# Effective Dirac neutrino mass operator in the Standard Model with a local
Abelian extension

Diego<EMAIL_ADDRESS>and David
<EMAIL_ADDRESS>
Instituto de Física, Universidad de Antioquia
Calle 70 # 52-21, Apartado Aéreo 1226, Medellín, Colombia

###### Abstract

We present 48 types of solutions to the anomaly cancellation conditions of
local Abelian extensions of the Standard Model (SM) with right-handed singlet
chiral fermions. At least two of them acquire effective light Dirac neutrino
masses, while the others get heavy masses from the spontaneous symmetry
breaking of the local Abelian symmetry, forming a dark sector with multi-
component and multi-generational fermionic dark matter. The corresponding
effective Dirac neutrino mass operator can be realized at tree-level or
radiatively by introducing extra scalars, and in some cases after imposing
extra scotogenic conditions. The Dirac Zee model with Dirac fermionic dark
matter is presented as an example of model where the neutrino and dark matter
phenomenology are basically independent of each other.

##  1 Introduction

We can relate fermionic dark matter (DM) with neutrinos in two ways, through
the fermions inside the loops in scotogenic models [1] or through anomaly-free
gauge extensions of the SM with the extra chiral fermions required to cancel
out the anomalies [2]. With Dirac neutrinos, we can have the two ways
simultaneously. In fact, regarding neutrinos, the oscillation data is
compatible with both Majorana or Dirac neutrino masses. To explain Dirac
neutrino masses, right-handed neutrinos have to be introduced. Additionally,
an extra local symmetry is also required to guarantee proper total lepton
number conservation. Moreover, to have a consistent anomaly-free local
symmetry, extra right-handed singlet chiral fields are required in a dark
sector which may include dark matter candidates.

In this work we propose a general framework for the systematic study of Dirac
neutrino masses in the context of a gauge Abelian symmetry with multi-
component dark matter and massive gauge bosons, including the possibility of
dark photons.

By using an effective operator approach with until 10 extra singlet chiral
fields, we found a well defined set of 48 types of solutions with effective
Dirac neutrino masses, multi-component and multi-generational dark matter, and
either a dark photon or a heavy gauge boson mediator.

This work is organized as follows. The anomaly conditions for the gauge
Abelian symmetry are presented in section 2, and expressed in terms of the
charges of a set of right-handed singlet chiral fields. In section 3, a subset
of them are used as the right-handed neutrino component of the effective Dirac
neutrino mass operator. Then, the charge of the complex singlet scalar that
breaks spontaneously the gauge Abelian symmetry is fixed. In section 4 we
search for the solutions in which the remaining subset of right-handed singlet
chiral fermions obtain heavy masses from the singlet scalar vacuum expectation
value. There, they are classified in 48 types of solutions and their main
features are highlighted. In section 5 the solution with the least number of
chiral fields is discussed and used as an example of the realization of the
effective Dirac neutrino mass operator through SM scalar extensions. Finally,
in section 6 our conclusions are presented.

##  2 Anomaly conditions

We consider an extension of the SM with an additional Abelian gauge symmetry
and extra right-handed chiral fields singlets under the SM
$\operatorname{SU}(3)_{c}\otimes\operatorname{SU}(2)_{L}\otimes\operatorname{U}(1)_{Y}$
group. We assume that they do not form vector-like pairs so that all of them
are massless before the spontaneous symmetry breaking of the Abelian gauge
symmetry. On the other hand, we consider a set of $N$ integers,
$n_{1},n_{2},\ldots,n_{N}$, which satisfy the following Diophantine equations

$\displaystyle\sum_{\rho=1}^{N}n_{\rho}=0\qquad\text{and}\qquad\sum_{\rho=1}^{N}n_{\rho}^{3}=0\,.$
(1)

In [3] it is shown that

  * •

The extra gauge Abelian symmetry can be identified as one _dark_ symmetry,
$\operatorname{U}(1)_{D}$, with $N_{\text{chiral}}=N$ right-handed singlet
chiral fermions with $D$-charges $n_{1},n_{2},\ldots n_{N}$.

  * •

If the set of integers has one integer, $m$, repeated three times, the extra
gauge Abelian symmetry can be identified as one _active_ symmetry,
$\operatorname{U}(1)_{X}$, with $N_{\text{chiral}}=N-3$ right-handed singlet
chiral fermions with $X$-charges $n_{1},n_{2},\ldots n_{N_{\text{chiral}}}$.
The SM right-handed chiral fermions $X$-charges are denoted with the same name
of the field333$Q$ and $L$ are the $X$-charges of the fermion doublets
$Q^{\dagger}$ and $L^{\dagger}$, respectively.. They can be written in terms
of $m$ and a free parameter that we choose to be the $X$-charge of the
conjugate of the SM lepton doublet, $L$, such that

$\displaystyle u=$ $\displaystyle\frac{4L}{3}-m\,,$ $\displaystyle
d=m-\frac{2L}{3}$ $\displaystyle\,,$ $\displaystyle Q=$
$\displaystyle-\frac{L}{3}\,,$ $\displaystyle e=$ $\displaystyle m-2L\,,$
$\displaystyle h=$ $\displaystyle L-m\,,$ (2)

where $h$ is the SM Higgs $X$-charge, and $Q$ is the $X$-charge of the
conjugate of the SM quark doublet.

