# Dirac dark matter, dark radiation, and the type-II seesaw mechanism in
alternative $U(1)_{X}$ standard model

Nobuchika Okada<EMAIL_ADDRESS>Department of Physics and Astronomy, University
of Alabama, Tuscaloosa, Alabama 35487, USA  Osamu Seto
<EMAIL_ADDRESS>Institute for the Advancement of Higher
Education, Hokkaido University, Sapporo 060-0817, Japan  Department of
Physics, Hokkaido University, Sapporo 060-0810, Japan

###### Abstract

We propose an extra $U(1)_{X}$ model with an alternative charge assignment for
right-handed right-handed neutrinos. The type-II seesaw mechanism by a triplet
Higgs field is promising for neutrino mass generation because of the
alternative charge assignment. The small vacuum expectation value (VEV) of an
additional Higgs doublet naturally leads to a very small VEV of the triplet
Higgs field, and as a result, the smallness of neutrino mass can be
understood. With the minimal Higgs field for $U(1)_{X}$ with the charge $1$,
right-handed neutrinos are candidates for Dirac dark matter (DM) and dark
radiation (DR). We have derived and imposed the LHC bound, the DR constraint
and the bound from DM direct searches in the wide range of parameter space.
Among various $U(1)_{X}$ choices, the DM direct search bound is found to be
weakest for $U(1)_{R}$ where the constraints from thermal DM and non-
negligible DR can be compatible. Such a number of the effective neutrino
species would be interesting from the viewpoint of the so-called Hubble
tension.

††preprint: EPHOU-22-006

##  I Introduction

The standard model (SM) of particle physics has been constructed on the basis
of the gauge principle and the spontaneous symmetry breaking by the Higgs
mechanism at the vacuum. The introduction of an extra $U(1)$ gauge interaction
is one of the promising and well-defined extensions of the SM. The $B-L$
(baryon number minus lepton number) appears to be an accidental global
symmetry in the SM, indicating that this might be a gauge symmetry in a UV
completion of the theory Pati:1973uk ; Davidson:1978pm ; Mohapatra:1980qe ;
Mohapatra:1980 . At the same time, the cancellation of an anomaly for a chiral
gauge theory is critical. When the gauge symmetry is extended from the SM
gauge group $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ to $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}\times U(1)_{B-L}$, the number of right-handed
neutrinos and those charge are limited to two choices by the anomaly
cancellation conditions. One is three right-handed neutrinos with each
$U(1)_{B-L}$ charge $-1$, which has been usually considered - and might be
called - the standard assignment. Under this assignment, it is straightforward
to explain observed neutrino masses by the type-I seesaw mechanism
Minkowski:1977sc ; Yanagida:1979as ; GellMann:1980vs ; Mohapatra:1979ia  with
right-handed neutrinos, by introducing the scalar field with the $B-L$ charge
$2$ that generates the Majorana masses of right-handed neutrinos. In the other
charge assignment, two out of three right-handed neutrinos have the
$U(1)_{B-L}$ charge of $-4$ and the one has the charge of $+5$, which is
sometimes called the alternative assignment Montero:2007cd .

A similar extra $U(1)$ extended model can be constructed based on the
$U(1)_{R}$ gauge symmetry where only right-handed fermions are charged while
left-handed ones are not charged Jung:2009jz . The $U(1)_{R}$ charged
particles are same as in the $U(1)_{B-L}$ model up to chirality, neutrino
masses can be generated by the type-I seesaw mechanism at tree level
Nomura:2017tih  or loop level Nomura:2017ezy . The implication due to the
chirality dependence Jung:2009jz ; Jana:2019mez ; Das:2021esm , the axial-
vector coupling Seto:2020jal , $U(1)_{R}$ charged Higgs scalars Ko:2013zsa ;
Ko:2012hd , and $U(1)_{R}$ interacting dark matter (DM) Chao:2017rwv ;
Seto:2020udg ; Okada:2020evk ; Nagao:2020azf , can be found in literature. The
anomaly-free most general gauged $U(1)$ extension of the SM is defined as a
linear combination of the SM $U(1)_{Y}$ and the $U(1)_{B-L}$ gauge groups.
With this convenient parametrization of $U(1)_{X}$ Appelquist:2002mw ;
Oda:2015gna ; Das:2016zue , we can study a wide class of extra $U(1)$ models
including the representative model $U(1)_{B-L}$ and $U(1)_{R}$. In this paper,
we consider $U(1)_{X}$ models with the alternative $U(1)_{B-L}$ assignment.

Under the alternative $B-L$ charge assignment, it is nontrivial for the type-I
seesaw mechanism to generate neutrino masses, because right-handed neutrinos
cannot form Yukawa coupling with lepton doublets and the SM Higgs doublet due
to the charge mismatch Ma:2014qra ; Sanchez-Vega:2015qva ; Nomura:2017jxb ;
Singirala:2017cch ; Geng:2017foe ; Okada:2018tgy ; Das:2018tbd ; Das:2017deo ;
Das:2019fee ; Choudhury:2020cpm ; Asai:2020xnz . Rather, the so-called type-II
seesaw mechanism by a $SU(2)$ triplet Higgs seems to be a simple way of
nonvanishing neutrino mass generation Schechter:1980gr ; Magg:1980ut ;
Cheng:1980qt , because a Dirac neutrino mass terms are not necessary there
Mahapatra:2020dgk ; Ghosh:2021khk .

In this paper we propose the triplet Higgs models in the extra $U(1)_{X}$
model with the alternative charge for right-handed neutrinos. The triplet
Higgs fields have to be charged under the extra $U(1)_{X}$ to have Yukawa
interactions with lepton doublets which are also charged under the $U(1)_{X}$.
Simultaneously, the triplet Higgs field requires the introduction of another
doublet Higgs to develop the vacuum expectation value (VEV) by a scalar
trilinear term. This second Higgs doublet cannot couple with the SM fermions
due to the charge mismatch. In this construction, the charge assignment
requires that the Higgs sector must be the combination of so-called type-I two
Higgs doublet model (THDM) Branco:2011iw  and triplet Higgs model. The
smallness of the generated neutrino mass via the type-II seesaw mechanism is a
consequence of the smallness of the second Higgs VEV. In a philosophical
sense, this is a neutrinophilic Higgs model Ma:2000cc  in the bosonic sector.
Since the small triplet Higgs VEV is naturally realized in the wide range of
parameter space, we may expect it easier to observe the Majorana nature
through the same sign dilepton signal from the doubly-charged Higgs boson in
collider experiments. The simplest way to realize the $U(1)_{X}$ breaking is
to introduce an SM singlet scalar with $U(1)_{X}$ charge $1$ whose VEV
generates not only the mass of pseudoscalar but also the mass of right-handed
neutrino DM. In the minimal extension, the singlet scalar plays three roles.
This model with only one $U(1)_{X}$ charged scalar predicts a Dirac right-
handed neutrino and one massless right-handed neutrino at the renormalizable
level.111For a model with nonrenormalizable terms, see Ref. Asai:2020xnz .
This could be interesting from the viewpoint of cosmology, because those
states are candidates of DM and dark radiation (DR), respectively. The DR has
been constrained by cosmological observations Planck:2018vyg , and the DR
constraint on the $U(1)_{B-L}$ model has been studied Heeck:2014zfa ;
FileviezPerez:2019cyn . On the other hand, DR is interesting, because it could
relax the so-called Hubble tension222For a review, see for example Ref.
DiValentino:2021izs  which is the discrepancy between the current Hubble
parameter $H_{0}$ inferred from the cosmic microwave background by Planck
Planck:2018vyg  and that measured by low-$z$ observations such as the SH0ES
collaboration Riess:2019cxk . The preferred value of the number of effective
neutrino species $N_{\mathrm{eff}}$ is evaluated as $3.2\lesssim
N_{\mathrm{eff}}\lesssim 3.5$ Planck:2018vyg  and $3.2\lesssim
N_{\mathrm{eff}}\lesssim 3.4$ Seto:2021xua ; Seto:2021tad  for different data
sets.

