Top and Higgs Flavor Changing Neutral Couplings in two Higgs Doublets Model

We study various channels of top and Higgs Flavor Changing Neutral Couplings (FCNC) in the two Higgs doublet Model with natural flavor conservation (2HDM-I and 2HDM-II). We update known results about t\to c \gamma, c Z, cg and comment on t\to c h0. The decays $t\to c h0$ as well as {h0,H0}\to \bar{t}c are sensitive both to the bottom Yukawa coupling as well as to the trilinear scalar couplings h0H+H- and H0H+H-. After imposing unitarity constraints as well as vacuum stability conditions on scalar sector parameters, in 2HDM-II we found that for large $\tan\beta\ga 40$ and rather light charged Higgs mass $M_{H+}\la 150$ GeV, the maximum values allowed for Br(t\to c h0), {\rm Br}(H0\to \bar{t}c) and {\rm Br}(h0\to \bar{t}c) are: $8\times 10^{-5}$, $10^{-3}$ and $10^{-4}$ respectively. For charged Higgs mass in the range [200,300] GeV, which can accommodate $B\to X_s\gamma$ constraint if one takes into account large theoretical uncertainties, the branching ratio of both H0\to\bar{t}c and h0\to \bar{t}c can still be slightly larger than $10^{-5}$. For A0\to \bar{t}c, its branching ratio is smaller than $\approx 10^{-7}$ in both 2HDM-I and 2HDM-II. We study also the top-charm associated production at e+e- colliders and its \gamma\gamma option as well as at muon colliders. It is found that the cross section of \gamma\gamma\to \bar{t}c can be of the order $0.01\to 0.1$ fb near threshold region, while the cross section of e+e-\to \bar{t}c is well below $10^{-2}$ fb. The situation is slightly better for muon colliders where a few fb cross section can be reached for large \tan\beta and low center of mass energy \sqrt{s}\la 500 GeV.