Top and Higgs flavor changing neutral couplings in two Higgs doublets model

We study various channels of top and Higgs flavor changing neutral couplings in the two Higgs doublet model with natural flavor conservation (2HDM-I and 2HDM-II). We update known results about t -> c gamma,cZ,cg and comment on t -> ch(0). The decays t -> ch(0) as well as {h(0),H-0}->(t) over barc are sensitive both to the bottom Yukawa coupling as well as to the trilinear scalar couplings h(0)H(+)H(-) and (HH+H-)-H-0. After imposing unitarity constraints as well as vacuum stability conditions on scalar sector parameters, in 2HDM-II we found that for large tan beta greater than or similar to 40 and rather light charged Higgs mass M-H +/-greater than or similar to 150 GeV, the maximum values allowed for Br(t -> ch(0)), Br(H-0 ->(t) over barc), and Br(h(0)-><<(t)over bar>c) are: 8x10(-5), 10(-3), and 10(-4) respectively. For charged Higgs mass in the range [200,300] GeV, which can accommodate B -> X-s gamma constraint if one takes into account large theoretical uncertainties, the branching ratio of both H-0 ->(t) over barc and h(0)->(t) over barc can still be slightly larger than 10(-5). For A(0)->(t) over barc, its branching ratio is smaller than approximate to 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 ->(t) over barc can be of the order 0.01 -> 0.1 fb near threshold region, while the cross section of e(+)e(-)->(t) over barc 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 root s less than or similar to 500 GeV.