Higgs-like scalar and $t\rightarrow cZ$ decays in the 2HDM type III with CP violation

In models with more than one doublet, the neutral scalars can change the flavor of fermions when CP symmetry is non conserved in scalar sector. The rare top decays with vector boson and up type quark as a final state have contributions from flavor-changing neutral scalars at one loop level. The 2HDM type III with CP violation allows that the top decays to neutral scalar and up type quark at tree level. We compute the contributions from the scalar sector in 2HDM type III with CP violation to rare top decays $t\rightarrow cZ$ and $t\rightarrow ch_1$, where $h_1$ is the lighest neutral Higgs.The branching ratio for $t\rightarrow c Z$ and $t\rightarrow c h_1$ are obtained in the allowed regions for the model parameters. In particular, the $BR(t\rightarrow c Z)$ can increase its value up to $10^{-3}$ for model parameters such that $\tan\beta= 1$ and masses for the additional Higgs bosons in the order of $m_{h_2,h_3,H^\pm}\sim 0.5$ TeV. Meanwhile the $BR(t\rightarrow c h_1)$ can reaches a values of the order of $\sim10^{-1}$. To constrain the model parameters, we realized a $\chi^2$ analysis at 90 $\%$ C.L. by using the reported values at LHC of signal strengths $R_{XX}$ for the Higgs-like scalar decays at tree and one loop levels. We obtain allowed regions for mixing angles of the neutral scalar fields in the CP violation context as a function of $\tan\beta$.