Covariant energy density functionals with and without tensor couplings at the Hartree-Bogoliubov level F. Mercier1J.-P. Ebran2 3and E. Khan1

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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
2
with the use of both Markov-Chain-Monte-Carlo method
[16, 17] and simplex minimization [18]. Constraints on
binding energy, radii, spin-orbit splittings and level gaps,
as well as several infinite matter constraints are consid-
ered. The effect of tensor coupling at the Hartree level is
studied in detail, in both spherical and axially deformed
systems.
This work is organized as follows : Sec. II introduces
the theoretical framework, while Sec. III focuses on the
fitting procedure. General results of the minimization
process are given in Sec. IV, and proper applications to
infinite matter, binding energies, radii, effect of deforma-
tion, SO splittings, gaps and densities are shown in Sec.
V.
II. LAGRANGIAN AND EQUATIONS OF
MOTION
We treat nuclei as collections of structure-less nucleons
whose strong interactions are described in terms of (effec-
tive) mesons exchange, and with electromagnetic interac-
tions mediated by photons. As spin-1/2 and isospin-1/2
fermions, nucleons are described by the isospin doublet
Ψ(x) = ψn(x)
ψp(x); Ψ(x) = ψ
n(x)γ0ψ
p(x)γ0,(1)
where ψi(x) (i=neutron, proton) is a Dirac spinor field
and γµ=0 the corresponding Dirac matrix. The effective
mesons and the photon are represented by a boson field
Φ(JΠ,T )
b(x) carrying the quantum numbers (JΠ, T ) (total
angular momentum, parity and isospin). They dictate
the behaviour of the bosons under Lorentz transforma-
tions and rotations in isospin space.
The equations of motion for the nucleonic and mesonic
(and photonic) degrees of freedom are obtained from a
covariant Lagrangian (density), which can be split into
two sectors,
L=LNN +Lbos.(2)
The Lagrangian in the nucleon-nucleon (NN) sector,
LNN, includes all the terms bilinear in the nucleon field
Ψ, i.e.
LNN = Ψ µµMgbΓΦ(JΠ,T )
bOτΨ,(3)
where M is the nucleon mass (we take M=Mp=Mn)
and gbparametrizes the coupling between the boson (me-
son or photon) field Φ(JΠ,T )
b(x) and the spinor bilinear
ΨΓOτΨ(x). The coupling constants gb, and in par-
ticular their density dependency, are discussed below in
section III A. The matrix Γ generically refers to one el-
ement of the set {14, γ5, γµ, γ5γµ, σµν }, where 14is the
four-by-four identity matrix, γa(a= 0,1,2,3,5) are the
Dirac matrices and σµν i
4[γµ, γν], forming a basis
for the space of four-by-four complex matrices. The ma-
trix Oτ∈ {12, τi}(i= 1,2,3) is either the two-by-two
identity matrix or one of the three Pauli matrices τiin
isospin space. All possible independent spinor bilinears
can be formed with the Γ and Oτmatrices, namely
five isoscalar bilinears :
i) Ψ1412Ψ (scalar),
ii) Ψγ512Ψ (pseudoscalar),
iii) Ψγµ12Ψ (4-vector),
iv) Ψγ5γµ12Ψ (pseudo 4-vector),
v) Ψσµν 12Ψ (rank-2 antisymmetric tensor),
five isovector bilinears :
i) Ψ14~τΨ (scalar),
ii) Ψγ5~τΨ (pseudoscalar),
iii) Ψγµ~τΨ (4-vector),
iv) Ψγ5γµ~τΨ (pseudo 4-vector),
v) Ψσµν~τΨ (rank-2 antisymmetric tensor),
where we use arrows for isovectors. Hereafter, we omit
the identity matrices 14and 12.
The spinor bilinears are coupled to bosonic fields
Φ(JΠ,T )
b(x) and derivatives thereof to eventually yield a
scalar-isoscalar contribution to Lagrangian (3). In addi-
tion to the electromagnetic coupling between protons
Lγ
NN =eΨγµAµ
1τ3
2Ψ(x),(4)
where Aµis the electromagnetic 4-potential and ethe el-
ementary proton charge, cEDFs only include the minimal
set of nucleon(N)-meson couplings yielding a satisfactory
description of nuclear bulk properties. A standard choice
involves (i) the σ,ωand ρ(effective) mesons, with quan-
tum numbers respectively equal to (0+,0), (1,0) and
(1,1), and therefore respectively represented by a scalar
isoscalar field σ(x), a 4-vector isoscalar field ωµ(x) and a
4-vector isovector field ~ρµ(x) and (ii) the N-σscalar cou-
pling as well as the N-ωand N-ρvector couplings, which
respectively read
Lσ
NN =gσΨσΨ(x),(5a)
Lω
NN =gωΨγµωµΨ(x),(5b)
Lρ
NN =gρΨγµ~ρµ? ~τΨ(x),(5c)
where ?refers to the scalar product in isospin space, while
gi(i=σ, ω, ρ) stands for the (density-dependent) N-i
coupling constant, to be adjusted. It should be noted
that, in general, a pseudovector coupling between the
nucleon and the pion is not considered when the cEDF
is treated at the so-called relativistic mean field (RMF)
level, where the exchange contributions are not computed
explicitly. Indeed, if the reflection symmetry is preserved
at the RMF level, the N-πcoupling yields a null direct
contribution.
