Skyrme functional from a three-body pseudopotential of second order in gradients: Formalism for central terms

Background: In one way or another, all modern parametrizations of the nuclear energy density functional (EDF) do not respect the exchange symmetry associated with Pauli's principle. It has been recently shown that this practice jeopardizes multireference (MR) EDF calculations by contaminating the energy with spurious self-interactions that, for example, lead to finite steps or even divergences when plotting it as a function of collective coordinates [J. Dobaczewski et al., Phys. Rev. C 76, 054315 (2007); D. Lacroix et al., Phys. Rev. C 79, 044318 (2009)]. As of today, the only viable option to bypass these pathologies is to rely on EDF kernels that enforce Pauli's principle from the outset by strictly and exactly deriving from a genuine, i.e., density-independent, Hamilton operator.

Purpose: The objective is to build cutting-edge parametrizations of the EDF kernel deriving from a pseudopotential that can be safely employed in symmetry restoration and configuration mixing calculations.

Methods: We wish to develop the most general Skyrme-like EDF parametrization containing linear, bilinear, and trilinear terms in the density matrices with up to two gradients, under the key constraint that it derives strictly from an effective Hamilton operator. While linear and bilinear terms are obtained from a standard one-body kinetic energy operator and a (density-independent) two-body Skyrme pseudopotential, the most general three-body Skyrme-like pseudopotential containing up to two gradient operators is constructed to generate the trilinear part. The present study is limited to central terms. Spin orbit and tensor will be addressed in a forthcoming paper.

Results: The most general central Skyrme-type zero-range three-body interaction is built up to second order in derivatives. The complete trilinear EDF, including time-odd and $T=1$ pairing parts, is derived along with the corresponding normal and anomalous fields entering the Hartree-Fock-Bogoliubov equations of motion. Its building blocks are the same local densities that the standard Skyrme functional is constructed from. The central three-body pseudopotential is defined out of six independent parameters. Expressions for bulk properties of symmetric, isospin-asymmetric, and spin-polarized homogeneous nuclear matter, as well as associated Landau parameters, are given.

Conclusions: This study establishes a first step towards a new generation of nuclear EDFs that respect Pauli's principle and that can be safely used in predictive and spuriosity-free SR and MR EDF calculations.