Right-handed neutrino dark matter in the classically conformal U(1)' extended Standard Model

TL;DR

This paper explores right-handed neutrino dark matter within a classically conformal U(1)' extended Standard Model, analyzing constraints from vacuum stability, collider bounds, and relic density, and finds viable parameter space with suppressed direct detection signals.

Contribution

It introduces a Z2-parity stabilized right-handed neutrino dark matter candidate in a conformal U(1)' model, constrained by multiple experimental and theoretical considerations.

Findings

Allowed Z' mass is above ~3.5 TeV.

RHN DM annihilates mainly via Z' portal.

Direct detection cross section is below current bounds.

Abstract

We consider the dark matter (DM) scenario in the context of the classically conformal U(1)' extended standard model (SM), with three right-handed neutrinos (RHNs) and the U(1)' Higgs field. The model is free from all the U(1)' gauge and gravitational anomalies in the presence of the three RHNs. We introduce a $Z_2$-parity in the model, under which an odd-parity is assigned to one RHN, while all the other particles is assigned to be $Z_2$-even, and hence the $Z_2$-odd RHN serves as a DM candidate. In this model, the U(1)' gauge symmetry is radiatively broken through the Coleman-Weinberg mechanism, by which the electroweak symmetry breaking is triggered. There are three free parameters in our model, the U(1)' charge of the SM Higgs doublet ($x_H$), the new U(1)' gauge coupling ($g_X$), and the U(1)' gauge boson ($Z'$) mass ($m_{Z'}$), which are severely constrained in order to solve the electroweak vacuum instability problem, and satisfy the LHC Run-2 bounds from the search for $Z'$ boson resonance. In addition to these constraints, we investigate the RHN DM physics. Because of the nature of classical conformality, we find that a RHN DM pair mainly annihilates into the SM particles through the $Z'$ boson exchange. This is the so-called $Z'$-portal DM scenario. Combining the electroweak vacuum stability condition, the LHC Run-2 bounds, and the cosmological constraint from the observed DM relic density, we find that all constrains complementarily work to narrow down the allowed parameter regions, and, especially, exclude $m_{Z'} \lesssim 3.5$ TeV. For the obtained allowed regions, we calculate the spin-independent cross section of the RHN DM with nucleons. We find that the resultant cross section well below the current experimental upper bounds.

Tables

Table 1. Table 1 : Particle contents of the U(1) ′ extended SM with Z 2 subscript 𝑍 2 Z_{2} parity. In addition to the SM particle contents, three generations of RHNs ν R i superscript subscript 𝜈 𝑅 𝑖 \nu_{R}^{i} ( i = 1 , 2 , 3 𝑖 1 2 3 i=1,2,3 denotes the generation index) and U(1) ′ Higgs field Φ Φ \Phi are introduced. Under Z 2 subscript 𝑍 2 Z_{2} parity, the only one RHN ν R 3 superscript subscript 𝜈 𝑅 3 \nu_{R}^{3} is odd, while the other particles, including ν R 1 superscript subscript 𝜈 𝑅 1 \nu_{R}^{1} and ν R 2 superscript subscript 𝜈 𝑅 2 \nu_{R}^{2} , are even.

	SU(3)c	SU(2)L	U(1)Y	U(1)′			$Z_2$
$q_L^i$	3	2	$+1/6$	$x_q$	=	$\frac{1}{3}{x}_H+\frac{1}{6}{x}_{\Phi }$	+
$u_R^i$	3	1	$+2/3$	$x_u$	=	$\frac{4}{3}{x}_H+\frac{1}{6}{x}_{\Phi }$	+
$d_R^i$	3	1	$-1/3$	$x_d$	=	$-\frac{2}{3}{x}_H+\frac{1}{6}{x}_{\Phi }$	+
${\ell }_L^i$	1	2	$-1/2$	$x_{\ell }$	=	$-x_H-\frac{1}{2}{x}_{\Phi }$	+
${\nu }_R^{1,2}$	1	1	$0$	$x_{\nu }$	=	$-\frac{1}{2}{x}_{\Phi }$	+
${\nu }_R^3$	1	1	$0$	$x_{\nu }$	=	$-\frac{1}{2}{x}_{\Phi }$	$-$
$e_R^i$	1	1	$-1$	$x_e$	=	$-2x_H-\frac{1}{2}{x}_{\Phi }$	+
$H$	1	2	$+1/2$	$x_H$	=	$x_H$	+
$\Phi$	1	1	$0$	$x_{\Phi }$	=	$x_{\Phi }$	+

Equations

$$D_{\mu}\equiv\partial_{\mu}-i\left(\begin{array}[]{cc}Y_{1}&Y_{X}\end{array}\right)\left(\begin{array}[]{cc}g_{1}&g_{1X}\\ g_{X1}&g_{X}\end{array}\right)\left(\begin{array}[]{c}B_{\mu}\\ B_{\mu}^{\prime}\end{array}\right),$$

$${\mathcal{L}}_{\mathrm{Yukawa}}$$

$$V={\lambda }_H{\left(H^{†}H\right)}^2+{\lambda }_{\Phi }{\left({\Phi }^{†}\Phi \right)}^2+{\lambda }_{\mathrm{mix}}\left(H^{†}H\right)\left({\Phi }^{†}\Phi \right),$$

$$V(\varphi )=\frac{{\lambda }_{\Phi }}{4}{\varphi }^4+\frac{{\beta }_{\Phi }}{8}{\varphi }^4\left(\ln \left[\frac{{\varphi }^2}{v_{\varphi }^2}\right]-\frac{25}{6}\right),$$

$${\beta }_{\Phi }$$

$${\lambda }_{\Phi }=\frac{11}{6}{\beta }_{\Phi },$$

$${\lambda }_{\Phi }=\frac{1}{3!}{\frac{d^{4}V(\varphi )}{d{\varphi }^4}|}_{\varphi =v_{\varphi }}.$$

$$m_{Z^{\prime }}=\sqrt{(x_{\Phi }{g}_X{v}_{\varphi }{)}^2+(x_H{g}_X{v}_h{)}^2}\simeq x_{\Phi }{g}_X{v}_{\varphi },m_{N^{1,2}}=\sqrt{2}{Y}_M^{1,2}{v}_{\varphi },m_{\mathrm{DM}}=\sqrt{2}{Y}_M^3{v}_{\varphi },$$

$$m_{\varphi }^2$$

$$V(h)=\frac{{\lambda }_H}{4}{h}^4+\frac{{\lambda }_{\mathrm{mix}}}{4}{v}_{\varphi }^2{h}^2,$$

$$m_h^2={\frac{d^{2}V}{d{h}^2}|}_{h=v_h}=|{\lambda }_{\mathrm{mix}}|v_{\varphi }^2=2{\lambda }_H{v}_h^2.$$

$${\lambda }_{\Phi }=\frac{11}{6}{\beta }_{\Phi }+\frac{|{\lambda }_{\mathrm{mix}}|}{2}{\left(\frac{v_h}{v_{\varphi }}\right)}^2\simeq \frac{1}{2v_{\varphi }^4}\left(\frac{11}{8{\pi }^2}{m}_{Z^{\prime }}^4+m_h^2{v}_h^2\right).$$

