Probing the symmetry energy at high densities

Outline:

• Why is the symmetry energy so uncertain at supra-saturation densities? Can the symmetry energy become super-soft or even negative at high densities?

--- Some observations by a non-expert of many-body theories

• Indications of a super-soft symmetry energy at supra-saturation densities from transport model analyses of the FOPI/GSI experimental data on pion production

• Can neutron stars be stable with a super-soft or negative symmetry energy at supra-saturation densities?

--- Some observations and attempts by a non-expert of astrophysics

& collaborators:De-Hua Wen and Chang Xu, Texas A&M University-CommerceLie-Wen Chen, Shanghai Jiao-Tung UniversityZhigang Xiao and Ming Zhang, Tsinghua University, ChinaGao-Chan Yong, Institute of Modern Physics, China

Bao-An Li

Esym (ρ) predicted by microscopic many-body theories

Sym

met

ry e

ner

gy

(MeV

)

Density

Effective field theory

DBHFRMF

BHF

Greens function

Variational

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

( ) ( ) ( )sym pureneutron nuclearE E E

Why is the symmetry energy so uncertain especially at high densities?

Based on the Fermi gas model (Ch. 6) and properties of nuclear matter (Ch. 8) of the textbook: Structure of the nucleus by M.A.

Preston and R.K. Bhaduri

Kinetic Isoscalar Isovector

Correlation function

Nuclear Structure based on Correlated Realistic Nucleon-Nucleon Potentials R. Roth, T. Neff, H. Hergert, H. Feldmeier, Nucl. Phys. A745 (2004) 3-33

Tensor correlations in the Unitary Correlation Operator Method Thomas Neff, Hans Feldmeier, Nucl. Phys. A713 (2003) 311-371

Correlation function

At saturation densityUsing Paris potential

2

pure neutron matter symmetric nuclear matter2

1( ) ( ) ( )

2sym

EE E E

I. Bombaci and U. LombardoPRC 44, 1892 (1991)

Using the Reid93 interaction

PRC68, 064307 (2003)

The most important contributions of nuclear force

Lecture notes of R. Machleidt at 2005 RIKEN summer school

In-medium properties of the short-range tensor force

G.E. Brown and Mannque Rho, PLB 237, 3 (1990)

G.E. Brown and Mannque Rho, PRL 66, 2720 (1991), Phys Rep. 396, 1 (2004)

Gogny central force

Correlation function in Fermi gas

Gale-Bertsch-Das Gupta’s parameterization of the K-dependent part of the isoscalar potential

Can the symmetry energy becomes negative at high densities?Yes, when the short-range repulsive tensor force in isosinglet n-p pairs dominates At high densities, the energy of pure neutron matter becomes lower than symmetric matter leading to negative symmetry energy

This

In phenomenological models where there is no explicit tensor force, the symmetry energystarts decreasing when the Lane potential becomes negative

Gogny-Hartree-Fock

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).

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

Why is the symmetry energy so uncertain especially at high densities?

Some observations of a non-expert:

Poor knowledge on • short-range NN correlations• in-medium properties of the short-range tensor force• --------

Can the symmetry energy becomes super-soft or even negative at high densities?

• There seems to be NO first principle forbidding it• It happens when the repulsive short-range tensor force due to the ρ-meson exchange in the n-p singlet channel dominates. • Using a 20% reduction of the ρ-meson mass as required to reproduce the half-lift of 14C, the symmetry energy becomes negative above about 4ρ0

• The Lane potential Un-Up flips sign when the Esym starts decreasing with density

Momentum and density dependence of the symmetry (isovector) potential

Lane potential extracted from n/p-nucleus scatterings and (p,n) charge exchange reactions provides only a constraint at ρ0:

Lane 1

1

kin

( ) / 2 ,

28 6 MeV, 0.1 0.2

for E 100 MeV

n p R kin

R

U U U V E

V

P.E. Hodgson, The Nucleon Optical Model, World P.E. Hodgson, The Nucleon Optical Model, World Scientific, 1994 Scientific, 1994

G.W. Hoffmann and W.R. Coker, PRL, 29, 227 (1972).G.W. Hoffmann and W.R. Coker, PRL, 29, 227 (1972).

G.R. Satchler, Isospin Dependence of Optical Model G.R. Satchler, Isospin Dependence of Optical Model Potentials, Potentials, in Isospin in Nuclear Physics, in Isospin in Nuclear Physics, D.H. Wilkinson (ed.), (North-Holland, Amsterdam,1969)D.H. Wilkinson (ed.), (North-Holland, Amsterdam,1969)

/n p isoscalar LaneU U U

Isospin fractionation in heavy-ion reactions

lowlow (high)(high) density region is more neutron-density region is more neutron-rich withrich with stiffstiff (soft)(soft) symmetry energysymmetry energy

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

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

Pion ratio probe of symmetry energy

at supra-normal densities

0

nn 0 1 5 a) Δ(1232) resonance model pp 5 1 0 in first chance NN scatterings: np(pn) 1 4 1 (negelect rescattering and reabsorption)

