# Muon $(g-2)$ and Thermal WIMP DM in ${\rm U(1)}_{L_{\mu}-L_{\tau}}$ Models

Seungwon Baek<EMAIL_ADDRESS>Department of Physics, Korea University,
Seoul 02841, Korea  Jongkuk Kim<EMAIL_ADDRESS>School of Physics,
KIAS, Seoul 02455, Korea  P. Ko<EMAIL_ADDRESS>School of Physics, KIAS, Seoul
02455, Korea

###### Abstract

${\rm U(1)}_{L_{\mu}-L_{\tau}}\equiv{\rm U(1)}_{X}$ model is anomaly free
within the Standard Model (SM) fermion content, and can accommodate the muon
$(g-2)$ data for $M_{Z^{\prime}}\sim O(10-100)$ MeV and $g_{X}\sim(4-8)\times
10^{-4}$. WIMP type thermal dark matter (DM) can be also introduced for
$M_{Z^{\prime}}\sim 2M_{\rm DM}$, if DM pair annihilations into the SM
particles occur only through the $s$-channel $Z^{\prime}$ exchange. In this
work, we show that this tight correlation between $M_{Z^{\prime}}$ and $M_{\rm
DM}$ can be completely evaded both for scalar and fermionic DM, if we include
the contributions from dark Higgs boson ($H_{1}$). Dark Higgs boson plays a
crucial role in DM phenomenology, not only for generation of dark photon mass,
but also opening new channels for DM pair annihilations into the final states
involving dark Higgs boson, such as dark Higgs pair as well as
$Z^{\prime}Z^{\prime}$ through dark Higgs exchange in the $s$-channel, and co-
annihilation into $Z^{\prime}H_{1}$ in case of inelastic DM. Thus dark Higgs
boson will dissect the strong correlation $M_{Z^{\prime}}\sim 2M_{\rm DM}$,
and much wider mass range is allowed for $U(1)_{X}$-charged complex scalar and
Dirac fermion DM, still explaining the muon $(g-2)$. We consider both generic
$U(1)_{X}$ breaking as well as ${\rm U(1)}_{X}\rightarrow Z_{2}$ (and also
into $Z_{3}$ only for scalar DM case).

###### pacs:

††preprint: KIAS-P22019

##  I Introduction

Recently, the Fermilab Muon $g-2$ Collaboration reported its first measurement
of the muon anomalous magnetic moment Muong-2:2021ojo . Their measurement is
consistent with data measured by Brookhaven E821 Collaboration Muong-2:2006rrc
. The combined FNAL and BNL data is given by

$\Delta a_{\mu}\equiv\frac{1}{2}(g_{\mu}-2)=\left(25.1\pm 5.9\right)\times
10^{-10}.$ (1)

The $4.2\sigma$ deviation from the SM expectation Czarnecki:2002nt ;
Melnikov:2003xd ; Davier:2010nc ; Hagiwara:2011af ; Aoyama:2012wk ;
Gnendiger:2013pva ; Kurz:2014wya ; Colangelo:2014qya ; Masjuan:2017tvw ;
Colangelo:2017fiz ; Davier:2017zfy ; Keshavarzi:2018mgv ; Hoferichter:2018kwz
; Colangelo:2018mtw ; Gerardin:2019vio ; Hoferichter:2019mqg ; Davier:2019can
; Bijnens:2019ghy ; Colangelo:2019uex ; Keshavarzi:2019abf ; Aoyama:2019ryr ;
Blum:2019ugy ; Aoyama:2020ynm  may request for new physics contributions to
the $\Delta a_{\mu}$, although the recent lattice results by the BMW
Collaboration reduces the deviation to $1.6\sigma$ Borsanyi:2020mff .

The simplest way to resolve this $\Delta a_{\mu}$ anomaly is to use
$\mathcal{O}(10^{1\pm 1})$ MeV-scale boson and its coupling to the muon with
size $\sim\mathcal{O}(10^{-4})$. If this new particle also couples to dark
matter of the Universe, it can affect the DM annihilation into the SM
particles in the early universe, and can affect thermal relic density of DM,
as well as behaves as dark radiation if they are light enough and in thermal
equilibrium with the SM sector.

There have been comprehensive study on the $U(1)_{L_{\mu}-L_{\tau}}$ gauge
extension of the SM He:1990pn ; He:1991qd  in the context of the $\Delta
a_{\mu}$ Baek:2001kca ; Ma:2001md ; Altmannshofer:2014pba ;
Altmannshofer:2016oaq ; Biswas:2016yan ; Huang:2021nkl ; Chen:2021vzk ;
Zhou:2021vnf ; Buras:2021btx ; Borah:2021mri ; Costa:2022oaa , the lepton
universality test Chun:2018ibr  and leptophilic DM Baek:2008nz ; Patra:2016shz
; Zu:2021odn  for the PAMELA $e^{+}$ excess PAMELA:2008gwm .

One can also address some $B$ anomalies if one introduces more particles with
nonzero ${\rm U(1)}_{L_{\mu}-L_{\tau}}$ charges Altmannshofer:2014cfa ;
Crivellin:2015mga ; Crivellin:2015lwa ; Ko:2017yrd ; Chen:2017usq ;
Baek:2017sew ; Baek:2018aru ; Baek:2019qte ; Biswas:2019twf ; Han:2019diw ;
Kumar:2020web ; Chao:2021qxq ; Borah:2021khc ; Ko:2021lpx  beyond DM. In this
work, we will ignore $B$ anomalies, and consider only the $\Delta a_{\mu}$ and
thermal WIMP DM. Then it is widely accepted that the $U(1)_{L_{\mu}-L_{\tau}}$
model can accommodate both the $\Delta a_{\mu}$ and thermal DM either (i) with
$M_{Z^{\prime}}\sim(2-3)$MeV, $M_{\rm DM}\sim O(10-100)$MeV 111We shall call
$Z^{\prime}$ dark photon, since it couples to the
$U(1)_{L_{\mu}-L_{\tau}}$-charged DM., which is very predictive
Foldenauer:2018zrz ; Holst:2021lzm ; Drees:2021rsg , or (ii) with DM with
rather large $U(1)$ charge Asai:2020qlp .

However, in this paper, we will show that these conclusions can be evaded if
we include the contributions of dark Higgs boson, the remnant of the dark
photon mass generation by spontaneous breaking through the nonzero VEV
$\langle\Phi\rangle\neq 0$. The role of dark Higgs boson in DM physics has
been largely ignored in the literature. Still there are some study on this
issue by the present authors in various directions: galactic center (GC)
$\gamma$-ray excess Ko:2014gha ; Ko:2014loa ; Baek:2014kna ; Ko:2015ioa , DM
bound state formations affecting relic density calculation Ko:2019wxq , DM
searches at high energy colliders such as ILC Ko:2016xwd ; Kamon:2017yfx , LHC
or 100 TeV $pp$ colliders Baek:2015lna ; Ko:2016ybp ; Dutta:2017sod ;
Ko:2018mew , $p$-wave annihilations of light DM (both scalar and Dirac fermion
DM) thereby evading the stringent constraints from BBN and CMB Baek:2020owl ,
and Higgs-portal assisted Higgs inflation Kim:2014kok  (see Refs. Ko:2016yfb ;
Ko:2018qxz  for review on these issues, and phenomenological constraints on
dark Higgs boson in Refs. Clarke:2013aya ; Ibe:2021fed ). The underlying point
is that DM phenomenology can change drastically if dark Higgs boson is
included. In this paper we demonstrate that this is indeed the case and that
one can completely evade the $s$-channel annihilation condition, $M_{\rm
DM}\sim M_{Z^{\prime}}/2$, for the correct DM thermal relic density.

##  II ${\rm U}(1)_{L_{\mu}-L_{\tau}}\equiv{\rm U}(1)_{X}$ Models

The ${\rm U(1)}_{L_{\mu}-L_{\tau}}\equiv{\rm U}(1)_{X}$ extensions of the SM
is known to be anomaly free without introducing additional chiral fermions
He:1990pn ; He:1991qd . The minimal set up for $\Delta a_{\mu}$ and thermal
WIMP dark matter is to introduce dark Higgs $\Phi$ for $Z^{\prime}$ mass and
scalar/fermion DM $X,\chi$, with the following ${\rm U(1)}_{X}$ charge
assignments:

$Q_{X}(\mu,\nu_{\mu},\tau,\nu_{\tau},X,\chi,\Phi)=(1,1,-1,-1,Q_{X},Q_{\chi},Q_{\Phi}),$
(2)

where $X$ and $\chi$ are complex scalar and Dirac fermion DM 222We take these
DM carry the same $U(1)_{X}$ charges as the SM muon for simplicity. This
assumption can be easily relaxed., and $\Phi$ is dark Higgs that breaks ${\rm
U}(1)_{X}$ spontaneously with its nonzero VEV:
$\Phi(x)=\frac{1}{\sqrt{2}}\left(v_{\Phi}+\phi(x)\right)$. The model
Lagrangian and various formulae in the interaction and mass bases are given in
Appendix A.

Figure 1:  Regions inside the yellow and Green shaded areas by the $\Delta
a_{\mu}$ are allowed at 1$\sigma$ and 2$\sigma$ C.L.. Cyan, black, and orange
regions are excluded by other experimental bounds. Above green solid line is
ruled out by the Borexino experiment. Region inside the orange area can
resolve the Hubble tension. We take two Benchmark Points (BP)
$(M_{Z^{\prime}},g_{X})$ as BPI =$(11.5\,{\rm MeV},4\times 10^{-4})$ and BPII
= $(100\,{\rm MeV},8\times 10^{-4})$.

In this model, the $Z^{\prime}$ contribution to $\Delta a_{\mu}$ at one-loop
is given by Baek:2001kca ; Ma:2001md ; Baek:2008nz ; Altmannshofer:2014pba

$\Delta
a_{\mu}=\frac{\alpha_{X}}{2\pi}\int_{0}^{1}dx\frac{2M_{\mu}^{2}x^{2}(1-x)}{x^{2}M_{\mu}^{2}+(1-x)M_{Z^{\prime}}^{2}},$
(3)

where $\alpha_{X}=g^{2}_{X}/4\pi$ with the $g_{X}$ being the $U(1)_{X}$ gauge
coupling constant. Taking $g_{X}\sim(4-8)\times 10^{-4}$, Eq. (3) can resolve
the discrepancy in Eq. (1) in the $M_{Z^{\prime}}<M_{\mu}$ limit. The region
for heavier $M_{Z^{\prime}}$ is excluded by the neutrino trident events in the
muon-neutrino scattering with a nucleus $N$,
$\nu_{\mu}N\rightarrow\nu_{\mu}N\mu^{+}\mu^{-}$ Altmannshofer:2014cfa . Data
measured by CHARM-II CHARM-II:1990dvf  and CCFR CCFR:1991lpl  Collaborations
provide a stringent constraint, which basically excludes the parameter region
with $M_{Z^{\prime}}>O(1)$ GeV Altmannshofer:2014pba  333 However, if one
extends the particle content, $U(1)_{L_{\mu}-L_{\tau}}$ models can easily
accommodate the $\Delta a_{\mu}$ at one loop levels involving those new
particles even for the heavy $Z^{\prime}$ case. See Refs.
Altmannshofer:2016oaq ; Patra:2016shz ; Huang:2021nkl ; Zu:2021odn ;
Chen:2021vzk ; Borah:2021jzu ; Singirala:2021gok ; Zhou:2021vnf ;
Buras:2021btx ; Qi:2021rhh ; Borah:2021mri  for explicit model constructions
in this direction.. In our study, we take $2\sigma$ exclusion limit from the
CCFR data. There are also constraints on $(M_{Z^{\prime}},g_{X})$ plane from
BaBar and LHC searches for the $4\mu$ channel, Borexino neutrino oscillation
data, and $\Delta N_{\rm eff}$, which are summarized in the Appendix B.

Considering all of the experimental bounds, the remaining parameter space for
the $\Delta a_{\mu}$ in case of light $Z^{\prime}$ is depicted in Fig. 1. In
the following, we shall take two benchmark points $(11.5\,{\rm MeV},4\times
10^{-4})$ [BPI] and $(M_{Z^{\prime}},g_{X})=(100\,{\rm MeV},8\times 10^{-4})$
[BPII]. Note that the Hubble tension can be relaxed in case of [BPI] with the
help of light $Z^{\prime}$ contributing to some amount of dark radiation
Kamada:2015era ; Escudero:2019gzq . In the main text, we show the results for
[BPI] only, relegating those for [BPII] in Appendix C.

