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-CommerceAndrew Steiner, INT, University of WashingtonChe Ming Ko, Texas A&M UniversityLie-Wen Chen, Xiao-Hua Li and Bao-Jun Chai, Shanghai Jiao Tong UniversityChang Xu, Nanjing UniversityXiao Han and Gao-Feng Wei, Xi’an Jiao Tong University

Symmetry Energy and Neutron-Proton Effective Mass Splitting

in Neutron-Rich Nucleonic Matter

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

ρ ρ

ρ

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)

Isospin diffusion data:M.B. Tsang et al., PRL. 92, 062701 (2004); T.X. Liu et al., PRC 76, 034603 (2007)

Hartree-Fock calculationsA. Steiner and B.A. Li, PRC72, 041601 (05)

PREX?

J.R. Stone

implication

Transport model calculationsB.A. Li and L.W. Chen, PRC72, 064611 (05)

112Sn+124Sn

Nuclear constraining the radii of neutron stars

APR: K0=269 MeV.The same incompressibility for symmetric nuclear matter of K0=211 MeV for x=0, -1, and -2

Bao-An Li and Andrew W. Steiner, Phys. Lett. B642, 436 (2006)

Nuclear limits

● .

Astronomers discover the fastest-spinning neutron-star

Science 311, 1901 (2006).

Chen, Ko and Li, PRL (2005)

Agrawal et al.PRL (2012)

Time Line

W.G. Newton, talk at NN2012

Upper limit

Lower limit

Thanks to the hard work of many of you

Community averages with physically meaningful error bars?

0 0( ) 31.6 MeV and L( ) 62.4 MeV

albeit without physically meaningful error barssymE

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

Microphysics governing the E ( ) and L( ) according to the HVH theorem sym

C. Xu, B.A. Li, L.W. Chen and C.M. Ko, NPA 865, 1 (2011)

The Lane potentialHigher order in isospin asymmetry

Relationship between the symmetry energy and the mean-field potentials

Lane potential

Symmetry energy

Isoscarlar effective mass

kinetic isoscalar isovector

Using K-matrix theory, the conclusion is independent of the interaction

Both U0 (ρ,k) and Usym(ρ,k) are density and momentum dependent

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

GognyGogny

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 distributionsXiao-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

Time Line

Prediction for CREX

CREX

2016

±2

* * 0 0 0*

00

0*

( ) 3 ( ) 12 ( )1( )

( ) 1 2[ ( ) 1]

symn p

F

mL Em m mmm Em

0 0What can we learn if both E ( ) and L( ) are well determined?sym

*

0( ) 0.7 0.05mm

At the mean-field level:

Constraining the n-p effective mass splitting

* *0For E ( )=31 MeV, if L 85 MeV then m msym n p

Symmetry energy and single nucleon potential MDI used in the IBUU04 transport model

12'

'0 0 0 0

, 3 , ' 3 '2 2 2 2

0

0

0

1 2 2,

( , , , , ) ( ) ( ) ( ) (1 ) 81

2 2( , ') ( , ')' '1 ( '

' , ( ) 121 , ( ) 96 ,

) / 1 ( ') /

2112 1 1

u l

l u

BU p A A B

C Cf r p f r pd p d pp p

B BA A

x x x x x

x K MeVx

p

xx

p

ρ

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).

softsoft

stiff

stiff

Single nucleon potential within the HF approach using a modified Single nucleon potential within the HF approach using a modified Gogny force:Gogny force:

Density ρ/ρ0

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

Default: Gogny force

Potential energy density

Usym,1 (ρ,p) and Usym,2 (ρ,p) in the MDI potential used in IBUU04 transport

model

What is the Equation of State of neutron-rich nucleonic matter?

18 18

12

12

12

3

0 )) (, (( ) sn ymp

nn

p pE E E

( , )n pE

symmetry energy

ρ=ρn+ρp0

1

density

Isospin asymmetry

Symmetric matter

ρ n=ρ p

Energy per nucleon in symmetric matter

Energy per nucleon in asymmetric matter

δIsospin asymmetry

???

2

pure neutron matter symmetric nuclear matter2

1( ) ( ) ( )2sym

EE E E

The axis of new opportunities

???

???

???

14 30Normal density of nuclear matter 2.7 10 g/cm

Examples

Sym

met

ry e

nerg

y (M

eV)

Density

Effective field theory

(Kaiser et al.) DBHFRMF BHF

Greens function

Variational many-body

A.E. L. Dieperink et al., Phys. Rev. C68 (2003) 064307

Essentially , all models and interactions available have been used to predict the Esym (ρ)

More examples: Skyrme Hartree-Fock and Relativistic Mean-Field predictions

23 RMFmodels

ρ

L.W. Chen, C.M. Ko and B.A. Li, Phys. Rev. C72, 064309 (2005); C76, 054316 (2007).

