Dense QCD in a Finite Volume

We study the properties of QCD at high baryon density in a finite volume where color superconductivity occurs. We derive exact sum rules for complex eigenvalues of the Dirac operator at a finite chemical potential, and show that the Dirac spectrum is directly related to the color superconducting gap $\Delta$. Also, we find a characteristic signature of color superconductivity: an $X$-shaped spectrum of partition function zeros in the complex quark mass plane near the origin, reflecting the $Z(2{)}_L\times Z(2{)}_R$ symmetry of the diquark pairing. Our results are universal in the domain ${\Delta }^{-1}\ll L\ll m_{\pi }^{-1}$ where $L$ is the linear size of the system and $m_{\pi }$ is the pion mass at high density.

Figure 1

The spectra of the Lee-Yang zeros in the complex $m$-plane in the vicinity of $m=0$ in the thermodynamic limit $V_4\to \infty$: (a) $\mu \gg {\Lambda }_{\mathrm{QCD}}$ (exactly obtained in our analysis), (b) $\mu =0$ [23], and (c) $\mu >{\mu }_c$ with the critical ${\mu }_c$ of chiral symmetry restoration obtained from the random matrix theory [9]. The density of the zeros fades away as $m\to 0$ in case (a), while that remains constant in case (b).