Probing for chiral $Z^{\prime}$ gauge boson through scattering measurement experiments

We consider a general $U(1)_{X}$ extension of the SM to investigate the chiral scenario introducing three RHNs $(N_{R})$ and an SM-singlet scalar $(\Phi)$ field. Three generations of RHNs are introduced to cancel gauge and mixed gauge-gravity anomalies. The corresponding field content is given in Tab. 1 where general $U(1)_{X}$ charges, being independent of generations, are written as $\tilde{x}_{f}$ before anomaly cancellation and $f$ stands for the three generations of quarks $(q_{L}^{\alpha},u_{R}^{\alpha},d_{R}^{\alpha})$ and leptons $(\ell_{L}^{\alpha},e_{R}^{\alpha},N_{R}^{\alpha})$ respectively where $\alpha(=1,~{}2,~{}3)$ is the generation index. The gauge and mixed gauge-gravity anomaly cancellation conditions in terms of the general charges are written below:

	$\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm SU}(3)_{C}\right]^{2}$	$\displaystyle\ :$	$\displaystyle 2\tilde{x}_{q}-\tilde{x}_{u}-\tilde{x}_{d}$	$\displaystyle\ =\ 0~{},$
	$\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm SU}(2)_{L}\right]^{2}$	$\displaystyle\ :$	$\displaystyle 3\tilde{x}_{q}+\tilde{x}_{\ell}$	$\displaystyle\ =\ 0~{},$
	$\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm U}(1)_{Y}\right]^{2}$	$\displaystyle\ :$	$\displaystyle\tilde{x}_{q}-8\tilde{x}_{u}-2\tilde{x}_{d}+3\tilde{x}_{\ell}-6\tilde{x}_{e}$	$\displaystyle\ =\ 0~{},$
	$\displaystyle\left[{\rm U}(1)_{X}\right]^{2}\otimes{\rm U}(1)_{Y}$	$\displaystyle\ :$	$\displaystyle{\tilde{x}_{q}}^{2}-{\tilde{2x}_{u}}^{2}+{\tilde{x}_{d}}^{2}-{\tilde{x}_{\ell}}^{2}+{\tilde{x}_{e}}^{2}$	$\displaystyle\ =\ 0~{},$
	$\displaystyle\left[{\rm U}(1)_{X}\right]^{3}$	$\displaystyle\ :$	$\displaystyle{6\tilde{x}_{q}}^{3}-{3\tilde{x}_{u}}^{3}-{3\tilde{x}_{d}}^{3}+{2\tilde{x}_{\ell}}^{3}-{\tilde{x}_{\nu}}^{3}-{\tilde{x}_{e}}^{3}$	$\displaystyle\ =\ 0~{},$
	$\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm grav.}\right]^{2}$	$\displaystyle\ :$	$\displaystyle 6\tilde{x}_{q}-3\tilde{x}_{u}-3\tilde{x}_{d}+2\tilde{x}_{\ell}-\tilde{x}_{\nu}-\tilde{x}_{e}$	$\displaystyle\ =\ 0~{},$		(1)

respectively. The Yukawa interactions between the fermions and the scalars $(H,\Phi)$ can be written following the $\mathcal{G}_{\rm SM}\otimes$ $U(1)_{X}$ gauge symmetry as

$${\cal L}^{\rm Yukawa}=-Y_{u}^{\alpha\beta}\overline{q_{L}^{\alpha}}Hu_{R}^{\beta}-Y_{d}^{\alpha\beta}\overline{q_{L}^{\alpha}}\tilde{H}d_{R}^{\beta}-Y_{e}^{\alpha\beta}\overline{\ell_{L}^{\alpha}}\tilde{H}e_{R}^{\beta}-Y_{\nu}^{\alpha\beta}\overline{\ell_{L}^{\alpha}}HN_{R}^{\beta}-\frac{1}{2}Y_{N}^{\alpha}\Phi\overline{(N_{R}^{\alpha})^{c}}N_{R}^{\alpha}+{\rm H.c.}~{},$$

(2)

where $H$ is the SM Higgs doublet and we transform it into $\tilde{H}$ following $i\tau^{2}H^{*}$ where $\tau^{2}$ is the second Pauli matrix. Hence, using Eq. 2 and following charge neutrality, we express the following relations between the general $U(1)_{X}$ charges of the particles as