Density

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 LIn 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?

Nuclear force

weakE&M

Stable neutron star @ ϐ-equilibrium

Requiring simultaneous solutions in both gravity and strong interaction!Grand Unified Solutions of Fundamental Problems in Nature!

Connecting Quarks with the Cosmos: Eleven Science Questions for the New Century, Committee on the Physics of the Universe, National Research Council

• 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?

Size of the pasta phase and symmetry energy

W.G. Newton, M. Gearheart and Bao-An LiThThe Astrophysical Journal (2012) in press.

Pasta

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

3 30 0(0.048[ / ( )] ( / )() 1 2 )symsymE Ex x

The critical proton fraction for direct URCA process to happen is XThe critical proton fraction for direct URCA process to happen is Xpp=0.14 for npe=0.14 for npeμμ matter obtained from energy-momentum conservation on the proton Fermi surfacematter obtained from energy-momentum conservation on the proton Fermi surface

Slow cooling: modified URCA:Slow cooling: modified URCA:( , ) ( , )

( , ) ( , )e

e

n n p p n p e

p n p n n p e

Faster cooling by 4 to 5 orders of Faster cooling by 4 to 5 orders of magnitude: direct URCAmagnitude: direct URCA

e

e

n p e

p n e

Consequence: long surface Consequence: long surface thermal emission up to a few thermal emission up to a few million yearsmillion years

B.A. Li, Nucl. Phys. A708, 365 (2002).

Direct URCA kaon condensation allowed

Neutron bubbles formationtransition to Λ-matter

Isospinseparationinstability

E(ρ,δ)= E(ρ,0)+Esym(ρ)δ2

Bao-An Li, Phys. Rev. Lett. 88 (2002) 192701

Z.G. Xiao et al, Phys. Rev. Lett. 102 (2009) 062502

TOV equation: a condition at hydrodynamical equilibrium

Gravity

Nuclear pressure

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

For npe matter

dP/dρ<0 if E’sym is big and negative (super-soft)P. Danielewicz, R. Lacey and W.G. Lynch, Science 298, 1592 (2002))

A degeneracy: matter content (EOS) and gravity

in determining properties of neutron starsSimon DeDeo, Dimitrios Psaltis Phys. Rev. Lett. 90 (2003) 141101

• Neutron stars are among the densest objects with the strongest gravity

• General Relativity (GR) may break down at strong-field limit

• There is no fundamental reason to choose Einstein’s GR over alternative gravity theories

Uncertain range of EOS

Gravity

Nuclear pressure

In GR, Tolman-Oppenheimer-Volkoff (TOV) equation: a condition for hydrodynamical equilibrium

Dimitrios Psaltis, Living Reviews in Relativity, 11, 9 (2008)

??

??????

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

In terms of the gravitational potentialYukawa 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

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)

Yasunori Fujii, Nature 234, 5-7 (1971); G.W. Gibbons and B.F. Whiting, Nature 291, 636 - 638 (1981)

Do we really know gravity at short distance? Not at all!

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).

A low-field limit of several alternative gravity theories

Lower limit to support neutrons stars with a super-soft

symmetry energy

Upper limits

22 / g

EOS including the Yukawa contribution

Supersoft Symmetry Energy Encountering Non-Newtonian Gravity in Neutron StarsDe-Hua Wen, Bao-An Li and Lie-Wen Chen, PRL 103, 211102 (2009)

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

SoftSoft

Stiff

Stiff

Soft E sym

Stiff Esym

density

Symmetry energy

n/p ratio at supra-normal densities

Central density

π-/ π+ probe of dense matter

2( , ) ( ,0) ( )symE E E

n/p ?

Circumstantial Evidence for a Super-soft Symmetry Energy at Supra-saturation Densities

Z.G. Xiao, B.A. Li, L.W. Chen, G.C. Yong and M. Zhang, Phys. Rev. Lett. 102 (2009) 062502

A super-soft nuclear symmetry energy is favored by the FOPI data!!!

W. Reisdorf et al. NPA781 (2007) 459 Data:

Calculations: IQMD and IBUU04

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 dominatesAt 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

3 30 0(0.048[ / ( )] ( / )() 1 2 )symsymE Ex x

Soft

Super-Soft

M. Kutschera et al., Acta Physica Polonica B37 (2006)

Lunch conversation with Prof. Dr. Dieter Hilscher on a sunny day in 1993 at HMI in Berlin

neutrons

protons

Ratio of neutrons in the tworeaction systems

The first PRL paper connecting the symmetry energywith heavy-ion reactions

Mechanism for enhanced n/p ratio of pre-equilibrium nucleons