††thanks: E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Leptogenesis, Fermion Masses and Mixings in a SUSY $SU(5)$ GUT with $D_{4}$
Flavor Symmetry

M. Miskaoui  M. A. Loualidi  LPHE, Modeling and Simulations, Faculty of
Science, Mohammed V University in Rabat, 10090 Rabat, Morocco

###### Abstract

We propose a model of fermion masses and mixings based on $SU(5)$ grand
unified theory (GUT) and a $D_{4}$ flavor symmetry. This is the first
comprehensive study of a 4D $SU(5)$ GUT with a flavor symmetry that does not
contain a triplet irreducible representation. The Yukawa matrices of quarks
and charged leptons are obtained after integrating out heavy messenger fields
from renormalizable superpotentials while neutrino masses originate from the
type I seesaw mechanism. The group theoretical factors from 24- and
45-dimensional Higgs fields lead to ratios between the Yukawa couplings in
agreement with data, while the dangerous proton decay operators are highly
suppressed. By performing a numerical fit, we find that the model captures
accurately the mixing angles, the Yukawa couplings and the $CP$ phase of the
quark sector at the GUT scale. The neutrino masses are generated at the
renormalizable level with the prediction of trimaximal mixing while an
additional effective operator is required to account for the baryon asymmetry
of the universe (BAU). The model is remarkably predictive because only the
normal neutrino mass ordering and the lower octant of the atmospheric angle
are allowed while the $CP$ conserving values of the Dirac neutrino phase
$\delta_{CP}$ are excluded. Moreover, the predicted values of the effective
Majorana mass $m_{\beta\beta}$ can be tested at future neutrinoless double
beta decay experiments. An analytical and a numerical study of the BAU via the
leptogenesis mechanism is performed. We focused on the regions of parameter
space where leptogenesis from the lightest right-handed neutrino is
successfully realized. Strong correlations between the parameters of the
neutrino sector and the observed BAU are obtained.

$SU(5)$ SUSY GUT, Fermion masses and mixings, Flavor symmetries, Leptogenesis.

##  I Introduction

During the last two decades, neutrino oscillation experiments have presented
vigorous measurements of the neutrino mass-squared differences and their
mixing angles111The global analysis of all available oscillation data can be
found in [3, 4, 5] [1, 2], conflicting the zero mass prediction of the
standard model (SM) of electroweak interactions. Besides the precise
measurement of the oscillation parameters, the $CP$ violation and the flavor
pattern in the quark and lepton sectors along with the fermion mass
hierarchies are not firmly established within the SM. Therefore, theoretical
investigations beyond the SM are requisite to explain the origin of the
smallness of neutrino masses and their mixing as well as the flavor structure
of the other fermions. Currently, the less-known oscillation parameters are
the atmospheric angle—the octant of $\theta_{23}$ is unknown;
$\theta_{23}<\pi/4$ or $\theta_{23}>\pi/4$—and the Dirac $CP$ phase
$\delta_{CP}$. The $CP$ violation is of particular interest especially after
the T2K collaboration excluded some values of $\delta_{CP}$ giving rise to a
large improvement of the observed antineutrino oscillation probability at
$3\sigma$ confidence level [6]. Moreover, $CP$ violation is one of the
essential ingredients among the three conditions presented by Sakharov to
explain the observed BAU through the baryogenesis mechanism222There are
several baryogenesis models using different scenarios to explain the BAU, for
a review see Refs. [8, 9, 10, 11, 12]. [7]. The other two conditions being the
baryon number violation and the deviation from thermal equilibrium. The reason
to search for this in the lepton sector is due to the fact that the SM
predictions for the $CP$ violation—which is encoded in the CKM phase
$\delta_{CKM}$—is insufficient to generate the observed BAU and thus, new
sources for $CP$ violation beyond the SM are required.

An interesting approach to successfully produce the observed excess of matter
over antimatter in the universe is through the leptogenesis mechanism [13],
which relies on the right-handed (RH) Majorana neutrinos introduced in the
context of type I seesaw mechanism [14, 15, 16, 17, 18]. In practical terms,
this approach requires lepton number violation which arise naturally in type I
seesaw models via the Majorana masses of the RH neutrinos. Then, a lepton
asymmetry (equally baryon minus lepton $(B-L)$ asymmetry) is generated by the
out of-thermal-equilibrium and $CP$ violating decays of these RH neutrinos
that is eventually converted into a primordial baryon asymmetry by means of
the SM sphaleron processes [19]. As a result, the three Sakharov conditions
are satisfied in this scenario, which is remarkable considering that
leptogenesis connects high energy scales where the BAU takes place and
neutrino oscillations that take place at low energy scales.

Grand unified theories are the most attractive high-energy completions of the
SM that can bridge the experimentally accessible low energies and the
ultraviolet models of new physics [20, 21, 22, 23, 24]. The group-theoretical
structure of these GUTs present an elegant framework where the SM gauge
couplings and representations of SM fields are unified. When combined with
supersymmetry (SUSY) [25], GUTs provide a more powerful explanation to some of
the open questions in the SM as well as a solution to some of the problems
that are not addressed in the minimal non-SUSY GUTs [26]. The simplest
realization of such a combination is provided by the SUSY $SU(5)$ model where
the SM quarks and leptons are assigned in joint irreducible representations;
the conjugate five $F=\bar{5}$ and the ten $T=10$, therefore providing a
natural explanation for charge quantization while the unification of the gauge
couplings occurs at a scale of approximately $2\times 10^{16}$ $\mathrm{GeV}$
[27, 28, 29, 30, 31].

As a result of unification, the masses of down quarks and charged leptons are
generated from a common renormalizable operator leading, after GUT symmetry
breaking, to Yukawa couplings of same order of magnitude; $y_{e}=y_{d},$
$y_{\mu}=y_{s},$ and $y_{\tau}=y_{b}$. It is well-known that these equalities
are acceptable for the third generation but fails for the remaining ones
because of their conflict with the experimental data. The Georgi Jarlskog (GJ)
relations $m_{\mu}/m_{s}=3$ and $m_{e}/m_{d}=1/3$ generated in the $SU(5)$ GUT
model from a specific renormalizable operator involving a 45-dimensional Higgs
$H_{45}$ presented a first example solution to this issue [32]. In other
words, after $SU(5)$ symmetry breaking, this operator leads to different
Clebsch-Gordan (CG) coefficients for the down quarks and charged leptons
Yukawa couplings and subsequently the GJ relations. Moreover, to test the
validity of the GUT Yukawa couplings in a model of low-energy SUSY such as the
minimal supersymmetric standard model (MSSM), an accurate incorporation of
SUSY threshold effects is necessary especially in the case of large or medium
$\tan\beta$ [33, 34, 35, 36]. However, there have been some studies showing
that the threshold corrections may be ignored when the running of fermion
masses to the GUT scale are included, see, e.g., [37, 38]. As an alternative
to the GJ relations, the introduction of Higgs fields in the adjoint, 45 or
75-dimensional representations of $SU(5)$ gives rise to non-trivial CG factors
with new predictions for the first two generation Yukawa coupling ratios that
are preferred phenomenologically when compared to the GJ relations. These new
ratios arise from effective operators which are themselves resulted from
several renormalizable operators involving couplings between matter, Higgs and
heavy messenger fields that can be integrated out around the GUT scale, see
for instance Refs. [39, 40] for ratios derived from dimension 5 and dimension
6 operators in the context of SUSY $SU(5)$.

On the other hand, neutrinos are massless in SUSY $SU(5)$ model which implies
that the oscillation phenomenon can not be explained within its minimal
realization. The simplest way to address this issue is by introducing RH
singlet fermions to generate neutrino masses via the type I seesaw mechanism,
while the mixing angles can be determined by invoking the well-known approach
of flavor symmetries. Non-abelian discrete symmetries333For reviews on the use
of non-abelian discrete symmetries, see for instance [41, 42, 43, 44, 45].,
which act on matter particles belonging to different families, are in
particular a powerful tool that are used in the context of GUTs to explain the
mass hierarchies and the mixing of all fermions [46]. Groups with triplet
representations are in particular very popular in flavor model building. For
example, the discrete symmetry $A_{4}$ is widely used in $SU(5)$ flavor models
to explain the patterns of neutrino masses and their mixing; see for instance
Refs. [47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57]. On the other hand,
discrete groups with doublet representations like $S_{3}$ and $D_{4}$ are less
employed in 4D $SU(5)$ flavor models. However, as we will show in the present
study, the $D_{4}$ can provide good results regarding the neutrino flavor
structure by allowing the three generations of neutrinos to be unified into
the representations $1$ and $2$ instead of $3$444As shown in Ref. [58], one of
the interesting properties of models with the $D_{4}$ group is that it
predicts the $\mu-\tau$ symmetry in a natural manner [59, 60, 61, 62, 63] by
using minimal set of flavon fields..

In this work, we build a predictive model based on SUSY $SU(5)$ GUT
supplemented by a $D_{4}$ flavor symmetry suitable for addressing the
abovementioned questions. In particular, we show that our construction leads
to results for the pattern of fermion masses and mixings that are consistent
with the current experimental data. In fact, this is the first
phenomenological analysis of the fermion mass and mixing structures within
SUSY $SU(5)$ using the dihedral group555A SUSY $SU(5)$ model based on $D_{4}$
symmetry was considered before in Ref. [64], however, the phenomenological
implications of both the lepton and quark sectors were lacking. Here we
provide a comprehensive numerical study for the fermion flavor structure.
$D_{4}$. Besides this discrete group, we have added a $U(1)$ symmetry to
engineer the invariance of the superpotentials in the quark and lepton
sectors, and also to prevent dangerous operators that mediate rapid proton
decay. Apart from the usual SUSY $SU(5)$ superfield spectrum, let us describe
the building blocks of the present model which can be classified into four
sets: _(a)_ a renormalizable messenger sector with messenger fields $X_{i}$
associated to the down-quark, the charged lepton and neutrino sectors and
$Y_{i}$ associated to the up-quark, _(b)_ two additional higher dimensional
Higgs fields in the 24 and 45 GUT representations required for gauge symmetry
breaking and obtain ratios of the Yukawa couplings compatible with the data,
_(c)_ several gauge singlets—the so-called flavon fields—carrying quantum
numbers under the flavor symmetry $D_{4}\times U(1)$ needed to break the
flavor symmetry and structure the fermions mass matrices, and _(d)_ three
right-handed neutrinos $N_{i=1,2,3}^{c}$ responsible for generating the tiny
neutrino masses via the type I seesaw mechanism as well as the BAU through the
leptogenesis mechanism666For leptogenesis models based on type I seesaw
mechanism in the framework of $SU(5)$ GUT see, for instance, Ref. [65, 66, 67,
68, 69] and references therein. The introduction of all of the above fields
with the requirement to keep the effective superpotentials invariant is highly
controlled by the group theoretical structure of the $D_{4}\times U(1)$ flavor
symmetry.

In the charged sector, the messenger fields $X_{i}$ and $Y_{i}$ are coupled to
the matter fields, the flavon fields and the 24 and 45 Higgs fields. When
$X_{i}$ and $Y_{i}$ are integrated out we obtain the effective operators
responsible for the quark and lepton Yukawa couplings, which is then followed
by the spontaneous breaking of the flavor and gauge symmetries after the
flavons and Higgs fields of the $SU(5)$—that is the $5$, $\bar{5}$, $24$, and
$45$ dimensional Higgs fields denoted respectively as $H_{5}$, $H_{\bar{5}}$,
$H_{24}$ and $H_{45}$—acquire nonzero vacuum expectation values (VEVs). On the
one hand, the specific VEV alignments of the flavons break the $D_{4}\times
U(1)$ symmetry and help shape the fermions mass matrices, leading eventually
to the appropriate flavor structure of the quarks and leptons. On the other
hand, the CG factors obtained from the VEV structures of $H_{24}$ and $H_{45}$
lead to the following double ratio of the Yukawa coupling of the first and
second generation $\frac{y_{\mu}}{y_{s}}\frac{y_{d}}{y_{e}}\simeq 10.12$ which
is consistent with experimental data [70].

In the chargeless sector, the neutrino masses are generated at the
renormalizable level through the type I seesaw formula
$m_{\nu}=m_{D}m_{M}^{-1}m_{D}^{T}$ where $m_{D}$ and $m_{M}$ are the Dirac and
the Majorana mass matrices respectively. The obtained neutrino mass matrix
$m_{\nu}$ is described by only three parameters leading to strong constraints
among the physical parameters. Moreover, $m_{\nu}$ is invariant under a
particular remnant $Z_{2}$ symmetry which is commonly referred to as a magic
symmetry [71], indicating that $m_{\nu}$ is diagonalized by the well-known
trimaximal mixing ($TM_{2}$) matrix—that depends on an arbitrary angle
$\theta$ and an arbitrary phase $\sigma$—which is consistent with the observed
neutrino mixing angles [72, 73, 74, 75, 76, 77, 78]. However, the leptogenesis
mechanism can not be induced at the renormalizable level given that the
neutrino Yuwaka coupling matrix is proportional to the identity matrix which
leads to a vanishing lepton asymmetry. Therefore, we show that by introducing
one nonrenormalizable operator as a correction to the neutrino Yukawa coupling
matrix, our model can accommodate successfully the observed BAU via
leptogenesis777This is a known requirement in models predicting the
tribimaximal mixing at the leading order, see for instance Refs. [79, 80]. The
difference in the present model is that the leading order is giving by the
trimaximal mixing.. Our main results in the neutrino sector are:

  * •

only the normal hierarchy (NH) for neutrino mass spectrum is allowed,

  * •

only the lower octant of the atmospheric angle is allowed,

  * •

the $CP$ conserving values of the Dirac $CP$ phase $\delta_{CP}$ are excluded,

  * •

the predicted values of the effective Majorana mass in neutrinoless double
beta decay ($0\nu\beta\beta$) are testable at future $0\nu\beta\beta$
searches, and

  * •

the correlation between the BAU parameter denoted as $Y_{B}$ and the neutrino
sector parameters satisfies the experimental bound of the baryon asymmetry
from the Planck collaboration [81].