	$\displaystyle-\frac{1}{2}x_{H}$	$\displaystyle=$	$\displaystyle-\tilde{x}_{q}+\tilde{x}_{u}\ =\ \tilde{x}_{q}-\tilde{x}_{d}\ =\ \tilde{x}_{\ell}-\tilde{x}_{e}=\ -\tilde{x}_{\ell}+\tilde{x}_{\nu},$
	$\displaystyle 2x_{\Phi}$	$\displaystyle=$	$\displaystyle-2\tilde{x}_{\nu}~{}.$		(3)

The general $U(1)_{X}$ charges of the fermions can be obtained solving Eqs. (1) and (3) which finally can be expressed using the scalar charges $x_{H}$ and $x_{\Phi}$ respectively. Simply the anomaly free charge assignment of the general $U(1)_{X}$ can be expressed in terms of a linear combination of two anomaly free scenarios, namely, $U(1)_{Y}$ of the SM and $B-L$ charges. Finally, we find that the left and right handed fremions under the general $U(1)_{X}$ scenario have different charges and hence they interact differently with the neutral, BSM gauge boson $Z^{\prime}$ in the model. We take $x_{\Phi}=1$ without the loss of generality which corresponds to the $U(1)_{B-L}$ and $U(1)_{R}$ scenarios with $x_{H}=0$ [101, 102, 103, 104, 105] and $x_{H}=2$ [106, 107, 108, 109, 110], respectively. $U(1)_{B-L}$ is a vector like scenario where left and right handed fermions of same type are equally charged under the $U(1)$ extension. In case of $U(1)_{R}$ scenario, we find that left handed fermions do not interact with the $Z^{\prime}$.

Using the general form of the $U(1)$ charges of the charged fermions, we notice that for $x_{H}=-1$ the $U(1)$ charge of the right handed electron $(e_{R})$ becomes zero and, as a result, it has no direct interaction with the $Z^{\prime}$ whereas other fermions will interact with the $Z^{\prime}$, manifesting the chiral nature of the model. In similar fashion, we find that for $x_{H}=-0.5$ the general $U(1)_{X}$ charge of the right handed up-type quark $u_{R}$ is zero implying no direct interaction with the $Z^{\prime}$, whereas other fermions will have nonzero general $U(1)_{X}$ charges allowing direct interactions with the $Z^{\prime}$. Similar behavior could be observed when $x_{H}=1$ where general $U(1)_{X}$ charge of right handed down type quark $(d_{R})$ is zero resulting into no direct interaction with the $Z^{\prime}$. Detailed charge assignments for these combinations of $x_{H}$ and $x_{\Phi}$ are given in Tab. 1. We consider two more scenarios with $x_{H}=0.5$ and $2$, setting $x_{\Phi}=1$, where all the charged fermions interact with the $Z^{\prime}$ manifesting the chiral behavior of the model.

The scalar sector of this scenario can be explored by introducing the renormalizable potential of this model and that can be given by

$\displaystyle V\ =\ m_{H}^{2}(H^{\dagger}H)+\lambda_{H}(H^{\dagger}H)^{2}+m_{\Phi}^{2}(\Phi^{\dagger}\Phi)+\lambda_{\Phi}(\Phi^{\dagger}\Phi)^{2}+\lambda_{\rm mix}(H^{\dagger}H)(\Phi^{\dagger}\Phi)~{},$

(4)

where $H$ and $\Phi$ can be separately approximated in the analysis of scalar potential, by taking $\lambda_{\rm mix}$ to be very small. After the breaking of general $U(1)_{X}$ and electroweak symmetries, the scalar fields develop vacuum expectation values (VEVs) as follows

$\displaystyle\braket{H}\ =\ \frac{1}{\sqrt{2}}\begin{pmatrix}v\\ 0\end{pmatrix}~{},\quad{\rm and}\quad\braket{\Phi}\ =\ \frac{v_{\Phi}}{\sqrt{2}}~{},$

(5)

where $v=246$ GeV is marked as the electroweak scale VEV at the potential minimum and $v_{\Phi}$ is a free parameter. After the general $U(1)_{X}$ symmetry is broken, the mass of the BSM neutral gauge boson can be evolved setting $x_{\Phi}=1$ as

$$M_{Z^{\prime}}=2g_{X}v_{\Phi}~{},$$

(6)

in the limit of $v_{\Phi}\gg v$. Here, $g_{X}$ is the general $U(1)_{X}$ coupling and the $Z^{\prime}$ mass is a free parameter.

