Methods beyond mean field: particle-vibration coupling

1. Workshop on “Nuclear magic numbers: new features far from stability” May 3rd-5th, 2010 CEA/SPhN (Saclay,France) Methods beyond mean field: particle-vibration coupling G. Colò

2. Z N The problem of the single-particle states The description in terms of indipendent nucleons lies at the basis of our understanding of the nucleus, but in many models the s.p. states are not considered (e.g., liquid drop, geometrical, or collective models). There are models in which there is increasing effort to describe the details of the s.p. spectroscopy (e.g., the shell model). It is fair to say that we miss a theory which can account well for the s.p. spectroscopy of (medium-heavy) stable and exotic nuclei. Cf. T. Otsuka, A. Schwenk…

3. Topics of this talk • How well can we discuss s.p. spectra using energy density functionals and/or extensions ? • What is the status of modern particle-vibration coupling (PVC) calculations ? • Role of PVC in the description of excited states (e.g., giant resonances).

4. Co-workers • P.F. Bortignon, M. Brenna, K. Mizuyama (Università degli Studi and INFN, Milano, Italy) • M. Grasso, N. Van Giai (IPN Orsay, France) • H. Sagawa (The University of Aizu, Japan)

5. Topic 1 : PVC for s.p. states (vs. mean-field or EDF)

6. 1-body density matrix Slater determinant Energy density functionals (EDFs) • The minimization of E can be performed either within the nonrelativistic or relativistic framework → Hartree-Fock or Hartree equations • In the former case one often uses a two-body effective force and defines a starting Hamiltonian; in the latter case a Lagrangian is written, including nucleons as Dirac spinors and effective mesons as exchanged particles. • 8-10 free parameters (typically). Skyrme/Gogny vs. RMF/RHF. • The linear response theory describes the small oscillations, i.e. the Giant Resonances (GRs) or other multipole strength → (Quasiparticle) Random Phase Approximation or (Q)RPA • Self-consistency !

7. Difference between self-consistent mean field (SCMF) and energy density functionals In the self-consistent mean field (SCMF) one starts really from an effective Hamiltonian Heff = T + Veff, and THEN builds < Φ | Heff | Φ > and defines this as E. In DFT, one builds directly E[ρ]. → More general !

8. Are the present functionals general enough ? Importance of the tensor terms. Cf. T. Otsuka et al. (tensor terms added to Skyrme forces: MSU, Milano, Warsaw, Bordeaux/Lyon/Saclay, Orsay) The most relevant effect concerns the spin-orbit splittings.

9. W. Zou, G.C., Z. Ma, H. Sagawa, P.F. Bortignon, PRC 77, 014314 (2008)

10. J. Phys. G: Nucl. Part. Phys. 37 (2010) 064013 Different approaches to the s.p. spectroscopy • EDF: • The energy of the last occupied state is given by ε=E(N)-E(N-1). • This is not a simple difference between different values of the same energy functional, because the even and odd nuclei include densities with different symmetry properties (odd nuclei include time-odd densities). • The above equation can be extended to the “last occupied state with given quantum numbers”. • THE MAIN LIMITATION IS THAT THE FRAGMENTATION OF THE S.P. STRENGTH CANNOT BE DESCRIBED.

11. NPA 553, 297c (1993) A. Gade et al., PRC 77 (2008) 044306 Experiment: (e,e’p), as well as (hadronic) transfer or knock-out reactions, show the fragmentation of the s.p. peaks. • Problems: • Ambiguities in the definition: use of DWBA ? Theoretical cross section have ≈ 30% error. • Consistency among exp.’s. • Dependence on sep. energy ? S ≡ Spectroscopic factor

12. + + … = • PVC: • In principle it is a many-body approach. • A set of closed equations for G, Π(0), W, Σ, Γ can be written (v12 given). • The Dyson equation reads • in terms of the one-body Green’s function • We assume the self-energy: 2nd order PT: ε + <Σ(ε)> Particle-vibration coupling

13. Second-order perturbation theory In most of the cases the coupling is treated phenomenologically. In, e.g., the original Bohr-Mottelson model, the phonons are treated as fluctuations of the mean field δU and their properties are taken from experiment. No treatment of spin and isospin. • THE MAIN LIMITATION: • A LOT OF UNCONTROLLED APPROXIMATIONS HAVE BEEN MADE WHEN IMPLEMENTING THIS THEORY IN THE PAST !

14. C. Mahaux et al., Phys. Rep. 120, 1 (1985)

15. + … + = G P. Papakonstantinou et al., Phys. Rev. C 75, 014310 (2006) W • For electron systems it is possible to start from the bare Coulomb force: • In the nuclear case, the bare VNN does not describe well vibrations ! Phys. Stat. Sol. 10, 3365 (2006) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA

16. RPA microscopic Vph • RMF + PVC calculations by P. Ring et al.: they also approximate the phonon part. • The most “consistent” calculations which are feasible at present start from Hartree or Hartree-Fock with Veff, by assuming this includes short-range correlations, and add PVC on top of it. • Very few ! • Pioneering Skyrme calculation by V. Bernard and N. Van Giai in the 80s (neglect of the velocity-dependent part of Veff in the PVC vertex, approximations on the vibrational w.f.)

