# Dirac dark matter, neutrino masses, and dark baryogenesis.

Diego<EMAIL_ADDRESS>Instituto de Física, Universidad de
Antioquia, Calle 70 No 52-21, Medellín, Colombia  Instituto de Física Gleb
Wataghin, UNICAMP, 13083-859, Campinas, SP, Brazil  Andrés
<EMAIL_ADDRESS>Instituto de Física, Universidad de
Antioquia, Calle 70 No 52-21, Medellín, Colombia  Walter
<EMAIL_ADDRESS>Department of Physics, Loyola University
Chicago, 1032 W. Sheridan Road, Chicago, IL, 60660, USA

###### Abstract

We present a gauged baryon number model as an example of models where all new
fermions required to cancel out the anomalies help to solve phenomenological
problems of the standard model (SM). Dark fermion doublets, along with the
iso-singlet charged fermions, in conjunction with a set of SM-singlet
fermions, participate in the generation of small neutrino masses through the
Dirac-dark Zee mechanism. The other SM-singlets explain the dark matter in the
Universe, while their coupling to an inert singlet scalar is the source of the
$CP$ violation. In the presence of a strong first-order electroweak phase
transition, this ``dark'' $CP$ violation allows for a successful electroweak
baryogenesis mechanism.

##  1 Introduction

In this work, we present a self-contained framework to explain the main
phenomenological problems of the standard model (SM) by using a dark sector
with a gauged baryon or lepton number Abelian symmetry, both of which allows
for a wide range of $Z^{\prime}$ masses and gauge couplings [1, 2]. In this
kind of models, fermionic dark matter serves as the source of the $CP$
violation required to generate the baryon asymmetry of the Universe [3, 4]. We
also search for all the solutions to the anomaly conditions compatible with
the effective Dirac-neutrino mass operator, focusing on their scotogenic
realizations. In this setup, we can have a dark sector that contains proper
dark matter candidates and sources both the baryon asymmetry of the Universe
and the smallness of neutrino masses.

The minimal field content for a gauged baryon or lepton number requires the
inclusion of an electroweak-scale set of fields consisting of a vector-like
fermion doublet and a vector-like iso-singlet [5, 6, 7], which behave as
_anomalons_ of the Abelian symmetry [3, 4]. The corresponding chiral fields
acquire masses from the Abelian gauge symmetry's spontaneous symmetry breaking
(SSB) at some higher energy scale. This condition fixes the absolute value of
the baryon or lepton number charge of the complex SM-singlet scalar
responsible for the SSB.

Extra SM-singlets are required to cancel out the anomalies. In this work, we
show that the anomaly conditions for a gauged baryon or lepton number symmetry
can be expressed in terms of the well known set of Diophantine equations

$\displaystyle\sum_{\alpha=1}^{N}z_{\alpha}=$ $\displaystyle 0\,,$
$\displaystyle\sum_{\alpha=1}^{N}z_{\alpha}^{3}=$ $\displaystyle 0\,,$ (1)

where $z_{\alpha}$ can be chosen as integers without loss of generality. The
data set with the full set of solutions with $N\leq 12$ and $|z_{\alpha}|\leq
30$ [8], can be used directly here. From there, we can choose the solutions
with at least two repeated charges to be assigned to the right-handed
neutrinos in the effective Dirac-neutrino mass operator with the complex SM-
singlet scalar to some nonzero power.

We further filter out solutions for which the extra SM-singlet fermions can
get masses from the SSB of the Abelian symmetry. We found 951
$\operatorname{U}(1)_{B}$ solutions that are classified into 25 types of
multi-component [9] and multi-flavor [10] dark matter models. We focus on the
phenomenology of the minimal $\operatorname{U}(1)_{B}$ solution, which allows
for effective Dirac-neutrino masses, and contains just a Dirac-fermionic dark
matter (DM) particle. The DM fields play a leading role as the source of the
$CP$ violation required for generating the baryon asymmetry [3] through the
Yukawa coupling with one extra SM-singlet scalar (with the same Abelian charge
as the scalar responsible for the SSB). The potential for this scalar and the
Higgs leads to the strong first-order electroweak phase transition (EWPT)
required to have successful electroweak baryogenesis [11]. This SM-singlet
scalar can acquire a vacuum expectation value (VEV) at high temperatures,
increasing the tree-level potential barrier and allowing for a stronger EWPT.

The dark $CP$ violation is diffused in front of the bubble walls, creating a
chiral asymmetry for the DM particle. The asymmetry is transferred to the SM
sector by the time-like component of the $Z^{\prime}$ background, which serves
as the baryon number chemical potential for the SM quarks through the
anomalous currents of the $Z^{\prime}$ [3, 4], resulting in a net baryon
asymmetry inside the bubble. Since $CP$ is spontaneously violated in the dark
sector, the model is safe against contributions to quark electric dipole
moments.

The required heavy anomalons can also be used to realize the effective Dirac-
neutrino mass operator through the Dirac-Dark Zee radiative seesaw mechanism.
In this way, the dark sector can also explain the smallness of the neutrino
masses. As in the original Dirac-Zee model, the extra scalar content is just a
set of iso-singlet charged scalars [12], which does not affect the anomaly
cancellation conditions.

Since the right-handed neutrinos can be thermalized in the early universe
through their $Z^{\prime}$ interactions with the SM quarks, we need to remain
consistent with cosmological constraints on the effective number of
relativistic degrees of freedom $N_{\text{eff}}$ [13]. This constraint goes in
the same direction as the constraints from direct detection cross section of
the Dirac-fermionic dark matter with the nucleon, requiring large vacuum
expectation values for the scalar field responsible for the SSB of
$\operatorname{U}(1)_{B}$. However, we still have room to explain the SM's
three more critical phenomenological problems, which can be fully explored in
future experiments.

The structure of this article is the following, in Section 2, we present the
general framework to define a gauged baryon o lepton number Abelian symmetry.
In Section 3, we focus on the minimal $\operatorname{U}(1)_{B}$ model with a
Dirac-dark Zee mechanism to generate small neutrino masses and a Dirac-
fermionic dark matter candidate. In Section 4, we establish the conditions to
have successful electroweak baryogenesis and perform the random scan to obtain
the models which satisfy all the experimental constraints, including the
proper $\Delta N_{\text{eff}}$. Finally, we present our conclusions in Section
5.

##  2 Local $\operatorname{U}(1)_{X}$ extension of the standard model

We extend the standard model (SM) by adding a general local
$\operatorname{U}(1)_{X}$ symmetry and a set of new chiral fields with
$X$-charges as displayed in Table 1, including $N^{\prime}$ SM-singlet chiral
fermions, two iso-singlet charged fermions $e^{\prime}_{R}$ and
$e^{\prime\prime}_{L}$, and two $\operatorname{SU}(2)_{L}$ doublets with
components
$L^{\prime}_{L}=\begin{pmatrix}N^{\prime}_{L}&e^{\prime}_{L}\end{pmatrix}^{\operatorname{T}}$
and
$L^{\prime\prime}_{R}=\begin{pmatrix}N^{\prime\prime}_{R}&e^{\prime\prime}_{R}\end{pmatrix}^{\operatorname{T}}$.
It is worth noting that, without this minimal set of extra chiral fields with
non-zero hypercharge, it is not possible to have zero lepton (quark)
$X$-charges, as required by a gauged baryon (lepton) number symmetry [5, 7].