We consider another scenario where the SM is extended by a general $U(1)_{X}$ gauge group with three generations of RHNs. The field content of the model is given in Tab. 2. The general U$(1)$ charges of the charged fermion fields are same for all generations, $\alpha(=1,2,3)$ where $\alpha$ is the generation index. The general $U(1)_{X}$ charges of the fields are written as $\tilde{x}_{f}$ before anomaly cancellation and $f$ stands for the three generations of quarks $(q_{L}^{\alpha},u_{R}^{\alpha},d_{R}^{\alpha})$ and leptons $(\ell_{L}^{\alpha},e_{R}^{\alpha})$ respectively. We introduce two $SU(2)$ doublet Higgs fields $(H_{1,2})$ where one is the SM like $(H_{1})$ and the other one is the BSM $(H_{2})$ Higgs. Due to different general $U(1)_{X}$ charge assignments $H_{1}$ does not couple with the BSM fermions. We introduce three SM-singlet scalar fields $(\Phi_{1,2,3})$ which are differently charged under general $U(1)_{X}$ gauge group. Due to the general $U(1)_{X}$ gauge symmetry, first two generations of the RHNs have charge $-4$ each and the third generation RHN has $+5$ charge under the general $U(1)_{X}$ gauge group [111]. The RHNs with semi non-universal $U(1)$ charges in this model help to cancel gauge and mixed gauge-gravity anomalies. As the result, we call this model an alternative general $U(1)_{X}$ scenario. Following the gauge and mixed gauge-gravity anomaly cancellation conditions, we relate the general $U(1)_{X}$ charges of the charged fermions as

	$\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm SU}(3)_{C}\right]^{2}$	$\displaystyle\ :$	$\displaystyle 2\tilde{x}_{q}-\tilde{x}_{u}-\tilde{x}_{d}$	$\displaystyle\ =\ 0~{},$
	$\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm SU}(2)_{L}\right]^{2}$	$\displaystyle\ :$	$\displaystyle 3\tilde{x}_{q}+\tilde{x}_{\ell}$	$\displaystyle\ =\ 0~{},$
	$\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm U}(1)_{Y}\right]^{2}$	$\displaystyle\ :$	$\displaystyle\tilde{x}_{q}-8\tilde{x}_{u}-2\tilde{x}_{d}+3\tilde{x_{\ell}}-6\tilde{x_{e}}$	$\displaystyle\ =\ 0~{},$
	$\displaystyle\left[{\rm U}(1)_{X}\right]^{2}\otimes{\rm U}(1)_{Y}$	$\displaystyle\ :$	$\displaystyle{\tilde{x}_{q}}^{2}-{2\tilde{x}_{u}}^{2}+{\tilde{x}_{d}}^{2}-{\tilde{x}_{\ell}}^{2}+{\tilde{x}_{e}}^{2}$	$\displaystyle\ =\ 0~{},$
	$\displaystyle\left[{\rm U}(1)_{X}\right]^{3}$	$\displaystyle\ :$	$\displaystyle 3({6\tilde{x}_{q}}^{3}-{3\tilde{x}_{u}}^{3}-{3\tilde{x}_{d}}^{3}+{2\tilde{x}_{\ell}}^{3}-{\tilde{x}_{e}}^{3})-2\tilde{x}_{\nu}^{3}-\tilde{x}_{\nu}^{\prime^{3}}$	$\displaystyle\ =\ 0~{},$
	$\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm grav.}\right]^{2}$	$\displaystyle\ :$	$\displaystyle 3(6\tilde{x}_{q}-3\tilde{x}_{u}-3\tilde{x}_{d}+2\tilde{x}_{\ell}-\tilde{x}_{e})-2\tilde{x}_{\nu}-\tilde{x}_{\nu}^{\prime}$	$\displaystyle\ =\ 0~{},$		(7)

