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-
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].
 P. Wurm for the CERES Collaboration, Nucl. Phys. A590, 103c (1995).
 M. Masera for the HELIOS Collaboration, Nucl. Phys. A590, 93c (1995).
 M. Naruki et al., Phys. Rev. Lett. 96, 092301 (2006).
 D. Trnka et al., Phys. Rev. Lett. 94, 192303 (2005).
 H. Geissel et al., Phys. Lett. B549, 64 (2002).
 K. Suzuki et al. Phys. Rev. Lett. 92, 072302 (2004).
 M. Harada and K. Yamawaki, Phys. Rev. Lett. 86, 757 (2001); Phys. Rep. 381, 1 (2003).
 G. E. Brown and M. Rho, Phys. Rep. 398, 301 (2004).
 T. Hatsuda and S. H. Lee, Phys. Rev. C46, R34 (1992); T. Hatsuda, S. H. Lee and H. Shiomi, Phys. Rev. C52, 3364
 M. Asakawa and C. M. Ko, Phys. Rev. C48, R526 (1993).
 Y. Koike, Phys. Rev. C51, 1488 (1995).
 X. Jin, D. B. Leinweber, Phys. Rev. C52, 3344 (1995).
 F. Klingl, N. Kaiser and W. Weise, Nucl. Phys.A 624, 527 (1997).
 V. Bernard and U. G. Meissner, Nucl. Phys. A489, 647 (1988).
 Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); Phys. Rev. 124, 246 (1961).
 S. P. Klevansky Rev. Mod. Phys. 64, 649 (1992).
 I.N. Mishustin, L.M. Satarov and W. Greiner Phys. Rep. 391, 363 (2004).
 M. Buballa, Phys. Rep. 407, 205 (2005).
 C. Ratti, M.A. Thaler and W. Weise Phys. Rev. D73, 14019 (2006).
 L.S. Celenza, B. Huang and C.M. Shakin Phys. Rev. C59, 1030 (1999).
 L. Giusti, F. Rapuano, M. Talevi and A. Vladikas, Nucl. Phys. B 538, 249 (1999).