arXiv:nucl-th/0703040v1 13 Mar 2007
In-medium omega meson mass and quark condensate in a Nambu Jona-Lasinio model
constrained by recent experimental data
R. Huguet, J.C. Caillon and J. Labarsouque
Universit´ e Bordeaux 1 ; CNRS/IN2P3 ;
Centre d’Etudes Nucl´ eaires de Bordeaux-Gradignan, UMR 5797
Chemin du Solarium, BP120, 33175 Gradignan, France
We have determined the relation between the in-medium ω meson mass and quark condensate in
the framework of a Nambu Jona-Lasinio model constrained by some recent experimental data on
the meson properties in nuclei. In addition to the usual four-quark interactions, we have included
eight-quark terms in the Lagrangian. The parameters of this model have been determined using the
meson properties in the vacuum as well as in the medium. More particularly, we have constrained
both the in-medium pion decay constant to the value measured in experiments on deeply bound
pionic atoms and the in-medium ω meson mass to the experimental value obtained either by the
TAPS collaboration or by the E325 experiment at KEK. Our results are compared to several
scaling laws and in particular to that of Brown and Rho.
PACS numbers: 12.39.Fe; 12.39.Ki; 14.40.-n; 21.65.+f
Keywords: Nambu-Jona-Lasinio model, in-medium ω meson mass, Brown and Rho scaling
These last years, much attention has been focussed on the modification of hadron properties in nuclear environment
and more particularly in the sector of the light vector mesons. The hope is that this modification could shed some
light on prominent features of QCD at low energy. In particular, the knowledge of the dependence of the in-medium
vector meson mass on the quark condensate is essential to a better understanding of the role played by the chiral
structure of the QCD vacuum.
Experimentally, an indirect indication of the modification of hadron properties in the medium has been provided by
the dilepton production measurements in relativistic heavy-ion collisions, like for example, experiments from CERES
and HELIOS collaborations. However, the interpretation in terms of a reduction of the ρ mass is still controversial.
Recently, new experiments using proton-induced nuclear reactions, or γ−A reactions have provided a more clear
experimental signature of the in-medium modifications of the ω mesons. In particular, the modification in nuclei of
the ω meson has been investigated in photoproduction experiments by the TAPS collaboration and its mass was
found to be m∗
−5(syst) MeV at 0.6 times the saturation density of nuclear matter. The same order
of magnitude, a 9% decrease of the in-medium ω mass at saturation, has been observed by Naruki et al. in 12 GeV
proton-nucleus reactions (E325/KEK).
On the other hand, experimental indications of the in-medium modification of the quark condensate, ?qq?, can be
obtained, for example, in experiments on deeply bound pionic atoms. Indeed, by deducing the isovector πN interaction
parameter in the pion-nucleus potential from the binding energy and width of deeply bound 1s states of π−in heavy
nuclei, the in-medium pion decay constant, f∗
π, can be extracted[5, 6]. The quark condensate is then connected to f∗
through the Gell-Mann-Oaks-Renner relation. The observed enhancement of the isovector πN interaction parameter
over the free πN value indicates a reduction of the pion decay constant in the medium which was found to be
π= 0.64 at saturation density of nuclear matter.
¿From a theoretical point of view, starting from the assumption of Harada and Yamawaki on the ”vector manifes-
tation” of chiral symmetry in which a hidden local symmetry theory is matched to QCD, Brown and Rho proposed
that, up to the saturation density, the vector meson mass in medium, m∗
relation : m∗
densates). In quite different frameworks, like, for example, in finite density QCD sum rule calculations[9, 10, 11, 12, 13]
or in the Nambu Jona-Lasinio model (NJL), the relation between the in-medium vector meson mass and quark
condensate is not so clear and thus more complicated to handle.
The recent experimental data, like those previously mentioned, should provide stringent tests for the models and for
the relation between the in-medium ω meson mass and quark condensate. An indication on the consequences of these
new constraints could be obtained by enforcing them in quark models incorporating the most prominent features of
QCD. In this context, the NJL model appears as a good candidate since it allows a dynamical description of both
the breaking of chiral symmetry and of the modification of the in-medium ω meson mass.
In this work, we have determined the dependence of the in-medium ω meson mass on the quark condensate in a
V, scales according to the approximative
V/mV ∼ [?qq?/?qq?0]1/2(where ?qq? and ?qq?0are respectively the in-medium and vacuum quark con-
NJL model constrained by in-medium meson properties in accordance with recent experimental data. In addition to
the usual four-quark interactions, we have included eight-quark terms in the NJL Lagrangian. The parameters of this
model have been determined using the meson properties in the vacuum as well as in the medium through the pion
decay constant and ω meson mass. More particularly, the in-medium pion decay constant has been constrained by
the value obtained in an experiment on deeply bound pionic atoms and we have fixed the in-medium ω meson mass
to the experimental values determined by the TAPS collaboration or by the E325 experiment at KEK. These in-
medium changes of meson properties arise from dynamical chiral symmetry restoration at the quark mean-field-RPA
level as well as from more complicated quark-gluon excitations usually parametrized in terms of many-body hadronic
interactions. Considering the importance of the role played by the chiral structure of the QCD vacuum, concerning
the ω meson mass and the pion decay constant, we made the assumption that, to leading order in nuclear density, the
main contribution comes from dynamical chiral symmetry restoration at the quark mean-field-RPA level. Our results
will be compared to several scaling laws and in particular to that of Brown and Rho.
