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

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. Z N Cf. T. Otsuka, A. Schwenk…

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).

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) Co-workers

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

Energy density functionals (EDFs) Slater determinant 1-body density matrix 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 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 !

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 H eff = T + V eff, and THEN builds and defines this as E. In DFT, one builds directly E[ρ]. → More general !

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.

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

Different approaches to the s.p. spectroscopy J. Phys. G: Nucl. Part. Phys. 37 (2010) 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.

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

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

THE MAIN LIMITATION: A LOT OF UNCONTROLLED APPROXIMATIONS HAVE BEEN MADE WHEN IMPLEMENTING THIS THEORY IN THE PAST ! 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 δ U and their properties are taken from experiment. No treatment of spin and isospin.

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

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

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

208 Pb PVCEDF 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 !

How to compare EDF and PVC ? ωnωn 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.

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 Removing uncontrolled approximations Our main result: the (t 1,t 2 ) part of Skyrme tend to cancel quite significantly the (t 0,t 3 ) part.

40 Ca (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

TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA 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 ? Still to be done…

The low-lying 2 + state is absent in 40 Ca. It can give a shift to the d 5/2 state in 42,44 Ca and change the above pattern: will this be in the direction of experiment ??

TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA StateS(th.)S(exp.) 1f5/ ±0.16 2p1/ p3/ f7/ ± Ca (neutron states) Spectroscopic factors As discussed above, there is no clear matching between the experimental and the theoretical definitions of these quantities. Theory: well defined ! Experiment ?

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

Topic 2 : PVC for excited states (Giant Resonances)

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

Second RPA: Γ + n = Σ X ph |ph -1 > - Y ph |p -1 h> + X php’h’ |ph -1 p’h ’-1 > - Y php’h’ |p -1 hp’ -1 h’> 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 ++ A+Σ(E) B -B -A-Σ*(-E) Σ php’h’ (E) = Σ α V ph,α (E-E α +iη) -1 V α,p’h’

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

RPAcontinuum 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).

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

ZN The measured total width (Γ exp =230 keV) is well reproduced. The accuracy of the symmetry restoration (if V Coul =0) can be established. The isobaric analog state: a stringent test

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.

Backup slides

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

Hedin equations (natural units)

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

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|s 1/2 > + β |d 3/2  2 + > If β is dominant this implies a decay on the |3/2 + > state with the same B(E2) of the 2 + state in 132 Sn. This is not the case if α is dominant. γ from deep inelastic reactions ↔ or from decay of trapped ions