$$V_{\mathrm{eff}}\supset \frac{{\lambda }_{\mathrm{mix}}}{4}{h}^2{\varphi }^2+\frac{{\beta }_{{\lambda }_{\mathrm{mix}}}}{8}{h}^2{\varphi }^2\left(\ln \left[{\varphi }^2\right]+C\right),$$

$${\beta }_{{\lambda }_{\mathrm{mix}}}$$

$$\displaystyle\frac{\partial^{4}V_{\rm eff}}{\partial h^{2}\partial\phi^{2}}\Big{|}_{h=0,\phi=v_{\phi}}=\lambda_{\rm mix},$$

$$V_{\mathrm{eff}}\supset \frac{{\lambda }_{\mathrm{mix}}}{4}{h}^2{\varphi }^2+\frac{{\beta }_{{\lambda }_{\mathrm{mix}}}}{8}{h}^2{\varphi }^2\left(\ln \left[\frac{{\varphi }^2}{v_{\varphi }}\right]-3\right).$$

$$\Delta m_h^2$$

$$\delta =\frac{m_h^2}{2|\Delta m_h^2|}.$$

$${\Omega }_{\mathrm{DM}}{h}^2$$

$$\frac{dY}{dx}$$

$$s$$

$$H(m_{\mathrm{DM}})$$

$$s{Y}_{\mathrm{EQ}}$$

$$⟨\sigma v⟩$$

$$\sigma (s)$$

$${\Gamma }_{Z^{\prime }}$$

$${\Omega }_{\mathrm{DM}}{h}^2$$

$$\frac{d\sigma }{d{M}_{\ell \ell }}=\sum _{a,b}{\int }_{\frac{M_{\ell \ell }^2}{E_{\mathrm{CM}}^2}}^{1}d{x}_1\frac{2M_{\ell \ell }}{x_1{E}_{\mathrm{CM}}^2}{f}_a(x_1,M_{\ell \ell }^2)f_b\left(\frac{M_{\ell \ell }^2}{x_1{E}_{\mathrm{CM}}^2},M_{\ell \ell }^2\right)\hat{\sigma }(\bar{q}q\to Z^{\prime }\to {\ell }^{+}{\ell }^{-}),$$

$$\hat{\sigma }(\bar{u}u\to Z^{\prime }\to {\ell }^{+}{\ell }^{-})$$

$$\hat{\sigma }(\bar{d}d\to Z^{\prime }\to {\ell }^{+}{\ell }^{-})$$

$$R_{\sigma }$$

$$v_{\varphi }^2$$

Full text

**Right-handed neutrino dark matter in the classically conformal U(1)′ extended Standard Model **

Satsuki Oda${}^{~{}a,b}$ 111satsuki.oda@oist.jp, Nobuchika Okada${}^{~{}c}$ 222okadan@ua.edu, and Dai-suke Takahashi${}^{~{}a,b}$ 333daisuke.takahashi@oist.jp

*a**Okinawa Institute of Science and Technology Graduate University (OIST),

Onna, Okinawa 904-0495, Japan*

*b**Research Institute, Meio University,

Nago, Okinawa 905-8585, Japan

*c**Department of Physics and Astronomy, University of Alabama,

Tuscaloosa, Alabama 35487, USA *

We consider the dark matter (DM) scenario in the context of the classically conformal U(1)′ extended standard model (SM), with three right-handed neutrinos (RHNs) and the U(1)′ Higgs field. The model is free from all the U(1)′ gauge and gravitational anomalies in the presence of the three RHNs. We introduce a $Z_{2}$-parity in the model, under which an odd-parity is assigned to one RHN, while all the other particles is assigned to be $Z_{2}$-even, and hence the $Z_{2}$-odd RHN serves as a DM candidate. In this model, the U(1)′ gauge symmetry is radiatively broken through the Coleman-Weinberg mechanism, by which the electroweak symmetry breaking is triggered. There are three free parameters in our model, the U(1)′ charge of the SM Higgs doublet ($x_{H}$), the new U(1)′ gauge coupling ($g_{X}$), and the U(1)′ gauge boson ($Z^{\prime}$) mass ($m_{Z^{\prime}}$), which are severely constrained in order to solve the electroweak vacuum instability problem, and satisfy the LHC Run-2 bounds from the search for $Z^{\prime}$ boson resonance. In addition to these constraints, we investigate the RHN DM physics. Because of the nature of classical conformality, we find that a RHN DM pair mainly annihilates into the SM particles through the $Z^{\prime}$ boson exchange. This is the so-called $Z^{\prime}$-portal DM scenario. Combining the electroweak vacuum stability condition, the LHC Run-2 bounds, and the cosmological constraint from the observed DM relic density, we find that all constrains complementarily work to narrow down the allowed parameter regions, and, especially, exclude $m_{Z^{\prime}}\lesssim 3.5$ TeV. For the obtained allowed regions, we calculate the spin-independent cross section of the RHN DM with nucleons. We find that the resultant cross section well below the current experimental upper bounds.

1 Introduction

There are important missing pieces in the Standard Model (SM), for example, a candidate for the dark matter (DM), and tiny neutrino masses and their flavor mixings. The SM should be extended so as to supplement these missing pieces. The so-called seesaw mechanism is a natural way to reproduce the tiny neutrino masses [1, 2, 3, 4, 5], where heavy Majorana right-handed neutrinos (RHNs) are introduced. The minimal gauged $B-L$ model [6, 7, 8, 9, 10, 11] is one of the simplest extensions of the SM with an extra gauge symmetry, in which the accidentally anomaly-free global $B-L$ (baryon number minus lepton number) in the SM is gauged. Three RHNs play an essential roll to cancel the gauge and gravitational anomalies of the model. Associated with the $B-L$ symmetry breaking, the RHNs acquire their Majorana masses, and hence the seesaw mechanism is automatically implemented. The minimal $B-L$ model can be generalized to the so-called minimal U(1)′ model [12]. Here, the U(1)′ gauge group is defined as a linear combination of the U(1)B-L and the SM U(1)Y gauge groups, so that the U(1)′ model is anomaly-free.

In our previous work [13, 14], we have investigated the minimal U(1)′ model with classically conformal invariance.444 See Refs. [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52] for recent work on new physics models with classically conformal invariance. In this model, the U(1)′ gauge symmetry is radiatively broken through the Coleman-Weinberg (CW) mechanism [53]. Given a negative mixing quartic coupling between the SM Higgs and the U(1)′ Higgs fields, once the U(1)′ Higgs field develops a vacuum expectation value (VEV), a negative mass squared of the SM Higgs doublet is generated, and thus the electroweak symmetry breaking is naturally triggered. In this model context, we have investigated the electroweak vacuum instability problem in the SM. Employing the renormalization group (RG) equations at the two-loop level and the central values for the world average masses of the top quark ($m_{t}=173.34$ GeV [54]) and the Higgs boson ($m_{h}=125.09$ GeV [55]), we have performed parameter scans to identify the parameter region for resolving the electroweak vacuum instability problem. We have also investigated the ATLAS and CMS search limits at the LHC Run-2 (2015) for the U(1)′ gauge boson ($Z^{\prime}$) [56, 57], and identified the allowed parameter regions in our model. Combining the constraints from the electroweak vacuum stability and the LHC Run-2 results, we have found a lower bound on the $Z^{\prime}$ boson mass. We also have calculated self-energy corrections to the SM Higgs doublet field through the heavy states, the right-handed neutrinos and the $Z^{\prime}$ boson, and have found the naturalness bound as $m_{Z^{\prime}}\lesssim 6$ TeV, in order to reproduce the right electroweak scale for the fine-tuning level better than 10%.