We are interested in _chiral_ solutions: the ones in which there is no $D$ or
$X$ opposite sign charges for the right-handed chiral fermions. This is
motivated by the fact that there are only chiral fermions in the SM.

The simple solution to obtain an active symmetry, $\operatorname{U}(1)_{X}$,
is just to have $m=1$ and three right-handed chiral fermions. When $h=0$, such
that $L=1$, this is the well known $\operatorname{U}(1)_{B-L}$ set of $N$
integers satisfying eq. (1): $(-1,\allowbreak-1,\allowbreak-1,\allowbreak
1,\allowbreak 1,\allowbreak 1)$, and any multiplicative factor of it:
$(-m,\allowbreak-m,\allowbreak-m,\allowbreak m,\allowbreak m,\allowbreak
m)$444This solutions cannot be interpreted as a dark symmetry because it would
contain three pairs of vector-like singlet fermions.. Note that we can make
the direct sum of this simple solution with any other solution with
$N_{\text{chiral}}$ integers that does not contains $m$, to obtain an active
gauge Abelian symmetry with three extra right handed chiral fermions with
$X$-charges $-m$ [4]. The same procedure can be used to obtain active
symmetries for any chiral solution of eq. (1), by identifying a subset of its
integers as $m$. For example, from the well known solution $(1,1,1,-4,-4,5)$
[5, 6] of the eq. (1), we can build the following extra active solutions with
their corresponding $m$

$\displaystyle(1,1,1,-4,-4,5)\oplus(-m,-m,-m,m,m,m)=$
$\displaystyle(1,1,1,-4,-4,5,-m,-m,-m,m,m,m),\text{ with $m\neq\pm 1$}$
$\displaystyle(1,1,1,-4,-4,5)\oplus(5,5,-5,-5)=$
$\displaystyle(1,1,1,-4,-4,5,5,5,-5,5),\text{ with $m=5$}$
$\displaystyle(1,1,1,-4,-4,5)\oplus(-4,4)=$
$\displaystyle(1,1,1,-4,-4,-4,4,5),\text{ with $m=-4$}\,.$ (3)

Finally, for any given solution, we also have a new solution from the direct
sum with itself. For example

$\displaystyle(1,1,1,-4,-4,5)\oplus(1,1,1,-4,-4,5)=$
$\displaystyle(1,1,1,1,1,1,-4,-4,-4,-4,5,5),\text{ with $m=1$ or $m=-4$}.$ (4)

In the next section we will apply phenomenological conditions upon all the
solutions with $5\leq N\leq 12$ integers obtained with the method explained in
[7] and implemented in the python package anomalies
[8]555https://pypi.org/project/anomalies/. The corresponding dataset with
$390\,074$ solutions to eqs. (1), with a maximum absolute value of 30, is in
[9]. They are complemented with the direct sums as in eqs. (2) and (4) for
each solution while keeping the maximum number of integers until 12.

##  3 Effective Dirac neutrino mass operator

In this section, we look for anomaly-free Abelian gauge extensions of the SM,
with right-handed singlet chiral fermions, realizing the effective Dirac
neutrino mass operator at a given dimension [10, 11]. In the two-component
spinor notation, this can be written as

$\displaystyle\mathcal{L}_{\text{eff}}=h_{\nu}^{\alpha
i}\,\left(\nu_{R\alpha}\right)^{\dagger}\,\epsilon_{ab}\,L_{i}^{a}\,H^{b}\left(\frac{S^{*}}{\Lambda}\right)^{\delta}+\text{H.c.},\qquad\text{with
$i=1,2,3$}\,,$ (5)

and $\delta=1,2,\ldots$ for dimension $4+\delta$ operators. Here
$h_{\nu}^{\alpha i}$ are dimensionless induced couplings with at least a
rank-2 matrix structure, $\nu_{R\alpha}$ are at least two right-handed
neutrinos ($\alpha=1,2,\ldots$) with equal Abelian charge $\nu$ extracted from
the full set of right-handed singlet chiral fields, $L_{i}$ are the lepton
doublets with Abelian charge $-L$, $H$ is the SM Higgs doublet with Abelian
charge $h=L-m$, $S$ is the complex singlet scalar responsible for the
spontaneous symmetry breaking (SSB) of the anomaly-free gauge symmetry with
Abelian charge

$\displaystyle s=-(\nu+m)/\delta\,.$ (6)

Finally, $\Lambda$ is a scale of new physics.

In general, after the SSB, several discrete symmetry are left, which may
guarantee the stability of at least one potential DM candidate [12]. $m\neq 0$
define an active symmetry where the collider restrictions on $Z^{\prime}$ are
rather strong because its allowed couplings to both light quarks and light
leptons, while $m=0$ corresponds to a dark symmetry with the corresponding
dark photon.

