Compatibility of Exotic States with Neutron Star Observation
Chang Ho Hyun∗)
Institute of Basic Science, Sungkyunkwan University, Suwon 440-746, Korea
Department of Physics, Seoul National University, Seoul 151-742, Korea
We consider the effect of hard core repulsion in the baryon-baryon interaction at short
distance to the properties of a neutron star. We obtain that, even with hyperons in the
interior of a neutron star, the neutron star mass can be as large as ∼ 2M⊙.
§1. Introduction
Reports on recent observations of pulsars in various binary systems show that
the maximum mass of a neutron star can be large as (1.7 ∼ 2.1)M⊙.
1), 2) Most of
them still have large uncertainty, but a few are within the above range with relatively
small error bars. The possibility of large mass of a neutron star thus has led to a
claim that exotic states of matter at high densities are not necessary in the neutron
star as far as its mass is concerned,3) but there also appeared a counter argument
that the large mass does not necessarily rule out the exotic states.4) There are many
sources of uncertainties at high densities, e.g. state of matter, constituent particles
and their interactions, but the information available to reduce the uncertainties is
not sufficient yet.
We revisit the neutron star mass problem with a simple phenomenological ap-
proach. One fixed point of nuclear matter physics is the nuclear saturation density;
its properties such as density, binding energy, symmetry energy, and compression
modulus are fairly well constrained. We describe these saturation properties in terms
of quantum hadrodynamics (QHD).5) The other fixed point we choose is the hard
core repulsion at short range. Though it is a kind of artifact adopted to describe the
nucleon-nucleon data, its role is clear in many phenomena of nuclear physics. The
effect of hard core can be parametrized with an excluded volume in the estimation
of thermodynamic variables.6), 7), 8) It has been more frequently employed to explain
the phase transition in the relativistic heavy ion collision environment, and could
describe well the transition from hadronic to quark-gluon plasma phase.9)
In this work, by including hard cores in the interaction of baryons, we explore
the bulk properties of the neutron star, and compare the result with the recent
observation of heavy neutron star masses. This paper is outlined as follows. In the
next section, we briefly address the basic formalism of QHD with hard core. The next
section comes up with numerical results, and brief concluding remarks are drawn in
the following section.
∗) e-mail address: hch@meson.skku.ac.kr
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.0504v1
2 C. H. Hyun
§2. Formalism
We employ the QHD Lagrangian,
ψ̄B (i∂ · γ −mB + gσBσ − gωBγ0ω0 − gρBτ3γ0b30)ψB
mN (gσNσ)
(gσNσ)
l=e, µ
ψ̄l(i∂ · γ −ml)ψl, (2.1)
where the baryon species B includes octet baryons, and σ, ω0 and b30 are non-
vanishing meson fields in the mean field approximation. When we account for the
forbidden region due to hard core, the baryon density is redefined as
1 + vevρ′
, (2.2)
where ρ′ is the density in the case of point particle and vev the excluded volume.
We assume vev =
πr30 where r0 is the radius of hard core, which is treated as a free
parameter in our consideration. Consistency with thermodynamic relations and self-
consistency conditions alter the form of state variables (pressure, chemical potential,
energy density and etc) and equation of motion of σ-meson field from those of point
particle ones. The explicit formulas and equations can be found in old6), 7), 8) and
recent10), 11) publications.
Three meson-nucleon coupling constants gσN , gωN and gρN and two σ-meson
self interaction coefficients b and c are fitted to five saturation properties, the sat-
uration density (0.17 fm−3), binding energy (16.0 MeV), symmetry energy (32.5
MeV), compression modulus (300 MeV) and nucleon effective mass (0.75m∗N ), with
a given hard core radius r0. Meson-hyperon coupling constants are determined by
quark counting rules, gMY = gMN
q=u,d nqY /3, where gMY is the meson-hyperon
coupling constant, nqY is the number of u and d quarks in a hyperon species Y
and gMN is the meson-nucleon coupling constant. As for the hard core radius of
hyperons, we assume the same value as that of the nucleon for simplicity.
Table I summarizes the parameters determined from the given saturation prop-
erties and hard core radii.
r0 (fm) gσN gωN gρN b (×10
3) c (×103)
0 8.44 8.92 7.76 3.97 4.00
0.2 8.43 8.91 7.72 3.80 4.37
0.3 8.39 8.89 7.64 3.38 5.26
0.4 8.30 8.85 7.47 2.51 7.11
0.5 8.16 8.78 7.19 0.93 10.47
Table I. Meson-nucleon coupling constants and coefficients b and c fitted to a set of saturation
properties described in the text with a given r0 value.
