Effect of particle-vibration coupling on the single-particle states: a consistent study

within the Skyrme framework

G. Colò

JAPAN-ITALY EFES Workshop

Torino, Sep. 6-8 2010

• P.F. Bortignon (Università degli Studi and INFN, Milano, Italy)

• H. Sagawa (The University of Aizu, Japan)

Co-workers

Topic of this talk

• What is the status of modern particle-vibration coupling (PVC) calculations ?

• How well can we reproduce s.p. spectra ?

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 is increasing effort to try to describe the s.p. spectroscopy. The shell model can describe well the np-nh couplings, less well the coupling with shape fluctuations.

There is an open debate to which extent density-functional methods can describe the s.p. spectroscopy (without dynamical effects).

Z N

Energy density functionals (EDFs)

EE effHH

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.

• 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 !

EDFs vs. many-body approaches

In the standard theory of quantum many-particle systems one has a more general picture in which dynamical effects play a significant role.

A set of closed equations for G, Π(0), W, Σ, Γ can be written (v12 given). They can be found e.g. in the famous paper(s) by L. Hedin in the case of the Coulomb force – they hold more generally (except…).

The Dyson equation reads

in terms of the one-body Green’s functionEDF = static limit

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) 044306

In the Dyson equation

we assume the self-energy is given by the coupling with RPA vibrations

In a diagrammatic way

2nd order PT:

ε + <Σ(ε)>

+ + … =

Particle-vibration coupling

Particle-vibration coupling (PVC) for nuclei

P. Papakonstantinou et al., Phys. Rev. C 75, 014310 (2006)

• 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)

+ … + =

W

G

• THE MAIN PROBLEM (in the nuclear case):

A LOT OF UNCONTROLLED APPROXIMATIONS HAVE BEEN MADE WHEN IMPLEMENTING THE 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 δU and their properties are taken from experiment. No treatment of spin and isospin.

One calculates the expressions corresponding to the diagrams using standard rules.

The signs of the denominators are such that “as a rule” particle states (hole states) close to the Fermi energy are shifted downwards (upwards).

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

Despite quantitative differences old calculations agree that one needs to introduce dynamical effects to explain the density of s.p. levels. This is associated to the effective mass.

• 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.

RPA

microscopic Vph• 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 Veff in the PVC vertex, approximations on the vibrational w.f.)

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.

The whole phonon wavefunction is considered, and all the terms of the Skyrme force enter the p-h matrix elements

A consistent study within the Skyrme framework

Our main result: the (t1,t2) part of Skyrme tend to cancel quite significantly the (t0,t3) part. We have also compared with the Landau-Migdal approximation.

40Ca (neutron states)

Δεi is the expectation value of Re Σi

Use of the Landau-Migdal approximation

The (t0,t3) part of the interaction, reads, for IS phonons

The velocity-dependent part is, using the Landau-Migdal approximation

Exact = -0.95 MeV

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) = 0.95 MeV

σ(including PVC) = 0.62 MeV

208Pb (neutron states)

• The tensor contribution plays a role in this case. In principle all parameters should be refitted after PVC.

• The r.m.s. difference between experiment and theory is:

σ(HF+tensor) = 1.51 MeV

σ(including PVC) = 1.21 MeV

• Do we learn in this case by looking at isotopic trends ? We need pairing and proper continuum treatment if we wish to go towards weakly bound systems (cf. K. Mizuyama talk).

• 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…

A few conclusions

• The aim of this contribution consists in showing the feasibility of fully MICROSCOPIC calculations including the particle-vibration coupling.

• Our approach takes into account Skyrme forces consistently. Results are (slightly) better than simple EDFs.

• We can also discuss s.p. spectroscopic factor.

• Technical progress in our calculations is still underway.

Backup slides

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 !

Hedin’s equations

(natural units)

A reminder on effective mass(es)

E-mass: m/mE k-mass: m/mk

The (time-consuming) calculations is much simplified if the interaction is simply a density-dependent delta-force:

And

becomes

Contribution of phonons with different multipolarity

Upper panel: particle states. Lower panel: hole states.

Signs fixed by energy denominators ω-Eint+iη

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.