Field | $\operatorname{SU}(2)_{L}$ | $\operatorname{U}(1)_{Y}$ | $\operatorname{U}(1)_{X}$
---|---|---|---
$u_{Ri}$ | $\mathbf{1}$ | $2/3$ | $u$
$d_{Ri}$ | $\mathbf{1}$ | $-1/3$ | $d$
$\left(Q_{i}\right)^{\dagger}$ | $\mathbf{2}$ | $-1/6$ | $Q$
$\left(L_{i}\right)^{\dagger}$ | $\mathbf{2}$ | $1/2$ | $L$
$e_{R_{i}}$ | $\mathbf{1}$ | $-1$ | $e$
$\left(L^{\prime}_{L}\right)^{\dagger}$ | $\mathbf{2}$ | $1/2$ | $-x^{\prime}$
$e^{\prime}_{R}$ | $\mathbf{1}$ | $-1$ | $x^{\prime}$
$L^{\prime\prime}_{R}$ | $\mathbf{2}$ | $-1/2$ | $x^{\prime\prime}$
$\left(e^{\prime\prime}_{L}\right)^{\dagger}$ | $\mathbf{1}$ | $1$ | $-x^{\prime\prime}$
$\chi_{\alpha}$ | $\mathbf{1}$ | $0$ | $z_{\alpha}$
Table 1: Fermion content and its quantum numbers. $i=1,2,3$,
$\alpha=1,2,\ldots,N^{\prime}$.

There are several conditions that we need to impose on the charge assignment
to achieve the goals of our model:

  1. 1.

The model must be anomaly free.

  2. 2.

At least two of the new SM-singlet chiral fields $\chi_{\alpha}$ must
correspond to the right-handed neutrinos associated with light Dirac neutrino
masses. We require that these fields have the same $\operatorname{U}(1)_{X}$
charge.

  3. 3.

Two SM-singlet chiral fields, say $\chi_{R}$ and $\chi_{L}$, can form the
Dirac-DM particle that also participates in the baryon asymmetry generation.
These fields couple to a heavy scalar field $\Phi$ that spontaneously breaks
$\operatorname{U}(1)_{X}$ and provides mass to the chiral fields. For
simplicity, we assume that the same scalar field provides the mass for the
heavy doublets $L^{\prime}_{L}$ and $L^{\prime\prime}_{R}$ (and the heavy iso-
singlets $e^{\prime}_{R}$ and $e^{\prime\prime}_{L}$). These requirements
impose the following conditions on the $X$-charges

$\displaystyle q_{\Phi}=$ $\displaystyle
q_{\chi_{R}}+q_{(\chi_{L})^{\dagger}}\,,$ (2a) $\displaystyle q_{\Phi}=$
$\displaystyle
q_{(L^{\prime}_{L})^{\dagger}}+q_{L^{\prime\prime}_{R}}=-\left[q_{e^{\prime}_{R}}+q_{(e^{\prime\prime}_{L})^{\dagger}}\right]=-x^{\prime}+x^{\prime\prime}\,.$
(2b)

The anomaly cancellation conditions on
$\left[\operatorname{SU}(3)_{c}\right]^{2}\operatorname{U}(1)_{X}$,
$\left[\operatorname{SU}(2)_{L}\right]^{2}\operatorname{U}(1)_{X}$,
$\left[\operatorname{U}(1)_{Y}\right]^{2}\operatorname{U}(1)_{X}$, allow us to
express three of the $X$-charges in terms of the others

$\displaystyle u=$
$\displaystyle-e-\frac{2}{3}L-\frac{1}{9}\left(x^{\prime}-x^{\prime\prime}\right)\,,$
$\displaystyle d=$ $\displaystyle
e+\frac{4}{3}L-\frac{1}{9}\left(x^{\prime}-x^{\prime\prime}\right)\,,$
$\displaystyle Q=$
$\displaystyle-\frac{1}{3}L+\frac{1}{9}\left(x^{\prime}-x^{\prime\prime}\right)\,,$
(3)

while the $\left[\operatorname{U}(1)_{X}\right]^{2}\operatorname{U}(1)_{Y}$
anomaly condition reduces to

$\displaystyle(e+L)(x^{\prime}-x^{\prime\prime})=0\,.$ (4)

Note that the vector-like solution $x^{\prime\prime}=x^{\prime}$ leads to the
same solution as the SM extension with only extra singlet chiral fermions with
no hypercharge in Ref. [14]. This kind of solution is incompatible with a
gauged baryon or lepton number and will not be considered here. To cancel out
the $\left[\operatorname{U}(1)_{X}\right]^{2}\operatorname{U}(1)_{Y}$ anomaly,
we choose instead

$\displaystyle e=-L\,,$ (5)

so that [4, 15]

$\displaystyle
Q=-u=-d=-\frac{1}{3}L+\frac{1}{9}\left(x^{\prime}-x^{\prime\prime}\right).$
(6)

Notice that, because of Eqs. (5) and (6), the $X$-charge of the Higgs must
always be zero in these scenarios.

The gravitational anomaly,
$\left[\operatorname{SO}(1,3)\right]^{2}\operatorname{U}(1)_{Y}$, and the
cubic anomaly, $\left[\operatorname{U}(1)_{X}\right]^{3}$, can be written as
the following system of Diophantine equations, respectively,

$\displaystyle\sum_{\alpha=1}^{N}z_{\alpha}=$ $\displaystyle 0\,,$
$\displaystyle\sum_{\alpha=1}^{N}z_{\alpha}^{3}=$ $\displaystyle 0\,,$ (7)

where $N=N^{\prime}+5$ and

$\displaystyle z_{N^{\prime}+1}=$ $\displaystyle-x^{\prime}\,,$ $\displaystyle
z_{N^{\prime}+2}=$ $\displaystyle x^{\prime\prime}\,,$ $\displaystyle
z_{N^{\prime}+2+i}$ $\displaystyle=L\,,\quad i=1,2,3$ (8)

for which the complete set of solutions can be easily calculated [16]444The
vector-like solution for $\operatorname{U}(1)_{L}$ and
$\operatorname{U}(1)_{B}$ in [4], in the proposed ordering, are respectively
$(1,\allowbreak 1,\allowbreak 1,\allowbreak q,\allowbreak-q-
N_{g},\allowbreak-q,\allowbreak
q+N_{g},\allowbreak-1,\allowbreak-1,\allowbreak-1)$ and $(q,\allowbreak-
N_{g}-q,\allowbreak-q,\allowbreak N_{g}+q)$ with massless neutrinos in the
last case. While the chiral $\operatorname{U}(1)_{L}$ solution in [15] is
$(2,\allowbreak 2,\allowbreak 2,\allowbreak-3,\allowbreak
6,\allowbreak-4,\allowbreak-5,\allowbreak-3,\allowbreak 0,\allowbreak
1,\allowbreak 1,\allowbreak 1)$. .

Notice that any solution of Eqs. (7) can be readily interpreted as gauged
baryon number symmetry, $\operatorname{U}(1)_{B}$, if we do not assign any
integer of the solution to $L$, so that it remains zero. The simplest solution
to have $\operatorname{U}(1)_{B}$ is for $N^{\prime}=2$ with a massive SM-
singlet Dirac fermion. Then, in addition to fixing $L=0$, we can choose
$z_{3}=-z_{1}=-x^{\prime}$ and $z_{4}=-z_{2}=x^{\prime\prime}$ [7]. However,
in this kind of solution, neutrinos are still massless, as in the SM. We are
interested here in solutions where the right-handed neutrinos are also charged
under $\operatorname{U}(1)_{B}$ or $\operatorname{U}(1)_{L}$

To have a gauge lepton number symmetry $\operatorname{U}(1)_{L}$, however, we
require two additional conditions on the solution of of Eqs. (7): (a) the set
must have three repeated integers whose value must be assigned to $L$; and,
since Eq. (6) can be rewritten as

$\displaystyle 9\,Q=-\sum_{\alpha=N^{\prime}+1}^{N^{\prime}+5}z_{\alpha}\,,$
(9)

(b) the corresponding subset of integers $z_{\alpha}$ in (2) must add to zero.