The tree-level realization of the renormalizable operator for Dirac neutrino
masses through the Higgs mechanism is obtained when $\delta=0$, such that
$\nu=-m$. Therefore, it is only possible for active symmetries. The simple
realization is for the $\operatorname{U}(1)_{B-L}$ solution $(-1,-1,-1,1,1,1)$
with $m=1$, $\nu=-1$, and $s=3$ to avoid a Majorana mass for $\nu_{R}$.
Therefore, the remnant $Z_{3}$ symmetry guarantees the Diracness of the tree-
level neutrino masses. However, such a solution imply very tiny Yukawa
coupling $h_{\nu}^{\alpha i}\sim\mathcal{O}(10^{-11})$, and therefore we do
not consider this kind of solutions here because their lack of explanation for
the smallness of neutrino masses.

For $\delta>0$ we can have tree-level [4] or radiative [3] realizations of the
non-renormalizable effective Dirac neutrino mass operator. The specific
realization can imply additional phenomenological conditions besides eq. (6)
[3]. However, there are realizations that only involve extra scalars beyond
the SM [13] which do not affect the anomaly cancellation conditions.
Therefore, any found solution satisfying eq. (6) can be embedded into a SM
scalar extension which realize the effective Dirac neutrino mass operator,
like the tree-level Type-II Dirac seesaw [4]. In this way, we can focus into
find the self-consistent solutions with at least two repeated charges, $\nu$,
which are compatible with the effective Dirac neutrino mass operator (5). This
self-consistent solutions are defined such that the only particles beyond the
SM are $\nu_{R\alpha}$, $S$ and the set of singlet chiral fermions which
acquires masses from $\langle S\rangle$ and that can play the role of (multi-
component) dark matter. The solutions until $N=9$ were already found in [4] in
the context of type-II Dirac seesaw realizations of the effective Dirac
neutrino mass operator for $\delta=1$. Here, we extend the search of self-
consistent solutions in the more general framework of the effective operator
until $N=12$ and all the relevant values for $\delta$.

##  4 Dark sector with multi-component and multi-generational dark matter

Starting from the extended dataset for the solutions with $N$ integers to the
Diophantine equations (1), we apply the following steps

  * •

Check that the solution has two (three) repeated integers to be identified as
$\nu$ and fix $N_{\nu}=2$ ($N_{\nu}=3$).

  * •

For $\delta=1,2,\ldots$ and all the possible combinations for $m$ and $\nu$ in
the solution, including $m=0$, find the $s$ value compatible with the
effective Dirac neutrino mass operator of D-$(4+\delta)$ according to eq. (6).

  * •

Interpret the integers in the solution that are different from $m$ and $\nu$
as the $D$-charges for $m=0$ or the $X$-charges for $m\neq 0$, of a set of
singlet chiral fermions: $\psi_{i}$, $i=1,\ldots,N_{\text{chiral}}-N_{\nu}$.
Then select the solutions for which the condition

$\displaystyle|n_{i}+n_{j}|=|s|,$ (7)

which guarantees that all the singlet chiral fermions, $\psi_{i}$, acquire
masses after the spontaneous symmetry breaking of the gauge Abelian symmetry
through $\langle S\rangle$.

We found $1\,122$ solutions where all the singlet chiral fermions have either
one effective mass through the operator in eq. (5) or from the vacuum
expectation value of $S$ after the spontaneous symmetry breaking of either the
dark or the active Abelian gauge symmetry. As shown in Fig. 1, most of the
solutions are for $\delta=1$ and $N=10$.

Figure 1: Distribution of solutions with $N$ integers to the Diophantine
equations (1) which allow the effective Dirac neutrino mass operator at
D-$(4+\delta)$ for at least two right-handed neutrinos and have non-vanishing
Dirac o Majorana masses for the other singlet chiral fermions in the solution.

Concerning the solutions with multi-component DM, we also explore the cases
which feature at least two DM candidates with _unconditional_ stability [14,
4]. This happens when the remnant symmetry $\mathbb{Z}_{|s|}$ contains two
subgroups $\mathbb{Z}_{p}\otimes\mathbb{Z}_{q}$ with $p$ and $q$ coprimes and
$|s|=pq$, which guarantee the stability of each lightest state under
$\mathbb{Z}_{p}$ and $\mathbb{Z}_{q}$ respectively. For the two DM candidates
associated to the set of chiral fields $\psi_{i}$ and $\chi_{j}$, we consider
below the following two possibilities for $|s|$ [14, 4]

  * •

$\mathbb{Z}_{6}\cong\mathbb{Z}_{2}\otimes\mathbb{Z}_{3}$: solutions with at
least a set of chiral fields with
$\psi_{i}\sim\left[\omega_{6}^{2}\vee\omega_{6}^{4}\right]$ under
$\mathbb{Z}_{6}$, and at least a set of chiral fields with
$\chi_{i}\sim\omega_{6}^{3}$ under $\mathbb{Z}_{6}$,

  * •

$\mathbb{Z}_{14}\cong\mathbb{Z}_{2}\otimes\mathbb{Z}_{7}$: solutions with at
least a set of chiral fields with
$\psi_{i}\sim\left[\omega_{14}^{2}\vee\omega_{14}^{6}\vee\omega_{14}^{8}\vee\omega_{14}^{10}\vee\omega_{14}^{12}\right]$
under $\mathbb{Z}_{14}$ and at least a set of chiral fields with
$\chi_{i}\sim\omega_{14}^{7}$ under $\mathbb{Z}_{14}$,

where $\omega_{|s|}=\operatorname{e}^{i\,2\pi/|s|}$.