The so-called weakly interacting massive particle (WIMP) is one of the most promising candidates of the DM in our Universe, which is in thermal equilibrium in the early Universe. Among many possibilities, a simple way to introduce a WIMP DM in the minimal U(1)′ model has been proposed in [58] (see also [59]), where $Z_{2}$-parity is introduced and an odd-parity is assigned to one RHN, while all the other particles is assigned to be $Z_{2}$-even. We adapt this scheme in our minimal U(1)′ model with the classically conformal invariance, and the $Z_{2}$-odd RHN is a DM candidate, while the other two RHNs are utilized for the seesaw mechanism. Note that only two RHNs are sufficient to reproduce the neutrino oscillation data, and the observed baryon asymmetry in the Universe through leptogenesis [60]. This system is called the minimal seesaw [61, 62]. In our model, there are two ways for the RHN DM to interact with the SM particles. One is mediated by the $Z^{\prime}$ boson ($Z^{\prime}$-portal) and the other is by the two Higgs bosons (Higgs portal) which are two mass eigenstates consisting of the SM Higgs and the U(1)′ Higgs bosons. Recently, the $Z^{\prime}$-portal DM scenarios [63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91] have been intensively investigated, while the Higgs portal RHN DM scenarios [58, 92, 93] have been analyzed in detail.

In this paper, we consider the classically conformal U(1)′ extended SM with the RHN DM. As we mentioned above, the allowed parameter regions in the classically conformal model are severely constrained in order to solve the electroweak vacuum instability problem, and to satisfy the LHC limits from the search for $Z^{\prime}$ boson resonance. In addition to these constraints, we will investigate the RHN DM physics. Because of the nature of classical conformality, we find the mass mixing between the SM Higgs and the U(1)′ Higgs bosons is very small, so that the RHN DM pair annihilation process mediated by the Higgs bosons is highly suppressed. Therefore, we focus on the study of the $Z^{\prime}$-portal RHN DM [80, 89], and identify allowed parameter regions to reproduce the observed DM relic density from the Planck 2015 result [94]. We will show that the DM physics, LHC phenomenology, and the electroweak vacuum stability condition complementarily work to narrow down the allowed parameter regions. For the identified allowed regions, we also calculate the spin-independent cross section of the RHN DM with nucleons and compare our results with the current upper bounds from the direct DM search experiments.

This paper is organized as follows: In the next section, we introduce the classically conformal U(1)′ extended SM with $Z^{\prime}$-portal RHN DM. We briefly review our previous work on the classically conformal U(1)′ model [13, 14]. In Sec. 3, we calculate the relic density of the $Z^{\prime}$-portal RHN DM. In Sec. 4, we study the $Z^{\prime}$ boson production at the LHC Run-2 (2016) [95, 96], and obtained the constraints on the model parameter space from the search results of the $Z^{\prime}$ boson resonance by the ATLAS and the CMS Collaborations. In Sec. 5, we combine all the results in the previous sections and narrow allowed regions. In Sec. 6, for the allowed parameter regions, we calculate the spin-independent cross section of the RHN DM with nucleons. The last section is devoted to conclusions.

2 The classically conformal U(1)′ extended SM with RHN DM

In this section, we will briefly review the results in Ref. [14]. Although the model is extended to incorporate the RHN DM, the results presented here are essentially the same as those in Ref. [14].

2.1 The model

The model we will investigate is the anomaly-free U(1)′ extension of the SM with the classically conformal invariance, which is based on the gauge group SU(3)${}_{C}\times$SU(2)${}_{L}\times$U(1)${}_{Y}\times$U(1)′. The particle contents of the model are listed in Table 1. In addition to the SM particle content, three generations of RHNs $\nu_{R}^{i}$ and a U(1)′ Higgs field $\Phi$ are introduced. We also introduce the $Z_{2}$ parity [58], and assign an odd parity to one RHN $\nu_{R}^{3}$, while the other particles, including $\nu_{R}^{1}$ and $\nu_{R}^{2}$, have even parity. The conservation of $Z_{2}$ parity ensures the stability of $\nu_{R}^{3}$, which is a unique candidate of the DM in our model.

The covariant derivative, which is relevant to U(1)${}_{Y}\times$ U(1)′, is defined as

[TABLE]

where $Y_{1}$ ($Y_{X}$) is U(1)Y (U(1)′ ) charge of a particle, and the gauge coupling $g_{X1}$ and $g_{1X}$ are introduced associated with a kinetic mixing between the two U(1) gauge bosons. In order to reproduce observed fermion masses and flavor mixings, we introduce the following Yukawa interactions:

[TABLE]

where $\tilde{H}\equiv i\tau^{2}H^{*}$, and the fourth and fifth terms in the right-handed side are for the seesaw mechanism to generate neutrino masses. Without loss of generality, the Majorana Yukawa couplings in the fifth term are already diagonalized in our basis. Because of the $Z_{2}$ parity, only two generation RHNs are involved in the neutrino Dirac Yukawa couplings and hence the neutrino Dirac mass matrix is 2 by 3. Once the U(1)′ Higgs field $\Phi$ develops a VEV, the U(1)′ symmetry is broken and the Majorana mass terms for the RHNs are generated. After the electroweak symmetry breaking, the seesaw mechanism [1, 2, 3, 4, 5] is automatically implemented, except that only two generation RHNs are relevant. This system is the minimal seesaw [61, 62], which possesses a number of free parameters $Y_{\nu}^{ij}$ and $Y_{M}^{j}$ ($i=1,2,3$, $j=1,2$) enough to reproduce the neutrino oscillation data with a prediction of one massless eigenstate.

In the particle contents, the two parameters ($x_{H}$ and $x_{\Phi}$) reflect the fact that the U(1)′ gauge group can be defined as a linear combination of the SM U(1)Y and the U(1)B-L gauge groups. Since the U(1)′ gauge coupling $g_{X}$ is a free parameter of the model and it always appears as a product $x_{\Phi}g_{X}$ or $x_{H}g_{X}$, we fix $x_{\Phi}=2$ without loss of generality throughout this paper. This convention excludes the case that U(1)′ gauge group is identical with the SM U(1)Y. The choice of $(x_{H},x_{\Phi})=(0,2)$ corresponds to the U(1)B-L model. Another example is $(x_{H},x_{\Phi})=(-1,2)$, which corresponds to the SM with the so-called U(1)R symmetry. When we choose $(x_{H},x_{\Phi})=(-16/41,2)$, the beta function of $g_{X1}$ ($g_{1X}$) at the 1-loop level has only terms proportional to $g_{X1}$ ($g_{1X}$) [13]. This is the orthogonal condition between the U(1)Y and U(1)′ at the 1-loop level, under which $g_{X1}$ and $g_{1X}$ do not evolve once we have set $g_{X1}=g_{1X}=0$ at an energy scale.

