Phase structure, collective modes, and the axial anomaly in dense QCD

Using a general Ginzburg-Landau effective Lagrangian, we study the topological structure and low-lying collective modes of dense QCD having both chiral and diquark condensates, for two and three massless flavors. As we found earlier, the QCD axial anomaly acts as an external field applied to the chiral condensate in a color superconductor and, as a new critical point emerges, leads to a crossover between the broken chiral symmetry and color superconducting phases. At intermediate densities where both chiral and diquark condensates are present, we derive a generalized Gell-Mann–Oakes-Renner relation between the masses of pseudoscalar bosons and the magnitude of the chiral and diquark condensates. We show explicitly the continuity of the ordinary pion at low densities to a generalized pion at high densities.

Figure 1

Phase structure in three- and two-flavor systems without coupling between the chiral and diquark condensates ($\gamma =\lambda =0$). Phase boundaries with a second-order transition are denoted by a single line, and with a first-order transition by a double line.

Figure 2

Phase structure in the three-flavor system with $\gamma >0$ and $\lambda =0$. Phase boundaries with a second-order transition are denoted by a single line, and with a first-order transition by a double line. On the left, C is a critical end point [$0<\gamma <(c/3)\sqrt{\beta /b}$], and on the right, a triple point [$\gamma >(c/3)\sqrt{\beta /b}$].

Figure 3

Phase structure in the two-flavor system for $b>0$ and $\lambda >0$. Left diagram: Case with a tetracritical point ($\frac{1}{2}\sqrt{b\beta }>\lambda >0$). The second-order line between NG and COE is characterized by $\alpha =2a\lambda /b$, and that between CSC and COE by $\alpha =a\beta /2\lambda$. Right diagram: Case with a bicritical point ($\lambda >\frac{1}{2}\sqrt{b\beta }$). The first-order line between NG and CSC is characterized by $\alpha =a\sqrt{\beta /b}$.

Figure 4

Phase structure of the two-flavor system with $b<0$ and $\lambda >0$. The second-order line between NG and COE is given by $\alpha =\lambda (b-\sqrt{b^2-4fa})/f$. The first-order line between CSC and COE is given by $\alpha =(\beta /2\lambda )[a-3(b-4{\lambda }^2/\beta {)}^2/(16f)]$, while the first-order line between NG and CSC is given by ${\alpha }^2=\beta [6bfa+(b^2-4fa{)}^{3/2}-b^3]/(6f^2)$.

Figure 5

Schematic phase structure with two light (up and down) quarks and a medium-heavy (strange) quark. The arrows show how the critical point and the phase boundaries move as the strange-quark mass increases towards the two-flavor limit.

Figure 6

Mass spectra of pions in the intermediate density region: (a) without $\sigma$-$d$ mixing ($\gamma =\lambda =0$); (b) with $\sigma$-$d$ mixing ($\gamma \ne 0$, $\lambda \ne 0$); and (c) in addition with finite quark masses ($m_q\ne 0$). Explicit expressions for the $m_{\pi }^2$’s are given in the text.

Figure 7

Mass spectra of the ${\eta }^{\prime }$ meson in the intermediate density region when (a) $\sigma$-$d$ mixing does not exist ($\gamma =\lambda =0$); (b) $\sigma$-$d$ mixing is considered ($\gamma \ne 0,\lambda \ne 0$); (c) a finite quark mass is considered ($m_q\ne 0$) in addition. Here $m_{{\eta }^{\prime }}^2=36c_0{\sigma }^3/f_{{\eta }^{\prime }}^2$, $m_{{\eta }_{1,2}^{\prime }}^2(m_q=0)=12(x_1\mp \sqrt{x_1^2-x_2})$, and $\Delta m_{{\eta }_{1,2}^{\prime }}^2=12(x_3\mp (x_1{x}_3-x_4)/\sqrt{x_1^2-x_2})m_q$.