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

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.

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 iγµ∂µ−M−gbΓΦ(JΠ,T )

bOτΨ,(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

1−τ3

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.

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

2MΩµν +ΓT

2M~

Rµν ? ~τ!Ψ#(x),

(6)

where ΓT

i(i=ω, ρ) stands for the (density-independent)

N-itensor coupling constant, Ωµν =∂µων−∂νωµand

Rµν =∂µ~ρν−∂ν~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

2∂µσ∂µσ−m2

σσ2−1

4Ωµν Ωµν −m2

ωωµωµ

−1

4~

Rµν ?~

Rµν −m2

ρ~ρµ? ~ρµ−1

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) + λU(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· ∇ +βM?(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). M∗stands 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

dρV

ρSσ+dgω

dρV

ρVω+τ3

dgρ

dρV

ρT V ρ. (13)

This equation involves the scalar, vector and isovector

densities

ρS=V∗βVT,(14a)

ρV=V∗VT,(14b)

ρT V =V∗τ3VT.(14c)

The new contribution σ0·T(r) coming from the tensor

couplings modiﬁes the Dirac eﬀective mass. It reads

T=−ΓT

M∇ω−τ3

ΓT

M∇ρ. (15)

At the RHB level, the bosonic degrees of freedom’s

equations of motion reduce to Klein-Gordon equations:

−∇2+m2

σσ=−gσρS,(16a)

−∇2+m2

ωω=gωρV+ΓT

M∇·jT ω,(16b)

−∇2+m2

ρρ=gρρT V +ΓT

M∇·jT ρ,(16c)

−∇2A=eρp,(16d)

with the isoscalar and isovector tensor currents

jT ω =V∗βσ0VT,(17a)

jT ρ =V∗βσ0τ3VT,(17b)

and the proton density

ρp≡ V∗1−τ3

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)

in the particle-particle channel. By assuming a sim-

ple Gaussian ansatz p(k) = e−a2k2, 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 −Eσ−Eω−Eρ−EC−ER+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

Γi≡gi(ρsat),(23a)

ξ≡ρV(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(x−1)] ,(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 =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 N−ω

and N−ρtensor 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

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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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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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