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CHINESE JOC:RNXL OF PHYSICS
OCTOBER 1997
VOL. 35, NO. .5
The Structure of Solitonic Quark Stars
Jiunn-Wei Chen*, Hong-Yee Chiu’, and W-Y. P. Hwang’
‘Department of Physzcs, National Tazwan University,
Taipei, Taiwan 106, R. 0. C.
2Code 914, Goddard Space Flight Center, Greenbelt, MD 20?‘71, U.S.A
(Received February 24, 1997)
We show that, through introduction of quantized scalar and fermion fields in curved
space-time, generalized mass and Tolman-Oppenheimer-Volkoff (TOV) equations may
be obtained as a means of incorporating scalar and fermionic matter fields (such as
quark matter or neutron matter). This provides a convenient framework to treat the
problem of solitonic quark stars, stars consisting of quark matter confined to a phase
which is separated from the hadronic phase (e.g., neutron matter) by both the volume
energy and the surface energy. We demonstrate that solitonic quark stars dominated by
the volume energy reduce to quark stars in the conventional sense while those dominated
by the surface energy reduce to fermionic soliton stars obtained by T. D. Lee and others.
In addition, the Chandrwekhar limit becomes quite arbitrary and the possibility for the
existence of quark stars is significantly enhanced, should we allow the surface energy as
an important factor for separating the two phases.
PACS. 97.60.Sm - Other objects believed to be disintegrating or collapsing.
PACS. 97.60.-s - Late stages of stellar evolution.
I. Introduction
Stable stellar configurations, i.e., the end products of stellar evolutions, are believed to
be assemblages of electrically neutral objects. During the old days (i.e., before the presentday particle physics is universally accepted), electrons, protons, and neutrons (bound in
stable light nuclei) are regarded as the fundamental building blocks of matter. Thus,
dwarfs (stellar configurations consisting of hydrogen and/or helium atoms), neutron stars
(consisting primarily of neutron matter, which is nuclear matter of a particular kind), and
black holes (gravitationally collapsed physical systems) are commonly accepted as the only
three possible end products for evolution of the various stellar objects.
In the standard model of the present-day particle physics, however, the building
blocks of matter include:
(1) quarks (and antiquarks) of six distinct flavors (up, down, strange, charm, bottom,
and top) and of three possible colors;
(2) leptons (electrons, muons, r-leptons, and associated neutrinos) and antileptons;
(3) gauge bosons or mediators of fundamental forces (y, W*, Z”, and gluons); and
543
0 1997 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
544
THE STRUCTURE OF SOLITONIC QUARK STARS
VOL. 35
(4) scalar fields (which may account for the origin of masses).
It is therefore natural to ask whether there exist stable stellar configurations other than
dwarfs, neutron stars, and black holes.
The central density of a typical neutron star is often greater than several times the
nuclear matter density P~.~. (= 2 x 10r4gm/cm3). At such high density, neutron matter
may fuse into quark matter consisting of up (u) and down (d) quarks. Furthermore, the
Fermi energies Ed and t, for such high-density quark matter are in fact higher than the
mass of the strange quark s so that the (Ed) quark matter should in turn convert itself into
a (&s)-symmetric quark matter (the “strange matter”) via the weak reactions such as
u + d -+ ‘u. + s, u -+ s + e+ + v,, and d + u $ e- $ P,. It is also sometime argued that the
phase transition into strange matter softens the equation of state, making it more likely for
supernovae explosion to take place.
Should a quark star (i.e., a star consisting of quark matter of a certain kind) exist or
a dense (neutron) star with a quark core exist, the characteristics of the star is not expected
to differ dramatically from that of a neutron star, making it very difficult, if not impossible,
to distinguish such star from a neutron star. On the other hand, the so-called neutron star
candidates (X-ray pulsars) may well be quark stars rather than neutron stars, unless a
clever way to distinguish between the two scenarios can be found.
On the other hand, scalar fields, or composites of Lorentz scalar in nature, are
indispensible building blocks of matter, which may explain why building blocks of matter
acquire masses at the fundamental level. In the present-day particle dynamics, such fields
interact with the matter fields very strongly at very high temperature (e.g., greater than
TeV) but such interaction becomes only very feeble at ordinary temperatures. It is therefore
plausible to conjecture that scalar nuggets (likely of topological nature), produced in the
very early universe, may expand and cool, becoming very weakly interacting with the rest of
the universe. In other words, such scalar-field soliton stars synthesized in the early universe
could well be the candidate for “cold dark matter”.