This paper is organized as follows. In Sec. II, we describe the extra
$U(1)_{X}$ model with the alternative charge assignment for right-handed
neutrinos and the generation of neutrino masses by the type-II seesaw
mechanism. The mass spectrum of the particles, especially the various Higgs
particles and right-handed neutrinos will be derived. We also summarize the
present experimental constraints on the model. In Sec. III we provide the
relevant formula for discussion of DM and DR. In Sec. IV we present the
interesting parameter region in terms of the current experimental and
cosmological constraints for the different parameter sets of the $U(1)_{X}$
model. Section V is devoted to our summary.

##  II Model

###  II.1 $U(1)_{X}$ model with a triplet scalar

| $SU(3)_{C}$ | $SU(2)_{L}$ | $U(1)_{Y}$ | $U(1)_{X}$
---|---|---|---|---
$q_{L}^{i}$ | $\mathbf{3}$ | $\mathbf{2}$ | $\frac{1}{6}$ | $\frac{1}{6}x_{H}+\frac{1}{3}$
$u_{R}^{i}$ | $\mathbf{3}$ | $\mathbf{1}$ | $\frac{2}{3}$ | $\frac{2}{3}x_{H}+\frac{1}{3}$
$d_{R}^{i}$ | $\mathbf{3}$ | $\mathbf{1}$ | $-\frac{1}{3}$ | $-\frac{1}{3}x_{H}+\frac{1}{3}$
$l_{L}^{i}$ | $\mathbf{1}$ | $\mathbf{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}x_{H}-1$
$e_{R}^{i}$ | $\mathbf{1}$ | $\mathbf{1}$ | $-1$ | $-x_{H}-1$
$\nu_{R}^{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $0$ | $-4$
$\nu_{R}^{2}$ | $\mathbf{1}$ | $\mathbf{1}$ | $0$ | $-4$
$\nu_{R}^{3}$ | $\mathbf{1}$ | $\mathbf{1}$ | $0$ | $5$
$\Phi_{1}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\frac{1}{2}$ | $\frac{1}{2}x_{H}+1$
$\Phi_{2}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\frac{1}{2}$ | $\frac{1}{2}x_{H}$
$\Delta_{3}$ | $\mathbf{1}$ | $\mathbf{3}$ | $1$ | $x_{H}+2$
$\Phi_{X}$ | $\mathbf{1}$ | $\mathbf{1}$ | $0$ | $1$
Table 1:  In addition to the SM particle content ($i=1,2,3$), three right-
handed neutrinos $\nu_{R}^{i}$ ($i=1,2,3$), one Higgs doublet $\Phi_{1}$, one
triplet Higgs $\Delta_{3}$, and one $U(1)_{X}$ Higgs field $\Phi_{X}$ are
introduced. $x_{H}$ is a real free parameter in the $U(1)_{X}$ charge unfixed
by the anomaly-free conditions.

Our model is based on the gauge group $SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}\times U(1)_{X}$. The particle content is listed in Table 1. Under
these gauge groups, three generations of right-handed neutrinos ($\nu_{R}^{i}$
with $i$ running $1,2,3$) have to be introduced for the anomaly cancellation.
We consider the alternative $U(1)_{B-L}$ charge assignment for right-handed
neutrinos. The $U(1)_{X}$ symmetry is defiend as the linear combination of
$U(1)_{Y}$ and $U(1)_{B-L}$, and the $U(1)_{X}$ charge is parametrized by the
relative $U(1)_{Y}$ charge $x_{H}$ normalized by the $U(1)_{B-L}$ charge. For
this charge assignment, right-handed neutrinos can not have Yukawa interaction
with left-handed leptons $l_{L}^{i}$ and the SM Higgs field $\Phi_{2}$ due to
the mismatch of the charge. Instead of introducing two doublet Higgs fields
and two singlet Higgs fields to generate Majorana masses of right-handed
neutrinos and to form Dirac neutrino masses for the type-I seesaw
Okada:2018tgy , we introduce, as an economical way, one triplet Higgs field
$\Delta_{3}$ and one doublet field $\Phi_{1}$, which can generate neutrino
masses by type-II seesaw, and one singlet field $\Phi_{X}$ to break the
$U(1)_{X}$ symmetry. Moreover, the SM singlet scalar with $U(1)_{X}$ charge
$1$ generates not only the mass of pseudoscalar but also the mass of right-
handed neutrino DM. Thus, this extension of the scalar sector is the minimal
and unique. The additional $\Phi_{1}$ does not have any Yukawa coupling to the
SM fermion as in the type-I THDM but is necessary to have a trilinear term of
$\Delta_{3}$.

The Yukawa couplings with those additional Higgs fields are given by

$\displaystyle\mathcal{L}_{\mathrm{Yukawa}}\supset-\frac{1}{\sqrt{2}}Y_{\Delta}^{ij}\overline{l_{L}^{i~{}C}}\cdot\Delta
l_{L}^{j}-\sum_{i=1,2}Y_{\nu_{R}^{i}}\Phi_{X}^{\dagger}\overline{\nu_{R}^{3~{}C}}\nu_{R}^{i}+\mathrm{H.c.},$
(1)

where the superscript $C$ denotes the charge conjugation, the dot denotes the
antisymmetric product of $SU(2)$, $Y_{\Delta}$ and $Y_{\nu_{R}^{i}}$ are
Yukawa couplings between left-handed lepton doublets and right-handed
neutrinos, respectively. After the triplet Higgs field develops a VEV
$v_{\Delta}$, left-handed neutrino masses are generated as

$\displaystyle(\mathcal{M}_{\nu})_{ij}=Y_{\Delta}^{ij}v_{\Delta}.$ (2)

After $\Phi_{X}$ develops a VEV $v_{X}$, the Yukawa interactions between
$\nu_{R}^{i}$ with $\Phi_{X}$ give the Dirac mass of right-handed neutrinos as

