Traditionally the core of neutron stars has been modeled as a uniform fluid of neutron-rich nuclear matter in equilibrium with respect to the weak interaction (β-stable matter). Nevertheless, due to the large value of the density, new hadronic degrees of freedom are expected to appear in addition to nucleons. Hyperons, baryons with a strangeness content, are an example of these new degrees of freedom. Contrary to terrestrial conditions, where hyperons are unstable and decay into nucleons through the weak interaction, the equilibrium conditions in neutron stars can make the inverse process happen. Hyperons may appear in the inner core of neutron stars at densities of about 2 − 3ρ. At such densities, the nucleon chemical potential is large enough to make the conversion of nucleons into hyperons energetically favorable. This conversion relieves the Fermi pressure exerted by the baryons and makes the equation of state (EoS) softer, as it is illustrated schematically in panel (a) of the Figure. As a consequence (see panel (b)) the mass of the star, and in particular the maximum mass (M_{max}, is reduced to values which are incompatible with the recent measurements of unusually high masses of the millisecond pulsars PSR J1903+0327 (1.667 +/- 0.021M_{⊙})) [1], PSR J1614-2230 (1.97 +/- 0.04M_{⊙}) [2], and PSR J0348+0432 (2.01 +/- 0.04M_{⊙}) [3].

Therefore, in view of these new and severe observational constraints, which rule out almost all currently proposed EoS with hyperons, a natural question arises: can hyperons, or strangeness in general, still be present in the neutron star interior if the maximum mass is reduced to values smaller than 2M_{⊙}, although their presence is energetically favorable? This question is at the origin of the so-called “hyperon puzzle”, whose solution is not easy and it is presently a subject of very active research. The solution of this problem requires a mechanism (or mechanisms) that could eventually provide the additional repulsion needed to make the EoS stiffer and, therefore the value of M_{max} compatible with the current observational limits. Three different mechanisms that could provide such additional repulsion have been proposed. They are: (i) the inclusion of a repulsive hyperon-hyperon interaction through the exchange of vector mesons or density dependent couplings, (ii) the inclusion of repulsive hyperonic three-body forces, or (iii) the possibility of a phase transition to deconfined quark matter at densities below the hyperon threshold. In the following we briefly comment these three possible solutions.

*Hyperon-hyperon repulsion*

This solution has been mainly explored in the context of relativistic mean field models and it is based on the well-known fact that, in a meson-exchange model of nuclear forces, vector mesons generate repulsion at short distances. If the interaction of hyperons with vector mesons is repulsive enough then it could provide the required stiffness to explain the current pulsar mass observations.

*Hyperonic three-body forces*

It is well known that the inclusion of three-nucleon forces in the nuclear Hamiltonian is fundamental to reproduce properly the properties of few-nucleon systems as well as the empirical saturation point of symmetric nuclear matter in calculations based on non-relativistic many-body approaches. Therefore, it seems natural to think that three-body forces involving one or more hyperons (i.e., nucleon-nucleon-hyperon, nucleon-hyperon-hyperon and hyperon-hyperon-hyperon) could also play an important role in the determination of the neutron star matter EoS, and contribute to the solution of the hyperon puzzle. These forces could eventually provide, as in the case of the three-nucleon ones, the additional repulsion needed to make the EoS stiffer at high densities and, therefore, make the maximum mass of the star compatible with the recent observations.

*Quarks in neutron stars*

Several authors have suggested that an early phase transition from hadronic mater to deconfined quark matter at densities below the hyperon threshold could provide a solution to the hyperon puzzle. Therefore, massive neutron stars could actually be hybrid stars with a stiff quark matter core.

The interested reader is referred to the extensive literature on this topic for further information.

**References
**[1] D. J. Champion et al., Science 320, 1309 (2008); P. C. C. Freire, arXiv:0907.32199; arXiv:1011.5809.

[2] P. Demorest et al., Nature 467, 1081 (2010).

[3] J. Antoniadis et al., Science 340 6131 (2013).

[4] R. A. Hulse and J. H, Taylor, Astrophys. J. Lett. 195 L51 (1975).

*Author: Isaac Vidaña (CFisUC, Department of Physics, University of Coimbra)*