# Dequantization of electric charge: Probing scenarios of cosmological multi-
component dark matter

Duong Van Loi  Nguyen Manh Duc111Leave of absence from Hanoi–Amsterdam High
School for the Gifted, Trung Hoa, Cau Giay, Hanoi 100000, Vietnam Phung Van
Dong  Corresponding author; email<EMAIL_ADDRESS>Phenikaa
Institute for Advanced Study and Faculty of Basic Science, Phenikaa
University, Yen Nghia, Ha Dong, Hanoi 100000, Vietnam

###### Abstract

Since the electric charge in the standard model is theoretically not
quantized, we may have a variant of it, called dark charge. Similar to the
electric charge, the dark charge neither commutes nor closes algebraically
with $SU(2)_{L}$. The condition of algebraic closure leads to a novel gauge
extension, $SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)_{N}$, where $Y$ and $N$
determine the electric and dark charges, respectively, apart from the color
group. We argue that the existence of the dark charge, thus $N$, leads to the
novel scenarios of multi-component dark matter, in general. The dark matter
stability is determined by a residual (or dark charge) gauge symmetry
isomorphic to an even $Z_{k}$ discrete group, where $k$ is specified dependent
on the value of the neutrino dark charge. This residual symmetry divides the
standard model particles into distinct classes, which possibly accommodate
dark matter, but the dark matter candidate cannot decay due to the color and
electric charge conservation. We analyze in detail three specific models
according to $k=2,4,6$ and determine the simplest dark matter candidates. For
small $U(1)_{N}$ coupling, the two-component dark matter scenarios implied by
the dark charge successfully explain the dark matter relic density and the
recent XENON1T excess, as well as the beam dump, neutrino scattering, and
astrophysical bounds. Otherwise, for large $U(1)_{N}$ coupling, we have multi-
WIMPs coexisted beyond the weak scale.

###### pacs:

12.60.Cn, 95.35.+d

##  I Introduction

The standard model of particle physics Zyla:2020zbs  has been enormously
successful in describing most of the observed phenomena. However, the explicit
evidences of neutrino oscillation RevModPhys.88.030501 ; RevModPhys.88.030502
and dark matter existence Hinshaw:2012aka ; Aghanim:2018eyx  cannot be
explained within the framework of the standard model. This proves that the
standard model must be extended in order to account for the new physics
beyond.

The existing experiments of the dark matter relic density, direct/indirect
detections, and particle colliders have not revealed any detail of particle
picture of dark matter. Obviously, many dedicated theories often assume dark
matter to be composed of a single particle kind, such as a WIMP Jungman:1995df
; Bertone:2004pz , while there is no any reason why only a single particle
kind of dark matter is existed. Since the constituent of dark matter is still
eluded, a scheme of multi-component dark matter seems to be naturally in
comparison to the rich structure of normal matter within an atom. Indeed, the
possibility that dark matter consists of more than one type of particle is
very attractive Berezhiani:1989fp ; Berezhiani:1990sy ; Boehm:2003ha ;
Ma:2006uv ; Zurek:2008qg ; Fukuoka:2010kx ; Batell:2010bp ; Chialva:2012rq ;
Biswas:2013nn ; Bhattacharya:2013hva ; Bian:2013wna ; Belanger:2014bga ;
Bian:2014cja ; Esch:2014jpa ; DiFranzo:2016uzc ; DuttaBanik:2016jzv ;
Karam:2016rsz ; Bhattacharya:2016ysw ; Arcadi:2016kmk ; Borah:2017xgm ;
Ahmed:2017dbb ; Bhattacharya:2017fid ; Bhattacharya:2018cgx ;
Bhattacharya:2018cqu ; Aoki:2018gjf ; DuttaBanik:2018emv ; Barman:2018esi ;
YaserAyazi:2018lrv ; Chakraborti:2018aae ; Chakraborti:2018lso ; Borah:2019epq
; Elahi:2019jeo ; Yaguna:2019cvp , with intriguing results for galaxy
structure Fan:2013yva ; Fan:2013tia , multiple gamma-ray line and boosted dark
matter Agashe:2014yua ; Kong:2014mia ; Alhazmi:2016qcs ; Kim:2016zjx ;
Giudice:2017zke ; Chatterjee:2018mej ; Kim:2018veo , dark matter self-
interaction Elbert:2014bma ; Tulin:2017ara , as well as motivated with
neutrino mass generation Heeck:2012bz ; Aoki:2012ub ; Aoki:2013gzs ;
Kajiyama:2013rla ; Karam:2015jta ; Bernal:2018aon ; Bonilla:2018ynb ;
Borah:2019aeq ; Bhattacharya:2019fgs ; Biswas:2019ygr ; Bhattacharya:2019tqq .

We wish to suggest that the structure of dark matter and the origin of
neutrino mass can be simultaneously understood from a very fundamental aspect
of the standard model, the electrodynamics. As shown in VanDong:2020cjf ,
since the electric charge ($Q$) is theoretically not fixed, there may exist a
variant of it, interpreted to be a physical dark charge ($D$), which coexists
with the electric charge, driving the model extension with implication for
neutrino mass and dark matter. As an indication, the most $U(1)$ gauge
extensions that mix with the hypercharge ($Y$) actually imply such a dark
charge to be a dequantization version of the usual electric charge. Analogous
to the electric charge, the dark charge neither commutes nor closes
algebraically with the $SU(2)_{L}$ weak isospin, $T_{i}$ ($i=1,2,3$). By
algebraic closure condition, or symmetry (flipping) principle, this leads to
the full gauge symmetry, $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}\otimes
U(1)_{N}$, where the last two $U(1)$’s factors determine the electric charge
and the dark charge, i.e. $Y=Q-T_{3}$ and $N=D-T_{3}$, respectively, where the
QCD group has been included for completeness. Additionally, to cancel the
$U(1)_{N}$ anomalies, the right-handed neutrino singlets must be included,
which have a dark charge identical to that of the usual neutrinos, whereas
both kinds of the neutrinos have vanished electric charge VanLoi:2020kdk .
These neutrino singlets obtain large Majorana masses according to the
$N$-charge breaking, while they couple to the usual neutrinos via the Higgs
field. As a result, appropriate small neutrino masses are induced in terms of
a canonical seesaw mechanism. Especially, the seesaw scale that breaks the
$N$-charge, combined with the weak scale which breaks the weak isospin
$T_{i}$, defines a residual discrete symmetry of the dark charge $D=T_{3}+N$,
which manifestly stabilizes dark matter candidates.222Dark matter stability
mechanism is different from the matter parity as a residual gauge symmetry
Dong:2013wca ; Dong:2015yra ; Alves:2016fqe ; Dong:2016gxl ; Dong:2017zxo ;
Dong:2018aak ; Nam:2020twn ; VanDong:2020bkg ; VanLoi:2019eax ; VanLoi:2020xcq
, since the dark charge does not commute with the weak isospin, while the
matter parity induced by $B-L$ breaking does.

In the literature, to make dark matter candidates co-existed (i.e. co-
stabilized) responsible for multi-component dark matter, discrete groups that
are larger than a parity, e.g. a non-simple $Z_{2}\otimes Z^{\prime}_{2}$
group, have often been assumed by ad hoc. We show in this work that such
stability mechanism originates from the residual gauge symmetry of the dark
charge, as mentioned, and that it generically leads to novel scenarios of
multi-component dark matter. Indeed, depending on the value of the neutrino
dark charge, $\delta$, the residual gauge symmetry may be correspondingly
isomorphic to a discrete group $Z_{k}$, where $k$ is even integer. Such a
residual symmetry divides the standard model particles into various classes,
which may accommodate dark matter candidates to be simultaneously stabilized.
We analyze in detail three concrete models according to $k=2,4,6$. The model
with $k=2$ implies a single dark matter candidate, while the remaining two
cases, as well as models with larger $k$, provide multi-component dark matter.
We prove that the dark matter candidates can have an arbitrary mass, not
necessarily lighter than the charged leptons and quarks, although both the
dark matter and the normal matter transform nontrivially under the same
residual symmetry, a consequence of the electric and color charge
conservation. This is opposite to the previous announcement VanDong:2020cjf
as well as differing from the most extensions for dark matter. However, the
dark matter candidates in the model with $k=4$, except for the case of scalar
dark matter, must be lighter than the $W$ boson, due to the dark field self-
interactions that potentially lead to a dark matter instability. As multi-
component dark matter is commonly implied, we choose the models with $k=4,6$
for further experimental investigation. We figure out the viable parameter
regime satisfying the dark matter abundance, direct detections, particle
colliders, as well as astrophysical bounds.

The rest of this work is organized as follows: In Sec. II, we construct the
model for a generic dark charge. In Sec. III, we determine the residual
discrete symmetry, showing that it may be isomorphic to a $Z_{k}$ with even
$k$ value. Then, the simplest candidates of multi-component dark matter are
identified. In Sec. IV, we present the phenomenology of the model, including
neutrino mass, $Z^{\prime}$ constraint, $U(1)_{N}$ running coupling, and the
schemes with $k=4,6$ responsible for two-component dark matter are
experimentally investigated. Finally, we summarize the results in Sec. V.

##  II  A model of dark charge

In the nature, the electric charges of fundamental particles come in discrete
amounts, equal to integer multiples of an elementary electric charge, called
electric charge quantization. However, if the electric charge is theoretically
not quantized, i.e. not discrete, it is dequantized, i.e. arbitrary or
continuous. This item is eventually associated with the theories, such as the
electrodynamics and the standard model, that do not explain the quantization
of electric charge. The former theory obeys the charge quantization, if a
magnetic monopole is presented, proposed by Dirac long ago Dirac:1931kp , but
the monopole has not been observed yet. The latter theory implies the charge
quantization, if all hidden symmetries, e.g. $L_{a}-L_{b}$ for
$a,b=e,\mu,\tau$ and $B-L$, are violated, since otherwise they make the
hypercharge, thus the electric charge, free, such that $Y\rightarrow
Y+x_{ab}(L_{a}-L_{b})+y(B-L)$ is always allowed, called dequantization effect
Babu:1989ex ; Foot:1990uf .

Indeed, in the standard model with the gauge symmetry,

$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y},$ (1)

the electric charge, $Q$, is related to the hypercharge, $Y$, through the weak
isospin, $T_{i}$ ($i=1,2,3$), by

$Q=T_{3}+Y.$ (2)

The standard model does not predict the quantization of the electric charge
$Q$ because—while the value of $T_{3}$ is quantized due to the non-Abelian
nature of the Lie algebra $SU(2)_{L}$—the value of $Y$ is completely arbitrary
on the theoretical ground of $U(1)_{Y}$ group, since $[Y,Y]=0$ for any $Y$.
For phenomenological purpose, $Y$ is often chosen to describe the observed
electric charge values, while does not explain them. Moreover, in the
theoretical side, the value of $Y$ can be constrained by the anomaly
cancellation and the fermion mass generation for each family, but it is
dequantized when including the three families or right-handed neutrinos into
account Pisano:1996ht ; Doff:1998we ; deSousaPires:1998jc ;
deSousaPires:1999ca ; VanDong:2005ux . The reason is that the standard model
always contains anomaly-free hidden symmetries, such as $L_{a}-L_{b}$ for
$a,b=e,\mu,\tau$ or $B-L$ for including the right-handed neutrinos, that
subsequently make $Y$ free, as mentioned.

We consider the latter by introducing the three right-handed neutrinos,
$\nu_{aR}$, to the model and assume $Y(\nu_{aR})=\delta$. The anomaly
cancellation and the fermion mass generation require $Y(e_{aR})=\delta-1$,
$Y(l_{aL})=\delta-1/2$, $Y(u_{aR})=(2-\delta)/3$, $Y(d_{aR})=-(1+\delta)/3$,
and $Y(q_{aL})=(1-2\delta)/6$, where $l_{aL}\equiv(\nu_{aL}\ e_{aL})^{T}$ and
$q_{aL}\equiv(u_{aL}\ d_{aL})^{T}$ VanDong:2020cjf ; VanLoi:2020kdk .333We
redefine $a$ (and $b$) to be a family index, that labels both leptons and
quarks, without confusion. We call $\delta$ to be the parameter of
dequantization. Namely, for $\delta=0$, the hypercharge and the electric
charge are properly recovered as in the normal sense, so we relabel
$Y=Y|_{\delta=0}$ and $Q=Q|_{\delta=0}$ for convenience in reading. Whereas,
for $\delta\neq 0$, we obtain the new charges as corresponding variants of the
electric charge and the hypercharge, by which we define $D=Q|_{\delta\neq 0}$,
called dark charge, and $N=Y|_{\delta\neq 0}$, called hyperdark charge. One
has a similar relation to Eq. (2),

$D=T_{3}+N.$ (3)

As the electric charge, the dark charge $D$ neither commutes nor closes
algebraically with the weak isospin, which indicates to a novel gauge
extension, such as

$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)_{N},$ (4)

where $N$ thus determines $D$ in the same way that $Y$ does so for $Q$,
respectively.

It is easily realized that $N$ was studied in the literature as a combination
of $xY+y(B-L)$ for $x=1$ and $y=-\delta$, since both $Y,B-L$ are anomaly-free.
However, the interpretation of the dark charge $D$ as well as this combination
as a dequantization of the electric charge did not appear until our proposal
VanDong:2020cjf ; VanLoi:2020kdk . An important result is that $D$ implies
dark matter stability; additionally, the dark matter can be unified with
normal matter in $SU(2)_{L}$ multiplet, because the dark charge $D$ is
noncommutative, which differs from the most extensions. Let us stress that
dark matter must be stabilized over the cosmological timescale. For instance,
if a dark matter particle with a mass between the weak and TeV scales decays
to two lighter particles, the relevant coupling strength that couples these
fields must be around $\sim 10^{-20}$. Indeed, the current bound of dark
matter lifetime is at least a billion times longer than our universe’s age,
leading to a corresponding coupling strength $\sim 10^{-25}$ Fermi-LAT:2012ugx
. Such tiny coupling, much smaller than the Dirac neutrino coupling $\sim
10^{-12}$, is not naturally stabilized against quantum contributions, which
might only be understood by symmetries.

