A microscopic study of pion condensation within Nambu–Jona-Lasinio model BA-TH/624-10 PHY-12545-TH-2009 $^{1}$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. PACS numbers: 12.38.Aw, 11.10.Wx, 11.30.Rd, 12.38.Gc on these analyses, we conclude that π c condensation is forbidden in realistic neutral quark matter within this effective model. I. INTRODUCTION 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 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 III. PION CONDENSATION AT FINITE µ I AND AT FINITE µ 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. L = ¯ e ( iγ$_{µ}$∂ µ + µ$_{e}$γ$_{0}$ ) e + ¯ ψ ( iγ$_{µ}$∂ µ + ˆ µγ$_{0}$ − m ) ψ + G [ ( ¯ ψψ ) 2 + ( ¯ ψiγ$_{5}$⃗ τψ ) 2 ] , (1) The NJL Lagrangian of the model is given for a two- flavor quark matter at a finite chemical potential [10] II. THE MODEL arXiv:1001.0471v1 [hep-ph] 4 Jan 2010 ∗ Electronic address: anglani@anl.gov