# Radiative Linear Seesaw Model, Dark Matter and $U(1)_{B-L}$

Weijian Wang a<EMAIL_ADDRESS>Zhi-Long Han b
<EMAIL_ADDRESS>a Department of Physics, North China Electric
Power University, Baoding 071003,China
b School of Physics, Nankai University, Tianjin 300071, China

###### Abstract

In this paper we propose a radiated linear seesaw model where the naturally
small term $\mu_{L}$ are generated at one-loop level and its soft-breaking of
lepton number symmetry attributes to the spontaneous breaking(SSB) of B-L
gauge symmetry. The value of $B-L$ charges for new particles are assigned to
satisfy the anomalies cancelation. It is founded that some new particles may
have exotic values of $B-L$ charge such that there exists residual
$Z_{2}\times Z_{2}^{\prime}$ symmetry even after SSB of $B-L$ gauge symmetry.
The $Z_{2}\times Z_{2}^{\prime}$ discrete symmetry stabilizes the these
particles as dark matter candidates. In the model, two classes of inert
fermions and scalars with different $B-L$ charges are introduced, leading to
two-component dark matter candidates. The lepton flavor violation processes,
the relic density of dark matter, the direct detection of dark matter and the
phenomenology at LHC are investigated.

##  I Introduction

The origin of tiny but no-zero neutrino masses observed by neutrino
oscillation experimentsneu1  remains so far a mystery, and so provides us an
opportunity to search the new physics beyond the standard model(SM). Perhaps
the simplest scenario which may understand the neutrino puzzle is to introduce
the Majorana mass that breaks the global $B-L$ symmetry though the dimension-5
Weinberg operator $\lambda LL\Phi\Phi/\Lambda$wein . This effective operator
can be realized though various pathways which depends on the new physics
scales the associated massages lie at. For instance in the case of widely
known type-I seesaw mechanismtype1  with right-handed Majorana ingredients
$N_{R}$ as massages, one needs the super-heavy masses for $N_{R}$ i.e
$10^{14\sim 16}$ GeV to fit the observed sub-eV neutrino mass. The right-
handed neutrinos are too heavy to be detected at future experiments. In
contrast, in the so-called low-scale scenarios, the small neutrino mass is not
only due to the massage state with heavy mass but also to another naturally
small mass parameter which breaks the lepton number symmetry. This is the
basic ideas behind many schemes including type-II seesaw modeltyp2 , inverse
seesaw modelinverse  and linear seesaw modellinear ; linear2 . In these
models, the mass of massage particles can be lowered down to TeV or even
hundreds GeV, a scale to be explored at collider experiments.

On the other hand, the Planck data has shown that $26\%$ of the energy density
of our universe is occupied by dark matter. In the view of particle physics,
the weakly interacting massive particles(WIMPs) are the most promising dark
matter candidates. In recent years, a class of models are proposed to
incorporate the neutrino mass puzzle and the existence of dark matter in a
unified framework. In these models, the neutrino masses are generated at loop
level and the dark matter is naturally contained as a inert particle, where
the $Z_{2}$ symmetry or a $U(1)_{X}$ symmetry is used to guarantee the
stability of dark matter. The radiated generation of neutrino mass has been
realized at one-loop level1loop ; Ma:2006km ; 1loope , two-loop level2loop ;
Ma:2007gq ; Kanemura:2011mw ; Kanemura:2014rpa ; Aoki:2014cja  and three-loop
level3loop . The systematic analysis of one and two-loop realization for with
possible topologies are performed in RefBonnet:2012kz .

In this paper, we proposed a radiated linear seesaw model where the lepton
number violation is due to the spontaneous symmetry breaking(SSB) of
$U(1)_{B-L}$ gauge symmetry, while the naturally small mass parameter is
generated at one-loop level. The linear seesaw model was fist studied in the
left-right theory with gauge group $SU(3)\times SU(2)_{L}\times
SU(2)_{R}\times U(1)_{B-L}$linear , and subsequently inspired on $SO(10)$
theory in the presence of gauge singletslinear2 . In linear seesaw scenario,
$\Psi_{R}$ and $\Psi_{L}$ are added onto SM so that the Lagrangian is given by

$L=M_{D}\bar{\nu}_{L}\Psi_{R}+M_{\Psi}\bar{\Psi}_{R}\Psi_{L}+\mu_{L}\tilde{\nu}_{L}\Psi_{L}+h.c$
(1)

The neutrino mass matrix in the basis of ($\nu_{L},\Psi_{R}^{c},\Psi_{L})$ is

$M_{\nu}=\left(\begin{array}[]{ccc}0&M_{D}&\mu_{L}\\\ M_{D}^{T}&0&M_{\Psi}\\\
\mu_{L}^{T}&M_{\Psi}^{T}&0\end{array}\right)$ (2)

The light neutrino mass is given by $m_{\nu}\sim\mu_{L}M_{D}M_{\Psi}^{-1}$.
Note that the $\mu_{L}$ violates the lepton number symmetry and plays the role
of a naturally small parameter. Thus it seems natural that there exists a
suppression mechanism where the $\mu_{L}$ term is generated via loop diagram
meanwhile the soft-breaking of lepton number symmetry may attribute to the SSB
of $B-L$ gauge symmetry. Moreover, when the WIMPs as the dark matter
candidates are involved in the loop diagram, we can reasonably assume they are
generated by a vacuum expectation value with respect to the SSB of $B-L$ gauge
symmetry at TeV scale. These are the main motivations of this work.

Following the spirit of Ref.Kanemura:2014rpa , the value of $B-L$ charges
should be carefully assigned since the anomalies cancelation must be
satisfied. It is founded that some new particles may have exotic values of
$B-L$ charge such that there exists residual $Z_{2}\times Z_{2}^{\prime}$
symmetry even after SSB of $B-L$ gauge symmetry. The $Z_{2}\times
Z_{2}^{\prime}$ discrete symmetry stabilizes the these particles from decaying
to SM ingredients. Thus the lightest particle with the same exotic value of
$B-L$ charge can be a dark matter candidate. In practise, we introduce two
classes of inert fermions and scalars to realize the model, leading to two
component dark matter candidates.

The existence of new fermions and scalars provides rich phenomenon. Tiny
neutrino masses are explained with one-loop induced linear-seesaw-like
mechanism. The charged-scalar mediates lepton flavor violation (LFV) of
charged leptons. The relic density and the direct detection of the two
component dark matter are investigated. The properties of discovered SM Higgs
will be changed by the new particles. And these new particles provide plenty
of new signatures at LHC. Especially, multi-lepton signals with missing
transverse energy $\cancel{E}_{T}$ can be used to test our model. We find that
our model can satisfy current constraints from phenomenons mentioned above.

The rest of paper is organised as follows. In Sec. II, we introduce the
realization of radiative linear seesaw and multi-component dark matter from
gauged $U(1)_{B-L}$. In Sec. III, we discuss the phenomenon of lepton flavor
violation, dark matter and collider signatures. Conclusions are given in Sec.
IV.

##  II Model

###  II.1 Model Setup

Particles | $\Psi_{R}$ | $\Psi_{L}$ | $N_{R}$ | $N_{R}^{\prime}$ | $N_{R}^{\prime\prime}$ | $\eta_{1}$ | $s_{1}$ | $\eta_{2}$ | $s_{2}$ | $\sigma$
---|---|---|---|---|---|---|---|---|---|---
$SU(2)_{L}$ | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1
$U(1)_{Y}$ | 0 | 0 | 0 | 0 | 0 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | 0 | 0
$U(1)_{B-L}$ | -1 | 0 | $-\frac{1}{2}$ | $x$ | $-1-x$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $x$ | $x$ | 1
Table 1: New particles content: $G_{SM}\times U(1)_{B-L}$

In our model the neutrino masses are generated via the diagram depicted in
Fig. 1. The new particles content and their charge assignment are listed in
Table. 1. We add $N_{\Psi}$ generation Weyl fermions $\Psi_{Ri}$, $\Psi_{Li}$,
$N_{1}$ right-handed Majorana neutrinos $N_{R\alpha}$, and $N_{2}$ pairs of
right-handed Majorana neutrinos $N_{R}^{\prime}$ and $N_{R}^{\prime\prime}$ to
the SM where $i,\alpha$ and $\beta$ are the generation indices. All the new
fermions are singlets under SM gauge group. Five new scalars
$\eta_{1},s_{2},\eta_{2},s_{2}$ and $\sigma$ are also added to SM. Because the
new fermions are all SM singlets, the $B-L$ gauge symmetry satisfies all
anomaly cancellations except for $[U(1)_{B-L}]\times[Gravity]^{2}$ and
$[U(1)_{B-L}]^{3}$ Ma:2001kg . Considering the conditions for the absence of
$[U(1)_{B-L}]\times[Gravity]^{2}$ and $[U(1)_{B-L}]^{3}$ anomaly, one has