The full fermion content, supplied with the hyperdark charge, under the gauge
symmetry in Eq. (4) is

$\displaystyle l_{aL}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\nu_{aL}\\\
e_{aL}\end{array}\right)\sim\left(1,2,-\frac{1}{2},\delta-\frac{1}{2}\right),$
(7) $\displaystyle q_{aL}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}u_{aL}\\\
d_{aL}\end{array}\right)\sim\left(3,2,\frac{1}{6},\frac{1}{6}-\frac{\delta}{3}\right),$
(10) $\displaystyle\nu_{aR}$ $\displaystyle\sim$
$\displaystyle(1,1,0,\delta),$ (11) $\displaystyle e_{aR}$ $\displaystyle\sim$
$\displaystyle(1,1,-1,\delta-1),$ (12) $\displaystyle u_{aR}$
$\displaystyle\sim$ $\displaystyle(3,1,2/3,2/3-\delta/3),$ (13) $\displaystyle
d_{aR}$ $\displaystyle\sim$ $\displaystyle(3,1,-1/3,-\delta/3-1/3),$ (14)

where $a=1,2,3$ is a family index and $\delta$ is arbitrarily nonzero,
$\delta\neq 0$. The left and right fermion $f_{L,R}$ have the same dark
charge, which coincides with the hyperdark charge value of the right component
above, since $D(f)=D(f_{R})=N(f_{R})$ due to $T_{3}(f_{R})=0$.

It is easily verified that all the anomalies are cancelled out for each
family, independent of $\delta$, as shown in Appendix of VanLoi:2020kdk .
There, we also indicated that the anomalies are cancelled out for different
dequantizations, such as $U(1)_{N}\rightarrow U(1)_{N_{1}}\otimes
U(1)_{N_{2}}\otimes\cdots\otimes U(1)_{N_{p}}$ according to
$\delta\rightarrow\delta_{1},\delta_{2},\cdots,\delta_{p}$, respectively.
Multi-component dark matter is naturally recognized, since each $U(1)_{N}$’s
factor presents an independent stability mechanism (see below). However,
within one factor, the $U(1)_{N}$ breaking also supplies the schemes of multi-
component dark matter. That said, the multi-component dark matter is a generic
result of the electric charge dequantization. Hereafter, only the latter case
is taken into account.

To break the gauge symmetry and generate appropriate masses, the scalar
content is introduced as

$\phi=\left(\begin{array}[]{c}\phi^{+}\\\
\phi^{0}\end{array}\right)\sim\left(1,2,\frac{1}{2},\frac{1}{2}\right),\hskip
14.22636pt\chi\sim(1,1,0,-2\delta).$ (15)

Here, the new field $\chi$ is necessarily to break $U(1)_{N}$, producing
Majorana $\nu_{R}$ mass via a coupling to $\nu_{R}\nu_{R}$. Additionally,
$\phi$ is the usual Higgs doublet, which has the dark charge identical to the
electric charge, such that both of these charges are not broken by the weak
vacuum. The vacuum expectation values (VEVs) are given by

$\langle\phi\rangle=\left(\begin{array}[]{c}0\\\
\frac{v}{\sqrt{2}}\end{array}\right),\hskip
14.22636pt\langle\chi\rangle=\frac{\Lambda}{\sqrt{2}},$ (16)

satisfying

$\Lambda\gg v=246\text{ GeV},$ (17)

for consistency with the standard model.

##  III  Residual dark charge

The gauge symmetry is broken via two stages,

$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)_{N}$
---
$\downarrow\Lambda$
$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}\otimes R_{N}$
$\downarrow v$
$SU(3)_{C}\otimes U(1)_{Q}\otimes R_{D}.$

In the first stage, only $U(1)_{N}$ is broken by the VEV of $\chi$, because of
$N\langle\chi\rangle=-\sqrt{2}\delta\Lambda\neq 0$. But, this breaking is not
completed. Indeed, a residual symmetry of $N$ is defined as $R_{N}=e^{i\alpha
N}$, since it is a $U(1)_{N}$ transformation. $R_{N}$ conserves the vacuum,
$R_{N}\langle\chi\rangle=\langle\chi\rangle$, leading to
$e^{-i2\alpha\delta}=1$, or $\alpha=k\pi/\delta$ for $k$ integer. We deduce

$R_{N}=e^{ik\pi N/\delta}=(-1)^{kN/\delta}.$ (18)

In the second stage, the VEV of $\phi$ breaks all $SU(2)_{L}\otimes
U(1)_{Y}\otimes R_{N}$, except for the color group, because of
$T_{i}\langle\phi\rangle=\frac{1}{2}\sigma_{i}\langle\phi\rangle\neq 0$ and
$Y\langle\phi\rangle=N\langle\phi\rangle=(0\ v/2\sqrt{2})^{T}\neq 0$. However,
the breaking is also not completed. Indeed, a conserved charge after the
breaking must take the form,

$X\equiv a_{i}T_{i}+bY+cN,$ (19)

that annihilates the weak vacuum, $X\langle\phi\rangle=0$. This yields
$a_{1}=a_{2}=0$ and $a_{3}=b+c$, hence

$\displaystyle X$ $\displaystyle=$ $\displaystyle b(T_{3}+Y)+c(T_{3}+N)$ (20)
$\displaystyle=$ $\displaystyle bQ+cD,$

where $Q=T_{3}+Y$ and $D=T_{3}+N$ are defined as before. Additionally, $Q,D$
commute, $[Q,D]=0$, and separately conserve the weak vacuum (similar to $X$),
i.e. $Q\langle\phi\rangle=D\langle\phi\rangle=0$. Hence, the residual symmetry
$X$ is Abelian by itself and factorized into,

$U(1)_{X}=U(1)_{Q}\otimes U(1)_{D},$ (21)

according to a transformation, $e^{iX}=e^{ibQ}e^{icD}$, multiplied by those of
$Q,D$, respectively.

The first factor $U(1)_{Q}$ is well known to be the electromagnetic symmetry,
which is the residual symmetry of $SU(2)_{L}\otimes U(1)_{Y}$, obviously
conserving the $\chi$ vacuum, since $Q\langle\chi\rangle=0$. The second factor
$U(1)_{D}$ is a residual symmetry of $SU(2)_{L}\otimes R_{N}$, which similar
to $R_{N}$, it must conserve the $\chi$ vacuum. In other words,
$e^{icD}\langle\chi\rangle=\langle\chi\rangle$ leads to $e^{-i2c\delta}=1$,
implying $c=\alpha=k\pi/\delta$. This restricts $U(1)_{D}$ to $R_{D}$ defined
by

$R_{D}=e^{ik\pi D/\delta}=(-1)^{kD/\delta},$ (22)

to be a residual symmetry of the dark charge $D=T_{3}+N$, thus of
$SU(2)_{L}\otimes U(1)_{N}$.

It is clear that $R_{D}$ is shifted from $R_{N}$ by the weak vacuum, not
commuted with the standard model symmetry, a consequence of existence of
noncommutative dark charge. $R_{D}$ would distinguish particles with different
weak isospin values and give a potential unification of normal matter and dark
matter in the same gauge multiplet VanDong:2020cjf . This feature compares
with supersymmetry, which does so for both kinds of matter (with different
spin values) in supermultiplet. $R_{D}$ differs from the most $U(1)$
extensions in the literature, which actually lead to commutative discrete
symmetries, like $R_{N}$, by contrast.

For comparison we collect the electric and dark charges as well as the $R_{D}$
values of all fields in Table 1, where $A$ commonly denotes all the gauge
fields corresponding to the gauge symmetry in Eq. (4), except for the $W$
boson. Additionally, the generation and left/right chirality indices have been
omitted since the relevant fields have the same $Q$, $D$, and $R_{D}$ values,
respectively.

Table 1: $Q$, $D$, and $R_{D}$ values of fields. Field | $Q$ | $D$ | $R_{D}$
---|---|---|---
$\nu$ | $0$ | $\delta$ | $(-1)^{k}$
$e$ | $-1$ | $\delta-1$ | $(-1)^{k(\delta-1)/\delta}$
$u$ | $2/3$ | $(2-\delta)/3$ | $(-1)^{k(2-\delta)/3\delta}$
$d$ | $-1/3$ | $-(1+\delta)/3$ | $(-1)^{-k(\delta+1)/3\delta}$
$\chi$ | $0$ | $-2\delta$ | $1$
$\phi^{+},W^{+}$ | $1$ | $1$ | $(-1)^{k/\delta}$
$\phi^{0},A$ | $0$ | $0$ | $1$

From Table 1, it is clear that if $k=0$, then $R_{D}=1$ for all fields and
every $\delta$, which is the identity transformation. To search for the
residual group structure, we find the minimal value of $|k|\neq 0$ that still
satisfies $R_{D}=1$ for all fields. Such value of $k$ depends on $\delta$ and
obeys the following equations simultaneously,

$\displaystyle k$ $\displaystyle=$ $\displaystyle 2n_{1},$ (23) $\displaystyle
k(\delta-1)/\delta$ $\displaystyle=$ $\displaystyle 2n_{2},$ (24)
$\displaystyle k(\delta-2)/3\delta$ $\displaystyle=$ $\displaystyle 2n_{3},$
(25) $\displaystyle k(\delta+1)/3\delta$ $\displaystyle=$ $\displaystyle
2n_{4},$ (26) $\displaystyle k/\delta$ $\displaystyle=$ $\displaystyle
2n_{5},$ (27)

where $n_{1,2,3,4,5}=0,\pm 1,\pm 2,\cdots$ are integer. From the first
equation, $k$ must be even integer, i.e. $k=\pm 2,\pm 4,\pm 6,\cdots$.

First, considering the case of $|k|=2$, we get $n_{1}=\pm 1$ and subsequent
relations

$n_{2}=2-3n_{4},\hskip 14.22636ptn_{3}=1-2n_{4},\hskip
14.22636ptn_{5}=3n_{4}-1.$ (28)

We also obtain a corresponding value of $\delta$, denoted by

$\delta_{2}=\frac{1}{3n_{4}-1}=-1,\frac{1}{2},-\frac{1}{4},\frac{1}{5},-\frac{1}{7},\frac{1}{8},-\frac{1}{10},\frac{1}{11},\cdots.$
(29)

Similarly, considering the next cases with $|k|=4$ and 6, we obtain the
corresponding values of $\delta$, denoted by

$\displaystyle\delta_{4}$ $\displaystyle=$
$\displaystyle\frac{2}{3n_{4}-2}=-1,2,-\frac{2}{5},\frac{1}{2},-\frac{1}{4},\frac{2}{7},\frac{1}{5},-\frac{2}{11},\cdots,$
(30) $\displaystyle\delta_{6}$ $\displaystyle=$
$\displaystyle\frac{1}{n_{4}-1}=\pm
1,\pm\frac{1}{2},\pm\frac{1}{3},\pm\frac{1}{4},\pm\frac{1}{5},\pm\frac{1}{6},\pm\frac{1}{7},\cdots,$
(31)

respectively. Notice that $\delta_{4}$ contains in part the values of
$\delta_{2}$ which correspondingly reduce to the first case and must be
omitted. Moreover, $\delta_{6}$ also includes values of $\delta_{2}$ and a
part of $\delta_{4}$ which reduce to the first or the second case and must be
eliminated too.

In other words, there are different values of $\delta$ resulting in $R_{D}=1$
for all fields according to the minimal solution of $|k|=2$, $4$, or $6$,
except the identity $k=0$. This implies that the residual symmetry $R_{D}$ is
automorphic to a discrete group, $Z_{2}$, $Z_{4}$, or $Z_{6}$, respectively,
providing dark matter candidates. To be concrete, in Table 2, we show three
specific cases, giving the results as expected. Notice that unlike the minimal
value of $k$ according to $R_{D}=1$ for all fields, we have restored $k$,
without confusion, to be arbitrarily integer characterizing transformations of
(22) within each residual group $R_{D}$.

Table 2: $R_{D}$ values of all fields when $\delta$ is fixed, where $k$ is arbitrarily integer. Field | $R_{D}\rightarrow Z_{2}$ | $R_{D}\rightarrow Z_{4}$ | $R_{D}\rightarrow Z_{6}$
---|---|---|---
| $(\delta=-1)$ | $(\delta=2)$ | $(\delta=1)$
$\nu$ | $(-1)^{k}$ | $(-1)^{k}$ | $(-1)^{k}$
$e$ | $1$ | $(-1)^{k/2}$ | $1$
$u$ | $(-1)^{-k}$ | $1$ | $(-1)^{k/3}$
$d$ | $1$ | $(-1)^{-k/2}$ | $(-1)^{-2k/3}$
$\chi$ | $1$ | $1$ | $1$
$\phi^{+},W^{+}$ | $(-1)^{-k}$ | $(-1)^{k/2}$ | $(-1)^{k}$
$\phi^{0},A$ | $1$ | $1$ | $1$

Of course, there are values of $\delta$ that lead to the residual symmetries
higher than the given ones, such as $Z_{8}$, $Z_{10}$, $Z_{12}$, and so forth.
They would imply more the stabilized dark matter components than the chosen
ones. Such generalization to a higher discrete symmetry is not trivial, but
not discussed further in this work.

###  III.1 The model with $\delta=-1$

The second column of Table 2 indicates $R_{D}=1$ for all fields and for the
minimal value of $|k|=2$, except $k=0$. Hence, the residual symmetry $R_{D}$
is automorphic to

$Z_{2}=\\{1,g\\},$ (32)

where $g\equiv e^{-i\pi D}=(-1)^{-D}$ and $g^{2}=1$.

Because the spin parity, $P_{S}=(-1)^{2s}$, is always conserved by the Lorentz
symmetry, we conveniently multiply the residual symmetry with the spin parity
group $S=\\{1,P_{S}\\}$ with $P_{S}^{2}=1$, to perform a new group
$Z_{2}\otimes S$, which has an element

$g_{S}=g\times P_{S}=(-1)^{-D+2s},$ (33)

satisfying $g_{S}^{2}=1$. Therefore, we have

$\mathcal{Z}_{2}=\\{1,g_{S}\\}$ (34)

to be an invariant subgroup of $Z_{2}\otimes S$ and decompose

$Z_{2}\otimes S\cong[(Z_{2}\otimes S)/\mathcal{Z}_{2}]\otimes\mathcal{Z}_{2}.$
(35)

Since $[(Z_{2}\otimes S)/\mathcal{Z}_{2}]=\\{\\{1,g_{S}\\},\\{g,P_{S}\\}\\}$
is conserved if $g_{S}$ is conserved, we can omit it, interpreting the
$\mathcal{Z}_{2}$ group as a residual symmetry instead of $Z_{2}$.

The field representations under $\mathcal{Z}_{2}$ are summarized in Table 3.
Notice that $\mathcal{Z}_{2}$ has two one-dimensional irreducible
representations, $\underline{1}^{(1)}$ and $\underline{1}^{(2)}$,
corresponding to $g_{S}=1$ and $-1$, respectively, as usual.