##  III Scalar DM ($X$)

###  III.1 Generic Case: $Q_{X}/Q_{\Phi}\neq\pm 1,\pm 1/2,\pm 1/3,etc.$

Let us first consider complex scalar DM with a generic $Q_{X}/Q_{\Phi}$. Then
the gauge invariant and renormalizable scalar DM Lagrangian is given by

$\mathcal{L}_{\rm
DM}=|D_{\mu}X|^{2}-m^{2}_{X}|X|^{2}-\lambda_{HX}|X|^{2}\left(|H|^{2}-\frac{v^{2}_{H}}{2}\right)-\lambda_{\Phi
X}|X|^{2}\left(|\Phi|^{2}-\frac{v^{2}_{\Phi}}{2}\right)$ (4)

where $D_{\mu}X=(\partial_{\mu}+ig_{X}Q_{X}Z^{\prime}_{\mu})X$. Here we assume
that $Q_{X}=1$, and $Q_{\Phi}$ is chosen in such a way that there are no gauge
invariant operators up to dim-5 that would make the DM $X$ decay into the SM
particles, so that DM particle would be stable or long-lived enough
Baek:2013qwa ; Ko:2022kvl . This case we call “generic” 444The cases with
$Q_{\Phi}=2$ or $3$ belong to special cases, since there are extra operators
that make DM stable. See the following discussions.. The $\lambda_{\Phi H}$
allows the $CP$-even neutral components of $\Phi$ and $H$, $\phi$ and $h$
respectively, to mix. The dark(SM)-Higgs-like mass eigenstate is denoted as
$H_{1(2)}$. Using the mixing angle $\alpha\in[-\pi/4,\pi/4]$, they are written
as $H_{1}=\phi\cos\alpha-h\sin\alpha,H_{2}=\phi\sin\alpha+h\cos\alpha$. In
this work we assume $M_{H_{1}}<M_{H_{2}}(=125\,{\rm GeV})$. See Appendix A for
more details.

In Fig. 2, we depict the Feynman diagrams relevant to the thermal relic
density of complex scalar DM $X$. In this case, the important channels for the
correct DM relic abundance turn out to be $XX^{\dagger}\to
H_{1}H_{1},Z^{\prime}Z^{\prime}$. Note that the channels with $H_{1}$ in the
final states [Fig. 2, (Bottom)] or in the $s$-channel propagator [Fig. 2,
(Top) (b)] would be possible only if the dark Higgs is included. Since the
${\rm U}(1)_{X}$ gauge coupling $g_{X}$ is small, the $XX^{\dagger}\rightarrow
Z^{\prime}Z^{\prime}$ with $t,u$-channel [Fig. 2 (c),(d) or the contact
interaction (a)] would not be efficient enough to dilute away DM particles, as
noticed in previous works. The large enhancement of the annihilation cross
sections is possible either by producing longitudinally polarized $Z^{\prime}$
[Fig. 2 (Top) (b)] involving the $s$-channel dark Higgs boson propagator
Baek:2015fea , or by taking large value of $\lambda_{\Phi X}$ [Fig. 2
(Bottom)]. Light dark Higgs boson $H_{1}$ also contributes to DM scattering on
nucleons, and the stringent bounds from various direct detection experiments
should be imposed. Explicit expressions and detailed discussions on
$\langle\sigma v\rangle_{Z^{\prime}Z^{\prime},H_{1}H_{1}}$ and $\sigma_{\rm
SI}$ are given in AppendixC1.

Figure 2:  (Top) Feynman diagrams for Complex scalar DM annihilating to a pair
of $Z^{\prime}$ bosons. (Bottom) Feynman diagrams for Complex scalar DM
annihilating to a pair of $H_{1}$ bosons.

Figure 3: Top: relic abundance of complex scalar DM as functions of
$\lambda_{\Phi X}$ for [BPI] for $M_{X}=1,~{}10,100,~{}1000$GeV, respectively.
We assumed $Q_{\Phi}=1.1$, $M_{H_{1}}=1$GeV, and $\sin\alpha=10^{-4}$. Solid
(Dashed) lines represent the region where bounds on DM direct detection are
satisfied (ruled out). Bottom: the preferred parameter space in the
$(M_{H_{1}},\lambda_{\Phi X})$ plane for $\lambda_{HX}=0$.

In the Top of Fig. 3, we show the DM relic abundance $\Omega_{\rm DM}h^{2}$ as
functions of $\lambda_{\Phi X}$ in case of [BPI], for $M_{X}=1$ (Red), $10$
(Blue),$100$ (Purple), $1000$ (Brown) GeV. The gray horizontal line
corresponds to $\Omega_{\rm DM}h^{2}=0.12$. The solid (dashed) region is
allowed (excluded) by DM direct detection experiments. In the Bottom of Fig.
3, we show the contours for $\Omega_{\rm DM}h^{2}=0.12$ in the
$(M_{H_{1}},\lambda_{\Phi X})$ plane for the same choices of $M_{X}$. Note
that there is ample parameter space for DM mass beyond $M_{Z^{\prime}}\sim
2M_{X}$ that can reproduce the correct thermal relic density, which is one of
the main findings of this work. Notice that (Top)(b) of Fig. 2 contributes
dominantly to the total cross section of $XX^{\dagger}\to
Z^{\prime}Z^{\prime}$. For BPI, the relic density is determined mainly by
$XX^{\dagger}\rightarrow H_{1}H_{1},Z^{\prime}Z^{\prime}$ below $M_{H_{1}}\sim
2M_{X}$. And above the $H_{1}$ resonance ,$M_{H_{1}}>2M_{X}$, the relic
density is mostly determined by $XX^{\dagger}\rightarrow Z^{\prime}Z^{\prime}$
since $XX^{\dagger}\rightarrow H_{1}H_{1}$ channel is kinematically forbidden.
As we can see from Eqs. (33) and (34), the dominant terms of the annihilation
cross sections are not very sensitive to the change of $M_{H_{1}}$ for
$M_{H_{1}}<2M_{X}$, while $\langle\sigma v_{\rm
rel}\rangle_{Z^{\prime}Z^{\prime}}\propto\lambda_{\Phi X}^{2}/M^{4}_{H_{1}}$
for $M_{H_{1}}>2M_{X}$. These facts account for the behavioral change in the
plot below and above the resonance. Similar plots for the [BPII] are shown in
AppendixC1, Fig. 7.

###  III.2 Local $Z_{2}$ scalar DM: $(Q_{X},Q_{\Phi})=(1,2)$

Now let us consider a special case $Q_{\Phi}=2$ and $Q_{X}=1$. In this case,
DM Lagrangian would have one more gauge invariant operator at renormalizable
level:

$\Delta{\cal L}_{\rm DM}=-\mu(X^{2}\Phi^{\dagger}+H.c.)$ (5)

to the generic case, Eq. (4). Then ${\rm U}(1)_{X}$ will be broken into its
subgroup $Z_{2}$ ($X\rightarrow-X$) after $\Phi$ gets nonzero VEV, á la
Krauss-Wilczek mechanism Krauss:1988zc . Such local $Z_{2}$ scalar DM model
with dark photon has been studied in the context of GC $\gamma$-ray excess
Baek:2014kna ; Okada:2019sbb  and XENON1T excess Baek:2020owl , respectively.
After the ${\rm U(1)}$ symmetry is broken by $v_{\Phi}\neq 0$, the $\mu-$term
is written as

$\mu\left(X^{2}\phi^{\dagger}+H.c.\right)=\frac{1}{\sqrt{2}}\mu
v_{\Phi}(X^{2}_{R}-X^{2}_{I})\left(1+\frac{\phi}{v_{\Phi}}\right),$ (6)

where $X=\left(X_{R}+iX_{I}\right)/\sqrt{2}$. This term gives rise to the mass
splitting between $X_{R}$ and $X_{I}$:

$\displaystyle M^{2}_{R}$ $\displaystyle=$ $\displaystyle
M^{2}_{X}+\sqrt{2}\mu v_{\Phi},\quad M^{2}_{I}=M^{2}_{X}-\sqrt{2}\mu
v_{\Phi}.$ (7)

Assuming $\mu>0$, we will take the lighter state $X_{I}$ as DM. The mass
splitting is represented by dimensionless quantity,
$\Delta\equiv(M_{R}-M_{I})/M_{I}$. Notice that dark photon interaction with DM
is off-diagonal(or inelastic):

$\displaystyle\mathcal{L}\supset
g_{X}Z^{\prime\mu}\left(X_{R}\partial_{\mu}X_{I}-X_{I}\partial_{\mu}X_{R}\right).$
(8)

For the benchmark point [BPI], there are two dominant DM annihilation
channels: $X_{I}X_{I}\to Z^{\prime}Z^{\prime},H_{1}H_{1}$, similarly to the
generic case discussed in the previous subsection. The corresponding Feynman
diagrams are similar to those in Fig. 2 with appropriate change of $X$ fields.
The corresponding Feynman diagrams are depicted in Fig. 8 of Appendix C2,
along with the relevant expressions for $\langle\sigma v\rangle$.

Figure 4: Top: Relic abundance of local $Z_{2}$ scalar DM as functions of
$\lambda_{\Phi X}$ for [BPI] and different values of mass splittings
($\Delta$). We take $\lambda_{HX}=0$, $M_{H_{1}}=10$GeV, and
$s_{\alpha}=10^{-4}$. All the curves satisfy the DM direct detection bound.
Bottom: The preferred parameter space in the $(M_{H_{1}},\lambda_{\Phi X})$
plane for different values of $\Delta$. The gray area is excluded by the
perturbative condition.

In Fig. 4, we show the DM relic density as functions of $\lambda_{\Phi X}$ for
the [BPI] for $M_{I}=1,10,10^{2},10^{3}$GeV and $\Delta=1,0.1,0.01$ with fixed
$M_{H_{1}}=10$GeV and $\sin\alpha=10^{-4}$. The new parameter $\mu$ makes the
prediction for the relic density significantly different from the generic case
we considered in the previous section. For the light DM we can obtain the
correct relic density via $X_{I}X_{I}\to Z^{\prime}Z^{\prime}$ while for the
heavy DM the channel $X_{I}X_{I}\to H_{1}H_{1}$ becomes relevant as well. In
Top of Fig. 4, as $\Delta$ increases the DM coupling with $H_{1}$ becomes
stronger, reducing the relic abundance. Note also that we can see bump-like
shape. This happens due to the cancellation between $\lambda_{\Phi X}v_{\Phi}$
and $\mu$. In Bottom of Fig. 4, we again see that the $X_{I}X_{I}\to
H_{1}H_{1},Z^{\prime}Z^{\prime}$ dominates below the resonance region
$M_{H_{1}}\sim 2M_{X}$ while $X_{I}X_{I}\to Z^{\prime}Z^{\prime}$ takes over
above the resonance. Near the resonance region co-annihilation $X_{I}X_{R}\to
Z^{\prime}H_{1}$ contributes to the relic density. However, this effect is
sub-dominant. If we took even smaller $\Delta$ such as $\Delta<10^{-3}$, we
could get the correct relic density for the heavier DM upto a few TeV through
$X_{I}X_{I}(X_{R}X_{R})\to H_{1}H_{1},Z^{\prime}Z^{\prime}$ channels.

For the DM direct detection, there are two processes in this case: one is
elastic scattering and the other is inelastic scattering process. In the
parameter space yielding the correct DM relic density, one has
$\delta\geq\mathcal{O}(100)$keV, so that inelastic scattering process does not
happen. Thus, we consider only the Higgs-mediated elastic scattering process,
the cross section of which is given in Appendix C2. It turns out that the
parameters chosen in Fig. 4 satisfy the bounds on the DM direct detections.

Another special case for the scalar DM case is for $Q_{\Phi}=3Q_{X}=3$, for
which ${\rm U}(1)_{X}\rightarrow Z_{3}$. In this case too, one can accommodate
both $\Delta a_{\mu}$ and thermal WIMP DM for $M_{Z^{\prime}}\sim(10-100)$
MeV, $g_{X}\sim 10^{-4}$ with a much wider range of DM mass due to the semi-
annihilation channels, $XX\rightarrow X^{\dagger}H_{1}$, as well as
$XX^{\dagger}\rightarrow H_{1}H_{1}$. More detailed discussion on this case
can be found in Appendix C3.

