Mass Mixing between QCD Axions

We introduce a novel level crossing phenomenon in the mass mixing between the QCD axions, one canonical QCD axion and one $Z_{\mathcal N}$ axion. The level crossing can take place at or slightly before the QCD phase transition critical temperature, depending on the ratio of the axion decay constants $\sim1.69$. The cosmological evolution of the mass eigenvalues in these two scenarios is similar; however, the transition of axion energy density differs significantly. Finally, we estimate the relic density of the QCD axion dark matter in this context. Additionally, this level crossing may have some interesting cosmological implications.