Effective field theory: A complete relativistic nuclear model

Abstract

We analyzed the results for finite nuclei and infinite nuclear and neutron matter using the standard $\sigma-\omega$ model and with the effective field theory. For the first time, we have shown here quantitatively that the inclusion of self-interaction of the vector mesons and the cross-interaction of all the mesons taken in the theory explain naturally the experimentally observed softness of equation of state without loosing the advantages of standard $\sigma-\omega$ model for finite nuclei. Recent experimental observations support our findings and allow us to conclude that without self- and cross-interactions the relativistic mean field theory is incomplete.

Full text

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arXiv:nucl-th/0308050v1 19 Aug 2003
Effective field theory: A complete relativistic nuclear model
P. Arumugam, B.K. Sharma, P.K. Sahu, and S.K. Patra
Institute of Physics, Sachivalaya Marg, Bhubaneswar - 751 005, India.
(Dated: August 19 2003)
We analyzed the results for finite nuclei and infinite nuclear and neutron matter using the standard
σ−ω model and with the effective field theory. For the first time, we have shown here quantitatively
that the inclusion of self-interaction of the vector mesons and the cross-interaction of all the mesons
taken in the theory explain naturally the experimentally observed softness of equation of state
without loosing the advantages of standard σ − ω model for finite nuclei. Recent experimental
observations support our findings and allow us to conclude that without self- and cross-interactions
the relativistic mean field theory is incomplete.
PACS numbers: 21.10.Dr, 21.10.Tg, 21.60.-n, 21.60.Fw
In the quest for an unified model describing both finite
nuclei and nuclear matter, the quantum hadrodynam-
ics (QHD) has been a successful tool for the past few
decades. QHD is the field theory of the nuclear many-
body problem using hadron degrees of freedom. Models
based on relativistic QHD takes care ab initio of many
natural phenomena which are practically absent or have
to be included in an ad hoc manner in the non-relativistic
formalism. One of the first successful models based on
QHD with relativistic mean field (RMF) was constructed
by Walecka [1] with vector and scalar meson fields. Later
on, to get reasonable incompressibility and to get good
results for finite nuclei, cubic and quartic nonlinearities of
the σ meson (standard nonlinear σ−ω model) were added
[2]. These models were proposed to be renormalizable
and that constraint limited the scalar interactions to a
quartic polynomial and disallowed the scalar-vector cross
interactions and vector-vector self-interactions. However,
the coupling constants are not assigned with their bare
(experimental) values but with some effective value to
have proper results for finite nuclei. Hence the renormal-
izability of the Lagrangian gets compromised by the use
of effective coupling constants.
Inspired by effective field theory (EFT), Furnstahl,
Serot and Tang [3] abandoned the idea of renormaliz-
ability and extended the RMF theory by allowing other
nonlinear scalar-vector and vector-vector interactions in
addition to tensor couplings [3, 4, 5, 6, 7]. The EFT
contains all the non-renormalizable couplings consistent
with the underlying symmetries of QCD. The effective
Lagrangian is obtained by employing suitable expansion
scheme to truncate the infinite number of terms. In such
scheme the ratios Φ/M, W/M, |∇Φ|/M2and |∇W|/M2
are the useful expansion parameters [3, 4, 5, 8, 9] where Φ
and W are scalar and vector meson fields respectively and
M is the nucleon mass. With the help of the concept of
naturalness (i.e., all coupling constants are of the order of
unity when written in appropriate dimensionless form),
it is then possible to compute the contributions of the
different terms in the expansion and to truncate the ef-
fective Lagrangian at a given level of accuracy [3, 4, 6, 7].
None of the couplings should be arbitrarily dropped out
to the given order without a symmetry argument. Ref-
erences [6, 7, 8, 9] have shown that it suffices to go to
fourth order in the expansion. At this level one recovers
the standard nonlinear σ−ω model plus a few additional
couplings, with thirteen free parameters in all. These pa-
rameters have been fitted (parameter sets G1 and G2) to
reproduce some observables of magic nuclei [3]. The fits
display naturalness, and the results are not dominated
by the last terms retained. This evidence confirms the
utility of the EFT concepts and justifies the truncation
of the effective Lagrangian at the first lower orders.
Recent applications of the models based on EFT in-
clude studies of pion-nucleus scattering [10] and of the nu-
clear spin-orbit force [11], as well as calculations of asym-
metric nuclear matter at finite temperature with the G1
and G2 sets [12]. In a previous work [8] we have analyzed
the impact of each one of the new couplings introduced in
the EFT models on the nuclear matter saturation proper-
ties and on the nuclear surface properties. In Ref. [9] we
have looked for constraints on the new parameters by de-
manding consistency with Dirac-Brueckner-Hartree-Fock
(DBHF) [13] calculations and the properties of finite nu-
clei. Using EFT we successfully explained the properties
of the drip-line nuclei as well as the symmetric and asym-
metric infinite nuclear matter including the neutron star
and compared with other theoretical calculations [14].
Very recently [15], the flow of matter in heavy-ion colli-
sions is analyzed to determine the pressures attained at
densities ranging from two to five times the saturation
density of nuclear matter. This experimental determi-
nation of the equation of state (EOS) of dense matter
motivated us to study the applicability of various rela-
tivistic models at extreme conditions. In this letter we
present the observations based on the results of our EFT
calculations, the standard nonlinear σ − ω model and
some of the recent experiments.
The description of EFT and the field equations for
nuclear matter and finite nuclei can be found in Refs.
[3, 4]. The field equations were derived [3] from an energy
density functional containing Dirac baryons and classical