In this way, the previously found solutions for SM-singlet chiral extensions
of the SM with effective Dirac-neutrino masses [8] can be used directly here.
There, a SM-singlet scalar $\Phi$ appears in the effective Dirac-neutrino mass
operator [17], in terms of Weyl fermions,

$\displaystyle\mathcal{L}_{\text{eff}}=h_{\nu}^{\alpha
i}\,\left(\nu_{R\alpha}\right)^{\dagger}\,\epsilon_{ab}\,L_{i}^{a}\,H^{b}\left(\frac{\Phi^{*}}{\Lambda}\right)^{\delta}+\text{H.c.},\qquad\text{with
$i=1,2,3$}\,,$ (10)

and $\delta=1,2,\ldots$ for dimension-$d=4+\delta$ operators. We only consider
solutions with at least two repeated charges, $\nu$, to be assigned to the
right-handed neutrinos, $\nu_{R\alpha}$,

$\displaystyle\chi_{1}\to\nu_{R1},\cdots,\chi_{N_{\nu}}\to\nu_{R\,N_{\nu}},\qquad
2\leq N_{\nu}\leq 3\,,$ (11)

and with the scalar $\Phi$ providing mass to all pairs of SM-singlet chiral
fields, like the ones in Eq. (2a). Since all the solutions in [8] have at
least one massive SM-singlet Dirac fermion, it is always possible to reassign
it to $x^{\prime}$ and $x^{\prime\prime}$ as in Eq. (2b), so that all the
solutions in [8] satisfy

$\displaystyle q_{\Phi}=$ $\displaystyle\frac{L-\nu}{\delta}\,,$
$\displaystyle|q_{\Phi}|=$ $\displaystyle|-x^{\prime}+x^{\prime\prime}|,$
$\displaystyle|q_{\Phi}|=$
$\displaystyle|\chi_{\alpha}+\chi_{\beta}|\qquad\alpha,\beta=N_{\nu},\ldots,N^{\prime}\,.$
(12)

Then, we can reinterpret each one of the dark symmetries,
$\operatorname{U}(1)_{D}$, in [8] the as a gauged baryon number symmetry,
$\operatorname{U}(1)_{B}$555The type of solutions with $m=0$ in Table 1 of
[8].; while the active symmetry solutions $\operatorname{U}(1)_{X}$ in [8],
which also require three repeated charges, need to be checked against the
extra conditions from Eq. (2b) and Eq. (9), with $Q=0$, before they can be
identified as a gauged lepton number symmetry, $\operatorname{U}(1)_{L}$.

We are interested in the case of a scotogenic scenario where the dark sector
participates in the radiative neutrino mass loop. To realize the effective
Dirac-neutrino mass operator with $d=5$, we consider the Dirac-Dark Zee
topology as shown in the diagram in Fig. 1. In order to have a rank-2 light
Dirac-neutrino mass matrix, we require the addition of two sets of two iso-
singlet charged scalars $\sigma^{-}_{\alpha}$ and
${\sigma^{\prime}}^{-}_{\alpha}$, ($\alpha=1,2$) with $X$-charges $\sigma$ and
$\sigma^{\prime}$ respectively. Then, we have

$\displaystyle{\sigma}=$ $\displaystyle L+x^{\prime}\,,$
$\displaystyle{\sigma^{\prime}}=$ $\displaystyle\nu+x^{\prime}\,,$
$\displaystyle q_{\Phi}=$ $\displaystyle\sigma-\sigma^{\prime}\,,$ (13)

and we can always recover the conditions in Eq. (12) for $\delta=1$. In this
way, this topology can be realized for all the $\delta=1$ solutions in [8].
The same happens for any other topology that realizes the effective Dirac-
neutrino mass operator since the addition of the scalars does not affect the
anomaly cancellation. In fact, once the anomaly conditions are satisfied, any
extra fields that are required to allow for the mass operator must be either
scalar of vector-like fermions.

Figure 1: Diagram for the Dirac-dark Zee model.

##  3 An explicit implementation for $\operatorname{U}(1)_{B}$

In this work, we consider a specific solution to the conditions from Section 2
in order to realize neutrino masses, dark matter, and baryogenesis. We analyze
the $\operatorname{U}(1)_{X}$ solution $(5,5,-2,-3,1,-6)$ in Ref. [8] (ordered
according to Eq. (2)), which allows the effective Dirac-neutrino mass operator
at $d=5$. We can reinterpret this as a local $\operatorname{U}(1)_{B}$ with
$Q=5/9$, $x^{\prime}=-1$, $x^{\prime\prime}=-6$ and the particle content shown
in Table 2.

Field | $SU(2)_{L}$ | $U(1)_{Y}$ | $U(1)_{B}$
---|---|---|---
$u_{Ri}$ | $\mathbf{1}$ | $2/3$ | $u=1/3$
$d_{Ri}$ | $\mathbf{1}$ | $-1/3$ | $d=1/3$
$\left(Q_{i}\right)^{\dagger}$ | $\mathbf{2}$ | $-1/6$ | $Q=-1/3$
$\left(L_{i}\right)^{\dagger}$ | $\mathbf{2}$ | $1/2$ | $L=0$
$e_{R}$ | $\mathbf{1}$ | $-1$ | $e=0$
$\left(L^{\prime}_{L}\right)^{\dagger}$ | $\mathbf{2}$ | $1/2$ | $-x^{\prime}=-3/5$
$e^{\prime}_{R}$ | $\mathbf{1}$ | $-1$ | $x^{\prime}=3/5$
$L^{\prime\prime}_{R}$ | $\mathbf{2}$ | $-1/2$ | $x^{\prime\prime}=18/5$
$\left(e^{\prime\prime}_{L}\right)^{\dagger}$ | $\mathbf{1}$ | $1$ | $-x^{\prime\prime}=-18/5$
$\nu_{R,1}$ | $\mathbf{1}$ | $0$ | $-3$
$\nu_{R,2}$ | $\mathbf{1}$ | $0$ | $-3$
$\chi_{R}$ | $\mathbf{1}$ | $0$ | $6/5$
$\left(\chi_{L}\right)^{\dagger}$ | $\mathbf{1}$ | $0$ | $9/5$
$H$ | $\mathbf{2}$ | $1/2$ | $0$
$S$ | $\mathbf{1}$ | $0$ | $3$
$\Phi$ | $\mathbf{1}$ | $0$ | $3$
${\sigma}^{-}_{\alpha}$ | $\mathbf{1}$ | $1$ | $3/5$
${\sigma^{\prime}}^{-}_{\alpha}$ | $\mathbf{1}$ | $-1$ | $-12/5$
Table 2: Fermion (top) and scalar (bottom) content and its quantum numbers,
$i=1,2,3$, $\alpha=1,2$ with proper normalized baryon number charges with a
global factor $-5/3$.

