aainstitutetext: Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117
Heidelberg, Germanybbinstitutetext: Institut für Theoretische Physik,
Westfälische Wilhelms-Universität Münster,
Wilhelm-Klemm-Str. 9, 48149 Münster, Germany

# Anomaly-free dark matter models with one-loop neutrino masses and a gauged
U(1) symmetry

T. de Boer b M. Klasen b and S. Zeinstra111Now with Demcon Multiphysics.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We systematically study and classify scotogenic models with a local U(1) gauge
symmetry. These models give rise to radiative neutrino masses and a stable
dark matter candidate, but avoid the theoretical problems of global and
discrete symmetries. We restrict the dark sector particle content to up to
four scalar or fermionic SU(2) singlets, doublets or triplets and use
theoretical arguments based on anomaly freedom, Lorentz and gauge symmetry to
find all possible charge assignments of these particles. The U(1) symmetry can
be broken by a new Higgs boson to a residual discrete symmetry, that still
stabilizes the dark matter candidate. We list the particle content and charge
assignments of all non-equivalent models. Specific examples in our class of
models that have been studied previously in the literature are the U(1)D
scotogenic and singlet-triplet scalar models breaking to $Z_{2}$. We also
briefly discuss the new phenomenological aspects of our model arising from the
presence of a new massless dark photon or massive $Z^{\prime}$ boson as well
as the additional Higgs boson.

††preprint: MS-TP-23-44

##  1 Introduction

The discovery of neutrino oscillations indicates that the neutrino masses are
non-zero, but extremely small Zyla:2020zbs ; Esteban:2020cvm . While the mass
generation of most Standard Model (SM) particles can be understood within the
SM Higgs mechanism following the discovery of a scalar boson of mass 125 GeV
by ATLAS ATLAS:2012yve and CMS CMS:2012qbp at the LHC, the SM can not
convincingly explain this smallness of the neutrino masses. On the other hand,
cosmological observations at many different length scales provide ample
evidence for the existence of cold Dark Matter (DM) in the Universe
Zyla:2020zbs ; Klasen:2015uma , although its exact nature remains an open
question. In radiative seesaw models, these two open ends are intimately
connected. There, neutrinos interact with the Higgs boson indirectly, and
neutrino masses are generated at the loop level Bonnet:2012kz ; Cai:2017jrq .
The particles in the loop can then contain one or more DM candidates in the
form of Weakly Interacting Massive Particles (WIMPs) Restrepo:2013aga . The
scotogenic model proposed by Ernest Ma Ma:2006km , which connects a second,
inert Higgs doublet with additional, sterile neutrinos, is a well-known
example. It has been widely studied in the literature in its original
formulation Klasen:2013jpa ; Vicente:2014wga ; Toma:2013zsa ; deBoer:2020yyw ;
deBoer:2021pon and more generalized versions Escribano:2020iqq . Radiative
seesaw models containing other multiplets have also been studied Esch:2016jyx
; Esch:2018ccs ; Fiaschi:2018rky ; deBoer:2021xjs .

In order to prevent a tree-level seesaw contribution and to guarantee DM
stability, a discrete $Z_{2}$ symmetry is often imposed on the new multiplets.
This particular choice is, however, not very well justified theoretically. A
similar role can, e.g., be played by a $Z_{3}$ Ma:2007gq ; Belanger:2012zr ;
Aoki:2014cja , $Z_{4}$ BenTov:2012tg or a higher $Z_{n}$ Belanger:2014bga
symmetry, which can even be linked to an $A_{4}$ neutrino family symmetry
Ma:2008ym ; BenTov:2012tg ; Ma:2015roa . In addition, there exist several
theoretical arguments why discrete symmetries should have a dynamical origin.
One motivation is that the spontaneous breaking of discrete symmetries to non-
trivial subgroups leads to the formation of domain walls, which have severe
cosmological problems Zeldovich:1974uw . These can be avoided in models with a
global U(1) symmetry, which have indeed been used to generate neutrino masses
at the one- Arhrib:2015dez ; Fiaschi:2019evv and two-loop Lindner:2011it ;
Bonilla:2016diq level. When the global U(1) is spontaneously broken, a
discrete $Z_{n}$ remains AristizabalSierra:2014irc . However, a second reason
why global symmetries should be of dynamical origin is that they are violated
by quantum gravity effects Ali:2020znq . This argument applies even to
continuous global symmetries Bekenstein:1972ky ; Abbott:1989jw ;
Kallosh:1995hi .

It is therefore the aim of this paper to systematically study and classify
scotogenic models, where the tree-level seesaw mechanism is forbidden by and
the stability of DM is obtained from a local U(1) symmetry. Following the
principle of Occham’s razor, we focus on models where the neutrino masses are
generated at the one-loop level and which are anomaly-free without the
postulation of additional particles. In Secs. 2 and 3 we give detailed
arguments on how our models are constructed. Our main result, i.e. the list of
models is given in table format in Sec. 3. The corresponding classification of
models with a discrete $Z_{2}$ symmetry and up to four dark multiplets has
been performed in Ref. Restrepo:2013aga . Specific examples in our class of
models that have been studied previously in the literature are the U(1)D
scotogenic Ma:2013yga and singlet-triplet scalar Brdar:2013iea models breaking
to $Z_{2}$. Models that rely on assumptions different from ours are, of
course, also possible, e.g. models with a gauged U(1)B-L symmetry breaking to
$Z_{n}$ Li:2010rb ; Ho:2016aye ; Bonilla:2018ynb ; Calle:2018ovc ;
Jana:2019mgj ; Dasgupta:2019rmf or with a gauged U(1)L symmetry breaking to
$Z_{3}$ Ma:2019coj ; Ma:2021fre , but they are either not anomaly-free or
require a larger number of new multiplets. A study similar to ours for Dirac
neutrinos has been performed in Ref. Bernal:2021ezl .

The addition of a local U(1) symmetry gives rise to a new vector boson, which
is a massless dark photon Ackerman:2008kmp in case of an unbroken symmetry or
a massive $Z^{\prime}$ boson Langacker:2008yv ; Fuks:2007gk ; Jezo:2014kla ;
Jezo:2014wra ; Bonciani:2015hgv ; Altakach:2020ugg ; Buras:2021btx ;
Farzan:2017xzy ; Klasen:2016qux ; Okada:2016gsh ; Camargo:2019mml if the
U$(1)$ is spontaneously broken. Even with the SM fields uncharged under the
new U$(1)$, the new gauge boson can be searched for via the kinetic mixing
portal. Giving mass to the $Z^{\prime}$ boson by the Higgs mechanism also
gives rise to a new scalar boson and modified Higgs couplings. In Sec. 4 we
briefly discuss the main aspects of the phenomenology. Finally we draw our
conclusions in Sec. 5.

##  2 Theoretical considerations

Figure 1: Topologies T3, T1-1, T1-2, T1-3 of scotogenic neutrino mass
generation at one loop Restrepo:2013aga .

Majorana neutrino masses can be generated through the effective $d=5$ Weinberg
operator Weinberg:1979sa

$\displaystyle\mathcal{L}\supset-\frac{c_{\alpha\beta}}{\Lambda}\left(L^{\alpha}H\right)\left(L^{\beta}H\right)+{\rm
h.c.},$ (1)

where $L^{\alpha,\beta}$ is the left-handed Weyl fermion denoting the SM
lepton doublet of flavor $\alpha,\beta$, $H$ is the SM Higgs doublet,
$\Lambda$ is the mass scale of the new particles and $c_{\alpha\beta}$ is
obtained by integrating out the new fields. After electroweak symmetry
breaking (EWSB), this operator turns into a Majorana mass term for the SM
neutrinos

$\displaystyle\mathcal{L}\supset-\frac{c_{\alpha\beta}v_{H}^{2}}{2\Lambda}\nu_{L}^{\alpha}\nu_{L}^{\beta}+{\rm
h.c.},$ (2)

which is suppressed by the scale of new physics. Tree level realizations of
the Weinberg operator are the well known type I-III seesaw mechanisms.
Radiative seesaw models are the one (or more) loop realizations Bonnet:2012kz
; Cai:2017jrq , which naturally suppress the neutrino mass and decrease the
scale of new physics from the GUT scale to the TeV scale.

The possible realizations of this operator at one loop with a viable DM
candidate have been systematically classified in Ref. Restrepo:2013aga with
the assumptions that the number of new fields is $\leq 4$ and that they are
singlets under SU(3)C and singlets, doublets or triplets under SU(2)L. A
discrete $Z_{2}$ symmetry was imposed to prevent tree-level neutrino masses
and to stabilize the DM. The four different possible topologies are shown in
Fig. 1.

The objective of this work is to replace the $Z_{2}$ symmetry by a gauged
U$(1)_{X}$ symmetry, which may or may not be broken. In order to keep the
models minimal, we increase the particle content of Refs. Bonnet:2012kz ;
Restrepo:2013aga only by the neutral extra gauge boson $Z^{\prime}$ and, for a
spontaneously broken U(1)X, by an additional scalar field $\zeta$ of charge
$X_{\zeta}$. This scalar must be a singlet under the SM gauge group, as it
should only break the new gauge symmetry.

###  2.1 Gauge invariance of the Weinberg operator

Several theoretical constraints must be taken into account when using a U(1)X
gauge symmetry to stabilize DM in radiative seesaw models. The first condition
is that the Weinberg operator should be allowed and gauge invariant under the
full gauge group
SU(3)${}_{C}\times$SU(2)${}_{L}\times$U(1)${}_{Y}\times$U(1)X. This holds even
if U(1)X is spontaneously broken (see below) and implies in particular for the
charges of the left-handed SM leptons $X_{L}$ and of the SM Higgs boson
$X_{H}$ under the new gauge group U(1)X that

$\displaystyle X_{L}+X_{H}$ $\displaystyle=0.$ (3)

Here we assume that the new symmetry does not distinguish between different
generations.

###  2.2 Gauge invariance of the SM Yukawa interactions

Second, the SM Yukawa interactions

$\displaystyle\mathcal{L}\supset-\frac{y_{d}}{\sqrt{2}}H^{\dagger}Qd_{R}^{c}-\frac{y_{u}}{\sqrt{2}}HQu_{R}^{c}-\frac{y_{e}}{\sqrt{2}}H^{\dagger}Le_{R}^{c}+{\rm
h.c.}$ (4)

should remain gauge invariant after adding the U(1)X symmetry. This implies
for the U(1)X charges of the left-handed quark and lepton doublets $Q,L$ and
the right-handed quark and lepton singlets $u_{R},d_{R},e_{R}$ that

$\displaystyle X_{Q}-X_{H}+X_{d_{R}^{c}}$ $\displaystyle=0,$ (5)
$\displaystyle X_{Q}+X_{H}+X_{u_{R}^{c}}$ $\displaystyle=0,$ (6)
$\displaystyle X_{L}-X_{H}+X_{e_{R}^{c}}$ $\displaystyle=0.$ (7)

Together with Eq. (3), this simplifies to

$\displaystyle X_{d_{R}^{c}}$ $\displaystyle=-X_{Q}-X_{L},$ (8) $\displaystyle
X_{u_{R}^{c}}$ $\displaystyle=-X_{Q}+X_{L},$ (9) $\displaystyle X_{e_{R}^{c}}$
$\displaystyle=-2X_{L}.$ (10)

###  2.3 Anomaly freedom

Third, we require our models to be anomaly-free with the given particle
content. The conditions for the gauge (and gravity) anomalies to cancel are
listed in Tab. 1.

