Abstract
A viable minimal model with spontaneous CP violation in the framework of Two Higgs Doublet Models is proposed. The model is based on a generalised Branco-Grimus-Lavoura model with a flavoured $\mathbb{Z}_2$ symmetry, under which two of the quark families are even and the third one is odd. The lagrangian respects CP invariance, but the vacuum has a CP violating phase, which is able to generate a complex CKM matrix, with the strength of CP violation compatible with experiment. In this model, the presence of scalar mediated flavour changing neutral couplings is deeply connected to the generation of a complex CKM matrix from the vacuum phase. Concerning the scalar sector, the most relevant aspect is that the new scalar particles are necessarily lighter than 1 TeV. A complete analysis of the model including the most relevant constraints is performed, showing that it is viable and that it has definite implications for the observation of New Physics signals in, for example, flavour changing Higgs decays or the discovery of the new scalars at the LHC. We give special emphasis to processes like $t\to h c,h u$, as well as $h\to bs, bd$, which are relevant for the LHC and the ILC.
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