200 400 600 800 1000 1200  s [GeV] 10 100 1000 Cross section [fb] H$^{+}$H - WW e e Signal + missing energy WW s = 1 TeV S$^{+}$S - AKS model -1 -0.5 0 0.5 1 cos - 10 100 Diff. cross section [fb] H$^{+}$H - e e Signal + missing energy s = 1 TeV S$^{+}$S - AKS model FIG. 4: left : The cross sections of the signal, e $^{+}$e − → S $^{+}$S − → τ $^{+}$τ − (+ missing energy), in the AKS model as a function of the collision energy √ s . right : The differential cross section of the signal for √ s = 1 TeV as a function of the angle of the direction of the outgoing τ − and the beam axis of incident electrons. In addition to the rate from the signal, those from backgrounds such as τ $^{+}$τ $^{−}$, τ $^{+}$τ $^{−}$νν and H $^{+}$H − are also shown. Because h 1 , 2 e ∼ O (1), the contribution from the t -channel N α R exchange diagrams to the production cross section of S $^{+}$S − dominate that from the Drell-Yan diagrams [14]. The cross section is about 87 fb for m$_{S}$ ± = 400 GeV at √$_{s}$ = 1 TeV. As the decay branching ratio of S ± → H $^{±}$η 0 is 100% and that of H ± → τ $^{±}$ν is also almost 100% because of the Type-X THDM interaction for tan β = 10, the final state of the signal is τ $^{+}$τ $^{−}$ννη $^{0}$η0 with almost the same rate as the parent S $^{+}$S − production. The main SM backgrounds are τ $^{+}$τ − and τ $^{+}$τ $^{−}$νν . The pair production of the doublet like charged Higgs boson H $^{+}$H − can also be the background. As the signal rate dominantly comes from the t -channel diagram, it becomes larger for larger √$_{s}$$_{, while the main backgrounds except for}$ τ τ ν$_{e}$ν$_{e}$ are smaller because they are dominantly s -channel processes (Fig. 4 ( left )). At √$_{s}$ = 1 TeV, the rate of the signal without cut is already large enough as compared to those of the backgrounds. It is expected that making appropriate kinematic cuts will improve the signal background ratio to a considerable extent. The √$_{s}$ scan will help us to confirm that the signal rate comes from the t -channel diagrams. Fig.4 ( right ) shows the differential cross section of the signal at √$_{s}$ = 1 TeV as a function of cos θ$_{τ}$ − , where θ$_{τ}$ − is the angle between the direction of the outgoing τ − and the beam axis of incident electrons. The distribution of the background from τ τ is asymmetric, so that the angle cut for larger cos θ$_{τ}$ − reduces the backgrounds. 14