We find that due to general $U(1)_{X}$ charges the SM charged fermions interact differently with the $Z^{\prime}$, manifesting the chiral nature of the model. The second Higgs doublet $H_{2}$ interacts with the SM lepton doublet $(\ell^{\alpha}_{L})$ and first two generations of the RHNs $(N_{R}^{1,2})$ due to the general $U(1)_{X}$ symmetry. Hence, the Dirac Yukawa mass term for $N_{R}^{1,2}$ can be generated. On the other hand, the corresponding Majorana mass term for $N_{R}^{1,2}$ can be generated from the Dirac Yukawa coupling with $\Phi_{1}$ followed by the general $U(1)_{X}$ symmetry breaking. The third generation of the RHN, $N_{R}^{3}$, has no Dirac Yukawa coupling involving any of the doublet Higgs fields being prohibited by the general $U(1)_{X}$ charge assignments. Therefore, it does not participate in the neutrino mass generation mechanism at the tree level, however, it can have Yukawa interaction with the $\Phi_{2}$ which further generates a Majorana mass term for $N_{R}^{3}$ after the general $U(1)_{X}$ symmetry breaking. Finally, we write the Yukawa interaction among the BSM sector as

$\displaystyle-\mathcal{L}_{\rm int}$

$\displaystyle\ \supset\$

$\displaystyle\sum_{\alpha=1}^{3}\sum_{\beta=1}^{2}Y_{1}^{\alpha\beta}\overline{\ell_{L}^{\alpha}}H_{2}N_{R}^{\beta}+\frac{1}{2}\sum_{\alpha=1}^{2}Y_{2}^{\alpha}\Phi_{1}\overline{(N_{R}^{\alpha})^{c}}N_{R}^{\alpha}+\frac{1}{2}Y_{3}\Phi_{2}\overline{(N_{R}^{3})^{c}}N_{R}^{3}+\rm{H.c.}~{},$

(8)

taking $Y_{2}$ being diagonal without the loss of generality. As in the previous case, we can solve the gauge and mixed gauge-gravity anomalies to estimate the charges of the SM particles in Tab. 2.

The scalar potential of this scenario can be given by

	$\displaystyle V$	$\displaystyle\ =\$	$\displaystyle m_{H_{1}}^{2}(H_{1}^{\dagger}H_{1})+\lambda_{H_{1}}(H_{1}^{\dagger}H_{1})^{2}+m_{H_{2}}^{2}(H_{2}^{\dagger}H_{2})+\lambda_{H_{2}}(H_{2}^{\dagger}H_{2})^{2}$		(9)
			$\displaystyle+m_{\Phi_{1}}^{2}(\Phi_{1}^{\dagger}\Phi_{1})+\lambda_{1}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+m_{\Phi_{2}}^{2}(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}$
			$\displaystyle+m_{\Phi_{3}}^{2}(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{3}(\Phi_{3}^{\dagger}\Phi_{3})^{2}+(\mu\Phi_{3}(H_{1}^{\dagger}H_{2})+{\rm H.c.})$
			$\displaystyle+\lambda_{4}(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{2})+\lambda_{5}(H_{1}^{\dagger}H_{2})(H_{2}^{\dagger}H_{1})+\lambda_{6}(H_{1}^{\dagger}H_{1})(\Phi_{1}^{\dagger}\Phi_{1})$
			$\displaystyle+\lambda_{7}(H_{1}^{\dagger}H_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{8}(H_{1}^{\dagger}H_{2})(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{9}(H_{2}^{\dagger}H_{2})(\Phi_{1}^{\dagger}\Phi_{1})$
			$\displaystyle+\lambda_{10}(H_{1}^{\dagger}H_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{11}(H_{1}^{\dagger}H_{2})(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{12}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})$
			$\displaystyle+\lambda_{13}(\Phi_{2}^{\dagger}\Phi_{2})(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{14}(\Phi_{3}^{\dagger}\Phi_{3})(\Phi_{1}^{\dagger}\Phi_{1})~{}.$

Choosing suitable parametrization for the scalar fields in this scenario to develop their respective VEVs, we can write

