Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 191–194, 2002

191

A Relativistic Point Coupling Model for Nuclear Structure Calculations
T. Bürvenich,a D. G. Madland,b J. A. Maruhn,∗,a and P.-G. ReinhardcaInstitut für Theoretische Physik, Universität Frankfurt, Robert-Mayer-Strasse 10, 60325 Frankfurt,
Germanyb
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87544, USAc
Institut für Theoretische Physik II, Universität Erlangen-Nürnberg, Staudtstrasse 7, 91058 Erlangen,Germany
Received: November 13, 2001; In Final Form: November 13, 2001
A relativistic point coupling model is discussed focusing on a variety of aspects. In addition to the coupling using
various bilinear Dirac invariants, derivative terms are also included to simulate finite-range effects. The formalism
is presented for nuclear structure calculations of ground state properties of nuclei in the Hartree and Hartree-Fockapproximations. Different fitting strategies for the determination of the parameters have been applied and the qualityof the fit obtainable in this model is discussed. The model is then compared more generally to other mean-fieldapproaches both formally and in the context of applications to ground-state properties of known and superheavy
nuclei. Perspectives for further extensions such as an exact treatment of the exchange terms using a higher-order
Fierz transformation are discussed briefly.

1. Introduction
Relativistic mean field (RMF) models are quite successful indescribing ground state properties of finite nuclei and nuclear
matter properties. They describe the nucleus as a system of
Dirac nucleons that interact in a relativistic covariant manner via
mean meson fields (for a review see Reference 1) or via mean
nucleon fields2, 3 whose explicit forms sometimes derive solely
from the meson field approaches.4 The meson fields are of finite
range (FR) due to meson exchange whereas the nucleon fields
are of zero range (contact interactions or point couplings PC) together with derivative terms that simulate finite interaction range
to some extent.
In this work we use mean nucleon fields constructed withcontact interactions (point couplings) to represent the system
of interacting Dirac nucleons. We choose this approach for thefollowing reasons: (a) possible physical constraints introduced
by explicit use of the Klein-Gordon approximation to describemean meson fields, in particular that of the (fictitious) sigma
meson, are avoided and instead the effects of the various incompletely understood and higher order processes are assumed to
be lumped into appropriate coupling constants and terms of the
Lagrangian, as explained in Reference 2, (b) the use of point
couplings allows not only (standard) relativistic Hartree calculations to be performed, but also relativistic Hartree-Fock calculations5, 6 by use of Fierz relations (up to fourth order7 ), and(c) the use of point couplings, because of their success in the
Nambu-Jona-Lasino model for the low-momentum domain of
QCD,8 is perhaps the best way to test for naturalness of the coupling constants in the seminal Weinberg expansion9 highlighting
the role of power counting and chiral symmetry in weakening Nbody forces. That is, two-nucleon forces are stronger than threenucleon forces, which are stronger than four-nucleon forces,..., resulting in a sequence making nuclear physics tractable.
If the dimensionless coupling constants of the corresponding
Lagrangian are of order 1 (natural) then QCD scaling and chiral symmetry apply to finite nuclei. Finally, (d), the RMF-PC
model allows one to investigate its relationship to nonrelativistic
point-coupling approaches like the Skyrme-Hartree-Fock (SHF)
approach and the RMF-FR approach to contrast the importanceand roles of the different features these models have, as well as
to obtain new insights.
It is important to note here that one can also view RMF-PC as∗Corresponding author.

de.