##  IV Local $Z_{2}$ fermion DM: $Q_{\Phi}=2Q_{\chi}=2$

For Dirac fermion DM model 555The generic case for Dirac fermion DM is
relegated to Appendix C4, since the results are already obtained in the
previous papers as described in Introduction., let us consider a special case
$Q_{\chi}=2Q_{\Phi}=2$, for which the DM Lagrangian at renormalizable level is
modified as

$\mathcal{L}_{\rm
DM}=\overline{\chi}(i\not{D}-m_{\chi})\chi-\left(y_{\Phi}\overline{\chi^{C}}\chi\Phi^{\dagger}+H.c.\right).$
(9)

Again the symmetry breaking pattern is ${\rm U}(1)_{X}\to Z_{2}$ (local
$Z_{2}$ fermion DM) due to the nonzero $v_{\Phi}$. This model is a dark gauge
model for inelastic fermion DM, and has been studied in the context of DM
bound state formation in Ref. Ko:2019wxq  and the XENON1T excess in Ref.
Baek:2020owl , respectively. In this model the light dark Higgs contribution
to the DM self-interaction and the relic density has been considered in
Kamada:2018zxi .

After ${\rm U}(1)_{X}$ symmetry breaking with nonzero $y_{\Phi}$, the original
Dirac fermion $\chi$ is decomposed into two Majorana fermions ($\chi_{R}$ and
$\chi_{I}$) with mass splitting $\propto v_{\Phi}$:

$\delta\equiv M_{R}-M_{I}=2y_{\Phi}v_{\Phi}.$ (10)

Assuming $y_{\Phi}>0$, we have $\delta>0$, and the lighter state $\chi_{I}$
becomes Majorana fermion DM, with $\chi_{R}$ being its excited state. Then the
Lagrangian of DM is written as

$\mathcal{L}_{\rm
DM}=\frac{1}{2}\sum_{i=R,I}\bar{\chi}_{i}\left(i\partial_{\mu}\gamma^{\mu}-M_{i}\right)\chi_{i}-i\frac{g_{X}}{2}Z^{\prime}_{\mu}\left(\bar{\chi}_{R}\gamma^{\mu}\chi_{I}-\bar{\chi}_{I}\gamma^{\mu}\chi_{R}\right)-\frac{1}{2}y_{\Phi}\left(c_{\alpha}H_{1}+s_{\alpha}H_{2}\right)\left(\bar{\chi}_{R}\chi_{R}-\bar{\chi}_{I}\chi_{I}\right).$
(11)

First let us consider thermal relic density within the standard freeze-out
mechanism. Note that the new singlet scalar $\Phi$ plays a crucial role for
the fermionic DM in this model to be thermal WIMP. And DM mass can be very
heavy up to $\sim O({\rm a~{}few})$ TeV. This result is in sharp contrast with
the results obtained in the literature where the $Z^{\prime}$ mass is assumed
to be generated by Stückelberg mechanism, ignoring the dark Higgs boson. In
particular the (co-)annihilation channel involving $H_{1}$ plays a dominant
role in determining the relic density. Note that these annihilation channels
are completely missing in the usual approach.

Figure 5: Feynman diagrams of local $Z_{2}$ fermion DM (co-)annihilating into
a pair of $Z^{\prime}$ bosons and $H_{1}$ bosons (Top), and $Z^{\prime}+H_{1}$
(Bottom).

The corresponding Feynman diagrams are described in Fig. 5.

DM direct detection in local $Z_{2}$ fermion DM case similar to the local
$Z_{2}$ scalar DM case (see Appendix C5 for the detail). Imposing this
constraint, we get the plots in the bottom of Fig. 6, where the dashed lines
are excluded by DM direct detection experiments. Still there is ample
parameter space for heavy DM mass $M_{I}$, far beyond $M_{I}\sim
M_{Z^{\prime}}/2\sim O(10-100)$MeV.

In the Top of Fig. 6, we plot $\Omega_{\rm DM}h^{2}$ as functions of $\Delta$
in case of the [BPI], for $M_{H_{1}}=5$GeV and different DM masses
$M_{I}=1,10,10^{2},10^{3}$GeV. The most dominant contribution comes from
$\chi_{I}\chi_{I}\to Z^{\prime}Z^{\prime},H_{1}H_{1}$ and $\chi_{I}\chi_{R}\to
H_{1}Z^{\prime}$. Solid lines denote the region which satisfy the DM direct
detection bounds. Note that the smaller splitting $\Delta$ is required for
heavy fermion DM to satisfy the relic abundance. In case of [BPI], co-
annihilation is important when DM is heavy and the mass splitting is small,
since the t-channel diagram of the co-annihilation also has $y_{\Phi}$
coupling. In case of [BPII], co-annihiation is not that important for the
correct $\Omega h^{2}$, since $v_{\Phi}$ is large. In the (Bottom) of Fig. 6,
we show the contours of $\Omega h^{2}=0.12$ in the $(M_{H_{1}},\Delta)$ plane
for the same choices of $M_{I}$. Similar plots for the [BPII] are shown in
AppendixC5, Fig. 11.

Figure 6: Top: Dark matter relic density as functions of mass splitting
$\Delta$ for [BPI] and for different values of DM mass,
$M_{I}=1,10,100,1000$GeV. Solid (Dashed) lines denote the region where bounds
on DM direct detection are satisfied (ruled out). Bottom: Preferred parameter
space in the $(M_{H_{1}},\Delta)$ plane for different DM masses. The gray
region is ruled out by the perturbativity condition on $\lambda_{\Phi}$.

##  V Conclusions

In this work, we considered the $\Delta a_{\mu}$ and thermal dark matter, both
scalar and Dirac fermion DM, in the $U(1)_{L_{\mu}-L_{\tau}}$ extensions of
the SM. $\Delta a_{\mu}$ can be accommodated for
$M_{Z^{\prime}}\simeq\mathcal{O}(10)$MeV and $g_{X}\simeq 10^{-4}$, for which
thermal DM could be achieved near the $Z^{\prime}$ resonance region only with
$M_{Z^{\prime}}\sim 2M_{\rm DM}$, if we do not include the dark Higgs boson. A
noble feature of this work is that we have included the contributions of the
dark Higgs boson which were ignored in the earlier literature. Details of the
DM phenomenology depend on the $U(1)$ charge assignments to the DM and the
dark Higgs ($\Phi$) fields. $U(1)_{L_{\mu}-L_{\tau}}$ symmetry can be broken
generically or in a special way into $Z_{2}$ (inelastic scalar or fermion DM
models) or $Z_{3}$ scalar DM model. New DM (co-)annihilation channels
involving the dark Higgs boson can open ${\rm DM}+{\rm\overline{DM}}\to
H_{1}H_{1},H_{1}Z^{\prime}$ as well as ${\rm DM}+{\rm\overline{DM}}\to
H_{1}\to Z^{\prime}Z^{\prime}$. In the latter process, there is an enhancement
in the longitudinal $Z^{\prime}$ pair production. Thanks to these newly open
channels, the DM mass range becomes much wider from GeV to $\mathcal{O}$(a
few) TeV, dissecting the tight correlation between $M_{Z^{\prime}}$ and
$M_{\rm DM}$: $M_{Z^{\prime}}\sim 2M_{\rm DM}$. Our analysis clearly shows
that DM phenomenology with a massive dark photon can not be complete without
including the dark Higgs boson.

## Acknowledgements

This work is in part supported by KIAS Individual Grants, Grant No. PG021403
(PK) and Grant No. PG074202 (JK) at Korea Institute for Advanced Study, and by
National Research Foundation of Korea (NRF) Grant No. NRF-2018R1A2A3075605
(SB) and No. NRF-2019R1A2C3005009 (PK), funded by the Korea government (MSIT).