$\displaystyle 3+(-\frac{1}{2})N_{1}+xN_{2}+(-1-x)N_{2}+(-1)N_{\psi}=0$ (3)
$\displaystyle
3+(-\frac{1}{2})^{3}N_{1}+x^{3}N_{2}+(-1-x)^{3}N_{2}+(-1)^{3}N_{\psi}=0$ (4)

After solving the anomaly free condition, one obtains

$N_{1}=2,\quad\quad N_{2}=1,\quad\quad N_{\psi}=1,\quad\quad
x=\frac{\sqrt{2}-1}{2}$ (5)

Thus we have the some inert particles classified into two parts. In the first
class, there are two Majorana right-handed neutrinos $(N_{R1}$, $N_{R2})$ and
the inert scalars $(\eta_{1},s_{1})$ with their B-L charge being
$-\frac{1}{2}$. In the second class, we obtain a pair of Majorana right-handed
neutrinos $(N_{R}^{\prime},N_{R}^{\prime\prime})$ along with the inert scalars
$(\eta_{2},s_{2})$ whose the B-L charges are irrational numbers
($\frac{\sqrt{2}-1}{2}$ or $\frac{-\sqrt{2}-1}{2}$). One notices that the new
particles with both $-\frac{1}{2}$ and the irrational numbers can not decay
into SM particles. Therefore the lightest particles belonging to the same
class is stable and can be regarded as a dark matter candidate. The relevant
Lagrangian for Yukawa sector is given by

$\begin{split}-L_{Y}=&y_{l}\overline{L_{l}}\psi_{R}i\tau_{2}\Phi^{\ast}+y^{\prime}\overline{\psi_{L}}\psi_{R}\sigma+h_{\alpha}\overline{N_{R\alpha}}\psi_{L}s_{1}+f_{\alpha
l}\overline{L_{l}^{c}}N_{R\alpha}^{c}i\tau_{2}\eta_{1}^{\ast}+\frac{1}{2}Y_{\alpha}\overline{N_{R\alpha}^{c}}N_{R\alpha}\sigma\\\
&+h\overline{N_{R}^{\prime}}\psi_{L}s_{2}+f_{l}\overline{L_{l}^{c}}N_{R}^{\prime\prime
c}i\tau_{2}\eta_{2}^{\ast}+\frac{1}{2}Y\overline{N_{R}^{\prime\prime
c}}N_{R}^{\prime}\sigma+h.c\end{split}$ (6)

Without losing generality, we work in the basis where the mass term of
$N_{R1,2}$ is diagonal. As for the mass term of $N_{R}^{\prime}$ and
$N_{R}^{\prime\prime}$, one can redefine the fields as

$\chi_{1}=\frac{1}{\sqrt{2}}(N_{R}^{\prime}+N_{R}^{\prime\prime})\quad\quad\quad\chi_{2}=\frac{i}{\sqrt{2}}(N_{R}^{\prime}-N_{R}^{\prime\prime})$
(7)

so that

$\frac{1}{2}Y\overline{N_{R}^{\prime\prime
c}}N_{R}^{\prime}\sigma\rightarrow\frac{1}{2}Y(\overline{\chi_{1}^{c}}\chi_{1}+\overline{\chi_{2}^{c}}\chi_{2})\sigma$
(8)

Now we get two Majorana neutrino eigenstates having the same masses. Note that
there is no interplay Yukawa terms between $N_{R1,2}$ and
$(N_{R}^{\prime},N_{R}^{\prime\prime})$ because of the the B-L charges
assignment they have.

The scalar potential in our model is given by

$\begin{split}V(\Phi,\sigma,\eta_{1},s_{1},\eta_{2},s_{2})=&-\mu_{\Phi}^{2}\Phi^{{\dagger}}\Phi+\lambda_{\Phi}(\Phi^{{\dagger}}\Phi)^{2}-\mu_{\sigma}^{2}|\sigma|^{2}+\lambda_{\sigma}|\sigma|^{4}\\\
&+\mu_{\eta_{1}}^{2}\eta_{1}^{{\dagger}}\eta_{1}+\lambda_{\eta_{1}}(\eta_{1}^{{\dagger}}\eta_{1})^{2}+\mu_{\eta_{2}}^{2}\eta_{2}^{{\dagger}}\eta_{2}+\lambda_{\eta_{2}}(\eta_{2}^{{\dagger}}\eta_{2})^{2}\\\
&+\mu_{s_{1}}^{2}|s_{1}|^{2}+\lambda_{s_{1}}|s_{1}|^{4}+\mu_{s_{2}}^{2}|s_{2}|^{2}+\lambda_{s_{2}}|s_{2}|^{4}+\lambda_{s_{1}s_{2}}|s_{1}|^{2}|s_{2}|^{2}\\\
&+\lambda_{\eta_{1}\Phi}(\Phi^{{\dagger}}\Phi)(\eta_{1}^{{\dagger}}\eta_{1})+\lambda^{\prime}_{\eta_{1}\Phi}(\eta_{1}^{{\dagger}}\Phi)(\Phi^{{\dagger}}\eta_{1})+\lambda_{\eta_{2}\Phi}(\Phi^{{\dagger}}\Phi)(\eta_{2}^{{\dagger}}\eta_{2})+\lambda^{\prime}_{\eta_{2}\Phi}(\eta_{2}^{{\dagger}}\Phi)(\Phi^{{\dagger}}\eta_{2})\\\
&+\lambda_{\eta_{1}\eta_{2}}(\eta_{1}^{{\dagger}}\eta_{1})(\eta_{2}^{{\dagger}}\eta_{2})+\lambda_{\eta_{1}\eta_{2}}^{\prime}(\eta_{1}^{{\dagger}}\eta_{2})(\eta_{2}^{{\dagger}}\eta_{1})\\\
&+\lambda_{s_{1}\Phi}|s_{1}|^{2}(\Phi^{{\dagger}}\Phi)+\lambda_{s_{1}\eta_{1}}|s_{1}|^{2}(\eta_{1}^{{\dagger}}\eta_{1})+\lambda_{s_{1}\eta_{2}}|s_{1}|^{2}(\eta_{2}^{{\dagger}}\eta_{2})\\\
&+\lambda_{s_{2}\Phi}|s_{2}|^{2}(\Phi^{{\dagger}}\Phi)+\lambda_{s_{2}\eta_{1}}|s_{2}|^{2}(\eta_{1}^{{\dagger}}\eta_{1})+\lambda_{s_{2}\eta_{2}}|s_{2}|^{2}(\eta_{2}^{{\dagger}}\eta_{2})\\\
&+\lambda_{\sigma\Phi}|\sigma|^{2}(\Phi^{{\dagger}}\Phi)+\lambda_{\sigma\eta_{1}}|\sigma|^{2}(\eta_{1}^{{\dagger}}\eta_{1})+\lambda_{\sigma\eta_{2}}|\sigma|^{2}(\eta_{2}^{{\dagger}}\eta_{2})\\\
&+\lambda_{s_{1}\sigma}|s_{1}|^{2}|\sigma|^{2}+\lambda_{s_{2}\sigma}|s_{2}|^{2}|\sigma|^{2}+(\mu_{1}s_{1}^{{\dagger}}\Phi^{{\dagger}}\eta_{1}+\mu_{2}s_{2}^{{\dagger}}\Phi^{{\dagger}}\eta_{2}+h.c)\end{split}$
(9)

where $\mu_{\Phi}^{2}$, $\mu_{\sigma}^{2}$, $\mu_{\eta_{1}}^{2}$,
$\mu_{\eta_{2}}^{2}$, $\mu_{s_{1}}^{2}$ and $\mu_{s_{2}}^{2}$ are taken as
positive values and the value coupling constants $\mu_{1}$ and $\mu_{2}$ in
trilinear terms can be set as positive by re-phasing $s_{1}$ and $s_{2}$.
Notice that there is no terms like $s_{1}\sigma^{2}$ or $s_{2}\sigma^{2}$
appearing in the scalar potential. This has two fold meanings: First, the
inert scalars $\eta_{1,2}$ and $s_{1,2}$ do not acquire the VEV after the SSB
of $\Phi$ and $\sigma$; Second, there exists a residual $Z_{2}\times
Z_{2}^{\prime}$ symmetry under which all the inert particles are odd even
after the breakdown of B-L symmetry. Therefore the residual $Z_{2}\times
Z_{2}^{\prime}$ symmetry stabilizes the inert particles, makes them to be two
component dark matter candidates.

