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Recently the D0 collaboration reports a direct search for a charged
Higgs boson produced by $q \bar q$ annihilation and decaying to $t \bar
b$ final state, in the $180 \leq M_{H^+} \leq 300$ GeV mass range, using
0.93 fb$^{-1}$ of data collected at center-of-mass energy
$\sqrt{s}=1.96$ TeV . The analysis lead to upper limits on the
production cross section in the two Higgs doublet model types I, II and
III . We discuss different scenarios for the two Higgs doublet model
type III taking into account the previous bounds obtained in the
literature on the parameter space of the model. We simulate three
different scenarios and we compare them with the data reported by D0
collaboration. Also, searches in the charged Higgs boson mass smaller than top quark mass region have been performed, using the production
cross section of top quark pairs at the Tevatron and we discuss the
possible limits on the charged Higgs boson masss obtained from
measurements of the ratio $R_\sigma=\sigma_{t \bar > > t}^{l+jets}/\sigma_{t \bar t}^{dileptons}$ within the framework of the
two Higgs doublet model type III.