Compatibility of Exotic States with Neutron Star Observation 3
§3. Numerical result
Fig. 1 shows the binding energy per a nucleon in the symmetric nuclear matter
with different hard core radii. Though the saturation properties are the same re-
gardless of r0 values, the equation of state becomes stiffer at high densities with a
larger r0 value.
 0  0.5  1  1.5  2  2.5
r0 = 0.0
    0.2
    0.4
    0.5
Fig. 1. Binding energy of a nucleon in the symmetric nuclear matter with different hard core radii.
The equation of state of neutron star matter is determined self-consistently by
the baryon number conservation, charge neutrality, β-equilibrium of baryons and
leptons, and equations of motion of meson fields. Once the equation of state is
determined, the mass-radius relation of a neutron star can be obtained by solving
Tolman-Oppenheimer-Volkoff (TOV) equation. Table II shows the maximum mass of
a neutron star, corresponding radius and central density with nucleons only (columns
of “np”) and with hyperons (columns of “npY ”). Consistent with the behavior of
the equation of state at high densities in Fig. 1, the maximum mass becomes larger
with a larger r0 value. For np case, when r0 = 0.5 fm, the increase amounts to
10% of the maximum mass without hard core. With hyperons, the maximum mass
increases to the range of large mass in recent observations when r0 & 0.3 fm.
n p n p Y
r0 (fm) M (M⊙) R (km) ρcent (ρ0) M (M⊙) R (km) ρcent (ρ0)
0 2.10 10.9 6.4 1.53 11.3 6.1
0.2 - - - 1.58 11.4 6.1
0.3 2.14 11.1 6.2 1.70 11.5 5.9
0.4 2.20 11.4 5.9 1.97 12.2 5.2
0.5 2.34 11.7 5.4 - - -
Table II. Maximum mass M in units of solar mass, and corresponding radius R in km and central
density ρcent in unit of the saturation density ρ0.
4 C. H. Hyun
§4. Conclusion
We investigated the maximum mass of a neutron star in a simple phenomeno-
logical approach where the hard-core repulsion is included in the QHD model. The
hard core radius is treated as a free parameter, and the meson-nucleon coupling con-
stants are fixed identical saturation properties. We obtained the equation of state
of neutron star matter that satisfies thermodynamic equations and self-consistency
conditions. Solving TOV equation, we obtained the mass-radius relation of a neu-
tron star. Our result shows that the maximum mass with hyperons can be as large
as observed masses with a hard core radius r0 & 0.3 fm. These values of r0 are in the
range of hard core radius 0.3 ∼ 0.6 fm in well-known hard-core potential models such
as Hamada-Johnston12) or Reid.13) More investigations are necessary to figure out
the uncertainties. For instance, the hard core size of hyperons can matter. The effect
of hard cores to the formation of other exotic states such as meson condensation or
deconfined quark phases is also worthy to be studied.
To conclude, the our result shows that the hyperon matter, which is known to
give the biggest effect to the mass-radius relation of a neutron star among possible
exotic states in the interior of a neutron star, is not necessarily incompatible with
the observed mass.
Acknowledgments
The author thanks the Yukawa Institute for Theoretical Physics at Kyoto Uni-
versity, where this work was initiated and developed during the YKIS2006 on ”New
Frontiers on QCD”. Author is grateful to Shung-ichi Ando for reading the manuscript.
This works was supported by the Basic Research Program of Korea Science & En-
gineering Foundation (R01-2005-000-10050-0).
References
1) D. J. Nice et al., Astrophys. J. 634 (2005), 1242
2) D. Page and S. Reddy, Ann. Rev. Nucl. Part. Sci. 65 (2006), 327
3) F. Özel, Nature 441 (2006), 1115
4) T. Klahn et al., nucl-th/0609067.
5) B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16 (1986), 1
6) D. H. Rischke, M. I. Gorenstein, H. Stöcker and W. Greiner, Z. Phys. C 51 (1991), 485
7) S. Kagiyama, A. Nakamura and T. Omodaka, Z. Phys. C 53 (1992), 163
8) J. Cleymans, J. Stalnacke and E. Suhonen, Z. Phys. C 55 (1992), 317
9) S. Kagiyama et al., Eur. Phys. J. C 25 (2002), 453
10) P. K. Panda et al., Phys. Rev. C 65 (2002), 065206
11) R. M. Aguirre and A. L. De Paoli, Phys. Rev. C 68 (2003), 055804
12) T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962), 382
13) R. V. Reid, Ann. Phys. (N.Y.) 50 (1968), 411
http://arxiv.org/abs/nucl-th/0609067
	Introduction
	Formalism
	Numerical result
	Conclusion

  We consider the effect of hard core repulsion in the baryon-baryon
interaction at short distance to the properties of a neutron star. We obtain
that, even with hyperons in the interior of a neutron star, the neutron star
mass can be as large as $\sim 2 M_\odot$.

