compat=1.1.0

# Neutrino mass and $(g-2)_{\mu}$ with dark $U(1)_{D}$ symmetry

Arnab Dasgupta<EMAIL_ADDRESS>Institute of Convergence
Fundamental Studies , Seoul-Tech, Seoul 139-743, Korea  Sin Kyu Kang
<EMAIL_ADDRESS>School of Liberal Arts , Seoul-Tech, Seoul 139-743,
Korea  Myeonghun Park<EMAIL_ADDRESS>School of Liberal Arts ,
Seoul-Tech, Seoul 139-743, Korea

###### Abstract

We propose an extension of the Standard model (SM) for radiative neutrino mass
by introducing a dark $U(1)_{D}$ gauge symmetry. The kinetic mixing between
the SM gauges and the dark $U(1)_{D}$ gauge arises at 1-loop mediated by new
inert scalar fields. We show that the tiny neutrino mass and dark matter
candidates are naturally accommodated. Motivated by the recent measurement of
$(g-2)_{\mu}$ indicating $4.2~{}\sigma$ deviation from the SM prediction, we
examine how the deviation $\Delta a_{\mu}$ can be explained in this model.

##  I Introduction

The recent measurement of the muon anomalous magnetic moment, $a_{\mu}$ =
$(g-2)_{\mu}/2$, by the E989 experiment at Fermilab Abi:2021gix  shows a
discrepancy with respect to the theoretical prediction of the Standard Model
(SM) Aoyama:2020ynm

$\displaystyle a^{\rm FNAL}_{\mu}=116592040(54)\times 10^{-11}$ (1)
$\displaystyle a^{\rm SM}_{\mu}=116591810(43)\times 10^{-11}$ (2)

which when combined with the previous Brookhaven determination Bennett:2004pv
of

$a^{\rm BNL}_{\mu}=116592089(63)\times 10^{-11}$ (3)

leads to a 4.2 $\sigma$ observed excess of $\Delta a_{\mu}=251(59)\times
10^{-11}$. The status of the SM calculation of the muon anomalous magnetic
moment $(g-2)_{\mu}$ has been updated recently in Aoyama:2020ynm ;
Zyla:2020zbs , although the hadronic contributions are still challenging to
reliably estimate Blum:2013xva . The apparent discrepancy with a
$4.2~{}\sigma$ deviation between the SM prediction and the measured value of
$(g-2)_{\mu}$ is a long-standing puzzle which may point to new physics.
Inspired by the latest Fermilab measurements, several literatures that
explains the discrepancy with updating possible theoretical models
Arcadi:2021cwg ; Zhu:2021vlz ; Han:2021gfu ; Baum:2021qzx ; Bai:2021bau ;
Das:2021zea ; Ge:2021cjz ; Brdar:2021pla ; Buen-Abad:2021fwq ; Zu:2021odn ;
Amaral:2021rzw ; Ahmed:2021htr ; Abdughani:2021pdc ; VanBeekveld:2021tgn ;
Cox:2021gqq ; Wang:2021bcx ; Gu:2021mjd ; Cao:2021tuh ; Yin:2021mls ;
Han:2021ify ; Aboubrahim:2021rwz ; Yang:2021duj ; Ferreira:2021gke ;
Wang:2021fkn ; Li:2021poy ; Cadeddu:2021dqx ; Calibbi:2021qto ; Chen:2021vzk ;
Escribano:2021css ; Chun:2021dwx ; Athron:2021iuf .

In this paper, we study how the recent measurement of $(g-2)_{\mu}$ can be
explained in a model for radiative generation of tiny neutrino masses. We
extend the scotogenic model by introducing a dark $U(1)_{D}$ gauge symmetry.
The spontaneous symmetry breaking of dark $U(1)_{D}$ for a massive a “dark
photon” induces $\mathbb{Z}_{2}$ symmetry, an essential ingredient in
scotogenic scenarios for tiny neutrino masses through 1-loop contribution. The
neutral component of a new inert scalar doublet is responsible for dark matter
(DM). In this model, kinetic mixing between the SM neutral gauge bosons and
the dark photon is naturally generated through 1-loop mediated by new scalar
fields. The introduction of a few new particles can reconcile the discrepancy
between the recent measurement and the SM prediction. In this paper, we will
examine the contributions of new scalars and dark photon to $(g-2)_{\mu}$ and
see how they can accommodate the $4.2~{}\sigma$ deviation of $(g-2)_{\mu}$.