T. D. Lee and his colleagues [I] were among the first ones who considered the possible existence of soliton stars. Subsequently, one of us [2] has also examined the possible
importance of such soliton stars in astronomy.
In the present paper, we first describe how the generalized mass and Tolman Oppenheimer-Volkoff (TOV) equations may be obtained as a means of incorporating scalar
and fermionic matter fields (such as quark matter or neutron matter). We then use the
framework to treat exclusively the problem of solitonic quark stars, stars consisting of quark
matter confined to a phase which is separated from the hadronic phase (for, e.g., neutron
matter) by both the volume energy and the surface energy.
II. General ideas
To study possible stellar configurations, we are interested in setting up the framework
to describe the behavior of a system of interacting scalar and fermion fields in curved spacetime.
The curved space-time manifold may be characterized by the spherical coordinates
(r,r,G):
JIUNN-WE1 CHEN, HONG-YEE CHIU, AND W-Y. HWANG
VOL. 35
545
ds2 = gwudxpx"
= -e2udt2
+ e2”dr2 + r2(de2 + sin28dp2).
(1)
In the present paper, we consider only stable spherical star solutions. Thus, we assume
that both u and v are functions of the “radial” coordinate T.
The background field x(x”) has the same quantum numbers as the physical vacuum.
It is Lorentz scalar in nature. It may be described by the lagrangian density [33
L, =
-;x+x+ - U(X)>
U(x) = B +
ix2 + ix3 t dx4’
(24
B
> 0.
Pb)
Here B is the bag constant. The potential U(x) is chosen such that it has the absolute
minimum U = 0 at x = X~ and another local minimum U = B at x = 0. Such choice of the
potential admits the (non-topological) soliton solutions which look like bubbles (“bags”)
with x z 0 inside the bubble and x = xm outside.
To study the question of soliton stars, we may introduce, on top of the background
scalar field x, the complex scalar fields 4(x”),
J% = -4+‘“$,, - U&(4+@,
/.L; >> p2 >
0.
(3)
w
Here the choice (4b) is used to ensure that the fields C$ exist only inside a bubble as they
get extremely heavy when they get outside.
In general, the fields 4 may be endowed with some internal group structure, such
as the case that they may transform like a triplet under color SU(3), or transform like a
doublet (or other multiplet) under SU(2) of the Glashow-Salam-Weinberg SU(2) x U(1)
electroweak theory. This aspect makes it possible to create scalar nuggets (of nontrivial
global topology in nature) during the very early universe. Of course, the fields 4 may also
be gauged, yielding the so-called “Higgs mechanism” (as a way to describe their interaction
with gauge fields).
Quarks may be described as quantized Dirac fields in curved space-time manifold.
One may introduce the space-time dependent 7 matrices:
THE STRUCTURE OF SOLITONIC QUARK STARS
546
q-z” = ~!f%w,, - SPW - QWP >>
so
VOL. 35
(54
that the lagrangian density for a Dirac field $(L+) may be given by
w
_fxm >> mp-
Here we have chosen the quark mass outside a bubble as m + fXOC, which is much greater
than, e.g., the proton mass mp, a typical hadron energy scale. Thus, again, quarks always
choose to stay inside a bubble (“bag”) in order to minimize the total energy of the system.
(If such bubble does not exist, quarks will first dig a bubble and then stay inside it.)
To obey Pauli exclusion principle for fermions, we may adopt the Thomas-Fermi
approximation to treat contributions from fermion fields. In other words, the system at
a given space-time point may be treated locally as an ideal Fermi gas. We may then
employ the many-body technique to improve such approximation. The method enables
us to compute the pressure Pf due to Fermion fields as a function of the Fermion energy
density of, Pf = Pf(pf).
It is a routine task to compute the Einstein tensor G,, from the given metric gfiLV
and, on the other hand, to express the overall energy-momentum tensor Tp,, in terms of
the various fields. To illustrate the issue, we consider the stable spherical star solutions
and, to this end, we assume that both x and 4 depend only on the “radial” coordinate T.
The (tt)-component of the energy-momentum tensor allows us to define the overall energy
density p(r):
(7)
so that, if we define the overall “mass” within the “radius” T,
m(T)
E
i(l-
ee2’),
(8)
the (tt)-component of the Einstein equation, Gtt = 87rTtt, becomes
dm(4
dr
= 4xr2p( T, m)
Note that we may use Eq. (8) t o e1 iminate v in favor of m so that the energy density p also
depends on m(r).