We consider the following chirally invariant two-flavor NJL Lagrangian up to eight-quark interaction terms :
L = q [iγµ∂µ− m0]q + g1
?(qq)2+ (qiγ5τq)2?(qγµq)2+ g4
?(qq)2+ (qiγ5τq)2?− g2(qγµq)2
where q denotes the quark field with two flavor (Nf= 2) and three color (Nc= 3) degrees of freedom and m0is the
diagonal matrix of the current quark masses (here in the isospin symmetric case). The second and third terms of
Eq.1 represent local four-quark interactions while those proportional to g3and g4are eight-quark interactions. Let us
recall that, in two-flavor models, the t’Hooft six fermion interaction term can be rewrited in terms of the four-quark
interactions considered here. We have not considered the term (qγµq)4since, as in the nucleonic NJL model,
it leads to a violation of the causality at high density.
The Dirac equation for a quark in mean-field approximation is given by :
iγµ∂µ− m0− 2g2γ0?qγ0q? + 2g1?qq? + 2g3?qq??qγ0q?2+ 4g4?qq?3?
which defines a dynamical constituent-quark mass :
q = 0,(2)
m = m0− 2g1
generated by a strong scalar interaction of the quark with the Dirac vacuum. In the gap equation (Eq.3), the quark
condensate ?qq? can be written as :
?qq? = −i
where here Tr denotes traces over color, flavor and spin. In Eq.4, S(k) represents the in-medium quark propagator
defined as :
γµkµ− m + iε+ iπγµkµ+ m
δ(k0− Ek)θ(kF− |k|),(5)
where Ek =
density by ρB=1
larization procedure. As many authors[14, 18], we introduce a three-momentum cutoff Λ which has the least impact
on medium parts of the regularized integrals, in particular at zero temperature. Thus, after the regularization
procedure, the quark condensate is given at each density by :
√k2+ m2and kF is the quark Fermi momentum. The baryonic density is related to the total quark
3ρq. The quark condensate is divergent due to the loop integrals and requires an appropriate regu-
?qq? = −NfNc
As usual, the ω meson mass is obtained by solving the Bethe-Salpeter equation in the quark-antiquark channel.
First, we define the quark-antiquark polarization operator in the ω channel by :
ω(q2) = −i
(2π)4Tr[iγµiS(p + q/2)iγνiS(p − q/2)]
where S(k) is given by Eq.5. Using the same regularization procedure as for the quark condensate, we obtain :
?1 − 4m2/κ2
The in-medium ω meson mass, m∗
ω, is then determined by the pole structure of the T-matrix, i.e. by the condition :
1 − 2
ω) = 0. (9)
We also need the pion mass and decay constant to adjust the model parameters. In the pseudo-scalar channel, the
polarization reads :
The in-medium π meson mass, m∗
π, and decay constant, f∗
π, are then given respectively by :
1 − 2
π) = 0,(12)
where the in-medium pion-quark-quark coupling constant, g∗
πqq, is obtained by :
We have six free parameters : the cutoff Λ, the bare quark mass m0, and the coupling constants g1, g2, g3and g4.
As usual, we have used for the fitting procedure the three following constraints : the pion mass mπ= 135 MeV, the
pion decay constant fπ= 92.4 MeV and the ω meson mass mω= 782 MeV in vacuum. Since additional constraints
like, for example, the value of the quark condensate in vacuum, do not allow to determine the cutoff Λ unambiguously,
as often done, we have considered several values for Λ or equivalently several values of the in-vacuum constituent
quark mass, mvac, values ranging between 400 MeV and 500 MeV. Such a rather large mass prevents the ω meson
to be unstable against decay into a quark-antiquark pair since m∗
that smaller constituent quark masses can be obtained in NJL models which take the confinement into account by
including Polyakov loops or using a confining interaction in the Lagrangian. As already mentioned, in addition
to these in-vacuum constraints, we have also chosen to take into account recent experimental results which provide
constraints in the medium. In particular, we have fixed f∗2
of nuclear matter) in accordance with what is obtained in experiments on deeply bound pionic atoms. Moreover,
the in-medium ω meson mass has been constrained to reproduce the experimental central value obtained either by
the TAPS collaboration, m∗
ω(ρB= 0.6ρ0) = 722 MeV, or by the E325 experiment at KEK, m∗
MeV. Thus, two families of parametrization sets denoted respectively by TAPS and KEK will be considered. Note
that once mπ, fπ, mω, f∗
The results are shown on Fig.1 where we have plotted m∗
ω/mωas a function of ?qq?/?qq?0for the two parametrization
sets TAPS and KEK. Note that ?qq?/?qq?0= 0.8 corresponds to a baryonic density close to the saturation one. The
shaded areas correspond to values of the ω meson mass for a constituent quark mass ranging from 400 MeV to 500
MeV. As we can see, these areas are rather narrow and the results are thus only weakly dependent on the value of
ωis always lower than 2m for every density. Note
π= 0.64 (where ρ0is the saturation density
ω(ρB= ρ0) = 711
ωand mvacare fixed, all the parameters are determined unambiguously.
