) = ϵSNM(n) + S(n)δ2. Here, the first term represents the energy per nucleon of symmetric +nuclear matter, while the second term accounts for the correction needed when δ ̸= 0. Therefore, +δ is a crucial parameter that distinguishes neutron stars (with δ >∼ 0.8) from most nuclei (with +δ <∼ 0.25). Given the relatively small values of the asymmetry δ for nuclei, in heavy-ion collision +experiments it is easier to constrain the coefficients of the EOS of symmetric matter, ϵSNM(nB). In +contrast, the energy contribution from the asymmetric term, also known as the symmetry energy, +constitutes a small fraction of the total energy of a nucleus even for neutron-rich heavy radioactive +isotopes (< 5% in the liquid drop model), and its determination requires precise measurements. +Furthermore, because the isospin effects in any observable tend to diminish with temperature, it +may be difficult to measure the symmetry energy at very high densities, which require high-energy +heavy-ion reactions. Therefore, symmetry energy is best probed in heavy-ion collisions of highly +asymmetric isotopes at low to intermediate energies. +Fig. 12 shows schematically the baryon density regions explored by different areas in nuclear +physics studies. Recent breakthroughs in astronomical observations with state-of-the-art instru- +ments led to the first detection of a binary neutron-star merger and the unprecedented radii mea- +surements of neutron stars with accurately known masses (see Section II C). The neutron star +mass-radius relationship provides an insight into the EOS at high densities above twice saturation +density (>∼ 2n0), as represented by the red arrow (labelled “Astro”) in the upper right corner. Labo- +ratory experiments, especially those using heavy-ion collisions, are essential to provide information +on the dependence of the EOS on density and the asymmetry (see also Section II A). High-energy +heavy-ion collisions can provide insight into the symmetric nuclear matter EOS as represented +by the gold right-pointing arrow (labeled “HIC(SNM)”), while current probes of the symmetry +energy are more suited for measurements of lower energy heavy-ion reactions (<∼ 600 AMeV) as +represented by the left-pointing gold arrow (labeled “HIC(asym)”). Many properties of nuclei, +such as masses and radii, have been shown to be mainly sensitive to densities around (2/3)n0, +however, with a careful selection of nuclear observables, the symmetry energy has been probed +over densities of 0.3n0 < nB < n0 using Pearson correlation methods [12, 335] (green left-pointing +arrow). Recent advances in chiral effective field theory (see Section II B) enabled extrapolations +of the EOS to be extended up to ≈ 1.5n0 [13], but the uncertainty increases exponentially with +density for densities that are higher than n0. It is not clear what is the maximum density up to +which such extrapolations can succeed. Finally, one of the most interesting regions is at very low +densities (<∼ 0.5n0), corresponding to the crust of a neutron star where matter is not uniform (see +Section II C). There, matter changes with increasing density from a Coulomb-dominated lattice to + +L +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.035 +nuclear pasta and, ultimately, to uniform matter. The density and nature of these transformations +are again dictated largely by the EOS. +Measurements made in heavy-ion collisions at intermediate energies, probing high densities +or, equivalently, small nucleon separations, will yield key insights into the nature of the nuclear +force, including the density-dependence of the nuclear symmetry energy. Experimental efforts to +determine the EOS for symmetric matter and the symmetry energy are described in Section III A +and III B, respectively. +Please note that all beam energies Elab quoted in this section are the single-beam kinetic energies +per nucleon, in units of AMeV or AGeV. (Alternatively, Elab is also sometimes denoted by other +authors as E/A, with units of MeV or GeV). Additionally, while many results are reported in +terms of their constraints on the incompressibility K0, one should refrain from interpreting them +as constraining the behavior of the EOS around the saturation density (see Section II A 2 for more +details). +A. +Experiments to extract the EOS of symmetric nuclear matter +Heavy-ion collision experiments worldwide have extensively studied the EOS of symmetric +nuclear matter at supra-saturation densities over the past four decades. Experiments based at +the Schwerionensynchrotron-18 (SIS-18) ring accelerator at the GSI Helmholtz Centre for Heavy +Ion Research (GSI) have probed Au+Au collisions at energies between Elab = 0.09–1.5 AGeV +(√sNN = 1.92–2.52 GeV), corresponding to fireball densities 1–2.5n0. Further experimental efforts +with Au+Au collisions were carried out at higher energies, Elab = 2–10 AGeV (√sNN = 2.70– +4.72 GeV), at the Alternating Gradient Synchrotron (AGS) at the Brookhaven National Laboratory +(BNL) to probe fireball densities 2.5–5n0. Complementing the densities reached at AGS-BNL is the +Beam Energy Scan (BES) program of the Solenoidal Tracker at RHIC (STAR) experiment at RHIC +in BNL, where high-statistics Au+Au collisions were performed at energies between Elab = 2.9– +30.0 AGeV (√sNN = 3–7.7 GeV) in the fixed-target mode. A selection of constraints on the EOS +extracted from the above experiments is shown in Fig. 9. Below, we describe the observables stud- +ied to extract the symmetric nuclear matter EOS, experiments probing the aforementioned density +ranges, and inferences for the hadronic transport codes. +1. +Measurements sensitive to the EOS +Collisions of heavy nuclei at relativistic energies lead to a rapid compression and heating of +matter trapped in the collision region, followed by its dynamic expansion and cooling (see Fig. 7). +The EOS governs both the compression as well as the expansion of the hot and dense nuclear matter, +which in turn affect measured particle distributions. For example, a stiffer EOS (characterizing +matter that is more incompressible) leads to a relatively smaller compression and, consequently, +smaller heating, but a faster transverse expansion. The smaller temperatures reached in the fireball +lead to smaller thermal dilepton and photon yields (see, e.g., [336–338]), while the faster expansion +manifests itself in relatively higher mean transverse momenta (see, e.g., [24]) and a shorter lifetime +of the fireball, the latter of which can be probed by a combination of the femtoscopic radii, R2 +out − +R2 +side, shown to be proportional to the duration of particle emission [339, 340]. +The EOS also plays a large role in the interplay between the initial geometry of the system, +the expansion of matter originating from nucleons trapped in the collision zone (participants), +and the propagation of nucleons which are either still incoming into the collision region or whose +trajectories do not directly cross the collision region (spectators). In systems colliding at beam +energies for which the speed of the fireball expansion is comparable with the speed of the spectators, + +36 +the resulting complex dynamical evolution affects the transverse expansion of the system and, +therefore, the angular particle distributions in the transverse plane dN/dφ. In particular, moments +of the angular momentum distribution, known as the collective flow coefficients and defined as +vn = +� dφ cos(nφ) (dN/dφ)/ +� dφ (dN/dφ), describe the collective motion of the system and are +highly sensitive to the EOS, as shown in numerous hydrodynamic [77, 341–346] and hadronic +transport [31, 67, 78, 79, 347, 348] models. At the same time, collective flow observables can be +measured with high precision, making them primary observables used to constrain the EOS. +In off-central collisions, the initial collision zone has an approximately elliptical shape, and the +pressure gradients within the collision zone are larger along its short axis. If the spectator nucleons +move out of the way before the fireball expands, the pressure gradients in the collision zone lead to +particle distributions around midrapidity which have maxima coincident with the reaction plane +(“in-plane” emission). If, however, the spectators stand in the way of the fireball expansion, this +leads to a preferential emission along the long axis of the collision zone (“out-of-plane” emission, +also referred to as “squeeze-out” due to the role that the spectators play in the expansion). The +preferential emission in either in-plane or out-of-plane direction is described by the second Fourier +coefficient of flow v2, also known as the elliptic flow, which is positive in the former case and +negative in the latter case (see the lower panel of Fig. 4). The magnitude of the elliptic flow, +as well as the energy at which v2 changes sign, are intrinsically connected to the stiffness of the +EOS: for example, a stiffer EOS results in both a faster expansion and a more forceful blocking by +spectators, which leads to a larger squeeze-out and a more negative v2. +The rapidity-dependence of the first Fourier coefficient of flow, the directed flow v1, is also +sensitive to the EOS as it measures the degree of spectator deflection due to the interaction with +the collision zone [349]. In the center-of-mass frame, the spectators from a nucleus moving in the +positive beam direction will be deflected to one side, while the spectators from the other nucleus, +moving in the negative beam direction, will be deflected to the opposite side, resulting in a positive +v1 at positive rapidities and a negative v2 at negative rapidities (here, the sign of v1 is a matter of +convention; see [58] for a more detailed explanation). The magnitude of the directed flow in each +region and, therefore, its slope at midrapidity are directly related to the EOS: for example, a softer +EOS leads to a smaller deflection and a smaller slope of v1 at midrapidity, where in particular a +sufficiently soft EOS can even lead to a negative slope of v1 [343, 350]. We note that spectators +are necessary to obtain substantial magnitudes of the slope of the directed flow, as can be seen by +its small values at high collision energies (see the upper panel of Fig. 4). +Beyond the collective flow phenomena, the EOS also has an effect on hadron production. In +particular, much attention has been given to production of hadrons in heavy-ion collision at energies +below the nominal production threshold in NN reactions (“sub-threshold” production), which +requires multiple sequential hadron-hadron collisions to occur. The probability of these collisions +is significantly higher in the high-density regions, and consequently the yield of sub-threshold +probes is expected to be substantially enhanced if higher densities are reached in the collision. +Of particular importance for the EOS studies is sub-threshold production of K+ mesons, which +undergo few final-state interactions with the nuclear medium and therefore mostly leave the fireball +unperturbed, making them a sensitive probe of the highest densities reached and, consequently, of +the nuclear EOS [87]. +2. +Experiments probing densities between 1–2.5n0 +As described above, sub-threshold particle yields can be used as probes of the EOS. In partic- +ular, due to their low in-medium cross-section, K+ mesons produced at energies lower than the +production threshold of Elab = 1.58 GeV (√sNN = 2.55 GeV) can carry unperturbed informa- + +37 +0.8 +1.0 +1.2 +1.4 +1.6 +Elab [GeV] +1 + 2 + 3 + 4 + 5 + 6 + 7 +(MK+/A)Au+Au / (MK+/A)C+C +0.8 +1.0 +1.2 +1.4 +1.6 +Elab [GeV] +1 + 2 + 3 + 4 + 5 + 6 + 7 +(MK+/A)Au+Au / (MK+/A)C+C +soft EOS, pot ChPT +hard EOS, pot ChPT +soft EOS, IQMD, pot RMF +hard EOS, IQMD, pot RMF +KaoS +soft EOS, IQMD, Giessen cs +hard EOS, IQMD, Giessen cs +❑HM +▲ SM +● FOPI +Au+Au +protons +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 + 0.25 +beam energy (A GeV) +0.4 +v2n(0.8) +FIG. 13. Left panel: Beam energy dependence of K+ yield ratios in inclusive Au+Au collisions and C+C +collisions between Elab = 0.8–1.5 AGeV (√sNN = 2.24–2.52 GeV). A comparison of RQMD [84] and IQMD [354] +model calculations indicates a soft EOS (K0 = 200 MeV, red symbols) instead of a hard EOS (K0 = +380 MeV, blue symbols) when compared to KaoS data [353] (black symbols). Figure from [351]. Right +panel: Beam-energy dependence of the elliptic flow for protons in Au+Au collisions at Elab = 0.4 AGeV +(√sNN = 2.07 GeV) (black symbols) as measured by the FOPI experiment [88]. +Comparison to IQMD +transport calculations with momentum dependence prefers a soft EOS (blue triangles) over a hard EOS (red +squares), yielding K0 = 190 ± 30 MeV. Figure from [67]. +tion on the fireball density and the stiffness of the EOS [351]. The Kaon Spectrometer (KaoS) +Experiment [352] at SIS18 in GSI studied the subthreshold production of K+ mesons at beam +energies between Elab = 0.6–2.0 AGeV (√sNN = 2.16–2.70 GeV), and established it as a sensitive +probe to the underlying EOS of the hot and dense nuclear matter. To reduce the experimental +and model uncertainties, the production of K+ mesons in a heavier Au+Au system was compared +with the production in a lighter C+C system [353]. Analyzing the experimental results together +with transport model calculations in the RQMD [84] and IQMD [354] model enabled extraction of the +EOS of symmetric nuclear matter characterized by an incompressibility of K0 = 200 MeV (see also +Fig. 9). Both models included effects due to the momentum-dependence of the EOS by including +K+/−N potentials, i.e., a repulsive mean field for K+ and an attractive mean field for K−, which +are required to reproduce the K+ and K− emission pattern [355] (see Fig. 13). +Collective behavior in heavy-ion collisions is likewise a very sensitive probe of the underlying +EOS and has been extensively studied since its discovery by the Plastic Ball spectrometer at the +Bevalac in Lawrence Berkeley National Laboratory [356, 357]. In particular, the elliptic flow v2 is +highly sensitive to both the initial geometry of the collisions and pressure gradients experienced +throughout the evolution of the created systems [77, 358]. The Four Pi (FOPI) Experiment at SIS18 +in GSI carried out extensive measurements of the beam energy dependence of the elliptic flow of +protons and light fragments (such as deuterons, tritons, and 3He) over the entire range of SIS18 +energies, Elab = 0.09–1.5 AGeV (√sNN = 1.92–2.52 GeV) [88, 359]. The nucler EOS extracted from +a comparison to IQMD simulations [67] is characterized by an incompressibility K0 = 190 ± 30 MeV +when momentum-dependent interactions are taken into consideration. This constraint is consistent +with the KaoS incompressibility inferences and suggests a soft EOS for symmetric nuclear matter +at 1-2.5n0 (see Fig. 13 and also Fig. 9). + +38 +K+ SMASH +K- SMASH +K+ UrQMD +K+ JAM +v2 +−0.08 +−0.04 +0 +0.04 +y - ycm +−1 +−0.5 +0 +0.4 < pT < 1.6 GeV/c +K+ +K- +−0.4 +−0.2 +0 +π+ SMASH +π- SMASH +π+ UrQMD +π+ JAM +v2 +−0.08 +−0.04 +0 +0.04 +y - ycm +−1 +−0.5 +0 +0.2 < pT < 1.6 GeV/c +π+ +π- +−0.4 +−0.2 +0 +p SMASH +p UrQMD +p JAM +Λ SMASH +Λ UrQMD +Λ JAM +v2 +−0.08 +−0.04 +0 +0.04 +y - ycm +−1 +−0.5 +0 +0.4 < pT < 2.0 GeV/c +√sNN = 3 GeV 10-40% +Au+Au collisions +p +Λ +v1 +−0.4 +−0.2 +0 +FIG. 14. Directed (v1, top) and elliptic (v2, bottom) flow of protons and lambda baryons (left panels), pions +(middle panels), and kaons (right panels) as a function of rapidity. Measurements from STAR [94] (symbols) +were performed with Au+Au collisions at √sNN = 3 GeV (Elab = 2.91 AGeV) and 10–40% centrality. +Results from UrQMD (blue bands), JAM (green bands), and SMASH (orange bands) hadronic transport models +were obtained using a relatively hard EOS at moderate densities (characterized by K0 = 300 in SMASH +and K0 = 380 MeV in JAM and UrQMD), with