PHYSICAL REVIEW C, VOLUME 65, 065206
Excluded volume effects in the quark meson coupling model

P. K. Panda,1 M. E. Bracco,2 M. Chiapparini,2 E. Conte,2 and G. Krein1
1Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, SP, Brazil

2Instituto de Fı´sica, Universidade do Estado do Rio de Janeiro,
Rua Sa˜o Francsico Xavier 524, 20559-900 Rio de Janeiro, RJ, Brazil

~Received 13 April 2002; published 20 June 2002!

Excluded volume effects are incorporated in the quark-meson coupling model to take into account in a
phenomenological way the hard-core repulsion of the nuclear force. The formalism employed is thermody-
namically consistent and does not violate causality. The effects of the excluded volume on in-medium nucleon
properties and the nuclear matter equation of state are investigated as a function of the size of the hard core. It
is found that in-medium nucleon properties are not altered significantly by the excluded volume, even for large
hard-core radii, and the equation of state becomes stiffer as the size of the hard core increases.

DOI: 10.1103/PhysRevC.65.065206 PACS number~s!: 12.39.Ba, 21.65.1f, 24.85.1p
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I. INTRODUCTION

The study of the properties of high-density and hig
temperature hadronic matter is of interest for understandin
wide range of phenomena associated with superdense
@1# and relativistic heavy-ion collisions@2#. One of the open
questions in this subject is the correct identification of
appropriate degrees of freedom to describe the diffe
phases of hadronic matter. Although this question will ev
tually be answered with first-principles calculations with t
fundamental theory of the strong interaction quantum ch
modynamics~QCD!, most probably through lattice QCD
simulations, presently one is still far from this goal and,
order to make progress, one must rely on model calculat
and make use of the scarce experimental information av
able. For matter at zero temperature and density close to
saturation density of nuclear matter, experiments seem to
dicate that the relevant degrees of freedom are the bar
and mesons. There is a long and successful history of ca
lations using models based on baryonic and mesonic deg
of freedom, such as potential models@3,4# and relativistic
field-theoretical models, generically known as quant
hadrodynamics~QHD! @5,6#. At densities several time
larger than the saturation density and/or high temperatu
one expects a phase of deconfined matter whose prope
are determined by the internal degrees of freedom of
hadrons. Early studies of deconfined matter@7# used the MIT
bag model@8#, in which the relevant degrees of freedom a
quarks and gluons confined by the vacuum pressure. On
other hand, at high densities, but not asymptotically hig
than the saturation density, the situation seems to be
complicated. The complication arises because of the po
bility of simultaneous presence of hadrons and deconfi
quarks and gluons in the system. Not much progress
been possible in this direction because of the necessity
model to be able to describe composites and constituen
the same footing—a step towards this direction is the form
ism developed in Ref.@9#.

An important step towards the formulation of a model
describe the different phases of hadronic matter in term
explicit quark-gluon degrees of freedom is the quark-me
0556-2813/2002/65~6!/065206~8!/$20.00 65 0652
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coupling ~QMC! model, proposed by Guichon@10# some
time ago and extensively investigated by Saito and
workers@11#—see also Ref.@12# for related studies. Matte
at low density and temperature is described as a system
nonoverlapping bags interacting through effective scalar-
vector-meson degrees of freedom, very much in the sa
way as in QHD@5,6#. The crucial difference is that in the
QMC, the effective mesons couple directly to the quarks
the interior of the baryons, with the consequence that
effective baryon-meson coupling constants become den
dependent. In addition, hadronic sizes are explicitly incor
rated through form factors calculable within the underlyi
quark model@13#. At very high density and/or temperatur
baryons and mesons dissolve and the entire system of de
fined matter, composed by quarks and gluons, becomes
fined within a single MIT bag@7#.

