1. Bao-An Li Collaborators:
Symmetry Energy of Neutron-Rich Matter on Earth and in Heaven
Bao-An Li
Collaborators:
Baojun Cai and W. G. Newton, Texas A&M University-Commerce, USA
Farrooh J. Fattoyev, Indiana University, USA
Plamen Krastev, Harvard University, USA
Larry Weinstein, Old Dominion University, USA
Or Hen, MIT, USA
Lie-Wen Chen, Shanghai Jiao Tong University, China
Jun Xu, Shanghai Institute of Applied Physics, China
Eli Piasetzky, Tel Aviv University, Israel

2. Outline What is nuclear symmetry energy? What do we know?
Why is it so uncertain especially at high densities?
(e.g., effects of isospin-dependent short-range correlations)
How to probe high-density symmetry energy with heavy-ion reactions?
(e.g., near-threshold pion production)
What are the astrophysics impacts of nuclear symmetry energy?
(e.g., 1. gravitational waves from spiraling binaries of neutron stars,
2. breaking the EOS-Gravity degeneracy in massive neutron stars) 2

3. EOS of cold, neutron-rich nucleonic matter
symmetry energy
Isospin asymmetry δ 12 12 12
Energy per nucleon in symmetric matter 18 18 3
Energy in asymmetric nucleonic matter
Symmetric matter
ρn=ρp ? ??
density
Strangeness
The axes of new opportunities
Isospin asymmetry
??? 3

4. Characterization of symmetry energy near normal density
The physical importance of L
In npe matter in the simplest model of neutron stars at ϐ-equilibrium
In pure neutron matter at saturation density of nuclear matter

5. Esym(ρ0)≈31.6±2.66 MeV L≈ 2 Esym(ρ0)=59±16 MeV
Constraints on Esym(ρ0) and L based on 29 analyses of data
Esym(ρ0)≈31.6±2.66 MeV
L≈ 2 Esym(ρ0)=59±16 MeV
L=2 Esym(ρ0) if Esym=Esym(ρ0)(ρ/ρ0)2/3
Bao-An Li and Xiao Han,
Phys. Lett. B727, 276 (2013).

6. ? ? ? ? What is the high-density symmetry energy?
(Examples of theoretical predictions)
Predictions using SHF or RMF energy density functionals
Predictions using microscopic theories ?
BHF
Constraints near ρ0 ? ? ?
M. B. Tsang et al.,
Phys. Rev. C86, (2012)
Phys. Rev. C 92, (2015)

7. Why is the symmetry energy so uncertain
Why is the symmetry energy so uncertain? (Besides the different many-body approaches used)
Isospin-dependence of short-range correlation due to the tensor force
Spin-isospin dependence of the 3-body force
Isospin dependence of pairing and clustering at low densities
(Valid only at the mean-field level)
Keith A. Brueckner, Sidney A. Coon, and Janusz Dabrowski, Phys. Rev. 168, 1184 (1968)
Correlation functions
Within a simple interacting Fermi gas model, the Fock term is
Vnp(T0) ? Vnp (T1)
fT0 ? fT1 , the tensor force in T0 channel makes them different

8. Tensor force induced (1) high-momentum tail in nucleon momentum distribution and (2) isospin dependence of SRC
Theory of Nuclear matter
H.A. Bethe
Ann. Rev. Nucl. Part. Sci., 21, (1971)
Fermi Sphere
O. Hen et al.,
Science 346, 614 (2014)

9. Proton skins in momentum space
Input single-nucleon momentum distribution from many-body theories
(e.g. SCGF: Self-consistent Greens’ function approach) and/or information
extracted from data based on the n-p dominance model
T. Otsuka et al.,
PRL 95, (2005);
PRL 97, (2006)
Proton skins in momentum space

10. Protons move much faster than neutrons in neutron-skins
Bao-Jun Cai, Bao-An Li and Lie-Wen Chen, PRC 94, (R) (2016)
Co-existence of neutron
skins in coordinate space
and proton skins in
momentum space
Extended-Thomas-Fermi
The average local momentum is defined via

11. Modified Gogny Hartree-Fock energy density functional incorporating SRC-induced
high momentum tail in the single nucleon momentum distribution
Kinetic
Zero-range two-body force
Three-body force
Momentum-dependent potential energy due to finite-range 2-body interaction
SRC-induced HMT in the single-nucleon
momentum distribution affects both the
kinetic energy and the momentum-dependent
part of the potential energy
arXiv: [nucl-th]
Bao-Jun Cai and Bao-An Li (2017)

