θ–PHYSICS: 0–vacuum versus π–vacuum

The behaviour of 0–vacuum and π–vacuum under CP symmetry is analyzed in two different families of quantum field theories: gauge theories in 4D and sigma models in 2D. In particular, the possibility of spontaneous CP symmetry breaking is analyzed by new continuum non-perturbative techniques. In the case of θ = 0–vacuum we find that the dependence of the vacuum energy density on θ around θ = 0 has not a first order cusp singularity, a missing requirement of the proof of the Vafa-Witten theorem. The result is based on the absence of Lee-Yang singularities for pure imaginary values of the θ–parameter. In the case of θ = π the result follows from the analysis of the nodal structure of the vacuum functional in the Schrodinger representation. The dynamics of both theories favors the localization of vacuum nodes at sphalerons instead of classical vacua as corresponds to parity even states. The existence of a dynamical level repulsion between parity even and parity odd states at low energies is the mechanism behind the absence of CP symmetry breaking in π–vacuum.