Susceptibilities to order $\alpha_s$ in the high density phase of QCD

We compute the free energy density of QCD to order $\alpha_s$ at very high density and non-zero quark masses. The counterterms needed to renormalise the theory to order $\alpha_s$ is same as the vacuum (non-zero density) theory. We investigate the response of the theory to non-zero quark masses and chemical potentials. We study quark number density and quark number susceptibility in the high density limit, where the ratio of the quark mass to the corresponding chemical potential is very small ($m/\mu\ll 1$). In this limit both number density and susceptibility contain a $\ln(m/\mu)$ contribution at order $\alpha_s$. We compute the scalar and pseudoscalar susceptibilities to order $\alpha_s$ in three flavour QCD at high density taking quark masses and chemical potentials to be degenerate and non-zero. At extremely high density, since $\alpha_s$ is very small, both the susceptibilities are found to be same in the chiral limit. This means that the scalar-pseudoscalar splitting is absent in the CFL phase.