###  II.2 Matrices of Scalar Particles

After the SSB, the scalar $\Phi$ and $\sigma$ is parameterized as

$\Phi=\left(\begin{array}[]{c}G^{+}\\\
\frac{v_{\phi}+\phi_{0}+iG_{\phi}}{\sqrt{2}}\end{array}\right)\quad\quad\quad\sigma=\frac{v_{\sigma}+\sigma_{0}+iG_{\sigma}}{\sqrt{2}}$
(10)

where $v_{\phi}\simeq 246$GeV is the VEV of the SM higgs doublet scalar and
$v_{\sigma}$ is responsible for the SSB of B-L symmetry Khalil:2006yi . The
Nambu-Godstone bosons $G^{+}$, $G_{\phi}$ and $G_{\sigma}$ are absorbed by the
longitudinal components of $W$, $Z$ and $Z^{\prime}$ gauge bosons. For
simplicity, we ignore the kinetic mixing between $U(1)_{Y}$ and $U(1)_{B-L}$
gauge boson Langacker:2008yv . Therefor the VEV $v_{\sigma}$ provides a mass
of $U(1)_{B-L}$ gauge boson $Z^{\prime}$ as
$M_{Z^{\prime}}=g_{B-L}v_{\sigma}$, where $g_{B-L}$ is the $U(1)_{B-L}$ gauge
coupling constant. For the extra gauge boson $Z^{\prime}$, LEP-II provides a
combined bound $M_{Z^{\prime}}/g_{B-L}>7~{}\textrm{TeV}$ Cacciapaglia:2006pk ,
which is just the lower bound on $v_{\sigma}$. Then we obtain the mass matrix
for CP-even scalars $\phi_{0}$ and $\sigma_{0}$

$M^{2}(\phi_{0},\sigma_{0})=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right)\left(\begin{array}[]{cc}M_{h}^{2}&0\\\
0&M_{H}^{2}\end{array}\right)\left(\begin{array}[]{cc}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{array}\right)$ (11)

where $h$ stands for the SM-like Higgs Aad:2012tfa ; Chatrchyan:2012ufa  and
$H$ is an extra CP-even Higgs boson Barger:2006sk ; Barger:2007im ;
Robens:2015gla  with the masses respectively as

$\displaystyle
M_{h}^{2}=\lambda_{\Phi}v_{\phi}^{2}+\lambda_{\sigma}v_{\sigma}^{2}-\sqrt{(\lambda
v_{\phi}^{2}-\lambda_{\sigma}v_{\sigma}^{2})^{2}+\lambda_{\sigma\Phi}^{2}v_{\phi}^{2}v_{\sigma}^{2}}$
(12) $\displaystyle
M_{H}^{2}=\lambda_{\Phi}v_{\phi}^{2}+\lambda_{\sigma}v_{\sigma}^{2}+\sqrt{(\lambda
v_{\phi}^{2}-\lambda_{\sigma}v_{\sigma}^{2})^{2}+\lambda_{\sigma\Phi}^{2}v_{\phi}^{2}v_{\sigma}^{2}}$
(13)

and mixing angle $\theta$ determined as

$\sin
2\theta=\frac{2\lambda_{\sigma\Phi}v_{\phi}v_{\sigma}}{M_{H}^{2}-M_{h}^{2}}$
(14)

On the other hand, the inert scalar $(\eta_{1},s_{1})$ and $(\eta_{2},s_{2})$
do not mix with $\Phi$ and $\sigma$ due to the residual $Z_{2}$ symmetry. The
mass matrix for inert scalar fields are

$M(\eta_{1},s_{1},\eta_{2},s_{2})=(\eta_{1}^{{\dagger}},s_{1}^{{\dagger}},\eta_{2}^{{\dagger}},s_{2}^{{\dagger}})\left(\begin{array}[]{cccc}M_{11}&M_{12}&0&0\\\
M_{21}&M_{22}&0&0\\\ 0&0&M_{33}&M_{34}\\\
0&0&M_{43}&M_{44}\end{array}\right)\left(\begin{array}[]{c}\eta_{1}\\\
s_{1}\\\ \eta_{2}\\\ s_{2}\end{array}\right)$ (15)

where

$\begin{split}&M_{11}=\mu_{\eta_{1}}^{2}+\frac{1}{2}\lambda_{\phi\eta_{1}}v_{\phi}^{2}+\frac{1}{2}\lambda_{\phi\eta_{1}}^{\prime}v_{\phi}^{2}+\frac{1}{2}\lambda_{\sigma\eta_{1}}^{2}v_{\sigma}^{2}\\\
&M_{22}=\mu_{s_{1}}^{2}+\frac{1}{2}\lambda_{s_{1}\phi}v_{\phi}^{2}+\frac{1}{2}\lambda_{s_{1}\sigma}v_{\sigma}^{2}\\\
&M_{33}=\mu_{\eta_{2}}^{2}+\frac{1}{2}\lambda_{\phi\eta_{2}}v_{\phi}^{2}+\frac{1}{2}\lambda_{\phi\eta_{2}}^{\prime}v_{\phi}^{2}+\frac{1}{2}\lambda_{\sigma\eta_{2}}^{2}v_{\sigma}^{2}\\\
&M_{44}=\mu_{s_{2}}^{2}+\frac{1}{2}\lambda_{s_{2}\phi}v_{\phi}^{2}+\frac{1}{2}\lambda_{s_{2}\sigma}v_{\sigma}^{2}\\\
&M_{12}=M_{21}=\frac{\mu_{1}}{\sqrt{2}}v_{\phi}\\\
&M_{34}=M_{43}=\frac{\mu_{2}}{\sqrt{2}}v_{\phi}\end{split}$ (16)

There is also no mixing between $(\eta_{1},s_{1})$ and $(\eta_{2},s_{2})$,
therefor a residual $Z_{2}^{\prime}$ symmetry between the two classes can be
realised. After diagonalizing the mass matrix, we obtain the mass eigenstates
of inert scalars as

$\left(\begin{array}[]{c}A_{1,2}^{0}\\\
H_{1,2}^{0}\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta_{1,2}&-\sin\theta_{1,2}\\\
\sin\theta_{1,2}&\cos\theta_{1,2}\end{array}\right)\left(\begin{array}[]{c}\eta_{1,2}^{0}\\\
s_{1,2}^{0}\end{array}\right)\quad,\quad\sin
2\theta_{1,2}=\frac{\sqrt{2}\mu_{1,2}v_{\phi}}{M_{A_{1,2}}^{2}-M_{H_{1,2}}^{2}}$
(17)

where

$M_{A_{1,2}}^{2}=\frac{1}{2}\Big{(}M_{\eta_{1,2}}^{2}+M_{s_{1,2}}^{2}+\sqrt{(M_{\eta_{1,2}}^{2}-M_{s_{1,2}}^{2})^{2}+2\mu_{1,2}^{2}v_{\phi}^{2}}\Big{)}$
(18)
$M_{H_{1,2}}^{2}=\frac{1}{2}\Big{(}M_{\eta_{1,2}}^{2}+M_{s_{1,2}}^{2}-\sqrt{(M_{\eta_{1,2}}^{2}-M_{s_{1,2}}^{2})^{2}+2\mu_{1,2}^{2}v_{\phi}^{2}}\Big{)}$
(19)

Here $M_{\eta_{1}}\equiv M_{11}$, $M_{s_{1}}\equiv M_{22}$,
$M_{\eta_{2}}\equiv M_{33}$ and $M_{s_{2}}\equiv M_{44}$.

###  II.3 Neutrino Mass

Figure 1: The one-loop diagrams for neutrino masses in our model.

