Accurate vacuum correction in finite nuclei A. Haga 1, H. Toki 1, S. Tamenaga 1, and Y. Horikawa 2 1.RCNP, Osaka Univ. 2.Juntendo Univ.

Δ2p Δ3p Haga et al., Phys. Rev. A, 66, (2002) Δp splitting anomalies in muonic atoms Haga et al., Phys. Rev. C, to be published

Relativistic picture of anti-protonic atoms

MFT ~ no sea approximation RHA ~ renormalization

Vacuum correction in QED e - e+e+ 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!

Relativistic Hartree in finite nucleus Dyson equation Heresatisfies the following expression, Multiplyingto each term of Dyson equation from left-hand side, one can get the Hartree equation;

Feynman diagram σ ω for σ-meson for ω-meson ρ s (r) ρ v (r)

Green Function CFCF CFCF ~ 0 ~ ∞ ~ ∞~ ∞ Finite

Finite density from vacuum ω ω ω + + ・・・

Numerical result 1 - Vector density - Parameter set HS Contribution from Dirac sea

Numerical result 1 - Vector density - Contribution from Dirac sea

Feynman diagrams in σ- ω- model ○Scalar density σ ω ＝ ・・・ ○Vector density ＝ ・・・

Vacuum contribution to scalar density

Numerical result 2 - Scalar density -

Parameter set HS Contribution from Dirac sea

Numerical result 2 - Scalar density - Accurate Local density Approximation

Summary  We have computed the accurate vacuum correction in finite nucleus.  Finite densities for vector and scalar forces have been calculated by the Green functional method on imaginary axis.  It has found that above calculation gives rise to non-negligible effect for vector density, while about a half of contribution of the local-density approximation for scalar density.

Application to the relativistic RPA the polarization function has poles at ω=ω ｎ. Actuary, the RPA integral equation is solved under the complex energy ω→ω+iε due to

Dirac equation on imaginary axis If we regard the effective mass and potential as constants, we find etc.

The radial Green function for Dirac particles Wronskian Regular boundary condition at x=0. x=∞.