Table 3: Field representations under the $\mathcal{Z}_{2}$ group. Field | $g_{S}$ | $\mathcal{Z}_{2}$
---|---|---
$\nu$ | $1$ | $\underline{1}^{(1)}$
$e$ | $-1$ | $\underline{1}^{(2)}$
$u$ | $1$ | $\underline{1}^{(1)}$
$d$ | $-1$ | $\underline{1}^{(2)}$
$\chi$ | $1$ | $\underline{1}^{(1)}$
$\phi^{+},W^{+}$ | $-1$ | $\underline{1}^{(2)}$
$\phi^{0},A$ | $1$ | $\underline{1}^{(1)}$

From Table 3, the model with $\delta=-1$ implies scenarios of a single dark
field, labeled $\Psi_{1}$, which transforms nontrivially under
$\mathcal{Z}_{2}$, i.e. $g_{S}=-1$, such as

$\Psi_{1}\sim(1,1,0,2d_{1})$ (36)

for a fermion or

$\Psi_{1}\sim(1,1,0,2d_{1}+1)$ (37)

for a scalar, where $d_{1}$ is integer. For simplicity, throughout this work,
we assume dark matter candidates to be spin-0 bosonic and/or spin-$1/2$
fermionic fields, transforming as the standard model singlet.444Introduction
of $SU(2)_{L}$ multiplets leads to a unification of normal field and dark
field by the gauge symmetry due to the noncommutative $R_{D}$, as mentioned.
This possibility is interesting, but is out of the scope of this work and
skipped. This result is similar to the previous interpretation obtained in
VanDong:2020cjf ; VanLoi:2020kdk . Therefore, we obtain the simplest dark
matter candidate to be either a fermion or a scalar with $d_{1}=0$.

We would like to stress that $\Psi_{1}$ can have an arbitrary mass, which does
not decay to the usual particles. First, all $\Psi_{1}$, $e$, $d$, $\phi^{+}$,
and $W^{+}$ are $\mathcal{Z}_{2}$ odd. Next, $\Psi_{1}$ is color and
electrically neutral, whereas the rest carry a color and/or an electric
charge. The color and electric charge conservation demands that $\Psi_{1}$
cannot decay to such a colored and/or electrically charged state. Indeed, by
contrast, suppose that $\Psi$ decays to a final state. This state must be
color and electrically neutral, as $\Psi_{1}$ is. Additionally, the
$\mathcal{Z}_{2}$ conservation requires that the final state must be combined
of an odd number of the odd fields $(e,d,\phi^{+},W^{+})$. Since $\phi^{+}$ is
eaten by $W^{+}$ and that $W^{+}$ decays to $(e^{+},\nu)$ or $(d^{c},u)$, the
final state includes only $(e,d)$ as possible old fields. Thus, the decay
process takes the form,

$\Psi_{1}\rightarrow
xe^{-}+\bar{x}e^{+}+yd+\bar{y}d^{c}+zu+\bar{z}u^{c}+\cdots,$ (38)

where the dots stand for the remaining fields (if any) that are electrically
and color neutral and $\mathcal{Z}_{2}$ even. The conservation laws imply, i)
$x+\bar{x}+y+\bar{y}=2n+1$ ($\mathcal{Z}_{2}$ odd), ii)
$-x+\bar{x}-y/3+\bar{y}/3+2z/3-2\bar{z}/3=0$ (electrically neutral), and iii)
$y+z-\bar{y}-\bar{z}=3n^{\prime}$ (color neutral), for $n,n^{\prime}$ integer.
The last two equations lead to $-x+\bar{x}-y+\bar{y}+2n^{\prime}=0$, which
combined with the first equation, yields a contradiction to be
$2(\bar{x}+\bar{y}+n^{\prime})=2n+1$. In other words, the symmetries
$SU(3)_{C}$, $U(1)_{Q}$, and $\mathcal{Z}_{2}$ jointly suppress the decay of
the $\Psi_{1}$ dark matter, if it has a mass larger than the ordinary odd
particles ($e,d,\phi^{+}$, $W^{+}$). Hence, the $\Psi_{1}$ dark matter can
have an arbitrary mass, and its stability is different from the most
extensions.

###  III.2 The model with $\delta=2$

From the third column of Table 2, it is clear that $R_{D}=1$ for all fields
with minimal value of $|k|=4$, except $k=0$. Thus, the residual symmetry
$R_{D}$ is automorphic to

$Z_{4}=\\{1,t,t^{2},t^{3}\\},$ (39)

where $t\equiv e^{i\pi D/2}=(-1)^{D/2}$ and $t^{4}=1$.

For convenience, we multiply the residual symmetry with the spin-parity group
$S=\\{1,P_{S}\\}$ to perform $Z_{4}\otimes S$, which has an invariant
subgroup,

$\mathcal{Z}_{4}=\\{1,t_{S},t_{S}^{2},t_{S}^{3}\\},$ (40)

where $t_{S}=t\times P_{S}=e^{i\pi(D/2+2s)}=(-1)^{D/2+2s}$ and $t_{S}^{4}=1$.
Therefore, we decompose

$Z_{4}\otimes S\cong[(Z_{4}\otimes S)/\mathcal{Z}_{4}]\otimes\mathcal{Z}_{4}.$
(41)

Since $[(Z_{4}\otimes
S)/\mathcal{Z}_{4}]=\\{\\{1,t_{S},t_{S}^{2},t_{S}^{3}\\},\\{t,P_{S},t^{2}\times
P_{S},t^{3}\\}\\}$ is conserved if $\mathcal{Z}_{4}$ is conserved, we can
consider $\mathcal{Z}_{4}$ to be a residual symmetry instead of $Z_{4}$ (note
that $P_{S}$ is always preserved). The field representations under
$\mathcal{Z}_{4}$ are given in Table 4 (third column). Here, $\mathcal{Z}_{4}$
has four one-dimensional irreducible representations, $\underline{1}^{(1)}$
according to $t_{S}=1$, $\underline{1}^{(2)}$ according to $t_{S}=i$,
$\underline{1}^{(3)}$ according to $t_{S}=-1$, and $\underline{1}^{(4)}$
according to $t_{S}=-i$, with $i$ to be imaginary unit.

Table 4: Field representations under $\mathcal{Z}_{4}$ and their transformations under $Z_{2}\otimes Z^{\prime}_{2}$. Field | $t_{S}$ | $\mathcal{Z}_{4}$ | $t^{2}_{S}$ | $[t_{S}]$ | $Z_{2}\otimes Z^{\prime}_{2}$
---|---|---|---|---|---
$\nu$ | $1$ | $\underline{1}^{(1)}$ | 1 | $\\{1,1\\}$ | $\underline{1}^{(1)}\otimes\underline{1}^{(1)}$
$e$ | $-i$ | $\underline{1}^{(4)}$ | $-1$ | $\\{-i,i\\}$ | $\underline{1}^{(2)}\otimes\underline{1}^{(2)}$
$u$ | $-1$ | $\underline{1}^{(3)}$ | 1 | $\\{-1,-1\\}$ | $\underline{1}^{(1)}\otimes\underline{1}^{(1)}$
$d$ | $i$ | $\underline{1}^{(2)}$ | $-1$ | $\\{i,-i\\}$ | $\underline{1}^{(2)}\otimes\underline{1}^{(2)}$
$\chi$ | $1$ | $\underline{1}^{(1)}$ | 1 | $\\{1,1\\}$ | $\underline{1}^{(1)}\otimes\underline{1}^{(1)}$
$\phi^{+},W^{+}$ | $i$ | $\underline{1}^{(2)}$ | $-1$ | $\\{i,-i\\}$ | $\underline{1}^{(2)}\otimes\underline{1}^{(2)}$
$\phi^{0},A$ | $1$ | $\underline{1}^{(1)}$ | 1 | $\\{1,1\\}$ | $\underline{1}^{(1)}\otimes\underline{1}^{(1)}$

It is noted that $\mathcal{Z}_{4}$ contains an invariant subgroup
$Z_{2}=\\{1,t^{2}_{S}\\}$ with the corresponding quotient group
$Z^{\prime}_{2}=\mathcal{Z}_{4}/Z_{2}=\\{\\{1,t^{2}_{S}\\},\\{t_{S},t^{3}_{S}\\}\\}\equiv\\{[1],[t_{S}]\\}$,
where each coset element $[x]$ consists of two elements of $\mathcal{Z}_{4}$,
the characteristic $x$ and other $t^{2}_{S}x$.555Hereafter, since the models
with $\delta=-1,2,1$ are distinct, the notations used for discrete groups and
their presentations may be identical, without confusion. However,
$\mathcal{Z}_{4}$ is not isomorphic to $Z_{2}\otimes Z^{\prime}_{2}$, and one
can verify that the fields transform the same under $Z_{2}$ and
$Z^{\prime}_{2}$, identical to the previous model with $\delta=-1$, because
$t^{2}_{S}$ is identical to $g_{S}$ while $[t_{S}]$ is always isomorphic to
$t^{2}_{S}$. They are also included in Table 4 as the last three columns, for
clarity.666Hereafter, although representations denoted by corresponding
dimensions as underlined may appear the same across different groups, they
must be distinguished and should not be confused, since they are always
supplied under their own group. That said, the $\mathcal{Z}_{4}$ basis is
necessarily to supply multi-component dark matter, while the $Z_{2}\otimes
Z^{\prime}_{2}$ group is basically reducible to the single dark matter. Extra
comment is that if one allows two factors $U(1)_{N_{1}}\otimes U(1)_{N_{2}}$
according to $\delta\rightarrow\delta_{1,2}$, as mentioned, they yield a
residual Klein group $Z^{(1)}_{2}\otimes Z^{(2)}_{2}$ with the factors
determined by independent dark charges $D_{1,2}$, respectively. This supplies
a scheme of two-component dark matter which transform nontrivially under the
$Z_{2}$’s factors, respectively, similar to the present $Z_{4}$ with only
$\delta$.

Table 5: Three kinds of dark field implied by $\mathcal{Z}_{4}$. Field | $\mathcal{Z}_{4}$ | Fermion | Scalar
---|---|---|---
$\Psi_{2}$ | $\underline{1}^{(2)}$ | $(1,1,0,4d_{2}-1)$ | $(1,1,0,4d_{2}+1)$
$\Psi_{3}$ | $\underline{1}^{(3)}$ | $(1,1,0,4d_{3})$ | $(1,1,0,4d_{3}+2)$
$\Psi_{4}$ | $\underline{1}^{(4)}$ | $(1,1,0,4d_{4}+1)$ | $(1,1,0,4d_{4}+3)$

From Table 4, the model with $\delta=2$ implies three kinds of dark field, as
determined in Table 5, where $d_{2,3,4}$ are integer. It is able to show that
$\mathcal{Z}_{4}$ is the smallest (i.e. $Z_{k}$) cyclic symmetry that allows a
scenario of two-component dark matter Yaguna:2019cvp ; Belanger:2014bga ;
Batell:2010bp . Among the solutions, we consider here the two scalar dark
fields to be the simplest candidates of dark matter, as resupplied in Table 6.

Table 6: The simplest dark matter candidates implied by $\mathcal{Z}_{4}$. Field | Spin | $t_{S}$ | $\mathcal{Z}_{4}$
---|---|---|---
$\Phi_{2}\sim(1,1,0,1)$ | $0$ | $i$ | $\underline{1}^{(2)}$
$\Phi_{3}\sim(1,1,0,2)$ | $0$ | $-1$ | $\underline{1}^{(3)}$

In this case of the model, the usual fields $e$, $u$, $d$, $\phi^{+}$, and
$W^{+}$ transform nontrivially under $\mathcal{Z}_{4}$, analogous to the dark
fields. Since $\phi^{+}$ is eaten by $W^{+}$, while $W^{+}$ decays to $\nu
e^{+}$ or $ud^{c}$, a potential decay of dark matter need not consider these
bosons in the product, similar to the previous model. So if dark matter
decays, the final state only includes $(e,u,d)$ plus others trivial under
$\mathcal{Z}_{4}$ and neutral under the electric and color charge. On the
other hand, dark matter that transforms as $\underline{1}^{(2)}$ or
$\underline{1}^{(4)}$ cannot decay to usual particles, since it is odd under
$t^{2}_{S}=g_{S}$, thus suppressed by the $t^{2}_{S}$ conservation similar to
the previous model. Dark matter that transforms as $\underline{1}^{(3)}$
cannot decay to a state that contains a single $e,u,d$ field, because of the
electric and color charge conservation. Also it cannot decay to a state that
contains any two of $e,u,d$, since the charge conservation restricts such
state to $e^{+}e^{-}$, $u^{c}u$, and $d^{c}d$, but this is not allowed by
$\mathcal{Z}_{4}$. Moreover, a final state that includes three of them is only
$eud^{c}$ or $udd$, but not allowed by $\mathcal{Z}_{4}$. And, a final state
that has four of them is only $euud$, $e^{+}ddd$, or others as combined of two
states of two $(e,u,d)$ fields above, but discarded by $\mathcal{Z}_{4}$.
Finally, a final state that contains more than four of them must be $eeuuu$ or
combine the above states due to the charge conservation, which is also
suppressed by $\mathcal{Z}_{4}$. Hence, $SU(3)_{C}$, $U(1)_{Q}$, and
$\mathcal{Z}_{4}$ do not permit any dark matter decaying to ordinary
particles. Notice that the dark fields may have seft-interactions, such as in
the case of the simplest dark fields above, which include a Lagrangian term,
$\Phi_{3}^{\dagger}\Phi_{2}^{2}$. Two dark fields are simultaneously
stabilized responsible for two-component dark matter if the mass of $\Phi_{3}$
is smaller than twice that of $\Phi_{2}$.

###  III.3 The model with $\delta=1$

From the last column of Table 2, we derive that $R_{D}=1$ for all fields with
the minimal value of $|k|=6$, except for $k=0$. Hence, the residual symmetry
is automorphic to

$Z_{6}=\\{1,p,p^{2},p^{3},p^{4},p^{5}\\},$ (42)

where $p\equiv e^{i\pi D}=(-1)^{D}$ and $p^{6}=1$. Following VanDong:2020cjf ;
VanDong:2020bkg , we factorize $Z_{6}$ into

$Z_{6}\cong Z_{2}\otimes Z_{3},$ (43)

where $Z_{2}$ is the invariant subgroup of $Z_{6}$, while $Z_{3}$ is the
quotient group of $Z_{6}$ by $Z_{2}$, given by

$\displaystyle Z_{2}$ $\displaystyle=$ $\displaystyle\\{1,p^{3}\\},$ (44)
$\displaystyle Z_{3}$ $\displaystyle=$
$\displaystyle\\{[1],[p^{2}],[p^{4}]\\}.$ (45)

Here each (coset) element of $Z_{3}$, denoted by $[y]$, contains two elements
of $Z_{6}$, the characteristic $y$ and the other $p^{3}y$ as multiplied by
$p^{3}$, thus $[1]=[p^{3}]=\\{1,p^{3}\\}$,
$[p^{2}]=[p^{5}]=\\{p^{2},p^{5}\\}$, and $[p^{4}]=[p]=\\{p,p^{4}\\}$ are well
understood.

