compat=1.1.0

Minimal seesaw and leptogenesis with the smallest modular finite group
Simone Marciano,222e-mail address<EMAIL_ADDRESS>Davide
Meloni,333e-mail address<EMAIL_ADDRESS>Matteo
Parriciatu,444e-mail address<EMAIL_ADDRESS>
Dipartimento di Matematica e Fisica, Università di Roma Tre
INFN Sezione di Roma Tre, Via della Vasca Navale 84, 00146, Roma, Italy

Abstract

> We propose a model for leptons based on the smallest modular finite group
> $\Gamma_{2}\cong S_{3}$ that, for the first time, accounts for both the
> hints of large low-energy CP-violation in the lepton sector and the matter-
> antimatter asymmetry of the Universe, generated by only two heavy right-
> handed neutrinos. These same states are also employed in a Minimal seesaw
> mechanism to generate light neutrino masses. Besides the heavy neutrinos,
> our particle content is the same as the Standard Model (SM), with the
> addition of one single modulus $\tau$, whose vacuum expectation value is
> responsible for both the modular and CP-symmetry breakings. We show that
> this minimalistic SM extension is enough to get an excellent fit to low
> energy neutrino observables and to the required baryon asymmetry $\eta_{B}$.
> Predictions for the neutrino mass ordering, effective masses in neutrinoless
> double beta decay and tritium decay as well as for the Majorana phases are
> also provided.

###### Contents

  1. 1 Introduction
  2. 2 Modular flavor symmetry at level $2$
  3. 3 The model
  4. 4 Numerical analysis and results
  5. 5 Leptogenesis
  6. 6 Conclusions
  7. A Beyond the $N_{1}-$dominated scenario
  8. B Modular forms and $S_{3}$ representations
  9. C Mapping to $\mathcal{D}$ and rescaling

##  1 Introduction

The flavor structure of the Standard Model (SM) is still an outstanding puzzle
in theoretical particle physics. The quite pronounced mass hierarchies between
the three generations of charged fermions in both quark and lepton sectors are
not addressed by the SM gauge symmetry, while the smallness of the neutrino
mass scale calls for an entirely different mechanism beyond the SM. The mixing
of the generations and CP-violation are also completely unconstrained by gauge
symmetry. As we enter the precision era of neutrino experiments, we face the
theoretical challenge to derive at least a major portion of the free
parameters from an organizing principle. In particular, the peculiar mixing
pattern of leptons (characterized by two large and one small angles), together
with the recent hints of CP violation in neutrino oscillations could be
understood with discrete symmetries through non-Abelian groups [1]. However,
this approach proved to be quite unsatisfactory due to the unconstrained
number of free parameters, and to the complicated scalar sector equipped with
a substantial number of flavons (spurion fields) which are needed to reproduce
the observed mixing [2]. A step forward was made by Feruglio in 2017 [3] where
modular symmetry takes the role of a flavor discrete symmetry in a bottom-up
approach. The Yukawa couplings of the SM become modular forms: pre-determined
functions of a complex field $\tau$ called “modulus”, which describes the
geometry of compactified extra-dimensions in superstring theory [4, 5]. As a
result, models based on modular symmetry are in general more predictive as
they require only a limited number of free parameters, and the flavon vacuum
alignment problem is replaced by the moduli stabilization, at least in top-
down approaches [6, 7, 8, 9]. Both the usual matter superfields and Yukawa
couplings transform in irreducible representations of the modular finite group
$\Gamma_{N}$ of a given level $N\in\mathbb{N}$. The appealing feature is that
for $N\leq 5$ these groups are isomorphic to the more familiar non-Abelian
discrete groups $S_{3},A_{4},S_{4}$ and $A_{5}$. As opposed to traditional
flavor models based on those groups, in modular symmetry a generic vacuum
expectation value (VEV) of the modulus breaks the modular group completely,
and the observables of the lepton sector are completely determined by the
value of this VEV, up to a limited number of free parameters in the
superpotential. Currently, in the literature there is a substantial number of
phenomenologically viable modular models for the lepton sector, some examples
are given by: $\Gamma_{2}\cong S_{3}$ [10, 11], $\Gamma_{3}\cong A_{4}$ [3,
10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31], $\Gamma_{4}\cong S_{4}$ [20, 32, 33, 34, 35, 36, 37, 38, 39, 40],
$\Gamma_{5}\cong A_{5}$ [37, 41, 42]. Models based on the double-cover
$\Gamma_{N}^{\prime}$ of the modular finite groups have also been put forward,
some examples are [43, 44, 45, 46, 47, 48, 49, 50]. For a more exhaustive list
of references, see the recent reviews [51, 52]. In addition to the flavor
puzzle, the source of the asymmetry between matter and antimatter in the early
universe remains an open issue. In this regard, once a seesaw mechanism is
introduced to give mass to neutrinos, leptogenesis [53, 54, 55, 56, 57, 58]
becomes almost inevitable. Indeed, Sakharov’s conditions [59] can be naturally
satisfied in this context; in fact, the seesaw mechanism predicts a violation
of lepton number through processes involving Majorana neutrinos, as well as
complex Yukawa couplings that induce CP violation. In addition, for a wide
range of parameter values, these processes can occur out of thermal
equilibrium.

In the supersymmetric modular symmetry business, a bunch of papers have
already discussed the issues related to leptogenesis; in particular, models
based on $\Gamma_{3}\cong A_{4}$ have been presented in [60, 61, 62, 63, 64,
27, 65, 66, 19], those on $\Gamma_{4}\cong S_{4}$ in [40, 38] while the double
covering group $\Gamma_{5}^{\prime}\cong A_{5}^{\prime}$ has been discussed in
[67].

In this paper, we contribute to the subject presenting a modular model based
on $S_{3}$ capable of explaining the low-energy lepton observables and CP-
violation as well as to account for the correct amount of the baryon asymmetry
of the Universe (BAU), $\eta_{B}$. Our construction is “minimal” in the sense
that only two right-handed neutrinos (RN) are added to the particle content of
the SM and that the unique modulus $\tau$ generates both the low energy CP-
violating phase and the high energy CP violation needed for the leptogenesis
mechanism to account for the BAU, without the addition of any other complex
phases in the Yukawa matrices. Our results for $\eta_{B}$ are obtained solving
the appropriate Boltzmann equations in the $N_{1}-$dominated scenario, that we
discuss in detail. A number of interesting predictions can be drawn from our
model, and are related to the effective masses in the neutrinoless double beta
decay and tritium decay (all below the current experimental values), as well
as to the values of the Majorana phases. The remaining sections of this paper
are organized as follows: in section 2 we recall the main ingredients of the
modular symmetry and its use in the supersymmetric (SUSY) context; section 3
is devoted to the analytic description of our model, which is subsequently
analyzed numerically in section 4; the relevant part of the present manuscript
related to leptogenesis is carefully described in section 5; in 6 we draw our
conclusions. Three appendices are supplied: in appendix A we discuss the
solution of the Boltzmann equations in presence of two sterile neutrinos and
justify the adopted $N_{1}-$dominated scenario; in appendix B we report our
choice for the arbitrary normalization of the level $N=2$ modular forms;
finally, in the last appendix C we show how to map the points outside the
fundamental domain into points belonging to it.

##  2 Modular flavor symmetry at level $2$

The modular group
$\overline{\Gamma}=\text{SL}(2,\mathbb{Z})/\\{\pm\mathbbm{1}\\}$ acts on the
modulus $\tau$, restricted to the upper-half complex plane, through the
transformation $\gamma\,:\,\tau\to\gamma(\tau)$,

$\gamma(\tau)=\frac{a\tau+b}{c\tau+d}\quad,\quad
a,b,c,d\in\mathbb{Z}\quad,\quad ad-bc=1\,,$ (2.1)

which is generated by $S$ and $T$ defined as:

$\begin{cases}S\,:\,\tau\rightarrow-\displaystyle\frac{1}{\tau}\\\ \\\
T\,:\,\tau\rightarrow\tau+1\end{cases}\quad\quad,\quad\quad\quad
S^{2}=(ST)^{3}=\mathbbm{1}\,.$ (2.2)

The generators (2.2) are represented in $\text{SL}(2,\mathbb{Z})$ by the
$2\times 2$ matrices $S=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$, and
$T=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$. A modular form is a holomorphic
function of $\tau$ which, under $\overline{\Gamma}$, transforms as:

$f(\gamma(\tau))=(c\tau+d)^{k}f(\tau)\,,$ (2.3)

where $k$ is a positive integer called “weight”.
While the group $\overline{\Gamma}$ is infinite and non-compact, compact and
finite groups can be constructed from its infinite normal subgroups
$\Gamma(N)$ for $N=1,2,3...$ defined as:

$\Gamma(N)=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\text{SL}(2,\mathbb{Z})\,\Big{|}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\equiv\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}(\text{mod
}N)\,\right\\}\,,$ (2.4)

where the natural number $N$ is called the level. In [3] Feruglio has shown
that it is always possible to find a basis where modular forms of a given
level transform in unitary representations of the finite groups
$\Gamma_{N}\equiv\overline{\Gamma}/\overline{\Gamma}(N)$ which, for $N\leq 5$,
are isomorphic to the non-Abelian discrete groups $S_{3},A_{4},S_{4},A_{5}$.
Here we focus on the lowest level $N=2$, which labels the smallest finite
modular group $\Gamma_{2}\cong S_{3}$. As reviewed in [11], modular forms of
level $2$ and (even) weight $k$ span a linear space of finite dimension
$k/2+1$, and a basis of lowest weight forms for all modular forms of level $2$
is provided by the $S_{3}$ doublet $Y_{1}(\tau),Y_{2}(\tau)$ given in appendix
B. Those can be employed to construct modular forms of higher weights using
the $S_{3}$ composition rules.