E-mail: maruhn@th.physik.uni-frankfurt.

an approach that lies in between the RMF-FR approach and thenonrelativistic Skyrme-Hartree-Fock (SHF) approach which isalso a well-developed self-consistent mean-field model that performs very well (for a review see Reference 10). Whereas SHF
is based upon density-dependent contact interactions with extensions to gradient terms, kinetic terms, and the spin-orbit interaction, RMF-FR is based upon a coupled field theory of Dirac nucleons and effective meson fields treated at the mean-field level,
where density dependence is modeled by nonlinear meson self
couplings and the role of gradient terms is taken over by the finite ranges of the mesons. The kinetic and spin-orbit terms areautomatically carried in both RMF models.1 Thus, a comparison of RMF-PC and SHF addresses the differences between inmedium Dirac and Schrödinger nucleons, that is, in kinetic andspin-orbit components, whereas a comparison of RMF-PC and
RMF-FR addresses the absence vs. presence of finite range andthe different treatments of density dependence. Herein we will
perform these comparisons using precisely the same fitting strategy as in recent SHF and RMF-FR adjustments11–13 except that
here we will in addition be guided by considerations of QCD
scaling and chiral symmetry, that is, naturalness.
In this paper we will present a new fit of the parameters, theapplication to superheavy nuclei together with a study of thespin-orbit force, which plays an essential role in that region. For
more details, see Reference 14.
2. The Model
2.1. The Lagrangian. The elementary building blocks ofthe point-coupling vertices are two-fermion terms of the generaltype(ψ̄Oτ Γψ),Oτ ∈ {1, τi },Γ ∈ {1, γµ , γ5 , γ5 γµ , σµν }(1)with ψ the nucleon field, τi the isospin matrices and Γ one of the
4 × 4 Dirac matrices. There thus is a total of 10 such building
blocks characterized by their transformation character in isospin
and in spacetime.
The interactions are then obtained as products of such expressions to a given order. The products are coupled, of course, to a
total isoscalar-scalar term. By “order” we mean the number ofsuch terms in a product, so that a second-order term corresponds
to a four-fermion coupling, and so on. In second order only theten elementary currents squared and contracted to scalars may
contribute, but at higher orders there is a proliferation of terms
because of the various possible intermediate couplings.

c 2002 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 6/30/2002

192 J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

0.8

0.8

0.4

0.4

Sm
144

Sn

Xe

Sn

132

136

Sn

124

Sn

120

Sn
100

112

Sr

Zr
90

Ni

88

Ni

56

Ca

Pb
214

58

Pb
202

16

Sn

Sn

120

124

Sn

112

90

88

Zr

Sr

Ni

Ca
48

58

Ca

Pb
208

16

4
3
2
1
0
-1
-2
-3
-4

Zr
90

40

48

Ca

Pb
208

16

Ca

4
3
2
1
0
-1
-2
-3
-4

O

[%]

40

2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0

O

Rdms [%]

2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0

2

2

1

1

0

0

ch

rrms [%]

Pb
214

Pb
208

Pb
202

Sn

Sn

124

112

Zr
90

Sr

88

58

56

48

Ni

-3

Ni

-3

Ca

-1
-2

Ca

-1
-2

40

Note that we use the nuclear physics convention for the isospin
where the neutron is associated with τ3 = +1 and the protonτ3 = −1.
As it stands this Lagrangian contains the nine coupling constants αS , αV , αTV , βS , γS , γV , δS , δV , and δTV . The subscripts
indicate the symmetry of the coupling: “S” stands for scalar,“V” for vector, and “T” for isovector, while the symbols referto the additional distinctions: α refers to four-fermion terms, δ
to derivative couplings, and β and γ to third- and fourth orderterms, respectively.
The model thus contains one or two free parameters morethan analogous RMF-FR models. This happens because most
RMF-FR models make the tacit assumption that the masses inthe ω- and ρ-field can be frozen at the experimental values of
the really existing mesons. The assumption is justified to theextent that the actual fits to observables are not overly sensitive
to these masses. In the RMF-PC model, however, experience
will still have to show whether the derivative-term coefficientscan be eliminated in a similar way, so that for the present work
all parameters are regarded as adjustable.
Similar to the RMF-FR approach, we consider the RMF-PC
approach as an effective Lagrangian for nuclear mean-field calculations at the Hartree level without anti-nucleon states (no-sea
approximation), and include pairing and center-of-mass corrections in the standard way.16, 17

