Three-body Force Effects on the Properties of Asymmetric Nuclear Matter

Zuo Wei Zuo Wei Institute of Modern Physics, Chinese Institute of Modern Physics, Chinese

Academy of Sciences, Lanzhou, ChinaAcademy of Sciences, Lanzhou, China

I. Bombaci, G.F.Burgio, U. Lombardo, H.-J. Schulze, A. Lejeune, J. F. Mathiot, B. A. Li , A. Li, Z. H. Li, L.G.Cao, C.W.Shen, J.M.Dong, H.F. Zhang, W. Scheid

From nucleon structure to nuclear structure and compact astrophysical objects

KITPC/ITP-CAS, Beijing, June 10 – July 20, 2012

Outline: 1. Introduction (Motivation) 2. Theoretical approaches 3. Results 4. Summary and conclusion

Properties of Asymmetric Nuclear Matter

Lattimer and Prakash, Science 304(2004)536.

Composition of neutron starComposition of neutron star & & Cooling of neutron Cooling of neutron starsstars

Lattimer et al., PRL66(1991)2701.

Double n/p ratio in HICDouble n/p ratio in HIC

B.A.Li, L.W.Chen,G.C.Yong, W. Zuo, PLB634(2006)378

Asymmetric nuclear matter at low densities extracted from HIC

Iso-scaling

Isospin diffusion

Conclusion: 0

0 7 1.1

0

.32( / ) ( ) 32( / )symE

Experiments at the NSCL/MSU ：Sn+Sn ， Ebeam/A=50 MeV

Z.-Q. Feng etal, PLB 683 (2010) 140

Z. G. Xiao etal, PRL 102(2009)062502

Asymmetric nuclear matter at high densities from HIC

P. Russotto etal, PLB. 697(2011)471FOPI Experiments at GSI ： Au+Au

Super-soft

Stiff

Conclusion: ?????

Between soft and stiff: γ=0.90.4

Theoretical Approaches

• Skyrme-Hartree-Fock• Relativistic Mean Field Theory• Relativistic Hartree-Fock

• Variational Approach• Green’s Function Theory • Brueckner Theory• Dirac-Brueckner Approach• Effective Field Theory

Symmetry energy predicted by SHF and RMF approachesSymmetry energy predicted by SHF and RMF approaches

L.W. Chen, C.M. Ko, B.A. Li, Phys. Rev. Lett. 94 (2005) 032701.B. A. Li ， L. W. Chen ， C. M. Ko ， Phys. Rep. 464 （ 2008 ） 113

SHF RMF

C. Fuchs and H. H. Wolter, EPJA30(2006)5 Dieperink et al., PRC67(2003)064307.

Symmetry energy predicted by various many-body Symmetry energy predicted by various many-body theories theories ---- ---- Extremely Large uncertainty at high densities!

Effective field theory

DBHF

BHFGreens function

Variational ?

Most recent results from BHF

Z.H. Li, U. Lombardo, H.-J. Schulze, Zuo et al., PRC74(2006)047304

Bethe-Goldstone Theory

• Bethe-Goldstone equation and effective G-matrix

→ Nucleon-nucleon interaction: ★ Two-body interaction : AV18 (isospin dependent) ★ Effective three-body force

→ Pauli operator :

→ Single particle energy :

→ “Auxiliary” potential : continuous choice

);,()()(

),();,(

21 21

212121

Gikk

kkkkQkkvvG

kkNNNN

effNN Vvv 32

2veffV3

2121 11),( knknkkQ

)()2/()( 22 kUmkk

Ak

kkkkGkkknkU ')]'()(['Re)'()('

Confirmation of the hole-line expansion of the EOS under

the contineous chioce (Song,Baldo,Lombardo,et al,PRL(1998))

Nuclear Matter Saturation Problem

The model of rigid nucleons interacting via realistic two-body forces fitting in-vacuum nucleon-nucleon scattering data can not reproduce the empirical saturation properties of nuclear matter (Coestor band, Coestor et al., PRC1(1970)765)

Effective NN interaction

Dirac-Brueckner approach [R.Machleidt, Adv. Nucl. Phys. 16 (1989) 189; Serot andWalecka, Int. Journ. Mod. Phys. E6(1997) 515]

There are on the market different ways on how to introduce the medium effects:

◆ Phenomenologocal three-body forceRijkijkijk VVV 2

--- Two or few adjustable parameters, which are adjusted simultaneously on nuclear saturation point and light nuclei (3He)properties.--- Extensively applied to neutron star physics within both variational approach [Wiringa et al., PRC38(1988) 1010; Akmal et al., PRC56 (1997)2261; C58(98) 1804.] and BHF approach [Baldo et al., Astron. Astrophys. 328(1997) 274.]