$\displaystyle\mathcal{L}_{\nu_{R}\mathrm{mass}}=$
$\displaystyle-(\nu_{R}^{1}{}^{\dagger}\,\,\nu_{R}^{2}{}^{\dagger}\,\,\nu_{R}^{3}{}^{T})\left(\begin{array}[]{ccc}0&0&Y_{\nu_{R}^{1}}\\\
0&0&Y_{\nu_{R}^{2}}\\\ Y_{\nu_{R}^{1}}&Y_{\nu_{R}^{2}}&0\\\
\end{array}\right)\frac{v_{X}}{\sqrt{2}}\left(\begin{array}[]{c}\nu_{R}^{1}{}\\\
\nu_{R}^{2}{}\\\ \nu_{R}^{3}{}^{*}\\\ \end{array}\right)$ (9) $\displaystyle=$
$\displaystyle-(\nu_{R}^{1}{}^{\dagger}\,\,\nu_{R}^{2}{}^{\dagger}\,\,\nu_{R}^{3}{}^{T})U^{\dagger}\left(\begin{array}[]{ccc}0&0&0\\\
0&0&m\\\ 0&m&0\\\ \end{array}\right)U\left(\begin{array}[]{c}\nu_{R}^{1}{}\\\
\nu_{R}^{2}{}\\\ \nu_{R}^{3}{}^{*}\\\ \end{array}\right),$ (16)

where we define

$\displaystyle U=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{m_{2}}{m}&-\frac{m_{1}}{m}&0\\\
\frac{m_{1}}{m}&\frac{m_{2}}{m}&0\\\ 0&0&1\\\ \end{array}\right),$ (20)
$\displaystyle m_{1}=$ $\displaystyle Y_{\nu_{R}^{1}}\frac{v_{X}}{\sqrt{2}},$
(21) $\displaystyle m_{2}=$ $\displaystyle
Y_{\nu_{R}^{2}}\frac{v_{X}}{\sqrt{2}},$ (22) $\displaystyle m=$
$\displaystyle\sqrt{m_{1}^{2}+m_{2}^{2}}=\sqrt{\frac{Y_{\nu_{R}^{1}}^{2}+Y_{\nu_{R}^{2}}^{2}}{2}}v_{X}.$
(23)

We find that one linear combination

$\displaystyle\nu_{R}=\frac{m_{2}}{m}\nu_{R}^{1}-\frac{m_{1}}{m}\nu_{R}^{2},$
(24)

is a massless state. Other two components are summarized as

$\displaystyle\mathcal{L}_{\nu_{R}\mathrm{mass}}=$
$\displaystyle-\left(\frac{m_{1}}{m}\nu_{R}^{1}{}^{\dagger}+\frac{m_{2}}{m}\nu_{R}^{2}{}^{\dagger}\,\,\,\nu_{R}^{3}{}^{T}\right)\left(\begin{array}[]{cc}0&1\\\
1&0\\\
\end{array}\right)m\left(\begin{array}[]{c}\frac{m_{1}}{m}\nu_{R}^{1}+\frac{m_{2}}{m}\nu_{R}^{2}\\\
\nu_{R}^{3}{}^{*}\\\ \end{array}\right)$ (29) $\displaystyle=$
$\displaystyle-\overline{\chi}m\chi,$ (30)

by composing a Dirac spinor as

$\displaystyle\chi=$
$\displaystyle\left(\begin{array}[]{c}\frac{m_{1}}{m}\nu_{R}^{1}+\frac{m_{2}}{m}\nu_{R}^{2}\\\
\nu_{R}^{3}{}^{*}\\\ \end{array}\right).$ (33)

There is one massless state $\nu_{R}$ and one Dirac fermion $\chi$. Thus,
$\nu_{R}$ behaves as DR and $\chi$ is a candidate for DM. It is worth noting
that $\chi$ has no direct coupling with the SM particles thanks to its
$U(1)_{X}$ charge assignment which guarantees the stability of the DM
candidate. This is in a remarkable contrast with right-handed neutrino DM in
the minimal $U(1)_{X}$ model with the standard charge assignment, where the
extra $Z_{2}$ parity has to be introduced by hand to stabilize DM Okada:2010wd
; Okada:2016gsh ; Okada:2016tci .333For a review on this class of models, see
e.g., Ref. Okada:2018ktp .

The gauge interactions of $\nu_{R}$ and $\chi$ can be read from those of
$\nu_{R}^{i}$

$\displaystyle\mathcal{L}_{\mathrm{int}}=$
$\displaystyle\overline{\nu_{R}^{i}}i\gamma^{\mu}\left(\partial_{\mu}-iq_{\nu_{R}^{i}}g_{X}X_{\mu}\right)\nu_{R}^{i}$
$\displaystyle=$ $\displaystyle
i\overline{\nu_{R}}\gamma^{\mu}\left(\partial_{\mu}-i(-4)g_{X}X_{\mu}\right)\nu_{R}+i\overline{\chi}\left(\gamma^{\mu}\partial_{\mu}-i((-4)P_{R}+(-5)P_{L})g_{X}X_{\mu}\right)\chi,$
(34)

while similarly, the $U(1)_{X}$ gauge interaction for an SM chiral fermion
($f_{L/R}$) can be read from the usual covariant derivative,

$\displaystyle\mathcal{L}_{\mathrm{int}}=\sum_{f_{L/R}}i\overline{f_{L/R}}\gamma^{\mu}\left(-iq_{f_{L/R}}\right)g_{X}X_{\mu}f_{L/R},$
(35)

where $q_{f_{L/R}}$ is a $U(1)_{X}$ charge of $f_{L/R}$ listed in Table 1.

The scalar potential is given by

$\displaystyle V=$ $\displaystyle V_{1}+V_{2},$ (36) $\displaystyle V_{1}=$
$\displaystyle+\hat{\mu_{1}}^{2}|\Phi_{1}|^{2}-\hat{\mu_{2}}^{2}|\Phi_{2}|^{2}$
$\displaystyle+\frac{1}{2}\lambda_{1}|\Phi_{1}|^{4}+\frac{1}{2}\lambda_{2}|\Phi_{2}|^{4}+\lambda_{3}|\Phi_{1}|^{2}|\Phi_{2}|^{2}+\lambda_{4}|\Phi_{1}^{\dagger}\Phi_{2}|^{2}$
$\displaystyle+\hat{\mu_{3}}^{2}\mathrm{Tr}(\Delta_{3}^{\dagger}\Delta_{3})+\frac{1}{2}\Lambda_{1}\left(\mathrm{Tr}(\Delta_{3}^{\dagger}\Delta_{3})\right)^{2}+\frac{1}{2}\Lambda_{2}\left(\left(\mathrm{Tr}(\Delta_{3}^{\dagger}\Delta_{3})\right)^{2}-\mathrm{Tr}[(\Delta_{3}^{\dagger}\Delta_{3})^{2}]\right)$
$\displaystyle+\left(\Lambda_{41}|\Phi_{1}|^{2}+\Lambda_{42}|\Phi_{2}|^{2}\right)\mathrm{Tr}(\Delta_{3}^{\dagger}\Delta_{3})+\Lambda_{51}\Phi_{1}^{\dagger}[\Delta_{3}^{\dagger},\Delta_{3}]\Phi_{1}+\Lambda_{52}\Phi_{2}^{\dagger}[\Delta_{3}^{\dagger},\Delta_{3}]\Phi_{2}$
$\displaystyle-\frac{\Lambda_{6}}{\sqrt{2}}(\Phi_{1}^{T}\cdot\Delta_{3}\Phi_{1}+\mathrm{H.c.}),$
(37) $\displaystyle V_{2}=$
$\displaystyle-\mu_{X}^{2}|\Phi_{X}|^{2}+\frac{1}{2}\lambda_{X}|\Phi_{X}|^{4}+\left(\lambda_{12}\Phi_{X}(\Phi_{1}^{\dagger}\Phi_{2})+\mathrm{H.c.}\right)$
$\displaystyle+\lambda_{X1}|\Phi_{X}|^{2}|\Phi_{1}|^{2}+\lambda_{X2}|\Phi_{X}|^{2}|\Phi_{2}|^{2}+\lambda_{X\Delta}|\Phi_{X}|^{2}\mathrm{Tr}(\Delta_{3}^{\dagger}\Delta_{3}).$
(38)

