Role of vacuum in relativistic nuclear model A. Haga 1, H. Toki 2, S. Tamenaga 2 and Y. Horikawa 3 1. Nagoya Institute of Technology, Japan 2. RCNP Osaka University, Japan 3. Juntendo University, Japan

Motivation Recently, it has been pointed out that, ・ the vacuum contribution is unnatural, and ・ the finite parameters fitted to experimental data encode the vacuum contribution. R. J. Furnstahl et al, Phys. Rev. C52 (1995); Nucl. Phys. A618(1997) etc. On the other hand ， ・ typical relativistic models give the small effective mass, but ・ the large effective mass is required from the beta-decay analysis and isoscalar giant quadrupole resonances (ISGQR). T. Niksic, et al., Phys. Rev. C71 (2005); Phys. Rev. C72 (2005) etc.

① the vacuum polarization gives the large effective mass automatically, and ② if we allow the large effective mass, the vacuum contribution becomes natural. There are the facts that, ① → Parameters in RMF might include the vacuum polarization inadequately. ② → The vacuum polarization can be treated explicitly. Motivation In this symposium, we show the vacuum-polarization effect both in the nuclear ground states and the nuclear excitations by fully- consistent RHA and RPA calculations.

Effective Lagrangian of the Walecka model with the vacuum contribution V F and Z F describe the vacuum effect of nucleons, Leading-order derivative expansion G. Mao, Phys. Rev. C67, (2003)

Leading-order derivative expansion Derivative expansion gives fairly good approximation to obtain the vacuum correction. (a) Vacuum correction to baryon density(b) Vacuum correction to scalar density A. Haga et al., Phys. Rev. C70 (2004)

Parameter sets used in the present study ; Relativistic effective mass

Strength of the meson fields is suppressed by the vacuum. Vacuum Total Scalar potential as a function of coupling constant g σ in nuclear matter. Nucleons

Fully-consistent RPA calculation RPA equation : Uncorrelated response function obtained by RHA mean-field potential Density partFeynman (vacuum polarization) part

Vacuum-polarization (Feynman) part Effective action AB Vacuum polarization is given by the functional derivatives of the effective action.

Decoupling of spurious state

Isoscalar giant quadrupole resonances (ISGQR) The model with the vacuum polarization reproduces the data on the ISGQR !

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.

Isoscalar giant monopole resonances (ISGMR) The centroids of the ISGMR does not shift as far as the compression modulus is kept the same, even if the vacuum polarization is included.

Isoscalar giant dipole resonances (ISGDR) The inclusion of the vacuum polarization shifts the ISGDR peaks to the lower energy.

Energy-weighted sum rules (EWSR) EWSR of B(EL) is approximately proportional to the relativistic effective mass:

Summary We have developed the fully-consistent RHA and 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. While the inclusion of the vacuum polarization affects the dipole compression mode, it does not affect the monopole ones if the compression modulus is kept the same. The EWSR is suppressed by including the vacuum polarization. The beta-decay and nuclear polarization analyses would also give us the evidence of the large effective mass, a role of the vacuum polarization.

Properties of the nuclear ground states In spite of the large differences of the scalar and vector potentials, the nuclear ground-state properties come out to be similar for each other.

Profiles of proton and neutron densities 40 Ca Proton density Neutron density

Profiles of proton and neutron densities 90 Zr Proton density Neutron density

Profiles of proton and neutron densities 208 Pb Proton density Neutron density

Scalar meson fieldVector meson field Scalar and vector mean-field potentials g σ = 6.05 g ω = 8.26 RHA (RHAT1) g σ = g ω = RMF (NL3) Why were small coupling constants required in RHA calculation?