Sensitivity of neutron radii in the $"" sup 208_Pb nucleus and a neutron star to nucleon-$ sigma_-$ rho_ coupling corrections in relativistic mean field theory

0.33 MB·English

by  G. Shen

Sensitivity of neutron radii in a 208Pb nucleus and a neutron star
to nucleon-sigma-rho coupling corrections in relativistic mean
field theory
G. Shen,1 J. Li,1 G. C. Hillhouse,1,2,∗ and J. Meng1,3,4,†
1School of Physics, Peking University, Beijing 100871
2Department of Physics, University of Stellenbosch, Stellenbosch, South Africa
5
0 3Institute of Theoretical Physics, Chinese Academy of Science, Beijing 100080
0
2 4Center of Theoretical Nuclear Physics, National Laboratory of

n
Heavy Ion Accelerator, Lanzhou 730000
a
J
5 (Dated: February 9, 2008)

1 Abstract
v
0
We study the sensitivity of the neutron skin thickness, S, in a 208Pb nucleus to the addition of
1
0
1 nucleon-sigma-rho coupling corrections to a selection (PK1, NL3, S271, Z271) of interactions in
0
5 relativistic mean field model. The PK1 and NL3 effective interactions lead to a minimum value of
0
/ S = 0.16 fm in comparison with the original value of S = 0.28 fm. The S271 and Z271 effective
h
t
- interactions yield even smaller values of S = 0.11 fm, which are similar to those for nonrelativistic
l
c
u mean field models. A precise measurement of the neutron radius, and therefore S, in 208Pb will
n
:
v place an important constraint on both relativistic and nonrelativistic mean field models. We also
i
X
study the correlation between the radius of a 1.4 solar-mass neutron star and S.
r
a
PACS numbers: 21.10.-k,21.10.Gv,26.60.+c,27.80.+w
∗e-mail: [email protected]
†e-mail: [email protected]
1
I. INTRODUCTION
Precise values for proton and neutron densities in nuclei, as well as their corresponding
root-mean-square (rms) radii, are very important for providing quantitative predictions in
nuclear physics and nuclear astrophysics. Our present understanding of nuclear phenomena
largely hinges on accurate and model independent determinations of charge densities and
radii via electron scattering. However, this picture of the nucleus will be more complete once
precise values for neutron radii become available. Currently, the situation regarding the ex-
perimental determination of neutron radii is unsatisfactory with errors typically being an
order of magnitude larger than those for proton radii [1]. What is more disconcerting, how-
ever, is the large variation in quoted values of the neutron radius for a single nucleus such as
208Pb [2, 3, 4, 5, 6, 7]. This dismal situation has prompted an experiment at Jefferson Lab-
oratory [8] to measure the neutron radius in 208Pb accurately and model independently via
parity-violating electron scattering, the results of which are expected to have a widespread
and important impact in nuclear physics.
Reliable neutron densities and radii are needed for quantitatively characterizing the bulk
properties of nuclear matter at normal nuclear densities, which in turn serve as valuable
input for: studying the dynamics of heavy-ion collisions, determining the equation-of-state
which underpins the evolution and structure of supernovae and neutron stars, studying the
propertiesofexoticneutron-richnucleifarfromthebeta-stabilityline, reducinguncertainties
inatomicparityviolationexperiments, aswellasforstudying pionicandantiprotonicatoms.
Recently, much attention [2, 3, 4, 5, 6, 7, 9, 10, 11] has been devoted to studying the
neutron skin thickness, S, in 208Pb, a quantity which is defined as the difference between the
rms radius of neutrons, < r2 >, and protons, < r2 >, i.e., S = < r2 >− < r2 >.
n p n p
A precise measurement opf the neutron radius of 2p08Pb, together with epxisting highp-precision
measurements of the proton radius [12], will yield a precise value for the neutron skin thick-
ness, thus providing one of the most stringent tests to date for current models of nuclear
structure. Indeed, the Parity Radius Experiment at Jefferson Laboratory aims to measure
the neutron radius in 208Pb to an unprecedented accuracy of 1% (±0.05 fm). Currently
relativistic and nonrelativistic models predict completely different values for the neutron ra-
dius and, hence, it is evident that the latter measurement will have an impact on deepening
our understanding of the dynamical basis of nuclear structure. In particular, nonrelativistic
2
Skyrme models predict S = 0.1 ∼ 0.2 fm [9, 10, 11], whereas present relativistic mean field
(RMF) models, on the other hand, give S = 0.2 ∼ 0.3 fm [9, 11]. Model-dependent anal-
yses of experimental data with hadronic probes yield values of S varying between 0.0 and
0.2 fm (see Table I), which seem to be more consistent with the predictions of nonrelativistic
models based on the Schr¨odinger equation. Hadronic measurements, however, suffer from
potentially serious theoretical systematic errors associated with uncertainties in nuclear re-
action mechanisms and, hence, one should be cautious about drawing conclusions regarding
the appropriateness of various dynamical models.
In this paper we focus on a relativistic description of nuclei and neutron stars within
the field theoretical framework of quantum hydrodynamics. In particular, we capitalize
on the success of the Walecka model and extensions thereof to successfully describe the
properties of nuclear matter and finite nuclei [13, 14, 15, 16, 17], as well as predict the
well-known central and spin-orbit potentials which are usually postulated in nonrelativistic
