On the temperature effects in QCD axion mass mixing

In this work, we extend the QCD axion mass mixing in the early Universe and investigate the temperature effects in the mixing. We explore the scenario where two $Z_N$ QCD axions undergo mass mixing during the QCD phase transition, yielding three distinct mixing scenarios: the mixing I, II, and III. These scenarios are realized through fine-tuning of the axion decay constants, the temperature parameters, as well as the value of $N$. We conduct a thorough analysis of the level crossing phenomena in these three mixing scenarios, detailing the conditions under which they occur. Notably, in the mixing I and II, the level crossing precedes the critical temperature of the QCD phase transition ($T_{\mathrm{QCD}}$), with minimal non-essential discrepancies in the cosmological evolution of the mass eigenvalues at $T_{\mathrm{QCD}}$. In contrast, the mixing III exhibits a unique double level crossings, occurring both before and at $T_{\mathrm{QCD}}$. Despite superficial similarities in axion evolution between the mixing II and III, we uncover fundamental differences between them. Additionally, we briefly address the transition in energy density between the two axions within our mixing scenarios. This work contributes to a deeper understanding of the role of the QCD axion in the early Universe and its potential implications for cold dark matter.