The rest of the paper is organized as follows. In Sec. II, we present the
particle content of the model as well as their transformation properties under
the $SU(5)\times D_{4}\times U(1)$ symmetry. In Sec. III, we first calculate
the mass matrices of the charged fermions and derive, through a numerical
study, the predictions of the mixing angles, the Yukawa couplings and the $CP$
phase of the quark sector at the GUT scale. We also comment the SUSY $SU(5)$
fast proton decay operators within our construction. Then, we make a detailed
study of the neutrino sector where, after gauge and flavor symmetry breaking,
the analytical expressions of the neutrino masses and mixing parameters as a
function of the model parameters are obtained. Following this, we perform a
numerical analysis of these parameters as well as the observables from non-
oscillatory experiments such as the effective Majorana mass $m_{\beta\beta}$
from $0\nu\beta\beta$ process. In Sec. IV, we carry out an analytical and a
numerical study of the BAU via the leptogenesis mechanism and show the
parameter space where the physical observables as well as the model parameters
are compatible with the observed BAU. In Sec. V, we give our conclusion. In
Appendix A, we describe the messenger sector of the model. In Appendix B, we
show explicitly that the contribution of the charged leptons to the lepton
asymmetry is highly suppressed to account for the BAU. In Appendix C, we
provide some tools on $D_{4}$ discrete group.

##  II Fermion masses and mixing within the $SU(5)\times D_{4}\times U(1)$
Model

###  II.1 Building blocks in $SU(5)\times D_{4}\times U(1)$ Model

In this section, we describe the different sectors of our $SU(5)\times
D_{4}\times U(1)$ GUT proposal and fix some notations. The chiral sector of
the minimal supersymmetric $SU(5)$ model involves matter and Higgs superfields
which are both supplemented by extra superfields in the present setup. The
usual matter superfields denoted as $T_{i}=(u_{i}^{c},e_{i}^{c},Q_{iL})$ and
$F_{i}=(d_{i}^{c},L_{i})$—with $i=1,2,3$ refers to the three generations of
matter—fit into the $10_{i}$ and $\bar{5}_{i}$ representations respectively,
while three right-handed neutrinos $N_{i}^{c}$ singlets of $SU(5)$ are added
in order to generate neutrino masses. Regarding the Higgs superfields, several
$SU(5)$ multiplets are needed to generate realistic fermion masses; most
importantly the $H_{5}$, $H_{\overline{5}}$, $H_{\overline{45}}$ and the
$H_{24}$. The low energy Higgs doublets $H_{u}$ and $H_{d}$ of the minimal
supersymmetric standard model (MSSM) arise from $H_{5}$ and a mixture of
$H_{\overline{5}}$ and $H_{\overline{45}}$ respectively. The 45-dimensional
Higgs is usually used to produce the GJ relations differentiating between the
(2-2) entry of the down quark and charged lepton mass matrices; however, it
has been shown in [39, 40] that there are many other options which
arepreferred compared to GJ relations. These alternativesarise from higher-
dimensional operators involving essentially higher dimensional Higgs
representations. In our proposal, we use $H_{\overline{45}}$ and $H_{24}$ to
produce the following ratios of the diagonal Yukawa couplings
$y_{e}/y_{d}=4/9$ and $y_{\mu}/y_{s}=9/2$ in good agreement with experiments
[70]; the different steps leading to these ratios is explained in details in
the next section. The 45-dimensional Higgs $H_{\overline{45}}$ satisfy the
following relations

$\displaystyle(H_{\overline{45}})_{c}^{ab}$ $\displaystyle=$
$\displaystyle-(H_{\overline{45}})_{c}^{ba}\quad,\quad(H_{\overline{45}})_{a}^{ab}=0$
$\displaystyle\left\langle(H_{\overline{45}})_{i}^{i5}\right\rangle$
$\displaystyle=$
$\displaystyle\upsilon_{45}\quad,\quad\left\langle(H_{\overline{45}})_{4}^{45}\right\rangle=-3\upsilon_{45}\
\text{with }i=1,2,3$ (II.1)

where $\upsilon_{45}$ is the VEV of $H_{\overline{45}}$. As for the adjoint
Higgs $H_{24}$ which is also responsible for breaking the $SU(5)$ group, it
develops its VEV along the direction

$\left\langle(H_{24})_{b}^{a}\right\rangle=\mathrm{diag}(1,1,1,\frac{-3}{2},\frac{-3}{2})\upsilon_{24}$
(II.2)

All the above superfields carry as well quantum numbers under the $D_{4}\times
U(1)$ group as depicted in Table (1). In this table, $F_{2,3}$ and
$N_{3,2}^{c}$ notations stand for $D_{4}$ doublet assignemnets
$(F_{2},F_{3})^{T}$ and $(N_{3},N_{2})^{T}$ respectively.

| $T_{1}$ | $T_{2}$ | $T_{3}$ | $F_{1}$ | $F_{2,3}$ | $N_{1}^{c}$ | $N_{3,2}^{c}$ | $H_{5}$ | $H_{\overline{5}}$ | $H_{\overline{45}}$ | $H_{24}$
---|---|---|---|---|---|---|---|---|---|---|---
$SU(5)$ | $10_{1}$ | $10_{2}$ | $10_{3}$ | $\overline{5}_{1}$ | $\overline{5}_{2,3}$ | $1_{1}^{\nu}$ | $1_{3,2}^{\nu}$ | $5_{H_{u}}$ | $\overline{5}_{H_{d}}$ | $\overline{45}_{H}$ | $24_{H}$
$D_{4}$ | $1_{+,-}$ | $1_{+,-}$ | $1_{+,+}$ | $1_{+,+}$ | $2_{0,0}$ | $1_{+,+}$ | $2_{0,0}$ | $1_{+,+}$ | $1_{+,-}$ | $1_{+,+}$ | $1_{+,+}$
$U\mathbb{(}1\mathbb{)}$ | $6$ | $12$ | $4$ | $13$ | $13$ | $-5$ | $-5$ | $-8$ | $4$ | $-16$ | $0$
Table 1: The $SU(5)\times D_{4}$ representations and $U(1)$ charges of the
matter, RH neutrinos and Higgs superfields.

On the other hand, the $D_{4}\times U(1)$ invariance requires the introduction
of several flavon fields in all the sectors of the model. In the up-quark
sector, only the top quark mass arises from a renormalizable coupling, the up
and charm quark masses are derived from higher dimensional Yukawa couplings
involving five flavon fields denoted as $\xi_{i=1,...,5}$. In the down quark
and charged lepton sector, four flavon fields denoted as $\phi$, $\varphi$,
$\Omega$ and $\Phi$ are needed for $D_{4}\times U(1)$ invariant
superpotential. When these flavons acquire their VEVs, they break the $D_{4}$
group and lead to appropriate mass matrices of down quarks and charged
leptons.

Flavons | $\xi_{1}$ | $\xi_{2}$ | $\xi_{3}$ | $\xi_{4}$ | $\xi_{5}$ | $\phi$ | $\varphi$ | $\Omega$ | $\Phi$
---|---|---|---|---|---|---|---|---|---
$D_{4}$ | $1_{+,+}$ | $1_{+,+}$ | $1_{+,-}$ | $1_{+,+}$ | $1_{+,-}$ | $1_{+,-}$ | $1_{+,-}$ | $2_{0,0}$ | $2_{0,0}$
$U\mathbb{(}1\mathbb{)}$ | $-4$ | $-10$ | $-2$ | $-16$ | $-8$ | $-9$ | $-14$ | $-21$ | $-9$
Table 2: The $D_{4}\times U(1)$ quantum numbers of the flavons used in the
quark and charged lepton sectors.

In the neutrino sector, the type I seesaw mechanism is responsible for the
light neutrino masses. In our proposal, five flavons are required for $U(1)$
invariance; three of them which are denoted as $\rho_{1}$, $\rho_{2}$ and
$\rho_{3}$ are assigned into different $D_{4}$ singlets while the remaining
two denoted as $\digamma$ and $\Gamma$ are transforming as $D_{4}$ doublets.

Flavons | $\rho_{1}$ | $\rho_{2}$ | $\rho_{3}$ | $\digamma$ | $\Gamma$
---|---|---|---|---|---
$D_{4}$ | $1_{+,+}$ | $1_{+,-}$ | $1_{-,-}$ | $2_{0,0}$ | $2_{0,0}$
$U\mathbb{(}1\mathbb{)}$ | $10$ | $10$ | $10$ | $10$ | $10$
Table 3: The $D_{4}\times U(1)$ quantum numbers of the flavons used in the
neutrino sector.

The flavons $\rho_{1}$ and $\digamma$ lead to the popular tribimaximal mixing
(TBM) matrix [82], while $\rho_{2}$, $\rho_{3}$ and $\Gamma$ are responsible
for the deviation of the neutrino mixing angles from their TBM values. The
quantum numbers under $D_{4}\times U(1)$ of these five flavons is as depicted
in Table (3).

###  II.2 Quark and charged lepton sectors

We start by the up-type quarks Yukawa matrix generated from higher order
operators derived from several renormalizable terms involving messenger fields
$Y_{i}$ and gauge singlet flavon fields $\xi_{i}$, see appendix A for more
details on $Y_{i}$ and Tables (1) and (2) to check the invariance under
$SU(5)\times D_{4}\times U(1)$ symmetry. After integrating out these messenger
fields888The renormalizable superpotentials for the quarks before integrating
out the messenger fields are given in appendix A. we obtain the invariant
effective superpotential for the up quarks

$\displaystyle W_{up}$ $\displaystyle=$
$\displaystyle\frac{y_{11}^{u}}{\Lambda}T_{1}T_{1}H_{5}\xi_{1}+\frac{y_{12}^{u}}{\Lambda}T_{1}T_{2}H_{5}\xi_{2}+\frac{y_{13}^{u}}{\Lambda}T_{1}T_{3}H_{5}\xi_{3}+\frac{y_{22}^{u}}{\Lambda}T_{2}T_{2}H_{5}\xi_{4}$
(II.3)
$\displaystyle+\frac{y_{23}^{u}}{\Lambda}T_{2}T_{3}H_{5}\xi_{5}+y_{33}^{u}T_{3}T_{3}H_{5}$

where $y_{ij}^{u}$ are the Yukawa coupling constants and $\Lambda$ is the
cutoff scale of the model which we take as the GUT scale. The $D_{4}$ flavor
symmetry is broken by the VEVs of the flavon fields as
$\left\langle\xi_{i}\right\rangle=\upsilon_{\xi_{i}}$ with $i=1,...,5$ while
the electroweak doublet $H_{u}$ contained in $H_{5}$ acquire its VEV as usual
$\left\langle H_{5}\right\rangle=\upsilon_{u}$. Assuming that the parameters
in $W_{up}$ are all real, the Yukawa matrix of up-type quarks can be written
as

$\mathcal{Y}_{up}=\left(\begin{array}[]{ccc}\frac{y_{11}^{u}}{\Lambda}\upsilon_{\xi_{1}}&\frac{y_{12}^{u}}{\Lambda}\upsilon_{\xi_{2}}&\frac{y_{13}^{u}}{\Lambda}\upsilon_{\xi_{3}}\\\
\frac{y_{12}^{u}}{\Lambda}\upsilon_{\xi_{2}}&\frac{y_{22}^{u}}{\Lambda}\upsilon_{\xi_{4}}&\frac{y_{23}^{u}}{\Lambda}\upsilon_{\xi_{5}}\\\
\frac{y_{13}^{u}}{\Lambda}\upsilon_{\xi_{3}}&\frac{y_{23}^{u}}{\Lambda}\upsilon_{\xi_{5}}&y_{33}^{u}\end{array}\right)=\left(\begin{array}[]{ccc}a_{11}&a_{12}&a_{13}\\\
a_{12}&a_{22}&a_{23}\\\ a_{13}&a_{23}&a_{33}\end{array}\right)$ (II.4)

We will now proceed with the down-type quarks and charged leptons generated
from the same Yukawa coupling
$10_{i}.\overline{5}_{j}.\overline{5}_{H_{d}}\equiv
T_{i}F_{j}H_{\overline{5}}$ where $i,j=1,2,3$ are the generation indices. The
superpotential leading to the Yukawa matrices is obtained from a
renormalizable superpotential that contains messenger fields denoted as
$X_{i}$. However, this time we need to add higher-dimensional Higgs
representations to differentiate between down quarks and charged lepton
masses, in particular we use the adjoint Higgs $H_{24}$ and the 45-dimensional
Higgs $H_{\overline{45}}$ for this purpose, see Eqs. (II.1) and (II.2).
Therefore, by using the superfield assignments in Tables (1) and (2) and
integrate out the messenger fields, we get the following invariant effective
superpotential for the down quarks and charged leptons

$W_{e,d}=\frac{y_{11}^{d}}{\left\langle
H_{24}\right\rangle^{2}}T_{1}F_{1}\phi\varphi
H_{\overline{5}}+\frac{y_{12}^{d}}{\left\langle
H_{24}\right\rangle^{2}}T_{1}F_{2,3}\Phi\varphi
H_{\overline{5}}+\frac{y_{22}^{d}}{\Lambda^{2}}T_{2}F_{2,3}\Phi
H_{24}H_{\overline{45}}+\frac{y_{33}^{d}}{\Lambda}T_{3}F_{2,3}\Omega
H_{\overline{5}}$ (II.5)

where $y_{ij}^{d}$ are the Yukawa coupling constants associated to the down
quarks and charged leptons. To illustrate how the adjoint Higgs contributes to
the entries of the Yukawa matrices and leading subsequently to a particular CG
factors that distinguish the down quarks Yukawa couplings from those of the
charged leptons,let us discuss the $y_{11}^{d}$ effective operator in (II.5).
This term is achieved by integrating out the heavy messenger fields from the
following renormalizable terms999The coupling constants are omitted in
$W_{e,d}^{Ren}$ for clarity.