◆ Microscopic three-body force

Microscopic Three-body Forces

N

R ,

,

)(b )(c

N

N

N

N

N

N

N

, , , ,

N

N

,

,

R,

)(a

,

, ,

Z-diagram

• Based on meson exchange approach• Be constructed in a consistent way with the adopted two-body

force---------microscopic TBF !• Grange et.al PRC40(1989)1040; W. Zuo etal., NPA706(2002)418

Effective Microscopic Three-body Force

• Effective three-body force effV3

231333213213

23133*

3321213

11,,',','

'1'1''dd4

1,','

rrrrrrrrrW

rrrrrTrrrrrV

n

nn

eff

→ Defect function: (r12)= (r12) – (r12)

★Short-range nucleon correlations (Ladder correlations) ★Evaluated self-consistently at each iteration

Effective TBF ---- Density dependent Effective TBF ---- Isospin dependent for asymmetric

nuclear matter

W. Zuo, A. Lejeune, U.Lombardo, J.F.Mothiot, NPA706(2002)418

EOS of SNM & saturation properties

(fm-3) EA (MeV) K (MeV)

0.19 –15.0 210

0.26 –18.0 230

Saturation properties:

TBF is necessary for reproducing the empirical saturation property of nuclear matter in a non-relativistic microscopic framework.

TBF based on Bonn B interaction

Li ZengHua ， H. J. Schulze, U. Lombardo, Wei Zuo ， PRC 77, 034316 (2008)

W. Zuo, A. Lejeune, U.Lombardo, J.F.Mothiot, EPJA 14 (2002) 469

Parabolic law

W. Zuo, Z.H.Li,A. Li, G.C.Lu, PRC 69(2004)064001

2( , , ) ( , ,0) ( , )A A symE T E T E T

Isospin dependence of the EOS of asymmetric nuclear matter

Density dependence of symmetry energyDensity dependence of symmetry energy

TBF effect Thermal effect

0.0 0.1 0.2 0.3 0.4 0.50

500

1000

1500

2000

2500

3000

K(

) (M

ev)

fm-3

0.0 0.1 0.2 0.3 0.4 0.5 0.60

500

1000

1500

2000

2500

3000

K()

(Me

v)

fm-3

Incompressibility of asymmetric nuclear matterIncompressibility of asymmetric nuclear matter

BHF(two-body force) BHF(two-body force + TBF)

M3Y interaction (Dao T. Khoa etal, NPA602(1996)98)

At the lowest mean field approximation, two problems of BHF approach for predicting nuclear s.p. properties:

1. At densities around the saturation density, the predicted optical potential depth is too deep as compared to the empirical value, and it destroy the Hugenholtz-Van Hove (HVH) theorem.

Solution: to include the effect of ground state correlations

J. P. Jeukenne et al., Phys. Rep. 25 (1976) 83

M. Baldo et al., Phys. Lett. 209 (1988) 135; 215 (1988) 19

2. At high densities, the predicted potential is too attractive and its momentum dependence turns out to be too weak for describing the experimental elliptic flow data.

P. Danielewicz, Nucl. Phys. A673 (2000) 375

Improvement in two aspects:

1. Extend the calculation of the effect of ground state correlations to asymmetric nuclear

W. Zuo et al., PRC 60 (1999) 024605

2. Include a microscopic three-body force (TBF) and the TBF-induced rearrangement contribution in calculating the s.p. properties

W. Zuo et al., NPA706 (2002) 418; PRC 74 (2006) 014317

Single Particle Potential beyond the mean field approximation:

1. Single particle potential at lowest BHF level

eff3

TBF

1( )

2 i jij k A

Vk ij ij n n

n

'

( ) ( ') Re ' [ ( ) ( ')] 'BHF Ak

U k n k kk G k k kk

);,()()(

),();,(

21 21

212121

Gikk

kkkkQkkvvG

kkNNNN

2. Ground state correlations

Full s.p. potential ：2( ) ( ) ( ) ( )BHF TBFU k U k U k U k

3. TBF rearrangement

Single particle potential at the BHF level

W. Zuo, I. Bombaci and U. Lombardo, PRC60(1999)024605W. Zuo, L.G. Gao, B.A. Li et al., Phys. Rev. C72 (2005)014005 .

In neutron rich matter : Up<Un at low momentaUp>Un at high enough momenta

W. Zuo, I. Bombaci, U. Lombardo, PRC 60 (1999) 024605

Pauli rearrangement contribution: Ground state correlationsPauli rearrangement contribution: Ground state correlations