As shown in Fig. 1, the tiny neutrino mass are generated by the linear seesaw
mechanism except that the $\mu_{L}$ terms are induced by a one-loop diagram.
The effective mass matrix for active neutrinos depicted in Fig. 1 is expressed
as

$M_{\nu ll^{\prime}}=M_{\nu ll^{\prime}}^{I}+M_{\nu ll^{\prime}}^{II}$ (20)

where

$\begin{split}&M_{\nu
ll^{\prime}}^{I}=\frac{v_{\phi}\sin\theta_{1}\cos\theta_{1}}{16\pi^{2}\sqrt{2}M_{\psi}}y_{l}\sum_{i=1}^{2}h_{i}f_{il^{\prime}}M_{i}\Big{[}\frac{M_{A_{1}^{2}}}{M_{i}^{2}-M_{A_{1}}^{2}}\ln\big{(}\frac{M_{A_{1}}^{2}}{M_{i}^{2}}\big{)}-\frac{M_{H_{1}^{2}}}{M_{i}^{2}-M_{H_{1}}^{2}}\ln\big{(}\frac{M_{H_{1}}^{2}}{M_{i}^{2}}\big{)}\Big{]}+(l\leftrightarrow
l^{\prime})\\\ &M_{\nu
ll^{\prime}}^{II}=\frac{v_{\phi}\sin\theta_{2}\cos\theta_{2}}{16\pi^{2}\sqrt{2}M_{\psi}}y_{l}hf_{l^{\prime}}M_{\chi}\Big{[}\frac{M_{A_{2}^{2}}}{M_{\chi}^{2}-M_{A_{2}}^{2}}\ln\big{(}\frac{M_{A_{2}}^{2}}{M_{\chi}^{2}}\big{)}-\frac{M_{H_{2}}^{2}}{M_{\chi}^{2}-M_{H_{2}}^{2}}\ln\big{(}\frac{M_{H_{2}}^{2}}{M_{\chi}^{2}}\big{)}\Big{]}+(l\leftrightarrow
l^{\prime})\end{split}$ (21)

where $M_{i}(i=1,2)$ denotes the masses for $N_{R1}$ and $N_{R2}$; $M_{\chi}$
denotes the masses for the eigenstates of $N_{R}^{\prime}$ and
$N_{R}^{\prime\prime}$. Tiny neutrino masses can be obtained using the
following benchmark points:

$\displaystyle\mu_{1}=\mu_{2}=0.1~{}\textrm{GeV},$ $\displaystyle
y=h=0.0028,~{}f=0.01,M_{\psi}=300~{}\textrm{GeV}$ $\displaystyle
M_{N_{R1}}=149.5~{}\textrm{GeV},$ $\displaystyle
M_{N_{R2}}=200~{}\textrm{GeV},M_{\chi}=150~{}\textrm{GeV}$ (22) $\displaystyle
M_{A^{0}_{1}}=300~{}\textrm{GeV},$ $\displaystyle
M_{\eta^{\pm}_{1}}=270~{}\textrm{GeV},M_{H^{0}_{1}}=1000~{}\textrm{GeV}$
$\displaystyle M_{A^{0}_{2}}=700~{}\textrm{GeV},$ $\displaystyle
M_{\eta^{\pm}_{2}}=690~{}\textrm{GeV},M_{H^{0}_{2}}=62~{}\textrm{GeV}$

with the index of Yukawa couplings suppressed for simplicity. Then we get
$M_{\nu}=0.0164~{}\textrm{eV}(\sqrt{\Delta m_{13}^{2}})$.

The benchmark point given above seems rather unusual since the $M_{\nu}$
becomes a rank-1 matrix as

$\begin{split}M_{\nu}\sim D\equiv\left(\begin{array}[]{ccc}1&1&1\\\ 1&1&1\\\
1&1&1\end{array}\right)\end{split}$ (23)

which is obviously not consistent with the results of neutrino oscillation
experiments. However, one reminds the expression of matrix (23) is just the
so-called flavor democratic model studied by many authors and related to some
flavor symmetrieszzx1 . The matrix $D$ can be diagonalized as

$V_{\nu}^{T}DV_{\nu}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&0&3\end{array}\right)\equiv\widehat{D}\quad\quad
V_{\nu}=\frac{1}{\sqrt{6}}\left(\begin{array}[]{ccc}\sqrt{3}&1&\sqrt{2}\\\
-\sqrt{3}&1&\sqrt{2}\\\ 0&-2&\sqrt{2}\end{array}\right)$ (24)

The unitary matrix $V_{\nu}$ corresponds to the democratic mixing pattern.
Since $\widehat{D}$ contains a dominant non-zero element at $(3,3)$ position,
the flavor democratic structure in $M_{\nu}$ in eqn(23) can be viewed as a
good approximation for our rank-2 neutrino mass matrix which exhibiting the
strong and normal order of neutrino mass spectrum. i.e $m_{1}=0\ll
m_{2}=\sqrt{\Delta m_{12}^{2}}\ll m_{3}=\sqrt{\Delta m_{13}^{2}}$. Thus the
benchmark point we take is reasonable. It is noted in the flavor basis the
democratic mixing matrix $V_{\nu}$ has already not been consistent with the
PMNS matrix $U_{PMNS}$ measured by experiments, however the $V_{\nu}$ can be
corrected by the charged lepton sector $V_{l}$ to fit the neutrino oscillation
dataccro .

The small values of $\mu_{1,2}$ lead to $\sin 2\theta_{1,2}=(3.8,7.2)\times
10^{-5}$, which plays a key rule in the suppression of tiny neutrino masses.
The choice of values of $\mu_{1,2}$ is mainly for phenomenological
consideration. First, in case of scalar DM, we have $Z-S-S^{*}$ coupling
proportional to $\sin^{2}\theta_{1,2}$. The spin independent elastic cross
section of DM requires $\sin\theta_{1,2}<0.05$Akerib:2013tjd , setting an
upper limit on $\mu_{1,2}\sim(10~{}\textrm{GeV})$ for electroweak(EW) scale
inert scalars. Second, in our benchmark point with inert particles at the EW-
scale and Yukawa couplings of the order $10^{-2}$ or $10^{-3}$, rich
phenomenon of new physics are expected for LHC and LFV processes. On the other
hand, larger values of $\mu_{1,2}$ is possible if we decrease the Yukawa
couplings to be more smaller values. But this will predict a too small branch
ratios for LFV processes. Too small Yukawa coupling also seems unnatural from
the viewpoint of model building. Another solution is to increase the mass of
$\psi$ or inert particle to TeV-scale, which is beyond the reach of LHC. For
the DM candidate $N_{R1}$($H^{0}_{2}$), its mass is set to be about half of
the mass of the s-channel mediator $H$(h). Then one obtains the large enough
DM annihilation cross section to account for the relic density. For the heavy
dirac fermion $\psi$ and inert doublet scalars $\eta_{1},A^{1}_{0}$, we choose
their masses around 300 GeV, so that they are testable at LHC. The other inert
scalars are around TeV-scale aiming to suppress the value of neutrino mass.

##  III Phenomenology

###  III.1 Lepton Flavor Violation

The Yukawa interactions of charged-scalar $\eta^{\pm}$ will contribute to the
LFV processes of charged leptons. Detail studies on LFV processes in
scotogenic models Ma:2006km  have been carried out in Ref.Toma:2013zsa ;
Vicente:2014wga . Currently, the most severe constraint coming from MEG
collaboration on muon radiative decay with an upper limit BR($\mu\to
e\gamma$)$<5.7\times 10^{-13}$ (90% C.L.) Adam:2013mnn . In our model, the
analytical branching ratio of $\mu\to e\gamma$ is calculated as Hisano:1995cp
; Toma:2013zsa :

$\mbox{BR}(\mu\to e\gamma)=\frac{3\alpha_{em}}{64\pi
G^{2}_{F}}\left|\sum_{i=1}^{2}\frac{f_{i\mu}f_{ie}^{*}}{M_{\eta^{+}_{1}}^{2}}F\left(\frac{M_{N_{i}}^{2}}{M_{\eta^{+}_{1}}^{2}}\right)+\frac{f_{l}f_{l}^{*}}{M_{\eta^{+}_{2}}^{2}}F\left(\frac{M_{N^{\chi}}^{2}}{M_{\eta^{+}_{2}}^{2}}\right)\right|^{2},$
(25)

where the loop function $F(x)$ is:

$F(x)=\frac{1-6x+3x^{2}+2x^{3}-6x^{2}\mbox{ln}x}{6(1-x)^{4}}.$ (26)

The benchmark point in Eqn.II.3 predicts BR$(\mu\to e\gamma)=9.8\times
10^{-14}$, which satisfies the current limit and is in the reach of future
sensitivity Baldini:2013ke . The limits on $\tau$ observables are less
stringent Aubert:2009ag ; Hayasaka:2010np . With the much natural Yukawa
structure in our benchmark point, the predicted
BR$(\tau\to\mu\gamma)=5.8\times 10^{-13}$ is far beyond the future
sensitivity. But on the other hand, hierarchal Yukawa structure
$|f_{ie}|\lesssim|f_{i\mu}|\lesssim|f_{i\tau}|$ with
$f_{i\tau}\sim\mathcal{O}(1)$ is still allowed from phenomenological point of
view. In this case, fermionic dark matter candidate $F$ annihilation in the
mass region between $2~{}\textrm{GeV}$ and $3~{}\textrm{TeV}$ through
$t$-channel exchange of $\eta$ can satisfy dark matter relic density bound
Vicente:2014wga . And as a consequence of hierarchal Yukawa structure, dark
matter $F$ annihilates mainly into third-family leptons: $\tau^{+}\tau^{-}$
and $\nu_{\tau}\bar{\nu}_{\tau}$.

