Chiral random matrix theory for colorful quark-antiquark condensates

In QCD at high density, the color-octet quark-antiquark condensate $\langle\overline\psi\gamma_0(\lambda^A)_C (\lambda^A)_F\psi\rangle$ is generally nonzero and dynamically breaks the $\mathrm{SU}(3)_C\times \mathrm{SU}(3)_L\times\mathrm{SU}(3)_R$ symmetry down to the diagonal $\mathrm{SU}(3)_V$. We evaluate this condensate in the mean-field approximation and find that it is of order $\mu\Delta^2\log(\mu/\Delta)$ where $\Delta$ is the BCS gap of quarks. Next we propose a novel non-Hermitian chiral random matrix theory that describes the formation of colorful quark-antiquark condensates. We take the microscopic large-$N$ limit and find that three phases appear depending on the parameter of the model. They are the color-flavor locked phase, the polar phase, and the normal phase. We rigorously derive the effective theory of Nambu-Goldstone modes and determine the quark-mass dependence of the partition function.