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2
scalar and vector mesons. According to Refs. [3, 4] theenergy density for finite nuclei can be written as
E(r) =
?
α
ϕ†
α
?
− iα·∇ + β(M − Φ) + W +1
2τ3R +1 + τ3
2
A −
i
2Mβα·
?
fv∇W +1
2fρτ3∇R + λ∇A
?
+
1
2M2(βs+ βvτ3)∆A
?
ϕα+
?1
1
2g2
?m2
2+κ3
3!
Φ
M+κ4
4!
Φ2
M2
?m2
s
g2
s
Φ2−ζ0
4!
1
g2
v
W4
+
1
2g2
s
?
1 + α1
Φ
M
?
?
(∇Φ)2−
v
?
R2−
1 + α2
Φ
M
?
(∇W)2−1
2
?
1 + η1
Φ
M+η2
2
Φ2
M2
?mv2
gv2W2
−
1
2g2
ρ
(∇R)2−1
2
1 + ηρΦ
M
ρ
g2
ρ
1
2e2(∇A)2+
1
3gγgvA∆W +
1
gγgρA∆R, (1)
where the index α runs over all occupied states ϕα(r) of
the positive energy spectrum, Φ ≡ gsφ0(r), W ≡ gvV0(r),
R ≡ gρb0(r), A ≡ eA0(r). gs, gv, gρand e are the cou-
pling constants corresponding to the fields φ0(r), V0(r),
b0(r) and A0(r) respectively and κ3, κ4, η1, η2, ζ0, fv
and fρare non-linear coupling constants.
The terms with gγ, λ, βsand βvtake care of effects re-
lated with the electromagnetic structure of the pion and
the nucleon (see Ref. [3]). Specifically, the constant gγ
concerns the coupling of the photon to the pions and the
nucleons through the exchange of neutral vector mesons.
The experimental value is g2
λ is needed to reproduce the magnetic moments of the
nucleons. It is defined by
γ/4π = 2.0. The constant
λ =1
2λp(1 + τ3) +1
2λn(1 − τ3), (2)
with λp= 1.793 and λn= −1.913 the anomalous mag-
netic moments of the proton and the neutron, respec-
tively. The terms with βsand βvcontribute to the charge
radii of the nucleon [3].
Variation of the energy density (1) with respect to ϕ†
and the meson fields gives the Dirac equation fulfilled
by the nucleons and the meson field equations [9]. The
α
Dirac equation corresponding to the energy density (1)
and the mean field equations for Φ, W, R and A can be
found in Ref. [9, 14]. The meson fields can also be inter-
preted as Kohn–Sham potentials [16] in the relativistic
case [17] and in this sense they include effects beyond
the Hartree approach like three-body and many-body in-
teractions through the nonlinear couplings [3, 4].
For infinite nuclear matter all of the gradients of the
fields in the energy density and field equations vanish.
Due to the fact that the solution of symmetric and asym-
metric nuclear matter in mean field depends on the ratios
g2
v[18], we have seven unknown parame-
ters. By imposing the values of the saturation density, to-
tal energy, incompressibility modulus and effective mass,
we still have three free parameters (the value of g2
fixed from the bulk symmetry energy coefficient J).
s/m2
sand g2
v/m2
ρ/m2
ρis
The baryon, scalar, isovector, proton and tensor den-
sities are same as for finite nuclei in the nuclear mat-
ter limit, i.e., ρ =
(2π)3
0
γ
(2π)3
0
d3k
√
momentum and k is the momentum at any density). The
expressions for pressure and energy density are
γ
?kf
d3k =
γ
6π2k3
fand ρs =
?kf
M∗
(k2+M∗2), and so on (here kf is the Fermi
P =
γ
3(2π)3
?
d3k
k2
E∗(k)+14!ζ0g2
wV4
0+1
2
?
1 + η1gσσ
M
+η2
2
g2
M2
σσ2
?
m2
ωV2
0
− m2
σσ2
?
1
2+κ3gσσ
3!M
+κ4g2
4!M2
σσ2
?
+1
2
?
1 + ηρgσσ
M
?
m2
ρb2
0, (3)
ǫ =
γ
(2π)3
?
d3kE∗(k) −1
4!ζ0g2
wV4
0−1
2
?
1 + η1gσσ
M
+η2
2
g2
M2
σσ2
?
m2
ωV2
0+1
2gρb0(ρp− ρn)
+ m2
σσ2
?
1
2+κ3gσσ
3!M
+κ4g2
4!M2
σσ2
?
+1
2
?
1 + ηρgσσ
M
?
m2
ρb2
0+ gωV0(ρp+ ρn) .(4)