$\displaystyle\braket{H_{1}}\ =\ \frac{1}{\sqrt{2}}\left(\begin{array}[]{c}v_{h_{1}}\\ 0\end{array}\right)~{},\;\braket{H_{2}}\ =\ \frac{1}{\sqrt{2}}\left(\begin{array}[]{c}v_{h_{2}}\\ 0\end{array}\right)~{},\braket{\Phi_{1}}\ =\ \frac{v_{1}}{\sqrt{2}}~{},\;\braket{\Phi_{2}}\ =\ \frac{v_{2}}{\sqrt{2}}~{},\;\braket{\Phi_{3}}\ =\ \frac{v_{3}}{\sqrt{2}}~{},~{}~{}~{}~{}$

(14)

with the condition, $\sqrt{v_{h_{1}}^{2}+v_{h_{2}}^{2}}=246~{}{\rm GeV}$. In this alternative general $U(1)_{X}$ extension of the SM, we consider negligibly small scalar quartic couplings among SM scalar doublet fields $H_{1,2}$ and SM-singlet scalar fields $\Phi_{1,2,3}$. As a result, this ensures higher order mixing between the RHNs after the general $U(1)_{X}$ breaking to be very strongly suppressed. In Eq. (9), we may consider $0<m_{\rm mix}^{2}=\mu v_{3}/\sqrt{2}\ll m_{\Phi_{3}}^{2}$ which further leads to $v_{h_{2}}\sim m_{\rm mix}^{2}v_{h_{1}}/m_{\Phi_{3}}^{2}\ll v_{h_{1}}$ [112].

Due to the presence of the general $U(1)_{X}$ gauge symmetry, the doublet scalar sector $H_{1,2}$ and singlet scalar sector $\Phi_{1,2,3}$ interact only through the coupling $\Phi_{3}(H_{1}^{\dagger}H_{2})+{\rm H.c.}$, however, this coupling has no significant effect to determine the VEVs $(v_{1,2,3})$ of the singlet scalar fields $(\Phi_{1,2,3})$, because there is already one collider constraint present in the form of $v_{1}^{2}+v_{2}^{2}+v_{3}^{2}\gg v_{h_{1}}^{2}+v_{h_{2}}^{2}$; one find that $\sqrt{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}$ should be typically larger than around $1$ TeV from various constraints in the light $Z^{\prime}$ case [11] and constraint from dilepton resonance in heavy $Z^{\prime}$ case [113, 114] since the value is related to the $Z^{\prime}$ boson mass and new gauge coupling. Therefore we arrange the parameters of the scalar potential in a way so that the VEVs of $\Phi_{1,2,3}$ will be almost same following $\mu<v_{1}$ whereas $\Phi_{3}$ can be considered as a spurion field which generate the mixing between $H_{1,2}$ in Eq. (9). Once the $\Phi_{3}$ acquires the VEV we get the mixing mass term between $H_{1,2}$ as $m_{\rm mix}=\frac{\mu v_{3}}{\sqrt{2}}$ which resembles the potential of two Higgs doublet model, however, due to the presence of the general $U(1)_{X}$ symmetry the SM-singlet fields $\Phi_{1,2,3}$ do not mix. As a result there are two existing physical Nambu-Goldstone (NG) bosons originating from the SM-singlet scalars. Due to the tiny quartic couplings and gauge couplings, the SM-singlet scalars become decoupled from the SM thermal bath in the early universe. Additionally, we consider that the singlet scalars are heavier than the neutral BSM gauge boson $Z^{\prime}$ preventing its decay into the NG bosons. The breaking of general U(1) gauge symmetry helps $Z^{\prime}$ to acquire the mass which copuld be given by

$\displaystyle M_{Z^{\prime}}=g_{X}\sqrt{64v_{1}^{2}+100v_{2}^{2}+9v_{3}^{2}+\frac{1}{4}x_{H}^{2}v_{h_{1}}^{2}+\left(-\frac{1}{2}x_{H}+3\right)^{2}v_{h_{2}}^{2}}\simeq g_{X}\sqrt{64v_{1}^{2}+100v_{2}^{2}+9v_{3}^{2}}~{}.$

(15)

which is a free parameter and the general $U(1)_{X}$ gauge coupling $g_{X}$ is also a free parameter. Due to the general $U(1)_{X}$ gauge structure $H_{2}$ only couples with $N_{R}^{1,2}$ making this case a neutrinophilic two Higgs Doublet Model (2HDM) framework [112, 115, 116, 117, 118].