## References

  * (1) Muon g-2 Collaboration, B. Abi et al., “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm,” Phys. Rev. Lett. 126 (2021) no. 14, 141801, arXiv:2104.03281 [hep-ex].
  * (2) Muon g-2 Collaboration, G. W. Bennett et al., “Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL,” Phys. Rev. D 73 (2006) 072003, arXiv:hep-ex/0602035.
  * (3) A. Czarnecki, W. J. Marciano, and A. Vainshtein, “Refinements in electroweak contributions to the muon anomalous magnetic moment,” Phys. Rev. D 67 (2003) 073006, arXiv:hep-ph/0212229. [Erratum: Phys.Rev.D 73, 119901 (2006)].
  * (4) K. Melnikov and A. Vainshtein, “Hadronic light-by-light scattering contribution to the muon anomalous magnetic moment revisited,” Phys. Rev. D 70 (2004) 113006, arXiv:hep-ph/0312226.
  * (5) M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, “Reevaluation of the Hadronic Contributions to the Muon g-2 and to alpha(MZ),” Eur. Phys. J. C 71 (2011) 1515, arXiv:1010.4180 [hep-ph]. [Erratum: Eur.Phys.J.C 72, 1874 (2012)].
  * (6) K. Hagiwara, R. Liao, A. D. Martin, D. Nomura, and T. Teubner, “$(g-2)_{\mu}$ and $\alpha(M^{2}_{Z})$ re-evaluated using new precise data,” J. Phys. G 38 (2011) 085003, arXiv:1105.3149 [hep-ph].
  * (7) T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, “Complete Tenth-Order QED Contribution to the Muon g-2,” Phys. Rev. Lett. 109 (2012) 111808, arXiv:1205.5370 [hep-ph].
  * (8) C. Gnendiger, D. Stöckinger, and H. Stöckinger-Kim, “The electroweak contributions to $(g-2)_{\mu}$ after the Higgs boson mass measurement,” Phys. Rev. D 88 (2013) 053005, arXiv:1306.5546 [hep-ph].
  * (9) A. Kurz, T. Liu, P. Marquard, and M. Steinhauser, “Hadronic contribution to the muon anomalous magnetic moment to next-to-next-to-leading order,” Phys. Lett. B 734 (2014) 144–147, arXiv:1403.6400 [hep-ph].
  * (10) G. Colangelo, M. Hoferichter, A. Nyffeler, M. Passera, and P. Stoffer, “Remarks on higher-order hadronic corrections to the muon g$-$2,” Phys. Lett. B 735 (2014) 90–91, arXiv:1403.7512 [hep-ph].
  * (11) P. Masjuan and P. Sanchez-Puertas, “Pseudoscalar-pole contribution to the $(g_{\mu}-2)$: a rational approach,” Phys. Rev. D 95 (2017) no. 5, 054026, arXiv:1701.05829 [hep-ph].
  * (12) G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, “Dispersion relation for hadronic light-by-light scattering: two-pion contributions,” JHEP 04 (2017) 161, arXiv:1702.07347 [hep-ph].
  * (13) M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, “Reevaluation of the hadronic vacuum polarisation contributions to the Standard Model predictions of the muon $g-2$ and ${\alpha(m_{Z}^{2})}$ using newest hadronic cross-section data,” Eur. Phys. J. C 77 (2017) no. 12, 827, arXiv:1706.09436 [hep-ph].
  * (14) A. Keshavarzi, D. Nomura, and T. Teubner, “Muon $g-2$ and $\alpha(M_{Z}^{2})$: a new data-based analysis,” Phys. Rev. D 97 (2018) no. 11, 114025, arXiv:1802.02995 [hep-ph].
  * (15) M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schneider, “Dispersion relation for hadronic light-by-light scattering: pion pole,” JHEP 10 (2018) 141, arXiv:1808.04823 [hep-ph].
  * (16) G. Colangelo, M. Hoferichter, and P. Stoffer, “Two-pion contribution to hadronic vacuum polarization,” JHEP 02 (2019) 006, arXiv:1810.00007 [hep-ph].
  * (17) A. Gérardin, H. B. Meyer, and A. Nyffeler, “Lattice calculation of the pion transition form factor with $N_{f}=2+1$ Wilson quarks,” Phys. Rev. D 100 (2019) no. 3, 034520, arXiv:1903.09471 [hep-lat].
  * (18) M. Hoferichter, B.-L. Hoid, and B. Kubis, “Three-pion contribution to hadronic vacuum polarization,” JHEP 08 (2019) 137, arXiv:1907.01556 [hep-ph].
  * (19) M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, “A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to $\mathbf{\boldsymbol{\alpha}(m_{Z}^{2})}$,” Eur. Phys. J. C 80 (2020) no. 3, 241, arXiv:1908.00921 [hep-ph]. [Erratum: Eur.Phys.J.C 80, 410 (2020)].
  * (20) J. Bijnens, N. Hermansson-Truedsson, and A. Rodríguez-Sánchez, “Short-distance constraints for the HLbL contribution to the muon anomalous magnetic moment,” Phys. Lett. B 798 (2019) 134994, arXiv:1908.03331 [hep-ph].
  * (21) G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub, and P. Stoffer, “Longitudinal short-distance constraints for the hadronic light-by-light contribution to $(g-2)_{\mu}$ with large-$N_{c}$ Regge models,” JHEP 03 (2020) 101, arXiv:1910.13432 [hep-ph].
  * (22) A. Keshavarzi, D. Nomura, and T. Teubner, “$g-2$ of charged leptons, $\alpha(M^{2}_{Z})$ , and the hyperfine splitting of muonium,” Phys. Rev. D 101 (2020) no. 1, 014029, arXiv:1911.00367 [hep-ph].
  * (23) T. Aoyama, T. Kinoshita, and M. Nio, “Theory of the Anomalous Magnetic Moment of the Electron,” Atoms 7 (2019) no. 1, 28.
  * (24) T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin, C. Jung, and C. Lehner, “Hadronic Light-by-Light Scattering Contribution to the Muon Anomalous Magnetic Moment from Lattice QCD,” Phys. Rev. Lett. 124 (2020) no. 13, 132002, arXiv:1911.08123 [hep-lat].
  * (25) T. Aoyama et al., “The anomalous magnetic moment of the muon in the Standard Model,” Phys. Rept. 887 (2020) 1–166, arXiv:2006.04822 [hep-ph].
  * (26) S. Borsanyi et al., “Leading hadronic contribution to the muon magnetic moment from lattice QCD,” Nature 593 (2021) no. 7857, 51–55, arXiv:2002.12347 [hep-lat].
  * (27) X. G. He, G. C. Joshi, H. Lew, and R. R. Volkas, “NEW Z-prime PHENOMENOLOGY,” Phys. Rev. D 43 (1991) 22–24.
  * (28) X.-G. He, G. C. Joshi, H. Lew, and R. R. Volkas, “Simplest Z-prime model,” Phys. Rev. D 44 (1991) 2118–2132.
  * (29) S. Baek, N. G. Deshpande, X. G. He, and P. Ko, “Muon anomalous g-2 and gauged L(muon) - L(tau) models,” Phys. Rev. D 64 (2001) 055006, arXiv:hep-ph/0104141.
  * (30) E. Ma, D. P. Roy, and S. Roy, “Gauged L(mu) - L(tau) with large muon anomalous magnetic moment and the bimaximal mixing of neutrinos,” Phys. Lett. B 525 (2002) 101–106, arXiv:hep-ph/0110146.
  * (31) W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, “Neutrino Trident Production: A Powerful Probe of New Physics with Neutrino Beams,” Phys. Rev. Lett. 113 (2014) 091801, arXiv:1406.2332 [hep-ph].
  * (32) W. Altmannshofer, M. Carena, and A. Crivellin, “$L_{\mu}-L_{\tau}$ theory of Higgs flavor violation and $(g-2)_{\mu}$,” Phys. Rev. D 94 (2016) no. 9, 095026, arXiv:1604.08221 [hep-ph].
  * (33) A. Biswas, S. Choubey, and S. Khan, “Neutrino Mass, Dark Matter and Anomalous Magnetic Moment of Muon in a $U(1)_{L_{\mu}-L_{\tau}}$ Model,” JHEP 09 (2016) 147, arXiv:1608.04194 [hep-ph].
  * (34) G.-y. Huang, F. S. Queiroz, and W. Rodejohann, “Gauged $L_{\mu}{-}L_{\tau}$ at a muon collider,” Phys. Rev. D 103 (2021) no. 9, 095005, arXiv:2101.04956 [hep-ph].
  * (35) J. Chen, Q. Wen, F. Xu, and M. Zhang, “Flavor Anomalies Accommodated in A Flavor Gauged Two Higgs Doublet Model,” arXiv:2104.03699 [hep-ph].
  * (36) S. Zhou, “Neutrino masses, leptonic flavor mixing, and muon $(g-2)$ in the seesaw model with the gauge symmetry,” Chin. Phys. C 46 (2022) no. 12, 011001, arXiv:2104.06858 [hep-ph].
  * (37) A. J. Buras, A. Crivellin, F. Kirk, C. A. Manzari, and M. Montull, “Global analysis of leptophilic Z’ bosons,” JHEP 06 (2021) 068, arXiv:2104.07680 [hep-ph].
  * (38) D. Borah, A. Dasgupta, and D. Mahanta, “TeV scale resonant leptogenesis with L$\mu$-L$\tau$ gauge symmetry in light of the muon g-2,” Phys. Rev. D 104 (2021) no. 7, 075006, arXiv:2106.14410 [hep-ph].
  * (39) F. Costa, S. Khan, and J. Kim, “A Two-Component Dark Matter Model and its Associated Gravitational Waves,” arXiv:2202.13126 [hep-ph].
  * (40) E. J. Chun, A. Das, J. Kim, and J. Kim, “Searching for flavored gauge bosons,” JHEP 02 (2019) 093, arXiv:1811.04320 [hep-ph]. [Erratum: JHEP 07, 024 (2019)].
  * (41) S. Baek and P. Ko, “Phenomenology of U(1)(L(mu)-L(tau)) charged dark matter at PAMELA and colliders,” JCAP 10 (2009) 011, arXiv:0811.1646 [hep-ph].
  * (42) S. Patra, S. Rao, N. Sahoo, and N. Sahu, “Gauged $U(1)_{L_{\mu}-L_{\tau}}$ model in light of muon $g-2$ anomaly, neutrino mass and dark matter phenomenology,” Nucl. Phys. B 917 (2017) 317–336, arXiv:1607.04046 [hep-ph].
  * (43) L. Zu, X. Pan, L. Feng, Q. Yuan, and Y.-Z. Fan, “Constraining $U(1)_{L_{\mu}-L_{\tau}}$ charged dark matter model for muon $g-2$ anomaly with AMS-02 electron and positron data,” arXiv:2104.03340 [hep-ph].
  * (44) PAMELA Collaboration, O. Adriani et al., “An anomalous positron abundance in cosmic rays with energies 1.5-100 GeV,” Nature 458 (2009) 607–609, arXiv:0810.4995 [astro-ph].
  * (45) W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, “Quark flavor transitions in $L_{\mu}-L_{\tau}$ models,” Phys. Rev. D 89 (2014) 095033, arXiv:1403.1269 [hep-ph].
  * (46) A. Crivellin, G. D’Ambrosio, and J. Heeck, “Explaining $h\to\mu^{\pm}\tau^{\mp}$, $B\to K^{*}\mu^{+}\mu^{-}$ and $B\to K\mu^{+}\mu^{-}/B\to Ke^{+}e^{-}$ in a two-Higgs-doublet model with gauged $L_{\mu}-L_{\tau}$,” Phys. Rev. Lett. 114 (2015) 151801, arXiv:1501.00993 [hep-ph].
  * (47) A. Crivellin, G. D’Ambrosio, and J. Heeck, “Addressing the LHC flavor anomalies with horizontal gauge symmetries,” Phys. Rev. D 91 (2015) no. 7, 075006, arXiv:1503.03477 [hep-ph].
  * (48) P. Ko, T. Nomura, and H. Okada, “Explaining $B\to K^{(*)}\ell^{+}\ell^{-}$ anomaly by radiatively induced coupling in $U(1)_{\mu-\tau}$ gauge symmetry,” Phys. Rev. D 95 (2017) no. 11, 111701, arXiv:1702.02699 [hep-ph].
  * (49) C.-H. Chen and T. Nomura, “Penguin $b\to s\ell^{\prime+}\ell^{\prime-}$ and $B$-meson anomalies in a gauged ${L_{\mu}-L_{\tau}}$,” Phys. Lett. B 777 (2018) 420–427, arXiv:1707.03249 [hep-ph].
  * (50) S. Baek, “Dark matter contribution to $b\to s\mu^{+}\mu^{-}$ anomaly in local $U(1)_{L_{\mu}-L_{\tau}}$ model,” Phys. Lett. B 781 (2018) 376–382, arXiv:1707.04573 [hep-ph].
  * (51) S. Baek and C. Yu, “Dark matter for $b\to s\mu^{+}\mu^{-}$ anomaly in a gauged $U(1)_{X}$ model,” JHEP 11 (2018) 054, arXiv:1806.05967 [hep-ph].
  * (52) S. Baek, “Scalar dark matter behind $b\to s\mu\mu$ anomaly,” JHEP 05 (2019) 104, arXiv:1901.04761 [hep-ph].
  * (53) A. Biswas and A. Shaw, “Reconciling dark matter, $R_{K^{(*)}}$ anomalies and $(g-2)_{\mu}$ in an ${L_{\mu}-L_{\tau}}$ scenario,” JHEP 05 (2019) 165, arXiv:1903.08745 [hep-ph].
  * (54) Z.-L. Han, R. Ding, S.-J. Lin, and B. Zhu, “Gauged $U(1)_{L_{\mu}-L_{\tau}}$ scotogenic model in light of $R_{K^{(*)}}$ anomaly and AMS-02 positron excess,” Eur. Phys. J. C 79 (2019) no. 12, 1007, arXiv:1908.07192 [hep-ph].
  * (55) N. Kumar, T. Nomura, and H. Okada, “A multi-charged particle model to explain muon $g-2$, flavor physics, and possible collider signature,” arXiv:2002.12218 [hep-ph].
  * (56) W. Chao, H. Wang, L. Wang, and Y. Zhang, “Dark matter, $Z^{\prime}$, vector-like quark at the LHC and $b\to s\mu\mu$ anomaly,” Chin. Phys. C 45 (2021) no. 8, 083105, arXiv:2102.07518 [hep-ph].
  * (57) D. Borah, M. Dutta, S. Mahapatra, and N. Sahu, “Lepton anomalous magnetic moment with singlet-doublet fermion dark matter in a scotogenic $U(1)_{L_{\mu}-L_{\tau}}$ model,” Phys. Rev. D 105 (2022) no. 1, 015029, arXiv:2109.02699 [hep-ph].
  * (58) P. Ko, T. Nomura, and H. Okada, “Muon $g-2$, $B\to K^{(*)}\mu^{+}\mu^{-}$ anomalies, and leptophilic dark matter in $U(1)_{\mu-\tau}$ gauge symmetry,” arXiv:2110.10513 [hep-ph].
  * (59) We shall call $Z^{\prime}$ dark photon, since it couples to the $U(1)_{L_{\mu}-L_{\tau}}$-charged DM.
  * (60) P. Foldenauer, “Light dark matter in a gauged $U(1)_{L_{\mu}-L_{\tau}}$ model,” Phys. Rev. D 99 (2019) no. 3, 035007, arXiv:1808.03647 [hep-ph].
  * (61) I. Holst, D. Hooper, and G. Krnjaic, “The Simplest and Most Predictive Model of Muon $g-2$ and Thermal Dark Matter,” arXiv:2107.09067 [hep-ph].
  * (62) M. Drees and W. Zhao, “U(1)L$\mu$$-$L$\tau$ for light dark matter, g$\mu$ $-$ 2, the 511 keV excess and the Hubble tension,” Phys. Lett. B 827 (2022) 136948, arXiv:2107.14528 [hep-ph].
  * (63) K. Asai, S. Okawa, and K. Tsumura, “Search for U(1)${}_{L_{\mu}-L_{\tau}}$ charged Dark Matter with neutrino telescope,” JHEP 03 (2021) 047, arXiv:2011.03165 [hep-ph].
  * (64) P. Ko, W.-I. Park, and Y. Tang, “Higgs portal vector dark matter for $\mathinner{\mathrm{GeV}}$ scale $\gamma$-ray excess from galactic center,” JCAP 09 (2014) 013, arXiv:1404.5257 [hep-ph].
  * (65) P. Ko and Y. Tang, “Galactic center $\gamma$-ray excess in hidden sector DM models with dark gauge symmetries: local $Z_{3}$ symmetry as an example,” JCAP 01 (2015) 023, arXiv:1407.5492 [hep-ph].
  * (66) S. Baek, P. Ko, and W.-I. Park, “Local $Z_{2}$ scalar dark matter model confronting galactic ${\mathrm{G}eV}$-scale $\gamma$-ray,” Phys. Lett. B 747 (2015) 255–259, arXiv:1407.6588 [hep-ph].
  * (67) P. Ko and Y. Tang, “Dark Higgs Channel for FERMI GeV $\gamma$-ray Excess,” JCAP 02 (2016) 011, arXiv:1504.03908 [hep-ph].
  * (68) P. Ko, T. Matsui, and Y.-L. Tang, “Dark matter bound state formation in fermionic Z2 DM model with light dark photon and dark Higgs boson,” JHEP 10 (2020) 082, arXiv:1910.04311 [hep-ph].
  * (69) P. Ko and H. Yokoya, “Search for Higgs portal DM at the ILC,” JHEP 08 (2016) 109, arXiv:1603.04737 [hep-ph].
  * (70) T. Kamon, P. Ko, and J. Li, “Characterizing Higgs portal dark matter models at the ILC,” Eur. Phys. J. C 77 (2017) no. 9, 652, arXiv:1705.02149 [hep-ph].
  * (71) S. Baek, P. Ko, M. Park, W.-I. Park, and C. Yu, “Beyond the Dark matter effective field theory and a simplified model approach at colliders,” Phys. Lett. B 756 (2016) 289–294, arXiv:1506.06556 [hep-ph].
  * (72) P. Ko and J. Li, “Interference effects of two scalar boson propagators on the LHC search for the singlet fermion DM,” Phys. Lett. B 765 (2017) 53–61, arXiv:1610.03997 [hep-ph].
  * (73) B. Dutta, T. Kamon, P. Ko, and J. Li, “Prospects for discovery and spin discrimination of dark matter in Higgs portal DM models and their extensions at 100 TeV $pp$ collider,” Eur. Phys. J. C 78 (2018) no. 7, 595, arXiv:1712.05123 [hep-ph].
  * (74) P. Ko, G. Li, and J. Li, “Impact of the 125 GeV Higgs boson on singlet fermion dark matter searches at the LHC,” Phys. Rev. D 98 (2018) no. 5, 055031, arXiv:1807.06697 [hep-ph].
  * (75) S. Baek, J. Kim, and P. Ko, “XENON1T excess in local $Z_{2}$ DM models with light dark sector,” Phys. Lett. B 810 (2020) 135848, arXiv:2006.16876 [hep-ph].
  * (76) J. Kim, P. Ko, and W.-I. Park, “Higgs-portal assisted Higgs inflation with a sizeable tensor-to-scalar ratio,” JCAP 02 (2017) 003, arXiv:1405.1635 [hep-ph].
  * (77) P. Ko, “Hidden Sector DM Models with Local Dark Gauge Symmetries: Higgs Portal DM Models and Beyond,” New Phys. Sae Mulli 66 (2016) no. 8, 966–977.
  * (78) P. Ko, “Particle, Astroparticle Physics and Cosmology in Dark Matter Models with Dark Gauge Symmetries,” J. Korean Phys. Soc. 73 (2018) no. 4, 449–465.
  * (79) J. D. Clarke, R. Foot, and R. R. Volkas, “Phenomenology of a very light scalar (100 MeV $<$ $m_{h}$ $<$ 10 GeV) mixing with the SM Higgs,” JHEP 02 (2014) 123, arXiv:1310.8042 [hep-ph].
  * (80) M. Ibe, S. Kobayashi, Y. Nakayama, and S. Shirai, “Cosmological constraints on dark scalar,” JHEP 03 (2022) 198, arXiv:2112.11096 [hep-ph].
  * (81) We take these DM carry the same $U(1)_{X}$ charges as the SM muon for simplicity. This assumption can be easily relaxed.
  * (82) CHARM-II Collaboration, D. Geiregat et al., “First observation of neutrino trident production,” Phys. Lett. B 245 (1990) 271–275.
  * (83) CCFR Collaboration, S. R. Mishra et al., “Neutrino tridents and W Z interference,” Phys. Rev. Lett. 66 (1991) 3117–3120.
  * (84) However, if one extends the particle content, $U(1)_{L_{\mu}-L_{\tau}}$ models can easily accommodate the $\Delta a_{\mu}$ at one loop levels involving those new particles even for the heavy $Z^{\prime}$ case. See Refs. Altmannshofer:2016oaq ; Patra:2016shz ; Huang:2021nkl ; Zu:2021odn ; Chen:2021vzk ; Borah:2021jzu ; Singirala:2021gok ; Zhou:2021vnf ; Buras:2021btx ; Qi:2021rhh ; Borah:2021mri  for explicit model constructions in this direction.
  * (85) A. Kamada and H.-B. Yu, “Coherent Propagation of PeV Neutrinos and the Dip in the Neutrino Spectrum at IceCube,” Phys. Rev. D 92 (2015) no. 11, 113004, arXiv:1504.00711 [hep-ph].
  * (86) M. Escudero, D. Hooper, G. Krnjaic, and M. Pierre, “Cosmology with A Very Light Lμ $-$ Lτ Gauge Boson,” JHEP 03 (2019) 071, arXiv:1901.02010 [hep-ph].
  * (87) S. Baek, P. Ko, and W.-I. Park, “Singlet Portal Extensions of the Standard Seesaw Models to a Dark Sector with Local Dark Symmetry,” JHEP 07 (2013) 013, arXiv:1303.4280 [hep-ph].
  * (88) P. Ko, C.-T. Lu, and U. Min, “Crossing two-component dark matter models and implications for 511 keV $\gamma$-ray and XENON1T excesses,” arXiv:2202.12648 [hep-ph].
  * (89) The cases with $Q_{\Phi}=2$ or $3$ belong to special cases, since there are extra operators that make DM stable. See the following discussions.
  * (90) S. Baek, “Dark matter and muon $(g-2)$ in local $U(1)_{L_{\mu}-L_{\tau}}$-extended Ma Model,” Phys. Lett. B 756 (2016) 1–5, arXiv:1510.02168 [hep-ph].
  * (91) L. M. Krauss and F. Wilczek, “Discrete Gauge Symmetry in Continuum Theories,” Phys. Rev. Lett. 62 (1989) 1221.
  * (92) N. Okada and O. Seto, “Inelastic extra $U(1)$ charged scalar dark matter,” Phys. Rev. D 101 (2020) no. 2, 023522, arXiv:1908.09277 [hep-ph].
  * (93) The generic case for Dirac fermion DM is relegated to Appendix C4, since the results are already obtained in the previous papers as described in Introduction.
  * (94) A. Kamada, K. Kaneta, K. Yanagi, and H.-B. Yu, “Self-interacting dark matter and muon $g-2$ in a gauged U$(1)_{L_{\mu}-L_{\tau}}$ model,” JHEP 06 (2018) 117, arXiv:1805.00651 [hep-ph].
  * (95) D. Borah, M. Dutta, S. Mahapatra, and N. Sahu, “Muon (g$-$2) and XENON1T excess with boosted dark matter in L$\mu$$-$L$\tau$ model,” Phys. Lett. B 820 (2021) 136577, arXiv:2104.05656 [hep-ph].
  * (96) S. Singirala, S. Sahoo, and R. Mohanta, “Light dark matter, rare B decays with missing energy in L$\mu$-L$\tau$ model with a scalar leptoquark,” Phys. Rev. D 105 (2022) no. 1, 015033, arXiv:2106.03735 [hep-ph].
  * (97) X. Qi, A. Yang, W. Liu, and H. Sun, “Scalar dark matter and Muon $g-2$ in a $U(1)_{L_{\mu}-L_{\tau}}$ model,” arXiv:2106.14134 [hep-ph].
  * (98) BaBar Collaboration, J. P. Lees et al., “Search for a muonic dark force at BABAR,” Phys. Rev. D 94 (2016) no. 1, 011102, arXiv:1606.03501 [hep-ex].
  * (99) CMS Collaboration, A. M. Sirunyan et al., “Search for an $L_{\mu}-L_{\tau}$ gauge boson using Z$\to 4\mu$ events in proton-proton collisions at $\sqrt{s}=$ 13 TeV,” Phys. Lett. B 792 (2019) 345–368, arXiv:1808.03684 [hep-ex].
  * (100) E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, “In the realm of the Hubble tension—a review of solutions,” Class. Quant. Grav. 38 (2021) no. 15, 153001, arXiv:2103.01183 [astro-ph.CO].
  * (101) G. Bellini et al., “Precision measurement of the 7Be solar neutrino interaction rate in Borexino,” Phys. Rev. Lett. 107 (2011) 141302, arXiv:1104.1816 [hep-ex].
  * (102) R. Harnik, J. Kopp, and P. A. N. Machado, “Exploring nu Signals in Dark Matter Detectors,” JCAP 07 (2012) 026, arXiv:1202.6073 [hep-ph].
  * (103) M. Bauer, P. Foldenauer, and J. Jaeckel, “Hunting All the Hidden Photons,” JHEP 07 (2018) 094, arXiv:1803.05466 [hep-ph].
  * (104) R. D. Young and A. W. Thomas, “Octet baryon masses and sigma terms from an SU(3) chiral extrapolation,” Phys. Rev. D 81 (2010) 014503, arXiv:0901.3310 [hep-lat].
  * (105) A. Crivellin, M. Hoferichter, and M. Procura, “Accurate evaluation of hadronic uncertainties in spin-independent WIMP-nucleon scattering: Disentangling two- and three-flavor effects,” Phys. Rev. D 89 (2014) 054021, arXiv:1312.4951 [hep-ph].
  * (106) CRESST Collaboration, A. H. Abdelhameed et al., “First results from the CRESST-III low-mass dark matter program,” Phys. Rev. D 100 (2019) no. 10, 102002, arXiv:1904.00498 [astro-ph.CO].
  * (107) DarkSide Collaboration, P. Agnes et al., “DarkSide-50 532-day Dark Matter Search with Low-Radioactivity Argon,” Phys. Rev. D 98 (2018) no. 10, 102006, arXiv:1802.07198 [astro-ph.CO].
  * (108) XENON Collaboration, E. Aprile et al., “Dark Matter Search Results from a One Ton-Year Exposure of XENON1T,” Phys. Rev. Lett. 121 (2018) no. 11, 111302, arXiv:1805.12562 [astro-ph.CO].
  * (109) XENON Collaboration, E. Aprile et al., “Light Dark Matter Search with Ionization Signals in XENON1T,” Phys. Rev. Lett. 123 (2019) no. 25, 251801, arXiv:1907.11485 [hep-ex].
  * (110) P. Ko and Y. Tang, “Self-interacting scalar dark matter with local $Z_{3}$ symmetry,” JCAP 05 (2014) 047, arXiv:1402.6449 [hep-ph].

