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, Ω, in the (h_L,h_R) plane, noticing how the instanton-induced interaction favors condensation in the scalar channel when the θ-angle, θ=a/f_a, vanishes. Increasing θ we find a phase transition where the scalar condensate rotates into a pseudo-scalar one. We present an analytical result for the topological susceptibility, χ, 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 χ=m_a^2 f_a^2, hence our result for χ gives an analytical result for m_a in the superconductive phase of high-density Quantum Chromodynamics.