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

Abstract

We consider the phase diagram of hadronic matter as a function of temperature, T , and baryon chemical potential, µ. Currently the dominant paradigm is a line of first order transitions which ends at a critical endpoint.In this work we suggest that spatially inhomogenous phases are a generic feature of the hadronic phase diagram at nonzero µ 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 quasi-long range order, analogous to smectic liquid crystals. We argue that there is a continuous line of first order transitions which separate spatially inhomogenous from homogenous 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 inhomogenous polymers, we suggest that instead there is a Lifshitz regime. Non-perturbative 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, √ s/A : 1 → 20 GeV. A quarkyonic phase only exists when quark excitations near the Fermi surface are (effectively) confined. While for three colors numerical simulations of lattice gauge theories at nonzero quark density are afflicted by the sign problem for three colors, they are possible for two colors [118-123]. Although the original argument for a quarkyonic phase was based upon the limit of a large number of colors [102], these simulations show that even for two colors, the expectation value of the Polyakov loop is small, indicating confinment, up to large values of the quark chemical potential [118-123]. Directly relevant to our analysis are the results of Bornyakov et al., who find that the string tension decreases gradually from its value in vacuum, to essentially zero at µ q ∼ 750 MeV: see Fig. 3 of Ref. [123]. This suggests

Indications of isotropic Lifshitz points in four dimensions

“…The flow equation of the parameters W and Z, due to their definition in Eqs. (16), (17), are obtained by selecting the coefficients of p 4 (for W) and of p 2 (for Z) in Eqs. (22), (23).…”

“…The LP theory finds application in various phenomenological contexts such as liquid crystals, or polymer mixtures, or high-Tc superconductors, or magnetic systems (for reviews see [5,6]), but also in the high energy sector, LPs turn out to be relevant in the formulation of emergent gravity [7][8][9][10] or higher spin gravity [11,12] or in the study of Lorentz invariance violation [13,14], as well as in the analysis of dense quark matter [15][16][17].…”

A Pedagogical Introduction to the Lifshitz Regime

“…This is true at the level of mean field theory. We show, however, that fluctuations dramatically affect the phase diagram [5]. In mean field theory, three phases meet at what is known as a Lifshitz point.…”

Inhomogeneous phases in the quark-meson model with explicit chiral-symmetry breaking

“…It is thus an interesting question, both in the chiral limit and away from it, whether these findings remain valid if fluctuation effects are taken into account. Investigations of such questions are presently subject of intese research [26,27], particularly within the functional renormalization-group approach [7,28,29] or by performing lattice simulations for lower-dimensional models [30,31].…”