Along the same line, we use the (rr)-component to define the pressure density
P(T, m):
VOL. 35
JIUNN-WE1 CHEN, HONG-YEE CHIU, AND W-Y. HWANG
P(,,m) = Pf - U(x) -
urn f
(
1 - J2m){%)2t(~)+(~)}~
547
(10)
The (rr)-component of the Einstein equation then yields
du
dr=
m+4rT3P
T ( T
(11)
- 277-l) ’
In the interior of a star (excluding the surface region), the energy-momentum conservation, T”Oip = 0, yields
(12)
(PfP)$-z.
We have, from Eqs. (11) and (la),
dP + P)(m
(p
- dr
+
T(T -
4RT3P)
’
2m)
(13)
which is the generalized Tolman-Oppenheimer-Volkoff (TOV) equation.
In the exterior of a star, we have p = 0 and P = 0 so that the Schwarzschild metric
dr2
1 + r’(d8’ f sin2Bdp2),
7
is the solution. Of course, we must handle with care the surface region in which neither of
Eqs. (13) and (14) applies.
Note that the generalized mass equation (9) and the generalized TOV equation (13)
differ from the conventional ones in that both the energy density p and the pressure P
depend on both T and m(T) in a nontrivial way, making it difficult to treat m as t h e
only variable (as in Lagrangian description). Nevertheless, the fact that such generalized
equations can be obtained suggests simple extension of the standard inside-out integration
procedure for stellar structure and evolution.
III. Sample numerical results
By suitably generalizing the energy density p(r) and the pressure density P(r), we
have shown that the stable spherical stars are governed by the stardard set of equations:
the generalized mass equation, the generalized TOV equation, and the equation of state
(EOS). For a given central energy density pc, this set of equations can easily be integrated
out until P(T = R) = 0, which defines the surface of the star. The star mass A4 is given by
M = m(r = R). This standard approach differs from what has been adopted by T. D. Lee
and his colleagues [l] and therefore may offer different insights concerning the structure of
solitonic quark stars.
Choosing suitable parameters for the potential U(x), we may adopt the approximate
soliton solution for x(r),
VOL. 35
JIUNN-WE1 CHEN, HONG-YEE CHIU, AND W-Y. HWANG
549
.____________---_____-_-_-----I
i
_,i
0
j
*
r
(Kmj
4
5
(0
FIG. 1. The structure of a solitonic quark star with B = 57 MeV/fm”, mx = 600TeV, xoo = mx,
and the central density pC = 5.1 x 10’6g/cm3.
(a) the star pressure P versus the distance T from the center of the star;
(b) same as (a) but only the behavior near the very narrow surface region is shown - note
that the scale for the radius is blown up by a factor of 10 23 in order to exhibit the structure;
(c) the energy density p versus the distance T;
(d) same as (c) near the very sharp surface region;
(e) the mass m(r) enclosed within the sphere of the radius r (from the center) as a function
of r;
(f) the normalized radius-mass ratio r/2Gm as a function of T, showing that the star is not
a black hole (i.e., with the ratio greater than unity everywhere).
-
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THE STRUCTURE OF SOLITONIC QUARK STARS
550
1
B=
lT$=
06
x_=
B = 57 MeV/fm3
rl;,== %
TeV
PC=
-
---
----
1;’
2.b
;b
r (Km)
solution
solution
I
II
p*
Lb
5.b
FIG. 2. There may be two solutions for the solitonic quark star for a given central density pc, as
the surface pressure P, due to the x field may have two intercepts with the pressure P due
to the (other) matter fields. For the critical central density p: = 6.53 x 10r5g/cm3, there
is only one solution; for pc < p:, there is no solution:
(a) the pressure P shown as a function of the distance T (from the center) for the two
solutions of the solitonic quark stars;
(b) the star mass M shown as a function of its central density pc (in units of g/cm3).
Figs. 1 (a)-(f) suggest that for the surface tension to play a significant role in stellar
structure the scalar particle must be unusually heavy, in the range of hundreds of Tel’. As
we are about to explore particle physics in the Tel’ range at Large Hadron Collider (LHC)
at CERN, it is perhaps a little premature to rule out such possibility on the ground of being
not natural enough.
In Figs. 2(a) and 2(b), we show that, with the same set of input parameters as for Figs.
1 (a)-(f), there may be two solutions for the solitonic quark star for a given central density
pc, as the surface pressure P, due to the x field may have two intercepts with the pressure
P due to the (other) matter fields. For the critical central density p: = 6.53 x 10*‘g/cm3,
there is only one solution; for pc < p:, there is no solution. Specifically, we show in Fig.