This paper is organized as follows. In section II, we present the setup and
the content of new particles by assigning quantum numbers appropriately. In
section III, we show how neutrino mass can arise from 1-loop and present the
kinetic mixing between SM gauge bosons and the dark photon. The contributions
of new scalars and the dark photon to $(g-2)_{\mu}$ are explicitly shown. In
section IV, numerical results and discussion are presented. We finally
conclude in section V.

##  II Model

To generate tiny neutrino masses radiatively and to have natural DM
candidates, we take the framework of the scotogenic model where two scalar
doublets $(\eta_{1},\,\eta_{2})$, two scalar singlets $(\phi,\,\chi)$ and two
vector-like neutral fermions $\Psi_{L,R}$ are introduced. The complete content
of the new fields introduced and their quantum number assignments under
$SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)_{D}$ gauge symmetry are shown in table
1. We note that the model is anomaly free.

Table 1: New particle content with quantum number assignment under an extended
gauge symmetry.
---
Field | SU(2)L | U(1)Y | U(1)D | 2S+1
$\eta_{1}$ | 2 | $\frac{\textbf{1}}{\textbf{2}}$ | -1 | 1
$\eta_{2}$ | 2 | $\frac{\textbf{1}}{\textbf{2}}$ | 1 | 1
$\phi$ | 1 | 0 | 1 | 1
$\chi$ | 1 | 0 | 2 | 1
$\Psi_{L}$ | 1 | 0 | 1 | 2
$\Psi_{R}$ | 1 | 0 | -1 | 2

The Lagrangian for the new fermions is given by

$\displaystyle{\cal L_{F}}\supset$ $\displaystyle
y_{1\nu}\overline{L}\tilde{\eta_{1}}\Psi_{R}+y_{2\nu}\overline{\Psi^{c}_{L}}\tilde{\eta_{2}}L+y_{\Psi_{L}}\chi^{\ast}\overline{\Psi}^{c}_{L}\Psi_{L}+y_{\Psi_{R}}\chi\overline{\Psi}^{c}_{R}\Psi_{R}+M_{\Psi}\overline{\Psi}_{L}\Psi_{R}+h.c.$
(4)

The Lagrangian for the entire scalar sector is given by

$\displaystyle{\cal L_{S}}\supset$ $\displaystyle
m^{2}_{H}H^{\dagger}H+m^{2}_{\phi}\phi^{\dagger}\phi+m^{2}_{\chi}\chi^{\dagger}\chi+m^{2}_{\eta_{1}}\eta_{1}^{\dagger}\eta_{1}+m^{2}_{\eta_{2}}\eta_{2}^{\dagger}\eta_{2}+\tilde{\mu_{2}}\tilde{\eta_{1}}\eta_{2}\chi+\tilde{\mu_{1}}\chi^{\ast}\phi\phi$
(5)
$\displaystyle+\lambda_{H}(H^{\dagger}H)^{2}+\lambda_{\phi}(\phi^{\dagger}\phi)^{2}+\lambda_{\chi}(\chi^{\dagger}\chi)^{2}+\lambda_{\eta_{1}}(\eta_{1}^{\dagger}\eta_{1})^{2}+\lambda_{\eta_{2}}(\eta_{2}^{\dagger}\eta_{2})^{2}$
$\displaystyle+\lambda_{\phi\eta_{1}}(\phi^{\dagger}\phi)(\eta_{1}^{\dagger}\eta_{1})+\lambda_{\phi\eta_{2}}(\phi^{\dagger}\phi)(\eta_{2}^{\dagger}\eta_{2})+\lambda_{\phi\chi}(\phi^{\dagger}\phi)(\chi^{\dagger}\chi)+\lambda_{\chi\eta_{1}}(\chi^{\dagger}\chi)(\eta_{1}^{\dagger}\eta_{1})$
$\displaystyle+\lambda_{\phi\eta_{2}}(\phi^{\dagger}\phi)(\eta_{2}^{\dagger}\eta_{2})+\lambda_{\eta_{1}\eta_{2}}(\eta_{1}^{\dagger}\eta_{1})(\eta_{2}^{\dagger}\eta_{2})+\lambda_{\eta_{1}\eta_{2}\phi}\tilde{\eta_{1}}\eta_{2}\phi^{2}+\lambda_{H\eta_{1}\phi\chi}H^{\dagger}\eta_{1}\phi^{\ast}\chi$
$\displaystyle+\lambda_{H\eta_{2}\phi\chi}H^{\dagger}\eta_{2}\phi\chi^{\ast}+(H^{\dagger}H)(\lambda_{H\eta_{1}}\eta_{1}^{\dagger}\eta_{1}+\lambda_{H\eta_{2}}\eta_{2}^{\dagger}\eta_{2}+\lambda_{H\chi}\chi^{\dagger}\chi+\lambda_{H\phi}\phi^{\dagger}\phi),$

where $H$ represents the SM Higgs scalar whose vacuum expectation value (VEV)
is denoted by $v$. In addition the SM Higgs, $\chi$ gets the VEV, which
spontaneously breaks $U(1)_{D}$, and then the scalar $\chi$ can be written as