In what follows, we will use the convenient notation111In
$Y_{\boldsymbol{\rho}}^{(a)}(\tau)$, the symbol $\boldsymbol{\rho}$ denotes
the irrep, the upper index $(a)$ signals how many times the basis
$Y_{1}(\tau),Y_{2}(\tau)$ was contracted to form
$Y_{\boldsymbol{\rho}}^{(a)}(\tau)$, thus having weight $2a$. Individual
components of the multiplets are simply given by $Y_{j}^{p}(\tau)$, where
$j=\\{1,2\\}$ and $p$ is an exponent.:

$Y^{(1)}_{\mathbf{2}}(\tau)\equiv\begin{pmatrix}Y_{1}(\tau)\\\
Y_{2}(\tau)\end{pmatrix}_{\mathbf{2}}\,.$ (2.5)

Making use of the composition rules listed in appendix B, the $3$ linearly
independent modular forms of weight $4$ are constructed in the following way:

$Y^{(1)}_{\mathbf{2}}(\tau)\otimes
Y^{(1)}_{\mathbf{2}}(\tau)=(Y_{1}^{2}(\tau)+Y_{2}^{2}(\tau))_{\mathbf{1}}\oplus\begin{pmatrix}Y_{2}^{2}(\tau)-Y_{1}^{2}(\tau)\\\
2Y_{1}(\tau)Y_{2}(\tau)\end{pmatrix}_{\mathbf{2}}\,,$ (2.6)

which, according to our notation, will be expressed as
$Y^{(2)}_{\mathbf{1}}(\tau)$ and $Y^{(2)}_{\mathbf{2}}(\tau)$, respectively.
Similarly, there are $4$ linearly independent modular forms of weight $6$:

$\begin{split}Y^{(2)}_{\mathbf{2}}(\tau)\otimes
Y^{(1)}_{\mathbf{2}}(\tau)=[3Y_{1}(\tau)Y_{2}^{2}(\tau)-Y_{1}^{3}(\tau)]_{\mathbf{1}}\oplus[Y_{2}^{3}(\tau)-3Y_{2}(\tau)Y_{1}^{2}(\tau)]_{\mathbf{1^{\prime}}}\oplus\\\
\oplus\begin{pmatrix}Y_{1}(\tau)(Y_{1}^{2}(\tau)+Y_{2}^{2}(\tau))\\\
Y_{2}(\tau)(Y_{1}^{2}(\tau)+Y_{2}^{2}(\tau))\end{pmatrix}_{\mathbf{2}}\,,\end{split}$
(2.7)

labelled in our notation as $Y^{(3)}_{\mathbf{1}}$,
$Y^{(3)}_{\mathbf{1^{\prime}}}$ and $Y^{(3)}_{\mathbf{2}}$, respectively.
Since the number of independent forms of weight $k$ grows with $k/2+1$, we
stop at weight $6$ to aim at a minimalistic number of free parameters in the
superpotential. One of the appealing features of modular invariance is that
the value of the modulus can be restricted to a fundamental domain defined as:

$\mathcal{D}=\left\\{\tau\in\mathbb{C}\,:\,\text{Im}\,\tau>0\,,\,\,\,\left|\text{Re}\,\tau\right|\leq\frac{1}{2}\,,\,\,\,|\tau|\geq
1\right\\}\,.$ (2.8)

This corresponds to the gray region $\mathcal{D}$ in figure C.1, appendix C.
Values outside $\mathcal{D}$ are redundant and can be mapped inside (2.8)
through the appropriate modular transformation (2.1). It is interesting to
observe that, in particular for $\Gamma_{2}\cong S_{3}$, the modular forms of
lowest weight $Y_{1}(\tau),Y_{2}(\tau)$ are intrinsically hierarchical for
$\tau\in\mathcal{D}$, see eq.(B.2); this allows us to define the appropriate
ratio:

$\frac{Y_{2}(\tau)}{Y_{1}(\tau)}\equiv\zeta=|\zeta|\,e^{i\,g}\,,$ (2.9)

where $g=g(\text{Re}\,\tau)$ is a real function depending on
$\text{Re}\,\tau$. The absolute value $|\zeta|$ satisfies $|\zeta|\lesssim 1$
for $\tau\in\mathcal{D}$ and is suppressed by
$e^{-\displaystyle\pi\text{Im}\,\tau}$, as shown in figure B.1. Thus, we can
use $Y_{2}(\tau)/Y_{1}(\tau)$ as an expansion parameter, independently of the
chosen normalization of the modular forms (more about that in appendix C).

###  2.1 The SUSY framework

The supersymmetric modular-invariant action, turning off gauge interactions
for simplicity, is written as:

$\mathcal{S}=\int d^{4}x\int d^{2}\theta
d^{2}\bar{\theta}\,K(\Phi,\bar{\Phi})+\left[\int d^{4}x\int
d^{2}\theta\,\mathcal{W}(\Phi)+\text{h.c.}\right]\,,$ (2.10)

where $\theta,\bar{\theta}$ are the superspace coordinates;
$\Phi=(\tau,\varphi)$ are chiral superfields, and $\varphi$ labels the usual
matter supermultiplets; $K(\Phi,\bar{\Phi})$ is the Kähler potential and
$\mathcal{W}(\Phi)$ is the superpotential. We assume that chiral superfields
transform under $\overline{\Gamma}$ as:

$\tau\to\gamma(\tau)=\displaystyle\frac{a\tau+b}{c\tau+d}\quad\quad,\quad\quad\varphi^{(I)}\to(c\tau+d)^{-k_{I}}\rho^{(I)}(\gamma)\varphi^{(I)}\,,$
(2.11)

where $\gamma\in\overline{\Gamma}$ and $\rho^{(I)}(\gamma)$ is the matrix for
the irreducible representation of $\Gamma_{N}$. Here the $k_{I}$ of the
superfields are called modular charges. The action (2.10) is invariant under
(2.11) provided that the superpotential $\mathcal{W}(\Phi)$ and the Kähler
potential transform as:

$\begin{cases}\mathcal{W}(\Phi)\to\mathcal{W}(\Phi)\\\ \\\
K(\Phi,\bar{\Phi})\to
K(\Phi,\bar{\Phi})+f(\Phi)+\bar{f}(\bar{\Phi})\end{cases}$ (2.12)

with $f(\Phi)$ being holomorphic. The superpotential $\mathcal{W}(\Phi)$ will
generally be a combination of Yukawa couplings $Y_{I_{1}...I_{n}}(\tau)$ and
chiral superfields

$\mathcal{W}(\Phi)=\sum(Y_{I_{1}...I_{n}}(\tau)\,\varphi^{(I_{1})}...\varphi^{(I_{n})})_{\mathbf{1}}\,.$
(2.13)

The Yukawa couplings are assumed to be modular forms which transform as:

$Y_{I_{1}...I_{n}}(\gamma(\tau))=(c\tau+d)^{k_{Y}}\rho(\gamma)Y_{I_{1}...I_{n}}(\tau)\,,$
(2.14)

with $k_{Y}\geq 2$ being an even integer in the case of $N=2$. The novelty
compared to classic non-Abelian discrete groups model building is that the
invariance of (2.13) is achieved by satisfying not only the existence of a
singlet contraction
$\rho\otimes\rho_{I_{1}}\otimes\rho_{I_{2}}...\otimes\rho_{I_{n}}\supset\mathbf{1}$
between all the irreps involved, but also that every operator in (2.13) must
be weightless, i.e. $k_{Y}=k_{I_{1}}+k_{I_{2}}+...+k_{I_{n}}$, the total
weight of the matter fields must be counterbalanced by appropriate Yukawa
modular forms of weight $k_{Y}$.
Finally, in most economical realizations of modular flavor models, the
invariant Kähler potential is given by:

$K(\Phi,\bar{\Phi})=-h\Lambda_{\tau}^{2}\log(-i\tau+i\bar{\tau})+\sum_{I}(-i\tau+i\bar{\tau})^{-k_{I}}{|\varphi^{(I)}|^{2}}\,,$
(2.15)

where $h>0$ and $\Lambda_{\tau}$ has mass-dimensions of one. Note that we used
the same symbol to denote the superfield and its scalar component.

In what follows, the vacuum expectation value (VEV) of the modulus $\tau$ is
the only source of flavor symmetry breaking, thus no additional flavons are
needed.222Here we do not deal with the dynamical justification for this VEV,
which is a subject of ongoing studies. See for examples [18], and [7, 9, 6,
8]. The scale of modular symmetry breaking can be located at the
compactification scale, which is supposed to be near the Planck scale. The
modular group is completely broken for every value of $\tau$ except for the
so-called symmetric points $\tau_{\text{sym}}=\\{\omega,i,i\infty\\}$ where
$\omega\equiv e^{2\pi i/3}$, which are left invariant under $ST$, $S$, and $T$
respectively [33].

###  2.2 gCP symmetry

In order to reduce the number of model free parameters (thus possibly making
the model itself more predictive), we assume a generalized CP symmetry (gCP).
This was made consistent with modular symmetry in [68] where it was shown that
under certain requirements for the group basis and the normalization of the
Yukawa forms, imposing a gCP symmetry consists in the reality condition of the
superpotential couplings.

Given this assumption, the only source of CP violation in gCP models is the
VEV of the complex modulus. Since $\tau$ appears in the expansions (B.2) of
appendix B through powers of $e^{2\pi i\tau}$, the CP violation is controlled
entirely by $\text{Re}\,\tau$ for $\tau\in\mathcal{D}$. This means that the
only complex phase in the mass matrices is represented by powers of $e^{2\pi
i\,\text{Re}\,\tau}$. Apart from the obvious $\text{Re}\,\tau=0$, the other
CP-conserving points consist in the boundary of $\mathcal{D}$. Note that all
the previously mentioned symmetric points are CP-conserving.

##  3 The model

Given that our main concern here is to find a viable seesaw model for
generating neutrino masses, for the charged-leptons sector we simply employ
the transformation properties under $\Gamma_{2}$ and a charge assignment
compatible with naturally hierarchical charged leptons, adopted from the
construction illustrated in [11]. In this respect, the superfields $E_{i}^{c}$
correspond to the three flavors of right-handed charged leptons (respectively
$\\{i=1,2,3\\}\equiv\\{e,\mu,\tau\\}$) with modular charges $4,0$ and $-2$,
respectively. The first two lightest left-handed $SU(2)_{\text{L}}$ doublets
are grouped into a doublet of $S_{3}$ while the heaviest family belongs to a
pseudo-singlet. Finally, the Higgs doublets transform as invariant singlets
with modular charges set to $0$. In table 1 we summarize the chiral
supermultiplets of our model, the transformation properties under
$\Gamma_{2}\cong S_{3}$ and the related modular charges. As a consequence, the
allowed terms in the superpotential are the following:

$\mathcal{W}_{e}^{H}=\alpha
E_{1}^{c}H_{d}(D_{\ell}Y_{\mathbf{2}}^{(3)})_{\mathbf{1}}+\beta
E_{2}^{c}H_{d}(D_{\ell}Y_{\mathbf{2}})_{\mathbf{1^{\prime}}}+\gamma
E_{3}^{c}H_{d}\ell_{3}+\alpha_{D}E_{1}^{c}H_{d}\ell_{3}Y_{\mathbf{1^{\prime}}}^{(3)}\,,$
(3.1)

and the resulting mass matrix (in the right-left basis) after electroweak
symmetry breaking reads333In the parentheses notation $(...)_{1,2}$ we denote
the two components of the corresponding doublet $Y_{\mathbf{2}}^{(a)}$.:

