Agradecimentos: 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 (INT), University of Washington (December 5–9, 2022) [900]. We thank the INT for its kind hospitality and stimulating research environment. K.A. thanks Hans-Rudolf Schmidt and Arnaud Le Fèvre, and M.S. thanks Daniel Cebra for insightful discussions. P.D. and B.T. thank Abdou Chbihi, Maria Colonna, Arnaud Le Fèvre, and Giuseppe Verde for discussing complementary international efforts.
A.S. thanks Jörg Aichelin, David Blaschke, Elena Bratkovskaya, Maria Colonna, Dan Cozma, Wick Haxton, Natsumi Ikeno, Gabriele Inghirami, Behruz Kardan, Declan Keane, Arnaud Le Fèvre, William Llope, Ulrich Mosel, Berndt Müller, Witold Nazarewicz, Grazyna Odyniec, Panagiota Papakonstantinou, Ralf Rapp, Peter Rau, Björn Schenke, Srimoyee Sen, Chun Shen, Christian Sturm, Giorgio Torrieri, and Wolfgang Trautmann for insightful discussions and comments on Sections 1–4. K.A. and M.S. thank Peter Senger and Richard Seto for helpful comments on the draft of Section 3.1. M.K. thanks Navid Abbasi, David Blaschke, Casey Cartwright, Saso Grozdanov, Ulrich Heinz, Gabriele Inghirami, Michal P. Heller, Jorge Noronha, Jacquelyn Noronha-Hostler, Dirk Rischke, Michal Spalinski, Misha Stephanov, and Giorgio Torrieri for helpful comments on Section 5.2. This work was supported in part by the INT’s U.S. Department of Energy grant No. DE-FG02-00ER41132. K.A. acknowledges support from the Bundesministerium für Bildung und Forschung (BMBF, German Federal Ministry of Education and Research) – Project-ID 05P19VTFC1 and Helmholtz Graduate School for Hadron and Ion Research (HGS-HIRe). Z.C. acknowledges support from the U.S. National Science Foundation grant PHYS-2110218. P.D. acknowledges support by the U.S. Department of Energy, Office of Science, under Grant DE-SC0019209. S.G. and I.T. acknowledge support from the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract No. DE-AC52-06NA25396, and by the Office of Advanced Scientific Computing Research, Scientific Discovery through Advanced Computing (SciDAC) NUCLEI program; S.G. is also supported by the Department of Energy Early Career Award Program, while the work of I.T. is additionally supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory under project number 20220541ECR. J.W.H. is supported by the U.S. National Science Foundation under grants PHY1652199, PHY2209318, and OAC2103680. M.K. is supported, in part, by the U.S. Department of Energy grant DE-SC0012447. C.-M.K. acknowledges support from the U.S. Department of Energy under Award No. DE-SC0015266. R.K., W.G.L., and M.B.T. are supported by the National Science Foundation under Grant No. PHY-2209145. B.-A.L. is supported in part by the U.S. Department of Energy, Office of Science, under Award Number DE-SC0013702, and the CUSTIPEN (China–U.S. Theory Institute for Physics with Exotic Nuclei) under the U.S. Department of Energy Grant No. DE-SC0009971. A.B.M. acknowledges support from the U.S. Department of Energy grant DE-FG02-93ER40773. W.G.N. is supported by the NASA grant 80NSSC18K1019 and the National Science Foundation grant 2050099. S.P. and O.S. are supported by the U.S. Department of Energy, Office of Science, grant no. DE-FG02-03ER41259. M.S. is supported by the Alexander von Humboldt Foundation and the U.S. Department of Energy grant DE-SC0020651. R.V. acknowledges support from the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Contract DE-AC52-07NA27344. H.W. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC-2094-390783311. H.Z. is supported by the National Science Centre, Poland , under grants No. 2021/41/B/ST2/02409 and 2020/38/E/ST2/00019, and by the Warsaw University of Technology project grants IDUB-POB-FWEiTE-3 and IDUB-POB POST-DOC PW Ver menos

Abstract: 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, endeav-ors 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 inter -mediate 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 temper-atures 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, capitaliz-ing 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 develop-ment 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 understand -ing 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 Ver menos