Relativistic mean field and RPA with negative energy states for finite nuclei

Akihiro Haga , Hiroshi Toki, Setsuo Tamenaga, Yoko Ogawa,

Research Center for Nuclear Physics (RCNP), Osaka UniversityYataro Horikawa

Department of Physics, Juntendo Univerty

INPC 6/6, 2007 at Tokyo International Forum

Introduction

It has been reported that the negative-energy state has an important role, 1. to remove the spurious state and to satisfy the current conservation for the transition

density (J. Dawson et al., PRC42, 2009 (1990)),2. to identify the giant resonances (P. Ring et al., NPA694, 249 (2001)),3. to give a new quenching of the Gamow-Teller strength (H. Kurasawa et al., PRL91, 062501 (2003)),4. to restore the gauge invariance in nuclear polarization correction of muonic atoms (A. Haga et al., PRC69, 044308 (2004)). ⇒ The no-sea approximation was made in these studies.

G. Mao el al. have indicated that nuclear properties are reasonably well even negative energy states are considered in terms of the derivative expansion, the RHA calculation (vacuum polarization).

G. Mao, H. Stocker, and W. Greiner, Int. J. Mod. Phys. E8, 389 (1999)G. Mao, Phys. Rev. C67, 044318 (2003) ⇒ 　 The effective mass should be larger than the conventional RMF predicts.

We have also developed the fully self-consistent finite calculation with the vacuum polarization within a framework of RHA and RPA;

A. Haga, S. Tamenaga, H. Toki, and Y. Horikawa, PRC70, 064322 (2004), A. Haga, H. Toki, S. Tamenaga, Y. Horikawa, and H.L.Yadav PRC72, 034301 (2005).

The treatment of negative energy states

The Dirac sea should be occupied.

The Dirac sea is unoccupied (no-sea approximation).

The mean field is constructed without negative energy nucleons.

Nucleons can be excited down in the negative energy states. (This contribution is important !)

The mean field is constructed with negative energy nucleons.

Nucleons are excited up from the negative energy states.

Introduction

It has been reported that the negative-energy state has an important role, 1. to remove the spurious state and to satisfy the current conservation for the transition

density (J. Dawson et al., PRC42, 2009 (1990)),2. to identify the giant resonances (P. Ring et al., NPA694, 249 (2001)),3. to give a new quenching of the Gamow-Teller strength (H. Kurasawa et al., PRL91, 062501 (2003)),4. to restore the gauge invariance in nuclear polarization correction of muonic atoms (A. Haga et al., PRC69, 044308 (2004)). ⇒ The no-sea approximation was made in these studies.

G. Mao el al. have indicated that nuclear properties are reasonably well even negative energy states are considered in terms of the derivative expansion, the RHA calculation (vacuum polarization).

G. Mao, H. Stocker, and W. Greiner, Int. J. Mod. Phys. E8, 389 (1999)G. Mao, Phys. Rev. C67, 044318 (2003) ⇒ 　 The effective mass should be larger than the conventional RMF predicts.

We have developed the fully self-consistent finite calculation with the negative-energy states within a framework of RHA and RPA;

A. Haga, S. Tamenaga, H. Toki, and Y. Horikawa, PRC70, 064322 (2004), A. Haga, H. Toki, S. Tamenaga, Y. Horikawa, and H.L.Yadav PRC72, 034301 (2005).

Example of the negative-energy effect ~ Baryon density ~

0ˆ ( ) Tr[ ( , ; )]2F

N C

dx i G x x

0[ *( ) ( )] ( , '; ) ( ')xi m x V x G x x x x

××××× 　 ××

××

CF

A. Haga, S. Tamenaga, H. Toki, and Y. Horikawa, PRC70, 064322 (2004),

Re

Im

Dirac Green’s function in a finite system ;

~ 0

~ 0

† 0( ) ( ) Tr[ ( , ; )]2

F iE

n nn i

dx x i G x x

Green’s function method

Renormalized with the counter term.

Scalar and vector mean-field potentials

The effective mass should be large in the model including negative energy contribution.

Scalar meson field Vector meson field

With negative-energy nucleons

Without negative-energy nucleons

With negative-energy nucleons

Without negative-energy nucleons

Scalar and vector potentials are reduced largely as the negative energy nucleons are explicitly considered.

In other words,

*( ) ( )Nm r m g r

Comparison with the local-density approximation and the derivative expansion of meson fields

Derivative expansion agrees with the rigorous calculation

very well !!

(b) Correction to scalar density(a) Correction to baryon density

Rigorous calculation is very time consuming!

Lagrangian density for nuclear part

Negative - energy contribution

Parametersgσ g2 g3 gω gρ fω g’ mσ,

For the ground state, this was studied by G. Mao, H. Stocker, and W. Greiner, Int. J. Mod. Phys. E8, 389 (1999); G. Mao, Phys. Rev. C67, 044318 (2003).

・・・　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　

Fully-consistent RPA calculation

])';,()';,([2

'

xyGyxGTrd

HB

HAAB

H

Uncorrelated polarization function obtained by the Green’s function ;

D F

BAABF

2

A B

ABF

Density part

Feynman part (Vacuum polarization) which is estimated by the derivative expansion;

A. Haga, H. Toki, S. Tamenaga, Y. Horikawa, and H. L.Yadav PRC72, 034301 (2005).

Nuclear excitations with the self-consistent RHA+RPA

Quadrupole resonances can be reproduced by enhanced nucleon effective mass

( , '; ) ( , '; )RPA Hi q q i q q 3 3

6

'( , ; ) ( , '; ) ( ', '; )

(2 ) H RPA

d kd ki q k iD k k i k q

The RPA equation (the BS equation) ;

0

2

4

6

8

10

12

14

16

18

0 10 20 30 40 50 60

RHAT (with VP)RHAT (without VP)

Negative energy contribution to GT quenching

Feynman term );,( qpGTvac

RHAT m*=0.80m

TM1 m*=0.63m NL3 m*=0.60m

Ikeda sum 95% (~5%) 89.0%(6.29%)

88.4%(6.80%)

208Pb

The rest comes from the excitations concerning with antinucleon states.

Excitation energy

Str

engt

h

No-sea approximationIn nuclear matter ~12%

(H. Kurasawa et al., PRL91, 062501 (2003)),

Summary We have developed the method to include the negative energy s

tates in the nuclear ground states and excited states for finite nuclei.

The negative energy contribution produces the repulsive scalar potential, and as a result, the large effective mass is provided.

The negative energy contribution is described by the derivative expansion method well. The nuclear excitations, in particular, quadrupole resonances can be reproduced with this method.

The large effective mass should be confirmed by the Dirac-Brueckner-Hartree-Fock calculation with the vacuum polarization. This study is in progress.