The equation of state of a system of fermions in a uniform magnetic field is obtained in terms of the thermodynamic quantities of the theory by using functional methods. It is shown that the breaking of the O(3) rotational symmetry by the magnetic field results in a pressure anisotropy, which leads to the distinction between longitudinal-and transverse-to-the-field pressures. A criterion to find the threshold field at which the asymmetric regime becomes significant is discussed. This threshold magnetic field is shown to be the same as the one required for the pure field contribution to the energy and pressures to be of the same order as the matter contribution. A graphical representation of the field-dependent anisotropic equation of state of the fermion system is given. Estimates of the upper limit for the inner magnetic field in self-bound stars, as well as in gravitationally bound stars with inhomogeneous distributions of mass and magnetic fields, are also found.

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A Comment [1] to our recent article [2], expressed criticism on our estimate of the maximum magnetic field that can exist inside a neutron star and on the pressure anisotropy we found for a magnetized gas of fermions. These two points are relevant for the physics of neutron stars. With our reply we attempt not only to address the Comment's criticisms, but for the sake of the reader's understanding, we also try to clarify the connection between our and some literature's results mentioned in the comment. I. MAXIMUM FIELD STRENGTHAs known, the inner magnetic fields of neutron stars are not directly accessible to observation, so their possible values can only be estimated with the help of heuristic methods. A widely accepted estimate of the maximum star's magnetic field ∼ 10 18 G was done in Ref.[3] using a macroscopic analysis based on the equipartition of the magnetic and gravitational energies (Virial Theorem), and assuming a constant density and a uniform magnetic field throughout the entire star. As stated in the Comment [1], this estimate is in agreement with numerical simulations that used various EoS of nuclear matter [4,5], although it should be mentioned that the results of Ref.[4] were later criticized [6] because in their calculations the authors omitted the compositional changes in the EoS due to Landau quantization. Same criticism is applicable to Ref.[5], since the magnetic field was not considered in the EoS they used. Ignoring the field in the EoS of nuclear matter is not consistent at strong fields because the EoS is substantially softened by the magnetic field for fields well below the proton critical field ∼ 10 20 G [6].In Ref.[2] we argued that if the star is a hybrid star, meaning its core is made of deconfined quark matter, while its outer layers are made of nuclear matter, the previous inner field estimates based on a constant mass distribution may not be reliable because in the hybrid star a much larger difference should exist between the matter densities in the outer layers and in the core. In Section IIB of our paper, we did two independent estimates of the maximum magnetic field of a hybrid star assuming a very dense core. The Comment criticism pertains to the first of these estimates. Like in [3], our first estimate was based on the equipartition of the gravitational and magnetic energies in a star of given mass and radius. The main difference with respect to [3] was that we considered an ad-hoc model with nonuniform matter and field magnitude distributions. The only purpose we had with this simple example was to illustrate that the maximum field estimate can change just by relaxing the assumption of uniform mass and field distributions that was used in [3]. In the second paragraph following Eq.(10) we wrote "we make no claim that this ad hoc model correctly describes the real way the field varies with the radius in a hybrid star". More importantly, this ad hoc model was not used for any of the main derivations of the paper, so it has no relevance for any of the paper's main results. Noti...

We investigate the equation of state (EoS) along the BCS-BEC crossover for a quark system described by a Nambu-Jona-Lasinio model with multi-fermion interactions. Together with attractive channels for particle-antiparticle (GS) and particle-particle (GD) interactions, a multi-fermion channel with coupling λ that accounts for the diquark-diquark repulsion is also considered. The chiral and diquark condensates are found in the mean-field approximation for different values of the coupling constants. The parameter values where the BCS-BEC crossover can take place are found, and the EoS is used to identify the stability region where the BEC regime has positive pressure. We discuss how the particle density and the repulsive diquark-diquark interaction affect the stability window in the GS − GD plane and find the profile of the pressure versus the density for various values of λ and GD. The effects of λ and GD in the BCS-BEC crossover tend to compensate each other, allowing for a feasible region of densities where the crossover can occur with positive pressure. These results, although mainly qualitative, should serve as a preliminary step in the microscopic analysis required to determine the feasibility of the BCS-BEC crossover and its realization in more realistic models of dense QCD that can be relevant for applications to neutron stars.

We study the evolution of the equation of state of a strongly interacting quark system as a function of the diquark interaction strength. We show that for the system to avoid collapsing into a pressureless boson gas at sufficiently strong diquark coupling strength, the diquark-diquark repulsion has to be self-consistently taken into account. In particular, we find that the tendency at zero temperature of the strongly interacting diquark gas to condense into the system ground state is compensated by the repulsion between diquarks if the diquark-diquark coupling constant is higher than a critical value λ C = 7.65. Considering such diquark-diquark repulsion, a positive pressure with no significant variation along the whole strongly interacting region is obtained. A consequence of the diquark-diquark repulsion is that the system maintains its BCS character in the whole strongly interacting region.