Introduction
Reports on recent observations of pulsars in various binary systems show that
the maximum mass of a neutron star can be large as (1.7 ∼ 2.1)M⊙.
1), 2) Most of
them still have large uncertainty, but a few are within the above range with relatively
small error bars. The possibility of large mass of a neutron star thus has led to a
claim that exotic states of matter at high densities are not necessary in the neutron
star as far as its mass is concerned,3) but there also appeared a counter argument
that the large mass does not necessarily rule out the exotic states.4) There are many
sources of uncertainties at high densities, e.g. state of matter, constituent particles
and their interactions, but the information available to reduce the uncertainties is
not sufficient yet.
We revisit the neutron star mass problem with a simple phenomenological ap-
proach. One fixed point of nuclear matter physics is the nuclear saturation density;
its properties such as density, binding energy, symmetry energy, and compression
modulus are fairly well constrained. We describe these saturation properties in terms
of quantum hadrodynamics (QHD).5) The other fixed point we choose is the hard
core repulsion at short range. Though it is a kind of artifact adopted to describe the
nucleon-nucleon data, its role is clear in many phenomena of nuclear physics. The
effect of hard core can be parametrized with an excluded volume in the estimation
of thermodynamic variables.6), 7), 8) It has been more frequently employed to explain
the phase transition in the relativistic heavy ion collision environment, and could
describe well the transition from hadronic to quark-gluon plasma phase.9)
In this work, by including hard cores in the interaction of baryons, we explore
the bulk properties of the neutron star, and compare the result with the recent
observation of heavy neutron star masses. This paper is outlined as follows. In the
next section, we briefly address the basic formalism of QHD with hard core. The next
section comes up with numerical results, and brief concluding remarks are drawn in
the following section.
∗) e-mail address: hch@meson.skku.ac.kr
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.0504v1
2 C. H. Hyun
§2. Formalism
We employ the QHD Lagrangian,
ψ̄B (i∂ · γ −mB + gσBσ − gωBγ0ω0 − gρBτ3γ0b30)ψB
mN (gσNσ)
(gσNσ)
l=e, µ
ψ̄l(i∂ · γ −ml)ψl, (2.1)
where the baryon species B includes octet baryons, and σ, ω0 and b30 are non-
vanishing meson fields in the mean field approximation. When we account for the
forbidden region due to hard core, the baryon density is redefined as
1 + vevρ′
, (2.2)
where ρ′ is the density in the case of point particle and vev the excluded volume.
We assume vev =
πr30 where r0 is the radius of hard core, which is treated as a free
parameter in our consideration. Consistency with thermodynamic relations and self-
consistency conditions alter the form of state variables (pressure, chemical potential,
energy density and etc) and equation of motion of σ-meson field from those of point
particle ones. The explicit formulas and equations can be found in old6), 7), 8) and
recent10), 11) publications.
Three meson-nucleon coupling constants gσN , gωN and gρN and two σ-meson
self interaction coefficients b and c are fitted to five saturation properties, the sat-
uration density (0.17 fm−3), binding energy (16.0 MeV), symmetry energy (32.5
MeV), compression modulus (300 MeV) and nucleon effective mass (0.75m∗N ), with
a given hard core radius r0. Meson-hyperon coupling constants are determined by
quark counting rules, gMY = gMN
q=u,d nqY /3, where gMY is the meson-hyperon
coupling constant, nqY is the number of u and d quarks in a hyperon species Y
and gMN is the meson-nucleon coupling constant. As for the hard core radius of
hyperons, we assume the same value as that of the nucleon for simplicity.
Table I summarizes the parameters determined from the given saturation prop-
erties and hard core radii.
r0 (fm) gσN gωN gρN b (×10
3) c (×103)
0 8.44 8.92 7.76 3.97 4.00
0.2 8.43 8.91 7.72 3.80 4.37
0.3 8.39 8.89 7.64 3.38 5.26
0.4 8.30 8.85 7.47 2.51 7.11
0.5 8.16 8.78 7.19 0.93 10.47
Table I. Meson-nucleon coupling constants and coefficients b and c fitted to a set of saturation
properties described in the text with a given r0 value.
