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Fluctuations in the quark-meson model for QCD with isospin chemical potential

We study the two-flavor quark-meson (QM) model with the functional renormalization group (FRG) to describe the effects of collective mesonic fluctuations on the phase diagram of QCD at finite baryon and isospin chemical potentials, $μ_B$ and $μ_I$. With only isospin chemical potential there is a precise equivalence between the competing dynamics of chiral versus pion condensation and that of collective mesonic and baryonic fluctuations in the quark-meson-diquark model for two-color QCD at finite baryon chemical potential. Here, finite $μ_B=3μ$ introduces an additional dimension to the phase diagram as compared to two-color QCD, however. At zero temperature, the ($μ_I,μ$)-plane of this phase diagram is strongly constrained by the "Silver Blaze problem." In particular, the onset of pion condensation must occur at $μ_I= m_π/2$, independent of $μ$ as long as $μ+ μ_I$ stays below the constituent quark mass of the QM model or the liquid-gas transition line of nuclear matter in QCD. In order to maintain this relation beyond mean field it is crucial to compute the pion mass from its timelike correlator with the FRG in a consistent way.