The fact that the same underlying quark model is used
different phases of hadronic matter makes the model v
attractive conceptually. Many applications and extensions
the model have been made in the last years—see Refs.@13–
19# and references therein. Of particular interest for the p
nomenology of finite nuclei was the introduction of
density-dependent bag constant by Jin and Jennings@14#.
These authors postulated different density dependences
the bag constant, in a way that the bag constant decreas
the nuclear density increases. One consequence of this is
large values for the scalar and vector mean fields at the s
ration density are obtained, leading to spin-orbit splittings
the single-particle levels of finite nuclei that are in bet
agreement with experiment than those obtained with
density-independent bag constant. Another consequence
smaller bag constant in medium is that the bag radius
comes considerably larger than in free space@14,16#. At satu-
ration density the nucleon radius increases 25% and at
sities three times higher the radius can increase as muc
50%. For higher densities the increase of radius is even m
dramatic. The consequences of changing the bag constan
nucleon sizes were investigated by Luet al. @17#.

A large increase of the bag radius naturally raises
question about the validity of the nonoverlapping bag pict
that underlies calculations of nuclear matter properties w
©2002 The American Physical Society06-1



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PANDA, BRACCO, CHIAPPARINI, CONTE, AND KREIN PHYSICAL REVIEW C65 065206
the model. At normal nuclear matter densities, the aver
distance between nucleons is of the order of 1.8 fm. The
fore, for densities larger than the normal density and b
radii larger than 1 fm there is a large probability that the ba
overlap significantly. However, before concluding that t
picture of independent nucleons breaks down, it is import
to recall that short-distance correlations are left out in a m
field calculation. These correlations are induced by the co
bined effects of the Pauli exclusion principle between id
tical nucleons and the hard core of the nucleon-nucleon
teraction that forbids scattering into occupied levels. T
success of the independent particle model of the nuclear s
model is due to the small size of hard core and the P
principle, which lead to a ‘‘healing’’ distance of the two
nucleon relative wave function that is smaller than the av
age distance between nucleons in medium@20#. In a model
like the QMC, where the finite size of the nucleons is ma
explicit through a bag structure, the incorporation of th
physics in the many-body dynamics is an interesting n
development. In the present paper we address this in a
nomenological way through an excluded volume approa
The prescription we use was developed in Ref.@21# for ideal
gases and further extended to relativistic field-theoretic m
els, such as QHD in Ref.@22#. In this approach, matter in th
hadronic phase is described by nonoverlapping rigid sphe
but when the density of matter is such that the relative d
tance between two spheres becomes smaller than the d
eter of the spheres, the excluded volume effect introduce
effective repulsion that mimics the hard-core repulsion of
nucleon-nucleon interaction.

Of course, at very high density the description of hadro
matter in terms of nonoverlapping bags should break do
In a purely geometrical view, one has the picture that o
the relative distance between two bags becomes m
smaller than the diameter of a bag, quarks and gluons sta
percolate and individual bags loose their identity. The d
sity at which this starts to happen is presently unkno
within QCD. In this respect, it is important not to confuse t
bag radius with the radius of the hard core of the nucle
nucleon force. Model studies@23# indicate that when the two
nucleons start to overlap, medium-range forces are gener
from the distortion of the quark distribution. The short-ran
repulsion, on the other hand, is due to the combined effec
the one-gluon exchange—mainly due to its spin-s
component—and the Pauli exclusion principle betwe
quarks of different nucleons that becomes efficient when
overlap of the two-nucleon wave functions is complete. A
though the described scenario might well not be ultimat
confirmed by a full QCD calculation, it seems, howev
clear that the two radii, the radius of the MIT bag and t
radius of the hard core of the nucleon-nucleon interacti
are of different sizes and have different origins in the phys
of hadron structure. In this sense, the radius of the exclu
volume will be taken to be smaller and unrelated to the
dius of the underlying MIT bag.

The excluded volume approach we use is thermodyna
cally consistent. Although the prescription can be exten
to take into account Lorentz contraction of the bags@24#, in
this initial exploratory investigation we use hard-sphe
06520
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bags, since a complete calculation would lead to mass
numerical calculations. In addition, as indicated by the inv
tigations in Ref.@24#, the effect of Lorentz contraction is
most important for light particles like pions. However, as w
will explicitly show, the approach does not lead to violatio
of causality for the density range where the model ma
sense.

The paper is organized as follows. In Sec. II we presen
short review of the excluded volume prescription of R
@22# and implement it to the QMC model. Numerical resu
are presented in Sec. III and our conclusions and pers
tives are discussed in Sec. IV.