1. The Pauli rearrangement is repulsive 2. It affects maily the s.p. potential at low momenta and vanishes repaidly above Fermi momentum 3. It distories the linear beta-dependence of the s.p. potential

Neutron-proton effective mass splitting in neutron-rich matter

M*n > M*p

11* d d1

d dF F

k k

m m Uk

m m k p k

neutrons

protons

Skyrme-like interactions:

mp* < mn* or mn* < mp*

B. A. Li et al., PRC69(2004)064602

Comparison to other predictions:

DBHF: mn* > mp*

Dalen et al., PRL95(2005)022302Z. Y. Ma et al., PLB 604 (2004)170F. Sammarruca et al., nucl-th/0411053

W. Zuo, L.G. Gao, B.A. Li et al., Phys. Rev. C72 (2005)014005 .

TBF rearrangment contribution to s.p. potential in symmetric nuclear matter

eff3

TBF

1( )

2 i jij k A

VU k ij ij n n

n

1. The TBF induces a strongly repulsive

rearrangement modification of the s. p.

potential at high densities and momenta.

2. The TBF rearrangement contribution is

strongly momentum dependent at high

densities and momenta.

Effective mass

Zuo, Lombardo, Schulze, Li, Phys. Rev. C74 (2006) 017304

The s. p. potentials at three different aproximations

S.p. potentials in SNM in three cases: 1. without the TBF; 2. including the TBF effect only via G-matrix; 3. including the full contribution of the TBF

W. Zuo, L.G. Gao, B.A. Li et al., Phys. Rev. C72 (2005)014005 .

S.p. potential including the TBF rearrangment contribution

Isospin dependence of the TBF rearrangment effect

1. Negligible at low densities around and

below the Fermi momentum.

2. Enhancement of the repulsion for neutrons

and the attraction for protons at high

densities

Symmetry potential Effective masses

1. Remarkable reduction of the neutron

and proton effective masses.

2. Suppression of the isospin splitting

in neutron-rich matter at high

densities.

As the asymmetry increases, the neutron depletion becomes smaller, while the proton

becomes larger, indicting that at a higher asymmetry, the many-body correlations induced by

the nucleon-nucleon interaction becomes stronger on protons and weaker on neutrons.

Momentum distribution in asymmetric nuclear matter

TBF effect on the 1S0 proton gap in neutron star matter

W. Zuo et al., PLB 595(2004)44

TBF suppresses strongly the 1S0 proton superfluidity in neutron stars1. It reduces the energy gap from ~1 to ~0.5 2. It suppresses largely the density region of the superfluidity

Proton superfludity: Going from SNM to PNM, the maximum gap value decreases and the density domain enlarges remarkably

W. Zuo et al., PRC75(2007)045806

3PF2 proton and neutron superfluidity in asymmetric nuclear matter Proton

W. Zuo et al., EurPhys. Lett. 84(2008)32001

Neutron

TBF effect on the 3PF2 neutron gap in neutron star matter and neutron stars

TBF enhances remarkably the 3PF2 neutron superfluidity in neutron star matter and in neutron stars

3PF2 gap in Neutron star matter

3PF2 gap in Neutron stars

W. Zuo et al., Phys. Rev. C78(2008)015805

A simple formula has been proposed for accurately predicting the Qα values of SHN

The increased stability of these SHN against alpha-decay with larger neutron number is primarily attributed to the effect of the symmetry energy.

Isospin dependence of Alpha-decay energies of SHNIsospin dependence of Alpha-decay energies of SHN

Summary • The TBF provides a repulsive contribution to the EOS of nuclear matter and

improves remarkably the predicted saturation properties.

• The empirical parabolic law for the EOS of ANM can be extended

to the highest asymmetry and to the finite-temperature case.

• The TBF leads to a strong enhancement of the stiffness of symmetry energy at high densities.

• The neutron-proton effective mass splitting in neutron-rich matter is

• The TBF induces a strongly repulsive and momentum-dependent rearrangement contribution to the s.p. potential at high densities.

m*n > m*p

Thank you !