Table 1: Conditions for gauge anomaly cancellation. $Y_{\psi}$ denotes the SM U(1)Y hypercharge, $X_{\psi}$ the new U(1)X charge of the left-handed Weyl fermion $\psi$. For SU(3)C we consider singlets $1_{3}$ and triplets $3_{3}$, while for SU(2)L we consider singlets $1_{2}$, doublets $2_{2}$ and triplets $3_{2}$. The sums run over all components. Anomaly | Constraint
---|---
SU$(3)_{C}^{2}\times$U$(1)_{Y}$ | $\sum_{\psi\in 3_{3}}Y_{\psi}=0$
SU$(3)_{C}^{2}\times$U$(1)_{X}$ | $\sum_{\psi\in 3_{3}}X_{\psi}=0$
SU$(2)_{L}^{2}\times$U$(1)_{Y}$ | $\sum_{\psi\in 2_{2}}Y_{\psi}+4\sum_{\psi\in 3_{2}}Y_{\psi}=0$
SU$(2)_{L}^{2}\times$U$(1)_{X}$ | $\sum_{\psi\in 2_{2}}X_{\psi}+4\sum_{\psi\in 3_{2}}X_{\psi}=0$
U$(1)_{Y}^{3}$ | $\sum_{\psi}Y_{\psi}^{3}=0$
U$(1)_{X}^{3}$ | $\sum_{\psi}X_{\psi}^{3}=0$
U$(1)_{Y}^{2}\times$U$(1)_{X}$ | $\sum_{\psi}Y_{\psi}^{2}X_{\psi}=0$
U$(1)_{X}^{2}\times$U$(1)_{Y}$ | $\sum_{\psi}X_{\psi}^{2}Y_{\psi}=0$
grav${}^{2}\times$U$(1)_{Y}$ | $\sum_{\psi}Y_{\psi}=0$
grav${}^{2}\times$U$(1)_{X}$ | $\sum_{\psi}X_{\psi}=0$

Using Eqs. (8)-(10), the SM contributions to the new gauge anomalies can be
expressed as:

$\displaystyle\text{SU}(3)_{C}^{2}\times\text{U}(1)_{X}$ $\displaystyle:\quad
0,$ (11) $\displaystyle\text{SU}(2)_{L}^{2}\times\text{U}(1)_{X}$
$\displaystyle:\quad 6\left[3X_{Q}+X_{L}\right],$ (12)
$\displaystyle\text{U}(1)_{X}^{3}$
$\displaystyle:\quad-6X_{L}^{2}\left[3X_{Q}+X_{L}\right],$ (13)
$\displaystyle\text{U}(1)_{Y}^{2}\times\text{U}(1)_{X}$
$\displaystyle:\quad-\frac{3}{2}\left[3X_{Q}+X_{L}\right],$ (14)
$\displaystyle\text{U}(1)_{X}^{2}\times\text{U}(1)_{Y}$ $\displaystyle:\quad
6X_{L}\left[3X_{Q}+X_{L}\right],$ (15)
$\displaystyle\text{grav}^{2}\times\text{U}(1)_{X}$ $\displaystyle:\quad 0.$
(16)

They must therefore either vanish or be canceled by contributions of the new
fields. In addition, the Witten anomaly must cancel, which is the case for an
even number of fermion doublets.

###  2.4 New fermions must be vectorlike

Since contributions to the anomalies from vector-like fermions cancel among
their left- and right-handed components, only fermions that are not vector-
like must be considered in more detail. As we will see, anomaly cancellation
requires all our new fermions to be vector-like.

Models with one new fermion (T3, T1-1)

If there is a single new fermion $\psi$, that is not a priori part of a
vector-like fermion, $\psi$ must be made vector-like. From the Witten anomaly
it is clear that if $\psi$ is a doublet, it must be made vector-like as we
must add a second doublet. The two Weyl fermions then combine to a single
Dirac fermion. If $\psi$ is a singlet or a triplet, the anomalies associated
with the SM hypercharge must cancel. As the SM and other new vector-like
fermions do not contribute to the hypercharge anomalies, $\psi$ must either be
vector-like or have zero hypercharge. However, if a singlet or triplet has
zero hypercharge, it must have a non-zero U$(1)_{X}$ charge, as otherwise
seesaw types I or III are possible. This holds also if the SM neutrino is
charged under U$(1)_{X}$, as $X_{L}=-X_{H}$. In this case the
$\text{grav}^{2}\times\text{U}(1)_{X}$ anomaly, which has no contributions
from the SM (cf. Eq. (16)), must cancel. This is only possible if $\psi$ is
vector-like.

Models with two new fermions (T1-2)

If there are two (or more) new fermions in the neutrino loop, at least one of
them must be a doublet and at least one must be a singlet or triplet, since
they must couple to the SM Higgs doublet in a gauge-invariant way. As the
doublet must be vector-like to cancel the Witten anomaly, it follows from the
arguments made above that the second fermion must also be vector-like.

Models with three new fermions (T1-3)

Again, at least one fermion must be a (vector-like) doublet in order to couple
to the SM Higgs boson. Several cases have to be distinguished. (i) In the
neutrino loop, three doublets can not couple to the two SM Higgs bosons in a
gauge-invariant way. (ii) If there are two vector-like doublets, the third
singlet/triplet must also be vector-like (see above). (iii) If the two
doublets are a priori not vector-like, they must have opposite hypercharge to
cancel the grav${}^{2}\times$U$(1)_{Y}$ and SU(2)${}_{L}^{2}\times$U$(1)_{Y}$
anomalies. These conditions also impose that the third singlet/triplet must be
vector-like or have zero hypercharge. With this result, the U(1)X anomaly
conditions then require the two doublets to be vector-like or have opposite
U(1)X charge, so that they can be identified with each other. It also follows
from above that the third singlet/triplet must also be vector-like. (iv) If
one fermion is a (vector-like) doublet and the other two are singlets or
triplets, the latter must be vector-like, identified with each other or have
zero hypercharge to cancel the SM hypercharge anomalies. Then there is no BSM
contribution to the $\text{U}(1)_{X}^{2}\times\text{U}(1)_{Y}$ anomaly. Since
Eq. (15) implies that the SM contributions must cancel, also the BSM
contributions must cancel among themselves. As singlets and triplets with zero
hypercharge must have a U$(1)_{X}$ charge in order to avoid seesaw types I and
III (see above), one finds that they must be vector-like or identified with
each other.

To sum up, we find that all new fermions must be vector-like or be combined
with another fermion to a vector-like fermion, whose components can form a
Dirac mass term.

###  2.5 SM fermions must be neutral

Since all new fermions must be vector-like, the expressions in Eqs. (12)-(15)
must all vanish. This is only possible with the two orthogonal solutions that
the new charges of the SM particles are either proportional to their SM
hypercharges or the $B-L$ charges Schwartz:2013pla

$\displaystyle X_{L}=-X_{e_{R}^{c}}=-X_{\nu_{R}^{c}}=-1,\qquad
X_{Q}=-X_{u_{R}^{c}}=-X_{d_{R}^{c}}=\frac{1}{3},$ (17)

where $X_{\nu_{R}^{c}}$ is the new charge of a right-handed neutrino. Since we
do not allow for a right-handed neutrino to avoid Dirac neutrino masses and
the tree-level seesaw mechanism, this possibility is ruled out. Note also that
the Weinberg operator violates the $B-L$ symmetry.

The only possibility is therefore to assign the SM particles a U(1)X charge
proportional to their hypercharge, which has a similar effect as gauge kinetic
mixing, since the SM particles then couple also to the new gauge boson. Since
for all topologies the SM hypercharge is conserved and $\zeta$ has zero
hypercharge, the charges of the new particles running in the neutrino loop are
shifted at each vertex by

$\displaystyle X_{\rm out}=X_{\rm in}+\lambda Y_{\rm SM}.$ (18)

One can then change the basis of the two U(1) groups and shift the U$(1)_{X}$
charges of the SM particles by $-\lambda Y_{\rm SM}$, which leaves us with the
case that the SM is uncharged under U$(1)_{X}$, whereas all new fields keep
their original charges.

###  2.6 New scalars and fermions must be charged

The stability of DM requires that neither the DM nor any other new particle in
the neutrino loop is uncharged under the new U(1)X symmetry. Otherwise the DM
particle could decay into SM particles either directly or through a diagram
resulting from cutting the loop at the propagators of the DM candidate and the
uncharged particle.

The $Z^{\prime}$ gauge boson of the abelian group U(1)X remains of course
uncharged, whereas the new scalar $\zeta$ must be charged in order to break
the new symmetry spontaneously.

###  2.7 Unbroken U(1)X symmetry

Since the SM is uncharged, all new particles in the neutrino loop can have the
same U$(1)_{X}$ charge. This is the case for an unbroken U(1)X symmetry, but
also if U(1)X is spontaneously broken through a vacuum expectation value (VEV)
$v_{\zeta}$ much smaller than the scale of new physics $\Lambda$. An even
number of new scalars $\zeta$ can then in principle couple to the neutrino
loop and lead to the higher-dimensional effective operators

$\displaystyle\mathcal{L}\supset-\frac{c_{\alpha\beta}}{\Lambda^{1+2n}}\left(L^{\alpha}H\right)\left(L^{\beta}H\right)|\zeta|^{2n}+{\rm
h.c.}$ (19)

However, when $\zeta$ obtains a VEV $v_{\zeta}\ll\Lambda$, these contributions
are suppressed by at least one power of $v_{\zeta}^{2}/\Lambda^{2}$, and in
the limit $v_{\zeta}\to 0$ only the Weinberg operator is relevant for neutrino
mass generation.

###  2.8 Broken U(1)X symmetry

In contrast, when $v_{\zeta}\approx\Lambda$, the higher-dimensional effective
operators are of equal importance as the Weinberg operator and must also be
taken into account. In this case we can still use the results of Ref.
Restrepo:2013aga as a complete classification.

However, mass dimension three vertices appearing in the topologies T1-1 and
T1-2 after the breaking of the U(1)X symmetry can violate the U$(1)_{X}$
charge by one unit of $X_{\zeta}$ through the vertex $H\phi\phi^{\prime}\zeta$
or similar terms with conjugate fields allowed by gauge (and Lorentz)
invariance. This may lead to additional charge assignments.

Propagators (appearing in all topologies) can also violate the U$(1)_{X}$
charge by one unit of $X_{\zeta}$ for fermions and by one or two units for
scalars through the vertices $\psi\psi^{\prime}\zeta$,
$\phi\phi^{\prime}\zeta$ and $\phi\phi^{\prime}\zeta\zeta$. The fields
$\psi(\phi)$ and $\psi^{\prime}(\phi^{\prime})$ must then be in conjugate
SU(3)${}_{C}\times$SU(2)${}_{L}\times$U(1)Y representations. For zero
hypercharge, one may also have $\psi=\psi^{\prime}$ and $\phi=\phi^{\prime}$,
which implies $X_{\psi/\phi}=X_{\zeta}/2$. In the following, we set
$X_{\zeta}=2$ without loss of generality, since one can always rescale the
U$(1)_{X}$ gauge coupling.

Note that in general the new scalars $\phi$ and fermions $\psi$ can not have
the same charge as $\zeta$. For scalars $\phi$, vertices like $H\phi\zeta$ for
doublets or $H\phi\zeta H$ for singlets/triplets would otherwise induce DM
mixing with the SM Higgs or DM decay after U(1)X symmetry breaking. This also
implies that the term $\phi\phi\zeta\zeta$ is not allowed. For fermions
$\psi$, the fermionic vertices in the neutrino loops always imply a scalar
with the same charge, which brings us back to the argument for scalars.