## Appendix

##  Appendix A Appendix A : Model Lagrangian and physical states after ${\rm
U(1)}_{L_{\mu}-L_{\tau}}$ symmetry breaking

Here let us give the explicit form of the model Lagrangian in the ${\rm
U(1)}_{L_{\mu}-L_{\tau}}\equiv{\rm U(1)}_{X}$ gauge symmetry.

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{SM}-\frac{1}{4}Z^{\prime\mu\nu}Z^{\prime}_{\mu\nu}-g_{X}Z^{\prime}_{\mu}\left(\bar{\ell}_{\mu}\gamma^{\mu}\ell_{\mu}-\bar{\ell}_{\tau}\gamma^{\mu}\ell_{\tau}+\bar{\mu}_{R}\gamma^{\mu}\mu_{R}-\bar{\tau}_{R}\gamma^{\mu}\tau_{R}\right)$
(12) $\displaystyle+$ $\displaystyle
D_{\mu}\Phi^{\dagger}D^{\mu}\Phi-\lambda_{\Phi}\left(\Phi^{\dagger}\Phi-\frac{v_{\Phi}^{2}}{2}\right)^{2}-\lambda_{\Phi
H}\left(\Phi^{\dagger}\Phi-\frac{v_{\Phi}^{2}}{2}\right)\left(H^{\dagger}H-\frac{v^{2}}{2}\right)+\mathcal{L}_{\rm
DM},$

where $g_{X}$ is the ${\rm U(1)}_{X}$ gauge coupling, and
$D_{\mu}\Phi=(\partial_{\mu}+ig_{X}Q_{\Phi}Z^{\prime}_{\mu})\Phi$.
$\mathcal{L}_{\rm DM}$ will depend on the spin and the ${\rm U(1)}_{X}$ charge
of dark matter, and will be specified case by case in the main text or in the
Appendix.