12. Symmetry energy gets softened at both low and high densities
Readjusting model parameters to reproduce the same saturation properties
of nuclear matter as well as Esym(ρ0)=31.6 MeV and L(ρ0)=58.9 MeV
The isospin-dependence of nuclear
incompressibility at ρ0
Symmetry energy gets softened
at both low and high densities
Data:
MDI+HMT-exp: -492
MDI+FFG:

13. How does the SRC affect the kinetic symmetry energy?
Chang Xu and Bao-An Li, arXiv:
Chang Xu, Ang Li and Bao-An Li,
JPCS 420, (2013).
Tensor force +
Repulsive core
Repulsive
core only
Esym>0

14. Can the symmetry energy become negative at high densities?
Yes, it happens when the tensor force due to ρ exchange in the T=0 channel dominates
At high densities, the energy of pure neutron matter can be lower than symmetric matter leading to negative symmetry energy
Example: proton fractions with interactions/models
leading to negative symmetry energy
M. Kutschera et al., Acta Physica Polonica B37 (2006)
Tensor force and/or 3-body force can make Esym negative
at high densities
Affecting the threshold
of hyperon formation,
kaon condensation, etc
3-body force effects
in Gogny or Skyrme HF
Super-Soft

15. Among the promising observables of high-density symmetry energy:
π -/π +, neutron-proton differential flow in heavy-ion collisions
Radii of neutron stars
Neutrino flux of supernova explosions
Strain amplitude and frequency of gravitational waves from spiraling neutron
star binaries and/or oscillations/rotations of deformed pulsars
A major scientific motivation of
Rare isotope beam facilities around the world
Neutron Star Interior Composition Explorer (NICER of NASA) and various
x-ray satellite (Chandra, LOFT, XMM-Newton, etc)
(3) Various gravitational wave detectors
EPJA, Vol. 50, No. 2 (2014)

16. The proton fraction x at ß-equilibrium
Isospin fractionation in heavy-ion reactions
low (high) density region is more
neutron-rich with stiff (soft) symmetry energy
The proton fraction x at ß-equilibrium
Slow cooling: modified URCA:
Faster cooling by 4 to 5 orders of magnitude: direct URCA
Bao-An Li, Phys. Rev. Lett. 88 (2002)

17. Probing the symmetry energy at supra-saturation densities
Stiff
Central density
density
π-/ π+ probe of dense matter
Soft Esym
Stiff Esym
n/p ?
n/p ratio at supra-normal densities

18. Calculations: IQMD and IBUU04
Circumstantial Evidence for a Super-soft Symmetry Energy at Supra-saturation Densities
W. Reisdorf et al.
NPA781 (2007) 459
Data:
Calculations: IQMD and IBUU04
Z.G. Xiao, B.A. Li, L.W. Chen, G.C. Yong and M. Zhang, Phys. Rev. Lett. 102 (2009)
A super-soft nuclear symmetry energy is favored by the FOPI data!!!

19. Signatures of high-density Esym in GW signals: Tidal deformability and mergers
Shibata, Keisuke; PRD73, (2006)
Tidal field Eij drives f-mode (quadrupole
deformation Qij)
Resulting energy transfer appears
as phase shift in gravitational waveform
Detectable cleanly f≈ Hz
Phase shift depends on one parameter:
tidal polarizability λ (or Love number k2)
Qij = -λEij
M,R M’ a
Flanagan, Hinderer, PRD77, (2008)
Hinderer et al, PRD81, (2010)

20. Signatures of S,L(n>n0) in GW signals: Tidal deformability and mergers
Fattoyev, Carvajal, Newton, Li, PRC87, (2013)
Detector sensitivities assuming
Optimally oriented, equal mass binary
at D=100 Mpc
Damour, Nagar, PRD81, (2010)
Damour, Nagar, Villain, PRD85, (2012)
Hinderer et al, PRD 81, (2010)
At 1.4MSUN, signature of the high density behavior of symmetry energy at limit of Adv LIGO sensitivity

21. A challenge: how can neutron stars be stable with a super-soft symmetry energy? If the symmetry energy is too soft, then a mechanical instability will occur when dP/dρ is negative, neutron stars will then all collapse while they do exist in nature
TOV equation: a condition at hydrodynamical equilibrium
Gravity
Nuclear pressure
For npe matter
P. Danielewicz, R. Lacey and W.G. Lynch,
Science 298, 1592 (2002))
dP/dρ<0 if E’sym is big and negative (super-soft)

22. • What is the nature of the dark energy? • How did the universe begin?
Connecting Quarks with the Cosmos:
Eleven Science Questions for the New Century,
National Research Council
• What is the dark matter?
• What is the nature of the dark energy?
• How did the universe begin?
• What is gravity?
• Are there additional spacetime dimensions?
• What are the masses of the neutrinos, and how have
they shaped the evolution of the universe?
• How do cosmic accelerators work and what are they
accelerating?
• Are protons unstable?
• Are there new states of matter at exceedingly high
density and temperature?
• How were the elements from iron to uranium made?
• Is a new theory of matter and light needed at the
highest energies?

