Mass mixing between QCD axions

We introduce a novel level crossing in the mass mixing between two QCD axions, one canonical QCD axion and one $Z_{\mathcal N}$ QCD axion. The level crossing can take place at the QCD phase transition critical temperature or slightly before it, depending on the ratio of the axion decay constants $\sim1.69$. The cosmological evolution of the mass eigenvalues in these two cases is similar, however, the transition of axion energy density is completely different. Finally, we estimate the relic density of the QCD axion dark matter. This level crossing may also have some cosmological implications.
Comment: 6 pages, 4 figures