the EOS used in SMASH becoming significantly softer at high +densities (see [58, 94] for more details). +3. +Experiments probing densities above 2.5n0 +Pioneering proton directed and elliptic flow measurements were performed in Au+Au colli- +sions for beam energies Elab = 2–10 AGeV (√sNN = 2.70–4.72 GeV) by the E895 [83, 90] and +E877 [82] experiments at AGS-BNL. Notably, it was observed that around Elab ≈ 4 AGeV +(√sNN ≈ 3.32 GeV), the proton v2 changes from a preferential out-of-plane emission, reflecting +a complex interplay between the spectators, the expanding collision zone, and the EOS, to an in- +plane emission (see the lower panel of Fig. 4). The experimental results were used in a comparison +with the pBUU transport model to extract the EOS for densities between 2–5n0, which constrained +the EOS to those described by values of the nuclear incompressibility between K0 = 210–300 MeV, +ruling out extremely soft and extremely hard EOSs [31] (see Fig. 9). This rather broad constraint +on K0 reflects the fact that the experimental results for the collective flow could not be reproduced +with one EOS. +The STAR Experiment at RHIC-BNL with its Beam Energy Scan (BES) program [360, 361] +performed Au+Au collisions for √sNN = 3–200 GeV. +In terms of the freeze-out temperature +and chemical potential, (Tfo, µfo), this allowed STAR to comprehensively scan the QCD phase +diagram from (80, 760) MeV to (166, 25) MeV, respectively. Probing the phase diagram at high +densities was possible at RHIC in part due to STAR’s capability to shift from a standard collider +to a fixed-target (FXT) mode, which was used to scan through the lower energies √sNN = 3– +13.7 GeV (Elab = 2.91–99.06 AGeV), thereby establishing a substantial overlap with the previously + +39 +discussed AGS experiments [362]. Recently, STAR measured collective flow (v1, v2) in collisions at +√sNN = 3.0 GeV [94] and √sNN = 4.5 GeV [93]. A comparison of results from the √sNN = 3.0 GeV +data (see Fig. 14) with UrQMD and JAM simulations indicates a relatively hard EOS (characterized by +K0 = 380 MeV) [94]; similarly, a recent Bayesian analysis of the STAR flow data based on a flexible +parametrization of the EOS used in the SMASH transport code results in a relatively hard EOS at +moderate densities (characterized by K0 = 300 ± 60 MeV) with a substantial softening at higher +densities [58]. However, both UrQMD and SMASH do not currently include momentum-dependent +interactions, which are crucial for a correct description of the transverse-momentum-dependence +of the elliptic flow [31]. Moreover, while the above models reproduce the proton v1, v2 well, none +of the models can simultaneously describe the flow of Lambda baryons and mesons (see Fig. 14). +4. +Challenges and opportunities +Experiments probing densities between 1–2.5n0 +The High Acceptance Di-Electron Spectrometer (HADES) Experiment [363] at SIS-18 in GSI +has performed collective flow measurements in Au+Au collisions at Elab = 1.23 AGeV (√sNN = +2.42 GeV). The high acceptance and high statistics of HADES measurements allow one to per- +form multi-differential studies of flow harmonics, ranging from v1 up to v6, which in turn enables +reconstruction of a full 3D-picture of the emission pattern in the momentum space [18, 148] (see +Fig. 15). In addition to the collective flow measurement capabilities, HADES can also precisely +measure the dielectron excess yield, which was used to extract the fireball temperature, finding +it to be 71.8 ± 2.1 MeV/kB [364]. These precise measurements of the fireball temperature and +the underlying dielectron spectra allow HADES to investigate the presence of a first-order phase +transition at SIS-18 energies and look for signs of a potential change of degrees of freedom [365]. +During its 2024 beam campaign, HADES will be in a unique position to measure the fireball +caloric curve and the beam energy dependence of the collective flow from Au+Au collisions at +Elab = 0.4–0.8 AGeV (√sNN = 2.07–2.24 GeV). Furthermore, there are ongoing efforts to establish +systematic consistency between results from FOPI and HADES, including understanding various +1 +− +0.5 +− +0 +0.5 +1 +cm +y +0.4 +− +0.3 +− +0.2 +− +0.1 +− +0 +0.1 +0.2 +2 +v +Protons +Centrality 20-30% +) +c +(GeV/ +tp +0.35 - 0.40 +0.55 - 0.60 +0.75 - 0.80 +0.95 - 1.00 +1.15 - 1.20 +1.35 - 1.40 +1.55 - 1.60 +1.75 - 1.80 +1.95 - 2.00 += 2.4 GeV +NN +s +Au+Au +HADES +Protons +1.0 < p +t < 1.5 GeV/c +Centrality 20-30% + = 2.4 GeV +NN +s +HADES +Au+Au + 0.0 + 0.2 + 0.4 + 0.6 + 0.8 +ycm +0.5 +1.0 +1.5 +2.0 + = 0 +φ +π +2 +1 +π +π +3 +2 + 0.0 + 0.2 +- 0.2 +- 0.4 +- 0.6 + 0.4 + 0.6 + 0.8 + = 0 +φ +π +2 +1 +π +π +2 +3 +ycm +FIG. 15. Left: Rapidity-dependence of proton elliptic flow (v2) in semi-central Au+Au collisions at Elab = +1.23 AGeV (√sNN = 2.42 GeV) for various pT bins (see legend) as measured by the HADES experiment. +Figure from [18]. +Right: 3D-picture of the proton angular emission pattern in momentum space (flow +coefficients from v1 to v6) for HADES data in semi-central Au+Au collisions at Elab = 1.23 AGeV (√sNN = +2.42 GeV). Figure from [148]. + +40 +detector-related effects. +This is highlighted by the observed discrepancy in pion multiplicities +between FOPI and HADES, which could be partially explained by different methods used by the +respective experiments to estimate the number of participant nucleons [366]. +The abundance of available and future data presents an opportunity to benchmark transport +model simulations with measurements from KaoS, FOPI, and HADES experiments by enabling +systematic studies of the symmetric nuclear matter EOS. A recent comparison between FOPI +measurements and dcQMD transport code [86], using the transverse rapidity [367] and flow spec- +tra [88] of protons and light clusters at Elab = 0.15–0.80 AGeV (√sNN = 1.95–2.24 GeV), has +further tightened the constraints on the nuclear EOS at the probed densities to one characterized +by an incompressibility K0 = 236 ± 6 MeV. The dcQMD analysis for the FOPI data is planned +to be extended up to Elab = 1.5 AGeV (√sNN = 2.52 GeV), probing densities above 2n0, by +taking into account an improved description of reaction dynamics through using more accurate +approximations for 3-body terms in the interaction and considering multi-pion decay channels for +the resonances [368]. +Moreover, perfect-fluid hydrodynamic calculations for binary-neutron-star mergers and heavy- +ion collisions at SIS-18 energies show that comparable temperatures (T ≈ 50 MeV) and densities +(nB ≈ 2n0) are reached in both systems [333, 334]. This has led to increasing efforts to use the +existing constraints on the EOS of symmetric nuclear matter from KaoS and FOPI experiments +in a multi-physics effort to constraint neutron star properties [17, 369, 370]. Such multi-physics +constraints are discussed in detail in Section IV. +Experiments probing densities above 2.5n0 +While collective flow can be used to deduce the geometry of the colliding system and its prop- +erties in an indirect way (see Figs. 14 and 15), a more direct method – femtoscopy – can provide +a direct handle on the space-time evolution of the fireball [371]. Here, the time of the particles’ +emission ∆τ is also a probe of the underlying EOS, with larger values of ∆τ corresponding to a +softer EOS [338] (see Fig. 16). Access to this information is provided by measurements of femto- +scopic radii Rlong, Rout, Rside [372], where the relation between Rout to Rside is strongly correlated +with ∆τ. The sensitivity of pion emission to the EOS has already been studied [338, 373], however, +experimental uncertainties are still too big to make precise comparisons with model calculations. +Ongoing studies of proton femtoscopy at STAR are expected to bring new, substantial references +for such investigations of the EOS. +In addition to studying the EOS of dense nuclear matter, the STAR BES program also aims to +search for a potential first-order phase transition from hadronic to partonic phase at higher baryonic +densities. This search can provide an input on collision energies at which hadronic transport models +should take into consideration new degrees of freedom. Among the explored observables, number- +of-constituent-quark (NCQ) scaling was used as an important evidence of creation of QGP at the +highest RHIC energy of √sNN = 200 GeV (Elab = 21, 300.0 AGeV) [374]. Recent results point to +the breaking of the NCQ scaling in Au+Au collisions at √sNN = 3 GeV (Elab = 2.91 AGeV) [94]. +Other observables that can hint at the possible existence of a first-order phase transition include +the thermodynamic susceptibilities of pressure, which are predicted to fluctuate in the vicinity of a +critical point and manifest as a specific behavior of higher-order moments of conserved quantities +(such as baryon number, strangeness, and electrical charge) with the beam energy [375, 376]. +STAR [377, 378] and HADES [379] have observed tentative non-monotonic behavior in the beam- +energy-dependence of the fourth-order net-proton cumulant (a proxy for the net-baryon number +cumulant) in Au+Au collisions at √sNN = 7.7–27.0 GeV (Elab = 2.91–400 AGeV). +The experimental effort to uncover the symmetric nuclear matter EOS will be further strength- +ened by the Compressed Baryonic Matter (CBM) experiment [49, 380] at the currently-under- +construction Facility for Antiproton and Ion Research (FAIR) in Darmstadt, Germany. The CBM + +41 +16 +18 +20 +22 + +Freeze-out time 〈t〉 (fm/c) +Au+Au 0%-10% π +2 +3 +4 +5 +6 +7 +8 +9 +- + Cascade + Hard EoS + CMF EoS + CMF_PT2 EoS + CMF_PT3 EoS +√s NN (GeV) +2 +3 +4 +5 +6 +7 +8 +9 +-12 +-8 +-4 +0 +4 +8 +12 +16 +Au+Au +0%-10% +|yππ|<0.35 +kT=(275±25) MeV/c +HADES π-π- +E895 π-π- +E866 π-π- +STAR π-π- +STAR π-π-+π+π+ +RO +2-RS +2 (fm2) +√sNN (GeV) +FIG. 16. Left: Beam energy dependence of the π− freeze-out time as extracted from UrQMD simulations for +different EOSs. The CMF PT2 and CMF PT3 EOS soften at low and high baryon density, respectively, by +introducing a first-order phase transition, and the pure cascade mode can be considered as an extremely +soft EOS. Figure from [338]. Right: Comparison of the beam energy dependence of R2 +out − R2 +side for π− +extracted from experiments and UrQMD simulations with different EOSs. Figure from [338]. +experiment, which at the time of writing is expected to become operational in 2028-29, aims to +use nucleus-nucleus collisions to precisely explore the QCD phase diagram with Au+Au collisions +in the energy range of √sNN = 2.9–4.9 GeV (Elab = 2.49–11.1 AGeV). Other particle beams, +such as Z = N species and protons can also be used at Elab = 15 AGeV (√sNN = 5.62 GeV) +and Elab = 30 AGeV (√sNN = 7.73 GeV), respectively. This will be enabled by using primary +heavy-ion beams from the Schwerionensynchrotron-100 (SIS-100) ring accelerator operating at an +intensity of 109 ions/s [381]. CBM will operate at unprecedentedly high peak interaction rates of up +to 10 MHz, which will be further complemented by a novel trigger-less data acquisition scheme and +online event selection. This will allow CBM to perform systematic, multi-differential measurements +of the dependence of observables on the beam energy and system size. The most promising observ- +ables to explore are: (i) event-by-event fluctuations, (ii) thermal radiation (photons and dileptons), +(iii) (multi-)strangeness, (iv) hypernuclei, and (v) charm production (recent physics performance +results can be found in [382, 383]). Moreover, the HADES Experiment will be moved to the SIS-100 +beamline in the CBM experimental cave to complement the overarching CBM physics program in +2031 [384]. The HADES detector, given its large polar angle acceptance (18◦ ≤ θ ≤ 85◦), will +perform reference measurements for CBM at lower SIS-100 energies. This will be done with light +collision systems, e.g., proton beams and heavy-ion beams with moderate particle multiplicities +(such as Ni+Ni or Ag+Ag collisions) [380]. Altogether, CBM represents an opportunity to link +the physics programs at SIS-18 and RHIC, thereby leading to a continuation of the Beam Energy +Scan program (see also the white paper on QCD Phase Structure and Interactions at High Baryon +Density: Continuation of BES Physics Program with CBM at FAIR [151]). +Overall, STAR-FXT and CBM-FAIR are capable of performing high-statistics multi-differential +measurements of the relevant EOS observables. However, a successful inference of the EOS depends +on comparisons to transport simulations. Although many transport codes are available for describ- +ing heavy-ion collisions in different energy ranges and extracting the underlying EOS (see [47] for +a review), currently none of the available codes can reproduce all proposed experimental observ- +ables (see, e.g., Fig. 14). A meaningful description of experimental data in the STAR-FXT and + +O +米O +米42 +CBM-FAIR range will require transport codes to incorporate physics allowing reproducing all of +the above-mentioned key measurements and more, see Section II A. +B. +Experiments to extract the symmetry energy +The energy contribution from the isospin dependence term, also known as the symmetry energy, +is a small fraction of the total energy of a nucleus even for neutron-rich heavy radioactive isotopes +(< 5% in the liquid drop model). However, due to the large isospin asymmetry in neutron stars, +the density dependence of the symmetry energy is very important, determining many neutron star +properties, including their size and the cooling pathways via neutrino emission. While experimental +inferences of the symmetry energy pose significant challenges, researchers have developed methods +to elucidate the relatively small effects that the asymmetry has on isospin-dependent observables, +e.g., by measuring ratios of neutron and proton observables or charged pion observables. Exper- +imental as well as theoretical systematic errors are further minimized by taking double ratios of +the same observable using two reactions that differ mainly in the neutron/proton content, as in +the measurement of isoscaling. To reach the widest range of asymmetry between reactions, intense +radioactive beams are necessary. Large experiments designed to measure symmetry energy can re- +quire large collaborations. However, small-scale experiments can likewise have an impact on some +of the outstanding problems. Consequently, many groups contribute to the diverse experimental +results. +1. +Experiments that probe low densities +At beam energies below Elab = 100 AMeV (√sNN = 1.93 GeV), the colliding nuclei overlap +briefly and then expand, with most of the detected particles being emitted during the expansion +stage. The rates of emission of neutrons and protons during the expansion are influenced by the +symmetry energy. Some nucleons emerge within fragments or clusters that are formed and emitted +throughout the reactions. Nearly all theory studies require the symmetry energy to be zero at zero +density. However, before matter reaches zero density, at low densities of (0.002 ≤ n/n0 ≤ 0.02) +many nucleons combine into clusters and preserve the information about the symmetry energy at +those low densities. In the estimation of Wada et al. [385, 386], clustering has a significant impact +on the symmetry energy below 0.03 nucleons/fm3, see Fig. 17 (we note here that this conclusion +depends on the definition of the symmetry energy). Overall, the presence of clusters changes the +characterization of the symmetry energy. Nonetheless, low-density clusterization is an important +ingredient in supernova matter and for the EOS in the neutrino sphere. It is also relevant to the +nature of proto-neutron star matter as it cools and the crust crystallizes [387]. +2. +Measurements to extract symmetry energy up to 1.5n0 +In the past decade, many studies have been conducted to extract the