Symmetry Energy and Neutron-Proton Effective Mass Splitting
in Neutron-Rich Nucleonic Matter
Bao-An Li
Texas A&M University-Commerce
Collaborators:
F. Fattoyev, J. Hooker, W. Newton and Jun Xu, TAMU-Commerce
Andrew Steiner, INT, University of Washington
Che Ming Ko, Texas A&M University
Lie-Wen Chen, Xiao-Hua Li and Bao-Jun Chai, Shanghai Jiao Tong University
Chang Xu, Nanjing University
Xiao Han and Gao-Feng Wei, Xi’an Jiao Tong University
Outline:
1. Why am I here?
Connection with the PREX-CREX experiments
2. Why is the symmetry energy is still so uncertain even at saturation density?
a) Decomposition of the symmetry energy Esym (ρ0) and its slope L according to
the Hugenholtz-Van Hove (HVH) theorem
b) An attempt to find out the most uncertain components of L from global
neutron-nucleus optical potentials
3. What can we say about the neutron-proton effective mass splitting if both
the Esym (ρ0) and L are well determined by PREX-CREX experiments?
Constraints from both isospin diffusion and n-skin in 208Pb
Isospin diffusion data:
Transport model calculations
B.A. Li and L.W. Chen, PRC72, 064611 (05)
ρ
M.B. Tsang et al., PRL. 92, 062701 (2004);
T.X. Liu et al., PRC 76, 034603 (2007)
112Sn+124Sn
ρρ
J.R. Stone
implication
PREX?
Hartree-Fock calculations
A. Steiner and B.A. Li, PRC72, 041601 (05)
Neutron-skin from nuclear scattering: V.E. Starodubsky and N.M. Hintz, PRC 49, 2118 (1994);
B.C. Clark, L.J. Kerr and S. Hama, PRC 67, 054605 (2003)
Nuclear constraining the radii of neutron stars
Bao-An Li and Andrew W. Steiner, Phys. Lett. B642, 436 (2006)
●
APR: K0=269 MeV.
The same incompressibility for symmetric nuclear
matter of K0=211 MeV for x=0, -1, and -2
.
Astronomers discover the fastest-spinning neutron-star
Science 311, 1901 (2006).
W.G. Newton, talk at NN2012
Chen, Ko and Li, PRL (2005)
Upper limit
Agrawal et al.
PRL (2012)
Lower limit
Time Line
Thanks to the hard work of many of you
Community averages with physically meaningful error bars?
E sym (  0 )  31.6 M eV and L(  0 )  62.4 M eV
albeit w ithout physically m eaningful err or bars
Why is the Esym(ρ) is still so uncertain
even at saturation density?
• Is there a general principle at some level,
independent of the interaction and many-body theory,
telling us what determines the Esym(ρ0) and L?
• If possible, how to constrain separately each component
of Esym(ρ0) and L?
Decomposition of the Esym and L according to the
Hugenholtz-Van Hove (HVH) theorem
1) For a 1-component system
at saturation density, P=0, then
2) For a 2-components system
at arbitrary density
M icrophysics governing the E
sym
(  ) and L(  ) according to the H V H theorem
The Lane potential
Higher order in isospin asymmetry
C. Xu, B.A. Li, L.W. Chen and C.M. Ko, NPA 865, 1 (2011)
Relationship between the symmetry energy and the mean-field potentials
Lane potential
Both U0 (ρ,k) and Usym(ρ,k) are density and momentum dependent
kinetic
isoscalar
isovector
Symmetry energy
Isoscarlar effective mass
Using K-matrix theory, the conclusion is independent of the interaction
Gogny HF
SHF
Usym,1 (ρ,p) in several models
R. Chen et al., PRC 85, 024305 (2012).
Usym,1 (ρ,p) in several models
Gogny
Usym,2 (ρ,p) in several models
Gogny
Gogny
Usym,2 (ρ,p) in several models
Providing a boundary condition on Usym,1 (ρ,p) and Usym,2 (ρ,p) at
saturation density from global neutron-nucleus scattering optical
potentials using the latest and most complete data base for n+A
elastic angular distributions
Xiao-Hua Li et al., PLB (2103) in press, arXiv:1301.3256
Providing a boundary condition on Usym,1 (ρ,p) and Usym,2 (ρ,p) at
saturation density from global neutron-nucleus scattering optical
potentials using the latest data base for n+A elastic angular
distributions
Xiao-Hua Li et al., PLB (2103) in press, arXiv:1301.3256
Constraints on Ln
from n+A elastic scatterings
Applying the constraints from neutron-nucleus scattering
Prediction for CREX
CREX
Time Line
W hat can w e learn if both E sym (  0 ) and L(  0 ) are w ell determ ined?
At the mean-field level:
m
*
m
m m
*
n
m
*
p
L (  0 )  3 E sym (  0 )  1 2
(  0 )  
EF (0 )
m
m
*
(  0 )  0.7  0.05
(0 )
1