The solutions can be classified in 48 types depending on the dimension of the
effective Dirac neutrino mass operator, D-$(4+\delta)$, for $\delta=1,2$
values666For larger $\delta$ there a few other solutions with similar
features; the number of Dirac or Majorana massive singlet fermions; the
multiplicity of generations for each one of them; and the unconditional
stability of at least two of the multi-component dark matter candidates. The
most representative solutions (with the lower possible maximum charge) for
each type, are presented in Table 1.

Solution | $N$ | $N_{\text{chiral}}$ | $m$ | $\nu$ | $\delta$ | $s$ | $N_{D}$ | $N_{M}$ | $G_{D}$ | $G_{M}$
---|---|---|---|---|---|---|---|---|---|---
$(1,-2,-3,5,5,-6)$ | $6$ | $6$ | $0$ | $5$ | $1$ | $-5$ | $2$ | $0$ | $1$ | $0$
$(3,3,3,-5,-5,-7,8)$ | $7$ | $4$ | $3$ | $-5$ | $2$ | $1$ | $1$ | $0$ | $1$ | $0$
$(1,-2,3,4,6,-7,-7,-7,9)$ | $9$ | $9$ | $0$ | $-7$ | $1$ | $7$ | $3$ | $0$ | $1$ | $0$
$(1,1,-4,-5,9,9,9,-10,-10)$ | $9$ | $9$ | $0$ | $9$ | $1$ | $-9$ | $3$ | $0$ | $2$ | $0$
$(1,2,-6,-6,-6,8,9,9,-11)$ | $9$ | $6$ | $-6$ | $9$ | $1$ | $-3$ | $2$ | $0$ | $1$ | $0$
$(1,-3,8,8,8,-12,-12,-17,19)$ | $9$ | $6$ | $8$ | $-12$ | $2$ | $2$ | $2$ | $1$ | $1$ | $1$
$(8,8,8,-12,-12,15,-17,-23,25)$ | $9$ | $6$ | $8$ | $-12$ | $2$ | $2$ | $2$ | $0$ | $1$ | $0$
$(1,-2,-2,3,3,-4,-4,6,6,-7)$ | $10$ | $10$ | $0$ | $6$ | $1$ | $-6$ | $3$ | $2$ | $2$ | $2$
$(1,-2,-2,3,4,-5,-5,7,7,-8)$ | $10$ | $10$ | $0$ | $-5$ | $1$ | $\boldsymbol{5}$ | $4$ | $0$ | $2$ | $0$
$(1,-2,-2,3,5,-6,-6,8,8,-9)$ | $10$ | $10$ | $0$ | $-6$ | $1$ | $6$ | $4$ | $0$ | $2$ | $0$
$(2,2,3,4,4,-5,-6,-6,-7,9)$ | $10$ | $10$ | $0$ | $2$ | $1$ | $\boldsymbol{-2}$ | $4$ | $2$ | $2$ | $2$
$(1,1,5,5,5,-6,-6,-6,-9,10)$ | $10$ | $10$ | $0$ | $1$ | $1$ | $-1$ | $4$ | $0$ | $3$ | $0$
$(2,2,4,4,-7,-7,-9,-9,10,10)$ | $10$ | $10$ | $0$ | $10$ | $2$ | $-5$ | $3$ | $0$ | $2$ | $0$
$(1,2,2,-3,6,6,-8,-8,-9,11)$ | $10$ | $10$ | $0$ | $-8$ | $1$ | $8$ | $4$ | $1$ | $2$ | $1$
$(1,-2,-3,5,6,-8,-9,11,11,-12)$ | $10$ | $10$ | $0$ | $11$ | $1$ | $-11$ | $4$ | $0$ | $1$ | $0$
$(1,1,-3,4,4,-7,8,-10,-10,12)$ | $10$ | $10$ | $0$ | $-10$ | $2$ | $5$ | $4$ | $0$ | $2$ | $0$
$(1,1,-2,-2,-4,6,-10,11,12,-13)$ | $10$ | $10$ | $0$ | $-2$ | $1$ | $2$ | $3$ | $2$ | $1$ | $2$
$(3,4,4,4,4,-5,-8,-8,-11,13)$ | $10$ | $10$ | $0$ | $-8$ | $1$ | $8$ | $2$ | $4$ | $1$ | $4$
$(4,4,5,6,6,-9,-10,-10,-11,15)$ | $10$ | $10$ | $0$ | $6$ | $1$ | $-6$ | $4$ | $0$ | $2$ | $0$
$(1,-2,-4,7,7,-10,-12,14,14,-15)$ | $10$ | $10$ | $0$ | $14$ | $1$ | $\boldsymbol{-14}$ | $3$ | $2$ | $1$ | $2$
$(1,2,2,-3,4,-6,12,-13,-14,15)$ | $10$ | $10$ | $0$ | $2$ | $1$ | $\boldsymbol{-2}$ | $4$ | $1$ | $1$ | $1$
$(1,4,4,-7,8,8,-9,-12,-12,15)$ | $10$ | $10$ | $0$ | $8$ | $1$ | $-8$ | $4$ | $2$ | $2$ | $2$
$(1,2,2,-9,-9,16,16,17,-18,-18)$ | $10$ | $10$ | $0$ | $-18$ | $1$ | $18$ | $3$ | $2$ | $2$ | $2$