Since $[p^{4}]=[p^{2}]^{2}=[p^{2}]^{*}$ and $[p^{2}]^{3}=[1]$, $Z_{3}$ is
completely generated by the element,

$[p^{2}]=[e^{i2\pi D}]=[\omega^{3D}],$ (46)

where $\omega\equiv e^{i2\pi/3}$ defines the cube root of unity. Whereas,
$Z_{2}$ is generated by the element,

$p^{3}=e^{i3\pi D}=(-1)^{3D}.$ (47)

It is noted that $3D$ is always integer, since $p^{6}=1$.

The theory conserves both the residual symmetries $Z_{2}$ and $Z_{3}$ after
symmetry breaking. Notice that $Z_{2}$ has two one-dimensional irreducible
representations, $\underline{1}^{(1)}$ according to $p^{3}=1$ and
$\underline{1}^{(2)}$ according to $p^{3}=-1$, whereas $Z_{3}$ has three one-
dimensional irreducible representations, $\underline{1}^{(1)}$ according to
$[p^{2}]=[1]=\\{1,1\\}$ or $\\{1,-1\\}\rightarrow 1$, $\underline{1}^{(2)}$
according to $[p^{2}]=[\omega]=\\{\omega,\omega\\}$ or
$\\{\omega,-\omega\\}\rightarrow\omega$, and $\underline{1}^{(3)}$ according
to $[p^{2}]=[\omega^{2}]=\\{\omega^{2},\omega^{2}\\}$ or
$\\{\omega^{2},-\omega^{2}\\}\rightarrow\omega^{2}$. Here, the presentations
of $Z_{3}$ are independent of the values $p^{3}=\pm 1$ that identify those of
$Z_{6}$ in a coset, hence the representations of $\mathcal{Z}_{3}$ are
determined by a homomorphism of $Z_{6}$ representations to those of $Z_{3}$,
such as $[p^{2}]=[r]=\\{r,\pm r\\}\rightarrow r$ corresponding to
$\underline{1}^{(1)}$, $\underline{1}^{(2)}$, and $\underline{1}^{(3)}$ for
$r=1$, $\omega$, and $\omega^{2}$, respectively.

The representations of all fields under the $Z_{2,3}$ groups are computed,
given in Table 7. It is clear that the $\underline{1}^{(3)}$ representation of
$Z_{3}$ is not presented for the existing fields. But, let us note that the
antiquarks transform as $\underline{1}^{(3)}$, since
$\underline{1}^{(2)*}=\underline{1}^{(3)}$ and vice versa.

Table 7: Field representations under $Z_{6}\cong Z_{2}\otimes Z_{3}$. Field | $p^{3}$ | $[p^{2}]=\\{p^{2},p^{5}\\}$ | $Z_{2}\otimes Z_{3}$
---|---|---|---
$\nu$ | $-1$ | $\\{1,-1\\}\rightarrow 1$ | $\underline{1}^{(2)}\otimes\underline{1}^{(1)}$
$e$ | $1$ | $\\{1,1\\}\rightarrow 1$ | $\underline{1}^{(1)}\otimes\underline{1}^{(1)}$
$u$ | $-1$ | $\\{\omega,-\omega\\}\rightarrow\omega$ | $\underline{1}^{(2)}\otimes\underline{1}^{(2)}$
$d$ | $1$ | $\\{\omega,\omega\\}\rightarrow\omega$ | $\underline{1}^{(1)}\otimes\underline{1}^{(2)}$
$\chi$ | $1$ | $\\{1,1\\}\rightarrow 1$ | $\underline{1}^{(1)}\otimes\underline{1}^{(1)}$
$\phi^{+},W^{+}$ | $-1$ | $\\{1,-1\\}\rightarrow 1$ | $\underline{1}^{(2)}\otimes\underline{1}^{(1)}$
$\phi^{0},A$ | $1$ | $\\{1,1\\}\rightarrow 1$ | $\underline{1}^{(1)}\otimes\underline{1}^{(1)}$

Furthermore, because the spin parity is always conserved, we conveniently
multiply the residual symmetry with the spin parity group, to form

$Z_{6}\otimes S\cong(Z_{2}\otimes S)\otimes Z_{3}.$ (48)

Here $Z_{2}\otimes S$ contains an invariant subgroup,

$P=\\{1,P_{M}\\},$ (49)

with

$P_{M}=p^{3}\times P_{S}=(-1)^{3D+2s}.$ (50)

Hence, we have

$Z_{6}\otimes S\cong[(Z_{2}\otimes S)/P]\otimes P\otimes Z_{3}.$ (51)

Since $(Z_{2}\otimes S)/P=\\{P,\\{p^{3},P_{S}\\}\\}$ is conserved if $P_{M}$
(or $P$) is conserved, we consider the remaining factor,

$Z_{6}\otimes S\supset P\otimes Z_{3}$ (52)

to be the relevant residual symmetry instead of $Z_{6}$. The quotient group,
$Z_{3}$, and its representations are remained, while $P$ has two one-
dimensional irreducible representations, $\underline{1}^{(1)}$ according to
$P_{M}=1$ and $\underline{1}^{(2)}$ according to $P_{M}=-1$. Thus,
representations of all fields under $P$ and $Z_{3}$ are (re)supplied in Table
8.

Table 8: Field representations under $P\otimes Z_{3}$. Field | $P_{M}$ | $[p^{2}]$ | $P\otimes Z_{3}$
---|---|---|---
$\nu$ | $1$ | $1$ | $\underline{1}^{(1)}\otimes\underline{1}^{(1)}$
$e$ | $-1$ | $1$ | $\underline{1}^{(2)}\otimes\underline{1}^{(1)}$
$u$ | $1$ | $\omega$ | $\underline{1}^{(1)}\otimes\underline{1}^{(2)}$
$d$ | $-1$ | $\omega$ | $\underline{1}^{(2)}\otimes\underline{1}^{(2)}$
$\chi$ | $1$ | $1$ | $\underline{1}^{(1)}\otimes\underline{1}^{(1)}$
$\phi^{+},W^{+}$ | $-1$ | $1$ | $\underline{1}^{(2)}\otimes\underline{1}^{(1)}$
$\phi^{0},A$ | $1$ | $1$ | $\underline{1}^{(1)}\otimes\underline{1}^{(1)}$

From Table 8, we see that the model with $\delta=1$ provides three kinds of
dark fields, in which the first kind transforms nontrivially under $P$ and
trivially under $Z_{3}$, the second kind transforms trivially under $P$ and
nontrivially under $Z_{3}$, and the last kind transforms nontrivially under
both $P$ and $Z_{3}$. Therefore, such kinds of dark fields are determined in
Table 9 as $\Psi_{5,6,7}$ fields, respectively, where $d_{5,6,7}$ are integer.

Table 9: Distinct kinds of dark fields implied by $P\otimes Z_{3}$. Field | $P\otimes Z_{3}$ | Fermion | Scalar
---|---|---|---
$\Psi_{5}$ | $\underline{1}^{(2)}\otimes\underline{1}^{(1)}$ | $(1,1,0,2d_{5})$ | $(1,1,0,2d_{5}+1)$
$\Psi_{6}$ | $\underline{1}^{(1)}\otimes\underline{1}^{(2)}$ | $(1,1,0,2d_{6}\pm 1/3)$ | $(1,1,0,2d_{6}\pm 2/3)$
$\Psi_{7}$ | $\underline{1}^{(2)}\otimes\underline{1}^{(3)}$ | $(1,1,0,2d_{7}\pm 2/3)$ | $(1,1,0,2d_{7}\pm 1/3)$

Among these solutions, we assume the simplest dark matter candidates
corresponding to the most minimal $N$ charges, summarized in Table 10. The
model responsible for two-component dark matter will be based upon $F_{5}$ and
$F_{6}$, which are self-interacted through a heavier dark field, $\Phi_{7}$.
The necessary condition for $F_{5}$ and $F_{6}$ to be co-stabilized is that
the net mass of $F_{5}$ and $F_{6}$ must be smaller than that of $\Phi_{7}$,
i.e. $m_{5}+m_{6}<m_{7}$.

Table 10: Simplest dark matter candidates implied by $P\otimes Z_{3}$. Field | Spin | $P_{M}$ | $[p^{2}]$ | $P\otimes Z_{3}$
---|---|---|---|---
$F_{5}\sim(1,1,0,0)$ | $1/2$ | $-1$ | 1 | $\underline{1}^{(2)}\otimes\underline{1}^{(1)}$
$F_{6}\sim(1,1,0,1/3)$ | $1/2$ | 1 | $\omega$ | $\underline{1}^{(1)}\otimes\underline{1}^{(2)}$
$\Phi_{7}\sim(1,1,0,-1/3)$ | $0$ | $-1$ | $\omega^{2}$ | $\underline{1}^{(2)}\otimes\underline{1}^{(3)}$

The phenomenology of this model is completely different from the $U(1)_{B-L}$
extension VanDong:2020bkg , since the ordinary particles transform
nontrivially under $P_{M}$ in the same with the dark fields (cf. Table 8).
Additionally, it happens similarly for $Z_{3}$, where the quarks are
nontrivial, as the dark fields. The electric and color charge conservation is
necessarily to ensure dark matter stability. And, the dark matter components
can have arbitrary masses, which need not be smaller than the $e,u,d$ masses.
Let us see.

Indeed, since $P_{M}$ is identical to $g_{S}$, the stability of the dark
matter component $F_{5}$ is always ensured by the $SU(3)_{C}$, $U(1)_{Q}$, and
$P$ symmetries, independent of its mass, analogous to the case of the dark
matter $\Psi_{1}$ in the model with $\delta=-1$. Whereas, the $SU(3)_{C}$ and
$Z_{3}$ symmetries are responsible for the stability of the dark matter
component $F_{6}$, where only quarks $u,d$ transform nontrivially under
$Z_{3}$ as $F_{6}$. Prove: Since $F_{6}$ as a dark matter is color neutral, it
cannot decay to any colored state, such as those that include a single quark.
The $SU(3)_{C}$ conservation requires a color-neutral final state if resulting
from a decay of $F_{6}$, by assumption. Hence, this final state containing
quarks must be composed of $q^{c}q$ or $qqq$. However, these combinations are
trivial under $Z_{3}$, implying the final state invariant under $Z_{3}$, which
cannot be a product of $F_{6}$ decay, as a contradiction. In other words, the
$SU(3)_{C}$ and $Z_{3}$ symmetries suppress the decay of $F_{6}$, even if
$F_{6}$ has a mass larger than that of quarks.

##  IV  Phenomenology

###  IV.1 Neutrino mass generation

The charged leptons and quarks gain appropriate masses through the Yukawa
couplings with the usual Higgs field $\phi$, similar to the standard model.

However, the advantage is relevant to the generation of neutrino masses.
Indeed, the neutrinos $\nu_{aL,R}$ possess the Yukawa terms,

$\mathcal{L}\supset
h^{\nu}_{ab}\bar{l}_{aL}\tilde{\phi}\nu_{bR}+\frac{1}{2}f^{\nu}_{ab}\bar{\nu}^{c}_{aR}\nu_{bR}\chi+H.c.$
(53)

Substituting the VEVs, $\nu_{R}$ obtains a Majorana mass matrix,
$m_{R}=-f^{\nu}\Lambda/\sqrt{2}$, while $\nu_{L,R}$ receives a Dirac mass
matrix, $m_{D}=-h^{\nu}v/\sqrt{2}$.

Since $v\ll\Lambda$, the observed neutrino ($\sim\nu_{L}$) mass is given by a
seesaw,

$m_{\nu}\simeq-
m_{D}m^{-1}_{R}m^{T}_{D}=h^{\nu}(f^{\nu})^{-1}(h^{\nu})^{T}\frac{v^{2}}{\sqrt{2}\Lambda},$
(54)

while the heavy neutrino ($\sim\nu_{R}$) has a mass proportional to $m_{R}$ at
$\Lambda$ scale.

The breaking of the hyperdark charge $N$ is necessary to generate small
neutrino masses, $m_{\nu}\sim v^{2}/\Lambda$, when compared with the weak
scale. This breaking or seesaw scale, $\Lambda$, is constrained by the
particle colliders below.

###  IV.2 Collider searches for $Z^{\prime}$

The $U(1)_{N}$ gauge boson, $Z^{\prime}$, obtains a mass,

$m_{Z^{\prime}}\simeq 2|\delta|g_{N}\Lambda,$ (55)

where $g_{N}$ is the $U(1)_{N}$ gauge coupling, and the contribution of small
mixing effects due to the kinetic mixing and the weak breaking is neglected.

$Z^{\prime}$ couples to the usual fermions via the Lagrangian,

$\mathcal{L}\supset-
g_{N}[N({f_{L}})\bar{f}_{L}\gamma^{\mu}f_{L}+N({f_{R}})\bar{f}_{R}\gamma^{\mu}f_{R}]Z^{\prime}_{\mu}.$
(56)

If $m_{Z^{\prime}}$ is smaller than the highest collision energy of the LEPII,
$\sqrt{s}=209$ GeV, it may be resonantly produced at the LEPII, and the
agreement with the standard model indicates that $g_{N}\lesssim 10^{-2}$, up
to a factor of $\delta$ Appelquist:2002mw .

If $m_{Z^{\prime}}>209$ GeV, the process $e^{+}e^{-}\to f\bar{f}$ gets an off-
shell contribution of $Z^{\prime}$, described by

$\displaystyle\mathcal{L}_{\mathrm{eff}}$ $\displaystyle\supset$
$\displaystyle\frac{g^{2}_{N}N({e_{L}})N({f_{L}})}{m^{2}_{Z^{\prime}}}(\bar{e}_{L}\gamma^{\mu}e_{L})(\bar{f}_{L}\gamma_{\mu}f_{L})$
(57) $\displaystyle+(LR)+(RL)+(RR).$

Taking $f=\mu,\tau$, the LEPII searched for such chiral structures, yielding a
strong limit for $Z^{\prime}$ mass per coupling,

$\frac{m_{Z^{\prime}}}{|\delta-1/2|g_{N}}\geq 6\ \mathrm{TeV},$ (58)

which translates to $\Lambda\geq 1.5|(2\delta-1)/\delta|$ TeV Carena:2004xs .