symmetry energy and +symmetry pressure [70], mostly at low densities. Since the nuclear EOS should give a good de- +scription of the properties of the nuclei, including the masses or binding energies, the large nuclear +mass data base provides a great resource to determine the symmetry energy at about (2/3)n0 from +(1) masses of double magic nuclei using Skyrme density functional [72], (2) nuclear masses using +density functional theory [71], and (3) the energies of isobaric analogue states [12]. +By nature, a heavy nucleus has excess neutrons which are needed to overcome the Coulomb +repulsion from the protons inside the nucleus. The symmetry-energy forces the excess neutrons + +43 +FIG. 17. The symmetry energy of clustered matter at very +low densities. Figure from [385]. +to the surface. +This surface layer of ex- +cess neutrons is referred to as a neutron +skin. +Thickness of this neutron skin re- +flects the symmetry pressure, or equiv- +alently the slope of the symmetry en- +ergy at the saturation density. The long- +awaited measurements of the neutron skin +of the 208Pb nucleus inferred from par- +ity violation in electron scattering were +recently published by the PREX collab- +oration [74, 388]. +The measured value +of 0.283 ± 0.071 fm, corresponding to the +symmetry pressure of 2.38±0.75 MeV/fm3 +at (2/3)n0, is rather large and disagrees +with most theoretical predictions. Subse- +quently, the PREX/CREX collaboration +measured the neutron skin of 48Ca to be +0.121 ± 0.026 (exp) ± 0.024 (model) fm +[389], which is much thinner than the +208Pb skin. +However, the 48Ca value is +much closer to the theoretical predictions. +The discrepancies between the two results and the expectations from models have not been re- +solved, and the CREX collaboration has not released official values for the slope of the symmetry +energy or symmetry pressure, even though there have been many attempts by outside groups to +resolve the apparent discrepancies between the two skin measurements [390–392]. +In the last few years, an alternative way to measure skin thickness has been proposed [393]. +In the limit of an exact charge symmetry, the proton radius of a given nucleus is identical to the +neutron radius of its mirror partner. Thus the neutron skin for a given nucleus may be determined +from the difference in proton radii measured in these mirror pairs. In reality, there are relativistic +and finite-size corrections, as well as corrections from the Coulomb force which breaks the isospin +symmetry. In principle, these corrections can be calculated within the energy density functional +theory. While larger neutron skins are expected in heavier nuclei due to the larger neutron excess, +making them a better probe, proton-rich mirrors of heavy nuclei is typically far beyond the limits +of existence. Thus this technique is limited to species of relatively low mass and isospin. Even with +the use of high-intensity isotope beams near the proton driplines, it is still a challenge to do such +experiments. The most recent result with this technique is from the 54Ni-Fe mirror pair [394]. +Complementary to structure experiments, heavy-ion collisions have probed the symmetry en- +ergy and pressure over a wide density range. +At incident energies below Elab = 100 AMeV +(√sNN ≤ 1.93 GeV), low densities (estimated to be around (1/3)n0) are reached when matter +expands after the initial impact and compression of the projectile and target. +Therefore, the +corresponding experimental observables primarily reflect the symmetry energy at sub-saturation +densities [6, 98]. The transport of neutrons and protons allows systems with isospin gradients to +equilibrate, where the degree of equilibration depends on the strength of the potential experienced +by the nucleons and the duration of transport. The technique of equilibration chronometry al- +lows the visualization of the time evolution of the neutron excess. Signatures of neutron-proton +equilibration obeying first-order kinetics are observed both in experimental data [395–398] and in +transport calculations +[399]. Since the equilibration depends on the neutron and proton chem- +ical potentials, this technique offers new experimental data to constrain the sub-saturation EOS +through comparisons with simulations [6, 47, 400]. + +25 +20 +15 +(MeV) +10 +5 +0 +10-3 +10-2 +10-1 +p(N +/fm3) +nucleon44 +Isoscaling was first observed in central 124Sn+124Sn and central 112Sn+112Sn collisions at beam +energy Elab = 50 AMeV (√sNN = 1.90 GeV) [401, 402]. Isoscaling describes a simple scaling law +governing the ratios of isotope yields from two systems which differ mainly in their neutron-proton +composition. It arises from the differences in the neutron and proton chemical potentials of the +two reactions and is, therefore, sensitive to the symmetry energy. The isospin diffusion, derived +from the isoscaling observable, reflects the driving forces arising from the asymmetry term of the +EOS [6, 400, 403, 404] and provides a measurement of the symmetry energy at around (1/3)n0 [70]. +Other observables used to study the symmetry energy with light charged particles include both +n/p and t/3He ratios and their double ratios obtained from two reactions with different isospin +content [47, 95, 96, 100, 405]. Due to the difficulties in measuring neutrons, neutron data is not +widely available. However, recent isoscaling measurements have allowed the construction of “pseudo +neutrons”, that is a reconstruction of neutron yields from light particle ratios such as t/3He [406]. In +particular, this method allows for a reconstruction of low-energy neutrons. However, due to the lack +of high-energy charged particles data, it is a challenge to reconstruct high-energy neutron spectra +in this way. Therefore, to study the symmetry energy at supranormal densities, neutron arrays +constructed with new advanced materials will be needed in the next generation of experiments. +In experiments utilizing central 124Sn+124Sn and central 112Sn+112Sn collisions at Elab = +120 AMeV (√sNN = 1.94 GeV) [407], the spectra of neutrons emitted to 90 degrees in the center- +of-mass frame are compared to the corresponding proton spectra. Transport calculations predict +that if the effective masses of neutrons and protons satisfy m∗ +n < m∗ +p, then fast neutrons coming +from the compressed participant region experience a more repulsive potential and a higher accel- +eration than do fast protons at the same momentum, resulting in an enhanced ratio of neutron +over proton (n/p) spectra at high energies. In contrast, calculations for m∗ +n > m∗ +p predict that the +effective masses enhance the acceleration of protons relative to neutrons, resulting in a lower n/p +spectral ratio. Bayesian analysis of the experimental results [98] compared to ImQMD calculations +shows that the values of the first two Taylor expansion coefficients of the symmetry energy, S0 +and L, depend on both the symmetry energy and to the effective mass splitting. More examples +of Bayesian analyses used to simultaneously constrain multiple parameters will be discussed in +Section IV B, where methods to extract multiple transport model input parameters are discussed. +3. +Selected constraints on the symmetry energy around 1.5n0 +Current constraints on the symmetry energy above saturation are obtained with large uncer- +tainties, mainly at densities around 1.5n0. This is the area of future opportunities, and we discuss +this in more detail here to illustrate the complexity of the experiments and analyses as well as the +central role played by transport models. +The nucleon elliptic flow is sensitive to the pressure generated in nuclear collisions and, there- +fore, to the EOS. Since a higher symmetry pressure will yield a larger magnitude of the elliptic flow +at midrapidity for neutrons than for protons, comparisons of the neutron and proton elliptic flows +provide sensitivity to the density-dependence of the symmetry energy [95]. The neutron and hydro- +gen elliptic flow from Au+Au collisions at a beam energy of Elab = 0.4 AGeV (√sNN = 2.07 GeV) +were measured in the FOPI-LAND and Asymmetric-Matter EOS (ASY-EOS) experiments, using +the Land Area Neutron Detector (LAND) for the measurement of the neutron flow. A comparison +of data to UrQMD simulations, shown in Fig. 18, was used to extract the dependence of the sym- +metry energy on density, parametrized as proportional to (nB/n0)γasy, and the symmetry energy +slope parameter L. The FOPI-LAND experiment reported γasy = 0.9 ± 0.4 and L = 83 ± 26 MeV +[68], whereas the ASY-EOS obtained γasy = 0.72 ± 0.19 and L = 72 ± 13 MeV [69], indicating +a moderately soft symmetry energy (see Fig. 18 and also Fig. 9). The analysis also illustrates + +45 +FIG. 18. Ratio of the elliptic flows of neutrons over +charged particles vn +2 /vch +2 +as a function of transverse +momentum per number of constituent nucleons pt/A +in Au+Au collisions at Elab = 0.4 AGeV (√sNN = +2.07 GeV). The comparison between ASY-EOS mea- +surements (square black symbols) and UrQMD trans- +port model calculations with a soft (pink dots) and +hard (green triangles) symmetry potentials shows a +preference for a soft symmetry energy; solid red line +indicates γasy = 0.75 ± 0.10. Figure from [69]. +the dependence of S0 and L on other in- +put parameters of the EOS, such as γasy. +A +subsequent comparison of data with dcQMD +model [408] gives a value of L = 85 ± 32 MeV +at n = 1.5n0. +In addition to the ASY-EOS experiment, +another effort that explores this density region +is the SAMURAI Pion-Reconstruction and Ion- +Tracker (SπRIT) experiment, performed with +radioactive tin isotopes at RIKEN, Japan. For +constraining the symmetry energy at supra- +saturation densities, pion yield ratios are con- +sidered as a unique observable since they do +not form composite particles with other parti- +cles. +This makes their yields independent of +clusterization processes which can affect the +symmetry energy (see Section III B 1). Further- +more, pion observables are predicted to be sen- +sitive to the nuclear EOS at high densities due +to their unique production mechanism: Above +Elab = 200 AMeV (√sNN = 1.97 GeV), some of +the interactions occurring in central collisions +are energetic enough to form excited ∆(1232) baryon resonances (through the NN ↔ N∆ scatter- +ing process), which then promptly decay into pions and nucleons. The high production threshold of +the ∆(1232) resonance ensures that pions originate from the early stages of the reaction, and there- +fore from regions characterized by a high density. The SπRIT collaboration measured charged pion +emission from systems characterized by a wide range of asymmetry [102] by colliding tin isotope +beams of 108,112,124,132Sn with isotopically enriched targets of 112,124Sn). +The production of π− strongly depends on n-n collisions in the high-density region, while π+ +production largely depends on p-p collisions (the production of π− and π+ is equally likely in n-p +collisions). It follows that the relative production of π− and π+ depends on the relative numbers +of neutrons and protons and, therefore, is sensitive to the symmetry energy in the high-density +region. Assuming a ∆-resonance model for pion production, one would expect that the pion yield +ratio Y (π−)/Y (π+) follows a (N/Z)2 dependence [95, 367]. However, the measured total pion yield +ratio follows N/Z with a best-fitted power index of 3.4, as shown in Fig. 19, where yield ratios +without a transverse momentum cut are depicted by yellow crosses with circle markers. The radius +of the circle in the center of each cross represents the experimental uncertainty, showcasing very +good experimental accuracy of the measurement in which systematic errors are reduced by taking +pion yield ratios. Moreover, comparisons of systems with different N/Z measured in the same ex- +periment reduces systematic errors [409]. The discrepancy between the theoretical expectation and +experimental data indicates the presence of dynamical factors beyond a simple ∆-resonance model, +while the large measured exponent suggests that the ratios are strongly affected by the symmetry +energy. When a transverse momentum cut of pT > 180 MeV/c is imposed, the result (represented +in Fig. 19 by blue crosses with circle markers) still shows the same (N/Z)3.4 dependence, suggesting +that effects due to the symmetry energy persist in high-momentum pions. Interestingly, current +transport models do not seem to be able to reproduce the strong N/Z dependence [102]. +While Fig. 19 shows the total yield, the left panel of Fig. 20 focuses on the pT -dependence of the +single ratio spectrum SR(π−/π+) = [dN(π−)/dpT ]/[dN(π+)/dpT ] for two extreme cases: reactions +of neutron rich (132Sn+124Sn) and of near-symmetric (108Sn+112Sn) systems. The data is compared + +E +1.3 +1.2 +y=0.75±0.10 +1.1 +2 +0.9 +stiff +0.8 +T +0.7 +0.6 +0.5 +soft +0.4 +0.3 +0.4 +0.5 +0.6 +0.7 +p/A (GeV/c)46 +1.2 +1.3 +1.4 +1.5 +1.6 +N/Z +1 +2 +3 +4 +5 +6 +Y( +)/Y( ++ ) +pT > 0 MeV/c +pT > 180 MeV/c +y = 1.1(N/Z)3.4 +y = 0.5(N/Z)3.4 +FIG. 19. Ratios of yields of π− over yields of π+ in +central (b < 3 fm) events for pions with pz > 0 in the +center-of-mass frame, plotted as a function of N/Z. +The yellow crosses show yield ratios with no transverse +momentum cut, while the blue crosses show yield ra- +tios for pT > 180 MeV/c. +The radius of the circle +inside each cross represents the statistical uncertainty +of the ratio. The dashed blue and dotted blue line cor- +responds to the best-fitted power functions of N/Z for +pT > 0 and pT > 180 MeV/c pion ratios, respectively. +Figure from [410]. +with the dcQMD model [145, 408], a Quantum +Molecular Dynamic transport model that in- +cludes total energy conservation and other ad- +vanced features. To extract the EOS, the dcQMD +model was used to predict single ratios with +12 different parameter sets in the (L, ∆m∗ +np) +space, forming a regular lattice; here, L is the +slope of the symmetry energy and ∆m∗ +np is the +neutron-proton effective mass splitting. +The +value of L in the lattice is either 15, 60, 106, +or 151 MeV and ∆m∗ +np/δ is either -0.33, 0, or +0.33. All other input parameters in the dcQMD +have been fixed by comparing to FOPI data, +as well as by comparing the predictions to the +total yield of the charged pions and the aver- +age pT obtained from the pion spectra. +De- +tails of the comparison can be found in Ref. +[101]. +The left panel in Fig. 20 shows a few +selected calculations and the measured single +ratios. +The (L, ∆m∗ +np) values for the solid +blue line are (60, −0.33δ), for the dashed blue +line are (60, 0.33δ), for the solid red line are +(151, −0.33δ), and for the dashed red line are +(151, 0.33δ). Coulomb effects dominate the low pT region, causing a steep rise in the measured +ratios at pT < 200 MeV/c. All calculations at pT < 200 MeV/c disagree with data, which could be +caused by inaccuracies in the simulation of Coulomb interactions or of the pion optical potential +above the saturation density. At pT > 200 MeV/c, the Coulomb and pion potential effects diminish +and the ratios should be good probes of the symmetry energy effects. +The predicted single ratios at pT > 200 MeV/c are interpolated with 2D cubic splines over +the (L, ∆m∗ +np) space, and the interpolated predictions are then compared to experimental mea- +surements through a chi-square analysis. The resultant multivariate constraint on L and ∆m∗ +np is +shown in the right panel of Fig. 20, where the green shaded region is the 1σ confidence interval +and the area enclosed by the two blue dashed curves is the 2σ confidence interval. The corre- +lation between ∆m∗ +np and L occurs because both parameters influence the nucleon momenta; L +influences the momenta through its isospin-dependent contribution to the nucleon potential