#  Universal predictions of Siegel modular invariant theories
near the fixed points

Gui-Jun Ding1 , Ferruccio Feruglio2 and Xiang-Gan Liu3
1Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China
2INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padua, Italy
3Department of Physics and Astronomy, University of California, Irvine, CA
92697-4575 USA

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

> We analyze a general class of locally supersymmetric, CP and modular
> invariant models of lepton masses depending on two complex moduli taking
> values in the vicinity of a fixed point, where the theory enjoys a residual
> symmetry under a finite group. Like in models that depend on a single
> modulus, we find that all physical quantities exhibit a universal scaling
> with the distance from the fixed point. There is no dependence on the level
> of the construction, the weights of matter multiplets and their
> representations, with the only restriction that electroweak lepton doublets
> transform as irreducible triplets of the finite modular group. Also the form
> of the kinetic terms, which here are assumed to be neither minimal nor
> flavor blind, is irrelevant to the outcome. The result is remarkably simple
> and the whole class of examined theories gives rise to five independent
> patterns of neutrino mass matrices. Only in one of them, the predicted
> scaling agrees with the observed neutrino mass ratios and lepton mixing
> angles, exactly as in single modulus theories living close to $\tau=i$.

##  1 Introduction

The flavour puzzle is one of the big unsolved problems in particle physics.
The observed regularities in the fermion mass spectrum and the mixing angles
still escape our understanding and their consistent description requires more
than twenty free parameters. One of the tools that have been exploited so far
to tackle the puzzle relies on flavour symmetries, acting in generation space
and supposed to reduce the number of flavour parameters, including those
associated with CP violation. In a bottom-up approach, the freedom of the
model builder is however disarmingly huge. Not only we can freely choose
flavour groups and representations of matter fields but, more importantly, we
should add an ad hoc symmetry-breaking sector to reconcile the approximate
predictions of the exact symmetry limit with the accurate experimental data.
The complicated architecture of such a sector often results in a limited
predictive power of the whole approach.

Modular invariance offers in principle some advantage over traditional flavour
symmetries of the above type [1]. The starting point is the symmetry-breaking
sector itself consisting, in the simplest case, of a single complex field, the
modulus, living in the upper half of the complex plane and transforming
nontrivially under the modular group. In a supersymmetric realization [2, 3],
modular invariance severely restricts the Yukawa couplings, which are required
to be modular forms, holomorphic functions of the modulus with appropriate
transformation properties under the modular group. Radiative corrections and
supersymmetry breaking effects are negligible in a large portion of the
parameter space [4]. Invariance under the modular group offers a simple,
axion-free solution to the strong CP problem [5]. Moreover, such a framework
is intimately related to the basic properties of superstring compactifications
[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27], allowing the bottom-up and top-down approaches to reinforce each
other [28]. An intense activity in both directions has been pursued in the
last years [29, 30].

These nice features are partially spoiled by the freedom associated with the
transformation law of the matter fields under the modular group and with the
choice of kinetic terms. The former requires the specification of a level $n$
and, for each matter multiplet $\varphi^{(I)}$, a weight $k_{I}$ and a
representation $\rho_{I}$ of the adopted finite modular group. The latter
affects the fermion mass spectrum [31], except for the case of minimal or
flavor universal Kähler potential. Some progress in dealing with this problem
has been made with the introduction of eclectic flavour symmetries [32, 33,
34, 35, 36, 37, 38], but the prize is the reintroduction of a nonminimal
symmetry-breaking sector [39, 40, 41, 42]. Even considering the lepton sector
alone, by exploiting the existing freedom a large number of models correctly
reproducing neutrino masses, lepton mixing angles, and predicting leptonic CP-
violating phases have been formulated [29, 30]. The variety of different
realizations allowed in a pure bottom-up approach has not allowed so far to
identify a unique scenario.

