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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

I.INTRODUCTION

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[1]

and HELIOS[2] collaborations. However, the interpretation in terms of a reduction of the ρ mass is still controversial.

Recently, new experiments using proton-induced nuclear reactions[3], or γ−A reactions[4] 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[4] 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.[3] 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[6] to be

f∗2

π= 0.64 at saturation density of nuclear matter.

¿From a theoretical point of view, starting from the assumption of Harada and Yamawaki[7] on the ”vector manifes-

tation” of chiral symmetry in which a hidden local symmetry theory is matched to QCD, Brown and Rho proposed[8]

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)[14], 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[15] 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

ω= 722+4

−4(stat)+35

π

π/f2

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-

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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[6] and we have fixed the in-medium ω meson mass

to the experimental values determined by the TAPS collaboration[4] or by the E325 experiment at KEK[3]. 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.

II.FORMALISM

We consider the following chirally invariant two-flavor NJL Lagrangian[15] up to eight-quark interaction terms :

L = q [iγµ∂µ− m0]q + g1

+g3

?(qq)2+ (qiγ5τq)2?(qγµq)2+ g4

?(qq)2+ (qiγ5τq)2?− g2(qγµq)2

?(qq)2+ (qiγ5τq)2?2,

(1)

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[16]. We have not considered the term (qγµq)4since, as in the nucleonic NJL model[17],

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

?

1 +g3N2

fN2

4g1

cρ2

B

+2g4

g1

?qq?2

?

?qq?,(3)

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

?

d4k

(2π)4TrS(k),(4)

where here Tr denotes traces over color, flavor and spin. In Eq.4, S(k) represents the in-medium quark propagator

defined as :

S(k) =

1

γµkµ− m + iε+ iπγµkµ+ m

Ek

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[18]. 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-

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?qq? = −NfNc

π2

?Λ

kF

mk2dk

Ek

. (6)

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 :

Πµν

ω(q2) = −i

?

d4p

(2π)4Tr[iγµiS(p + q/2)iγνiS(p − q/2)]

−gµν+qµqυ

q2

(7)

=

?

?

Πω(q2),

where S(k) is given by Eq.5. Using the same regularization procedure as for the quark condensate, we obtain :

Πω(q2) =NfNc

12π2q2

?4(Λ2+m2)

4(p2

F+m2)

?1 − 4m2/κ2

q2− κ2

?

1 +2m2

κ2

?

dκ2.(8)

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

?

g2− g3?qq?2?

Πω(q2= m∗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 :

Ππ(q2) =?qq?

m

+ NcNfq2I(q2),(10)

with

I(q2) =

1

8π2

?4(p2

F+m2)

4(Λ2+m2)

1

q2− κ2

?

1 −4m2

κ2dκ2.(11)

The in-medium π meson mass, m∗

π, and decay constant, f∗

π, are then given respectively by :

1 − 2

?

g1+g3N2

fN2

4

cρ2

B

+ 2g4?qq?2

?

Ππ(q2= m∗2

π) = 0,(12)

f∗

π= NcNfg∗

πqqmI(q2= m∗2

π),(13)

where the in-medium pion-quark-quark coupling constant, g∗

πqq, is obtained by :

g∗2

πqq=

?dΠπ(q2)

dq2

?−1

q2=m∗2

π

.(14)

III.RESULTS

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[18], we have considered several values for Λ or equivalently several values of the in-vacuum constituent

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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[19] or using a confining interaction[20] 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[6]. Moreover,

the in-medium ω meson mass has been constrained to reproduce the experimental central value obtained either by

the TAPS collaboration[4], m∗

ω(ρB= 0.6ρ0) = 722 MeV, or by the E325 experiment at KEK[3], 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

mvacused.

ωis always lower than 2m for every density. Note

π(ρB= ρ0)/f2

π= 0.64 (where ρ0is the saturation density

ω(ρB= ρ0) = 711

π, m∗

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 :

m∗

mω

ω

= [?qq?/?qq?0]α.(15)

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.

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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.

πm2

πwhere

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 [21] 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