At a high scale $\gg v\simeq 246$ GeV, $U(1)_{X}$ symmetry is spontaneously
broken by the VEV of $\Phi_{X}=v_{X}/\sqrt{2}=\mu_{X}^{2}/\lambda_{X}$. At the
$U(1)_{X}$ broken vacuum, the $U(1)_{X}$ gauge boson acquires the mass,

$\displaystyle m_{X}^{2}=g_{X}^{2}v_{X}^{2},$ (39)

as $\nu_{R}$ does in Eq. (16). Then, the scalar $\phi_{X}$ from $\Phi_{X}$
also has the mass $m_{\phi_{X}}^{2}=\lambda_{X}v_{X}^{2}$. Thus, the effective
scalar potential at a low-energy scale below the scale of $v_{X}$ is given by

$\displaystyle V=$
$\displaystyle+\mu_{1}^{2}|\Phi_{1}|^{2}-\mu_{2}^{2}|\Phi_{2}|^{2}-\left(\mu_{12}^{2}(\Phi_{1}^{\dagger}\Phi_{2})+\mathrm{H.c.}\right)$
$\displaystyle+\frac{1}{2}\lambda_{1}|\Phi_{1}|^{4}+\frac{1}{2}\lambda_{2}|\Phi_{2}|^{4}+\lambda_{3}|\Phi_{1}|^{2}|\Phi_{2}|^{2}+\lambda_{4}|\Phi_{1}^{\dagger}\Phi_{2}|^{2}$
$\displaystyle+\mu_{3}^{2}\mathrm{Tr}(\Delta_{3}^{\dagger}\Delta_{3})+\frac{1}{2}\Lambda_{1}\left(\mathrm{Tr}(\Delta_{3}^{\dagger}\Delta_{3})\right)^{2}+\frac{1}{2}\Lambda_{2}\left(\left(\mathrm{Tr}(\Delta_{3}^{\dagger}\Delta_{3})\right)^{2}-\mathrm{Tr}[(\Delta_{3}^{\dagger}\Delta_{3})^{2}]\right)$
$\displaystyle+\left(\Lambda_{41}|\Phi_{1}|^{2}+\Lambda_{42}|\Phi_{2}|^{2}\right)\mathrm{Tr}(\Delta_{3}^{\dagger}\Delta_{3})+\Lambda_{51}\Phi_{1}^{\dagger}[\Delta_{3}^{\dagger},\Delta_{3}]\Phi_{1}+\Lambda_{52}\Phi_{2}^{\dagger}[\Delta_{3}^{\dagger},\Delta_{3}]\Phi_{2}$
$\displaystyle-\frac{\Lambda_{6}}{\sqrt{2}}(\Phi_{1}^{T}\cdot\Delta_{3}\Phi_{1}+\mathrm{H.c.}),$
(40)

where the third term is generated from the third term in Eq. (38) by replacing
$\Phi_{X}$ with its VEV as $-\mu_{12}^{2}=\lambda_{12}v_{X}/\sqrt{2}$.
$\mu_{i}^{2}=\hat{\mu_{i}}^{2}+\lambda_{iX}v_{X}^{2}/2$, $\lambda_{i}$, and
$\Lambda_{i(j)}$ are coupling constants. Unlike the usual THDM, the
$(\Phi_{1}^{\dagger}\Phi_{2})^{2}$ term is absent due to the $U(1)_{X}$ gauge
symmetry. The $\mu_{12}^{2}$ terms generated by the VEV of $v_{X}$ is
essential to give the mass of a $CP$ odd Higgs boson and remove a dangerous
Nambu-Goldstone boson from the spectrum.

The stationary conditions are expressed as

$\displaystyle\mu_{1}^{2}$
$\displaystyle=\frac{2\mu_{12}^{2}v_{2}-v_{1}\left(\lambda_{1}v_{1}^{2}+v_{2}^{2}(\lambda_{3}+\lambda_{4})+v_{\Delta}^{2}(\Lambda_{41}-\Lambda_{51})-2\Lambda_{6}v_{\Delta}\right)}{2v_{1}},$
(41) $\displaystyle\mu_{2}^{2}$
$\displaystyle=\frac{\lambda_{2}v_{2}^{3}+v_{1}^{2}v_{2}(\lambda_{3}+\lambda_{4})+v_{2}v_{\Delta}^{2}(\Lambda_{42}-\Lambda_{52})-2\mu_{12}^{2}v_{1}}{2v_{2}},$
(42) $\displaystyle\mu_{3}^{2}$
$\displaystyle=\frac{\Lambda_{6}v_{1}^{2}-v_{\Delta}\left(\lambda_{1}v_{\Delta}^{2}+v_{1}^{2}(\Lambda_{41}-\Lambda_{51})+v_{2}^{2}(\Lambda_{42}-\Lambda_{52})\right)}{2v_{\Delta}},$
(43)

where $v_{1}$ and $v_{2}$ are the VEVs of $\Phi_{1}$ and $\Phi_{2}$,
respectively. Our notation satisfies $v=\sqrt{v_{1}^{2}+v_{2}^{2}}\simeq 246$
GeV. The condition (43) can be recast as

$\displaystyle\Lambda_{6}=\frac{v_{\Delta}\left(\lambda_{1}v_{\Delta}^{2}+v_{1}^{2}(\Lambda_{41}-\Lambda_{51})+v_{2}^{2}(\Lambda_{42}-\Lambda_{52})+2\mu_{3}^{2}\right)}{v_{1}^{2}}\simeq\frac{v_{\Delta}2\mu_{3}^{2}}{v_{1}^{2}},$
(44)

where, in the last approximation, we have assumed the condition for the type-
II seesaw mechanism, $\mu_{3}^{2}\gg v^{2}$. For a given $\Lambda_{6}$ and
$\mu_{3}^{2}$, our smaller $v_{1}$ results a smaller $v_{\Delta}$, which
naturally fits the $\rho$ parameter constraint and enhances the decay rate of
the doubly-charged Higgs boson to the same sign dilepton
$H^{\pm\pm}\rightarrow\ell_{i}^{\pm}\ell_{j}^{\pm}$ as we will show. In the
small $v_{\Delta}$ limit, the masses of scalars are given by