48

-0.8

Ca

-0.8

40

0.0
-0.4

O

0.0
-0.4

Pb

= ψ̄(iγµ ∂µ − m)ψ,1
1
L4f = − αS (ψ̄ψ)(ψ̄ψ) − αV (ψ̄γµ ψ)(ψ̄γ µ ψ)
2
2
1


− αTV (ψ̄τγµ ψ) · (ψ̄τγ µ ψ),
2
1
1
1
hot
L = − βS (ψ̄ψ)3 − γS (ψ̄ψ)4 − γV [(ψ̄γµ ψ)(ψ̄γ µ ψ)]2 ,
3
4
4
1
1
der
ν
L = − δS (∂ν ψ̄ψ)(∂ ψ̄ψ) − δV (∂ν ψ̄γµ ψ)(∂ν ψ̄γ µ ψ)221ν− δTV (∂ν ψ̄τγµ ψ) · (∂ ψ̄τγ µ ψ),21em
L = −eAµ ψ̄[(1 − τ3 )/2]γ µ ψ − Fµν F µν .
(2)
4

L

208

L = Lfree + L4f + Lhot + Lder + Lem ,free

2.2. Coupling Constants. In addition to the older
parametrization PC-LA2 that was based on a relatively small
set of data and partly relied on naturalness of the parameters to
restrict their freedom, we also fitted a new parameter set for theversion of the model given in eq 2. The parameters are fitted in
the same way as NL-Z2.18 The set of nine coupling constants
emerging from the fitting procedure with the lowest value ofχ2 is called PC-F1, and its performance is compared to that of
NL-Z2 in Figure 1.
Comparing the average errors between PC-F1 and NL-Z2, we
see slightly different trends. NL-Z2 is superior with respect tobinding energies and surface thicknesses. It does, however, perform less brilliantly concerning radii. The total χ2 of NL-Z2
is 132.7 which is 34% larger than that for PC-F1. The overall
performance of the point-coupling thus seems to be a bit better,although the difference is not dramatic.
Since we know that spin-orbit coupling is crucial for superheavy elements,18 we show in Figure 2 the relative errors fora selection of spin-orbit splittings in 16 O, 132 Sn, and 208 Pb. We
have taken care to choose splittings which can be deduced reliably from spectra of neighbouring odd nuclei.19 All RMF forces,
except for PC-LA, perform very well. We see now that the wellfitted point coupling model PC-F1 does as well as finite range
RMF. The ability to describe the spin-orbit force correctly isthus a feature of the relativistic approach. The force PC-LA
falls clearly below the others. The poor performance is related
to the too weak fields at large densities. The example demonstrates that one needs a sufficiently large set of observables topin down the nuclear mean field sufficiently well.
The three well performing models all have very similar spinorbit potentials whereas PC-LA has one 10% stronger whichis shifted a little bit to larger radii. This difference yields theobserved mismatch in the spin-orbit splittings.
2.3. Superheavy Elements. The upper panel of Figure 3
shows the relative errors in binding energies for the heaviest
even-even nuclei with known experimental masses (compareE [%]In analogy to the nonrelativistic Skyrme-force models, one
goes one step beyond zero range and complements the pointcoupling model by derivative terms in the Lagrangian, as e.g.
∂ν ψ̄τi Γ µj ψ. The derivative is understood to act on both ψ̄ and ψ.
In the present work we consider the following four-fermion
vertices: isoscalar-scalar, (ψ̄ψ)2 corresponding to the σ-field,
isoscalar-vector, (ψ̄γµ ψ)(ψ̄γ µ ψ) corresponding to the ω-field,
and isovector-vector, (ψ̄τγµ ψ) · (ψ̄τγ µ ψ), corresponding to theρ-field, together with their corresponding gradient couplings∂ν (...)∂ν (...).
These constitute a complete set of second-order scalar andvector currents whose coupling strengths in the corresponding
Lagrangian we wish to test for naturalness. We neglect all tensor couplings in the present work because they have had little effect in corresponding RMF-FR calculations.1 Finally, the pseudoscalar channel (π-meson) is not included here because it does
not contribute at the Hartree level.
Given the vast number of possible higher-order terms we begin with those that have already been demonstrated to be ofuse in the existing calculations with the RMF-FR and RMFPC approaches. These are the familiar nonlinear terms in thescalar coupling, (ψ̄ψ)3 and (ψ̄ψ)4 , as well as a nonlinear vector
term [(ψ̄γµ ψ)(ψ̄γ µ ψ)]2 as used in some RMF-FR15 and RMFPC2 models. Finally, of course, the electromagnetic field andthe free Lagrangian of the nucleon field must be included.
Combining all of these terms, we obtain the Lagrangian of
the point-coupling model asBürvenich

