Brazilian Journal of Physics, vol. 35, no. 3B, September, 2005 869
The Importance of Strange Mesons in Neutron Star Properties
R. Cavagnoli and D. P. Menezes
Depto de F´ısica - CFM - Universidade Federal de Santa Catarina - Florian ´opolis - SC - CP. 476 - CEP 88.040 - 900 - Brazil
Received on 31 May, 2005
In order to obtain the properties of compact stellar objects, appropriate equations of state have to be used. In the literature, strange meson fields, namely the scalar meson fieldσ∗(975)and the vector meson fieldφ(1020), had to be considered in order to reproduce the observed strongly attractiveΛΛinteraction. The introduction of these strange mesons makes the equations of state harder (EOS) due to the repulsive effect of theφ(1020) meson. In this work the inclusion of these mesons in the equation of state and their influence on the properties of the neutron stars are investigated.
I. INTRODUCTION
In the present work we use the relativistic non-linear Wa-lecka model (NLWM) [1], at zero temperature (T =0), with the lowest baryon octet{N,Λ,Σ,Ξ}inβequilibrium with the lightest leptons {e−,µ−} and compare the results with the
same model plus strange meson fields,σ∗(975)andφ(1020), which introduce strangeness to the interaction according to [2] and [3]. Strange meson fields, namely the scalar meson fieldσ∗(975)and the vector meson fieldφ(1020), had to be considered in order to reproduce the observed strongly attrac-tiveΛΛinteraction. This formalism applied to compact ob-jects like neutron stars, where the energies are such that allow the appearance of the eight lightest baryons. The motivation
for this study lies in our interest to describe the interaction between hadrons taking into account a growing number of effects in order to better describe it. Given the difficulty of making comparisons with experimental data we will, for now limit ourselves to verify if the inclusion of these mesons sig-nificantly alters some quantities like pressure and energy den-sity in the equation of state and the bulk properties of compact stars.
II. THE FORMALISM
The lagrangian density of the NLWM with the inclusion of the strange meson sector and leptons forβequilibrium is:
L
=8
∑
B=1 ¯ ΨB
h
γµ
³
i∂µ−gωBVµ−gρB~τ.~bµ
´
−(MB−gσBσ)
i
ΨB
+1 2
¡∂
µσ∂µσ−m2σσ2¢−3!1kσ3−4!1λσ4−14ΩµνΩµν+ 1 2m
2
ωVµVµ
−1
2~Bµν~B µν+1
2m 2
ρ~bµ~bµ
+1 2
¡∂
µσ∗∂µσ∗−m2σ∗σ∗2¢+1
2m 2
φφµφµ−
1 4SµνS
µν−
∑
Bgσ∗BΨ¯BΨBσ∗−
∑
B
gφBΨ¯BγµΨBφµ
+ 2
∑
l=1 ¯
Ψl(iγµ∂µ−Ml)Ψl, (1)
whereΩµν=∂µVν−∂νVµ,~Bµν=∂µ~bν−∂ν~bµ−gρ
³
~bµ×~bν´
and Sµν =∂µφν−∂νφµ, with B extending over the eight baryons,giBare the coupling constants of mesonsi,i=σ,ω,ρ with baryon B, and mi is the mass of meson i. λ and k are the weighs of the non-linear scalar terms and~τ is the
870 R. Cavagnoli and D. P. Menezes
giB are defined by giB =xBgi where xB=
p
2/3, for hy-perons, [4], xB=1 for the nucleons, and alsogσ=8.910, gω=10.626, gρ=8.208,gσ∗Λ=gσ∗Σ=5.11,gσ∗Ξ=9.38,
gφΛ=gφΣ=4.31,gφΞ=8.62,k=−6.426 10−4, λ=5.530 according to [5] and [6]. The strange mesons interact with hyperons only (gσ∗p=gσ∗n=gφp=gφn=0). The
mas-ses of baryons of the octect are: MN =938MeV (nucle-ons), MΛ=1116MeV, MΣ=1193MeV, MΞ =1318MeV and the meson masses are: mσ=512MeV,mω=738MeV, mρ=770MeV,mσ∗=975MeV,mφ=1020MeV. In order to
account for theβequilibrium in the star the leptons are also in-cluded in the lagrangian density of eq. (1) as a non-interacting Fermi gas. The masses of the leptons areMe− =0.511MeV
andMµ−=105.66MeV.
Applying the Euler-Lagrange equations to (1) and using the mean-field approximation (σ→ hσi=σ0, Vµ→ hVµi= δµ0V0and~bµ→ h~bµi=δµ0b0µ≡δµ0bµ), we obtain:
σ0=−
k 2m2
σσ
2 0−
λ 6m2
σσ
3 0+
∑
B gσ
m2
σxBρσB, (2)
V0=
∑
Bgω
m2
ωxBρB, (3)
b0=
∑
Bgρ
m2
ρxBτ3ρB, (4)
σ∗
0=
∑
Bgσ∗
m2
σ∗
xBρB, (5)
φ0=
∑
Bgφ
m2φxBρB, (6)
where
ρσB= MB∗
π2
Z KFB
0
p2d p p
p2+M∗
B
, (7)
ρB= 1 3π2K
3
FB , M
∗
B=MB−gσBσ0−gσ∗Bσ∗0, (8)
and the 0 subscripts added to the fields mean that a mean fi-eld approximation, where the meson fifi-elds were considered as classical fields was performed.
