Crystallography of three-flavor quark matter

Abstract

We analyze and compare candidate crystal structures for the crystalline color superconducting phase that may arise in cold, dense but not asymptotically dense, three-flavor quark matter. We determine the gap parameter $\Delta$ and free energy $\Omega (\Delta )$ for many possible crystal structures within a Ginzburg-Landau approximation, evaluating $\Omega (\Delta )$ to order ${\Delta }^6$. In contrast to the two-flavor case, we find a positive ${\Delta }^6$ term and hence an $\Omega (\Delta )$ that is bounded from below for all the structures that we analyze. This means that we are able to evaluate $\Delta$ and $\Omega$ as a function of the splitting between Fermi surfaces for all the structures we consider. We find two structures with particularly robust values of $\Delta$ and the condensation energy, within a factor of 2 of those for the CFL phase which is known to characterize QCD at asymptotically large densities. The robustness of these phases results in their being favored over wide ranges of density. However, it also implies that the Ginzburg-Landau approximation is not quantitatively reliable. We develop qualitative insights into what makes a crystal structure favorable, and use these to winnow the possibilities. The two structures that we find to be most favorable are both built from condensates with face-centered cubic symmetry: in one case, the $⟨ud⟩$ and $⟨us⟩$ condensates are separately face-centered cubic; in the other case $⟨ud⟩$ and $⟨us⟩$ combined make up a face-centered cube.