Spectra of positive- and negative-energy nucleons in finite nuclei

1. Spectra of positive- and negative-energy nucleons in finite nuclei G. Mao1,2, H. Stöcker2, and W. Greiner2 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, 044318 (2003); High Ene. Phys. Nucl. Phys. 27, 692 (2003).

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

3. . 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.

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

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

6. Tensor Couplings

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

8. The wave packet can be expanded as and are probability amplitudes

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

10. 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 · · and commute with and are eigenfunctions of

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

12. Nucleons Anti-nucleons where

13. In numerical solutions Nucleons: Anti-nucleons:

14. Vector fields change signs G-parity

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

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

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

18. Param: 9 ~ Set:

19. RMF RHA ----- -----

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

21. Charge densities

22. Vacuum contributions to the scalar density and baryon density RHAT

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

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

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

26. single particle spectra of neutrons and antineutrons

27. Proton and anti-proton potentials in Proton anti-proton NL1 54.1 750.0 RHA1 42.6 362.0 RHAT 46.6 396.8 at 0.9 fm

28. 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

29. r nucleon o anti-nucleon × nucleon–anti-nucleon pair 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)