The phase where $U(1)$ symmetry is spontaneously broken is described by

$\Phi(x)=\frac{1}{\sqrt{2}}\left(v_{\Phi}+\phi(x)\right),$

where $v_{\Phi}$ is VEV of dark Higgs field (see Table I). We shall call
$Z^{\prime}$ dark photon, since it couples to the scalar or fermion DM in our
work. The $Z^{\prime}$ boson mass is given by

$\displaystyle M_{Z^{\prime}}$ $\displaystyle=$ $\displaystyle
g_{X}|Q_{\Phi}|v_{\Phi}.$ (13)

For the two CP-even neutral scalar bosons, we can define the mixing matrix $O$
which connects the mass and the interaction eigenstates:

$\left(\begin{array}[]{c}\phi\\\
h\end{array}\right)=O\left(\begin{array}[]{c}H_{1}\\\
H_{2}\end{array}\right)\equiv\left(\begin{array}[]{cc}c_{\alpha}&s_{\alpha}\\\
-s_{\alpha}&c_{\alpha}\end{array}\right)\left(\begin{array}[]{c}H_{1}\\\
H_{2}\end{array}\right),$ (14)

where $s_{\alpha}(c_{\alpha})\equiv\sin\alpha(\cos\alpha)$. $(\phi,h)$ and
$(H_{1},H_{2})$ are interaction and mass eigenstates with masses $M_{H_{i}}$
(i=1,2), respectively. The mixing angle $\alpha$ is defined by

$\displaystyle\tan 2\alpha=\frac{\lambda_{\Phi
H}v_{\Phi}v_{H}}{\lambda_{H}v^{2}_{H}-\lambda_{\Phi}v^{2}_{\Phi}},$ (15)

where $v_{H}=246$ GeV is the VEV of the SM Higgs.

The mass matrix in the basis $(\phi,h)$ can be expressed in terms of the
physical parameters as follows:

$\left(\begin{array}[]{cc}2\lambda_{\Phi}v_{\Phi}^{2}&\lambda_{\Phi
H}v_{\Phi}v_{H}\\\ \lambda_{\Phi
H}v_{\Phi}v_{H}&2\lambda_{H}v_{H}^{2}\end{array}\right)=\left(\begin{array}[]{cc}M_{H_{1}}^{2}{c_{\alpha}^{2}}+M_{H_{2}}^{2}{s_{\alpha}^{2}}&(M_{H_{2}}^{2}-M_{H_{1}}^{2})c_{\alpha}s_{\alpha}\\\
(M_{H_{2}}^{2}-M_{H_{1}}^{2})c_{\alpha}s_{\alpha}&M_{H_{1}}^{2}{s_{\alpha}^{2}}+M_{H_{2}}^{2}{c_{\alpha}^{2}}\end{array}\right).$
(16)

Now we can take $M_{H_{i}}$ and $\sin\alpha$ as independent parameters. The
mixing angle $\alpha$ is constrained by Higgs decay data, and should be rather
small, $\sin\alpha\lesssim 10^{-3}{\rm(BPI)},10^{-2}{\rm(BPII)}$. Notice that
$H_{1}$ is mostly dark Higgs boson and $H_{2}$ is the SM Higgs-like state.
Therefore, we take $M_{H_{2}}=125.25$GeV. Since we are interested in the light
dark Higgs case ($M_{H_{1}}<M_{H_{2}}$) in our model, we can always take
$|\alpha|<\pi/4$. In the small mixing angle limit, we have $H_{1}\simeq\phi$
and $H_{2}\simeq h$.

The gauge interaction terms between $Z^{\prime}$ and SM particles are

$\displaystyle\mathcal{L}_{Z^{\prime}}\supset-
g_{X}\left(\bar{\mu}\gamma^{\mu}\mu-\bar{\tau}\gamma^{\mu}\tau+\bar{\nu}_{L\mu}\gamma^{\mu}\nu_{L\mu}-\bar{\nu}_{L\tau}\gamma^{\mu}\nu_{L\tau}\right)Z^{\prime}_{\mu}.$
(17)

After spontaneous symmetry breakings, the additional interactions with the
dark Higgs $\phi$ are given by

$\displaystyle\mathcal{L}_{\phi}\supset\frac{1}{2}g^{2}_{X}Q^{2}_{\Phi}Z^{\prime\mu}Z^{\prime}_{\mu}\phi^{2}+g^{2}_{X}Q^{2}_{\Phi}v_{\Phi}Z^{\prime\mu}Z^{\prime}_{\mu}\phi-\lambda_{\Phi}v_{\Phi}\phi^{3}-\lambda_{H}v_{H}h^{3}-\frac{\lambda_{\Phi
H}}{2}v_{\Phi}\phi h^{2}-\frac{\lambda_{\Phi H}}{2}v_{H}\phi^{2}h\ ,$ (18)

in the interaction basis.

We assume that the kinetic mixing between $Z^{\prime}$ and the hypercharge
gauge boson $B$ is zero at high scale. Nevertheless, small mixings of
$Z^{\prime}$ boson with the SM photon $A$ and the $Z$ boson are generated by
radiative one-loop level corrections involving $\mu$ and $\tau$ leptons.

$\displaystyle\mathcal{L}_{\epsilon}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon_{A}}{2}F_{\mu\nu}Z^{\prime\mu\nu}-\frac{\epsilon_{Z}}{2}Z_{\mu\nu}Z^{\prime\mu\nu},$
(19)

where $F_{\mu\nu}$ and $Z_{\mu\nu}$ are the field strength for the SM photon
and $Z$ boson, respectively. The canonical basis of gauge fields can be
recovered by the transformation

$\displaystyle A_{\mu}$ $\displaystyle\to$ $\displaystyle
A_{\mu}+\epsilon_{A}Z^{\prime}_{\mu},$ (20) $\displaystyle Z_{\mu}$
$\displaystyle\to$ $\displaystyle
Z_{\mu}-\epsilon_{Z}\frac{M^{2}_{Z^{\prime}}}{M^{2}_{Z}}Z^{\prime}_{\mu},$
(21) $\displaystyle Z^{\prime}_{\mu}$ $\displaystyle\to$ $\displaystyle
Z_{\mu}+\epsilon_{Z}Z_{\mu}.$ (22)

Through the mixing, we have interaction terms which reads Drees:2021rsg

$\displaystyle\mathcal{L}_{\rm\epsilon}$ $\displaystyle=$
$\displaystyle-\epsilon_{A}Z^{\prime}_{\mu}J^{\mu}_{\rm
em}+\frac{g}{\cos\theta_{W}}\epsilon_{Z}\frac{M^{2}_{Z^{\prime}}}{M^{2}_{Z}}Z^{\prime}_{\mu}J^{\mu}_{\rm
NC},$ (23) $\displaystyle\epsilon_{A}$ $\displaystyle=$
$\displaystyle-\frac{eg_{X}}{12\pi^{2}}\ln\left(\frac{M^{2}_{\tau}}{M^{2}_{\mu}}\right)\simeq-\frac{g_{X}}{70},$
(24)

The mixing between $Z^{\prime}$ and $Z$ boson is much suppressed by
$M^{2}_{Z^{\prime}}/M^{2}_{Z}$ when $M_{Z^{\prime}}\ll M_{Z}$. Therefore, this
effect is negligible for light $Z^{\prime}$ mass region.

The decay width of the $Z^{\prime}$ boson is

$\displaystyle\Gamma(Z^{\prime}\to\ell\bar{\ell})$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{X}M_{Z^{\prime}}}{12\pi}\left(1+\frac{2M^{2}_{\ell}}{M^{2}_{Z^{\prime}}}\right)\sqrt{1-\frac{4M^{2}_{\ell}}{M^{2}_{Z^{\prime}}}}\theta\left(M_{Z^{\prime}}-2M_{\ell}\right),$
(25) $\displaystyle\Gamma(Z^{\prime}\to\nu_{\ell}\bar{\nu}_{\ell})$
$\displaystyle=$ $\displaystyle\frac{g^{2}_{X}M_{Z^{\prime}}}{24\pi},$ (26)

where $\ell=\mu,\tau$ and $\theta(x)$ is the unit step function. $Z^{\prime}$
decay into a pair of DM is assumed to be kinematically forbidden for the DM
mass ranges chosen in this paper.

##  Appendix B Appendix B: Constraints on the ${\rm U(1)}_{L_{\mu}-L_{\tau}}$
Models

In this Section, we first check whether the preferred parameter space
$M_{Z^{\prime}}\sim\mathcal{O}(10-100)$ MeV and $g_{X}<1$ for the $\Delta
a_{\mu}$ is allowed after considering all the relevant experimental
constraints involving $Z^{\prime}$ boson.

###  B.1 B1. Muon $(g-2)$

The magnetic moment of muon is given by

$\displaystyle\vec{\mu}_{\mu}$ $\displaystyle=$ $\displaystyle
g_{\mu}\left(\frac{e}{2m}\right)\vec{S},$ (27)

where $g_{\mu}$ is the gyromagnetic ratio. Taking into account radiative
corrections, the anomalous magnetic moment of the muon is defined as

$\displaystyle a_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(g_{\mu}-2\right).$ (28)

In the ${\rm U(1)}_{L_{\mu}-L_{\tau}}$ models, the new gauge boson
$Z^{\prime}$ makes an additional contribution to the muon $(g-2)$ through the
vertex correction to the muon-photon coupling Baek:2001kca ; Baek:2008nz :

$\Delta
a_{\mu}=\frac{\alpha_{X}}{2\pi}\int_{0}^{1}dx\frac{2M_{\mu}^{2}x^{2}(1-x)}{x^{2}M_{\mu}^{2}+(1-x)M_{Z^{\prime}}^{2}},$
(29)

where $\alpha_{X}=g^{2}_{X}/4\pi$. Taking $g_{X}\sim(4-8)\times 10^{-4}$, this
effect resolves the discrepancy in the $M_{Z^{\prime}}<M_{\mu}$ limit.

If one extends the particle content by including exotic leptons and/or dark
matter, there will be additional contributions even for heavier $Z^{\prime}$
and one can accommodate $\Delta a_{\mu}$, which were discussed in the
literature Borah:2021mri ; Qi:2021rhh ; Singirala:2021gok ; Buras:2021btx ;
Zhou:2021vnf ; Borah:2021jzu ; Chen:2021vzk ; Zu:2021odn ; Huang:2021nkl ;
Patra:2016shz ; Altmannshofer:2016oaq .

###  B.2 B2. BaBar and LHC 4$\mu$ channels

Dark photon searches at electron beam dump experiments provide constraints on
our model parameter spaces. For example, BaBar searches for the process
$e^{+}e^{-}\to Z^{\prime}\mu^{+}\mu^{-}$ followed by
$Z^{\prime}\rightarrow\mu^{+}\mu^{-}$ BaBar:2016sci . BaBar bounds depend on
the branching ratio of the $Z^{\prime}$ decaying to DMs. In our paper, we
consider light $Z^{\prime}$ ($10-100$ MeV) and heavy heavy DM, and the
$Z^{\prime}$ decay into a pair of DM is kinematically forbidden in our case.
We take $2\sigma$ exclusion limit from the BaBar bounds, as shown in Fig. 1.

Measurements of the $Z\to 4\mu$ at the LHC also give stringent constraints on
our models. LHC has measured $pp\to Z\to 4\mu$ channels. In the analysis, four
muon events near an invariant mass near $Z$ boson mass are used. In
$U(1)_{L_{\mu}-L_{\tau}}$ model, the produced $Z$ boson might decay to
$Z^{\prime}$ and a muon pair. If $M_{Z^{\prime}}>2M_{\mu}$, the produced
$Z^{\prime}$ subsequently decays to a pair of muons. Therefore, the events are
sensitive to $pp\to Z\to 4\mu$ decay channel. We take $2\sigma$ exclusion
limit from the CMS data CMS:2018yxg .

