1. Trento, 14.05.09 Giessen-BUU: recent progress T. Gaitanos (JLU-Giessen) Model outline Relativistic transport (GiBUU) (briefly) The transport Eq. in relativistic hadrodynamics ground state in transport, momentum dependence in RMF Not trivial: ground state in transport, momentum dependence in RMF New ground state (GS) initialization for transport studies Results of GS simulations Application (Giant Monopol Resonances, GMR) Theoretical aspects and numerical results (Vlasov, Vlasov+coll.), comparison with expriments (excitation energy, widths of GMR) Momentum dependence in relativistic hadrodynamics Non-linear derivative terms in original Lagrangian of QHD arXiv: 0904.1130 [nucl-th] Final remarks Many thanks to GiBUU-group

2. Trento, 14.05.09 Outline of the model…

3. Trento, 14.05.09 The MF approach of Quantum-Hadro-Dynamics (QHD) Equations of motion for Dirac (  ) and boson fields (  ) in Mean-Field (MF) approach: The Energy-Momentum Tensor in MF: Infinite nuclear matter (  B (x,t)=const.)  no 4-derivatives, no Coulomb Finite systems: Local Density Appr. (LDA) 83 J.D. Walecka, Ann. Phys. (N.Y.) 83 (1974) 497 Finite systems: beyond LDA, space-like derivarites included (surface effects)

4. Trento, 14.05.09 The relativistic transport equation (BUU) Starting basis Starting basis The MF-approach of QHD in terms of the effective Dirac equation: Q.Li, J.Q. Wu, C.M. Ko, Phys. Rev. C39 (1989) 849 B. Blättel, V. Koch, U. Mosel, Rep. Prog. Phys. 56 (1993) 1 Wigner transform of the 1-b-density matrix („Wigner-matrix“):

5. Trento, 14.05.09 Numerical realization (Test-Particle Ansatz)… Discretization of the phase-space distribution f(x,p * ) in terms of N „test particles“ http://www.physik.uni-giessen.de/GiBUU/ Energy momentum tensor  energy density  + Nuclear ground state (initialization) „test particles“ initialized according to empirical density distributions (Wood-Saxon type for heavy nuclei, harmonic-osz. Type for light systems) Problem: density profiles not consistent with mean-field used in propagation  variational method of  =  [  ] in RMF  different density distr. for ground state  nucleus not in its „real“ groundstate, but in an „excited“ state  affects stability

6. Trento, 14.05.09 New initialization: method… Here: Relativistic Thomas-Fermi (RTF) model for spherical nuclei (Horst Lenske) variational method for energy density functional  [  p,  n ] Relativistic Thomas-Fermi (RTF) equations  Relativistic Thomas-Fermi (RTF) equations + meson field equations (for the different meson fields) RTF densities

7. Trento, 14.05.09 New initialization: relativistic fields (scalar, vector, etc) & stability… old initialization new initialization Fluctuations in different Lorentz-components of nuclear self energy (scalar, vector) V ~ (vector – scalar) fluctuates considerably! Almost perfect stability + agreement with ground state scalar part vector part

8. Trento, 14.05.09 New initialization: relativistic potential (scalar-vector) & stability… old initialization new initialization

9. Trento, 14.05.09 New initialization: relativistic fields (scalar, vector, isocvector, coulomb) & stability…

10. Trento, 14.05.09 New initialization: density distributions & stability… old initialization new initialization

11. Trento, 14.05.09 New initialization: Fermi energies in RTF… E F (protons) E F (neutrons)

12. Trento, 14.05.09 New initialization: Fermi energies in BUU… old initialization new initialization

13. Trento, 14.05.09 New initialization: Binding energy & rms-radius in BUU… Old initialization new initialization RTF-binding energy

14. Trento, 14.05.09 Ground state in BUU-II: Results (application to proton-induced reactions)

15. Trento, 14.05.09 Giant Resonances – preliminaries… highly collective modes Giant resonances = highly collective modes of nuclear excitationCollectivity Coherent Coherent super-position of many single-particle transitions from one shell to another Collective motion Collective motion of an appreciable fraction of nucleons of nucleus Monopol (L=0) Dipol (L=1) Oktupol (L=2) … (L>3)

16. Trento, 14.05.09 Giant Monopol Resonances (GMR) - Importance… Indirect determination of the nuclear compression modulus (important for EoS ~  sat ) E* Microscopic approaches (RPA) : Determine E* with several nuclear models compression modulus and NM properties (compression modulus) Excitation energy of GMR Compression modulus of NM Effective interactions 208 Pb 90 Zr Exp. data E GMR ~A -1/3

17. Trento, 14.05.09 Giant Monopol Resonances (GMR) in GiBUU (influence of init. method…) Vlasov calculations 0

18. Trento, 14.05.09 Giant Monopol Resonances (GMR) – excitation energy & width (Vlasov)… rms radius (fm)

19. Trento, 14.05.09 Giant Monopol Resonances (GMR) – excitation energy & width (Vlasov+coll.)…

20. Trento, 14.05.09 Giant Monopol Resonances (GMR) – excitation energy & width (Vlasov+coll.)… Vlasov Vlasov Vlasov+coll.

21. Trento, 14.05.09 Non-Linear derivatives in relativistic hadrodynamics – Motivation Starting basis Starting basis QHD Lagrangian (original Walecka, 1974) Dirac Equation: Dirac Equation: momenta and mass dressed by density dependent self energies Schrödinger equivalent optical potential: Schrödinger equivalent optical potential: linear increase with energy! Not consistent with Dirac phenomenology (Hama et al.) Microscopic Dirac-Brueckner: non-linear density AND density dependence

22. Trento, 14.05.09 Non-Linear Derivatives (NLD) in relativistic hadrodynamics – The NLD Lagrangian Non-linear derivative operators: Auxiliary field: Auxiliary field: structure not relevant (no rearrangement in nuclear matter) Mass term: Mass term: just to not re-normalize the standard QHD couplings Parameter  : Parameter  : ~hadronic mass scale, e.g.,  =1 GeV Starting basis Starting basis again QHD Lagrangian (original Walecka, 1974) Modified interaction: Modified interaction: non-linear operators in scalar and vector int. terms

23. Trento, 14.05.09 Non-Linear Derivatives (NLD) in relativistic hadrodynamics – Field equations Dirac-equation in nuclear matter: NLD Lagrangian NLD Lagrangian contains all higher order derivatives of the baryon field Generalized Euler-Lagrange (Noether-currents, etc…): Meson-field equations: Density and energy dependence of self energies Non-linear density dependence of meson fields, particularly, of the vector field

24. Trento, 14.05.09 Non-Linear Derivatives (NLD) in relativistic hadrodynamics – Results (density dependence)

25. Trento, 14.05.09 Non-Linear Derivatives (NLD) in relativistic hadrodynamics – Results (density dependence)

26. Trento, 14.05.09 Non-Linear Derivatives (NLD) in relativistic hadrodynamics – Results (density & energy dependence)

27. Trento, 14.05.09 Non-Linear Derivatives (NLD) in relativistic hadrodynamics – Results (energy dependence)

28. Trento, 14.05.09 Non-Linear Derivatives (NLD) in relativistic hadrodynamics – Results (energy dependence, optical potential)