###  2.9 Residual global symmetry

Before U$(1)_{X}$ breaking, the Lagrangian is invariant under the local gauge
transformation $e^{i\alpha(x)X_{\phi,\psi,\zeta}}$. For fixed
$\alpha={2\pi/X_{\zeta}}$, $\zeta$ transforms trivially, so that after
U$(1)_{X}$ breaking the Lagrangian is still invariant under the global
transformation $e^{i2\pi{X_{\phi,\psi}/X_{\zeta}}}$. For a fixed ratio
$X_{\phi}/X_{\zeta}=r$, the charges of the other fields vary only by $n\in Z$
units of $X_{\zeta}$. It is obvious that this variation does not affect the
global invariance of the Lagrangian, since $e^{i2\pi(r\pm n)}=e^{i2\pi r}$.
The ratio $r$, however, does, so that for $r={1\over 2},{1\over 3},{1\over 4}$
only a global $Z_{2,3,4}$ symmetry remains. Depending on the model, the
residual symmetry can also be larger, in particular a global $U(1)_{X}$. The
models with a residual $Z_{2}$ symmetry are similar to those in Ref.
Restrepo:2013aga apart from the new gauge and Higgs bosons and the fact that
all scalars are complex and all fermions vector-like. The residual symmetry
still stabilizes DM and prevents a tree-level seesaw mechanism.

##  3 One-loop scotogenic models with a local U(1) symmetry

We will now discuss the possible charge assignments in our models. Recall that
for all topologies and all choices of SU(2)L multiplets and SM hypercharge
parameter $\alpha$, we can assign the same U$(1)_{X}$ charge $\beta$ to the
new scalar and fermion fields. For a broken U(1)X with
$v_{\zeta}\approx\Lambda$, additional possibilities are found by allowing
U$(1)_{X}$ violation in vertices and propagators, leading to particle mixing.
These cases are discussed individually. At the end of each subsection, we give
a list of non-equivalent models of the respective topology. Models that are
inconsistent with direct detection bounds have been omitted.

###  3.1 T1-1

First, we can always have the assignment

$\displaystyle X_{\varphi}=X_{\phi^{\prime}}=X_{\psi}=X_{\phi}=\beta.$ (20)

Second, in this topology both vertices that couple to the SM Higgs boson can
violate U$(1)_{X}$ through an additional coupling of $\zeta$. We can therefore
also have

$\displaystyle X_{\varphi}\pm
X_{\zeta}=X_{\phi^{\prime}}=X_{\psi}=X_{\phi}=\beta.$ (21)

Third, if $\zeta$ couples to the propagators, one may find additional
assignments for a subset of SM hypercharge parameters $\alpha$, which we
discuss individually.

#### $\alpha=1$

$\phi^{\prime}$ has zero hypercharge. All new charge assignments are
equivalent to those already found once one redefines
$\phi^{\prime}\to\left(\phi^{\prime}\right)^{\dagger}$.

#### $\alpha=-1$

$\phi$ has zero hypercharge. All new charge assignments are equivalent to
those already found once one redefines $\phi\to\left(\phi\right)^{\dagger}$.

#### $\alpha=0$

Both $\psi$ and $\varphi$ have zero hypercharge. One finds one new non-
equivalent charge assignment given by

$\displaystyle
X_{\varphi}=-X_{\phi^{\prime}}=X_{\psi}=X_{\phi}=\pm\frac{X_{\zeta}}{2}.$ (22)

$\phi$ and $\phi^{\prime}$ also have opposite hypercharge. If they are in
addition in the same representation of SU$(2)_{L}$, mixing between $\phi$ and
$\left(\phi^{\prime}\right)^{\dagger}$ can be induced by the breaking of
U$(1)_{X}$. This allows for the non-equivalent charge assignments

$\displaystyle X_{\varphi}=X_{\phi^{\prime}}\pm
2X_{\zeta}=X_{\psi}=X_{\phi}=\pm\frac{X_{\zeta}}{2},$ (23) $\displaystyle
X_{\varphi}=X_{\phi^{\prime}}\mp
X_{\zeta}=X_{\psi}=X_{\phi}=\pm\frac{X_{\zeta}}{2},$ (24) $\displaystyle
X_{\varphi}\mp X_{\zeta}=X_{\phi^{\prime}}\mp
X_{\zeta}=X_{\psi}=X_{\phi}=\pm\frac{X_{\zeta}}{2}.$ (25)

If $\phi$ and $\phi^{\prime}$ are in the same representation of SU$(2)_{L}$
and have opposite U$(1)_{X}$ charge, we can identify them with each other by
defining $\phi^{\prime}=\phi^{\dagger}$. However, in order to have at least
two massive neutrinos (see below), there must be then two generations of
either $\psi$ or $\phi$. The latter case is equivalent to the case where the
fields are not identified with each other.

Table 2: Non-equivalent models of topology T1-1. $X_{\zeta}$ is normalized to 2. Model | $\alpha$ | $\varphi$ | $\phi^{\prime}$ | $\psi$ | $\phi$ | $X_{\varphi}$ | $X_{\phi^{\prime}}$ | $X_{\psi}$ | $X_{\phi}$
---|---|---|---|---|---|---|---|---|---
T1-1-A | $0$ | $1^{S}_{0}$ | $2^{S}_{-1}$ | $1^{F}_{0}$ | $2^{S}_{1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-1-A | $0$ | $1^{S}_{0}$ | $2^{S}_{-1}$ | $1^{F}_{0}$ | $2^{S}_{1}$ | $\beta\pm 2$ | $\beta$ | $\beta$ | $\beta$
T1-1-A | $0$ | $1^{S}_{0}$ | $2^{S}_{-1}$ | $1^{F}_{0}$ | $2^{S}_{1}$ | $1$ | $-1$ | $1$ | $1$
T1-1-A | $0$ | $1^{S}_{0}$ |  | $1^{F}_{0}$ | $2^{S}_{1}$ | $1$ |  | $1$ | $1$
T1-1-A | $0$ | $1^{S}_{0}$ | $2^{S}_{-1}$ | $1^{F}_{0}$ | $2^{S}_{1}$ | $1$ | $\pm 3$ | $1$ | $1$
T1-1-A | $0$ | $1^{S}_{0}$ | $2^{S}_{-1}$ | $1^{F}_{0}$ | $2^{S}_{1}$ | $3$ | $3$ | $1$ | $1$
T1-1-A | $\pm 2$ | $1^{S}_{\pm 2}$ | $2^{S}_{1,-3}$ | $1^{F}_{\pm 2}$ | $2^{S}_{3,-1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-1-A | $\pm 2$ | $1^{S}_{\pm 2}$ | $2^{S}_{1,-3}$ | $1^{F}_{\pm 2}$ | $2^{S}_{3,-1}$ | $\beta\pm 2$ | $\beta$ | $\beta$ | $\beta$
T1-1-B | $0$ | $1^{S}_{0}$ | $2^{S}_{-1}$ | $3^{F}_{0}$ | $2^{S}_{1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-1-B | $0$ | $1^{S}_{0}$ | $2^{S}_{-1}$ | $3^{F}_{0}$ | $2^{S}_{1}$ | $\beta\pm 2$ | $\beta$ | $\beta$ | $\beta$
T1-1-B | $0$ | $1^{S}_{0}$ | $2^{S}_{-1}$ | $3^{F}_{0}$ | $2^{S}_{1}$ | $1$ | $-1$ | $1$ | $1$
T1-1-B | $0$ | $1^{S}_{0}$ |  | $3^{F}_{0}$ | $2^{S}_{1}$ | $1$ |  | $1$ | $1$
T1-1-B | $0$ | $1^{S}_{0}$ | $2^{S}_{-1}$ | $3^{F}_{0}$ | $2^{S}_{1}$ | $1$ | $\pm 3$ | $1$ | $1$
T1-1-B | $0$ | $1^{S}_{0}$ | $2^{S}_{-1}$ | $3^{F}_{0}$ | $2^{S}_{1}$ | $3$ | $3$ | $1$ | $1$
T1-1-B | $\pm 2$ | $1^{S}_{\pm 2}$ | $2^{S}_{1,-3}$ | $3^{F}_{\pm 2}$ | $2^{S}_{3,-1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-1-B | $\pm 2$ | $1^{S}_{\pm 2}$ | $2^{S}_{1,-3}$ | $3^{F}_{\pm 2}$ | $2^{S}_{3,-1}$ | $\beta\pm 2$ | $\beta$ | $\beta$ | $\beta$
T1-1-C | $\pm 1$ | $2^{S}_{\pm 1}$ | $1^{S}_{0,-2}$ | $2^{F}_{\pm 1}$ | $1^{S}_{2,0}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-1-C | $\pm 1$ | $2^{S}_{\pm 1}$ | $1^{S}_{0,-2}$ | $2^{F}_{\pm 1}$ | $1^{S}_{2,0}$ | $\beta\pm 2$ | $\beta$ | $\beta$ | $\beta$
T1-1-D | $1$ | $2^{S}_{1}$ | $1^{S}_{0}$ | $2^{F}_{1}$ | $3^{S}_{2}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-1-D | $1$ | $2^{S}_{1}$ | $1^{S}_{0}$ | $2^{F}_{1}$ | $3^{S}_{2}$ | $\beta\pm 2$ | $\beta$ | $\beta$ | $\beta$
T1-1-D | $-1$ | $2^{S}_{-1}$ | $1^{S}_{-2}$ | $2^{F}_{-1}$ | $3^{S}_{0}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-1-D | $-1$ | $2^{S}_{-1}$ | $1^{S}_{-2}$ | $2^{F}_{-1}$ | $3^{S}_{0}$ | $\beta\pm 2$ | $\beta$ | $\beta$ | $\beta$
T1-1-F | $\pm 1$ | $2^{S}_{\pm 1}$ | $3^{S}_{0,-2}$ | $2^{F}_{\pm 1}$ | $3^{S}_{2,0}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-1-F | $\pm 1$ | $2^{S}_{\pm 1}$ | $3^{S}_{0,-2}$ | $2^{F}_{\pm 1}$ | $3^{S}_{2,0}$ | $\beta\pm 2$ | $\beta$ | $\beta$ | $\beta$
T1-1-G | $0$ | $3^{S}_{0}$ | $2^{S}_{-1}$ | $1^{F}_{0}$ | $2^{S}_{1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-1-G | $0$ | $3^{S}_{0}$ | $2^{S}_{-1}$ | $1^{F}_{0}$ | $2^{S}_{1}$ | $\beta\pm 2$ | $\beta$ | $\beta$ | $\beta$
T1-1-G | $0$ | $3^{S}_{0}$ | $2^{S}_{-1}$ | $1^{F}_{0}$ | $2^{S}_{1}$ | $1$ | $-1$ | $1$ | $1$
T1-1-G | $0$ | $3^{S}_{0}$ |  | $1^{F}_{0}$ | $2^{S}_{1}$ | $1$ |  | $1$ | $1$
T1-1-G | $0$ | $3^{S}_{0}$ | $2^{S}_{-1}$ | $1^{F}_{0}$ | $2^{S}_{1}$ | $1$ | $\pm 3$ | $1$ | $1$
T1-1-G | $0$ | $3^{S}_{0}$ | $2^{S}_{-1}$ | $1^{F}_{0}$ | $2^{S}_{1}$ | $3$ | $3$ | $1$ | $1$
T1-1-G | $\pm 2$ | $3^{S}_{\pm 2}$ | $2^{S}_{1,-3}$ | $1^{F}_{\pm 2}$ | $2^{S}_{3,-1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-1-G | $\pm 2$ | $3^{S}_{\pm 2}$ | $2^{S}_{1,-3}$ | $1^{F}_{\pm 2}$ | $2^{S}_{3,-1}$ | $\beta\pm 2$ | $\beta$ | $\beta$ | $\beta$
T1-1-H | $0$ | $3^{S}_{0}$ | $2^{S}_{-1}$ | $3^{F}_{0}$ | $2^{S}_{1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-1-H | $0$ | $3^{S}_{0}$ | $2^{S}_{-1}$ | $3^{F}_{0}$ | $2^{S}_{1}$ | $\beta\pm 2$ | $\beta$ | $\beta$ | $\beta$
T1-1-H | $0$ | $3^{S}_{0}$ | $2^{S}_{-1}$ | $3^{F}_{0}$ | $2^{S}_{1}$ | $1$ | $-1$ | $1$ | $1$
T1-1-H | $0$ | $3^{S}_{0}$ |  | $3^{F}_{0}$ | $2^{S}_{1}$ | $1$ |  | $1$ | $1$
T1-1-H | $0$ | $3^{S}_{0}$ | $2^{S}_{-1}$ | $3^{F}_{0}$ | $2^{S}_{1}$ | $1$ | $\pm 3$ | $1$ | $1$
T1-1-H | $0$ | $3^{S}_{0}$ | $2^{S}_{-1}$ | $3^{F}_{0}$ | $2^{S}_{1}$ | $3$ | $3$ | $1$ | $1$
T1-1-H | $\pm 2$ | $3^{S}_{\pm 2}$ | $2^{S}_{1,-3}$ | $3^{F}_{\pm 2}$ | $2^{S}_{3,-1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-1-H | $\pm 2$ | $3^{S}_{\pm 2}$ | $2^{S}_{1,-3}$ | $3^{F}_{\pm 2}$ | $2^{S}_{3,-1}$ | $\beta\pm 2$ | $\beta$ | $\beta$ | $\beta$