$\chi=v_{\chi}+h_{\chi}+i\xi_{\chi},$ (6)

where $v_{\chi}$ is VEV of $\chi$, and $h_{\chi},\xi_{\chi}$ denote the real
and imagiranry components of the field, respectively. After Ew symmetry as
well as $U(1)_{D}$ are spontaneously broken, we obtain the squared mass matrix
for the scalars $H,\chi$ given as

$\displaystyle
M_{H\chi}^{2}=\left(\begin{matrix}\lambda_{H}v^{2}&\lambda_{H\chi}vv_{\chi}\\\
\lambda_{H\chi}vv_{\chi}&\lambda_{\chi}v^{2}_{\chi}\end{matrix}\right).$ (7)

The squared mass matrix is diagnalized by the mixing matrix defined as

$\displaystyle\left(\begin{matrix}H^{0}\\\
\chi\end{matrix}\right)=\left(\begin{matrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{matrix}\right)\left(\begin{matrix}h_{1}\\\
h_{2}\end{matrix}\right),$ (8)

where the mixing angle $\theta$ is given as

$\tan
2\theta=\frac{2\lambda_{H\chi}v_{\chi}}{\lambda_{\chi}v^{2}_{\chi}-\lambda_{H}v^{2}}$
(9)

We take into account the kinetic mixing between the SM gauges and $U(1)_{D}$
gauge which arises via 1-loop. Then, the kinetic terms of the gauge fields
containing the kinetic mixing are given by

${\cal L_{\rm
kin}}=-\frac{1}{4}\hat{F}_{\mu\nu}\hat{F}^{\mu\nu}-\frac{1}{4}\hat{Z}_{\mu\nu}\hat{Z}^{\mu\nu}-\frac{1}{4}\hat{V}_{\mu\nu}\hat{V}^{\mu\nu}-\frac{\epsilon}{2}\hat{F}_{\mu\nu}\hat{V}^{\mu\nu}-\frac{\epsilon_{ZV}}{2}\hat{Z}_{\mu\nu}\hat{V}^{\mu\nu},$
(10)

where $\hat{F}_{\mu\nu},\hat{Z}_{\mu\nu}$ , and $\hat{V}_{\mu\nu}$ represent
the gauge field strengths for $U(1)_{Y},SU(2)_{L}$ , and $U(1)_{D}$,
respectively. The relevent piece of the covariant derivatives is given as

$-ieQA_{\mu}-i\left[\frac{g}{c_{W}}(T^{3}_{L}-s^{2}_{W}Q)-g_{D}s_{\theta_{2V}}Q_{D}\right]Z_{\mu}-i\left[\epsilon
eQ+g_{D}Q_{D}\right]Z^{\prime}_{\mu}$ (11)

where $Q$ and $Q_{D}$ are the charges of $U(1)_{Y}$ and $U(1)_{D}$,
respectively. The kinetic terms are diagonalized by the field redefinition
given as

$\displaystyle\left(\begin{matrix}\hat{A}_{\mu}\\\ \hat{Z}_{\mu}\\\
\hat{V}_{\mu}\end{matrix}\right)=\left(\begin{matrix}1&0&-\frac{\epsilon}{D}\\\
0&1&-\frac{\epsilon_{ZV}}{D}\\\
0&0&\frac{1}{D}\end{matrix}\right)\left(\begin{matrix}A_{\mu}\\\ Z_{\mu}\\\
Z^{\prime}_{\mu}\end{matrix}\right).$ (12)

Following Ref.Rueter:2020qhf , the mixing parameter $\epsilon$ generated from
the 1-loop diagram mediated by two scalars $\eta_{1},\eta_{2}$ as shown in
fig.1 and is given as

$\epsilon=\frac{eg_{D}}{48\pi^{2}}{\rm
ln}\left(\frac{M^{2}_{\eta_{1}}}{M^{2}_{\eta_{2}}}\right).$ (13)

$Z,A$$Z_{D}$$\eta_{1}$$\eta_{2}$ | $Z,A$$Z_{D}$$\eta_{1,2}$$\eta_{1,2}$
---|---
Figure 1: Feynman diagrams for the amplitudes generating the Kinetic mixing.

