Topological susceptibility and axion potential in two-flavor superconductive quark matter

We study the potential of the axion, $a$, of Quantum Chromodynamics, in the two-flavor color superconducting phase of cold and dense quark matter. We adopt a Nambu-Jona-Lasinio-like model. Our interaction contains two terms, one preserving and one breaking the $U(1)_A$ symmetry: the latter is responsible of the coupling of axions to quarks. We introduce two quark condensates, $h_L$ and $h_R$, describing condensation for left-handed and right-handed quarks respectively; we then study the loci of the minima of the thermodynamic potential, $\Omega$, in the $(h_L,h_R)$ plane, noticing how the instanton-induced interaction favors condensation in the scalar channel when the $\theta-$angle, $\theta=a/f_a$, vanishes. Increasing $\theta$ we find a phase transition where the scalar condensate rotates into a pseudo-scalar one. We present an analytical result for the topological susceptibility, $\chi$, in the superconductive phase, which stands both at zero and at finite temperature. Finally, we compute the axion mass and its self-coupling. In particular, the axion mass $m_a$ is related to the full topological susceptibility via $\chi=m_a^2 f_a^2$, hence our result for $\chi$ gives an analytical result for $m_a$ in the superconductive phase of high-density Quantum Chromodynamics.