###  B.3 B3. Neutrino trident production

Neutrino beam experiments such as such as CHARM-II CHARM-II:1990dvf  and CCFR
CCFR:1991lpl  have established neutrino trident production, the muon-neutrino
scattering on nucleus producing a muon pair,
$\nu_{\mu}N\to\nu_{\mu}N\mu^{+}\mu^{-}$ Altmannshofer:2014pba . The observed
scattering cross sections are

$\displaystyle\frac{\sigma_{\rm CHARM-II}}{\sigma_{\rm SM}}=1.58\pm
0.57\,,\qquad\frac{\sigma_{\rm CCFR}}{\sigma_{\rm SM}}=0.82\pm 0.28\,.$ (30)

The observation data is consistent with the SM prediction. Therefore, the
neutrino trident production would provide a stringent constraint, which
basically excludes the parameter region with $M_{Z^{\prime}}>O(1)$ GeV
Altmannshofer:2014pba . In our study, we take $2\sigma$ exclusion limit from
the CCFR data.

###  B.4 B4. Cosmic Microwave Background

There exists interaction between $Z^{\prime}$ and neutrinos in our model. And
we assume that $Z^{\prime}$ and DM were in thermal equilibrium with SM bath at
the early Universe. If $Z^{\prime}$ mass is much below the muon mass,
$Z^{\prime}$ decay process will provide energy and entropy into neutrinos. If
this happens after epoch of the neutrino decoupling corresponding to $T\sim
O(1)$ MeV, it gives additional entropy and energy. It spoils the sucessful Big
Bang Nucleosynthesis (BBN). This effect modifies $N_{\rm eff}$. We take
$N_{\rm eff}<3.4$ as exclusion limit. However, there exists a discrepancy
between the observation of today’s Hubble constant coming from Cosmic
Microwave Background and other astrophysical observation. It is so-called
Hubble tension. To alleviate the Hubble tension, increasing $N_{\rm eff}$ is
needed Escudero:2019gzq ; DiValentino:2021izs . We take one of our benchmark
point $(M_{Z^{\prime}},g_{X})=(11.5{\rm MeV},4\times 10^{-4})$ by allowing
$N_{\rm eff}>3.2$ to solve this tension.

###  B.5 B5. Borexino

Reactor neutrino experiments can provide useful constraints on $g_{X}$ and
$M_{Z^{\prime}}$ by neutrino-electron scattering process. Borexino
collaboration provides the most sensitive data Bellini:2011rx  on this
channel. The ${}^{7}{\rm Be}$ neutrinos, $\nu_{e}$, oscillate to $\nu_{\mu}$
and $\nu_{\tau}$ on the way to the Earth. At the detector, they are scattered
by electron via $Z^{\prime}$ exchange Harnik:2012ni ; Bauer:2018onh . And the
bound is shown in Fig. 1.

###  B.6 B6. Dark Higgs Decay

The dark Higgs decay width to a pair of lepton is given by

$\displaystyle\Gamma\left(H_{1}\to\ell^{+}\ell^{-}\right)$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{\ell}M_{H_{1}}}{8\pi
v^{2}_{H}}\sin^{2}\alpha\left(1-\frac{4M^{2}_{\ell}}{M^{2}_{H_{1}}}\right)^{3/2}.$
(31)

where $\ell=e,\mu,\tau$. If kinematically allowed, the dark Higgs can decay to
a pair of $Z^{\prime}$ boson which is given by

$\displaystyle\Gamma\left(H_{1}\to Z^{\prime}Z^{\prime}\right)$
$\displaystyle=$
$\displaystyle\frac{g^{2}_{X}c^{2}_{\alpha}}{32\pi}\frac{M^{3}_{H_{1}}}{M^{2}_{Z^{\prime}}}\left(1-\frac{4M^{2}_{Z^{\prime}}}{M^{2}_{H_{1}}}-\frac{12M^{4}_{Z^{\prime}}}{M^{4}_{H_{1}}}\right).$
(32)

In our interesting parameter space which corresponds to
$M_{Z^{\prime}}\sim\mathcal{O}(10)$MeV and $M_{H_{1}}\geq\mathcal{O}(0.1)$GeV,
the dark Higgs dominantly decays into a pair of $Z^{\prime}$ boson, followed
by $Z^{\prime}\rightarrow\nu_{\alpha}\overline{\nu}_{\alpha}$ with
$\alpha=\mu,\tau$. The produced neutrinos might shift $\Delta N_{\rm eff}$.
However, this effect is negligible since comoving number density $Y_{H_{1}}$
at the decoupling epoch is very small. For $M_{H_{1}}\leq 2M_{Z^{\prime}}$, we
require that lifetime of the dark Higgs should be less than $\sim O(1)$ sec to
avoid the bound from BBN.

##  Appendix C Appendix C: Collections of relevant formulae

In this Appendix, for completeness, we collect the relevant formulae for the
discussions on the DM phenomenology, such as ${\cal L}_{\rm DM}$, thermal
averaged DM annihilation cross sections, and direct detection cross sections
for the spin-independent DM-nucleon scattering. We also include details of a
few cases which are mentioned only briefly in the main text.

###  C.1 C1. Scalar DM: Generic Case

In this case, the DM Lagrangian is given in Eq. (4) along with Eq. (12) for
other sectors. Important annihilation channels that determine dark matter
relic abundance are $XX^{\dagger}\to
H_{1}H_{1},Z^{\prime}Z^{\prime},Z^{\prime}H_{1}$. The corresponding Feynman
diagrams are presented in Fig. 2. The thermal averaged cross section of
$XX^{\dagger}\to H_{1}H_{1}$ is

$\displaystyle\langle\sigma v_{\rm rel}(XX^{\dagger}\to H_{1}H_{1})\rangle$
$\displaystyle\simeq$ $\displaystyle\frac{\mathcal{S}}{16\pi
s}\left(\lambda_{\Phi
X}c^{2}_{\alpha}+\lambda_{HX}s^{2}_{\alpha}\right)^{2}\left(1-\frac{4M^{2}_{H_{1}}}{s}\right)^{1/2},$
(33)

where $\mathcal{S}$ is symmetric factor. The thermal averaged cross section of
$XX^{\dagger}\to Z^{\prime}Z^{\prime}$ is

$\displaystyle\langle\sigma v_{\rm rel}(XX^{\dagger}\to
Z^{\prime}Z^{\prime})\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{32\pi
s}\overline{|\mathcal{M}|^{2}}\left(1-\frac{4M^{2}_{Z^{\prime}}}{s}\right)^{1/2}.$
(34)

The $Z^{\prime}_{L}Z^{\prime}_{L}$ final state gives the dominant
contribution:

$\displaystyle\overline{|\mathcal{M}|^{2}}$ $\displaystyle\simeq$
$\displaystyle\frac{s^{2}}{4M^{4}_{Z^{\prime}}}(\lambda_{\Phi
X}v_{\Phi})^{2}\left|\left(\frac{c_{\alpha}\kappa_{1}}{s-M^{2}_{H_{1}}+i\Gamma_{H_{1}}M_{H_{1}}}+\frac{s_{\alpha}\kappa_{2}}{s-M^{2}_{H_{2}}+i\Gamma_{H_{2}}M_{H_{2}}}\right)\right|^{2},$
(35)

where

$\displaystyle\kappa_{1}$ $\displaystyle=$ $\displaystyle
6\lambda_{H}v_{H}s^{3}_{\alpha}-6\lambda_{\Phi}v_{\Phi}c^{3}_{\alpha}+3\lambda_{\Phi
H}v_{H}s_{\alpha}c^{2}_{\alpha}-3\lambda_{\Phi
H}v_{\Phi}c_{\alpha}s^{2}_{\alpha},$ (36) $\displaystyle\kappa_{2}$
$\displaystyle=$
$\displaystyle-6\lambda_{\Phi}v_{\Phi}c^{2}_{\alpha}s_{\alpha}-6\lambda_{H}v_{H}s^{2}_{\alpha}c_{\alpha}+\lambda_{\Phi
H}\left(2c^{2}_{\alpha}s_{\alpha}-s^{3}_{\alpha}\right)-\lambda_{\Phi
H}v_{H}\left(c^{3}_{\alpha}-2c_{\alpha}s^{2}_{\alpha}\right).$ (37)

The thermal DM can be detected by DM direct detection searches. Spin-
independent DM-nucleon scattering cross section is given by

$\displaystyle\sigma_{\rm SI}$ $\displaystyle=$
$\displaystyle\frac{\mu^{2}_{N}}{4\pi}\left(\frac{M_{N}}{M_{X}}\right)^{2}\frac{c^{4}_{\alpha}}{M^{4}_{H_{1}}}f^{2}_{N}\left[\lambda_{\Phi
X}\frac{v_{\Phi}}{v_{H}}t_{\alpha}\left(1-\frac{M^{2}_{H_{1}}}{M^{2}_{H_{2}}}\right)-\lambda_{HX}\left(t^{2}_{\alpha}+\frac{M^{2}_{H_{1}}}{M^{2}_{H_{2}}}\right)\right]^{2},$
(38)

where $f_{N}=0.327$ Young:2009zb ; Crivellin:2013ipa , $M_{N}$ is the nucleon
mass, and $\mu_{N}=M_{X}M_{N}/(M_{X}+M_{N})$. For $M_{\rm DM}\geq 1$GeV, The
most stringent bound comes from CRESST CRESST:2019jnq , DarkSide-50
DarkSide:2018kuk  and XENON1T XENON:2018voc ; XENON:2019gfn , which can be
evaded by choosing small enough $\sin\alpha$ and $\lambda_{HX}\approx 0$.

Figure 7:  The (Top) plots show the relic abundance of complex scalar DM for
$Q_{\Phi}=1.1$ as functions of dark Higgs mass $M_{H_{1}}$ for [BPI] (Left)
and [BPII] (Right). The (Bottom) plots show the relic density as functions of
$\lambda_{\Phi X}$ (Left) and the preferred parameter space in the
$(M_{H_{1}},\lambda_{\Phi X})$ plane for $\lambda_{HX}=0$ (Right) for [BPII] .
We take four different DM masses, $M_{X}=1,~{}10,100,~{}1000$GeV,
respectively. Solid (Dashed) lines represent the region where bounds on DM
direct detection are satisfied (ruled out).

In the top panel of Fig. 7, we show the $\Omega h^{2}$ as functions of
$M_{H_{1}}$ for two [BP]’s. The bottom panel is for the [BPII]: (Left) $\Omega
h^{2}$ as a function of $\lambda_{\Phi X}$, and (Right) the allowed parameter
space in the $(M_{H_{1}},\lambda_{\Phi X})$. Solid (Dashed) lines represent
the region where bounds on DM direct detection are satisfied (ruled out). It
is found that the scalar DM can be thermal WIMP in wide mass ranges, outside
$M_{Z^{\prime}}\sim(2-3)M_{X}$ due to the contributions from the dark Higgs
$\phi\simeq H_{1}$ that opens new contribution to $Z^{\prime}Z^{\prime}$ and
new annihilation channel into $H_{1}H_{1}$.

###  C.2 C2. Local $Z_{2}$ Scalar DM

Feynman diagrams for local $Z_{2}$ scalar DM model are depicted in Fig. 8:

Figure 8:  (Top) Feynman diagrams for local $Z_{2}$ scalar DM annihilatiing to
a pair of $Z^{\prime}$ bosons. (Bottom) Feynman diagrams for local $Z_{2}$
scalar DM annihilatiing to a pair of $H_{1}$ bosons, which is mostly dark
Higgs-like.

The thermal averaged cross section annihilating to a pair of $Z^{\prime}$ is

$\displaystyle\langle\sigma v_{\rm rel}(X_{I}X_{I}\to
Z^{\prime}Z^{\prime})\rangle$ $\displaystyle\approx$
$\displaystyle\frac{1}{32\pi
s}\overline{|\mathcal{M}|^{2}}\left(1-\frac{4M^{2}_{Z^{\prime}}}{s}\right)^{1/2}$
(39)

with

$\displaystyle\overline{|\mathcal{M}|^{2}}$ $\displaystyle\simeq$
$\displaystyle\frac{s^{2}}{v^{2}_{\Phi}}\left|\frac{\lambda_{1}c_{\alpha}}{s-M^{2}_{H_{1}}+i\Gamma_{H_{1}}M_{H_{1}}}+\frac{\lambda_{2}s_{\alpha}}{s-M^{2}_{H_{2}}+i\Gamma_{H_{2}}M_{H_{2}}}\right|^{2},$
(40)

where $\lambda_{1}=\left(\lambda_{\Phi
X}v_{\Phi}-\sqrt{2}\mu\right)c_{\alpha}-\lambda_{HX}v_{H}s_{\alpha}$ and
$\lambda_{2}=\left(\lambda_{\Phi
X}v_{\Phi}-\sqrt{2}\mu\right)s_{\alpha}+\lambda_{HX}v_{H}c_{\alpha}$. Again
the longitudinal $Z^{\prime}_{L}$ pair will give the dominant contributions
here. Neglecting $Z^{\prime}$ mass in the final states, the thermally averaged
annihilation cross sectionaround freeze-out temperature is

$\displaystyle\langle\sigma v_{\rm rel}(X_{I}X_{I}\to
Z^{\prime}Z^{\prime})\rangle$ $\displaystyle\approx$
$\displaystyle\frac{1}{8\pi}\frac{M^{2}_{I}}{v^{2}_{\Phi}}\left|\frac{\lambda_{1}c_{\alpha}}{s-M^{2}_{H_{1}}+i\Gamma_{H_{1}}M_{H_{1}}}+\frac{\lambda_{2}s_{\alpha}}{s-M^{2}_{H_{2}}+i\Gamma_{H_{2}}M_{H_{2}}}\right|^{2}.$
(41)

Note that one can see the longitudinal enhancement effect.