23. Competing Theories: Dark Matter or Modified Gravity
Observations hinting the existence of
Dark Matter: rotational curve,
Cluster dynamics, weak lensing,
collisionless passing of Bullet cluster, ….
Modified Newtonian Dynamics
Dark Matter?
Moti
Some modified gravity theories,
pass GR test at solar scale,
can explain all observations
including the Bullet Cluster
without using Dark Matter
R.H. Sanders,
Astron. Astrophys. 136, L21 (1984).
J. R. Brownstein, J. W. Moffat,
MNRAS.382:29-47,2007

24. gravity EOS Gravity-EOS Degeneracy in massive neutron stars ?
Strong-field gravity: Generality Relativity or Modified Gravity? ?
gravity
GR+[Modified Gravity]
Action S=Sgravity+Smatter
========
Massive neutron stars
Matter+[Dark Matter]+[Dark Energy]
EOS ?
Contents and stiffness of the EOS of super-dense matter?
For high-density neutron-rich nucleonic matter, the most
uncertain part of the EOS is the nuclear symmetry energy

25. An example of EOS-Gravity degeneracy in Massive Neutron Stars
Simon DeDeo, Dimitrios Psaltis
Phys. Rev. Lett. 90 (2003)
Dimitrios Psaltis, Living Reviews in Relativity, 11, 9 (2008)
Neutron stars are among the densest
objects with the strongest gravity
General Relativity (GR) may break down at
strong-field limit and “there is no fundamental
reason to choose Einstein’s GR over alternative
gravity theories”
Need at least 2 observables to break the
degeneracy
Uncertain range
of EOS
Stiff EOS: V. R. Pandharipande, Nucl. Phys. A 174, 641 (1971).
Soft EOS: R. B. Wiringa, V. Fiks, and A. Fabrocini, Phys. Rev. C38, 1010 (1988)
Scalar-Tensor theory with quadratic coupling:

26. Physics origin of the Yukawa term
In grand unification theories, conventional gravity has to be
modified due to either geometrical effects of extra space-time dimensions at short length, a new boson or the 5th force
String theorists have published TONS of papers
on the extra space-time dimensions
N. Arkani-Hamed et al., Phys Lett. B 429, 263–272 (1998); J.C. Long et al., Nature 421, 922 (2003);
C.D. Hoyle, Nature 421, 899 (2003)
Yukawa potential due to the exchange of a new boson proposed in the super-symmetric extension of the Standard Model of the Grand Unification Theory, or the fifth force
In terms of the gravitational potential
Yasunori Fujii, Nature 234, 5-7 (1971); G.W. Gibbons and B.F. Whiting, Nature 291, (1981)
The neutral spin-1 gauge boson U is a candidate, it is light and weakly interacting,
Pierre Fayet, PLB675, 267 (2009),
C. Boehm, D. Hooper, J. Silk, M. Casse and J. Paul, PRL, 92, (2004).

27. Influences of the Yukawa term in modified gravity on Neutron stars

28. Supersoft Symmetry Energy Encountering Non-Newtonian Gravity in Neutron Stars
De-Hua Wen, Bao-An Li and Lie-Wen Chen, PRL 103, (2009)
EOS including the Yukawa contribution
Mass-shedding limit

30. The rise, fall and reappearing of the 5th force --- evidence of a new boson of 17 MeV
Ep=1.1 MeV

31. The rise, fall and reappearing of the 5th force --- evidence of a new boson of 17 MeV

32. Supporting theories ---- nothing seems to be wrong with a new boson of 17 MeV (interaction range of 12 fm, it does not affect properties of nuclei but neutron stars)

33. Can nuclear physics explain the anomaly observed in the internal pair production in the Beryllium-8 nucleus?
Xilin Zhang, Gerald A. Miller, (Submitted on 14 Mar 2017), arXiv: [nucl-th]
Particle Physics Models for the 17 MeV Anomaly in Beryllium Nuclear Decays
Jonathan L. Feng, Bartosz Fornal, Iftah Galon, Susan Gardner, Jordan Smolinsky, Tim M. P. Tait, Philip Tanedo, Phys. Rev. D 95, (2017)
On going and planed new experiments