$\displaystyle m_{h/H}^{2}=$
$\displaystyle\left(\begin{array}[]{cc}\lambda_{1}v_{1}^{2}+\mu_{12}^{2}\frac{v_{2}}{v_{1}}&v_{1}v_{2}(\lambda_{3}+\lambda_{4})-\mu_{12}^{2}\\\
v_{1}v_{2}(\lambda_{3}+\lambda_{4})-\mu_{12}^{2}&\lambda_{2}v_{2}^{2}+\mu_{12}^{2}\frac{v_{1}}{v_{2}}\\\
\end{array}\right),$ (47) $\displaystyle m_{H_{2}}^{2}=$ $\displaystyle
m_{H_{2}^{\pm}}^{2}-\frac{1}{2}\left(\Lambda_{51}v_{1}^{2}+\Lambda_{52}v_{2}^{2}\right),$
(48) $\displaystyle m_{A_{1}}^{2}=$
$\displaystyle\mu_{12}^{2}\left(\frac{v_{1}}{v_{2}}+\frac{v_{2}}{v_{1}}\right),$
(49) $\displaystyle m_{A_{2}}^{2}=$ $\displaystyle m_{H_{2}}^{2},$ (50)
$\displaystyle m_{H_{1}^{\pm}}^{2}=$ $\displaystyle
m_{A_{1}}^{2}-\frac{\lambda_{4}}{2}v^{2},$ (51) $\displaystyle
m_{H_{2}^{\pm}}^{2}=$
$\displaystyle\frac{1}{2}\left(\Lambda_{41}v_{1}^{2}+\Lambda_{42}v_{2}^{2}+2\mu_{3}^{2}\right),$
(52) $\displaystyle m_{H^{\pm\pm}}^{2}=$ $\displaystyle
m_{H_{2}^{\pm}}^{2}+\frac{1}{2}\left(\Lambda_{51}v_{1}^{2}+\Lambda_{52}v_{2}^{2}\right).$
(53)

###  II.2 Summary of experimental constraints

####  II.2.1 $\rho$ parameter

The $\rho$ parameter in this model is given as

$\displaystyle\rho\equiv\frac{m_{W}^{2}}{m_{Z}^{2}c_{W}^{2}}=\frac{1+\frac{2v_{\Delta}^{2}}{v^{2}}}{1+\frac{4v_{\Delta}^{2}}{v^{2}}},$
(54)

which is experimentally constrained as $\rho=1.00038\pm 0.00020$ Zyla:2020zbs
. We find $v_{\Delta}\lesssim 0.78(2.6)$ GeV at the $2(3)$ sigma.

####  II.2.2 The Z boson decay width

Nonobservation of the exotic decay of the $Z$ boson constrains the mass of the
doubly-charged Higgs boson as $m_{Z}<2m_{H^{\pm\pm}}$ Kanemura:2013vxa .

####  II.2.3 Doubly-charged Higgs boson search at the LHC

The LHC bound on the doubly-charged Higgs boson has been derived as
$m_{H^{\pm\pm}}\gtrsim 880$ GeV for
$H^{\pm\pm}\rightarrow\ell^{\pm}\ell^{\pm}$ CMS:2012dun ; ATLAS:2017xqs  and
$m_{H^{\pm\pm}}\gtrsim 350$ GeV for $H^{\pm\pm}\rightarrow W^{\pm}W^{\pm}$
ATLAS:2018ceg ; ATLAS:2021jol .

####  II.2.4 Lepton flavor violation

In a triplet Higgs model, lepton flavor-violating decays of a charged lepton
are induced at tree level Chun:2003ej ; Kakizaki:2003jk ; Akeroyd:2009nu . The
branching ratio is given by Kakizaki:2003jk

$\mathrm{Br}(\ell_{i}\rightarrow\bar{\ell}_{j}\ell_{k}\ell_{l})=\frac{1}{64G_{F}^{2}m^{4}_{H^{\pm\pm}}}\left|\frac{(Y_{\Delta}^{ij})^{\dagger}Y_{\Delta}^{kl}}{2}\right|^{2}.$
(55)

The stringent bound is $\mathrm{Br}(\mu\rightarrow\bar{e}ee)<1.0\times
10^{-12}$ from the SINDRUM experiment SINDRUM:1987nra .

A flavor-violating radiative decay $\ell_{i}\rightarrow\ell_{j}\gamma$ is also
induced and its branching ratio for $\mu\rightarrow e\gamma$, which gives the
most stringent bound, is given by Kakizaki:2003jk

$\mathrm{Br}(\mu\rightarrow
e\gamma)=\frac{48\pi^{3}\alpha_{\mathrm{em}}}{G_{F}^{2}m^{4}_{H^{\pm\pm}}}\left|(Y_{\Delta}^{\dagger}Y_{\Delta})_{e\mu}\frac{1}{16\pi^{2}}\frac{3}{16}\right|^{2}.$
(56)

The MEG experiment has reported the latest result of
$\mathrm{Br}(\mu\rightarrow e\gamma)<4.2\times 10^{-13}$ MEG:2016leq .

Any of those lepton flavor-violating decay gives the lower bound on the VEV of
the triplet Higgs field as $v_{\Delta}>\mathcal{O}(1)$ eV Antusch:2018svb  for
$m_{H^{\pm\pm}}\lesssim 1$ TeV.

####  II.2.5 $X$ boson search at the LHC

In this section we evaluate the production cross section of this process at
the LHC for a choice of parameters consistent with the LHC constraints from
dilepton channel $pp\rightarrow X\rightarrow\ell\bar{\ell}$ CMS:2018ipm ;
ATLAS:2019erb  and the dijet constraints CMS:2018wxx ; ATLAS:2019bov . The
decay rates of $X$ are given by

$\displaystyle\sum_{f=\mathrm{quarks,leptons}}\Gamma_{X}(X\rightarrow
f\bar{f})$ $\displaystyle=\frac{g_{X}^{2}}{24\pi}m_{X}F(x_{H}),$ (57)
$\displaystyle\Gamma_{X}(X\rightarrow\nu_{R}\overline{\nu_{R}})$
$\displaystyle=\frac{g_{X}^{2}}{24\pi}m_{X}16,$ (58)
$\displaystyle\Gamma_{X}(X\rightarrow\chi\overline{\chi})$
$\displaystyle=\frac{g_{X}^{2}}{24\pi}\sqrt{m_{X}^{2}-4m_{\chi}^{2}}\left(41m_{X}^{2}-120m_{\chi}^{2}\right),$
(59)

with the auxiliary function Okada:2016tci

$F(x_{H})=13+16x_{H}+10x_{H}^{2}.$ (60)

