Axion Mass and the Ground State of Deconfining SU(2) Yang–Mills Thermodynamics

For the deconfinement phase of an SU(2) Yang–Mills theory, we compute the axion mass $m_A$ by appealing to the Veneziano–Witten formula. The topological susceptibility $\chi$ arises (i) from a precisely computable thermal ground-state contribution due to a center of a relevant (anti)caloron, and (ii) from contributions due to free thermal quasi-particles in the effective theory. Both (i) and (ii) are derived by using standard Euclidean thermal field theory techniques. While contribution (i) is positive and $\propto T^4$, contribution (ii) is negative, as demanded by reflection positivity, but negligible compared to contribution (i). As a consequence, practically from the critical temperature $T_c$ onward, a real-valued axion mass $m_A\left(T\right)=\sqrt{{\displaystyle \frac{2}{3}}}\pi {\displaystyle \frac{T^2}{M_P}}$ emerges when the Peccei–Quinn scale is assumed to be the Planck mass $M_P$, independently of the Yang–Mills scale that the axion associates with. We discuss why our results deviate from those found in the dilute instanton gas and interacting instanton liquid approximations, and from results obtained in lattice simulations. Assuming the universe is dark sector to be based on such ultralight axion species, which are nonrelativistic for $T\ll M_P$, we investigate the cosmological conditions for their global Bose condensation as the very early universe cooled to temperatures of the order of ${10}^9$eV.