II. EXCLUDED VOLUME IN THE QMC MODEL

Initially, for completeness and in order to make the pap
self-contained, we briefly recapitulate the excluded volu
prescription of Ref.@22#. Let us start with the ideal gas o
one-particle species with temperatureT, chemical potential
m, and volumeV. The pressure is related to the grand pa
tion functionZ as

P~T,m!5 lim
V→`

T
ln Z~T,m,V!

V
, ~1!

with Z defined as

Z~T,m,V!5 (
N50

`

e2mN/TZ~T,N,V!, ~2!

whereZ is the canonical partition function. The authors
Ref. @22# included the excluded volume effect starting fro
the canonical partition function as

Zexcl~T,N,V!5Z~T,N,V2v0N!Q~V2v0N!. ~3!

This ansatz is motivated by considering that the volumeV
for a system ofN particles is reduced to an effective volum
V2v0N, wherev0 is the volume of a particle. In a hadroni
gas,v0 can be interpreted as the region excluded by the h
core of the nucleon-nucleon interaction. For a spherical
gion, v054pr 3/3 with r the hard-core radius. Using Eq.~3!
into Eq. ~2!, the grand partition function becomes

Z excl~T,m,V!5 (
N50

`

e2mN/TZ~T,N,V2v0N!Q~V2v0N!.

~4!

There is a difficulty for evaluation of the sum overN par-
ticles in this equation because of the dependence of the a
able volume on the varying number of particles,N, because
Z(T,N,V2v0N) does not factor as a product as in the ca
of anN-independent volume. To overcome this difficulty th
authors in Ref.@22# have performed a Laplace transform
tion on the variableV in Eq. ~4! as

Z̃excl~T,m,j!5E
0

`

dVe2jVZ excl~T,m,V!. ~5!

Using Eq.~4! in this, and making the change of variable
6-2



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EXCLUDED VOLUME EFFECTS IN THE QUARK MESON . . . PHYSICAL REVIEW C65 065206
V5x1v0N, ~6!

one obtains

Z̃excl~T,m,j!5E
0

`

dxe2jxZ~T,m̃,x!5Z̃excl~T,m̃,j!,

~7!

where m̃5m2v0Tj. Now the integrand in this is factoriz
able ~for the present case of independent particles! and the
sum overN can be implemented. It is a simple exercise@21#
to show that the pressure of the system is given as@22#

P~T,m!5P8~T,m̃ !, ~8!

with

m̃5m2v0P~T,m!. ~9!

The meaning of Eq.~8! is that the pressure of the system a
with excluded volume and with chemical potentialm,
P(T,m), is equal to the pressure of a system without e
cluded volume but with an effective chemical potentialm̃

5m2v0P(T,m), denoted byP8(T,m̃). Note that once the
expression forP8(T,m̃) is known, the pressure of the syste
is given by an implicit function.

The baryon density, the entropy density, and the ene
density for the system are given by the usual thermodyna
cal expressions

r~T,m![S ]P

]m D
T

5
r8~T,m̃ !

11v0r8~T,m̃ !
, ~10!

S~T,m![S ]P

]T D
m

5
S8~T,m̃ !

11v0r8~T,m̃ !
, ~11!

e~T,m![TS2P1mr5
e8~T,m̃ !

11v0r8~T,m̃ !
. ~12!

These relations define a thermodynamically consistent
malism, since the fundamental thermodynamical relati
are fulfilled.

Next we apply this formalism to the QMC model fo
nuclear matter at zero temperature@10,11#. In the QMC
model, the nucleon in nuclear matter is assumed to be
scribed by a static MIT bag in which quarks interact w
scalars0 and vectorv0 mean mesonic fields. The meson
fields are meant to represent effective degrees of freed
not necessarily identified with real mesons. Therefore, si
the introduction of the excluded volume is to represent
hard-core nucleon-nucleon interaction, Eqs.~8!–~12! will be
applied to the baryons only. The same prescription has b
used in the application of the formalism to QHD in Ref.@22#.

In the QMC model, the pressure and energy density
ceive contributions from baryons and mesons and are g
as
06520
-

y
i-

r-
s

e-

m,
e
e

en

-
n

P5PB2
1

2
ms

2s0
21

1

2
mv

2 v0
2 , ~13!

e5eB1
1

2
ms

2s0
21

1

2
mv

2 v0
2 , ~14!

where PB and eB are the baryon contributions. As sa
above, the excluded volume will be applied only to the ba
onic contributions and, at zero temperature, one need
consider only Eqs.~10! and ~12!,

r5
r8

11v0r8
, ~15!

eB5
eB8

11v0r8
. ~16!

