1. Nuclear Structure and dynamics within the Energy Density Functional theory Denis Lacroix IPN Orsay Coll: G. Scamps, D. Gambacurta, G. Hupin M. Bender and Th. Duguet Outline: Low lying modes and Collective excitations Configuration mixing and spectroscopy Some open issues in beyond mean-field approaches

2. Generalities Pairing effect on nuclear dynamic 2n-transfer reactions 2n-break-up reactions Assié and Lacroix, PRL102 (2009) Scamps, Lacroix, PRC 87 (2013) Scamps, Lacroix, arXiv:1307.1909 Goal Systematic study of the pairing Influence on nuclear dynamics

3. Pair transfer: the nuclear structure and reaction perspective Nuclear reaction on a mesh TDHF is a standard tool : Slater Single-particle evolution Simenel, Lacroix, Avez, arXiv:0806.2714v2 BCS limit of TDHFB (also called Canonical basis TDHFB) Neglect Introduction of pairing: TDHFB Quasi-particle evolution (Active Groups: France, US, Japan…) TDHFB = 1000 * (TDHF) Less demanding than TDHFB Reasonable results for collective motion Sometimes more predictive than TDHFB Ebata, Nakatsukasa et al, PRC82 (2010) Scamps, Lacroix, Bertsch, Washiyama, PRC85 (2012)

4. Collective motion

5. Pairing effect on nuclear collective motion Comparison TDHF+BCS / QRPA Illustration with the GQR Strength distribution in deformed 34 Mg QRPA: C. Losa, et al PRC 81, (2010). QRPA TDHF+BCS Q 22 Q 20 Almost no difference between TDHF+BCS and TDHFB (QRPA) Main effect of pairing is to set the deformation

6. Systematic in Spherical nuclei Isoscalar GQR Isovector GQR 263 nuclei 324 nuclei 38 S Scamps, Lacroix, PRC88 (2013)

7. Systematic in deformed nuclei Scamps, Lacroix, arXiv:1401.5211 Excitation operators

8. Systematic in deformed nuclei: illustration QRPA: Yoshida, Nakatsukasa, PRC88 (2013) Collective energy Damping width

9. Systematic in deformed nuclei: fragmentation and damping prolate oblate Energy splitting: Damping is more complex: High order deformation is important Scamps, Lacroix, arXiv:1401.5211

10. Systematic in deformed nuclei: triaxial nuclei 54 triaxial nuclei Scamps, Lacroix, arXiv:1401.5211

11. Difficulties TDHF+BCS Exp QRPA (Bertsch, Terasaki, Engel) Low lying 2 + states Collective motion Mean-field Collective sector Low-lying sector

12. Lacroix, Ayik, Chomaz, Prog. Part and Nucl. Phys. (2004) Improving collective state description Standard RPA states Coupling to ph-phonon Coupling to 2p2h states

13. Lacroix, Ayik, Chomaz, Prog. Part and Nucl. Phys. (2004) Improving collective state description GQR in 208 Pb GQR in 40 Ca EWSR

14. Remaining difficulty Perturbative treatment of the coupling ? Within the EDF: Requires better defined techniques Requires to define power counting No cut-off with cut-off Most often UV divergent No real perturbative scheme See for instance: Moghrabi, Grasso, Phys. Rev. C 86 (2012)

15. Improving low-lying state description TDHF+BCS Exp QRPA Low lying 2 + states Prediction from TDHF+BCS: Implementing configuration mixing Bertsch et al, PRL99 (2007)

16. 74 Kr Mean-Field Energy 0+0+ 0+0+ 0+0+ 2+2+ 2+2+ 2+2+ 4+4+ 4+4+ 6+6+ 8+8+ Correlation Energy Single Reference (SR)- Mean-Field Ground state Multi- Ref. (MR)-GCM Beyond mean-field Configuration mixing within Energy Density Functional Restoration of broken symmetries (particle number, angular momentum, …) Excited state and spectroscopy … but we are starting from a functional theory framework Formal and practical difficulties Yes but

17. Single Reference (SR)- Mean-Field Ground state Multi- Ref. (MR)-GCM Beyond mean-field Towards systematic studies with mean-field Configuration mixing within Energy Density Functional M mesh points Lacroix et al, PRC79 (2009), Bender et al, PRC79 (2009), Duguet et al, PRC79 (2009) Problem due to the direct mapping Between Hamiltonian and EDF Connected to self-interaction and self-pairing A solution has been proposed (not for   ) This solution does not work For most complex calculations Before correction Corrected SIII force Angular momentum +particle number proj. Requires to come back to true interaction?

18. Difficulties  2,  3  2,  3  4  2,  3  4  5 Fitting the Equation of state is not so simple Requires at list a 4 body interaction Fitting both mean-field and pairing With the same interaction also not easy Energy density

19. Summary/Discussion: Systematic of collective motion and low lying excitations Recent progress in the development of transport model with pairing Damping is underestimated Low lying states are poorly described Mean energy, … are quite nicely reproduced Need for Beyond-Mean-field approach Perturbative techniques Non-perturbative techniques like GCM Beyond Mean-field approaches within a functional theory have to be applied with special caution.