All non-equivalent models of topology T1-1 are listed in Tab. 2. As in Ref.
Restrepo:2013aga , the field content is denoted as $L_{Y}^{\mathcal{L}}$,
where $L$ is the type of SU(2)L multiplet (1 for singlet, 2 for doublet, 3 for
triplet), $\mathcal{L}$ denotes scalars ($S$) or fermions ($F$), and $Y\equiv
2(Q-I_{3})$ is the hypercharge. Since $X_{\zeta}$ is set to 2, the parameter
$\beta$ can not take the values $0,\pm 2$. Models with scalar doublets of
charge $\pm 4$ are also not allowed, since the vertex $H\phi\zeta\zeta$ would
then induce mixing of the new scalar with the Higgs boson and make DM
unstable.

###  3.2 T1-2

First, we can always have the assignment

$\displaystyle X_{\psi}=X_{\phi}=X_{\phi^{\prime}}=X_{\psi^{\prime}}=\beta.$
(26)

Second, in this topology there is only one three-scalar vertex which can
violate U$(1)_{X}$. Thus $\zeta$ must also always couple to a propagator. This
leads to additional assignments for a subset of SM hypercharge parameters
$\alpha$, which we discuss individually.

#### $\alpha=-1$

$\psi^{\prime}$ and $\phi$ have zero hypercharge. This allows the new charge
assignment

$\displaystyle
X_{\psi}=X_{\phi}=-X_{\phi^{\prime}}=X_{\psi^{\prime}}=\pm\frac{X_{\zeta}}{2}.$
(27)

#### $\alpha=0$

$\psi$ and $\phi^{\prime}$ have zero hypercharge. We find the non-equivalent
charge assignment

$\displaystyle
X_{\psi}=-X_{\phi}=X_{\phi^{\prime}}=X_{\psi^{\prime}}=\pm\frac{X_{\zeta}}{2}.$
(28)

All non-equivalent models of topology T1-2 are listed in Tab. 3. Again, the
parameter $\beta\neq 0,\pm 2$, and the assignment $\beta=\pm 4$ may yield
problems with DM stability, if the SM Higgs boson mixes with a new scalar
doublet.

Table 3: Non-equivalent models of topology T1-2. $X_{\zeta}$ is normalized to 2. Model | $\alpha$ | $\psi$ | $\phi$ | $\phi^{\prime}$ | $\psi^{\prime}$ | $X_{\psi}$ | $X_{\phi}$ | $X_{\phi^{\prime}}$ | $X_{\psi^{\prime}}$
---|---|---|---|---|---|---|---|---|---
T1-2-A | $0$ | $1^{F}_{0}$ | $2^{S}_{1}$ | $1^{S}_{0}$ | $2^{F}_{1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-2-A | $0$ | $1^{F}_{0}$ | $2^{S}_{1}$ | $1^{S}_{0}$ | $2^{F}_{1}$ | $1$ | $-1$ | $1$ | $1$
T1-2-A | $-2$ | $1^{F}_{-2}$ | $2^{S}_{-1}$ | $1^{S}_{-2}$ | $2^{F}_{-1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-2-B | $0$ | $1^{F}_{0}$ | $2^{S}_{1}$ | $3^{S}_{0}$ | $2^{F}_{1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-2-B | $0$ | $1^{F}_{0}$ | $2^{S}_{1}$ | $3^{S}_{0}$ | $2^{F}_{1}$ | $1$ | $-1$ | $1$ | $1$
T1-2-B | $-2$ | $1^{F}_{-2}$ | $2^{S}_{-1}$ | $3^{S}_{-2}$ | $2^{F}_{-1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-2-D | $1$ | $2^{F}_{1}$ | $1^{S}_{2}$ | $2^{S}_{1}$ | $3^{F}_{2}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-2-D | $-1$ | $2^{F}_{-1}$ | $1^{S}_{0}$ | $2^{S}_{-1}$ | $3^{F}_{0}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-2-D | $-1$ | $2^{F}_{-1}$ | $1^{S}_{0}$ | $2^{S}_{-1}$ | $3^{F}_{0}$ | $1$ | $1$ | $-1$ | $1$
T1-2-F | $1$ | $2^{F}_{1}$ | $3^{S}_{2}$ | $2^{S}_{1}$ | $3^{F}_{2}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-2-F | $-1$ | $2^{F}_{-1}$ | $3^{S}_{0}$ | $2^{S}_{-1}$ | $3^{F}_{0}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-2-F | $-1$ | $2^{F}_{-1}$ | $3^{S}_{0}$ | $2^{S}_{-1}$ | $3^{F}_{0}$ | $1$ | $1$ | $-1$ | $1$

###  3.3 T1-3

First, we can always have the assignment

$\displaystyle X_{\Psi}=X_{\psi^{\prime}}=X_{\phi}=X_{\psi}=\beta.$ (29)

Second, in this topology none of the vertices can violate U$(1)_{X}$. Thus
$\zeta$ must always couple to two propagators. This leads to additional
assignments for

#### $\alpha=0$

Both $\Psi$ and $\phi$ have zero hypercharge. This allows the new charge
assignment

$\displaystyle
X_{\Psi}=-X_{\psi^{\prime}}=X_{\phi}=X_{\psi}=\pm\frac{X_{\zeta}}{2},$ (30)

If $\psi$ and $\psi^{\prime}$ are in the same representation of SU$(2)_{L}$,
mixing between $\psi$ and $\left(\psi^{\prime}\right)^{c}$ can be induced by
the breaking of U$(1)_{X}$. We find no new non equivalent charge assignment.
If $\psi$ and $\psi^{\prime}$ are in the same SU$(2)_{L}$ representation and
have opposite U$(1)_{X}$ charge, we can combine them into a vector-like
multiplet (specifically doublet) instead of making both fields vector-like.
However, in order to have at least two massive neutrinos, there must then be
two generations of either $\phi$ or $\psi$. The latter case is equivalent to
the case where the fields are not identified with each other.

Table 4: Non-equivalent models of topology T1-3. $X_{\zeta}$ is normalized to 2. Model | $\alpha$ | $\Psi$ | $\psi^{\prime}$ | $\phi$ | $\psi$ | $X_{\Psi}$ | $X_{\psi^{\prime}}$ | $X_{\phi}$ | $X_{\psi}$
---|---|---|---|---|---|---|---|---|---
T1-3-A | $0$ | $1^{F}_{0}$ | $2^{F}_{1}$ | $1^{S}_{0}$ | $2^{F}_{-1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-3-A | $0$ | $1^{F}_{0}$ | $2^{F}_{1}$ | $1^{S}_{0}$ | $2^{F}_{-1}$ | $1$ | $-1$ | $1$ | $1$
T1-3-A | $0$ | $1^{F}_{0}$ | $2^{F}_{1}$ | $1^{S}_{0}$ |  | $1$ | $-1$ | $1$ |
T1-3-B | $0$ | $1^{F}_{0}$ | $2^{F}_{1}$ | $3^{S}_{0}$ | $2^{F}_{-1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-3-B | $0$ | $1^{F}_{0}$ | $2^{F}_{1}$ | $3^{S}_{0}$ | $2^{F}_{-1}$ | $1$ | $-1$ | $1$ | $1$
T1-3-B | $0$ | $1^{F}_{0}$ | $2^{F}_{1}$ | $3^{S}_{0}$ |  | $1$ | $-1$ | $1$ |
T1-3-C | $\pm 1$ | $2^{F}_{\pm 1}$ | $1^{F}_{2,0}$ | $2^{S}_{\pm 1}$ | $1^{F}_{0,-2}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-3-D | $1$ | $2^{F}_{1}$ | $1^{F}_{2}$ | $2^{S}_{1}$ | $3^{F}_{0}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-3-D | $-1$ | $2^{F}_{-1}$ | $1^{F}_{0}$ | $2^{S}_{-1}$ | $3^{F}_{-2}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-3-F | $\pm 1$ | $2^{F}_{\pm 1}$ | $3^{F}_{2,0}$ | $2^{S}_{\pm 1}$ | $3^{F}_{0,-2}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-3-G | $0$ | $3^{F}_{0}$ | $2^{F}_{1}$ | $1^{S}_{0}$ | $2^{F}_{-1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-3-G | $0$ | $3^{F}_{0}$ | $2^{F}_{1}$ | $1^{S}_{0}$ | $2^{F}_{-1}$ | $1$ | $-1$ | $1$ | $1$
T1-3-G | $0$ | $3^{F}_{0}$ | $2^{F}_{1}$ | $1^{S}_{0}$ |  | $1$ | $-1$ | $1$ |
T1-3-H | $0$ | $3^{F}_{0}$ | $2^{F}_{1}$ | $3^{S}_{0}$ | $2^{F}_{-1}$ | $\beta$ | $\beta$ | $\beta$ | $\beta$
T1-3-H | $0$ | $3^{F}_{0}$ | $2^{F}_{1}$ | $3^{S}_{0}$ | $2^{F}_{-1}$ | $1$ | $-1$ | $1$ | $1$
T1-3-H | $0$ | $3^{F}_{0}$ | $2^{F}_{1}$ | $3^{S}_{0}$ |  | $1$ | $-1$ | $1$ |

All non-equivalent models of topology T1-3 are listed in Tab. 4. Again, the
parameter $\beta\neq 0,\pm 2$ and in case of a scalar doublet $\neq\pm 4$.

###  3.4 T3

First, we can always have the assignment

$\displaystyle X_{\phi^{\prime}}=X_{\phi}=X_{\psi}=\beta.$ (31)

Second, in this topology none of the vertices can violate U$(1)_{X}$. Thus
$\zeta$ must always couple to two propagators. This could lead to additional
assignments for

#### $\alpha=-1$

$\psi$ has zero hypercharge, $\phi$ and $\phi^{\prime}$ have opposite
hypercharge. However, even if $\phi$ and $\phi^{\prime}$ are in addition in
the same representation of SU$(2)_{L}$ and mixing between $\phi$ and
$\left(\phi^{\prime}\right)^{\dagger}$ is induced by the breaking of
U$(1)_{X}$, we find no additional possible charge assignments.