With the new $\operatorname{U}(1)_{Y}$ charged chiral fermions and the two
sets of charged iso-singlet scalars (${\sigma}^{+}_{\alpha}$,
${\sigma^{\prime}}^{-}_{\alpha}$), it is now possible to realize the
scotogenic Dirac-Dark Zee topology as displayed in the diagram in Fig. 1. The
new charge assignment allows for the following terms in the Lagrangian

$\displaystyle-\mathcal{L}$ $\displaystyle\supset
h_{a,\Phi}\left(\chi_{L}\right)^{\dagger}\chi_{R}\Phi^{*}+h_{b,\Phi}\left(e^{\prime}_{R}\right)^{\dagger}e^{\prime\prime}_{L}\Phi^{*}+h_{c,\Phi}(L^{\prime}_{L})^{\dagger}L^{\prime\prime}_{R}\Phi^{*}+h^{i\alpha}_{d}L^{\prime}_{L}L_{i}{\sigma}^{+}_{\alpha}$
$\displaystyle+h_{a,S}\left(\chi_{L}\right)^{\dagger}\chi_{R}S^{*}+h_{b,S}\left(e^{\prime}_{R}\right)^{\dagger}e^{\prime\prime}_{L}S^{*}+h_{c,S}(L^{\prime}_{L})^{\dagger}L^{\prime\prime}_{R}S^{*}$
$\displaystyle+h_{e}^{\alpha\beta}\nu_{R,\alpha}e^{\prime}_{R}{\sigma^{\prime}}^{+}_{\beta}+h_{g}H^{\dagger}\left(e^{\prime}_{R}\right)^{\dagger}L^{\prime}_{L}+h_{h}(L^{\prime\prime}_{R})^{\dagger}e^{\prime\prime}_{L}H+{\rm\,h.c.}\,,$
(14)

where $\alpha,\beta=1,\,2$ and $i=1,\,2,\,3$. The $h$'s are Yukawa couplings,
which we assume to be real parameters for the sake of simplicity, except the
parameter $h_{a,S}$ which remains complex and will lead to $CP$ violation in
the model as in the baryogenesis mechanism studied in Refs. [4, 3]. We will
further explore this mechanism in Sec. 4. Finally, the Lagrangian also
contains the scalar potential

$\displaystyle
V(H,S,\Phi,{\sigma}^{\pm}_{\alpha},{\sigma^{\prime}}^{\pm}_{\alpha})=$
$\displaystyle
V(H)+V(S)+V(\Phi)+V({\sigma}^{\pm}_{\alpha})+V({\sigma^{\prime}}^{\pm}_{\alpha})$
$\displaystyle+\left[\kappa^{\alpha\beta}_{S}S{\sigma}^{+}_{\alpha}{\sigma^{\prime}}^{-}_{\beta}+\kappa_{\Phi}^{\alpha\beta}\Phi{\sigma}^{+}_{\alpha}{\sigma^{\prime}}^{-}_{\beta}\
+\lambda^{\prime}_{S\Phi}\left(S^{*}\Phi\right)^{2}+\text{h.c.}\right]\,,$
(15)

where

$\displaystyle V(\phi_{i})=$
$\displaystyle\mu_{\phi_{i}}^{2}\phi_{i}^{\dagger}\phi_{i}+\lambda_{\phi_{i}}\left(\phi_{i}^{\dagger}\phi_{i}\right)^{2}+\lambda_{\phi_{i}\phi_{j}}\phi_{i}^{\dagger}\phi_{i}\phi_{j}^{\dagger}\phi_{j}+\left(\lambda_{\phi_{i}S\Phi}\phi_{i}^{\dagger}\phi_{i}S^{*}\Phi+\text{h.c.}\right)\,,$
(16)

with $\phi_{i}=H,S,\Phi,{\sigma}^{+}_{\alpha},{\sigma^{\prime}}^{-}_{\alpha}$,
and $i<j$ .

###  3.1 Symmetry breaking and spectrum

In this model, the scalar $\Phi$ develops a vacuum expectation value (VEV),
$\langle\Phi\rangle=v_{\Phi}/\sqrt{2}$ that remains constant as the Universe
evolves near the EWPT, where the SM Higgs develops its VEV, $\langle
H\rangle=v/\sqrt{2}$, with $v=246.2$ GeV. At zero temperature, the scalar
fields $S,\sigma_{\alpha}^{+},\sigma^{\prime-}_{\alpha}$ do not obtain a VEV.

At zero temperature, after the electroweak symmetry breaking, the scalars
$H=\begin{pmatrix}G^{+}&H^{0}\end{pmatrix}^{\operatorname{T}}$ and $\Phi$ are
mixed. In the basis $(h^{0},\Phi^{0})$, the mass mixing matrix is given by

$\displaystyle
m_{h}^{2}=\left(\begin{array}[]{cc}-3\lambda_{H}v^{2}+\mu_{H}^{2}-\frac{1}{2}v_{\Phi}^{2}\lambda_{H\Phi}&-vv_{\Phi}\lambda_{H\Phi}\\\
-vv_{\Phi}\lambda_{H\Phi}&-\frac{1}{2}\lambda_{H\Phi}v^{2}+\mu_{\Phi}^{2}-3v_{\Phi}^{2}\lambda_{\Phi}\\\
\end{array}\right)\,,$ (19)

where $H^{0}=(h^{0}+v+i\,G^{0})/\sqrt{2}$,
$\Phi=(\Phi^{0}+v_{\Phi}+i\,G^{\prime 0})/\sqrt{2}$ and $G^{\pm}$, $G^{0}$,
$G^{\prime 0}$ are Goldstone bosons. This matrix is diagonalized by a unitary
transformation

$\displaystyle Z^{h}m_{h}^{2}Z^{h\dagger}=m_{h,\text{diag}}^{2}\,,$ (20)

such that

$\displaystyle\begin{pmatrix}h^{0}\\\ \Phi^{0}\\\
\end{pmatrix}=Z^{h}\begin{pmatrix}h_{1}\\\ h_{2}\\\
\end{pmatrix}=\begin{pmatrix}\cos\alpha&\sin\alpha\\\
-\sin\alpha&\cos\alpha\\\ \end{pmatrix}\begin{pmatrix}h_{1}\\\ h_{2}\\\
\end{pmatrix}\,.$ (21)

The fields ${\sigma}^{+}_{\alpha}$ and ${\sigma^{\prime}}^{-}_{\alpha}$ mix
among themselves and generate four heavy charged scalars that will play an
important role in the generation of the neutrino masses via one-loop as
displayed in the Feynman diagram in Fig. 1. In the basis
$({\sigma}^{\pm}_{\alpha},{\sigma^{\prime}}^{\pm}_{\beta})$, the charged-
scalar mixing matrix is given by

$\displaystyle m_{H}^{\pm
2}=\left(\begin{array}[]{cc}\dfrac{1}{2}\lambda_{H,{\sigma}^{\pm}_{\alpha}}v^{2}+\mu_{{\sigma}^{+}_{\alpha}}^{2}&\dfrac{1}{\sqrt{2}}\kappa_{\Phi}^{\alpha\beta}v_{\Phi}\\\
\dfrac{1}{\sqrt{2}}(\kappa_{\Phi}^{\alpha\beta})^{T}v_{\Phi}&\dfrac{1}{2}\lambda_{H,{\sigma^{\prime}}^{\pm}_{\alpha}}v^{2}+\mu_{{\sigma^{\prime}}_{\alpha}}^{2}\\\
\end{array}\right)\,,$ (24)

which is diagonalized by a $Z^{+}$ matrix, such that

$\displaystyle Z^{+}m_{H}^{2\pm}Z^{+\dagger}=m^{2}_{H^{\pm},\text{diag}}\,,$
(25)

where

$\displaystyle\begin{pmatrix}{\sigma}^{+}_{1}\\\ {\sigma}^{+}_{2}\\\
{\sigma^{\prime}}^{+}_{1}\\\
{\sigma^{\prime}}^{+}_{2}\end{pmatrix}=\sum_{i,j=1}^{4}Z^{+}_{ji}H_{j}^{+}\,.$
(26)

Also, there is a heavy scalar $S$ that will play an important role in the
baryogenesis mechanism.

