a 3-dimensional high-density EOS parameter space [332] for two cases: for the
case where only the mass is observed (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. [355] and blue line using the radius
reported by Miller et al. [88]). The orange line is the upper limit of the
symmetry energy from analyzing the upper limit (68% confidence) of tidal
deformation of GW170817 [329]. 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. 6.3 shows the corresponding proton fractions in PSR
J0740+6620. The influence 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 $E_{\rm{sym}}(n_{B})$ around
$(2$-$3)n_{0}$ from these analyses are useful compared to the model
predictions shown in Fig. 6.1, much more precise constraints on the
$E_{\rm{sym}}(n_{B})$ above $2n_{0}$ are needed.

Pinning down the symmetry energy above $2n_{0}$ will be very challenging, but
achieving this goal will bring a great reward. For example, without a reliable
knowledge of the symmetry energy at suprasaturation 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 occur) at $\beta-$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 $\gtrsim 2n_{0}$. These efforts, in concert with
astrophysical research using high-precision X-rays from massive neutron stars
(e.g., NICER and STROBE-X [409]), GWs from new LIGO/VIRGO runs and from next-
generation detectors such

Figure 6.4: Correlation between the average nucleon effective mass and the
average nucleon density (top), and the distribution of nucleon effective
masses (bottom) in the reaction of 132Sn+124Sn at 10 fm/$c$ with a beam energy
$E_{\rm{kin}}=50\ A\rm{MeV}$ ($\sqrt{s_{{}_{\rm NN}}}=1.9\ \rm{GeV}$) and an
impact parameter $b=5\ \rm{fm}$, as simulated within the IBUU transport model
with an explicitly isospin-dependent single-nucleon potential. Figure from
Ref. [867].

as the Einstein Telescope and the Cosmic Explorer, and future detection of
post-merger high-frequency GWs, will better constrain $E_{\rm{sym}}(n_{B})$ at
densities around and above $2n_{0}$. For these efforts to be fruitful, it is
imperative to explore potential observables carrying undistorted information
on the symmetry energy above $2n_{0}$ from neutron stars and their mergers as
well as in high-energy heavy-ion reactions.

### 6.5 Density-dependence of neutron-proton effective mass splitting in
neutron-rich matter

The nucleon effective mass is a fundamental quantity characterizing the
propagation of a nucleon in a nuclear medium [868, 869, 870, 871], accounting
(to leading 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 $\Delta m_{np}^{*}$ have essential consequences for cosmology,
astrophysics, and nuclear physics through influencing, e.g., the equilibrium
neutron to proton ratio in the early universe and primordial nucleosynthesis
[872], properties of mirror nuclei [873], and the location of drip-lines
[874]. In heavy-ion reactions, $\Delta m_{np}^{*}$ is of importance for
isospin-sensitive observables [195, 875, 876, 877, 878, 879, 62, 880].

The momentum-dependence of the single-nucleon potential is normally
characterized by the nucleon effective mass $m^{*}_{\tau}$ that can be
decomposed into an isoscalar and an isovector component [881, 882, 196]. Due
to our poor knowledge of the momentum dependence of isovector interactions,
the isovector nucleon effective mass measured by using the neutron-proton
effective mass splitting $\Delta m^{*}_{np}$ [860] has not been constrained
well [883, 196]. Based on the HVH theorem, $\Delta m^{*}_{np}$ was found
approximately proportional to the isospin asymmetry $\delta$ of the medium,
with a coefficient depending on the density as well as momentum-dependence of
both the isoscalar and isovector nucleon potential [860]. Over the last
decade, significant efforts have been made to extract this coefficient at
$n_{0}$. A recent survey [884] of model analyses using data from mostly
nucleon-nucleus scattering and giant resonances of heavy nuclei suggests that
the $\Delta m^{*}_{np}$, scaled by the average nucleon mass in free space,
ranges from 0 to about $0.5\delta$ [296, 835, 885, 886, 860, 887, 888, 889,
890].

