Two-loop Neutrino Model with Exotic Leptons
Hiroshi Okada
Yuta Orikasa
We propose a two-loop induced neutrino mass model, in which we show some bench mark points to satisfy the observed neutrino oscillation, the constraints of lepton flavor violations, and the relic density in the co-annihilation system satisfying the current upper bound on the spin independent scattering cross section with nuclei. We also discuss new sources of muon anomalous magnetic moment.
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\begin{document}


%%%%%%%%%
\title{Two-loop Neutrino Model with Exotic Leptons}
%\preprint{KIAS-P14078}
%
\author{Hiroshi Okada}
\email{macokada3hiroshi@gmail.com}
\affiliation{Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 300}

\author{Yuta Orikasa}
\email{orikasa@kias.re.kr}
\affiliation{School of Physics, KIAS, Seoul 130-722, Korea}
\affiliation{Department of Physics and Astronomy, Seoul National University, Seoul 151-742, Korea}

\date{\today}




\begin{abstract}
We propose a two-loop induced neutrino mass model, in which  we show some bench mark points to satisfy the observed neutrino oscillation, the constraints of lepton flavor violations, and the relic density in the co-annihilation system satisfying the current upper bound on  the spin independent scattering cross section with nuclei. We also discuss new sources of muon anomalous magnetic moment.
\end{abstract}
\maketitle
\newpage


\section{Introduction}
 Nowadays the standard model (SM) becomes trustworthy to describe microscopic fundamental physics, since the SM Higgs has been discovered at the CERN Large Hadron Collider (LHC).
 However it has to be still extended in order to include dark matter (DM) and (minuscule but) massive neutrinos that make an allusion to their existence by overwhelming experimental evidences.
%%%
One of the economical and elegant solutions to resolve these controversial issues is to accommodate radiative seesaw models to SM, in which active neutrino masses are radiatively arisen and exotic fields are naturally introduced to induce such tiny masses.
Such an exotic field can be frequently identified as a DM candidate. In this sense, one might find that neutrinos have a strong correlation to the DM candidate.
%Due to the fascinating nature, there  exist the vast numbers of  papers along the though of this line~\cite{xxx}. 
%Especially, Ma model~\cite{Ma:2006km} is renowned as a minimal radiative seesaw model including fermionic or bosonic DM candidate.
%Even after the discovery of the neutral scalar boson with isospin doublet of the Standard Model (SM) around $125\ \rm{GeV}$ by the ATLAS \cite{Aad:2012tfa} and the CMS \cite{Chatrchyan:2012ufa} at the Large Hadron Collider (LHC), some particles such as dark matter candidates and minuscule but massive neutrinos are not directly found yet in spite of a lot of indirect signature by several experiments. Due to their missing parts, the theory provides a vast of possibilities. Along the though of this line, radiative seesaw models are one of the economical theory not also to involve a DM and the neutrinos but also correlate them strongly.
%%%
Following the landmarks~\cite{Zee, Cheng-Li, zee-babu, Krauss:2002px, Ma:2006km, Aoki:2008av, Gustafsson:2012vj},
a vast of literature on radiative seesaw model has recently arisen in Refs.~\cite{Hambye:2006zn, Gu:2007ug, Sahu:2008aw, Gu:2008zf, Babu:2002uu, AristizabalSierra:2006ri, AristizabalSierra:2006gb,
%%%
Nebot:2007bc, Bouchand:2012dx, Kajiyama:2013sza,McDonald:2013hsa, Ma:2014cfa, Schmidt:2014zoa, Herrero-Garcia:2014hfa,
Ahriche:2014xra,Long1, Long2, Aoki:2010ib, Kanemura:2011vm, Lindner:2011it, 
%%%
Kanemura:2011jj, Aoki:2011he, Kanemura:2011mw,
Schmidt:2012yg, Kanemura:2012rj, Farzan:2012sa, Kumericki:2012bf, Kumericki:2012bh, Ma:2012if, Gil:2012ya, Okada:2012np,
Hehn:2012kz, Baek:2012ub, Dev:2012sg, Kajiyama:2012xg, Kohda:2012sr, Aoki:2013gzs, Kajiyama:2013zla, Kajiyama:2013rla, Kanemura:2013qva, Dasgupta:2013cwa, Baek:2013fsa, Okada:2014vla, Ahriche:2014cda, Ahriche:2014oda,Chen:2014ska,
Kanemura:2014rpa, Okada:2014oda, Fraser:2014yha, Okada:2014qsa, Hatanaka:2014tba, Baek:2015mna, Jin:2015cla,
Culjak:2015qja, Okada:2015nga, Geng:2015sza, Okada:2015bxa, Geng:2015coa, Ahriche:2015wha, Restrepo:2015ura, Kashiwase:2015pra, Nishiwaki:2015iqa, Wang:2015saa, Okada:2015hia, Ahriche:2015loa,
%%%%
Ahn:2012cg, Ma:2012ez, Kajiyama:2013lja, Hernandez:2013dta, Ma:2014eka, Aoki:2014cja, Ma:2014yka, Ma:2015pma, 
Ma:2013mga, radlepton1, radlepton2, Okada:2014nsa, Brdar:2013iea,
%%%
Bonnet:2012kz,Sierra:2014rxa, 
Davoudiasl:2014pya, Lindner:2014oea,Okada:2014nea, MarchRussell:2009aq, King:2014uha}



In this paper we employ new fermions and bosons in addition to the SM-like Higgs boson, in which the leading 
neutrino masses can be induced at the two-loop level where the relevant Lagrangian is controlled by an additional global $U(1)$ symmetry.
And the effective tri-linear coupling between the SM-like Higgs and an isospin triplet boson, which is needed to have a massive CP-odd neutral boson, is also generated at the one-loop level through such exotic fields after the global $U(1)$ symmetry spontaneously.

%%%
Following the paper of \cite{Arina:2012aj}, the neutral component can be a DM candidate if there is enough mass difference between two neutral fermions to evade the constraint of the direct detection searches  via the SM neutral gauge boson (Z).
The mass difference arises from the type-II like term after acquiring the vacuum expectation values (VEVs) of the isospin triplet boson.  
However since the mass difference is very tiny because its VEV be less than a few GeV constrained by the electroweak precision test, we have to work on the co-annihilation system to obtain the observed relic density, if we focus on rather lighter DM mass that is less than 80 GeV.

%%
Heavy charged lepton masses themselves are constrained by the LEP  and LHC experiment, and the mass difference between the DM mass and its lepton is also constrained by the electroweak precision test. As a result, the allowed range of the DM mass can be highly restricted.

%%%
We have two new sources to explain the deviation of anomalous magnetic moment to SM.
However either of them that has a strong correlation to the neutrino masses cannot reach the sizable value of the anomalous magnetic moment. This is because the neutrino oscillation requires rather large off-diagonal neutrino mass matrix elements, which tends to be in conflict with the anomalous magnetic moment. Notice here that constraints of the lepton flavor violations (LFVs) that always emerge in such radiative models are not so strong to restrict the Yukawa couplings related to the neutrino masses.



This paper is organized as follows.
In Sec.~II, we show our model building including Higgs masses, neutrino mass, LFV, muon anomalous magnetic  moment, and DM.
In Sec.~III, we show our numerical results. We conclude in Sec.~VI.









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Model}

%\subsection{Model setup}

\begin{table}[thbp]
\centering {\fontsize{10}{12}
\begin{tabular}{|c||c|c|c|c|}
\hline Fermion & $L_L$ & $ e_{R} $ & $L'_{L(R)}$  & $e'_{L(R)}$  
  \\\hhline{|=#=|=|=|=|$}
$(SU(2)_L,U(1)_Y)$ & $(\bm{2},-1/2)$ & $(\bm{1},-1)$ & $(\bm{2},-1/2)$   & $(\bm{1},-1)$
\\\hline
%$U(1)_L$ & $+1$ & $-1$  & $-1$ & $+1$ & $-1$ & $0$ & $0$  & $0$  \\\hline
%%%
%$U(1)'$ & $-1/2$ & $-1/2$ & $-1/2$ & $-1/2$ & $3/2$  \\\hline
%$U(1)'$ & $\ell$ & $\ell$ & $\ell$ & $\ell$ & $\ell-\eta_2$ & $\ell-\eta_2$  \\\hline
$U(1)_{} $ & $-1$ & $-1$ &  $-3/2$ &  $-3/2$    \\\hline
%$Z_2$ & $+$ & $+$ &  $-$  \\\hline
%%%
\end{tabular}%
} \caption{Lepton sector; notice the three (or two) flavor index of each field $L_L$, $e_R$, $L'_{L(R)}$ and  $e'_{L(R)}$ is abbreviated.} 
\label{tab:1}
\end{table}

\begin{table}[thbp]
\centering {\fontsize{10}{12}
\begin{tabular}{|c||c|c|c|c|c|}
\hline Boson  & $\Phi$   & $\eta$    & $\varphi$   & $\Delta$    & $S$ 
  \\\hhline{|=#=|=|=|=|=|}
$(SU(2)_L,U(1)_Y)$ & $(\bm{2},1/2)$  & $(\bm{2},1/2)$   & $(\bm{1},0)$   & $(\bm{3},1)$   & $(\bm{1},0)$ \\\hline
%$U(1)_L$ & $0$ & $0$ & $0$  & $0$  \\\hline
%%%
%$U(1)' $  & $0$ & $0$ & $1$  & $-1$   & $2$  \\\hline
%$U(1)' $  & $0$ & $0$ & $\eta_2$  & $\eta_2-2\ell$   & $-2\ell$  \\\hline
$U(1)_{} $ & $0$ & $3/2$ &  $1$ & $3$ &  $1/2$    \\\hline
%$Z_2$ & $+$ & $-$ &  $+$  \\\hline
%%%
\end{tabular}%
} 
\caption{Boson sector }
\label{tab:2}
\end{table}

In this section, we review our model, in which
the particle contents for leptons and bosons are respectively shown in Tab.~\ref{tab:1} and Tab.~\ref{tab:2}. 
We add vector-like fermions  of $L'_{L(R)}$ with isospin doublet and $e'_{L(R)}$ with isospin singlet, but $-3/2$ charge under the global symmetry to the SM fields. Each of the exotic field needs (at least) two flavors in order to satisfy current neutrino oscillation data~\cite{pdf}. 
Moreover, an introduction of $e'$ is requested only to have mass difference between the neutral component and the charged component of $L'$ in our model. Otherwise our DM (neutral component of $L'$) does not satisfy the observed relic density $\Omega h^2\approx$0.12~\cite{Ade:2013zuv} due to an enhancement of its thermal averaged cross section with co-annihilation system, as we will discuss later.
%%%
As for new bosons, we introduce
two neutral isospin singlet scalars $\varphi$  and $S$  with  $1$ and $1/2$ global charge for each,
an  isospin doublet scalar $\eta$ with  the $3/2$ global charge, and an  isospin triplet scalar $\Delta$ with the $3$ global charge.
Note here that $\Phi$ is neutral under the global charge not to couple to our physical Goldstone boson.
%%%
Then we assume that $\Phi$, $\varphi$, and $\Delta$ have  VEV, which are symbolized by $v/\sqrt2$, $v'/\sqrt2$, and  $v_\Delta/\sqrt2$ respectively.
%%%
%{\it Notice here that $v_\Delta$ is induced not through spontaneously but radiatively at one-loop level as we will discuss later. }
VEV of $\varphi$ spontaneously breaks  the global symmetry down. 
%Otherwise the $\mathbb{Z}_2$ symmetry which guarantees DM stability is spontaneously broken. 
Even after the global $U(1)$ breaking as well as the electroweak breaking, a remnant discrete  symmetry $Z_2$ remains, which is understood as an accidental symmetry. This $Z_2$ symmetry plays a role in  assuring the stability of our  DM candidate; neutral component of $L'_{L(R)}$.


The relevant  Lagrangian for Yukawa sector and scalar potential under these assignments
are given by
\begin{align}
-\mathcal{L}_{Y}
&=
(y_\ell)_a \bar L_{L_a} \Phi e_{R_a} + (y_{L})_{ai} \bar L_{L_a} L'_{R_i} S + (y'_{L})_{ij} \bar L'^c_{L_i}(i\tau_2) \Delta L'_{L_j}
+ (y'_{R})_{ij} \bar L'^c_{R_i}(i\tau_2) \Delta L'_{R_j}\nn\\
&
+(y'_{LR})_{ij}\bar L'_{L_i} \Phi e'_{R_j} 
+(y'_{RL})_{ij}\bar L'_{R_i} \Phi e'_{L_j} 
+ (y_S)_{ib}\bar e'_{L_i} e_{R_b} S^*\nn\\
&
+(M_L)_{ij}  \bar L'_{L_i} L'_{R_j}  +(M_R)_{ij}  \bar e'_{L_i} e'_{R_j} +\rm{h.c.} \label{Lag:Yukawa}\\ 
%%%
\mathcal{V}
&=
 m^2_\Phi |\Phi|^2 + m^2_\eta |\eta|^2 + m^2_\varphi |\varphi|^2  + m^2_S |S|^2  + m^2_\Delta {\rm Tr}[|\Delta|^2]\nn\\
&+\mu_S (S^2 \varphi^* +{\rm h.c.}) +\mu_\eta (\eta^T(i\tau_2)\Delta^\dag \eta +{\rm h.c.})
+\lambda_0(\eta^\dag\Phi S\varphi+{\rm h.c.})
 \nn\\
&+
\lambda_{\Phi}|\Phi|^4 + \lambda_{\Phi\eta}|\Phi|^2|\eta|^2  + \lambda'_{\Phi\eta}|\Phi^\dag\eta|^2 + \lambda_{\Phi\varphi}|\Phi|^2|\varphi|^2
+\lambda_{\Phi S}|\Phi|^2|S|^2+\lambda_{\Phi\Delta}|\Phi|^2{\rm Tr}[|\Delta|^2] \nn\\
&+
\lambda'_{\Phi\Delta}\sum_{i}^{1-3}(\Phi^\dag \tau_i\Phi){\rm Tr}[\Delta^\dag \tau_i \Delta]
%%% %%%%
+
\lambda_{\eta}|\eta|^4 + \lambda_{\eta\varphi}|\eta|^2|\varphi|^2
+\lambda_{\eta S}|\eta|^2|S|^2+\lambda_{\eta\Delta}|\eta|^2{\rm Tr}[|\Delta|^2] \nn\\
&+
\lambda'_{\eta\Delta}\sum_{i}^{1-3}(\eta^\dag \tau_i\eta){\rm Tr}[\Delta^\dag \tau_i \Delta]
%%% %%%%
+
\lambda_{\varphi}|\varphi|^4 
+\lambda_{\varphi S}|\varphi|^2|S|^2+\lambda_{\varphi\Delta}|\varphi|^2{\rm Tr}[|\Delta|^2] \nn\\
%%% %%%%
&+
\lambda_{S}|S|^4 
+\lambda_{S\Delta}|S|^2{\rm Tr}[|\Delta|^2] + \lambda_{\Delta}({\rm Tr}[|\Delta|^2])^2 
+\lambda'_\Delta Det[\Delta^\dag \Delta]
%%%
%+ ({\rm trivial\ quartic\ couplings})
,
\label{HP}
\end{align}
where $\tau_i(i$=1-3) is Pauli matrix, each of the index $a(b)$ and $i(j)$ that runs $1$-$3$ and $1$-$2(3)$ represents the number of generations,  and the first term of $\mathcal{L}_{Y}$ can generates the (diagonalized) charged-lepton masses.
We work on the basis where all the coefficients are real and positive for our brevity. 
%The couplings $\lambda_1$, $(\lambda_2)_{iiii}\ (i=1\,,2)$, $\lambda_{8}$, and  $\lambda_{9}$ have to be positive to stabilize the Higgs potential.
%%%


\subsection{Scalar sector}
After the EW symmetry breaking, each of scalar field has nonzero mass.
We parametrize  these scalar fields as 
\begin{align}
&\Phi =\left[
\begin{array}{c}
\phi^+\\
\phi^0
\end{array}\right],\
%%%
\eta =\left[
\begin{array}{c}
\eta^+\\
\eta^0
\end{array}\right],\
%%%
\Delta =\left[
\begin{array}{cc}
\frac{\Delta^+}{\sqrt2} & \Delta^{++}\\
\Delta^0 & -\frac{\Delta^+}{\sqrt2}
\end{array}\right].
\label{component}
\end{align}
And the neutral components of the above fields and the singlet scalar field can be expressed as
\begin{align}
\phi^0&=\frac1{\sqrt2}(v + h + ia),\ \eta^0=\frac1{\sqrt2}(\eta_R+i \eta_I),\ 
 \Delta^0=\frac1{\sqrt2}(v_\Delta+\Delta_R+i \Delta_I),\nn\\
 %%%
 \varphi &= \frac1{\sqrt2}(v'+\rho)e^{i G/v'}, S = \frac1{\sqrt2}(S_R+i S_I),
\label{Eq:neutral}
\end{align}
where $h$ is the SM-like Higgs, and $v$ and $v_\Delta$ is related to the Fermi constant $G_F$ by $v^2+2 v^2_\Delta=1/(\sqrt{2}G_F)\approx(246$ GeV)$^2$.