en- +ergy, and ∆m∗ +np influences the momenta via its isospin-dependent impact on the nucleonic kinetic +energy. Either increasing L or decreasing ∆m∗ +np will increase the energies of neutrons relative to +protons. This increases the numbers of n-n collisions relative to p-p collisions at energies above +the pion production threshold and enhances the production of π− relative to that for π+. +4. +Challenges and opportunities +Experiment and theory +Currently, there are few experiments that aim at inferring the symmetry energy and symmetry +pressure from heavy-ion collisions probing densities of 1–2n0. Furthermore, the available constraints +have very large uncertainties, especially for the symmetry pressure. It is worth noting that heavy- +ion collision experiments do not measure the symmetry energy or pressure directly, but rather +they depend on comparisons with transport model simulations that describe the dynamics of the + +47 +0 +100 +200 +300 +400 +pT (MeV/c) +100 +101 +Single Ratio (SR) +132Sn + 124Sn, E/A = 270 MeV +0 +100 +200 +300 +400 +pT (MeV/c) +108Sn + 112Sn, E/A = 270 MeV + L(MeV) m * +np +60 -0.33 +60 0.33 +151 -0.33 +151 0.33 +50 +100 +150 +L (MeV) +−0.3 +−0.2 +−0.1 +0.0 +0.1 +0.2 +0.3 +∆m∗ +np/δ +1-σ +2-σ +FIG. 20. Left panel: Single pion spectral ratios for 132Sn+124Sn (top) and 108Sn+112Sn (bottom) reac- +tions with four selected dcQMD predictions overlaid [86]. Right panel: Correlation constraint between L +and ∆m∗ +np/δ, extracted from pion single ratios at pT > 200 MeV/c in collisions of both neutron-deficient +108Sn+112Sn and neutron-rich 132Sn+124Sn systems. The light blue shaded region (dashed blue lines) cor- +responds to 68% (95%) confidence interval [101]. +collisions [47]. +The large uncertainties in available constraints mainly arise from the intrinsic +uncertainties of the transport models and the accuracy of determining the parameters used as an +input in these models. For example, a general feature of low-energy heavy-ion collisions is that +more nucleons are emitted in light clusters than are emitted as free neutrons and protons, while +the reverse is true of most transport model simulations of these reactions. Theoretical approaches +to this issue have been proposed (see Section II A), but are rarely implemented to model the +coalescence of nucleons in the medium into the observed distribution of clusters, and therefore it is +not clear to what extent these approaches are valid. The current inaccuracy in cluster production +complicates and limits the scientific conclusions that can be drawn by comparing data to transport +theory, and therefore improving the accuracy of cluster production in transport theory would be a +very significant achievement, enabling more stringent constraints on the symmetry energy. +It is important to quantify major sources of systematic uncertainties in the transport model +implementations and in the model parameters. Due to the quality as well as technical details of +solutions adopted in different models, it may not be realistic to establish all uncertainties for all +transport models. Nonetheless, developing methods to validate transport models and performing +these validations remains a primary goal for the Transport Model Evaluation Project (TMEP) +collaboration, and it is essential to extracting reliable constraints on the EOS from heavy-ion +collisions (see Section II A). +The current capabilities at FRIB, using beam energies up to Elab = 200 AMeV (√sNN = +1.97 GeV), allow for exploration of densities up to 1.5n0, and the neutron excess can be varied +over a wide range by changing the composition of the rare isotope beams and targets, allowing +to more closely recreate the matter found in extreme astrophysical environments (e.g., neutron +stars). From the dense collision region in heavy-ion collisions, pions and free nucleons are emitted +with high transverse momentum. The relative yields of these particles, especially as a function of +energy, as well as particle elliptic flow contain information about the dense collision zone and thus +can be used to constrain the EOS that governs supra-saturation matter. Individual efforts based +on small-scale experiments, which are the strength of the field, have provided a diversity of results. +However, in order to take advantage of multiple-parameter Bayesian analyses, described below, +and given the tight allotment of the expensive (and coveted) beam time, future experiments should +utilize detectors that provide large coverage. The development of the time projection chamber + +48 +(TPC) detector at FRIB is essential to measure both pions and charged particles. The detector +can be coupled with an upgraded or a new Large Neutron Array (LANA) to measure both charged +particles and neutrons. Additionally, putting the TPC detector at the target position of the High +Rigidity Spectrometer (HRS) enables coincident measurement of projectile-like fragments. The +determination of the centrality and the reaction plane, required, e.g., for the elliptic flow studies, +would benefit from a construction of a 4π detector placed around the target when the silicon +detectors are used to identify the charged particles before the completion of the HRS. Such a +detector would have to measure the energies of the emitted fragments and nucleons as well as their +multiplicities with minimal energy losses. +Reaching higher densities requires the energy upgrade to Elab = 400 AMeV (√sNN = 2.07 GeV). +With the capability for producing high-intensity rare isotope beams with a wide range of asymme- +tries, FRIB400 is essential for the U.S. effort to lead in the determination of the density-dependence +of the symmetry energy [22]. +The beams available at FRIB, being complementary to those that can be accelerated at Eu- +ropean facilities, may represent a unique opportunity to conduct nuclear transport investigations +also by the international nuclear physics community. As described above, the development of de- +tector arrays with high isotopic resolution over a wide dynamic range, from light particles to heavy +fragments, provides the prospect of measuring observables (especially in the context of isospin +diffusion and drift as well as in collective motion phenomena) that can amplify the sensitivity to +the symmetry energy. Coupled with its capability to use high-quality radioactive beams, FRIB +may represent a focus of interest for a wider community, stimulating the need for international +discussions and collaborations in the coming years. Such an interest may concern also theoretical +physicists that have been collaborating with FRIB colleagues within the Transport Model Evalu- +ation Project (TMEP) initiative, aimed at improving investigations of the isospin-dependent EOS +with comparisons to experimental observables (see also Section II A). +Multiple Parameter Bayesian analysis +The EOS is only one of many input parameters in transport models used to simulate heavy-ion +collisions. Often, multiple measurements probing different parts of the collisions are needed to +constrain other parameters of these models, such as the momentum-dependence of the isovector +mean-field potential, or the in-medium isospin-dependent cross sections. However, constraining +transport model parameters with experimental results is a delicate endeavor. The outcomes of +nuclear collisions are influenced by a multitude of processes, and therefore the experimentally +measured final stage observables can depend simultaneously on values of multiple parameters. +However, carefully chosen observables may only be sensitive to just a few specific parameters. The +full extent of the dependence of a given transport model on input parameters can only be tested +empirically after performing a complete series of simulations of heavy-ion collisions. +Bayesian statistical methods provide means to quantify the relation between observable values +and physical parameters. They also provide a systematic way of constraining multiple nuclear +properties and utilizing prior knowledge from different experiments, prior constraints from other +sources, and results from new experimental measurements. For example, in the n/p ratio exper- +iment mentioned above, measuring the yield ratios of neutron and protons spectra, a Bayesian +analysis comparing the experimental results [98] to ImQMD calculations determines both ∆m∗ +np and +the relationship between S0 and L, even though the uncertainties are large. More precise measure- +ments in the future will enable a better resolution. +In the long term, it is important to develop Bayesian analyses of multiple observables to de- +termine multiple parameters simultaneously. +As an example, in the SπRIT experiment many +observables have been measured with four reaction systems. Eight observables in total, including +the directed flow, elliptical flow, and the stopping observable from different reactions, are fitted si- + +49 +FIG. 21. Posterior distribution obtained from a Bayesian analysis of ImQMD simulation results and experi- +mental data from SπRIT experiments [410]. Eight available observables are used for Bayesian analysis. The +values for median and 68% confidence interval of the marginalized distribution are tabulated on the upper +right-hand side of the figure. Figure from [410]. +multaneously by varying five transport model input parameters (two pertaining to the shape of the +symmetry energy term in the nuclear matter EOS, two pertaining to the nuclear effective masses, +and one pertaining to the nuclear in-medium cross-section). The posterior distribution shows a +weak constraining power on the symmetry energy terms, but a strong sensitivity to effective masses +and in-medium cross-section [410], see Fig. 21. +The posterior parameter distributions are generated from repeated sampling of transport model +predictions for hundreds of thousands of times, each with different parameter values. If carried out +directly, this process would consume an unreasonable amount of computational resources. This can +be alleviated with an effective, efficient, and capable model emulator which emulates the behavior +of transport models at all points of the allowed parameter space from predictions at just a few tens +of parameter values. Gaussian processes are readily available and commonly used in emulators, +but the procedures for tuning hyperparameters vary across analyses. Numerous heuristics and cost +functions are proposed for the optimization of hyperparameters, and one can also marginalize over +all nuisance parameters with a Markov chain Monte Carlo. + +50 +(MeV) +S +30 +150 +(MeV) +100 +50 +1.0 +Nu/ +0.8 +Nu/ +1.00 +0.75 +0.25 +0.00 +n +-0.25 +30 +40 +50 +50 100 150 +0.8 +1.0 0.75 1.00 -0.25 0.00 +0.25 +So (MeV) +L (MeV) +ms /mN +m,/mN +n50 +IV. +THE EQUATION OF STATE FROM COMBINED CONSTRAINTS +Nuclear +Neutron star +Isospin diffusion in HICs +Masses and radii +Dipole polarizability +Tidal deformability +Spectral ratios of light clusters +Moment of inertia +Nuclear masses and radii +Gravitational binding energy +Isobaric analog states +Cooling of young neutron stars +n/p ratios in HICs +Bulk oscillation modes +Neutron skins +Crust cooling +Mirror nuclei +Pulsar glitches +Giant resonances +Lower and upper limits on neutron star spin periods +Flow of particles in HICs +Torsional crust oscillations +Charged pion ratios in HICs +Crust-core interface modes +TABLE I. Illustrative list of nuclear and astrophysical observables. +There have been many attempts to extract the equation of state (EOS) as a function of density +from both nuclear experiments and astronomical observations. In Table I, we provide an illustrative +list of relevant experimental and observational measurements. Importantly, these observables probe +the EOS at different densities: a few probe the EOS near the saturation density n0, but many probe +densities that are significantly higher or lower. For example, nuclear structure typically probes +densities that are somewhat lower than n0, while analyses of heavy-ion collisions or properties of +neutron stars probe larger density ranges, as schematically illustrated in Figs. 12 and 22. +Comparing constraints based on different measurements allows one to test their consistency and +ultimately find tight constraints on the EOS over the full range of densities that can be probed +either by experiments or by astronomical observations. Techniques of Bayesian inference or Pearson +correlation analyses are well-suited to this endeavor and can provide more readily useful and +Astro (M,R,Λ)� +HIC� +Astro: crust observables� +Chiral EFT� +Nuclear masses, � +Δrnp, αD, ….� +FIG. 22. An ensemble of EOSs that range over crust and +core uncertainties consistently. Ranges over which differ- +ent nuclear and astrophysical probes provide information +about the neutron star EOS are indicated. Figure modified +from [411]. +testable +information +on +the +density- +dependence of the EOS than, e.g., statis- +tical comparisons or combining the Taylor +expansion coefficients (such as S0, L, and +Ksym) obtained from individual analyses. +Key to this approach is the determination +of the density that each experimental ob- +servable most accurately probes. +Away +from that density, weaker constraints on +the EOS are possible, but the analysis is +more complex. +In this section, we review the variety of +observables that have been used to place +constraints on the EOS; heavy-ion collision +experiments, which produce many of these +constraints, are described in Section III. +We then discuss recent attempts at com- +bining various constraints that result in +meaningful EOSs with quantified uncer- +tainties. + +？??? +Outer Crust +Outer +Inner Core +Neutron Drip +Inner Crust +Crust/Core Transition +Core +1036. +pressure (Dyne/cm2) +1034 +1032 +1030 +J, L, Ksym +in1! +n2 +1011 +1012 +1013 +1014 +1015 +1016 +energy density (g/cm3)51 +A. +Constraints +As discussed in Section III, experiments are often designed to explore certain aspects of the EOS. +Accordingly, we classify the constraints obtained from laboratory measurements as sensitive to +either the symmetric nuclear matter EOS or the symmetry energy. In addition to the experimental +inferences, constraints on the EOS can be also obtained from neutron star observations as well as +from chiral EFT theory at low densities. The list of constraints discussed here is not exhaustive. +Rather, it represents a slice of widely acknowledged constraints at the moment of writing. We note +that some of the constraints reviewed here have already been presented in Sections II and III, to +which we refer when appropriate. +Symmetric matter constraints from laboratory experiments +Some properties of the symmetric nuclear matter are fairly well-known near n0. For exam- +ple, the generally accepted values of n0 and binding energy at saturation E0 are 0.16 fm−3 and +−16 MeV [198, 203], respectively, to within 4%. The incompressibility parameter, K0, has been +determined from giant monopole resonance (GMR) experiments [412] to be 231±5 MeV. However, +subsequent GMR measurements of the Sn isotopes cast larger uncertainties on K0 [213]. While +these larger uncertainties are consistent with values of K0 determined from heavy-ion collision +experiments [67, 353, 413], we note here that these experiments derive their constraints on K0 +based on density functionals that are parametrized with K0, but used to describe the high-density +behavior of the EOSs (i.e., these experiments do not probe the incompressibility at saturation; see +also a similar discussion in Section II A 2). Measurements of the collective flow from high energy +Au+Au collisions have constrained the EOS for symmetric nuclear matter at densities spanning +(1–4.5)n0 [31, 58, 66, 67] (see the left panel in Fig. 9), as described in Sections II A 2 and III A. +Symmetry energy constraints from laboratory