$M_{\ell}=\begin{pmatrix}\alpha(Y^{(3)}_{\mathbf{2}})_{1}&\alpha(Y^{(3)}_{\mathbf{2}})_{2}&\alpha_{D}Y^{(3)}_{\mathbf{1^{\prime}}}\\\
\beta Y_{2}&-\beta Y_{1}&0\\\ 0&0&\gamma\end{pmatrix}_{\text{RL}}v_{d}\,,$
(3.2)

where $v_{d}$ is the VEV of $H_{d}$ and the modular forms of weight $6$ are
given by (2.7). One can recast the matrix (3.2) in a form suitable for getting
perturbative expressions of the eigenvalues; in order to do that, we define
$A\equiv\alpha_{D}/\alpha$, $B\equiv(\beta/\alpha)Y_{1}^{-2}$,
$C\equiv(\gamma/\alpha)Y_{1}^{-3}$ and $\zeta$ is the expansion parameter
defined in (2.9). Thus, we get:

$M_{\ell}=v_{d}\alpha
Y_{1}^{3}\begin{pmatrix}(1+\zeta^{2})&\zeta(1+\zeta^{2})&-A\zeta(3-\zeta^{2})\\\
B\zeta&-B&0\\\ 0&0&C\end{pmatrix}_{\text{RL}}\,.$ (3.3)

Assuming $\frac{\beta}{\alpha}\sim\frac{\gamma}{\alpha}\approx 1$ and using
the $q$-expansions (B.2) we estimate $|B|,|C|\gg A$ for $\tau\in\mathcal{D}$.
It is now easy to obtain the approximated eigenvalues in powers of $\zeta$ as
follows:

$\displaystyle
m_{e}=v_{d}\alpha\left(|Y_{1}^{3}|+\frac{3}{2}|Y_{1}^{3}||\zeta|^{2}+\mathcal{O}(\zeta^{3})\right)$
(3.4) $\displaystyle
m_{\mu}=v_{d}\alpha\left(|Y_{1}|+\frac{1}{2}|Y_{1}||\zeta|^{2}+\mathcal{O}(\zeta^{3})\right)$
(3.5) $\displaystyle
m_{\tau}=v_{d}\alpha\left(1+\frac{9A^{2}}{2}|Y_{1}^{6}||\zeta|^{2}+\mathcal{O}(\zeta^{3})\right)\,.$
(3.6)

The hierarchy $(m_{\tau},m_{\mu},m_{e})\sim m_{\tau}(1,|Y_{1}|,|Y_{1}|^{3})$
naturally arises considering that $|Y_{1}|\approx 7/100$. Given the non-
diagonal structure of the charged-leptons mass matrix (3.3), a non-trivial
contribution to the PMNS matrix arises; considering that
$A/|C|^{2}\ll|\zeta|$, the unitary matrix $U_{\ell}$ that diagonalises
$M_{\ell}^{\dagger}M_{\ell}$ can be expressed as follows:

$U_{\ell}\sim\begin{pmatrix}1-\displaystyle\frac{|\zeta|^{2}}{2}&-\overline{\zeta}&0\\\
\zeta&1-\displaystyle\frac{|\zeta|^{2}}{2}&0\\\
0&0&1\end{pmatrix}+\mathcal{O}(\zeta^{3})\,,$ (3.7)

up to $\mathcal{O}(1)$ coefficients. In particular, we see that $U_{\ell}$
would provide a non-negligible correction to the solar angle, for which we
expect only a small $\mathcal{O}(\zeta)$ contribution for $\text{Im}\,\tau$
sufficiently large. All in all, the hierarchical texture of the charged-
leptons is provided by the modular invariance through our construction. The
mass spectrum is completely determined by five dimensionless real parameters,
$\text{Re}\,\tau$, $\text{Im}\,\tau$, $\alpha_{D}/\alpha$, $\beta/\alpha$ and
$\gamma/\alpha$ and one global scale $v_{d}\alpha$.

###  3.1 Neutrino sector

| $E_{1}^{c}$ | $E_{2}^{c}$ | $E_{3}^{c}$ | $D_{\ell}$ | $\ell_{3}$ | $H_{d,u}$ | $N^{c}$
---|---|---|---|---|---|---|---
$\scriptstyle{SU(2)_{L}\times U(1)_{Y}}$ | $(\mathbf{1},+1)$ | $(\mathbf{1},+1)$ | $(\mathbf{1},+1)$ | $(\mathbf{2},-1/2)$ | $(\mathbf{2},-1/2)$ | $(\mathbf{2},\mp 1/2)$ | $(\mathbf{1},0)$
$\Gamma_{2}\cong S_{3}$ | $\mathbf{1}$ | $\mathbf{1^{\prime}}$ | $\mathbf{1^{\prime}}$ | $\mathbf{2}$ | $\mathbf{1^{\prime}}$ | $\mathbf{1}$ | $\mathbf{2}$
$k_{I}$ | $4$ | $0$ | $-2$ | $2$ | $2$ | $0$ | $2$

Table 1: Chiral supermultiplets, transformation properties under
$\Gamma_{2}\cong S_{3}$ and modular charges. For the sign convention of the
modular charges, we refer to our choice in eq. (2.11).

To keep the number of free parameters at minimum, we adopt the Minimal seesaw
scenario [69, 70] consisting in the introduction of two SM singlets right-
handed Majorana fields $N^{c}_{1},N^{c}_{2}$. As listed in table 1, these are
assigned to an $S_{3}$ doublet $N^{c}\sim\mathbf{2}$. The invariant
superpotential of the neutrino sector is given by:

$\begin{split}\mathcal{W}_{\nu}=gH_{u}N^{c}D_{\ell}Y^{(2)}_{\mathbf{2}}+g^{\prime}H_{u}(N^{c}Y^{(2)}_{\mathbf{2}})_{\mathbf{1^{\prime}}}\ell_{3}+g^{\prime\prime}H_{u}(N^{c}D_{\ell})_{\mathbf{1}}Y^{(2)}_{\mathbf{1}}+\\\
+\Lambda[(N^{c}N^{c})_{\mathbf{2}}Y^{(2)}_{\mathbf{2}}+\lambda(N^{c}N^{c})_{\mathbf{1}}Y^{(2)}_{\mathbf{1}}]\,,\end{split}$
(3.8)

where $g,g^{\prime},g^{\prime\prime},\lambda$ are dimensionless free
parameters and $\Lambda$ is the Majorana right-handed mass scale. As mentioned
in section 2.2, all these couplings (as well as the ones from the charged-
leptons sector) are real due to the imposed gCP symmetry. The only complex
parameter is the modulus $\tau$.

Exploiting the composition rules of $S_{3}$ given in appendix B, the Dirac
mass matrix reads:

$M_{D}=gv_{u}\begin{pmatrix}-(Y_{2}^{2}-Y_{1}^{2})+\displaystyle\frac{g^{\prime\prime}}{g}(Y_{1}^{2}+Y_{2}^{2})&2Y_{1}Y_{2}&\displaystyle\frac{g^{\prime}}{g}(2Y_{1}Y_{2})\\\
2Y_{1}Y_{2}&(Y_{2}^{2}-Y_{1}^{2})+\displaystyle\frac{g^{\prime\prime}}{g}(Y_{1}^{2}+Y_{2}^{2})&-\displaystyle\frac{g^{\prime}}{g}(Y_{2}^{2}-Y_{1}^{2})\end{pmatrix}_{\text{RL}}\,,$
(3.9)

where $v_{u}$ denotes the VEV of the neutral scalar component of $H_{u}$. The
Majorana mass matrix of right-handed neutrinos is given by:

$\mathcal{M}_{R}=\Lambda\begin{pmatrix}-(Y_{2}^{2}-Y_{1}^{2})+\lambda(Y_{1}^{2}+Y_{2}^{2})&2Y_{1}Y_{2}\\\
2Y_{1}Y_{2}&(Y_{2}^{2}-Y_{1}^{2})+\lambda(Y_{1}^{2}+Y_{2}^{2})\end{pmatrix}_{\text{RR}}\,.$
(3.10)

The mass matrix of the light Majorana neutrinos is then obtained from the well
known formula of type-I seesaw:

$m_{\nu}=-M_{D}^{T}\mathcal{M}_{R}^{-1}M_{D}\,.$ (3.11)

Note that, as a consequence of the Minimal seesaw, the matrix $m_{\nu}$ has
rank $2$, i.e. in the limit of exact SUSY our model predicts that the lightest
neutrino is massless.444Regarding the stability of $m_{\textit{lightest}}=0$
against the quantum corrections, it has been shown that the condition still
holds at one-loop for a running between the seesaw and electroweak scales. The
two-loop effect is vanishingly small [71]. From (3.11) it follows that the
mass-scale of light neutrinos is completely determined by the global parameter
$g^{2}v_{u}^{2}/\Lambda$ and, from the discussion below eq. (3.7), we conclude
that the low-energy neutrino mixing and mass-splittings will be mainly
dictated by five parameters:
$\text{Re}\,\tau,\text{Im}\,\tau,g^{\prime}/g,g^{\prime\prime}/g,\lambda$.
Note that, among these, the real and imaginary part of $\tau$ already appear
in the charged-leptons mass matrix.

##  4 Numerical analysis and results

Parameter | Best-fit value and $1\sigma$ range
---|---
| NO | IO
$r\equiv\Delta m^{2}_{\text{sol}}/|\Delta m^{2}_{\text{atm}}|$ | $0.0295\pm 0.0008$ | $0.0298\pm 0.0008$
$\sin^{2}\theta_{12}$ | $0.303^{+0.012}_{-0.012}$ | $0.303^{+0.012}_{-0.011}$
$\sin^{2}\theta_{13}$ | $0.02225^{+0.00056}_{-0.00059}$ | $0.0223^{+0.00058}_{-0.00058}$
$\sin^{2}\theta_{23}$ | $0.451^{+0.019}_{-0.016}$ | $0.569^{+0.016}_{-0.021}$
$J_{\text{CP}}$ | $-0.027_{-0.010}^{+0.010}$ | $-0.032_{-0.007}^{+0.007}$
$m_{e}/m_{\mu}$ | $0.0048\pm 0.0002$
$m_{\mu}/m_{\tau}$ | $0.0565\pm 0.0045$
Table 2: Neutrino observables and their 1$\sigma$ ranges, from NuFIT 5.2 [72]
using the dataset with SK atmospheric data [73]. Here “Normal ordering” and
“Inverted ordering” for the mass spectrum are indicated with NO and IO,
respectively. The mass ratios of charged-leptons are taken from [3]. Here we
defined $\Delta m^{2}_{\text{sol}}\equiv m_{2}^{2}-m_{1}^{2}$, $|\Delta
m^{2}_{\text{atm}}|\equiv|m_{3}^{2}-(m_{1}^{2}+m_{2}^{2})/2|$. The Jarlskog
invariant is defined in (4.2) and can be expressed as
$J_{\text{CP}}=c_{12}s_{12}c_{23}s_{23}c_{13}^{2}s_{13}\sin\delta_{\text{CP}}$.
To extract its best-fit value and $1\sigma$ uncertainty, we referred to its
one-dimensional $\chi^{2}$ projection from NuFIT 5.2. The charged-leptons mass
ratios are taken from [3]. The running of low-energy neutrino observables can
be neglected since in our fit we do not obtain a quasi-degenerate spectrum
[74].