17. 208Pb PVC EDF The r.m.s. deviations between theory and experiment are 0.9 MeV for this EDF implementation and range between 0.7 and 1.2 MeV for PVC calculation. Lack of systematics !

18. How to compare EDF and PVC ? ωn Since the phonon wavefunction is associated to variations (i.e., derivatives) of the denisity, one could make a STATIC approximation of the PVC by inserting terms with higher densities in the EDF.

19. Removing uncontrolled approximations We have implemented a version of PVC in which the treatment of the coupling is exact, namely we do not wish to make any approximation in the vertex. All the phonon wavefunction is considered, and all the terms of the Skyrme force enter the p-h matrix elements Our main result: the (t1,t2) part of Skyrme tend to cancel quite significantly the (t0,t3) part.

20. 40Ca (neutron states) • The tensor contribution is in this case negligible, whereas the PVC provides energy shifts of the order of MeV. • The r.m.s. difference between experiment and theory is: • σ(HF+tensor) = 1.40 MeV • σ(including PVC) = 0.96 MeV

21. Still to be done… • Do we learn in this case by looking at isotopic trends ? • We have a quite large model space of density vibrations. Do we miss important states which couple to the particles ? • Do we need to go beyond perturbation theory ? • Is the Skyrme force not appropriate ? TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA

22. The low-lying 2+ state is absent in 40Ca. It can give a shift to the d5/2 state in 42,44Ca and change the above pattern: will this be in the direction of experiment ??

23. Spectroscopic factors As discussed above, there is no clear matching between the experimental and the theoretical definitions of these quantities. Theory: well defined ! Experiment ? 40Ca (neutron states) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA

24. A reminder on effective mass(es) E-mass: m/mE k-mass: m/mk TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA

25. Topic 2 : PVC for excited states (Giant Resonances)

26. Kamerdzhiev et al. 120Sn spreading width Photoabsorbtion cross section ↔ GDR Berman-Fultz Continuum-RPA → escape width Γ↑ Γexp = Γ↑ + Γ↓

27. A+Σ(E) B -B -A-Σ*(-E) Σphp’h’ (E) = Σα Vph,α(E-Eα+iη)-1Vα,p’h’ Second RPA: Γ+n = Σ Xph |ph-1> - Yph |p-1h> + Xphp’h’ |ph-1p’h’-1> - Y php’h’ |p-1hp’-1h’> The theory is formally sound (e.g., EWSRs are conserved). Handling an explicit 2p-2h basis is feasible only in light nuclei. Projecting the SRPA equations in the 1p-1h space*, one gets a RPA-like equation. Σ (E) = + + *Cf. the talk by M. Grasso

28. A+Σ(E) B -B -A-Σ*(-E) Σphp’h’ (E) = Σα Vph,α(E-Eα+iη)-1Vα,p’h’ The state α is not a 2p-2h state but 1p-1h plus one phonon Σphp’h’(E) = Pauli principle ! Re and Im Σ cf. G.F.Bertsch et al., RMP 55 (1983) 287

29. RPA continuum coupling 1p-1h-1 phonon coupling This effective Hamiltonian can be diagonalized and from its eigenvalues and eigenvectors one can extract the response function to a given operator O. It is possible to extract at the same time to calculate the branching ratios associated with the decay of the GR to the A-1 nucleus in the channel c (hole state).

30. N. Paar, D. Vretenar, E. Khan, G.C., Rep. Prog. Phys. 70, 691 (2007)

31. Z N The isobaric analog state: a stringent test The measured total width (Γexp=230 keV) is well reproduced. The accuracy of the symmetry restoration (if VCoul=0) can be established.

32. Conclusions • The aim of this contribution consists in making an overview of the existing MICROSCOPIC calculations including the particle-vibration coupling. • Few calculations exist (on top of Skyrme-HF or RMF). They seem to perform better than EDF. • The problem of the s.p. spectroscopy is indeed quite open ! • Technical progress is underway … • Still to come: the unambiguous definition of spectroscopic factors, calculations based on the bare force and … • Schemes to include PVC for the description of the GR lineshape do exist.

34. Backup slides

35. G.C., H. Sagawa, S. Fracasso, P.F. Bortignon, Phys. Lett. B 646 (2007) 227. The introduction of the tensor force improves the results. The same parameters of the tensor force have been used in Ca, Pb.

37. Hedin equations (natural units)

38. IV dipole IS quadrupole The spreading width is due to the coupling of the simple 1p-1h configurations (or 2 quasiparticle) with more complex states.

39. Call for more exclusive measurements • Simple measure of the energy of a s.p. state cannot give hints on his wavefunction. • Decay measurements can. In this case we focus on γ-decay. |1/2+> = α|s1/2> + β |d3/22+> If β is dominant this implies a decay on the |3/2+> state with the same B(E2) of the 2+ state in 132Sn. This is not the case if α is dominant. γ from deep inelastic reactions ↔ or from decay of trapped ions