$g_{X}$ has been constrained to be small in previous studies. As in Ref.
Das:2019fee , since the total $X$ boson decay width is very narrow, we use the
narrow width approximation to evaluate the $X$ boson production cross section

$\displaystyle\sigma(pp\rightarrow X)$ $\displaystyle=2\sum_{q,\bar{q}}\int
dx\int dyf_{q}(x,Q)f_{\bar{q}}(x,Q)\hat{\sigma}(\hat{s}),$ (61)
$\displaystyle\hat{\sigma}(\hat{s})$
$\displaystyle=\frac{4\pi^{2}}{3}\frac{\Gamma_{X}(X\rightarrow
q\bar{q})}{m_{X}}\delta(\hat{s}-m_{X}^{2}),$ (62)

where $f_{q}$ and $f_{\bar{q}}$ are the parton distribution function (PDF) for
a quark and antiquark, $\hat{s}=xys$ is the invariant mass squared of
colliding quarks for the center of mass energy $s$. The factor $2$ in Eq. (61)
counts two ways of $q$ coming from which proton out of two colliding protons.
Since the most severe bound is from the dilepton channel ($\ell=e,\mu$), we
calculate $\sigma(pp\rightarrow X)\mathrm{Br}(X\rightarrow\ell\bar{\ell})$
with

$\mathrm{Br}(X\rightarrow\ell\bar{\ell})=\frac{8+12x_{H}+5x_{H}^{2}}{4F(x_{H})},$
(63)

and compare it with the ATLAS results ATLAS:2019erb . We employ PDFs of CTEQ6L
Pumplin:2002vw  with a factorization scale $Q=m_{X}$ for simplicity. Following
the manner to obtain a suitable $k$-factor presented in Ref. Okada:2016gsh ,
we scale our result by a $k$-factor of $k=0.947$ to match the recent ATLAS
analysis in our calculation. The results will be presented after we discuss
cosmological constraints.

###  II.3 Decay of doubly-charged Higgs boson

The decay rates for principal decay modes of $H^{\pm\pm}$ are Aoki:2011pz ;
BhupalDev:2013xol

$\displaystyle\Gamma(H^{\pm\pm}\rightarrow\ell_{i}^{\pm}\ell_{j}^{\pm})$
$\displaystyle\simeq
S_{ij}\frac{|Y_{\Delta}^{ij}|^{2}}{2}\frac{m_{H^{\pm\pm}}}{4\pi},$ (64)
$\displaystyle\Gamma(H^{\pm\pm}\rightarrow W^{\pm}W^{\pm})$
$\displaystyle\simeq\frac{g^{4}v_{\Delta}^{2}m^{3}_{H^{\pm\pm}}}{64\pi
m_{W}^{4}}\left(\frac{3m_{W}^{4}}{m_{H^{\pm\pm}}^{4}}-\frac{m_{W}^{2}}{m_{H^{\pm\pm}}^{2}}+\frac{1}{4}\right),$
(65)

with

$\displaystyle S_{ij}=\left\\{\begin{array}[]{c}1\\\ \frac{1}{2}\\\
\end{array}\qquad\mathrm{for}\quad\begin{array}[]{c}i\neq j\\\ i=j\\\
\end{array}\right..$ (70)

Those of other minor modes, which are not relevant for our later discussion,
can be found in Ref. Aoki:2011pz .

In the left panel of Fig. 1, we show the decay branching ratio
$\mathrm{Br}(H^{\pm\pm}\rightarrow\ell^{\pm}\ell^{\pm})$ with the light blue
curve and $\mathrm{Br}(H^{\pm\pm}\rightarrow W^{\pm}W^{\pm})$ with the orange
curve. Here, we have taken $m_{H^{\pm\pm}}=900$ GeV and substituted the
formula of neutrino mass (2) into (64). Then, the magnitude of neutrino masses
is taken to be the scale of atmospheric neutrino mass difference of
$\mathcal{O}(0.1)$ eV. For $v_{\Delta}\gtrsim 10^{-4}$ GeV, reflecting the
Majorana nature of neutrino mass generated by the type-II seesaw mechanism,
the lepton number-violating mode $H^{\pm\pm}\rightarrow\ell^{\pm}\ell^{\pm}$
is dominant. The right panel shows that, in our model, a small $v_{\Delta}$
can be realized by a sub-GeV scale $v_{1}$. This is a distinctive feature of
our model. While the tiny $v_{\Delta}$ and, as the result, the smallness of
neutrino mass comes from the small dimensionful parameter $\Lambda_{6}$ in the
minimal Higgs triplet model, the smallness of neutrino mass is a result of the
smallness of $v_{1}$ and an energy scale of $\Lambda_{6}$ being much smaller
than the EW scale is not necessarily required. In this sense, our $\Phi_{1}$
plays the same role as the so-called neutrinophilic Higgs field only in the
scalar sector.

|
---|---
Figure 1: Left: The decay branching ratio of $H^{\pm\pm}$ for
$m_{H^{\pm\pm}}=900$ GeV. The light blue and orange curves indicate
$\mathrm{Br}(H^{\pm\pm}\rightarrow\ell^{\pm}\ell^{\pm})$ and
$\mathrm{Br}(H^{\pm\pm}\rightarrow W^{\pm}W^{\pm})$, respectively.
$v_{\Delta}\simeq 10^{-4}$ GeV is the critical value at which the dominant
mode changes. Right: The $v_{1}$ dependence of $v_{\Delta}$. This is for
$\Lambda_{6}=10^{3}$ GeV and $m_{H^{\pm\pm}}=900$ GeV. A very small
$v_{\Delta}$ can be easily achieved with a sub-GeV scale $v_{1}$.

##  III Cosmology

Next, we consider cosmological constraints and implication of our model.

###  III.1 Dark radiation

With one $U(1)_{X}$ charged scalar $\Phi_{X}$, out of three right-handed
neutrinos, there is one massless state $\nu_{R}$ and one Dirac fermion $\chi$.
We at first consider the DR constraint due to thermal production of $\nu_{R}$.

