1. Nuclear density functional theory with a semi-contact 3-body interaction Denis Lacroix IPN Orsay Outline Infinite matter Results Energy density function and symmetry breaking Symmetry restoration and configuration mixing: difficulties within EDF Problems with    terms in Beyond Mean-Field calculation. Proposal of a new EDF based on semi-contact three-body interaction Collaboration: M. Bender, K. Bennaceur, T. Duguet. G. Hupin

2. Mean-Field Theories Configuration Mixing within Energy Density Functional Sly4-Bender,Bertsch,Heenen, PRL (2005) E exp – E th (MeV) E exp - E th 20 40 60 100 80 120 140 Neutron number N Sly4-Bertsch,Sabbey, Uusnakki PRC (2005) Energy scale! 300 1000 1600 20 40 60 100 80 120 140 160 Neutron number N (Skyrme, Gogny, …) To make the functional predictive, we use and abuse of symmetry breaking Pairing Correlations Odd-even effects Pairing Correlations Odd-even effects Quadrupole Correlation - Rotational Bands Quadrupole Correlation - Rotational Bands Surface Vibration Surface Vibration Translation Rotation Particle number …

3. Single Reference (SR)- Mean-Field Configuration Mixing within Energy Density Functional Sly4-Bender,Bertsch,Heenen, PRL (2005) E exp – E th (MeV) E exp - E th 20 40 60 100 80 120 140 Neutron number N Sly4-Bertsch,Sabbey, Uusnakki PRC (2005) Energy scale! 300 1000 1600 20 40 60 100 80 120 140 160 Neutron number N (Skyrme, Gogny) Multi- Ref. (MR)-GCM Bender et al, PRC74 (2006) 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

4. GCM is not so easy to use in EDF M=9 19 39 59 79 99 M=9 M=99 Divergence Jump Multi- Ref. (MR)-GCM M: number of Mesh points Application of conf. mixing in EDF Needs to be regularized

5. Practical and conceptual difficulties in Configuration Mixing within EDF (I) Lacroix, et al, PRC79 (2009), (II) Bender et al PRC79 (2009), (III) Duguet et al, PRC79 (2009). Correction is possible Before correction Corrected SIII force with Example: particle number projection or with Projection on Good particle number Some uncontrolled and not understood spurious contributions persists M. Bender (private communication)

6. Duguet et al, PRC79 (2009). The EDF energy will depend on the integration contour and becomes ill defined We can use  2  3  4  5 but not    The specific problem of the density dependent term   Example: particle number projection Transition density Is a complex number Is multivalued in the complex plane

7. Problem:    is very useful and practical Interacting fermions in different regimes of density Fictitious equation of state Low density regime Saturation density regime At low density Bulgac, Forbes, … High density regime Lee-Yang formula Unitary gas Galitskii formula At saturation

8. Strategy : restart from true interaction to mimic density dependent term Nuclear EDF phenomenology Effective interaction 1-body, 2-body, 3-body ? Strategy 1 Hamiltonian 1-body, 2-body, 3-body Many-body technology Single-Ref. Multi-Ref. Back to the original strategy Constraints: - start from a Hamiltonian - no   Finite range ? Zero-range (requires many terms)

9. Back to the construction of effective Hamiltonian With finite range interaction Goal: mimic the proper dependence of the energy around saturation i.e. Semi-contact 3-body interaction Jacobi coordinate (r,R) Idea: take a zero range in R 2-body interaction c 1 and c 2 are free parameters Infinite matter case Convenient only for Need for at least 3-body interaction

10. Semi-contact 3-body interaction Effective Hamiltonian with 2-body and 3-body Semi-contact 3-body interaction Jacobi coordinate (r,R) Idea: take a zero range in R Our starting point: Lacroix, Bennaceur, Phys. Rev. C91 (2015) and (antisymmetrization)

11. EDF based on semi-contact 3-body interaction Our starting point: Lacroix, Bennaceur, Phys. Rev. C91 (2015) Infinite matter case

12. EDF based on semi-contact 3-body interaction EDF = 2-body + 3-body (semi-contact) zero-range or finite-range Lacroix, Bennaceur, Phys. Rev. C91 (2015). Illustration I : Skyrme (2-body)+3-body Properties after a global fit:

13. EDF based on semi-contact 3-body interaction EDF = 2-body + 3-body (semi-contact) zero-range or finite-range Illustration II : Gogny (2-body)+3-body Lacroix, Bennaceur, Phys. Rev. C91 (2015). Properties after a global fit:

14. Some remarks: towards application to nuclei Complete expression Two difficulties : First step : simplified interaction Neglect the exchange with the third particle Equivalent to the result of a density dependent finite range 2-body interaction is not on the mesh Exchange with the third particles

15. Simplified 2-body interaction: some remarks Can we still mimic a    behavior ? Yes 2-body functional

16. Simplified 2-body interaction: some remarks The functional is then equivalent to the one derived from the effective density dependent 2-body interaction: Same as F. Chappert (PhD) with  =1 Chappert, Pillet, Girod, Berger, Phys. Rev. C92 (2015)

17. Summary In LDA only functional of  2  3  4  5 could be used, not    Lacroix et al, PRC79 (2009), Bender et al, PRC79 (2009), Duguet et al, PRC79 (2009) To mimic density-dependent term, we proposed a new 3-body interaction Gives a good description on infinite systems In different spin isospin channels M=9 19 39 59 79 99 M=9 M=99 Divergence Jump Combining Nuclear EDF and conf. Mean-field leads to specific problems Several solutions have been proposed -regularization of divergences -New EDF with 2-body density matrix Future dev: application to finite systems One solution is to go back to effective Hamiltonian And keep consistency between mean-field and pairing