Session C9: Minisymposium: Toward a Universal Density Functional Theory for Nuclei I

Towards a Universal Density Functional Theory for nuclei : challenges to overcome

In the study of medium to heavy mass nuclei, nuclear Density Functional Theory (DFT), based on the self-consistent Hartree-Fock-Bogoliubov (HFB) method and its extensions, is the theoretical tool of choice. As more exotic beams facilities are being built or proposed to be built around the world, DFT is on the edge of becoming a predictive theory for all nuclei but the lightest. This is not only true for ground state properties, such as binding energies, radii or multipoles of the density, but also for low energy excited states of different types and for the calculation of their decay probabilities. These decisive advances are coming to life thanks to the development of better energy functionals and thanks to the increase of computer resources. However, the needed accuracy and predictive power still leaves much to be desired and DFT is facing important challenges in its quest for the truly universal energy density functional which should be able to describe properties of finite nuclei in all possible exotic modes as well as extended asymmetric nucleonic matter. In this talk, I will elaborate on those challenges before discussing the results of two ongoing studies aiming at tackling some of them. The first one is an attempt to construct the pairing part of the nuclear functional starting from the bare nucleon-nucleon interaction. Indeed, and despite its major role, the nature of pairing correlations in nuclei is largely unknown and has mostly relied on pure phenomenology so far. The long-term goal is to identify the in-medium effects at play in the pairing channel and model them via isoscalar and isovector density-dependences and/or gradient corrections. I will describe a way to realize the first step of such a program, that is, to set up a functional which is able to reproduce the pairing properties generated by the full realistic $AV18$ bare nucleon-nucleon force in finite nuclei. The second study deals with the conceptual problem one faces when defining the Particle Number Projected (PNP) HFB method within the context of DFT. Indeed, one manipulates in this case an ill-defined functional presenting divergences and jumps whenever a single particle state crosses the Fermi surface. I will discuss how those divergences and jumps are related to a spurious ``self-pairing'' interaction between paired nucleons and how one can identify and remove the corresponding spurious contributions to the projected energy.   

Density Functional Theory for Fermions close to the Unitary Regime

We consider interacting Fermi systems close to the unitary regime and compute the corrections to the energy density that are due to a large scattering length and a small effective range. Our approach exploits the universality of the density functional and determines the corrections from the analyical results for the harmonically trapped two-body system. The corrections due to the finite scattering length compare well with the result of Monte Carlo simulations. We also apply our results to symmetric neutron matter.   

Energy Functionals from Low-Momentum Potentials

The nonperturbative nature of conventional inter-nucleon interactions is strongly scale or resolution dependent, and can be radically modified by using the renormalization group to lower the momentum cutoff of the two-nucleon potential. Recent calculations demonstrate that using low-momentum potentials (``$V_{{\rm low\ }k}$'') with consistent three-body forces leads to saturating nuclear matter at the Hartree-Fock level, with rapidly converging perturbative corrections in the particle-particle channel $^a$. With these interactions, the density matrix expansion (DME) becomes a natural tool for the microscopic construction of a universal energy functional for nuclei $^b$. By varying the cutoff, the resolution dependence of the functional can be studied. The use of sharp momentum cutoffs in $V_{{\rm low\ }k}$ complicates the application of the DME in coordinate space. This problem is resolved with the recent generalization of $V_{{\rm low\ }k}$ to smooth cutoff regulators. \newline \newline $^a$S.~K.~Bogner, A.~Schwenk, R.~J.~Furnstahl and A.~Nogga, Nucl.\ Phys.\ A {\bf 763}, 59 (2005). \\ $^b$J. W. Negele and D. Vautherin, Phys.\ Rev.\ C {\bf 5}, 1472 (1972).   

The Nuclear Equation of State and Its Applications to Neutron Stars

One of the most challenging problems in both theoretical and experimental nuclear physics is to understand the nature of matter under extreme conditions of density and pressure. Observations of neutron star properties impose important constraints on the equation of state of dense matter, as the latter is the basic input quantity that enters the structure equations of these compact objects. Continuing with our systematic study of the effective nucleon- nucleon interactions in dense and isospin-asymmetric hadronic environment, we will present predictions of neutron star masses and radii obtained from our relativistic equation of state. We use realistic nucleon-nucleon potentials defined in the framework of the meson-exchange potential model and the Dirac-Brueckner approach. We will provide an overview of theoretical predictions and recent observational data. This broad outlook will help us gauge the quality of our tools and determine the importance of mechanisms beyond the present model.