###  III.2 Dark Matter

In our model, a multi-component dark matter scenario is possible due to the
residual $Z_{2}^{\prime}$ symmetry between two sets of new scalars and
fermions. For instance, we choose the lightest fermion $N_{R1}$ (refer as $F$)
in set one and the lightest scalar $H_{2}^{0}$ (refer as $S$) in set two as
dark matter candidates. These dark matter candidates must satisfy two
experimental constrains : (1) the dark matter relic density observed by Plank
Ade:2013zuv  $\Omega_{DM}h^{2}=0.1193\pm 0.0014$; (2) cross section for direct
detection of dark matter scattering off nucleon set by LUX Akerib:2013tjd .

Theoretical calculation of dark matter relic density is well described in
Bertone:2004pz , and it is calculated with the help of packages Feynrules
feynrules  and micrOMEGAs micromegas  in our analysis. Because the $t$-channel
Yukawa portal may suffer constrains from LFV or neutrino masses, and in our
benchmark point, the $t$-channel contribution to relic density is less than
1%, thus we will focus on the $s$-channel $h/H/Z^{\prime}$ portal for
simplicity. Firstly, the relic density of fermion/scalar dark matter for one
dark matter candidate is presented in Fig. 2, where we neglect the conversion
$F\bar{F}\leftrightarrow SS^{*}$ between the two dark matter candidates. For
the fermion dark matter $F$, the light $h$ portal can not acquire a sufficient
annihilation cross section, while the heavy $H$ portal is still promising when
$M_{F}\sim M_{H}/2$, which is because the suppression of large
$v_{\sigma}=8~{}\textrm{TeV}$. Anyway, the $Z^{\prime}$ portal can easily
satisfy the relic density when $M_{F}\sim M_{Z^{\prime}}/2$. For the scalar
dark matter $S$, it is dominantly made from the singlet scalar $s_{2}^{0}$,
due to small mixing $\theta_{2}$. The relic density can easily be attained
when $M_{S}\sim M_{h}/2$ and $M_{S}\sim M_{H}/2$, while the $Z^{\prime}$
portal is not promising, mainly because the small $B-L$ charge of $S$ and
suppression of heavy $M_{Z^{\prime}}$.

Figure 2: Relic density of fermion (left) and scalar (right) dark matter as a
function of $M_{DM}$ for one dark matter candidate. Here, we set the relevant
scalar interaction coupling
$\lambda_{\eta_{2}\Phi}=\lambda^{\prime}_{\eta_{2}\Phi}=\lambda_{s_{2}\Phi}=-0.001$,
$\lambda_{\sigma\eta_{2}}=\lambda_{s_{2}\sigma}=-0.001$. We also fix
$M_{h}=125~{}\textrm{GeV}$, $M_{H}=300/600~{}\textrm{GeV}$, $\sin\theta=0.3$,
$M_{Z^{\prime}}=4~{}\textrm{TeV}$, and $v_{\sigma}=8~{}\textrm{TeV}$.

Secondly, we take into account the conversion of two component dark matter
$F\bar{F}\leftrightarrow SS^{*}$, which can be mediated by $s$-channel
$h/H/Z^{\prime}$, where the $H$-portal is expected to be the dominant one.
Therefor, $HSS^{*}$ and $HF\bar{F}$ are the two most relevant couplings to
study conversion. For simplicity, we further assume
$\lambda_{\sigma\eta_{2}}=\lambda_{s_{2}\sigma}=\lambda$, which determines
$HSS^{*}$ coupling, and fix other parameters as discussed in FIG. 2 if not
mentioned.

The dependence of $F/S$ relic density on $\lambda$ is shown in FIG. 3. For the
fermion dark matter, when $M_{F}>M_{S}$, the larger $\lambda$ the larger
$F\bar{F}\leftrightarrow SS^{*}$ annihilation rate, and therefor the smaller
the relic density. It is clear that the $\Omega_{F}h^{2}$ can differ by about
one order of magnitude between $\lambda=-0.001$ and $\lambda=-0.05$. But when
$M_{F}<M_{S}$, the effect of conversion $F\bar{F}\to SS^{*}$ is quite small.
Noticing for $\lambda=-0.05$, the $H\to SS^{*}$ can greatly enhance the total
decay width of $H$, which causes the increase of $\Omega_{F}h^{2}$ around
$M_{F}\sim M_{H}/2$. For the scalar dark matter, the increase of $\lambda$
will decrease the relic density significantly due to the increase of $SS^{*}$
annihilation. But, the friction of conversion $SS^{*}\to F\bar{F}$ keeps the
same, since the Yukawa coupling $HF\bar{F}$ is fixed by $M_{F}$. The arguments
are true if we exchange the roles of $F$ and $S$, and fix $HSS^{*}$ coupling
while verify $HF\bar{F}$ coupling.

Figure 3:  The dependence of $F/S$ relic density on $\lambda$.

Another aspect of conversion is the masses of the two dark matter. The
left(right) of FIG. 4 shows the $F(S)$ relic density for
$M_{S(F)}=60,150,300~{}\textrm{GeV}$ with $\lambda=-0.02$. In case of fermion
dark matter, it is clear that the smaller the $M_{S}$ the larger
$F\bar{F}\rightarrow SS^{*}$ annihilation rate, and therefore the smaller the
relic density. For relative heavy $M_{S}=150,300~{}\textrm{GeV}$, the
conversion has tiny effect on the $F$ relic density when $M_{F}<M_{H}/2$. In
case of scalar dark matter, the dependence of $\Omega_{S}h^{2}$ on $M_{F}$ is
a little complicated, since the $HF\bar{F}$ coupling is directly related with
the $M_{F}$. For $M_{S}<80~{}\textrm{GeV}$, the effect of conversion is
relatively small, and one expect that the larger the $M_{F}$ the smaller the
$S$ relic density, which is mainly caused by the increase of $HF\bar{F}$
coupling. In medium mass region $80<M_{S}<200~{}\textrm{GeV}$, the conversion
effect would be dominant, thus the smaller the $M_{F}$ the smaller the $S$
relic density. In the high mass region, the conversion effect is comparable to
$HF\bar{F}$ coupling effect, which makes the dependence of $\Omega_{S}h^{2}$
on $M_{F}$ nonlinear. In a word, the $HSS^{*}$ and $HF\bar{F}$ couplings play
vital importance in dark matter conversion. The conversion can take place in
both direction $F\bar{F}\to SS^{*}$ and $SS^{*}\to F\bar{F}$ when $M_{F}\sim
M_{S}$, which can be obtained when both $F$ and $S$ are mainly annihilation
through $H$-portal. If not the case, only the conversion of heavier one into
light one is relevant Esch:2014jpa .

Figure 4:  The effect of two component dark matter conversion on $F/S$ relic
density for fixed $\lambda=-0.02$.

