A Private SUSY 4HDM with FCNC in the Up-sector

We present a private SUSY Higgs model with four Higgs doublets, where each fermion type (up, down, and charged leptons) obtain their masses from a different Higgs doublet $H_f\hspace{0.1cm}( f = u_1, d, e)$. After imposing the conditions for anomaly cancellation, one finds that the remaining doublet $H_{u_2}$, must have the same hypercharge as $H_{u_1}$, and thus can only couple to up-type quarks, which opens the possibility to have FCNC's in this sector. We discuss the Lagrangian and the Higgs potential of the model, in order to identify the Higgs mass eigenstates and their interactions, with Yukawa matrices of the texture type. After imposing LHC constraints on the Higgs properties, we identify viable regions of parameter space, which we use to evaluate the decay $t \rightarrow ch$, finding that it can reach typically $B.R. (t \rightarrow ch) \approx \mathcal{O}(10^{-5} )$. These rates are compared with current bounds from LHC, finding that some cases are already ruled out, but some cases could only be tested at future LHC stages.