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 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.

Figure 1

The gap equation. The labels $\alpha$, $\beta$ represent the external color indices and $i$, $j$ represent the external flavor indices. All the other color-flavor indices are contracted. ${\Delta }_{CF}$ (and ${\Delta }_{CF}^{†}$) are matrices of the form (21) and carry the same color and flavor indices as the neighboring propagators. The dashed lines represent the propagator $(i\partial ̸-\mathrm{\mu ̸}{)}^{-1}$ and the solid lines represent $(i\partial ̸+\mathrm{\mu ̸}{)}^{-1}$. Evaluating the gap equation involves substituting (21) for ${\Delta }_{CF}$, doing the contraction over the internal color-flavor indices, and evaluating the loop integrals in momentum space.

Figure 2

Examples of contributions to the free energy. The five diagrams depict a ${\Pi }_{ud}$ contribution to ${\alpha }_3{\Delta }_3^{*}{\Delta }_3$, a ${\mathcal{J}}_{udud}$ contribution to ${\beta }_3({\Delta }_3^{*}{\Delta }_3{)}^2$, a ${\mathcal{J}}_{udus}$ contribution to ${\beta }_{32}{\Delta }_3^{*}{\Delta }_3{\Delta }_2^{*}{\Delta }_2$, a ${\mathcal{K}}_{ududud}$ contribution to ${\gamma }_3({\Delta }_3^{*}{\Delta }_3{)}^3$ and a ${\mathcal{K}}_{udusus}$ contribution to ${\gamma }_{322}{\Delta }_3^{*}{\Delta }_3({\Delta }_2^{*}{\Delta }_2{)}^2$.

Figure 3

${\overline{\beta }}_{32}(\varphi )={\overline{J}}_{32}(\varphi )$ for the two-plane wave crystal structure with condensate (80). $\varphi$ is the angle between ${\mathbf{q}}_2$ and ${\mathbf{q}}_3$. For more complicated crystal structures, ${\overline{\beta }}_{32}$ is given by the sum in (59), meaning that it is a sum of ${\overline{J}}_{32}(\varphi )$ evaluated at various values of $\varphi$ corresponding to the various angles between a vector in $\{{\mathbf{q}}_2\}$ and a vector in $\{{\mathbf{q}}_3\}$.

Figure 4

${\overline{\gamma }}_{322}(\varphi )$ for the two-plane wave crystal structure with condensate (80). $\varphi$ is the angle between ${\mathbf{q}}_2$ and ${\mathbf{q}}_3$. ${\overline{\gamma }}_{322}(0)=0.243$ and ${\overline{\gamma }}_{322}(\varphi )$ increases monotonically with $\varphi$.

Figure 5

Free energy $\Omega$ vs $\Delta$ for the CubeX crystal structure, described in Table , at four values of $M_s^2/\mu$. From top curve to bottom curve, as judged from the left half of the figure, the curves are $M_s^2/\mu =240$, 218.61, 190, and 120 MeV, corresponding to $\alpha =0.233$, 0.140, 0, $-0.460$. The first-order phase transition occurs at $M_s^2/\mu =218.61\mathrm{MeV}$. The values of $\Delta$ and $\Omega$ at the minima of curves like these are what we plot in Figs. 6, 7.

Figure 6

Gap parameter $\Delta$ versus $M_s^2/\mu$ for three-flavor crystalline color superconducting phases with various crystal structures. The crystal structures are described in Table . For comparison, we also show the CFL gap parameter and the gCFL gap parameters ${\Delta }_1$, ${\Delta }_2$ and ${\Delta }_3$ [11, 12]. Recall that the splitting between Fermi surfaces is proportional to $M_s^2/\mu$, and that small (large) $M_s^2/\mu$ corresponds to high (low) density.

Figure 7

Free energy $\Omega$ versus $M_s^2/\mu$ for the three-flavor crystalline color superconducting phases with various crystal structures whose gap parameters are plotted in Fig. 6. The crystal structures are described in Table . Recall that the gCFL phase is known to be unstable, meaning that in the regime where the gCFL phase free energy is plotted, the true ground state of three-flavor quark matter must be some phase whose free energy lies below the dashed line. We see that the three-flavor crystalline color superconducting quark matter phases with the most favorable crystal structures that we have found, namely, 2Cube45z and CubeX described in (87, 90), have sufficiently robust condensation energy (sufficiently negative $\Omega$) that they are candidates to be the ground state of three-flavor quark matter over a wide swath of $M_s^2/\mu$, meaning over a wide range of densities.

Figure 8

The CubeX crystal structure of Eq. (90). The figure extends from 0 to $a/2$ in the $x$, $y$ and $z$ directions. Both ${\Delta }_2(\mathbf{r})$ and ${\Delta }_3(\mathbf{r})$ vanish at the horizontal plane. ${\Delta }_2(\mathbf{r})$ vanishes on the darker vertical planes, and ${\Delta }_3(\mathbf{r})$ vanishes on the lighter vertical planes. On the upper (lower) dark cylinders and the lower (upper) two small corners of dark cylinders, ${\Delta }_2(\mathbf{r})=+3.3\Delta$ (${\Delta }_2(\mathbf{r})=-3.3\Delta$). On the upper (lower) lighter cylinders and the lower (upper) two small corners of lighter cylinders, ${\Delta }_3(\mathbf{r})=-3.3\Delta$ (${\Delta }_3(\mathbf{r})=+3.3\Delta$). Note that the largest value of $|{\Delta }_I(\mathbf{r})|$ is $4\Delta$, occurring along lines at the centers of the cylinders. The lattice spacing is $a$ when one takes into account the signs of the condensates; if one looks only at $|{\Delta }_I(\mathbf{r})|$, the lattice spacing is $a/2$. $a$ is given in (88). In (90) and hence in this figure, we have made a particular choice for the relative position of ${\Delta }_3(\mathbf{r})$ versus ${\Delta }_2(\mathbf{r})$. We show in Appendix  that one can be translated relative to the other with no cost in free energy.