Finally, we discuss the constrains from direct detection of dark matter. The
Current experiment constraints assume the existence of only one dark matter
specie. However two-component dark matter candidates are predicted in our
model. Therefore the contribution of cross section on nucleon for each specie
should be rescaled by the fraction factor of relic density. We define the
fraction of the mass density of $i$-th dark matter in case of multi-component
dark matter Cao:2007fy ; Aoki:2012ub :

$\epsilon_{i}=\frac{\Omega_{i}h^{2}}{\Omega_{CDM}h^{2}},$ (27)

where $i=F,S$ in our consideration. Thereafter, the up limit of direct
detection is:

$\frac{\epsilon_{F}}{M_{F}}\sigma_{F-N}+\frac{\epsilon_{S}}{M_{S}}\sigma_{S-N}<\frac{\sigma_{exp}}{M_{DM}}$
(28)

Here, $\sigma_{F-N}(\sigma_{S-N})$ denotes the scattering cross section of
$F(S)$ with a nucleon $N$. The benchmark point in Eqn. II.3 gives the spin-
independent scattering cross section $\sigma_{F-N}^{SI}=1.10\times
10^{-46}~{}\textrm{cm}^{2}$ ($\sigma_{S-N}^{SI}=1.62\times
10^{-44}~{}\textrm{cm}^{2}$) with $\Omega_{F}h^{2}=1.12\times 10^{-1}$
($\Omega_{S}h^{2}=2.64\times 10^{-4}$). Although the bare $\sigma_{S-N}^{SI}$
is larger than the LUX upper constraint $1.1\times
10^{-47}\textrm{cm}^{2}/\textrm{GeV}$Akerib:2013tjd , the contribution of
scalar $S$ to the scattering on nucleon is suppressed because of its small
faction $\epsilon_{S}=2.21\times 10^{-3}$. The value of expression on left of
Eqn. 28 is $1.3\times 10^{-48}\textrm{cm}^{2}/\textrm{GeV}$, which is smaller
than current LUX bound. Thus, the fermion dark matter is dominant in this
scenario, while the scalar dark matter must be less than $4\%$ to escape
current LUX bound.

###  III.3 Collider Signatures

As shown in our bench mark point (Eqn. II.3), the new particles are all at the
electroweak scale, which makes them testable at LHC. Interactions between
these new particles and SM Higgs $h$ will of course modify the properties of
$h$, thus give us some indirect hints. Nowadays the most precise measurement
of $M_{h}$ is the combined results of ATLAS and CMS Aad:2015zhl :

$M_{h}=125.09\pm 0.21(\mbox{stat.})\pm 0.11(\mbox{sys.})~{}\textrm{GeV}$ (29)

Apparently, the extra of new scalars and fermions would change the decay rates
of SM Higgs $h$. For instance, mixing between $h$ and additional scalar
singlet $H$ will modify tree-level $h$ decays. And the additional charged-
scalars $\eta^{\pm}_{1,2}$ will contribute to the loop-induced decays as
$h\to\gamma\gamma$ Arhrib:2012ia .

It’s well known that, for Higgs-portal dark matter, upper limits on Higgs
invisible decay is also interpreted an upper limits on dark matter-nucleon
scattering cross section Baek:2014jga . Direct measurement of Higgs invisible
decay in associated with $Z$ by ATLAS set an upper limit of 75% at 95% C.L.
Aad:2014iia . Combined analysis with Higgs signal strength give a more tight
upper limit of 37% at 95% C.L. Bechtle:2014ewa ; Corbett:2015ksa ; Aad:2015gba
. In the future, the weak boson fusion channel might have the ability to probe
invisible decay to 2-3% with $3000~{}\textrm{fb}^{-1}$ at LHC
Bernaciak:2014pna . Light dark matter candidates in our model will contribute
the the invisible decays of $h$ in our model. In case of scalar dark matter,
the branching ratio of Higgs invisible decay, i.e., BR($h\to SS^{*}$) is
$1.7\%$ for $M_{S}=62~{}\textrm{GeV}$ with
$\lambda_{s_{2}\Phi}=\lambda_{s_{2}\sigma}=-0.001$, $\sin\theta=0.3$. On the
other hand for fermion dark matter, BR($h\to F\bar{F}$) is $1.7\times 10^{-6}$
for $M_{F}=62~{}\textrm{GeV}$ with $v_{\sigma}=8~{}\textrm{TeV}$,
$\sin\theta=0.3$, although it might not be favored by the constrains on relic
density of dark matter.

Figure 5: Branching ratios of $H$ as a function of $M_{H}$ for
$\sin\theta=0.3$(left) and $\sin\theta=0.01$(right) in our benchmark point in
Eqn. II.3.

Then we discuss the mixing between $h$ and $H$. The analysis of signal
strength of $h$ constrains $\sin^{2}\theta<0.23$ at 95% C.L. Giardino:2013bma
; Falkowski:2013dza . Direct search of $H$ in the $ZZ$ and $WW$ channel now
has push this limit down to $\sin^{2}\theta<0.1$ with no new physics
contribute to the decays of $H$ Pelliccioni:2015hva . Future hadron collider,
i.e., HL-LHC, has the ability to probe $\sin^{2}\theta\sim 4\times 10^{-2}$,
and lepton collider, i.e., CEPC, could reach $\sin^{2}\theta\sim 2\times
10^{-3}$ Dawson:2013bba . In Fig. 5, we show the branching ratios of $H$ for
two values of $\sin\theta(0.3,0.01)$. For a relatively large mixing angle
$\sin\theta=0.3$, the heavy neutral Higgs $H$ decays dominantly into SM
particles. The branching ratio of invisible decay $H\to SS^{*}$ can reach
$10\%$ for $M_{H}\sim 165~{}\textrm{GeV}$, and we expect it becomes dominant
when $M_{H}<160~{}\textrm{GeV}$. The branching ratios of $H$ decaying into
other new physical particles are below $10^{-3}$ in this case. While for
$M_{H}\gg M_{W}$, it is well known that decays of $H$ into vector bosons are
determined by their Goldstone nature, which implies:

$\mbox{BR}(H\to hh)\approx\mbox{BR}(H\to ZZ)\approx\frac{1}{2}\mbox{BR}(H\to
WW)$ (30)

The asymptotic behavior of this relation is clear shown on left picture of
Fig. 5. On the other aspect, for a relatively small mixing angle
$\sin\theta=0.01$, decays of $H$ into SM particles will be suppressed and
decays into new particles will be greatly enhanced. $H\to SS^{*}$ is dominant
when $M_{H}<400~{}\textrm{GeV}$. The branching ratio of $\psi\bar{\psi}$ will
reach about 0.25 when $M_{H}\sim 600~{}\textrm{GeV}$, which is comparable with
$H\to W^{+}W^{-}$. In this case, $H\to F\bar{F}$ is below $10\%$ and
$H\to\eta^{+}\eta^{-}/A_{1}^{0}A_{1}^{0*}$ is below $2\%$.

The heavy neutral Higgs $H$ is testable for large mixing angle $\theta$. For
example, the promising channels to probe heavy neutral Higgs $H$ would be
$ZZ\to 4l$, $ZZ\to 2l2\nu$, $ZZ\to 2l2j$, $ZZ\to 2l2\tau$, $WW\to 2l2\nu$,
$WW\to l\nu 2j$, $hh\to 4b$, and $hh\to 2b2\gamma$ Buttazzo:2015bka . At the
same time, we would like to mention that the heavy Higgs $H$ could enhance the
di-Higgs production $hh$ Baglio:2012np  by a factor of 18 comparing to
standard model case Chen:2014ask . For small mixing angle $\theta$, production
of $H$ will be suppressed by this small $\theta$, thus makes it challenging to
probe directly at colliders.

Figure 6: Total decay width of $Z^{\prime}$ as a function of $M_{Z^{\prime}}$
(for fixed values of $g_{B-L}$), and $g_{B-L}$ (for fixed values of
$M_{Z^{\prime}}$).

Next we review the properties of $U(1)_{B-L}$ gauge boson $Z^{\prime}$. With
about $20~{}\textrm{fb}^{-1}$ data at $8~{}\textrm{TeV}$ LHC, bound on
$Z^{\prime}$ has been push up to $2.95~{}\textrm{TeV}$ by CMS through the
ratio $R_{\sigma}=\sigma(pp\to Z^{\prime}\to\ell^{+}\ell^{-})/\sigma(pp\to
Z\to\ell^{+}\ell^{-})$, where $\ell=e,\mu$ Khachatryan:2014fba . In our
benchmark point, we choose $M_{Z^{\prime}}=4~{}\textrm{TeV}$ and $g_{B-L}=0.5$
($v_{\sigma}=8\textrm{TeV}$), which can safely satisfy current experimental
limits and can be tested at $14~{}\textrm{TeV}$ LHC with
$100~{}\textrm{fb}^{-1}$ Basso:2008iv ; Basso:2010pe . Fig. 6 shows the total
decay width of $Z^{\prime}$ as a function of $M_{Z^{\prime}}$ and $g_{B-L}$.
Depending on $g_{B-L}$, $\Gamma_{Z^{\prime}}$ varies from a few to hundreds of
GeV. For such large $\Gamma_{Z^{\prime}}$, it can be directly measured by the
leptonic final states at LHC Basso:2008iv ; Basso:2010pe .