35. Summary
The study of nuclear symmetry energy and its astrophysical impacts is Exciting!
Symmetry energy at supra-saturation densities is the most uncertain part of the EOS of dense neutron-rich nucleonic matter
Symmetry energy has significant effects on
terrestrial experiments and astrophysical
observables

36. May the 5th Force be with You!
Symmetry Energy
neutron
SRC
Hard
protonn or
Soft?
Dark Matter or Modified Gravity?

37. What is the high-density symmetry energy?
Indications of experiments (model dependent)
P. Russotto et al. (ASY-EOS Collaboration),
Phys. Rev. C94, (2016).
Z.G. Xiao et al, based on GSI/FOPI data
Phys. Rev. Lett. 102, (2009).

38. Effects of isospin-dependent SRC on the kinetic symmetry energy of quasi-nucleons
Chang Xu and Bao-An Li, arXiv:
Chang Xu, Ang Li and Bao-An Li,
JPCS 420, (2013).
While the Fermi momentum for PNM
is higher than that for SNM at the same density in the mean-field models,
if more than 15% nucleons are in the
high-momentum tail of SNM due to the
tensor force for n-p T=0 channel, the kinetic symmetry energy becomes negative
Tensor force +
Repulsive core
Repulsive
core only
Reduced Density

39. Pion ratio probe of symmetry energy at supra-normal densitiesGC
Coefficients2
Pion ratio probe of symmetry energy at supra-normal densities 39

40. Average kinetic energies of neutrons and protons in nuclei
(1) Light nuclei: Predictions of the Variational Many-Body theory with AV18+UX interaction
R. B.Wiringa, R. Schiavilla, S. C. Pieper, and J. Carlson, PRC 89, (2014).
(2) Low-order correlation operator aproximation by Jan Ryckebusch et all.
With SRC
no SRC
J. Phys G. 42, ( 2015).

41. Phenomenological nucleon momentum distribution n(k) including SRC effects guided by microscopic theories and experimental findings
The n(k) is not directly measurable, but some of its features are observable.
B.J. Cai and B.A. Li, PRC92, (R) (2015); PRC93, (2016).
Isospin-dependent
depletion of Fermi sea
Isospin-dependent
high momentum tail
All parameters are fixed by
Jlab data: HMT in SNM=25%, 1.5% in PNM,
Contact C for SNM from deuteron wavefunction
(3) Contact C in PNM from microscopic theories
All parameters are assumed to have a linear
dependence on isospin asymmetry as indicated
by SCGF and BHF calculations

42. The high-momentum tail in deuteron scales as 1/K4
O. Hen, L. B. Weinstein, E. Piasetzky, G. A. Miller, M. M. Sargsian and Y. Sagi, PRC92, (2015).
VMB calculations
VMB calculations
Ultracold
6Li and 40K
Ultracold
6Li and 40K
=k/kF
K’=K/KF
Stewart et al. PRL 104, (2010)
Kuhnle et al. PRL 105, (2010)

43. EOS and contact C of pure neutron matter
B.J. Cai and B.A. Li,
PRC92, (R) (2015).
density
The contact C of PNM is derived from its EOS
Using Tan’s adiabatic sweep theorem
n(k)=C/K4
S. Tan, Annals of Physics 323 (2008)

44. Proton fraction in neutron stars at β-equilibrium

45. The x parameter is introduced to mimic
Symmetry energy and single nucleon potential MDI used in the IBUU04 transport model
The x parameter is introduced to mimic
various predictions on the symmetry energy
by different microscopic nuclear many-body
theories using different effective interactions.
It is the coefficient of the 3-body force term
stiff ρ
soft
Default: Gogny force
Density ρ/ρ0
Potential energy density
Single nucleon potential within the HF approach using a modified Gogny force:
C.B. Das, S. Das Gupta, C. Gale and B.A. Li, PRC 67, (2003).
B.A. Li, C.B. Das, S. Das Gupta and C. Gale, PRC 69, ; NPA 735, 563 (2004).