Through the energy-momentum tensor, we obtain:
εa= 1 π2
Ã
∑
i=B,l
ZKFi
0
p2d p
q
p2+M∗2 i ! +m 2 ω 2 V 2 0+ m2 ρ 2 b 2 0+ m2 σ 2 σ 2 0+ k 6σ 3 0+ λ 24σ 4 0+
m2σ∗
2 σ
∗2 0 +
m2φ 2 φ
2
0, (9)
Pa= 1 3π2
∑
i=B,l
ZKFi
0
p4d p
q
p2+M∗2 i
+
m2ω 2 V
2 0+
m2ρ 2 b
2 0−
m2σ 2 σ 2 0− k 6σ 3 0− λ 24σ 4 0−
m2σ∗
2 σ ∗2 0 + m2 φ 2 φ 2
0. (10)
In a neutron star, charge neutrality and baryon number must be conserved quantities. Moreover, the conditions of chemical equilibrium hold. In terms of the chemical potentials of the constituent particles, these conditions read:
µn=µp+µe−, µµ−=µe−, µΣ0=µΞ0=µΛ=µn, µΣ−=µΞ−=µn+µe−,
µΣ+=µp=µn−µe−. (11)
III. RESULTS
The inclusion of hyperons and strange mesons alters the equations of state and the particle fractions, as can be seen
from Figs 1 and 2.
From Fig. 1 we notice that the inclusion of the hyperons softens the equations of state in comparison with the EOS obtained only with nucleons and leptons. The inclusion of the strange mesons hardens these equations a little at higher energy densities. This indicates that the influence of the strange mesons is significant at higher densities, what can be easily seen in Fig. 2, where we notice a difference in the frac-tions of heavier hyperons, at densities above 5ρ0, whereρ0is the saturation density of the nuclear matter.
ma-Brazilian Journal of Physics, vol. 35, no. 3B, September, 2005 871
0 0.5 1 1.5 2 2.5 3
0 1 2 3 4 5 6 7 8
P (fm
-4 )
ε (fm-4
)
(II) (III) (I)
with (σ* , φ) (III) no (σ* , φ) (II) no hyperons, no (σ* , φ) (I)
FIG. 1: Pressure versus energy density.
0.0001 0.001 0.01 0.1 1
0 1 2 3 4 5 6 7 8 Yi
ρ/ρ0
e
-µ
-n
p
Λ Σ - Σ0
Σ+ Ξ
-Ξ0
no (σ* , φ)
0.0001 0.001 0.01 0.1 1
0 1 2 3 4 5 6 7 8
ρ/ρ0
e
-µ
-n
p
Λ Σ - Σ0
Σ+
Ξ
-Ξ0
with (σ* , φ)
FIG. 2: Particle fractionsYi=ρi/ρ, i = B,l, as a function of the total baryon densityρin units ofρ0.
ximum gravitational mass and with the maximum radius are shown for two possible EOS: without the strange mesons and with them. In these cases the crust of the stars were not
inclu-ded.
The observed values for the mass of the neutron stars lie between 1.2 to 1.8Msun. Our results are in the expected range. From table I and Fig. 3, one can see that the differences in the star properties with and without strange meson are not
8 9 10 11 12 13 14
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
R (Km)
M / Msun
no (σ* , φ) with (σ* , φ)
FIG. 3: Radius versus gravitational mass for the families of stars obtained within the NLWM and within the NLWM plus the strange mesons sector after the TOV equations were solved.
No (σ∗,φ) With (σ∗,φ) Mmax/Msun 1.93 1.92
R(Km) 11.30 10.91
ε0(fm−4) 6.39 7.01 Rmax(Km) 13.33 13.33
M/Msun 1.39 1.39 ε0(fm−4) 1.75 1.78
TABLE I: Neutron star properties.ε0is the central energy density.
very relevant. Nevertheless, the constitution of the stars at large densities are somewhat different. At about four times the saturation density (see Fig. 2) the inclusion of the strange mesons start playing its role in the constitution of the stars. At this high energy a phase transition to a deconfined phase of quarks or to a system with kaon condensates can already take place. These two possibilities are certainly more impor-tant to the properties of neutron stars than a system containing strange mesons. The influence of the inclusion of the strange mesons in protoneutron stars with temperatures around 30 to 40MeV and the their importance when trapped neutrinos are included are under investigation.
ACKNOWLEDGMENTS
This work was partially supported by CNPq (Brazil).
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