The LHC searched for dilepton signals $pp\to f\bar{f}$ for $f=e,\mu,\tau$
mediated by $Z^{\prime}$, providing a mass bound about $m_{Z^{\prime}}=4$ TeV
for $Z^{\prime}$ coupling similar to $Z$, which indicates to a bound for
$\Lambda\sim m_{Z^{\prime}}/2g\sim 3$ TeV, where $g$ is the $SU(2)_{L}$
coupling Aaboud:2017buh .

All these imply a bound on the new physics scale to be $\Lambda\sim
m_{Z^{\prime}}/g_{N}>\mathcal{O}(1)$ TeV. Here, note that $Z^{\prime}$ does
not decay to the dark fields. By contrast, if $Z^{\prime}$ decays to some dark
fields, such bound is more relaxed.

It is noted that although $\Lambda$ is constrained, $m_{Z^{\prime}}$ may be
small as $g_{N}$ is small, and vice versa.

###  IV.3 $U(1)_{N}$ running coupling

Notice that the interactions of dark fields with $U(1)_{N}$ gauge portal as
well as dark field self-couplings govern the dark matter observables. Here the
interaction of a $X$ dark field with $Z^{\prime}$ is given by the Lagrangian
$\mathcal{L}\supset\bar{X}(i\gamma^{\mu}D_{\mu}-m_{X})X$ or
$\mathcal{L}\supset(D^{\mu}X)^{\dagger}(D_{\mu}X)-m^{2}_{X}X^{\dagger}X$ for a
dark fermion or scalar, respectively, where
$D_{\mu}=\partial_{\mu}+ig_{N}N_{X}Z^{\prime}_{\mu}$ is the covariant
derivative, and $N_{X}$ is the hyperdark charge of the dark field. Due to the
cyclic property of a discrete symmetry, $N_{X}$ may be arbitrary, similar to
the ones given via $d_{1,2,\cdots,7}$ above. One may wonder that the theory is
not well defined for large $N_{X}$.

First of all, opposite to the non-Abelian charges, the Abelian charges are
completely arbitrary, as also mentioned from outset. The Lagrangian of a
$U(1)_{N}$ gauge theory always conserves a scaling symmetry,

$g_{N}\rightarrow g^{\prime}_{N}=g_{N}/c,\hskip 14.22636ptN\rightarrow
N^{\prime}=cN,$ (59)

where $g^{\prime}_{N}$ and $N^{\prime}$ are the coupling and charge after
transformation by any value of $c$. Thus, one may work in a basis, such that
$g_{N}$ is infinitesimal, while $N$ is rarely big, leaving the physics
unchanged. Further, the running of $g_{N}$ along with energy scale $\mu$
satisfies the RG equation,

$\mu\partial g_{N}/\partial\mu=\beta(g_{N})=-(g^{3}_{N}/16\pi^{2})b_{N},$ (60)

where the beta function,

$b_{N}=-\frac{2}{3}\sum_{L}N^{2}_{L}-\frac{2}{3}\sum_{R}N^{2}_{R}-\frac{1}{3}\sum_{S}N^{2}_{S},$
(61)

is summed over the left and right chiral fermions and scalars, respectively.
The RG equation preserves the above scaling symmetry, implying a theory with
large $N$ behaving similar to normal ones, as a result. A feature of the
Abelian theory is $b_{N}<0$ for every $N$. Hence, the coupling $g_{N}$
decreases when $\mu$ decreases. Given that the theory is to be well defined at
a Planck or grand unification scale, i.e. $g_{N}N\sim 1$, it works
perturbatively and is predictive below such a large scale.777An exception is
that the hyperdark charge may not be unified with the normal charges in a GUT
framework, since it necessarily results from a dequantization of electric
charge. However, such charge possibly originate from a string Faraggi:2011xu
or a flipped GUT Huong:2016kpa .

As a consequence, we divide the dark matter phenomenology into two cases. If
$g_{N}$ is small, the relevant dark matter may have a large hyperdark charge.
This would imply a large signal strength for dark matter scattering with
electrons, shown below (see also VanDong:2020bkg  for $B-L$ model). By
contrast, if $g_{N}$ is large, the charge $N$ should be small as of usual
particles. In this case, the $Z^{\prime}$ portal effectively governs the WIMP
components, co-existed beyond the weak scale, also shown below (see also
Nam:2020twn  for noncommutative $B-L$ model).

###  IV.4  Small $U(1)_{N}$ coupling: Probing the XENON1T excess

Recently the XENON1T collaboration announced an excess in electronic recoil
energy ranging from 1 to 7 keV, peaked about 2.3 keV, with a statistical
signification above 3$\sigma$, with 285 events produced above the expected
background of $232\pm 15$ events Aprile:2020tmw .

This excess may be explained within the scenarios of two-component dark matter
obtained in this work, given that one dark matter component is boosted with a
velocity of $0.1c$ order Kannike:2020agf ; Fornal:2020npv ; Primulando:2020rdk
; Su:2020zny ; Cao:2020bwd ; Alhazmi:2020fju ; DelleRose:2020pbh ; Ko:2020gdg
; Dey:2020sai . Additionally, the $U(1)_{N}$ coupling constant, $g_{N}$, is
small enough to evade the low energy experimental bounds, simultaneously this
enhances the fast dark matter charge, thus its signal strength as measured by
the XENON1T VanDong:2020bkg .

In the model with $\delta=2$, both the dark matter components $\Phi_{2,3}$
populate our galaxy, possessing densities set by their annihilation to the
usual leptons and quarks through the $U(1)_{N}$ gauge boson, $Z^{\prime}$,
portal as well as the self-annihilation between them by the self-couplings,
$\Phi^{\dagger}_{3}\Phi^{2}_{2}$ and
$(\Phi^{\dagger}_{2}\Phi_{2})(\Phi^{\dagger}_{3}\Phi_{3})$. If $\Phi_{2,3}$
have masses $m_{2,3}$, respectively, such that $m_{2}<m_{3}<2m_{2}$,
$\Phi_{3}$ is boosted in annihilation
$\Phi_{2}\Phi_{2}\rightarrow\Phi_{3}Z^{\prime}$, while $\Phi_{2}$ is boosted
in annihilation $\Phi_{3}\Phi_{3}\rightarrow\Phi_{2}\Phi_{2}$. Whereas, if
$m_{3}<m_{2}$, only the component $\Phi_{3}$ is boosted through the
annihilations,
$\Phi_{2}\Phi_{2}\rightarrow\Phi_{3}\Phi_{3},\Phi_{3}Z^{\prime}$. Above,
$Z^{\prime}$ has a mass to be smaller than those of $\Phi_{2,3}$ due to the
small $g_{N}$. The boosted dark matter is either both $\Phi_{2,3}$ or only
$\Phi_{3}$ depending on their relative masses.

In the model with $\delta=1$, the dark matter component $F_{5}$ contributes to
the whole cold dark matter, set by its annihilation to $F_{6}$ via the self-
interaction $F_{5}F_{6}\Phi_{7}$, where $m_{6}<m_{5}$. The component $F_{6}$
has vanished relic density due to its annihilations to $Z^{\prime}$ as well as
the usual particles. But $F_{6}$ is presently produced, boosted in the
annihilation of the cold dark matter $F_{5}$ due to the $F_{5}F_{6}\Phi_{7}$
self-interaction similar to the thermal process. In this model, only the
boosted dark matter component is $F_{6}$.

In all the mentioned models, the boosted dark matter subsequently scatter on
electrons in the XENON1T, causing the observed effect, enhanced by the large
fast dark matter charge. The next phenomenology happens quite the same. Hence,
we take only the model with $\delta=1$ into account, hereafter.

Table 11: Couplings of $Z^{\prime}$ with usual fermions. $f$ | $g^{Z^{\prime}}_{V}(f)$ | $g^{Z^{\prime}}_{A}(f)$
---|---|---
$\nu_{e},\nu_{\mu},\nu_{\tau}$ | $-3g_{N}/4$ | $g_{N}/4$
$e,\mu,\tau$ | $-g_{N}/4$ | $-g_{N}/4$
$u,c,t$ | $-g_{N}/12$ | $g_{N}/4$
$d,s,b$ | $5g_{N}/12$ | $-g_{N}/4$

The Lagrangian of the considering model includes

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\bar{F}_{5}(i\gamma^{\mu}\partial_{\mu}-m_{5})F_{5}+\bar{F}_{6}(i\gamma^{\mu}D_{\mu}-m_{6})F_{6}$
(62)
$\displaystyle+(D^{\mu}\Phi_{7})^{\dagger}(D_{\mu}\Phi_{7})-m^{2}_{7}\Phi^{\dagger}_{7}\Phi_{7}+\bar{e}(i\gamma^{\mu}D_{\mu}-m_{e})e$
$\displaystyle+(y\bar{F}_{5}F_{6}\Phi_{7}+H.c.),$

where $D_{\mu}=\partial_{\mu}+ig_{N}NZ^{\prime}_{\mu}$ is the covariant
derivative. Note that $F_{5,6}$ are vectorlike, while $e$ is chiral (cf. Table
11), and that the dark field masses, $m_{7}>m_{5}+m_{6}$ and $m_{5}>m_{6}$, as
mentioned. We have fixed $F_{5}\sim(1,1,0,0)$, thus it has only self-couplings
to other dark fields and gravity interaction with normal matter, as desirable.
Whereas, $F_{6}$ and $\Phi_{7}$ may possess generic hyperdark charges, i.e.

$F_{6}\sim(1,1,0,1/3+2n),\hskip 14.22636pt\Phi_{7}\sim(1,1,0,-1/3-2n),$ (63)

for $n$ integer, which differ from the minimal basic charges in Table 10 by
even numbers, similar to the ones in Table 9. This is always allowed by the
cyclic property of residual $Z_{3}$ and $P$ groups and obviously they do not
change the representations of the fields.

The cold dark matter $F_{5}$ obtains a correct density by its $t$-channel
annihilation to the second dark matter, i.e. $F_{5}F_{5}\to F_{6}F_{6}$,
exchanged by the $\Phi_{7}$ dark field, as depicted by the left diagram in
Fig. 1. $F_{6}$ strongly couples to the $U(1)$ gauge boson $Z^{\prime}$, by
which it has a vanished density given by the right diagram in Fig. 1. Note
that $F_{6}$ also annihilates to the usual particles via $Z^{\prime}$ portal,
but it is small, compared to the given channel when its hyperdark charge is
large. Presently, $F_{6}$ is boosted via a process similar to the left diagram
in Fig. 1, and subsequently this fast dark matter scatters on electrons in the
XENON1T experiment via the diagram in Fig. 2, by the $Z^{\prime}$ portal. The
mildly boosted (i.e., $v\sim 0.1c$) phenomenon happens when $m_{5}\approx
m_{6}$ and we assume $m_{5,6}>m_{e}$ to satisfy the BBN and CMB bounds
Sabti:2019mhn .

Figure 1: Dark matter conversion and annihilation of the second dark matter.
The second process has an extra $u$-channel, but undepicted, for simplicity.

Figure 2: Scatter of the boosted dark matter with electron.

By the process in Fig. 2, the electronic recoil energy approaches
$E_{R}<2m_{e}v^{2}$ for $m_{6}>m_{e}$ and in agreement with the XENON1T if
$F_{6}$ is boosted with $v^{2}=(m_{5}^{2}-m^{2}_{6})/m^{2}_{5}\sim 1\%$
Kannike:2020agf . The transferred momentum is $|q|=E_{R}/v\sim 40$ keV. In the
limit of $Z^{\prime}$ mass to be much larger than $|q|$, the $F_{6}$-$e$
scattering cross section is given by Joglekar:2019vzy

$\sigma_{e}=\frac{g_{N}^{4}(1/3+2n)^{2}m_{e}^{2}}{8\pi m_{Z^{\prime}}^{4}}.$
(64)

The flux of the fast dark matter $F_{6}$ comes from all the sky due to $F_{5}$
cold dark matter annihilations in our galaxy. Hence, the relation between the
number of signal events and the scattering cross section is derived
Fornal:2020npv ; Agashe:2014yua

$\frac{N_{\text{sig}}}{100}=\frac{16\sigma_{e}}{3\
\text{pb}}\left(\frac{1\text{ MeV}}{m_{5}}\right)^{2},$ (65)

where $m_{5}>m_{e}$ up to GeV and the 1 MeV does not mean a benchmark value.
Noting that $m_{Z^{\prime}}\simeq 2g_{N}\Lambda$ for the model with
$\delta=1$, we deduce

$\frac{N_{\text{sig}}}{100}\simeq 6.4\times
10^{-14}\left(\frac{1}{3}+2n\right)^{2}\left(\frac{m_{e}}{m_{5}}\right)^{2}\left(\frac{3\text{
TeV}}{\Lambda}\right)^{4}.$ (66)

Required to explain the number of signal events, $N_{\mathrm{sig}}\sim 100$,
the hyperdark charge of dark matter is large,

$|n|\sim 1.98\times
10^{6}\left(\frac{m_{5}}{m_{e}}\right)\left(\frac{\Lambda}{3\
\mathrm{TeV}}\right)^{2}>1.98\times 10^{6},$ (67)

where both $\Lambda$ and $m_{5}$ have a lower bound, $\Lambda>3$ TeV and
$m_{5}>m_{e}$, as mentioned. Additionally, from Eq. (67), the dark matter
charge required is increased, when the dark matter mass $m_{5}$ and/or the new
physics scale $\Lambda$ are larger.

In this case, we approximate the beta function through
$b_{N}\simeq-(5/3)(1/3+2n)^{2}$, yielding the RG equation,

$\mu\frac{\partial
g^{\prime}_{N}}{\partial\mu}\simeq\frac{5}{48\pi^{2}}g^{\prime 3}_{N},$ (68)

where $g^{\prime}_{N}=|1/3+2n|g_{N}$ slides as the usual couplings. Defining
the hyperdark charge coupling strength, $\alpha^{\prime}_{N}=g^{\prime
2}_{N}/4\pi$, we obtain the solution,

$\frac{1}{\alpha^{\prime}_{N}}=\frac{1}{\alpha^{\prime}_{NG}}-\frac{5}{6\pi}\ln\frac{\mu}{\mu_{G}},$
(69)

where G indicates to those at the large (GUT or Planck) scale. Imposing a
perturbative limit for the hyperdark charge interaction at the large scale,
say $\alpha^{\prime}_{NG}=1$ at $\mu_{G}=10^{16}$ GeV, it leads to
$\alpha^{\prime}_{N}\simeq 0.08$ at scale $\mu\sim 1$ MeV of interest, or in
other words,

$|1/3+2n|g_{N}\simeq 1.$ (70)

Combining this equation with the condition $N_{\mathrm{sig}}=100$ required to
explain the excess, we get

$m_{Z^{\prime}}\simeq 1.81\times 10^{3}(g_{N}/m_{5})^{1/2}\text{ MeV}^{3/2}.$
(71)

From Eq. (70) and Eq. (67), we obtain a bound for the $U(1)_{N}$ gauge
coupling,

$g_{N}\lesssim 2.52\times 10^{-7}.$ (72)

From Eq. (71) and Eq. (72), the mass of the $Z^{\prime}$ boson is
correspondingly bounded by

$m_{Z^{\prime}}\lesssim 1.28\text{ MeV},$ (73)

since $m_{5}>m_{e}$. This bound is very close to the sensitive limit of beam
dump experiments, which detect the signal of the decay $Z^{\prime}\rightarrow
e^{+}e^{-}$, that requires $m_{Z^{\prime}}>2m_{e}\simeq 1$ MeV Ilten:2018crw ;
Bauer:2018onh . That said, $Z^{\prime}$ is always viable below such mass limit
since the beam dump experiments are not able to find it, in spite of what size
$g_{N}$ is.