Figure 1. Errors in percent for the observables binding energy, diffraction radius, surface thickness, and rms charge radius for PC-F1 (filled
diamonds) and NL-Z2 (open squares) are seen on the left. The rightpanels show the absolute mean errors for the corresponding observables, where the dashed lines indicate the chosen relative errors ∆O in
the fitting procedure.

A Relativistic Point Coupling Model

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

60

60
16

O (1p)Sn (2d)208
Pb (2d)

30

30

0

-10

-10

-20

-20

NL-Z2

NL3

0

NL3

10

NL-Z2

ls

20

10

PC-F1

208

PC-F1

[%]

40

20

O (1p)Sn (2d)Pb (3p)208
Pb (2f)

132

50

40

PC-LA

[%]

50

16

neutrons

132

PC-LA

protonsls

193

Figure 2. The percentage error in ls-splittings for protons (left) and for neutrons (right). The experimental errors are smaller than the size of thesymbols used in these figures. The lines serve to guide the eye.
Z = 98Z = 100

Z = 102Z = 104 Z = 106 Z = 108

0.4

E [%]

0.2

0.0

-0.2

PC-F1
PC-LA
NL-Z2
NL3

-0.4

144

148

152

156

144

148

152

156

148

152

156

152

154

156

144

148

152

156

144

148

152

156

148

152

156

152

154

156

0.32

2

0.3

0.28

0.26

0.24

Neutron Number N

Figure 3. Deviation in % of the calculated energies from the experimental values (upper figure) and ground-state deformations (lower figure) inaxially deformed and reflection symmetric calculations. The errors for the binding energies are smaller than the size of the symbols used in this figure.
The symbol with error bars indicates the measured ground-state deformation together with its uncertainty of 254 No (Ref. 22, 23).

with a similar figure in Reference 20). The lower panel delivers as complementing information the ground-state deformations expressed in terms of the dimensionless quadrupole deformation β2 . The calculations were performed allowing axiallysymmetric deformation assuming reflection-symmetric shapes.
The agreement is remarkable. All forces (with some exceptions for PC-LA) produce only small deviations which stay wellwithin the given error band. This is a gratifying surprise because we are here 40–50 mass units above the largest nucleus
included in the fit. It is to be noted that most SHF forces do not
perform so well and have general tendency to underbinding forsuperheavy nuclei.20 There are also (small but) systematic differences between the RMF models. NL321 generally overbinds
a little while NL-Z2 and PC-F1 tend to underbind. All forces
show yet unresolved isovector trends. The increase of the binding energy with increasing neutron number is too small. These
trends are already apparent for known nuclei (details for PC-F1
not shown in this paper). The reasons for all these trends are not
yet understood. Finally, mind the kinks visible for the Z = 98

and Z = 100 isotopes at neutron number N = 152 which hint at asmall (deformed) shell closure there.
All forces predict strong prolate ground-state deformations
for these superheavy nuclei (β2 ≈ 0.26–0.31). The trends look
similar for all forces. The largest deformations appear at N = 148
and/or N = 150. But there are systematic differences in detail:NL-Z2 has always larger ground-state deformations than theother forces, while PC-F1, PC-LA, and NL3 show much similar
deformations. The difference is probably related to the surface
energy: NL-Z2 has a lower surface energy than NL3. The symbol with error bars at Z/N = 102/152 in Figure 3 corresponds tothe measured ground-state deformation of 254 No (Ref. 22, 23).
This deformation is overestimated by all forces, PC-LA andNL3 stay within the error bars, though. The error ranges from 6
to 13% which is still acceptable.
The prediction of new magic shell closures in superheavy elements varies amongst the mean field models.24 For protons one
has a competition between Z = 114, 120, and 126. For neutrons
one finds N = 172 and 184. The finite-range RMF models agree