In total, the parameter space of our model consists of only eight
dimensionless parameters, as discussed above.555The overall mass scales
$\alpha v_{d}$ and $g^{2}v_{u}^{2}/\Lambda$ are easily recovered through the
values of $m_{\tau}$ and $|\Delta m_{\text{atm}}^{2}|$ and thus are not
introduced into the fit. In doing so, we used for $m_{\tau}$ the value
reported in [26] and $|\Delta m^{2}_{\text{atm}}|/(10^{-3}\text{
eV}^{2})=2.507^{+0.026}_{-0.027}\,\textbf{(NO)},\,\,\,\,2.486^{+0.025}_{-0.028}\,\textbf{(IO)}$
from NuFIT 5.2. To verify that our model is able to reproduce the seven
experimental dimensionless observables reported in table 2, we performed a
$\chi^{2}$ analysis for which we used the Gaussian approximation

$\chi^{2}(p_{i})=\sum_{j=1}^{7}\left(\frac{q_{j}(p_{i})-q_{j}^{\text{b-f}}}{\sigma_{j}}\right)^{2}\,,$
(4.1)

for the mixing
$\\{\sin^{2}\theta_{12},\sin^{2}\theta_{13},\sin^{2}\theta_{23}\\}$, the mass
ratios $\\{m_{e}/m_{\mu},m_{\mu}/m_{\tau},r\\}$ and for the invariant
$J_{\text{CP}}$. Here $q_{j}(p_{i})$ are the observables obtained from our
model with the set of dimensionless parameters
$p_{i}=\\{\beta/\alpha,...,g^{\prime}/g,...,\lambda\\}$ taken as input. On the
other hand, $q_{j}^{\text{b-f}}$ and $\sigma_{j}$ are the best-fit values and
the $1\sigma$ symmetrized uncertainties taken from table 2. The Jarlskog
invariant $J_{\text{CP}}$ parameterises the amount of CP violation in the
lepton sector and is defined as [72]:

$J_{\text{CP}}=\text{Im}\,[U_{11}U_{12}^{*}U_{21}^{*}U_{22}]\,,$ (4.2)

where $U\equiv U_{\ell}^{\dagger}U_{\nu}$ is the PMNS matrix [75, 76] for the
lepton mixing, which in the case of Majorana neutrinos can be written in the
standard convention:

$U=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{\text{CP}}}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{\text{CP}}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{\text{CP}}}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{\text{CP}}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{\text{CP}}}&c_{23}c_{13}\end{pmatrix}\text{diag}(e^{-i\alpha_{1}},e^{-i\alpha_{2}},1)\,,$
(4.3)

where $c_{ij}\equiv\cos\theta_{ij}$ and $s_{ij}\equiv\sin\theta_{ij}$ and
$\alpha_{1},\alpha_{2}$ are the two Majorana phases. Note that, if the
lightest neutrino is massless (as in our case), only one Majorana phase is
physical.

It must be noted that, given the CP symmetry of the model which consists in
the invariance under $\tau\to-\tau^{*}$ (a reflection across the imaginary
axis), we obtain two sets of points distinguished only by $\pm\text{Re}\,\tau$
with the CP violating (CPV) phases given by
$\\{\pm\delta_{\text{CP}},\pm\alpha_{1},\pm\alpha_{2}\\}$. For thoroughness,
we reported both sets in figure 1. The results of our fit666We explored the
parameter space with an algorithm inspired by ref. [77] which mimics the
Brownian motion of a particle in a potential that depends on the figure of
merit $l(p_{i})=\sqrt{\chi^{2}(p_{i})}$, where $p_{i}$ is the array of free
dimensionless parameters mentioned in the text. are given in table 3, where
the $1\sigma$ ranges correspond to the increase of one unit of the one
dimensional $\chi^{2}$ projections from $\chi^{2}_{\text{min}}$.

| Best-fit and $1\sigma$ range
---|---
$\text{Re}\,\tau$ | $\pm 0.244_{-0.067}^{+0.012}$
$\text{Im}\,\tau$ | $1.132_{-0.297}^{+0.027}$
$\beta/\alpha$ | $0.92_{-0.03}^{+0.85}$
$\gamma/\alpha$ | $-1.20_{-2.14}^{+0.06}$
$\log_{10}(\alpha_{D}/\alpha)$ | $-13.4_{-76.3}^{+13.2}$
$g^{\prime}/g$ | $2.76_{-0.23}^{+0.21}$
$g^{\prime\prime}/g$ | $-2.53_{-0.03}^{+0.13}$
$\log_{10}(|\lambda|)$ | $-12.2_{-59.2}^{+10.9}$
$v_{d}\,\alpha$, [GeV] | $1.08_{-0.69}^{+0.06}$
$v_{u}^{2}\,g^{2}/\Lambda$ [eV] | $3.46_{-1.65}^{+0.55}$
$\sin^{2}\theta_{12}$ | $0.305_{-0.011}^{+0.011}$
$\sin^{2}\theta_{13}$ | $0.0221_{-0.0005}^{+0.0006}$
$\sin^{2}\theta_{23}$ | $0.448_{-0.016}^{+0.014}$
$r$ | $0.0296_{-0.0008}^{+0.0006}$
$m_{e}/m_{\mu}$ | $0.0048_{-0.0002}^{+0.0001}$
$m_{\mu}/m_{\tau}$ | $0.0574_{-0.0050}^{+0.0032}$
Ordering | NO
$J_{\text{CP}}$ | $-0.018_{-0.002}^{+0.002}$
$\alpha_{1}/\pi$ | $0$
$\alpha_{2}/\pi$ | $\pm 0.112_{-0.014}^{+0.792}$
$m_{1}$ [meV] | $0$
$m_{2}$ [meV] | $8.620_{-0.123}^{+0.095}$
$m_{3}$ [meV] | $50.806_{-0.021}^{+0.016}$
$\textstyle\sum_{i}m_{i}$ [eV] | $0.0594_{-0.0001}^{+0.0001}$
$|m_{\beta\beta}|$ [meV] | $3.61_{-0.09}^{+0.09}$
$m_{\beta}^{\text{eff}}$ [meV] | $8.90_{-0.09}^{+0.10}$
$d_{\text{FT}}$ | 3.03
$\chi^{2}_{\text{min}}$ | 0.98
Table 3:  Best fit values of the free parameters and observables with their
$1\sigma$ ranges, obtained as described in the text.

The fit is excellent, with $\chi^{2}_{\text{min}}=0.98$ and most dimensionless
free parameters being $\mathcal{O}(1)$ with only two exceptions, related to
the parameters $\alpha_{D}/\alpha$ (which can take values arbitrarily close to
zero with a spread reaching $\mathcal{O}(1)$ values, all inside its $1\sigma$
range) and $\lambda$, which takes values arbitrarily close to zero as selected
by the fit procedure. The degree of “fine-tuning” for the free parameters of
the model, has been evaluated by means of a normalised Altarelli-Blankenburg
measure $d_{\text{FT}}$ [78]:

$d_{\text{FT}}=\frac{\displaystyle\sum_{i}\left|\frac{\text{par}_{i}}{\delta\text{par}_{i}}\right|}{\displaystyle\sum_{i}\left|\frac{\text{obs}_{i}}{\sigma_{i}}\right|}\,.$
(4.4)

Here $|\text{par}_{i}|$ is the best-fit value of the free parameter and
$|\delta\text{par}_{i}|$ is the shift of the parameter from the best-fit value
which causes $\chi^{2}-\chi^{2}_{\text{min}}=1$, while keeping all the other
parameters fixed to their best-fit values. Intuitively, a model can be
considered fine-tuned if it takes even a very small shift from the best-fit
value of some parameters to produce a large variation in the $\chi^{2}$. The
measure (4.4) is normalized by the absolute sum of the ratios between the
observables best-fit values and their uncertainties. For the fit shown in
table 3 we obtained $d_{\text{FT}}\sim\mathcal{O}(1)$ which suggests that our
model is not affected by a fine-tuning much larger than the one which already
affects the input data.

Let us now turn our attention to the predictions of our model. First of all,
we found that the Inverted Ordering of the neutrino mass eigenstates is
strongly disfavoured compared to the Normal Ordering (NO); equally relevant is
the fact that, as anticipated, the lightest neutrino mass is exactly
vanishing. This allows us to easily understand the numerical results on the
Majorana effective mass of neutrinoless double-beta decay and the effective
neutrino mass of tritium decay. In the first case, the usual definition
(inserting $\alpha_{1}=0$) gives:

$|m_{\beta\beta}|=\big{|}\sum_{i}m_{i}U_{ei}^{2}\big{|}=\sqrt{\Delta
m^{2}_{\text{atm}}}\big{|}\sqrt{r}\,s_{12}^{2}c_{13}^{2}+\sqrt{1+r}\,s_{13}^{2}e^{-i(\delta_{\text{CP}}-\alpha_{2})}\big{|}\sim\mathcal{O}(1)\,\text{meV}\,,$
(4.5)

while in the latter case:

$m_{\beta}^{\text{eff}}=\sqrt{\sum_{i}m_{i}^{2}|U_{ei}|^{2}}=\sqrt{\Delta
m^{2}_{\text{atm}}}\,\sqrt{s_{13}^{2}+r(1-c_{13}^{2}c_{12}^{2})}\sim\mathcal{O}(10)\,\text{meV}\,,$
(4.6)

in perfect agreement with the results shown in table 3. Finally, the non-
vanishing neutrino mass eigenstates lie in the region of tens of meV, implying
$\sum_{i}m_{i}\sim 0.06$ eV, perfectly compatible with the most recent upper
bound of $0.115\,\text{eV}$ ($95\,\%\,\text{C.L.}$) from [79]. This, again, is
mainly dictated by the NO scenario with vanishing lightest neutrino mass:

$\sum_{i=1}^{3}m_{i}=\sqrt{\Delta m^{2}_{\text{atm}}}(\sqrt{r}+\sqrt{1+r})\sim
0.06\,\text{eV}\,.$ (4.7)

In figure 1 some correlations between observables (both fitted and predicted
ones) and free parameters are displayed. All points in the plots have been
selected by our algorithm and satisfy the relation $\sqrt{\chi^{2}}\leq 5$.
The yellow point corresponds to the minimum $\chi^{2}$ reported in table 3
while the black one is the CP-transformed value of the modulus, which produces
the inversion of all the CPV phases in the mass matrices. The orange lines and
bands correspond to the experimental best fit and 1 $\sigma$ ranges,
respectively, taken from table 2. For illustrative purposes, we also report
points located outside the fundamental domain.