All non-equivalent models of topology T3 are listed in Tab. 5. Again, the
parameter $\beta\neq 0,\pm 2$ and in case of a scalar doublet $\neq\pm 4$.
Concrete examples in this class are the gauged scotogenic model T3-B
($\alpha=-1,\beta=1$) Ma:2013yga ; Hagedorn:2018spx and the gauged singlet-
triplet scalar and doublet fermion model T3-A ($\alpha=-2,\beta=1$)
Brdar:2013iea .

Table 5: Non-equivalent models of topology T3. $X_{\zeta}$ is normalized to 2. Model | $\alpha$ | $\phi^{\prime}$ | $\phi$ | $\psi$ | $X_{\phi^{\prime}}$ | $X_{\phi}$ | $X_{\psi}$
---|---|---|---|---|---|---|---
T3-A | $0$ | $1^{S}_{0}$ | $3^{S}_{2}$ | $2^{F}_{1}$ | $\beta$ | $\beta$ | $\beta$
T3-A | $-2$ | $1^{S}_{-2}$ | $3^{S}_{0}$ | $2^{F}_{-1}$ | $\beta$ | $\beta$ | $\beta$
T3-B | $1,-3$ | $2^{S}_{1,-3}$ | $2^{S}_{3,-1}$ | $1^{F}_{2,-2}$ | $\beta$ | $\beta$ | $\beta$
T3-B | $-1$ | $2^{S}_{-1}$ | $2^{S}_{1}$ | $1^{F}_{0}$ | $\beta$ | $\beta$ | $\beta$
T3-C | $1,-3$ | $2^{S}_{1,-3}$ | $2^{S}_{3,-1}$ | $3^{F}_{2,-2}$ | $\beta$ | $\beta$ | $\beta$
T3-C | $-1$ | $2^{S}_{-1}$ | $2^{S}_{1}$ | $3^{F}_{0}$ | $\beta$ | $\beta$ | $\beta$
T3-E | $0,-2$ | $3^{S}_{0,-2}$ | $3^{S}_{2,0}$ | $2^{F}_{1,-1}$ | $\beta$ | $\beta$ | $\beta$

##  4 Phenomenological considerations

The models proposed above give rise to a wide variety of new phenomena. By
construction and most importantly, they generate (at least two) non-vanishing
neutrino masses Bonnet:2012kz ; Cai:2017jrq and include at least one viable
dark matter (DM) candidate Restrepo:2013aga . Similarly to the models with a
$Z_{2}$ symmetry, the values of the neutrino masses, the DM particle type,
relic density, direct, indirect and collider detection prospects, as well as
the predicted lepton flavor violation (LFV) rates are in general quite model-
dependent Klasen:2013jpa ; Vicente:2014wga ; Toma:2013zsa ; Esch:2016jyx ;
Esch:2018ccs ; Fiaschi:2018rky ; Escribano:2020iqq ; deBoer:2020yyw ;
deBoer:2021xjs ; deBoer:2021pon .

Replacing the $Z_{2}$ by a gauged U(1)X symmetry introduces a new gauge boson,
which may mix with the SM U(1)Y boson through a renormalizable kinetic mixing
operator Okun:1982xi ; Galison:1983pa ; Holdom:1985ag ; Foot:1991kb ;
Zhang:2018fbm . It may also obtain a mass by either the Stückelberg
Stueckelberg:1938hvi ; Kors:2004dx ; Kribs:2022gri ; Feldman:2007wj or the
Higgs mechanism Englert:1964et ; Guralnik:1964eu ; Higgs:1964pj . In the
latter case, the massive $Z^{\prime}$-boson will be accompanied by a new
physical scalar boson after spontaneous symmetry breaking.

The phenomenology of the dark gauge and Higgs sectors can to some extent be
discussed independently of the matter sector. A massless dark photon
necessarily requires another, massive DM particle. A massive dark gauge boson
can in principle itself constitute DM, if it is sufficiently long-lived and
its mass is sufficiently small, although we do not consider this case here
Fabbrichesi:2020wbt . Similarly, the dark Higgs boson might constitute DM, if
it is lighter than the dark photon, but again we do not consider this case
here Mondino:2020lsc .

###  4.1 Massless dark photons

Replacing the $Z_{2}$ with a gauged U(1)X symmetry introduces a new, a priori
massless gauge boson. Since only the new scalars and fermions are charged
under this U(1)X group, while all SM particles are uncharged, this corresponds
to the addition of a “dark” photon. When the fundamental Lagrangian of a model
contains several abelian gauge groups, gauge kinetic mixing occurs, since the
field strength tensors $F^{\mu\nu}$ are individually gauge invariant and
products of different field strength tensors are allowed by gauge invariance
Okun:1982xi ; Galison:1983pa ; Holdom:1985ag . The Lagrangian

$\displaystyle\mathcal{L}\supset$
$\displaystyle-\frac{1}{4}F_{Y}^{\mu\nu}F_{Y\mu\nu}-\frac{\epsilon}{2}F_{Y}^{\mu\nu}F_{X\mu\nu}-\frac{1}{4}F_{X}^{\mu\nu}F_{X\mu\nu}+\left(\begin{matrix}A_{X\mu}&A_{Y\mu}\end{matrix}\right)\left(\begin{matrix}g_{X}&0\\\
0&g_{Y}\end{matrix}\right)\left(\begin{matrix}j_{X}^{\mu}\\\
j_{Y}^{\mu}\end{matrix}\right)$ (32)

then depends on the kinetic mixing parameter $\epsilon$ and the SM and DM
interaction currents $j_{X,Y}^{\mu}$ coupling to the gauge fields $A_{X,Y\mu}$
with strengths $g_{X,Y}$. It can be written with diagonal kinetic terms by
choosing (a) either the dark photon $A_{X\mu}$ to not couple to the
hypercharge current $j_{Y}^{\mu}$ or (b) the hypercharge field $A_{Y\mu}$ to
not couple to the dark current $j_{X}^{\mu}$, but not both. When all
contributions are included, physical processes do not depend on the choice of
basis, and the kinetic mixing effects do not show up in electromagnetic and
weak interactions if only SM particles are involved in the calculations
Foot:1991kb ; Zhang:2018fbm .

Massless dark photons necessarily induce long-range interactions between the
DM particles, which in our models must all be charged under U(1)X, independent
of their spin and SM quantum numbers, and come with equal numbers of positive
and negative charges. Annihilations can be suppressed, if the dark matter mass
is sufficiently high and the dark fine-structure constant
$\alpha_{X}=g_{X}^{2}/(4\pi)$ is sufficiently small. Furthermore, the correct
relic abundance can be obtained if the DM also couples to SM weak
interactions, as long as the kinetic mixing with ordinary photons is
sufficiently small. The primary limit on $\alpha_{X}$ then comes from the
demand that the DM be effectively collisionless in galactic dynamics, which
implies $\alpha_{X}<10^{-3}$ for TeV-scale DM. These values are also
compatible with constraints from structure formation and primordial
nucleosynthesis Ackerman:2008kmp . They have recently been updated and
combined with other constraints from stars, supernovae, precision atomic
physics, LFV as well as collider physics Fabbrichesi:2020wbt . When
interpreted in terms of effective field theory, LFV, primordial
nucleosynthesis, star cooling and other phenomena set limits on the scale of
the dimension-six operators describing DM interactions in the 1-15 TeV range
Dobrescu:2004wz . The rotation of the mixing term in Eq. (32) leaves the
photon coupled to the dark sector particles with strength $\epsilon
e/\sqrt{1-\epsilon^{2}}$. The dark sector particles are then interpreted as
millicharged particles. Searches are accordingly parameterized in terms of
their mass and electromagnetic coupling modulated by the mixing angle, which
is then constrained to be below ${\cal O}(0.1)$ in the GeV–TeV region probed
by LEP and the LHC, but orders of magnitude smaller in the regions below and
above from cosmological constraints Fabbrichesi:2020wbt .

###  4.2 Symmetry breaking and massive $Z^{\prime}$ boson

The U$(1)$ gauge symmetry can be broken through the addition of a new scalar.
For this purpose we take a complex scalar $\zeta$, which we allow to develop a
vev $v_{\zeta}$. If $v_{\zeta}$ lies at the same scale as the masses of the
new particles, all charge assignments that were listed in the overview in
Tabs. 2-5 are possible. If $v_{\zeta}\ll\Lambda$, models with $n\neq 0$ in the
Weinberg operator Eq. (19) are suppressed which might yield too small neutrino
masses. Especially in the case of $v_{\zeta}=0$, only charge assignments where
all new fields in the neutrino loop have the same U$(1)_{X}$ charge are
possible. With the Standard Model Higgs doublet $H$ and the new scalar singlet
$\zeta$ having charges of 0 and 2 under the new U$(1)_{X}$ group respectively,
the scalar potential is given by

$\displaystyle
V=-m_{H}^{2}(H^{\dagger}H)-m_{\zeta}^{2}(\zeta^{\dagger}\zeta)+\frac{\lambda_{H}}{2}(H^{\dagger}H)^{2}+\frac{\lambda_{\zeta}}{2}(\zeta^{\dagger}\zeta)^{2}+\lambda_{H\zeta}(H^{\dagger}H)(\zeta^{\dagger}\zeta).$
(33)

The first step is to find the minimum of the potential, which should be the
case when $H$ and $\zeta$ obtain their vevs. We denote the vevs of these
fields by

$\displaystyle\langle H\rangle=\left(\begin{matrix}0\\\
\frac{v_{H}+h}{\sqrt{2}}\end{matrix}\right),\qquad\langle\zeta\rangle=\frac{v_{\zeta}+\sigma}{\sqrt{2}}.$
(34)

Minimizing the potential yields

$\displaystyle v_{H}^{2}=$
$\displaystyle\frac{2\lambda_{H\zeta}m_{\zeta}^{2}-2\lambda_{\zeta}m_{H}^{2}}{\lambda_{H\zeta}^{2}-\lambda_{\zeta}\lambda_{H}},$
(35) $\displaystyle v_{\zeta}^{2}=$
$\displaystyle\frac{2\lambda_{H}m_{\zeta}^{2}-2\lambda_{H\zeta}m_{H}^{2}}{\lambda_{\zeta}\lambda_{H}-\lambda_{H\zeta}^{2}}.$
(36)

The mass matrix for the scalar fields expressed in terms of the VEV’s is given
by

$\displaystyle
M=\left(\begin{matrix}\lambda_{H}v_{H}^{2}&\lambda_{H\zeta}v_{H}v_{\zeta}\\\
\lambda_{H\zeta}v_{H}v_{\zeta}&\lambda_{\zeta}v_{\zeta}^{2},\end{matrix}\right).$
(37)

The eigenstate dominated by the SM Higgs is the one measured at LHC
ATLAS:2012yve ; CMS:2012qbp . Searches for modified Higgs couplings and a
second Higgs boson constrain the scalar sector Schabinger:2005ei ;
Dawson:2017jja ; Bechtle:2020pkv ; Choi:2013zqz ; Cheung:2015dta ;
Cheung:2015cug ; ATLAS:2015ciy .