Imposing the classically conformal invariance, the scalar potential is given by

[TABLE]

where the mass terms are forbidden by the conformal invariance. If $\lambda_{\rm mix}$ is negligibly small, we can analyze the Higgs potential separately for $\Phi$ and $H$ as a good approximation. This will be justified in the following subsections. When the Majorana Yukawa couplings $Y_{M}^{i}$ are negligible compared to the U(1)′ gauge coupling, the $\Phi$ sector is identical with the original CW model [53], so that the radiative U(1)′ symmetry breaking will be achieved. Once $\Phi$ develops a VEV through the CW mechanism, the tree-level mass term for the SM Higgs doublet is effectively generated through $\lambda_{\rm mix}$ in Eq. (2.3). Taking $\lambda_{\rm mix}$ negative, the induced mass squared for the Higgs doublet is negative and, as a result, the electroweak symmetry breaking is driven in the same way as in the SM.

2.2 Radiative U(1)′ gauge symmetry breaking

Assuming $\lambda_{\rm mix}$ is negligibly small, we first analyze the U(1)′ Higgs sector. Without mass terms, the Coleman-Weinbeg potential [53] at the 1-loop level is found to be

[TABLE]

where $\phi/\sqrt{2}=\Re[\Phi]$, and we have chosen the renormalization scale to be the VEV of $\Phi$ ($\langle\phi\rangle=v_{\phi}$). Here, the coefficient of the 1-loop quantum corrections is given by

[TABLE]

where in the last expression, we have used $\lambda_{\Phi}^{2}\ll(x_{\Phi}g_{X})^{4}$ as usual in the CW mechanism and set $g_{X1}=g_{1X}=0$ at $\langle\phi\rangle=v_{\phi}$, for simplicity. The stationary condition $\left.dV/d\phi\right|_{\phi=v_{\phi}}=0$ leads to

[TABLE]

and this $\lambda_{\Phi}$ is nothing but a renormalized self-coupling at $v_{\phi}$ defined as

[TABLE]

For more detailed discussion, see Ref. [32].

Associated with this radiative U(1)′ symmetry breaking (as well as the electroweak symmetry breaking), the U(1)′ gauge boson ($Z^{\prime}$ boson), the Majorana RHNs $\nu_{R}^{1,2}$, and the RHN DM particle $\nu_{R}^{3}$ acquire their masses as

[TABLE]

where $v_{h}=246$ GeV is the SM Higgs VEV, and we have used $x_{\Phi}v_{\phi}\gg x_{H}v_{h}$, which will be verified below. In this paper, we assume degenerate masses for $\nu_{R}^{1,2}$, ($Y_{M}^{1}=Y_{M}^{2}=y_{M}$, equivalently, $m_{N^{1,2}}=m_{N}$), for simplicity. The U(1)′ Higgs boson mass is given by

[TABLE]

where $y_{\rm DM}=Y_{M}^{3}$. When the Yukawa couplings are negligibly small, this equation reduces to the well-known relation derived in the original paper by Coleman-Weinberg [53]. For a sizable Majorana mass, this formula indicates that the potential minimum disappears, so that there is an upper bound on the RHN mass for the U(1)′ symmetry to be broken radiatively. This is in fact the same reason why the CW mechanism in the SM Higgs sector fails to break the electroweak symmetry when the top Yukawa coupling is large as observed. In order to avoid the destabilization of the U(1)′ Higgs potential, we simply set $m_{Z^{\prime}}^{4}\gg m_{N}^{4}$ in the following analysis, while $m_{\rm DM}\simeq m_{Z^{\prime}}/2$ as we will find in the next section. Note that this condition does not mean that the Majorana RHNs must be very light, even though a factor difference between $m_{Z^{\prime}}$ and $m_{N}$ is enough to satisfy the condition. For simplicity, we set $y_{M}=0$ at $v_{\phi}$ in the following RG analysis as an approximation.

2.3 Electroweak symmetry breaking

Let us now consider the SM Higgs sector. In our model, the electroweak symmetry breaking is achieved in a very simple way. Once the U(1)′ symmetry is radiatively broken, the SM Higgs doublet mass is generated through the mixing quartic term between $H$ and $\Phi$ in the scalar potential in Eq. (2.3),

[TABLE]

where we have replaced $H$ by $H=1/\sqrt{2}\,(0,\,h)$ in the unitary gauge. Choosing $\lambda_{\rm mix}<0$, the electroweak symmetry is broken in the same way as in the SM [25, 26]. However, we should note that a crucial difference from the SM is that, in our model, the electroweak symmetry breaking originates from the radiative breaking of the U(1)′ gauge symmetry. At the tree level, the stationary condition $V^{\prime}|_{h=v_{h}}=0$ leads to the relation $|\lambda_{\rm mix}|=2\lambda_{H}(v_{h}/v_{\phi})^{2}$, and the Higgs boson mass $m_{h}$ is given by

[TABLE]

In the following RG analysis, this is used as the boundary condition for $\lambda_{\rm mix}$ at the renormalization scale $\mu=v_{\phi}$. Note that since $\lambda_{H}\sim 0.1$ and $v_{\phi}\gtrsim 10$ TeV by the large electron-positron collider (LEP) constraint [97, 98, 99], $|\lambda_{\rm mix}|\lesssim 10^{-5}$, which is very small.

In our discussion about the U(1)′ symmetry breaking, we neglected $\lambda_{\rm mix}$ by assuming it to be negligibly small. Here we justify this treatment. In the presence of $\lambda_{\rm mix}$ and the Higgs VEV, Eq. (2.6) is modified as

[TABLE]

Considering the current LHC Run-2 bound from search for $Z^{\prime}$ boson resonances [95, 96], $m_{Z^{\prime}}\gtrsim 4$ TeV, we find that the first term in the parenthesis in the last equality is 5 orders of magnitude greater than the second term, and therefore we can analyze the two Higgs sectors separately.