1  2[
m
m
*
(  0 )  1]
Constraining the n-p effective mass splitting
F o r E sym (  0 )= 3 1 M eV , if L  8 5 M eV th en m n  m
*
*
p
Symmetry energy and single nucleon potential MDI used in the IBUU04 transport model
ρ
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
soft
Default: Gogny force
Density ρ/ρ0
Potential energy density
Single nucleon potential within the HF approach using a modified Gogny force:
U (  ,  , p ,  , x )  Au ( x )

2C
0
,

 , '  
d
1
2
3
p '
 '

 
 Al ( x )
 B(
) (1  x 
0
0
0
f  ( r , p ')
1  ( p  p ')
2
, Al ( x )   1 2 1 
/ 
2

2 Bx
 1
2 C
0
, '

d
3
p '
2
)  8 x
  1

  1  0
B
f  ' ( r , p ')
1  ( p  p ')
, Au ( x )   9 6 
2 Bx
 1
2
/ 
2
, K 0  2 1 1M e V
C.B. Das, S. Das Gupta, C. Gale and B.A. Li, PRC 67, 034611 (2003).
B.A. Li, C.B. Das, S. Das Gupta and C. Gale, PRC 69, 034614; NPA 735, 563 (2004).
'
Usym,1 (ρ,p) and Usym,2 (ρ,p) in the MDI
potential used in IBUU04 transport model
 E
What is the 1Equation
of State ofE (neutron-rich
nucleonic matter?
    E (  ) 
 E ( )
)
2
sym
2 
pure neutron matter
2
symmetry energy
symmetric nuclear matter
Isospin asymmetry δ
12

 n   p 

E (  n ,  p )  E 0 (  n   p )  E s ym (  ) 
      



12
12
E (  n ,  p )
Energy per nucleon in symmetric matter
18
18
3
Energy per nucleon in asymmetric matter
N ormal density of nuclear matter  0  2.7  10 g/cm
14
3
density
0
Isospin asymmetry
ρ=ρn+ρp
Essentially , all models and interactions available have been used to predict the Esym (ρ)
Symmetry energy (MeV)
Examples
BHF
Greens function
Variational
many-body
Density
A.E. L. Dieperink et al., Phys. Rev. C68 (2003) 064307
More examples:
Skyrme Hartree-Fock and Relativistic Mean-Field predictions
ρ
23 RMF
models
Density
L.W. Chen, C.M. Ko and B.A. Li, Phys. Rev. C72, 064309 (2005); C76, 054316 (2007).
Among interesting questions regarding nuclear symmetry energy:
• Why
is the density dependence of symmetry energy so uncertain especially at
high densities?
• What are the major underlying physics determining the symmetry energy?
• What is the symmetry free-energy at finite temperature?
• What is the EOS of low-density clustered matter? How does it depend on the
isospin asymmetry of the system? Linearly or quadratically? Can we still define
a symmetry energy for clustered matter? What are the effects of n-p pairing on
low density EOS?
• How to constrain the symmetry energy at various densities using terrestrial
nuclear experiments and/or astrophysics observations?
Current Situation:
• Many experimental probes predicted
• Major progress made in constraining the symmetry energy around and below ρ0
• Interesting features found about the EOS of low density n-rich clustered matter
• Several sensitive astrophysical observables identified/used to constrain Esym
• High-density behavior of symmetry energy remains contraversial
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
Many other astrophysical observables, e.g., radii, core-crust transition density,
cooling rate, oscillation frequencies and damping rate, etc of neutron stars
Neutron stars as a natural testing ground of grand unification theories of
fundamental forces?
Connecting Quarks with the Cosmos: Eleven Science Questions for the New
Century, Committee on the Physics of the Universe, National Research Council
weak
E&M
Nuclear force
Stable neutron star
@
ϐ-equilibrium
• What is the dark matter?
• What is the nature of the dark energy?
• How did the universe begin?
• What is gravity?
• 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?
• Are there additional spacetime dimensions?
• How were the elements from iron to uranium made?
• Is a new theory of matter and light needed at the
highest energies?
Requiring simultaneous solutions in both gravity and strong interaction!
Grand Unified Solutions of Fundamental Problems in Nature!
Size of the pasta phase and symmetry energy
W.G. Newton, M. Gearheart and Bao-An Li
ThThe Astrophysical Journal (2012) in press.
Torsional crust oscillations
M. Gearheart, W.G. Newton, J. Hooker and Bao-An Li,
Monthly Notices of the Royal Astronomical Society, 418, 2343 (2011).
The proton fraction x at ß-equilibrium in proto-neutron stars is determined by
x  0 .0 4 8[ E sym (  ) / E sym (  0 )] (  /  0 )(1  2 x )
3
3
The critical proton fraction for direct URCA process to happen is Xp=0.14 for npeμ
matter obtained from energy-momentum conservation on the proton Fermi surface
Slow cooling: modified URCA:

