Fluctuations in cool quark matter and the phase diagram of quantum chromodynamics

We consider the phase diagram of hadronic matter as a function of temperature, T, and baryon chemical potential, mu. Currently the dominant paradigm is a line of first order transitions which ends at a critical endpoint. In this work we suggest that spatially inhomogeneous phases are a generic feature of the hadronic phase diagram at nonzero mu and low T. Familiar examples are pion and kaon condensates. At higher densities, we argue that these condensates connect onto chiral spirals in a quarkyonic regime. Both of these phases exhibit the spontaneous breaking of a global U(1) symmetry and quasilong range order, analogous to smectic liquid crystals. We argue that there is a continuous line of first order transitions which separate spatially inhomogeneous from homogeneous phases, where the latter can be either a hadronic phase or a quark-gluon plasma. While mean field theory predicts that there is a Lifshitz point along this line of first order transitions, in three spatial dimensions strong infrared fluctuations wash out any Lifshitz point. Using known results from inhomogeneous polymers, we suggest that instead there is a Lifshitz regime. Nonperturbative effects are large in this regime, where the momentum dependent terms for the propagators of pions and associated modes are dominated not by terms quadratic in momenta, but quartic. Fluctuations in a Lifshitz regime may be directly relevant to the collisions of heavy ions at (relatively) low energies, root s / A:1 -> 20 GeV.