1. ExtreMe Matter Institute Workshop and XXVI Max Born Symposium, 9 - 11 July 2009
Roberto Anglani
Dipartimento di Fisica, Università di Bari, Italia
We study aspects of the pion condensation in two-flavor neutral quark matter using the Nambu--Jona-Lasinio model of QCD at finite density. We investigate the
role of electric charge neutrality, and explicit symmetry breaking via quark mass, both of which control the onset of the charged pion condensation. We show that
the equality between the electric chemical potential and the in-medium pion mass, as a threshold, persists even for a composite pion system in the medium, pro-
vided the transition to the pion condensed phase is of the second order. Moreover, we find that the pion condensate in neutral quark matter is extremely fragile to
the symmetry breaking effect via a current quark mass m, and is ruled out for m > 10 keV.
8. At each value of (μ,μe) we compute the chiral and the pion condensate by mini-
mization of the thermodynamical potential. The solid line represents the first order
transition from the πc condensed phase to the chiral symmetry broken phase with-
out πc condensate. The bold dot is the critical end point for the first order transition,
after which the second order transition sets in. The dashed line indicates the first
order transition between the two regions where the chiral symmetry is broken and
restored respectively. The dot-dashed line is the neutrality line which is obtained
by requiring the global electrical neutrality condition. The neutrality line shows
the impossibility to find a pion condensate in the “physical limit”.
9. The solid line represents the border between the two regions where the
chiral symmetry is broken and restored. The bold dot is the critical end point
of the first order transition. The shaded region indicates the region where the
πc condensation occurs. In the chiral limit (mπ=0) our results are in good
agreement with those obtained by Ebert and Klimenko [8]. There exist two
critical values of the quark chemical potential μc1 and μc2 corresponding to the
onset and the vanishing of the πc condensation respectively. Increasing the
current quark mass results in the shrinking of the shaded region till the point
μc1 = μc1 for mπ = 9 MeV, corresponding to a current quark mass of m = 10 keV.
The gapless πc condensation is extremely fragile against the symmetry
breaking effect by a current quark mass. [8] JPG 2006
10. The in-medium pion masses defined as the poles of the pion propaga-
tors in the rest frame, are computed in the random phase approximation to
the Bethe-Salpeter equation (BSe) at N=0. The finite isospin density is re-
sponsible for the splitting of the masses of the charged pions. The excita-
tion gaps for the charged pions are: Mπ+ + μe and Mπ- - μe as positive and
negative solutions of the BSe in the boson frequency. Provided the tran-
sition to the pion condesed phase is of the second order, the thresh-
old “in the medium” remains Mπ- = μe.
[same result, in the vacuum, obtained with the ChiPT analysis of Son and Stephanov, PRL 2001]
The results of this work are published in: Phys. Rev. D 79, 034032 (2009)
in collaboration with H. Abuki, R. A., R. Gatto, M. Pellicoro and M. Ruggieri.
A study of pion condensation within Nambu-Jona-Lasinio model
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1. In 1970’s Migdal [1] suggested the
possibility of pion condensation in nu-
clear medium. Thereafter many studies
of “in-medium” pion properties [2] have
been performed, due to the important
consequences in neutron star physics [3],
subnuclear physics [4], supernovae [5]
and heavy ion collisions [6]. These
analysis considered pions as elemen-
tary object, but we know that they can
be considered as Nambu-Goldstone
bosons generated by chiral symmetry
breaking. Hence the internal structure of
the pion can be sensitive to the QCD
vacuum modifications.
2. The present study revisits the possibil-
ity of charged pion (πc) condensation in
the medium starting from a microscopic
model where quarks are constituents of
pions.
[1] Migdal, Zh. Eksp. Teor. Fiz. 1971; [2] Kunihiro et al., NPA 1989;
[3] Toki et al., NPA 1989; [4] Scalapino et al., Astroph.J. 1975;
[5] Ishizuka et al., JPG 2008, [6] Zimanyi et al., PRL 1979.
3. Nambu--Jona-Lasinio (NJL) model is
naturally conceived to reproduce the
global QCD symmetries as chiral sym-
metry.