2.4 Solving the electroweak vacuum instability

In the SM with the observed Higgs boson mass of $m_{h}=125.09$ GeV [55], the RG evolution of the SM Higgs quartic coupling shows that the running coupling becomes negative at the intermediate scale $\mu\simeq 10^{10}$ GeV [100] for $m_{t}=173.34$ GeV [54], and hence the electroweak vacuum is unstable. In our U(1)′ extended SM, however, there is a parameter region to solve this electroweak vacuum instability problem [13, 14].555 In the absence of the classical conformal invariance, the electroweak vacuum instability problem has been investigated in Refs. [101, 102, 103, 84]. There are only three free parameters in our model, $x_{H}$, $v_{\phi}$, and $g_{X}$, which are also interpreted as $x_{H}$, $m_{Z^{\prime}}$, and $\alpha_{g_{X}}=g_{X}^{2}/(4\pi)$. Inputs of the couplings at $v_{\phi}$ are determined by these three parameters. In Fig. 11, we show the RG evolution of the SM Higgs quartic coupling in our model (solid line), along with the SM result (dashed line). Here, we have taken $x_{H}=-0.575$, $m_{Z^{\prime}}=4$ TeV and $\alpha_{g_{X}}=0.01$, which corresponds to $v_{\phi}=5.64$ TeV and $g_{X}(v_{\phi})=0.354$, as an example. The Higgs quartic coupling remains positive all the way up to the Planck mass scale, so the electroweak vacuum instability problem is solved.

In order to identify a parameter region to resolve the electroweak vacuum instability, we perform parameter scans for the free parameters $x_{H}$, $v_{\phi}$ and $g_{X}$. In this analysis, we impose several conditions on the running couplings at $v_{\phi}\leq\mu\leq M_{\rm P}$ ($M_{\rm P}=2.44\times 10^{18}$ GeV is the reduced Planck mass): stability conditions of the Higgs potential ($\lambda_{H},\lambda_{\Phi}>0$), and the perturbative conditions that all the running couplings remain in the perturbative regime, namely, $g_{i}^{2}$ ($i=1,2,3$), $g_{X}^{2}$, $g_{X1}^{2}$, $g_{1X}^{2}<4\pi$ and $\lambda_{H}$, $\lambda_{\Phi}$, $\lambda_{\rm mix}<4\pi$. For theoretical consistency, we also impose a condition that the 2-loop beta functions are smaller than the 1-loop beta functions (see Ref. [14] for detail). In Fig. 11, we show the result of our parameter scans in the three-dimensional parameter space of ($m_{Z^{\prime}},\alpha_{g_{X}},x_{H}$). As a reference, we show a horizontal plane corresponding to the orthogonal case $x_{H}=-16/41$. There is no overlapping of the plane with the resultant parameter regions to resolve the electroweak vacuum instability.

2.5 Naturalness bounds from SM Higgs mass corrections

Once the classically conformal symmetry is radiatively broken by the CW mechanism, the masses for the $Z^{\prime}$ boson and the Majorana RHNs are generated, and they contribute to self-energy corrections of the SM Higgs doublet. If the U(1)′ gauge symmetry breaking scale is very large, the self-energy corrections may exceed the electroweak scale and require us to fine-tune the model parameters in reproducing the correct electroweak scale. See [104] for related discussions. As heavy states, we have the RHNs and $Z^{\prime}$ boson, whose masses are generated by the U(1)′ gauge symmetry breaking.

Since the original theory is classically conformal and defined as a massless theory, the self-energy corrections to the SM Higgs doublet originate from corrections to the mixing quartic coupling $\lambda_{\rm mix}$. Thus, what we calculate to derive the naturalness bounds are quantum corrections to the term $\lambda_{\rm mix}h^{2}\phi^{2}$ in the effective Higgs potential

[TABLE]

where the logarithmic divergence and the terms independent of $\phi$ are all encoded in $C$. Here, the major contributions to quantum corrections are from the $Z^{\prime}$ boson loops:

[TABLE]

where the first term is from the one-loop diagram, and the second one is from the two-loop diagram [25, 26] involving the $Z^{\prime}$ boson and the top quark. By adding a counter-term, we renormalize the coupling $\lambda_{\rm mix}$ with the renormalization condition,

[TABLE]

where $\lambda_{\rm mix}$ is the renormalized coupling. As a result, we obtain

[TABLE]

Substituting $\phi=v_{\phi}$, we obtain the SM Higgs self-energy correction as

[TABLE]

For the stability of the electroweak vacuum, we impose $\Delta m_{h}^{2}\lesssim m_{h}^{2}$ as the naturalness. The most important contribution to $\Delta m_{h}^{2}$ is the first term of Eq. (2.17) generated through the one-loop diagram with the $Z^{\prime}$ gauge boson, and the second term becomes important in the case of the U(1)B-L model, where $x_{H}=0$.

If $\Delta m_{h}^{2}$ is much larger than the electroweak scale, we need a fine-tuning of the tree-level Higgs mass ($|\lambda_{\rm mix}|v_{\phi}^{2}/2$) to reproduce the correct SM Higgs VEV, $v_{h}=246$ GeV. We simply evaluate a fine-tuning level as

[TABLE]

Here, $\delta=0.1$, for example, indicates that we need to fine-tune the tree-level Higgs mass squared at the 10% accuracy level.

3 Relic density of the RHN DM

In this section, we calculate the thermal relic density of the RHN DM and identify the model parameter region to be consistent with the Planck 2015 measurement [94] (68% confidence level):

[TABLE]

In our model, the RHN DM particles mainly annihilate into the SM particles through the $s$-channel process mediated by the U(1)′ gauge boson $Z^{\prime}$.

The Boltzmann equation of the RHN DM is given by

[TABLE]

where temperature of the Universe is normalized by the mass of the RHN DM $x=m_{\rm DM}/T$, $H(m_{\rm DM})$ is the Hubble parameter at $T=m_{\rm DM}$, $s$ is the entropy density, $Y=n/s$ is the yield of the RHN DM which is defined by the ratio of the number density $n$ to $s$, $Y_{\rm EQ}$ is the yield in the thermal equilibrium, and $\langle\sigma v\rangle$ is the thermal averaged product of the RHN DM annihilation cross section $\sigma$ and relative velocity $v$. Explicit formulas of these are summarized as follows:

[TABLE]

where $g_{\rm DM}=2$ is the number of degrees of the freedom for the RHN DM, $g_{*}$ is the effective total number of degrees of freedom for particles in thermal equilibrium (in this paper, we set $g_{*}=106.75$ for the SM particles), and $K_{2}$ is the modified Bessel function of the second kind. The thermally-averaged annihilation cross section times velocity is given by

[TABLE]

where the reduced cross section is defined as $\hat{\sigma}(s)=2(s-4m_{\rm DM}^{2})\sigma(s)$ with the total cross section $\sigma(s)$, $K_{1}$ is the modified Bessel function of the first kind. The total cross section of the RHN DM annihilation process $\nu_{R}^{3}\nu_{R}^{3}\to Z^{\prime}\to f\bar{f}$ ($f$ denotes the SM fermion)666 Although there are also other annihilation processes, such as $\nu_{R}^{3}\nu_{R}^{3}\to\phi\phi$, $\nu_{R}^{3}\nu_{R}^{3}\to\phi Z^{\prime}$ and $\nu_{R}^{3}\nu_{R}^{3}\to Z^{\prime}Z^{\prime}$ (see, for example, Ref. [105]), all these cross sections are estimated to be much less than 1 pb, which is a typical cross section to reproduce $\Omega_{\rm DM}h^{2}\simeq 0.1$, for $\alpha_{g_{X}}\sim 0.01$ (see Figs. 5 and 6), $y_{\rm DM}\sim g_{X}$, and $m_{\rm DM}\sim 1$ TeV.

is calculated as

[TABLE]

where the total decay width of $Z^{\prime}$ boson is given by

[TABLE]

Here, we have neglected all SM fermion masses except for the top quark mass $m_{t}$.