Now we turn to the gauge sector. In case of two abelian gauge groups, one has
to introduce gauge kinetic mixing Holdom:1985ag . Through some (non-
orthogonal) basis transformations, the effect of kinetic mixing can be shifted
to an off diagonal gauge coupling. After spontaneous symmetry breaking, the
gauge bosons obtain their masses. The covariant derivative for
SU$(2)_{L}\times$U$(1)_{Y}\times$U$(1)_{X}$ can be written as

$\displaystyle D_{\mu}=\partial_{\mu}-i\left(g_{Y}A_{\mu}+\frac{-\epsilon
g_{Y}}{\sqrt{1-\epsilon^{2}}}A^{\prime}_{\mu}\right)Y-i\frac{g_{X}}{\sqrt{1-\epsilon^{2}}}A^{\prime}_{\mu}X-ig\tau^{a}W^{a}_{\mu}$
(38)

after rotating the kinetic mixing term $\epsilon$ to an upper triangular gauge
coupling matrix (see e.g. Ref. Zhang:2018fbm ). $X,Y$ and $\tau^{a}$ are the
generators of U$(1)_{X}$, U$(1)_{Y}$ and SU$(2)_{L}$ respectively. The part of
the Lagrangian relevant for the mass of the gauge bosons is given by

$\displaystyle\mathcal{L}\supset\left|D_{\mu}H\right|^{2}+\left|D_{\mu}\zeta\right|^{2}.$
(39)

Expanding this around the VEV’s we obtain the neutral gauge boson mass matrix

$\displaystyle\mathcal{L}\supset$
$\displaystyle\frac{v_{H}^{2}}{8}\left(g_{Y}A_{\mu}+\frac{-\epsilon
g_{Y}}{\sqrt{1-\epsilon^{2}}}A^{\prime}_{\mu}-gW^{3}_{\mu}\right)^{2}+\frac{v_{\zeta}^{2}}{2}\left(\frac{g_{X}}{\sqrt{1-\epsilon^{2}}}A^{\prime}_{\mu}X_{\zeta}\right)^{2}$
(40)
$\displaystyle=\left(\begin{matrix}A_{\mu}&W^{3}_{\mu}&A^{\prime}_{\mu}\end{matrix}\right)\left(\begin{matrix}\frac{g_{Y}^{2}v_{H}^{2}}{8}&-\frac{gg_{Y}v_{H}^{2}}{8}&-\frac{\epsilon
g_{Y}^{2}v_{H}^{2}}{8\sqrt{1-\epsilon^{2}}}\\\
-\frac{gg_{Y}v_{H}^{2}}{8}&\frac{g^{2}v_{H}^{2}}{8}&\frac{\epsilon
gg_{Y}v_{H}^{2}}{8\sqrt{1-\epsilon^{2}}}\\\ -\frac{\epsilon
g_{Y}^{2}v_{H}^{2}}{8\sqrt{1-\epsilon^{2}}}&\frac{\epsilon
gg_{Y}v_{H}^{2}}{8\sqrt{1-\epsilon^{2}}}&\frac{\epsilon^{2}g_{Y}^{2}v_{H}^{2}}{8(1-\epsilon^{2})}+\frac{g_{X}^{2}v_{\zeta}^{2}X_{\zeta}^{2}}{2(1-\epsilon^{2})}\end{matrix}\right)\left(\begin{matrix}A^{\mu}\\\
W^{3\mu}\\\ A^{\prime\mu}\end{matrix}\right).$ (41)

Diagonalizing this matrix yields one vanishing eigenvalue (for the photon) and
the masses for the $Z$ and $Z^{\prime}$ boson which are, up to order
$\epsilon^{2}$, given by

$\displaystyle m_{Z}^{2}=$ $\displaystyle
M_{\tilde{Z}}^{2}\left(1-\epsilon^{2}\sin^{2}\theta_{w}\frac{M_{\tilde{Z}}^{2}}{M_{X}^{2}-M_{\tilde{Z}}^{2}}\right),$
(42) $\displaystyle m_{Z^{\prime}}^{2}=$ $\displaystyle
M_{X}^{2}\left(1+\epsilon^{2}\left(1+\sin^{2}\theta_{w}\frac{M_{\tilde{Z}}^{2}}{M_{X}^{2}-M_{\tilde{Z}}^{2}}\right)\right),$
(43)

with the Weinberg angle $\theta_{w}$ and

$\displaystyle M_{\tilde{Z}}^{2}=$
$\displaystyle\frac{(g^{2}+g_{Y}^{2})v_{H}^{2}}{4}=\frac{g^{2}v_{H}^{2}}{4\cos^{2}\theta_{w}},$
(44) $\displaystyle M_{X}^{2}=$ $\displaystyle
g_{X}^{2}X_{\zeta}^{2}v_{\zeta}^{2}.$ (45)

The $Z^{0}$ boson mass has been experimentally measured by e.g. the LEP
experiments and matches the Standard Model prediction. As the shift in the
$Z^{0}$ mass depends on the kinetic mixing and the $Z^{\prime}$ mass, this can
be used to constrain the viable parameter space. The mixing also modifies the
couplings, allowing for further tests of the model.

A general review of the phenomenology for heavy $Z^{\prime}$ bosons can be
found in Ref. Langacker:2008yv . Dedicated experimental analyses on invisible
$Z^{\prime}$ decays have set limits in the MeV to 10 GeV range Lees:2017lec ;
NA64:2019imj . Model-independent limits on the kinetic mixing have been
obtained in Ref. Hook:2010tw , and the phenomenology of a similar setup to
ours has been discussed in Lao:2020inc . Collider constraints have been
examined in Ref. Aboubrahim:2022qln . A recent review of the theory and
phenomenology of massless and massive dark photons and different limits can be
found in Ref. Fabbrichesi:2020wbt .

###  4.3 Massive neutrino generations

In this section we discuss how many generations of the new fields are required
in order to allow for a least two massive neutrinos. In Ref. Bonnet:2012kz the
formulas for the neutrino masses arising from each topology are given. We
first assume only one generation of each new field. In this case a simple
pattern emerges in the neutrino mass formulas. Assuming that the neutrinos
have Yukawa interactions with the new fields given by the couplings
$y,y^{\prime}$, the neutrino mass matrix is

$\displaystyle\left(M_{\nu}\right)_{\alpha_{\beta}}\propto
y_{\alpha}y^{\prime}_{\beta}+y^{\prime}_{\alpha}y_{\beta}=:A_{\alpha\beta}+A_{\alpha\beta}^{T}.$
(46)

The proportionality factor depends on the masses and couplings of the new
fields. It is easy to see that the matrix $A$ has rank 1. The rank of
$M_{\nu}$ and thus the number of massive neutrinos can be estimated to be
$\leq 2$. As the entries of the Yukawa couplings are not fixed a priori, the
neutrino matrix will actually have rank two unless two fields in the loop are
identified with each other, which yields $y=y^{\prime}$.222Strictly speaking
some entries of $y$ and $y^{\prime}$ could be exactly the same or have a
special relation to one another. This should not pose a problem since such a
scenario would be extremely fine-tuned or would require additional symmetries.
Some models allow for the case where two fields are identified with each
other. If one does so, the models predict only one massive neutrino unless one
introduces several generations of one of the new fields.

###  4.4 Dark matter

The dark matter candidate can be both fermionic and scalar, whichever new
(stable) field is the lightest. We do not consider $Z^{\prime}$ dark matter
here. With $\mathcal{O}(1)$ couplings and electroweak interactions, the dark
sector is in thermal equilibrium in the early Universe, and the relic density
is determined by the well-known freeze-out process. The dark matter self
annihilation depends both on the model and the dark matter candidate and
generally involves new particles in the $t$\- and $u$-channels as well as
neutral gauge and Higgs bosons in the $s$-channel as well as four-point
interactions for scalar dark matter.

The dark matter particle can be searched for by standard WIMP searches such as
indirect detection MAGIC:2016xys ; HESS:2022ygk ; IceCube:2023ies and direct
detection XENONCollaboration:2023orw ; PandaX-II:2020oim ; LUX-ZEPLIN:2022xrq
; PICO:2019vsc . The dark sector particles can also be searched for at
colliders. The dark matter phenomenology for the $Z_{2}$ models has been
studied in a number of models Klasen:2013jpa ; deBoer:2020yyw ; deBoer:2021pon
; Esch:2016jyx ; Esch:2018ccs ; Fiaschi:2018rky ; deBoer:2021xjs . For the
U$(1)$ models, the main qualitative differences are the additional
annihilation and scattering channels involving the $Z^{\prime}$ gauge boson
and potentially the $\zeta$ Higgs boson. An analysis of direct and indirect
detection dominated by kinetic mixing with a comparable setup to ours can be
found in Refs. Mambrini:2011dw ; Lao:2020inc . It is worth noting that for
$v_{\zeta}\sim\mathcal{O}(100\text{keV})$ the dark matter candidate can have a
just slightly heavier partner, thus allowing for inelastic scattering Tucker-
Smith:2001myb .

###  4.5 Further considerations

Radiative seesaw models generally allow for LFV to occur through similar
diagrams as the neutrino loop Toma:2013zsa ; Vicente:2014wga ; deBoer:2020yyw
; deBoer:2021pon ; Fiaschi:2018rky . This allows for an additional way to test
these models MEG:2016leq ; SINDRUM:1987nra .

Dark photons also contribute to the anomalous magnetic moment, given that
their masses are not too heavy, and have therefore been discussed as an
explanation for the muon magnetic anomaly $(g_{\mu}-2)$ Muong-2:2023cdq .
However this explanation is excluded by the experimental limits Lees:2017lec ;
NA64:2019imj assuming invisible dark photon decays.

With new particles added, the running of the couplings is modified, which
might allow for gauge coupling unification. With the new particles being color
singlets, SU$(3)_{c}$ running is not changed while the SU$(2)_{L}$ beta
function becomes larger with the addition of new particles. Therefore the
scale where SU$(3)_{c}$ and SU$(2)_{L}$ cross is lowered with respect to the
SM only case. This has been observed in Ref. Hagedorn:2016dze for the $Z_{2}$
case and shown for the models presented in this paper in Refs. deBoer:MSc ;
Zeinstra:PhD . To be precise, if unification occurs, it happens at scales
around $\mathcal{O}(10^{12}\text{GeV})$ to $\mathcal{O}(10^{14}\text{GeV})$.
Unification scales at this order tend to be in conflict with proton decay
limits Nath:2006ut ; Super-Kamiokande:2020wjk .

##  5 Summary and outlook

Minimal extensions of the Standard Model can introduce viable dark matter
candidates and generate neutrino masses. To stabilize the dark sector a new
symmetry must be introduced. This symmetry can be a local U$(1)$ symmetry
which is theoretically better motivated than the often used discrete or global
symmetries. We followed previous work and allowed for up to four dark fields
in addition to the new gauge boson and the scalar Higgs fields breaking the
new U$(1)$ gauge group. Arguments based on anomaly cancellation, dark matter
stability and the neutrino topologies restrict the possible charge
assignments. Given our assumptions, the Standard Model must be uncharged under
the new gauge group and all new fermions must be vector like. All $Z_{2}$
models from Ref. Restrepo:2013aga can be promoted to models with a gauged
U$(1)$ and many models allow for different charge assignments. We listed the
particle content and charge assignments of all possible models.

The new vector boson gives rise to a rich phenomenology complementing the
known phenomenology of the $Z_{2}$ models. The dark gauge boson, coupling to
the SM simply through the kinetic mixing portal, can be massless or obtain a
mass through spontaneous symmetry breaking. The kinetic mixing portal gives a
model independent way of testing the modified gauge sector. In case of a
massive $Z^{\prime}$ boson an additional scalar field can be introduced in
order to break the new U$(1)$ symmetry giving rise to signatures in the Higgs
sector.

As a further direction, one could address the coincidence of scales inherent
to the scotogenic models. It is assumed, that all explicit scales in these
models are at 100 GeV or TeV scale. As there are a number of independent mass
parameters, it is hard to explain why they all are rather close to each other.
In a classically scale invariant setting where scale symmetry is broken
dynamically Coleman:1973jx , all scales would be related to the symmetry
breaking scale. This would give an explanation for the coincidence of scales
and also address the hierarchy problem Bardeen:1995kv . Scale invariant
radiative seesaw models have been presented in Refs. Ahriche:2016cio ;
Ahriche:2016ixu . The U$(1)_{X}$ gauge contribution to the effective potential
can easily drive dynamical symmetry breaking making our models especially
attractive for a scale-invariant setting.