experiments +In the past decade, many studies have been conducted to extract the symmetry energy and +the symmetry pressure, and some of the widely-known constraints are plotted in the right panel +of Fig. 9, which includes both the usual EOS constraint bands as well as symbols located at +densities which a novel analysis in Ref. [70] identified as the most sensitive densities for a given +measurement. At (2/3)n0, precise symmetry energy constraints have been obtained from studies +on nuclear masses using Skyrme density functional forms for the EOS. These are labeled in the +right panel of Fig. 9 as “mass(Skyrme)” [72] and “mass(DFT)” [71], respectively. In this density +region there are also precise constraints obtained from the energies of isobaric analogue states [12], +indicated in the right panel of Fig. 9 by a data point labelled as “IAS”. The dipole polarizability +αD, reflecting the response of a nucleus to the presence of an external electric field, also helps to +constrain the symmetry energy at low densities. Constraints on the symmetry pressure Psym, which +is proportional to the derivative of the symmetry energy with respect to density, have been recently +provided by the measurements of the neutron skin of 208Pb in the Lead Radius EXperiment (PREX +and PREX-II) [74, 388, 414] and of the neutron skin of 48Ca in the Calcium Radius EXperiment +(CREX) [389–391], both at Jefferson Lab, which use parity-violating weak neutral interactions to +probe the neutron distribution in 208Pb and 48Ca. A range of other scattering experiments have +measured the neutron skins of a number of neutron-rich isotopes and likewise used them to constrain +the symmetry energy [7, 415, 416]. Giant dipole resonances and polarizabilities [9, 417, 418] in +neutron-rich isotopes provide another source of information about the symmetry energy [335, 419– +424], as do mirror nuclei [393, 394]. At densities far from (2/3)n0, heavy-ion collisions have been +used to probe the symmetry energy, as is described in Sections II A 2 and III B 3, and shown in the +right panel of Fig. 9. + +52 +Constraints from astronomical observations +The bulk properties of neutron stars (such as their maximum mass, radii, tidal deformabilities, +moments of inertia, limits on the rotation frequency, and binding energy) depend strongly on the +distribution of matter throughout the star, therefore providing a measure of the EOS integrated +over the range of densities present in the star. The mass-radius relationship has a one-to-one corre- +spondence to the neutron star EOS [425], and it is known that the radius, the tidal deformability, +and the moment of inertia provide the strongest constraints on the EOS above 2n0 [251, 426], +while the maximum measured mass of neutron stars constrains the EOS at the highest densities. +Together, the tidal deformability measured in GW170817, the mass of J0740+6620, and the two +mass-radius measurements of NICER, discussed in Section II C, form the current gold standard in +measuring the neutron star properties using astronomical observations. +A number of astronomical observables also probe the neutron star crust physics, which results +in constraints on the pure neutron matter EOS, and in particular on the symmetry energy. This +is because the neutrons provide the hydrostatic pressure that supports the inner crust, and the +interplay between these neutrons and the lattice of nuclei that makes up the crust determines the +crust-core boundary as well as the possible nuclear pasta shapes that appear near that boundary. +The crust physics also depends, more weakly, on the symmetric matter EOS. The nuclear EOS +at subsaturation densities, down to where the neutron drip begins (nB ≈ 10−4n0), is therefore an +essential ingredient in crust models. +Due to the complexity of crust physics, extracting rigorous EOS constraints from observations +of crust-associated neutron star behavior is in its early stages, and it is an area where substantial +progress can be made over the next decade. Here we list some constraints on the symmetry energy +as an illustration of this potential, but, at the same time, we note that they are very tentative and +do not have well-quantified errors; indeed, some of them are mutually exclusive, emphasizing the +need to make progress in applying microscopic nuclear physics models to these observations. +Constraints on the symmetry energy and its slope can be obtained from studying the following +phenomena: A study of the cooling of the neutron star in the Cas A supernova remnant [312], +which has been observed to cool on a timescale of decades, implies that the neutron star core may +have superfluid properties [265, 427, 428]. Studying the temperatures of the population of neutron +stars whose surface X-ray emission is observable leads to constraints on the neutron star masses +and radii and the composition of the core [429]. Constraints from quasi-periodic oscillations in +the X-ray tail of gamma-ray flares from soft gamma-ray repeaters [285, 430, 431], which could be +a signature of torsional oscillations of the crust. Potential measurements of the crustal moment +of inertia from glitches – sudden changes in rotation frequency – of radio pulsars and some X-ray +pulsars [432–437]. +Limits on the longest and shortest observed periods of neutron stars probe +physics such as the magnetic field evolution in the crust [438] and the development of rotation- +induced instabilities in oscillations such as r-modes [439, 440]. During the last few seconds of an +inspiral prior to the merger of two neutron stars or a neutron star with a black hole, tidal forces +may shatter the crust, causing a gamma-ray flare: in this scenario, coincident timing of the flare +with the gravitational wave signal measures the resonant frequency of crust-core interface modes +and sets constrains on the symmetry energy [276, 441]. The cooling of the crusts of quiescent +low-mass X-ray binaries promises to provide a source of constraints on the composition and size of +the neutron star crust and the extent of nuclear pasta phases therein [283, 442–444]. The expected +accurate measurement of the moment of inertia of pulsar J0737-3039a [445] will set constraints on +the EOS competitive with the current radius constraints [446–450]. The heat capacity of a neutron +star core can be measured by using inferences of the core temperature of transiently-accreting +neutron stars, and strongly suggests that a core dominated by a color-flavor-locked quark phase is +ruled out [451]. Some such objects are observed to have efficient cooling in the core, constraining +superfluid gap models and the symmetry energy [452]. + +53 +Constraints from nuclear theory +In recent years, many-body nuclear theory such as chiral effective field theory (χEFT), discussed +in Section II B, has made significant progress to be considered as the canonical nuclear matter +EOS at low densities with rigorous uncertainty quantification [155–158]. Even though the theory +is developed mainly for densities below saturation, it has been extended to 2n0, and it is a popular +constraint for studies that focus mainly on astronomical observations. +B. +EOS obtained by combining various constraint sets +Each of the nuclear and astrophysical observables discussed above provides vital information +about the EOS over some density range, that can be combined with other constraints to globally +constrain the density-dependence of the EOS from sub-saturation to supra-saturation densities. +Such analysis techniques are relatively new, but several of such global constraints now exist, and +a selection of studies is briefly described below to illustrate their potential. +Beloin et al. [429] used relativistic mean-field models of the nuclear interaction to model the +structure and cooling of neutron stars. +This analysis consistently combines nuclear data, neu- +tron star mass-radius measurements, and neutron star cooling measurements within a Bayesian +framework. +Legred et al. [251] performed nonparametric EOS inference based on Gaussian processes. It +combines information from X-ray, radio, and gravitational wave observations of neutron stars. +Their results are plotted in Fig. 23 and labelled as “Legred et al.”. +These Bayesian analyses +incorporate astrophysical data and provide constraints on the neutron star EOS at higher densities. +Drischler et al. [453] performed a Bayesian analysis of correlated effective field theory truncation +errors based on order-by-order calculations up to next-to-next-to-next-to-leading order in the χEFT +expansion. The neutron star matter pressure calculated with these EOS is shown in Fig. 23 and +labeled as “Drischler et al.”. +Huth et al. [17] combined nuclear theory via χEFT calculations (constraining the EOS below +1.5n0), EOS inferences from heavy-ion collisions via the FOPI (constraining the symmetric matter +EOS up to 2n0) and ASY-EOS (constraining the symmetry energy at around 1.5n0) experiments, +Huth et al. +Drischler et al. +Legred et al. +Pressure [MeV/fm3] +0.1 +1 +10 +100 +baryon density nB/n0 +0.5 +1 +1.5 +2 +2.5 +3 +FIG. 23. The pressure of neutron star matter as a func- +tion of number density nB, as obtained by Huth et al. [17], +Drischler et al. [453], and Legred et al. [251] at 95%, 95%, +and 90% confidence interval, respectively. +and astrophysical data on bulk neutron star +properties (constraining the total neutron +star EOS above 2n0). The EOS models were +extended to high densities using a speed- +of-sound model. +The results are shown in +Fig. 23 and labeled as “Huth et al.”. +Yue et al. [454] constructed neutron star +models using a Skyrme energy-density func- +tional, which allowed them to consistently +calculate the neutron skin of +208Pb and +combine constraints from heavy-ion colli- +sions, measurements of the neutron skin, and +astrophysical constraints within the same +model. +Neill et al. [411] followed a similar strat- +egy as the example above, using Skyrme +models which were extended to the crust. +This allowed them to combine neutron skin +measurements, NICER/LIGO observations, + +54 +a crust observable (the resonant frequency of the crustal i-mode), and nuclear mass data to con- +strain both the core and crust properties as well as the EOS. By calculating all these quantities +using the same underlying Skyrme energy density functional (and polytrope parametrizations at +the highest densities), some poorly controlled modeling uncertainties were eliminated. This work +demonstrated the complementarity of different observables: within the particular model used, nu- +clear masses constrain mainly the zeroth and first symmetry energy expansion coefficients, S0 +and L, the crust observable has the largest impact on the inferred values of L and the second +expansion coefficient Ksym, and the neutron star radius and tidal deformability have the largest +impact on the inferred values of Ksym and two polytrope parameters. Thus, when combined, dif- +ferent observables provide complementary information that can contribute to a complete picture +of the EOS. The ranges of these overlapping data are depicted in Fig. 22. +Without crust observables, neutron star radii and tidal deformabilities tend to give weaker +constraints on Ksym and stronger constraints on L (see the analysis of a large number of studies +in [455]). +However, the relative constraints on the symmetry energy parameters change when +the used priors include the criterion that the crust is stable and incorporate potential data from +crust observables [411]. While this result is model-dependent and correlations with higher-order +symmetry parameters need to be investigated, it demonstrates the way in which crust observables +could significantly contribute to constraining the EOS, and motivates the need for improving models +to consistently combine crust and core observables with nuclear data. +One of the defining strengths of the global constraint analysis is that one or more additional +constraint(s) can be always included as long as an assessment of the corresponding statistical and +systematic uncertainties is also provided. Moreover, the more data from nuclear and astrophysical +observables can be meaningfully included in such EOS inferences, the greater the ability to deliver +a robust EOS. Therefore, constraining the EOS by combining various inferences is highly promising +and, furthermore, well-suited for the coming era of multi-differential observables from heavy-ion +collisions and multi-messenger astronomical observations. +V. +CONNECTIONS TO OTHER AREAS OF NUCLEAR PHYSICS +A. +Applications of hadronic transport +In addition to the use of transport codes to study fundamental nuclear physics, their ability +to describe the transport and interactions of particles in a material also make them valuable for +applications that benefit society. +Examples include the design of nuclear physics experiments, +detector development and simulations of detector performance, as well as medical applications and +radiation shielding in accelerator and space exploration. Some of these uses are outlined here. +Transport models are widely used to simulate particle emission from nucleus-nucleus collisions. +In these simulations, the four-momentum of every emitted particle is tracked, making it possible +to generate double differential distributions, the particle spectra at various emission angles. These +distributions are particularly important for applications. +Most transport models are optimized for describing physics in certain energy ranges. The type +of code required can be tailored to the desired application. For example, some models perform best +at energies of a few hundred MeV and below, which is near the peak of the cosmic ray flux [456], +while others are more applicable for GeV-scale energies and above, in the tail of the cosmic ray +flux. Over the entire experimental energy range, from intermediate energies through the highest +collider energies, transport codes have been successfully employed to design complex detectors, +optimize experimental setups, and carry out analyses of experimental data, including assessing the +detector efficiencies and background contributions. + +55 +1. +Detector design +In high-energy experiments, the code packages most commonly used for detector development +and data analysis are Geant3 [457], Geant4 [458], and FLUKA [459]. Most of these simulation +packages use cascade codes, that is codes without mean-field potentials, to describe particle trans- +port through matter. Modern transport codes that can cover a wide range of energies such as +PHITS (Particle and Heavy-Ion Transport code System) can, however, provide a more complex +description of particle transport [47]. +Aside from heavy-ion collisions, transport simulations play an important role for a variety of fun- +damental physics experiments. For example, long-baseline neutrino experiments need to determine +the incoming neutrino energy in order to extract the neutrino mixing parameters, CP violating +phases, and neutrino mass ordering [460]. However, because the neutrino beam is generated from +fixed-target proton-nucleus interactions producing secondary π and K mesons with neutrino decay +products, there are large uncertainties on the energy of the interacting neutrinos. The neutrino +energy must be reconstructed from the measurement of the final state [461, 462] which is often mod- +eled by simple Monte-Carlo cascade approaches. Reliable transport descriptions could significantly +improve these studies for the Deep Underground Neutrino Experiment (DUNE) [463], as well as +for the ongoing experiments NuMI Off-axis νe Appearance (NOvA) [464] and Tokai to Kamioka +(T2K) [465]. Other experiments that require transport simulations of backgrounds include dark +matter searches [466], semi-inclusive electron scattering such as (e,e′p) on nuclear targets at Jef- +ferson Lab to search for color transparency, short-range correlations [467], and hadronization in a +nuclear medium [468]. +2. +Space exploration, radiation therapy, and nuclear data +Transport models can also be used in applications relevant to space exploration to understand +and mitigate the harmful effects of the space radiation environment on electronics and astronauts. +Collisions of galactic cosmic rays (GCRs) with nuclei, whether in the Earth’s atmosphere or in +the material of a spacecraft above it, can generate showers of particles, including pions, muons, +neutrinos, electrons, and photons as well as protons and neutrons. GCRs cover a wide range of +energies, from tens or hundreds of MeV up to the TeV scale, and ion species, spanning elements +1 ≤ Z < 28 [475], making it challenging to determine all their potential effects in a given material. +The penetrating power of the initial GCRs and the secondaries generated by their interaction +with matter can have a serious impact on the safety and viability of space exploration. The 1% +of GCR primaries which are heavier than Helium nuclei can pose an especially serious problem, +given that the damage they inflict scales as Z2. The secondary particles generated from GCR +interactions with spacecraft materials [476] such as aluminum, polyethylene, and composites can +harm astronauts and disrupt or even disable electronic systems. Moreover, spacecraft shielding +designed to reduce the GCR flux is itself a target that can increase the secondary flux. Because +of the wide variety of possible shielding materials and thicknesses, transport models are essential +to determine the sensitivity of the secondaries (regarding both their flux and composition) to +different shielding configurations, as well as the subsequent harmful impact of those secondaries on +electronic systems [477] and humans [478]. A pictorial overview of applications transport modelling +and nuclear data in space missions is given in Fig. 24. +Due to the lack of data at the appropriate energies, simulations of space radiation effects have +large uncertainties. The space research community has generally relied on phenomenological nu- +clear reaction models such as the Double Differential Fragmentation model (DDFRG) [479], which +consists of a sum of multiple exponential distributions with parameters fit to data. Many of these + +56 +models employ abrasion-ablation models [480, 481] (where abrasion and ablation refer to parti- +cle removal in ion-ion interactions and nuclear de-excitation following abrasion, respectively) or +semi-empirical parametrizations, see Ref. [482]. +Researchers modeling these interactions could +benefit from transport codes discussed in this White Paper. The use of hadronic transport models +such as the Ultrarelativistic Quantum Molecular Dynamics (UrQMD) code [482], which was shown +to correctly predict proton and deuteron yields from the BNL Alternating Gradient Synchrotron +measurements of collisions of protons on Be and Au targets at Elab = 15 AGeV [483, 484], see +Fig. 25, could significantly advance simulations of collisions relevant for space exploration. For +further information about the needs for space applications, see Refs. [485, 486]. +Similar transport modeling needs arise in charged particle therapy for medical applications such +as cancer treatment. In this case, the ion beam is tuned to penetrate the tissue at the tumor location +so that the Bragg peak, or maximal dose, is delivered to the tumor site while minimizing the spread +of the charged particle beams into surrounding tissue due to target fragmentation and secondary +scattering [487]. Transport models can play an important role in improving the effectiveness and +safety of charged particle therapy in cancer treatment [487, 488]. Moreover, if ions such as carbon +are used instead of protons, the beam may also fragment and spread in the body. These interactions +are also studied employing abrasion-ablation models. Better models of projectile fragmentation +are needed to determine the effect of ion beams on normal tissue. Recently, the Stochastic Mean +Field (SMF) [489] and Boltzmann-Langevin One Body (BLOB) [490] models have been coupled with +Geant4 for studies related to radiation therapy [491]. +The nuclear information required for applications falls under the general umbrella term of +FIG. 24. Some of the applications of hadronic transport calculations and nuclear data in space exploration +research (counterclockwise from top left): energy production in outer space, such as with the TOPAZ +nuclear reactor [469] or the proposed fission surface power system on the Moon KRUSTY [470]; nuclear +thermal rocket propulsion [471]; planetary exploration [472]; and dose and shielding calculations of ions +passing through electronics and humans (left: a heavy-ion interaction with a shielding structure [473], right: +particle spectra calculated for an incident solar minimum GCR iron spectrum [474]). + +Nuclear reactors +Dose/Shielding calculations for electronics & humans +in outer space +Cosmic ray track +Geant4 +104 +FLUKA +PHITS +Metal + Metal +neutrons +10 +3DHZETRN (N=1) +3DHZETRN (N=22) + Glass +102 +0 g/cm? +Gate +10 +Field +10° +oxide +-Drain +Source +protons (xo.1) +Depletion +10° +Positive region +10 +He (x0.05) +well +10° +- Funneling +10 +TOPAZ +region +Substrate +105 +10-2 +10-1 +100 +10l +102 +103 +104 +Kinetic energy (MeV/n) +Nuclear propulsion +Planetary exploration +Passive (n, xy) +rays +Cosmic +Fast +Natural +KRUSTY +neutrons +radioactivity57 +−1.0−0.50.0 0.5 1.0 1.5 2.0 2.5 3.0 +y +10−5 +10−4 +10−3 +10−2 +10−1 +100 +101 +102 +103 +dN +dy +p+Be +UrQMD, protons +UrQMD, deuterons +E802, protons +E802, deuterons +−1.0−0.50.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 +y +p+Au, minb., 14.6 A GeV +FIG. 25. Rapidity distributions of protons and deuterons in +minimum bias p+Be (left) and p+Au (right) collisions at +a beam energy of Elab = 14.6 AGeV. Blue dashed and red +solid lines show results for protons and deuterons, respec- +tively, obtained from the UrQMD model, compared to data +from the E892 experiment (blue and red dots for protons +and deuterons, respectively) [484]. Figure from Ref. [483]. +“nuclear data”. The Geant3, Geant4, and +FLUKA codes all utilize information taken +directly from nuclear data libraries. How- +ever, standard nuclear databases cover +almost exclusively neutron-induced reac- +tions, while few charged-particle data are +available. +In addition, the energy range +covered by these databases typically only +extends to 20 MeV. In higher energy +databases such as the GSI-ESA-NASA +database [482], there are essentially no +data for light ions beyond Elab = 3 AGeV +and scant data for heavy ions beyond a +few hundred AMeV [482]. Transport mod- +els such as Quantum Molecular Dynamics +(QMD) [492] and PHITS [493, 494] have +been used to simulate higher-energy colli- +sions to fill the gaps in data. Experiments +at nuclear accelerators are needed to verify +these calculations. +The +US +Nuclear +Science +Advisory +Committee (NSAC) has been charged to “assess challenges, opportunities and priorities for ef- +fective stewardship of nuclear data”. +As part of the development of the Long Range Plan for +nuclear science, town halls involving different sub-fields of the US nuclear physics community have +adopted nuclear data resolutions, including a recommendation to identify cross-cutting opportuni- +ties with other programs. We suggest that one of these opportunities is the use of transport codes +to advance and enhance high-energy applications, such as space research and advanced medical +treatments. +B. +Hydrodynamics +Relativistic hydrodynamics (Landau-Lifshitz hydrodynamics [495]) can be defined as the effec- +tive field theory (EFT) describing fluids on energy scales much smaller than the fluid tempera- +ture [496]. Hydrodynamic equations encode the evolution of conserved charges, such as energy +density and electric charge density, in spacetime. Solving the hydrodynamic equations requires +the EOS as a crucial ingredient. Thus, in turn, hydrodynamics can be used to constrain the EOS. +For example, this has been done or is of relevance for the quark-gluon plasma (QGP) generated +in heavy-ion collisions [329, 497, 498] (recently, prospects of constraints on the EOS for nuclear +matter forming the primordial QGP emerged [499]), for (proto)neutron stars [500–505], as well as +for neutron star mergers [506–511]. +1. +Status +Hydrodynamics has had a great success describing nuclear matter generated in heavy-ion col- +lisions over a wide range of energies [512, 513]. Remarkably, hydrodynamics applies to various +system sizes accessible in heavy-ion collisions [514], with ALICE, ATLAS, and CMS experiments +showing collective fluid behavior in proton-ion [515–517] and even proton-proton collisions [518– +520], which was also successfully reproduced hydrodynamically [521]. Collective behavior in small + +58 +systems was also observed at RHIC by the PHENIX and STAR experiments [522, 523]. +It was realized early on that first-order hydrodynamics (in Landau or Eckart frame) is causality +violating and unstable [524]. At this time, the standard solution to this problem is the M¨uller- +Israel-Stewart (MIS) theory [525–527], or versions thereof [528, 529], which are used in most hy- +drodynamic codes modeling heavy-ion collisions. In the MIS theory, transient modes are added as +regulators ensuring a causal time evolution [530]. The behavior of such transient modes depends on +the way they are introduced [528, 530]. Since they are generally not associated with any conserved +quantities, their behavior is not what hydrodynamics aims to describe. MIS thus relies on these +transient modes to decay sufficiently fast for the observables to behave hydrodynamically. This +poses a problem for MIS at early times in a heavy-ion collision, when the regulator transient modes +are still present, because observables sensitive to the early times may reflect the physics of these +regulators. In addition, the causality violation [531] and stability [532] in these setups has to be +monitored when modeling, for example, heavy-ion collisions. +Alternatively, a more direct approach to constructing causal viscous hydrodynamics is based +on the realization that hydrodynamics is causal when considered in a general frame (and not, +e.g., Landau frame or Eckart frame). In that case it is not necessary to introduce any regulator +or auxiliary fields, as the differential equations governing the hydrodynamic fields (temperature, +fluid velocity, and chemical potential) are hyperbolic (i.e., there exists a solution for all times) and +their time evolution is causal by construction. This leads to the Bemfica-Disconzi-Noronha-Kovtun +(BDNK) theory [533–538], which is capable of, for example, modeling neutron star mergers [537]. +BDNK also has a practical use in constructing manifestly causal numerical codes solving hyperbolic +equations. Note, however, that BDNK is merely a causal formulation of hydrodynamics, and thus +BDNK is still not expected to be a good approximation at early times. +Finally, a rigorous field-theory formulation of hydrodynamics was achieved, which expresses +it as an EFT based on a generating functional [539–545], for a summary see Ref. [546]. +This +approach employs the Schwinger-Keldysh formalism of thermal field theory. As applications of +this formulation, effects of stochastic interactions on hydrodynamic correlation functions [547] +as well as a theory of non-linear diffusion were derived, taking into account large hydrodynamic +fluctuations (for example, leading to the dependence of transport coefficients on fluctuations of the +hydrodynamic fields) [548, 549]. +2. +Range of applicability +Many factors influence whether a system may be described hydrodynamically. Most impor- +tantly, like any EFT, hydrodynamics requires a separation of scales between the microscopic physics +and the scales on which the system is described. Let us focus on two remarkable results regarding +the range of applicability of hydrodynamics: +1) The unreasonable effectiveness of hydrodynamics far away from equilibrium. +2) Systematic extensions of hydrodynamics, extending its range of applicability. +Regarding 1), the applicability of hydrodynamics has been historically tied to a requirement +of a near-equilibrium state, near-isotropy, and small gradients. Astonishingly, heavy-ion collisions, +where neither of these three conditions is met, were successfully described hydrodynamically, which +is often referred to as the unreasonable effectiveness of hydrodynamics [550]. As a possible expla- +nation, hydrodynamic attractors were proposed [551] in a supersymmetric conformal theory, and +subsequently studied for heavy-ion collisions [552–561]. +The underlying reason for the attrac- +tor behavior is proposed to be kinematic, i.e., owed to a fast expansion in the boost-invariant +plasma [554, 562, 563]. Since systems cease to be boost invariant as the collision energy is lowered, + +59 +FIG. 26. Example of an extension of the regime of applica- +bility of hydrodynamics: spin hydrodynamics. While stan- +dard hydrodynamics is valid at small frequencies and mo- +menta (labeled as “pure hydro regime”, indicated by cyan +blue region), in the presence of spin degrees of freedom spin +hydrodynamics is valid in an extended regime (labeled as +“spin hydro regime”, indicated by a pink region). This is +facilitated by adding the slowly relaxing spin modes (green +curves) to the spectrum of standard hydrodynamic shear +(red solid curve) and sound (blue solid curve) modes. HY- +DRO+ is constructed in a similar way by adding modes +which relax slower and slower when approaching the criti- +cal point in the QCD phase diagram, bearing implications +for the EOS [564, 565]. Figure adapted from [566]. +this may pose a challenge to the develop- +ment of hydrodynamic attractors in nu- +clear matter at low to intermediate ener- +gies. +Within a certain class of models (in- +spired by the gauge/gravity correspon- +dence), the position-space hydrodynamic +expansion (in proper time) around the +Bjorken flow within the MIS theory was +shown to diverge factorially [551]. +In +contrast, for the same theory Fourier- +transformed into the momentum space +there is a finite convergence radius lim- +ited by the branch point singularity closest +to the origin in the complex momentum +plane [567–571]. This inspired the formu- +lation of hydrodynamics far from equilib- +rium via resummation [572]. +Regarding 2), a standard method to ex- +tend the regime of validity of hydrodynam- +ics is to add one or several mode(s). In +fact, promoting the shear tensor to an aux- +iliary field (regulator) adds a mode to the +spectrum of first-order hydrodynamic for- +mulation yielding the MIS model. In an- +other crucial example, critical fluctuations +need to be taken into account near the crit- +ical point in the QCD phase diagram, and +a set of slow modes can be added to the hydrodynamic modes yielding HYDRO+ [573, 574], which +in turn bears implications for the EOS and the speed of sound [564, 565]. Furthermore, Lambda +hyperon polarization data [575] indicates that the QGP is highly vortical and polarized [576– +578], which motivated the inclusion of spin in various hydrodynamic descriptions [566, 579–588] +(see, e.g., Fig. 26). Within a different systematic extension of hydrodynamics, dynamical elec- +tromagnetic fields can be added, leading to versions of magnetohydrodynamics which couple the +hydrodynamic conservation equations to Maxwell’s equations [589–591], and which can also include +the chiral anomaly [592–594], relevant for neutron stars [594]. Finally, another natural extension of +hydrodynamics is the simultaneous inclusion of multiple conserved