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The nuclear equation of state (EOS) is at the center of numerous theoretical and experimental efforts in nuclear physics. With advances in microscopic theories for nuclear interactions, the availability of experiments probing nuclear matter under conditions not reached before, endeavors to develop sophisticated and reliable transport simulations to interpret these experiments, and the advent of multi-messenger astronomy, the next decade will bring new opportunities for determining the nuclear matter EOS, elucidating its dependence on density, temperature, and isospin asymmetry. Among controlled terrestrial experiments, collisions of heavy nuclei at intermediate 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 temperature, 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, respectively. Collisions of neutron-rich isotopes further bring the opportunity to probe effects due to the isospin asymmetry. However, capitalizing on the enormous scientific effort aimed at uncovering the dense nuclear matter EOS, both at RHIC and at FRIB as well as at other international facilities, depends on the continued development of state-of-the-art hadronic transport simulations. This white paper highlights the essential role that heavy-ion collision experiments and hadronic transport simulations play in understanding strong interactions in dense nuclear matter, with an emphasis on how these efforts can be used together with microscopic approaches and neutron star studies to uncover the nuclear EOS., Comment: White paper prepared for the 2023 Long Range Plan. v3: Updated version as published in Progress in Particle and Nuclear Physics. Note: the published version does not include the executive summary; in the updated arXiv version, the executive summary is included as an appendix. v4: Corrected list of authors

The fundamental properties of dense nuclear matter, as it exists in the core of massive stellar objects, are still largely unknown. The investigation of the high-density equation of state (EOS), which determines mass and radii of neutron stars and the dynamics of neutron star mergers, is in the focus of astronomical observations and of laboratory experiments with heavy-ion collisions. Moreover, the microscopic degrees-of-freedom of strongly interacting matter at high baryon densities are also unknown. While Quantum-Chromo-Dynamics (QCD) calculations on the lattice find a smooth chiral crossover between hadronic matter and the quark-gluon plasma for high temperatures at zero baryon chemical potential, effective models predict a 1st order chiral transition with a critical endpoint for matter at large baryon chemical potentials. Up to date, experimental data both on the high-density EOS and on a possible phase transition in dense baryonic matter are very scarce. In order to explore this terra incognita, dedicated experimental programs are planned at future heavy-ion research centres: the CBM experiment at FAIR, and the MPD and BM@N experiments at NICA. The research programs and the layout of these experiments will be presented. The future results of these laboratory experiments will complement astronomical observations concerning the EOS, and, in addition, will shed light on the microscopic degrees of freedom of QCD matter at neutron star core densities., Comment: 21 pages, 15 figures, ICNFP 21, to be published in Physica Scripta

The Compressed Baryonic Matter (CBM) experiment at the future Facility for Antiproton and Ion Research (FAIR) in Darmstadt is designed to investigate the properties of high-density QCD matter with multi-differential measurements of hadrons and leptons, including rare probes like multi-strange anti-hyperons and charmed particles. The research program covers the study of the high-density equation-of-state of nuclear matter, and the exploration of the QCD phase diagram at large baryon-chemical potentials, including the search for quark matter and the critical endpoint of a hypothetical 1st order phase transition. The CBM setup comprises detector systems for the identification of charged hadrons, electrons, and muons, for the determination of collision centrality and the orientation of the reaction plane, and a free-streaming data read-out and acquisition system, which allows online reconstruction and selection of events up to reaction rates of 10 MHz. In this article, emphasis is placed on the measurement of muon pairs in Au-Au collisions at FAIR beam energies, which are unique probes to determine the temperature of the fireball, and, hence, to search for a caloric curve of QCD matter. Simultaneously, the subthreshold production of charmonium can be studied via its dimuon decay, in order to shed light on the microscopic structure of QCD matter at high baryon densities. The CBM setup with focus on dimuon measurements and results of the corresponding physics performance studies will be presented., Comment: 8 pages, 9 figures

