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Using the well-known low-energy effective Lagrangian of QCD --valid for small(non-vanishing) quark masses and a large number of colors-- we study in detailthe regions of parameter space where $CP$ is spontaneously broken/unbroken fora vacuum angle $\theta= \pi$. In the $CP$-broken region there are first orderphase transitions as one crosses $\theta=\pi$, while on the (hyper)surfaceseparating the two regions, there are second order phase transitions signaledby the vanishing of the mass of a pseudo Nambu-Goldstone boson and by adivergent QCD topological susceptibility. The second order point sits at theend of a first order line associated with the $CP$ spontaneous breaking, in theappropriate complex parameter plane. When the effective Lagrangian is extendedby the inclusion of an axion these features of QCD imply that standardcalculations of the axion potential have to be revised when the QCD parametersfall in the above mentioned $CP$-broken region, in spite of the fact that theaxion solves the strong-$CP$ problem. These latter results could be of interestfor axionic dark matter calculations if the topological susceptibility of pureYang-Mills theory falls off sufficiently fast when temperature is increasedtowards the QCD deconfining transition.

Spontaneous $CP$ breaking in QCD and the axion potential: an effective Lagrangian approach