Models of Supersymmetry (SUSY) based on a minimal representation of the Higgs sector, involving only two Higgs-doublet fields, are flawed by the presence of a dimensionful parameter entering the fundamental Langrangian that bears no connection with the energy scale where SUSY ensures unification among the fundamental forces of nature and that is unrelated to any of the fields entering the model. Nevertheless, most of the phenomenological studies carried out so far for the Large Hadron Collider (LHC) and a future Linear Collider (LC) have been based on the Minimal Supersymmetric Standard Model (MSSM). New analyses now need to be performed in the case of non-minimal SUSY scenarios such as the Next-to-MSSM (NMSSM). The matter is manifestly pressing if one further recalls that SUSY-induced CP-violation can lead to dramatic effects onto the Higgs mass spectra as well as its production and decay channels relevant to the physics of such machines. This has already been proven to be true in the MSSM and its impact may well be even more spectacular in the case of the NMSSM. We intend to study the phenomenological consequences of a non-minimal Higgs sector in presence of explicit CP-violation at the above mentioned colliders.
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