Covariant energy density functionals with and without tensor couplings at the
Hartree-Bogoliubov level
F. Mercier,1J.-P. Ebran,2, 3 and E. Khan1
1IJCLab, Universit´e Paris-Saclay, IN2P3-CNRS, F-91406 Orsay Cedex, France
2CEA,DAM,DIF, F-91297 Arpajon, France
3Universit´e Paris-Saclay, CEA, Laboratoire Mati`ere en Conditions Extrˆemes, 91680, Bruy`eres-le-Chˆatel, France
Background: The study of additional terms in functionals is relevant to better describe nuclear structure phe-
nomenology. Among these terms, the tensor one is known to impact nuclear structure properties, especially in
neutron-rich nuclei. However, its effect has not been studied on the whole nuclear chart yet.
Purpose: The impact of terms corresponding to the tensor at the Hartree level, is studied for infinite nuclear
matter as well as deformed nuclei, by developing new density-dependent functionals including these terms. In
particular, we study in details the improvement such a term can bring to the description of specific nuclear
observables.
Methods: The framework of covariant energy density functional is used at the Hartree-Bogoliubov level. The
free parameters of covariant functionals are optimized by combining Markov-Chain-Monte-Carlo and simplex
algorithms.
Results: An improvement of the RMS binding energies, spin-orbit splittings and gaps is obtained over the nuclear
chart, including axially deformed ones, when including tensors terms. Small modifications of the potential energy
surface and densities are also found. In infinite matter, the Dirac mass is shifted to a larger value, in better
agreement with experiments.
Conclusions: Taking into account additional terms corresponding to the tensor terms in the vector-isoscalar
channel at the Hartree level, improves the description of nuclear properties, both in nuclei and in nuclear matter.
I. INTRODUCTION
The covariant Energy Density Functional (cEDF) ap-
proach achieved great success in describing finite nuclei
and infinite nuclear matter properties [1]. The covari-
ant formulation provides a natural mechanism for the
appearance of central and spin-orbit (SO) parts of the
interaction in terms of combinations of scalar and vec-
tor potentials. This allows to treat these terms on equal
footing, in a more economical way.
The tensor force is of particular importance for the
nucleon-nucleon interaction, first recognized to be re-
sponsible for the deuteron binding energy [2] and non-
zero electric quadrupole moment of the deuteron [3]. To-
day, the impact of the tensor term has been studied in
details for interactions, both covariant [4–6] or not [7–
9]. It is expected that this term acts on the SO splitting
between single-nucleon levels. Indeed, the latter mainly
depends on the Dirac effective mass, which is linked to
the scalar potential; introducing tensor terms increases
the Dirac mass, while keeping reliable description of SO
splittings.
In a covariant framework, the nucleon-nucleon interac-
tion can be introduced by meson exchange and the ten-
sor terms are defined as derivative terms in the vector-
isoscalar (ω) and vector-isovector (ρ) channels. Since
derivative terms are the simplest terms to be added to
a functional, tensor terms can also be considered as the
next relevant contribution to an EDF based interaction.
Historically, the first appearance of explicit tensor cou-
plings in RMF framework can be found in [4], with non-
linear coupling for the scalar-scalar degree of freedom in
spherical nuclei. This study showed a negligible impact of
the ρtensor coupling, while the ωone seemed to improve
slightly the fit of the interaction, with an increased effec-
tive mass. Many studies were then carried out to extend
these calculations to the deformed case at the Hartree
level [10]. Numbers of specific studies have been done to
understand the effect of tensor terms on e.g. spin-orbit
splittings [11–13], shell gaps [14], surface thickness [15],
pseudo spin-orbit splitting [5], nuclear matter properties
[6].
The full treatment of the tensor term would require
the inclusion of the Fock term. However, this precludes
from making large scale calculations on the nuclear chart,
due to the complexity of a tensor covariant Hartree-Fock
approach. Indeed, a study of the interplay of the tensor
terms together with pairing and deformation, in a covari-
ant approach, is still lacking. This can be undertaken as
the Hartree level, where the tensor terms rather acts as
an extension of the functionnal than a full treatment of
this term. Nevertheless, such a study can give hints of
the behavior of the tensor effect over the nuclear chart.
Moreover, a known problem with relativistic functionals
is the low value of the effective Dirac mass M?=M+S,
usually around M?/M ≈0.6, instead of the empirically
determined M?/M ≈0.75. The inclusion of a tensor
term allows to partially decouple the scalar and vector
part of the interaction an should allow for a better de-
scription of the effective mass, by decreasing the value of
the scalar potential.
In this work, new parametrizations of cEDF, with den-
sity dependent coupling constant, are introduced at the
Hartree-Bogoliubov level. The corresponding free param-
eters are optimized by means of least-square procedure
arXiv:2210.11142v1 [nucl-th] 20 Oct 2022