we have considered are allowed by the experimental observations.

![Feynman diagram for the production of $H^{+}_1H^{-}_1$ at $e^{+}e^{-}$ collider. The right panel diagram represents the contribution of the RH neutrinos in the pair-production process. ](ee_delta_delta.pdf){width="4.5in"}

. \[fig:feyn\]

The pair-production of the charged Higgs at LHC is through the s-channel process mediated by $\gamma$, $Z$ and $Z_R$ bosons which gives small production cross-section. In a lepton collider, on the other hand, there is an additional t-channel process mediated by the neutrinos as shown in Fig. \[fig:feyn\]. Owing to the large couplings of the charged singlet with the right-handed leptons and the small masses of the right-handed neutrinos in this model, this t-channel process will be the major pair-production channel. The masses of the right-handed neutrinos used in our analysis were taken as $$M_{N_1} = 17~\text{eV},~M_{N_2} = 6.8~\text{KeV},~ M_{N_3} = 8.2~\text{KeV},$$ for which the values of the Yukawa coupling $\lambda'_R \sim \mathcal{O}(1)$. We thus study the pair production of the charged Higgs at 1 TeV run of the International Linear Collider (ILC) [@ilc] and 3 TeV run by Compact Linear Collider (CLIC) [@clic].

![Production cross section of $H^{+}_1 H^{-}_1$ at $e^{+}e^{-}$ collider for different center-of-mass energies.[]{data-label="fig:prod"}](hc1_hc1_prod.pdf){width="3.3in" height="3.0in"}

We include the relevant vertices 