Nuclear ground states and excitations in relativistic nuclear models

1. Nuclear ground states and excitations in relativistic nuclear models Akihiro Haga, Setsuo Tamenaga, Hiroshi Toki, Research Center for Nuclear Physics (RCNP), Osaka University Yataro Horikawa Juntendo University Chiral07 @Osaka, November 14, 2007

2. Introduction • It has been reported that the negative-energy state has an important role; 1. In removing the spurious state and satisfying the current conservation for the transition density (J. Dawson et al., PRC42, 2009 (1990)), 2. In centroidenergies of the giant resonances (P. Ring et al., NPA694, 249 (2001)), 3. In quenching of the Gamow-Teller strength (H. Kurasawa et al., PRL91, 062501 (2003)), 4. In 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 + 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).

3. Treatment of Negative Energy States The Dirac sea is unoccupied (no-sea approximation). The mean field is constructed without negative energy nucleons. The nucleons can be excited toward the negative energy states. (This contribution is important !) The Dirac sea should be occupied. The nucleons can be excited toward only the positive energy states. The mean field is constructed with negative energy nucleons.

4. Introduction • It has been reported that the negative-energy state has an important role; 1. In removing the spurious state and satisfying the current conservation for the transition density (J. Dawson et al., PRC42, 2009 (1990)), 2. In centroidenergies of the giant resonances (P. Ring et al., NPA694, 249 (2001)), 3. In quenching of the Gamow-Teller strength (H. Kurasawa et al., PRL91, 062501 (2003)), 4. In 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 + 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).

5. 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,

6. Example of the exact treatment ~ Baryon density ~ ~ 0 Green’s function method Dirac Green’s function in a finite system ; ××××× ×× CF ×× ~ 0 Renormalized with the counter term. A. Haga, S. Tamenaga, H. Toki, and Y. Horikawa, PRC70, 064322 (2004),

7. Comparison with the local-density approximation and the leading-order derivative expansion (a) Vacuum correction to baryon density (b) Vacuum correction to scalar density Leading order of derivative expansion gives fairly good approximation to obtain the vacuum correction !!

8. Lagrangian density for nucleus Vacuumcontribution ・・・　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 Parameters gσ 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).

9. Scalar and vector mean-field potentials Scalar meson field Vector meson field With negative-energy nucleons Without negative-energy nucleons Without negative-energy nucleons With negative-energy nucleons Scalar and vector potentials are not able to be large as far as the negative energy nucleons are seriously considered. In other word, The effective mass should be large in the model including negative energy contribution. m* ~ 0.8mN

10. Fully-consistent RPA calculation Uncorrelated polarization tensor obtained by RHA Green function ; Feynman part (Vacuum polarization) which is estimated by the lowest order of derivative expansion; Density part A B A. Haga, H. Toki, S. Tamenaga, Y. Horikawa, and H. L.Yadav PRC72, 034301 (2005). A. Haga, H. Toki, S. Tamenaga, and Y. Horikawa, nucl-th/0601041 (2005)

11. T1/2 ~ 0.6 s Nuclear excitations and Effective mass The RPA response (the BS equation) ; The conventional relativistic RPA calculation could not reproduce the data of the ISGQR and of the β-decay rate… T1/2 ~ 0.3 s T1/2 ~ 2.0 s The model with vacuum polarization can reproduce the data well !

12. Other responses and other nuclei Enhanced effective mass provides the good agreement with the experimental data

13. Vacuum polarization in σ-ω model Net Potential in Positive-energy state M σ+ω ω -M -ω Large attractive potential is provided in Negative-energy state→Polarize the Vacuum To obtain the stable nucleus, Net Potential in Negative-energy state σ σ-ω Small Potentials are required

14. Summary • We have shown that the lowest-order derivative expansion of effective action in relativistic nuclear model gives a good description of the vacuum polarization of nucleon-antinucleon field. • We have developed the fully consistent RHA + RPA method including the vacuum polarization. Adjusting of parameters with nuclear matter and ground state properties, the effective mass should be large as m* ~ 0.8mN in typical relativistic models. • Then, we have obtained better result in the ISGQR and the β-decay rate. This is mainly due to the large effective mass. • The large effective mass causes the shortage of the spin-orbit force in the nuclear medium. In this point, we expect to be compensated by the surface pion condensation and the momentum dependent force generated by the tensor term, the Fock term and theparticle-vibration effect.