After the anomaly cancellation conditions are imposed, we notice that $Z^{\prime}$ in the above models can interact with the left and right handed SM fermions differently manifesting chiral nature of the models. Fixing $x_{\Phi}=1$ in Case-I we find that the chiral nature is the same as the Case-II. Therefore we write the interactions between the fermions with the $Z^{\prime}$ in the following as

$\displaystyle\mathcal{L}_{\rm int}=-g_{X}(\overline{f}\gamma_{\mu}q_{f_{L}}P_{L}f+\overline{f}\gamma_{\mu}q_{f_{R}}P_{R}f)Z_{\mu}^{\prime}~{},$

(16)

where $P_{L(R)}=(1\pm\gamma_{5})/2$, $q_{f_{L}}$ and $q_{f_{R}}$ are the corresponding general U$(1)$ charges of the left handed $(f_{L})$ and right handed $(f_{R})$ fermions. Hence we write vector coupling $(c_{V}=\frac{q_{f_{L}}+q_{f_{R}}}{2})$ and axial vector coupling $(c_{A}=\frac{q_{f_{L}}-q_{f_{R}}}{2})$ for the SM fermions following the charge assignments of the Cases-I and II in Tab. 3 fixing $x_{\Phi}=1$.

Hence, we notice that interactions between SM fermions and $Z^{\prime}$ are chiral in nature in general $U(1)_{X}$ extension of the SM. The partial decay width of $Z^{\prime}$ into different fermions can be calculated using Eq. (16) and we write down the expression as

$\displaystyle\Gamma(Z^{\prime}\to\bar{f}f)$

$\displaystyle=N_{C}\frac{M_{Z^{\prime}}g_{X}^{2}}{24\pi}\left[\left(q_{f_{L}}^{2}+q_{f_{R}}^{2}\right)\left(1-\frac{m_{f}^{2}}{M_{Z^{\prime}}^{2}}\right)+6q_{f_{L}}q_{f_{R}}\frac{m_{f}^{2}}{M_{Z^{\prime}}^{2}}\right]~{},$

(17)

where $m_{f}$ is the mass of different SM fermions and $q_{L,R}$ are the functions of $x_{H}$. Here, $N_{C}$ is the color factor being $1$ for the SM leptons and $3$ for the SM quarks. The light neutrinos $(\nu_{L})$ are considered to be massless due to their tiny mass and putting $q_{f_{R}}=0$ in Eq. 17, we obtain the partial decay width of $Z^{\prime}$ into a pair of one generation light neutrinos as

$\displaystyle\Gamma(Z^{\prime}\to\nu\nu)=\frac{M_{Z^{\prime}}g_{X}^{2}}{24\pi}q_{f_{L}}^{2}~{},$

(18)

were $q_{f_{L}}$ is a function of $x_{H}$. The $Z^{\prime}$ gauge boson can decay into a pair of heavy Majorana neutrinos if $Z^{\prime}$ is heavier than twice the mass of the heavy neutrinos. The corresponding partial decay width into single generation of heavy neutrino pair can be written as

$\displaystyle\Gamma(Z^{\prime}\to N_{R}^{\alpha}N_{R}^{\alpha})=\frac{m_{Z^{\prime}}g_{X}^{2}}{24\pi}q_{N_{R}}^{2}\left(1-4\frac{m_{N}^{2}}{M_{Z^{\prime}}^{2}}\right)^{\frac{3}{2}}~{},$

(19)

with $q_{N_{R}}$ is the general $U(1)_{X}$ charge of the heavy neutrinos which could be found from Tabs. 1 and 2, respectively and $m_{N}$ is the mass of the heavy neutrinos. If we consider that the RHNs are heavier than the half of the $M_{Z^{\prime}}$ then the decay of $Z^{\prime}$ into a pair of RHN is kinematically forbidden. We find that the $U(1)_{X}$ charges of the fermions in Cases-II are same as those in Case-I with $x_{\Phi}=1$. As a result we can utilize same bounds for both the cases.