Some nontrivial aspects of Poincaré and CPT invariance of flavor vacuum

We study the explicit form of Poincaré and discrete transformations of flavor states in a two-flavor scalar model, which represents the simplest example of the field mixing. Because of the particular form of the flavor vacuum condensate, we find that the aforementioned symmetries are spontaneously broken. The ensuing vacuum stability group is identified with the Euclidean group E(3). With the help of Fabri–Picasso theorem, we show that flavor vacua with different time labels and in different Lorentz frames are unitarily inequivalent to each other and they constitute a manifold of zero-flavor-charge states. Despite the spontaneous breakdown of Poincaré and CPT symmetries that characterizes such vacua, we provide arguments on the absence of Goldstone Bosons. We also prove that the phenomenologically relevant oscillation formula is invariant under these transformations. In particular we prove that flavor oscillation formula on flavor vacuum has the same form in all Lorentz frames, by means of general arguments, valid at all energy scales.