$Z^{\prime }$-portal right-handed neutrino dark matter in the minimal ${\mathrm{U}(1)}_X$ extended Standard Model

We consider a concise dark matter (DM) scenario in the context of a nonexotic U(1) extension of the Standard Model (SM), where a new ${\mathrm{U}(1)}_X$ gauge symmetry is introduced along with three generations of right-handed neutrinos (RHNs) and a SM gauge singlet Higgs field. The model is a generalization of the minimal gauged $\mathrm{U}(1{)}_{B-L}$ (baryon number minus lepton number) extension of the SM, in which the extra ${\mathrm{U}(1)}_X$ gauge symmetry is expressed as a linear combination of the SM ${\mathrm{U}(1)}_Y$ and $\mathrm{U}(1{)}_{B-L}$ gauge symmetries. We introduce a $Z_2$-parity and assign an odd-parity only for one RHN among all particles, so that this $Z_2$-odd RHN plays the role of DM. The so-called minimal seesaw mechanism is implemented in this model with only two $Z_2$-even RHNs. In this context, we investigate the physics of RHN DM, focusing on the case where this DM particle communicates with the SM particles through the ${\mathrm{U}(1)}_X$ gauge boson ($Z^{\prime }$ boson). This “$Z^{\prime }$-portal RHN DM” scenario is controlled by only three free parameters: the ${\mathrm{U}(1)}_X$ gauge coupling (${\alpha }_X$), the $Z^{\prime }$ boson mass ($m_{Z^{\prime }}$), and the ${\mathrm{U}(1)}_X$ charge of the SM Higgs doublet ($x_H$). We consider various phenomenological constraints to identify a phenomenologically viable parameter space. The most important constraints are the observed DM relic abundance and the latest LHC Run-2 results on the search for a narrow resonance with the dilepton final state. We find that these are complementary with each other and narrow the allowed parameter region, leading to the lower mass bound of $m_{Z^{\prime }}\gtrsim 2.7\mathrm{TeV}$.

Figure 1

The relic abundance of the $Z^{\prime }$-portal RHN DM as a function of its mass ($m_{\mathrm{DM}}$) for $m_{Z^{\prime }}=4\mathrm{TeV}$. In the left panel, we have fixed $x_H=0$ (the minimal $B-L$ model limit) and shown the relic abundance for various values of the gauge coupling, ${\alpha }_X=0.025$, 0.027, 0.028, and 0.030 (solid lines from top to bottom). In the right panel, we have fixed ${\alpha }_X=0.027$ and shown the relic abundance for various values of $x_H=-0.8$, 0, 0.5, and 1.0 (solid lines from bottom to top). The two horizontal lines denote the range of the observed DM relic density, $0.1183\le {\mathrm{\Omega }}_{\mathrm{DM}}{h}^2\le 0.1213$, in Eq. (5).

Figure 2

The lower bounds on ${\alpha }_X$ as a function of $m_{Z^{\prime }}$ for various values of $x_H$, to satisfy the cosmological constraint of $0.1183\le {\mathrm{\Omega }}_{\mathrm{DM}}{h}^2\le 0.1213$. The solid lines from top to bottom correspond to $x_H=-3$, $+1$, $-2$, 0, and $-1$, respectively. As the input $x_H$ value is going away from the point of $x_H=-0.8$, the lower bound on ${\alpha }_X$ is increasing.

Figure 3

Left: The cross section as a function of the $Z_{SSM}^{\prime }$ mass (solid line) with $k=1.28$, along with the ATLAS results in 2016 [51] and 2015 [61] from the combined dielectron and dimuon channels. Right: The cross sections calculated for various values of ${\alpha }_X$ with $k=1.28$, for the minimal $B-L$ model limit ($x_H=0$). The solid lines from left to right correspond to ${\alpha }_X={10}^{-5}$, ${10}^{-4.5}$, ${10}^{-4}$, ${10}^{-3.5}$, ${10}^{-3}$, ${10}^{-2.5}$, ${10}^{-2}$, and ${10}^{-1.5}$, respectively.

Figure 4

Left: The cross-section ratio as a function of the $Z_{SSM}^{\prime }$ mass (solid line) with $k=1.61$, along with the CMS results from 2015 [62] and 2016 [52] from the combined dielectron and dimuon channels. Right: The cross section ratios calculated for various values of ${\alpha }_X$ with $k=1.61$ for $x_H=0$. The solid lines from left to right correspond to ${\alpha }_X={10}^{-4.5}$, ${10}^{-4}$, ${10}^{-3.5}$, ${10}^{-3}$, ${10}^{-2.5}$, ${10}^{-2}$, and ${10}^{-1.75}$, respectively.

Figure 5

The lower bound on $m_{Z^{\prime }}/g_X$ as a function of $x_H$. We have employed the final LEP 2 data [55] at the 95% confidence level.

Figure 6

Left: The upper bounds on ${\alpha }_X$ as a function of $m_{Z^{\prime }}$ for $x_H=-1$, 0, and $+1$ from top to bottom, respectively, for the solid and dashed lines. The solid lines denote the bounds from the ATLAS results [51] while the dashed lines denote the bounds from the CMS results [52]. Right: The upper bounds on ${\alpha }_X$ after combining the ATLAS and CMS results shown in the left panel. The solid lines correspond to the combined upper bounds for $x_H=-1$, 0, and $+1$ from top to bottom, respectively. The perturbativity bounds of Eq. (17) for $x_H=-1$, 0, and $+1$ are shown as the horizontal dashed-dotted lines from top to bottom, respectively.

Figure 7

Allowed parameter region for the $Z^{\prime }$-portal RHN DM scenario. The top-left panel shows the results for the minimal $B-L$ model limit ($x_H=0$). The (black) solid line denotes the lower bound on ${\alpha }_X$ as a function of $m_{Z^{\prime }}$ to reproduce the observed DM relic abundance. The lower dashed line (in red) shows the upper bound on ${\alpha }_X$ obtained from the search results for $Z^{\prime }$-boson resonance at the LHC. The shaded region is the final result after combining the cosmological and the LHC constraints, leading to the lower mass bound of $m_{Z^{\prime }}\gtrsim 3.6\mathrm{TeV}$. For a comparison, we have also shown the upper long-dashed line (in red) obtained in Ref. [40] by using the LHC results in 2015. The LEP upper bound in Eq. (16) is depicted as the dotted line. We also show the perturbativity bound on ${\alpha }_X$ as the dashed-dotted line. The top-right, bottom-left, and bottom-right panels are the same as the top-left panel, but $x_H=-1$, $-2$, and $+1$, respectively.

Figure 8

Allowed parameter region for the $Z^{\prime }$-portal RHN DM scenario for $m_{Z^{\prime }}=4\mathrm{TeV}$. The (black) solid line shows the cosmological lower bound on ${\alpha }_X$ as a function of $x_H$. The dashed line (in red) shows the upper bound on ${\alpha }_X$ obtained from the combined ATLAS and CMS bounds. The shaded region is the final result for the allowed parameter space after combining the cosmological and the LHC constraints, leading to the allowed range of $-2.1\le x_H\le 0.3$. The LEP bound appears above the plot range. The dashed-dotted line denotes the theoretical upper bound on ${\alpha }_X$ in Eq. (17).