A microscopic study of pion condensation within Nambu--Jona-Lasinio model

0.23 MB·English

by  R. Anglani

BA-TH/624-10
PHY-12545-TH-2009
A microscopic study of pion condensation within Nambu–Jona-Lasinio model
R. Anglani1,2,3,∗
1Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA
2Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Bari, I-70126 Bari, Italy
3Dipartimento di Fisica, Universit`a di Bari, I-70126 Bari, Italy
We have studied the phenomenology of pion condensation in 2-flavor neutral quark matter at
finite density with Nambu–Jona-Lasinio (NJL) model of QCD. We have discussed the role of the
bare quark mass m and the electric chemical potential µe in controlling the condensation. The
central result of this work is that the onset for π-condensed phase occurs when |µe| reaches the
value of the in-medium pion mass Mπ provided the transition is of the second order, even for a
composite pion system in the medium. Finally, we have shown that the condensation is extremely
fragile with respect to the explicit chiral symmetry breaking via a finite current quark mass.
0
1 PACSnumbers: 12.38.Aw,11.10.Wx,11.30.Rd,12.38.Gc
0
2
I. INTRODUCTION on these analyses, we conclude that πc condensation is
n
forbidden in realistic neutral quark matter within this
a
J effective model.
Thepossibilityofpioncondensationinnuclearmedium
4
was suggested by A. B. Migdal for the first time in the