By solving the Boltzmann Eq. (3.2) numerically, we find the asymptotic value of the yield $Y(\infty)$, and the present DM relic density is given by

[TABLE]

where $s_{0}=2890$ cm*-3* is the entropy density of the present Universe, and $\rho_{c}/h^{2}=1.05\times 10^{-5}$ GeV/cm3 is the critical density. Our analysis involves four parameters, namely $\alpha_{g_{X}}$, $m_{Z^{\prime}}$, $m_{\rm DM}$ and $x_{H}$. For $m_{Z^{\prime}}=4$ TeV and $x_{H}=-0.575$, we show in Fig. 2 the resultant RHN DM relic density as a function of the RHN DM mass $m_{\rm DM}$, along with the range of the observed DM relic density, $0.1183\leq\Omega_{\rm DM}h^{2}\leq 0.1213$ [94] (two horizontal dashed lines). The solid lines from top to bottom show the resultant RHN DM relic densities for various values of the gauge coupling, $\alpha_{g_{X}}=0.002$, $0.00235$, $0.003$, $0.004$ and $0.005$. The plots indicate the lower bound on $\alpha_{g_{X}}\geq 0.00235$ for $m_{Z^{\prime}}=4$ TeV and $x_{H}=-0.575$ in order to reproduce the observed relic density. In addition, we can see that the enhancement of the RHN DM annihilation cross section via the $Z^{\prime}$ boson resonance is necessary to satisfy the cosmological constraint and hence, $m_{\rm DM}\simeq m_{Z^{\prime}}/2$.

4 Collider constraints on the U(1)′ $Z^{\prime}$ boson

The ATLAS and the CMS Collaborations have searched for $Z^{\prime}$ boson resonance at the LHC Run-1 with $\sqrt{s}=8$ TeV, and continued the search at the LHC Run-2 with $\sqrt{s}=13$ TeV. The most stringent bounds on the $Z^{\prime}$ boson production cross section times branching ratio have been obtained by using the dilepton final state. For the so-called sequential SM $Z^{\prime}$ ($Z^{\prime}_{SSM}$) model [106], where the $Z^{\prime}_{SSM}$ boson has exactly the same couplings with the SM fermions as those of the SM $Z$ boson, the latest cross section bounds from the LHC Run-2 results lead to lower bounds on the $Z^{\prime}_{SSM}$ boson mass as $m_{Z^{\prime}_{SSM}}\geq 4.05$ TeV in the ATLAS 2016 results [95] and $m_{Z^{\prime}_{SSM}}\geq 4.0$ TeV in the CMS 2016 results [96], respectively. We interpret these ATLAS and CMS results into the U(1)′ $Z^{\prime}$ boson case and derive constraints on $x_{H}$, $\alpha_{g_{X}}$ and $m_{Z^{\prime}}$.

We calculate the dilepton production cross section for the process $pp\to Z^{\prime}+X\to\ell^{+}\ell^{-}+X$. The differential cross section with respect to the invariant mass $M_{\ell\ell}$ of the final state dilepton is described as

[TABLE]

where $f_{a}$ is the parton distribution function for a parton $a$, and $E_{\rm CM}=13$ TeV is the center-of-mass energy of the LHC Run-2. In our numerical analysis, we employ CTEQ5M [107] for the parton distribution functions. In the case of the U(1)′ model, the cross sections for the colliding partons are given by

[TABLE]

where the total decay width of the $Z^{\prime}$ boson is given in Eq. (3.6). By integrating the differential cross section over a range of $M_{\ell\ell}$ set by the ATLAS and CMS analyses, respectively, we obtain the cross section as a function of $x_{H}$, $\alpha_{g_{X}}$ and $m_{Z^{\prime}}$, which are compared with the lower bounds obtained by the ATLAS and CMS Collaborations.

In interpreting the ATLAS and the CMS results for the U(1)′ $Z^{\prime}$ boson, we follow the strategy in [80]. We first analyze the sequential SM $Z^{\prime}$ model to check the consistency of our analysis with the one by the ATLAS and the CMS Collaborations. With the same couplings as the SM, we calculate the differential cross section of the process $pp\to Z^{\prime}_{SSM}+X\to\ell^{+}\ell^{-}+X$ like Eq. (4.1). According to the analysis by the ATLAS Collaboration at the LHC Run-2, we integrate the differential cross section for the range of 120 GeV$\leq M_{\ell\ell}\leq 6000$ GeV [95] and obtain the cross section of the dilepton production process as a function of the $Z^{\prime}_{SSM}$ boson mass. Our result is shown as a solid line in Fig. 33, along with the plots presented by the ATLAS Collaboration [95] (Here we have also shown the ALTAS 2015 result [56] for comparison. We can see that the ATLAS 2016 result has dramatically improved the bound obtained by the ATLAS 2015 result.). In Fig. 33, the experimental upper bounds on the $Z^{\prime}$ boson production cross section are depicted as the horizontal solid (red) curves. The theoretical $Z^{\prime}$ boson production cross section presented in [95] is shown as the diagonal dashed line, and the lower limit of the $Z^{\prime}_{SSM}$ boson mass is found to be $4.05$ TeV, which can be read off from the intersection point of the theoretical prediction (diagonal dashed line) and the experimental cross section bound (horizontal lower solid (red) curve). In order to take into account the difference of the parton distribution functions used in the ATLAS analysis and our analysis, and QCD corrections of the process, we have scaled our resultant cross section by a factor $k=1.16$ in Fig. 33, with which we can obtain the same lower limit of the $Z^{\prime}_{SSM}$ boson mass as $4.05$ TeV. We can see that our result (solid line) in Fig. 33 with the factor of $k=1.16$ is very consistent with the theoretical prediction (diagonal dashed line) presented by the ATLAS Collaboration. We use this factor in the following analysis for the U(1)′ $Z^{\prime}$ production process, when we interpret the ATLAS 2016 result.

We apply the same strategy and compare our results for the $Z^{\prime}_{SSM}$ model with those in the CMS 2016 results [96]. According to the analysis by the CMS Collaboration, we integrate the differential cross section for the range of $0.95\;m_{Z^{\prime}_{SSM}}\leq M_{\ell\ell}\leq 1.05\;m_{Z^{\prime}_{SSM}}$ [96] and obtain the cross section. In the CMS analysis, the limits are set on the ratio of the $Z^{\prime}_{SSM}$ boson cross section to the $Z/\gamma^{*}$ cross section:

[TABLE]

where the $Z/\gamma^{*}$ production cross sections in the mass window of $60$ GeV$\leq M_{\ell\ell}\leq 120$ GeV are predicted to be $1928$ pb at the LHC Run-2 [96]. Our result for the $Z^{\prime}_{SSM}$ model is shown as the solid line in Fig. 33, along with the plot presented in [96] (Here we have also shown the CMS 2015 result [57] for comparison. We can see that the CMS 2016 result has dramatically improved the bound obtained by the CMS 2015 result.). The analyses in this CMS paper leads to the lower limits of the $Z^{\prime}_{SSM}$ boson mass as $4.0$ TeV, which is read off from the intersection point of the theoretical prediction (diagonal dashed line) and the experimental cross section bound (horizontal lower solid (red) curve). In order to obtain the same lower mass limits, we have scaled our resultant cross section by a factor $k=1.42$ in Fig. 33. With this $k$ factor, our result (solid line) is very consistent with the theoretical prediction (diagonal dashed line) presented in Ref. [96]. We use this $k$ factor in our analysis to interpret the CMS result for the U(1)′ $Z^{\prime}$ boson case.