%$\eta$ is the inert doublet and the mass of $\eta$ should be positive. In our model, $\eta$ mass is generated by the mixing between $\eta$ and $\varphi$. Consequently, the mixing should be positive at the symmetry breaking scale, 
%\begin{eqnarray}\lambda_{\eta\varphi}>0. \label{inert1}\end{eqnarray}
%And the quartic couplings satisfy the following inert conditions\cite{Barbieri:2006dq}, 
%\begin{eqnarray}\lambda_\Phi>0,\  \lambda_\eta>0,\  
%\lambda_{\Phi\eta}+\lambda'_{\Phi\eta}-\mid\lambda''_{\Phi\eta}\mid>-2\sqrt{\lambda_\Phi\lambda_\eta}. \label{inert2}\end{eqnarray}

The CP even Higgs boson mass matrix with VEV in the basis of ($\Delta_R,h,\rho$) is given by
\begin{align}
(M^{2})^{vev}_{\rm CP-even}  = \left(%
\begin{array}{ccc}
\frac{\mu_{\rm eff} v^2}{\sqrt2 v_\Delta} +2\lambda_{\Delta}v_\Delta^2 
&\left[(\lambda_{\Phi\Delta}+\lambda'_{\Phi\Delta})v_\Delta-\sqrt2\mu_{\rm eff}\right]v 
& \lambda_{\Delta\varphi} v'v_\Delta \\
\left[(\lambda_{\Phi\Delta}+\lambda'_{\Phi\Delta})v_\Delta-\sqrt2\mu_{\rm eff}\right]v  
 & 2 \lambda_{\Phi}v^2
 & \lambda_{\Phi\varphi} vv'  \\
 \lambda_{\Delta\varphi} v'v_\Delta & \lambda_{\Phi\varphi} vv'  & 2\lambda_{\varphi} v'^2 \\
\end{array}\right),
% \!=\! \left(\begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array}\right)
%\left(\begin{array}{cc} m^2_{h_{\rm SM}} & 0 \\ 0 & m^2_{H}  \end{array}\right)
%\left(\begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta &\cos\theta \end{array}\right), 
\end{align}
and $(M^{2})^{vev}_{\rm CP-even}$ is diagonalized by 3 $\times$ 3 orthogonal mixing matrix $O_R$ as
$O_R (M^{2})^{vev}_{\rm CP-even} O_R^T=$ diag.$(m_{h_1}^2, m_{h_{\rm SM}}^2, m_{h_3}^2)$.
Here
 $h_{\rm SM}$ is the SM Higgs and $h_1$ and $h_3$ are additional Higgses.
eigenstate. 
%The mixing angle $\theta$ is given by \be\tan 2\theta=\frac{ -2\lambda_{\Phi\varphi}v v'}{v'^2(4\lambda_\varphi - \lambda_{\Phi\varphi})}.\label{II.12}\ee in terms of the mass eigenstates $h_{\rm SM}$ and $H$ as
%\begin{eqnarray}
% h &=& h_{\rm SM}\cos\theta + H\sin\theta, \nn\\
% \rho &=&- h_{\rm SM}\sin\theta + H\cos\theta.\label{eq:mass_weak}
% \end{eqnarray}
% The mixing angle $\theta$ is generally constrained by $h_{\rm SM} \rightarrow \gamma\gamma$ process at LHC,
% however we can avoid such a constraint, since  we expect $v/v'\le(0.1)<<1$, as can be seen in Eq.(\ref{II.12}), which  is negligible.
%%%



The CP odd Higgs boson mass matrix with VEV in the basis of ($\Delta_I,a$) is given by
\begin{align}
(M^{2})^{vev}_{\rm CP-odd}  = \left(%
\begin{array}{cc}
\frac{\mu_{\rm eff} v^2 }{\sqrt2 v_\Delta}& -\sqrt2 \mu_{\rm eff} v \\
 -\sqrt2 \mu_{\rm eff} v
&  2\sqrt2 \mu_{\rm eff} v_\Delta \\
\end{array}\right),
% \!=\! \left(\begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array}\right)
%\left(\begin{array}{cc} m^2_{h_{\rm SM}} & 0 \\ 0 & m^2_{H}  \end{array}\right)
%\left(\begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta &\cos\theta \end{array}\right), 
\end{align}
and $(M^{2})^{vev}_{\rm CP-odd}$ is diagonalized by 2 $\times$ 2 orthogonal mixing matrix $O_I$ as
$O_I (M^{2})^{vev}_{\rm CP-odd}  O_I^T=$ diag.$(0,m_{a}^2)$, where $m_{a}^2=\frac{\mu_{\rm eff}(v^2+4 v^2_\Delta)}{\sqrt2 v_\Delta}$ and
the massless mode is absorbed by the neutral gauge boson $Z$ to be massive.
%Here $h_{\rm SM}$ is the SM Higgs and $h_2$ and $h_3$ are additional Higgses. 




The CP even inert Higgs boson mass matrix in the basis of ($\eta_R,S_R$) is given by
\begin{align}
(M^{2})^{inert}_{\rm CP-even}  = \left(%
\begin{array}{cc}
\frac{2m^2_\eta-2\sqrt2\mu_\eta v_\Delta + \lambda_{\eta\varphi}v'^2+(\lambda_{\Phi\eta}+\lambda'_{\Phi\eta})v^2
+(\lambda_{\eta\Delta}+\lambda'_{\eta\Delta})v_\Delta^2 }{2}
&\frac{\lambda_0 v v'}{2}  \\
\frac{\lambda_0 v v'}{2}
& \frac{2m^2_S +2\sqrt2\mu_S v' + \lambda_{\varphi S}v'^2+\lambda_{\Phi S} v^2
+\lambda_{S\Delta} v_\Delta^2 }{2} \\
\end{array}\right),
% \!=\! \left(\begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array}\right)
%\left(\begin{array}{cc} m^2_{h_{\rm SM}} & 0 \\ 0 & m^2_{H}  \end{array}\right)
%\left(\begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta &\cos\theta \end{array}\right), 
\end{align}
and $(M^{2})^{inert}_{\rm CP-even} $ is diagonalized by 2 $\times$ 2 orthogonal mixing matrix $V_R$ as
$V_R (M^{2})^{inert}_{\rm CP-even}  V_R^T=$ diag.$(m_{H_1}^2,m_{H_2}^2)$.
%Here $h_{\rm SM}$ is the SM Higgs and $h_2$ and $h_3$ are additional Higgses. 




The CP odd inert Higgs boson mass matrix in the basis of ($\eta_I,S_I$) is given by
\begin{align}
(M^{2})^{inert}_{\rm CP-odd}  = \left(%
\begin{array}{cc}
\frac{2m^2_\eta + 2\sqrt2\mu_\eta v_\Delta + \lambda_{\eta\varphi}v'^2+(\lambda_{\Phi\eta}+\lambda'_{\Phi\eta})v^2
+(\lambda_{\eta\Delta}+\lambda'_{\eta\Delta})v_\Delta^2 }{2}
&\frac{\lambda_0 v v'}{2}  \\
\frac{\lambda_0 v v'}{2}
& \frac{2m^2_S -2\sqrt2\mu_S v' + \lambda_{\varphi S}v'^2+\lambda_{\Phi S} v^2
+\lambda_{S\Delta} v_\Delta^2 }{2} \\
\end{array}\right),
% \!=\! \left(\begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array}\right)
%\left(\begin{array}{cc} m^2_{h_{\rm SM}} & 0 \\ 0 & m^2_{H}  \end{array}\right)
%\left(\begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta &\cos\theta \end{array}\right), 
\end{align}
and $(M^{2})^{inert}_{\rm CP-odd} $ is diagonalized by 2 $\times$ 2 orthogonal mixing matrix $V_R$ as
$V_I (M^{2})^{inert}_{\rm CP-odd}  V_I^T=$ diag.$(m_{A_1}^2,m_{A_2}^2)$.
%Here $h_{\rm SM}$ is the SM Higgs and $h_2$ and $h_3$ are additional Higgses. 




The singly charged Higgs boson mass matrix with VEV in the basis of ($\Delta^+,\phi^+$) is given by
\begin{align}
(M^{2})^{vev}_{\rm singly}  = \left(%
\begin{array}{cc}
\frac{(\sqrt2\mu_{\rm eff}-\lambda'_{\Phi\Delta} v_\Delta) v^2 }{2 v_\Delta} 
& \left(\frac{\lambda'_{\Phi\Delta}v_\Delta}{\sqrt2}- \mu_{\rm eff}\right) v \\
 \left(\frac{\lambda'_{\Phi\Delta}v_\Delta}{\sqrt2}- \mu_{\rm eff}\right) v
&  (\sqrt2\mu_{\rm eff}-\lambda'_{\Phi\Delta} v_\Delta) v_\Delta \\
\end{array}\right),
% \!=\! \left(\begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array}\right)
%\left(\begin{array}{cc} m^2_{h_{\rm SM}} & 0 \\ 0 & m^2_{H}  \end{array}\right)
%\left(\begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta &\cos\theta \end{array}\right), 
\end{align}
and $(M^{2})^{vev}_{\rm singly}$ is diagonalized by 2 $\times$ 2 unitary mixing matrix $O_C$ as
$O_C (M^{2})^{vev}_{\rm singly} O_C^\dag=$ diag.$(0,m_{C}^2)$, where $m_{C}^2=\frac{(\sqrt2\mu_{\rm eff}-\lambda'_{\Phi\Delta}v_\Delta)(v^2+2 v^2_\Delta)}{2 v_\Delta}$ and
the massless mode is absorbed by the charged gauge boson $W^\pm$ to be massive.
The singly charged inert  boson mass eigenstate is given by
\begin{align}
m^2_{\eta^\pm}=\frac{2m^2_\eta+\lambda_{\eta\varphi}v'^2+\lambda_{\Phi\eta}v^2 +(\lambda_{\eta\Delta}-\lambda'_{\eta\Delta}) v^2_\Delta}{2}.
\end{align}

The doubly charged boson mass eigenstate is given by
\begin{align}
m^2_{\Delta^{\pm\pm}}=\left(\frac{\mu_{\rm eff}}{\sqrt2 v_\Delta}-\lambda'_{\Phi\Delta}\right)v^2.
\end{align}





\subsection{Effective trilinear coupling of  $\mu_{\rm eff}$}
In our model, the term $\mu_{\rm eff}\Phi^T(i\tau_2)\Delta^\dag\Phi$ is forbidden at the leading order,
 but induced at the one-loop level mediated by inert neutral bosons $\eta^0$ and $S$ as  depicted in the lower part of Fig.~\ref{neut-mass}.
 The effective trilinear coupling of  $\mu_{\rm eff}$ is given by
 \begin{align}
\mu_{\rm eff}&=\frac{(\lambda_0 v')^2\mu_\eta}{6}\int_0^1\frac{dx_1dx_2dx_3\delta(x_1+x_2+x_3-1)}{(4\pi)^2}\\
&
\left[
\frac{(V_R^T)_{2i}(V_R)_{i1} (V_R^T)_{1j}(V_R)_{j2} (V_R^T)_{1k}(V_R)_{k1}}{x_1 m^2_{H_i}+x_2 m^2_{H_j}+x_3 m^2_{H_k}}
-
\frac{(V_I^T)_{2i}(V_I)_{i1} (V_I^T)_{1j}(V_I)_{j2} (V_I^T)_{1k}(V_I)_{k1}}{x_1 m^2_{A_i}+x_2 m^2_{A_j}+x_3 m^2_{A_k}}
\right],\nn
 \end{align}
 where each of ($i,j,k$) runs form 1 to 2.





