We study the signatures of a two Higgs doublet model of Davidson and Logan. The model includes an extra Higgs doublet with the vacuum expectation value (VEV) much smaller than the one of the standard model like Higgs. The smaller VEV is related to the origin of the small neutrino mass in the two Higgs doublet model. In the model, a single non-standard model like Higgs production of weak gauge boson fusion is suppressed due to the smallness of the vacuum expectation value. In contrast to the single Higgs production, the cross section of the Higgs pair production due to gauge boson fusion is not suppressed. Using the model, we compute the charged Higgs and neutral Higgs pair production cross section in W Z annihilation channel. In the two Higgs doublet model, the charged Higgs H^+ decays into a pair of the charged anti-lepton and right-handed neutrino. The neutral Higgs boson decays into right-handed neutrino and left-handed anti-neutrino pair which is invisible. A single charged anti-lepton and three neutrinos are the products of the subsequent decays of the charged Higgs and the neutral Higgs. W Z pair production gives the background for the signal through the decays W^+ ->nu l^+ and Z -> nu nubar. By multiplying the charged and neutral Higgses production cross section with the lepton flavor specific decay branching fractions of charged Higgs, we define a measurement which characterizes the present model. We numerically compute the measurement and find the sizable deviation from the standard model prediction.
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