Compatibility of Exotic States with Neutron Star Observation 3
§3. Numerical result
Fig. 1 shows the binding energy per a nucleon in the symmetric nuclear matter
with different hard core radii. Though the saturation properties are the same re-
gardless of r0 values, the equation of state becomes stiffer at high densities with a
larger r0 value.
 0  0.5  1  1.5  2  2.5
r0 = 0.0
    0.2
    0.4
    0.5
Fig. 1. Binding energy of a nucleon in the symmetric nuclear matter with different hard core radii.
The equation of state of neutron star matter is determined self-consistently by
the baryon number conservation, charge neutrality, β-equilibrium of baryons and
leptons, and equations of motion of meson fields. Once the equation of state is
determined, the mass-radius relation of a neutron star can be obtained by solving
Tolman-Oppenheimer-Volkoff (TOV) equation. Table II shows the maximum mass of
a neutron star, corresponding radius and central density with nucleons only (columns
of “np”) and with hyperons (columns of “npY ”). Consistent with the behavior of
the equation of state at high densities in Fig. 1, the maximum mass becomes larger
with a larger r0 value. For np case, when r0 = 0.5 fm, the increase amounts to
10% of the maximum mass without hard core. With hyperons, the maximum mass
increases to the range of large mass in recent observations when r0 & 0.3 fm.
n p n p Y
r0 (fm) M (M⊙) R (km) ρcent (ρ0) M (M⊙) R (km) ρcent (ρ0)
0 2.10 10.9 6.4 1.53 11.3 6.1
0.2 - - - 1.58 11.4 6.1
0.3 2.14 11.1 6.2 1.70 11.5 5.9
0.4 2.20 11.4 5.9 1.97 12.2 5.2
0.5 2.34 11.7 5.4 - - -
Table II. Maximum mass M in units of solar mass, and corresponding radius R in km and central
density ρcent in unit of the saturation density ρ0.
4 C. H. Hyun
§4. Conclusion
We investigated the maximum mass of a neutron star in a simple phenomeno-
logical approach where the hard-core repulsion is included in the QHD model. The
hard core radius is treated as a free parameter, and the meson-nucleon coupling con-
stants are fixed identical saturation properties. We obtained the equation of state
of neutron star matter that satisfies thermodynamic equations and self-consistency
conditions. Solving TOV equation, we obtained the mass-radius relation of a neu-
tron star. Our result shows that the maximum mass with hyperons can be as large
as observed masses with a hard core radius r0 & 0.3 fm. These values of r0 are in the
range of hard core radius 0.3 ∼ 0.6 fm in well-known hard-core potential models such
as Hamada-Johnston12) or Reid.13) More investigations are necessary to figure out
the uncertainties. For instance, the hard core size of hyperons can matter. The effect
of hard cores to the formation of other exotic states such as meson condensation or
deconfined quark phases is also worthy to be studied.
To conclude, the our result shows that the hyperon matter, which is known to
give the biggest effect to the mass-radius relation of a neutron star among possible
exotic states in the interior of a neutron star, is not necessarily incompatible with
the observed mass.
Acknowledgments
The author thanks the Yukawa Institute for Theoretical Physics at Kyoto Uni-
versity, where this work was initiated and developed during the YKIS2006 on ”New
Frontiers on QCD”. Author is grateful to Shung-ichi Ando for reading the manuscript.
This works was supported by the Basic Research Program of Korea Science & En-
gineering Foundation (R01-2005-000-10050-0).
References
1) D. J. Nice et al., Astrophys. J. 634 (2005), 1242
2) D. Page and S. Reddy, Ann. Rev. Nucl. Part. Sci. 65 (2006), 327
3) F. Özel, Nature 441 (2006), 1115
4) T. Klahn et al., nucl-th/0609067.
5) B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16 (1986), 1
6) D. H. Rischke, M. I. Gorenstein, H. Stöcker and W. Greiner, Z. Phys. C 51 (1991), 485
7) S. Kagiyama, A. Nakamura and T. Omodaka, Z. Phys. C 53 (1992), 163
8) J. Cleymans, J. Stalnacke and E. Suhonen, Z. Phys. C 55 (1992), 317
9) S. Kagiyama et al., Eur. Phys. J. C 25 (2002), 453
10) P. K. Panda et al., Phys. Rev. C 65 (2002), 065206
11) R. M. Aguirre and A. L. De Paoli, Phys. Rev. C 68 (2003), 055804
12) T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962), 382
13) R. V. Reid, Ann. Phys. (N.Y.) 50 (1968), 411
http://arxiv.org/abs/nucl-th/0609067
	Introduction
	Formalism
	Numerical result
	Conclusion