Figure 1: Correlations between observables and free parameters. All the
plotted points were accepted by the algorithm used for the fit and satisfy
$\sqrt{\chi^{2}}\leq 5$. The yellow point corresponds to the minimum
$\chi^{2}$. The black point is the CP-transformed value of the modulus, which
produces the inversion of all the CPV phases in the mass matrices. Orange
lines and bands correspond to the experimental best fit and 1$\sigma$ ranges
taken from table 2. Points located outside the fundamental domain are
displayed for the purpose of illustration.

##  5 Leptogenesis

We now discuss the mechanism of leptogenesis. It provides an elegant
explanation for the matter-antimatter asymmetry of the Universe, attributing
it to a lepton number asymmetry generated by the decays of heavy right-handed
neutrinos. This asymmetry is then converted into a baryon asymmetry through
non-perturbative processes referred to as _sphalerons_ [80, 56, 81].

As commonly done, we express the baryon asymmetry by the parameter $\eta_{B}$,
as

$\eta_{B}\equiv\frac{n_{B}-n_{\bar{B}}}{n_{\gamma}}\bigg{|}_{0}\;,$ (5.1)

where $n_{B},n_{\bar{B}}$ and $n_{\gamma}$ refer to the number densities of
the baryons, antibaryons and photons, while the subscript “0” means “at
present time”.

We have worked so far in the limit of exact supersymmetry. We will assume that
the SUSY breaking scale $m_{SUSY}$ is above the mass scale of the right-handed
neutrinos of our model, so that we can avoid considering the superparticles
contribution to the final baryon asymmetry, as well as all the complications
related to the gravitino densities [82, 83, 58, 84]. Nevertheless, it is
imperative to exercise caution regarding the magnitude of $m_{\text{SUSY}}$,
as arbitrarily large values may introduce complications. This is discussed in
depth in Ref. [85]. Notably, the lepton masses undergo a correction which is
proportional to the ratio $m_{\text{SUSY}}/\mathcal{F}$, where $\mathcal{F}$
represents the characteristic scale at which the supersymmetry-breaking sector
communicates with the visible sector. This scale can be chosen in proximity to
the Planck scale $M_{pl}$. As we will discuss in detail in the following, our
model provides right-handed neutrinos with masses of order
$\mathcal{M}\sim\mathcal{O}(10^{12})$ GeV. Thus, we could choose
$m_{\text{SUSY}}\simeq 10^{14}$ GeV and $\mathcal{F}\simeq 10^{18}$ GeV so
that the corrections to the low energy observables do not affect our results,
and we can proceed in discussing the leptogenesis without considering any
massive superparticle.
In this section, we will focus on the so-called $N_{1}$ dominated scenario
(N1DS) [86, 87, 88], even though our model involves one additional sterile
state $N_{2}$. Then, we will solve numerically the semi-classical Boltzmann
Equations [89, 90, 55] for the baryon asymmetry evolution, and we will present
a realization that provides the value of the baryon-to-photon ratio $\eta_{B}$
inside the $3\sigma$ experimental allowed region [91].

###  5.1 Thermal leptogenesis: basic physics

The lepton number violating decays of the sterile Majorana neutrinos $N_{i}$
are the main ingredients for the thermal leptogenesis recipe [92, 83]. They
depend, in general, on the mass of the heavy neutrinos, as well as their
Yukawa couplings with the charged leptons and the Higgs doublet. In order to
generate a lepton asymmetry, we first need a thermal production of a
distribution of $N_{i}$ through scattering in the early Universe at
temperatures around $T\sim M_{N_{i}}$, where we indicate with $M_{N_{i}}$ the
mass of the $N_{i}$ sterile neutrino. As the temperature drops below the mass
$M_{N_{i}}$, these sterile neutrinos undergo decays via processes that violate
the lepton number by one unit ($\Delta L=1$), shown in figure 2 and, if these
decays are slow enough compared to the expansion rate of the Universe, the
$N_{i}$ abundance does not decrease according to the Boltzmann equilibrium
statistics $\propto e^{-M_{N_{i}}/T}$, as the equilibrium demanded. This is
how a net lepton asymmetry is generated. Thus, the crucial ingredient is to
estimate the mass of the heavy neutrinos. To this aim, it is useful to recast
the Majorana and Dirac mass matrices of eqs. (3.9) and (3.10) in the following
way:

$M_{D}=\;v_{u}\,gY_{1}^{2}\,\begin{pmatrix}\left(1+g^{\prime\prime}/g\right)+\left(g^{\prime\prime}/g-1\right)\zeta^{2}&2\,\zeta&2\,\zeta\,g^{\prime}/g\\\
2\,\zeta&\left(1+g^{\prime\prime}/g\right)\zeta+\left(g^{\prime\prime}/g-1\right)&(1-\zeta^{2})\,g^{\prime}/g\end{pmatrix}_{\text{RL}}\;,$
(5.2)
$\mathcal{M}_{R}=\;\Lambda\,Y_{1}^{2}\,\begin{pmatrix}\lambda(1+\zeta^{2})-(\zeta^{2}-1)&2\,\zeta\\\
2\,\zeta&\lambda(\zeta^{2}+1)+(\zeta^{2}-1)\end{pmatrix}_{\text{RR}}\;,$ (5.3)

where $\zeta=Y_{2}/Y_{1}$. With the modulus $\tau$ in the fundamental domain,
and for small values of $\text{Re}\,\tau$, it reads:

$\zeta\simeq 8\,\sqrt{3}\,t^{2}\,\left(1+i\,\pi\,\text{Re}\tau\right)\,,\quad
t\equiv e^{-(\pi\,\text{Im}\tau)/2}\,,$ (5.4)

where only terms up to $\mathcal{O}(t^{2})$ are considered.
Therefore, with $|\zeta|\ll 1$ and all the free parameters close to their best
fit values quoted in table 3, it is evident that the mass scale of the right-
handed neutrinos is basically $\mathcal{M}_{N}\sim\Lambda Y_{1}^{2}$ and the
Yukawas $\mathcal{Y}\sim g\,Y_{1}^{2}$. In other words, both the mass scale of
the Majorana neutrinos and the Yukawas get suppressed by the modular form
$Y_{1}$, which can be expanded as:

$Y_{1}(\tau)\simeq\dfrac{7}{100}+\dfrac{42}{25}\,t^{2}\,\left(1+2\,i\,\pi\,\text{Re}\tau\right)+\mathcal{O}(t^{3})\,.$
(5.5)

Since the mass-scale of light neutrinos is completely determined by the global
parameter $g^{2}\,v_{u}^{2}/\Lambda$, we can estimate $\Lambda$. By imposing
that the dimensionless parameter $g$ assumes a value of order
$\mathcal{O}(1)$, we get $\Lambda\sim\mathcal{O}(10^{14})$ GeV. This, along
with the suppression provided by the modular form $Y_{1}$, implies a mass
scale $\mathcal{M}_{N}$ for the Majorana neutrinos of approximately $10^{12}$
GeV.

$N_{i}$$H_{u}$$l_{\alpha}$

$N_{i}$$H_{u}$$l$$l_{\alpha}$$H_{u}$$N_{j}$

$N_{i}$$l,\bar{l}$$H_{u}$$N_{j}$$H_{u}$$l_{\alpha}$

Figure 2: CP-violating $N_{i}$ decay.

In principle, the interference between the tree-level and the loop
contributions in figure 2 produces CP violating decays that can generate
asymmetries in all flavors $\alpha$. Flavor effects [93, 94, 95] become
relevant when the rate of interaction $\Gamma_{\beta}$ of the processes
involving a lepton with flavor $\beta$ ($\beta=e,\mu,\tau$) are rapid enough
to discern distinct lepton flavors. Under these circumstances, the coherence
of leptons and antileptons as flavor superpositions is lost during
leptogenesis. Consequently, it becomes necessary to adopt a flavor basis to
calculate the baryon asymmetry. Typically, the temperature below which the
flavor effect cannot be neglected is such that the interaction rate
$\Gamma_{\beta}$ for a charged lepton with flavor $\beta$ becomes faster than
the expansion rate $H(T)$ of the Universe. The interaction rate can be
estimated [96, 97] as:

$\Gamma_{\beta}\simeq 10^{-2}\,\alpha_{\beta}^{2}\,T\;,$ (5.6)

where $\alpha_{\beta}$ is the charged-lepton Yukawa coupling. For a certain
flavor-dependent $T\lesssim T_{\beta}$, the condition $\Gamma_{\beta}(T)>H(T)$
is satisfied, so that the typical temperatures below which the leptogenesis
becomes sensitive to lepton flavors are $T_{e}\simeq 4\cdot 10^{4}$ GeV,
$T_{\mu}\simeq 2\cdot 10^{9}$ GeV and $T_{\tau}\simeq 5\cdot 10^{11}$ GeV [98,
99, 100]. The amount of CP asymmetry generated by the interference between the
tree and one loop decay diagrams of figure 2 can be described by the
$\epsilon_{i}$ parameters, defined as:

$\displaystyle\epsilon_{i}\,=$
$\displaystyle\sum_{\alpha}\dfrac{\Gamma(N_{i}\rightarrow
L_{\alpha}\,H_{u})-\Gamma(N_{i}\rightarrow\overline{L_{\alpha}}\,H_{u}^{\ast})}{\Gamma(N_{i}\rightarrow
L_{\alpha}\,H_{u})+\Gamma(N_{i}\rightarrow\overline{L_{\alpha}}\,H_{u}^{\ast})}=$
(5.7) $\displaystyle=$
$\displaystyle\,\sum_{j\not=i}\dfrac{\text{Im}\left[\left(\mathcal{Y}^{\dagger}\mathcal{Y}\right)^{2}_{ji}\right]}{8\pi(\mathcal{Y}^{\dagger}\mathcal{Y})_{ii}}\left[f(x_{ij})+g(x_{ij})\right]\,\quad\text{with}\;i=1,2$

with $x_{ij}=M_{j}^{2}/M_{i}^{2}$ and the loop functions [101, 102]:

$\displaystyle f(x_{ij})\,=$
$\displaystyle\sqrt{x_{ij}}\Big{[}1-(1+x_{ij})\log\left(1+\frac{1}{x_{ij}}\right)+\frac{1}{1-x_{ij}}\Big{]}\,,$
(5.8) $\displaystyle g(x_{ij})\,=$
$\displaystyle\dfrac{\sqrt{x_{ij}}(1-x_{ij})}{(1-x_{ij})^{2}+x_{ij}(\mathcal{Y}^{\dagger}\mathcal{Y})^{2}_{jj}/(16\pi^{2})}\;,$

which are related to the vertex correction and the self energy correction,
respectively.
The lepton number violation produced by the sterile neutrino decays can be
depleted by scatterings and inverse processes, which are often referred to as
_wash-out processes_. If they occur out-of-equilibrium, _i.e._ they are not
very fast compared to the Hubble expansion rate of the Universe $H(T)$ at
temperature $T=M_{N_{i}}$, a net asymmetry can survive.
Typically, we can quantify the departure from thermal equilibrium by the
_decay parameter_ which, in terms of the decay rates $\Gamma_{i}$ of the
$N_{i}$ sterile neutrino, reads as follows:

$K_{i}\equiv\dfrac{\Gamma_{i}}{H(T=M_{i})}=\dfrac{\widetilde{m_{i}}}{m^{\ast}}\;,$
(5.9)

where the _effective neutrino mass_ [103] is defined as:

$\widetilde{m_{i}}=\dfrac{\left(\mathcal{Y}^{\dagger}\mathcal{Y}\right)_{ii}\,v_{u}^{2}}{M_{i}}\;,$
(5.10)

and $m^{\ast}\simeq 1.1\cdot 10^{-3}$ eV. We can distinguish three different
regimes: $K\ll 1$, $K\simeq 1$ and $K\gg 1$, which we identify as the _weak_ ,
_intermediate_ and _strong_ wash-out regime, respectively. Using our best fit
values of the free parameters of the model, we deduce we are always in a
strong wash-out regime; thus, the contributions to the leptogenesis from the
$\Delta L=1$ and $\Delta L=2$ processes777We refer to the relevant lepton
number violating processes in the thermal plasma: the $\Delta L=1$ scatterings
between $N_{i}$ and leptons mediated by the Higgs boson, involving the top
quark or electroweak gauge bosons, and the $\Delta L=2$ scatterings between
the leptons and the Higgs boson with an intermediate $N_{i}$ heavy neutrino.
can be safely neglected, as discussed in [55, 104, 105]. Therefore, from now
on, we will only consider the decays and the inverse decays. Let us now
discuss the relevant Boltzmann Equation (BE) for the leptogenesis in our
model, with two heavy sterile neutrinos $N_{1,2}$ with masses $M_{1}\lesssim
M_{2}$. Assuming the reheating temperature $T_{\text{rh}}>M_{1,2}$, both the
sterile neutrinos $N_{1,2}$ are thermally produced [106, 80, 107, 108] in the
Early Universe, and the set of classical kinetic equations [89] could be
written as:888It is convenient to write the BE for the $B-L$ asymmetry
evolution, because all the electroweak processes conserve it, including those
mediated by the sphalerons.

$\displaystyle\dfrac{\text{d}N_{N_{i}}}{\text{d}z}=$ $\displaystyle-
D_{i}\left(N_{N_{i}}-N_{N_{i}}^{eq}\right)\,,\quad\text{with }i=1,2$ (5.11)
$\displaystyle\dfrac{\text{d}N_{B-L}}{\text{d}z}=$
$\displaystyle-\sum_{i=1}^{2}\epsilon_{i}\,D_{i}\left(N_{N_{i}}-N_{N_{i}}^{eq}\right)-\sum_{i=1}^{2}W_{i}\,N_{B-L}$

with $z=M_{1}/T$. Here, $D_{i}$ is the decay term while $W_{i}$ is the washout
factor; futhermore, the $N_{i}$’s are the number densities of the RH sterile
neutrinos, while $N_{B-L}$ is the amount of $B-L$ asymmetry, both normalized
by the comoving volume [56]. With this choice, we automatically take into
account the expansion of the Universe, which was not possible if we were to
employ the number density $n_{X}$ of the particle species. The equilibrium
abundances of $N_{i}$’s are given by
$N_{i}^{eq}=\frac{3}{8}z_{i}^{2}\mathcal{K}_{2}(z_{i})$, with
$\mathcal{K}_{2}$ the Bessel function of the second kind. In the literature,
the comoving volume $R^{3}_{\ast}(t)$ is usually chosen as to contain one
photon at the time $t_{\ast}$, before the onset of the leptogenesis [105]:

$N_{X}(t)=n_{X}(t)R^{3}_{\ast}(t)\,,$ (5.12)

with

$R_{\ast}(t_{\ast})=\left(n_{\gamma}^{eq}(t_{\ast})\right)^{-1/3}\,,$ (5.13)

so that $N_{\gamma}(t_{\ast})=1$. Using $z_{i}=z\sqrt{x_{1i}}$, we can write
the decay term $D_{i}$ and the washout factor $W_{i}$ as:

$\displaystyle D_{i}=$
$\displaystyle\,K_{i}\,x_{1i}\,z\,\braket{1/\gamma_{i}}\,,$ (5.14)
$\displaystyle W_{i}\simeq W_{i}^{ID}=$
$\displaystyle\,\dfrac{1}{4}\,K_{i}\sqrt{x_{1i}}\mathcal{K}_{1}(z_{i})z_{i}^{3}\,,$

where $W_{i}^{ID}$ is the inverse decay term which contributes to the washout
of the lepton asymmetry, and $\braket{1/\gamma_{i}}$ is the thermally averaged
dilation factor, given as the ratio of the modified Bessel functions of the
second kind:

$\Biggl{\langle}\dfrac{1}{\gamma_{i}}\Biggr{\rangle}=\dfrac{\mathcal{K}_{1}(z_{i})}{\mathcal{K}_{2}(z_{i})}\,.$
(5.15)

The final $B-L$ asymmetry $N_{B-L}^{f}$ is given by:

$N_{B-L}^{f}=N_{B-L}(z=\infty)=N_{B-L}^{in}e^{-\sum_{i}\int
dz^{\prime}W_{i}(z^{\prime})}+N_{B-L}^{lep}\,,$ (5.16)

where $N_{B-L}^{lep}$ is the $B-L$ produced via thermal leptogenesis and
$N_{B-L}^{in}$ indicates the possible pre-existing asymmetry [109, 110], at an
initial temperature $T_{in}$ before the leptogenesis onset. The potential pre-
existing $B-L$ asymmetry could introduce further constraints on the parameter
space of the model [107, 111, 112, 113]. In order to consider a scenario of
pure leptogenesis from RH neutrino decays, we will fix $N_{B-L}^{in}=0$
throughout the rest of the paper.

During the expansion of the Universe, the $B-L$ asymmetry is reprocessed in a
baryon asymmetry through sphaleron processes. If sphalerons decouple before
the electroweak phase transition, the amount of $B$ asymmetry produced is
[114]:

$N^{f}_{B}=\dfrac{(8\,N_{F}+4\,N_{H})}{(22\,N_{F}+13\,N_{H})}N^{f}_{B-L}\;,$
(5.17)

where $N_{F}=3$ and $N_{H}=2$ stand for the number of fermion generations and
the number of Higgs doublets, respectively. The predicted baryon-to-photon
ratio has to be compared with the value $\eta_{B}$ measured at recombination
[91]. The relation is:

$\eta_{B}=\dfrac{N_{B}^{f}}{f}\,,$ (5.18)

where $f=2387/86$ is the dilution factor calculated assuming standard photon
production from the onset of leptogenesis till recombination [89].

###  5.2 Validity of the $N_{1}-$dominated scenario

In presence of a hierarchical mass spectrum $M_{1}\lesssim M_{2}$, only the
contribution from the lightest sterile neutrino to the leptogenesis is
commonly considered, the so-called $N_{1}-$dominated scenario. However, as
widely discussed in literature [86, 87, 88], there exists a parameter space
for which such an assumption is no longer valid, and the heaviest sterile
neutrino contribution to the final baryon asymmetry cannot be neglected. An
important parameter that can be used to test whether the $N_{1}-$dominated
scenario sets in is the parameter $\delta_{i}=(M_{i}-M_{1})/M_{1}$. This is
related to $x_{1i}$ by $\sqrt{x_{1i}}=1+\delta_{i}$, and it will tell us
whether the CP-asymmetry can undergo resonant enhancement [55, 57, 115, 116,
117]. In fact, it can be noted that, when dealing with an almost degenerate
mass spectrum, the self-energy loop function $g(x_{ij})$ in eq.(5.8) can be
reformulated in terms of $\delta_{j}$ in the following manner:

$g(\delta_{j})\simeq\,\dfrac{\delta_{j}}{2\left[(\delta_{j})^{2}+(\mathcal{Y}^{\dagger}\mathcal{Y})^{2}_{jj}/(64\pi^{2})\right]}\;.$
(5.19)

Therefore, if the Yukawa matrix and the mass splittings are independent
quantities, the resonance enhancement occurs if the condition [101]

$\delta_{j}^{2}\simeq(\mathcal{Y}^{\dagger}\mathcal{Y})^{2}_{jj}/(64\pi^{2})$
(5.20)

holds. As already noted, our model provides Yukawa couplings suppressed by the
modular form $Y_{1}$ while the mass splitting parameter $\delta_{2}$ belongs
to the range $\left[0.7,2.3\right]$, as our evaluation shows in figure 3.
Given that the right-hand side of eq.(5.20) is approximately on the order of
$\mathcal{O}(10^{-5})$, we safely deduce that the resonance condition is never
satisfied in the best fit regions.999Notice that the resonance would have
probably ruined the prediction of the model regarding the baryon-to-photon
ratio $\eta_{B}$; indeed, with heavy sterile neutrinos with masses around
$10^{12}$ GeV, the resonance would imply an enhancement of the CP-violating
parameter such that the final baryon asymmetry exceeds the observed value by
many order of magnitudes (see _e.g._ [118]). We can now discuss the validity
of the $N_{1}-$dominated scenario in presence of another heavy neutrino
$N_{2}$. In general, one should solve eqs.(5.11) taking into account the wash-
out effects due to both $N_{1}$ and $N_{2}$ sterile neutrinos. The strength of
the $N_{2}$ wash-out effects is encoded into $W_{2}^{ID}$, see eq.(5.14),
which depends on the $\delta_{2}$ parameter. There exists a specific value
$\delta_{2}^{\ast}$ above which the influence of $W_{2}^{ID}$ becomes
irrelevant compared to $W_{1}^{ID}$; in that case, leptogenesis would be
predominantly driven by the presence of the lightest neutrino, from which the
name of $N_{1}-$dominated scenario. Conversely, for
$\delta_{2}<\delta_{2}^{\ast}$, the effect of $W_{2}^{ID}$ becomes relevant.