\subsection{Inert conditions}
To forbid VEVs  for our inert bosons $\eta^0$ and $S$, the quartic couplings of $\lambda_{\eta}$ and $\lambda_S$ has to be always positive.
To achieve the situation up to one-loop level, we have to satisfy the following conditions at least:
\begin{align}
0\le \lambda_{\eta}^{\rm total}=\lambda_\eta+\delta\lambda_\eta^{(1)},\
0\le \lambda_{S}^{\rm total}=\lambda_S+\delta\lambda_S^{(1)},
\label{eq:cond-inert}
\end{align}
where
\begin{eqnarray}
\delta\lambda_\eta^{(1)}=-10\left| \mu_\eta\right|^4 F_0\left(\eta,\Delta\right),
\end{eqnarray}
\begin{eqnarray}
\delta\lambda_S^{(1)}= -8\left| \mu_S\right|^4 F_0\left(S,\varphi\right),
\end{eqnarray}
with
\begin{align}
F_0(f_1,f_2)=\frac{1}{(4\pi)^2}\int_0^1dxdy\delta(x+y-1)\frac{xy}{(x m^2_{f_1} + y m^2_{f_2})^2}.
\end{align}



\subsection{Fermion Sector}
%%%
Let us fist define the exotic  fermion as follow:
\begin{align}
L'_{L(R)}\equiv 
\left[
\begin{array}{c}
N'\\
E'^-
\end{array}\right]_{L(R)}.
\end{align}
{\it Neutral exotic fermion}:
Then the mass matrix for the neutral fermion in the basis of $[N'_L,N'^c_R]$ is given by 
\begin{align}
M_N=
%[\bar N'^c_L,\bar N'_R]
\left[\begin{array}{cc}
M_L & m \\
m &  M_L \\
\end{array}\right],
%%%
%\left[\begin{array}{c}N'_L\\ N'^c_R\end{array}\right],
\end{align}
where $m\equiv y'_L v_\Delta/\sqrt2$ ($m<<M_L$) and we assume to be one generation case with positive real for our simple analysis.
%%%
$M_N$ is diagonalized by 2 $\times$ 2 unitary mixing matrix $V_N$ as
$V_N M_N V_N^T=$ diag.$(M_L-m, M_L + m)$, where $V_N$ is written as a maximal mixing form%%%
\begin{align}
V_N=\frac{1}{\sqrt2}
\left[\begin{array}{cc}
i & -i \\
1 &1 \\
\end{array}\right],
\end{align}
and we define the mass eigenstate of the neutral fermion [$N_1,N^c_2$] then we have the following relations:
\begin{align}
\left[\begin{array}{c}N'_L\\ N'^c_R\end{array}\right]
=
\frac{1}{\sqrt2}
\left[\begin{array}{c}
i  N_1+N^c_2 \\
-iN_1 +N^c_2 \\
\end{array}\right].
%%%
\end{align}
Furthermore we redefine these fields as 
$\psi_1\equiv N_1+N^c_1$ and $\psi_2\equiv N^c_2+N_2$, then we obtain the Majorana fields $\psi_1$ and $\psi_2$.
We summarize the relations between $\psi$ and N below
\begin{align}
P_L \psi_1&=N_1,\ P_R \psi_1=N^c_1,\ P_L\psi_2=N^c_2,\ P_R \psi_2=N_2,\\
\bar \psi_1 P_R &=\bar N_1,\ \bar \psi_1 P_L =\bar N^c_1,\ \bar \psi_2 P_R =\bar N^c_2,\ \bar \psi_2 P_L =\bar N_2.
\end{align}
The lighter field $N_1$ can be a DM candidate, but the mass difference between them $2m$ is expected to be tiny because $m$ is originated from $v_\Delta$. As a result, we have to consider the co-annihilation system  (at least) including $\psi_1$ and $\psi_2$ to obtain the relic density.
Notice here that the lowest bound on $m$ comes from the inelastic scattering through $Z$ boson and should be heavier than ${\cal O}$(100) keV~\cite{Arina:2012aj}.   






{\it Singly charged exotic fermion}:
The mass matrix for the singly charged fermion in the basis of $[E', e']$ is given by 
\begin{align}
M_E=
%[\bar N'^c_L,\bar N'_R]
\left[\begin{array}{cc}
M_L & m'_e \\
m'_e &  M_R \\
\end{array}\right],
%%%
%\left[\begin{array}{c}N'_L\\ N'^c_R\end{array}\right],
\end{align}
where we assume $m'_e \equiv y'_{LR} v/\sqrt2\approx y'_{RL} v/\sqrt2$ ($m'_e<<M_L M_R$) with one generation. Note here that the mass matrix can be taken positive real without loss of generality.
%%%
$M_E$ is diagonalized by 2 $\times$ 2 unitary mixing matrix $V_C$ as
\begin{align}
V_C M_E V_C^T= \frac12{\rm diag.}\left(M_L + M_R-\sqrt{(M_L-M_R)^2+4 m'_e}, M_L + M_R+\sqrt{(M_L-M_R)^2+4 m'_e}\right),%\equiv(M_{E_1},M_{E_2})
\end{align}
where $V_C$ is as%%%
\begin{align}
V_C=
\left[\begin{array}{cc}
c_E & s_E \\
-s_E &c_E \\
\end{array}\right],\
s_{2E}
%=\frac{2 m'_e}{M_{E_2}-M_{E_1}}
=\frac{m'_e}{\sqrt{(M_L-M_R)^2+4 m'_e}},
\end{align}
and $s_E\equiv \sin\theta_E$ and $c_E\equiv \cos\theta_E$.
Then we define the mass eigenstate of the charged fermion [$E_1,E_2$], we have the following relations:
\begin{align}
\left[\begin{array}{c}E'\\ e'\end{array}\right]
=
\left[\begin{array}{c}
c_E E_1- s_E E_2  \\
s_E E_1+c_E E_2 \\
\end{array}\right].
%%%
\end{align}

{\it Relation between neutral and charged fermion}:
Since the DM mass ($M_X$) can be approximately given by $M_L$ and $m'_e<<M_L,M_R$,
we  have the relation from the charged fermions approximately:
\begin{align}
M_{E_1}\approx  M_X,\ M_{E_2}\approx  M_R.
\end{align}
On the other hand if the mass difference between $X$ and $E_1$ is enough tiny,
%lighter than  about 10 \% of the DM mass,
$E_1$ also participates in the co-annihilation system of DM. 
%that spoils the DM scenario due to the enhancement of the thermal cross section of $E_1$.
%%%
%\footnote{Because $E_1$ has iso-spin double, it interacts with Z and W boson. The cross section through such interactions is very large, and the relic density becomes too small.}
%%%
%In our case, such a situation is always evaded unless we work on ${\cal O}$(10) GeV mass range of DM.
The upper bound on $M_E$ is derived from the electroweak precision data, which is typically written in term of $\Delta S,\Delta T,\Delta U$ parameters. The most stringent bound comes from $\Delta T$ that suggests the mass difference between $M_X$ and $M_E$ should be smaller than 45 GeV~\cite{Arina:2012aj}:
\begin{align}
M_{E_1}-M_X\lesssim 45\ {\rm GeV}.
\end{align}
The mass of $E_1$ is also constrained by the LHC searches and the lower bound is around 300 GeV if $E_1$ decays into the SM charged-lepton + missing(=DM), which is an analogous analysis of the slepton searches~\cite{Okada:2014qsa}.
However $E_1$ cannot decay into the SM charged-lepton + missing kinematically but decay into the SM charged lepton + two missing fermions( active neutrinos  + DM) through the lighter mass of $H_1$ or $A_1$, which is assumed to be heavier than the mass of $E_1$ in our case. Then only the constraint comes from the LEP experiment that suggests~\cite{j.beringer}
\begin{align}
100\ {\rm GeV} \lesssim M_{E_1}\ {\rm and}\  15\ {\rm GeV}\ \lesssim M_{E_1}-M_X.
\label{eq:lep-bound}
\end{align}





%%%%%%%%%%%%%%%%%%%
\begin{figure}[tbc]
\begin{center}
\includegraphics[scale=0.7]{neutmass.pdf}
\caption{Feynman diagram for the neutrino mass.}
\label{neut-mass}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%




\subsection{Neutrino mass matrix}
The neutrino mass matrix can be generated at two-loop level  as depicted in Fig.~\ref{neut-mass}, and its form is given by% follows~\cite{Ma:2006km, Hehn:2012kz}:
\begin{align}
({\cal M}_\nu)_{ab}=
-\frac{1}{2(4\pi)^2}
&\sum_{k}^{1,2}(y_L)_{a,k}(y_L)_{b,k}\sum_{j}^{1,2}
\left[
(V^T_{I,2j})^2
\left(\frac{M_{\psi_{1,k}}X_{1,j} }{X_{1,j}-1}\ln X_{1,j}-\frac{M_{\psi_{2,k}}X_{2,j} }{X_{2,j}-1}\ln X_{2,j}\right)\right.\nn\\
&\left.
-(V^T_{R,2j})^2
\left(\frac{M_{\psi_{1,k}}Y_{1,j} }{Y_{1,j}-1}\ln Y_{1,j}-\frac{M_{\psi_{2,k}}Y_{2,j} }{Y_{2,j}-1}\ln Y_{2,j}\right)
\right],   \label{eq:neut-theory}
\end{align}
where $X_{i,j}\equiv \left(\frac{M_{\psi_{i,k}}}{m_{A_j}}\right)^2$  and $X_{i,j}\equiv \left(\frac{M_{\psi_{i,k}}}{m_{H_j}}\right)^2$ .
Remind here that two flavor of $E_k$(k=1-2) is introduced to obtain the current neutrino oscillation data.
 $(\mathcal{M}_\nu)_{ab}$ can be generally diagonalized by the Maki-Nakagawa-Sakata mixing matrix $V_{\rm MNS}$ (MNS) as
\begin{align}
(\mathcal{M}_\nu)_{ab} &=(V_{\rm MNS} D_\nu V_{\rm MNS}^T)_{ab},\quad D_\nu\equiv (m_{\nu_1},m_{\nu_2},m_{\nu_3}),
\\
V_{\rm MNS}&=
\left[\begin{array}{ccc} {c_{13}}c_{12} &c_{13}s_{12} & s_{13} e^{-i\delta}\\
 -c_{23}s_{12}-s_{23}s_{13}c_{12}e^{i\delta} & c_{23}c_{12}-s_{23}s_{13}s_{12}e^{i\delta} & s_{23}c_{13}\\
  s_{23}s_{12}-c_{23}s_{13}c_{12}e^{i\delta} & -s_{23}c_{12}-c_{23}s_{13}s_{12}e^{i\delta} & c_{23}c_{13}\\
  \end{array}
\right],
\end{align}
where we neglect the Majorana phase as well as Dirac phase $\delta$ in the numerical analysis for simplicity.
The following neutrino oscillation data at 95\% confidence level~\cite{pdf} is given as
\begin{eqnarray}
&& 0.2911 \leq s_{12}^2 \leq 0.3161, \; 
 0.5262 \leq s_{23}^2 \leq 0.5485, \;
 0.0223 \leq s_{13}^2 \leq 0.0246,  
  \\
&& 
%  m_{\nu_2} ({\rm eV}) = 0.0087,  \; 
  \ |m_{\nu_3}^2- m_{\nu_2}^2| =(2.44\pm0.06) \times10^{-3} \ {\rm eV}^2,  \; 
 % m_{\nu_3} ({\rm eV}) = 0.0502 .
  \ m_{\nu_2}^2- m_{\nu_1}^2 =(7.53\pm0.18) \times10^{-5} \ {\rm eV}^2, \nn
  \label{eq:neut-exp}
  \end{eqnarray}
where we assume one of three neutrino masses is zero with normal ordering in our analysis below.



\subsection{Muon anomalous magnetic moment and Lepton flavor violations}
%%%
The muon anomalous magnetic moment (muon $g-2$) has been 
measured at Brookhaven National Laboratory. 
The current average of the experimental results is given by~\cite{bennett}
\begin{align}
a^{\rm exp}_{\mu}=11 659 208.0(6.3)\times 10^{-10}. \notag
\end{align}
It has been well known that there is a discrepancy between the
experimental data and the prediction in the SM. 
The difference $\Delta a_{\mu}\equiv a^{\rm exp}_{\mu}-a^{\rm SM}_{\mu}$
was calculated in Ref.~\cite{discrepancy1} as 
\begin{align}
\Delta a_{\mu}=(29.0 \pm 9.0)\times 10^{-10}, \label{dev1}
\end{align}
and it was also derived in Ref.~\cite{discrepancy2} as
\begin{align}
\Delta a_{\mu}=(33.5 \pm 8.2)\times 10^{-10}. \label{dev2}
\end{align}
The above results given in Eqs. (\ref{dev1}) and (\ref{dev2}) correspond
to $3.2\sigma$ and $4.1\sigma$ deviations, respectively. 

In our model, we have new contributions to $\Delta a_\mu$ coming from $y_L$ and $y_S$ terms. 
These contributions are calculated as
\begin{align}
\Delta a_\mu &\approx
%\sum_{\alpha=1}^3%%%\sum_{k}^{1-2}
\frac{(y_L)_{21}^2%(y_L)_{2k}
m^2_\mu}{4 (4\pi)^2}
\sum_{a}^{1-2}
\left[  (V^T_R)^2_{2a} G(H_a,E_1)+  (V^T_I)^2_{2a} G(A_a,E_1) \right]\nn\\
%%%
&
+%\sum_{\alpha=1}^3%%%\sum_{k}^{1-2}
\frac{(y_S)_{22}^2%(y_L)_{2k}
m^2_\mu}{4 (4\pi)^2}
\sum_{a}^{1-2}
\left[  (V^T_R)^2_{2a} G(H_a,E_2)+  (V^T_I)^2_{2a} G(A_a,E_2) \right]
\label{eq:muon-g-2}
\end{align}
%%%%%%%
where
\begin{align}
G(f_1,f_2)&\approx
\left.
 \int_0^1 dx \int_0^{1-x}dy
\frac{x(y+z)}{x m^2_{f_1}+(y+z)m^2_{f_2}}\right|_{z=1-x-y}\nn\\
&=\frac{2 m_{f_1}^6 + 3 m_{f_1}^4 m_{f_2}^2 - 6 m_{f_1}^2 m_{f_2}^4 + m_{f_2}^6 + 6 m_{f_1}^4 m_{f_2}^2\ln\left(\frac{m_{f_2}^2}{m_{f_1}^2}\right) }{6(m_{f_1}^2-m_{f_2}^2)^4}.
\end{align}



\begin{table}[t]
\begin{tabular}{c|c|c} \hline
Process & $(i,j)$ & Experimental bounds ($90\%$ CL) \\ \hline
%%%%%%%
$\mu^{-} \to e^{-} \gamma$ & $(2,1)$ &
	$\text{Br}(\mu \to e\gamma) < 5.7 \times 10^{-13}$  \\
$\tau^{-} \to e^{-} \gamma$ & $(3,1)$ &
	$\text{Br}(\tau \to e\gamma) < 3.3 \times 10^{-8}$ \\
$\tau^{-} \to \mu^{-} \gamma$ & $(3,2)$ &
	$\text{Br}(\tau \to \mu\gamma) < 4.4 \times 10^{-8}$  \\ \hline
\end{tabular}
\caption{Summary of $\ell_i \to \ell_j \gamma$ process and the lower bound of experimental data~\cite{Adam:2013mnn}.}
\label{tab:Cif}
\end{table}



Our relevant lepton flavor violation process ($\ell_i\to\ell_j\gamma$) comes from the same terms of anomalous magnetic moment at the one-loop level in principle. Each  of flavor dependent process has to satisfy the current upper bound, as can be seen in Table~\ref{tab:Cif}.
However the contribution from $y_S$ can be always negligible assuming the diagonal $y_S$. This is because this term does not contribute to the neutrino masses. Hence we consider the contribution from $y_L$ only.
Then the branching form is given as 
\begin{align}
\text{Br}(\ell_i\to\ell_j\gamma)&\approx
\sum_{k}^{1-2}\frac{6 |(y_L)_{jk} (y_L)_{ik} |^2}{(16\pi^2{\rm G_F})^2}
\left|(V_R^T)^2_{2,a}g(H_a,E_k)+(V_I^T)^2_{2,a}g(A_a,E_k)\right|^2,
\label{eq:g-2}
\end{align}
where 
\begin{align}
g(f_1,f_2)&\approx 
\left.
\int_0^1 dx \int_0^{1-x}dy
\frac{xy}{x m^2_{f_1}+(y+z)m^2_{f_2}}\right|_{z=1-x-y},
\end{align}
and ${\rm G_F}$ is Fermi constant.





