Thermal Pions ns Isospin Chemical Potential Effects

The density corrections, in terms of the isospin chemical potential $\mu_I$, to the mass of the pions are investigated in the framework of the SU(2) low energy effective chiral invariant lagrangian. As a function of temperature and $\mu_I =0$, the mass remains quite stable, starting to grow for very high values of $T$, confirming previous results. However, the dependence for a non-vanishing chemical potential turns out to be much more dramatic. In particular, there are interesting corrections to the mass when both effects (temperature and chemical potential) are simultaneously present. At zero temperature the $\pi ^{\pm}$ should condensate when $\mu_{I} = \mp m_{\pi}$. This is not longer valid anymore at finite $T$. The mass of the $\pi_0$ acquires also a non trivial dependence on $\mu_I$ at finite $T$.