Spontaneous Violation of $CP$ Symmetry in the Strong Interactions

Some time ago Dashen [Phys. Rev. D 3, 1879 (1971)] pointed out that spontaneous $CP$ violation can occur in the strong interactions. I show how a simple effective Lagrangian exposes the remarkably large domain of quark mass parameters for which this occurs. I close with some warnings for lattice simulations.

Figure 1

The schematic phase diagram of quark-gluon dynamics as a function of the two lightest quark masses. The shaded regions exhibit spontaneous $CP$ breaking. On the diagonal line with $m_u=m_d$ there are three degenerate pions due to isospin symmetry. The neutral pion mass vanishes on the boundary of the $CP$ violating phase. The asymptotes of the boundaries are given by the strange quark mass. In the $CP$ conserving phases, the vacuum fluctuates about the indicated values for $\Sigma$.