Schr¨odinger-equation-based models [18, 19, 20, 21]. More specifically, we supplement exist-
ing relativistic mean field Lagrangian densities with two new isospin dependent higher order
correction terms relating to the coupling of the nucleon current to sigma- and rho-meson
fields. These terms can be associated with multi-meson exchange processes occurring in the
inner higher-density region of nuclei. In low-energy nucleon-nucleon scattering models, the
contribution to the scattering amplitudes is usually dominated by single-meson exchange
with an effective mass of less than 1 GeV. However, it has been demonstrated that the
inclusion of two-meson exchange at low energies provides a major improvement in the de-
scription of nucleon-nucleon scattering data [22]. In general there is no unique way for
introducing nonlinear isovector-nucleon couplings in relativistic mean field models. In this
paper we introduce the simplest structure generating the nonlinear isospin-dependent terms.
Values for the various combinations of the new coupling constants are extracted by fitting
to the properties of nuclear matter – such as saturation density, binding energy per nucleon,
nuclear incompressibility and the symmetry energy – and the corresponding values for the
neutron skin thickness in 208Pb are extracted for a selection of RMF models. In particular,
we focus on the PK1 [23], NL3 [24], S271 and Z271 [25] RMF parameterizations.
One of the issues we wish to address in the paper is to find ways of adjusting the neutron
radius in 208Pb by adding new terms to the Lagrangian density of existing relativistic mean
field models, while at the same time keeping the value of proton radius (which is accurately
3
determined experimentally) fixed. In particular, the question arises as to whether it is
possible for RMF models to produce values of the neutron radius which are in the same
range as those for nonrelativistic models.
Inrecent analyses, a linear relationship between the neutron skin in 208Pb andthe nuclear
symmetry energy at saturation density was proposed [10, 11, 26]. We also investigate the
relationship between the latter quantities for our new RMF models. The new interaction
terms considered in this paper influence the density dependence of symmetry energy, which
in turn affects the neutron skin in 208Pb. In addition, we extrapolate from normal to dense
neutron matter and study the correlation between the radius of a neutron star and the
neutron radius in 208Pb. Indeed, Horowitz and Piekarewicz [27] have performed such a
correlation analysis for a wide range of RMF models and they concluded that, whereas the
radius of a 0.5 solar-mass neutron star can be inferred from the radius of 208Pb, the radius of
a 1.4 solar-mass neutron star is not uniquely constrained by a measurement of the neutron
skin thickness. In this paper we perform a similar analysis for the PK1, NL3, S271 and Z271
effective interactions supplemented with our new terms, but for simplicity we only consider
the mass and radius of a 1.4 solar-mass neutron star.
This paper is organized in the following sequence. A brief formulation of our new RMF
model in relation to existing RMF models for finite nuclei and neutron star matter is pre-
sented in Sec. II. The extraction of parameter sets for our new RMF models is discussed
in Sec. III. In Sec. IV results are presented and discussed for the neutron radius in 208Pb.
Values of neutron radii in 208Pb for different parameter sets are then correlated with the
radius of a 1.4 solar-mass neutron star. Finally, in Sec. V, we summarize the main points
of this paper.
II. HIGHERORDERCORRECTIONSTORMFMODELSFORSPHERICALNU-
CLEI AND NEUTRON STARS
The basic physics underlying RMF models and their application in nuclear physics can
be found in Refs. [28, 29, 30]. In this section we present our new RMF model and indicate
its relation to existing RMF models. The basic ansatz of the RMF theory is a Lagrangian
density whereby nucleons are described as Dirac particles which interact via the exchange
4
of sigma- (σ), omega- (ω), and rho- (ρ) mesons, and also the photons (A), namely:
1+τ
L = ψ iγµ∂ −m−g σ −g γµω −g γµ~τ ·~ρ −eγµ 3A ψ
µ σ ω µ ρ µ 2 µ
(cid:20) (cid:21)
1 1
+ ∂µσ∂ σ − m2σ2 −U(σ)
2 µ 2 σ
1 1
− ωµνω + m2ωµω +U(ω)
4 µν 2 ω µ
1 1
− ~ρµν ·~ρ + m2~ρµ ·~ρ
4 µν 2 ρ µ
1
− AµνA (1)
µν
4
where the field tensors of the vector mesons and the electromagnetic field take the following
form:
ωµν = ∂µων −∂νωµ,
Aµν = ∂µA −∂νA ,
ν µ
~ρµν = ∂µ~ρν −∂ν~ρµ −g ~ρµ ×~ρν . (2)
ρ
The nonlinear self-couplings for σ and ω mesons are respectively:
U(σ) = 1g σ3 + 1g σ4, U(ω) = 1c (ωµω )2, (3)
3 2 4 3 4 3 µ
with the self-coupling constants g , g , and c .
2 3 3
To modify the density dependence of nuclear matter symmetry energy, Horowitz and
Piekarewicz [25] introduced the following nonlinear omega-rho coupling term:
L = 4Λ g2ρ~ ·ρ~µg2ω ωµ. (4)
HP v ρ µ ω µ
In addition to considering the above term, in this paper we introduce, for the first time,
the following two new couplings of the nucleon current to sigma- and rho-meson fields:
L = −Γ ψ¯g γµ(g σ/m) ~τ ·ρ~ ψ (5)
1 1 ρ σ µ
L = −Γ ψ¯g γµ(g σ/m)2~τ ·ρ~ ψ (6)
2 2 ρ σ µ
i.e., the total Lagrangian density of interest is:
L′ = L +L + L + L . (7)
HP 1 2
5
The classical variational principle leads to the following equations of motion:
[α·p + V(r) + β(m + S(r))]ψi = εiψi (8)
for the nucleon spinors, where
V(r) = β{g /ω + g (1 + Γ (g σ/m) + Γ (g σ/m)2)~τ ·/ρ~ + e(1 + τ3) /A }
ω µ ρ 1 σ 2 σ µ 2 µ (9)