\subsection{Dark matter}
First of all, we discuss the direct detection searches reported by  the experiment of LUX~\cite{Akerib:2013tjd}.
As mentioned before, the inelastic scattering process through $Z$ boson is always evaded to retain the mass difference with ${\cal O}$(100) keV, which is generated via $y'_Lv_\Delta$ in our model.
We have Higgs portal process and our scattering cross section with nucleon is given by
\begin{align}
\sigma_N\approx 0.082\frac{m^4_N |y'_L|^2}{\pi}\left[\sum_{i}^{1-3}\frac{(O_R^T)_{1i}(O^T_R)_{2i} }{m^2_{h_i}}\right]^2,
\end{align}
where $h_2\equiv h_{\rm SM}$ and the mass of neutron, which is symbolized by $m_N$, is around 939 GeV.
LUX suggests that $\sigma_N$ should be less than ${\cal O}(10^{-45})$ cm$^2$ at ${\cal O}$(10) GeV mass range of DM.






{\it Relic density of DM}: 
Before the serious analysis to the relic density, we fix a situation on DM(which is denoted by $\psi_1\equiv X$) as follows.
We focus on the region of $50\ {\rm GeV} \lesssim M_X\lesssim 80\ {\rm GeV}$ to realize a fine perspective(or simple) analysis, and suppose a co-annihilation system on DM, in which there exist  degenerated fermions $\psi_2$ and $E_1$ that could affect to the relic density ($\Omega h^2\approx0.12$) reported by Planck~\cite{Ade:2013zuv}.
%in a co-annihilation system among.
%Since the mass of our dark matter $X(\equiv\psi_1)$ degenerates into that of $\psi_2$ and  $E_1$, we have to consider co-annihilation system to obtain the observed relic density ($\Omega h^2\approx0.12$) reported by Planck~\cite{Ade:2013zuv}.
Its complete analysis and formula can be found in Ref.~\cite{Griest:1990kh} and \cite{Edsjo:1997bg}. 
%%%
Our relevant processes for the thermal averaged cross section comes from the co-annihilations and annihilation for the degenerated fields
of $\Psi_{i,1} E_1\to\ell\nu_L$ via W boson, and $E_1\bar E_1\to f\bar f$ via t-channel mediated by $H_a$ and $A_a$  that comes from $y_L$ and s-channel mediated by  Z boson. Here $f$ represents all the SM fermions that are kinematically allowed.
As subdominant modes, there exist
 $X\bar X\to\nu_L\bar\nu_L$ via t- and u-channels mediated by $H_a$ and $A_a$, $X\bar\psi_2\to f\bar f$ via s-channel mediated by Z boson, $X\bar X\to2G$ via t- and u-channels mediated by $\psi_2$, 
and $X\bar X\to f\bar f$ via s-channel mediated by three CP-even neutral bosons $h_i$ from $y'_L$.
%Those details are found in Appendix.
%%%
Once the total cross section $\sigma_{ij}v_{\rm rel}$ is given in al the above processes, the effective annihilation
cross section is given by 
\begin{equation}
\sigma_{\mathrm{eff}}v_{\rm rel}=\sum_{i=1}^k\sum_{j=1}^k\frac{g_{i}g_{j}}
{g_{\mathrm{eff}}^2}(\sigma_{ij}v_{\rm rel})
\left(1+\Delta_i\right)^{3/2}\left(1+\Delta_j\right)^{3/2}
e^{-(\Delta_i+\Delta_j)x},
\label{eq:coann}
\end{equation}
where $g_{\mathrm{eff}}$ is the effective degree of freedom 
\begin{equation}
g_{\mathrm{eff}}=\sum_{i=1}^kg_{i}\left(1+\Delta_i\right)^{3/2}e^{-\Delta_ix},
\end{equation}
and $\Delta_i\equiv (m_i-m_1)/m_1$ is the mass difference between DM
and the other degenerate particles, $g_i=2$ is the degree of freedom
for each Majorana particle $\chi_i$, $x=m_1/T$ and $\sigma_{ij}v$ is
(co-)annihilation cross section between $i$ and $j$.
Then the formula of  relic density is found as
\begin{align}
\Omega h^2\approx \frac{1.07\times10^9}
{g^{1/2}_* M_{\rm pl}[{\rm GeV}] \int_{x_f}^\infty \left(\frac{a_{\rm eff}}{x^2}+6\frac{b_{\rm eff}}{x^3} \right)},
\end{align}
where $g_*\approx 100$ is the total number of effective relativistic degrees of freedom at the time of freeze-out,
$M_{\rm pl}=1.22\times 10^{19}[{\rm GeV}] $ is Planck mass, $x_f\approx25$, and $ a_{\rm eff}$ and $ a_{\rm eff}$ are derived by
expanding $\sigma_{\mathrm{eff}}v_{\rm rel}$  in terms of $v_{\rm rel}$ as
\begin{align}
\sigma_{\mathrm{eff}}v_{\rm rel}\approx a_{\rm eff}+ b_{\rm rel} v^2_{\rm eff}.
\end{align}
Notice here that the mass of DM is assumed to be less than the mass of $W^\pm$ boson($\approx81$ GeV),
because the cross section is too large to satisfy the relic density once the $W^\pm$-boson final modes are open.





\section{Numerical results}
Now that all the formulae have been provided, we have a numerical analysis.
First of all, we fix the following parameters in the scalar sector:
\begin{align}
& v_\Delta \approx 0.0007686\ [{\rm GeV}],\ v' \approx 319.7\ [{\rm GeV}],\
 \mu_S \approx 278.0\ [{\rm GeV}],\ \mu_\eta \approx 715.2\ [{\rm GeV}],\ m_\eta \approx 371.8\ [{\rm GeV}],\nn\\
& m_S \approx 341.6\ [{\rm GeV}],\ 
\lambda_\Phi \approx 0.1300,\
\lambda_{\Delta_1} \approx 0.7226,\ \lambda_{\Delta_2} \approx 0.3587,\ 
\lambda_\eta \approx 0.9717,\
 \lambda_\varphi \approx 0.6274,\nn\\
 & \lambda_S \approx 0.9333,\
  \lambda_{\Phi\Delta} \approx 0.8641,\
\lambda'_{\Phi\Delta} = 0.7313,\
\lambda_{\eta\Delta}
%\[Lambda]\[CapitalDelta]\[Eta]1 \
\approx 0.9876,\
\lambda_{\eta\Delta}'\approx
%\[Lambda]\[CapitalDelta]\[Eta]2 = \
0.7700,\
%%%
 \lambda_{\varphi\Delta}\approx
%\[Lambda]\[CapitalDelta]\[CurlyPhi] = \
0.2470,\nn\\ 
&\lambda_{S\Delta}\approx
%\[Lambda]\[CapitalDelta]s = \
0.7253,\
\lambda_{\Phi\eta}\approx
%\[Lambda]\[CapitalPhi]\[Eta]1 = \
0.6639,\
\lambda_{\Phi\eta}'\approx
%\[Lambda]\[CapitalPhi]\[Eta]2 = \
0.9232,\
\lambda_{\Phi\varphi}\approx
%\[Lambda]\[CapitalPhi]\[CurlyPhi] = \
6.912\times 10^{-6},\ 
\lambda_{S\Phi}\approx
%\[Lambda]\[CapitalPhi]s = \
0.7578,\nn\\
&\lambda_{\eta\varphi}\approx
%\[Lambda]\[Eta]\[CurlyPhi] 
 0.3144,\
\lambda_{S\eta}\approx
%\[Lambda]\[Eta]s = 
0.9915,\ 
\lambda_{S\varphi}\approx
%\[Lambda]\[CurlyPhi]s = \
0.6321,\
\lambda_{0}\approx
%\[Lambda]0 = 
0.6693,\
\mu_{\rm eff} \approx 0.002806\ [{\rm GeV}],
\end{align}
where these above values satisfy the inert conditions in Eq.~(\ref{eq:cond-inert}).
Then we obtain the physical values as follows:
\begin{align}
&m_{h_{\rm SM}}\approx125.5\ [{\rm GeV}],\ m_{h_{3}}\approx368.1\ [{\rm GeV}],\ m_{h_1}\approx395.3\ [{\rm GeV}],\nn\\ 
&m_{H_1}\approx442.2\ [{\rm GeV}],\ m_{H_{2}}\approx551.7\ [{\rm GeV}],\ m_{A_1}\approx204.9\ [{\rm GeV}],\ m_{A_2}\approx454.6\ [{\rm GeV}],\nn\\ 
& m_a\approx395.3\ [{\rm GeV}],\  m_C\approx422.3\ [{\rm GeV}],\  m_{\eta^\pm}\approx417.6\ [{\rm GeV}],\  m_{\Delta^{\pm\pm}}\approx447.8\ [{\rm GeV}],\nn\\
&
O_R\approx
\left[\begin{array}{ccc}
-1 & 7.4\times 10^{-6} & 2.1\times 10^{-7}  \\
-7.4\times 10^{-6} & -1  & 4.8\times 10^{-6} \\
 2.1\times 10^{-7} & 4.8\times 10^{-6} &1   \\
\end{array}\right],\
O_I=
\left[\begin{array}{cc}
6.2\times 10^{-6}   & 1 \\
-1 &6.2\times 10^{-6}   \\
\end{array}\right],\nn\\
%%%
&
V_R\approx
\left[\begin{array}{cc}
-0.97 & 0.25   \\
 0.25 & 0.97  \\
\end{array}\right],\
V_I\approx
\left[\begin{array}{cc}
-0.16 & 0.99   \\
 0.99 & 0.16  \\
\end{array}\right],\
V_C\approx
\left[\begin{array}{cc}
4.4\times 10^{-6}  & 1   \\
 -1 & 4.4\times 10^{-6}   \\
\end{array}\right].
%%%
\end{align}
We search the other physical values with these values, where we take 
\begin{align}
& 0.9\le y'_L\le \sqrt2,\ 
10^{-6}\le (y_L)_{2,3}\le 10^{-5},
\ 50\ {\rm GeV} \le M_X \le 80\ {\rm GeV},\nn\\ %M_X\le
& M_{E_1} \le M_X+45\ {\rm GeV}, \
 300 \ {\rm GeV} \le [M_{\psi_{1,2}},M_{\psi_{2,k}}]\le 500\ {\rm GeV} ,
\end{align}
in our numerical parameter spaces
to reproduce neutrino oscillation data, LFVs, and relic density of DM by using these above values.
Notice here that we solve the other $(y_L)_{i\neq2,j\neq3}$  by comparing the theoretical form and the central experimental value in Eqs.~(\ref{eq:neut-theory})-(\ref{eq:neut-exp}), and $M_{E_2}$ is assumed to be large enough to decouple these phenomenologies .
%we find some parameter sets . %%%
Here we show four benchmark points (BPs) as shown in Table~\ref{bench-mark}, in which we obtain ${\rm Br}(\tau\to e\gamma)$ and ${\rm Br}(\tau\to \mu\gamma)$ for all the BPs are respectively ${\cal O}(10^{-14}\sim10^{-13})$ and ${\cal O}(10^{-12}\sim10^{-11})$ that are completely safe at the current bounds. On the other hand ${\rm Br}(\mu\to e\gamma)$ is rather close to the current upper bound.
%%%
However the BP$_1$ and  BP$_2$ does not satisfy the LEP bound in Eq. (\ref{eq:lep-bound}), since the DM mass is less than 100 GeV while the mass difference between $M_X$ and $M_{E_1}$ is greater than 15 GeV. As a result, only the BP$_3$ and BP$_4$ are complete solutions to satisfy all the data that we discuss.


As for the direct detection searches, our elastic scattering cross section form reduces to $1.1\times 10^{-53} (y'_L)^2$ [cm$^2$] at the range of ${\cal O}$(10) GeV DM mass, which is always  below the current upper bound of the LUX experiment.
% Notice here that the scattering cross section  does not depend  on the DM mass at the range where $m_N<<M_X$.

As for the anomalous magnetic moment, $\Delta a_\mu={\cal O}(10^{-14})$ at most  is obtained by the contribution from $y_L$ in the first term of Eq.~(\ref{eq:g-2}), which is much below the sizable value in Eq.~(\ref{dev1}) or  Eq.~(\ref{dev2}) . It originates from the fact that $y_L$ cannot enlarge due to the the constraints of neutrino oscillation data and LFV of $\mu \to e\gamma$.
On the other hand the contribution from $y_S$ in the second term of Eq.~(\ref{eq:g-2}) reaches the lower experimental value in the perturbative limit(i.e., $y_S=4\pi$), since this term can be independent of such kind of constraints.
In this limit we obtain, for example,  $\Delta a_\mu\approx2\times10^{-9}$ with $M_{E_2}=$600 GeV.
%, we can obtain the sizable value if $y_S={\cal O}$(1).










\begin{center}
\begin{table}[t]
\begin{tabular}{c||c|c|c|c|c|c|c||c|c|c}\hline\hline  
%&\multicolumn{5}{c||}{Lepton Fields} & \multicolumn{2}{c}{Scalar Fields} \\\hline
%& ~$y'_L$~ 
& ~$|(y_L)_{1,1}|$~ & ~$|(y_L)_{1,2}|$~ &~$|(y_L)_{1,3}|$~&~$|(y_L)_{2,1}|$~&  ~$|(y_L)_{2,2}|$~ & ~$|(y_L)_{2,3}|$~ & ~$\frac{M_{X}}{\rm GeV}$ 
&~$\frac{M_{E_1}}{\rm GeV}$  & ~$\Omega h^2$ & ~${\rm Br}(\mu\to e\gamma)$ \\\hline 
${\rm BP}_1$ %& $1.0$ 
& $0.0025$  & $0.031$ & $0.036$ & $1.6\times10^{-4}$ & $2.7\times10^{-4}$ &$4.5\times10^{-6}$
& ${62.27}$&${79.55}$ &$0.119$
& $8.1\times 10^{-14}$ \\\hline 
${\rm BP}_2$  % &$1.4$ 
& $0.0029$  & $0.028$ & $0.032$ & $2.5\times10^{-5}$ & $3.9\times10^{-5}$ &$4.3\times10^{-6}$
& ${59.65}$&${76.49}$ &$0.120$
& $9.6\times 10^{-14}$ \\\hline
${\rm BP}_3$  % &$1.1$ 
& $0.0023$  & $0.029$ & $0.034$ & $5.4\times10^{-5}$ & $9.5\times10^{-5}$ &$4.5\times10^{-6}$
& ${78.67}$&${103.6}$ &$0.112$
& $7.3\times 10^{-14}$ \\\hline
${\rm BP}_4$  % &$1.1$ 
& $0.0014$  & $0.0216$ & $0.019$ & $2.9\times10^{-5}$ & $5.1\times10^{-5}$ &$4.7\times10^{-6}$
& ${79.7}$&${100.5}$ &$0.125$
& $7.8\times 10^{-15}$ \\\hline 
%%%
\end{tabular}
\caption{Four bench mark points(BPs), where ${\rm Br}(\tau\to e\gamma)$ and ${\rm Br}(\tau\to \mu\gamma)$ for all the BPs are respectively ${\cal O}(10^{-14}\sim10^{-13})$ and ${\cal O}(10^{-12}\sim10^{-11})$ that are completely safe.
%Note here that $M_{E_1}\approx 100$ GeV for all the BPs.
}
\label{bench-mark}
\end{table}
\end{center}




\section{Conclusions}
We have studied a two-loop induced radiative neutrino model, in which we have shown some allowed bench mark points to satisfy
the observed neutrino masses, LFVs, and the relic density of DM in the co-annihilation system satisfying the current upper bound  on the spin independent scattering with nucleon as well as LEP.
We have also shown a new source ($y_S$) to marginally obtain the sizable value of the anomalous magnetic moment  in the perturbative limit.