$(1,-3,-6,7,-10,11,-16,18,18,-20)$ | $10$ | $10$ | $0$ | $18$ | $2$ | $-9$ | $4$ | $0$ | $1$ | $0$
$(1,-4,5,-6,-6,10,-14,15,20,-21)$ | $10$ | $10$ | $0$ | $-6$ | $1$ | $6$ | $4$ | $0$ | $1$ | $0$
$(2,-3,-6,7,12,-14,-14,17,20,-21)$ | $10$ | $10$ | $0$ | $-14$ | $1$ | $\boldsymbol{14}$ | $4$ | $1$ | $1$ | $1$
$(3,6,6,-7,8,8,-14,-14,-17,21)$ | $10$ | $10$ | $0$ | $-14$ | $1$ | $\boldsymbol{14}$ | $4$ | $1$ | $2$ | $1$
$(8,8,9,10,10,-13,-18,-18,-27,31)$ | $10$ | $10$ | $0$ | $-18$ | $1$ | $\boldsymbol{18}$ | $4$ | $1$ | $2$ | $1$
$(1,1,1,-2,-2,-5,-5,6,6,7,-8)$ | $11$ | $8$ | $1$ | $-2$ | $1$ | $1$ | $3$ | $0$ | $2$ | $0$
$(1,-2,-2,-2,-3,4,4,-5,6,7,-8)$ | $11$ | $8$ | $-2$ | $4$ | $1$ | $-2$ | $3$ | $1$ | $1$ | $1$
$(1,1,2,2,2,-4,-4,7,-8,-9,10)$ | $11$ | $8$ | $2$ | $-4$ | $1$ | $2$ | $2$ | $2$ | $1$ | $2$
$(2,2,2,-4,-4,-5,7,-8,9,10,-11)$ | $11$ | $8$ | $2$ | $-4$ | $1$ | $2$ | $3$ | $0$ | $1$ | $0$
$(1,-2,-3,-3,-3,5,5,-7,8,10,-11)$ | $11$ | $8$ | $-3$ | $5$ | $2$ | $-1$ | $3$ | $0$ | $1$ | $0$
$(3,3,3,-4,-4,7,7,-8,-9,-9,11)$ | $11$ | $8$ | $3$ | $-9$ | $2$ | $3$ | $3$ | $0$ | $2$ | $0$
$(1,3,5,-6,-6,-6,8,-9,12,12,-14)$ | $11$ | $8$ | $-6$ | $12$ | $1$ | $-6$ | $3$ | $1$ | $1$ | $1$
$(1,-2,6,6,6,-7,8,-9,-12,-12,15)$ | $11$ | $8$ | $6$ | $-12$ | $1$ | $\boldsymbol{6}$ | $3$ | $0$ | $1$ | $0$
$(1,3,3,6,6,6,-7,-10,-12,-12,16)$ | $11$ | $8$ | $6$ | $-12$ | $1$ | $\boldsymbol{6}$ | $2$ | $2$ | $1$ | $2$
$(1,-2,-2,-2,3,3,4,4,-5,-5,-5,6)$ | $12$ | $9$ | $-5$ | $-2$ | $1$ | $\boldsymbol{7}$ | $3$ | $0$ | $2$ | $0$
$(1,1,-3,4,5,5,5,-6,-7,-7,-8,10)$ | $12$ | $9$ | $5$ | $-7$ | $1$ | $2$ | $3$ | $2$ | $1$ | $2$
$(1,1,1,-2,4,-7,-7,-7,8,9,9,-10)$ | $12$ | $9$ | $-7$ | $9$ | $1$ | $-2$ | $2$ | $3$ | $1$ | $3$
$(1,1,-3,-3,-5,-5,-5,7,7,7,9,-11)$ | $12$ | $9$ | $-5$ | $7$ | $1$ | $-2$ | $3$ | $2$ | $2$ | $2$
$(1,-3,-3,-3,4,6,7,9,-10,-10,-10,12)$ | $12$ | $9$ | $-3$ | $-10$ | $1$ | $13$ | $3$ | $0$ | $1$ | $0$
$(1,1,1,3,3,-5,7,7,-11,-11,-11,15)$ | $12$ | $9$ | $1$ | $-11$ | $1$ | $10$ | $3$ | $1$ | $2$ | $1$
$(1,1,1,3,5,5,-5,5,-9,-9,-13,15)$ | $12$ | $9$ | $5$ | $-9$ | $2$ | $2$ | $2$ | $3$ | $1$ | $3$
$(1,-2,-2,3,6,-10,-10,-10,13,14,14,-17)$ | $12$ | $9$ | $-10$ | $14$ | $1$ | $-4$ | $4$ | $2$ | $2$ | $2$
$(1,-3,9,-11,-13,-13,-13,15,15,15,21,-23)$ | $12$ | $9$ | $-13$ | $15$ | $1$ | $-2$ | $3$ | $1$ | $1$ | $1$
Table 1:  Set of charges satisfying the Diophantine equations (1) together
with the conditions enumerated in the text, for solutions with
$N_{\textbf{chiral}}$ massive fermions, including the Dirac neutrinos masses
allowed by the effective operator of D-$4+\delta$, the solution with
unconditional stability through $\mathbb{Z}_{|s|}$ is highlighted with a bold
font for the charge of $|s|=\boldsymbol{6},\boldsymbol{14}$. $N_{D}$ and
$N_{M}$ are the number of massive Dirac and Majorana singlet fermions in each
solution, while $G_{D}$ and $G_{M}$ are the corresponding number of maximum
generations in the set.