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3
?
?
?
?
?
?
?
?
?
?
FIG. 1:
ratio of nuclear matter density with its saturation value (ρ0),
using the parametrizations NL3 [9], TM1 [21], G1, and G2 [3]
and in a DBHF calculation [13].
Scalar (Us) and vector (Uv) potentials against the
Here γ=2 for pure neutron matter and γ=4 for symmet-
ric nuclear matter. The asymmetry of nuclear matter is
defined by the parameter α. For the symmetric matter,
α=0 and for the neutron matter, α=1.
While examining the effect of nonlinear coupling in nu-
clear matter [9] using various nuclear force parameters,
the NL3 parameter set [19] (considered as a representa-
tive of the standard nonlinear σ−ω parametrization) has
been found to fail in following the DBHF results even
at slightly higher densities. It is well known that the
DBHF theory in relativistic framework explains well the
nuclear matter at higher densities (∼ 2ρ0) [20, 21]. In
contrast to the standard σ − ω model, the EFT calcula-
tions at high density regimes yield results in accordance
with DBHF. This scenario is depicted in Fig. 1, where we
present the results of calculations with TM1 parameter
set [21] also. In the calculation with TM1, only a quar-
tic vector self-interaction term is included apart from the
terms in NL3. This inclusion was done arbitrarily with-
out considering the underlying QCD symmetries or the
naturalness. However, with this self-interaction term, the
TM1 give better results at higher densities. This demon-
strates the importance of self-interactions at higher den-
sities and exposes the inadequacy of the standard non-
linear σ − ω model. This argument is further supported
by Fig. 2 in which the variation of binding energy per
particle (E/A) is plotted as a function of ρ/ρ0. The NL3
parameter set gives a much too stiff EOS whereas the
other parameter sets give a softer EOS which is consis-
tent with the observed neutron star masses [5] and radii
(See Table I) and measurements of kaon production in
heavy-ion collisions [22].
The recent experimental observations [15, 22] rule out
any strongly repulsive nuclear EOS and has confirmed
the predictions made above. The zero-temperature EOS
for symmetric nuclear matter derived experimentally is
shown in Fig. 3 along with the results obtained from
?
?
?
?
?
?
?
?
FIG. 2: EOS for the same cases as in Fig. 1.
TABLE I: Upper panel: The surface energy coefficient Es
and surface thickness t (in fm). Middle panel: Energy per
nucleon E/A (in MeV), charge radius rch (in fm) and spin-
orbit splittings ∆ESO (in MeV) of the least-bound nucleons.
Lower panel: the neutron star radius R (in km) and the mass
ratio M/M⊙.
TM1
18.51 18.36 18.06
1.91 1.99
−8.15 −8.08 −7.97 −7.97
2.662.73
5.6 6.4
5.66.3
−8.65 −8.64 −8.67 −8.68
3.463.48
5 6.1
5.2 6.3
−8.71 −8.69 −8.71 −8.68
4.274.28
1.4 1.6
−7.87 −7.87 −7.87 −7.86
5.54 5.52
0.70.8
1.4 1.6
13.91 15.47 13.93 10.04 10.0–12.0
2.82.782.16
NL3G1 G2
17.8 16.5–21.0
2.08
Exp.
Es
t
E/A
rch
∆ESO (n,1p)
1.98 2.2–2.5
−7.98
2.73
16O
2.722.72
5.9
5.9
66.2
6.3 (p,1p) 5.9
48Ca E/A
−8.67
3.47rch
∆ESO (n,1d)
3.44
5.8
6.2
3.44
5.63.6
4.3 (p,1d)6
90ZrE/A
rch
∆ESO (n,2p)
−8.71
4.26 4.28
1.8
4.28
1.8 0.5
208Pb E/A
−7.87
5.50rch
∆ESO (n,3p)
5.5
0.9
1.8
5.5
0.9
1.8
0.9
1.3(p,2d)
R
M/M⊙
2.041.5–2.5
different calculations.
calculations based on NL3 deviate drastically from the
experimental observation and the EFT calculations with
G1 upto some extent and G2 to an excellent extent, agree
with the experiment. A similar situation prevails in the
EOS of neutron matter and can be seen in Fig. 4. From
the figures it is very clear that NL3 calculations are not
suitable for nuclear matter as the formalism is incomplete
without the self- and cross-interaction terms.
other hand the EFT calculations with G1 and G2 param-
eter sets explain the situation in nuclear matter naturally
without any forced changes in parameters or in formal-
ism and the fit with the experiment is outstanding. The
experimental data are explained reasonably good only by
In Fig.3 we can see that the
On the

