7 FIG. 9: The gravitational mass of the hybrid star is plotted as a function of a) the star radius and b) the central density, for different values of µ 0 for Λ 2 with a = 0 . 17. FIG. 10: Mass of the maximum mass star configuration as a function of the parameter µ 0 for Λ 2 with a = 0 . 17. of the consequences of increasing the cutoff is a faster decrease the constituent s quark mass with density and, therefore, the onset of s quark at lower densities, giving rise to a larger pressure for the same chemical potential. The phase transition to a deconfined quark phase occurs at smaller densities and pressures and the density dis- continuity at the phase transition is smaller. For cutoff Λ$_{2}$ ( a = 0 . 17), stars with a quark core are obtained for a choice of the parameter µ$_{0}$ < 360 MeV. The maximum mass of these stars is between (1.46-1.51) M$_{⊙}$ , and is compatible with most of the compact star observations. However, the highly massive stars PSR B1516 + 02B [34], and PSR J1748-2021B [35], in case they are confirmed, would not reproduced. According to Baldo et al the instability of compact stars within the NJL model is probably due to the lack We have studied the possibility of formation of sta- ble compact stars with a quark core within the su(3) NJL model with a chemical potential dependent ultra- violet cutoff. We use a su(3) NJL model parametriza- tion which describes the vacuum properties of low mass mesons (pions and kaons) and choose the parametriza- tion of the cutoff so that it increases with density. One Fig. 12 shows the u and s quark condensates as a func- tion of the chemical potential for both cutoff parametriza- tions. For the cutoff Λ$_{1}$ the module of the condensates starts to increase for µ ≳ 475 MeV. This is not the case Λ$_{2}$: the module of the condensates decreases with den- sity leading the system to chiral symmetry restauration. Therefore, the cutoff proposed in this work is physically favored to cutoff Λ$_{1}$. We believe the behavior of Λ$_{1}$ at high densities is due to the high value of the cutoff Λ$_{1}$ at these densities. It is seen from Table II that a smaller parameter µ$_{0}$ and a harder cutoff Λ gives rise to a smaller maximum mass. The occurrence of a quark core reduces a lot the maximum mass but we can still get a reasonable value, ∼ 1 . 45 M$_{⊙}$ , which is consistent with the observed maximum neutron star masses, except for the still not confirmed, highly massive compact stars, the millisec- ond pulsars PSR B1516 + 02B [34], and PSR J1748- 2021B [35] with masses well above 2 M$_{⊙}$ . The µ$_{0}$ range with a stable quark core changes depend- ing on the parameters used to Λ$_{2}$. In Fig. (9) are shown the different configurations for the different values of µ$_{0}$ in the case of Λ$_{2}$ with a = 0 . 17. The value of maxi- mum mass decreases if µ$_{0}$ increases. We have plotted the mass of the maximum mass configuration as a func- tion µ$_{0}$ in Fig. 10. The limit of stability corresponds to the minimum in this plot. In Fig. 11 the different star configurations for different values of a and the same µ$_{0}$ = 347 MeV, are shown. We conclude that increasing a decreases the mass of the maximum mass configuration and the corresponding radius, because the EOS becomes softer. Model/Cutoff M$_{max}$ R ϵ$_{i}$ ϵ$_{f}$ ϵ$_{c}$ ( M ⊙ ) (km) (fm − $^{4}$) (fm − $^{4}$) (fm − $^{4}$) Λ 1 ( µ 0 = 400MeV) 1.701 11.14 5.97 8.26 M Λ 1 ( µ 0 = 350MeV) 1.674 11.64 4.69 6.22 M Λ 2 ( µ 0 = 400MeV) 1.621 12.00 3.84 5.29 M Λ 2 ( µ 0 = 350MeV) 1.456 10.56 2.60 3.42 7.62 GM1 with hyperons 1.705 11.11 5.99 348 350 352 354 356 358 360 362 1.43 1.44 1.45 1.46 1.47 1.48 1.49 $_{0}$ (MeV) M max /M sun 10 12 14 0 0.5 1 1.5 R (km) M/M sun 2 4 6 8 10 1 1.1 1.2 1.3 1.4 1.5 / c M/M sun $_{0}$ = 346.5 MeV $_{0}$ = 348 MeV $_{0}$ = 350 MeV $_{0}$ = 357 MeV Hadron phase Quark phase b) a) TABLE II: Maximum gravitational mass and radius of the hybrid stars (hs) and quark stars (qs) obtained with different parametrizations of the cutoff Λ and two values of the transi- tion chemical potential µ 0 . In the last line the values for the maximum mass neutron star (ns) obtained with GM1 with hyperons [27]. M indicates inside the mixed phase. IV. SUMMARY