Regarding the new charged fermions shown in Table (2), we have that, in the
basis $E_{Li}=(e^{\prime}_{L}\,,e^{{}^{\prime\prime}}_{L})$,
$E_{Ri}^{\dagger}=(e^{{}^{\prime}}_{R}\,,e^{{}^{\prime\prime}}_{R})^{\dagger}$,
the charged-fermion mixing matrix is given by

$\displaystyle
m_{e^{{}^{\prime}}}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}h_{f}\,v&h_{c,\Phi}\,v_{\Phi}\\\
h_{b,\Phi}\,v_{\Phi}&h_{h}\,v\\\ \end{array}\right)\,.$ (29)

This matrix can be diagonalized by the biunitary transformation

$\displaystyle
V^{L}m_{e^{{}^{\prime}}}(U^{R})^{\dagger}=m_{e^{\prime}_{n}}^{\text{diag}}\,,$
(30)

where the mass eigenstates
$e^{\prime}_{n}=(e^{\prime}_{L}\,,(e^{\prime}_{R})^{\dagger})_{n}$ are defined
by

$\displaystyle e^{\prime}_{Ln}=$ $\displaystyle
V^{L}_{ni}E_{Li}=\begin{pmatrix}\cos\theta_{L}&\sin\theta_{L}\\\
-\sin\theta_{L}&\cos\theta_{L}\end{pmatrix}\begin{pmatrix}e^{\prime}_{L}\\\
e^{{}^{\prime\prime}}_{L}\end{pmatrix}\,,\quad(e^{\prime}_{R})^{\dagger}_{n}=U^{R}_{ni}(E_{Ri})^{\dagger}=\begin{pmatrix}\cos\theta_{R}&\sin\theta_{R}\\\
-\sin\theta_{R}&\cos\theta_{R}\end{pmatrix}\begin{pmatrix}e_{R}^{{}^{\prime}\dagger}\\\
e^{{}^{\prime\prime}\dagger}_{R}\end{pmatrix}\,,$ (31)

where $\theta_{L,R}$ are the two mixing angles.

Also, in this model, there are two new neutral dark Dirac fermions

$\displaystyle N=$ $\displaystyle\begin{pmatrix}N^{\prime}_{L}\\\
N^{\prime\prime}_{R}\end{pmatrix}\,,$ $\displaystyle\chi=$
$\displaystyle\begin{pmatrix}\chi_{L}\\\ \chi_{R}\end{pmatrix}\,,$ (32)

such that $N$ will be a heavy fermion and $\chi$ will be the candidate for the
DM particle.

Finally, the mixing matrix for the neutral gauge sector is given by

$\displaystyle
m_{Z}^{2}=\left(\begin{array}[]{ccc}\frac{1}{4}g_{1}^{2}\,v^{2}&-\frac{1}{4}g_{1}g_{2}\,v^{2}&0\\\
-\frac{1}{4}g_{1}g_{2}\,v^{2}&\frac{1}{4}g_{2}^{2}\,v^{2}&0\\\
0&0&9g_{B}^{2}\,v_{\Phi}^{2}\\\ \end{array}\right)\,,$ (36)

where $g_{1},g_{2},g_{B}$ are the $\operatorname{U}(1)_{Y}$,
$\operatorname{SU}(2)_{L}$ and $\operatorname{U}(1)_{B}$ gauge couplings.
After diagonalization, the mass eigenstates are given by three neutral gauge
bosons $(\gamma,\,Z,\,Z^{\prime})$, where $\gamma$ is the photon field, $Z$ is
the SM gauge boson, and $Z^{\prime}$ is a new gauge boson with a mass
$m_{Z^{\prime}}=3g_{B}v_{\Phi}$.

###  3.2 Dirac neutrino masses

When the scalar fields $H$ and $\Phi$ acquire VEVs, neutrinos obtain Dirac
masses via the five-dimensional effective operator in Eq. (10), whose one-loop
realization is shown in Fig. 1. The diagram yields the mass matrix

$\displaystyle\mathcal{M}_{i\alpha}=$
$\displaystyle\sum_{\beta=1}^{2}h_{d}^{i\beta}\times\Lambda_{\beta}\times
h_{e}^{\alpha\beta}\,,$ (37)

where $\Lambda_{\beta}$ is the loop factor given by

$\displaystyle\Lambda_{\beta}=$
$\displaystyle\dfrac{1}{16\pi^{2}}\sum_{j=1}^{4}Z^{+}_{j,\alpha}\,Z^{+}_{j,\alpha+2}\sum_{n=1}^{2}V^{L}_{n1}\,U^{R}_{n1}\,m_{e^{\prime}_{n}}\times\left[\dfrac{m_{e^{\prime}_{n}}^{2}\ln(m_{e^{\prime}_{n}}^{2})-m_{H_{j}^{+}}^{2}\ln(m_{H_{j}^{+}}^{2})}{\left(m_{e^{\prime}_{n}}^{2}-m_{H_{j}^{+}}^{2}\right)}\right]\,,$
(38)

where $Z^{+}_{i,j}$, $V^{L}_{ij}$ and $U^{R}_{ij}$ are the rotation matrices
defined in Eqs. (26) and (31). $m_{e^{\prime}_{n}}$, $m_{H_{j}^{+}}$ are the
masses of the dark electrons and the charged scalars that are rotating in the
loop, respectively.

Neutrino oscillation data at $3\sigma$ [18] allow us to set the values of the
Yukawa couplings in the Eq. (3). In the basis where $\nu_{R}^{\alpha}$ are
mass eigenstates, the mass matrix (37) can be written as [19]

$\mathcal{M}_{ik}=(U_{\text{PMNS}})_{ik}\,(m_{\nu})_{k}\,,$ (39)

where $U_{\text{PMNS}}$ is the Pontecorvo-Maki-Nakagawa-Sakata matrix [20] and
$(m_{\nu})_{k}$ are the neutrino mass eigenvalues. Comparing the Eqs. (37) and
(39), we have 10 unknown parameters, $h_{e}^{\alpha\beta},h_{d}^{i\alpha}$,
and 9 equations. We can further simplify our analysis by imposing normal
ordering for neutrino masses, $m_{\nu 1}<m_{\nu 2}<m_{\nu 3}$666Alternatively,
we could choose inverted ordering for the masses, $m_{\nu 3}<m_{\nu 1}<m_{\nu
2}$, and the analysis would proceed the same way., and leave the couplings
$h_{e}^{\alpha\beta}$ as free parameters. We obtain the following relations:

$\displaystyle h_{e}^{\alpha\beta}$ $\displaystyle=$
$\displaystyle\text{free}\,,$ $\displaystyle h_{d}^{i1}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{\Lambda_{1}}\left(\dfrac{h_{e}^{23}m_{\nu
2}U_{i2}-h_{e}^{22}m_{\nu
3}U_{i3}}{h_{e}^{13}h_{e}^{22}-h_{e}^{12}h_{e}^{23}}\right),$ $\displaystyle
h_{d}^{i2}$ $\displaystyle=$
$\displaystyle+\dfrac{1}{\Lambda_{2}}\left(\dfrac{h_{e}^{13}m_{\nu
2}U_{i2}-h_{e}^{12}m_{\nu
3}U_{i3}}{h_{e}^{13}h_{e}^{22}-h_{e}^{12}h_{e}^{23}}\right)\,.$ (40)

By construction, these Yukawa couplings will reproduce the current neutrino
oscillation data [18].

###  3.3 Dark matter

In this work, the Dirac fermion $\chi\,=\,\left(\chi_{L},\,\chi_{R}\right)$ is
the DM candidate. The processes involved in the calculation of the DM relic
abundance include $\chi\bar{\chi}\to l_{i}\bar{l_{i}}$, $\chi\bar{\chi}\to
q_{i}\bar{q_{i}}$, $\chi\bar{\chi}\to\nu_{i}\bar{\nu_{i}}$, $\chi\bar{\chi}\to
h_{k}h_{k}$, $\chi\bar{\chi}\to h_{k}Z$, $\chi\bar{\chi}\to h_{k}Z^{\prime}$,
$\chi\bar{\chi}\to Z^{\prime}Z^{\prime}$.