## Acknowledgments

This work has been supported by the DFG through the Research Training Network
2149 “Strong and weak interactions - from hadrons to dark matter”.

## References

  * (1) Particle Data Group collaboration, _Review of Particle Physics_ , _PTEP_ 2020 (2020) 083C01.
  * (2) I. Esteban, M.C. Gonzalez-Garcia, M. Maltoni, T. Schwetz and A. Zhou, _The fate of hints: updated global analysis of three-flavor neutrino oscillations_ , _JHEP_ 09 (2020) 178 [2007.14792].
  * (3) ATLAS collaboration, _Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC_ , _Phys. Lett. B_ 716 (2012) 1 [1207.7214].
  * (4) CMS collaboration, _Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC_ , _Phys. Lett. B_ 716 (2012) 30 [1207.7235].
  * (5) M. Klasen, M. Pohl and G. Sigl, _Indirect and direct search for dark matter_ , _Prog. Part. Nucl. Phys._ 85 (2015) 1 [1507.03800].
  * (6) F. Bonnet, M. Hirsch, T. Ota and W. Winter, _Systematic study of the d=5 Weinberg operator at one-loop order_ , _JHEP_ 07 (2012) 153 [1204.5862].
  * (7) Y. Cai, J. Herrero-García, M.A. Schmidt, A. Vicente and R.R. Volkas, _From the trees to the forest: a review of radiative neutrino mass models_ , _Front. in Phys._ 5 (2017) 63 [1706.08524].
  * (8) D. Restrepo, O. Zapata and C.E. Yaguna, _Models with radiative neutrino masses and viable dark matter candidates_ , _JHEP_ 11 (2013) 011 [1308.3655].
  * (9) E. Ma, _Verifiable radiative seesaw mechanism of neutrino mass and dark matter_ , _Phys. Rev. D_ 73 (2006) 077301 [hep-ph/0601225].
  * (10) M. Klasen, C.E. Yaguna, J.D. Ruiz-Alvarez, D. Restrepo and O. Zapata, _Scalar dark matter and fermion coannihilations in the radiative seesaw model_ , _JCAP_ 04 (2013) 044 [1302.5298].
  * (11) A. Vicente and C.E. Yaguna, _Probing the scotogenic model with lepton flavor violating processes_ , _JHEP_ 02 (2015) 144 [1412.2545].
  * (12) T. Toma and A. Vicente, _Lepton Flavor Violation in the Scotogenic Model_ , _JHEP_ 01 (2014) 160 [1312.2840].
  * (13) T. de Boer, M. Klasen, C. Rodenbeck and S. Zeinstra, _Absolute neutrino mass as the missing link to the dark sector_ , _Phys. Rev. D_ 102 (2020) 051702 [2007.05338].
  * (14) T. de Boer, R. Busse, A. Kappes, M. Klasen and S. Zeinstra, _Indirect detection constraints on the scotogenic dark matter model_ , 2105.04899.
  * (15) P. Escribano, M. Reig and A. Vicente, _Generalizing the Scotogenic model_ , _JHEP_ 07 (2020) 097 [2004.05172].
  * (16) S. Esch, M. Klasen, D.R. Lamprea and C.E. Yaguna, _Lepton flavor violation and scalar dark matter in a radiative model of neutrino masses_ , _Eur. Phys. J. C_ 78 (2018) 88 [1602.05137].
  * (17) S. Esch, M. Klasen and C.E. Yaguna, _A singlet doublet dark matter model with radiative neutrino masses_ , _JHEP_ 10 (2018) 055 [1804.03384].
  * (18) J. Fiaschi, M. Klasen and S. May, _Singlet-doublet fermion and triplet scalar dark matter with radiative neutrino masses_ , _JHEP_ 05 (2019) 015 [1812.11133].
  * (19) T. de Boer, R. Busse, A. Kappes, M. Klasen and S. Zeinstra, _New constraints on radiative seesaw models from IceCube and other neutrino detectors_ , _Phys. Rev. D_ 103 (2021) 123006 [2103.06881].
  * (20) E. Ma, _Z(3) Dark Matter and Two-Loop Neutrino Mass_ , _Phys. Lett. B_ 662 (2008) 49 [0708.3371].
  * (21) G. Belanger, K. Kannike, A. Pukhov and M. Raidal, _$Z_{3}$ Scalar Singlet Dark Matter_, _JCAP_ 01 (2013) 022 [1211.1014].
  * (22) M. Aoki and T. Toma, _Impact of semi-annihilation of $\mathbb{Z}_{3}$ symmetric dark matter with radiative neutrino masses_, _JCAP_ 09 (2014) 016 [1405.5870].
  * (23) Y. BenTov, X.-G. He and A. Zee, _An $A_{4}$ x $Z_{4}$ model for neutrino mixing_, _JHEP_ 12 (2012) 093 [1208.1062].
  * (24) G. Bélanger, K. Kannike, A. Pukhov and M. Raidal, _Minimal semi-annihilating $\mathbb{Z}_{N}$ scalar dark matter_, _JCAP_ 06 (2014) 021 [1403.4960].
  * (25) E. Ma, _Dark Scalar Doublets and Neutrino Tribimaximal Mixing from A(4) Symmetry_ , _Phys. Lett. B_ 671 (2009) 366 [0808.1729].
  * (26) E. Ma, _Radiative Mixing of the One Higgs Boson and Emergent Self-Interacting Dark Matter_ , _Phys. Lett. B_ 754 (2016) 114 [1506.06658].
  * (27) Y.B. Zeldovich, I.Y. Kobzarev and L.B. Okun, _Cosmological Consequences of the Spontaneous Breakdown of Discrete Symmetry_ , _Zh. Eksp. Teor. Fiz._ 67 (1974) 3.
  * (28) A. Arhrib, C. Bœhm, E. Ma and T.-C. Yuan, _Radiative Model of Neutrino Mass with Neutrino Interacting MeV Dark Matter_ , _JCAP_ 04 (2016) 049 [1512.08796].
  * (29) J. Fiaschi, M. Klasen, M. Vargas, C. Weinheimer and S. Zeinstra, _MeV neutrino dark matter: Relic density, lepton flavour violation and electron recoil_ , _JHEP_ 11 (2019) 129 [1908.09882].
  * (30) M. Lindner, D. Schmidt and T. Schwetz, _Dark Matter and neutrino masses from global U(1) B-L symmetry breaking_, _Phys. Lett. B_ 705 (2011) 324 [1105.4626].
  * (31) C. Bonilla, E. Ma, E. Peinado and J.W.F. Valle, _Two-loop Dirac neutrino mass and WIMP dark matter_ , _Phys. Lett. B_ 762 (2016) 214 [1607.03931].
  * (32) D. Aristizabal Sierra, M. Dhen, C.S. Fong and A. Vicente, _Dynamical flavor origin of $\mathbb{Z}_{N}$ symmetries_, _Phys. Rev. D_ 91 (2015) 096004 [1412.5600].
  * (33) P. Ali, A. Eichhorn, M. Pauly and M.M. Scherer, _Constraints on discrete global symmetries in quantum gravity_ , _JHEP_ 05 (2021) 036 [2012.07868].
  * (34) J.D. Bekenstein, _Nonexistence of baryon number for black holes. ii_ , _Phys. Rev. D_ 5 (1972) 2403.
  * (35) L.F. Abbott and M.B. Wise, _Wormholes and Global Symmetries_ , _Nucl. Phys. B_ 325 (1989) 687.
  * (36) R. Kallosh, A.D. Linde, D.A. Linde and L. Susskind, _Gravity and global symmetries_ , _Phys. Rev. D_ 52 (1995) 912 [hep-th/9502069].
  * (37) E. Ma, I. Picek and B. Radovčić, _New Scotogenic Model of Neutrino Mass with $U(1)_{D}$ Gauge Interaction_, _Phys. Lett. B_ 726 (2013) 744 [1308.5313].
  * (38) V. Brdar, I. Picek and B. Radovcic, _Radiative Neutrino Mass with Scotogenic Scalar Triplet_ , _Phys. Lett. B_ 728 (2014) 198 [1310.3183].
  * (39) T. Li and W. Chao, _Neutrino Masses, Dark Matter and B-L Symmetry at the LHC_ , _Nucl. Phys. B_ 843 (2011) 396 [1004.0296].
  * (40) S.-Y. Ho, T. Toma and K. Tsumura, _Systematic $U(1)_{B\textendash L}$ extensions of loop-induced neutrino mass models with dark matter_, _Phys. Rev. D_ 94 (2016) 033007 [1604.07894].
  * (41) 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].
  * (42) J. Calle, D. Restrepo, C.E. Yaguna and O. Zapata, _Minimal radiative Dirac neutrino mass models_ , _Phys. Rev. D_ 99 (2019) 075008 [1812.05523].
  * (43) S. Jana, P.K. Vishnu and S. Saad, _Minimal realizations of Dirac neutrino mass from generic one-loop and two-loop topologies at $d=5$_, _JCAP_ 04 (2020) 018 [1910.09537].
  * (44) A. Dasgupta, S.K. Kang and O. Popov, _Radiative Dirac neutrino mass, neutrinoless quadruple beta decay, and dark matter in B-L extension of the standard model_ , _Phys. Rev. D_ 100 (2019) 075030 [1903.12558].
  * (45) E. Ma, _Leptonic Source of Dark Matter and Radiative Majorana or Dirac Neutrino Mass_ , _Phys. Lett. B_ 809 (2020) 135736 [1912.11950].
  * (46) E. Ma, _Gauged lepton number, Dirac neutrinos, dark matter, and muon g-2_ , _Phys. Lett. B_ 819 (2021) 136402 [2104.10324].
  * (47) N. Bernal, J. Calle and D. Restrepo, _Anomaly-free Abelian gauge symmetries with Dirac scotogenic models_ , _Phys. Rev. D_ 103 (2021) 095032 [2102.06211].
  * (48) L. Ackerman, M.R. Buckley, S.M. Carroll and M. Kamionkowski, _Dark Matter and Dark Radiation_ , _Phys. Rev. D_ 79 (2009) 023519 [0810.5126].
  * (49) P. Langacker, _The Physics of Heavy $Z^{\prime}$ Gauge Bosons_, _Rev. Mod. Phys._ 81 (2009) 1199 [0801.1345].
  * (50) B. Fuks, M. Klasen, F. Ledroit, Q. Li and J. Morel, _Precision predictions for $Z^{\prime}$ \- production at the CERN LHC: QCD matrix elements, parton showers, and joint resummation_, _Nucl. Phys. B_ 797 (2008) 322 [0711.0749].
  * (51) T. Ježo, M. Klasen, F. Lyonnet, F. Montanet, I. Schienbein and M. Tartare, _Can new heavy gauge bosons be observed in ultra-high energy cosmic neutrino events?_ , _Phys. Rev. D_ 89 (2014) 077702 [1401.6012].
  * (52) T. Jezo, M. Klasen, D.R. Lamprea, F. Lyonnet and I. Schienbein, _NLO+NLL limits on $W^{\prime}$ and $Z^{\prime}$ gauge boson masses in general extensions of the Standard Model_, _JHEP_ 12 (2014) 092 [1410.4692].
  * (53) R. Bonciani, T. Jezo, M. Klasen, F. Lyonnet and I. Schienbein, _Electroweak top-quark pair production at the LHC with $Z^{\prime}$ bosons to NLO QCD in POWHEG_, _JHEP_ 02 (2016) 141 [1511.08185].