charges, in particular, baryon +number B, strangeness S, and electric charge Q (BSQ charges) [532]. This renders transport coef- +ficients matrix-valued, which means that gradients in one charge may lead to diffusion of another +charge [595, 596]. +Modern hydrodynamics has been developed in close relation to the gauge/gravity correspon- +dence (a.k.a., AdS/CFT or holography). This development, which notably yielded the only consis- +tent theoretical description of fluids with η/s as low as found in heavy-ion data [597–599], began +with the insight that a lower bound on entropy production per degree of freedom (η/s) for a cer- +tain class of theories is related to black branes in the Anti de Sitter (AdS) spacetime [600]. The +fluid/gravity correspondence [601] as a systematic construction tool led to the discovery of the +chiral vortical effect [602, 603] and the re-discovery of the chiral magnetic effect [603, 604]. Holo- +graphic models are also suitable for exploration of plasmas at high densities [605], phase transitions +(in particular a holographic version of the QCD critical point [606]), neutron stars [607], taking + +[w(k)l = frequency scale +non-conserved +quantities +disappear fast +fast modes +(transient) +Non-hydro regime +T +IWsound(k)l +[Wshear(k)| +approximately +[Wspin,I (k) +conserved +charges diffuse +[W spin,(k)I +himois +spin +relaxdtion +rate +Spin hydro regime +S +conserved charges +diffuse slowly +Pure hydro regime +0 +k = wave number60 +into account finite coupling [608, 609], and for investigating the far-from-equilibrium regime of +holographic plasmas [610–612]. +3. +Challenges and opportunities +Given the recent developments described above, there are strong reasons to assume that hydro- +dynamics either is valid or can be extended to be valid for the description of dense nuclear matter +at intermediate energy scales, even in small systems with large gradients, far from equilibrium, +and near the QCD critical point. Such (extended) versions of hydrodynamics may well overlap +with the regime of validity of hadronic transport simulations, which needs to be studied. Here, +in particular, further development of hybrid approaches using both hydrodynamics and hadronic +transport will contribute to a better description of intermediate energy heavy-ion collisions. +The way ahead will require pushing forward the development of the rigorous theoretical for- +mulation of hydrodynamics, as well as testing its applicability with exactly solvable models (e.g., +constructed using the gauge/gravity correspondence) and, most importantly, against experimental +data. By continuing the development of hydrodynamics in parallel with gauge/gravity models, +the proposed versions of spin hydrodynamics can be tested and constructed rigorously using the +correspondence; the same statement also applies to versions of magnetohydrodynamics. In the +context of (magneto)hydrodynamics, one may also explore the interplay of multiple conserved +charge currents and anomalous currents, leading to novel transport phenomena [593, 613–617]. +For an efficient modeling of heavy-ion collisions (as well as neutron stars and neutron star merg- +ers), the BDNK approach needs to be developed and implemented in standard codes for data +analysis. At high densities, it becomes necessary to describe the propagation of multiple conserved +charges, in particular, the BSQ charges [532]. Consequently, the initial state used in numerical +hydrodynamic simulations must be modified to include BSQ degrees of freedom [618–620]. Sim- +ilarly, the EOS [621, 622] and the exact charge conservation when particles are formed (see, e.g., +Ref. [623, 624]) need to take into account BSQ charges. Beyond describing all conserved charges, +theoretical consistency on one hand and the need to describe systems far-from-equilibrium on the +other hand both necessitate a rigorous treatment of hydrodynamic fluctuations, which has been +done using a deterministic approach to fluctuations [625–628]. As a viable future complementary +approach, hydrodynamic fluctuations can be included using the Schwinger-Keldysh formulation +of hydrodynamics [546]. These goals are in line with two recent white papers: Snowmass The- +ory Frontier: Effective Field Theory Topical Group Summary [629] and Snowmass White Paper: +Effective Field Theories for Condensed Matter Systems [630]. +VI. +EXPLORATORY DIRECTIONS +A. +Dense nuclear matter EOS meeting extreme gravity and dark matter in supermassive +neutron stars +Do we need an independent determination of the nuclear EOS using terrestrial experiments in +the era of high-precision multi-messenger astronomy? While it is often emphasized that combined +data analyses of heavy-ion reactions and neutron star observations within a unified EOS theory +framework are a powerful tool to study the EOS (see Section IV), the independent extraction of +the nuclear EOS from heavy-ion reactions alone is fundamentally important. This assertion is +motivated by a well-known degeneracy [631] between the EOS of dense matter (including hadronic +and/or quark matter, and dark matter) and strong-field gravity in studies aimed at understanding + +61 +properties of super-massive neutron stars, the minimum mass of black holes, and properties of dark +matter [632–639]. +In the Astro2020 Science White Paper on Extreme Gravity and Fundamental Physics [640], +future gravitational wave (GW) observations are envisioned to enable unprecedented and unique +science related to +• The nature of gravity: Can we prove Einstein wrong? What building-block principles and +symmetries in nature invoked in the description of gravity can be challenged? +• The nature of dark matter: Is dark matter composed of particles, dark objects, or modifica- +tions of gravitational interactions? +• The nature of compact objects: Are black holes and neutron stars the only astrophysical +extreme compact objects in the Universe? What is the EOS of densest matter? +An independent determination of the EOS of dense nuclear matter from terrestrial experiments, +which are free from gravitational effects, will address the question of whether exotic physics, such +as modified gravity, is necessary to describe the behavior and phenomena of supermassive stars. +Thus constraining the EOS from heavy-ion collision experiments will help realize the astrophysical +science goals. +The fundamental questions listed above are among the eleven greatest physics questions for the +new century identified by the U.S. National Research Council in 2003 [641]. While gravity was the +first force discovered in nature, the quest to unify it with other fundamental forces remains elusive, +partially because of its apparent weakness at short distances [642, 643]. Moreover, while Einstein’s +general relativity (GR) theory for gravity has successfully passed all observational tests so far, it is +still not fully tested in the strong-field domain [644]. Searches for evidence of possible deviations +from GR are at the forefront of several fields in natural sciences. It is fundamentally important to +test whether GR will break down at the strongest possible gravitational fields reachable. For this +goal, supermassive neutron stars are among the ideal testing sites [645, 646]. However, as already +mentioned above, their properties can be accounted for by either modifying gravity, adding dark +matter, and/or adjusting the nuclear EOS. Thus, an independent inference of the nuclear EOS +from terrestrial experiments is fundamentally important for breaking the degeneracy between the +EOS of supermassive neutron stars and the strong-field gravity. +There are already some indications that the EOS of dense neutron-rich matter may play a +significant role in understanding the nature of gravity [647–650]. Effects of the nuclear symmetry +energy on the gravitational binding energy [651], surface curvature, and red shift [652], which are +normally used to measure the strength of gravity of massive stars in GR, as well as examples of +mass-radius relations in several classes of modified gravity theories are reviewed briefly in Ref. [653]. +More precise information about the dense nuclear matter EOS from terrestrial experiments will +enable further progress in this direction. +B. +Nuclear EOS with reduced spatial dimensions +Nuclear systems under constraints, with high degrees of symmetry and/or collectivity, may +be considered as effectively moving in spaces with reduced spatial dimensions. +Historically, in +developing modern methodologies, the spatial dimension d has been considered to be either a +continuous or a discrete variable. Many exciting and fundamentally new experimental discoveries +in reduced dimensions have been made in recent years in material sciences (e.g., the graphene [654] +and topological insulator [655, 656]) and cold atom physics (e.g., the superfluidity in a strongly +correlated 2D Fermi gas [657] and the generalized hydrodynamics in a strongly interacting 1D Bose +gas [658]). + +62 +The EOS of neutron-rich matter in spaces with reduced dimensions can be linked to that in the +conventional 3-dimensional (3D) space by the ϵ-expansion (ϵ = d − 4) [659–661]. The latter is a +perturbative approach that has been successfully used in treating second-order phase transitions +and related critical phenomena in solid state physics and, more recently, in studying the EOS of +cold atoms in 1D, 2D, and two-species Fermi and/or Bose gases with mixed dimensions [662, 663]. +The energy per nucleon E(nB, δ, d) in cold nuclear matter of dimension d at density nB and isospin +asymmetry δ can be expanded around nB = n0, δ = 0, and d = 3. In cold symmetric nuclear +matter, the E(nB, δ = 0, d) is predicted to decrease with decreasing d, indicating that nuclear +matter with a smaller d tends to be more bound but, at the same time, saturates at a higher +3D-equivalent density. The symmetry energy was also found to become smaller in spaces with +lower dimensions compared to the conventional 3D case [664]. +Can we find or make 1D and/or 2D nuclear systems in our 3D world? Can nucleons in neutron- +skins of heavy nuclei be considered as living in spaces with reduced spatial dimensions, and if so, +can we discover the related effects in heavy-ion collisions? Can some of the objects (e.g., lasagna) in +the predicted pasta phase [291, 312, 665] of the neutron star crust be described as nuclear systems +with 1D, 2D, or fractional dimensions? What are the roles of the dimension-dependent EOS in +multi-dimensional models of late stellar evolution? If 1D/2D simulations using 3D EOS do not +lead to supernova explosions, what will happen if the corresponding 1D/2D EOSs are used instead? +Answers to these questions may provide new perspectives on the EOS of neutron-rich matter in +3D and help solve some of the unresolved puzzles. +C. +Interplay between nucleonic and partonic degrees of freedom: SRC effects on nuclear +EOS, heavy-ion reactions, and neutron stars +Short-range correlations (SRCs) in nuclei, that is correlations in the nuclear ground-state wave +function, are mostly due to isosinglet neutron-proton pairs that have temporally fluctuated into +a high-relative-momentum state with approximately zero total center-of mass-momentum and a +spatial separation of about 1 fm [128, 129, 666–669]. Subnucleonic degrees of freedom are expected +to play an important role in understanding SRC-related phenomena. Altered quark momentum +distributions in nucleons embedded in nuclei with respect to those in free nucleons, known as +the EMC effect, have been studied extensively since 1983 [670]. +SRCs have been proposed as +one of the two leading causes of the EMC effect [671, 672]. Recent experiments found that the +strength of SRCs and the EMC effect are strongly correlated [673, 674] and that they both depend +strongly on the isospin asymmetry of the nuclei. Moreover, strong evidence was found that only +the momentum distributions of quarks in SRC nucleon pairs in nuclei are modified with respect +to free nucleons. Furthermore, the distributions of quarks in protons of neutron-rich nuclei are +modified more than in neutrons, implying that, on average, u quarks are modified more than d +quarks in neutron-rich nuclei [674], in analogy to an earlier finding that SRCs make protons mover +faster than neutrons in neutron-rich nuclei [675]. These phenomena reflect profound QCD effects +in the nuclear medium. Studying the flavour- and spin-dependence of nucleon structure functions +is at the forefront of QCD and is a major science driver of future EIC experiments. An example +of a correlation formed on short-range QCD length scales are quark-quark correlations known as +diquarks. It was recently proposed that diquark formation across two nucleons via the attractive +QCD quark-quark potential is the underlying QCD-level source of SRCs in nuclear matter and the +cause of the EMC effect [676]. +The SRC-related effects have consequences for the nuclear structure, high-energy quark dis- +tributions (the EMC effect), and high-density nuclear matter, including its EOS and in-medium +nucleon-nucleon scattering cross sections. Better understanding of SRC effects in dense neutron- + +63 +rich medium through heavy-ion reactions may have important ramifications. The profound conse- +quences of SRCs on the nuclear matter EOS can be easily seen when one considers the well-known +Hugenholtz-Van Hove (HVH) theorem, which was derived by assuming there are sharp Fermi sur- +faces for nucleons. The theorem provides intrinsic connections among the nuclear symmetry energy, +momentum dependences of both isoscalar and isovector nucleon potentials, and the corresponding +nucleon isoscalar effective mass and neutron-proton effective mass splitting in neutron-rich mat- +ter [677]. However, due to the SRCs nucleons do not have sharp Fermi surfaces, but extended +high-momentum tails, as evidenced by many experiments at the Jefferson Laboratory (JLAB) and +Brookhaven National Laboratory (BNL), see, e.g., Refs. [127–130] for recent reviews. Therefore, +the above relations may be completely altered by the isospin-dependence of SRCs induced by the +nuclear tensor force, which is much stronger in the symmetric nuclear matter than in pure neu- +tron matter [135, 678, 679]. Consequently, the composition of the symmetry energy itself (e.g., +the ratio of its kinetic over potential parts) may also be very different from the one without con- +sidering the SRCs [131]. Most of the parametrizations of the nuclear symmetry energy used so +far in both nuclear physics and astrophysics adopt the kinetic symmetry energy predicted by the +free Fermi gas model. However, such parametrizations neglect SRC effects that may lead to a +reduced or even negative kinetic symmetry energy. This effect originates in the fact that SRCs +are dominated by isosinglet neutron-proton pairs. As the system becomes more neutron-rich, an +increasingly larger fraction of protons, compared to neutrons, are found in the high momentum +tail [128, 129, 674]. Consequently, the kinetic symmetry energy is reduced compared to the free +Fermi gas model prediction [131–138]. +Interesting indications have been found very recently of SRC effects on the cooling of protoneu- +tron stars, the formation of baryon resonances, dark matter, and nuclear pasta as well as on tidal +deformation and mass-radius correlation in neutron stars [680–687], and also on several features of +nuclear matter and heavy-ion reactions [688–695]. However, much more work remains to be done +to systematically and consistently address the SRC-related issues in hadronic transport simulations +(see Section II A). Investigations of SRC effects on the nuclear EOS using heavy-ion collisions at +FRIB and FRIB400 will complement the ongoing and planned SRC research programs at JLAB, +GSI, and EIC at BNL. Together, these efforts