content_image/436078.jpg
# A microscopic study of pion condensation within Nambu–Jona-Lasinio model 

$^{1}$Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA 

$^{2}$Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Bari, I-70126 Bari, Italy 

$^{3}$Dipartimento di Fisica, Universit` a di Bari, I-70126 Bari, Italy 

R. Anglani 1, 2, 3,$_{∗}$ 

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 µ e in controlling the condensation. The central result of this work is that the onset for π -condensed phase occurs when | µ e | reaches the value of the in-medium pion mass M π 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. 

The possibility of pion condensation in nuclear medium was suggested by A. B. Migdal for the first time in the 1970s [1, 2]. Thereafter, many studies [3] of “in-medium” pion properties have been performed due to the impor- tant consequences in subnuclear physics [4], in the physics of neutron stars [5], supernovas [6], and the heavy ion col- lisions [7]. These analysis considered the pion as an ele- mentary object but we know that they can be considered as Nambu-Goldstone bosons generated by chiral symme- try breaking. Hence the internal structure and the mass of the pion can be sensitive to the Quantum Chromody- namics (QCD) vacuum modifications in the finite density environment [8]; moreover the finite baryon density, and even the isospin density arising from the neutrality condi- tion, can modify the structure of the QCD ground state, and it can in turn produce significant modifications of the pion properties in the medium. 

In this work [9], we present the possibility of a charged pion ( π $^{c}$) condensation starting from a microscopic model which is built with quarks as the constituents of pions, and which exhibits chiral symmetry restoration at the fi- nite quark chemical potential µ or temperature T . We derive an appropriate criterion for π c condensation, that we use to show the fragility of the π c condensation with respect to explicit chiral symmetry breaking via a finite current quark mass. The conditions for the onset of π c condensation at finite density are investigated using the Nambu–Jona-Lasinio (NJL) model of QCD. We find that the threshold for π c condensation for noninteracting ele- mentary pion gas is µ$_{e}$ = M$_{π}$ − ( − µ$_{e}$ = M$_{π}$ + ) for positive (negative) µ$_{e}$ , where µ$_{e}$ is the electric chemical potential and M$_{π}$ the in-medium pion mass. Finally, we clarify the effect of the current quark mass in the electrically neutral ground state, and the effect of a non-vanishing µ$_{e}$ in presence of a finite current quark mass. Based 

PACS numbers: 12.38.Aw, 11.10.Wx, 11.30.Rd, 12.38.Gc 

## I. INTRODUCTION 

on these analyses, we conclude that π c condensation is forbidden in realistic neutral quark matter within this effective model. 

The NJL Lagrangian of the model is given for a two- flavor quark matter at a finite chemical potential [10] 

<img src='content_image/436078.jpg'>

Here e denotes the electron field, and ψ is the quark spinor with Dirac, color and flavor indices (implicitly summed). The bare quark mass is m = m$_{u}$ = m$_{d}$ and G the coupling constant. The electrical chemical poten- tial µ$_{e}$ is necessary to keep the system electrically neu- tral [10], while µ$_{I}$ serves as the isospin chemical poten- tial in the hadron sector, µ$_{I}$ = − µ$_{e}$ . The quark chemi- cal potential matrix ˆ µ is defined in flavor-color space as ˆ µ = diag( µ − 2 $_{3}$µ$_{e}$ , µ + 1 $_{3}$µ$_{e}$ ) ⊗ 1$_{c}$ , where 1$_{c}$ denotes the identity matrix in color space, µ is the quark chemical po- tential related to the conserved baryon number. In the mean field approximation (MFA), we examine the pos- sibility that the ground state develops condensation in the σ = G 〈 ¯ ψψ 〉 and/or π = G 〈 ¯ ψiγ$_{5}$ τ ψ 〉 channels, where τ = { τ$_{1}$, τ$_{2}$, τ$_{3}$ } denotes the Pauli matrices. Due to the absence of driving forces, we find 〈 π$_{3}$ 〉 is always zero. For convenience we denote M = m − 2 σ and N = 2 √ π 2 1 + π 2 $_{2}$. In the numerical analyses, we have fixed Λ = 651 MeV and G = 2 . 12 / Λ 2 so that the model reproduces f$_{π}$ = 92 MeV, 〈 ¯ uu 〉 = − (250 MeV) $^{3}$, and m$_{π}$ = 139 MeV in the vacuum with m = 5 . 5 MeV. 

## III. PION CONDENSATION AT FINITE µ I AND AT FINITE µ 

The pion condensation “in the vacuum” ( µ = 0) has been investigated with the chiral perturbation ap- proach [11] and within NJL model [12]. In both studies 

## II. THE MODEL 