194 J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002
in predicting a doubly magic 292 120172 . Precisely the same result emerges from PC-F1. This doubly magic nucleus is thus acommon feature of relativistic models. For the density profile
of 292 120172 , we observe a central depression in accordance withother mean-field approaches.18, 25, 26
In deformed calculations done in the way as described in Reference 20, we obtain, again in agreement with other relativistic
models, deformed shell closures at Z = 104 for the protons andN = 162 for the neutrons. The nuclei in that region of the nuclear
chart have deformations with β2 ≈ 0.2–0.3. Thus also in the deformed case, these different types of RMF models agree wellconcerning their predictions of shell structure for superheavy elements.
3. Summary and Prospects
In summary one may thus conclude that the use of relativistic point-coupling models produces results of similar quality as
the finite-range ones, even regarding the surface properties, withdifferences, of course, noticeable in the details. It should be
noted that larger differences appear in quantities not discussed
here, such as the density distributions. While this does not leadto new conclusions for superheavy nuclei at this moment, thismodel opens the way to valuable new insights for two reasons:(1) It can be expanded to include a full treatment of exchange
terms. In recent work7 we have shown that it is possible togeneralize the Fierz transformation to higher-order terms, allowing the reformulation of the exchange contributions in terms ofdensities and currents. Although the expressions arising in thisway are quite complicated, their numerical evaluation should bestraightforward.
(2) The examination of naturalness of the coupling constants
may be a helpful way to make fits more unique. Following
Manohar and Georgi27 we can scale a generic Lagrangian termof the physical series asl  m  µ

n
π
ψψ
∂ , mπL ∼ −clmn 2
fπ2 Λ2 ,
(3)
fπ
Λ
fπ Λ
where ψ and π are nucleon and pion fields, respectively, fπ
and mπ are the pion decay constant, 92.5 MeV, and pion mass,139.6 MeV, respectively, Λ = 770 MeV is the ρ meson mass,and (∂µ , mπ ) signifies either a derivative or a power of the pionmass. Dirac matrices and isospin operators (we use t here
rather than τ) have been ignored. Chiral symmetry demands28∆ = l +n − 2 ≥ 0, such that the series contains only positive powers of (1/Λ). If the theory is natural,27, 29 the Lagrangian should
lead to dimensionless coefficients clmn of order unity. Thus, allinformation on scales ultimately resides in the clmn . If they arenatural, QCD scaling works.
The nine QCD-scaled coupling constants of the parameter set
used in this paper turn out to be all natural. So far as we areaware, this is the first complete set of natural QCD-scaled coupling constants, with order up to Λ−2 , that has been obtainedwith unconstrained least-squares parameter adjustment to measured ground-state observables.
The main problem with this type of effective field theory
has been the unambiguous extraction of the coupling constants,
where changing one affects the contributions from all the others, making it difficult to judge the physical appropriateness ofthe corresponding interaction terms of the Lagrangian. The naturalness of the coupling constants, however, appears to provide
an additional physical constraint that substantially reduces theseambiguities. Furthermore, this physical constraint derives fromlow-momentum QCD scaling and chiral symmetry.
Acknowledgement. The authors would like to thank M. Bender, J. Friar, W. Greiner, P. Möller, and A. J. Sierk for many valuable discussions. This work was supported in part by the Bundesministerium für Bildung und Forschung (BMBF), ProjectBürvenich
No. 06 ER 808, by Gesellschaft für Schwerionenforschung(GSI), and the U. S. Department of Energy.
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