If kinematically allowed, the DM annihilation cross section into the dark
Higgs boson is also possible:

$\displaystyle\langle\sigma v_{\rm rel}(X_{I}X_{I}\to H_{1}H_{1})\rangle$
$\displaystyle\approx$ $\displaystyle\frac{1}{32\pi s}\left(\lambda_{\Phi
X}c^{2}_{\alpha}+\lambda_{HX}s^{2}_{\alpha}\right)^{2}\sqrt{1-\frac{4M^{2}_{H_{1}}}{s}}$
(42)

In the local $Z_{2}$ scalar DM case, there are two processes in DM direct
detection. One is elastic scattering and the other is inelastic scattering
process. In our interesting parameter space where
$\delta\geq\mathcal{O}(100)$keV, inelastic scattering process does not happen.
Thus, we will concentrate on elastic scattering process mediated by both the
dark and the SM Higgs bosons. In this case, the spin-independent elastic DM-
nucleon scattering is given by

$\displaystyle\sigma_{\rm SI}$ $\displaystyle=$
$\displaystyle\frac{\mu^{2}_{N}}{4\pi}\left(\frac{M_{N}}{M_{I}}\right)^{2}\frac{c^{4}_{\alpha}}{M^{4}_{H_{1}}}f^{2}_{N}\left[\left(\lambda_{\Phi
X}-\frac{\sqrt{2}\mu}{v_{\Phi}}\right)\frac{v_{\Phi}}{v_{H}}t_{\alpha}\left(1-\frac{M^{2}_{H_{1}}}{M^{2}_{H_{2}}}\right)-\lambda_{HX}\left(t^{2}_{\alpha}+\frac{M^{2}_{H_{1}}}{M^{2}_{H_{2}}}\right)\right]^{2}.$
(43)

We choose small enough $\sin\alpha(\simeq\tan\alpha)$ and $\lambda_{HX}$ in
order to avoid the strong constraints from direct detections.

Figure 9:  (Left) Relic abundance of local $Z_{2}$ scalar DM in case of
[BPII]. We take $\lambda_{HX}=0$, $M_{H_{1}}=10$GeV, and $s_{\alpha}=10^{-4}$.
All the lines satisfy the DM direct detection bound. (Right) Relic abundance
of local $Z_{2}$ scalar DM in the $(M_{H_{1}},\lambda_{\Phi X})$ plane.

###  C.3 C3. Local $Z_{3}$ scalar DM : $(Q_{X},Q_{\Phi})=(1,3)$

There is another special case for the complex scalar DM: $Q_{\Phi}=3$ and
$Q_{X}=1$, for which ${\rm U(1)}_{X}\rightarrow Z_{3}$ Ko:2014loa ; Ko:2014nha
by Krauss-Wilczek mechanism. The relevant Lagrangian is given by

$\mathcal{L}_{\rm
DM}=D^{\mu}X^{\dagger}D_{\mu}X-m^{2}_{X}X^{\dagger}X-\lambda_{HX}X^{\dagger}X\left(H^{\dagger}H-\frac{v^{2}_{H}}{2}\right)-\lambda_{\Phi
X}X^{\dagger}X\left(\Phi^{\dagger}\Phi-\frac{v^{2}_{\Phi}}{2}\right)\\\
+\lambda_{3}\left(X^{3}\phi^{\dagger}+H.c.\right).$

In this case there appears a new mechanism, semi-annihilations $XX\rightarrow
X^{\dagger}H_{1},X^{\dagger}Z^{\prime}$ in addition to the usual annihilation
channels $XX^{\dagger}\rightarrow Z^{\prime}(\phi,h)\rightarrow$ (SM
particles) Ko:2014nha . Thus the mass scale of the complex scalar DM $X$ can
be in a wide range, evading the mass relation, $M_{Z^{\prime}}\sim(2-3)M_{X}$,
that was derived in the case with $Z^{\prime}$ only and without $H_{1}$. In
Fig. 10, we show for different choices of $\lambda_{\Phi X}$ $\Omega h^{2}$ as
functions of $\lambda_{3}$ which controls the strength of semi-annihilation.
Since $g_{X}\sim O(10^{-4})$ is very small, the channel $XX\rightarrow
X^{\dagger}Z^{\prime}$ is not important compared to $XX\rightarrow
X^{\dagger}H_{1}$. Therefore the parameter $\lambda_{3}$ controlling the
channel $XX\rightarrow X^{\dagger}H_{1}$ becomes most important. This is why
two plots in the left and the right are almost identical for our choice of
parameters. Once again, we observe that the dark Higgs can modify DM
phenomenology significantly and the allowed mass range for the complex scalar
DM $X$ can be very far from $M_{Z^{\prime}}\sim 2M_{X}$ for a special choice
of dark charges, $Q_{X}=1=Q_{\Phi}/3$.

Figure 10:  Relic abundance of $Z_{3}$ scalar DM for the [BPI] (Left) and the
[BPII] (Right), respectively. Here we fixed $\lambda_{HX}=0$ for simplicity.

###  C.4 C4. Fermion DM: Generic Case

###  C.5 Generic Case: $Q_{\Phi}\neq 2Q_{\chi}=2$

For the generic case of Dirac fermion DM, the DM Lagrangian is given by

$\displaystyle\mathcal{L}_{\rm DM}$ $\displaystyle=$
$\displaystyle\overline{\chi}(i\not{D}-m_{\chi})\chi.$ (44)

In this case there is no direct renormalizable interactions between $\chi$ and
$\Phi$ for generic $Q_{\Phi}$ (except for $Q_{\Phi}=2$), which is in sharp
contrast to the scalar DM cases we discussed earlier. If we assume $g_{X}\sim
O(10^{-4})$ for the muon $(g-2)$, DM pair annihilation cross sections for
$\chi\bar{\chi}\rightarrow Z^{\prime}Z^{\prime},Z^{\prime}H_{1}$ are $\sim
O(g_{X}^{4})$. Therefore they are too small for the correct thermal relic
density of DM. Only $\chi\bar{\chi}\rightarrow Z^{\prime}\rightarrow$(SM
particles) turns out to can reproduce the correct relic density only in the
resonance region with $M_{Z^{\prime}}\sim(2-3)M_{\chi}$, as observed in
previous works and described in the Introduction.

The thermal averaged DM annihilation cross section is given by

$\displaystyle\langle\sigma v_{\rm
rel}(\chi\bar{\chi}\to\ell\bar{\ell})\rangle$ $\displaystyle\simeq$
$\displaystyle\frac{g^{4}_{X}Q^{2}_{\chi}}{2\pi}\frac{\left(2M^{2}_{\chi}+M^{2}_{\ell}\right)}{\left(4M^{2}_{\chi}-M^{2}_{Z^{\prime}}\right)^{2}+\Gamma^{2}_{Z^{\prime}}M^{2}_{Z^{\prime}}}\sqrt{1-\frac{M^{2}_{\ell}}{M^{2}_{\chi}}},$
(45) $\displaystyle\langle\sigma v_{\rm
rel}(\chi\bar{\chi}\to\nu_{\ell}\bar{\nu_{\ell}})\rangle$
$\displaystyle\simeq$
$\displaystyle\frac{g^{4}_{X}Q^{2}_{\chi}}{2\pi}\frac{M^{2}_{\chi}}{\left(4M^{2}_{\chi}-M^{2}_{Z^{\prime}}\right)^{2}+\Gamma^{2}_{Z^{\prime}}M^{2}_{Z^{\prime}}},$
(46)

where $\ell=\mu,\tau$. Since $g_{X}$ is $\mathcal{O}(10^{-4})$ in order to
explain $\Delta a_{\mu}$ anomaly with $M_{Z^{\prime}}\sim(10-100)$ MeV, the
total annihilation cross section is too small to get the correct relic
density. The correct DM relic abundance reaches in the resonance region with
$M_{\chi}\sim M_{Z^{\prime}}/2$.

If kinematically allowed, there are additional DM annihilation channels which
are given by

$\displaystyle\langle\sigma v_{\rm rel}(\chi\bar{\chi}\to
Z^{\prime}Z^{\prime})\rangle$ $\displaystyle\simeq$
$\displaystyle\frac{g^{4}_{X}Q^{4}_{\chi}}{4\pi}\frac{M^{2}_{\chi}-M^{2}_{Z^{\prime}}}{\left(M^{2}_{Z^{\prime}}-2M^{2}_{\chi}\right)^{2}}\sqrt{1-\frac{M^{2}_{Z^{\prime}}}{M^{2}_{\chi}}}$
(47) $\displaystyle\langle\sigma v_{\rm rel}(\chi\bar{\chi}\to
Z^{\prime}H_{1})\rangle$ $\displaystyle\simeq$
$\displaystyle\frac{c^{2}_{\alpha}g^{4}_{X}Q^{2}_{\Phi}Q^{2}_{\chi}}{256\pi}\frac{\left[16M_{\chi}^{4}-8M_{\chi}^{2}(M^{2}_{H_{1}}-5M^{2}_{Z^{\prime}})+\left(M^{2}_{H_{1}}-M^{2}_{Z^{\prime}}\right)^{2}\right]}{M_{\chi}^{4}\left[\left(4M^{2}_{\chi}-M_{Z^{\prime}}^{2}\right)^{2}+M^{2}_{Z^{\prime}}\Gamma^{2}_{Z^{\prime}}\right]}$
(48)
$\displaystyle\times\sqrt{M^{4}_{H_{1}}+\left(4M^{2}_{\chi}-M^{2}_{Z^{\prime}}\right)^{2}-2M^{2}_{H_{1}}\left(M^{2}_{Z^{\prime}}+4M^{2}_{\chi}\right)}$

However, relic density of DM is satisfied near resonance area even though we
consider additional DM annihilation channels.

###  C.6 C5. Local $Z_{2}$ fermion DM

For the direct detection in the local $Z_{2}$ fermion DM case, there are also
two processes as in the local $Z_{2}$ scalar DM case: one is elastic
scattering through (dark) Higgs exchanges, and the other is inelastic
scattering process through $Z^{\prime}$ exchange with the kinetic mixing. Here
again the inelastic scattering is negligible for $\delta\gtrsim O(10)$keV.
Therefore we impose only the Higgs-mediated elastic scattering, the explicit
form of which is given by

$\displaystyle\sigma_{\rm SI}$ $\displaystyle=$
$\displaystyle\frac{\mu^{2}_{N}}{\pi}\Delta^{2}\left(\frac{M_{I}M_{N}}{v_{H}v_{\Phi}}\right)^{2}f^{2}_{N}s^{2}_{\alpha}c^{2}_{\alpha}\left(\frac{1}{M^{2}_{H_{1}}}-\frac{1}{M^{2}_{H_{2}}}\right)^{2},$
(49)

where $\Delta\equiv\left(M_{R}-M_{I}\right)/M_{I}$.

Figure 11:  (Top) Dark matter relic density as functions of dark Higgs mass
$M_{H_{1}}$ for [BPI] (Left) and [BPII] (Right) (Bottom-Left) Dark matter
relic density as functions of $\Delta$ for [BPII], and (Bottom-right)
Preferred parameter region in the $(\Delta,M_{H_{1}})$ plane. Solid (Dashed)
lines denote the region where bounds on DM direct detection are satisfied
(ruled out).