For practical calculations, it is convenient to parametrizer8
in terms of akF according to

r85
g

6p2
kF

3 . ~17!

This allows one to write the QMC expressions forPB8 andeB8
as

PB85
1

3

g

2p2 F1

4
kF

3AkF
21M* 22

3

8
M* 2kFAkF

21M* 2

1
3

8
M* 4lnS kF1AkF

21M* 2

M*
D G , ~18!

eB85r8AkF
21M* 22PB8 . ~19!

In these,M* is the in-medium nucleon mass calculated w
the MIT bag. Its value is determined by solving the MIT ba
equations for quarks coupled to the mean fieldss0 andv0.
In order to completely determineM* , and therefore the
nuclear matter properties~15! and ~16!, one needss0 and
v0. The scalar mean field is determined self-consisten
from the minimization condition at densityr:

]e

]s0
50, ~20!

which leads to

s05
1

11v0r8

S~s0!

ms
2

, ~21!

with
6-3



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PANDA, BRACCO, CHIAPPARINI, CONTE, AND KREIN PHYSICAL REVIEW C65 065206
S~s0!52
1

p2

]M*

]s0
F kFAkF

21M* 2

2M* 2lnS kF1AkF
21M* 2

M*
D G . ~22!

The vector mean fieldv0 is obtained from its equation o
motion as

v05
3gv

q

mv
2

r. ~23!

Solution of Eq.~21! proceeds as follows. For a givenr,
we use Eq.~15! to obtainr8, and from thisr8 we obtainkF
of Eq. ~17!. The derivative]M* /]s0 can be performed ex
plicitly, S(s0) is then known, and the transcendental eq
tion for s0 is easily solved numerically. The results are p
sented in the following section.

III. RESULTS AND DISCUSSIONS

We start fixing the free-space bag properties. We use z
quark masses only and use two values for the bag radiuR
50.6 fm andR50.8 fm. There are two unknowns,z0 and
the bag constantB. These are obtained as usual by fitting t
nucleon massM5939 MeV and enforcing the stability con
dition for the bag. The values obtained forz0 and B are
displayed in Table I.

Next we proceed to nuclear matter properties. We w
consider two versions of the model. In the first one, the b
constantB is fixed at its vacuum value, and in the second o
the bag constant changes accordingly to the original Jin
Jennings@14# ansatz, namely,

TABLE I. Parameters used in the calculation.

mq(MeV) R(fm) B1/4(MeV) z0 ms(MeV) mv(MeV)

0 0.6 211.3 3.987 550 783
0 0.8 170.3 3.273 550 783
06520
-
-

ro

ll
g
e
d

B* 5B expS 2
4gs

Bs

MN
D , ~24!

wheregs
B is an additional parameter andB is the value of the

bag constant in vacuum. In this work we usegs
B52.8, which

is the same as in Ref.@17#.
The quark-meson coupling constantsgs

q andgv53gv
q are

fitted to obtain the correct saturation properties of nucl
matter, EB[E/A2M5e/r2M5215.7 MeV at r5r0
50.15 fm23. We take the standard values for the mes
masses,ms5550 MeV andmv5783 MeV. We present re-
sults for three different values of the hard core,r 50.4 fm,

FIG. 1. The energy per nucleon of nuclear matter as a func
of r/r0 for different hard-core radii. All curves are for the same s
of quark-meson coupling constants. The upper panel of the fig
corresponds toR50.8 fm and the lower one is forR50.6 fm.
tura-
TABLE II. The quark-s andv-nucleon coupling constants, in-medium nucleon properties at the sa
tion density, and the nuclear matter incompressibility forR50.6 fm.

Hard-core gs
q gv R* /R M* /MN x* /x K

radius~fm! ~MeV!