$W_{e,d}^{Ren}\supset F_{1}\phi X_{1}+\overline{X}_{1}\varphi
X_{2}+\overline{X}_{2}H_{\overline{5}}T_{1}+X_{1}H_{24}\overline{X}_{1}+X_{2}H_{24}\overline{X}_{2}$
(II.6)

After integrating out $X_{1,2}$ and $\overline{X}_{1,2}$ from the first three
terms in (II.6), we are left with the first operator in (II.5);
$T_{1}F_{1}\phi\varphi H_{\overline{5}}$. On the other hand, the last two
terms in II.6 are responsible for the appearance of the square of the Higgs
adjoint VEV $\left\langle H_{24}\right\rangle^{2}$ in the denominator of the
effective operator. Specifically, the masses of the messenger pairs $X_{1,2}$
and $\overline{X}_{1,2}$ are achieved when $H_{24}$ acquire its VEV101010We
assume for simplicity that the VEV of the adjoint Higgs is around the GUT
scale as well as the cutoff scale; $\Lambda\equiv\left\langle
H_{24}\right\rangle=M_{GUT}\simeq 2\times 10^{16}\mathrm{GeV}$. $\left\langle
H_{24}\right\rangle$ with the group structure given in Eq. (II.2) which is
then followed by integrating out $X_{1}\overline{X}_{1}$ and
$X_{2}\overline{X}_{2}$ to obtain eventually the first effective operator in
(II.5). According to the group structure of $\left\langle
H_{24}\right\rangle$, the down quark mass is multiplied by the inverse of the
CG factors in the first three entries of the adjoint Higgs VEV in Eq. (II.2)
which is just $1$ in this case, while the electron mass is multiplied by the
inverse of the fourth and fifth components such that the resulting CG
coefficient is $\frac{-2}{3}\times\frac{-2}{3}=\frac{4}{9}$. The same
discussion holds for the second effective operator in (II.5) while for the
third effective operator the CG factors arise in the numerator. For
completeness, when the flavon fields acquire their VEVs in accordance with the
following alignment

$\left\langle\phi\right\rangle=\upsilon_{\phi}\quad,\quad\left\langle\varphi\right\rangle=\upsilon_{\varphi}\quad,\quad\left\langle\Phi\right\rangle=(\upsilon_{\Phi},0)^{T}\quad,\quad\left\langle\Omega\right\rangle=(0,\upsilon_{\Omega})^{T}$
(II.7)

we end up with the Yukawa matrices of the down-type quarks $\mathcal{Y}_{d}$
and charged leptons $\mathcal{Y}_{e}$ expressed as

$\displaystyle\mathcal{Y}_{d}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{y_{11}^{d}\upsilon_{\varphi}\upsilon_{\phi}}{\upsilon_{24}^{2}}&\frac{y_{12}^{d}\upsilon_{\varphi}\upsilon_{\Phi}}{\upsilon_{24}^{2}}&0\\\
0&\frac{y_{22}^{d}\upsilon_{24}\upsilon_{\Phi}}{\Lambda^{2}}&0\\\
0&0&\frac{y_{33}^{d}\upsilon_{\Omega}}{\Lambda}\end{array}\right)=\left(\begin{array}[]{ccc}b_{11}&b_{12}&0\\\
0&b_{22}&0\\\ 0&0&b_{33}\end{array}\right)$ (II.14)
$\displaystyle\mathcal{Y}_{e}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{4}{9}\frac{y_{11}^{d}\upsilon_{\phi}}{\upsilon_{24}^{2}}&0&0\\\
\frac{4}{9}\frac{y_{12}^{d}\upsilon_{\varphi}\upsilon_{\Phi}}{\upsilon_{24}^{2}}&\frac{9}{2}\frac{y_{22}^{d}\upsilon_{24}\upsilon_{\Phi}}{\Lambda^{2}}&0\\\
0&0&\frac{y_{33}^{d}\upsilon_{\Omega}}{\Lambda}\end{array}\right)=\left(\begin{array}[]{ccc}\frac{4}{9}b_{11}&0&0\\\
\frac{4}{9}b_{12}&\frac{9}{2}b_{22}&0\\\ 0&0&b_{33}\end{array}\right)$ (II.21)

These Yukawa matrices imply diagonal Yukawa couplings $y_{d}=b_{11}$,
$y_{s}=b_{22}$, $y_{b}=b_{33}$, $y_{e}=\frac{4}{9}b_{11}$,
$y_{\mu}=\frac{9}{2}b_{22}$ and $y_{\tau}=b_{33}$ where $y_{d}$, $y_{s},$ and
$y_{b}$ stand for the eigenvalues of the down quark Yukawa matrix
$\mathcal{Y}_{d}$, and $y_{e}$, $y_{\mu},$ and $y_{\tau}$ stand for the
eigenvalues of the charged lepton Yukawa matrix $\mathcal{Y}_{e}$. Thus, we
find that for the third family Yukawa coupling we have the well-known $b-\tau$
unification; $y_{\tau}=y_{b}$ which is still compatible with experimental
constraints [39], while for the first two families—instead of the GJ
relation—we find alternative GUT predictions with modified CG factors given as

$\frac{y_{e}}{y_{d}}=\frac{4}{9}\quad,\quad\frac{y_{\mu}}{y_{s}}=\frac{9}{2}$
(II.22)

To check the validity of these ratios, there are two particular constraints
for their GUT values developed in reference [70]; these are given by

$\frac{(1+\bar{\eta}_{l})y_{e}}{(1+\bar{\eta}_{q})y_{d}}\approx
0.41_{-0.06}^{+0.02}\quad,\quad\frac{(1+\bar{\eta}_{l})y_{\mu}}{(1+\bar{\eta}_{q})y_{s}}\approx
4.36\pm 0.23$ (II.23)

where $\bar{\eta}_{l}$ and $\bar{\eta}_{q}$ denote the threshold correction
parameters while the numerical values represent the $1\sigma$ uncertainties.
From these relations, one can derive the following double ratio at the GUT
scale independent of the threshold corrections [70]

$\frac{y_{\mu}}{y_{s}}\frac{y_{d}}{y_{e}}\approx 10.7_{-0.8}^{+1.8}$ (II.24)

As a result, we find in the present model that the ratios between Yukawa
couplings of the first two generations given in Eq. (II.22) give rise to the
relation $\frac{y_{\mu}}{y_{s}}\frac{y_{d}}{y_{e}}\simeq 10.12$ which is
consistent with the double ratio at the GUT scale given in Eq. (II.24).

  * •

Numerical estimates

Our model predicts the Yukawa couplings and mixing parameters at the GUT scale
which we assumed to be also the flavor symmetry breaking scale. To compare the
obtained spectrum of our model with the data extrapolated at the GUT scale,
the experimental values must run up to the GUT scale taking into account the
SUSY parameters; $\tan\beta$ and SUSY threshold correction effects. Such an
analysis has been performed in Ref. [70], where the extracted CKM parameters
and all Yukawa couplings at the GUT scale for $\tan\beta=5$ and $\tan\beta=10$
with $M_{SUSY}=1$ $\mathrm{TeV}$ and $\eta_{b}=-0.2437$ are given in Table (3)
of Ref. [47]. In our model, recall that the mass matrices of the charged
fermions are generally expressed as

$m_{up}=\upsilon_{u}\mathcal{Y}_{up}\qquad,\qquad
m_{down}=\upsilon_{d}\mathcal{Y}_{d}\qquad,\qquad
m_{lep}=\upsilon_{d}\mathcal{Y}_{e}$ (II.25)

where $\mathcal{Y}_{up}$, $\mathcal{Y}_{d}$ and $\mathcal{Y}_{e}$ refer to the
Yukawa matrices of up quarks, down quarks and charged leptons obtained in Eqs.
(II.4), (II.14) and (II.21), respectively. Recall also that the ratio between
the Higgs VEVs $\upsilon_{u}$ and $\upsilon_{d}$ is defined as
$\tan\beta=\frac{\upsilon_{u}}{\upsilon_{d}}$ while the SM Higgs VEV is
$\upsilon=\sqrt{\upsilon_{u}^{2}+\upsilon_{d}^{2}}=174$ $\mathrm{GeV}$, we
have

$m_{up}=\upsilon\sin\beta\mathcal{Y}_{up}\qquad,\qquad
m_{down}=\upsilon\cos\beta\mathcal{Y}_{d}\qquad,\qquad
m_{lep}=\upsilon\cos\beta\mathcal{Y}_{e}$ (II.26)

We assume in our calculations that all Yukawa coupling constants are real
except for $y_{16}$ which is complex and leads to the complex nature of the
parameter $b_{12}=\left|b_{12}\right|e^{i\epsilon}$, and subsequently to the
$CP$ violation in the quark sector. Moreover, the Yukawa matrices in Eqs.
(II.4) and (II.14) involve the following independent parameters
$\left\\{a_{11},a_{12},a_{13},a_{22},a_{23},a_{33},b_{11},b_{12},\left|b_{22}\right|,b_{33}\text{
and }\epsilon\right\\}$, which we need to fix in order to perform our
numerical analysis at the GUT scale.

Model parameters | Values | Model parameters | Values
---|---|---|---
$a_{11}$ | $0.1189\times 10^{-4}$ | $b_{11}$ | $0.44929\times 10^{-4}$
$a_{12}$ | $0.1530\times 10^{-2}$ | $\left|b_{12}\right|$ | $0.240406\times 10^{-3}$
$a_{13}$ | $0.2487\times 10^{-2}$ | $b_{22}$ | $0.9545\times 10^{-2}$
$a_{22}$ | $0.2076\times 10^{-2}$ | $b_{33}$ | $0.0705$
$a_{23}$ | $0.01857$ | $\epsilon$ | $\frac{\pi}{3}$
$a_{33}$ | $0.5218$ |  |
Table 4: Parameters of the quark and charged leptons Yukawa matrices at the
GUT scale with $\tan\beta=10$.

Next, we fix these input parameters for two cases: $\tan\beta=10$ and
$\tan\beta=5$. For $\tan\beta=10$, the numerical values are reported in Table
(4). The values reported in this table are fixed respecting the fact that the
magnitude of the flavon VEVs are smaller than the flavor symmetry breaking
scale; $\upsilon_{flavons}<M_{GUT}$, while we fix the phase $\epsilon$ to the
value $\frac{\pi}{3}$ which yields the correct experimental fit of the
$CP$-violating Dirac phase of the quark sector.The above estimates concerning
the input parameters produce the values of the physical quantities—namely the
quark mixing angles, the Yukawa couplings and the $CP$ phase—at the GUT scale;
these numerical values are as reported in Table (5).

Observables | values | Observables | values | Observables | values
---|---|---|---|---|---
$\theta_{12}^{q}/^{\circ}$ | $13.0433$ | $y_{u}/10^{-6}$ | $2.88221$ | $y_{d}/10^{-6}$ | $4.33494$
$\theta_{13}^{q}/^{\circ}$ | $0.18015$ | $y_{c}/10^{-3}$ | $1.40921$ | $y_{s}/10^{-5}$ | $9.79486$
$\theta_{23}^{q}/^{\circ}$ | $2.05405$ | $y_{t}$ | $0.51988$ | $y_{b}/10^{-3}$ | $7.015$
$\delta_{CP}^{q}/^{\circ}$ | $69.166^{\circ}$ |  |  |  |
Table 5: The predictions for the Yukawa eigenvalues, the mixing angles and the
$CP$ phase of the quark sector for $\tan\beta=10$.

We repeat the same numerical fit for $\tan\beta=5$, where the input parameters
and the output for the physical parameters are reported in Tables (6) and (7),
respectively.

Model parameters | Values | Model parameters | Values
---|---|---|---
$a_{11}$ | $0.12397\times 10^{-4}$ | $b_{11}$ | $0.226238\times 10^{-4}$
$a_{12}$ | $0.15866\times 10^{-3}$ | $\left|b_{12}\right|$ | $0.11827\times 10^{-3}$
$a_{13}$ | $0.25912\times 10^{-2}$ | $b_{22}$ | $0.47008\times 10^{-3}$
$a_{22}$ | $0.21459\times 10^{-2}$ | $b_{33}$ | $0.036597$
$a_{23}$ | $0.019344$ | $\epsilon$ | $\frac{\pi}{3}$
$a_{33}$ | $0.5435$ |  |
Table 6: Parameters of the quark and charged leptons Yukawa matrices at the GUT scale with $\tan\beta=5$. Observables | values | Observables | values | Observables | values
---|---|---|---|---|---
$\theta_{12}/^{\circ}$ | $13.0295$ | $y_{u}/10^{-6}$ | $2.92027$ | $y_{d}/10^{-6}$ | $4.16836$
$\theta_{13}/^{\circ}$ | $0.18019$ | $y_{c}/10^{-3}$ | $1.42946$ | $y_{s}/10^{-5}$ | $9.21045$
$\theta_{23}/^{\circ}$ | $2.05418$ | $y_{t}$ | $0.5333$ | $y_{b}/10^{-3}$ | $6.953$
$\delta_{CP}^{q}/^{\circ}$ | $69.18^{\circ}$ |  |  |  |
Table 7: The predictions for the Yukawa eigenvalues, the mixing angles and the
$CP$ phase of the quark sector for $\tan\beta=5$

This fit has been performed using the Mixing Parameter Tools package [83]. The
obtained values are in a good agreement with the GUT scale data for both
$\tan\beta=5$ and $\tan\beta=10$ [70, 47].

