### Two-Higgs-Doublet type-II and -III models and $t\to c h$ at the LHC

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Based on the updated $8$ TeV LHC data for the Higgs searches, we study the constraints of generic two-Higgs-doublet model (2HDM) type-III and the impacts of the new Yukawa couplings. For comparisons, we revisit the analysis in 2HDM type-II. For understanding the influence of all involving free parameters and realizing their correlations, we employ $\chi$-square fitting approach by including theoretical and experimental constraints, such as $B\to X_s \gamma$, $B_q-\bar B_q$ mixing, S, T and U oblique parameters, the production of standard model Higgs and its decay to $\gamma\gamma$, $WW^*/ZZ^*$, $\tau^+ \tau^-$, etc. The errors of analysis are taken at $68\%$, $95.5\%$ and $99.7\%$ confidence level. Due to the new Yukawa couplings being associated with $\cos(\alpha-\beta)$ and $\sin(\alpha-\beta)$, we find that the allowed regions for $\sin\alpha$ and $\tan\beta$ in type-III could be broader when the dictated parameter $\chi_F$ is positive; however, for negative $\chi_F$, the limits are more strict than those in type-II model. By using the constrained parameters, we find that the deviation from the SM in $h\to Z\gamma$ could be of ${\cal O}(10\%)$. Additionally, we also study the top-quark flavor changing processes induced at the tree level in type-III model and find that when all current experimental data are considered, we get $Br(t\to c(h, H) )< 10^{-3}$ for $m_h=125.36$ and $m_H=150$ GeV and $Br(t\to cA)$ slightly exceeds $10^{-3}$ for $m_A =130$ GeV.