Charged Higgs pair production in association with the ${Z}^{0}$ boson at electron-positron colliders

In this paper, the production of the charged Higgs pair associated with the ${Z}^{0}$ boson is analyzed in the minimal extension of the standard model, the so-called two-Higgs-doublet model. The process ${e}^{+}{e}^{-}\to {H}^{+}{H}^{-}{Z}^{0}$ is calculated at the tree level including all the possible diagrams in the two-Higgs-doublet model. The numerical analysis is performed in consideration of the current experimental constraints and various scenarios for the free parameters of the model. The results are presented as a function of the center-of-mass energy, the charged Higgs mass (${m}_{{H}^{±}}$), and the ratio of the vacuum expectation values (${t}_{\beta }$). The unpolarized cross section, taking into account the results in the flavor physics, gets up to 0.278 fb for ${m}_{{H}^{±}}=175\text{}\text{}\mathrm{GeV}$, and it declines with decreasing ${m}_{{H}^{±}}$ in type I. However, it gets down to 0.073 fb for ${m}_{{H}^{±}}=500\text{}\text{}\mathrm{GeV}$ in type II. Further, the calculation is also carried out in the nonalignment scenario and low-${m}_{{h}^{0}}$ scenario. The effect of the polarized incoming ${e}^{+}$ and ${e}^{-}$ beams shows that the cross section is enhanced by a factor up to 2.5 at $P\left(+0.60,-0.80\right)$ polarization configuration. Decay channels of the charged Higgs, possible final states of the process, and some differential distributions belonging to the charged Higgs and ${Z}^{0}$ boson are examined for each scenario. The analysis shows that some channels have a higher branching ratio such as ${H}^{+}\to t\overline{b}$, ${H}^{+}\to {W}^{+}{h}^{0}$, and ${H}^{+}\to {W}^{+}{H}^{0}$. These decay channels are essential in the charged Higgs searches at the lepton colliders for the scenarios interested. The detection of the charged Higgs is a powerful sign for the extended scalar sectors, and the results show the potential of a future lepton collider.