  * •

Comments on proton decay

Before we turn to the neutrino sector, we give brief comments on proton decay
which is one of the most important predictions in GUTs. It is well known that,
in the framework of the minimal SUSY $SU(5)$, the fast proton decay comes from
the contributions of the dimension four $TFF$ and dimension five $TTTF$ baryon
number violating operators111111For a brief review on proton decay coming from
dimension four and five operators, see for example the appendix C of Ref.
[50]. These operators lead to proton lifetime lower than the limit provided by
the Super-Kamiokande experiment [84]. It has been shown that the $d=4$
operators can be prevented by imposing the usual $R$ symmetry like in the case
of the MSSM [85]. On the other hand, the $d=5$ operators
$T_{i}T_{j}T_{k}F_{l}$ are generically induced via the exchange of color
triplet Higgsino [86, 87]. This issue of Higgsino-mediated proton
decay121212The proton decay via dimensional five operators is mediated by the
heavy color triplet Higgsino, and is obtained after integrating out the
colored Higgs triplet, for more details see for example Refs. [88, 89, 90]. is
intimately connected with the so-called doublet-triplet splitting problem—that
is the problem of differentiating between the masses of the Higgs triplets and
the Higgs doublets contained in the five dimensional Higgs of the $SU(5)$
GUT—the most effective ways proposed in the literature to resolve this
splitting problem is provided by the missing partner (MP) and the double
missing partner (DMP) mechanisms [91, 92, 93, 94].

In our model, the renormalizable operators $T_{i}F_{j}F_{k}$ are forbidden
because they transform nontrivially under the $U(1)$ symmetry and thus, the
proton stability at dimension 4 is guaranteed. Moreover, the $d=4$ operators
that may arise from the coupling with the flavon fields present in the model
are also prevented by the $U(1)$ symmetry as can be checked easily from the
field assignments in Tables (1-3).
For the $d=5$ operators, recall first that the usual tree-level Yukawa
couplings of the first and second generations are prevented by the $U(1)$
symmetry, and thus the usual operators $T_{i}T_{j}T_{k}F_{l}$ are absent in
our model. On the other hand, since the quark and charged lepton masses are
generated from effective operators that involve flavon fields—see Eqs. (II.3)
and (II.5)—proton decay can arise from higher dimension-7 and dimension-8
operators of the form

$\frac{1}{M_{T}}T_{i}T_{j}T_{k}F_{l}\
\frac{f_{1}}{M}\frac{f_{2}}{M}(\frac{f_{3}}{M})^{n}$ (II.27)

where $M_{T}$ denotes the mass of the colored Higgs triplet, M stands for the
cutoff scale $\Lambda$ or the VEV of the adjoint Higgs, $n=0,1$ and $f_{i}$
stands for the flavon fields $\xi_{i=1,...,5}$, $\phi$, $\varphi$, $\Phi$, and
$\Omega$. As an illustration, after inegrating out the colored Higgs triplet
from the first operator of Eq. (II.3) and the last operator of Eq. (II.5), we
obtain the dimension-7 operator

$\frac{1}{M_{T}}\frac{1}{\Lambda^{2}}T_{1}T_{1}T_{3}F_{2,3}\xi_{1}\Omega$
(II.28)

Since $M_{T}$ and $\Lambda$ are both expected to be at the GUT scale, it is
straightforward to realize that contribution of the operator (II.28) to proton
decay is sufficiently suppressed. The same discussion holds for all the
operators involving flavon fields.

###  II.3 Neutrino masses and mixing

The fermion sector of the supersymmetric $SU(5)$-GUT model is extended by
three right-handed neutrino superfields $N_{i=1,2,3}^{c}$ transforming as
gauge singlets $1_{i}$, and carrying quantum numbers under the $D_{4}\times
U(1)$ flavor group. Therefore, the light active neutrino masses are generated
through the famous type I seesaw mechanism. In our set up with the quantum
numbers of matter and Higgs superfields given in Table (1), five flavon
superfields are required for $U(1)$ invariance in such a way that they all
couple only to the Majorana mass term $N_{i}^{c}N_{j}^{c}$ and they all carry
the same $U(1)$ charge $q_{U(1)}=10$. Three of these flavons denoted as
$\rho_{1}$, $\rho_{2}$ and $\rho_{3}$ are assigned to the $D_{4}$ singlets
$1_{+,+}$, $1_{+,-}$ and $1_{-,-}$ respectively, while the remaining two
denoted as $\Gamma$ and $\digamma$ are assigned to the $D_{4}$ doublet
$2_{0,0}$. Thus, by using the quantum numbers in Tables (1) and (2), the
superpotential invariant under the $SU(5)\times D_{4}\times U(1)$ group is
given by

$\displaystyle\mathcal{W}_{\nu}$ $\displaystyle=$
$\displaystyle\lambda_{1}N_{1}^{c}F_{1}H_{5}+\lambda_{2}N_{3,2}^{c}F_{2,3}H_{5}+\lambda_{3}N_{1}^{c}N_{1}^{c}\rho_{1}+\lambda_{4}N_{3,2}^{c}N_{3,2}^{c}\rho_{1}$
(II.29)
$\displaystyle+\lambda_{5}N_{1}^{c}N_{3,2}^{c}\digamma+\lambda_{6}N_{1}^{c}N_{3,2}^{c}\Gamma+\lambda_{7}N_{3,2}^{c}N_{3,2}^{c}\rho_{2}+\lambda_{8}N_{3,2}^{c}N_{3,2}^{c}\rho_{3}$

where $\lambda_{i=1,...,8}$ are Yukawa coupling constants. The first two terms
in $\mathcal{W}_{\nu}$ are the Dirac Yukawa terms leading to the Dirac mass
matrix $m_{D}$ while the remaining couplings give rise to the Majorana mass
matrix $m_{M}$. The Higgs doublet develops its VEV as usual $\left\langle
H_{u}\right\rangle=\upsilon_{u}$, while the VEV alignments of the flavons are
chosen as follows

$\left\langle\rho_{1}\right\rangle=\upsilon_{\rho_{1}}\quad,\quad\left\langle\rho_{2}\right\rangle=\upsilon_{\rho_{2}}\quad,\quad\left\langle\rho_{3}\right\rangle=\upsilon_{\rho_{3}}\quad,\quad\left\langle\digamma\right\rangle=(\upsilon_{\digamma},\upsilon_{\digamma})^{T}\quad,\quad\left\langle\Gamma\right\rangle=(0,\upsilon_{\Gamma})^{T}$
(II.30)

By using the tensor product of $D_{4}$ irreducible representations given in
Eqs. (VII.64) and (VII.65) and imposing the constraint
$\lambda_{8}\upsilon_{\rho_{3}}=-\lambda_{7}\upsilon_{\rho_{2}}=\left(\lambda_{6}\upsilon_{\Gamma}\right)/2$,
we find that the Dirac and Majorana mass matrices have the following forms

$m_{D}=\upsilon_{u}\left(\begin{array}[]{ccc}\lambda_{1}&0&0\\\
0&\lambda_{2}&0\\\ 0&0&\lambda_{2}\end{array}\right)\quad,\quad
m_{M}=M_{R}\left(\begin{array}[]{ccc}a&b&b+\mathrm{k}\\\ b&\mathrm{k}&c\\\
b+\mathrm{k}&c&0\end{array}\right)$ (II.31)

where $M_{R}$ is the mass scale of the heavy RH Majorana neutrinos and
$a=\frac{\lambda_{3}\upsilon_{\rho_{1}}}{M_{R}}$,
$b=\frac{\lambda_{5}\upsilon_{\digamma}}{M_{R}}$,
$c=\frac{\lambda_{4}\upsilon_{\rho_{1}}}{M_{R}}$ and
$\mathrm{k}=\frac{\lambda_{6}\upsilon_{\Gamma}}{M_{R}}$. Assuming that
$\lambda_{1}=\lambda_{2}$ and $c=a+b$, the usual canonical seesaw formula
$m_{\nu}=m_{D}m_{M}^{-1}m_{D}^{T}$ yields the total neutrino mass matrix

$m_{\nu}=\frac{m_{0}}{P}\left(\begin{array}[]{ccc}-\left(a+b\right)^{2}&\left(a+b\right)\left(b+\mathrm{k}\right)&b^{2}-\mathrm{k}^{2}-b\left(\mathrm{k}-a\right)\\\
\left(a+b\right)\left(b+\mathrm{k}\right)&-\left(b+\mathrm{k}\right)^{2}&-a^{2}-ab+b^{2}+\mathrm{k}b\\\
b^{2}-\mathrm{k}^{2}-b\left(\mathrm{k}-a\right)&-a^{2}-ab+b^{2}+\mathrm{k}b&a\mathrm{k}-b^{2}\end{array}\right)$
(II.32)

where $m_{0}=\frac{\left(\lambda_{1}\upsilon_{u}\right)^{2}}{M_{R}}$ and
$P=\left(a+2b+\mathrm{k}\right)\left(a\mathrm{k}-a^{2}+b^{2}-\mathrm{k}^{2}\right)$.
In the limit where $\mathrm{k}\rightarrow 0$, this matrix obeys the well-known
$\mu-\tau$ symmetry [59, 60, 61, 62, 63], and thus it is diagonalized by the
TBM matrix which is $CP$ conserving and predicts $\theta_{13}=0$ and
$\theta_{23}=\pi/4$.Therefore, the presence of $\mathrm{k}$ is necessary to
break the $\mu-\tau$ symmetry and produce a small deviation from the TBM
pattern. Without loss of generality, only the parameter $\mathrm{k}$ is taken
to be complex—$\mathrm{k}\rightarrow\left|\mathrm{k}\right|e^{i\phi_{k}}$
where $\phi_{k}$ is a $CP$ violating phase—which is sufficient to ensure $CP$
violation in the lepton sector. On the other hand, the matrix $m_{\nu}$ enjoys
the magic symmetry property which refers to the equality of the sum of each
row and the sum of each column in the neutrino mass matrix [71]. This property
implies that the neutrino matrix is diagonalized by the well-known trimaximal
mixing matrix $\mathcal{U}_{TM_{2}}$ so that
$m_{\nu}^{\mathrm{diag}}=\mathcal{U}_{TM_{2}}^{\dagger}m_{\nu}\mathcal{U}_{TM_{2}}$
with

$\mathcal{U}_{TM_{2}}=\left(\begin{array}[]{ccc}\sqrt{\frac{2}{3}}\cos\theta&\frac{1}{\sqrt{3}}&\sqrt{\frac{2}{3}}\sin\theta
e^{-i\sigma}\\\
-\frac{\cos\theta}{\sqrt{6}}-\frac{\sin\theta}{\sqrt{2}}e^{i\sigma}&\frac{1}{\sqrt{3}}&\frac{\cos\theta}{\sqrt{2}}-\frac{\sin\theta}{\sqrt{6}}e^{-i\sigma}\\\
-\frac{\cos\theta}{\sqrt{6}}+\frac{\sin\theta}{\sqrt{2}}e^{i\sigma}&\frac{1}{\sqrt{3}}&-\frac{\cos\theta}{\sqrt{2}}-\frac{\sin\theta}{\sqrt{6}}e^{-i\sigma}\end{array}\right)$
(II.33)

The full mixing matrix is given by
$\mathcal{U}_{\nu}=\mathcal{U}_{TM_{2}}\mathcal{U}_{P}$ where
$\mathcal{U}_{P}=\mathrm{diag}(1,e^{i\frac{\alpha_{21}}{2}},e^{i\frac{\alpha_{31}}{2}})$
is a diagonal matrix that contains the Majorana phases $\alpha_{21}$ and
$\alpha_{31}$. The parameters $\theta$ and $\sigma$ are respectively an
arbitrary angle and a phase which will be related to the neutrino oscillation
parameters; the observed neutrino mixing angles $\theta_{ij}$ and the Dirac
$CP$ phase $\delta_{CP}$. The diagonalization of the neutrino matrix II.32 by
$\mathcal{U}_{TM_{2}}$ induces relations between our model parameters and the
trimaximal mixing parameters $\sigma$ and $\theta$, we find

$\tan
2\theta=\frac{\sqrt{3}\left|\mathrm{k}\right|\sqrt{b^{2}\cos^{2}\phi_{\mathrm{k}}+a^{2}\sin^{2}\phi_{k}}}{2ab-b\left|\mathrm{k}\right|\cos\phi_{k}}\quad,\quad\tan\sigma=\frac{-a}{b}\tan\phi_{k}$
(II.34)

As a result, the eigenmasses of $m_{\nu}$ are as follows

$\begin{array}[]{c}\left|m_{1}\right|=\frac{m_{0}}{\sqrt{(a-b)^{2}-\left|\mathrm{k}\right|(a-b)\cos\phi_{k}+(\left|\mathrm{k}\right|^{2}/4)}}\quad,\quad\left|m_{2}\right|=\frac{m_{0}}{\sqrt{(a+2b)^{2}+2\left|\mathrm{k}\right|(a+2b)\cos\phi_{k}+\left|\mathrm{k}\right|^{2}}}\\\
\left|m_{3}\right|=\frac{m_{0}}{\sqrt{(a+b)^{2}-\left|\mathrm{k}\right|(a+b)\cos\phi_{k}+(\left|\mathrm{k}\right|^{2}/4)}}\end{array}$
(II.35)

where the denominators of these masses corresponds to ratios of the right-
handed neutrino masses and their mass scale $M_{R}$

$\begin{array}[]{c}\frac{\left|M_{1}\right|}{M_{R}}=\sqrt{(a-b)^{2}-\left|\mathrm{k}\right|(a-b)\cos\phi_{k}+(\left|\mathrm{k}\right|^{2}/4)}\quad,\quad\frac{\left|M_{2}\right|}{M_{R}}=\sqrt{(a+2b)^{2}+2\left|\mathrm{k}\right|(a+2b)\cos\phi_{k}+\left|\mathrm{k}\right|^{2}}\\\
\frac{\left|M_{3}\right|}{M_{R}}=\sqrt{(a+b)^{2}-\left|\mathrm{k}\right|(a+b)\cos\phi_{k}+(\left|\mathrm{k}\right|^{2}/4)}\end{array}$
(II.36)

Regarding the mixing angles, it is well-known that the total lepton mixing
matrix is derived from the product between two matrices;
$U_{PMNS}=\mathcal{U}_{l}^{\dagger}\mathcal{U}_{\nu}$ where $\mathcal{U}_{l}$
is the matrix that diagonalizes the charged lepton mass matrix while
$\mathcal{U}_{TM_{2}}$ is as described above. In the present model however, it
is easy to check that the charged lepton mixing angles $\theta_{ij}^{l}$
derived from the diagonalization of $m_{l}=\upsilon_{d}\mathcal{Y}_{e}$—where
$\mathcal{Y}_{e}$ is given in Eq. (II.21)—are all equals to zero; thus, they
do not affect the neutrino mixing angles derived from $\mathcal{U}_{\nu}$.
Therefore, by using the PDG standard parametrization of the PMNS matrix [95],
the reactor, solar and atmospheric angles are expressed as

$\sin^{2}\theta_{13}=\frac{2}{3}\sin^{2}\theta~{},~{}\sin^{2}\theta_{12}=\frac{1}{3-2\sin^{2}\theta}~{},~{}\sin^{2}\theta_{23}=\frac{1}{2}-\frac{3\sin
2\theta}{2\sqrt{3}(3-2\sin^{2}\theta)}\cos\sigma\text{.}$ (II.37)

Before we perform a numerical analysis of neutrino masses and model
parameters, we should notice that the above masses and mixing parameters are
valid at the GUT scale. Therefore, to match these parameters with the
experimental values (of the mixing angles, the $CP$ phase and the mass squared
differences), their evolution from the GUT scale to low energy must be carried
out. However, although the final values are model dependent, it was
illustrated in Ref. [83] that for SUSY models, if $\tan\beta$ is small, the
RG-induced effects on the above parameters are controllable and can be safely
neglected. Accordingly, since the type I seesaw mechanism is related to
physics at very high energy scales, we can work in a scenario with small
neutrino Yukawa couplings in such a way that their contribution to the RG
evolution can be neglected [70].