Notice that $m_{Z^{\prime}}$ and $g_{N}$ are further constrained by the
neutrino-electron scattering, horizontal branch stars, and supernova 1987A
experiments, similar to the $B-L$ model VanDong:2020bkg , which is signified
as follows. The neutrino-electron scattering due to the $Z^{\prime}$ exchange
yielding a bound $g_{N}\lesssim 10^{-6}$ for $m_{Z^{\prime}}\sim$ MeV is
obviously suppressed by the condition (72). The experiment with horizontal
branch stars studies energy loss, carried by $Z^{\prime}$, implying a bound
$m_{Z^{\prime}}>0.34$ MeV for $g_{N}$ satisfying (72) but above the excluded
regime $g_{N}\nsim 10^{-8}$–$10^{-11}$ as constrained by the supernova 1987A.
The BBN supplies a lower limit on $g_{N}$ which would exclude the viable lower
region with $g_{N}\lesssim 10^{-11}$, according to $m_{Z^{\prime}}\sim$ MeV.
In what follows, we need discuss the last one.

Since dark matter components $F_{5,6}$ are above the BBN bound at MeV, as
given, we need only examine two cases of $Z^{\prime}$ mass. If the
$Z^{\prime}$ mass is also above this BBN bound, no sub-MeV hidden states
exist. Hence, no new relativistic degrees of freedom are presented during the
BBN. Only constraint from the BBN is that $Z^{\prime}$ decays into the
standard model fast enough, larger than the Hubble rate, in order to avoid
late dissociation process. Thus, the decay $Z^{\prime}\to
e^{+}e^{-},\nu\nu^{c}$ before $T\sim$ MeV requires $g_{N}\gtrsim
10^{-9}(\mathrm{MeV}/m_{Z^{\prime}})^{1/2}\sim 10^{-9}$. If $Z^{\prime}$ is
below the BBN bound, $Z^{\prime}$ also does not contribute to the allowed
number of relativistic degrees of freedom during the BBN at the standard model
temperature. This is because the decay $Z^{\prime}\to\nu\nu^{c}$ is fast
similar to the above case, thus $g_{N}\gtrsim 10^{-9}$. There is no new long-
lived state present during the BBN, and due to this fast decay the kinetic
equilibrium between $Z^{\prime}$ and the standard model through the reaction
$Z^{\prime}e^{\pm}\leftrightarrow\gamma e^{\pm}$ is not maintained for
temperature dropped below the $Z^{\prime}$ mass, $m_{Z^{\prime}}\sim$ MeV. In
summary, in addition to the XENON1T constraints in (73) and (72), the
astrophysical and cosmological experiments restrict

$m_{Z^{\prime}}>0.34\ \mathrm{MeV}\ \mathrm{and}\ g_{N}\gtrsim 10^{-8}.$ (74)

Last, but not least, let us evaluate the dark matter relic densities as
stated. It is suitably to assume $m_{6}>m_{Z^{\prime}}$, as above mentioned.
Hence, $F_{6}$ is obviously annihilated to $Z^{\prime}$ via the right diagram
in Fig. 1. The annihilation cross section is

$\langle\sigma v_{\mathrm{rel}}\rangle_{F_{6}F_{6}\rightarrow
Z^{\prime}Z^{\prime}}\simeq\frac{g^{\prime 4}_{N}}{16\pi m^{2}_{6}}\gg 1\
\mathrm{pb},$ (75)

since $g^{\prime}_{N}\simeq 1$ and that $m_{6}$ is radically below the weak
scale. The $F_{6}$ density is negligible,

$\Omega_{F_{6}}h^{2}\simeq\frac{0.1\ \mathrm{pb}}{\langle\sigma
v_{\mathrm{rel}}\rangle_{F_{6}F_{6}\to Z^{\prime}Z^{\prime}}}<2\times
10^{-8},$ (76)

for $m_{F_{6}}<10$ GeV.

Hence, the cold dark matter abundance is approximated by the $F_{5}$
contribution,

$\Omega_{\text{DM}}h^{2}=\Omega_{F_{5}}h^{2}+\Omega_{F_{6}}h^{2}\simeq\Omega_{F_{5}}h^{2}.$
(77)

Applying the Feynman rules for the left diagram in Fig. 1, we obtain the
annihilation cross-section for $F_{5}$,

$\langle\sigma v_{\mathrm{rel}}\rangle_{F_{5}F_{5}\rightarrow
F_{6}F_{6}}\simeq\frac{|y|^{4}(m_{5}+m_{6})^{2}}{8\pi(m_{7}^{2}-m_{6}^{2}+m_{5}^{2})^{2}}\left(1-\frac{m_{6}^{2}}{m_{5}^{2}}\right)^{1/2}.$
(78)

Using the approximation $m_{6}\approx m_{5}$, the data
$\Omega_{\text{DM}}h^{2}\simeq 0.1\ \mathrm{pb}/\langle\sigma
v_{\mathrm{rel}}\rangle_{F_{5}F_{5}\to F_{6}F_{6}}\simeq 0.12$ Zyla:2020zbs
is recovered if

$|y|\simeq 4.95\times
10^{-4}\left(\frac{m_{7}}{2m_{5}}\right)\left(\frac{m_{5}}{m_{e}}\right)^{1/2}\left(\frac{\delta
m^{2}}{m^{2}_{5}}\right)^{-1/8}.$ (79)

The dark matter self-coupling depends on $m_{7}/m_{5}$ and $m_{5}/m_{e}$,
having a lower bound $|y|>10^{-3}$, for $m_{7}>2m_{5}$ and $m_{5}>m_{e}$,
where note that $\delta
m^{2}/m^{2}_{5}\equiv(m^{2}_{5}-m^{2}_{6})/m^{2}_{5}\sim 1\%$. As an instance,
we make a contour of the dark matter self-coupling $y$ and the $F_{5}$ mass
$m_{5}$ in Fig. 3.

Figure 3: Dark matter self-coupling and mass contoured for the correct
abundance, taking $m_{7}=2m_{5}$.

Note that $F_{5}$ may annihilate to $Z^{\prime}$ via 1-loop diagrams with both
$\Phi_{7}$ and $F_{6}$ running in the loop. The amplitude is then evaluated as
$M_{\mathrm{rad}}\sim\frac{y^{2}m_{5}}{m^{2}_{6,7}}\frac{g^{\prime
2}_{N}}{16\pi^{2}}$, which is substantially smaller than the given tree-level
amplitude, $M_{\mathrm{tree}}\sim\frac{y^{2}m_{5,6}}{m^{2}_{7}}$, due to the
loop factor suppression $\sim 1/16\pi^{2}$ and $g^{\prime}_{N}\simeq 1$. The
tree-level annihilation of $F_{5}$ to $F_{6}$ sets the relic density.

###  IV.5  Large $U(1)_{N}$ coupling: Implication for co-existed WIMPs

WIMP is a natural solution of the thermal dark matter paradigm, which
contributed to the thermal bath of the early universe, then decoupled, i.e.
freeze-out, with a present-day density when its annihilation rate into usual
particles matches the Hubble expansion rate. Many standard model extensions
imply existence of a single WIMP, such as supersymmetry and extradimension
Jungman:1995df ; Bertone:2004pz , as well as gauge approach Dong:2013ioa ;
Dong:2014esa ; Dong:2014wsa ; Dong:2015rka ; Dong:2015dxw ; Huong:2016ybt ;
Huong:2018ytz ; Huong:2019vej ; Dinh:2019jdg ; Dong:2016sat ; Dong:2017ayu ;
VanDong:2020nwb , but this work adds that the dequantization of electric
charge unravels a picture of structured WIMPs. It is noted that the
considering versions of dark matter actually contain two simultaneously
stabilized WIMPs. Hence, it is sufficiently to investigate one of them, say
the model with $\delta=1$ for the candidates in Table 10, which have the
minimal basic hyperdark charges and large $U(1)_{N}$ gauge coupling. We also
assume the $F_{5,6}$ masses to be beyond the weak scale.

Dark matter pair annihilation in the present case includes those given
similarly to Fig. 1, plus the extra $s$-channel diagrams (not depicted) for
$F_{6}$ annihilation to the standard model particles (leptons, quarks, Higgs
and gauge bosons) exchanged by $Z^{\prime}$, namely $F_{6}F^{c}_{6}\to
ff^{c},ZH$ where $f$ indicates to the usual leptons and quarks. Notice that
$F_{5}$ does not annihilate to the usual particles, since its dark charge
vanishes.

It is straightforward to determine the annihilation cross section of $F_{6}$,
such as

$\displaystyle\langle\sigma
v_{\mathrm{rel}}\rangle_{F_{6}F_{6}\to\mathrm{all}}$ $\displaystyle\simeq$
$\displaystyle\frac{\theta(m_{6}-m_{Z^{\prime}})g_{N}^{4}m_{6}^{2}}{324\pi(m^{2}_{Z^{\prime}}-2m^{2}_{6})^{2}}\left(1-\frac{m^{2}_{Z^{\prime}}}{m^{2}_{6}}\right)^{3/2}$
(80)
$\displaystyle+\frac{g_{N}^{2}m_{6}^{2}\sum_{f}N_{C}(f)\\{[g_{V}^{Z^{\prime}}(f)]^{2}+[g_{A}^{Z^{\prime}}(f)]^{2}\\}}{9\pi(m_{Z^{\prime}}^{2}-4m_{6}^{2})^{2}}$
$\displaystyle+\frac{g_{N}^{2}g_{Z^{\prime}ZH}^{2}m^{2}_{6}}{144\pi(m_{Z^{\prime}}^{2}-4m_{6}^{2})^{2}m^{2}_{Z}},$

where $g^{Z^{\prime}}_{V,A}(f)$ were supplied in Table 11, while the
$HZZ^{\prime}$ coupling is given by $g_{Z^{\prime}ZH}\simeq-g_{N}v/2$.

Since the dark matter components $F_{5,6}$ now have arbitrary masses above the
weak scale, the conversion between $F_{5,6}$ is either $F_{5}F_{5}^{c}\to
F_{6}F_{6}^{c}$ if $m_{5}>m_{6}$ or $F_{6}F_{6}^{c}\to F_{5}F_{5}^{c}$ if
$m_{5}<m_{6}$. The former process is described by the left diagram in Fig. 1,
in which the result in Eq. (78) properly applies for this case. The latter, or
inverse process, is given by a $t$-channel diagram similar to the previous
process by replacement $F_{5}\leftrightarrow F_{6}$ with the common $\Phi_{7}$
mediator, implying an annihilation cross section,

$\langle\sigma v_{\mathrm{rel}}\rangle_{F_{6}F_{6}\to
F_{5}F_{5}}\simeq\frac{|y|^{4}(m_{5}+m_{6})^{2}}{8\pi(m_{7}^{2}-m_{5}^{2}+m_{6}^{2})^{2}}\left(1-\frac{m_{5}^{2}}{m_{6}^{2}}\right)^{1/2}.$
(81)

It is noted that for the case $m_{6}>m_{5}$, the annihilation cross sections
in Eqs. (80) and (81) set the $F_{6}$ density. However, $F_{5}$ never
annihilates after the temperature of the universe falls below the $F_{6}$
mass. Thus, it has a large density, overpopulating the universe. This case
should be discarded. Hence, we consider only $m_{5}>m_{6}$. And, the $F_{5}$
density is governed by its annihilation to $F_{6}$, i.e.

$\Omega_{F_{5}}h^{2}\simeq\frac{0.1\ \mathrm{pb}}{\langle\sigma
v_{\mathrm{rel}}\rangle_{F_{5}F_{5}\to F_{6}F_{6}}},$ (82)

given by Eq. (78), while the $F_{6}$ density is determined by its annihilation
to the $Z^{\prime}$ and/or standard model particles,

$\Omega_{F_{6}}h^{2}\simeq\frac{0.1\ \mathrm{pb}}{\langle\sigma
v_{\mathrm{rel}}\rangle_{F_{6}F_{6}\to\mathrm{all}}},$ (83)

obtained by Eq. (80). The total dark matter density is
$\Omega_{\mathrm{DM}}h^{2}=\Omega_{F_{5}}h^{2}+\Omega_{F_{6}}h^{2}$, where
both the components contribute, unlike the above model for XENON1T.

The parameters that significantly govern the dark matter observables are the
dark field self-coupling $y$, the $U(1)_{N}$ gauge coupling $g_{N}$, and the
new physics scale $\Lambda$, besides the dark matter masses $m_{5,6}$ and the
mediator mass $m_{7}$. Let us make contours of
$\Omega_{\mathrm{DM}}h^{2}=0.12$ inspired by the experiment Zyla:2020zbs  as
functions of $m_{5,6}$ according to several choices of the remaining
parameters. Namely, in Fig. 4 the upper panel is plotted for $x\equiv
m_{7}/(m_{5}+m_{6})=1.1$, $g_{N}=0.2$, $\Lambda=3$ TeV, and $y=0.8$ and 1
corresponding to each curve as marked in the panel. Whereas, also in Fig. 4,
the lower panel is plotted for $y=0.8$, $x=1.1$, $g_{N}=0.2$, and the curves
according to the values $\Lambda=3$ and $3.5$ TeV, respectively.

Figure 4:  Total dark matter density contoured as functions of component dark
matter masses for different choices of $y,\Lambda$; the value of other
parameters relevant to each panel is detailedly supplied in the text.

Note that the disconnected regions on each curve are due to a $Z^{\prime}$
resonance in the density, at which $m_{6}=\frac{1}{2}m_{Z^{\prime}}$, which
reduces the density to zero, thus suppressed for correct abundance.