Figure 3: Contour plots showing the values assumed by $\delta_{2}$ in the best
fit regions. For a clearer and more concise data visualization, all the fitted
points from section 4 were mapped into $\mathcal{D}$, see appendix C for
details.

Usually, in a strong wash-out regime, a value $\delta_{2}^{\ast}\gtrsim
1.5\div 5$ is found [86, 88]. In our model, $\delta_{2}^{\ast}\simeq 1$, so
that the $N_{1}-$dominated scenario is valid for
$\text{Re}\tau\in[-0.44,-0.22]$ (which includes our best fit point) where
$\delta_{2}\gtrsim\delta_{2}^{\ast}$, while in the other regions of the
parameter space we cannot a priori neglect the wash-out effects of the $N_{2}$
sterile neutrinos. In appendix A we illustrate that in the absence of specific
correlations among the parameters, the contribution of the heavier sterile
neutrino to leptogenesis is negligible for $\delta_{2}\gtrsim 1.5$.

###  5.3 Numerical analysis

Figure 4: Pink band: dependence of the final baryon-to-photon asymmetry
$\eta_{B}$ on the rescaling parameter “$r$” ($\Lambda\to r\,\Lambda$) for a
fixed set of the theory’s free parameters, satisfying $\sqrt{\chi^{2}}\leq 2$.
The observed value of $\eta_{B}$ [91] is shown with the blue line. The
unflavored leptogenesis is provided for $10^{-1}<r<10^{2}$, and it turns out
to be successful for $16.8\leq r\leq 22.1$. In particular, at the best fit,
$\eta_{B}$ in agreement with the observation for $r=\bar{r}=19.9$.

We solve the full Boltzmann Equations presented in eqs.(5.11) for the best fit
points that satisfy $\sqrt{\chi^{2}}\leq 2$. With the heavy sterile neutrino
masses at $\mathcal{O}(10^{12})$ GeV, the model provides $\eta_{B}\lesssim
10^{-10}$, slightly below the observed value. However, since only the ratio
$g^{2}\,v_{u}^{2}/\Lambda$ is constrained by the masses of the light
neutrinos, the mass scale of the Majorana neutrinos can be regarded as a free
parameter of the model. We can safely rescale $\Lambda\rightarrow r\,\Lambda$
and $g\rightarrow\sqrt{r}\,g$ without altering the results of the fit in table
3.101010A similar work has been done in [119], in a context of a minimal
extended seesaw, with an additional singlet field. Notice that varying the
mass-scale of the right-handed neutrinos have relevant consequences. For
$r\lesssim 10^{-1}$ the mass-scale of the sterile neutrinos is lowered down to
$\mathcal{O}(10^{11})$, below which the _flavored regime_ sets in; on the
other hand, if $r\gtrsim 10^{2}$ the _resonant regime_ occurs. We confine the
analysis to the range $r\in\left[10^{-1},10^{2}\right]$, focusing on the
unflavored scenario that involves solely the lightest sterile neutrino [53,
56, 54, 105, 104, 83, 120]. The results are shown in figure 4, where the pink
band represents the dependence of the final baryon-to-photon asymmetry
$\eta_{B}$ on the parameter $r$ for a fixed set of the theory’s free
parameters, satisfying $\sqrt{\chi^{2}}\leq 2$; in _blue_ , instead, we report
the observed value of $\eta_{B}$ (due to the small uncertainty on this
observation, the $1\sigma$ allowed range is basically indistinguishable from
the shown dotted line). In particular, the best fit realization of our model
provides a successful leptogenesis if $\Lambda\rightarrow\bar{r}\Lambda$,
where $\bar{r}=19.9$. It is important to observe that this selection does not
compromise the quality of our fit; in fact, it modifies $g$ by a
multiplicative factor of approximately $\sqrt{\bar{r}}\sim 4$, thereby
maintaining $g$ as an order $\mathcal{O}(1)$ parameter, as for most of the
other parameters of the model.

##  6 Conclusions

In the present work, we have addressed the problem of finding a modular
invariant minimalistic construction based on the smallest modular finite group
$\Gamma_{2}\cong S_{3}$ that was compatible with low energy neutrino
oscillation data and, with the same model parameters, able to accommodate the
measured value of the baryon asymmetry of the Universe (BAU). Our attempts
have been guided by the requirement to minimally extend the Standard Model
particle content; with the aim to naturally generate small neutrino masses, we
relied on a seesaw mechanism built with the help of only two heavy right-
handed neutrinos. A straightforward prediction of our model is that the
lightest neutrino mass is zero at tree level. Having imposed a generalized CP-
symmetry, not only we were able to restrict the number of free parameters to
ten (eight dimensionless, of which two come from
$\tau=\text{Re}\,\tau+i\,\text{Im}\,\tau$, and two global scales) but we have
also secured that the modulus $\tau$ is the only source of CP-violation. We
showed that, even within this limitation, this was enough to reproduce the
current experimental value of the leptonic Jarlskog invariant within
$1\sigma$. Furthermore, the model strongly favours the Normal Ordering of
neutrino masses, a prediction that can be tested in upcoming experiments [121,
122, 123, 124]. The predictions for the Majorana effective mass of the
neutrinoless double-beta decay and the effective neutrino mass of tritium
decay are both substantially within the current upper bounds from [125, 126],
and are both analitically understood from the constraint on the lightest
neutrino mass, which is in turn a consequence of the Minimal seesaw scenario.

Since the only source of CP violation in our model is dictated by
$\text{Re}\,\tau$, it was highly non-trivial to generate a sufficient amount
of BAU through the leptonic asymmetry $\eta_{B}$; we have demonstrated,
indeed, that this is the case in our model. Given that in the presence of a
mild hierarchical heavy mass spectrum with $\delta_{2}\,\gtrsim\,1$ only the
contribution from the lightest sterile neutrino to the leptogenesis is
commonly considered, we solved the appropriate Boltzmann Equations in the
$N_{1}$-dominated scenario and shown that a mild rescaling of the Majorana
right-handed mass scale $\Lambda$ (not fixed by the fit) is enough to
guarantee the required $\eta_{B}$ value. Given the minimality of our
$\Gamma_{2}\cong S_{3}$ construction, we consider this feature a relevant
result in the model building based on modular invariant theories.

## Acknowledgements

The authors would like to thank J. Penedo for useful discussions.

##  Appendix A Beyond the $N_{1}-$dominated scenario

Figure A.1:  Top left (right) panel: evolutions of the efficiency factors
$k_{i}^{f}$ in terms of $K_{1}$ with $\delta_{2}=1,\,0.1,\,0.01$
($\delta_{2}=3,\,10,\,20$), with $K_{2}\simeq 25$. The efficiency factors of
$N_{1}$ and $N_{2}$ are shown in blue and red dashed lines, respectively.
Bottom left (right) panel: $|N_{B-L}|$ as a function of $z=M_{1}/T$ and other
relevant quantities such as $N_{i}$ and $W_{i}^{ID}$, with $\delta_{2}=0.3$
($\delta_{2}=2$), $N_{i}^{in}=3/4$ and $K_{1}=K_{2}=10$. The colour code for
$N_{i},W_{i}^{ID}$ is highlighted in the legend. Solid red line shows
$|N_{B-L}|$ for a pure $N_{1}-$dominated scenario. The solid blue line shows
the asymmetry generated by $N_{1}$, partially depleted by the wash-out effects
of $N_{1}$ and $N_{2}$, given by $W_{1}^{ID}$ and $W_{2}^{ID}$, respectively.
The solid black line indicates $|N_{B-L}|$ produced by $N_{2}$, undergoing
$W_{1,2}^{ID}$ washout.

In this appendix we firstly illustrate that, in the absence of specific
correlations among the decay parameters, the feasibility of the
$N_{1}$-dominated scenario hinges crucially on the value of the $\delta_{2}$
parameter, defined in section 5.2. This is shown in figure A.1. Then, we
discuss the concrete example related to the best fit realization of our model.
In presence of two right-handed sterile neutrinos, the solution of the BEs in
eq.(5.11) can be written as [127]:

$N_{B-L}^{f}=\sum_{i}\epsilon_{i}k^{f}_{i}\,\,,\quad i=1,2$ (A.1)

with $\epsilon_{i}$ the usual CP-violating parameters (see eqs.(5.7) and
(5.8)) and $k_{i}^{f}$ the _efficiency factors_. Without strongly hierarchical
Yukawa couplings, for quasi-degenerate heavy neutrinos we can assume
$\epsilon_{1}\simeq\epsilon_{2}$. Thus, the contribution of each $N_{i}$ to
the final baryon asymmetry is governed by the efficiency factor $k_{i}^{f}$,
which can be written as:

$\displaystyle k_{i}^{f}\equiv
k_{i}(z=\infty)=-\int_{z_{in}\to\,0}^{{z_{fin}\to\,\infty}}\dfrac{\text{d}N_{i}}{\text{d}z^{\prime}}\exp{\left[-\sum_{i}\int_{z^{\prime}}^{z}W_{i}(z^{\prime\prime})\text{d}z^{\prime\prime}\right]}\text{d}z^{\prime}\,.$
(A.2)

We focus on the strong washout regime, for which the Yukawa couplings are
strong enough to let any species reach the equilibrium even if starting from a
vanishing initial density. Therefore, the efficiency factors can be computed
by inserting into the general formula (A.2) the equilibrium values
$\frac{\text{d}N^{\text{eq}}_{i}}{\text{d}z^{\prime}}$ for both rate
abundances. Also, the $k_{i}^{f}$ factors depend on the washout factors
$K_{1},K_{2}$, as well as on the $\delta_{2}=(M_{2}-M_{1})/M_{1}$ factor which
accounts for the mass difference between the RH neutrinos. In the left (right)
upper panel of figure A.1 we plotted the $k_{i}^{f}$ factors for
$\delta_{2}=1,\,0.1,\,0.001$ ($\delta_{2}=3,\,10,\,20$) values. In order to
highlight the effects of the washout, we assumed $K_{2}\simeq 25$.

Figure A.2:  $|N_{B-L}|$ as a function of $z=M_{1}/T$ for the best fit
realization with $r=\bar{r}=19.9$ (see section 5.3 for further details). The
colour code is the same as the bottom panels of figure A.1. A zero initial
abundance $N_{i}^{in}$ is assumed.