FIG. 1: In-medium ω meson mass as a function of the quark condensate. The shaded areas correspond to values obtained for
a constituent quark mass in the range 400 < mvac < 500 MeV. The full lines represent the scaling laws given by Eq.15 for
α =1/3, 1/2 and 1.
In order to determine an approximate form for the relation between the vector meson mass and the quark condensate,
we have considered scaling laws of the general form :
Any value of α can be considered but we have chosen to show here the results for α = 1/2 which corresponds to the
Brown and Rho scaling and for neighboring values: α = 1/3 and α = 1.
The full lines on Fig.1 represent the scaling laws given by Eq.15 for α = 1/3, 1/2 and 1. A rather good agreement
with the case α = 1/2 corresponding to the Brown and Rho scaling law is obtained for the TAPS parametrization set
while the KEK result clearly favours α = 1/3. Assuming Eq.15 for the scaling law, this result can be understood since
to leading order (?qq?/?qq?0)α≃ 1 − α(1 − ?qq?/?qq?0) and ?qq?/?qq?0≃ 1 − βρ/ρ0with β = σNρ0/f2
σN is the πN sigma term. The TAPS or KEK results can then be used to determine the product αβ obtained by
eliminating the quark condensate in m∗
ω/mω. Using the value σN≃ 35 MeV obtained in the NJL model for m ≃ 450
MeV, the TAPS and KEK results provide respectively α close to 1/2 and 1/3 in agreement with our full calculation.
On the other hand, the experimental values of the in-medium ω meson mass are not determined unambiguously. In
particular, the TAPS collaboration obtained a rather large uncertainty including statistical and systematical errors.
By taking into account such an uncertainty in our calculation, we have found that the scaling law (Eq.15) with α = 1
is clearly ruled out but the case α = 1/3 is not totally excluded. However, note that the central value of the TAPS
result clearly favours α = 1/2.
FIG. 2: In-medium ω meson mass as a function of the baryonic density for the TAPS (solid curve) and KEK (dashed curve)
parametrization sets. For comparison the scaling laws (15) with α = 1/2 and 1/3 are also shown (dot-dashed curves).
Let us now determine how the quark condensate and the ω meson mass depend on the baryonic density for central
values of the constituent quark mass, i.e. for mvac= 450 MeV, respectively for the TAPS and KEK parametrizations.
The parameters of the NJL model are then Λ = 575 MeV, m0= 5.6 MeV, g1Λ2= 2.53, g2Λ2= 5.20, g3Λ8= 62.5
and g4Λ8= 2.27 for the TAPS parametrization and Λ = 575 MeV, m0 = 5.6 MeV, g1Λ2= 2.58, g2Λ2= 4.12,
g3Λ8= 12.4 and g4Λ8= 1.22 for the KEK one. Let us mention that, whatever the parametrization considered,
the model provides a quark condensate in the vacuum ?uu?1/3= −240 MeV in good agreement with the lattice
calculations: ?uu?1/3= −(231± 4± 8±6) MeV  and a critical density close to four times the saturation one. On
Fig.2, we have plotted m∗
for α = 1/2 (dashed curve) and α = 1/3 (dot-dashed curve), as a function of the dimensionless baryonic density
ρB/ρ0. As we can see, the density dependences of the in-medium ω meson mass obtained with the TAPS and KEK
parametrizations lead to a drop close to 10% at saturation density (9% for KEK and 12% for TAPS). As already
ω/mω(solid curves) for the two parametrizations TAPS and KEK as well as [?qq?/?qq?0]α
discussed, to a good level of approximation, the in-medium ω meson mass determined using the KEK result varies as
a function of the baryonic density like the third root of the quark condensate. On the other hand, using the TAPS
result, the density dependence of the in-medium ω meson mass is not very much different from the square root of the
quark condensate reflecting the fact that the results follow approximately the Brown and Rho scaling law up to the
We have determined the in-medium ω meson mass and quark condensate in a NJL model with eight quark interaction
terms. The parameters of this model have been determined using the meson properties in the vacuum but also in
the medium through the value of the pion decay constant obtained in experiments on deeply bound pionic atoms as
well as the ω meson mass measured either by the TAPS collaboration or by the E325/KEK experiment. When the
in-medium ω meson mass is constrained to the experimental data obtained by the TAPS collaboration, the Brown
and Rho scaling law is approximately recovered. On the other hand, when the KEK result is used, the in-medium ω
meson mass varies rather like the third root of the quark condensate. However, in both cases, this corresponds to a
drop of the ω meson mass at saturation density close to 10%, a result which is lower than those found in QCD sum
rule calculations where a decrease close to 15-25% is generally obtained[9, 12, 13].
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