The search for a universal energy density functional (EDF) [1], that would be able to provide a spectroscopic-quality [2] description of atomic nuclei, is at the focus of the present-day studies in nuclear structure. Recently, several advanced phenomenological analyses were aimed at improving the standard relativistic [3] or nonrelativistic local [4,2,5,6,7] or nonlocal functionals [8]. A significant ongoing effort is also devoted to microscopic derivations of the functionals (see, for example, Refs. [9,10,11].) One of the central points of the current EDF studies is the question: to what extent can the finite-range effective interactions be approximated as quasi-local density functionals?
The framework to build the quasi-local theory was set up in the seminal paper by Negele and Vautherin (NV) [12], which introduced the so-called density matrix expansion (DME) method. Later, other methods to achieve the same goal, like the semiclassical expansion [13,14] were also proposed and studied. The original NV expansion, for the scalar density, allowed for treating only the even-order terms in relative coordinates, and thus was applicable only to even-power gradient densities. In the present study, we revisit the NV expansion by adding the odd-power gradient densities through the local gauge-invariance condition. There are two other important differences of the present approach with respect to the NV original, namely, (i) we treat the spin (vector) densities analogously to the scalar densities and (ii) we apply a different DME for the direct terms. These differences are motivated by the effective-theory [15] interpretation of the DME advocated in the present study. The ultimate test of these ideas can only be obtained by analysing microscopic properties of nuclear densities [11].
We present a complete set of expressions in all spin-isospin channels, applicable to arbitrary finite-range central, spin-orbit, and tensor interactions. Our study is focused on applying the NV expansion to the Gogny interaction [16,17]. This allows us to look at the correct scale of the interaction range that properly characterizes nuclear low-energy phenomena. By performing expansion up to second order (or next-to-leading order, NLO), one obtains the local Skyrme functional [18,19]. In this way, we establish a firm link between two different, local and nonlocal, very successful functionals.
The paper is organized as follows. In Section 2, we introduce reformulation of the NV expansion in terms of particles without spin. Discussion of this example allows us to present details of the approach without complications that otherwise could have obscured the main ideas. In Sections 3 and 4, we present results with spin and isospin degrees of freedom reintroduced, with Section 4 containing applications of the formalism to the Gogny force. Conclusions are given in Section 5, and A contains the discussion of spin and isospin polarized nuclear matter. Preliminary results of the present study were published in Ref. [20].