1. Nuclear excitations in relativistic nuclear models
1Akihiro Haga, 1Hiroshi Toki,
1Setsuo Tamenaga, and 2Yataro Horikawa
1. RCNP Osaka University
2. Juntendo University
8/1-8/3

2. Introduction
The relativistic nuclear model has succeed to describe the nuclear ground-state properties, mainly due to the large spin-orbit force arising from the (small) relativistic effective mass.
Recently, however, it is reported that there are discrepancies in the nuclear excitations calculated by the relativistic model, and it presumably originated by the relativistic effective mass.
In the conventional relativistic model, the negative-energy state is not used for the construction of ground state (no-sea approximation). This treatment is crucial to lead to small effective mass.
We thought important to reconsider the negative-energy contribution (vacuum polarization)

3. What is the vacuum polarization?
Vacuum of electron-positron field is polarized by electromagnetic field generated by nuclear charge.
Vacuum of nucleon-anti-nucleon field
is also polarized by nuclear force!
e - e+

4. Vacuum polarization in Walecka modelω
σ+ω
potential for the positive-energy state -ω σ
σ-ω
potential for the negative-energy state

5. Instability of vacuum polarization
Nucleons
Total
No solution
Vacuum
Scalar potential as a function of coupling constant gσ
in nuclear matter.
Effective mass,
should be large!!

6. Computation of the Dirac-sea effect – Derivative expansion –
One-loop vacuum contribution to Lagrangian:
is expanded in terms of the derivatives of meson fields as,
Then, the functional coefficients are given as,
A. Haga et. al., Phys. Rev. C70, (2004)

7. Effective Lagrangian of the Walecka model with the vacuum contribution
Tensor coupling term
G. Mao, Phys. Rev. C67, (2003)
A. Haga, nucl-th/
Leading-order derivative expansion
Parameters, gσ , gω , gρ , mσ… are selected with nuclear matter properties and ground-state properties of finite spherical nuclei (binding energy and charge radius).

8. Fully-consistent RPA calculation
RPA equation
: Uncorrelated polarization tensor obtained by RHA Green function
Density part
Feynman (vacuum polarization) part

9. Vacuum-polarization (Feynman) part
Effective action
Mesons at initial and final vertices
Vacuum polarization is given by the functional derivatives of the effective action. A B
A. Haga et al., Phys. Rev. C72 (2005)

10. Isoscalar giant quadrupole resonances
The model with the vacuum polarization reproduces the data on the ISGQR !

11. Excitation energies of ISGQR as a function of the relativistic effective mass.
The relativistic effective mass m*/m~0.8 is required to reproduce experimental ISGQR energies.

12. RHAT RHAT’ RHA TM1 NL3 g’ 0.69 0.89 0.96 0.55 Two-body Interaction;
fπ=0.08, mπ=139MeV
g’ is obtained by fitting with the peak of the Gamow-Teller resonance in 208Pb.
Then, the value of g’ is model-dependent as,
RHAT
RHAT’
RHA
TM1
NL3 g’
0.69
0.89
0.96
0.55
g’ is sensitive to the value of
which is the strength of tensor force.

13. RHAT TM1 NL3 Ikeda sum 90.5% 89.0% 88.4% GTR Distribution
Fitting g’ with the GTR peak, 19.2MeV, the shape becomes very similar.
RHAT
TM1
NL3
Ikeda sum
90.5%
89.0%
88.4%
Exp.
60-70%
Cut Off energy 50MeV.
The rest comes from the excitations larger than 50MeV＋antinucleon states.

14. Magnetic dipole (M1) resonance
Using the same interaction （and g’） as GTR analysis,
RHAT can reproduce the excitation energy !
On the other hand,
RHAT
TM1
NL3
Strength
27.6
28.9
31.5
Exp.17.5
Strength is still overestimated though it is suppressed by the vacuum effect.

15. Shell structure for unstable nucleiν π
ＮＬ３
ＲＨＡＴ
1s1/2
1p3/2
1p1/2
1d5/2
1d3/2
2s1/2
1f7/2
1f5/2
2p3/2
2p1/2
1g9/2
76Fe
RHA + BCS calculation
ν1f７/2
π1f7/2
NL3: ΔE = MeV (0.734)
RHAT: ΔE = MeV (0.710)
76Fe

16. β-decay life times in 78Ni
T1/2= 2.0 sec (NL3)
T1/2= 0.6 sec (RHAT) f0
Exp. T1/2~ 0.3 sec

17. Summary
We have developed the calculation method of the vacuum polarization in relativistic Hartree approximation and fully-consistent RHA + RPA calculation using the derivative-expansion method.
The RHA calculation produces the enhanced effective mass naturally, because the inclusion of vacuum effect makes meson fields weak.
We have found that the relativistic effective mass is about 0.8 to reproduce the ISGQR excitation energies.
The quenching of the Gamow-Teller sum rule still remains with the vacuum polarization. In addition, we found the quenching of the M1 resonance caused by the vacuum polarization.
The inclusion of the vacuum polarization affects the shell structure around the Fermi level. As a result, the beta-decay life time in 78Ni is improved by this effect.
The beta-decay and nuclear polarization analyses in unstable nucleus give us the evidence of the large effective mass, that is, a role of the vacuum polarization.