] 1970s[1,2]. Thereafter,manystudies[3]of“in-medium” II. THE MODEL
h pion properties have been performed due to the impor-
p
tantconsequencesinsubnuclearphysics[4],inthephysics
- The NJL Lagrangian of the model is given for a two-
p ofneutronstars[5],supernovas[6],andtheheavyioncol- flavor quark matter at a finite chemical potential [10]
e lisions [7]. These analysis considered the pion as an ele-
h mentaryobjectbutweknowthattheycanbeconsidered L = e¯(iγµ∂µ+µeγ0)e+ψ¯(iγµ∂µ+µˆγ0−m)ψ
[
1 atrsyNbarmeabkuin-Gg.olHdsetnocneetbhoesionntsergneanlesrtartuecdtubryecahnirdalthsyemmmases- +Gh(cid:0)ψ¯ψ(cid:1)2+(cid:0)ψ¯iγ5~τψ(cid:1)2i , (1)
v of the pion can be sensitive to the Quantum Chromody-
Here e denotes the electron field, and ψ is the quark
1 namics(QCD)vacuummodificationsinthefinitedensity
spinor with Dirac, color and flavor indices (implicitly
7
environment [8]; moreover the finite baryon density, and
summed). The bare quark mass is m = m = m and
4 u d
eventheisospindensityarisingfromtheneutralitycondi-
0 G the coupling constant. The electrical chemical poten-
tion, can modify the structure of the QCD ground state,
1. and it can in turn produce significant modifications of tial µe is necessary to keep the system electrically neu-
tral [10], while µ serves as the isospin chemical poten-
0 I
the pion properties in the medium.
0 tial in the hadron sector, µI = −µe. The quark chemi-
1 Inthiswork[9],wepresentthepossibilityofacharged cal potential matrix µˆ is defined in flavor-color space as
v: pion(πc)condensationstartingfromamicroscopicmodel µˆ = diag(µ− 32µe,µ+ 13µe)⊗1c, where 1c denotes the
i which is built with quarks as the constituents of pions, identitymatrixincolorspace,µisthequarkchemicalpo-
X andwhichexhibitschiralsymmetryrestorationatthefi- tential related to the conserved baryon number. In the
r nite quark chemical potential µ or temperature T. We mean field approximation (MFA), we examine the pos-
a
derive an appropriate criterion for πc condensation, that sibility that the ground state develops condensation in
we use to show the fragility of the πc condensation with the σ = Ghψ¯ψi and/or π = Ghψ¯iγ5τψi channels, where
respect to explicit chiral symmetry breaking via a finite τ = {τ1,τ2,τ3} denotes the Pauli matrices. Due to the
current quark mass. The conditions for the onset of πc absenceofdrivingforces,wefindhπ3iisalwayszero. For
condensation at finite density are investigated using the conveniencewedenoteM =m−2σandN =2 π2+π2.
p 1 2
Nambu–Jona-Lasinio(NJL)modelofQCD.Wefindthat In the numerical analyses, we have fixed Λ = 651 MeV
the threshold for πc condensation for noninteracting ele- and G = 2.12/Λ2 so that the model reproduces fπ = 92
mentarypiongasisµe =Mπ− (−µe =Mπ+)forpositive MeV, hu¯ui = −(250MeV)3, and mπ = 139 MeV in the
(negative) µ , where µ is the electric chemical potential vacuum with m=5.5 MeV.
e e
and M the in-medium pion mass. Finally, we clarify
π
the effect of the current quark mass in the electrically
neutral ground state, and the effect of a non-vanishing III. PION CONDENSATION AT FINITE µI AND
AT FINITE µ
µ in presence of a finite current quark mass. Based
e
The pion condensation “in the vacuum” (µ = 0)
has been investigated with the chiral perturbation ap-
∗Electronicaddress: [email protected] proach [11] and within NJL model [12]. In both studies
2
FIG. 2: Phase diagram of neutral matter in (µ, mπ) plane.
IV. THE ROLE OF THE CURRENT MASS AND
THE PHASE DIAGRAM (µ,µe)
Inthepastyears,alargenumberoftheanalysisabout
πc condensation have been performed in the chiral limit.
Inthissectionweinvestigatetheroleofthefinitecurrent
quark mass in πc condensation. In order to do this, we
set the cutoff Λ and the coupling G to the values spec-
FIG.1: TheconstituentquarkmassM,thepioncondensate ified above and we treat m as a free parameter. As a
N,andmesonmassesasafunctionofµatT =0intheneutral consequence, the pion mass at µ=T =0, mπ in the fol-
phase for a toy value of the current quark mass m=10 keV. lowing,isafreeparameteraswell. InFig.2wereportthe
phase diagram in (µ, m ) plane in the neutral case. The
π
solid line represents the border between the two regions
where chiral symmetry is broken and restored. The bold
thethresholdfortheonsetofthecondensationisfoundto
dot is the critical endpoint of the first order transition.
be µe =mπ, i.e., when the absolute value of the electric Theshadedregionindicatestheregionwhereπc conden-
chemical potential equals the vacuum pion mass.
sation occurs. In the chiral limit (m = 0) our results
π
We now consider neutral quark matter at µ 6= 0 and are in good agreement with those obtained in Ref. [16].
T = 0 and we want to study the relation between the Indeed, thereexisttwocriticalvaluesofthequarkchem-
thresholdofπc condensationatfinitedensityandthein- ical potential, µ and µ , corresponding to the onset
c1 c2
mediumpionmassesintheneutralgroundstate(forfur- andvanishingofπccondensation,respectively. Whenthe
ther recent studies at finite baryon and/or isospin chem- current quark mass increases, a shrinking of the shaded
ical potential see also Refs. [13, 14]). In Ref. [10], it is region occurs till the point µ ≡ µ for mc ∼ 9 MeV,
c1 c2 π
shown that at the physical point m = 5.5 MeV, there is corresponding to a current quark mass of m ∼ 10 keV.
no room for πc condensation in the neutral phase (sim- Hence the gapless πc condensation is extremely fragile
ilar results obtained also in Ref. [15]). Even though the with respect to the symmetry breaking effect of the cur-
picturechangeswhenthecurrentmassislowered,forour rent quark mass. As a final investigation, in Fig. 3 we
discussionitisenoughtostatethatweconsideracurrent report the phase diagram of quark matter in the (µ, µ )
e
quark mass of the order of 10 keV. In Fig. 1 we plot M plane when the current quark mass is tuned to m = 5.5
andN intheneutralphaseasafunctionofµ. Inthisfig- MeV. At each value of (µ, µ ) we compute the chiral
e
ureMπ0,Mπ± denotethein-mediumpionmassesdefined and pion condensates by minimization of the thermody-
by the poles of the pion propagators in the rest frame, namical potential. The solid line represents the first or-
computed in the randomt phase approximation (RPA) der transition from the πc condensed phase to the chiral
to the Bethe-Salpeter (BS) equation. The positive and symmetry broken phase without the πc condensate. The
negativesolutionsoftheBSequationinω correspondsto bolddotisthecriticalendpointforthefirstordertransi-
theexcitationgapsforπ+ andπ−,whichare(M +µ ) tion, afterwhichthesecondordertransitionsetsin. The
π+ e
and(Mπ−−µe),respectively. FromFig.1wenoticethat dashed line indicates the first order transition between
the transition to the pion condensed phase is of second the two regions where chiral symmetry is broken and
order and it occurs at the point where Mπ− =µe. For a restored, respectively. The dot-dashed line is the neu-
more detailed mathematical discussion of the numerical trality line µneut =µ (µ) which is obtained by requiring
e e
results shown in Fig. 1 we refer to [9]. the global electrical neutrality condition, ∂Ω/∂µ = 0.
e
3
the onset of the charged pion condensation. We show
that the equality between the electric chemical poten-
tial and the “in-medium” pion mass, |µ | = M , as a
e π
threshold, persists even for a composite pion system in
the medium, provided the transition to the pion con-
densedphaseisofthesecondorder(thesamequalitative
picture has been found also in a recent analysis in the
massive Gross-Neveu model [13]). Moreover, we have
found that the pion condensate in neutral quark matter
is extremely fragile to the symmetry breaking effect via
a finite current quark mass m, and is ruled out for m
larger than the order of 10 keV.
A final comment is in order. In this study the con-
tribution to finite baryon density comes only from the
FIG. 3: Phase diagram in (µ, µe) plane at m=5.5 MeV.
constituent quarks and not from nucleons. The presence
of nucleons may make the pion condensation even less
favorable. The reason can be found in the Tomozawa-
The neutrality line manifestly shows the impossibility of
Weinberg pion-nucleon interaction, which is isospin odd,
finding a πc condensate, in this physical situation.
giving rise to a repulsive pion self-energy at the physi-
calsituationwheretheneutrondensityislargerthanthe
V. CONCLUSIONS proton density (see Ref. [17]). For this reason an inter-
esting investigation may consider the pion condensation
with the presence of nucleon degrees of freedom within
We have studied the pion condensation in two-flavor
NJL model (see Ref. [18] for useful prescriptions).
neutral quark matter using the Nambu–Jona-Lasinio
model of QCD at finite density. We have investigated This work was supported in part by the U.S. Depart-
the role of electric charge neutrality, and explicit sym- mentofEnergy,OfficeofNuclearPhysics,undercontract
metry breaking via quark mass, both of which control no. DE-AC02-06CH11357.
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