In table 2, we give the decay branching ratios of $Z^{\prime}$ in our
benchmark point. The dominant decay channels of $Z^{\prime}$ are
$q\bar{q},l\bar{l}$, and $\nu_{L}\bar{\nu}_{L}$, while all of the new particle
final states only account for about $20\%$. A distinct feature of $U(1)_{B-L}$
gauge boson $Z^{\prime}$ is the definite relation between quark and lepton
final states:

$\mbox{BR}(Z^{\prime}\to q\bar{q}):\mbox{BR}(Z^{\prime}\to l\bar{l})\simeq
2:3$ (31)

after summing over all flavors. This relation can be used to distinguish
$U(1)_{B-L}$ gauge boson $Z^{\prime}$ from $Z^{\prime}$ in other models
Langacker:2008yv . More practical on experiment, the B-L nature of
$Z^{\prime}$ can be tested if BR$(Z^{\prime}\to
b\bar{b})/$BR$(Z^{\prime}\to\mu^{+}\mu^{-})=1/3$ is confirmed Kanemura:2011mw
. In our model with only left-handed light neutrinos, the dominant invisible
decay channel of $Z^{\prime}$ is BR$(Z^{\prime}\to\nu_{L}\bar{\nu}_{L})$,
which is half of BR$(Z^{\prime}\to l\bar{l})$. Further with dark matter
candidate in our model, $Z^{\prime}$ invisible decays get additional
contributions from $Z^{\prime}$ into dark matter pairs. For instance,
BR$(Z^{\prime}\to\mbox{inv.})$ could be 0.2457, 0.1990, 0.1964 for $FF$, $FS$
and $SS$ dark matter separately. So precise measurement of
BR$(Z^{\prime}\to\mbox{inv.})$ will shed light on the nature of dark matter.

$q\bar{q}$ | $l\bar{l}$ | $\nu_{L}\bar{\nu}_{L}$ | $\psi\bar{\psi}$ | $N_{R1}\bar{N}_{R1}$ | $N_{R2}\bar{N}_{R2}$ | $\chi_{1}\bar{\chi_{1}}$ | $\chi_{2}\bar{\chi_{2}}$
---|---|---|---|---|---|---|---
0.25 | 0.38 | 0.19 | 0.063 | 0.0077 | 0.0077 | 0.016 | 0.032
$HH$ | $hh$ | $A^{0}_{1}A_{1}^{0*}$ | $H^{0}_{1}H_{1}^{0*}$ | $A^{0}_{2}A_{2}^{0*}$ | $H^{0}_{2}H_{2}^{0*}$ | $\eta_{1}^{+}\eta_{1}^{-}$ | $\eta_{2}^{+}\eta_{2}^{-}$
0.030 | 0 | 0.0076 | 0.0051 | 0.0010 | 0.0013 | 0.0076 | 0.0010
Table 2: Branching ratios of $Z^{\prime}$ in our benchmark point. Here, we set
$M_{H}=300~{}\textrm{GeV}$ and $\sin\theta=0$ for simplicity. Figure 7:
Branching ratios of $\psi$ as a function of $M_{\psi}$. We have fixed
$\sin\theta=0.3$, $M_{H}=300~{}\textrm{GeV}$ and $y=h=0.01$.

Another interesting feature of our model is the existence of heavy Dirac
fermion $\psi$, thus there are no lepton number violation (LNV) decays as
$\psi\to W^{-}l^{+}$. For $M_{\psi}<M_{h}$, the Higgs decay into a pair of
light and heavy neutrinos, $h\to\bar{\nu}\psi+\bar{\psi}\nu$ will open, which
could increase $\Gamma_{h}$ by up to almost 30% and significantly affects
Higgs searches at the LHC Chen:2010wn . In this paper, we consider
$M_{\psi}>M_{h}$. Therefor, decay channels of $\psi$ could be $W^{+}l^{-}$,
$Z\nu$, $h\nu$, and if kinematically allowed $H\nu$, $A^{0}_{i}N_{j}$,
$A^{0}_{i}\chi_{j}$, $H^{0}_{i}N_{j}$, $H^{0}_{i}\chi_{j}(i,j=1,2)$ are also
possible. Due to tiny mixing angle $\theta_{1,2}$, branching ratios of
$\psi\to A^{0}_{i}N_{j},A^{0}_{i}\chi_{j}$ are negligible. In Fig. 7, we show
the branching ratios of $\psi$. It is clear that $\psi$ will decays dominantly
into standard model final states for comparable Yukawa couplings of $y$ and
$h$. Approximately for $M_{\psi}\gg M_{W}$, we have:

$\frac{1}{\cos^{2}\theta}\mbox{BR}(\psi\to h\nu)\approx\mbox{BR}(\psi\to
Z\nu)\approx\frac{1}{2}\mbox{BR}(\psi\to W^{+}l^{-})$ (32)

Decays of $\psi$ into new physical particles is small in this case.
BR$(\psi\to H^{0}_{2}\chi)$ is about $10\%$, once is kinematically opened.
BR$(\psi\to H\nu)$ is suppressed by $\sin^{2}\theta$, thus it is always much
smaller. As shown in table 2, BR$(Z^{\prime}\to\psi\bar{\psi})\simeq 0.063$
for one generation in our model, so $\psi\bar{\psi}$ can be produced through
$Z^{\prime}$ portal. A possible promising signature is the tri-lepton channel
Basso:2008iv :

$pp\to Z^{\prime}\to\psi\bar{\psi}\to W^{+}l^{-}+W^{-}l^{+}\to
2l^{\pm}l^{\mp}jj+\cancel{E}_{T}$ (33)

The cross section of this tri-lepton signal is about 0.017 fb in our benchmark
point, so the tri-lepton is only promising at future high-luminosity LHC. The
mass of $\psi$ can be reconstructed using the transverse mass of two opposite
sign leptons with missing transverse momentum Basso:2008iv . Another feature
of $\psi$ is the possible large mixing with $\nu_{L}$ comparing to canonical
type-I seesaw type1 . As discussed in Sec. II.3, the mixing $V_{\nu\psi}$
between $\nu_{L}$ and $\psi_{L}$ is $M_{D}/\sqrt{M_{D}^{2}+M_{\psi}^{2}}$. For
$M_{D}\sim\mathcal{O}(1)\textrm{GeV}$,
$M_{\psi}\sim\mathcal{O}(100)\textrm{GeV}$,
$V_{\nu\psi}\sim\mathcal{O}(10^{-2})$. Thereafter, $\psi$ could be largely
associated produced with charged leptons through $W$ Bambhaniya:2014kga :

$\displaystyle pp\to W^{*}\to l^{\pm}\psi\to l^{\pm}+W^{\pm}l^{\mp}\to
l^{\pm}l^{\mp}jj$ (34) $\displaystyle pp\to W^{*}\to l^{\pm}\psi\to
l^{\pm}+W^{\pm}l^{\mp}\to l^{\pm}l^{\mp}l^{\pm}\cancel{E}_{T}$ (35)

The production cross section
$\sigma(l^{\pm}\psi)=350\times|V_{i\psi}|^{2}~{}\textrm{fb}$ in our benchmark
point. And it might be promising at 14TeV LHC with about
$100~{}\textrm{fb}^{-1}$. Testability of this heavy Dirac neutrino $\psi$ is
less promising than the heavy Majorana neutrinos with same mixing scale, since
the latter could rise LNV signatures Han:2006ip ; del Aguila:2007em ;
Atre:2009rg ; Deppisch:2015qwa .

Finally, we discuss the decays of inert scalars and fermions. $N_{R1}$ and
$H^{0}_{2}$ are a dark matter candidate as in our benchmark point in Eqn.
II.3. Decays of $N_{R2}$ are dominated by $N_{R2}\to l^{\pm}\eta_{1}^{\mp*}\to
l^{\pm}l^{\mp}N_{R1}$ and $N_{R2}\to\nu A_{1}^{0*}\to\nu\nu N_{R1}$ through
the Yukawa coupling $f$. Decays of $\chi_{i}$ are $\chi_{i}\to
H^{0}_{2}\psi^{*}$ with the off-shell $\psi^{*}$ further decaying into
$W^{+}l^{-}/Z\nu/h\nu/H^{0}_{2}\nu$. $A_{1}^{0}$ and $\eta_{1}^{\pm}$ mainly
decay through the Yukawa coupling $f$, which leads to $A_{1}^{0}\to\nu N_{Ri}$
and $\eta_{1}^{\pm}\to l^{\pm}N_{Ri}$. The heavy $Z_{2}$ odd scalar
$H_{1}^{0}$ decays into $\psi N_{Ri}$ through Yukawa coupling $h_{\alpha}$ and
into $hA_{1}^{0}$ through trilinear coupling $\mu_{1}$. Similar, decays of
$A^{0}_{2}$ are $A^{0}_{2}\to\nu\chi_{i}$ and $A^{0}_{2}\to hH_{2}^{0}$, while
decays of $\eta_{2}^{\pm}$ are $\eta_{2}^{\pm}\to l^{\pm}\chi_{i}$ and
$\eta_{2}^{\pm}\to W^{\pm}H^{0}_{2}$.

