Spectra of positive- and negative-energy nucleons in finite nuclei G. Mao 1,2, H. Stöcker 2, and W. Greiner 2 1) Institute of High Energy Physics Chinese Academy of Sciences 2) Frankfurt University, Germany I. Introduction II. RHA for Finite Nuclei III. Numerical Results IV. Summary and Outlook 1. G.Mao, H. Stöcker, and W. Greiner, Int. J. Mod. Phys. E8, 389 (1999); AIP Conf. Proc. 597, 112 (2001). 2. G. Mao, Phys. Rev. C67, (2003); High Ene. Phys. Nucl. Phys. 27, 692 (2003).

× × × × × × × × × × × × × × × × × ×... 1p 1s E. nucleon × nucleon–anti-nucleon pair shell model states vacuum (1) potential of nucleons (2) potential of anti-nucleons due to G-parity,vector fields change signs estimation based on no-sea approximation, param. dep. 1. Auerbach et al., PLB182, 221 (1986). 2. Reinhard et al., ZPA323, 13 (1986). × × × × ×

Investigate the properties of quantum vacuum in the medium. A verification for the application of relativistic Quantum Field Theory in a many-body system. Determine the individual scalar and vector potential Build a basis for the study of anti-matter and anti-nuclei....

Relativistic Hartree Approach nucleonanti-nucleon.. valence-nucleon contribution Dirac-sea contribution describing bound states of nucleons and anti-nucleons consistently other densities similar

II. RHA for Finite Nuclei Quantum Hadrodynamics B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16, 1(1986) here and

Tensor Couplings

Dirac equation In static nuclear matter particle, posi. ene. particle, neg. ene.

and are probability amplitudes The wave packet can be expanded as

antiparticle, posi. ene. antiparticle, neg. ene. and are the annihilation and creation operators for the particles and antiparticles One can expand the wave packet of antiparticles analogous to that of particles. In quantum field theory:

In finite nuclei, the Dirac equation can be written as The field operator can be expanded according to nucleons and anti-nucleons : quantum number Spherical Nuclei andcommute with and are eigenfunctions of · ·

P spherical spinor Inserting into the Dirac equation, one gets coupled equations for and

Nucleons Anti-nucleons where

In numerical solutions Nucleons: Anti-nucleons:

Vector fields change signs G-parity

Orthonormalization of wave functions matrix equation From the Dirac equation one can have From above equations one obtains

Meson-field equations valence-nucleon contribution Dirac-sea contribution other densities similar

eff. pot. deri. term. total derivative baryon number is conserved

Param: Set: 9 ~

RMF RHA -----

s.o. splitting inshell fluc. Tensor couplings enlarge by a factor of 2 Binding Energy are improved Dirac-sea effects are enhanced

Charge densities

Vacuum contributions to the scalar density and baryon density RHAT

Relative amplitude to the baryon density 16 O: < 4.0 % 40 Ca: < 2.3 % 208 Pb: < 0.6 % RHA1

Scalar and Vector potentials for S and V RHAT larger than RHA1 about 20 MeV

ameliorated evidently deepened 20~30 MeV single particle spectra of protons and antiprotons

single particle spectra of neutrons and antineutrons

Proton and anti-proton potentials in Proton anti-proton NL RHA RHAT at 0.9 fm

IV. Summary and Outlook 1. RHA including tensor couplings describing bound states of positive- and negative-energy nucleons in finite nuclei consistently. 2. Parameters fitted to the properties of spherical nuclei from RHA is about half of RMF RHAT: effect of tensor couplings is increased by a factor of 2 is deepened 20~30 MeV · · ·

1. N. Auerbach, A.S. Goldhaber, M.B. Johnson, L.D. Miller and A. Picklesimer, PLB 182, 221(1986) 2. Y. Jin and D.S. Onley, PRC 38, 813(1988) nucleon o anti-nucleon × nucleon–anti-nucleon pair r