%\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.5cm}
%\hspace{0.2cm} {\bf Acknowledgments}
\section*{Acknowledgments}
\vspace{0.5cm}
Authors thank to Dr. Kei Yagyu for fruitful discussions.
H.O. expresses his sincere gratitude toward all the KIAS members, Korean cordial persons, foods, culture, weather, and all the other things.
This work was supported by the Korea Neutrino Research Center which is established by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. 2009-0083526) (Y.O.).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\if0
\begin{appendix}
The cross section of $(\sigma_{ij})v_{\rm}$ is composed of each of the cross section as 
\begin{align}
(\sigma_{ij})v_{\rm}=(\sigma_{ij})v_{\rm}(X\bar X\to\nu_L\bar\nu_L)+(\sigma_{ij})v_{\rm}(X\bar\psi_2\to f\bar f)+
(\sigma_{ij})v_{\rm}(X\bar X\to f\bar f),
\end{align}
where each of the cross section is given by
\begin{align}
(\sigma_{ij})v_{\rm}
\end{align}



1)...$X\bar X\to\nu_L\bar\nu_L$ via t- and u-channels mediated by $H_a$ and $A_a$ that comes from $y_L$, \\
2)...$X\bar\psi_2\to f\bar f$ ($f$ is all the SM fermions) via s-channel mediated by Z boson from kinetic term,\\
3)...$X\bar X\to f\bar f$ via s-channel mediated by three CP-even neutral bosons $h_i$ from $y'_L$.

The cross section of $(\sigma_{ij})v_{\rm}$ is 

Each of the invariant mass squared to explain the relic density  is given by




\end{appendix}
\fi


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  %10 citations counted in INSPIRE as of 28 Jan 2014

  %\cite{Kajiyama:2013rla}
\bibitem{Kajiyama:2013rla} 
  Y.~Kajiyama, H.~Okada and T.~Toma,
  %``Multicomponent dark matter particles in a two-loop neutrino model,''
  Phys.\ Rev.\ D {\bf 88}, 015029 (2013)
  [arXiv:1303.7356].
  %%CITATION = ARXIV:1303.7356;%%
  %4 citations counted in INSPIRE as of 14 Nov 2013

%\cite{Kanemura:2013qva}
\bibitem{Kanemura:2013qva} 
  S.~Kanemura, T.~Matsui and H.~Sugiyama,
  %``Loop Suppression of Dirac Neutrino Mass in the Neutrinophilic Two
	%Higgs Doublet Model,''
  Phys.\ Lett.\ B {\bf 727}, 151 (2013)
  [arXiv:1305.4521 [hep-ph]].
  %%CITATION = ARXIV:1305.4521;%%
  %8 citations counted in INSPIRE as of 30 Jul 2014




%\cite{Dasgupta:2013cwa}
\bibitem{Dasgupta:2013cwa} 
  B.~Dasgupta, E.~Ma and K.~Tsumura,
  %``WIMP Dark Matter and Neutrino Mass from Peccei-Quinn Symmetry,''
  Phys.\ Rev.\ D {\bf 89}, 041702 (2014)
  [arXiv:1308.4138 [hep-ph]].
  %%CITATION = ARXIV:1308.4138;%%
  %7 citations counted in INSPIRE as of 30 Jul 2014




\bibitem{Baek:2013fsa} 
  S.~Baek, H.~Okada and T.~Toma,
  %``Two loop neutrino model and dark matter particles with global B-L
	%symmetry,''
  JCAP {\bf 1406}, 027 (2014)
  [arXiv:1312.3761 [hep-ph]].
  %%CITATION = ARXIV:1312.3761;%%
  %9 citations counted in INSPIRE as of 30 Jul 2014
 





  %\cite{Okada:2014vla}
\bibitem{Okada:2014vla} 
  H.~Okada,
  %``Two loop Induced Dirac Neutrino Model and Dark Matters with Global $U(1)'$ Symmetry,''
  arXiv:1404.0280 [hep-ph].
  %%CITATION = ARXIV:1404.0280;%%
  

%\cite{Ahriche:2014cda}
\bibitem{Ahriche:2014cda} 
  A.~Ahriche, C.~S.~Chen, K.~L.~McDonald and S.~Nasri,
  %``Three-loop model of neutrino mass with dark matter,''
  Phys.\ Rev.\ D {\bf 90}, no. 1, 015024 (2014)
  [arXiv:1404.2696 [hep-ph]].
  %%CITATION = ARXIV:1404.2696;%%
  %18 citations counted in INSPIRE as of 22 mar 2015

%\cite{Ahriche:2014oda}
\bibitem{Ahriche:2014oda} 
  A.~Ahriche, K.~L.~McDonald and S.~Nasri,
  %``A Model of Radiative Neutrino Mass: with or without Dark Matter,''
  JHEP {\bf 1410}, 167 (2014)
  [arXiv:1404.5917 [hep-ph]].
  %%CITATION = ARXIV:1404.5917;%%
  %15 citations counted in INSPIRE as of 22 mar 2015

%\cite{Chen:2014ska}
\bibitem{Chen:2014ska} 
  C.~-S.~Chen, K.~L.~McDonald and S.~Nasri,
  %``A Class of Three-Loop Models with Neutrino Mass and Dark Matter,''
  Phys.\ Lett.\ B {\bf 734}, 388 (2014)
  [arXiv:1404.6033 [hep-ph]].
  %%CITATION = ARXIV:1404.6033;%%





%\cite{Kanemura:2014rpa}
\bibitem{Kanemura:2014rpa} 
  S.~Kanemura, T.~Matsui and H.~Sugiyama,
  %``Neutrino Mass and Dark Matter from Gauged $U(1)_{B-L}$ Breaking,''
  Phys.\ Rev.\ D {\bf 90}, 013001 (2014)
  [arXiv:1405.1935 [hep-ph]].
  %%CITATION = ARXIV:1405.1935;%%




%\cite{Okada:2014oda}
\bibitem{Okada:2014oda} 
  H.~Okada and Y.~Orikasa,
  %``X-ray line in Radiative Neutrino Model with Global $U(1)$ Symmetry,''
  Phys.\ Rev.\ D {\bf 90}, no. 7, 075023 (2014)
  [arXiv:1407.2543 [hep-ph]].
  %%CITATION = ARXIV:1407.2543;%%
  %9 citations counted in INSPIRE as of 10 Dec 2014




%\cite{Fraser:2014yha}
\bibitem{Fraser:2014yha} 
  S.~Fraser, E.~Ma and O.~Popov,
  %``Scotogenic Inverse Seesaw Model of Neutrino Mass,''
  Phys.\ Lett.\ B {\bf 737}, 280 (2014)
  [arXiv:1408.4785 [hep-ph]].
  %%CITATION = ARXIV:1408.4785;%%

%\cite{Okada:2014qsa}
\bibitem{Okada:2014qsa} 
  H.~Okada, T.~Toma and K.~Yagyu,
  %``Inert Extension of the Zee-Babu Model,''
  Phys.\ Rev.\ D {\bf 90}, no. 9, 095005 (2014)
  [arXiv:1408.0961 [hep-ph]].
  %%CITATION = ARXIV:1408.0961;%%



%\cite{Hatanaka:2014tba}
\bibitem{Hatanaka:2014tba} 
  H.~Hatanaka, K.~Nishiwaki, H.~Okada and Y.~Orikasa,
  %``A Three-Loop Neutrino Model with Global $U(1)$ Symmetry,''
  Nucl.\ Phys.\ B {\bf 894}, 268 (2015)
  [arXiv:1412.8664 [hep-ph]].
  %%CITATION = ARXIV:1412.8664;%%
  %7 citations counted in INSPIRE as of 21 Apr 2015





%\cite{Baek:2015mna}
\bibitem{Baek:2015mna} 
  S.~Baek, H.~Okada and K.~Yagyu,
  %``Flavour Dependent Gauged Radiative Neutrino Mass Model,''
  JHEP {\bf 1504}, 049 (2015)
  [arXiv:1501.01530 [hep-ph]].
  %%CITATION = ARXIV:1501.01530;%%
  %7 citations counted in INSPIRE as of 26 juin 2015

%\cite{Jin:2015cla}
\bibitem{Jin:2015cla} 
  L.~G.~Jin, R.~Tang and F.~Zhang,
  %``A three-loop radiative neutrino mass model with dark matter,''
  Phys.\ Lett.\ B {\bf 741}, 163 (2015)
  [arXiv:1501.02020 [hep-ph]].
  %%CITATION = ARXIV:1501.02020;%%

%\cite{Culjak:2015qja}
\bibitem{Culjak:2015qja} 
  P.~Culjak, K.~Kumericki and I.~Picek,
  %``Scotogenic R$\nu$MDM at three-loop level,''
  Phys.\ Lett.\ B {\bf 744}, 237 (2015)
  [arXiv:1502.07887 [hep-ph]].
  %%CITATION = ARXIV:1502.07887;%%
  %2 citations counted in INSPIRE as of 26 juin 2015

%\cite{Okada:2015nga}
\bibitem{Okada:2015nga} 
  H.~Okada,
  %``Two Loop Radiative Seesaw and X-ray line Dark Matter with Global U(1) Symmetry,''
  arXiv:1503.04557 [hep-ph].
  %%CITATION = ARXIV:1503.04557;%%
  %3 citations counted in INSPIRE as of 26 juin 2015

%\cite{Geng:2015sza}
\bibitem{Geng:2015sza} 
  C.~Q.~Geng and L.~H.~Tsai,
  %``Study of Two-Loop Neutrino Mass Generation Models,''
  arXiv:1503.06987 [hep-ph].
  %%CITATION = ARXIV:1503.06987;%%
  %1 citations counted in INSPIRE as of 26 Jun 2015
 
%\cite{Okada:2015bxa}
\bibitem{Okada:2015bxa} 
  H.~Okada, N.~Okada and Y.~Orikasa,
  %``Radiative Seesaw in Minimal 3-3-1 Model,''
  arXiv:1504.01204 [hep-ph].
  %%CITATION = ARXIV:1504.01204;%%
  %3 citations counted in INSPIRE as of 26 juin 2015
  
%\cite{Geng:2015coa}
\bibitem{Geng:2015coa} 
  C.~Q.~Geng, D.~Huang and L.~H.~Tsai,
  %``Comment on "A three-loop radiative neutrino mass model with dark matter" [Phys. Lett. B 741 (2015) 163],''
  Phys.\ Lett.\ B {\bf 745}, 56 (2015)
  [arXiv:1504.05468 [hep-ph]].
  %%CITATION = ARXIV:1504.05468;%%
  %1 citations counted in INSPIRE as of 26 Jun 2015

%\cite{Ahriche:2015wha}
\bibitem{Ahriche:2015wha} 
  A.~Ahriche, K.~L.~McDonald, S.~Nasri and T.~Toma,
  %``A Model of Neutrino Mass and Dark Matter with an Accidental Symmetry,''
  Phys.\ Lett.\ B {\bf 746}, 430 (2015)
  [arXiv:1504.05755 [hep-ph]].
  %%CITATION = ARXIV:1504.05755;%%
  %4 citations counted in INSPIRE as of 26 juin 2015

%\cite{Restrepo:2015ura}
\bibitem{Restrepo:2015ura} 
  D.~Restrepo, A.~Rivera, M.~S\'anchez-Pel\'aez, O.~Zapata and W.~Tangarife,
  %``Radiative neutrino masses in the singlet-doublet fermion dark matter model with scalar singlets,''
  arXiv:1504.07892 [hep-ph].
  %%CITATION = ARXIV:1504.07892;%%

%\cite{Kashiwase:2015pra}
\bibitem{Kashiwase:2015pra} 
  S.~Kashiwase, H.~Okada, Y.~Orikasa and T.~Toma,
  %``Two Loop Neutrino Model with Dark Matter and Leptogenesis,''
  arXiv:1505.04665 [hep-ph].
  %%CITATION = ARXIV:1505.04665;%%
  
  %\cite{Nishiwaki:2015iqa}
\bibitem{Nishiwaki:2015iqa} 
  K.~Nishiwaki, H.~Okada and Y.~Orikasa,
  %``A Three Loop Neutrino Model with Isolated $k^{\pm\pm}$,''
  arXiv:1507.02412 [hep-ph].
  %%CITATION = ARXIV:1507.02412;%%
  %2 citations counted in INSPIRE as of 23 Aug 2015
  
%\cite{Wang:2015saa}
\bibitem{Wang:2015saa} 
  W.~Wang and Z.~L.~Han,
  %``A Radiative Linear Seesaw, Dark Matter and $U(1)_{B-L}$,''
  arXiv:1508.00706 [hep-ph].
  %%CITATION = ARXIV:1508.00706;%%
  %1 citations counted in INSPIRE as of 23 ao\UTF{00FB}t 2015

  
  %\cite{Okada:2015hia}
\bibitem{Okada:2015hia} 
  H.~Okada and K.~Yagyu,
  %``Three-loop Neutrino Mass Model with Doubly Charged Particles from Iso-Doublets,''
  arXiv:1508.01046 [hep-ph].
  %%CITATION = ARXIV:1508.01046;%%

%\cite{Ahriche:2015loa}
\bibitem{Ahriche:2015loa} 
  A.~Ahriche, K.~L.~McDonald and S.~Nasri,
  %``A Radiative Model for the Weak Scale and Neutrino Mass via Dark Matter,''
  arXiv:1508.02607 [hep-ph].
  %%CITATION = ARXIV:1508.02607;%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% emphasizing non-Alelian discrete symmetry %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


  
 \bibitem{Ahn:2012cg} 
  Y.~H.~Ahn and H.~Okada,
  %``Non-zero $\theta_{13}$ linking to Dark Matter from Non-Abelian
	%Discrete Flavor Model in Radiative Seesaw,''
  Phys.\ Rev.\ D {\bf 85}, 073010 (2012)
  [arXiv:1201.4436 [hep-ph]].
  %%CITATION = ARXIV:1201.4436;%%

