B. Electron-electron collisions As already stated, the ILC has a further advantage to test radiative seesaw models via the experiment at the e $^{−}$e− collision option, where dimension five operator of e $^{−}$e$^{−}$φ $^{+}$φ$^{+}$, which is the sub-diagram of the loop diagrams for neutrino mass matrix. This direct test of the dimension five operator is essential to identify the radiative seesaw models. The Majorana nature in the Zee-Babu model is in the lepton number violating coupling constant µ of k $^{++}$ω $^{−}$ω$^{−}$, which generates the dimension five operator of e $^{−}$e$^{−}$ω $^{+}$ω+ at the tree level via the s -channel k −− exchange diagram. The cross section of e $^{−}$e− → ω $^{−}$ω− is given by where the total width Γ$_{k}$ of k ±± is computed as about 168 GeV in our scenario in Eq.(11). On the other hand, in the Ma model and the AKS model, the operator comes from the t -channel right-handed neutrino exchange diagram. The cross section is evaluated as t min 128 πs α =1 t − m N α R u − m N α R ∣$^{2}$ where n is the number of generation of right-handed neutrinos, φ − represents the Z$_{2}$ -odd charged scalar boson ξ − in the Ma model and S − in the AKS model. The constants c α represent ˆ h α e or h α e in the Ma model or the AKS model, respectively. We note that due to the Majorana nature of the t -channel diagram, we obtain much larger cross section in the e $^{−}$e− collision than in the e $^{+}$e − collision in each model assuming the same collision energy. The mass matrix of left-handed neutrinos is generated at the one, two and three loop levels in the Ma model, the Zee-Babu model and the AKS model, respectively. Therefore, the coupling constants can be basically hierarchical among the models, so are the cross sections. For the typical scenarios in these models, the cross sections are shown in Fig 5. The rate of ω $^{−}$ω− in the Zee-Babu model can be larger than several times 100 fb for 800 GeV < ∼ √$_{s}$ < ∼ 1 . 5 TeV. It becomes maximal (several times pb) at √$_{s}$ ∼ m$_{k}$ , and above that asymptotically reduces by 1 /s . The maximal value of the cross section is sensitive to the value of g$_{ee}$ and µ . In the parameter sets where these coupling constants are smaller the cross section becomes smaller. The signal should be like-sign dilepton with a missing energy. On the other hand, in the Ma model, production cross sections of e $^{−}$e− → ξ $^{−}$ξ− are smaller 15 σ ( e $^{−}$e− → φ $^{−}$φ$^{−}$) = ∫ t max t min dt 1 128 πs ∣ n ∑ α =1 ( | c $^{α}$| $^{2}$m$_{N}$ α $_{R}$) ( 1 t − m 2 N α R + 1 u − m 2 N α R )∣ ∣ , (15) σ ( e $^{−}$e− → ω $^{−}$ω$^{−}$) = 1 8 π √ 1 − 4 m 2 ω s µ $^{2}$g2 ee ( s − m 2 $_{k}$) 2 + m 2 $_{k}$Γ 2 k , (14)