# Compact stars with a quark core within NJL model 

C. H. Lenzi 1 , $^{2}$, A. S. Schneider$^{3}$, C. Providˆencia$^{2}$, R. M. Marinho Jr.1 

$^{1}$Departamento de F´ısica, Instituto Tecnol´ ogico de Aeron´ autica, Campo Montenegro, S˜ ao Jos´ e dos Campos, SP, 12228-900, Brazil 

$^{2}$Centro de F´ısica Computacional, Department of Physics, University of Coimbra, Rua Larga, Coimbra, 3004-516, Portugal 

3 Department of Physics, Indiana University, Swain Hall West 117, 727 East Third Street Bloomington, Indiana 47405 

(Dated: November 2, 2018) 

An ultraviolet cutoff dependent on the chemical potential as proposed by Casalbuoni et al is used in the su(3) Nambu-Jona-Lasinio model. The model is applied to the description of stellar quark matter and compact stars. It is shown that with a new cutoff parametrization it is possible to obtain stable hybrid stars with a quark core. A larger cutoff at finite densities leads to a partial chiral symmetry restoration of quark s at lower densities. A direct consequence is the onset of the s quark in stellar matter at lower densities and a softening of the equation of state. 

Compact stars are complex systems which may contain exotic matter such as hyperons, kaon condensation, a non-homogenous mixed quark-hadron phase or, in their core, a pure quark phase [1, 2]. 

The hadronic phase has been successfully described within a relativistic mean-field theory with the inclu- sion of hyperons (for a review see [1]). The quark phase has frequently been described by the schematic MIT bag model [3, 4] or by the Nambu-Jona-Lasinio (NJL) model [5, 6]. The NJL model contains some of the basic sym- metries of QCD, namely chiral symmetry. It has been very successful in describing the vacuum properties of low lying mesons and predicts at sufficiently high densi- ties/temperatures a phase transition to a chiral symmet- ric state [7–10]. However, it is just an effective theory that does not take into account quark confinement. 

The authors of [11] have studied the possible existence of deconfined quark matter in the interior of neutron stars using the NJL model to describe the quark phase and could show that within this model typical neutron stars do not possess any deconfined quark matter in their cen- ter. It was shown that as soon as quark matter appears the star becomes unstable and collapses into a black-hole. It was also pointed out that the large constituent strange quark mass obtained with NJL over a wide range of den- sities was the cause of this behavior. In [12] it was shown that for warm neutrino free stellar matter a small quark core could appear at finite temperature. The reason can be traced back to a faster reduction of the s quark con- stituent mass at low densities and, therefore, the onset of the s quark at lower densities. For warm stellar matter with an entropy per particle equal or below 2, with or without trapped neutrinos no quark core was obtained [13]. 

Over the last decade, it has been realized that strong interacting matter at high density and low temperature may possess a large assortment of phases. Different pos- 

PACS numbers: 

## I. INTRODUCTION 

sible patterns for color superconductivity [14] have been conjectured (for a review see e.g. [15, 16] and references therein quoted). Very recently, a new phase of QCD, named quarkyonic phase, characterized by chiral sym- metry and confinement has been predicted [17]. We will not consider these phases in the present work. 

It was shown in [18] that at very large densities the standard NJL model is not able to reproduce the correct QCD behavior of the gap parameter in the quark color flavor locked (CFL) phase. In order to solve this prob- lem Casalbuoni et al have introduced a ultraviolet cutoff dependent on the baryonic chemical potential [18]. The dependence of a parameter of the model on the chemical potential changes the thermodynamics of the model and has to be dealt with care [19]. Within su(2) NJL Baldo et al have investigated whether a cutoff dependent on the chemical potential could solve the problem of star insta- bility with the onset of the quark phase and concluded that this was not a solution [20]. The question that may be raised is whether within the su(3) NJL a different be- havior occurs due to the a different behavior of the s quark constituent mass with density. We will show that a larger cutoff at finite baryonic densities will move the onset of the s quark to smaller densities due to a faster decrease of the s quark constituent mass with density. 

After a review of the standard su(3) NJL model we will introduce in section II the parametrization of the cutoff dependence on the chemical potential, the ther- modynamic consistency of the modified model and the β -equilibrium conditions. In section III we discuss the star stability and the dependence of the maximum mass configuration on the cutoff. In the last section we draw some conclusions. 