S(r) = g σ
 σ
and
(m2 − ∇2) σ = −g ρ − g σ2 − g σ3 −g ρ~ ·~jµ(Γ (g σ/m) + 2Γ (g2σ/m2))
σ σ s 2 3 ρ µ 1 σ 2 σ
(m2 − ∇2)ωµ = g jµ −c ωµ(ωµω )−8Λ g2g2ρ~ ·ρ~µωµ
v ω 3 µ v ρ ω µ
 (10)

(m2 − ∇2)ρ~µ = g ~jµ(1 + Γ (g σ/m) + Γ (g σ/m)2)−8Λ g2g2ω ωµρ~µ
 ρ ρ 1 σ 2 σ v ρ ω µ
− ∇2Aµ = ejµ
 p




forthe mesons and photons. The nucleon spinors provide the relevant source terms:
ρ = A ψ ψ
s i=1 i i
jµ = PAi=1ψiγµψi


~jµ = PAi=1ψiγµ~τψi
jpµ = PAi=1ψiγµ1+2τ3ψi



where the summations run overthe valPence nucleons only. The present method neglects
the contribution of negative energy states, i.e. the so-called no sea approximation. The
above non-linear equations are solved by iteration within the context of the mean field
approximation whereby the meson field operators are replaced by their expectation values.
A. Spherical nuclei
Since spherical nuclei such as 208Pb respect time reversal symmetry, there are no currents
in the nucleus and the spatial vector components ω, ρ and A vanish. One is left with
the time-like components ω , ρ~ and A . Charge conservation guarantees that only the
0 0 0
3-components of the isovector ρ survive. As the system obeys rotational symmetry, the
0,3
nucleon potentials and the meson-field sources depend only on the radial coordinate r. The
Dirac spinor, ψ(r) is characterized by the angular momentum quantum numbers l,j,m,
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the isospin t = +(−) for protons (neutrons) respectively, and any other relevant quantum
numbers i, i.e.
iGlij(r)Yl (θ,φ)
ψ(r) = r jm χ (11)
 Filj(r)(~σ ·rˆ)Yl (θ,φ)  t
r jm
 
where Yl (θ,φ) are the usual spinor spherical harmonics, Glj(r)/r and Flj(r)/r denote
jm i i
the upper- and lower-component radial wave functions, and χ represents the isospin wave
t
functions specified by the isospin t. The radial wave functions are normalized as:
∞
dr(| Glj(r) |2 + | Flj(r) |2) = 1. (12)
i i
Z0
The following phase convention is adopted for the vector spherical harmonics:
(~σ ·rˆ)Yl = −Yl′ ,
jm jm
where
l+1, j = l+1/2,
l′ = 2j −l =

l−1, j = l−1/2.