0 5.98 8.95 0.9934 0.7757 0.8659 257
0.4 5.93 8.81 0.9936 0.7789 0.8684 285

B5constant 0.5 5.87 8.66 0.9939 0.7824 0.8711 316
0.6 5.76 8.38 0.9942 0.7887 0.8759 372

0 4.32 9.87 1.0849 0.7388 0.8882 268
0.4 4.26 9.72 1.0844 0.7426 0.8909 297

B5B* 0.5 4.20 9.57 1.0839 0.7468 0.8938 330
0.6 4.09 9.29 1.0830 0.7543 0.8988 386
6-4



7

EXCLUDED VOLUME EFFECTS IN THE QUARK MESON . . . PHYSICAL REVIEW C65 065206
TABLE III. Same as Table II forR50.8 fm.

Hard-core gs
q gv R* /R M* /MN x* /x K

radius~fm! ~MeV!

0 5.74 8.19 0.9930 0.8034 0.8342 249
0.4 5.69 8.06 0.9932 0.8060 0.8371 277

B5constant 0.5 5.64 7.91 0.9935 0.8088 0.8404 30
0.6 5.54 7.64 0.9938 0.8139 0.8461 361

0 4.14 9.34 1.0799 0.7609 0.8594 261
0.4 4.09 9.20 1.0795 0.7640 0.8627 290

B5B* 0.5 4.03 9.05 1.0792 0.7675 0.8661 322
0.6 3.93 8.79 1.0785 0.7737 0.8723 378
a

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on

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o l of
0.5 fm, and 0.6 fm, and for two values of bag radii,R
50.6 fm and 0.8 fm. The pairr 50.6 fm, R50.6 fm rep-
resents the situation that the size of the hard core is the s
as of the bag and is included for illustrative purposes.

Initially, we investigate the effect of the excluded volum
on the binding energy per particle for the values ofr andR
mentioned above. The results for the different values ofr and
R are shown in Fig. 1, where we plotEB as a function of the
nuclear densityr. In this figure the coupling constantsgs

q

andgv for a givenR are the same for the different values
r. As expected, the effect of an effective repulsion due to
hard core is clearly seen in this figure. The effect obviou
increases as the size of hard core increases. At the satur

FIG. 2. The energy per nucleon of nuclear matter as a func
of r/r0 for different hard-core radii. The quark-meson coupli
constants are refitted such as to obtain the correct saturation p
The upper panel of the figure corresponds toR50.8 fm and the
lower one is forR50.6 fm.
06520
me

e
y
ion

density, the largest value of the effective repulsion is of
order of 4 MeV. The effect is not as dramatic as one co
expect. For comparison with another repulsive effect,
mention that Fock terms@13# give 5 MeV repulsion for the
binding energy.

We now readjust the coupling constantsgs
q and gv such

as to obtain the correct saturation binding energy of nuc
matter for the different values ofr and R. Our aim is to
investigate the changes in the properties of nuclear ma
and in-medium nucleon properties due to the hard co
Tables II and III present the values of the coupling consta
and the ratios of in-medium to free-space bag radiiR* /R,
nucleon massesM* /M , and bag eigenvaluesx* /x. The
tables also show the changes in the incompressibility
different hard-core radii. The results are such that nucle

n

int.
FIG. 3. The effective radius of the nucleon as a function ofr/r0

corresponding to the different hard-core radii. The upper pane
the figure corresponds toR50.8 fm and the lower one is forR
50.6 fm.
6-5



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he

PANDA, BRACCO, CHIAPPARINI, CONTE, AND KREIN PHYSICAL REVIEW C65 065206
FIG. 4. The in-medium nucleon mass as a function ofr/r0

corresponding to the different hard-core radii. The upper pane
the figure corresponds toR50.8 fm and the lower one is forR
50.6 fm.

FIG. 5. Thes field as a function ofr/r0 corresponding to the
different hard-core radius. The upper panel of the figure co
sponds toR50.8 fm and the lower one is forR50.6 fm.
06520
properties are not changed significantly, being at most at
level of 2%. The incompressibility is a little more sensitiv
than nucleon properties to the extra repulsion induced by
hard core, but the increase is at most 120 MeV.