  * •

Numerical analysis

The fact that the neutrino mass ordering remains unknown requires the
investigation of the two possible options: either $\Delta m_{31}^{2}$ $>0$
referred to as normal mass hierarchy or $\Delta m_{32}^{2}$ $<0$ known as the
inverted mass Hierarchy (IH). In the latter case that implies
$m_{3}<m_{2}<m_{1}$, it is easy to deduce from the first relation in Eq.
(II.37) as well as the $3\sigma$ region of the reactor angle from Ref. [3]
that the parameter $\theta$ lies in the interval $0.1763\leq\theta\leq
0.1920$.

Figure 1: In the IH case, the allowed points in the parameter space of
($\Delta m_{21}^{2},$ $\theta$) with the color code indicating $\Delta
m_{32}^{2}$.

On the other hand, by requiring the values of the mass-squared differences
$\Delta m_{ij}^{2}$ within their $3\sigma$ experimental ranges and
constraining the sum of neutrino masses to be $\sum m_{i}<0.12$ $\mathrm{eV}$
[81], we use the eigenmasses in Eq. (II.35) to plot in Fig. (1) the
correlation among $\Delta m_{21}^{2},$ $\Delta m_{32}^{2}$ and the parameter
$\theta$. We find that $\theta$ lies in the interval
$0.398\lesssim\theta\lesssim 0.579$ which implies that both
$\sin^{2}\theta_{13}$ and $\sin^{2}\theta_{12}$ fall far outside their
$3\sigma$ experimental range. For this reason, the IH scheme is excluded in
our model.

As regards to the NH scheme, we rewrite the masses $m_{2}$ and $m_{3}$in terms
of the lightest neutrino mass $m_{1}$ and the mass squared differences as
$m_{2}=\sqrt{m_{1}^{2}+\Delta m_{21}^{2}}$ and $m_{3}=\sqrt{m_{1}^{2}+\Delta
m_{31}^{2}}$. Moreover, by using Eqs. (II.35) and (II.37), these masses can
also be expressed as a function of the free parameters $a$, $b$, k and
$\phi_{k}$ to which ascribe the smallness of neutrino masses. Thus, we allow
$a$, $b$, and $\mathrm{k}$to vary in the range $[-1,1]$ while we allow
$\phi_{k}$ to vary in the range $[0,\pi]$. In Fig. (2), the trimaximal mixing
parameters $\sigma$ (top left panel) and $\theta$ (top right panel) are
projected on the planes $\left(\sin^{2}\theta_{23},\mathrm{k}\right)$ and
$\left(\sin^{2}\theta_{13},\mathrm{k}\right)$ respectively. As inputs, the
angle $\theta$ is allowed to vary in the range
$\left[0\rightarrow\pi/2\right]$ while the phase $\sigma$ is randomly varied
in the range $\left[0\rightarrow 2\pi\right]$. From the left panel of Fig.
(2), we observe from the scattered points that for the atmospheric angle only
the lower octant ($\sin^{2}\theta_{23}<0.5$) is allowed. Therefore, an
important prediction of the current model is that it excludes the maximal as
well as the higher octant of the atmospheric angle. Moreover, the $3\sigma$
allowed intervals of the oscillation parameters restrict the range of $\sigma$
as well as the range of the parameter k

$0.57565\lesssim\sigma\lesssim
1.57073\qquad,\qquad-0.52597\lesssim\mathrm{k}\lesssim 0.55115$ (II.38)

Notice by the way that the parameter $\mathrm{k}$ is responsible for the
deviation from the TBM values of the mixing angles. This deviation is encoded
in the parameter $\theta$ which is easily seen when we set $\theta\rightarrow
0$ in Eq. (II.37) resulting to restore the TBM values. From the top right
panel of Fig. (2), we find that the range of $\theta$ is also restricted to
$0.17548\lesssim\theta\lesssim 0.19129$ while the range of the reactor angle
remains almost unchanged compared to its $3\sigma$ allowed range.

Figure 2: The allowed points of the trimaximal mixing parameters $\sigma$ (top
left panel) and $\theta$ (top right panel) projected on the planes
$\left(\sin^{2}\theta_{23},\mathrm{k}\right)$ and
$\left(\sin^{2}\theta_{13},\mathrm{k}\right)$ respectively. The bottom panel
shows the allowed points in the parameter space of ($a$, $b$) with the color
code indicating the phase $\phi_{k}$.

In Fig. (2), the bottom panel shows the correlation between the parameters $a$
and $b$ with the color code showing the phase $\phi_{k}$. By taking into
account the $3\sigma$ experimental ranges of $\Delta m_{ij}^{2}$,
$\sin\theta_{ij}$ and $\delta_{CP}$ from the most recent global fit by NuFIT
collaboration [3], and the current cosmological upper bound on the sum of the
three light neutrino masses given by $\sum m_{i}<0.12$ $\mathrm{eV}$, we find
that the range of the phase $\phi_{k}$ gets more restricted compared to its
input range; $0.55275\lesssim\phi_{k}\lesssim 2.56095$. Regarding the Dirac
$CP$ phase $\delta_{CP}$, the results reported by the T2K long-baseline
experiment showed strong hints for $CP$ violation in neutrino oscillations
while $CP$ conservation is disfavored at $2\sigma$ level [6]. One approach to
estimate the magnitude of $\delta_{CP}$ is by means of the Jarlskog invariant
parameter defined as $J_{CP}=Im(\mathcal{U}_{e1}\mathcal{U}_{\mu
1}^{\ast}\mathcal{U}_{\mu 2}\mathcal{U}_{e2}^{\ast})$. In the PDG standard
parametrization, this parameter is exhibited in terms of the three mixing
angles and the Dirac $CP$ phase as follows [95]

$J_{CP}=\frac{1}{8}\sin 2\theta_{12}\sin 2\theta_{13}\sin
2\theta_{23}\cos\theta_{13}\sin\delta_{CP}$ (II.39)

while in the case of the trimaximal mixing, it takes a simpler form given
by$J_{CP}^{TM}=\left(1/6\sqrt{3}\right)\sin 2\theta\sin\sigma$. By matching
$J_{CP}^{TM}$ with the Eq. (II.39), we find a correlation between the Dirac
$CP$ phase, the arbitrary phase $\sigma$, and the atmospheric angle

$\sin 2\theta_{23}\sin\delta_{CP}=\sin\sigma$ (II.40)

Taking into account the fact that atmospheric angle is well determined
experimentally as well as the fact that the range of $\sigma$ given in Eq.
(II.38) excludes the exact value of $n\pi$ with $n$ can be any integer, it is
easy to deduce analytically that the $CP$ conserving values of $\delta_{CP}$
are not allowed which implies that the present model admits only the $CP$
violating values of $\delta_{CP}$.

###  II.4 Neutrino masses from non-oscillatory experiments

Constraining the absolute neutrino mass scale is one of the most important
purposes of the forthcoming neutrino experiments. This scale can be probed by
various non-oscillatory neutrino experiments. Cosmological observations are in
particular a powerful tool to probe the total sum of neutrino masses. Indeed,
in the framework of $\Lambda$CDM model with three massive active neutrinos,
the latest Planck data combined with baryon acoustic oscillations (BAO)
measurements provided an upper bound on the sum of neutrino masses of $\sum
m_{i}<0.12$ $\mathrm{eV}$ [81]. Another way to probe this scale is through
direct neutrino mass determination where the study of the electron energy
spectrum near its endpoint region is up to date the most sensitive method to
determine the electron antineutrino mass. The effective electron neutrino mass
is defined in terms of the three neutrino mass eigenvalues $m_{i}$ and the
flavor mixing parameters $U_{ei}$ as
$m_{\beta}=\left(\sum_{i}\left|U_{ei}\right|^{2}m_{i}^{2}\right)^{1/2}$.
Currently, the most valid bounds on $m_{\beta}$ are presented by the KATRIN
experiment which provides an upper limit on the electron antineutrino mass of
$1.1$ $\mathrm{eV}$ [96] and eventually aims at a sensitivity of $0.2$
$\mathrm{eV}$ [97].

Figure 3: Left: Prediction for the absolute neutrino masses and their sum
$\sum m_{i}$ as a function of $m_{1}$. Right: $m_{\beta}$ as a function of
$m_{1}$ where the gray dashed vertical region is disfavored by Planck+BAO
while the gray dashed horizontal region is the limit on $m_{\beta}$ from
KATRIN collaboration.

Using the upper limit $\sum m_{i}<0.12$ $\mathrm{eV}$ and the neutrino
oscillation parameters ($\theta_{ij}$ and $\Delta m_{ij}^{2}$) within their
currently allowed $3\sigma$ ranges as well as the restricted interval of our
model parameters given in the previous subsection, we show in the left panel
of Fig. (3) the three neutrino masses $m_{1}$, $m_{2}$ and $\ m_{3}$ given in
Eq. (II.35), and their sum $\sum m_{i}$ as a function of the lightest neutrino
mass $m_{1}$. We find that our model predicts the following ranges

$\displaystyle 0.064759$ $\displaystyle\lesssim$ $\displaystyle\sum
m_{i}\left(\mathrm{eV}\right)\lesssim 0.100929\quad,\quad 0.005033\lesssim
m_{1}\left(\mathrm{eV}\right)\lesssim 0.021934$ $\displaystyle 0.009851$
$\displaystyle\lesssim$ $\displaystyle m_{2}\left(\mathrm{eV}\right)\lesssim
0.023530\quad,\text{\quad}0.049614\lesssim
m_{3}\left(\mathrm{eV}\right)\lesssim 0.055478$ (II.41)

As a result, the predicted values of $\sum m_{i}$ around the lower bound $\sim
0.064759$ $\mathrm{eV}$ are consistent with normal mass hierarchy which
requires131313This bound is obtained by taking the best fit values of the mass
squared differences $\Delta m_{21}^{2}$ and $\Delta m_{31}^{2}$ from Ref. [3]
with any value of the lightest neutrino mass $m_{1}$ obtained in Eq. (II.41).
$\sum m_{i}\left(\mathrm{eV}\right)\gtrsim 0.065431$.This lower bound of $\sum
m_{i}$ may be achieved in the forthcoming experiments with further
cosmological data such as CORE+BAO aiming to reach a $0.062$ $\mathrm{eV}$
sensitivity on the sum of the three active neutrino masses [98].
In the right panel of Fig. (3), we show the correlation between $m_{\beta}$
and the lightest neutrino mass $m_{1}$. The orange region is achieved by
varying all the input parameters ($\Delta m_{ij}^{2}$, $a$, $b$,
$\left|\mathrm{k}\right|$ and $\phi_{k}$) in their $3\sigma$ ranges while the
red points stands for our model prediction. We find that the effective
electron neutrino mass is given by

$\mathrm{\ }0.0100158\lesssim m_{\beta}\left(\mathrm{eV}\right)\lesssim
0.023765$ (II.42)

It is clear that our predictions for $m_{\beta}$ are too small when compared
to the anticipated future $\beta$-decay experiments sensitivities such as
KATRIN ($\sim 0.2$ eV) [97], HOLMES ($\sim 0.1$ eV) [99], and Project 8 ($\sim
0.04$ eV) [100]. If the actual electron neutrino mass would be measured by one
of these experiments the neutrino sector of the present model will be ruled
out. Otherwise, the obtained values could be probed by new experimental
projects that must aim to reach improved sensitivities around $0.01$
$\mathrm{eV}$.