46. N+π NN NΔ Probing the symmetry energy at high densities
π -/π + in heavy-ion collisions
Neutron-proton differential flow & n/p ratio in heavy-ion coll.
Neutrino flux of supernova explosions
Strength and frequency of gravitational waves
Where does the Esym information get in, get out or get lost in pion production?
*1 Isospin fractionation, i.e., the nn/pp ratio at high density is determined by the Esym(ρ)
*2 The isovector potential for Delta resonance is completely unknown
*5.Pion mean-field (dispersion relation), S and/or P wave
and their isospin dependence are poorly known, existing
studies are inconclusive.
How does the pion mean-field affect
the Delta production threshold?
Other final state interactions?
NN NΔ
3. How to define the isospin
asymmetry of the N+Δ matter?
4. The lifetime of Δ controls the
effects of its mean field on pions
N+π

47. Bao-Jun Cai, F. J. Fattoyev, Bao-An Li and W. G
Bao-Jun Cai, F. J. Fattoyev, Bao-An Li and W.G. Newton, PRC 92, (2015).
Xρ=1

48. In the pi-N molecule model, assuming pions have no mean-field,the Delta isovector potential is linked to the nucleon isovector potential
Bao-An Li, PRL88, (2002) and NPA 365 (2002)
To study effects of the completely unknown Delta isovector potential, multiply the
above with a Delta-probing-factor:

50. Delta isovector potential has NO effect on the high-energy spectrum!

51. WHY? Delta lifetime and mass distribution in Au+Au collisions at b=0
In-medium Delta width, H. Lenske et al.
Delta lifetime and mass distribution
in Au+Au collisions at b=0
WHY?
High-mass Delta produced in energetic collisions decays too quickly
to feel any mean-field effect! Only long-lived low mass Deltas have the
time to feel mean-field effects.

52. Bao-An Li, Phys. Rev. C 92, (2015)
Energetic pions are still sensitive to the
Esym(ρ) in deeply sub-threshold collisions

53. Upper limits on the strength α and range λ of the Yukawa term
Torsion
balance
Upper limits on the strength α and range λ of the Yukawa term
M.I. Krivoruchenko et al., PRD 79, (2009)
E.G. Adelberger et al., PRL 98, (2007)
D.J. Kapner et al., PRL 98, (2007)
Serge Reynaud et al., Int. J. Mod. Phys. A20, (2005)

54. Constraining the energy dependence of symmetry potential at saturation density
RMF
Isovector optical potential from
nucleon-nucleus scattering
J. W. Holt, N. Kaiser, G. A. Miller
Phys. Rev. C 93, (2016)

55. Constraining the radii of neutron stars
Bao-An Li and Andrew W. Steiner, Phys. Lett. B642, 436 (2006) ● .
Nuclear limits
APR: K0=269 MeV.
The same incompressibility for symmetric nuclear matter of K0=211 MeV for x=0, -1, and -2

56. What to we add and modify?
Extended+ Thomas-Fermi Approximation (ETF+) considering the isospin-dependent SRC and effectively ħ4 and higher order terms
What to we add and modify?
Mimic effects of ħ4 and higher terms
H.Krivine and J. Treiner, PLB 88, 212 (1979)
X. Campi and S. Stringari, NPA 337, 313 (1980)
M. Barranco, M. Pi and X. Vinas, PLB124, 131 (1983)
Isospin-dependent SRC constrained by data
ΦJ=1 for sharp Fermi spheres, it is larger than 1
with SRC-induced high momentum tails
Kinetic energy density in infinite matter

57. Confirmation by Microscopic Many-Body Theories
1. Isaac Vidana, Artur Polls, Constanca Providencia
PRC84, (R) (2011)
Brueckner--Hartree--Fock approach using the Argonne V18 potential
plus the Urbana IX three-body force
2. Arianna Carbone, Artur Polls, Arnau Rios, EPL 97, (2012)
Self-Consistent Green’s Function Approach with Argonne Av18, CDBonn, Nij1, N3LO interactions
3. Alessandro Lovato, Omar Benhar et al., extracted from results already published in
Phys. Rev. C83:054003,2011
Using Argonne V’6 interaction
4. A. Rios, A. Polls, W. H. Dickhoff
PRC 89, (2014).
Ladder Self-Consistent Green Function
Fermi gas
They all included the tensor force and
many-body correlations using different techniques
with tensor correlation

58. Size of the high-momentum tail
data
Predictions are model dependent:
10-25%
Experimental indication:
P2N(∞)≈20-30% in symmetric matter

59. Dominance of the isosinglet (T=0) interaction
Symmetry energy
At saturation density
Paris potential
Self-consistent
Green’s function
BHF
I. Bombaci and U. Lombardo PRC 44, 1892 (1991)
PRC68, (2003)