will reveal new knowledge about the spin-isospin +dependence of three-body and tensor forces in dense neutron-rich matter. At short distances, these +forces are mostly due to the ρ-meson exchange [696]. The in-medium ρ-meson mass, determined +by QCD, may be significantly different from its free-space value [696–698]. Such modification in +the ρ-meson mass has been found to significantly affect the high-density behavior of the nuclear +symmetry energy [131, 699]. However, effects of the QCD quark-quark potential and the modified +tensor or three-body force on in-medium nucleon-nucleon cross sections remain to be explored. +D. +High-density symmetry energy above 2n0 +Section III B has primarily focused on the physics of nuclear symmetry energy up to ≈ 2n0. This +is because of substantial experimental challenges for measuring the symmetry energy using more +energetic beams. +However, at higher densities, but below the hadron-quark transition density, +there are also many interesting issues to be addressed [97, 705]. Therefore, it is worthwhile to +explore possible future directions to attack this problem (see also the white paper on QCD Phase +Structure and Interactions at High Baryon Density: Continuation of BES Physics Program with +CBM at FAIR [151]). +Experiments at FRIB400, FAIR, and other high-energy rare isotope beam facilities around the +world are expected to provide tremendous resolving power for determining the symmetry energy +at densities >∼ 2n0. While both the magnitude Esym(n0) and slope L of the symmetry energy + +64 +1 +2 +3 +0 +30 +60 +90 +120 +150 + Neutron Star + Observations +Esym (MeV) +ρ� +ρ0 + NL-RMF(18) + PC-RMF(3) + RHF(2) + DD-RMF(2) + Gogny-HF(2) + SHF(33) +� +� +� +� + BHF (Vidana) + BHF (Z.H. Li) + DBHF (Fuchs) + DBHF (Sammarruca) + Chiral EFT-N3LO450 + Chiral EFT-N3LO600 + VMB-APR + VMB-FP + VMB-WFF1 + VMB-WFF2 + VMB-WFF3 +FIG. 27. Left: Symmetry energy as a function of baryon density as obtained within 60 example models, +selected from 6 classes of over 520 phenomenological models and/or energy density functionals. +Right: +Symmetry energy as a function of baryon density as obtained within 11 examples from microscopic and/or +ab initio theories. Thick blue lines are the upper and lower boundaries of symmetry energy from analyses +of neutron star observables. Figure from Refs. [700–704]. +at n0 have been relatively well determined (Esym(n0) ≈ 31.7 ± 3.2 MeV and L ≈ 58 ± 19 MeV +[506, 700, 706, 707], in very good agreement with χEFT calculations [13, 166, 206, 253]), the +curvature Ksym and skewness Jsym of the nuclear symmetry energy are still poorly known. In +particular, Ksym is most critical for determining the crust-core transition density and pressure in +neutron stars [708–710]. Besides the importance for astrophysics, an experimental determination +20 +30 +40 +50 +60 +70 +80 +Esym(2ρ0) (MeV) +FOPI-LAND (2011) +ASY-EOS (2016) +Xie & Li (2019) +Zhang & Li (2019) +Zhou et al. (2019) +d’Etivaux (2019) +Nakazat & Suzuki (2019) +Symmetry energy at 2ρ0 from analyzing terrestrial and astrophysical data +2021 fiducial value=51 ± 13 MeV +Xie & Li (2020) +Tsang et al. (2020) +Yue et al.(2021) +Heavy-ion reactions +Neutron stars +Zhang et al. (2020) +FIG. 28. +Symmetry energy at twice the saturation +density from both heavy-ion reactions and neutron +stars. Figure modified from figures in Refs. [700, 711]. +of the high-density behavior of the nuclear sym- +metry energy will provide important guidance +for developing high-density nuclear many-body +theories. +Indeed, the density region explored +in heavy-ion reactions at BES, HADES, and in +the future at FRIB400 and FAIR is mostly be- +yond the current validity range of χEFT, and +it is also where the EOSs predicted by vari- +ous nuclear many-body theories, especially the +symmetry energy contributions, start to diverge +broadly (see Fig. 27). +Recent neutron star observations have led +to some progress in constraining the symme- +try energy at suprasaturation densities. Shown +in Fig. 28 is a compilation of recent results on +the symmetry energy at 2n0 from two analyses + +65 +of heavy-ion reactions at GSI and nine independent analyses of neutron star properties by sev- +eral groups. These analyses give a mean value of Esym(2n0) ≈ 51 ± 13 MeV at 68% confidence +level, as indicated by the green line. Interestingly, χEFT+MBPT calculations predict a value of +Esym(2n0) ≈ 45±3 MeV [158]. Similarly, quantum Monte Carlo calculations using local interactions +derived from the χEFT up to next-to-next-to-leading order predict a value of Esym(2n0) ≈ 46 ± 4 +MeV [184]. Evidently, the mean value of Esym(2n0) from the analyses mentioned above is consis- +tent with the χEFT predictions, albeit with large uncertainties. As noted before, 2n0 is near the +upper validity limit of the current χEFT theories. Thus, more precise measurements of Esym(2n0) +will help to test χEFT predictions. +Inspecting the results shown in Fig. 28 shows clearly that more work is necessary to reduce the +error bars. Most of the neutron star constraints are extracted from radii and tidal deformations of +canonical neutron stars with masses around 1.4M⊙. These observables are known to be sensitive +mostly to the values of pressure around (1-2)n0 in neutron stars, and therefore their constraints +on Esym(nB) around and above 2n0 are not strong. +Observables from more massive neutron stars were expected to place stronger constraints on +the high-density symmetry energy. To illustrate how the recent NICER+XMM-Newton’s mea- +surements of both the radius and mass of PSR J0740+6620 can influence the constraint on the +symmetry energy at densities above 2n0, the upper panel of Fig. 29 shows the extracted lower +limits of Esym(nB) obtained from directly inverting the TOV equation within a 3-dimensional +high-density EOS parameter space [238] for two cases: for the case where only the mass is ob- +served (green line), and for the case where both the mass and radius are observed (red line using +the 68% confidence lower radius limit reported by Riley et al. [264] and blue line using the radius +reported by Miller et al. [255]). The orange line is the upper limit of the symmetry energy from +0 +20 +40 +60 +80 +100 +120 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0.00 +0.03 +0.06 +0.09 +0.12 + Causality & M +max +=2.01 M +sun + Causality & R +2.01 +=11.41 km + Causality & R +2.01 +=12.2 km + Causality & L +1.4 +=427 + + +E +sym + [MeV] +11.1% + + +X +p +r/r0 +FIG. 29. Constraints on the high-density sym- +metry energy and proton fraction in neutron +stars from analyzing the tidal polarizability of +GW170817 and NICER’s observation of PSR +J0740+6620. Figure from Ref. [238]. +analyzing the upper limit (68% confidence) of tidal +deformation of GW170817 [236]. The upper limits +of the symmetry energy from the upper radius limits +reported by both Riley et al. and Miller et al. are +far above the upper limit of symmetry energy from +GW170817. +The lower panel in Fig. 29 shows the correspond- +ing proton fractions in PSR J0740+6620. The in- +fluence of knowing both the mass and radius of +this most massive neutron star currently known is +seen by comparing the green line with the red or +blue line, while the difference between the red and +blue lines indicates the systematic error from the +two independent analyses of the same observational +data. Although the estimates of Esym(nB) around +(2-3)n0 from these analyses are useful compared to +the model predictions shown in Fig. 27, much more +precise constraints on the Esym(nB) above 2n0 are +needed. +Pinning down the symmetry energy above 2n0 +will be very challenging, but achieving this goal will +bring a great reward. For example, without a reli- +able knowledge of the symmetry energy at suprasat- +uration densities, the density profile of the proton +fraction in the core of neutron stars (which has to +be higher than about 11% for the fast cooling to oc- + +66 +cur) at β−equilibrium is not determined. Consequently, whether the fast cooling of protoneutron +stars occurs through the direct URCA process remains uncertain. Heavy-ion reactions, especially +with high-energy radioactive beams, will provide the much-needed data to calibrate nuclear many- +body theories and constrain nuclear symmetry energy at densities >∼ 2n0. These efforts, in concert +with astrophysical research using high-precision X-rays from massive neutron stars (e.g., NICER +and STROBE-X [317]), GWs from new LIGO/VIRGO runs, and future detection of post-merger +high-frequency GWs, will better constrain Esym(nB) at densities around and above 2n0. For these +efforts to be fruitful, it is imperative to explore potential observables carrying undistorted infor- +mation on the symmetry energy above 2n0 from neutron stars and their mergers as well as in +high-energy heavy-ion reactions. +E. +Density-dependence of neutron-proton effective mass splitting in neutron-rich matter +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +<ρ/ρ0> +0.6 +0.7 +0.8 +0.9 +1 + (GeV) +0.2 +0.4 +0.6 +0.8 +1 +1.2 +m +* +τ (GeV) +0 +200 +400 +600 +dN/dm (GeV) +−1 +n +p +n +p +132Sn+ +124Sn, E/A=50 MeV, b=5 fm +t=10 fm/c +FIG. 30. +Correlation between the average nu- +cleon effective mass and the average nucleon +density (top), and the distribution of nucleon +effective masses (bottom) in the reaction of +132Sn+124 Sn at 10 fm/c with a beam energy +Elab = 50 AMeV and an impact parameter +b = 5 fm, as simulated within the IBUU trans- +port model with an explicitly isospin-dependent +single-nucleon potential. Figure from Ref. [712]. +The nucleon effective mass is a fundamental +quantity characterizing the propagation of a nucleon +in a nuclear medium [713–716], accounting (to lead- +ing order) for effects such as the space-time non- +locality of the effective nuclear interactions or Pauli +exchange effects. +The magnitude and sign of the +difference (splitting) between the effective masses +of neutrons and protons ∆m∗ +np have essential con- +sequences for cosmology, astrophysics, and nuclear +physics through influencing, e.g., the equilibrium +neutron to proton ratio in the early universe and +primordial nucleosynthesis [717], properties of mir- +ror nuclei [718], and the location of drip-lines [719]. +In heavy-ion reactions, ∆m∗ +np is of importance for +isospin-sensitive observables [100, 107, 720–725]. +The momentum-dependence of the single-nucleon +potential is normally characterized by the nucleon +effective mass m∗ +τ that can be decomposed into an +isoscalar and an isovector component [108, 726, 727]. +Due to our poor knowledge of the momentum de- +pendence of isovector interactions, the isovector nu- +cleon effective mass measured by using the neutron- +proton effective mass splitting ∆m∗ +np [706] has not +been constrained well [108, 728]. Based on the HVH +theorem, ∆m∗ +np was found approximately propor- +tional to the isospin asymmetry δ of the medium, +with a coefficient depending on the density as well +as momentum-dependence of both the isoscalar and +isovector nucleon potential [706]. Over the last decade, significant efforts have been made to extract +this coefficient at n0. A recent survey [729] of model analyses using data from mostly nucleon- +nucleus scattering and giant resonances of heavy nuclei suggests that the ∆m∗ +np, scaled by the +average nucleon mass in free space, ranges from 0 to about 0.5δ [208, 677, 706, 730–735]. +While experimental efforts to better constrain ∆m∗ +np at n0 using heavy-ion reactions with in- +termediate energy stable beams are ongoing (see, e.g., Ref. [98]), future experiments at FRIB +and FRIB400 will enable more sensitive probes of not only ∆m∗ +np at n0, but also of its density- + +67 +FIG. 31. Top: Density dependence of the nuclear sym- +metry energy for two typical symmetry energy func- +tionals used in the IBUU simulations. Bottom: Density +dependence of the isospin asymmetry δ in 132Sn+124 +Sn collisions at 20 fm/c with a beam energy of 400 +MeV/A and an impact parameter of 1 fm, and in the +core of neutron stars at β-equilibrium (inset). Taken +from Refs. [95, 736]. +dependence (which cannot be probed by the +nucleon-nucleus +scattering +and +giant +reso- +nances) up to about 2n0. +As an illustration, +shown in Fig. 30 are the density dependence +of the average nucleon effective mass (top) +and the distribution of the nucleon effective +masses (bottom) during a typical FRIB reac- +tion as simulated [712] within the IBUU trans- +port model with an explicitly isospin-dependent +single-nucleon potential [106, 720]. +From the +top panel, it is seen that the neutron-proton ef- +fective mass splitting is positive and increases +with the density up to about 1.3n0, consistent +with recent χEFT calculations [208]. To reach +higher densities, more energetic beams are re- +quired. +Heavy-ion reactions at FRIB400 will extend +the ranges of both density and isospin asymme- +try of the medium formed. Shown in the lower +panel of Fig. 31 are the isospin asymmetry δ +as a function of density during a typical reac- +tion at FRIB400 (main) and in neutron stars +at β-equilibrium (inset), calculated using the +same two typical symmetry energy functionals, +shown in the upper panel. The δ-nB relations in +both systems show the same isospin fractiona- +tion phenomena, e.g., reaching a higher isospin +asymmetry when a density functional with a +lower symmetry energy is used. One can also +see that generally, the low-density regions are +more neutron-rich than the high-density regions. These δ-nB relations are the fundamental origins +of all isospin-sensitive observables in both heavy-ion reactions and neutron stars. +A number of observables in heavy-ion reactions have been proposed as promising messengers of +the underlying momentum-dependence of the isovector potential and the corresponding neutron- +proton effective mass splitting, see, e.g., [20, 737, 738] for reviews. The momentum dependence of +the single-nucleon potential affects the reaction dynamics directly through the equations of motion +and indirectly through the scattering term of nucleons. As the in-medium nucleon-nucleon cross +section is proportional to the square of the reduced effective mass of the two colliding nucleons, +the nucleon effective mass will affect the nuclear stopping power (which is also described in the +literature in terms of the nucleon mean free path especially for nucleon-nucleus scattering) [739]. +Consequently, the reaction dynamics and observables of heavy-ion reactions are expected to bear +useful information about the density-dependence of the neutron-proton effective mass splitting in +neutron-rich matter. The challenge is to find such observables that are both robust and sensitive +to the variations of the neutron-proton effective mass splitting with density. The nucleon effective +mass affects also transport properties of neutron stars, see, e.g., Refs. [233, 740–744]. Neutron +star observables, e.g., neutrino emission and torsional oscillations of neutron stars, may also pro- +vide useful information about the density-dependence of neutron-proton effective mass splitting in +neutron-rich matter. Explorations of these issues are invaluable. + +68 +ACKNOWLEDGMENTS +This White Paper has benefited from talks and discussions at the workshop on Dense nuclear +matter equation of state in heavy-ion collisions that took place at the Institute for Nuclear Theory, +University of Washington (December 5-9, 2022). +K.A. thanks Hans-Rudolf Schmidt and Arnaud Le F`evre, and M.S. thanks Daniel Cebra for +insightful discussions. P.D. and B.T. thank Abdou Chbihi, Maria Colonna, Arnaud Le F`evre, and +Giuseppe Verde for discussing complementary international efforts. +K.A. and M.S. thank Peter Senger and Richard Seto for helpful comments on the draft of +Section III A. 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