The evolution of the relic abundance of DM follows the standard WIMP freeze-
out mechanism [21]. In the early radiation-dominated era of the Universe,
$\chi$ was in thermal equilibrium with the primordial plasma. As the Universe
cooled down to a temperature below the DM mass, the DM annihilation rate was
overtaken by the expansion of the Universe, $\Gamma\ll H$, and a relic density
of DM was frozen-out. The current relic abundance of DM depends on the
thermally-averaged annihilation cross-section $\langle\sigma
v\rangle\,\approx\,a\,+\,bv^{2}\,+\,\mathcal{O}(v^{4})$, which yields [21, 22]

$\Omega_{\chi}h^{2}=\frac{2.04\times
10^{9}x_{f}}{M_{\mathrm{Pl}}\sqrt{g_{*}(m_{\chi})}\left(a+3b/x_{f}\right)}\,,$
(41)

where $g_{*}(m)$ is the effective number of degrees of freedom, $x_{f}\equiv
m/T$ and $h$ is today's Hubble parameter in units of $100~{}{\rm km/s/Mpc}$.

Figure 2: SI independent DM-nucleon interactions: Vector(left) and scalar
(right) portals.

This model allows for direct detection signatures since the DM scatters with
nucleons through the $t$-channel with the $Z^{{}^{\prime}}$ and scalars
$h_{k}$ gauge boson as shown in Fig. 2. The spin-independent (SI) cross-
section for the interaction via the $Z^{{}^{\prime}}$ gauge field is given by
[23]

$\displaystyle\sigma^{\text{SI}}_{N}=\dfrac{1}{4\pi}\dfrac{M_{N}^{2}\,m_{\chi}^{2}}{(M_{N}\,+\,m_{\chi})^{2}}\dfrac{g_{B}^{4}}{M_{Z^{\prime}}^{4}}\,,$
(42)

where $M_{N}$ is the nucleon mass (neutron or proton). We only show here the
vector interaction because it is the dominant process. This was verified
numerically in Sec. 3.3.1.

####  3.3.1 Dark matter results

To study the phenomenology of this specific $\operatorname{U}(1)_{B}$ model,
we have performed a random scan of the parameter space, varying the free
parameters as described in Table 3.

Parameter | Range
---|---
$v_{\Phi}/$GeV | $10^{2}-2\times 10^{4}$
$g_{B}$ | $10^{-6}-1$
$h_{\\{a,b,c\\},\\{S,\Phi\\}}$, $h_{g}$, $h_{h}$ | $10^{-4}-1$
$h_{e}$ | $10^{-5}-1$
$\kappa_{S}^{\alpha\alpha}/$GeV, $\kappa_{\Phi}^{\alpha\alpha}/$GeV | $10^{-4}-1$
$(\mu_{S}^{2}$, $\mu_{\sigma_{1}^{\alpha}}^{2}$, $\mu_{\sigma_{2}^{\alpha}}^{2})/$GeV2 | $10^{6}-10^{10}$
$\lambda_{\Phi}$, $\lambda_{S}$ | $10^{-4}-1$
$\lambda_{\Phi H}$, $\lambda_{SH}$, $\lambda_{{\sigma}H}$, $\lambda_{{\sigma^{\prime}}H}$ | $10^{-5}-0.1$
Table 3: Scan ranges for free parameters of this model.

We implemented the model in SARAH [24, 25, 26, 27, 28], and coupled it to the
SPheno [29, 30] routines to obtain the spectrum. To compute the DM relic
density, we used MicrOMEGAs 5.0.4 [31], which includes the channels discussed
above and some special processes such as coannihilations and resonances [32].
Out all points in the parameter space, we selected the models that yield the
current value for the DM relic density, $\Omega_{\chi}h^{2}=0.1200\pm 0.0012$
to $3\,\sigma$ [33] and reproduce the neutrino parameters, following the
analysis described in Sec. 3.2. For those models, we computed the SI DM-
nucleus scattering cross-section (42), and checked it against the current
experimental bounds from PandaX-4T [34], XENON1T [35], and prospective
constraints from LZ [36] and DARWIN [37]. Additionally, we have filtered out
those points that are inconsistent with monojet searches at the LHC [38]. The
results are shown in Fig. 3.

Figure 3: The SI cross-section (blue dots) and the current experimental
constraints from PandaX-4T [34], XENON1T [35], and prospects from LZ [36] and
DARWIN [37]. We also show the Neutrino Coherent Scattering (NCS) [39, 40, 41,
42].

In Fig. 3, the models under the red continuous line and $m_{\chi}\gtrsim 10$
(GeV) survive the current experimental limits imposed by PandaX-4T [34] and
could be explored in the near future by the LZ [36] and DARWIN [37]
experiments. It is noteworthy that there are points in the figure that fall
into the neutrino floor (yellow region). They could be confused with neutrino
coherent scattering (NCS) with nucleons, and they need a special analysis that
is beyond the scope of this work.

An additional component of our analysis is the calculation of the spin-
dependent (SD) WIMP-neutron cross-section using MicrOMEGAs. We found that all
models under the red line have a SD cross-section $\sigma^{\text{SD}}\leq
10^{-43}$ pb, which falls below the experimental constraints from XENON1T
[43]. Furthermore, the SD cross-section obtained is below the prospects as LZ
[36] and DARWIN [37].

##  4 Dark $CP$ violation and electroweak baryogenesis

The third goal of this work is to generate the baryonic asymmetry in the
Universe. The standard lore lists the conditions for a theory of baryogenesis
as the Sakharov conditions [44]: Violation of charge $C$, charge-parity $CP$,
and there must exist processes that occur after exiting thermal equilibrium
and violate baryon number, $B$. To satisfy these conditions, we will adapt the
mechanism presented in [3, 4] in which $CP$ violation occurs in the dark
sector and is mediated to the SM sector by the new $Z^{\prime}$ gauge boson.
Baryogenesis results from the dynamics of the same hidden-sector fields that
are also responsible for dark matter and neutrino masses. The goal of this
section, rather than presenting a new mechanism for baryogenesis, is to show
that models of scotogenic Dirac neutrino masses and dark matter can easily
accommodate electroweak baryogenesis in the same fashion as [3, 4]777Ref. [11]
presented another mechanism where $CP$ violation occurs in the hidden sector
and there is a Yukawa coupling between the dark and SM fermions..

In the scalar sector of the model, the key fields in the mechanism are the
scalars $H=h/\sqrt{2}$ and $s=|S|$. A strong first-order phase transition is
incorporated into this scenario through the evolution of the VEVs of these two
fields, whose potential can be rewritten as [11]888We assume that the field
$\Phi$, which has a much larger VEV is integrated out and does not play any
role in the baryogenesis mechanism beyond providing mass terms in the
Lagrangian.