There, the list of $N$ integers in column `Solution' satisfy the Diophantine
equations (1) for an active symmetry if $m\neq 0$ or for a dark symmetry if
$m=0$ (in such a case the number of new singlet chiral fermions is increased
by 3). In each solution there are $N_{\text{chiral}}$ singlet chiral fermions,
from which $N_{\nu}$ can develop effective Dirac neutrino masses from the
operator in eq. (5) with D-$(4+\delta)$. The remaining singlet chiral field
acquires Dirac or Majorana masses from the Yukawa couplings with the complex
singlet scalar, $S$. Finally, the solutions with unconditional stability are
highlighted with a bold font in the column $s$ of Table 1. The last four
columns correspond to the number of massive Dirac (Majorana) fermions $N_{D}$
($N_{M}$) and the maximum number of generations for the Dirac (Majorana)
fermions $G_{D}$ ($G_{N}$) in each solution.

There are not types of solutions with a single massive Dirac or Majorana
singlet fermion in several generations: $N_{DM}=1$ and $G_{DM}>1$, which would
point out to a pure scotogenic realization of the Dirac neutrino mass operator
without extra fermionic DM candidates. In fact, in general, most of the
solutions feature multi-component dark matter with the lighter of all them
protected by at least one remnant discrete symmetry, $\mathbb{Z}_{|s|}$. Each
DM candidate can be the lighter in a sector with several generations, and for
some cases highlighted with a bold $s$, with at least two of them protected by
$\mathbb{Z}_{p}\otimes\mathbb{Z}_{q}$ with $p$ and $q$ coprimes and $|s|=pq$ .

Figure 2: Number of massive Dirac and Majorana fermions of the full set of
solutions in Fig. 1. The discs magenta and green (yellow and cyan) are for
dark (active) symmetries which allows the effective Dirac neutrino mass
operator for at least two right-handed neutrinos for D-$(4+\delta)$, with
$\delta=1,2$ respectively. Similarly, the blue and red points are the type of
solution which satisfy the unconditional stability conditions for at least two
DM candidates.

In Fig. 2 it is illustrated the variety in which the number of Majorana and
Dirac massive singlet fermions appears in the several types of solutions. We
can have types of solutions with until four Majorana and two Dirac fermions
for dark Abelian gauge symmetries of D-5 (magenta discs) and four Dirac
fermions and two Majorana ones for dark or active Abelian gauge symmetries of
D-5 (magenta and yellow discs respectively).

Each massive singlet fermion can belong to a set of generations with the same
quantum numbers but different masses. In Fig. 3 it is shown the way in which
each dark matter candidate is grouped in generations for the several types of
solutions. From a phenomenological point of view, this is important because
this allows to have the rank of the effective neutrino mass matrix
sufficiently high in scotogenic realizations of Dirac neutrino mass operator
[3]. Moreover, the interplay between several generations can boost the direct
and indirect signals of the DM candidate [15]. We can have types of solutions
with until four generations of Majorana singlet fermions or with until three
generations of Dirac singlet fermions, for dark symmetries of D-5 (magenta
discs in the plot).

Note also that dark symmetries with $\delta=1$ and unconditional stability,
represented as blue dots in figs. 2 and 3, are typically associated to a high
number of massive singlet Dirac fermions, fig. 2, in sectors of at most two
generations of massive Dirac or Majorana singlet fermions, fig. 3. On the
other hand, active symmetries with $\delta=1$ and unconditional stability,
represented as red dots in figs. 2 and 3, are associated to at least two Dirac
singlet fermions and less than three Majorana singlet fermions, fig. 2,
without extra generations, fig. 3.

Figure 3: Same as Fig. 2 but for number generations of massive Dirac and
Majorana fermions of the full set of solutions in Fig. 1.

In any scotogenic realization of the effective Dirac neutrino mass operator, a
set of the heavy singlet fermions must participate directly in the generation
of the light neutrino masses. In this way, the corresponding fermion Yukawa
couplings are also constrained by neutrino physics. Extra conditions are
required to check if a solution can have a scotogenic realization. In fact,
all the solutions until $N=9$ for the one-loop scotogenic realization of the
effective Dirac neutrino mass operator at D-5 and D-6 were presented in [3].
As an example, the solution in the first row of Table 1 was analyzed there,
corresponding to one scotogenic realization of the D-5 operator at one-loop
through a dark symmetry with two independent Dirac fermionic dark matter
candidates (magenta disc with coordinates $(0,2)$ in Figure 2). In general,
for each type of solution we can have extra DM sectors which may be
independent of the neutrino physics. It is worth noticing that in the absence
of scotogenic conditions, the realization of the effective Dirac neutrino mass
operator is always possible for SM extensions with only extra scalars at tree-
level [16, 4], one-loop [13, 17] and two and three-loops [13]. Therefore, it
is well motivated to make a systematic study of multi-component Dirac or
Majorana fermionic DM, independent of the specific realization of the
effective Dirac neutrino mass operator. Moreover, since the neutrino physics
is in general independent of the DM phenomenology, we expect that the General
Neutrino Interactions (GNI) can be large and well suited to be explored in
neutrino experiments [17]. Additionally, the dark symmetry models contains a
dark photon which can be searched in both neutrino and direct detection
experiments. Alongside, the active symmetry contains a $Z^{\prime}$ with both
quark and lepton couplings which have strong dilepton signals at the LHC. It
turns out that the $Z^{\prime}$ portal also allows us to probe the models with
$m\neq 0$ through modifications of the cosmological history of the Universe,
namely, via additional contributions from the right handed neutrinos to the
effective number of relativistic degrees of freedom $N_{\text{eff}}$, with
constraints that are competitive with the ones obtained from colliders [18].

