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

<!--HTML--><p>After a quick review of the U(1) and strong-CP problems in QCD I will turn to some recent work on spontaneous CP violation in QCD at vacuum angle $\theta = \pi$ and to its possible relevance for determining the axion potential at temperatures up to the deconfinement temperature $T_c$.<br /> Using an effective Lagrangian approach, valid in the double large-N and chiral limit (with $r_i \sim m_i N/\Lambda$ fixed), we determine the regions in the multidimensional parameter space of the $r_i$ where CP is spontaneously broken/unbroken at $\theta=\pi$. The two regions are separated by an hypersurface on which lines of first order transitions end,&nbsp; there is a massless pseudoscalar meson (in spite of chiral symmetry being explicitly broken), and the topological susceptibility of QCD diverges.<br /> When we add to the model a generic axion field (in order to ensure CP at all values of $\theta$) the above considerations have a bearing on the shape of the axion potential near the boundary of its periodicity interval. This may have implications on the calculation of axionic dark matter abundance if some $r_i$ have a marked temperature dependence as one approaches $T_c$.</p>