Introduction

One of the big challenges of the current research in nuclear structure physics is the search for a universal energy density functional (EDF) [1]. Among different possible approaches to this search, the consideration of a local or quasi-local EDF based on the density-matrix expansion (DME) is in recent years the object of intense studies [2,3,4,5,6,7]. These aim at improving the classic work of Negele and Vautherin [8,9] and better theoretical understanding based on the effective theory [10,11] and on the framework of the density functional theory [12].

In this work, we focus on the recent new expansion of the nuclear energy density in higher-order derivatives of densities, presented in Ref. [2]. There, following the effective-theory approach, a Skyrme-like quasi-local next-to-next-to-next-to-leading order (N$^{3}$LO) EDF was proposed, without direct references to the effective interactions but with terms of the EDF constrained only by symmetry principles. Such an EDF is phenomenological in the sense that it depends on the coupling constants, which must be fitted to available experimental data, see recent Refs. [13,14] on fitting the second-order (NLO) Skyrme functionals. Fits of the N$^{3}$LO EDF are much more complicated because of strong inter-dependencies of the coupling constants and instabilities [15] occurring in certain regions of the parameter space.

To better recognize the structure and properties of the N$^{3}$LO EDF, in the present study we analyze the relationships between the quasi-local functionals and pseudopotentials, which are two-body interactions expanded in the series of powers of relative momenta. Such effective sixth-order interactions were employed by Haxton to describe the bare interaction within an effective theory [16]. For the standard Skyrme force [17,18], which is the NLO pseudopotential, relations between the pseudopotential and the EDF are well known, see Ref. [19] for the complete analysis.

At present, it is not clear whether it is most advantageous to relate the nuclear EDF to the Hartree-Fock (HF) average energy of some two-body interaction or not. When the EDF is generated from such a interaction as done here, one avoids the problems related to self-interactions [20,21] and beyond-mean-field applications, see, e.g., Refs. [22,23]. One should notice that when quasi-local functionals are derived from non-local ones through the DME, the relation to pseudopotentials is usually broken [4]. By providing the full analysis at N$^{3}$LO of relations between the EDF and pseudopotential, the present work builds the baseline for future investigation of the problem.

The paper is organized as follows. In Sec. 2 we construct the pseudopotential in two alternative forms and list all its terms up to N$^{3}$LO. We also evaluate the constraints imposed by the gauge symmetry. In Sec. 3 we discuss the procedure of HF averaging to obtain the EDF from the pseudopotential. In particular, in Sec. 3.1 we derive the general relations connecting the parameters of the Galilean-invariant pseudopotential to the coupling constants of the EDF, whereas in Sec. 3.2 we derive the constraints for the case of conserved gauge symmetry. In Sec. 4 we reduce our results to the case of the conserved spherical, space-inversion, and time-reversal symmetries. After formulating the conclusions of the present study in Sec. 5, in Appendices A-C we present derivations related to the time-reversal invariance and hermiticity of the pseudopotential, we list results pertaining to the gauge-invariant pseudopotentials, and we give relations between the two alternative forms of pseudopotentials. Results obtained in the present work that are too voluminous to be published in the printed form are collected in the supplemental material [24].