Natural $Z^{\prime }$-portal Majorana dark matter in alternative U(1) extended standard model

We consider a nonexotic gauged ${\mathrm{U}(1)}_X$ extension of the Standard Model (SM), where the ${\mathrm{U}(1)}_X$ charge of a SM field is given by a linear combination of its hypercharge and baryon-minus-lepton ($B-L$) number. All the gauge and mixed gauge-gravitational anomalies are canceled in this model with the introduction of three right-handed neutrinos (RHNs). Unlike the conventional minimal ${\mathrm{U}(1)}_X$ model, where a universal ${\mathrm{U}(1)}_X$ charge of $-1$ is assigned to three RHNs, we consider an alternative charge assignment, namely, two RHNs ($N_R^{1,2}$) have ${\mathrm{U}(1)}_X$ charge $-4$ while one RHN ($N_R$) has a $+5$ charge. With a minimal extension of the Higgs sector, the three RHNs acquire their Majorana masses associated with ${\mathrm{U}(1)}_X$ symmetry breaking. While $N_R^{1,2}$ have Yukawa coupling with the SM lepton doublets and play an essential role for the “minimal seesaw” mechanism, $N_R$ is isolated from the SM particles due to its ${\mathrm{U}(1)}_X$ charge and hence it is a natural candidate for the dark matter (DM) without invoking additional symmetries. In this model context, we investigate the $Z^{\prime }$-portal RHN DM scenario, where the RHN DM communicates with the SM particles through the ${\mathrm{U}(1)}_X$ gauge boson ($Z^{\prime }$ boson). We identify a narrow parameter space by combining the constraints from the observed DM relic abundance, the results of the search for a $Z^{\prime }$ boson resonance at the Large Hadron Collider Run-2, and the gauge coupling perturbativity up to the Planck/grand unification scale. A special choice of ${\mathrm{U}(1)}_X$ charges for the SM fields allows us to extend the model to $\mathrm{SU}(5)\times \mathrm{U}(1{)}_X$ grand unification. In this scenario, the model parameter space is more severely constrained, which will be explored at future high energy collider experiments.

Figure 1

The DM relic abundance as a function of the DM mass ($m_{\mathrm{DM}}$) for a fixed $m_{Z^{\prime }}=3\mathrm{TeV}$ and $x_H=-0.8$. The solid line from top to bottom correspond to the resultant relic abundances for ${\alpha }_X=2.0\times {10}^{-5}$, $5.06\times {10}^{-5}$, $5.0\times {10}^{-4}$, and $2.0\times {10}^{-3}$, respectively. The two horizontal dashed lines indicate the range of the observed DM relic density, $0.1183\le {\mathrm{\Omega }}_{\mathrm{DM}}{h}^2\le 0.1213$.

Figure 2

The lower bounds on ${\alpha }_X$ as a function of $m_{Z^{\prime }}$ for various values of $x_H$, in order to reproduce the observed DM relic abundance. The solid, dashed, and dotted lines correspond to $x_H=2$, 0 and $-1.2$, respectively. The dashed and dotted lines are well overlapping.

Figure 3

Allowed parameter regions for fixed $x_H$ values in (${\alpha }_X$, $m_{Z^{\prime }}$)-plane. The solid lines are the cosmological lower bounds on ${\alpha }_X$ as a function of $m_{Z^{\prime }}$. The (red) dashed and (red) dot-dashed lines are the upper bounds on ${\alpha }_X$ from the LHC and LEP results, respectively. The perturbativity bounds on ${\alpha }_X$ are depicted by the dotted lines. The (green) shaded regions satisfy all the constrains. The top-left (right) panel shows the result for a fixed $x_H=-1.2$ ($x_H=0$), and the bottom panel shows the result for $x_H=2$. The LEP bound for $x_H=-1.2$ lies outside the range shown in the plot.

Figure 4

Allowed parameter region [(green) shaded] for fixed $m_{Z^{\prime }}=3\mathrm{TeV}$ in (${\alpha }_X$, $x_H$)-plane. The solid line is the cosmological lower bound on ${\alpha }_X$ as a function of $x_H$, while the (red) dashed line is the upper bound on ${\alpha }_X$ obtained from the LHC Run-2 results. The dotted line denotes the perturbativity bound. The LEP bound lies outside of the range shown in this figure.

Figure 5

RG running of the SM gauge couplings (left) and the SM Higgs quartic coupling (right). RG equations employed in our analysis are listed in the Appendix.

Figure 6

Allowed parameter region ((green) shaded) for the $\mathrm{SU}(5)\times \mathrm{U}(1{)}_X$ grand unification with $Z^{\prime }$-portal RHN DM. The line codings are same as in Fig. 3.