Since $F_{5}\sim(1,1,0,0)$ is sterile, it does not interact with detectors in
direct detection experiment. Effect of dark matter in the direct detection
comes only from a potential scatter of the $F_{6}$ component with nuclei in a
large detector.888Since this WIMP is non-relativistic, yielding a small
electronic recoil energy, it cannot explain the XENON1T excess. Indeed, the
effective Lagrangian describing the $F_{6}$-quark interaction can be derived
from the $t$-channel exchange diagram by the new gauge boson $Z^{\prime}$ to
be

$\mathcal{L}_{\text{eff}}\supset\bar{F}_{6}\gamma^{\mu}F_{6}\left[\bar{q}\gamma_{\mu}(\alpha_{q}P_{L}+\beta_{q}P_{R})q\right],$
(84)

where $P_{L,R}=(1\mp\gamma_{5})/2$, and

$\alpha_{u,d}=-\beta_{u}/2=\beta_{d}/4=-g_{N}^{2}/18m_{Z^{\prime}}^{2}.$ (85)

Hence, we obtain the spin-independent (SI) cross-section for the scattering of
$F_{6}$ on a nucleon as

$\sigma^{\text{SI}}_{F_{6}-\text{nucleon}}=\frac{4m^{2}_{\text{nucleon}}}{\pi
A^{2}}[\lambda_{p}Z+\lambda_{n}(A-Z)]^{2},$ (86)

where

$\displaystyle\lambda_{p}=[2(\alpha_{u}+\beta_{u})+\alpha_{d}+\beta_{d}]/8,$
$\displaystyle\lambda_{n}=[\alpha_{u}+\beta_{u}+2(\alpha_{d}+\beta_{d})]/8.$
(87)

Above, $Z$ is the nucleus charge and $A$ is the total number of nucleons in
the nucleus. Because the dark matter components partially contribute to the
total density, the effective SI cross-section of $F_{6}$ is given by

$\sigma^{\text{SI}}_{\text{eff}}(F_{6})=\frac{\Omega_{F_{6}}h^{2}}{\Omega_{\text{DM}}h^{2}}\sigma^{\text{SI}}_{F_{6}-\text{nucleon}}.$
(88)

Figure 5: The SI $F_{6}$-nucleon effective scattering cross-section limit as a
function of $F_{6}$ mass according to the several choices of $\Lambda$, where
the excluded region (light brown) lies above the experimental (green and
yellow) bands.

Taking $A=131$ and $Z=54$ according to the Xe nucleus and
$m_{\text{nucleon}}\simeq 1$ GeV, in Fig. 5 we plot the SI cross-section of
$F_{6}$ corresponding to the previous choices of $\Lambda$ and $y=0.8$,
$x=1.1$, $g_{N}=0.2$. We always assume that the two dark matter components
give the correct relic density. The XENON1T experimental bounds Aprile:2017iyp
; Aprile:2018dbl  have also been included to the plot. Combining the results
in Fig. 4 lower panel and Fig. 5, the viable dark matter mass regime is either
$0.6\text{ TeV}<m_{5}<1.1\text{ TeV}$ and $0.57\text{ TeV}<m_{6}<0.62\text{
TeV}$ for $\Lambda=3$ TeV, or $0.78\text{ TeV}<m_{5}<0.98\text{ TeV}$ and
$0.68\text{ TeV}<m_{6}<0.71\text{ TeV}$ for $\Lambda=3.5$ TeV, but always
ensuring that $m_{5}>m_{6}$.

##  V Conclusion

The dequantization of electric charge leads to the existence of a
noncommutative dark charge and that the full gauge symmetry of the model takes
the form, $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)_{N}$. This
setup not only yields suitable neutrino masses, but also implies the novel
schemes of multi-component dark matter. The dark matter stability is ensured
by the residual dark charge identical to an even $Z_{k}$ symmetry, a remnant
of the gauge symmetry after symmetry breaking. Additionally, the dark matter
components can have an arbitrary mass, despite the fact that they and normal
particles transform nontrivially under the discrete symmetry. With this
mechanism, we have pointed out the simplest models for two-component dark
matter. Additionally, we have proved that they can address the XENON1T anomaly
recently observed as the second dark matter component is boosted in
annihilation of the first dark matter component. Alternatively, these models
can contain two WIMPs with mass above the weak scale, satisfying the relic
density and direct detection bounds.

## Acknowledgments

We would like to thank Dr. Cao H. Nam (Phenikaa University) for the useful
discussions. This research is funded by Vietnam National Foundation for
Science and Technology Development (NAFOSTED) under grant number
103.01-2019.353.

