A recent calculation of the nuclear energy density functional from chiral two- and three-nucleon forces is extended to the isovector terms pertaining to different proton and neutron densities. An improved density-matrix expansion is adapted to the situation of small isospin-asymmetries and used to calculate in the Hartree-Fock approximation the density-dependent strength functions associated with the isovector terms. The two-body interaction comprises of long-range multi-pion exchange contributions and a set of contact terms contributing up to fourth power in momenta. In addition, the leading order chiral three-nucleon interaction is employed with its parameters fixed in computations of nuclear few-body systems. With this input one finds for the asymmetry energy of nuclear matter the value MeV, compatible with existing semi-empirical determinations. The strength functions of the isovector surface and spin-orbit coupling terms come out much smaller than those of the analogous isoscalar coupling terms and in the relevant density range one finds agreement with phenomenological Skyrme forces. The specific isospin- and density-dependences arising from the chiral two- and three-nucleon interactions can be explored and tested in neutron-rich systems.
Isovector part of nuclear energy density functional
from chiral two- and three-nucleon forces111Work supported in part by BMBF, GSI and the DFG cluster of excellence: Origin and Structure of the Universe.
Physik Department T39, Technische Universität München, D-85747 Garching, Germany
PACS: 12.38.Bx, 21.30.Fe, 21.60.-n, 31.15.Ew
The nuclear energy density functional approach is the many-body method of choice in order to calculate the properties of medium-mass and heavy nuclei in a systematic manner [1, 2]. Parameterized non-relativistic Skyrme functionals [3, 4] as well as relativistic mean-field models [5, 6] have been widely and successfully used for such nuclear structure calculations. In a complementary approach one attempts to constrain the analytical form of the functional and the values of its couplings from many-body perturbation theory and the underlying two- and three-nucleon interaction. Switching from conventional hard-core NN-potentials to low-momentum interactions [7, 8] is essential in this respect, because the nuclear many-body problem formulated in terms of the latter becomes significantly more perturbative.
In many-body perturbation theory the contributions to the energy are written in terms of density-matrices convoluted with the finite-range interaction kernels, and are therefore highly non-local in both space and time. In order to make such functionals numerically tractable in heavy open-shell nuclei it is necessary to develop simplified approximations for these functionals in terms of local densities and currents. In such a construction the density-matrix expansion comes prominently into play as it removes the non-local character of the exchange (Fock) contribution to the energy by mapping it onto a generalized Skyrme functional with density-dependent couplings. For some time the prototype for that has been the density-matrix expansion of Negele and Vautherin , but recently Gebremariam, Duguet and Bogner  have developed an improved version for spin-unsaturated nuclei. They have demonstrated that phase-space averaging techniques allow for a consistent expansion of both the spin-independent (scalar) part as well as the spin-dependent (vector) part of the density-matrix.
By applying these new techniques a microscopically constrained nuclear energy density functional has been derived from the chiral NN-potential at next-to-next-to-leading order (NLO) in ref. by Gebremariam, Bogner and Duguet. These authors have proposed that the density-dependent couplings associated with the pion-exchange interactions should be added to a standard Skyrme functional (with several adjustable parameters). In the sequel it has been demonstrated in ref. that this new energy density functional gives numerically stable results and that it exhibits a small but systematic reduction of the -measure compared to standard Skyrme functionals (without any pion-exchange terms).