Another possible portal to probe the scale of neutrino masses comes from
experiments exploring the nature of neutrinos which is also one of the present
objectives in the field of neutrino physics. Up to now, the probe of the
Majorana nature of neutrinos is available only through $0\nu\beta\beta$ decay.
This is a process that violates lepton number L by two units, and since there
are no SM interactions that violates L, the discovery of $0\nu\beta\beta$
would have interesting implications for model building beyond the SM such as
the existence of a new mechanism for mass generation compared to the charged
fermions obtaining their masses via the Higgs mechanism. The $0\nu\beta\beta$
decay amplitude is proportional to the effective Majorana mass
$\left|m_{\beta\beta}\right|$ defined as
$\left|m_{\beta\beta}\right|=\left|\sum_{i}U_{ei}^{2}m_{i}\right|$, and may be
expressed in terms of our model parameters and the parameters of the
$\mathcal{U}_{\nu}$ mixing matrix

$\left|m_{\beta\beta}\right|=\left|\frac{2m_{1}}{3}\cos^{2}\theta+\frac{1}{3}\sqrt{m_{1}^{2}+\Delta
m_{21}^{2}}e^{\frac{i}{2}\alpha_{21}}+\frac{2}{3}\sin^{2}\theta\sqrt{m_{1}^{2}+\Delta
m_{31}^{2}}e^{\frac{i}{2}(\alpha_{31}-2\sigma)}\right|$ (II.43)

Notice that the relevance of the absolute mass scale in $0\nu\beta\beta$
experiments arise from the dependence of $\left|m_{\beta\beta}\right|$ on
$m_{i}$.

Figure 4: $\left|m_{\beta\beta}\right|$ as a function of $m_{1}$. The vertical
gray dashed region indicates the upper limit on the sum of the three light
neutrino masses from Planck+BAO data.

Although $0\nu\beta\beta$ decay has not been observed, there are dozens of
running and forthcoming experiments around the world setting as their
objective the detection of this process. The current limits on
$\left|m_{\beta\beta}\right|$ come from the KamLAND-Zen [101], CUORE [102],
GERDA [103] and EXO [104] experiments corresponding to
$\left|m_{\beta\beta}\right|<(0.061-0.165)\mathrm{eV}$,
$\left|m_{\beta\beta}\right|<(0.075-0.35)\mathrm{eV}$,
$\left|m_{\beta\beta}\right|<(0.079-0.180)\mathrm{eV}$ and
$\left|m_{\beta\beta}\right|<(0.078-0.239)\mathrm{eV}$ respectively. Fig. (4)
shows the correlation between $\left|m_{\beta\beta}\right|$ and the lightest
neutrino mass $m_{1}$ for normal mass hierarchy. This plot is obtained by
varying the oscillation parameters in their $3\sigma$ range while the Majorana
phases are varied in the range $[0\rightarrow 2\pi]$. The horizontal dashed
lines represent the limits on $\left|m_{\beta\beta}\right|$ from current
$0\nu\beta\beta$ decay experiments while the vertical gray region is
disfavored by the Planck+BAO data. From this figure, we extract our range of
the effective Majorana mass

$0.000715\lesssim\left|m_{\beta\beta}\right|\left(\mathrm{eV}\right)\lesssim
0.022028$ (II.44)

The predictions for $\left|m_{\beta\beta}\right|$ are far from the current
sensitivities mentioned above, on the other hand, the next-generation
experiments such as GERDA Phase II, CUPID, nEXO and SNO+-II will cover the
values of $\left|m_{\beta\beta}\right|$ in Eq. (II.44) as they aim for
sensitivities around
$\left|m_{\beta\beta}\right|\sim\left(0.01-0.02\right)\mathrm{eV}$ [105],
$\left|m_{\beta\beta}\right|\sim\left(0.006-0.017\right)\mathrm{eV}$ [106],
$\left|m_{\beta\beta}\right|\sim\left(0.008-0.022\right)\mathrm{eV}$ [107] and
$\left|m_{\beta\beta}\right|\sim\left(0.02-0.07\right)\mathrm{eV}$ [108]
respectively.

##  III Leptogenesis in SUSY $SU(5)\times D_{4}\times U(1)$ model

In this section, we investigate the generation of the baryon asymmetry of the
universe within our SUSY $SU(5)\times D_{4}\times U(1)$ model in the normal
neutrino mass hierarchy. In this scenario, the presence of three RH neutrinos
as the key ingredients for small neutrino masses can also produce the BAU
through the leptogenesis mechanism. In this case, a lepton asymmetry $Y_{L}$
(equally $B-L$ asymmetry $Y_{B-L}$) is generated through the out-of-
equilibrium $CP$ violating decays of RH neutrinos $N_{i}^{c}$ (and their
supersymmetric partners in SUSY models) in the early universe. This lepton
asymmetry is then partially converted into the baryon asymmetry of the
universe $Y_{B}$ via ($B+L$) violating sphaleron transitions [19].

The excess of baryons over anti-baryons is evaluated through the baryon
asymmetry $Y_{B}$ relative to the entropy density $s$ or the baryon asymmetry
$\eta_{B}$ relative to the density of photons $n_{\gamma}$, defined
respectively as

$Y_{B}=\frac{n_{B}-n_{\overline{B}}}{s}\quad\quad,\quad\quad\eta_{B}=\frac{n_{B}-n_{\overline{B}}}{n_{\gamma}}$
(III.45)

where $n_{B}$ and $n_{\overline{B}}$ are the number densities of baryons and
anti-baryons. The experimental values of these parameters obtained from the
latest data from Planck satellite are given by $Y_{B}=(8.72\pm 0.08)\times
10^{-11}$ and $\eta_{B}=\left(6.13\pm 0.04\right)\times 10^{-10}$ [81]. In
order to perform an approximate estimation of $Y_{B}$, we use the following
two approaches:

  * •

(a) It is well-known that when the right-handed neutrino mass spectrum is
hierarchical, the contribution to the lepton asymmetry can be created only by
the decay of the lightest RH neutrino [13, 109, 110]. Since only the normal
hierarchy is allowed in our model, it is clear from Eqs. (II.35) and (II.36)
as well as from the Fig. (5) that $M_{3}$ is the lightest RH neutrino where
the masses of $M_{1}$ and $M_{2}$ in this model differ at most by a factor of
3 (with $3M_{3}<M_{2},M_{1})$.

  * •

(b) Since all Majorana masses are above $T=10^{12}(1+\tan^{2}\beta)$ for
$\tan\beta=5$ and $\tan\beta=10$—used in the charged fermion sector to fit the
experimental data—we perform our study in the one flavor approximation where
all charged leptons are out-of-equilibrium and there is no difference between
them at the time leptogenesis takes place.

Figure 5: The RH neutrino masses ($M_{1}$, $M_{2}$ and $M_{3}$) as a function
of the Majorana mass scale $M_{R}$.

Taking this two points into consideration, the magnitude of $B-L$ asymmetry
generated by $N_{3}$ can be parameterized as follows [111]

$Y_{B-L}=-\left(\varepsilon_{N_{3}}Y_{N_{3}}^{eq}+\varepsilon_{\tilde{N}_{3}}Y_{\tilde{N}_{3}}^{eq}\right)\eta_{33}$
(III.46)

where $\varepsilon_{N_{3}}$ ($\varepsilon_{\tilde{N}_{3}}$) is the $CP$
asymmetry produced in the decay of $N_{3}^{c}$ ($\tilde{N}_{3}^{c}$),
$\eta_{33}$ is the efficiency factor141414The efficiency factor in the general
formula of the $B-L$ asymmetry is written as the sum over all the lepton
flavors $\sum\limits_{\alpha,\beta}\eta_{\alpha\beta}$. In the present study
where we employ the one flavor approximation, $\eta_{33}$ is the only
efficiency factor relevant for leptogenesis. related to the washout of the
$CP$ asymmetries due to $N_{3}^{c}$ ($\tilde{N}_{3}^{c}$) decays, and
$Y_{N_{3}}^{eq}$ is the number density of $N_{3}$ over the entropy density
($n_{N_{3}}/s$) defined as [111]:
$\left.Y_{N_{3}}^{eq}\right|_{T>>M_{3}}=\frac{135\zeta(3)}{4\pi^{4}g_{\ast}}$
where $\zeta(3)$ denotes the Riemann zeta function and $g_{\ast}$ is the
number of spin-degrees of freedom in thermal equilibrium; in MSSM
$g_{\ast}=228.75$.
The first component to consider in our calculation is the source of the $CP$
asymmetry given by the $CP$ violating parameters $\varepsilon_{N_{i}}$ and
$\varepsilon_{\tilde{N}_{i}}$ in $N_{i}^{c}$ and $\tilde{N}_{i}^{c}$ decays,
averaged over the different decay channels $N_{i}^{c}\rightarrow LH_{u},$
$\tilde{L}\tilde{H}_{u}$ and $\tilde{N}_{i}^{c}\rightarrow\tilde{L}H_{u},$
$L\tilde{H}_{u}$ respectively. These RH neutrinos and their superpartners
decay, with decay rates that reads respectively as

$\Gamma_{N}=\Gamma\left(N_{i}^{c}\rightarrow
LH_{u}\right)+\Gamma\left(N_{i}^{c}\rightarrow\tilde{L}\tilde{H}_{u}\right)\text{
and
}\Gamma_{\tilde{N}}=\Gamma\left(\tilde{N}_{i}^{c}\rightarrow\tilde{L}H_{u}\right)+\Gamma\left(\tilde{N}_{i}^{c}\rightarrow
L\tilde{H}_{u}\right)$ (III.47)

while the $CP$ violating parameters are given by

$\varepsilon_{N_{i}}=\frac{\Gamma_{N}-\bar{\Gamma}_{N}}{\Gamma_{N}+\bar{\Gamma}_{N}}\text{
and
}\varepsilon_{\tilde{N}_{i}}=\frac{\Gamma_{\tilde{N}}-\bar{\Gamma}_{\tilde{N}}}{\Gamma_{\tilde{N}}+\bar{\Gamma}_{\tilde{N}}}$
(III.48)

In SUSY models, the effects from the superparticles produce relatively small
corrections to the BAU [112]. Therefore, by ignoring supersymmetry
breaking151515For cases where SUSY can not be ignored see [113, 114].—as a
result of which the RH neutrinos and their superpartners have equal masses
$M_{N_{i}}=M_{\tilde{N}_{i}}$, equal decay rates
$\Gamma_{N_{i}}=\Gamma_{\tilde{N}_{i}}$ and equal $CP$ asymmetries
$\varepsilon_{N_{i}}=\varepsilon_{\tilde{N}_{i}}$ [111]—we can factorize by
the $CP$ asymmetry in Eq. (III.46) as
$Y_{B-L}=-\varepsilon_{N_{3}}\left(Y_{N_{3}}^{eq}+Y_{\tilde{N}_{3}}^{eq}\right)\eta_{33}$.
Likewise, when the equilibrium densities for leptons and sleptons are equal
$Y_{N_{i}}^{eq}\approx Y_{\tilde{N}_{i}}^{eq}$, we find that the $B-L$
asymmetry parameter $Y_{B-L}$ is enhanced by a factor of $2$. Bringing
together all these effects, the $CP$ asymmetry can be explicitly expressed in
the one flavor approximation as

$\varepsilon_{N_{3}}=\frac{1}{8\pi}\sum_{j=1,2}\frac{Im\left[\left(\mathcal{Y}_{\nu}\mathcal{Y}_{\nu}^{\dagger}\right)_{j3}^{2}\right]}{\left(\mathcal{Y}_{\nu}\mathcal{Y}_{\nu}^{\dagger}\right)_{33}}f\left(\frac{M_{j}}{M_{3}}\right)$
(III.49)

where
$f\left(x\right)=\sqrt{x}\left(1-\left(1+x\right)\ln\left[\left(1+x\right)/x\right]\right)$
and __ $\mathcal{Y}_{\nu}$__ is the neutrino Yukawa coupling matrix in the
basis where the Majorana mass matrix $m_{M}$ and the Yukawa matrix of the
charged leptons $\mathcal{Y}_{e}$ are both diagonal. However, as explained in
the Appendix B, the lepton asymmetry is produced when the neutrino Yukawa
coupling matrix is given by
$\mathcal{U}_{\nu}^{\dagger}Y_{D}\mathcal{U}_{l}$—$\mathcal{U}_{\nu}=\mathcal{U}_{TM}\mathcal{U}_{P}$
being the matrix that diagonalizes $m_{M}$ and $\mathcal{U}_{l}$ the matrix
that diagonalizes $\mathcal{Y}_{e}$—lead to a highly suppressed $CP$ asymmetry
$\left|\varepsilon_{N_{3i}}\right|\sim\mathcal{O}(10^{-12}-10^{-10})$. In
order to meet the requirements of a successful leptogenesis that produces the
experimental values of the baryon asymmetry parameter $Y_{B}$, we add a
correction to the leading order Dirac Yukawa matrix in Eq. (II.31). To account
for this correction, we introduce a new flavon field $\omega$ which transforms
as $1_{+-}$ under $D_{4}$ with zero $U(1)$ charge, we have

$\delta W_{D}=\frac{\lambda_{9}}{\Lambda}N_{3,2}^{c}F_{2,3}H_{5}\omega$
(III.50)

where $\lambda_{9}$ is a complex coupling constant
$\lambda_{9}=\left|\lambda_{9}\right|e^{i\phi_{\omega}}$. This effective
superpotential is obtained from the following renormalizable superpotential

$W_{D}^{ren}=N_{3,2}^{c}F_{2,3}X_{5}+\overline{X}_{5}H_{5}\omega$ (III.51)

where $X_{5}$ is a messenger field that transforms as $SU(5)$ quintet, $D_{4}$
singlet $1_{+-}$ and has a $U(1)$ charge equals to $-8$. The contribution of
$\delta W_{D}$ is small and will not provide any considerable effect in the
obtained neutrino masses and mixing. When the flavon field $\omega$ acquires
its VEV as $\left\langle\omega\right\rangle=\upsilon_{\omega}$, we end up with
the total Yukawa mass matrix161616Notice that the total light neutrino mass
matrix involving the small correction $\delta Y_{D}$ is almost similar to the
one in Eq. (II.29) and yields approximatively to the same neutrino
phenomenology.