$\displaystyle
V(h,s)\,=\,\frac{\lambda_{H}}{4}\left(h^{2}-v^{2}\right)^{2}+\frac{\lambda_{S}}{4}\left(s^{2}-w^{2}\right)^{2}+\frac{\lambda_{SH}}{2}h^{2}s^{2}\,,$
(43)

where $v$ and $w$ are the VEVs for $h$ and $s$, respectively, at the minimum
of the potential. We require two stable minima, $(0,\,w_{0})$ and $(v,\,w)$,
for this potential and a tree-level barrier between them. Furthermore, the
global minimum at zero temperature must be at $v_{\text{EW}}\equiv
v(0)=246\,{\rm GeV}$ and $w(0)=0$. Following the potential analysis in [45],
we find that these conditions can be satisfied if

$\displaystyle\lambda_{SH}>0,\quad\lambda_{H}\lambda_{S}-\frac{1}{4}\lambda_{SH}^{2}<-\frac{\lambda_{SH}m_{s}^{2}}{2v^{2}}\,.$
(44)

At high temperatures, $\Phi$ breaks the $\operatorname{U}(1)_{B}$ symmetry and
the global minimum of the potential is given by $\left(0,w_{0}(T)\right)$ and
the electroweak symmetry is exact. As the temperature decreases, the
electroweak minimum forms with $(v(T),w(T))$. At the critical temperature
$T_{c}$, both minima are degenerate. For lower temperatures, $T<T_{c}$, the
electroweak minimum becomes the global minimum. The finite-temperature
effective potential is given by

$\displaystyle V_{T}(h,s)\,=\,$
$\displaystyle\frac{\lambda_{H}v_{c}^{4}}{4}\left(\frac{h^{2}}{v_{c}^{2}}+\frac{s^{2}}{w_{c}^{2}}-1\right)^{2}+\frac{\lambda_{H}v_{c}^{2}}{m_{s,c}^{2}w_{0,c}^{4}}h^{2}s^{2}+(T^{2}-T_{c}^{2})(c_{h}h^{2}+c_{s}s^{2})\,,$
(45)

where the subscript $c$ denotes the quantity at $T=T_{c}$. The coefficients
$c_{s}$ and $c_{h}$ correspond to one-loop thermal corrections and are given
by

$\displaystyle c_{h}\,=\,$
$\displaystyle\frac{1}{48}\left(9g_{2}^{2}+3g_{1}^{2}+12y_{t}^{2}+24\lambda_{H}+\lambda_{HS}\right)\,,\quad
c_{s}\,=\,\frac{1}{12}\left(3\lambda_{S}+2\lambda_{HS}\right)\,.$ (46)

An additional condition, to ensure that the global minimum for this potential
is the broken one when $T=0$, is

$\frac{c_{h}}{c_{s}}>\sqrt{\frac{\lambda_{h}}{\lambda_{s}}}.$ (47)

Using the thin-wall approximation [46], the nucleation temperature, $T_{n}$,
is defined by the condition [11]

$\exp\left(-S_{3}/T_{n}\right)\,=\,\frac{3}{4\pi}\left(\frac{H(T_{n})}{T_{n}}\right)^{4}\left(\frac{2\pi\,T_{n}}{S_{3}}\right)^{\frac{3}{2}}\,,$
(48)

where $S_{3}$ is the Euclidean action of the bubble and $H(T)$ is the Hubble
rate. In this approximation, to describe the bubble wall profile, we use the
ansatz in which the space dependence of the fields is given by

$h(z)\,=\,\frac{1}{2}v(T_{n})\left(1-\tanh\left(z/L_{w}\right)\right)\,,\qquad
s(z)\,=\,\frac{1}{2}s_{0}\left(1+\tanh\left(z/L_{w}\right)\right)\,,$ (49)

where $z$ is the direction normal to the wall and $L_{w}$ is the wall width
and $s_{0}\equiv w_{0}(T_{n})$. At the first-order phase transition, bubbles
nucleate and expand through the primordial plasma, causing perturbations on
the particle and antiparticle densities.

$P$ and $CP$ violation is incorporated by adding a term 999In the same spirit
of [4], we could add other terms with different powers of $S,\,S^{*}$ or
$\Phi$, however, we keep only this one for simplicity. We also ignore the
backreaction of $\delta V$ into the spontaneous breaking of $U(1)_{B}$.

$\delta V(S,\Phi)\,=\,\tilde{\lambda}_{\Phi S}\Phi^{*2}S^{2}\,,$ (50)

which is invariant under $\operatorname{U}(1)_{B}$ but generates a term
$\delta V\sim\langle\Phi\rangle^{2}S^{2}$ after $\Phi$ acquires a VEV. This
term in the potential is minimized by setting the phase of $S$ equal to
$\pi/2$. The chiral fields, $\chi_{L}$ and $\chi_{R}$, couple to both scalar
fields $\Phi$ and $S$. After the spontaneous breaking of
$\operatorname{U}(1)_{B}$, those chiral fields acquire an effective mass

$M_{\chi}\,=\ m_{\chi}+\lambda e^{i\theta}s\,,$ (51)

where $m_{\chi}=h_{a,\Phi}\langle\Phi\rangle$ and we have parametrized the
Yukawa coupling $h_{a,S}\,=\,\lambda e^{i\theta-i\pi/2}$. The parameters
$m_{\chi}$ and $\tilde{\lambda}_{\Phi S}$ are taken to be real, which is
possible due to the freedom to redefine the fields. However, we cannot
eliminate the complex phase in the second term. This phase violates $CP$ and
will lead to opposite signs in the perturbations of particles and
antiparticles, resulting in a net asymmetry in the interior of the bubble,
which is not washed out if the condition $v(T_{n})/T_{n}>1$ is satisfied [11].

The evolution of the particle and anti-particle distribution functions is
obtained from the Boltzmann equations, which are recast as the diffusion
equation for the rescaled chemical potential,
$\xi_{i}(z)\,\equiv\,{\mu_{i}(z)}/{T}\,=\,{6}\left(n_{i}-\overline{n}_{i}\right)/{T^{3}}$,

$-D_{L}\xi^{\prime\prime}_{\chi_{L}}-v_{w}\xi^{\prime}_{\chi_{L}}+\Gamma_{L}(\xi_{\chi_{L}}-\xi_{\chi_{R}})\,=\,S_{\cancel{CP}}\,,$
(52)

where $D_{L}$ is the diffusion constant for $\chi_{L}$, which is related to
the scattering rate $\Gamma_{L}$ by $D_{L}=\langle
v_{p_{z}}^{2}\rangle/3\Gamma_{L}$. Here, $\langle\,\rangle$ means thermal
average. $S_{\cancel{CP}}$ is $CP$-violating source that results from the
variation of $\theta$ [47],

$\displaystyle S_{\cancel{CP}}\,=\,-\frac{\lambda}{2}\frac{v_{w}D_{h}}{\langle
v_{p_{z}}^{2}\rangle
T}\left\langle\frac{|p_{z}|}{\omega^{2}}\right\rangle\left(M_{\chi}^{2}\,\theta^{\prime}\right)^{\prime\prime}\,,$
(53)

where $v_{w}$ is the wall's velocity and

$\displaystyle\quad\langle
v_{p_{z}}^{2}\rangle\,=\,\frac{3x+2}{x^{2}+3x+2}\,,\quad\left\langle\frac{|p_{z}|}{\omega^{2}}\right\rangle\,=\,\frac{(1-x)e^{-x}+x^{2}E_{1}(x)}{4m_{\chi}^{2}K_{2}(x)}\,,\quad
x\equiv m_{\chi}/T\,,$
$\displaystyle\left(M_{\chi}^{2}\theta^{\prime}\right)^{\prime\prime}\,=\,\frac{m_{\chi}s_{0}\lambda\left(-2+\cosh\left(\frac{2z}{L_{w}}\right)\right)\sin\theta}{L_{w}^{3}\cosh^{4}\left(\frac{z}{L_{w}}\right)}\,.$
(54)

$E_{1}(x)$ is the error function and $K_{2}(x)$ is the modified Bessel
function of the second kind.

