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

Abstract

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 $<us>\sim \Delta \exp (i{\mathbf{q}}_2\cdot \mathbf{r})$ and $<ud>\sim \Delta \exp (i{\mathbf{q}}_3\cdot \mathbf{r})$ whose position-space dependence is that of two plane waves with wave vectors ${\mathbf{q}}_2$ and ${\mathbf{q}}_3$ 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 $\Delta \to 0$ limit as expected, find that it correctly predicts that $\Delta$ decreases with increasing angle between ${\mathbf{q}}_2$ and ${\mathbf{q}}_3$ meaning that the phase with ${\mathbf{q}}_2\parallel {\mathbf{q}}_3$ has the lowest free energy, and find that the Ginzburg-Landau approximation is conservative in the sense that it underestimates $\Delta$ at all values of the angle between ${\mathbf{q}}_2$ and ${\mathbf{q}}_3$.

16 pages, 6 figures. Small changes only. Version to appear in Phys. Rev. D

References in corpus (3)

• Two lectures on color superconductivity
• Chromomagnetic instability in dense quark matter
• Novel Phases at High Density and their Roles in the Structure and Evolution of Neutron Stars

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