$\mathcal{Y}_{D}=Y_{D}+\delta Y_{D}=\frac{m_{D}}{\upsilon_{u}}+\delta
Y_{D}=\left(\begin{array}[]{ccc}\lambda_{1}&0&0\\\ 0&\lambda_{1}&0\\\
0&0&\lambda_{1}\end{array}\right)+\kappa
e^{i\phi_{\omega}}\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\
0&1&0\end{array}\right)$ (III.52)

where $\kappa=\frac{2\left|\lambda_{9}\right|\upsilon_{\omega}}{\Lambda}$ is a
free parameter which should be small ($\kappa<<1$) in order to produce the
correct BAU. Taking into account this correction, the total Yukawa neutrino
mass matrix is defined as
$\mathcal{Y}_{\nu}=\mathcal{U}_{\nu}^{\dagger}\mathcal{Y}_{D}$. Thus, after
calculating the product $\mathcal{Y}_{\nu}\mathcal{Y}_{\nu}^{\dagger}$ in the
basis where the Majorana mass matrix is diagonal, the $CP$ asymmetry parameter
$\varepsilon_{N_{3}}$ corresponding to the lightest RH neutrino $N_{3}$ is
given approximately by

$\varepsilon_{N_{3}}=\frac{\kappa^{2}}{9\pi}\cos^{2}\phi_{\omega}\left[2\left(\sin^{2}(2\theta)\sin^{2}(\sigma-\frac{\alpha_{31}}{2})\right)f\left(\frac{\tilde{m}_{1}}{\tilde{m}_{3}}\right)+\left(\left(\sin^{2}\left(\sigma+\frac{(\alpha_{21}-\alpha_{31})}{2}\right)\sin^{2}\theta\right)\right)f\left(\frac{\tilde{m}_{2}}{\tilde{m}_{3}}\right)\right]$
(III.53)

where $\tilde{m}_{i}$ are the washout mass parameters expressed as
$\tilde{m}_{i}=\upsilon_{u}^{2}\frac{\left(\mathcal{Y}_{\nu}\mathcal{Y}_{\nu}^{\dagger}\right)_{ii}}{M_{i}}$.

The second component to address in this computation is the efficiency factor
$\eta_{33}$. A good approximation is to consider the region of RH neutrino
masses smaller than $10^{14}$ $\mathrm{GeV}$, preventing possible washout
effects from $\Delta L=2$ scattering processes. In this case, the efficiency
factor $\eta_{33}$ can be expressed approximately as a function of the washout
mass parameter $\tilde{m}_{3}$ as171717In the region where the lightest RH
neutrino mass is of order $\mathcal{O}$($10^{14}\mathrm{GeV)}$ or higher, the
efficiency factor $\eta_{33}$ takes into account the effect of washout
processes. Therefore, it can be computed by solving numerically the full
system of Boltzmann equations. [111]

$\eta_{33}\approx\left(\frac{3.3\times
10^{-3}\text{{eV}}}{\tilde{m}_{3}}+\left(\frac{\tilde{m}_{3}}{0.55\times
10^{-3}\text{{eV}}}\right)^{1.16}\right)^{-1}\quad\text{with\quad}\tilde{m}_{3}=\upsilon_{u}^{2}\frac{\left(\mathcal{Y}_{\nu}\mathcal{Y}_{\nu}^{\dagger}\right)_{33}}{M_{3}}$
(III.54)

Notice here that the smallness of the parameter $\kappa<<\lambda_{1}$ implies
that the washout mass parameters become approximately identical
$\tilde{m}_{i}\approx m_{i}$ and hence $\tilde{m}_{3}\approx m_{3}$. Moreover,
since the neutrino mass $m_{3}$ has values close to $0.5\times 10^{-1}$ eV as
given in Eq. (II.41), then the efficiency factor $\eta_{33}$ in our model is
roughly $\eta_{33}\approx 0.5\times 10^{-2}$.
Let us now derive the expression of the baryon asymmetry parameter $Y_{B}$.
This parameter is related to lepton asymmetry $Y_{B-L}$ given in Eq. (III.46)
through sphaleron transitions, we have [115]

$Y_{B}\approx\left(\frac{8N_{f}+4N_{H}}{22N_{f}+13N_{H}}\right)Y_{B-L}=-2\left(\frac{8N_{f}+4N_{H}}{22N_{f}+13N_{H}}\right)Y_{N_{3}}^{eq}\varepsilon_{N_{3}}\eta_{33}$
(III.55)

where $N_{f}=3$ is the number of fermion generations and $N_{H}=2$ is the
number of Higgs doublets in the MSSM. From this, we deduce the amount of
baryon asymmetry generated in the present model

$Y_{B}\approx-1.266\times 10^{-3}\varepsilon_{N_{3}}\eta_{33}$ (III.56)

Therefore, $Y_{B}$ in our model depends on the trimaximal parameters $\theta$
and $\sigma$, the light neutrino masses $m_{i}$, the Majorana phases
$\alpha_{31}$ and $\alpha_{21}$, as well as $\kappa$ and the phase
$\phi_{\omega}$ coming from the extra contribution in the Dirac mass matrix.
Using the ranges of the parameters $\theta$ and $\sigma$ restricted by the
neutrino oscillation data, we show in the left panel of Fig. (6) the
correlation between the baryon asymmetry parameter $Y_{B}$ and the parameters
$\theta$ and $\sigma$. We observe that there are many scattered points that
correlate $\theta$ and $\sigma$ with the Planck bound on $Y_{B}$.

Figure 6: Left panel: $Y_{B}$ as a function of the trimaximal parameters
$\theta$ and $\sigma$. Right panel: $Y_{B}$ as a function of the parameter
$\kappa$ with the color code indicating the allowed range of the $CP$
asymmetry parameter $\varepsilon_{N_{3}}$. The horizontal blue line
corresponds to the Planck bound on $Y_{B}$.

In addition, since the parameters $\kappa$ and $\phi_{\omega}$ are not
controlled by the neutrino oscillation data, we allow them to vary in the
ranges $\left[-0.1\rightarrow 0.1\right]$ and $\left[0\rightarrow 2\pi\right]$
respectively. Then, we plot in the right panel of Fig. (6) the correlation
between $Y_{B}$ and $\kappa$ where the color palette corresponds to the
absolute value of the $CP$ asymmetry parameter $\varepsilon_{N_{3}}$. We find
that the observed baryon asymmetry correspond to $\kappa$ in the range
$\left[-0.1\rightarrow-0.0085\right]\cup\left[0.009\rightarrow 0.1\right]$ and
$\left|\varepsilon_{N_{3}}\right|$ in the range $[1.302\rightarrow
1.433]\times 10^{-5}$.

On the other hand, it is clear from the $CP$ asymmetry parameter
$\varepsilon_{N_{3}}$ in Eq. (III.53) that the source of $CP$ violation in the
lepton sector could arise from the interplay between the low energy $CP$
phases (Dirac and Majorana phases $\delta_{CP}$, $\alpha_{31}$ and
$\alpha_{21}$) and the high energy $CP$ phase $\phi_{\omega}$__ originated
from the complex coupling constant $\lambda_{9}$ in the Dirac mass matrix; see
Eq. (III.50). Therefore, we plot in Fig. (7), the baryon asymmetry parameter
$Y_{B}$ as a function of the low energy $CP$ phases ($\alpha_{31}$,
$\alpha_{21}$ and $\delta_{CP}$) and the high energy $CP$ phase
$\phi_{\omega}$ which is the key ingredient for generating the observed range
of $Y_{B}$.

Figure 7: The baryon asymmetry $Y_{B}$ as a function of Majorana phases
$\alpha_{21}$ (top left), $\alpha_{31}$ (top right), Dirac CP phase
$\delta_{CP}$ (bottom left) and the high energy CP phase $\phi_{\omega}$
(bottom right). The horizontal blue line corresponds to the Planck bound.

From Fig. (7), we observe that the ranges of the Majorana phases and the Dirac
phase are not constrained compared to their inserted intervals, nevertheless,
the scattered points are consistent with the Planck limit on $Y_{B}$. Thereby,
we find that the entire inserted ranges for the Dirac and Majorana phases are
consistent with the BAU while the $CP$ conserving values
$\alpha_{31},\alpha_{21}=0,\pi$ of Majorana phases are also allowed. However,
even in the case of these $CP$ conserving values, $CP$ violation is guaranteed
by the high energy $CP$ phase $\phi_{\omega}$. For this reason, we plot in the
bottom right panel of Fig. (7) the correlation between $Y_{B}$ and
$\phi_{\omega}$ where we find that $\phi_{\omega}$ vary within the range
$0\lesssim\phi_{\omega}\lesssim 6.279$, while the $CP$ conserving values
$\phi_{\omega}=\frac{\pi}{2}$ and $\phi_{\omega}=\frac{3\pi}{2}$ as well as
the regions around them are excluded (the sections of the blue line without
any points). Therefore, this source of $CP$ violation plays a crucial role in
generating the baryon asymmetry in the present model.
From another point of view, since both $Y_{B}$ and $m_{\beta\beta}$ depend on
the Majorana phases $\alpha_{31}$ and $\alpha_{21}$, there exists a
correlation between the effective Majorana mass $m_{\beta\beta}$ which governs
the $0\nu\beta\beta$ process, and the baryon asymmetry parameter $Y_{B}$.
Therefore, we show in the top left of Fig. (8) $Y_{B}$ as a function of
$m_{\beta\beta}$ where we observe that there are several points satisfying the
Planck limit on the baryon asymmetry parameter.

Figure 8: The baryon asymmetry $Y_{B}$ as a function of the effective Majorana
mass $m_{\beta\beta}$ (top left), the lightest RH neutrino mass $M_{3}$ (top
right) and the lightest neutrino mass $m_{1}$ (bottom). The horizontal blue
line corresponds to the Planck bound.

On the other hand, since the absolute mass scale of right-handed neutrinos
cannot be determined by the data of the neutrino oscillations and the BAU, we
plot in the top right panel of Fig. (8) the baryon asymmetry parameter $Y_{B}$
as a function of the lightest RH neutrino mass $M_{3}$ which is allowed to
vary in the range181818This particular interval of the RH neutrino mass
$M_{3}$ is considered while taking into account the two approximations
mentioned before Eq. (III.46). $2.6\times 10^{13}$ GeV$\lesssim M_{3}<10^{14}$
GeV. As for the bottom panel of Fig. (8), we plot the correlation between the
baryon asymmetry parameter $Y_{B}$ and the lightest neutrino mass $m_{1}$.
From both plots, we observe that there are several points within the Planck
limit on $Y_{B}$.

##  IV Summary and conclusion

In this work, we have presented a model with a $D_{4}$ family symmetry to
explain the fermion flavor structures in the framework of supersymmetric
$SU(5)$ grand unified theory. This is the first realization of a four-
dimensional $SU(5)$ GUT with a flavor symmetry that does not include triplet
irreducible representations. To establish a comprehensive study of this model,
we have enlarged the field content of the usual scalar and matter sectors of
SUSY $SU(5)$ GUT. Explicitly, we have added three RH neutrinos to generate
neutrino masses via the type I seesaw mechanism, heavy messenger fields to
make the model renormalizable at the GUT scale, higher dimensional Higgs
multiplets to produce realistic quark-lepton Yukawa coupling ratios, and gauge
singlet flavon fields to give rise to the observed fermion mass spectrum and
mixing through the spontaneous symmetry breaking of the flavor group.
Moreover, after adding these fields, an additional $U(1)$ symmetry is needed
to control the invariance of the superpotentials in the quark and lepton
sectors, and also to control the dangerous $d=4$ and $d=5$ proton decay
operators.

Integrating out the heavy messenger fields from the renormalizable
superpotentials gives rise to higher-dimensional effective operators
responsible for the fermion flavor structures. Moreover, to go beyond the
minimal $SU(5)$ relation $\mathcal{Y}_{e}^{T}=\mathcal{Y}_{d}$ as well as the
popular GJ relations which are disfavored by the experimental results, we have
considered the CG factors $y_{e}/y_{d}=4/9$ and $y_{\mu}/y_{s}=9/2$ which are
realized through the coupling of messenger fields with higher 24- and
45-dimensional Higgs fields and the flavon fields. This has led to the double
ratio $\frac{y_{\mu}}{y_{s}}\frac{y_{d}}{y_{e}}\simeq 10.12$ which is in good
agreement with the phenomenological value at GUT scale. We have performed a
numerical analysis in the down and charged lepton Yukawa sector where we have
fixed our model parameters—the free parameters in the entries of the Yukawa
matrices—and provided an accurate fit to the mixing angles, the Yukawa
couplings and the Dirac $CP$ phase of the quark sector at the GUT scale.

The small neutrino masses are generated via the type I seesaw mechanism where
the Dirac and Majorana mass matrices arise from renormalizable terms. The
resulting neutrino mass matrix is of the trimaximal mixing form which is
compatible with current neutrino data. By using the $3\sigma$ experimental
range of $\sin^{2}\theta_{13}$ for both neutrino mass hierarchies we derived
the range of the trimaximal mixing parameter $\theta$ where we found that only
the normal mass hierarchy is allowed. Therefore, we have carried out our
numerical study in this regime where we found that our model allows for
$\theta_{13}\neq 0$ and $\theta_{23}<\pi/4$ as well as excludes the conserving
values of the Dirac neutrino $CP$ phase $\delta_{CP}$. We have explored the
neutrino parameter space and showed numerically the predicted ranges of the
non-oscillatory observables $m_{\beta}$, $m_{\beta\beta}$ and $\Sigma m_{i}$
that fit the $3\sigma$ experimental range of the mixing angles and the mass
squared splittings. In particular, the predicted values of $m_{\beta\beta}$
are testable at future neutrinoless double beta decay experiments.