While experimental efforts to better constrain $\Delta m^{*}_{np}$ at $n_{0}$
using heavy-ion reactions with intermediate energy stable beams are ongoing
(see, e.g., Ref. [186]), future experiments at FRIB and FRIB400 will enable
more sensitive probes of not only $\Delta m^{*}_{np}$ at $n_{0}$, but also of
its density-dependence (which cannot be probed by the nucleon-nucleus
scattering and giant resonances) up to about $2n_{0}$. As an illustration,
shown in Fig. 6.4 are the density dependence of the average nucleon effective
mass (top) and the distribution of the nucleon effective masses (bottom)
during a typical FRIB reaction as simulated [867] within the IBUU transport
model with an explicitly isospin-dependent single-nucleon potential [194,
875]. From the top panel, it is seen that the neutron-proton effective mass
splitting is positive and increases with the density

Figure 6.5: Top: Density dependence of the nuclear symmetry energy for two
typical symmetry energy functionals used in the IBUU simulations. Bottom:
Density dependence of the isospin asymmetry $\delta$ in 132Sn+124Sn 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 $\beta$-equilibrium (inset). Figure
modified from Ref. [891].

up to about $1.3n_{0}$, consistent with recent $\chi$EFT calculations [296].
To reach higher densities, more energetic beams are required.

Heavy-ion reactions at FRIB400 will extend the ranges of both density and
isospin asymmetry of the medium formed. Shown in the lower panel of Fig. 6.5
are the isospin asymmetry $\delta$ as a function of density during a typical
reaction at FRIB400 (main) and in neutron stars at $\beta$-equilibrium
(inset), calculated using the same two typical symmetry energy functionals,
shown in the upper panel. The $\delta$-$n_{B}$ relations in both systems show
the same isospin fractionation 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 $\delta$-$n_{B}$ 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.,
[892, 893, 101] 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, especially for nucleon-nucleus scattering, in terms of the nucleon
mean free path) [894].

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. [895, 326, 896, 897, 898, 899]. Neutron star observables,
e.g., neutrino emission and torsional oscillations of neutron stars, may also
provide useful information about the density-dependence of neutron-proton
effective mass splitting in neutron-rich matter. Explorations of these issues
are invaluable.

##  Appendix A 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.

  * 1.

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 matter becomes very stiff at densities of the order of a few
times saturation density, leading to values of the speed of sound exceeding
$1/\sqrt{3}$ 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
matter. 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.

  * 2.

Among controlled terrestrial experiments, collisions of heavy nuclei at
intermediate beam kinetic 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. 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.

  * 3.

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. Given the imminent results from ongoing and future experimental
analyses, there is an urgent need for a theoretical research program that will
further inform and accelerate the development of hadronic transport models
used to extract the EOS from experimental data. Support for this program is
imperative to fully realize the potential of U.S. efforts leading the
exploration of the dense nuclear matter EOS.

## Acknowledgements

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, Grażyna 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,
Michał Spaliński, 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.

## Author’s contributions

Section 1 is primarily written by A.S., with input from P.D., B.-A.L., W.G.L.,
S.P, and M.B.T. Section 2.1 is primarily written by P.D., A.S., and H.W.
Section 2.2 is primarily written by C.D., S.G., J.W.H., and I.T. Section 2.3
is primarily written by W.G.N. Section 3.1 is primarily written by K.A., M.S,
and H.Z. Section 3.2 is primarily written by K.W.B., Z.C., R.K., A.B.M., and
M.B.T. Section 4 is primarily written by W.G.N. and M.B.T. Section 5.1 is
primarily written by R.V. and H.W. Section 5.2 is primarily written by M.K.
Section 6 is primarily written by B.-A.L. All primary authors participated in
discussions and provided in-depth comments on all versions of the manuscript.
A.S. is the primary editor of the manuscript.

The list of authors starts with the primary editor, followed by primary
authors listed in alphabetical order, which are then followed by endorsing
authors likewise listed in alphabetical order. The primary authors provided
the content of this White Paper, with individual contributions outlined above.
The endorsing authors are members of the nuclear physics community who support
the message of the White Paper. Many of the endorsing authors provided
extensive comments and valuable suggestions based on the first public version
of the manuscript.

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