Spontaneous CP Violation Jarlskog Invariant in SM ⊗ S3

In our days, CP (Charge Parity) violation in the Standar Model of fundamental interactions still remains as an open problem. It is well known that explicit CP violation may be included by impossing complex Yukawa couplings in the Yukawa sector or complex Higgs couplings in exttended Higgs sectors with more than one Higgs field. It is desirable to have a fundamental CP violation theory, in that sence, we analyse the diferent secenarios for Spontaneous CP violation in an exteded Higgs model with three Higgs fields and a discrete flavour permutational symmetry S3. Spontaneous CP violation effects contribute to the Higgs mass matrix, as well as, up and down quark mass matrices. This complex quark mass matrices allow us to study the conditions for a non-vanishing Jarlskog invariant J which provides a necessary and sufficient contribution for a spontaneous CPV coming from SM ⊗ S3