%\cite{Ma:2012ez}
\bibitem{Ma:2012ez} 
  E.~Ma, A.~Natale and A.~Rashed,
  %``Scotogenic $A_4$ Neutrino Model for Nonzero $\theta_{13}$ and Large
	%$\delta_{CP}$,''
  Int.\ J.\ Mod.\ Phys.\ A {\bf 27}, 1250134 (2012)
  [arXiv:1206.1570 [hep-ph]].
  %%CITATION = ARXIV:1206.1570;%%
    
%\cite{Kajiyama:2013lja}
\bibitem{Kajiyama:2013lja} 
  Y.~Kajiyama, H.~Okada and K.~Yagyu,
  %``$T_7$ Flavor Model in Three Loop Seesaw and Higgs Phenomenology,''
  JHEP {\bf 10}, 196 (2013)
  arXiv:1307.0480 [hep-ph].
  %%CITATION = ARXIV:1307.0480;%%

%\cite{Hernandez:2013dta}
\bibitem{Hernandez:2013dta} 
  A.~E.~Carcamo Hernandez, I.~d.~M.~Varzielas, S.~G.~Kovalenko, H.~P\"{a}s and I.~Schmidt,
  %``Lepton masses and mixings in a $A_{4}$ multi-Higgs model with radiative seesaw mechanism,''
  Phys.\ Rev.\ D {\bf 88}, 076014 (2013)
  [arXiv:1307.6499 [hep-ph]].
  %%CITATION = ARXIV:1307.6499;%%
  %2 citations counted in INSPIRE as of 25 Nov 2013

%\cite{Ma:2014eka}
\bibitem{Ma:2014eka} 
  E.~Ma and A.~Natale,
  %``Scotogenic $Z_2$ or $U(1)_D$ Model of Neutrino Mass with
	%$\Delta(27)$ Symmetry,''
  Phys.\ Lett.\ B {\bf 723}, 403 (2014)
  [arXiv:1403.6772 [hep-ph]].
  %%CITATION = ARXIV:1403.6772;%%

%\cite{Aoki:2014cja}
\bibitem{Aoki:2014cja} 
  M.~Aoki and T.~Toma,
  %``Impact of semi-annihilation of $\mathbb{Z}_3$ symmetric dark matter
	%with radiative neutrino masses,''
  JCAP {\bf 1409}, 016 (2014)
  [arXiv:1405.5870 [hep-ph]].
  %%CITATION = ARXIV:1405.5870;%%
  %4 citations counted in INSPIRE as of 07 Oct 2014

%\cite{Ma:2014yka}
\bibitem{Ma:2014yka} 
  E.~Ma,
  %``Syndetic Model of Fundamental Interactions,''
  Phys.\ Lett.\ B {\bf 741}, 202 (2015)
  [arXiv:1411.6679 [hep-ph]].
  %%CITATION = ARXIV:1411.6679;%%
  %4 citations counted in INSPIRE as of 22 mar 2015
  

%\cite{Ma:2015pma}
\bibitem{Ma:2015pma} 
  E.~Ma,
  %``Transformative A_4 Mixing of Neutrinos with CP Violation,''
  arXiv:1504.02086 [hep-ph].
  %%CITATION = ARXIV:1504.02086;%%
  %4 citations counted in INSPIRE as of 26 Jun 2015
  
  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% radiative generation of quark/charged lepton masses %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  
  %\cite{Ma:2013mga}
\bibitem{Ma:2013mga} 
  E.~Ma,
  %``Radiative Origin of All Quark and Lepton Masses through Dark Matter
	%with Flavor Symmetry,''
  Phys.\ Rev.\ Lett.\  {\bf 112}, 091801 (2014)
  [arXiv:1311.3213 [hep-ph]].
  %%CITATION = ARXIV:1311.3213;%%
  
%\cite{Okada:2013iba}
\bibitem{radlepton1} 
  H.~Okada and K.~Yagyu,
  %``Radiative Generation of the Lepton Mass,''
  Phys.\ Rev.\ D {\bf 89}, 053008 (2014)
  [arXiv:1311.4360 [hep-ph]].
  %%CITATION = ARXIV:1311.4360;%%
  %8 citations counted in INSPIRE as of 30 Jul 2014

%\cite{Baek:2014awa}
\bibitem{radlepton2} 
  S.~Baek, H.~Okada and T.~Toma,
  %``Radiative lepton model and dark matter with global $U(1)'$ symmetry,''
  Phys.\ Lett.\ B {\bf 732}, 85 (2014)
  [arXiv:1401.6921 [hep-ph]].
  %%CITATION = ARXIV:1401.6921;%%

%\cite{Okada:2014nsa}
\bibitem{Okada:2014nsa} 
  H.~Okada and K.~Yagyu,
  %``Radiative generation of lepton masses with the U(1)′ gauge symmetry,''
  Phys.\ Rev.\ D {\bf 90}, no. 3, 035019 (2014)
  [arXiv:1405.2368 [hep-ph]].
  %%CITATION = ARXIV:1405.2368;%%
  %11 citations counted in INSPIRE as of 22 Mar 2015
  

%\cite{Brdar:2013iea}
\bibitem{Brdar:2013iea} 
  V.~Brdar, I.~Picek and B.~Radovcic,
  %``Radiative Neutrino Mass with Scotogenic Scalar Triplet,''
  Phys.\ Lett.\ B {\bf 728}, 198 (2014)
  [arXiv:1310.3183 [hep-ph]].
  %%CITATION = ARXIV:1310.3183;%%
  %13 citations counted in INSPIRE as of 08 sept. 2015



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% operator analysis %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\cite{Bonnet:2012kz}
\bibitem{Bonnet:2012kz} 
  F.~Bonnet, M.~Hirsch, T.~Ota and W.~Winter,
  %``Systematic study of the d=5 Weinberg operator at one-loop order,''
  JHEP {\bf 1207}, 153 (2012)
  [arXiv:1204.5862 [hep-ph]].
  %%CITATION = ARXIV:1204.5862;%%
  %32 citations counted in INSPIRE as of 30 Jul 2014
  
%\cite{Sierra:2014rxa}
\bibitem{Sierra:2014rxa} 
  D.~Aristizabal Sierra, A.~Degee, L.~Dorame and M.~Hirsch,
  %``Systematic classification of two-loop realizations of the Weinberg operator,''
  JHEP {\bf 1503}, 040 (2015)
  [arXiv:1411.7038 [hep-ph]].
  %%CITATION = ARXIV:1411.7038;%%
  %7 citations counted in INSPIRE as of 22 Mar 2015
  
  
  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% accompanying conformal EWSB %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 
%\cite{Davoudiasl:2014pya}
\bibitem{Davoudiasl:2014pya} 
  H.~Davoudiasl and I.~M.~Lewis,
  %``Right-Handed Neutrinos as the Origin of the Electroweak Scale,''
  Phys.\ Rev.\ D {\bf 90}, no. 3, 033003 (2014)
  [arXiv:1404.6260 [hep-ph]].
  %%CITATION = ARXIV:1404.6260;%%
  %9 citations counted in INSPIRE as of 22 Mar 2015
  
  %\cite{Lindner:2014oea}
\bibitem{Lindner:2014oea} 
  M.~Lindner, S.~Schmidt and J.~Smirnov,
  %``Neutrino Masses and Conformal Electro-Weak Symmetry Breaking,''
  arXiv:1405.6204 [hep-ph].
  %%CITATION = ARXIV:1405.6204;%%

%\cite{Okada:2014nea}
\bibitem{Okada:2014nea} 
  H.~Okada and Y.~Orikasa,
  %``Classically Conformal Radiative Neutrino Model with Gauged B-L Symmetry,''
  arXiv:1412.3616 [hep-ph].
  %%CITATION = ARXIV:1412.3616;%%




  
  
  
  %%%% SUSY originated radiative neutrino model %%%%
  %\cite{MarchRussell:2009aq}
\bibitem{MarchRussell:2009aq} 
  J.~March-Russell, C.~McCabe and M.~McCullough,
  %``Neutrino-Flavoured Sneutrino Dark Matter,''
  JHEP {\bf 1003}, 108 (2010)
  [arXiv:0911.4489 [hep-ph]].
  %%CITATION = ARXIV:0911.4489;%%
  %34 citations counted in INSPIRE as of 07 Oct 2014
%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%  Systematic analysis of cock-tail type model %%%%
  %\cite{King:2014uha}
\bibitem{King:2014uha} 
  S.~F.~King, A.~Merle and L.~Panizzi,
  %``Effective theory of a doubly charged singlet scalar: complementarity of neutrino physics and the LHC,''
  arXiv:1406.4137 [hep-ph].
  %%CITATION = ARXIV:1406.4137;%%
  %1 citations counted in INSPIRE as of 31 Aug 2014

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%         radiative seesaw end          %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%\cite{Arina:2012aj}
\bibitem{Arina:2012aj} 
  C.~Arina, R.~N.~Mohapatra and N.~Sahu,
  %``Co-genesis of Matter and Dark Matter with Vector-like Fourth Generation Leptons,''
  Phys.\ Lett.\ B {\bf 720}, 130 (2013)
  [arXiv:1211.0435 [hep-ph]].
  %%CITATION = ARXIV:1211.0435;%%
  %25 citations counted in INSPIRE as of 08 sept. 2015


  \bibitem{pdf} 
  K.A. Olive et al. (Particle Data Group), Chin. Phys. C, 38, 090001 (2014).

%\cite{Ade:2013zuv}
\bibitem{Ade:2013zuv} 
  P.~A.~R.~Ade {\it et al.}  [Planck Collaboration],
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  Astron.\ Astrophys.\  (2014)
  [arXiv:1303.5076 [astro-ph.CO]].
  %%CITATION = ARXIV:1303.5076;%%
  %2596 citations counted in INSPIRE as of 08 Nov 2014




\bibitem{j.beringer}
J. Beringer, et al., Phys. Rev. D 86 (2012) 010001.


\bibitem{bennett}
  G.~W.~Bennett {\it et al.}  [Muon G-2 Collaboration],
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  Phys.\ Rev.\ D {\bf 73}, 072003 (2006)
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  %%CITATION = HEP-EX/0602035;%%


\bibitem{discrepancy1}
%\cite{Jegerlehner:2009ry}\bibitem{Jegerlehner:2009ry} 
  F.~Jegerlehner and A.~Nyffeler,
  %``The Muon g-2,''
  Phys.\ Rept.\  {\bf 477}, 1 (2009)
  [arXiv:0902.3360 [hep-ph]].
  %%CITATION = ARXIV:0902.3360;%%
  
%\cite{Benayoun:2011mm}
\bibitem{discrepancy2} 
  M.~Benayoun, P.~David, L.~Delbuono and F.~Jegerlehner,
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 [arXiv:1106.1315 [hep-ph]].
  %%CITATION = ARXIV:1106.1315;%%

%\cite{Adam:2013mnn}
\bibitem{Adam:2013mnn} 
  J.~Adam {\it et al.} [MEG Collaboration],
  %``New constraint on the existence of the $\mu^+ \to e^+\gamma$ decay,''
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  %%CITATION = ARXIV:1303.0754;%%
  %297 citations counted in INSPIRE as of 08 sept. 2015


%\cite{Akerib:2013tjd}
\bibitem{Akerib:2013tjd} 
  D.~S.~Akerib {\it et al.}  [LUX Collaboration],
  %``First results from the LUX dark matter experiment at the Sanford Underground Research Facility,''
  Phys.\ Rev.\ Lett.\  {\bf 112}, 091303 (2014)
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  %804 citations counted in INSPIRE as of 23 juin 2015




%\cite{Griest:1990kh}
\bibitem{Griest:1990kh} 
  K.~Griest and D.~Seckel,
  %``Three exceptions in the calculation of relic abundances,''
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  %688 citations counted in INSPIRE as of 14 juin 2015




%\cite{Edsjo:1997bg}
\bibitem{Edsjo:1997bg} 
  J.~Edsjo and P.~Gondolo,
  %``Neutralino relic density including coannihilations,''
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  %%CITATION = HEP-PH/9704361;%%
  %398 citations counted in INSPIRE as of 14 juin 2015