The thermal averaged cross section of $\nu_{R}$ for the temperature $T\ll
m_{X}$ is expressed as

$\langle\sigma v\rangle\simeq\sum_{f}\langle\sigma
v(f\bar{f}\leftrightarrow\nu_{R}\nu_{R})\rangle,$ (71)

with

$\displaystyle\langle\sigma v(u\bar{u}\leftrightarrow\nu_{R}\nu_{R})\rangle$
$\displaystyle=\frac{2g^{4}N_{c}T^{2}(x_{H}(17x_{H}+20)+8)}{9\pi m_{X}^{4}},$
(72) $\displaystyle\langle\sigma
v(d\bar{d}\leftrightarrow\nu_{R}\nu_{R})\rangle$
$\displaystyle=\frac{2g^{4}N_{c}T^{2}(x_{H}(5x_{H}-4)+8)}{9\pi m_{X}^{4}},$
(73) $\displaystyle\langle\sigma
v(\ell\bar{\ell}\leftrightarrow\nu_{R}\nu_{R})\rangle$
$\displaystyle=\frac{2g^{4}T^{2}(x_{H}(5x_{H}+12)+8)}{\pi m_{X}^{4}},$ (74)
$\displaystyle\langle\sigma
v(\nu\bar{\nu}\leftrightarrow\nu_{R}\nu_{R})\rangle$
$\displaystyle=\frac{2g^{4}T^{2}(x_{H}+2)^{2}}{\pi m_{X}^{4}},$ (75)

and the color factor $N_{c}=3$ for quarks. The invariant squared amplitude
before taking thermal averaging are listed in Appendix A. The decoupling
temperature $T_{\mathrm{dec}}$ of $\nu_{R}$ from the thermal bath is evaluated
by

$\left.\langle\sigma v\rangle
n_{\nu_{R}}\right|_{T=T_{\mathrm{dec}}}=H(T_{\mathrm{dec}}),$ (76)

where $H=\dot{a}/a$ is the cosmic expansion rate in the radiation dominated
universe described by

$\displaystyle H^{2}$ $\displaystyle=\frac{1}{3M_{P}^{2}}\rho_{r},$ (77)
$\displaystyle\rho_{r}$ $\displaystyle=\frac{\pi^{2}g_{*}}{30}T^{4},$ (78)

with $a$, $\rho_{r}$, and $g_{*}$ are the scale factor, the energy density of
radiation and the number of relativistic degrees of freedom, respectively.
$M_{P}\simeq 2.4\times 10^{18}$ is the reduced Planck mass. After $\nu_{R}$
decoupled from thermal bath at the decoupling temperature $T_{\mathrm{dec}}$,
the energy density of $\nu_{R}$, $\rho_{\nu_{R}}$, decreases as
$\rho_{\nu_{R}}\propto a^{-4}$. By parametrizing the energy density of
$\nu_{R}$ with $\Delta N_{\mathrm{eff}}$ as

$\rho_{\nu_{R}}=\Delta
N_{\mathrm{eff}}\frac{7}{4}\frac{\pi^{2}}{30}T_{\nu}^{4},$ (79)

the total radiation energy density except photons is expressed as

$N_{\mathrm{eff}}=N_{\mathrm{eff}}^{\mathrm{\nu}}+\Delta N_{\mathrm{eff}},$
(80)

where $T_{\nu}$ is the temperature of left-handed SM neutrinos and
$N_{\mathrm{eff}}^{\mathrm{\nu}}$ is the effective number of neutrinos in the
SM. For recent calculations of $N_{\mathrm{eff}}^{\mathrm{\nu}}$, see e.g.,
Refs. Escudero:2018mvt ; Bennett:2019ewm ; Escudero:2020dfa ; Akita:2020szl ;
Bennett:2020zkv .

###  III.2 Dark matter

####  III.2.1 Abundance

We estimate the thermal relic abundance of our Dirac DM $\chi$ by solving the
Boltzmann equation,

$\frac{dn}{dt}+3Hn=-\langle\sigma v\rangle(n^{2}-n_{\rm EQ}^{2}),$ (81)

where $n$ is the number density of $\chi$, $n_{\rm EQ}$ is its number density
at thermal equilibrium, $\langle\sigma v\rangle$ is the thermal averaged
products of the annihilation cross section and the relative velocity. The
amplitude squared integrated over the scattering angle $\theta$ is given by

$\sum_{i=f,\nu_{R}}\int\overline{|\mathcal{M}(i\bar{i}\leftrightarrow\chi\chi)|^{2}}d\cos\theta=\frac{2g_{X}^{4}}{3}\frac{s(F(x_{H})+16)\left(79m_{N}^{2}+41s\right)}{\left(m_{X}^{2}-s\right)^{2}+\Gamma_{X}^{2}m_{X}^{2}}$
(82)

where $s$ is the center-of-mass energy and the total decay width is given by

$\Gamma_{X}=\sum_{i=f,\nu_{R},\chi}\Gamma_{X}(X\rightarrow i\bar{i}),$ (83)

with each partial decay width (57), (58) and (59).

The resultant DM relic abundance is given by

$\displaystyle\Omega_{\chi}h^{2}=\frac{1.1\times
10^{9}x_{d}\mathrm{GeV}^{-1}}{\sqrt{8\pi g_{*}}M_{P}\langle\sigma v\rangle},$
(84)

where $x_{d}=m_{\chi}/T_{d}$ with the decoupling temperature $T_{d}$
Kolb:1990vq .

####  III.2.2 Direct DM detection bound

The DM $\chi$ with the mass $m_{\chi}$ can scatter off nucleons through the
$X$ boson exchange. The spin-independent (SI) weakly interacting massive
particle (WIMP)-nucleon cross section for the elastic scattering through
vector-vector couplings is given by Jungman:1995df

$\sigma_{SI}=\frac{\mu_{N}^{2}}{4\pi}\left(Zb_{p}+(A-Z)b_{n}\right)^{2},$ (85)

with

$\displaystyle\mu_{N}$ $\displaystyle=\frac{m_{\chi}m_{N}}{m_{\chi}+m_{N}},$
(86) $\displaystyle b_{p}$
$\displaystyle=\frac{g_{\chi}}{m_{X}^{2}}(2g_{u}+g_{d}),$ (87) $\displaystyle
b_{n}$ $\displaystyle=\frac{g_{\chi}}{m_{X}^{2}}(g_{u}+2g_{d}),$ (88)
$\displaystyle g_{\chi}$ $\displaystyle=4g_{X},$ (89) $\displaystyle g_{u}$
$\displaystyle=\left(\frac{1}{2}\left(\frac{x_{H}}{6}+\frac{2x_{H}}{3}\right)+\frac{1}{3}\right)g_{X},$
(90) $\displaystyle g_{d}$
$\displaystyle=\left(\frac{1}{2}\left(\frac{x_{H}}{6}-\frac{x_{H}}{3}\right)+\frac{1}{3}\right)g_{X},$
(91)

Here, $Z$ and $A$ are the atomic number and the mass number of a target.
$m_{N}$ is the mass of nucleus.

##  IV Benchmark points

|  |
---|---|---
|  |
Figure 2: Constraints summary. LHC(ATLAS) constraints (black),
$N_{\mathrm{eff}}=3.5\,(3.1)$ with red solid (dashed) curve, points satisfying
both $\Omega_{\chi}h^{2}\simeq 0.1$ and the constraints from DM direct
searches for $m_{N}=1$ TeV (blue) and $2$ TeV (green). Upper left: For the
$x_{H}=1$ case. Upper right: For the $x_{H}=0;U(1)_{B-L}$ case. Lower left:
For the $x_{H}=-1.2$ case. Lower right: For the $x_{H}=-2;U(1)_{R}$ case.

We examine the constraints on our model for several parameters by collider
experiments and cosmology.

