1 The Random Phase Approximation in Nuclear Physics  Lay out of the presentation: 1. Linear response theory: a brief reminder 2. Non-relativistic RPA (Skyrme) 3. Relativistic RPA (RMF) 4. Extension to QRPA 5. Beyond RPA.

2 Linear Response Theory  In the presence of a time-dependent external field, the response of the system reveals the characteristics of the eigenmodes.  In the limit of a weak perturbing field, the linear response is simply related to the exact two-body Green’s function.  The RPA provides an approximation scheme to calculate the two-body Green’s function..

3  Adding a time-dependent external field:.

4 First order response as a function of time.

5 Two-body Green’s Function and density-density correlation function.

6 Linear response function and Strength distribution

7 Main results:  The knowledge of the retarded Green’s function gives access to:  Excitation energies of eigenmodes (the poles)  Transition probabilities (residues of the response function)  Transition densities (or form factors), transition currents, etc… of each excited state.

8 TDHF and RPA (1)

9 TDHF and RPA (2) And by comparing with p.5

10 Residual p-h interaction

11 Analytic summation of single- particle continuum 1) u, w are regular and irregular solutions satisfying appropriate asymptotic conditions 2) This analytic summation is not possible if potential U is non-local.

12 Approximate treatments of continuum (1) T. Vertse, P. Curutchet, R.J. Liotta, Phys. Rev. C 42, 2605 (1990).

13 Approximate treatments of continuum (2)  Calculate positive-energy s.p. states with scattering asymptotic conditions, and sum over an energy grid along the positive axis, up to some cut-off  Sum over discrete states of positive energy calculated with a box boundary condition.

14 Transition densities and divergence of transition currents Solid: GQR Dashed: low-lying 2+ Dotted: empirical

15 Convection current distributions GQR in 208PbLow-lying 2+ in 208Pb

16 Finite temperature Applications: evolution of escape widths and Landau damping of IVGDR with temperature.

17 RPA on a p-h basis

18 A and B matrices

19 Restoration of symmetries  Many symmetries are broken by the HF mean- field approximation: translational invariance, isospin symmetry, particle number in the case of HFB, etc…  If RPA is performed consistently, each broken symmetry gives an RPA (or QRPA) state at zero energy (the spurious state)  The spurious state is thus automatically decoupled from the physical RPA excitations  This is not the case in phenomenological RPA.

20 Sum rules  For odd k, RPA sum rules can be calculated from HF, without performing a detailed RPA calculation.  k=1: Thouless theorem  k=-1: Constrained HF  k=3: Scaling of HF.

21 Phenomenological RPA  The HF mean field is replaced by a parametrized mean field (harmonic oscillator, Woods-Saxon potential, …)  The residual p-h interaction is adjusted (Landau- Migdal form, meson exchange, …)  Useful in many situations (e.g., double-beta decay)  Difficulty to relate properties of excitations to bulk properties (K, symmetry energy, effective mass, …).

22 Relativistic RPA on top of RMF

23 Fermi states and Dirac states

24 Single-particle spectrum

25 The Hartree polarization operator

26 Fermi and Dirac contributions

27 The RRPA polarization operator  Generalized meson propagator for density- dependent case (Z.Y. Ma et al., 1997).

28 Diagrammatic representation

29 RRPA and TDRMF  One can derive RRPA from the linearized version of the time-dependent RMF  At each time, one assumes the no-sea approximation, i.e., ones keeps only the positive energy states  These states are expanded on the complete set (at positive and negative energies) of states calculated at time t=0  This is how the Dirac states appear in RRPA. How important are they?  From the linearized TDRMF one obtains the matrix form of RRPA, but the p-h configuration space is much larger than in RPA!.

30 Effect of Dirac states on ISGMR

31 Effect of Dirac states on ISGQR

32 Effect of Dirac states on IVGDR

33 Including continuum in RRPA

34 QRPA (1)  The scheme which relates RPA to linearized TDHF can be repeated to derive QRPA from linearized Time-Dependent Hartree-Fock- Bogoliubov (cf. E. Khan et al., Phys. Rev. C 66, (2002))  Fully consistent QRPA calculations, except for 2- body spin-orbit, can be performed (M. Yamagami, NVG, Phys. Rev. C 69, (2004)).

35 QRPA (2)  If Vpp is zero-range, one needs a cut-off in qp space, or a renormalisation procedure a la Bulgac. Then, one cannot sum up analytically the qp continuum up to infinity  If Vpp is finite range (like Gogny force) one cannot solve the Bethe-Salpeter equation in coordinate space  It is possible to sum over an energy grid along the positive axis ( Khan - Sandulescu et al., 2002).

36 Pairing window method K. Hagino, H. Sagawa, Nucl. Phys. A 695, 82 (2001).

37 2+ states in 120Sn

38 2+ states in 120Sn, with smearing

39 3- states in 120Sn, with smearing

40 Beyond RPA (1)  Large amplitude collective motion: Generator Coordinate Method  RPA can describe escape widths if continuum is treated, and it contains Landau damping, but spreading effects are not in the picture  Spreading effects are contained in Second RPA  Some applications called Second RPA are actually Second TDA: consistent SRPA calculations of nuclei are still waited for.

41 Beyond RPA (2)  There exist models to approximate SRPA:  The quasiparticle-phonon model (QPM) of Soloviev et al. Recently, attempts to calculate with Skyrme forces (A. Severyukhin et al.)  The ph-phonon model: see G. Colo. Importance of correcting for Pauli principle violation  Not much done so far in relativistic approaches.

42 Beyond RPA (3)  Particle-vibration coupling

43 Effect of particle-vibration coupling

44 Acknowledgments Thanks to Wenhui LONG for Powerpoint tutoring. Thanks to Wenhui LONG for Powerpoint tutoring.