n  (n, p )  p  (n, p )  e   e
E(ρ,δ)= E(ρ,0)+Esym(ρ)δ2

p  (n, p )  n  (n, p )  e   e
Consequence: long surface
thermal emission up to a few
million years
Faster cooling by 4 to 5 orders of
magnitude: direct URCA

n  p  e
 e
p  n  e


Isospin
separation
instability
Direct URCA
kaon condensation allowed
e
Neutron bubbles formation
transition to Λ-matter
B.A. Li, Nucl. Phys. A708, 365 (2002).
Z.G. Xiao et al, Phys. Rev. Lett. 102 (2009) 062502
Bao-An Li, Phys. Rev. Lett. 88 (2002) 192701
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)
A degeneracy: matter content (EOS) and gravity
in determining properties of neutron stars
Simon DeDeo, Dimitrios Psaltis Phys. Rev. Lett. 90 (2003) 141101
Dimitrios Psaltis, Living Reviews in Relativity, 11, 9 (2008)
• Neutron stars are among the densest
objects with the strongest gravity
Gravity
??????
• General Relativity (GR) may break down at
strong-field limit
??
Nuclear pressure
Uncertain range
of EOS
• There is no fundamental reason to choose
Einstein’s GR over alternative gravity theories
In GR,
Tolman-Oppenheimer-Volkoff (TOV) equation:
a condition for hydrodynamical equilibrium
Do we really know gravity at short distance?
Not at all!
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)
In terms of the gravitational potential
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
A low-field limit of several alternative gravity theories
Yasunori Fujii, Nature 234, 5-7 (1971); G.W. Gibbons and B.F. Whiting, Nature 291, 636 - 638 (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, 101301 (2004).
Supersoft Symmetry Energy Encountering Non-Newtonian Gravity in Neutron Stars
De-Hua Wen, Bao-An Li and Lie-Wen Chen, PRL 103, 211102 (2009)
EOS including the Yukawa contribution
g
2
/
2
Promising Probes of the Esym(ρ) in Nuclear Reactions
At sub-saturation densities
 Global nucleon optical potentials from n/p-nucleus and (p,n) reactions
 Thickness of n-skin in 208Pb measured using various approaches
and sizes of n-skins of unstable nuclei from total reaction cross sections
 n/p ratio of FAST, pre-equilibrium nucleons
 Isospin fractionation and isoscaling in nuclear multifragmentation
 Isospin diffusion/transport
 Neutron-proton differential flow
 Neutron-proton correlation functions at low relative momenta
 t/3He ratio and their differential flow
Towards supra-saturation densities
 π -/π + ratio, K+/K0 ?
 Neutron-proton differential transverse flow
 n/p ratio of squeezed-out nucleons perpendicular to the reaction plane
 Nucleon elliptical flow at high transverse momentum
 t-3He differential and difference transverse flow
(1) Correlations of multi-observable are important
(2) Detecting neutrons simultaneously with charged particles is critical
B.A. Li, L.W. Chen and C.M. Ko, Physics Reports 464, 113 (2008)
Probing the symmetry energy at supra-saturation densities Symmetry energy
E (  ,  )  E (  , 0 )  E sy m (  ) 
2
Central density
density
π-/ π+ probe of dense matter
Stiff Esym
n/p ?
n/p ratio at supra-normal densities
Circumstantial Evidence for a Super-soft Symmetry Energy at Supra-saturation Densities
Data:
W. Reisdorf et al.
NPA781 (2007) 459
Calculations: IQMD and IBUU04
A super-soft nuclear symmetry energy is favored by the FOPI data!!!
Z.G. Xiao, B.A. Li, L.W. Chen, G.C. Yong and M. Zhang, Phys. Rev. Lett. 102 (2009) 062502
Can the symmetry energy become negative at high densities?
Yes, it happens when the tensor force due to rho 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)
x  0.048[ E sym (  ) / E sym (  0 )] (  /  0 )(1  2 x )
3
Super-Soft
3
Lunch conversation with Prof. Dr. Dieter Hilscher on a sunny day in 1993 at HMI in Berlin
Ratio of neutrons in the two
reaction systems
neutrons
protons
Mechanism for enhanced n/p ratio of pre-equilibrium nucleons
The first PRL paper connecting the symmetry energy
with heavy-ion reactions