2

2

2

)(5

5ZN

NZZ

NZN

R. Stock, Phys. Rep. 135 (1986) 259. b) Thermal model: (G.F. Bertsch, Nature 283 (1980) 281; A. Bonasera and G.F. Bertsch, PLB195 (1987) 521)

exp[2( ) / ]n p kT

H.R. Jaqaman, A.Z. Mekjian and L. Zamick, PRC (1983) 2782.

c) Transport models (more realistic approach): Bao-An Li, Phys. Rev. Lett. 88 (2002) 192701, and several papers by others

31 1( ) {ln ( ) ( )}

2

m mn p mnn p asy asy Coul m T n p

mp

mV V V kT b

m

GCCoefficients2

Circumstantial evidence for a super-soft symmetry energy at high densities

400 MeV/A

FOPI/GSI data on pion productionWilly Reisdorf et al., NPA781 (2007) 459 Transport model analysisZ. Xiao et al., PRL 102, 062502 (2009)

Au+Au

Is the super-soft symmetry energy

“unpleasant” or “unphysical”?Unpleasant !E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer, NPA627, 710 (1997); NPA635, 231 (1998).Repeated by several others in other papers

Unphysical !

Quoted by several people in a number of papers

Crazy!

TOV equationP(r+dr)

P(r)

Gravity

Nuclear pressure

Why ?The only reason stated is that “ if the symmetry

energy is too soft neutron stars will then collapse while they do exist in nature”

For npe matter

Do we really know gravity at the Fermi distance?

``It's remarkable that gravity, despite being the first to be discovered, is by far the most poorly understood force," says Nima Arkani-Hamed of Harvard University

Roland Pease, Nature 411, 986-988 (28 June 2001)

Annu. Rev. Nucl. Part. Sci. 2003. 53:77–121

Extra dimension at short length or a new Boson?

String theorists have published TONS of papers on the extra dimension

In terms of the gravitational potential Repulsive 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

The neutral spin-1 gauge boson U is a candidate, it can mediate the interaction among dark matter particles, e.g., Pierre Fayet, PLB675, 267 (2009), C. Boehm, D. Hooper, J. Silk, M. Casse and J. Paul, PRL, 92, 101301 (2004).

Arkani-Hamed, N., Dimopoulos, S. & Dvali, G. Phys Lett. B 429, 263–272 (1998). 

J.C. Long et al., Nature 421, 922-925 (2003);

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

C.D. Hoyle, Nature 421, 899–900 (2003)

Experimental constraints on the strength α and range λ of the

Yukawa term M.I. Krivoruchenko et al., PRD 79, 125023 (2009)E.G. Adelberger et al., PRL 98, 131104 (2007)D.J. Kapner et al., PRL 98, 021101 (2007)

Serge Reynaud et al., Int. J. Mod. Phys. A20, 2294 (2005)

Influences of the Yukawa term on Neutron stars

It has NO effect on finite nuclei

M.I. Krivoruchenko et al, PRD 79, 125023 (2009)

Partially constrained EOS for astrophysical studies

Danielewicz, Lacey and Lynch, Science 298, 1592 (2002))

EOS of MDIx1+WILB

22 / g

km 12 Rand 1.4Msun of NSa

support toGeV 50/ 222

g

716Hz ofpulsar a

support toGeV 150/ Maximum 222 g

Upper limit

Lower limit to support neutrons stars with a super-soft symmetry energy at high densities

Some thoughts and observations

• It may not be that crazy to think about a super-soft and/or even negative symmetry energy at supra-saturation densities especially if you are a string theorist!

• The FOPI/GSI pion data indicates a super-soft symmetry energy at high densities

• The high-density behavior of the nuclear symmetry energy relies on the short-range correlations and the in-medium properties of the short-range tensor force in the n-p singlet channel.

• NS can be stable even with a super-soft symmetry energy if one considers the possibilities of extra-dimensions, new bosons and/or a 5th force as proposed in string theories and super-symmetric extensions of the Standard Model

The moment of inertia provides a sensitive probe to determine g2/2

Symmetry energy and single nucleon potential 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 p

p p

B BA A

x x x x x

xK MeVx

p

xx

p

66666666666666

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 momentum dependence of the nucleon potential is a result of the non-localityof nuclear effective interactions and the Pauli exclusion principle

The x parameter is introduced to mimic various predictions on the symmetry energy by different microscopic nuclear many-body theories using different effective interactions

Default: Gogny force

W. Reisdorf et al. for the FOPI collaboration , NPA781 (2007) 459

IQMD: Isospin-Dependent Molecular Dynamics C. Hartnack, Rajeev K. Puri, J. Aichelin, J. Konopka, S.A. Bass, H. Stoecker, W. Greiner Eur.Phys.J. A1 (1998) 151-169

Near-threshold π-/π+ ratio as a probe of symmetry energy at supra-normal densities

lowlow (high)(high) density region is more neutron-rich density region is more neutron-rich withwith stiff stiff (soft)(soft) symmetry energysymmetry energy

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

Need a symmetry energy softer than the above to make the pion production region more neutron-rich!

2/3 0

00

2/3100 3(2corresponding t 1) ( )

5o ( )

8 FsymE E

Formation of dense, asymmetric nuclear matter

SoftSoft

Stif

fS

tiff

Soft Esym

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

Momentum dependence of the isoscalar potentialCompared with variational many-body theory

Constraining the radii of NON-ROTATING 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 lim

its

● .