For μ << ΛQCD quark-antiquark attractive
forces favor the appearance of a non
vanishing condensate
of some of the global symmetries of QCD, these effective models are helpful in-
struments of investigation. The Nambu–Jona-Lasinio (NJL) model is one of the
famous and studied effective model of QCD [54–58]. Purpose of the Chapter 1 is
to summarize the relevant features of the NJL model of quantum chromodynamics
and to develop the formalism that we have used in the investigation described in
the following Chapters of this work.
***
The Nambu–Jona-Lasinio model has a major advantage. Since the theory is con-
structed to reproduce exactly the QCD symmetries, is able to take into account
also the chiral symmetry which is fundamental for the exploration of lightest
hadron sector, where the dynamics of strong interactions is simplified. Indeed at
low density (quark chemical potential µ less than 300 MeV), the quark-antiquark
attractive forces generated by perturbative gluon exchanges and non-perturbative
istanton interactions favors the appearance of non-vanishing quark-antiquark con-
densates. The true vacuum of QCD turns out to be populated by a condensate of
quark-antiquark pairs, and is characterized by the order parameter
¯!! = 0. (1)
responsible for the breaking of chiral
symmetry and the generation of mass-
less Goldstone bosons that carry the
quantum #’s of pions.
4. The chiral symmetry breaking and res-
toration considerations allow to simplify
the dynamics of strong interactions in
low-density regimes.
5. A previous NJL study “in the vacuum”
has been performed in [7].
[7] He, Jin and Zhuang, PRD 2005
6. We study 2-flavor quark matter at finite
barion and isospin density
is a necessary condition for the condensation itself at zero temperature. As a mat
ter of fact if pions have to condense then it is necessary that their occupation num
ber is non vanishing. The interesting point is that this necessary condition turn
out to be sufficient as well, at least at µ = 0. Next in this Chapter we show that the
correspondence between gapless pion spectrum and onset of pion condensation
also occurs at finite quark density.
Finally let us notice that, the aforementioned condition |µe| > m! for the onse
of pion condensation has been obtained by means of the chiral perturbation theory
at µ = 0. In this kind of approach, the chiral SU(2) multiplet is an elementary field
and not a composite one. On the other hand the same threshold in the vacuum can
be obtained within the NJL model [120] where pions are defined by construction
as bound states of one quark and one antiquark.
3.2 The model and symmetry structure
We study the two flavor quark matter at finite chemical potential within the NJL
model.
The Lagrangian of the model is given by [95]
L = ¯e(i"µ#µ
+ µe"0)e+ ¯$ i"µ#µ
+ ˆµ"0 −m $
+G ( ¯$$)2
+( ¯$i"5%$)2
, (3.15
where e denotes the electron field, $ is the quark spinor with Dirac, color and
flavor indices (implicitly summed). m = mu = md is the bare quark mass and G
is the coupling constant. µe is the electric charge chemical potential needed to
sustain the system to be electrically neutral [95], while µI serves as the isospin
chemical potential in hadron sector, µI = −µe. The quark chemical potentia
matrix ˆµ is defined in flavor-color space as
ˆµ = diag(µ −
2
3
µe,µ +
1
3
µe)⊗1c
, where 1c denotes the identity matrix in color space, µ is the quark chemica
potential related to the conserved baryon number.
By performing the mean field approximation, we examine the possibility tha
the ground state develops the condensation in
& = G ¯$$ π = G ¯$i"5τ$ (3.16
m=mu=md is the bare quark mass and G is the cou-
pling constant. μe is the electric charge chemical po-
tential needed to sustain the system to be electri-
cally neutral, while μI serves as the isospin chemi-
cal potential in hadron sector, μI = −μe. The quark
chemical potential matrix μ is defined in flavor-color
space as μ=diag(μ-2μe/3, μ+μe/3) × 13.
By performing the mean field approxi-
mation, we examine the possibility that
the ground state develops the condensa-
tion in
σ = ¯ψψ π = ¯ψiγ5τψ
with N=2(π1
2+ π2
2)1/2.
7. We investigate how the bare quark
mass and the electric chemical poten-
tial can influence the pion condensation
in a significative way.
“Three days of Strong Interactions”, Wrocław (Poland)