As it is evident, as the wash-out parameter increases, the hierarchy between
the two efficiency factor becomes stronger, implying that the contribution to
the finally BAU from the heavier neutrino can be neglected, and the
$N_{1}-$dominated scenario is recovered. This is confirmed by the numerical
solutions of the BE in case of $\delta_{2}=0.3$ and $\delta_{2}=2$, shown in
figure A.1 in the left lower panel and in the right lower panel, respectively.
Usually, a good rule of thumb to check whether the $N_{1}-$dominated scenario
is a correct approximation consists in computing the $\delta_{2}$ parameter:
for $\delta_{2}\gtrsim 1.5\div 5$, the contribution of the heaviest sterile
neutrino $N_{2}$ to the leptogenesis can be safely neglected and the
hierarchical N1DS is realized. Nevertheless, in a realistic model one should
take into account possible correlations between the decay parameters $K_{1}$
and $K_{2}$ as well as possible hierarchies in the Yukawa couplings. Such
correlations could either establish a strong hierarchy between the two
efficiency factors $k_{i}^{f}$, even for relatively small values of the
$\delta_{2}$ parameter, or limit the accessibility of the N1DS to cases where
$\delta_{2}>5$.
In the case of our best fit realization analyzed in section 5.3, we have
$r=\bar{r}=19.9$ and a value of $\delta_{2}\simeq 1.7$; these, along with a
particular correlation between the two decay parameters $K_{1}\sim 10\,K_{2}$,
lead to a significant depletion of the asymmetry produced by $N_{2}$. Such a
situation is well depicted in figure A.2, where the $|B-L|$ evolution with
$z$, as well as the other relevant quantities such as $N_{i}$ and $W_{i}^{ID}$
is shown. In the figure, the red and blue solid lines correspond to the
asymmetry produced by $N_{1}$, partially depleted by the wash-out effect due
to $N_{1}$ and $N_{1}+N_{2}$, respectively. The black line indicates the
asymmetry produced by $N_{2}$, undergoing wash-out from $N_{1}+N_{2}$. The
particular correlation between the two decay parameters together with
$\delta_{2}\simeq 1.7$, provides a huge gap between the value at the plateau
reached by the black line and the red line. When the strength of the inverse
decays of $N_{1}$ reaches its maximum value, the asymmetry produced by $N_{2}$
is practically switched off, so it is efficiently washed-out. In other words,
the final dynamics of $|B-L|$ is governed only by the processes involving
$N_{1}$. This is clear by analysing the evolution of the red line and the blue
line: if one takes into account the asymmetry produced by $N_{1}$ subjected to
the $N_{2}$ wash-out along with the $N_{1}$ wash-out (blue line), the final
asymmetry coincides with the pure N1DS (red line).

##  Appendix B Modular forms and $S_{3}$ representations

The group $S_{3}$ is characterized by three irreducible representations: the
singlet $\mathbf{1}$, the pseudo-singlet $\mathbf{1^{\prime}}$ and the doublet
$\mathbf{2}$. We work in a basis where the $S_{3}$ doublet generators are
given by:

$\begin{split}\rho(S)=\frac{1}{2}\begin{pmatrix}-1&-\sqrt{3}\\\
-\sqrt{3}&1\end{pmatrix}\quad,\quad\rho(T)=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\\\ \\\
(\rho(S))^{2}=(\rho(T))^{2}=(\rho(S)\rho(T))^{3}=\mathbbm{1}\quad\,.\end{split}$
(B.1)

The level $N=2$ modular forms of lowest weight transform as a doublet and can
be expanded as:

$\begin{pmatrix}Y_{1}(\tau)\\\
Y_{2}(\tau)\end{pmatrix}_{\mathbf{2}}=\begin{pmatrix}\displaystyle\frac{7}{100}+\frac{42}{25}q+\frac{42}{25}q^{2}+\frac{168}{25}q^{3}+...\\\
\\\
\displaystyle\frac{14\sqrt{3}}{25}q^{1/2}(1+4q+6q^{2}+...)\end{pmatrix}\,,$
(B.2)

where $q\equiv e^{2\pi i\tau}$, and $\tau=\text{Re}\,\tau+i\,\text{Im}\,\tau$.
The normalization of our modular forms differs from the one chosen in [10],
and it is justified in [11]. We stress that to our knowledge there is no
physical prescription on the normalization of modular forms: it is an
arbitrary choice. Recent discussions on this issue can be found in refs. [128,
129]. Apart from that, as discussed in the text, the ratio
$|Y_{2}(\tau)/Y_{1}(\tau)|$ can be used as an expansion parameter for
$\tau\in\mathcal{D}$. This is shown in figure B.1 and the behaviour is
obviously independent of the chosen normalization.

Figure B.1: Visualization of $|\zeta|=|Y_{2}/Y_{1}|$ for $\tau\in\mathcal{D}$.

Labelling some $S_{3}$ pseudo-singlets with $y_{i}$ for $i=1,2,3...$, and with
$\psi_{1,2},\varphi_{1,2}$ two doublet components, the tensor decomposition
rules are given by:

$\displaystyle\begin{split}\mathbf{1^{\prime}}\otimes\mathbf{1^{\prime}}=\mathbf{1}\quad\sim\quad
y_{1}y_{2}\,,\\\
\mathbf{1^{\prime}}\otimes\mathbf{2}=\mathbf{2}\quad\sim\quad\begin{pmatrix}-y_{1}\psi_{2}\\\
y_{1}\psi_{1}\end{pmatrix}\end{split}\,,$ (B.3)

$\mathbf{2}\otimes\mathbf{2}=\mathbf{1}\oplus\mathbf{1^{\prime}}\oplus\mathbf{2}\,\begin{cases}\mathbf{1}\quad\sim\quad\psi_{1}\varphi_{1}+\psi_{2}\varphi_{2}\\\
\\\
\mathbf{1^{\prime}}\quad\sim\quad\psi_{1}\varphi_{2}-\psi_{2}\varphi_{1}\\\
\\\
\mathbf{2}\quad\sim\quad\begin{pmatrix}\psi_{2}\varphi_{2}-\psi_{1}\varphi_{1}\\\
\psi_{1}\varphi_{2}+\psi_{2}\varphi_{1}\end{pmatrix}\,.\end{cases}$ (B.4)

##  Appendix C Mapping to $\mathcal{D}$ and rescaling

Figure C.1: Examples of regions outside the fundamental domain $\mathcal{D}$.
The mapping is provided by combinations of the generators $S,T$ defined in
(2.2). Here $T^{\prime}\equiv T^{-1}$.

Points outside the fundamental domain (2.8) are redundant: some examples of
modular mappings to $\mathcal{D}$ are given in figure C.1. The act of mapping
into $\mathcal{D}$ implies that some free parameters of the superpotential may
have to be rescaled, as argued in [33] . The Kähler potential for the matter
fields is given by:

$\sum_{I}(-i(\tau-\bar{\tau}))^{-k_{I}}|\varphi^{(I)}|^{2}=\sum_{I}(2\text{Im}\,\tau)^{-k_{I}}|\varphi^{(I)}|^{2}\,,$
(C.1)

and it is modular-invariant since under a modular transformation the relevant
quantities transform as:

$\text{Im}\,\tau\quad\to\quad|c\tau+d|^{-2}\text{Im}\,\tau\quad\quad,\quad\varphi^{(I)}\quad\to\quad(c\tau+d)^{-k_{I}}\varphi^{(I)}\,.$
(C.2)

In order to read the canonical kinetic terms from the potential, the
superfields can be rescaled as:

$\varphi^{(I)}=\tilde{\varphi}^{(I)}(2\text{Im}\,\tau)^{k_{I}/2}\,,$ (C.3)

and the modular-invariant operators of the superpotential change as:

$\alpha_{i}\varphi^{(1)}\varphi^{(2)}\varphi^{(3)}Y_{k_{Y}}\quad\to\quad\alpha_{i}(2\text{Im}\,\tau)^{\frac{k_{1}+k_{2}+k_{3}}{2}}\tilde{\varphi}^{(1)}\tilde{\varphi}^{(2)}\tilde{\varphi}^{(3)}Y_{k_{Y}}\,,$
(C.4)

where $\alpha_{i}$ is the associated coupling and $Y_{k_{Y}}$ is the required
modular form of weight $k_{Y}$. Recall that from the modular constraints on
the superpotential we know that $k_{1}+k_{2}+k_{3}=k_{Y}$. If one performed a
numerical scan and extracted best-fit values for $\alpha_{i}$, namely
$\hat{\alpha}_{i}$, the kinetic rescaling reads:

$\hat{\alpha}_{i}={{\alpha}_{i}}{(2\text{Im}\,\tau)^{k_{Y}/2}}\,.$ (C.5)

Acting with a modular transformation, $\text{Im}\,\tau$ transforms as in
(C.2). As a consequence:

${{\alpha}_{i}}{(2\text{Im}\,\tau)^{k_{Y}/2}}\quad\to\quad\underbrace{{{\alpha}_{i}}{(2\text{Im}\,\tau)^{k_{Y}/2}}}_{\hat{\alpha_{i}}}|c\tau+d|^{-k_{Y}}\,.$
(C.6)

Thus, the coupling extracted from the numerical scan is mapped into
$\hat{\alpha}_{i}^{\prime}\equiv|c\tau+d|^{-k_{Y}}\hat{\alpha}_{i}$. By
modular invariance, the parameter sets $\\{\tau,\hat{\alpha}_{i}\\}$ and
$\\{\gamma(\tau),\hat{\alpha}_{i}^{\prime}\\}$ lead to the same observables,
as can be verified. As a common practice, the fitted parameters are the
dimensionless ratios
$\beta/\alpha,\gamma/\alpha,g^{\prime}/g,g^{\prime\prime}/g...$ etc. These are
mapped to:

$\frac{g_{i}}{g_{j}}\quad\to\quad\frac{g_{i}}{g_{j}}|c\tau+d|^{-k_{Y}^{i}+k_{Y}^{j}}\,,$
(C.7)

where $k_{Y}^{i},k_{Y}^{j}$ are the weights of the modular forms associated
with the $i$-th and $j$-th operators of the superpotential. In our model, all
the operators from the neutrino sector have the same $k_{Y}^{i,j}$, thus the
dimensionless ratios (C.7) are invariant under the modular mapping. The only
non-trivial rescaling of dimensionless parameters happens in the charged-
leptons sector where, in general, $k_{Y}^{i}\neq k_{Y}^{j}$ for $i\neq j$. The
neutrino global mass-scale $g^{2}v_{u}^{2}/\Lambda$ is mapped into
$|c\tau+d|^{-k_{Y}}g^{2}v_{u}^{2}/\Lambda$ where $k_{Y}=4$ in our case of eq.
(3.8).

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