The search for effective 4-Fermi interactions mediated by the $Z^{\prime}$ boson at the LEP leads to a lower bound on $m_{Z^{\prime}}/g_{X}$ [97, 98, 99]. Employing the limits from the final LEP 2 data [99] at 95% confidence level, we follow Ref. [98] and derive a lower bound on $m_{Z^{\prime}}/g_{X}$ as a function $x_{H}$. Our result is shown in Fig. 4.

5 Combined results

Now let us combine all the constraints that we have obtained in the previous sections from the RHN DM physics, collider phenomenology, and the electroweak vacuum stability. In Fig. 5, we show the allowed region in the ($m_{Z^{\prime}}$, $\alpha_{g_{X}}$)-plain for fixed $x_{H}=-0.575$, as an example. The shaded region indicates the parameter space for solving the electroweak vacuum instability. The (blue) right solid line shows the lower bound on $\alpha_{g_{X}}$ as a function of $m_{Z^{\prime}}$ to reproduce the observed DM relic density of the Planck result [94]. The (red) left solid (dashed) line shows the upper bound on $\alpha_{g_{X}}$ obtained from the search results for $Z^{\prime}$ boson resonance by the CMS [96] (ATLAS [95]) Collaboration. The (green) shaded region in between two solid lines satisfies all constraints. These three constraints are complementary to narrow down the allowed region to be 4 TeV $\lesssim m_{Z^{\prime}}\lesssim 8$ TeV and $0.009\lesssim\alpha_{g_{X}}\lesssim 0.017$. We also show the naturalness bounds for 10% (right dotted line) and 30% (left dotted line) fine-tuning levels.

In Fig. 6, we show allowed parameter regions in the ($x_{H}$, $\alpha_{g_{X}}$)-plain for various $m_{Z^{\prime}}$ values. Fig. 66 is for $m_{Z^{\prime}}=4$ TeV. The shaded region indicates the parameter space for solving the electroweak vacuum instability. The (blue) convex-downward solid line shows the lower bound on $\alpha_{g_{X}}$ as a function of $x_{H}$ to reproduce the observed DM relic density. The (red) convex-upward solid (dashed) line shows the upper bound on $\alpha_{g_{X}}$ obtained from the search results for $Z^{\prime}$ boson resonance by the CMS [96] (ATLAS [95]) Collaboration, and the (red) dashed-dotted lines also show the LEP bounds. The (green) shaded region in between two solid lines satisfies all constraints. These three constraints are complementary to narrow down the allowed region to be $-1.1\lesssim x_{H}\lesssim-0.4$ and $0.002\lesssim\alpha_{g_{X}}\lesssim 0.02$. We also show the naturalness bounds for 10% (dashed line) and 30% (dotted line) fine-tuning levels. Figs. 66, 66 and 66 are the same as Fig. 66, but $m_{Z^{\prime}}=3.75$ TeV, 3.5 TeV and 3 TeV, respectively. From Fig. 66, the allowed region to satisfy these three constraints indicates $-0.9\lesssim x_{H}\lesssim-0.5$ and $0.003\lesssim\alpha_{g_{X}}\lesssim 0.015$ for fixed $m_{Z^{\prime}}=3.75$ TeV. As $m_{Z^{\prime}}$ decreases, the LHC upper bound lines are shifted downward, while the DM lower bound line remains almost the same (it slightly moves to downward). Therefore, the allowed region between the LHC upper bounds and the DM lower bound narrows. On the other hand, the shaded region remains almost the same, so that the (green) shaded region disappears for $m_{Z^{\prime}}\lesssim 3.5$ TeV.

6 Direct detection of RHN DM

A variety of experiments are underway and also planned for directly detecting a DM particle through its elastic scattering off with nuclei.777 We can also consider an indirect detection of the RHN DM through cosmic rays from a pair annihilation of the RHN DMs. However, using the parameters in the allowed regions shown in Sec. 5, we have found that the pair annihilation cross section is much smaller than the current upper bounds obtained from, for example, the Fermi-LAT experiments [111].

In this section, we calculate the spin-independent elastic scattering cross section of the RHN DM particle via the Higgs bosons exchange,888 There is another process for the RHN DM to scatter off with nuclei via $Z^{\prime}$-boson exchange. Since the RHN DM is a Majorana particle, only its interaction with nuclei is spin-dependent. We have calculated this spin-dependent cross section to be $\sigma_{\rm SD}\sim 10^{-9}$ pb, which is far below the current upper bounds, $\sigma_{\rm SD}\lesssim 10^{-4}$ pb obtained from the LUX [112] and the IceCube [113] experiments.

and compare our results with the current experimental results and a prospective reach by future experiments.

From Eq. (2.8), the U(1)′ Higgs VEV $v_{\phi}$ is expressed as a function of $m_{Z^{\prime}}$, $\alpha_{g_{X}}$ and $x_{H}$:

[TABLE]

In Sec. 3, we have also shown that $m_{\rm DM}\simeq m_{Z^{\prime}}/2$ to satisfy the experimental relic density of the $Z^{\prime}$-portal RHN DM, which means $y_{\rm DM}\simeq m_{Z^{\prime}}/2\sqrt{2}v_{\phi}\simeq\sqrt{2\pi\alpha_{g_{X}}}$, and then Eq. (2.9) is approximately expressed as

[TABLE]

Using the SM Higgs boson mass in Eq. (2.11), the scalar mass matrix is found to be

[TABLE]

The mass eigenstates $h^{\prime}$ and $\phi^{\prime}$ are defined as

[TABLE]

with the mixing angle $\theta$ given by

[TABLE]

and their mass eigenvalues are given by

[TABLE]

Here, we have used the fact that except for the special case, $m_{h}^{2}\simeq m_{\phi}^{2}$, the mixing angle is always small because of the suppression by $v_{h}/v_{\phi}$ with $v_{h}=246$ GeV and $v_{\phi}\gtrsim 10$ TeV. Thus, the mass eigenstate $h^{\prime}$ is the SM-like Higgs boson, while $\phi^{\prime}$ is the U(1)′-like Higgs boson.