Particles | $\eta_{1}^{+}\eta_{1}^{-}$ | $\eta^{\pm}_{1}A_{1}^{0}$ | $A_{1}^{0}A_{1}^{0*}$ | $\eta_{2}^{+}\eta_{2}^{-}$ | $\eta^{\pm}_{2}A_{2}^{0}$ | $A_{2}^{0}A_{2}^{0*}$
---|---|---|---|---|---|---
$\sigma$ (in fb) | 5.8 | 16 | 3.6 | 0.089 | 0.31 | 0.075
Table 3: Production cross sections for inert scalar doublets.

The inert scalar doublets can be pair production through DY process. In Table
3, we list the production cross sections for inert scalar doublets. Many
signatures can be risen from the inert particles. In Ma’s scotogenic model
Ma:2006km , promising signals of the doublet scalar on collider are multi-
lepton final states with missing transverse energy $\cancel{E}_{T}$
Dolle:2009ft ; Miao:2010rg ; Gustafsson:2012aj . Similar signals can also be
produced in our model, for example:

$\displaystyle 2l+\cancel{E}_{T}$ $\displaystyle:$
$\displaystyle\eta^{+}_{1}\eta^{-}_{1}\to l^{+}N_{R1}+l^{-}N_{R1}$
$\displaystyle:$ $\displaystyle\eta^{+}_{2}\eta^{-}_{2}\to
W^{+}H^{0*}_{2}+W^{-}H^{0}_{2}\to
l^{+}\nu_{l}H^{0*}_{2}+l^{-}\bar{\nu}_{l}H^{0}_{2}$ $\displaystyle
3l+\cancel{E}_{T}$ $\displaystyle:$ $\displaystyle\eta^{+}_{1}A^{0}_{1}\to
l^{+}N_{R2}+\nu N_{R1}\to l^{+}l^{\pm}l^{\mp}N_{R1}+\nu N_{R1}$
$\displaystyle:$ $\displaystyle\eta^{\pm}_{2}A^{0}_{2}\to
W^{\pm}H^{0(*)}_{2}+hH^{0}_{2}\to
l^{\pm}\nu_{l}H^{0(*)}_{2}+l^{+}\nu_{l}l^{-}\bar{\nu}_{l}H^{0}_{2}$
$\displaystyle 4l+\cancel{E}_{T}$ $\displaystyle:$
$\displaystyle\eta^{+}_{1}\eta_{1}^{-}\to l^{+}N_{R2}+l^{-}N_{R1}\to
l^{+}l^{\pm}l^{\mp}N_{R1}+l^{-}N_{R1}$ $\displaystyle:$ $\displaystyle
A^{0}_{2}H^{0*}_{2}\to hH^{0}_{2}H^{0*}_{2}\to
l^{+}l^{-}l^{+}l^{-}H^{0}_{2}+H^{0*}_{2}$

With much different decay topologies between our and Ma’s model, it would be
distinguishable even with same signals. Apart from these multi-lepton signals,
there are also some other interesting signals in our model, i.e.:

$\displaystyle 2l^{\pm}jj+\cancel{E}_{T}$ $\displaystyle:$
$\displaystyle\eta^{\pm}_{2}A^{0}_{2}\to W^{\pm}H^{0(*)}_{2}+hH^{0}_{2}\to
l^{\pm}\nu_{l}H^{0(*)}_{2}+l^{\pm}\nu_{l}jjH^{0}_{2}$ (39) $\displaystyle
l^{\pm}b\bar{b}+\cancel{E}_{T}$ $\displaystyle:$
$\displaystyle\eta^{\pm}_{2}A^{0}_{2}\to W^{\pm}H^{0(*)}_{2}+hH^{0}_{2}\to
l^{\pm}\nu_{l}H^{0(*)}_{2}+b\bar{b}H^{0}_{2}$ (40)

The lepton number violation signal $2l^{\pm}jj+\cancel{E}_{T}$ suffers much
lower SM background, thus might make this signal very promising on LHC. The
$l^{\pm}b\bar{b}+\cancel{E}_{T}$ has a relatively large production rate due to
$h\to b\bar{b}$ dominant in $h$ decay, so it might also be promising.

##  IV Conclusions

In usual canonical seesaw mechanisms, it requires the heavy states with the
scale of masses being grand unification scale to generated the small neutrino
mass. In linear seesaw scenario with $m_{\nu}\simeq\mu_{L}M_{D}/M_{\Psi}$, the
neutrino masses suffers a two fold suppression by both lepton number symmetry
violating term $\mu_{L}$ and heavy mass $M_{\Psi}$. The linear seesaw model
can lower the seesaw scale such that new physics may arise at TeV scale. In
this work, we construct a radiated linear seesaw model where the naturally
small term $\mu_{L}$ are generated at one-loop level and its soft-breaking of
lepton number symmetry attributes to the SSB of B-L symmetry at TeV scale. To
satisfy the anomalies cancelation, the value of B-L charges for inert
particles are found to be exotic such that there exists residual $Z_{2}\times
Z_{2}^{\prime}$ symmetry even after SSB of $B-L$ gauge symmetry. It is shown
that the residual symmetry stabilizes the inert particles as dark matter
candidates. In our model, we introduce two no-interplay classes of inert
particles to realize the model such that the lightest inert particles
belonging in each class play is the dark matter matter candidate. Therefore we
have propose a two-component dark matter model. The seesaw scale of radiated
linear seesaw scale can be as low as a few hundred GeV, leading to interesting
phenomenology.

Given a benchmark point at electro-weak scale, we illustrate the main
prediction of our model. For the Yukawa coupling $f_{li}(f_{l})$ at 0.01
order, our benchmark point predicts Br$(\mu\to e\gamma)\sim 10^{-13}$, an
order slightly under the current constraints and in the reach of the
forthcoming experiment. The two component dark matter candidates are realized
in our model. To account for the observed relic density, the annihilation of
dark matter are dominant by the $s$-channel scalars $h/H$ or gauge boson
$Z^{\prime}$. For the fermion DM, we find that the $h$-channel is excluded.
But it is still allowed for the $H$-channel and $Z^{\prime}$-channel. On the
contrary, for the scalar DM, the $Z^{\prime}$-channel is excluded while the
$h/H$-channel is allowed. And the heavy Higgs $H$ also plays a vital import
rule in the conversion between fermion and scalar dark matter. Collider
signatures of our model are also very rich. The precise measurements of SM
Higgs $h$ will put tight constrain light scalar DM and heavy scalar $H$. With
a relatively large mixing angle $\sin\theta=0.3$, the $H\to ZZ,W^{+}W^{-}$
channels are testable at LHC. For the extra $Z$ boson and heavy lepton $\psi$,
the tri-lepton channel of $Z\to\bar{\psi}\psi$ is promising at HL-LHC. With a
larger cross section, the associated production of $l^{\pm}\psi$ may be more
promising. The inert doublet scalar can also produce multi-lepton channels.
And some distinct channels, as $2\ell^{\pm}jj+\cancel{E}_{T}$,
$\ell^{\pm}bb+\cancel{E}_{T}$, can be used to distinguish our model.

Finally, we would like to mention that the radiated linear seesaw model we
proposed is the minimal version where only one $\Psi$ fermion mediator is
included. In this scenario, $M_{\nu}$ is a rank-2 mass matrix and the lightest
neutrino must be massless. However, more complicated scenarios exist,
corresponding to other solutions of the anomaly free condition. Then one may
obtain the rank-3 neutrino mass matrix. Such scenarios predicts more new
particles with different $B-L$ charges, the model construction and the
phenomenology deserve us further study.

## Acknowledgments

We would like to thank Ran Ding for the help on the analysis of dark matter.
The work of Weijian Wang is supported by Special Fund of Theoretical Physics,
Grant No. 11447117 and Fundamental Research Funds for the Central
Universities.

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