\end{thebibliography}


\end{document}








1509.04068.extracted_bib
\def\be{\begin{equation \def\ee{
1509.04068.extracted_bib
{Zee A.~Zee, %``A Theory Of Lepton Number Violation, Neutrino Majorana Mass, And %Oscillation,'' Phys.\ Lett.\ B {93, 389 (1980) [Erratum-ibid.\ B {95, 461 (1980)].
1509.04068.extracted_bib
{Cheng-Li T.~P.~Cheng and L.~F.~Li, %``Neutrino Masses, Mixings and Oscillations in SU(2) x U(1) Models of Electroweak Interactions,'' Phys.\ Rev.\ D {22, 2860 (1980). %%CITATION = PHRVA,D22,2860;%%
1509.04068.extracted_bib
{zee-babu A.~Zee, %``Quantum Numbers Of Majorana Neutrino Masses,'' Nucl.\ Phys.\ B {264, 99 (1986); %%% K.~S.~Babu, %``MODEL OF 'CALCULABLE' MAJORANA NEUTRINO MASSES,'' Phys.\ Lett.\ B {203, 132 (1988). %\cite{Krauss:2002px
1509.04068.extracted_bib
{Krauss:2002px L.~M.~Krauss, S.~Nasri and M.~Trodden, %``A Model for neutrino masses and dark matter,'' Phys.\ Rev.\ D {67, 085002 (2003) [arXiv:hep-ph/0210389]. %%CITATION = PHRVA,D67,085002;%% %\cite{Ma:2006km
1509.04068.extracted_bib
{Ma:2006km E.~Ma, %``Verifiable radiative seesaw mechanism of neutrino mass and dark matter,'' Phys.\ Rev.\ D {73, 077301 (2006) [hep-ph/0601225]. %%CITATION = HEP-PH/0601225;%% %327 citations counted in INSPIRE as of 09 Dec 2013 %\cite{Aoki:2008av
1509.04068.extracted_bib
{Aoki:2008av M.~Aoki, S.~Kanemura and O.~Seto, %``Neutrino mass, Dark Matter and Baryon Asymmetry via TeV-Scale %Physics %without Fine-Tuning,'' Phys.\ Rev.\ Lett.\ {102, 051805 (2009) [arXiv:0807.0361]. %%CITATION = PRLTA,102,051805;%% %\cite{Gustafsson
1509.04068.extracted_bib
{Gustafsson:2012vj M.~Gustafsson, J.~M.~No and M.~A.~Rivera, %``The Cocktail Model: Neutrino Masses and Mixings with Dark Matter'' Phys.\ Rev.\ Lett.\ {110, 211802 (2013) arXiv:1212.4806 [hep-ph]. %3 citations counted in INSPIRE as of 19 Sep 2013 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%% works on radiative seesaw model %%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\cite{Hambye:2006zn
1509.04068.extracted_bib
{Hambye:2006zn T.~Hambye, K.~Kannike, E.~Ma and M.~Raidal, %``Emanations of Dark Matter: Muon Anomalous Magnetic Moment, Radiative Neutrino Mass, and Novel Leptogenesis at the TeV Scale,'' Phys.\ Rev.\ D {75, 095003 (2007) [hep-ph/0609228]. %%CITATION = HEP-PH/0609228;%% %54 citations counted in INSPIRE as of 03 Oct 2014 %\cite{Gu:2007ug
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{Gu:2007ug P.~-H.~Gu and U.~Sarkar, %``Radiative Neutrino Mass, Dark Matter and Leptogenesis,'' Phys.\ Rev.\ D {77, 105031 (2008) [arXiv:0712.2933 [hep-ph]]. %%CITATION = ARXIV:0712.2933;%% %10 citations counted in INSPIRE as of 20 Mar 2013 %\cite{Sahu:2008aw
1509.04068.extracted_bib
{Sahu:2008aw N.~Sahu and U.~Sarkar, %``Extended Zee model for Neutrino Mass, Leptogenesis and Sterile Neutrino like Dark Matter,'' Phys.\ Rev.\ D {78, 115013 (2008) [arXiv:0804.2072 [hep-ph]]. %%CITATION = ARXIV:0804.2072;%% %17 citations counted in INSPIRE as of 06 Jan 2015 %\cite{Gu:2008zf
1509.04068.extracted_bib
{Gu:2008zf P.~-H.~Gu and U.~Sarkar, %``Radiative seesaw in left-right symmetric model,'' Phys.\ Rev.\ D {78, 073012 (2008) [arXiv:0807.0270 [hep-ph]]. %%CITATION = ARXIV:0807.0270;%% %4 citations counted in INSPIRE as of 20 Mar 2013 %%%%%%%% This part is added following the Dr. Sierra's notification %%%%%%%% %\cite{Babu:2002uu
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{Babu:2002uu K.~S.~Babu and C.~Macesanu, %``Two loop neutrino mass generation and its experimental consequences,'' Phys.\ Rev.\ D {67, 073010 (2003) [hep-ph/0212058]. %%CITATION = HEP-PH/0212058;%% %78 citations counted in INSPIRE as of 13 juil. 2015 %\cite{AristizabalSierra:2006ri
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{AristizabalSierra:2006ri D.~Aristizabal Sierra and D.~Restrepo, %``Leptonic Charged Higgs Decays in the Zee Model,'' JHEP {0608, 036 (2006) [hep-ph/0604012]. %%CITATION = HEP-PH/0604012;%% %15 citations counted in INSPIRE as of 13 juil. 2015 %\cite{AristizabalSierra:2006gb
1509.04068.extracted_bib
{AristizabalSierra:2006gb D.~Aristizabal Sierra and M.~Hirsch, %``Experimental tests for the Babu-Zee two-loop model of Majorana neutrino masses,'' JHEP {0612, 052 (2006) [hep-ph/0609307]. %%CITATION = HEP-PH/0609307;%% %60 citations counted in INSPIRE as of 13 Jul 2015 %%%%%%%% The End of this part %%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% others (e.g., collider specific issues) %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\cite{Nebot:2007bc
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{Nebot:2007bc M.~Nebot, J.~F.~Oliver, D.~Palao and A.~Santamaria, %``Prospects for the Zee-Babu Model at the CERN LHC and low energy experiments,'' Phys.\ Rev.\ D {77, 093013 (2008) [arXiv:0711.0483 [hep-ph]]. %%CITATION = ARXIV:0711.0483;%% %61 citations counted in INSPIRE as of 04 juil. 2015 %\cite{Bouchand:2012dx
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{Bouchand:2012dx R.~Bouchand and A.~Merle, %``Running of Radiative Neutrino Masses: The Scotogenic Model,'' JHEP {1207, 084 (2012) [arXiv:1205.0008 [hep-ph]]. %%CITATION = ARXIV:1205.0008;%% %21 citations counted in INSPIRE as of 30 Jul 2014 %\cite{Kajiyama:2013sza
1509.04068.extracted_bib
{Kajiyama:2013sza Y.~Kajiyama, H.~Okada and K.~Yagyu, %``Electron/Muon Specific Two Higgs Doublet Model,'' Nucl.\ Phys.\ B {887, 358 (2014) [arXiv:1309.6234 [hep-ph]]. %%CITATION = ARXIV:1309.6234;%% %20 citations counted in INSPIRE as of 22 mar 2015 %\cite{McDonald:2013hsa
1509.04068.extracted_bib
{McDonald:2013hsa K.~L.~McDonald, %``Probing Exotic Fermions from a Seesaw/Radiative Model at the LHC,'' JHEP {1311, 131 (2013) [arXiv:1310.0609 [hep-ph]]. %%CITATION = ARXIV:1310.0609;%% %\cite{Ma:2014cfa
1509.04068.extracted_bib
{Ma:2014cfa E.~Ma, %``Vanishing Higgs one-loop quadratic divergence in the scotogenic %model and beyond,'' Phys.\ Lett.\ B {732, 167 (2014) [arXiv:1401.3284 [hep-ph]]. %%CITATION = ARXIV:1401.3284;%% %4 citations counted in INSPIRE as of 30 Jul 2014 %\cite{Schmidt:2014zoa
1509.04068.extracted_bib
{Schmidt:2014zoa D.~Schmidt, T.~Schwetz and H.~Zhang, %``Status of the Zee\UTF{2013Babu model for neutrino mass and possible tests at a like-sign linear collider,'' Nucl.\ Phys.\ B {885, 524 (2014) [arXiv:1402.2251 [hep-ph]]. %%CITATION = ARXIV:1402.2251;%% %11 citations counted in INSPIRE as of 22 Mar 2015
1509.04068.extracted_bib
{Herrero-Garcia:2014hfa J.~Herrero-Garcia, M.~Nebot, N.~Rius and A.~Santamaria, %``The Zee?Babu model revisited in the light of new data,'' Nucl.\ Phys.\ B {885, 542 (2014) [arXiv:1402.4491 [hep-ph]]. %%CITATION = ARXIV:1402.4491;%% %7 citations counted in INSPIRE as of 08 Nov 2014 %\cite{Ahriche:2014xra
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{Ahriche:2014xra A.~Ahriche, S.~Nasri and R.~Soualah, %``Radiative Neutrino Mass Model at the $e^{-e^{+$ Linear %Collider,'' Phys.\ Rev.\ D {89, 095010 (2014) [arXiv:1403.5694 [hep-ph]]. %%CITATION = ARXIV:1403.5694;%% %7 citations counted in INSPIRE as of 30 Jul 2014 %\cite{Long:2014fja
1509.04068.extracted_bib
{Long1 H.~N.~Long and V.~V.~Vien, %``Neutrino mixing with nonzero $\theta_{13$ in Zee-Babu model,'' Int.\ J.\ Mod.\ Phys.\ A {29, no. 13, 1450072 (2014) [arXiv:1405.1622 [hep-ph]]. %%CITATION = ARXIV:1405.1622;%% %\cite{VanVien:2014apa
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{Long2 V.~Van Vien, H.~N.~Long and P.~N.~Thu, %``Neutrino mixing and CP violation phases in Zee-Babu model,'' arXiv:1407.8286 [hep-ph]. %%CITATION = ARXIV:1407.8286;%% %%%%%%%%%% END %%%%%%%%%%% %\cite{Aoki:2010ib
1509.04068.extracted_bib
{Aoki:2010ib M.~Aoki, S.~Kanemura, T.~Shindou and K.~Yagyu, %``An R-parity conserving radiative neutrino mass model without %right-handed neutrinos,'' JHEP {1007, 084 (2010) [Erratum-ibid.\ {1011, 049 (2010)] [arXiv:1005.5159 [hep-ph]]. %%CITATION = ARXIV:1005.5159;%% %\cite{Kanemura:2011vm
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{Kanemura:2011vm S.~Kanemura, O.~Seto and T.~Shimomura, %``Masses of dark matter and neutrino from TeV scale spontaneous %$U(1)_{B-L$ breaking,'' Phys.\ Rev.\ D {84, 016004 (2011) [arXiv:1101.5713 [hep-ph]]. %%CITATION = ARXIV:1101.5713;%% %\cite{Lindner:2011it
1509.04068.extracted_bib
{Lindner:2011it M.~Lindner, D.~Schmidt and T.~Schwetz, %``Dark Matter and Neutrino Masses from Global $U(1)_{B-L$ Symmetry %Breaking,'' Phys.\ Lett.\ B {705, 324 (2011) [arXiv:1105.4626 [hep-ph]]. %%CITATION = ARXIV:1105.4626;%% %\cite{Kanemura:2011jj
1509.04068.extracted_bib
{Kanemura:2011jj S.~Kanemura, T.~Nabeshima and H.~Sugiyama, %``Neutrino Masses from Loop-Induced Dirac Yukawa Couplings,'' Phys.\ Lett.\ B {703, 66 (2011) [arXiv:1106.2480 [hep-ph]]. %%CITATION = ARXIV:1106.2480;%% %\cite{Aoki:2011he
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{Aoki:2011he M.~Aoki, J.~Kubo, T.~Okawa and H.~Takano, %``Impact of Inert Higgsino Dark Matter,'' Phys.\ Lett.\ B {707, 107 (2012) [arXiv:1110.5403 [hep-ph]]. %%CITATION = ARXIV:1110.5403;%% %\cite{Kanemura:2011mw
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{Kanemura:2011mw S.~Kanemura, T.~Nabeshima and H.~Sugiyama, %``TeV-Scale Seesaw with Loop-Induced Dirac Mass Term and Dark Matter %from $U(1)_{B-L$ Gauge Symmetry Breaking,'' Phys.\ Rev.\ D {85, 033004 (2012) [arXiv:1111.0599 [hep-ph]]. %%CITATION = ARXIV:1111.0599;%%
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{Schmidt:2012yg D.~Schmidt, T.~Schwetz and T.~Toma, %``Direct Detection of Leptophilic Dark Matter in a Model with %Radiative Neutrino Masses,'' Phys.\ Rev.\ D {85, 073009 (2012) [arXiv:1201.0906 [hep-ph]]. %%CITATION = ARXIV:1201.0906;%% %\cite{Kanemura:2012rj
1509.04068.extracted_bib
{Kanemura:2012rj S.~Kanemura and H.~Sugiyama, %``Dark matter and a suppression mechanism for neutrino masses in the Higgs triplet model,'' Phys.\ Rev.\ D {86, 073006 (2012) [arXiv:1202.5231 [hep-ph]]. %\cite{Farzan:2012sa
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{Farzan:2012sa Y.~Farzan and E.~Ma, %``Dirac neutrino mass generation from dark matter,'' Phys.\ Rev.\ D {86, 033007 (2012) [arXiv:1204.4890 [hep-ph]]. %%CITATION = ARXIV:1204.4890;%% %17 citations counted in INSPIRE as of 30 Jul 2014 %\cite{Kumericki:2012bf
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{Kumericki:2012bf K.~Kumericki, I.~Picek and B.~Radovcic, %``Critique of Fermionic R\nuMDM and its Scalar Variants,'' JHEP {1207, 039 (2012) [arXiv:1204.6597 [hep-ph]]. %%CITATION = ARXIV:1204.6597;%% %15 citations counted in INSPIRE as of 30 Jul 2014 %\cite{Kumericki:2012bh
1509.04068.extracted_bib
{Kumericki:2012bh K.~Kumericki, I.~Picek and B.~Radovcic, %``TeV-scale Seesaw with Quintuplet Fermions,'' Phys.\ Rev.\ D {86, 013006 (2012) [arXiv:1204.6599 [hep-ph]]. %%CITATION = ARXIV:1204.6599;%% %26 citations counted in INSPIRE as of 30 Jul 2014 %\cite{Ma:2012if
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{Ma:2012if E.~Ma, %``Radiative Scaling Neutrino Mass and Warm Dark Matter,'' Phys.\ Lett.\ B {717, 235 (2012) [arXiv:1206.1812 [hep-ph]]. %%CITATION = ARXIV:1206.1812;%% %21 citations counted in INSPIRE as of 30 Jul 2014 %\cite{Gil:2012ya
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{Gil:2012ya G.~Gil, P.~Chankowski and M.~Krawczyk, %``Inert Dark Matter and Strong Electroweak Phase Transition,'' Phys.\ Lett.\ B {717, 396 (2012) [arXiv:1207.0084 [hep-ph]]. %%CITATION = ARXIV:1207.0084;%% %22 citations counted in INSPIRE as of 30 Jul 2014 %\cite{Okada:2012np
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{Okada:2012np H.~Okada and T.~Toma, %``Fermionic Dark Matter in Radiative Inverse Seesaw Model with U(1)_{B-L,'' Phys.\ Rev.\ D {86, 033011 (2012) arXiv:1207.0864 [hep-ph]. %%CITATION = ARXIV:1207.0864;%% %\cite{Hehn:2012kz
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{Hehn:2012kz D.~Hehn and A.~Ibarra, %``A radiative model with a naturally mild neutrino mass hierarchy,'' Phys.\ Lett.\ B {718, 988 (2013) [arXiv:1208.3162 [hep-ph]]. %%CITATION = ARXIV:1208.3162;%% %1 citations counted in INSPIRE as of 10 Mar 2013 %\cite{Baek:2012ub
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{Baek:2012ub S.~Baek, P.~Ko, H.~Okada and E.~Senaha, %``Can Zee-Babu model implemented with scalar dark matter explain both Fermi/LAT 130 GeV $\gamma$-ray excess and neutrino physics ?,'' JHEP {1409, 153 (2014) [arXiv:1209.1685 [hep-ph]]. %%CITATION = ARXIV:1209.1685;%% %22 citations counted in INSPIRE as of 29 Dec 2014 %\cite{Dev:2012sg
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{Dev:2012sg P.~S.~B.~Dev and A.~Pilaftsis, %``Minimal Radiative Neutrino Mass Mechanism for Inverse Seesaw Models,'' Phys.\ Rev.\ D {86, 113001 (2012) [arXiv:1209.4051 [hep-ph]]. %%CITATION = ARXIV:1209.4051;%% %5 citations counted in INSPIRE as of 10 Mar 2013 %\cite{Kajiyama:2012xg