The spin-independent elastic scattering cross section with nucleon is given by

[TABLE]

where $\mu_{\rm DM,N}=m_{N}m_{{\rm DM}}/(m_{N}+m_{{\rm DM}})$ is the reduced mass of the RHN DM-nucleon system with the nucleon mass $m_{N}=0.939$ GeV, and

[TABLE]

is the nuclear matrix element accounting for the quark and gluon contents of the nucleon. In evaluating $f_{T_{q}}$, we use the results from the lattice QCD simulation [114]: $f_{T_{u}}+f_{T_{d}}\simeq 0.056$ and $|f_{T_{s}}|\leq 0.08$. For conservative analysis, we take $f_{T_{s}}=0$ in the following. Using the trace anomaly formula, $\sum_{q=u,d,s}f_{T_{q}}+f_{TG}=1$ [115, 116, 117, 118, 119], we obtain $f_{N}^{2}\simeq 0.0706\;m_{N}^{2}$. Using Eqs. (6.1), (6.2) and (6.13), $\sigma_{\rm SI}$ is expressed as a function of only two free parameters: $\alpha_{g_{X}}$ and $m_{Z^{\prime}}$.

For a fixed $x_{H}=-0.575$, the resultant spin-independent cross section $\sigma_{\rm SI}$ as a function of $m_{Z^{\prime}}$ is depicted in Fig. 7. Here, for a fixed $m_{Z^{\prime}}$ value, $\alpha_{g_{X}}$ is taken from the shaded region in Fig. 5 to solve the electroweak vacuum instability problem. The (green) shaded region in between around 3.5 TeV and 9 TeV corresponds to the (green) shaded parameter region in Fig. 5, which satisfies all three constraints, the electroweak vacuum stability condition, the LHC Run-2 bound, and the cosmological constraint from the observed RHN DM relic density. The (red) upper solid (dashed) line shows the XENON1T [108] (LUX 2016 [109]) upper bound on $\sigma_{\rm SI}$ as a function of $m_{Z^{\prime}}\simeq 2m_{\rm DM}$, and the (red) dotted line shows the prospective reach for the upper bound on $\sigma_{\rm SI}$ in the next-generation successor of the LUX experiment, the LUX-ZEPLIN (LZ) DM experiment [110]. Our resultant spin-independent cross section appears below the future reach.

In Fig. 8, we show the resultant $\sigma_{\rm SI}$ in the ($x_{H}$, $\sigma_{\rm SI}$)-plain for various $m_{Z^{\prime}}$ values, corresponding to the parameter regions shown in Fig. 6. Fig. 88 shows our results for $m_{Z^{\prime}}=4$ TeV. The shaded regions indicate the parameter space for solving the electroweak vacuum instability. The (green) shaded region in the range of $-1.1\lesssim x_{H}\lesssim-0.4$ corresponds to the (green) shaded region in Fig. 66, which satisfies all three constraints, the electroweak vacuum stability condition, the LHC Run-2 bound, and the cosmological constraint from the observed RHN DM relic density. The (red) upper solid (dashed) line shows the XENON1T [108] (LUX 2016 [109]) upper bound on $\sigma_{\rm SI}$, and the (red) dotted line shows the prospective reach for the upper bound on $\sigma_{\rm SI}$ in the LZ DM experiment [110]. Figs. 88, 88 and 88 are the same as Fig. 88, but for $m_{Z^{\prime}}=3.75$ TeV, 3.5 TeV and 3 TeV corresponding to Fig. 66, 66 and 66, respectively. Fig. 88 has a (green) shaded region in the range of $-0.9\lesssim x_{H}\lesssim-0.5$ to satisfy the three constraints, while Figs. 88 and 88 have no such region.

7 Conclusions

We have considered the DM scenario in the context of the classically conformal U(1)′ extended SM, with three RHNs and the U(1)′ Higgs field. The model is free from all the U(1)′ gauge and gravitational anomalies in the presence of the three RHNs. We have introduced a $Z_{2}$-parity in the model, under which an odd-parity is assigned to one RHN, while all the other particles are assigned to be $Z_{2}$-even. In our model, the $Z_{2}$-odd RHN serves as a stable DM candidate, while the other two RHNs are utilized for the the minimal seesaw mechanism in order to reproduce the neutrino oscillation data and the observed baryon asymmetry in the Universe through leptogenesis. In this model, the U(1)′ gauge symmetry is radiatively broken through the CW mechanism, by which the electroweak symmetry breaking is triggered. There are three free parameters in our model, the U(1)′ charge of the SM Higgs doublet ($x_{H}$), the new U(1)′ gauge coupling ($\alpha_{g_{X}}$), and the U(1)′ gauge boson ($Z^{\prime}$) mass ($m_{Z^{\prime}}$).

In this model context, we have first investigated a possibility to resolve the electroweak vacuum instability with the current world average of the experimental data, $m_{t}=173.34$ GeV and $m_{h}=125.09$ GeV. By analyzing the RG evolutions of the couplings of the model at the two-loop level, we have performed a parameter scan for the three parameters, $m_{Z^{\prime}}$, $\alpha_{g_{X}}$ and $x_{H}$, and have identified parameter regions which can solve the electroweak instability problem and keep all coupling values in the perturbative regime up to the Planck mass scale. We have found that the resultant parameter regions are very severely constrained. Next, we have calculated the thermal relic density of the RHN DM and identified the model parameter region to reproduce the observed DM relic density of the Planck 2015 measurement. In our model, the RHN DM particles mainly annihilate into the SM particles through the $s$-channel process mediated by the $Z^{\prime}$ boson. We have obtained the lower bound on $\alpha_{g_{X}}$ as a function of $m_{Z^{\prime}}$ and $x_{H}$ from the observed DM relic density. We have also considered the LHC Run-2 bounds from the search for the $Z^{\prime}$ boson resonance by the recent ATLAS and CMS analysis, which lead to the upper bounds on $\alpha_{g_{X}}$ as a function of $m_{Z^{\prime}}$ and $x_{H}$. The LEP results from the search for effective 4-Fermi interactions mediated by the $Z^{\prime}$ boson can also constrain the model parameter space, but the LEP constraints are found to be weaker than those obtained from the LHC Run-2 results. Finally, we have combined all the constraints. The cosmological constraint on the RHN DM yields the lower bound on $\alpha_{g_{X}}$ as a function of $m_{Z^{\prime}}$ and $x_{H}$, while the upper bound on $\alpha_{g_{X}}$ is obtained from the LHC Run-2 results, so that these constraints are complementary to narrow the allowed parameter regions. We have found that only small portions in these allowed parameter regions can solve the electroweak vacuum instability problem. In particular, no allowed region to satisfy all constraints exists for $m_{Z^{\prime}}\lesssim 3.5$ TeV. For the obtained allowed regions, we have calculated the spin-independent cross section of the RHN DM with nucleons. We have found that the resultant cross section well below the current experimental upper bounds.

Acknowledgements

The work of D.-s.T. and S.O. was supported by Advanced Medical Instrumentation unit [Sugawara unit] and Mathematical and Theoretical Physics unit [Hikami unit], respectively, of the Okinawa Institute of Science and Technology Graduate University. The work of N.O. was supported in part by the United States Department of Energy (DE-SC0013680).