Since the low energy $CP$ violation which manifest itself in the mixing matrix
in the form of the Dirac and Majorana phases is not sufficient to describe the
BAU, we have added an extra effective operator in the neutrino sector to
produce the observed BAU via the leptogenesis mechanism. This operator which
involves a new flavon field $\omega$ is obtained, as in the quark sector, by
integrating out heavy messenger fields. Its contribution serves as a
correction that perturbs the structure of the Dirac mass matrix while the high
energy $CP$ phase $\phi_{\omega}$ that arises from the complex coupling
constant in this operator is a new source of $CP$ violation. Therefore, we
have performed a numerical study to estimate the values of the $CP$ asymmetry
parameter $\varepsilon_{N_{3i}}$ that are consistent with the baryon asymmetry
parameter $Y_{B}$. We have focused on the unflavored leptogenesis
approximation scenario under which the range of the lightest RH neutrino mass
is given by $M_{3}\left(\mathrm{GeV}\right)\in\left[2.6\times
10^{13}\rightarrow 10^{14}\right]$. We found that the $CP$ asymmetry parameter
$\varepsilon_{N_{3}}$ is mainly related to the high energy $CP$ phase
$\phi_{\omega}$. Therefore, we showed through scatter plots that the $CP$
conserving values $\phi_{\omega}=\frac{\pi}{2}$ and
$\phi_{\omega}=\frac{3\pi}{2}$ as well as the regions around them are
excluded, while the lepton asymmetry parameter $\varepsilon_{N_{3i}}$ must be
of order $\left|\varepsilon_{N_{3i}}\right|\sim O(10^{-5})$ to satisfy the
Planck limit on $Y_{B}$.

##  V Appendix A: Messenger sector

In this Appendix we discuss the renormalizable superpotentials of all the
fermions including their Feynman diagrams to obtain the higher dimensional
operators relevant for the Yukawa mass matrices. The complete list of the
messenger field content including their $SU(5)$ and $D_{4}$ representations as
well as their $U\mathbb{(}1\mathbb{)}$ charges is given in Table (8).

Figure 9: Diagrams Diagrams inducing effective operators for the up-type quark
sector. The last diagram implies that the top quark mass originate from the
renormalizable operator $T_{3}T_{3}H_{5}.$

To be precise, the messenger fields $Y_{i}$ are relevant for the up quark
sector while $X_{i}$ are involved in the down quark, the charged lepton and
the Dirac neutrino sectors.

Messenger fields $X_{1}$ $X_{2}$ $\ \ \ \ X_{3}$ $X_{4}$ $X_{5}$ $Y_{1}$
$Y_{2}$ $SU(5)$ $5$ $5$ $\ \ \ \ \ 5$ $5$ $5$ $10$ $10$ $D_{4}$ $1_{+,-}$
$1_{+,+}$ $\left(\begin{array}[]{c}0\\\ X_{3}\end{array}\right)$ $1_{+,-}$
$1_{+,-}$ $1_{+,-}$ $1_{+,-}$ $U(1)$ $-4$ $10$ $-13$ $8$ $-8$ $2$ $-4$

Table 8: Messenger fields relevant in our model and their $D_{4}$
representations as well as their $U(1)$ charges. The messenger fields $X_{i}$
live in $5$-dimensional representation while $Y_{i}$ live in $10$-dimensional
representation. We assume that their mass is around the GUT scale.

The renormalizable superpotential invariant under $D_{4}\times U(1)$
associated to the up quarks reads as

$\displaystyle W_{up}^{Ren}$ $\displaystyle=$ $\displaystyle
H_{5}T_{1}Y_{1}+\overline{Y}_{1}T_{1}\xi_{1}+H_{5}T_{1}Y_{1}+\overline{Y}_{1}T_{2}\xi_{2}+H_{5}T_{1}Y_{1}$
(V.57)
$\displaystyle+\overline{Y}_{1}T_{3}\xi_{3}+H_{5}T_{2}Y_{2}+\overline{Y}_{2}T_{2}\xi_{4}+H_{5}T_{2}Y_{2}+\overline{Y}_{2}T_{3}\xi_{5}$

where we have omitted the coupling constants from all terms for simplicity.
The couplings in this superpotential are illustrated by the Feynman diagrams
provided in Fig. (9). After integrating out the messenger fields $Y_{i}$ and
$\bar{Y}_{i}$ from $W_{up}^{Ren}$ we obtain the effective superpotential
responsible for the masses of the up quarks given in Eq. (II.3).

As for the down-type quark and charged lepton sector, the renormalizable
superpotential involving the five-plets messenger fields $X_{i}$ and
$\overline{X}_{i}$ is given by

$\displaystyle W_{d,e}^{Ren}$ $\displaystyle=$ $\displaystyle F_{1}\phi
X_{1}+X_{1}H_{24}\overline{X}_{1}+\overline{X}_{1}\varphi
X_{2}+X_{2}H_{24}\overline{X}_{2}+\overline{X}_{2}H_{\overline{5}}T_{1}+F_{2,3}\Phi
X_{1}$ $\displaystyle+X_{1}H_{24}\overline{X}_{1}+\overline{X}_{1}\varphi
X_{2}+X_{2}H_{24}\overline{X}_{2}+\overline{X}_{2}H_{\overline{5}}T_{1}+F_{2,3}H_{24}X_{3}$
$\displaystyle+\overline{X}_{3}\Phi
X_{1}+\overline{X}_{1}H_{\overline{45}}T_{2}+F_{2,3}\Omega
X_{4}+\overline{X}_{4}H_{\overline{5}}T_{3}$

while the renormalizable terms relevant for the effective operator responsible
for generating a successful BAU is given as follows

$W_{D}^{ren}=N_{3,2}^{c}F_{2,3}X_{5}+\overline{X}_{5}H_{5}\omega$ (V.59)

Once more, integrating out the these heavy messenger fields give rise to the
effective superpotentials of the down quarks, the charged leptons and the
Dirac neutrino; see Eqs. (II.5) and (III.50). The mass terms of the messenger
fields takes the form
$W_{MF}=M_{X_{i}}X_{i}\overline{X}_{i}+M_{Y_{j}}Y_{j}\overline{Y}_{j}$ where
$i=1,2,..5$ and $j=1,2$. In fact, $\overline{X}_{i}$ and $\overline{Y}_{j}$
are the corresponding fields of $X_{i}$ and $Y_{j}$, they are hosted by the
$SU(5)$ representations $\overline{5}$ and $10$ respectively, and they have
the same $D_{4}$ representations as their partners but with opposite $U(1)$
charges. The Feynman diagrams relevant for the superpotetials (V) and (V.59)
are illustrated in Fig. (10).

Figure 10: Diagrams inducing effective operators for the down-type quark and
lepton sector. The last diagram is associated to the small Dirac correction.

##  VI Appendix B: $CP$ asymmetry from the contribution of the charged lepton
mixing

In this appendix, we show that the contribution of the charged lepton mixing
matrix to the Yukawa mass matrix in Eq. (II.31), before adding the correction
$\delta Y_{D}$, cannot accommodate the observed value of the BAU. Starting
with the $CP$ asymmetry formula in Eq. (III.49) which can be explicitly
expressed in the one flavor approximation as

$\varepsilon_{N_{3}^{c}}=\frac{1}{8\pi}\left\\{\frac{Im\left[\left(\mathcal{Y}_{\nu}\mathcal{Y}_{\nu}^{\dagger}\right)_{13}^{2}\right]}{\left(\mathcal{Y}_{\nu}\mathcal{Y}_{\nu}^{\dagger}\right)_{33}}f\left(\frac{M_{1}}{M_{3}}\right)+\frac{Im\left[\left(\mathcal{Y}_{\nu}\mathcal{Y}_{\nu}^{\dagger}\right)_{23}^{2}\right]}{\left(\mathcal{Y}_{\nu}\mathcal{Y}_{\nu}^{\dagger}\right)_{33}}f\left(\frac{M_{2}}{M_{3}}\right)\right\\}$
(VI.60)

where $\mathcal{Y}_{\nu}=\mathcal{U}_{TM_{2}}^{\dagger}Y_{D}\mathcal{U}_{l}$
is the neutrino Yukawa coupling matrix in the basis where the Majorana mass
matrix $m_{M}$ and the Yukawa matrix of the charged leptons $\mathcal{Y}_{e}$
are both diagonal—see section II—with191919The diagonalization of
$\mathcal{Y}_{e}$ given in Eq. (II.21) is obtained as a function of $b_{11}$,
$b_{12}$ and $b_{22}$. Then, we replace these parameters by their numerical
values given in Table (4) to produce $\mathcal{U}_{l}$.

$Y_{D}=\frac{m_{D}}{\upsilon_{u}}=\lambda_{1}\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\
0&0&1\end{array}\right)\quad,\quad\mathcal{U}_{l}=\left(\begin{array}[]{ccc}\frac{8b_{11}-81b_{22}}{b_{12}\sqrt{\left(\frac{8b_{11}-81b_{22}}{b_{12}}\right)^{2}+64}}&\frac{8}{\sqrt{\left(\frac{8b_{11}-81b_{22}}{b_{12}}\right)^{2}+64}}&0\\\
0&1&0\\\ 0&0&1\end{array}\right)$ (VI.61)

For our calculations, the values of the free parameters $b_{ij}$ are fixed by
their values in the case of $\tan\beta=5$ as reported in Table (6). As a
result, by inserting the expression of $\mathcal{Y}_{\nu}$ in Eq. (VI.60), we
find that the $CP$ asymmetry parameter $\varepsilon_{N_{3}^{c}}$ depends on
the coupling constant $\lambda_{1}$, the trimaximal mixing parameters $\theta$
and $\sigma$, the Mojorana $CP$ phases $\alpha_{31}$ and $\alpha_{21}$ as well
as the light neutrino masses $m_{i=1,2,3}$. Approximately,
$\varepsilon_{N_{3}^{c}}$ is given as

$\displaystyle\varepsilon_{3}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{1}^{2}10^{-6}}{8\pi}[(1.548\,5(1-\cos^{2}\theta+\sin^{2}\theta)^{2}\sin^{2}(\frac{\alpha_{31}}{2})\cos^{2}\sigma\sin^{2}\sigma)f\left(\frac{m_{1}}{m_{3}}\right)$
(VI.62)
$\displaystyle+(2.322\,6\sin^{2}\theta(\cos\sigma\cos\left(\frac{1}{2}(\alpha_{21}-\alpha_{31})\right)+\sin\sigma\sin\left(\frac{1}{2}(\alpha_{21}-\alpha_{31})\right))^{2})f\left(\frac{m_{2}}{m_{3}}\right)]$

Assuming that the coupling constant $\lambda_{1}$ is of order one and taking
into account the obtained regions of the parameters $\theta$ and $\sigma$, the
Majorana $CP$ phases $\alpha_{31}$ and $\alpha_{21}$ as well as the neutrino
masses $m_{i=1,2,3}$, the baryon asymmetry parameter
$\left|\varepsilon_{N_{3}^{c}}\right|$ is up to order
$\mathcal{O(}10^{-12}-10^{-10}\mathcal{)}$. However, as discussed in section
III, to generate the observed baryon asymmetry, the parameter
$\left|\varepsilon_{N_{3}^{c}}\right|$ must be of order
$\mathcal{O(}10^{-5}\mathcal{)}$. Therefore, the charged lepton contribution
to the $CP$ asymmetry parameter $\left|\varepsilon_{N_{3}^{c}}\right|$ is too
small and subsequently the baryon asymmetry parameter $Y_{B}$ is strongly
suppressed.

##  VII Appendix C: Some aspects of the dihedral group $D_{4}$

The dihedral group $D_{4}$ is a finite group that is generated by the
reflection $t$ and the $45^{\circ}$ rotation $s$ satisfying $s^{4}=t^{2}=I$
and $tst=s^{-1}$. A rotation followed by a reflection is different than a
reflection followed by a rotation which means that the two generators $s$ and
$t$ do not commute with each other. This non-abelian group has $5$ irreducible
representations $R_{i=1,...,5}$: one doublet denoted as $2_{0,0}$, and four
singlets denoted as $1_{+,+}$(the trivial singlet), $1_{+,-}$ $1_{-,+}$ and
$1_{-,-}$. The indices of these representations represent their characters
under the two generators $t$ and $s$ as in the following table

$\chi_{R_{i}}$ | $\chi_{2_{0,0}}$ | $\chi_{1_{+,+}}$ | $\chi_{1_{+,-}}$ | $\chi_{1_{-,+}}$ | $\chi_{1_{-,-}}$
---|---|---|---|---|---
$t$ | $0$ | $+1$ | $+1$ | $-1$ | $-1$
$s$ | $0$ | $+1$ | $-1$ | $+1$ | $-1$
(VII.63)

The squares of the dimensions of these irreducible representations are related
to the order $8$ of the $D_{4}$ group through the formula
$8=1_{+,+}^{2}+1_{+,-}^{2}+1_{-,+}^{2}+1_{-,-}^{2}+2_{0,0}$. Let us now turn
to the tensor products among the irreducible representations of $D_{4}$. The
tensor product between two $D_{4}$ doublets is decomposed into a sum of the
four singlet representations of $D_{4}$ as

$\left(\begin{array}[]{c}x_{1}\\\
x_{2}\end{array}\right)_{2_{0,0}}\otimes\left(\begin{array}[]{c}y_{1}\\\
y_{2}\end{array}\right)_{2_{0,0}}=\left(x_{1}y_{2}+x_{2}y_{1}\right)_{1_{+,+}}\oplus\left(x_{1}y_{1}+x_{2}y_{2}\right)_{1_{+,-}}\oplus\left(x_{1}y_{2}-x_{2}y_{1}\right)_{1_{-,+}}\oplus\left(x_{1}y_{1}-x_{2}y_{2}\right)_{1_{-,-}}$
(VII.64)

while the tensor products among the singlet representations can be expressed
as

$1_{d,e}\otimes 1_{f,g}=1_{df,eg}\qquad\text{with}\qquad d,e,f,g=\pm$ (VII.65)

For more details on the $D_{4}$ group, see, e.g., [42].

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