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Quark matter under strong magnetic fields

We revisit three of the mathematical formalisms used to describe magnetized quark matter in compact objects within the MIT and the Nambu-Jona-Lasinio models and then compare their results. The tree formalisms are based on 1) isotropic equations of state, 2) anisotropic equations of state with different parallel and perpendicular pressures and 3) the assumption of a chaotic field approximation that results in a truly isotropic equation of state. We have seen that the magnetization obtained with both models is very different: while the MIT model produces well-behaved curves that are always positive for large magnetic fields, the NJL model yields a magnetization with lots of spikes and negative values. This fact has strong consequences on the results based on the existence of anisotropic equations of state. We have also seen that, while the isotropic formalism results in maximum stellar masses that increase considerably when the magnetic fields increase, maximum masses obtained with the chaotic field approximation never vary more than 5.5$\%$. The effect of the magnetic field on the radii is opposed in the MIT and NJL models: with both formalisms, isotropic and chaotic field approximation, for a fixed mass, the radii increase with the increase of the magnetic field in the MIT bag model and decrease in the NJL, the radii of quark stars described by the NJL model being smaller than the ones described by the MIT model.