2(a) the pressure P as a function of the distance T (from the center) for the two solutions
of the solitonic quark stars and in Fig. 2(b) the star mass M shown as a function of its
central density pc.
Note that the star solutions in the second branch, shown in a dashed curve in Fig.
2(b), are not stable, as a small perturbation to squeeze the star to a higher central density
would make the system collapse indefinitely. Similarly, those stars in the first branch, which
has a central density beyond the critical density corresponding to the first maximum star
mass (the Chandrasekhar limit [S]) but below the valley region, are also not stable. The
stable star with its central density below the Chandrasekhar limit behaves very much like
a neutron star but, due to the introduction of the surface energy, the Chandrasekhar limit
becomes quite arbitrary, contrary to conventional neutron stars or quark stars. Note that
adoption of the equation of state for quark matter also leads to a new stable region which
consists of solitonic quark stars with a central density about (lo2 - 104) times those in the
._,-i
i
VOL. 35
JIUNN-WE1 CHEN, HONG-YEE CHILI, AND W-Y. HWANG
5
B = 5 7 XLeV/fm’
x-= m,
4-
- q
- - - mr
$-
2 -
=
30 GeV
= 6 0 0 TeV
/
- _ -
,___________________-_-___________---__i
FIG. 3. The structure of the solitonic quark star varies with the mass mx of the scalar field, which
determines the strength of the surface energy compared to the volume energy 8 (which is
taken as the value given in the MIT bag model). Two values of mx, 30GeV and 600TeV,
are used for illustration:
(a) The star mass A4 shown as a function of its central density pe (in units of g/cm3). Note
that the density lower bounds for producing the s quark and the c quark are indicated by
the arrows;
(b) The star radius R shown as a function of the central density pc;
(c) the normalized ratio R/2GM shown as a function of the central density pc.
first stable region. We believe that search of quark stars in this region could offer exciting
new possibilities.
The structure of the solitonic quark star varies with the mass m, of the scalar
field, which, together with the value of xoo, determines the strength of the surface energy
compared to the volume energy B (which is taken as the value given in the MIT bag model).
In Figs. 3 (a)-(c), two values of m,, 30 GeV and 600TeV, are used for illustration, with
552
VOL. 35
THE STRUCTURE OF SOLITONIC QUARK STARS
L
B = 57 MeV/fm'
--_
.---- x------______
__
:i;.
I"
-
1c
2
-
,,,,I
1
i 0
.,s
io '
,,'
;0'
ij '
m,, x_ (Tex’)
m,
(TeL?
(b)
(a)
Cc)
FIG. 4. (a) The critical star mass M, (i.e. the Chandrasekhar limit or the maximum star mass for
the varying central density pe) is shown as a function of mx or of xoo (whichever applies).
The solid curve is obtained with the assumption of mx = xW, while the long dashed curve
with mx fixed and the short dashed curve with xm fixed.
(b) The overall volume energy Eg and the surface energy Es for the star of the maximally
possible (Chandrasekhar) mass M = M,, are shown as functions of m,.
(c) The central density pc of the Chandrasekhar star (with M = M,,) is shown as a function
of m,.
the former (smaller) value leading to stars controlled by the volume energy while the latter
by the surface energy. In 3(a), the star mass M is shown as a function of its central density
pc. Note that the density lower bounds for producing the s quark and the c quark are
indicated by the arrows. In 3(b), the star radius R is shown as a function of the central
density pr and in 3(c) the normalized ratio R/2GM is shown as a function of the central
density pc.
In Fig. 4(a), the critical star mass M, (i.e. the Chandrasekhar limit [6] or the
JIUNN-WEI CHEN, HONG-YEE CHIC, AND W-Y. HWANG
VOL. 35
553
FIG. 5. The Chandrasekhar limit M,, for quark stars with B = 0 is shown as a function of mX.
Here it is assumed, for the sake of simplicity, that quarks are massless. It is seen that the
Chandrasekhar limit decreases linearly with m, in the bi-logarithmic plot.
maximum star mass for the varying central density pc) is shown as a function of m, or
of X~ (whichever applies). The solid curve is obtained with the assumption of m, = xoo,
while the long dashed curve with m, fixed and the short dashed curve with xc0 fixed. In
Fig. 4(b), the overall volume energy EB and the surface energy Es for the star of the
maximally possible (Chandrasekhar) mass M = Ivf,, are shown as functions of m,. In
Fig. 4(c), the central density pc of the Chandrasekhar star (with M = MC,.) is shown as a
function of m,.
