Rishi Sharma — Page 5

arXiv2006

The Crystallography of Three-Flavor Quark Matter

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 two 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 and condensates are separately face centered cubic; in the other case and combined make up a face centered cube.

hep-ph↗

arXiv2006

Testing the Ginzburg-Landau approximation for three-flavor crystalline color superconductivity

It is an open challenge to analyze the crystalline color superconducting phases that may arise in cold dense, but not asymptotically dense, three-flavor quark matter. At present the only approximation within which it seems possible to compare the free energies of the myriad possible crystal structures is the Ginzburg-Landau approximation. Here, we test this approximation on a particularly simple "crystal" structure in which there are only two condensates $ \sim Δ\exp(i {\bf q_2}\cdot {\bf r})$ and $ \sim Δ\exp(i {\bf q_3}\cdot {\bf r})$ whose position-space dependence is that of two plane waves with wave vectors ${\bf q_2}$ and ${\bf q_3}$ at arbitrary angles. For this case, we are able to solve the mean-field gap equation without making a Ginzburg-Landau approximation. We find that the Ginzburg-Landau approximation works in the $Δ\to 0$ limit as expected, find that it correctly predicts that $Δ$ decreases with increasing angle between ${\bf q_2}$ and ${\bf q_3}$ meaning that the phase with ${\bf q_2}\parallel {\bf q_3}$ has the lowest free energy, and find that the Ginzburg-Landau approximation is conservative in the sense that it underestimates $Δ$ at all values of the angle between ${\bf q_2}$ and ${\bf q_3}$.

hep-ph↗