BCS-BEC crossover in a relativistic superfluid and its significance to quark matter

The character change of a superfluid state due to the variation of the attractive force is investigated in the relativistic framework with a massive fermion. Two crossovers are found. One is a crossover from the usual BCS state to the Bose-Einstein condensation (BEC) of bound fermion pairs. The other is from the BEC to the relativistic Bose-Einstein condensation (RBEC) of nearly massless bound pairs where antiparticles as well as particles dominate the thermodynamics. Possible realization of the BEC and RBEC states in the quark matter is also pointed out.

Figure 1

(a) Critical temperature $T_{\mathrm{c}}$ normalized by the Fermi energy $E_{\mathrm{F}}=\sqrt{m^2+p_{\mathrm{F}}^2}$ (thick solid line) as a function of the coupling $G/G_0$. $G_0$ is defined by $G_0={\pi }^2/{\Lambda }^2$. For other lines, see the text. (b) Chemical potential $\mu /E_{\mathrm{F}}$ as a function of the coupling $G/G_0$. The dashed line represents the level where $\mu =m$; $m/E_{\mathrm{F}}\simeq 0.89$ in the present case. The dotted line corresponds to one half of the bound boson’s mass in the vacuum. (c) The ratios of $N_{\mathrm{F}},N_{\overline{\mathrm{F}}},N_{\mathrm{B}}$ and $N_{\overline{\mathrm{B}}}$ to the fixed $N_{\mathrm{total}}$ as functions of $G/G_0$. $N_{\mathrm{MF}}/N_{\mathrm{total}}$ and $N_{\mathrm{fluc}}/N_{\mathrm{total}}$ are also plotted. The line for $N_{\mathrm{MF}}$ is behind that for $N_{\mathrm{F}}$ for $G/G_0\lesssim 1.07$ because of the absence of antifermions.