In this work, we ignore the term proportional to $\epsilon_{ZV}$ which is in
general much smaller than $\epsilon$.

##  III Radiative neutrino mass and New contributions to $(g-2)_{\mu}$

Let us first study how the light neutrino mass can be radiatively generated.
From the field redefinition of the neutral fermions whose mass terms are
presented in Eq.(4) given as

$\displaystyle\left(\begin{matrix}\Psi_{L}\\\
\Psi_{R}\end{matrix}\right)=\left(\begin{matrix}\cos\theta_{N}&\sin\theta_{N}\\\
-\sin\theta_{N}&\cos\theta_{N}\end{matrix}\right)\left(\begin{matrix}N_{1}\\\
N_{2}\end{matrix}\right),$ (14)

we can obtain mass eigenvalues of the new neutral fermions after $\chi$ gets
VEV given as

$\displaystyle
M_{N_{1}(2)}=\frac{1}{2}[y_{\Psi_{L}}v_{\chi}+y_{\Psi_{R}}v_{\chi}\mp\sqrt{M_{\Psi}^{2}-4y_{\Psi_{L}}y_{\Psi_{R}}v_{\chi}^{2}}].$
(15)

As for the scalar doublet the mass matrix can be simplified by assuming the
$\phi$ only mixes with the lightest doublet hence the mixing matrix takes the
following form

$m^{2}_{R,I}=\begin{pmatrix}\mu^{2}_{\eta_{1}}&\lambda_{H\eta_{1}\phi\chi}vv_{\chi}\\\
\lambda_{H\eta_{1}\phi\chi}vv_{\chi}&\mu^{2}_{\phi}\pm\mu_{1}v_{\chi}\end{pmatrix}.$
(16)

Consequently, the projection to the mass basis is given as

$\displaystyle\eta_{R,I}$
$\displaystyle=\cos(\theta_{R,I})\xi_{1R,I}+\sin(\theta_{R,I})\xi_{2R,I}$
$\displaystyle\phi_{R,I}$
$\displaystyle=-\sin(\theta_{R,I})\xi_{1R,I}+\cos(\theta_{R,I})\xi_{2R,I}$
(17)

The 1-loop Feynman diagram for the generation of neutrino mass is shown in
Fig. 2. The neutrino mass is explicitly given by

$\displaystyle(M_{\nu})_{\alpha\beta}$ $\displaystyle=\sum_{k}y_{\alpha
k}y_{\beta k}\Lambda_{k}$ (18) $\displaystyle\Lambda_{k}$
$\displaystyle=\frac{M_{N_{k}}}{16\pi^{2}}\left[\cos^{2}(\theta_{R})F(x_{1R})+\sin^{2}(\theta_{R})F(x_{2R})\right.$
$\displaystyle-\left.\cos^{2}(\theta_{I})F(x_{1I})-\sin^{2}(\theta_{I})F(x_{2I})\right],$
(19)

where $F(x_{i},x_{j})$ is defined as

$\displaystyle F(x)$ $\displaystyle=\frac{x}{x-1}\ln[x],$ (20)

with $x_{i}=\frac{m_{i}^{2}}{M_{N_{2}}^{2}}$. For the purpose of numerical
calculation, We take Casas-Ibarra (CI) parametrisation Casas:2001sr  for the
matrix of the Yukawa coupling satisfying the neutrino data given as

$\displaystyle y_{i\alpha}\ =\
\left(UD_{\nu}^{1/2}R^{\dagger}\Lambda^{1/2}\right)_{i\alpha}\,,$ (21)

where $R,UD_{\nu}$ and $\Lambda$ are an arbitrary complex orthogonal matrix,
the neutrino mixing matrix, diagonal light neutrino mass matrix , and diagonal
loop factor given in eq.(III). For the sake of simplicity, we take $R$ to be
identity. As for $U$ and $D_{\nu}$, we take experimental results for the case
of normal hierarchy given in Zyla:2020zbs .

Now, let us consider the contributions of dark sector to $(g-2)_{\mu}$. In our
model, new particles are assumed to be heavy except for the dark photon
$Z^{\prime}$ and a light dark scalar $h_{2}$ whose masses are order of a few
100 MeV. Then, sizable corrections to $(g-2)_{\mu}$ arises from the
interactions

$\displaystyle(e\epsilon)\overline{\mu}\gamma^{\mu}\mu Z^{\prime}_{\mu},$
$\displaystyle(\sin\theta y_{\mu})\overline{\mu}\mu h_{2}$ (22)

The $(g-2)_{\mu}$ has new three main contributions which can be categorized as
follows:

  1. 1.