## References

  * (1) Particle Data Group collaboration, _Review of Particle Physics_ , _PTEP_ 2020 (2020) 083C01.
  * (2) T. Kajita, _Nobel lecture: Discovery of atmospheric neutrino oscillations_ , _Rev. Mod. Phys._ 88 (2016) 030501.
  * (3) A. B. McDonald, _Nobel lecture: The sudbury neutrino observatory: Observation of flavor change for solar neutrinos_ , _Rev. Mod. Phys._ 88 (2016) 030502.
  * (4) WMAP collaboration, _Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results_ , _Astrophys. J. Suppl._ 208 (2013) 19 [1212.5226].
  * (5) Planck collaboration, _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6 [1807.06209].
  * (6) G. Jungman, M. Kamionkowski and K. Griest, _Supersymmetric dark matter_ , _Phys. Rept._ 267 (1996) 195 [hep-ph/9506380].
  * (7) G. Bertone, D. Hooper and J. Silk, _Particle dark matter: Evidence, candidates and constraints_ , _Phys. Rept._ 405 (2005) 279 [hep-ph/0404175].
  * (8) Z. G. Berezhiani and M. Y. Khlopov, _Cosmology of Spontaneously Broken Gauge Family Symmetry_ , _Z. Phys. C_ 49 (1991) 73.
  * (9) Z. G. Berezhiani and M. Y. Khlopov, _Physics of cosmological dark matter in the theory of broken family symmetry. (In Russian)_ , _Sov. J. Nucl. Phys._ 52 (1990) 60.
  * (10) C. Boehm, P. Fayet and J. Silk, _Light and heavy dark matter particles_ , _Phys. Rev. D_ 69 (2004) 101302 [hep-ph/0311143].
  * (11) E. Ma, _Supersymmetric Model of Radiative Seesaw Majorana Neutrino Masses_ , _Annales Fond. Broglie_ 31 (2006) 285 [hep-ph/0607142].
  * (12) K. M. Zurek, _Multi-Component Dark Matter_ , _Phys. Rev. D_ 79 (2009) 115002 [0811.4429].
  * (13) H. Fukuoka, D. Suematsu and T. Toma, _Signals of dark matter in a supersymmetric two dark matter model_ , _JCAP_ 1107 (2011) 001 [1012.4007].
  * (14) B. Batell, _Dark Discrete Gauge Symmetries_ , _Phys. Rev. D_ 83 (2011) 035006 [1007.0045].
  * (15) D. Chialva, P. Dev and A. Mazumdar, _Multiple dark matter scenarios from ubiquitous stringy throats_ , _Phys. Rev. D_ 87 (2013) 063522 [1211.0250].
  * (16) A. Biswas, D. Majumdar, A. Sil and P. Bhattacharjee, _Two Component Dark Matter : A Possible Explanation of 130 GeV $\gamma-$ Ray Line from the Galactic Centre_, _JCAP_ 1312 (2013) 049 [1301.3668].
  * (17) S. Bhattacharya, A. Drozd, B. Grzadkowski and J. Wudka, _Two-Component Dark Matter_ , _JHEP_ 10 (2013) 158 [1309.2986].
  * (18) L. Bian, R. Ding and B. Zhu, _Two Component Higgs-Portal Dark Matter_ , _Phys. Lett._ B728 (2014) 105 [1308.3851].
  * (19) G. Bélanger, K. Kannike, A. Pukhov and M. Raidal, _Minimal semi-annihilating $Z_{N}$ scalar dark matter_, _JCAP_ 06 (2014) 021 [1403.4960].
  * (20) L. Bian, T. Li, J. Shu and X.-C. Wang, _Two component dark matter with multi-Higgs portals_ , _JHEP_ 03 (2015) 126 [1412.5443].
  * (21) S. Esch, M. Klasen and C. E. Yaguna, _A minimal model for two-component dark matter_ , _JHEP_ 09 (2014) 108 [1406.0617].
  * (22) A. DiFranzo and G. Mohlabeng, _Multi-component Dark Matter through a Radiative Higgs Portal_ , _JHEP_ 01 (2017) 080 [1610.07606].
  * (23) A. Dutta Banik, M. Pandey, D. Majumdar and A. Biswas, _Two component WIMP–FImP dark matter model with singlet fermion, scalar and pseudo scalar_ , _Eur. Phys. J._ C77 (2017) 657 [1612.08621].
  * (24) A. Karam and K. Tamvakis, _Dark Matter from a Classically Scale-Invariant $SU(3)_{X}$_, _Phys. Rev. D_ 94 (2016) 055004 [1607.01001].
  * (25) S. Bhattacharya, P. Poulose and P. Ghosh, _Multipartite Interacting Scalar Dark Matter in the light of updated LUX data_ , _JCAP_ 04 (2017) 043 [1607.08461].
  * (26) G. Arcadi, C. Gross, O. Lebedev, Y. Mambrini, S. Pokorski and T. Toma, _Multicomponent Dark Matter from Gauge Symmetry_ , _JHEP_ 12 (2016) 081 [1611.00365].
  * (27) D. Borah, A. Dasgupta, U. K. Dey, S. Patra and G. Tomar, _Multi-component Fermionic Dark Matter and IceCube PeV scale Neutrinos in Left-Right Model with Gauge Unification_ , _JHEP_ 09 (2017) 005 [1704.04138].
  * (28) A. Ahmed, M. Duch, B. Grzadkowski and M. Iglicki, _Multi-Component Dark Matter: the vector and fermion case_ , _Eur. Phys. J. C_ 78 (2018) 905 [1710.01853].
  * (29) S. Bhattacharya, P. Ghosh, T. N. Maity and T. S. Ray, _Mitigating Direct Detection Bounds in Non-minimal Higgs Portal Scalar Dark Matter Models_ , _JHEP_ 10 (2017) 088 [1706.04699].
  * (30) S. Bhattacharya, P. Ghosh and N. Sahu, _Multipartite Dark Matter with Scalars, Fermions and signatures at LHC_ , _JHEP_ 02 (2019) 059 [1809.07474].
  * (31) S. Bhattacharya, A. K. Saha, A. Sil and J. Wudka, _Dark Matter as a remnant of SQCD Inflation_ , _JHEP_ 10 (2018) 124 [1805.03621].
  * (32) M. Aoki and T. Toma, _Boosted Self-interacting Dark Matter in a Multi-component Dark Matter Model_ , _JCAP_ 1810 (2018) 020 [1806.09154].
  * (33) A. Dutta Banik, A. K. Saha and A. Sil, _Scalar assisted singlet doublet fermion dark matter model and electroweak vacuum stability_ , _Phys. Rev._ D98 (2018) 075013 [1806.08080].
  * (34) B. Barman, S. Bhattacharya and M. Zakeri, _Multipartite Dark Matter in $SU(2)_{N}$ extension of Standard Model and signatures at the LHC_, _JCAP_ 1809 (2018) 023 [1806.01129].
  * (35) S. Yaser Ayazi and A. Mohamadnejad, _Scale-Invariant Two Component Dark Matter_ , _Eur. Phys. J._ C79 (2019) 140 [1808.08706].
  * (36) S. Chakraborti, A. Dutta Banik and R. Islam, _Probing Multicomponent Extension of Inert Doublet Model with a Vector Dark Matter_ , _Eur. Phys. J._ C79 (2019) 662 [1810.05595].
  * (37) S. Chakraborti and P. Poulose, _Interplay of Scalar and Fermionic Components in a Multi-component Dark Matter Scenario_ , _Eur. Phys. J._ C79 (2019) 420 [1808.01979].
  * (38) D. Borah, A. Dasgupta and S. K. Kang, _Two-component dark matter with cogenesis of the baryon asymmetry of the Universe_ , _Phys. Rev._ D100 (2019) 103502 [1903.10516].
  * (39) F. Elahi and S. Khatibi, _Multi-Component Dark Matter in a Non-Abelian Dark Sector_ , _Phys. Rev._ D100 (2019) 015019 [1902.04384].
  * (40) C. E. Yaguna and O. Zapata, _Multi-component scalar dark matter from a $Z_{N}$ symmetry: a systematic analysis_, _JHEP_ 03 (2020) 109 [1911.05515].
  * (41) J. Fan, A. Katz, L. Randall and M. Reece, _Double-Disk Dark Matter_ , _Phys. Dark Univ._ 2 (2013) 139 [1303.1521].
  * (42) J. Fan, A. Katz, L. Randall and M. Reece, _Dark-Disk Universe_ , _Phys. Rev. Lett._ 110 (2013) 211302 [1303.3271].
  * (43) K. Agashe, Y. Cui, L. Necib and J. Thaler, _(In)direct Detection of Boosted Dark Matter_ , _JCAP_ 10 (2014) 062 [1405.7370].
  * (44) K. Kong, G. Mohlabeng and J.-C. Park, _Boosted dark matter signals uplifted with self-interaction_ , _Phys. Lett. B_ 743 (2015) 256 [1411.6632].
  * (45) H. Alhazmi, K. Kong, G. Mohlabeng and J.-C. Park, _Boosted Dark Matter at the Deep Underground Neutrino Experiment_ , _JHEP_ 04 (2017) 158 [1611.09866].
  * (46) D. Kim, J.-C. Park and S. Shin, _Dark Matter “Collider” from Inelastic Boosted Dark Matter_ , _Phys. Rev. Lett._ 119 (2017) 161801 [1612.06867].
  * (47) G. F. Giudice, D. Kim, J.-C. Park and S. Shin, _Inelastic Boosted Dark Matter at Direct Detection Experiments_ , _Phys. Lett. B_ 780 (2018) 543 [1712.07126].
  * (48) A. Chatterjee, A. De Roeck, D. Kim, Z. G. Moghaddam, J.-C. Park, S. Shin et al., _Searching for boosted dark matter at ProtoDUNE_ , _Phys. Rev. D_ 98 (2018) 075027 [1803.03264].
  * (49) D. Kim, K. Kong, J.-C. Park and S. Shin, _Boosted Dark Matter Quarrying at Surface Neutrino Detectors_ , _JHEP_ 08 (2018) 155 [1804.07302].
  * (50) O. D. Elbert, J. S. Bullock, S. Garrison-Kimmel, M. Rocha, J. Oñorbe and A. H. Peter, _Core formation in dwarf haloes with self-interacting dark matter: no fine-tuning necessary_ , _Mon. Not. Roy. Astron. Soc._ 453 (2015) 29 [1412.1477].
  * (51) S. Tulin and H.-B. Yu, _Dark Matter Self-interactions and Small Scale Structure_ , _Phys. Rept._ 730 (2018) 1 [1705.02358].
  * (52) J. Heeck and H. Zhang, _Exotic Charges, Multicomponent Dark Matter and Light Sterile Neutrinos_ , _JHEP_ 05 (2013) 164 [1211.0538].
  * (53) M. Aoki, M. Duerr, J. Kubo and H. Takano, _Multi-Component Dark Matter Systems and Their Observation Prospects_ , _Phys. Rev. D_ 86 (2012) 076015 [1207.3318].
  * (54) M. Aoki, J. Kubo and H. Takano, _Two-loop radiative seesaw mechanism with multicomponent dark matter explaining the possible $\gamma$ excess in the Higgs boson decay and at the Fermi LAT_, _Phys. Rev. D_ 87 (2013) 116001 [1302.3936].
  * (55) Y. Kajiyama, H. Okada and T. Toma, _Multicomponent dark matter particles in a two-loop neutrino model_ , _Phys. Rev. D_ 88 (2013) 015029 [1303.7356].
  * (56) A. Karam and K. Tamvakis, _Dark matter and neutrino masses from a scale-invariant multi-Higgs portal_ , _Phys. Rev._ D92 (2015) 075010 [1508.03031].
  * (57) N. Bernal, D. Restrepo, C. Yaguna and O. Zapata, _Two-component dark matter and a massless neutrino in a new $B-L$ model_, _Phys. Rev. D_ 99 (2019) 015038 [1808.03352].
  * (58) C. Bonilla, S. Centelles-Chuliá, R. Cepedello, E. Peinado and R. Srivastava, _Dark matter stability and Dirac neutrinos using only Standard Model symmetries_ , _Phys. Rev. D_ 101 (2020) 033011 [1812.01599].
  * (59) D. Borah, R. Roshan and A. Sil, _Minimal two-component scalar doublet dark matter with radiative neutrino mass_ , _Phys. Rev._ D100 (2019) 055027 [1904.04837].
  * (60) S. Bhattacharya, P. Ghosh, A. K. Saha and A. Sil, _Two component dark matter with inert Higgs doublet: neutrino mass, high scale validity and collider searches_ , 1905.12583.
  * (61) A. Biswas, D. Borah and D. Nanda, _Type III seesaw for neutrino masses in U(1) B-L model with multi-component dark matter_, _JHEP_ 12 (2019) 109 [1908.04308].
  * (62) S. Bhattacharya, N. Chakrabarty, R. Roshan and A. Sil, _Multicomponent dark matter in extended $U(1)_{B-L}$: neutrino mass and high scale validity_, _JCAP_ 04 (2020) 013 [1910.00612].
  * (63) P. Van Dong, _Flipping principle for neutrino mass and dark matter_ , _Phys. Rev. D_ 102 (2020) 011701 [2003.13276].
  * (64) D. Van Loi, C. H. Nam, N. H. Tan and P. Van Dong, _Dark charge vs electric charge_ , 2004.06005.
  * (65) P. Dong, H. Hung and T. Tham, _3-3-1-1 model for dark matter_ , _Phys. Rev. D_ 87 (2013) 115003 [1305.0369].
  * (66) P. V. Dong, _Unifying the electroweak and B-L interactions_ , _Phys. Rev. D_ 92 (2015) 055026 [1505.06469].
  * (67) A. Alves, G. Arcadi, P. Dong, L. Duarte, F. S. Queiroz and J. W. F. Valle, _Matter-parity as a residual gauge symmetry: Probing a theory of cosmological dark matter_ , _Phys. Lett. B_ 772 (2017) 825 [1612.04383].
  * (68) P. Dong, D. Huong, D. Loi, N. Nhuan and N. Ngan, _Phenomenology of the $SU(3)_{C}\otimes SU(2)_{L}\otimes SU(3)_{R}\otimes U(1)_{X}$ gauge model_, _Phys. Rev. D_ 95 (2017) 075034 [1609.03444].
  * (69) P. V. Dong, D. T. Huong, F. S. Queiroz, J. W. F. Valle and C. A. Vaquera-Araujo, _The Dark Side of Flipped Trinification_ , _JHEP_ 04 (2018) 143 [1710.06951].
  * (70) P. Van Dong, D. T. Huong, D. A. Camargo, F. S. Queiroz and J. W. F. Valle, _Asymmetric Dark Matter, Inflation and Leptogenesis from $B-L$ Symmetry Breaking_, _Phys. Rev. D_ 99 (2019) 055040 [1805.08251].
  * (71) C. H. Nam, D. Van Loi, L. X. Thuy and P. Van Dong, _Multicomponent dark matter in noncommutative $B-L$ gauge theory_, _JHEP_ 12 (2020) 029 [2006.00845].
  * (72) P. Van Dong, C. H. Nam and D. Van Loi, _Canonical seesaw implication for two-component dark matter_ , _Phys. Rev. D_ 103 (2021) 095016 [2007.08957].
  * (73) D. Van Loi, P. Van Dong and D. Van Soa, _Neutrino mass and dark matter from an approximate B $-$ L symmetry_, _JHEP_ 05 (2020) 090 [1911.04902].
  * (74) D. Van Loi, C. H. Nam and P. Van Dong, _Dark matter in the fully flipped 3-3-1-1 model_ , 2012.10979.
  * (75) P. A. M. Dirac, _Quantised singularities in the electromagnetic field,_ , _Proc. Roy. Soc. Lond. A_ 133 (1931) 60.
  * (76) K. S. Babu and R. N. Mohapatra, _Quantization of Electric Charge From Anomaly Constraints and a Majorana Neutrino_ , _Phys. Rev. D_ 41 (1990) 271.
  * (77) R. Foot, G. C. Joshi, H. Lew and R. R. Volkas, _Charge quantization in the standard model and some of its extensions_ , _Mod. Phys. Lett. A_ 5 (1990) 2721.
  * (78) F. Pisano, _A Simple solution for the flavor question_ , _Mod. Phys. Lett._ A11 (1996) 2639 [hep-ph/9609358].
  * (79) A. Doff and F. Pisano, _Charge quantization in the largest leptoquark bilepton chiral electroweak scheme_ , _Mod. Phys. Lett._ A14 (1999) 1133 [hep-ph/9812303].
  * (80) C. A. de Sousa Pires and O. P. Ravinez, _Charge quantization in a chiral bilepton gauge model_ , _Phys. Rev._ D58 (1998) 035008 [hep-ph/9803409].
  * (81) C. A. de Sousa Pires, _Remark on the vector - like nature of the electromagnetism and the electric charge quantization_ , _Phys. Rev._ D60 (1999) 075013 [hep-ph/9902406].
  * (82) P. V. Dong and H. N. Long, _Electric charge quantization in SU(3)(C) x SU(3)(L) x U(1)(X) models_ , _Int. J. Mod. Phys._ A21 (2006) 6677 [hep-ph/0507155].
  * (83) Fermi-LAT collaboration, _Fermi LAT Search for Dark Matter in Gamma-ray Lines and the Inclusive Photon Spectrum_ , _Phys. Rev. D_ 86 (2012) 022002 [1205.2739].
  * (84) T. Appelquist, B. A. Dobrescu and A. R. Hopper, _Nonexotic Neutral Gauge Bosons_ , _Phys. Rev. D_ 68 (2003) 035012 [hep-ph/0212073].
  * (85) M. Carena, A. Daleo, B. A. Dobrescu and T. M. P. Tait, _$Z^{\prime}$ gauge bosons at the Tevatron_, _Phys. Rev. D_ 70 (2004) 093009 [hep-ph/0408098].
  * (86) ATLAS collaboration, _Search for new high-mass phenomena in the dilepton final state using 36 fb -1 of proton-proton collision data at $\sqrt{s}=13$ TeV with the ATLAS detector_, _JHEP_ 10 (2017) 182 [1707.02424].
  * (87) A. E. Faraggi and V. M. Mehta, _Proton Stability and Light $Z^{\prime}$ Inspired by String Derived Models_, _Phys. Rev. D_ 84 (2011) 086006 [1106.3082].
  * (88) D. T. Huong and P. V. Dong, _Left-right asymmetry and 750 GeV diphoton excess_ , _Phys. Rev. D_ 93 (2016) 095019 [1603.05146].
  * (89) XENON collaboration, _Excess electronic recoil events in XENON1T_ , _Phys. Rev. D_ 102 (2020) 072004 [2006.09721].
  * (90) K. Kannike, M. Raidal, H. Veermäe, A. Strumia and D. Teresi, _Dark Matter and the XENON1T electron recoil excess_ , _Phys. Rev. D_ 102 (2020) 095002 [2006.10735].
  * (91) B. Fornal, P. Sandick, J. Shu, M. Su and Y. Zhao, _Boosted Dark Matter Interpretation of the XENON1T Excess_ , _Phys. Rev. Lett._ 125 (2020) 161804 [2006.11264].
  * (92) R. Primulando, J. Julio and P. Uttayarat, _Collider Constraints on a Dark Matter Interpretation of the XENON1T Excess_ , _Eur. Phys. J. C_ 80 (2020) 1084 [2006.13161].
  * (93) L. Su, W. Wang, L. Wu, J. M. Yang and B. Zhu, _Atmospheric Dark Matter and Xenon1T Excess_ , _Phys. Rev. D_ 102 (2020) 115028 [2006.11837].
  * (94) Q.-H. Cao, R. Ding and Q.-F. Xiang, _Searching for sub-MeV boosted dark matter from xenon electron direct detection_ , _Chin. Phys. C_ 45 (2021) 045002 [2006.12767].
  * (95) H. Alhazmi, D. Kim, K. Kong, G. Mohlabeng, J.-C. Park and S. Shin, _Implications of the XENON1T Excess on the Dark Matter Interpretation_ , _JHEP_ 05 (2021) 055 [2006.16252].
  * (96) L. Delle Rose, G. Hütsi, C. Marzo and L. Marzola, _Impact of loop-induced processes on the boosted dark matter interpretation of the XENON1T excess_ , _JCAP_ 02 (2021) 031 [2006.16078].
  * (97) P. Ko and Y. Tang, _Semi-annihilating $Z_{3}$ dark matter for XENON1T excess_, _Phys. Lett. B_ 815 (2021) 136181 [2006.15822].
  * (98) U. K. Dey, T. N. Maity and T. S. Ray, _Prospects of Migdal Effect in the Explanation of XENON1T Electron Recoil Excess_ , _Phys. Lett. B_ 811 (2020) 135900 [2006.12529].
  * (99) N. Sabti, J. Alvey, M. Escudero, M. Fairbairn and D. Blas, _Refined Bounds on MeV-scale Thermal Dark Sectors from BBN and the CMB_ , _JCAP_ 01 (2020) 004 [1910.01649].
  * (100) A. Joglekar, N. Raj, P. Tanedo and H.-B. Yu, _Relativistic capture of dark matter by electrons in neutron stars_ , _Phys. Lett._ B (2020) 135767 [1911.13293].
  * (101) P. Ilten, Y. Soreq, M. Williams and W. Xue, _Serendipity in dark photon searches_ , _JHEP_ 06 (2018) 004 [1801.04847].
  * (102) M. Bauer, P. Foldenauer and J. Jaeckel, _Hunting All the Hidden Photons_ , _JHEP_ 07 (2018) 094 [1803.05466].
  * (103) P. V. Dong, T. P. Nguyen and D. V. Soa, _3-3-1 model with inert scalar triplet_ , _Phys. Rev. D_ 88 (2013) 095014 [1308.4097].
  * (104) P. V. Dong, N. T. K. Ngan and D. V. Soa, _Simple 3-3-1 model and implication for dark matter_ , _Phys. Rev. D_ 90 (2014) 075019 [1407.3839].
  * (105) P. V. Dong, D. T. Huong, F. S. Queiroz and N. T. Thuy, _Phenomenology of the 3-3-1-1 model_ , _Phys. Rev. D_ 90 (2014) 075021 [1405.2591].
  * (106) P. V. Dong, C. S. Kim, D. V. Soa and N. T. Thuy, _Investigation of Dark Matter in Minimal 3-3-1 Models_ , _Phys. Rev. D_ 91 (2015) 115019 [1501.04385].
  * (107) P. Van Dong, N. T. K. Ngan, T. D. Tham, L. D. Thien and N. T. Thuy, _Phenomenology of the simple 3-3-1 model with inert scalars_ , _Phys. Rev. D_ 99 (2019) 095031 [1512.09073].
  * (108) D. T. Huong and P. V. Dong, _Neutrino masses and superheavy dark matter in the 3-3-1-1 model_ , _Eur. Phys. J. C_ 77 (2017) 204 [1605.01216].
  * (109) D. T. Huong, P. V. Dong, N. T. Duy, N. T. Nhuan and L. D. Thien, _Investigation of Dark Matter in the 3-2-3-1 Model_ , _Phys. Rev. D_ 98 (2018) 055033 [1802.10402].
  * (110) D. T. Huong, D. N. Dinh, L. D. Thien and P. Van Dong, _Dark matter and flavor changing in the flipped 3-3-1 model_ , _JHEP_ 08 (2019) 051 [1906.05240].
  * (111) D. N. Dinh, D. T. Huong, N. T. Duy, N. T. Nhuan, L. D. Thien and P. Van Dong, _Flavor changing in the flipped trinification_ , _Phys. Rev. D_ 99 (2019) 055005 [1901.07969].
  * (112) P. V. Dong and D. T. Huong, _Left-right model for dark matter_ , _Commun. in Phys._ 28 (2018) 21 [1610.02642].
  * (113) P. V. Dong, D. Q. Phong, D. V. Soa and N. C. Thao, _The economical 3-3-1 model revisited_ , _Eur. Phys. J. C_ 78 (2018) 653 [1706.06152].
  * (114) P. Van Dong and D. Van Loi, _Asymmetric matter from $B-L$ symmetry breaking_, _Eur. Phys. J. C_ 80 (2020) 1137 [2001.03862].
  * (115) XENON collaboration, _First Dark Matter Search Results from the XENON1T Experiment_ , _Phys. Rev. Lett._ 119 (2017) 181301 [1705.06655].
  * (116) XENON collaboration, _Dark Matter Search Results from a One Ton-Year Exposure of XENON1T_ , _Phys. Rev. Lett._ 121 (2018) 111302 [1805.12562].