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 〈us〉 ∼ ∆exp(iq2 · r) and 〈ud〉 ∼ ∆exp(iq3 · r) whose position-space dependence is that of two plane waves with wave vectors q2 and q3 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 ∆ → 0 limit as expected, find that it correctly predicts that ∆ decreases with increasing angle between q2 and q3 meaning that the phase with q2 ‖ q3 has the lowest free energy, and find that the Ginzburg-Landau approximation is conservative in the sense that it underestimates ∆ at all values of the angle between q2 and q3.