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{Kajiyama:2012xg Y.~Kajiyama, H.~Okada and T.~Toma, %``Light Dark Matter Candidate in B-L Gauged Radiative Inverse %Seesaw,'' Eur.\ Phys.\ J.\ C {73, 2381 (2013) [arXiv:1210.2305 [hep-ph]]. %%CITATION = ARXIV:1210.2305;%% %13 citations counted in INSPIRE as of 30 Jul 2014 %\cite{Kohda:2012sr
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{Kohda:2012sr M.~Kohda, H.~Sugiyama and K.~Tsumura, %``Lepton number violation at the LHC with leptoquark and diquark,'' Phys.\ Lett.\ B {718, 1436 (2013) [arXiv:1210.5622 [hep-ph]]. %%CITATION = ARXIV:1210.5622;%% %15 citations counted in INSPIRE as of 05 juil. 2015 %2-loop radiative seesaw %\cite{Aoki:2013gzs
1509.04068.extracted_bib
{Aoki:2013gzs M.~Aoki, J.~Kubo and H.~Takano, %``Two-loop radiative seesaw mechanism with multicomponent dark matter %explaining the possible \UTF{0160\UTF{00C3 excess in the Higgs boson decay and %at the Fermi LAT,'' Phys.\ Rev.\ D {87, no. 11, 116001 (2013) [arXiv:1302.3936 [hep-ph]]. %%CITATION = ARXIV:1302.3936;%% %24 citations counted in INSPIRE as of 30 Jul 2014 %\cite{Kajiyama:2013zla
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{Kajiyama:2013zla Y.~Kajiyama, H.~Okada and K.~Yagyu, %``Two Loop Radiative Seesaw Model with Inert Triplet Scalar Field,'' Nucl.\ Phys.\ B {874, 198 (2013) [arXiv:1303.3463 [hep-ph]]. %%CITATION = ARXIV:1303.3463;%% %10 citations counted in INSPIRE as of 28 Jan 2014 %\cite{Kajiyama:2013rla
1509.04068.extracted_bib
{Kajiyama:2013rla Y.~Kajiyama, H.~Okada and T.~Toma, %``Multicomponent dark matter particles in a two-loop neutrino model,'' Phys.\ Rev.\ D {88, 015029 (2013) [arXiv:1303.7356]. %%CITATION = ARXIV:1303.7356;%% %4 citations counted in INSPIRE as of 14 Nov 2013 %\cite{Kanemura:2013qva
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{Kanemura:2013qva S.~Kanemura, T.~Matsui and H.~Sugiyama, %``Loop Suppression of Dirac Neutrino Mass in the Neutrinophilic Two %Higgs Doublet Model,'' Phys.\ Lett.\ B {727, 151 (2013) [arXiv:1305.4521 [hep-ph]]. %%CITATION = ARXIV:1305.4521;%% %8 citations counted in INSPIRE as of 30 Jul 2014 %\cite{Dasgupta:2013cwa
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{Dasgupta:2013cwa B.~Dasgupta, E.~Ma and K.~Tsumura, %``WIMP Dark Matter and Neutrino Mass from Peccei-Quinn Symmetry,'' Phys.\ Rev.\ D {89, 041702 (2014) [arXiv:1308.4138 [hep-ph]]. %%CITATION = ARXIV:1308.4138;%% %7 citations counted in INSPIRE as of 30 Jul 2014
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{Baek:2013fsa S.~Baek, H.~Okada and T.~Toma, %``Two loop neutrino model and dark matter particles with global B-L %symmetry,'' JCAP {1406, 027 (2014) [arXiv:1312.3761 [hep-ph]]. %%CITATION = ARXIV:1312.3761;%% %9 citations counted in INSPIRE as of 30 Jul 2014 %\cite{Okada:2014vla
1509.04068.extracted_bib
{Okada:2014vla H.~Okada, %``Two loop Induced Dirac Neutrino Model and Dark Matters with Global $U(1)'$ Symmetry,'' arXiv:1404.0280 [hep-ph]. %%CITATION = ARXIV:1404.0280;%% %\cite{Ahriche:2014cda
1509.04068.extracted_bib
{Ahriche:2014cda A.~Ahriche, C.~S.~Chen, K.~L.~McDonald and S.~Nasri, %``Three-loop model of neutrino mass with dark matter,'' Phys.\ Rev.\ D {90, no. 1, 015024 (2014) [arXiv:1404.2696 [hep-ph]]. %%CITATION = ARXIV:1404.2696;%% %18 citations counted in INSPIRE as of 22 mar 2015 %\cite{Ahriche:2014oda
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{Ahriche:2014oda A.~Ahriche, K.~L.~McDonald and S.~Nasri, %``A Model of Radiative Neutrino Mass: with or without Dark Matter,'' JHEP {1410, 167 (2014) [arXiv:1404.5917 [hep-ph]]. %%CITATION = ARXIV:1404.5917;%% %15 citations counted in INSPIRE as of 22 mar 2015 %\cite{Chen:2014ska
1509.04068.extracted_bib
{Chen:2014ska C.~-S.~Chen, K.~L.~McDonald and S.~Nasri, %``A Class of Three-Loop Models with Neutrino Mass and Dark Matter,'' Phys.\ Lett.\ B {734, 388 (2014) [arXiv:1404.6033 [hep-ph]]. %%CITATION = ARXIV:1404.6033;%% %\cite{Kanemura:2014rpa
1509.04068.extracted_bib
{Kanemura:2014rpa S.~Kanemura, T.~Matsui and H.~Sugiyama, %``Neutrino Mass and Dark Matter from Gauged $U(1)_{B-L$ Breaking,'' Phys.\ Rev.\ D {90, 013001 (2014) [arXiv:1405.1935 [hep-ph]]. %%CITATION = ARXIV:1405.1935;%% %\cite{Okada:2014oda
1509.04068.extracted_bib
{Okada:2014oda H.~Okada and Y.~Orikasa, %``X-ray line in Radiative Neutrino Model with Global $U(1)$ Symmetry,'' Phys.\ Rev.\ D {90, no. 7, 075023 (2014) [arXiv:1407.2543 [hep-ph]]. %%CITATION = ARXIV:1407.2543;%% %9 citations counted in INSPIRE as of 10 Dec 2014 %\cite{Fraser:2014yha
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{Fraser:2014yha S.~Fraser, E.~Ma and O.~Popov, %``Scotogenic Inverse Seesaw Model of Neutrino Mass,'' Phys.\ Lett.\ B {737, 280 (2014) [arXiv:1408.4785 [hep-ph]]. %%CITATION = ARXIV:1408.4785;%% %\cite{Okada:2014qsa
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{Okada:2014qsa H.~Okada, T.~Toma and K.~Yagyu, %``Inert Extension of the Zee-Babu Model,'' Phys.\ Rev.\ D {90, no. 9, 095005 (2014) [arXiv:1408.0961 [hep-ph]]. %%CITATION = ARXIV:1408.0961;%% %\cite{Hatanaka:2014tba
1509.04068.extracted_bib
{Hatanaka:2014tba H.~Hatanaka, K.~Nishiwaki, H.~Okada and Y.~Orikasa, %``A Three-Loop Neutrino Model with Global $U(1)$ Symmetry,'' Nucl.\ Phys.\ B {894, 268 (2015) [arXiv:1412.8664 [hep-ph]]. %%CITATION = ARXIV:1412.8664;%% %7 citations counted in INSPIRE as of 21 Apr 2015 %\cite{Baek:2015mna
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{Baek:2015mna S.~Baek, H.~Okada and K.~Yagyu, %``Flavour Dependent Gauged Radiative Neutrino Mass Model,'' JHEP {1504, 049 (2015) [arXiv:1501.01530 [hep-ph]]. %%CITATION = ARXIV:1501.01530;%% %7 citations counted in INSPIRE as of 26 juin 2015 %\cite{Jin:2015cla
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{Jin:2015cla L.~G.~Jin, R.~Tang and F.~Zhang, %``A three-loop radiative neutrino mass model with dark matter,'' Phys.\ Lett.\ B {741, 163 (2015) [arXiv:1501.02020 [hep-ph]]. %%CITATION = ARXIV:1501.02020;%% %\cite{Culjak:2015qja
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{Culjak:2015qja P.~Culjak, K.~Kumericki and I.~Picek, %``Scotogenic R$\nu$MDM at three-loop level,'' Phys.\ Lett.\ B {744, 237 (2015) [arXiv:1502.07887 [hep-ph]]. %%CITATION = ARXIV:1502.07887;%% %2 citations counted in INSPIRE as of 26 juin 2015 %\cite{Okada:2015nga
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{Okada:2015nga H.~Okada, %``Two Loop Radiative Seesaw and X-ray line Dark Matter with Global U(1) Symmetry,'' arXiv:1503.04557 [hep-ph]. %%CITATION = ARXIV:1503.04557;%% %3 citations counted in INSPIRE as of 26 juin 2015 %\cite{Geng:2015sza
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{Geng:2015sza C.~Q.~Geng and L.~H.~Tsai, %``Study of Two-Loop Neutrino Mass Generation Models,'' arXiv:1503.06987 [hep-ph]. %%CITATION = ARXIV:1503.06987;%% %1 citations counted in INSPIRE as of 26 Jun 2015 %\cite{Okada:2015bxa
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{Okada:2015bxa H.~Okada, N.~Okada and Y.~Orikasa, %``Radiative Seesaw in Minimal 3-3-1 Model,'' arXiv:1504.01204 [hep-ph]. %%CITATION = ARXIV:1504.01204;%% %3 citations counted in INSPIRE as of 26 juin 2015 %\cite{Geng:2015coa
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{Geng:2015coa C.~Q.~Geng, D.~Huang and L.~H.~Tsai, %``Comment on "A three-loop radiative neutrino mass model with dark matter" [Phys. Lett. B 741 (2015) 163],'' Phys.\ Lett.\ B {745, 56 (2015) [arXiv:1504.05468 [hep-ph]]. %%CITATION = ARXIV:1504.05468;%% %1 citations counted in INSPIRE as of 26 Jun 2015 %\cite{Ahriche:2015wha
1509.04068.extracted_bib
{Ahriche:2015wha A.~Ahriche, K.~L.~McDonald, S.~Nasri and T.~Toma, %``A Model of Neutrino Mass and Dark Matter with an Accidental Symmetry,'' Phys.\ Lett.\ B {746, 430 (2015) [arXiv:1504.05755 [hep-ph]]. %%CITATION = ARXIV:1504.05755;%% %4 citations counted in INSPIRE as of 26 juin 2015 %\cite{Restrepo:2015ura
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{Restrepo:2015ura D.~Restrepo, A.~Rivera, M.~S\'anchez-Pel\'aez, O.~Zapata and W.~Tangarife, %``Radiative neutrino masses in the singlet-doublet fermion dark matter model with scalar singlets,'' arXiv:1504.07892 [hep-ph]. %%CITATION = ARXIV:1504.07892;%% %\cite{Kashiwase:2015pra
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{Kashiwase:2015pra S.~Kashiwase, H.~Okada, Y.~Orikasa and T.~Toma, %``Two Loop Neutrino Model with Dark Matter and Leptogenesis,'' arXiv:1505.04665 [hep-ph]. %%CITATION = ARXIV:1505.04665;%% %\cite{Nishiwaki:2015iqa
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{Nishiwaki:2015iqa K.~Nishiwaki, H.~Okada and Y.~Orikasa, %``A Three Loop Neutrino Model with Isolated $k^{\pm\pm$,'' arXiv:1507.02412 [hep-ph]. %%CITATION = ARXIV:1507.02412;%% %2 citations counted in INSPIRE as of 23 Aug 2015 %\cite{Wang:2015saa
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{Wang:2015saa W.~Wang and Z.~L.~Han, %``A Radiative Linear Seesaw, Dark Matter and $U(1)_{B-L$,'' arXiv:1508.00706 [hep-ph]. %%CITATION = ARXIV:1508.00706;%% %1 citations counted in INSPIRE as of 23 ao\UTF{00FBt 2015 %\cite{Okada:2015hia
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{Okada:2015hia H.~Okada and K.~Yagyu, %``Three-loop Neutrino Mass Model with Doubly Charged Particles from Iso-Doublets,'' arXiv:1508.01046 [hep-ph]. %%CITATION = ARXIV:1508.01046;%% %\cite{Ahriche:2015loa
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{Ahriche:2015loa A.~Ahriche, K.~L.~McDonald and S.~Nasri, %``A Radiative Model for the Weak Scale and Neutrino Mass via Dark Matter,'' arXiv:1508.02607 [hep-ph]. %%CITATION = ARXIV:1508.02607;%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% emphasizing non-Alelian discrete symmetry %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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{Ahn:2012cg Y.~H.~Ahn and H.~Okada, %``Non-zero $\theta_{13$ linking to Dark Matter from Non-Abelian %Discrete Flavor Model in Radiative Seesaw,'' Phys.\ Rev.\ D {85, 073010 (2012) [arXiv:1201.4436 [hep-ph]]. %%CITATION = ARXIV:1201.4436;%% %\cite{Ma:2012ez
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{Hernandez:2013dta A.~E.~Carcamo Hernandez, I.~d.~M.~Varzielas, S.~G.~Kovalenko, H.~P\"{as and I.~Schmidt, %``Lepton masses and mixings in a $A_{4$ multi-Higgs model with radiative seesaw mechanism,'' Phys.\ Rev.\ D {88, 076014 (2013) [arXiv:1307.6499 [hep-ph]]. %%CITATION = ARXIV:1307.6499;%% %2 citations counted in INSPIRE as of 25 Nov 2013 %\cite{Ma:2014eka
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{Aoki:2014cja M.~Aoki and T.~Toma, %``Impact of semi-annihilation of $\mathbb{Z_3$ symmetric dark matter %with radiative neutrino masses,'' JCAP {1409, 016 (2014) [arXiv:1405.5870 [hep-ph]]. %%CITATION = ARXIV:1405.5870;%% %4 citations counted in INSPIRE as of 07 Oct 2014 %\cite{Ma:2014yka
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{Ma:2013mga E.~Ma, %``Radiative Origin of All Quark and Lepton Masses through Dark Matter %with Flavor Symmetry,'' Phys.\ Rev.\ Lett.\ {112, 091801 (2014) [arXiv:1311.3213 [hep-ph]]. %%CITATION = ARXIV:1311.3213;%% %\cite{Okada:2013iba
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{Davoudiasl:2014pya H.~Davoudiasl and I.~M.~Lewis, %``Right-Handed Neutrinos as the Origin of the Electroweak Scale,'' Phys.\ Rev.\ D {90, no. 3, 033003 (2014) [arXiv:1404.6260 [hep-ph]]. %%CITATION = ARXIV:1404.6260;%% %9 citations counted in INSPIRE as of 22 Mar 2015 %\cite{Lindner:2014oea
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{Lindner:2014oea M.~Lindner, S.~Schmidt and J.~Smirnov, %``Neutrino Masses and Conformal Electro-Weak Symmetry Breaking,'' arXiv:1405.6204 [hep-ph]. %%CITATION = ARXIV:1405.6204;%% %\cite{Okada:2014nea
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{King:2014uha S.~F.~King, A.~Merle and L.~Panizzi, %``Effective theory of a doubly charged singlet scalar: complementarity of neutrino physics and the LHC,'' arXiv:1406.4137 [hep-ph]. %%CITATION = ARXIV:1406.4137;%% %1 citations counted in INSPIRE as of 31 Aug 2014 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% radiative seesaw end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\cite{Arina:2012aj
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{Arina:2012aj C.~Arina, R.~N.~Mohapatra and N.~Sahu, %``Co-genesis of Matter and Dark Matter with Vector-like Fourth Generation Leptons,'' Phys.\ Lett.\ B {720, 130 (2013) [arXiv:1211.0435 [hep-ph]]. %%CITATION = ARXIV:1211.0435;%% %25 citations counted in INSPIRE as of 08 sept. 2015
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