Sound velocity peak and conformality in isospin QCD

We study zero temperature equations of state (EOS) in isospin QCD within a quark-meson model which is renormalizable and hence eliminates high density artifacts in models with the ultraviolet cutoff (e.g., NJL models). The model exhibits a crossover transition of pion condensations from the Bose-Einstein-Condensation regime at low density to the Bardeen-Cooper-Schrieffer regime at high density. The EOS stiffens quickly and approaches the quark matter regime at density significantly less than the density for pions to spatially overlap. The sound velocity develops a peak in the crossover region, and then gradually relaxes to the conformal value $1/3$ from above, in contrast to the perturbative QCD results which predicts the approach from below. In the context of QCD computations, this opposite trend is in part due to the lack of gluon exchanges in our model, and also due to the non-perturbative power corrections arising from the condensates. We argue that with large power corrections the trace anomaly can be negative. In quantitative level, our EOS is consistent with the lattice results in the BEC regime but begins to get stiffer at higher density. The sound velocity peak also appears at higher density. The BCS gap in our model is $\Delta \simeq 300$ MeV in the quark matter domain, and naive application of the BCS relation for the critical temperature $T_c \simeq 0.57\Delta$ yields the estimate $T_c \simeq 170$ MeV, in good agreement with the lattice data.