Nuclear Structure and dynamics within the Energy Density Functional theory

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

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

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 Introduction of pairing: TDHFB Quasi-particle evolution (Active Groups: France, US, Japan…) BCS limit of TDHFB (also called Canonical basis TDHFB) TDHFB = 1000 * (TDHF) Neglect Less demanding than TDHFB Reasonable results for collective motion Ebata, Nakatsukasa et al, PRC82 (2010) Sometimes more predictive than TDHFB Scamps, Lacroix, Bertsch, Washiyama, PRC85 (2012)

4. Collective motion

5. Pairing effect on nuclear collective motion Comparison TDHF+BCS / QRPA Strength distribution in deformed 34Mg Illustration with the GQR Q20 Q22 QRPA TDHF+BCS QRPA: C. Losa, et al PRC 81, (2010). Almost no difference between TDHF+BCS and TDHFB (QRPA) Main effect of pairing is to set the deformation

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

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

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

9. Systematic in deformed nuclei: fragmentation and damping Damping is more complex: Energy splitting: oblate prolate 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 Low-lying sector Collective sector Low lying 2+ states Collective motion Mean-field Exp TDHF+BCS QRPA (Bertsch, Terasaki, Engel)

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

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

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

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

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

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

18. Difficulties Fitting the Equation of state is not so simple r2, r3 ,r4 Energy Requires at list a 4 body interaction r2, r3 density Fitting both mean-field and pairing With the same interaction also not easy r2, r3 ,r4 , r5 Energy density

19. Summary/Discussion: Recent progress in the development of transport model with pairing Systematic of collective motion and low lying excitations Mean energy, … are quite nicely reproduced Damping is underestimated Low lying states are poorly described 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.