Mediated by dark scalar $\chi$ through SM Higgs Dark scalar mixing.

  2. 2.

Mediated by the dark photon through kinetic mixing.

  3. 3.

Mediated by the inert-double and the neutral fermion.

The 1-loop contribution mediated by the dark scalar $h_{2}$ to $(g-2)_{\mu}$
is given byQueiroz:2014zfa .

$\displaystyle\Delta a_{\mu}(h_{2})$ $\displaystyle=$
$\displaystyle\frac{y^{2}_{\mu}s^{2}_{\theta}}{4\pi^{2}}\frac{m^{2}_{\mu}}{M^{2}_{h_{2}}}\int^{1}_{0}dx\frac{x^{2}(2-x)}{(1-x)(1-\xi_{h_{2}}^{2}x)+\xi_{h_{2}}^{2}x}$
(23) $\displaystyle=$
$\displaystyle\frac{y^{2}_{\mu}s^{2}_{\theta}}{4\pi^{2}}\frac{m^{2}_{\mu}}{M^{2}_{h_{2}}}\left[{\rm
ln}\left(\frac{M_{h_{2}}}{m_{\mu}}\right)-\frac{7}{12}\right],$

where $\xi_{h_{2}}=m_{\mu}/M_{h_{2}}$. The 1-loop contribution mediated by the
dark photon $Z^{\prime}$ to $(g-2)_{\mu}$ is given by

$\Delta
a_{\mu}(Z^{\prime}_{\mu})=\frac{e^{2}\epsilon^{2}}{8\pi^{2}}\frac{m^{2}_{\mu}}{M^{2}_{Z^{\prime}}}\int^{1}_{0}dx\frac{2x^{2}(1-x)}{(1-x)(1-\xi_{Z^{\prime}}^{2}x)+\xi_{Z^{\prime}}^{2}x}$
(24)

where $\xi_{Z^{\prime}}=m_{\mu}/M_{Z^{\prime}}$.

$\nu_{\alpha}$$\nu_{\beta}$$\langle H\rangle$$\langle
H\rangle$$\langle\chi\rangle$$N_{i}$$\eta_{i}$$\phi$$\phi$$\eta_{i}$ Figure 2:
Neutrino Mass production.

Along with these there is another contribution coming from the neutral fermion
and the charged scalar which is given as

$\displaystyle\Delta a_{\mu}(\eta,N)$
$\displaystyle=\sum_{ik}\frac{y^{\eta_{i}*}_{\mu k}y^{\eta_{i}}_{\mu
k}}{16\pi^{2}}\left[\frac{1}{6(\alpha_{ik}-1)^{4}}\left(-2\alpha^{3}_{ik}-3\alpha^{3}_{ik}+6\alpha_{ik}-1+6\alpha^{2}_{ik}\ln[\alpha_{ik}]\right)\right],$

where $\alpha_{ik}=m^{2}_{N_{k}}/m^{2}_{\eta_{i}}$. It is worthwhile to note
that the contribution mediated by the dark scalar is negligibly small because
the yukawa coupling for the muon is already of the order $10^{-4}$ and small
mixing between the dark scalar and the SM Higgs gives another suppression.

##  IV Numerical Result and discussion

Ignoring the contributions from the light dark scalar, the total contribution
of $\Delta a_{\mu}$ is given by the addition of the 1-loop contributions
mediated by the dark photon, and by inert scalar bosons accompanied by neutral
fermions as follows:

$\displaystyle\Delta a_{\mu}$ $\displaystyle=\Delta a_{\mu}(Z^{\prime})+\Delta
a_{\mu}(\eta,N).$ (25)

We note that $\Delta a_{\mu}(\eta,N)$ has a overall minus sign which ends up
contributing negatively and needs to be compensated by the contribution coming
from the kinetic mixing $\Delta a_{\mu}(Z^{\prime})$.