In the recent work  the calculation of the nuclear energy density functional has been continued and extended with improved (chiral) two- and three-nucleon interactions as input. For the two-body interaction the NLO chiral NN-potential has been used in ref.. It consists of long-range multi-pion exchange terms and two dozen low-energy constants which parameterize the short-distance part of the NN-interaction. The actual calculation in ref. has been performed with the version NLOW developed in ref. by lowering the cut-off scale to MeV. This value coincides with the resolution scale below which evolved low-momentum NN-potentials become nearly model-independent and exhibit desirable convergence properties in perturbative many-body calculations [7, 8, 15]. The (low-momentum) two-body interaction NLOW has been supplemented in ref. by the leading order (NLO) chiral three-nucleon interaction with its parameters , and determined in computations of nuclear few-body systems [15, 16]. With this input the nuclear energy density functional has been derived to first order in many-body perturbation theory, i.e. in the Hartree-Fock approximation. For the effective nucleon mass and the strength functions and of the (isoscalar) surface and spin-orbit coupling terms reasonable agreement with results of phenomenological Skyrme forces has been found (in the relevant density range). However, as indicated in particular by the nuclear matter equation of state , an improved description of the energy density functional requires at least the treatment of the two-nucleon interaction to second order in many-body perturbation theory.
The purpose of the present paper is to extend the calculation of the nuclear energy density functional in ref. to isospin-asymmetric many-nucleon systems with different proton and neutron densities. The additional isovector terms play an important role in the description of long chains of stable isotopes and for nuclei far from stability. Our paper is organized as follows. In section 2 we recall the improved density-matrix expansion of Gebremariam, Duguet and Bogner  whose Fourier transform to momentum space provides the adequate technical tool to calculate the nuclear energy density functional in a diagrammatic framework. In section 3 we present the two-body contributions to the various density-dependent strength functions , , , and , separately for the finite-range pion-exchange and the zero-range contact interactions. Section 4 comprises the corresponding analytical expressions for the three-body contributions grouped into contact , -exchange () and -exchange () terms. Finally, we discuss in section 5 our numerical results and add some concluding remarks.
2 Density-matrix expansion and isovector part of energy density functional
The starting point for the construction of an explicit nuclear energy density functional is the bilocal density-matrix as given by a sum over the orbitals occupied by protons and neutrons: . According to Gebremariam, Duguet and Bogner  it can be expanded in relative and center-of-mass coordinates, and , with expansion coefficients determined by local proton and neutron densities. These are the particle densities , the kinetic energy densities and the spin-orbit densities (for definitions in terms of the orbitals , see section 2 in ref.). The Fourier transform of the expanded density-matrix with respect to both coordinates defines in momentum space a medium insertion:
for the inhomogeneous isospin-asymmetric many-nucleon system. Here, denotes the third Pauli isospin-matrix and we have displayed only the (relevant) terms proportional to differences of proton and neutron densities: , , . The local Fermi momenta are related to the (particle) densities in the usual way: , , . When working to quadratic order in deviations from isospin symmetry (i.e. proton-neutron differences) it is sufficient to use an average Fermi momentum in the prefactors of and .
Up to second order in proton-neutron differences and spatial gradients the isovector part of the nuclear energy density functional takes the form:
Here, is the interacting part of the asymmetry energy of (homogeneous) nuclear matter. The non-interacting (kinetic energy) contribution to the asymmetry energy is included in the nuclear energy density functional through the kinetic energy density term, , with MeV the (free) nucleon mass. The strength function of the isovector surface term has the decomposition:
where comprises all those contributions for which the factor originates directly from the momentum dependence of the interactions in an expansion up to order . The Fourier transformation in eq.(1) converts this factor into . The second last term in eq.(2) describes the isovector spin-orbit interaction in nuclei. Depending on the sign and size of its strength function the spin-orbit potentials for protons and neutrons are differently composed from the gradients of the local proton and neutron densities.
3 Two-body contributions
In this section the two-body contributions to the various strength functions , , , and are worked out. We follow closely section 3 in ref. where the input two-body interaction, the chiral nucleon-nucleon potential NLOW , has been described in sufficient detail. In the (first-order) Hartree-Fock approximation the finite-range multi-pion exchange interactions lead in combination with the density-matrix expansion (i.e. by employing the product of two medium insertions ) to the following two-body contributions to the strength functions:
with the (isoscalar minus isovector) combination of the central, spin-spin and tensor NN-potentials in momentum space:
In the integrands of eqs.(4,9) the momentum transfer variable is to be set to . The double-prime in eq.(7) denotes a second derivative and we have given the numerical value for resulting from the (negative) curvature of the isovector central potential shown in Fig. 1 of ref..
