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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

In order to provide particle identification (PID) of charged hadrons at the future high-rate Compressed Baryonic Matter (CBM) experiment the TOF group has developed a large-area Time-of-Flight (ToF) wall equipped with Multi-gap Resistive Plate Chambers (MRPC) capable to operate at incident particle fluxes of up to 30 kHz/cm$^2$. Prior to its destined operation at the Facility for Antiproton and Ion Research (FAIR) - starting in 2025 - this high-rate timing MRPC technology will be used for physics research at two scientific pillars of the FAIR Phase-0 program: the end-cap TOF upgrade of the STAR experiment at RHIC and the mTOF wall of the mCBM experiment at SIS18. At STAR, the fixed-target program of the Beam Energy Scan II (BES-II) will rely on 108 CBM MRPC detectors for forward PID at trigger rates of up to 2 kHz. At mCBM, high-performance benchmark runs of Lambda-baryon production at top SIS18 energies and CBM design interaction rates of 10 MHz will become feasible with a PID backbone consisting of 25 CBM MRPC detectors. Apart from the physics perspective, these pre-FAIR involvements will help gathering experience in operating the final CBM TOF wall comprising about 1500 MRPC detectors and 110,000 readout channels. The status of the FAIR phase 0 program will be discussed., Comment: 8 pages, 7 figures, Prepared for submission to JINST

In the CBM (Compressed Baryonic Matter) experiment constructed at the Facility for Anti-proton and Ion Research (Fair) at GSI, Darmstadt, Germany, MRPC(Multi-gap Resistive Plate Chamber) is adopted to construct the large TOF (Time-of-Flight) system to achieve an unprecedented precision of hadron identification, benefiting from its good time resolution, relatively high efficiency and low building price. We have developed a kind of double-ended readout strip MRPC. It uses low resistive glass to keep good performance of time resolution under high-rate condition. The differential double stack structure of 2x4 gas gaps help to reduce the required high voltage to half. There are 24 strips on one counter, and each is 270mm long, 7mm wide and the interval is 3mm. Ground is placed onto the MRPC electrode and feed through is carefully designed to match the 100 Ohm impedance of PADI electronics. The prototype of this strip MRPC has been tested with cosmic ray, a 98% efficiency and 60ps time resolution is gotten. In order to further examine the performance of the detector working under higher particle flux rate, the prototype has been tested in the 2014 October GSI beam time and 2015 February CERN beam time. In both beam times a relatively high rate of 1 kHz/cm2 was obtained. The calibration is done with CBM ROOT. A couple of corrections has been considered in the calibration and analysis process (including time-walk correction, gain correction, strip alignment correction and velocity correction) to access actual counter performances such as efficiency and time resolution. An efficiency of 97% and time resolution of 48ps are obtained. All these results show that the real-size prototype is fully capable of the requirement of the CBM-TOF, and new designs such as self-sealing are modified into the strip counter prototype to obtain even better performance.