In our analysis, we assume that the mass of the dark photon is around 100 MeV,
whereas the mass of the lightest inert neutral scalar is around
$\geq\mathcal{O}(500)$ GeV which is the favorable regime for the good dark
matter candidate. For the yukawa coupling $y_{\nu}$, we take to be
$\mathcal{O}(1)$ so that we can achieve the tiny neutrino mass of order
atmospheric mass scale by tuning the trilinear coupling $\mu_{i}$ small. But
we keep the mass splitting between $\eta_{DMR}$ and $\eta_{DMI}$ to be above
$\mathcal{O}$(keV) to circumvent the constraint coming from direct detection
resulting from $Z-$ mediation. Under the assumptions above, we see that the
contribution from the 1-loop mediated by the dark photon is dominant over the
others.

To calculate $\Delta a_{\mu}$ we scan the ranges of the input parameter given
as

$\displaystyle 1{\rm GeV}$ $\displaystyle\lesssim m_{\eta_{1}}\lesssim
1000{\rm GeV},$ $\displaystyle 0.1$ $\displaystyle\lesssim g_{D}\lesssim 1,$
$\displaystyle m_{\eta_{2}}=4m_{\eta_{1}},$ $\displaystyle
m_{\phi}=5m_{\eta_{1}},$ $\displaystyle m_{N_{i}}=m_{\eta_{1}}.$

Note that the parameter $\epsilon$ can be calculated from eq.(13). In Fig. 3,
we present how $\Delta a_{\mu}$ is predicted in terms of model parameters. The
upper left panel shows the predictions of $\Delta a_{\mu}(Z^{\prime})$ in the
parameter space $(m_{Z_{D}}/g_{D},\epsilon)$, whereas the upper right panel
shows the predictions of $\Delta a_{\mu}(\eta,N)$ in the parameter space
$(m_{\eta},y^{\dagger}_{\nu}y_{\nu})$. The two red points correspond to the
bench mark points presented in Table 2. The lower panel shows how $\Delta
a_{\mu}$ is composed of the two contributions. As can be seen, $\Delta
a_{\mu}(Z^{\prime})$ is dominant over $\Delta a_{\mu}(\eta,N)$ for the ranges
of our parameter space we scan. The black curve in the plot corresponds to the
$4.2~{}\sigma$ deviation from the SM prediction. We have checked that the
bench mark points present in the Table 2 can accommodate the tiny neutrino
mass or order of atmospheric mass scale.

|
---|---

Figure 3: Predictions of $\Delta a_{\mu}(Z^{\prime})$ in terms of $(m_{Z_{D}}/g_{D},\epsilon)$ (upper right) and those of $\Delta a_{\mu}(\eta,N)$ in terms of $(m_{\eta},y^{\dagger}_{\nu}y_{\nu})$ (upper left). Predictions of $\Delta a_{\mu}$ along with $\Delta a_{\mu}(Z^{\prime})$ and $\Delta a_{\mu}(\eta,N)$. Two red points correspond to the two bench mark points presented in Table 2 and the black curve correspond to $4.2\sigma$ deviation. Parameters | BP1 | BP2
---|---|---
$g_{D}$ | 0.92 | 0.98
$v_{\chi}$ | 10 MeV | 50 MeV
$\mu_{1}$ |  $1.14$ (keV) |  $460.8$ (eV)
$m_{\eta_{1}}$ | 500 GeV | 800 GeV
$m_{\eta_{2}}$ | 2 TeV | 3.2 TeV
$m_{\phi}$ | 2.5 TeV | 3.6 TeV
$\lambda_{H\chi}$ | 0.01 | 0.01
$\lambda_{\chi}$ | 0.001 | 0.001
$\lambda_{H\eta_{1}\phi}$ | 0.4 | 0.4
$\lambda_{H\eta_{2}\phi}$ | 0 | 0
Table 2: Two bench mark points accommodating the tiny neutrino mass around
atmospheric scale, dark matter relic density and the $4.2~{}\sigma$ deviation
of $(g-2)_{\mu}$.

##  V conclusion

We have considered a dark $U(1)_{D}$ extension of the SM gauge symmetry to
achieve the tiny neutrino mass and to have dark matter candidate. In this
model, the kinetic mixing between the SM gauges and the $U(1)_{D}$ gauge
naturally arises at 1-loop mediated by new inert scalar fields. The light
neutral inert scalar boson can be a good dark matter candidate. Motivated by
the recent measurement of $(g-2)_{\mu}$ indicating $4.2~{}\sigma$ deviation
from the SM prediction, we have studied how the deviation $\Delta a_{\mu}$ can
be explained in this model. The 1-loop mediated by dark photon with mass of
order 100 MeV as well as inert scalar fields accompanied by the neutral
fermions can accommodate the recent measurement of $(g-2)_{\mu}$ at Fermilab.

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