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Functional renormalization group at finite density and Bose condensation

We discuss the functional renormalization group and pion condensation in the presence of a finite isospin chemical potential $μ_I$. We calculate the phase diagram as function of temperature $T$ and $μ_I$. While the exact effective average action is invariant under certain gauge transformations, the effective action in the local-potential approximation is not. As a consequence, the critical chemical potential $μ_I^c$ for Bose-Einstein condensation at T=0 is no longer equal to the mass of the condensing mode. We discuss possible solutions to this problem.