In addition there are the two-body contributions from the zero-range contact potential of the chiral NN-interaction NLOW. The corresponding expression in momentum space includes constant, quadratic, and quartic terms in momenta and it can be found in section 2.2 of ref.. The Hartree-Fock contributions from the NN-contact potential to the strength functions read:
The 24 low-energy constants , and are determined (at the cut-off scale of MeV) in fits to empirical NN-phase shifts and deuteron properties . Their numerical values have been extracted from the pertinent NN-scattering code and are listed in section 3 of ref.. Let us mention that the contributions proportional to and in eqs.(10-14) have also been worked out in appendix B of ref. and we find agreement with their results. The terms proportional to as well as the master formulas eqs.(4-9) for the finite-range contributions are new.
4 Three-body contributions
In this section the three-body contributions to the strength functions , , , and are worked out. We employ the leading order chiral three-nucleon interaction  which consists of a contact piece (with parameter ), a -exchange component (with parameter ) and a -exchange component (with parameters , and ). In order to treat the three-body correlations in isospin-asymmetric inhomogeneous nuclear many-body systems we assume (as done in ref.) that the relevant product of density-matrices can be represented in momentum space in a factorized form by . Such a factorization ansatz respects by construction the correct nuclear matter limit, but it involves approximations in comparison to more sophisticated treatments outlined in section 4 of ref.. Actually, the present approach is similar to the method DME-I introduced in ref.. In comparison to ref. the diagrammatic calculation of the isovector terms gets essentially modified only by relative isospin factors occurring at various places. However, their pattern is rather complex and therefore it is preferable to write out each (non-vanishing) contribution individually. We give for each diagram only the final result omitting all technical details related to extensive algebraic manipulations, expansions, and solving elementary integrals.
The three-body contribution from the contact interaction is represented by the left diagram in Fig. 1. One finds a contribution to the asymmetry energy:
which depends quadratically on the density and is equal with opposite sign to the contribution to the energy per particle . This property follows from the form of the underlying energy density as it is determined by the Pauli exclusion principle and the symmetry under exchange. Due to the momentum-independence of the three-body contact interaction the contributions to the other strength functions vanish.
Next, we consider the three-body contributions from the -exchange component of the chiral 3N-interaction as represented by the right diagram in Fig. 1. Putting in three medium insertions one finds the following analytical expressions:
with the abbreviation . Note that there is no contribution to the isovector spin-orbit coupling strength , essentially because the -exchange does not generate any.
4.3 Hartree diagram proportional to
We continue with the three-body contributions from the -exchange Hartree diagram shown in the left part of Fig. 2. Again putting in three medium insertions one derives the following analytical results:
which depend only on the two isoscalar coupling constants and . The isovectorial (spin-dependent) -vertex gets eliminated by a vanishing spin-trace (over the left nucleon ring). The vanishing contributions to and from the -exchange three-body Hartree diagram are particularly remarkable, in view of the fact that their isoscalar counterparts ( and in eqs.(24,25) of ref.) are quite sizeable. The actual calculation shows that the isospin-structure of the -vertex excludes the desired coupling of the gradient to the vectors and .
4.4 Fock diagram proportional to
Finally, there are the three-body contributions from the -exchange Fock diagram shown in the right part of Fig. 2. For this diagram the occurring integrals over three Fermi spheres cannot be solved analytically in all cases. After a somewhat tedious calculation of the separate pieces proportional to , and one finds the following results for the Fock contributions to the strength functions:
with the auxiliary functions:
A double-index notation has been introduced for partial derivatives multiplied by powers of the variables and :
which applies in the same way to the functions and .
with the auxiliary function:
with given in eq.(29). A good check of all formulas collected in this section is provided by their Taylor-expansion in . Despite the superficial opposite appearance the leading term in the -expansion is . In several cases it is even a higher power of . The full Taylor series in has however a small radius of convergence