Giessen-BUU: recent progress

1. Giessen-BUU: recent progress T. Gaitanos (JLU-Giessen) • Model outline • Relativistic transport (GiBUU) • (briefly) The transport Eq. in relativistic hadrodynamics • 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. Outline of the model…

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

4. Starting basis The MF-approach of QHD in terms of the effective Dirac equation: • Wigner transform of the 1-b-density matrix („Wigner-matrix“): The relativistic transport equation (BUU) 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

5. Numerical realization (Test-Particle Ansatz)… http://www.physik.uni-giessen.de/GiBUU/ • Discretization of the phase-space distribution f(x,p*) in terms of N „test particles“ + • Energy momentum tensor  energy density e • 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 e=e[r] in RMF  different density distr. for ground state •  nucleus not in its „real“ groundstate, but in an „excited“ state •  affects stability

6. Relativistic Thomas-Fermi (RTF) equations + meson field equations (for the different meson fields) RTF densities New initialization: method… • Here: Relativistic Thomas-Fermi (RTF) model for spherical nuclei (Horst Lenske) • variational method for energy density functional e[rp,rn]

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

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

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

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

11. EF (protons) EF (neutrons) New initialization: Fermi energies in RTF…

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

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

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

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

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

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

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

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

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

21. Non-Linear derivatives in relativistic hadrodynamics – Motivation • Starting basis QHD Lagrangian (original Walecka, 1974) • Dirac Equation: momenta and mass dressed by density dependent self energies • 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. Starting basis again QHD Lagrangian (original Walecka, 1974) Non-Linear Derivatives (NLD) in relativistic hadrodynamics – The NLD Lagrangian • Modified interaction: non-linear operators in scalar and vector int. terms • Non-linear derivative operators: • Auxiliary field: structure not relevant (no rearrangement in nuclear matter) • Mass term: just to not re-normalize the standard QHD couplings • Parameter L: ~hadronic mass scale, e.g., L=1 GeV

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

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

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

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

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

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