In Fig. 5, the Chandrasekhar limit M,, for solitonic quark stars with B = 0 i s
shown as a function of m,. Here it is assumed, for the sake of simplicity, that quarks are
massless. It is seen that the Chandrasekhar limit begins at a gigantic value of about 1012
solar masses for a massless x field and decreases linearly with mx in the bi-logarithmic plot.
Furthermore, this figure offers understanding of the fermionic soliton stars obtained by T.
D. Lee and his colleagues [l], as to why the Chandrasekher limit is quite arbitrary for stars
dominated by the surface energy.
In Figs. 1-5, we have for the sake of simplicity neglected the interactions among
quarks, as dictated by quantum chromodynamics (QCD) - by setting czs = 0. In previous
studies such as in [5], it was seen that introduction of strong interactions among quarks
makes the possible existence of quark stars a suspect, unless CYS or the bag constant B is
sufficiently small. With introduction of the surface energy for separating the quark-gluon
and hadron phases, we find that the situation could be significantly different, depending
on the relative importance of the surface energy. A sample numerical study is illustrated
through Figs. 6(a)-(c) and 7.
With some modification, we may consider a neutron star with a quark core. The
equation of state for neutron matter may be taken from the nuclear force studies. To this
end, we examine in Figs. 6 (a)-(c) th e relative stability between neutron matter and strange
matter, with B = 57MeV/f m3 and cyS = 0.8 for the strange matter. Specifically, we show
.
900:
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THE STRUCTURE OF SOLITONIC QUARK STARS
554
100
’
/
do
100
P (MeV/fm3)
(a)
I
100
500
P (MeV/fm3)
(b)
FIG. 6. Relative stability between neutron matter and strange matter, with B = 57MeV/fm3 and
cry, = 0.8 for the strange matter.
(a) Gibb’s free energy per baryon G/N vs pressure P for neutron matter (solid curve) and
strange matter (dashed curve).
(b) The density-pressure relation (equation of state) of such matter which is in neutron
phase (P < 320MeV/fm3) or strange phase (P > 320MeV/fm3). (c) The baryon number
density n is shown as a function of P.
in Fig. 6(a) Gibb’s free energy per baryon G/N versus pressure P for neutron matter
(solid curve) and strange matter (dashed curve); in Fig. 6(b) the density-pressure relation
(equation of state) of such matter which is in neutron phase (P < 320MeV/fm3) or strange
phase (P > 320MeV/fm3); and in Fig. 6(c) the baryon number density n is shown as a
function of P.
In Fig. 7, we show the star mass as a function of its central density pC for stars which
consist of strange matter and neuton matter (whichever is more stable). &h.J. is the density
which the phase transition from neutron matter into strange matter occurs. There is little
JIUNN-WE1 CHEN, HONG-YEE CHIU, AND W-Y. HWANG
VOL. 35
555
PC
FIG. 7. The star mass versus its central density pc (in units of g/cm3) for stars which consist of
strange matter and neuton matter (whichever is more stable). pph.t, is the density which
the phase transition from neutron matter into strange matter occurs.
doubt that the possibility for the existence of pure quark stars, or of neutron stars with a
quark core, is significantly enhanced, should we allow the surface energy as an important
factor for separating the two phases.
IV. Summary
In the present paper, we have shown that, through introduction of quantized scalar
and fermion fields in curved space-time, generalized mass and Tolman-Oppenheimer-Volkoff
(TOV) equations may be obtained as a means of incorporating scalar and fermionic matter
fields (such as quark matter or neutron matter). This provides a convenient framework to
treat the problem of solitonic quark stars, stars consisting of quark matter confined to a
phase which is separated from the hadronic phase (for, e.g., neutron matter) by both the
volume energy and the surface energy. Our numerical results demonstrate that solitonic
quark stars dominated by the volume energy reduce to quark stars in the conventional sense
[5] while those dominated by the surface energy reduce to fermionic soliton stars obtained
by T. D. Lee and his colleagues [l). In addition, the Chandrasekhar limit [S] becomes
quite arbitrary and the possibility for the existence of quark stars is significantly enhanced,
should we allow the surface energy as an important factor for separating the two phases.
Acknowledgement
This work was initiated during the visit.of one of us (Chiu) to National Taiwan
University. This work is supported in part by the National Science Council of R.O.C.
under the grants NSC84-2112-M002-037 and NSC86-2112-M002-OlOY.
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THE STRUCTURE OF SOLITONIC QUARK STARS
VOL. 35
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[ 3 ] R. Friedbergand T. D. Lee, Phys. Rev. D15, 1694 (1977); 16, 1096 (1977).
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