The first case is $x_{H}=0$, which corresponds to $U(1)_{B-L}$. On the upper-
right panel in Fig. 2, we show the LHC constraints drawn by black solid curve
in a $(m_{X},g_{X})$ plane and parameter points where desired thermal DM
abundance $\Omega_{\chi}h^{2}\simeq 0.1$ is reproduced without confronting the
latest DM direct search XENON1T(2018) XENON:2018voc . The blue and green
curves correspond to fixed DM masses of $1$ TeV and $2$ TeV, respectively. The
sharp drops around $m_{X}=2$ TeV and $4$ TeV respectively appear by the $X$
boson resonance in the annihilation processes. End points of those curves are
due to the constraints from DM direct search experiments. In $\chi$ DM
parameter space, only the vicinity of $s$-channel $X$ resonance is allowed,
due to the stringent DM direct search bound. Red solid (dashed) curve
represent the predicted $N_{\mathrm{eff}}$ of $3.5\,(3.1)$ by taking the
contribution of $\nu_{R}$ radiation into account. The LHC provides a more
stringent limit for a lighter $m_{X}$ mass region $m_{X}\lesssim 3.5$ TeV,
while the cosmological bound is significant for $m_{X}\gtrsim 3.5$ TeV.

For comparison, the different $x_{H}$ models are also displayed in Fig. 2. The
upper-left panel shows the results for $x_{H}=1$. In this case, both the LHC
and cosmological $N_{\mathrm{eff}}$ bounds are more severe than those of the
$U(1)_{B-L}$ case. Hence, a smaller $g_{X}$ than that in the $U(1)_{B-L}$
model are allowed. The LHC bound becomes most less stringent for $x_{H}=-1.2$
Okada:2016tci , which is shown in the lower-left panel. In this case, the
$N_{\mathrm{eff}}$ constraint is comparable or more stringent than the LHC
bound for all mass range of $m_{X}$ in the plot.

Finally, we consider the case of $x_{H}=-2$, which corresponds to $U(1)_{R}$.
This is a case that the constraint from direct DM search becomes weaker by the
destructive interference between proton and neutron in the cross section,
which can be easily seen by rewriting the SI WIMP-nucleon cross section (85)
as

$\sigma_{SI}\propto(A(x_{H}+4)+2x_{H}Z)^{2}.$ (92)

For atomic nucleus with $A\simeq 2Z$, the cancellation occurs around
$x_{H}\simeq-2$. Thus, a larger $g_{X}$ than that in the $U(1)_{B-L}$ model is
allowed. As can be seen in the lower-right panel of Fig. 2, $m_{X}$ is not
necessarily so close to $2m_{\chi}$, and more interestingly, there exist
parameter points that thermal DM with the mass of $2$ TeV and $\Delta
N_{\mathrm{eff}}=\mathcal{O}(0.1)$ can be simultaneously realized. We note
that another model for the simultaneous realization of thermal DM and DR was
proposed in Ref. Okada:2019sbb , where DM is flavored $U(1)$ interacting
scalar and the gauge boson is light.

##  V Summary

We have proposed a simple extension of the SM with an extra $U(1)_{X}$ gauge
symmetry with an alternative charge assignment for right-handed neutrinos. The
type-II seesaw mechanism is an economical way to generate appropriate neutrino
masses in this framework and an additional Higgs doublet to have a trilinear
interaction with the triplet Higgs field. Since the tadpole term is
proportional to the squared VEV of the additional doublet Higgs field, the
smallness of neutrino mass could be understood as the consequence of the
smallness of the additional Higgs VEV. Thus, without introducing a very small
dimensionful parameter in the Higgs potential, we can observe the dilepton
decay of the doubly-charged Higgs boson, which is evidence of nonconservation
of lepton number.

We have also derived the constraints for thermal DM and DR. Since we have
introduced only one SM singlet $U(1)_{X}$ breaking scalar, it predicts the
existence of one massless state and Dirac fermion in the right-handed neutrino
sector. The LHC sets a stringent bound on the model parameters for
$m_{X}\lesssim$ a few TeV, while the DR constraint is more significant for
$m_{X}\gtrsim$ a few TeV. The most stringent bound comes from the DM physics
in the wide range of parameter space. The null results of direct DM searches
impose the upper bound on the $U(1)_{X}$ gauge coupling. A case with
$x_{H}\simeq-2$ is exceptional because the direct DM search bound becomes
weaker due to cancellation in nucleon DM scattering cross section. Then,
thermal DM and $\Delta N_{\mathrm{eff}}=\mathcal{O}(0.1)$ can be consistent.
Such an $N_{\mathrm{eff}}$ would be interesting from the viewpoint of Hubble
tension.

Another important but unaddressed subject is baryogenesis, which is beyond the
scope of this paper. We, here, note that thermal leptogenesis by heavy-triplet
Higgs particles Ma:1998dx ; Hambye:2003ka ; Hambye:2005tk ; Chongdar:2021tgm
or Affelck-Dine baryogenesis by scalar condensations Barrie:2021mwi  appears
to be promising.

## Acknowledgments

This work is supported in part by the U.S. DOE Grant No. DE-SC0012447 (N.O.),
the Japan Society for the Promotion of Science (JSPS) KAKENHI Grants No.
19K03860, No. 19K03865 and No. 21H00060 (O.S.).

##  Appendix A Amplitude

We give explicit formulas of the invariant amplitude squared of scatterings
between the SM fermions and $\nu_{R}$.

$\displaystyle\int\overline{|\mathcal{M}(u\bar{u}\leftrightarrow\nu_{R}\nu_{R})|^{2}}d\cos\theta=\frac{8g^{4}s\left(m_{f}^{2}((7x_{H}+40)x_{H}+16)+s((17x_{H}+20)x_{H}+8)\right)}{27\left(\Gamma_{X}^{2}m_{X}^{2}+\left(m_{X}^{2}-s\right)^{2}\right)},$
(93)
$\displaystyle\int\overline{|\mathcal{M}(d\bar{d}\leftrightarrow\nu_{R}\nu_{R})|^{2}}d\cos\theta=\frac{8g^{4}s\left(m_{f}^{2}(16-x_{H}(17x_{H}+8))+s((5x_{H}-4)x_{H}+8)\right)}{27\left(\Gamma_{X}^{2}m_{X}^{2}+\left(m_{X}^{2}-s\right)^{2}\right)},$
(94)
$\displaystyle\int\overline{|\mathcal{M}(\ell\bar{\ell}\leftrightarrow\nu_{R}\nu_{R})|^{2}}d\cos\theta=\frac{8g^{4}s\left(m_{f}^{2}((7x_{H}+24)x_{H}+16)+s((5x_{H}+12)x_{H}+8)\right)}{3\left(\Gamma_{X}^{2}m_{X}^{2}+\left(m_{X}^{2}-s\right)^{2}\right)},$
(95)
$\displaystyle\int\overline{|\mathcal{M}(\nu\bar{\nu}\leftrightarrow\nu_{R}\nu_{R})|^{2}}d\cos\theta=\frac{8g^{4}s^{2}(x_{H}+2)^{2}}{3\left(\Gamma_{X}^{2}m_{X}^{2}+\left(m_{X}^{2}-s\right)^{2}\right)}.$
(96)

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