In general, we can see that multi-component and multi-generational DM
candidates are the trend for gauge Abelian extensions of the SM with massive
singlet chiral fermions compatible with the effective Dirac neutrino mass
operator of D-$(4+\delta)$, for $\delta>0$.

##  5 Simplest realization: Dirac Zee model with Dirac fermionic DM

As an illustration, consider the type of solution as the one in the second row
of Table 1: $(3,\allowbreak 3,\allowbreak
3,\allowbreak-5,\allowbreak-5,\allowbreak-7,\allowbreak 8)$. This features the
one with the least $N_{\text{chiral}}=4$, as summarized in Table 2. This
solution is the simplest of the types of solutions with
$\operatorname{U}(1)_{X}$ and D-6 ($\delta=2$), represented by the cyan dot
with coordinates $(0,1)$ in Figures 2 and 3.

The solution corresponds to an extension of the SM with $m=-3$ that for $h=0$
give rise to a local $\operatorname{U}(1)_{B-L}$ until a global factor of
$-1/3$: $u=d=-Q=1$ and $e=-L=-3$. This factor has been included in the column
$\operatorname{U}(1)_{B-L}$ of Table 2. The effective Dirac neutrino mass
operator of D-6 is compatible with two light Dirac neutrino masses through the
right-handed neutrino, $\nu_{R}$, with a $(B-L)$-charge of $\nu=-5/3$. Without
further conditions, this can be realized if we include two weak-singlet
charged scalars, denoted as $\sigma^{-}_{1}$ and $\sigma^{-}_{2}$ in Table 2.
Then, we can have the one-loop realization of the effective Dirac neutrino
mass operator at D-6 [13, 17] illustrated in Figure 4, which features the
Dirac Zee model. From the diagram there, we have $\sigma_{1}=-2L$ and
$\sigma_{2}=e+\nu=-2-2L$, with the corresponding $\operatorname{U}(1)_{B-L}$
charges in Table 2. The neutrino related phenomenology of the model has been
analysed in detail in [17], featuring large GNI along with charged lepton
flavour violating processes, and strong dilepton signals at the LHC.

Field | $SU(2)_{L}$ | $\operatorname{U}(1)_{Y}$ | $\operatorname{U}(1)_{X}$ | $\operatorname{U}(1)_{B-L}$
---|---|---|---|---
$Q_{i}$ | $\mathbf{2}$ | $1/6$ | $L/3$ | $1/3$
$u_{Ri}$ | $\mathbf{1}$ | $2/3$ | $4L/3-3$ | $1/3$
$d_{Ri}$ | $\mathbf{1}$ | $-1/3$ | $3-2L/3$ | $1/3$
$L_{i}$ | $\mathbf{2}$ | $-1/2$ | $-L$ | $-1$
$e_{Ri}$ | $\mathbf{1}$ | $-1$ | $3-2L$ | $-1$
$\nu_{R\alpha}$ | $\mathbf{1}$ | $0$ | $-5$ | $-5/3$
$\psi_{1}$ | $\mathbf{1}$ | $0$ | $-7$ | $-7/3$
$\psi_{2}$ | $\mathbf{1}$ | $0$ | $8$ | $8/3$
$H$ | $\mathbf{2}$ | $1/2$ | $L-3$ | $0$
$S$ | $\mathbf{1}$ | $0$ | $1$ | $1/3$
$\sigma_{1}^{-}$ | $\mathbf{1}$ | $-1$ | $2L$ | $2$
$\sigma_{2}^{-}$ | $\mathbf{1}$ | $-1$ | $(-2-2L)$ | $-8/3$
Table 2: $X$ and proper $B-L$ normalized charges for the first solution in
Table 1, $(3,\allowbreak 3,\allowbreak
3,\allowbreak-5,\allowbreak-5,\allowbreak-7,\allowbreak 8)$, for which $m=3$,
$\nu=-5$, $\delta=2$ and therefore from eq. (6), $s=1$. For the column
$\operatorname{U}(1)_{B-L}$ we fix $L=3$ and change $X\to X/3$. Moreover,
$i=1,2,3$ and $\alpha=1,2$. The electroweak quantum numbers are also shown for
each one of SM fields, the extra singlet chiral fermions, and the scalars in
the last four rows. Figure 4: Diagram with the $X$-charge flux of the fields
in the one-loop Dirac Zee model which realizes the effective Dirac neutrino
mass operator at D-6