After some tedious algebra, one gets the radial equation for the upper and lower compo-

nents, respectively [13]:
ǫ Glj(r) = − ∂ + κi Flj(r) + (m + S(r) + V(r))Glj(r)
i i ∂r r i i (13)
ǫ Flj(r) = (cid:0)+ ∂ + κi(cid:1)Glj(r) − (m + S(r) − V(r))Flj(r)
 i i ∂r r i i
(cid:0) (cid:1)
where 
−(j +1/2), for j = l+1/2,
κ =

+(j +1/2), for j = l−1/2,

and

V(r) = g ω + g (1 + Γ (g σ/m) + Γ (g σ/m)2)τ ρ + 1e(1 + τ )A ,
ω 0 ρ 1 σ 2 σ 3 0,3 2 3 0 (14)

S(r) = g σ.
 σ
The meson fields equations reduce to radial Laplace equations of the form:
∂2 2 ∂
− + m2 φ = S (15)
∂r2 r∂r φ φ
(cid:18) (cid:19)
7
where m are the meson masses for φ = σ, ω, ρ and m = 0 for the photon. The relevant
φ φ
source terms are:
−g ρ − g σ2 − g σ3 −g ρ ρ (Γ (g σ/m) + 2Γ (g2σ/m2)) for the σ field,
σ s 2 3 ρ 0,3 3 1 σ 2 σ
 g ρ − c ω3 − 8Λ g2g2ρ2 ω for the ω field,
ω υ 3 0 v ρ ω 0,3 0
S =  (16)
φ 
 g ρ (1 + Γ (g σ/m) + Γ (g σ/m)2)−8Λ g2g2ω2ρ for the ρ field,
 ρ 3 1 σ 2 σ v ρ ω 0 0,3
eρ (r) for the Coulomb field,
 c



wherethe various nucleon densities are given by:
4πr2ρ (r) = A (|G (r)|2 − |F (r)|2),
s i=1 i i
4πr2ρυ(r) = PAi=1(|Gi(r)|2 + |Fi(r)|2), (17)

4πr2ρ3(r) = PZi=1(|Gi(r)|2 + |Fi(r)|2) − Ni=1(|Gi(r)|2 + |Fi(r)|2),
4πr2ρc(r) = PZi=1(|Gi(r)|2 + |Fi(r)|2). P



The neutron and protoPn rms radii are directly related to the neutron and proton density
distributions via the following relationship:
ρ (r)r2d3r
R = < r2 > = i (18)
i i sR ρi(r)d3r
q
where ρ (r) (i = n,p) denotes the corresponding neRutron and proton baryon density distri-
i
butions.
The total binding energy of the system is:
E = E +E +E +E +E −mA
nucleon σ ρ ω c
1 1 1
= ǫ − d3r{g σρ (r) + g σ3 + g σ4}
i σ s 2 3
2 3 2
i Z
X
1 g σ g σ
− d3rg ρ {(1 + 2Γ σ + 3Γ ( σ )2)ρ (r) − 8Λ (g ω g ρ )2}
ρ 0,3 1 2 3 v ω 0 ρ 0,3
2 m m
Z
1 1 1
− d3r{g ω − c ω4}ρ (r)− d3rA ρ (r)−mA. (19)
2 ω 0 2 3 0 v 2 0 c
Z Z
Eq. (13) is solved self-consistently using the Runge-Kutta algorithm and the shooting
method,andtheLaplaceequations,givenbyEq.(15),aresolvedemployingaGreenFunction
method.
B. Neutron stars
Nextwediscuss theRMFLagrangiandensity usedfordescribing aneutronstarconsisting
of only nucleons and leptons (mainly electrons, e−, andmuons, µ−). The Lagrangiandensity
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associated with the leptons is:
L = ψ¯ (iγµ∂ −m )ψ , (20)
lepton λ µ λ λ
λ=e−,µ−
X
such that the full Lagrangian density under consideration consists of a hadronic part, given
by Eq. (7), plus a leptonic part, given by Eq. (20). Introducing the mean field and no-
sea approximations, the equations of motion for baryons and mesons can be derived. The
detailed formalism of which can be found in Ref. [32]. The Dirac equation represents the
equation of motion associated with the leptons. Based on chemical equilibrium and charge
neutral conditions, for a given total baryon density, the equations of motion are solved self-
consistently by iteration, and the meson fields, particle densities and Fermi momenta of each
species (protons, neutrons, electrons, and muons) are obtained simultaneously.
Thecanonicalenergy-momentumtensorisderivedbyinvokingtheinvarianceofspacetime
translation, namely:
∂L
Tµν = −gµνL + ∂νφ, (21)
∂(∂ φ)
µ
φ
X
where gµν is the Minkowski metric in rectilinear coordinates. For the case in which neutron
star matter can be considered to be a perfect fluid, the energy-momentum tensor is given
by [31]:
ǫ 0 0 0
0 p 0 0 
Tµν = . (22)
0 0 p 0 
 
