Tailoring new interactions in the nuclear many-body problem for beyond- mean-field models Marcella Grasso Tribute to Daniel Gogny

Outline Context: Energy Density Functional (EDF) theory. Mean-field-based models Beyond the mean field: Which interaction to use? Focus on the second-order EOS of nuclear matter 1) Regularization and adjustment of parameters 2) Description of the low-density limit in neutron matter 3) Renormalizability of the problem Conclusions and Perspectives This work is done in collaboration with: Jerry Yang, Bira van Kolck, Denis Lacroix, IPN Orsay Gianluca Colo’, Xavier Roca-Maza, Milano University

Nuclear structure, reactions and neutron stars Energy Density Functional (EDF) models Beyond-mean -field models (correlations). - Describing complex phenomena - Improving predictive power of models -NUMERICAL COMPLEXITY -DIVERGENCES -INTERACTION ? Phenomenological effective interactions adjusted at the mean- field level : double counting

EDF: calculations are currently done with Skyrme and Gogny forces - Double counting and ultraviolet divergences – some specific solutions exist, for instance a subtraction method -> PVC and SRPA (talk of D. Gambacurta) Within the EDF: designing interactions adapted for beyond mean-field models (cancellation of double counting and regularization of divergences) GENERAL OBJECTIVE

The mean-field approximation represents the leading order of the perturbative Dyson expansion for the many-body problem The total energy at first order is calculated by computing the direct and exchange following diagrams Illustration: equation of state (EOS) of matter

Going beyond the leading order in nuclear matter By including the 2 nd - order contribution in the EOS of nuclear matter: Interactions adjusted at the mean-field level. Double counting Zero-range forces -> ultraviolet divergences beyond the mean field (regularization is needed) Analyzing the renormalizability of the problem (independence on the chosen regularization) 12 3 Regularization techniques Power counting analysis Effective field theories (chiral interactions)

1 Nuclear matter. Regularization and adjustment of parameters (Skyrme interaction without spin-orbit) Spin-exchange operator Nine parameters to adjust

Equation of state of nuclear matter with a Skyrme-type interaction This second-order contribution diverges with a Skyrme-type interaction Moghrabi, Grasso, Colo’, PRL 105, (2010) Yang, Grasso, Roca-Maza, Colo’, Moghrabi, in preparation

Second-order contribution to the EOS v -> interaction G -> propagator Effective mass k F1 and k F2 -> Fermi momenta of the two nucleons In symmetric matter, neutron and proton Fermi momenta are the same:

Illustration: EOS of symmetric matter and cutoff regularization First order Second order Convenient change of variables: using the incoming and outgoing relative momenta k and k’ Then the propagator can be simplified and written as *

Having introduced the combinations of Skyrme parameters

Second-order contribution for symmetric matter (without the spin- orbit term). Sum of the two following terms (cutoff on k’)

Asymptotic behavior:

Illustration for symmetric matter and cutoff regularization First: computation of the second-order contribution Yang, Grasso, Roca-Maza, Colo’, Moghrabi, in preparation Second: adjustment of parameters (double counting and divergence). Benchmark EOS: SLy5 mean field Second -order EOS Only second -order term Set of parameters for each cutoff

Pressure and incompressibility PRESSURE INCOMPR. Yang, Grasso, Roca-Maza, Colo’, Moghrabi, in preparation

Finite part of the second-order EOS of neutron matter Different combinations of parameters

Same k N dependence as the second term in the Lee-Yang low-density expansion in (ak N ) a is the scattering length => fm in neutron matter Lee and Yang, Phys. Rev. 105, 1119 (1957)

2 Low-density for neutron matter (t0-t3 model) Spin-exchange operator Yang, Grasso, Lacroix, in preparation

Neutron matter at usual density scales. Example of Lyon-Saclay forces adjusted on the neutron EOS SLy5 -> Chabanat et al. NPA 627, 710 (1997); 635, 231 (1998), 643, 441 (1998) Akmal et al. -> PRC 58, 1804 (1998) … and what about very low densities? (Lee Yang k F dependence)

Low-density regime LOW DENSITY

Can we reproduce the low density with the mean field? Lee-Yang expansion (first terms) Mean field EOS (t0-t3 model) ? Yes, for α=1/3

We have to constrain the parameters in the following way:

It is possible to constrain the low-density behavior, with α=1/3, and to adjust x0 and x3 for reproducing a reasonable EOS for symmetric matter But the EOS of neutron matter is completely wrong at ordinary scales of densities

The second-order contribution has the k F 4 term Can we get simultaneously the low-density behavior (with a correct value of the scattering length a) and a reasonable EOS for usual densities ? Yang, Grasso, Lacroix, in preparation Direction: going to higher orders … but only second order seems to be not enough to correctly reproduce the EOS at both density scales (with the correct value of the scattering length)

3 Nuclear matter. Renormalizability (Skyrme interaction without spin-orbit) Spin-exchange operator

Is our problem renormalizable? Analysis done for symmetric matter Yang, Grasso, van Kolck, Moghrabi, in preparation Yes, if the theory is not dependent on the regularization (observables are independent of the cutoff) Objective: demanding renormalizability by a redefinition of the existing parameters A step towards the more general objective: searching for the correct power counting that indicates the proper hierarchy of allowed interactions

Demanding renormalizability through a redefinition of the existing Skyrme parameters at each order Finite Absorbed Divergent

Different contributions Absorbed Divergent

Yang, Grasso, van Kolck, Moghrabi, in preparation FIT

Conclusions and Perspectives 1.Interaction tailored for beyond mean field in the EDF framework 2.Regularization (cutoff and dimensional), renormalizability, low density in nuclear matter 3.Perspectives: power counting analysis, combining low-density and standard density scales. Continuing towards applications to nuclei