Vacuum Energy and Θ–vacuum

The highly non-trivial structure of the θ–vacuum encodes many of the fundamental properties of gauge theories. In particular, the response of the vacuum to the θ–term perturbation is sensitive to the existence of confinement, chiral symmetry breaking, etc. We analyze the dependence of the vacuum energy density on θ around two special values, θ = 0 and θ = π. The existence or not of singular behaviors associated to spontaneous breaking of CP symmetry in these vacua has been a controversial matter for years. We clarify this important problem by means of continuum non-perturbative techniques. The results show the absence of first order cusp singularities on the vacuum energy density at θ = 0 and θ = π for some gauge theories. This smooth dependence of the energy on θ might have implications for long standing cosmological problems like the baryonic asymmetry and the cosmological constant problem.