The novelty of this mechanism, as presented by the authors of [3, 4], is that
the chiral particles, for which the asymmetry is initially generated, do not
couple to the $\operatorname{SU}(2)_{L}$ current, but instead give rise to a
non-zero $\operatorname{U}(1)_{B}$ charge density in the proximity of the
wall. This results in a $Z^{\prime}$ background that couples to the SM fields
with $\operatorname{U}(1)_{B}$ charge,

$\langle
Z^{\prime}_{0}\rangle\,=\,\frac{g_{B}\,(q_{\chi_{L}}-q_{\chi_{R}})T_{n}^{3}}{6M_{Z^{\prime}}}\int_{-\infty}^{\infty}\operatorname{d}z_{1}\,\xi_{\chi_{L}}(z^{\prime})\,e^{-M_{Z^{\prime}}|z-z^{\prime}|}\,,$
(55)

which generates a chemical potential for the SM quarks 101010At $T=T_{c}$, the
heavy fields $L^{\prime}_{L}$ and $L^{\prime\prime}_{R}$ are already thermally
decoupled and the $\operatorname{U}(1)_{B}$ current in the plasma is anomalous
with respect to $\operatorname{SU}(2)_{L}$. This allows the generation of the
non-zero chemical potential [4].,

$\mu_{Q}(z)\,=\,\mu_{d_{R},u_{R}}(z)\,=\,3\times\frac{1}{3}\times g_{B}\langle
Z^{\prime}_{0}(z)\rangle.$ (56)

This chemical potential sources a thermal-equilibrium asymmetry in the quarks
[4], $\Delta n_{Q}^{\rm EQ}(z)\,\sim\,T_{n}^{2}\,\mu_{Q}(z)$.

Finally, the baryon-number asymmetry is then given by

$n_{B}\,=\,\frac{\Gamma_{\rm
sph}}{v_{w}}\int_{0}^{\infty}\operatorname{d}z\,n_{Q}^{\rm
EQ}(z)\,\exp\left(-\frac{\Gamma_{\rm sph}}{v_{w}}\,z\right)\,,$ (57)

where $\Gamma_{\rm sph}$ is the sphaleron rate. The baryon-to-photon-number
ratio is then obtained by

$\eta_{B}\,=\,\frac{n_{B}}{s(T_{n})},\quad
s(T)\,\equiv\,\frac{2\pi^{2}}{45}g_{*S}(T)\,T^{3}\,,$ (58)

where $g_{*S}(T)$ is the effective number of relativistic degrees of freedom.

Our goal is to find what regions of the parameter space yield a baryon-number
asymmetry approximately equal to the measured value [48],

$\displaystyle 0.82\times 10^{-10}<\eta_{B}<0.92\times 10^{-10}\,.$ (59)

Figure 4: The blue points are the models shown in Fig. 3 that fulfill the
relic abundance of DM and the neutrino masses. The black points are those that
are not excluded by direct detection of DM and give the observed baryon
asymmetry at the Universe. The green-shaded region is in tension with the
measured number of relativistic degrees of freedom [49]. The other shaded
areas show the exclusions coming from meson decays [50] (yellow), the hadronic
widths of $\Upsilon$ and $J/\psi$ (purple and orange), LHC dijet searches [51]
(red), and from non-detection of the anomaly-cancelling fermions [52] (gray).

We computed the baryon asymmetry for the $U(1)_{B}$ model presented in Sec. 3
for those points in the parameter space that are consistent with neutrino
masses and that produce the expected relic abundance of dark matter. For the
free parameters that are not involved in neutrino mass and dark matter
calculations, we performed a random scan in the ranges shown in Table 4.

The results are displayed in the plane $M_{Z^{\prime}}-g_{B}$ in Fig. 4. The
blue points are models that fulfill the relic abundance of DM and neutrino
masses (displayed also in Fig. 3). The black points are the models that, in
addition, yield the observed baryon asymmetry (59) and are not excluded by
direct detection of DM.

Parameter | Range
---|---
$\theta$ | $(-\pi/2\,,\pi/2)$
$s_{0}/{\rm GeV}$ | $100-500$
$T_{n}/{\rm GeV}$ | $100-200$
$L_{w}$ | $1/T_{n}-10/T_{n}$
$v_{w}$ | $0.05-0.5$
Table 4: Scan ranges for the free parameters that are involved in the
baryogenesis mechanism.

We found that, for $M_{Z^{\prime}}\gtrsim 800$ GeV, $\eta_{B}$ is always lower
than the experimental value. This behavior is understood from the fact that
the baryon asymmetry scales as $\sim g_{B}/M_{Z^{\prime}}$. For illustrative
purposes, we include in Table 5 the average values of the parameters that
yield the black dots in Fig. 4.

| $M_{Z^{\prime}}/{\rm GeV}$ | $\log_{10}g_{B}$ | $m_{\chi}/{\rm GeV}$ | $\theta$ | $\log_{10}\lambda_{SH}$ | $\log_{10}\lambda_{S}$ | $s_{0}/{\rm GeV}$ | $L_{w}\,T_{n}$ | $T_{n}/{\rm GeV}$ | $T_{c}/{\rm GeV}$ | $v_{w}$
---|---|---|---|---|---|---|---|---|---|---|---
Average | 86.49 | -3.1 | 272.0 | 0.385 | -1.50 | -2.38 | 346.5 | 6.45 | 111.6 | 131.2 | 0.252
Std. dev. | 80.0 | 0.94 | 184.2 | 1.274 | 0.79 | 1.20 | 93.6 | 2.70 | 10.3 | 6.1 | 0.13

Table 5: Average and standard deviation of the parameter values that yield the
right baryon asymmetry.

Models with low $v_{\Phi}$ and $m_{\chi}\gtrsim 10$ GeV are excluded by direct
detection of DM. This is understood because according to Eq. (42), the SI
cross-section scales as $1/v_{\Phi}^{4}$ which needs high values for
$v_{\Phi}$ to past the current experimental bounds of PandaX-4T. In addition,
in Fig. 4, we show the limits from the hadronic widths of $\Upsilon$ and
$J/\psi$, constraints from the LHC dijet searches [51], and constraints from
meson decays [50]. Finally, the right-handed neutrinos contribute to the
number of relativistic degrees of freedom. We computed this contribution [49]
and required that $\Delta N_{\text{eff}}<0.3$, as measured by the Planck
collaboration [33]. This results in $M_{Z^{\prime}}/g_{B}\gtrsim 30$ TeV,
which implies $v_{\Phi}\gtrsim 10$ TeV. This constraint imposes a strong
restriction over our model as it excludes, in Fig. 4, the green region. The
bottom line is that only the black points under the green continuous line
$M_{Z^{\prime}}/g_{B}=30$ TeV reproduce the relic abundance of DM, neutrino
masses, and the baryon asymmetry at the Universe while being allowed by direct
detection of DM and cosmological constraints on $\Delta N_{\text{eff}}$.

##  5 Conclusions

We have added an explanation to the smallness of Dirac-neutrino masses in the
framework of electroweak baryogenesis through the $Z^{\prime}$ boson of a
gauged baryon or lepton number, by using the required non SM-singlet fermions
fields as the scotogenic particles of the Dirac-dark Zee model. In addition to
generating Dirac masses for neutrinos and the expected relic density of dark
matter, the dark sector is responsible for providing the necessary amount of
$CP$ violation and the non-zero $Z^{\prime}$ background to generate the baryon
asymmetry in the Universe. We scanned the parameter space requiring that these
goals be achieved while being compatible with direct detection experiments and
big bang nucleosynthesis. We found that models with $800\,{\rm
GeV}\,\gtrsim\,M_{Z}^{\prime}\,\gtrsim\,{\rm 30\,{\rm MeV}}$ can satisfy all
conditions with dark matter masses in the WIMP range.

## Acknowledgments

We want to thank Yue Zhang for very valuable feedback and illustrative
conversations in the course of this work. The work of DR is supported by
Sostenibilidad UdeA, and the UdeA/CODI Grant 2020-33177, and FAPESP funding
Grant 2021/11383-4. W.T. is supported by the National Science Foundation under
Grant No. PHY-2013052.

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