3
In this work, we enrich standard cEDFs by including
the next simplest terms, i.e. the N-ωand N-ρ(Lorentz)
tensor couplings
Lω+ρ;T
NN ="Ψσµν ΓT
ω
2Mµν +ΓT
ρ
2M~
Rµν ? ~τ!Ψ#(x),
(6)
where ΓT
i(i=ω, ρ) stands for the (density-independent)
N-itensor coupling constant, µν =µωννωµand
~
Rµν =µ~ρνν~ρµare the ωand ρfield strength tensors,
respectively.
Lbos is the Lagrangian for the bosonic degrees of free-
dom, i.e. the meson fields and the electromagnetic 4-
potential:
Lbos =1
2µσµσm2
σσ21
4µν µν m2
ωωµωµ
1
4~
Rµν ?~
Rµν m2
ρ~ρµ? ~ρµ1
4(Fµν Fµν ),
(7)
with mi(i=σ, ω, ρ) the mass of the meson iand Fµν
the electromagnetic field strength tensor.
Treating Lagrangian (2) in the relativistic Hartree-
Bogoliubov (RHB) approximation, eventually yields the
equation of motion for a quasi-nucleon (in the Bogoliubov
sense) in the quantum state k:
hD(q)λ∆(q)
(q)h
D(q) + λU(q)
V(q)k
=Ek(q)U(q)
V(q)k
,
(8)
where Ekstands for the eigenenergy of the quasiparticle
(qp) kwhile the Dirac spinors Ukand Vkcontribute to
the qp wavefunction. The chemical potential is called λ
and qcollects a set of constrained collective coordinates
(e.g. deformation parameters, pairing gap, etc.).
It should be noted that other approximations are
needed to obtain Eq. (8), among which the no-sea ap-
proximation where the Dirac sea of states with negative
energies does not contribute to the densities and currents
[19], or the omission of the time-dependence of the meson
fields (it amounts to consider the exchange of mesons as
an instantaneous process, which can be justified by their
heavy mass compared to the typical relative momenta
between the two interacting nucleons), or the preser-
vation of time-reversal symmetry, with the consequence
that only the time-like component of the 4-vector fields
contribute (currents do not contribute).
The fields hDand ∆ are the RHB mean potential in
the particle-hole and particle-particle channels respec-
tively. More precisely, the single-nucleon Dirac Hamil-
tonian reads
hD=iα· +βM?(r) + σ0·T(r)+V(r).(9)
In the last equation, bold symbols refer to 3-vector in real
space and ·indicates the corresponding scalar product,
α=γ0γ,γis a 3-vector with components γi(i= 1,2,3),
β=γ0and σ0is a 3-vector with components σ0i(i=
1,2,3). Mstands for the Dirac effective mass
M?(r) = M+S(r),(10)
involving the nucleon scalar self-energy
S=gσσ. (11)
V(r) is the nucleon vector (time-like component) self-
energy
V=gωω+τ3gρρ+e1 + τ3
2A+V(R),(12)
where we have set ω(r)ω0(r) and ρ(r)ρ0
3(r) (only
the time-like component of 4-vectors and the third com-
ponent of isovectors contribute due to the set of approxi-
mations made, as discussed above) and where V(R)is the
so-called rearrangement term
V(R) =dgσ
V
ρSσ+dgω
V
ρVω+τ3
dgρ
V
ρT V ρ. (13)
This equation involves the scalar, vector and isovector
densities
ρS=VβVT,(14a)
ρV=VVT,(14b)
ρT V =Vτ3VT.(14c)
The new contribution σ0·T(r) coming from the tensor
couplings modifies the Dirac effective mass. It reads
T=ΓT
ω
Mωτ3
ΓT
ρ
Mρ. (15)
At the RHB level, the bosonic degrees of freedom’s
equations of motion reduce to Klein-Gordon equations:
2+m2
σσ=gσρS,(16a)
2+m2
ωω=gωρV+ΓT
ω
M·jT ω,(16b)
2+m2
ρρ=gρρT V +ΓT
ρ
M·jT ρ,(16c)
2A=p,(16d)
with the isoscalar and isovector tensor currents
jT ω =Vβσ0VT,(17a)
jT ρ =Vβσ0τ3VT,(17b)
and the proton density
ρp≡ V1τ3
2VT.(18)
Lagrangian (2) only contributes to the particle-hole
channel. It is complemented by a separable pairing force
in momentum space [20, 21] hk|V1S0|k0i=Gp(k)p(k0)
4
in the particle-particle channel. By assuming a sim-
ple Gaussian ansatz p(k) = ea2k2, the two parameters
Gand aare typically adjusted to reproduce the den-
sity dependence of the pairing gap at the Fermi sur-
face, obtained in nuclear matter with the Gogny D1S
parametrization [22].
