Introduction

The Skyrme force was introduced into nuclear physics more than half a century ago [1,2] and it is still a concept that is widely used in methods striving to determine properties of nuclei irrespective of their mass number and isospin. However, at present we understand this concept in a significantly different way than it was originally proposed. Indeed, instead of using this force as an effective interaction within the Hartree-Fock (HF) approximation, we rather focus on the underlying Skyrme energy density functional (EDF), without direct references to the effective interaction or HF approximation.

In electronic systems, the use of functionals of density is motivated by formal results originating from the Hohenberg-Kohn [3] and Kohn-Sham theorems [4], whereby exact ground-state energies of many-fermion systems can be obtained by minimizing a certain exact functional of the one-body density. This led to numerous extensions and applications, now collectively known under the name of density functional theory (DFT) [5,6,7].

The fact that properties of electronic systems are governed by the well-known Coulomb interaction allows for derivations of functionals from first principles, by which token this approach can proudly be called a theory. For nuclear systems, the luxury of knowing the exact interaction is not there, so the analogous approaches developed in this domain of physics carry the name of the EDF methods.

In this article we construct a phenomenological nuclear EDF based on strategies that are proper to effective theories [8]. There, guiding principles [9] are based on: (i) appropriate choice of effective fields, (ii) building effective Lagrangian or Hamiltonian densities restricted only by symmetry principles, (iii) employing ideas of power counting. In the low-energy nuclear structure, correct fields can probably be associated with nonlocal one-body nuclear densities. Then, functionals of densities acquire the meaning of effective Hamiltonian densities. Although a formal construction of power-counting schemes is not yet available, ideas based on the density matrix expansion (DME) [10,11,12,13,14,15,16] (see also a recent example of an application to electronic systems in Ref. [17]) can be used to propose expansions in terms of moments of effective nuclear interactions, or equivalently, in orders of derivatives acting on the one-body densities. This is precisely the strategy we are going to follow in the present study.

Effective field theories (EFT) were recently extensively applied in analyzing properties of nuclear systems. Here we are not able to give an even shortest possible review of this rapidly developing area of physics, but let us mention two specific examples.

Firstly, the nucleon-nucleon (NN) scattering properties were very successfully described by employing the effective nucleon-pion Lagrangian at next-to-next-to-next-to-leading order (N$^3$LO), see Ref. [18] and references cited therein. This showed that the EFT expansion is capable of grasping the main features of nuclear interactions at low energies, without explicitly invoking microscopic foundations in terms of, e.g., heavy meson exchanges.

Secondly, methods using the harmonic-oscillator effective operators have been developed up to N$^3$LO, to be employed within the shell-model approaches [19]. There, the N$^3$LO expansion was explicitly expressed in the form of pseudopotentials that contain derivatives up to sixth order. Evidently, such pseudopotentials are exact equivalents of higher-order Skyrme-like forces. When averaged within the HF approximation, they would lead to EDFs depending on derivatives of densities up to sixth order. This allows us to label our approach with the traditional name of the N$^3$LO expansion too.

There is also a recent significant effort in deriving nuclear EDFs directly from low-energy QCD within chiral perturbation theory, see, e.g., Refs. [20,21,22]. This may have a potential of providing new important insight into the precise structure of terms in the EDF, while at present we are bound to proceed phenomenologically, with only the symmetry constraints available, as is done in the present study.

In all rigorous EFT expansions, one strives to achieve convergence in describing physical observables by going to higher and higher orders of expansion. This is best illustrated by the so-called Lepage plots [19,23], where for theories cut at different orders, relative errors of observables are plotted as functions of energy. In nuclear EDF methods, this kind of convergence tests were never performed--simply because the functionals beyond the second order (NLO) of the standard Skyrme type had never been constructed or studied. The present study constitutes the first step towards this goal.

Our paper is organized as follows. In Sec. 2 we define basic building blocks for our construction, and then we construct local densities up to N$^3$LO. In Sec. 3 we construct terms in the EDF up to N$^3$LO and evaluate constraints imposed by the Galilean and gauge symmetries. In Sec. 4 we derive results for the case of conserved spherical, space-inversion, and time-reversal symmetries. After formulating conclusions of the present study in Sec. 5, in Appendix A we discuss general symmetry properties of the energy density, in Appendix B we present details of the adopted choice of the phase convention, and in Appendix C we list results pertaining to the Galilean and gauge symmetries.