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 ($m_{H^{\pm }}$), 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 $m_{H^{\pm }}=175\mathrm{GeV}$, and it declines with decreasing $m_{H^{\pm }}$ in type I. However, it gets down to 0.073 fb for $m_{H^{\pm }}=500\mathrm{GeV}$ 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 $\mathrm{P}(+0.60,-0.80)$ 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.

Figure 1

All the Feynman diagrams that contribute to the scattering process $e^{+}{e}^{-}\to H^{+}{H}^{-}{Z}^0$ at the tree level. The dashed-wavelike line represents the nature of the propagating particle; vector boson ($\gamma /Z$-boson) or scalar particle ($h^0/H^0/A^0$). The bold $\mathbf{H}$ represents any of the $h^0/H^0/A^0$ bosons.

Figure 2

Comparison among various polarization cases for two benchmark points in the nonalignment scenario. The polarization configurations are depicted in the figure. (left): $c_{\beta \alpha }=0.1$, $t_{\beta }=45$, $m_{H^0}=200\mathrm{GeV}$, and type I is set. ${\sigma }_{LR}$, ${\sigma }_{RL}$, and ${\sigma }_{UU}$ represent the possible helicity configuration of the incoming positron and electron beams. (right): The same caption as the left figure, but $c_{\beta \alpha }=0.01{(\frac{150}{m_{H^0}})}^2$, $m_{H^0}=425\mathrm{GeV}$, and type II is assumed.

Figure 3

The cross section is given for the unpolarized incoming beams as a function of the $m_{H^{\pm }}$ and $\sqrt{s}$ in the non-alignment scenario. The scan is done for $t_{\beta }=45$ and $m_{H^{\pm }}$ is calculated by varying the $m_{H^0}$. (left): $c_{\beta \alpha }=0.1$ and Type-I couplings, (right): $c_{\beta \alpha }=0.01{(\frac{150}{m_{H^0}})}^2$ and Type-II couplings are considered.

Figure 4

The unpolarized cross section at $\sqrt{s}=2.5\mathrm{TeV}$ as a function of the $m_{H^{\pm }}$ and $t_{\beta }$ in the nonalignment scenario; (left): Type-I Yukawa couplings with $c_{\beta \alpha }=0.1$, (right): Type-II Yukawa couplings with $c_{\beta \alpha }=0.01{(\frac{150}{m_H})}^2$ are set.

Figure 5

The cross section distributions in the low-$m_H$ scenario, in which $m_{h^0}=95\mathrm{GeV}$. (left): The process as a function of $\sqrt{s}$ is plotted for various polarizations depicted in the plot. (right): The cross section as a a function of light Higgs mass ($m_{h^0}$) and $\sqrt{s}$.

Figure 6

The cross section distributions are plotted in favor of the current experiments in the flavor physics. (left): The cross section of the process as a function of $\sqrt{s}$ for various polarized beam configurations is plotted, where the type-I Yukawa coupling scheme is set, $m_H=175\mathrm{GeV}$ and $t_{\beta }=10$. (right): The same caption as the left figure, but type II and $m_H=500\mathrm{GeV}$ are set.

Figure 7

The cross section distributions for the last scenario called the favored region in the recent experimental constraints. The distributions are at $\sqrt{s}=3\mathrm{TeV}$, $s_{\beta \alpha }=1$, and all the extra Higgs masses are taken as $m_H=m_{H^0}=m_{A^0}=m_{H^{\pm }}$. (left): Type-I Yukawa couplings. (right): Type-II Yukawa couplings are set.

Figure 8

Distributions are plotted for the branching ratios of the charged Higgs boson and $CP$-even $H^0$ boson in the nonalignment scenario. Decay channels are indicated in each figure. All the extra Higgs masses are taken as $m_H=m_{A^0}=m_{H^{\pm }}$, and $m_{H^0}$ is calculated with the help of Eq. (12). Type-I Yukawa couplings are assumed, and $c_{\beta \alpha }=0.1$ is set. (first row): Decays of $H^{\pm }$ are plotted. (second row): Decays of the $CP$-even $H^0$ boson are plotted.

Figure 9

Distributions for the branching ratios of the charged Higgs boson in the nonalignment scenario. Decay channels are indicated in each figure. The Higgs masses are taken as $m_H=m_{A^0}=m_{H^{\pm }}$, and $m_{H^0}$ is calculated with the help of Eq. (12). The calculation is carried out for $c_{\beta \alpha }=0.01{(\frac{150\mathrm{GeV}}{m_{H^0}})}^2$, and type-II Yukawa couplings are assumed.

Figure 10

Distributions for the branching ratios of the charged Higgs boson in the low-$m_H$ scenario. Distributions are given for $c_{\beta \alpha }=1$, $m_{H^0}=125\mathrm{GeV}$, and accordingly$m_{A^0}=m_{H^{\pm }}$ are calculated with the help of Eq. (12). (left): Type-I Yukawa couplings are assumed. (right): Type-II Yukawa couplings are assumed.

Figure 11

Distributions for the branching ratios of the charged Higgs boson in the region favored by the recent experimental constraints. Distributions are for $s_{\beta \alpha }=1$, and all the extra Higgs masses are taken as $m_H=m_{H^0}=m_{A^0}=m_{H^{\pm }}$. (left): Type-I Yukawa couplings are set. (right): Type-II Yukawa couplings are set.

Figure 12

The differential cross section as a function of the kinematical properties of $Z$ and $H^{\pm }$ bosons where $\sqrt{s}=3\mathrm{TeV}$, and the rest of the parameters are indicated in the text for each scenario. (left): $d\sigma /d|y_Z|$ distribution. (center): $d\sigma /d{p}_T^Z$ distribution. (right): $d\sigma /d|\mathrm{\Delta }{y}_{H^{-}{H}^{+}}|$ distribution.