The coupled nucleonic and bosonic equations of motion
are expanded in a harmonic oscillator basis, for which de-
tails can be found in App.A and App.B. Solving them,
yields the qp energies and wavefunctions, from which one
can compute nuclear observables (e.g. binding energies
and radii) as well as the canonical single-particle spec-
trum.
The total energy is computed as
E=Epart EσEωEρECER+Epair +ECM (19)
where Epart represents the sum over the particles ener-
gies obtained by diagonalizing the Hartree Hamiltonian.
The Eφ(φ=σ, ω, ρ) represent the mesonic field ener-
gies, ECthe energy of the Coulomb field, ERthe rear-
rangement energy, Epair the pairing energy and ECM the
centre of mass correction. The latter one is computed as
ECM =P2/2M. The tensor contributions are directly
taken into account in the mesonic field energies Eωand
Eρ. The RMS radius rRMS and charge radius rCare
defined as
rRMS =Zd3rρ(r)r2(20)
rC=qr2
p+ 0.64 (21)
where the factor 0.64 accounts for the finite size of the
proton.
Finally, it has been checked that all the results are
stable with N= 18 harmonic oscillator shells, which are
then considered for all the calculations.
III. OPTIMIZATION OF THE ENERGY
DENSITY FUNCTIONALS
A. Free parameters
The cEDF involves the N-meson coupling constants as
well as the nucleon and mesons masses as free parameters.
The scalar and vector coupling constants gi=gi(ρV(r))
(i=σ, ω, ρ) are taken as explicit functions of the vector
density [23] :
gi(ρV(r)) = Γihi(ξ), i =σ, ω, ρ, (22)
where
Γigi(ρsat),(23a)
ξρV(r)
ρsat
.(23b)
The hfunctions read
hi(ξ)ai
1 + bi(ξ+di)2
1 + ci(ξ+di)2, i =σ, ω, (24)
in the isoscalar channel (i=σ, ω) and
hρ(ξ)exp [aρ(x1)] ,(25)
in the isovector channel.
The parameters entering the definition of the density-
dependent coupling constants are not independent vari-
ables. A first obvious constraint requires that hi(1) = 1,
yielding the relation
ai=1 + ci(1 + di)2
1 + bi(1 + di)2, i =σ, ω. (26)
An additional constraint can be imposed on the second
derivative of the hfunctions in order to ensure that the
rearrangement contributions do not diverge at zero den-
sity: h00
i(0) = 0 (i=σ, ω), leading to
ci=1
3d2
i
, i =σ, ω. (27)
In summary, the coupling constants use 11 free param-
eters, i.e. the 6 parameters Γi,biand di(i=σ, ω) in
the isoscalar channel, the 2 parameters Γρand aρin the
isovector channel, the saturation density ρsat as well as
the 2 density-independent tensor couplings ΓT
i(i=ω, ρ).
The nucleon mass is set to M=Mn=Mp= 939 MeV,
the ωand ρmeson masses are fixed to their observed
value in free space, i.e. mω= 783 MeV and mρ= 763
MeV, while the σmeson mass remains a free parameter
to be adjusted.
In order to reliably assess the impact of the tensor cou-
plings on nuclear properties at the RHB level, two new
parametrizations will be derived, using the same fitting
protocol: one for a cEDF without the tensor couplings
(10 free parameters) and one for a cEDF with the Nω
and Nρtensor couplings (12 free parameters).
B. Experimental dataset
Before discussing the fitting protocol, we first present
the pool of empirical data selected to calibrate the
cEDFs. The database contains both infinite homoge-
neous nuclear matter and finite nuclei properties Oi, each
with an uncertainty ∆Oi, enabling a measure of the qual-
ity of the optimal parametrization via a sensitivity anal-
ysis.
A first set of constraints comes from the properties of
infinite nuclear matter, namely the isoscalar ones: energy
per particle of symmetric nuclear matter at equilibrium
E0, incompressibility K0, and the isovector ones: symme-
try energy coefficient at saturation density J, and Ksym.
These parameters characterize the nuclear equation of
摘要:

CovariantenergydensityfunctionalswithandwithouttensorcouplingsattheHartree-BogoliubovlevelF.Mercier,1J.-P.Ebran,2,3andE.Khan11IJCLab,UniversiteParis-Saclay,IN2P3-CNRS,F-91406OrsayCedex,France2CEA,DAM,DIF,F-91297Arpajon,France3UniversiteParis-Saclay,CEA,LaboratoireMatiereenConditionsExtr^emes,9168...

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