Laboratory experiments with high-energetic heavy-ion collisions offer the opportunity to explore fundamental properties of nuclear matter, such as the high-density equation-of-state, which governs the structure and dynamics of cosmic objects and phenomena like neutron stars, supernova explosions, and neutron star mergers. A particular goal and challenge of the experiments is to unravel the microscopic degrees-of-freedom of strongly interaction matter at high density, including the search for phase transitions, which may feature a region of phase coexistence and a critical endpoint. As the theory of strong interaction is not able to make firm predictions for the structure and the properties of matter high baryon chemical potentials, the scientific progress in this field is driven by experimental results. The mission of future experiments at FAIR and NICA, which will complement the running experimental programs at GSI, CERN, and RHIC, is to explore new diagnostic probes, which never have been measured before at collision energies, where the highest net-baryon densities will be created. The most promising observables, which are expected to shed light on the nature of high-density QCD matter, comprise the collective flow of identified particles including multi-strange (anti-) hyperons, fluctuations and correlations, lepton pairs, and charmed particles. In the following, the perspectives for experiments in the NICA energy range will be discussed., Comment: 23 pages, 21 figures, Physics introduction to the Technical Design Report on the Inner Tracking System of the MPD experiment at NICA

The Facility for Antiproton and Ion Research (FAIR) in Darmstadt will provide unique research opportunities for the investigation of fundamental open questions related to nuclear physics and astrophysics, including the exploration of QCD matter under extreme conditions, which governs the structure and dynamics of cosmic objects and phenomena like neutron stars, supernova explosions, and neutron star mergers. The physics program of the Compressed Baryonic Matter (CBM) experiment is devoted to the production and investigation of dense nuclear matter, with a focus on the high-density equation-of-state (EOS), and signatures for new phases of dense QCD matter. According to the present schedule, the CBM experiment will receive the first beams from the FAIR accelerators in 2025. This article reviews promising observables, outlines the CBM detector system, and presents results of physics performance studies., Comment: 16 pages, 13 figures. Physica Scripta 2020

The Compressed Baryonic Matter (CBM) experiment will investigate high-energy heavy-ion collisions at the international Facility for Antiproton and Ion Research (FAIR), which is under construction in Darmstadt, Germany. The CBM research program is focused on the exploration of QCD matter at neutron star core densities, such as study of the equation-of-state and the search for phase transitions. Key experimental observables include (multi-) strange (anti-) particles, electron-positron pairs and dimuons, particle correlations and fluctuations, and hyper-nuclei. In order to measure these diagnostic probes multi-differentially with unprecedented precision, the CBM detector and data acquisition systems are designed to run at reaction rates up to 10 MHz. This requires the development of fast and radiation hard detectors and readout electronics for track reconstruction, electron and muon identification, time-of-flight determination, and event characterization. The data are read-out by ultra-fast, radiation-tolerant, and free-streaming front-end electronics, and then transferred via radiation-hard data aggregation units and high-speed optical connections to a high-performance computing center. A fast and highly parallelized software will perform online track reconstruction, particle identification, and event analysis. The components of the CBM experimental setup will be discussed, and results of physics performance studies will be presented., Comment: 22 pages, 27 figures, 8th International Conference on New Frontiers in Physics (ICNFP 2019)

The Multi-Purpose Detector (MPD) at NICA collider has a substantial discovery potential concerning the exploration of the QCD phase diagram in the region of high net-baryon densities and moderate temperatures. The anisotropic transverse flow is one of the key observables to study the properties of dense matter created in heavy-ion collisions. The MPD performance for anisotropic flow measurements is studied with Monte-Carlo simulations of gold ions at NICA energies $\sqrt{s_{NN}}=4-11$ GeV using different heavy-ion event generators. Different combinations of the MPD detector subsystems are used to investigate the possible systematic biases in flow measurements, and to study effects of detector azimuthal non-uniformity. The resulting performance of the MPD for flow measurements is demonstrated for directed and elliptic flow of identified charged hadrons as a function of rapidity and transverse momentum in different centrality classes., Comment: 5 pages, 4 figures, to be published in the Proceedings of the Baldin ISHEPP XXIV (Dubna, 17-22 September 2018), EPJ Web of Conferences