Introduction

Since the discovery of the neutron by Chadwick in 1932, the quest for nuclear interactions has been driving an intense research in nuclear physics. Nuclei are quantum objects composed of correlated components, the nucleons, whose mutual interactions are governed by the low-energy limit of quantum chromodynamics. On the one hand, the contact between the field theory of quarks and gluons and low-energy nuclear structure has been set in the framework of chiral effective field theory [1,2]. On the other hand, the concept of an effective theory (ET) is powerful enough to embrace a larger set of nuclear structure models, including nuclear potentials employed in phenomenological self-consistent approaches [3].

One of the main features of nuclear forces in the vacuum is the fact that nucleons interact over distances that are comparable with their sizes. Therefore, even when contact or zero-range forces might be adopted for computational reasons, to obtain accurate description of nuclear observables the introduction of finite-range effects is mandatory. In the domain of phenomenological models, different approaches resort to some degree of approximation in describing how nucleons effectively interact in nuclear medium. In mean-field calculations, the use of finite-range interactions was pioneered by Brink and Boecker [4] and extended by Gogny [5,6], who argued that long and intermediate range of the nuclear in-medium effective force should be included explicitly. Nonetheless, the zero-range approximation was also widely and successfully employed in mean-field calculations, e.g., for determining binding energies and radii, using the zero-range Skyrme interactions [7,8]. In this case, the finite-range character of the interaction was mimicked through a dependence on relative momenta of the interacting nucleons.

At present, an increased availability of powerful computational resources makes the option of finite-range interaction more and more viable. Furthermore, we tend to regard phenomenological interactions as low-energy in-medium manifestations of the nuclear force, rather than competing and mutually-exclusive approximations of an underlying 'true' potential. This explains the intense ongoing work devoted to improving the form of the phenomenological energy density functionals and interactions [9,10,11,12,13], along with the recent progress in elaborating new parametrizations of existing ones [14,15,16]. The zero-range standard Skyrme interaction has been enriched with higher-order derivative terms [11], whereas extensions to three-body component of the force have been envisaged in both nonlocal [10] and contact [13] forces.

In a recent work [3], we showed that it is possible to apply in a consistent way the ET methodology to low-energy nuclear physics. There we presented a proof-of-principle study concerning the convergence, independence of regularization scale, and naturalness of a new class of effective interactions, introduced as regularized zero-range pseudopotentials. In another study [17], we introduced a regularized interaction that was a genuine force, that is, it was density independent, and built as an expansion in relative momenta. This latter interaction has proven to be able to capture the relevant physics in the mechanism of saturation but the effective mass. The absence of density-dependent term makes the regularized pseudopotentials suitable for beyond-mean-field calculations, without divergences at high momenta and other pathological behaviors affecting the functionals in multi-reference EDF calculations [18,19], that is, when standard techniques, like symmetry restoration or Generator Coordinate Method in its most general form, are employed.

In this paper we introduce the general formalism describing the finite-range regularized pseudopotential and nonlocal EDFs obtained by the averaging procedure over the uncorrelated nuclear wavefunction. By considering a finite-range interaction, we get a functional which is bilinear in densities that are built by coupling differential operator to the one-body density matrix. We consider here the expansion in derivatives up the sixth order, corresponding to the N$^3$LO functional. Such a functional is genuinely nonlocal, as compared to the quasilocal form of the EDF stemming from the zero-range pseudopotential. The derived EDF is richer in numbers of terms and variety of tensorial couplings of densities to bilinear scalars, as compared to its quasilocal version. This implies, in principle, a functional that has the potential of being more flexible and predictive, but it also implies a larger parameter space to be optimized to nuclear data. The derived regularized pseudopotentials, along with the corresponding nonlocal EDFs, are ready for the optimization procedure, which is the natural follow-up of this formal development, and is now being carried out using the new implementations built within the solver HFODD [20,21,22].

The contents of the paper is as follows. In Sec. 2 we present the construction of the nonlocal EDF from the finite-range pseudopotential. In particular, in Sec. 2.1 we introduce the main features of the finite-range pseudopotential and in Sec. 2.2 we present the nonlocal EDF up to sixth order in derivatives. In Sec. 3 we reduce our results to the case of infinite nuclear matter, whereas in Sec. 4 we compare the NLO EDF derived in the spherical-tensor formalism to that derived in the Cartesian representation. In Sec. 5 we consider a reduced version of the finite-range pseudopotential, for which only the central part of the force is taken into account and a simplified dependence on the relative momenta is assumed. Section 6 presents construction of higher-order spin-orbit (SO) and tensor terms in the Cartesian representation. Finally, our conclusions are presented in Sec. 7.