The effect of the hard core as a function of the nucle
density r on the binding energy is shown in Fig. 2. On
notices that the equation of state becomes stiffer as the
of the hard core increases. The ratiosR* /R, M* /M , and the
s0 field as functions ofr are shown in Figs. 3, 4, and 5
respectively. As found previously, the in-medium bag rad
decreases~increases! for a constant~in-medium changed!
bag parameter. Now, the change in the in-medium bag ra
decreases as the hard-core radius increases. This is clea
effect due to the fact that as the hard-core radius increa
one has less attraction, and the bag properties change le
Fig. 4 one sees the interesting feature that as the in-med
nucleon mass increases, the binding energy curve is st
when volume corrections are included, contrary to the c
without excluded volume. This is again an effect of ex
repulsion due to the hard core. That one gets less attrac
as the hard-core radius increases can be seen in Fig. 5, w
we plot s0 as function ofr for different combinations ofr
and R. The less attraction is simply due to the factor
1v0r in the denominator in Eq.~21!, which increases asv0
increases and makes the right-hand side of Eq.~21! to con-
tribute less tos0.

To conclude this section, we mention that for neutr
stars, for instance, one is interested in the equation of s
pressureP versus energye. One important question here is t
check whether causality is respected by such an equatio

of

-

FIG. 6. The pressure of the nuclear matter as a function of
energy density corresponding to the different hard-core radii. T
upper panel of the figure corresponds toR50.8 fm and the lower
one is forR50.6 fm.
6-6



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tion

s

EXCLUDED VOLUME EFFECTS IN THE QUARK MESON . . . PHYSICAL REVIEW C65 065206
state. Figure 6 presentsP versuse for different values ofr
andR. For comparison, the causal limitP5e is also shown
in the figure. Clearly seen is that all the cases studied h
respect the causal condition]P/]e<1, so that the speed o
sound remains lower than the speed of light. This resul
consistent with Ref.@25#, where it was shown that for rea
istic situations of temperatures below the QCD phase tra
tion, which is believed to be of the order of 200 MeV, th
excluded volume prescription used here@21,22# does not
lead to conflicts with causality.

IV. CONCLUSIONS AND PERSPECTIVES

In this paper we have incorporated excluded volume
fects in the quark-meson coupling model in a thermodyna
cally consistent manner. The excluded volume simulates
phenomenological way the short-range hard-core repul
of the nucleon-nucleon force, in the sense that it does
allow nucleons to occupy all space as they were pointli
The consequences for in-medium nucleon properties
saturation properties of nuclear matter due to the exclu
volume effects have been investigated for different bag
hard-core radii. The bag constant was allowed to chang
medium and differences with respect to a fixed bag cons
were studied. It was also shown that the prescription u
does not lead to violations of causality.

We found that the excluded volume induces an effect
repulsion that increases as the size of hard core increa
The repulsion is at most 4 MeV at the saturation dens
In-medium nucleon properties, such as bag radius
nucleon mass are not changed significantly, as compare
the changes when excluded volume effects are not taken
ar

is

n

06520
re

is

i-

f-
i-
a
n

ot
.
d
d
d
in
nt
d

e
es.
.
d
to
to

account. The changes are at most at the level of 2%.
incompressibility is a little more sensitive, but the increase
at most 120 MeV. The excluded volume also induces
effect that the binding energy curve as a function of t
nuclear density is stiffer as the in-medium nucleon mass
creases. This feature is contrary to the case without exclu
volume. It arises because of the extra repulsion due to
hard core that leads to a smaller sigma field and con
quently to less attraction.

The formalism of the present paper can be extended
several ways. We intend to incorporate taking into accoun
the Lorentz contraction of the bags. As indicated by the
thors of Ref.@24#, the effects of the Lorentz contraction
most important for light particles like pions. Also possib
extensions of the formalism presented here to finite temp
tures are currently under progress. As a final remark
should be clear that an excluded volume approach is by
means a complete replacement of explicit calculations
short-range correlation effects, such as through a Be
Goldstone type of approach@20#. There is one attempt to
include short-range quark-quark correlations in the QM
model @26# and its further investigation in the context of
Bethe-Goldstone approach is an interesting new direc
that should be undertaken in the near future.

ACKNOWLEDGMENTS

One of the authors~P.K.P.! would like to acknowledge
the IFT, São Paulo, for kind hospitality. This research wa
supported in part by CNPq and FAPESP~Grant No. 99/
08544-0!.
s,

J.

.

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