In the new realization of the Dark Zee model presented here, we have the
novelty that the model can also have a Dirac fermionic dark matter candidate.
In fact, the Lagrangian includes a new Yukawa term

$\displaystyle\mathcal{L}\supset$ $\displaystyle
y_{D}\psi_{1}\psi_{2}S^{*}+\text{H.c,}$ (8)

with two singlet chiral fermions of $(B-L)$-charges $-7/3$ and $8/3$
respectively. After the spontaneous symmetry breaking of the active gauge
symmetry, this give rise to a Dirac heavy fermion mass

$\displaystyle\mathcal{L}_{\text{DM}}=M_{\Psi}\overline{\Psi}\Psi\,,$ (9)

where
$\Psi=\begin{pmatrix}\left(\psi_{1}\right)^{\dagger}&\psi_{2}\end{pmatrix}^{\operatorname{T}}$
and $M_{\Psi}=y_{D}\langle S\rangle/\sqrt{2}$.

Therefore, in contrary to the scotogenic models, the DM phenomenology can be
studied independently of the neutrino physics. And therefore, previous
analysis of simplified dark matter models with Dirac fermionic dark matter in
extensions of the SM with $\operatorname{U}(1)_{B-L}$, fully apply here. In
particular, the annihilation cross section for
$\overline{\psi}\psi\to{Z^{\prime}}^{*}\to\overline{f}f$ is given in [19]. As
we implemented this model in SARAH [20], we can have the relic density
calculation directly from micrOMEGAs [21].

The corresponding spin-independent direct detection cross section of dark
matter per nucleon is given by [22, 19]

$\displaystyle\sigma_{\psi N}=$
$\displaystyle\frac{\mu_{N}^{2}}{\pi}\frac{g_{X}^{4}}{M^{4}_{Z^{\prime}}}\left(\psi_{1}-\psi_{2}\right)^{2}L^{2}\,,$
(10)

where $\psi_{1}=-7/3$, $\psi_{2}=8/3$ and $L=1$, are the
$\operatorname{U}(1)_{B-L}$ charges in Table 2, and

$\displaystyle\mu_{N}=\frac{M_{N}M_{\Psi}}{M_{N}+M_{\Psi}}\,,$ (11)

is the reduced mass.

The cross section can be written as

$\displaystyle\sigma_{\psi
N}^{\mathrm{SI}}\left(\mathrm{cm}^{2}\right)=4.8\times
10^{-47}\left(\frac{\mu_{N}}{1\mathrm{GeV}}\right)^{2}\left(\frac{0.1}{g_{X}}\right)^{4}\left(\frac{M_{Z^{\prime}}}{4\
\text{TeV}}\right)^{4}\left(\psi_{1}-\psi_{2}\right)^{2}L^{2}\mathrm{~{}cm}^{2}\,.$
(12)

We can illustrate the DM phenomenology for a specific benchmark point which is
compatible with collider, $\Delta N_{\text{eff}}$ [18] and the current direct
detection currents constraints [23]. We choose $g_{X}=0.1$, $M_{Z^{\prime}}=4\
\text{TeV}$, $m_{S}=5\ \text{TeV}$. For this specific benchmark point, the
main constraint arises from the current upper bound on $\Delta N_{\text{eff}}$
for which $M_{Z^{\prime}}\gtrsim 40\ \text{TeV}$. This kind of constraint has
the potential to probe the full model with the next generation of CMB
experiments [18]. From Figure 5 we can see that the proper relic density can
be obtained for $M_{\Psi}\approx 1840\text{ GeV}$ or $M_{\Psi}\approx
2050\text{ GeV}$

Figure 5: Dirac fermionic Dark matter relic density as a function of its mass
for $\operatorname{U}(1)_{B-L}$ with copuling $g_{B-L}=g_{X}=0.1$,
$M_{Z^{\prime}}=4\ \text{TeV}$, $m_{S}=5\ \text{TeV}$.

##  6 Conclusions

We found around one thousand solutions to the anomaly free conditions of local
Abelian extensions of the SM with $N$ integers and $N_{\text{chiral}}$ right
handed singlet fermions which include the ones forming the effective set of
Dirac neutrinos and a dark sector with massive fermions. We classify the
solutions in 48 types in Table 1 depending if the symmetry is dark
($N_{\text{chiral}}=N$) or active ($N_{\text{chiral}}=N-3$), the dimension of
the effective Dirac neutrino mass operator ($4+\delta$), the number of
independent dark matter candidates, and the number of generations of each
massive fermion. The scalar realizations of the effective Dirac neutrino mass
operator feature a set of parameters which explain independently the neutrino
oscillations and the phenomenology of a multi-component and multi-generational
dark matter sector. This allows for large GNI and charged lepton flavor
violation observables while keeping the features of the simplified dark matter
models. We have illustrated this separation explicitly with a simple one-loop
Dirac Zee model which has a Dirac fermionic dark matter candidate. We could
expect enhanced effects in scalar realizations of the effective Dirac neutrino
mass operator at two-loops and three-loops [13], which are not yet fully
explored in the literature.

## Acknowledgments

The work of DR is supported by Sostenibilidad UdeA, and the UdeA/CODI Grants
2017-16286 and 2020-33177.

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