 
0 0 0 p 
 
 
Comparing the ground-state expectation value of Eq. (21) with Eq. (22), via the field equa-
tions the energy density, ǫ, and the pressure, P, are given by:
1 1 1 1 3 1
ǫ = m2σ2 + g σ3 + g σ4 + m2ω2 + c ω4 + m2ρ2 +12g2g2Λ ω2ρ2
2 σ 3 2 4 3 2 ω 0 4 3 0 2 ρ 0,3 ρ ω v 0 0,3
1 kB 1 kλ
+ k2dk k2 +(m +g σ)2 + k2dk k2 + m2, (23)
π2 B σ π2 λ
B=n,pZ0 λ=e−,µ−Z0 q
X p X
1 1 1 1 1 1
P = − m2σ2 − g σ3 − g σ4 + m2ω2 + c ω4 + m2ρ2 +4g2g2Λ ω2ρ2
2 σ 3 2 4 3 2 ω 0 4 3 0 2 ρ 0,3 ρ ω v 0 0,3
1 kB k4 1 kλ k4
+ dk + dk , (24)
3π2 k2 +(m +g σ)2 3π2 k2 +m2
B=n,pZ0 B σ λ=e−,µ−Z0 λ
X X
p p
9
where k and k are the Fermi momenta of baryons and leptons respectively.
B λ
The mass and radius of a neutron star are obtained by employing the Oppenheimer-
Volkoff (OV) equations [33, 34]:
dp(r) [p(r)+ǫ(r)][M(r)+4πr3p(r)]
= − , (25)
dr r(r −2M(r))
r
M(r) = 4π ǫ(r′)r′2dr′, (26)
Z0
where r denotes the radial coordinate relative to the center of the star, p(r) and ǫ(r) are the
pressure and energy density at a radial point r in the star respectively, and M(r) represents
the mass of the sphere contained within a radius r. Since zero pressure cannot support a
neutron star from collapsing, we define the radius, R, of the star as that radius at which
the pressure is zero. The mass total mass, M(R), of the star is subsequently defined as the
mass contained within a sphere of radius R.
III. EXTRACTION OF PARAMETER SETS FOR NEW RMF MODELS
In this section we present the procedure and criteria for extracting values for our new
coupling constants Γ and Γ in Eqs. (5) and (6) respectively. In particular, we consider
1 2
the addition of these new isopsin-dependent higher order correction terms to the PK1 [23],
NL3 [24], S271 and Z271 [25] RMF models. The parameters sets for these interactions are
presented in Table II.
We start with the recent PK1 effective interaction [23]. The PK1 effective interaction
provides an excellent description of the properties of nuclear matter as well as for nuclei near
and far from the valley of β stability. For this interaction, symmetric nuclear matter satu-
rates at a Fermi momentum of 1.30 fm−1 with a binding energy per nucleon of −16.27 MeV
and an incompressibility of K = 283 MeV. For comparison with the results of the PK1
effective interaction, we also employ the NL3, S271, and Z271 effective interactions, which
have been used in Refs. [25, 27] to observe the effects of adding a nonlinear omega-rho cou-
pling term, given by Eq. (4), on the neutron skin thickness, S, in 208Pb and the radius of
1.4 solar-mass neutron star. The NL3 effective interaction has also been used extensively
to reproduce a variety of nuclear properties, such as binding energies, nuclear radii, nuclear
density distribution, single particle spectra, etc. The NL3 and S271 effective interactions
contain a sigma-meson self-coupling and Z271 includes both sigma- and omega-meson self-
10