  * (54) M.M. Altakach, T. Ježo, M. Klasen, J.-N. Lang and I. Schienbein, _Electroweak tt hadroproduction in the presence of heavy Z’ and W’ bosons at NLO QCD in POWHEG_ , _Phys. Rev. D_ 103 (2021) 115026 [2012.14855].
  * (55) A.J. Buras, A. Crivellin, F. Kirk, C.A. Manzari and M. Montull, _Global analysis of leptophilic Z’ bosons_ , _JHEP_ 06 (2021) 068 [2104.07680].
  * (56) Y. Farzan and M. Tortola, _Neutrino oscillations and Non-Standard Interactions_ , _Front. in Phys._ 6 (2018) 10 [1710.09360].
  * (57) M. Klasen, F. Lyonnet and F.S. Queiroz, _NLO+NLL collider bounds, Dirac fermion and scalar dark matter in the B–L model_ , _Eur. Phys. J. C_ 77 (2017) 348 [1607.06468].
  * (58) N. Okada and S. Okada, _$Z^{\prime}_{BL}$ portal dark matter and LHC Run-2 results_, _Phys. Rev. D_ 93 (2016) 075003 [1601.07526].
  * (59) D.A. Camargo, M. Klasen and S. Zeinstra, _Discovering heavy U(1)-gauged Higgs bosons at the HL-LHC_ , _J. Phys. G_ 48 (2020) 025002 [1903.02572].
  * (60) S. Weinberg, _Baryon and Lepton Nonconserving Processes_ , _Phys. Rev. Lett._ 43 (1979) 1566.
  * (61) M.D. Schwartz, _Quantum Field Theory and the Standard Model_ , Cambridge University Press (3, 2014).
  * (62) C. Hagedorn, J. Herrero-Garcia, E. Molinaro and M.A. Schmidt, _Phenomenology of the Generalised Scotogenic Model with Fermionic Dark Matter_ , _JHEP_ 11 (2018) 103 [1804.04117].
  * (63) L.B. Okun, _Limits of electrodynamics: paraphotons?_ , _Sov. Phys. JETP_ 56 (1982) 502.
  * (64) P. Galison and A. Manohar, _Two Z’s or not two Z’s?_ , _Phys. Lett. B_ 136 (1984) 279.
  * (65) B. Holdom, _Two U(1)’s and Epsilon Charge Shifts_ , _Phys. Lett. B_ 166 (1986) 196.
  * (66) R. Foot and X.-G. He, _Comment on Z Z-prime mixing in extended gauge theories_ , _Phys. Lett. B_ 267 (1991) 509.
  * (67) J.-X. Pan, M. He, X.-G. He and G. Li, _Scrutinizing a massless dark photon: basis independence_ , _Nucl. Phys. B_ 953 (2020) 114968 [1807.11363].
  * (68) E.C.G. Stueckelberg, _Interaction energy in electrodynamics and in the field theory of nuclear forces_ , _Helv. Phys. Acta_ 11 (1938) 225.
  * (69) B. Kors and P. Nath, _A Stueckelberg extension of the standard model_ , _Phys. Lett. B_ 586 (2004) 366 [hep-ph/0402047].
  * (70) G.D. Kribs, G. Lee and A. Martin, _Effective Field Theory of Stückelberg Vector Bosons_ , 2204.01755.
  * (71) D. Feldman, Z. Liu and P. Nath, _The Stueckelberg Z-prime Extension with Kinetic Mixing and Milli-Charged Dark Matter From the Hidden Sector_ , _Phys. Rev. D_ 75 (2007) 115001 [hep-ph/0702123].
  * (72) F. Englert and R. Brout, _Broken Symmetry and the Mass of Gauge Vector Mesons_ , _Phys. Rev. Lett._ 13 (1964) 321.
  * (73) G.S. Guralnik, C.R. Hagen and T.W.B. Kibble, _Global Conservation Laws and Massless Particles_ , _Phys. Rev. Lett._ 13 (1964) 585.
  * (74) P.W. Higgs, _Broken Symmetries and the Masses of Gauge Bosons_ , _Phys. Rev. Lett._ 13 (1964) 508.
  * (75) M. Fabbrichesi, E. Gabrielli and G. Lanfranchi, _The Dark Photon_ , 2005.01515.
  * (76) C. Mondino, M. Pospelov, J.T. Ruderman and O. Slone, _Dark Higgs Dark Matter_ , _Phys. Rev. D_ 103 (2021) 035027 [2005.02397].
  * (77) B.A. Dobrescu, _Massless gauge bosons other than the photon_ , _Phys. Rev. Lett._ 94 (2005) 151802 [hep-ph/0411004].
  * (78) R.M. Schabinger and J.D. Wells, _A Minimal spontaneously broken hidden sector and its impact on Higgs boson physics at the large hadron collider_ , _Phys. Rev. D_ 72 (2005) 093007 [hep-ph/0509209].
  * (79) S. Dawson and M. Sullivan, _Enhanced di-Higgs boson production in the complex Higgs singlet model_ , _Phys. Rev. D_ 97 (2018) 015022 [1711.06683].
  * (80) P. Bechtle, D. Dercks, S. Heinemeyer, T. Klingl, T. Stefaniak, G. Weiglein et al., _HiggsBounds-5: Testing Higgs Sectors in the LHC 13 TeV Era_ , _Eur. Phys. J. C_ 80 (2020) 1211 [2006.06007].
  * (81) S. Choi, S. Jung and P. Ko, _Implications of LHC data on 125 GeV Higgs-like boson for the Standard Model and its various extensions_ , _JHEP_ 10 (2013) 225 [1307.3948].
  * (82) K. Cheung, P. Ko, J.S. Lee and P.-Y. Tseng, _Bounds on Higgs-Portal models from the LHC Higgs data_ , _JHEP_ 10 (2015) 057 [1507.06158].
  * (83) K. Cheung, P. Ko, J.S. Lee, J. Park and P.-Y. Tseng, _Higgs precision study of the 750 GeV diphoton resonance and the 125 GeV standard model Higgs boson with Higgs-singlet mixing_ , _Phys. Rev. D_ 94 (2016) 033010 [1512.07853].
  * (84) ATLAS collaboration, _Constraints on new phenomena via Higgs boson couplings and invisible decays with the ATLAS detector_ , _JHEP_ 11 (2015) 206 [1509.00672].
  * (85) BaBar collaboration, _Search for Invisible Decays of a Dark Photon Produced in ${e}^{+}{e}^{-}$ Collisions at BaBar_, _Phys. Rev. Lett._ 119 (2017) 131804 [1702.03327].
  * (86) D. Banerjee et al., _Dark matter search in missing energy events with NA64_ , _Phys. Rev. Lett._ 123 (2019) 121801 [1906.00176].
  * (87) A. Hook, E. Izaguirre and J.G. Wacker, _Model Independent Bounds on Kinetic Mixing_ , _Adv. High Energy Phys._ 2011 (2011) 859762 [1006.0973].
  * (88) J. Lao, C. Cai, Z.-H. Yu, Y.-P. Zeng and H.-H. Zhang, _Fermionic and scalar dark matter with hidden $\mathrm{U}(1)$ gauge interaction and kinetic mixing_, _Phys. Rev. D_ 101 (2020) 095031 [2003.02516].
  * (89) A. Aboubrahim, M.M. Altakach, M. Klasen, P. Nath and Z.-Y. Wang, _Combined constraints on dark photons and discovery prospects at the LHC and the Forward Physics Facility_ , _JHEP_ 03 (2023) 182 [2212.01268].
  * (90) MAGIC, Fermi-LAT collaboration, _Limits to Dark Matter Annihilation Cross-Section from a Combined Analysis of MAGIC and Fermi-LAT Observations of Dwarf Satellite Galaxies_ , _JCAP_ 02 (2016) 039 [1601.06590].
  * (91) H.E.S.S. collaboration, _Search for Dark Matter Annihilation Signals in the H.E.S.S. Inner Galaxy Survey_ , _Phys. Rev. Lett._ 129 (2022) 111101 [2207.10471].
  * (92) IceCube collaboration, _Search for neutrino lines from dark matter annihilation and decay with IceCube_ , 2303.13663.
  * (93) (XENON Collaboration)††, XENON collaboration, _First Dark Matter Search with Nuclear Recoils from the XENONnT Experiment_ , _Phys. Rev. Lett._ 131 (2023) 041003 [2303.14729].
  * (94) PandaX-II collaboration, _Results of dark matter search using the full PandaX-II exposure_ , _Chin. Phys. C_ 44 (2020) 125001 [2007.15469].
  * (95) LUX-ZEPLIN collaboration, _First Dark Matter Search Results from the LUX-ZEPLIN (LZ) Experiment_ , _Phys. Rev. Lett._ 131 (2023) 041002 [2207.03764].
  * (96) PICO collaboration, _Dark Matter Search Results from the Complete Exposure of the PICO-60 C 3F8 Bubble Chamber_, _Phys. Rev. D_ 100 (2019) 022001 [1902.04031].
  * (97) Y. Mambrini, _The ZZ’ kinetic mixing in the light of the recent direct and indirect dark matter searches_ , _JCAP_ 07 (2011) 009 [1104.4799].
  * (98) D. Tucker-Smith and N. Weiner, _Inelastic dark matter_ , _Phys. Rev. D_ 64 (2001) 043502 [hep-ph/0101138].
  * (99) MEG collaboration, _Search for the lepton flavour violating decay $\mu^{+}\rightarrow\mathrm{e}^{+}\gamma$ with the full dataset of the MEG experiment_, _Eur. Phys. J. C_ 76 (2016) 434 [1605.05081].
  * (100) SINDRUM collaboration, _Search for the Decay mu+ — $>$ e+ e+ e-_, _Nucl. Phys. B_ 299 (1988) 1.
  * (101) Muon g-2 collaboration, _Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm_ , 2308.06230.
  * (102) C. Hagedorn, T. Ohlsson, S. Riad and M.A. Schmidt, _Unification of Gauge Couplings in Radiative Neutrino Mass Models_ , _JHEP_ 09 (2016) 111 [1605.03986].
  * (103) T. de Boer, _M.Sc. thesis, University of Münster_ , 2021.
  * (104) S. Zeinstra, _Ph.D. thesis, University of Münster_ , 2021.
  * (105) P. Nath and P. Fileviez Perez, _Proton stability in grand unified theories, in strings and in branes_ , _Phys. Rept._ 441 (2007) 191 [hep-ph/0601023].
  * (106) Super-Kamiokande collaboration, _Search for proton decay via $p\to e^{+}\pi^{0}$ and $p\to\mu^{+}\pi^{0}$ with an enlarged fiducial volume in Super-Kamiokande I-IV_, _Phys. Rev. D_ 102 (2020) 112011 [2010.16098].
  * (107) S.R. Coleman and E.J. Weinberg, _Radiative Corrections as the Origin of Spontaneous Symmetry Breaking_ , _Phys. Rev. D_ 7 (1973) 1888.
  * (108) W.A. Bardeen, _On naturalness in the standard model_ , in _Ontake Summer Institute on Particle Physics_ , 8, 1995.
  * (109) A. Ahriche, K.L. McDonald and S. Nasri, _The Scale-Invariant Scotogenic Model_ , _JHEP_ 06 (2016) 182 [1604.05569].
  * (110) A. Ahriche, A. Manning, K.L. McDonald and S. Nasri, _Scale-Invariant Models with One-Loop Neutrino Mass and Dark Matter Candidates_ , _Phys. Rev. D_ 94 (2016) 053005 [1604.05995].