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4
?
?
?
?
?
?
?
?
?
FIG. 3: Zero temperature EOS for symmetric nuclear matter.
The shaded region corresponds to experimental data [15]. The
EOS from calculations using NL3, G1 and G2 parameter sets
are represented by dashed, dotted and solid lines respectively.
?
?
?
?
?
?
?
?
?
?
?
FIG. 4: Zero temperature EOS for neutron matter. The up-
per and lower shaded regions correspond to experimental data
[15] with strong and weak density dependences, respectively.
Other cases are same as in Fig. 3.
the calculations of Akmal et al [23] which employs the
Argonne v18interaction. Such an interaction is not ap-
plied successfully in the case of finite nuclei. The EFT
calculations are proved to give very good results for fi-
nite nuclear properties [9] as well as for infinite nuclear
matter including neutron star [14]. In Table I we present
some of the sample results of EFT calculations. More
results for finite nuclei and nuclear matter can be found
in Ref. [9, 14]. In choosing between the parameter sets
G1 and G2 for further calculations we prefer G2. It is
worth to note that G2 presents positive values of Φ4cou-
pling constant (κ4), as opposed to G1 and to many of
the most successful RMF parametrizations, such as NL3.
Actually the negative value of κ4 is not acceptable be-
cause the energy spectrum then has no lower bound [24].
However such negative value is necessary in the standard
σ−ω model to get the results closer to the experimental
values. On the other hand to have positive value for κ4it
is not necessary to make two parameter sets as was done
in Ref. [21].
In conclusion, with the new experimental values for
EOS, the predictions of EFT are proved to be true. With
the inclusion of self- and cross-interactions and without
forcing any change of parameters or modifying the for-
malism the EFT calculations with G2 parameter set ex-
plain finite nuclei and infinite nuclear matter in a uni-
fied way with commendable level of accuracy in both
the cases. Any Lagrangian without all types of self- and
cross-interactions is incomplete and at present EFT can
be considered as a complete unified theory for finite nu-
clei as well as for infinite nuclear matter.
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