Bose-Einstein condensation and pion stars

Pion stars consisting of Bose-Einstein condensed charged pions have recently been proposed as a new class of compact stars. We use the two-particle irreducible effective action to leading order in the $1/N$-expansion to describe charged and neutrals pions as well as the sigma particle. Tuning the parameters in the Lagrangian correctly, the onset of Bose-Einstein condesation of charged pions is exactly at $\mu_I=m_{\pi}$, where $\mu_I$ is the isospin chemical potential. We calculate the pressure, energy density, and equation of state, which are used as input to the Tolman-Oppenheimer-Volkov equations. Solving these equations, we obtain the mass radius-relation for pion stars. Global electric charge neutrality is ensured by adding the contribution to the pressure and energy density from a gas of free relativistic leptons. We compare our results with those of recent lattice simulations and find good agreement. The masses of the pion stars are up to approximately 200 solar masses while the corresponding radii are of the order of $10^5$ km.