Despite our inability to designate a single successful theory, from all the
existent effective models we might have learned something significant about
the principle underlying the fermion sector. Indeed, the leptonic models
formulated so far suggest a preference for a value of the modulus near the
self-dual fixed point $\tau_{0}=i$ [43], which preserves the symmetry under
$\tau\to-1/\tau$. Such a preference is even more pronounced for the subset of
CP-invariant models, where the violation of CP is spontaneous and entirely
controlled by the modulus $\tau$. Indeed, if electroweak lepton doublets are
assigned to irreducible triplets of the finite modular group, a typical choice
allowing to minimize the number of free parameters, the behavior of modular
invariant models of lepton masses near the fixed points $\tau_{0}=i$ and
$\tau_{0}=-1/2+i~{}\sqrt{3}/2$ is universal [44]. All physical quantities
scale with the distance $|\tau-\tau_{0}|$ in a way that is largely independent
of the level $n$, the weight $k_{I}$, the specific representations $\rho_{I}$
and even the form of kinetic terms. Only a few patterns of neutrino mass
matrices can be realized, depending on the chosen fixed point. In particular,
near $\tau_{0}=i$ almost all the successful models predict a normal ordering
of neutrino masses, with mixing angles and neutrino mass differences that are
all accommodated by $|\tau-\tau_{0}|\approx 0.1$. The key feature under this
universal behavior is the residual $\mathbbm{Z}_{4}$ symmetry enjoyed by the
theory at the fixed point, spontaneously broken by $(\tau-\tau_{0})$. In the
bottom-up approach, the modulus $\tau$ is treated as a free parameter,
optimized to maximize the agreement with the data. Remarkably, fixed points
$\tau_{0}$ are extrema of a modular invariant energy density [45, 46].
Moreover, minima of the energy density close to but distinct from the fixed
points have been established in modular invariant theories [47, 48, 49, 50,
51]. Cosmological evolution can offer a mechanism for moduli trapping near the
points enjoying an enhanced symmetry [52, 53, 54]. In general,
$(\tau-\tau_{0})$ provides a useful expansion parameter to understand the
hierarchy among charged fermion masses and mixing angles [55, 56, 57, 58, 59,
60, 61, 62, 63, 64, 65, 66].

Theories with a single modulus might offer an oversimplified description of
the fermion mass spectrum. For example, in superstring compactifications
fermion masses typically depend on several moduli. It is thus natural to ask
whether a universal behavior persists in multi-moduli theories in the vicinity
of the fixed points, where a residual symmetry under a discrete group $G_{0}$
applies. Moreover, if such universality is indeed exhibited in these richer
theories, there is the chance of finding new realistic patterns for fermion
mass matrices. An attractive generalization of the upper half of the complex
plane and the modular group $SL(2,\mathbbm{Z})$ is the Siegel upper half
plane, where the Siegel modular group $Sp(2g,\mathbbm{Z})$ operates. The
Siegel upper half plane has complex dimension $g(g+1)/2$ and coincides with
the ordinary upper half complex plane when the genus $g$ is one. Higher genera
are realized in string theory compactification [67, 68, 69, 70, 71]. Bottom-up
realizations have been formulated in ref. [72], and their spontaneous CP
breaking has been analyzed in ref. [73].

The purpose of the present paper is to study the simplest possible such
generalization, at genus $g=2$, and its property in the vicinity of the fixed
points, which have been fully classified in [74, 75, 76]. We focus on the
lepton sector, adopting completely generic level $n$, weights $k_{I}$ of the
involved multiplets and representations $\rho_{I}$, with the only assumption
that the electroweak lepton doublets $L$ are assigned to an irreducible
triplet of the relevant modular group. We also allow the kinetic terms to be
the most general ones. These features are implemented in a CP-invariant
locally supersymmetric theory. There are strong indications that the four-
dimensional CP symmetry is a gauge symmetry [77, 78, 79], even starting from a
higher-dimensional theory where CP is not conserved. It has been conjectured,
as a general property of string theory, that CP is indeed a gauge symmetry of
the four-dimensional theory. In this context, CP can only be violated
spontaneously, by complex expectation values of fields, in our case the
moduli. The moduli will be restricted to a suitable region of the Siegel upper
half plane, invariant under a subgroup of $Sp(4,\mathbbm{Z})$ hosting three-
dimensional irreducible representations of the finite modular group. The
widest such a region has complex dimension two and provides a nontrivial
extension of the framework studied in ref. [43, 44].

In Section 2 we will review the Siegel modular group $Sp(4,\mathbbm{Z})$ and
the two-dimensional invariant regions of the Siegel upper half plane. We
identify the region that allows to assign lepton doublets to irreducible
triplets of the relevant finite modular group and we analyze the fixed points
belonging to this region. In Section 3 we define our CP-invariant locally
supersymmetric theory and we describe the requirements for modular invariance.
At the fixed points the theory enjoys a residual symmetry under a finite group
$G_{0}$. An important step is provided by a field redefinition that,
linearizing the action of $G_{0}$, considerably simplifies our task. In
Section 4, for each fixed point we identify the group $G_{0}$ and we provide
the decomposition of all irreducible triplets of the finite modular group
under $G_{0}$. This information is sufficient to find the pattern of neutrino
mass matrices, in the basis where kinetic terms are canonical and charged
lepton mass matrices diagonal, expressed as a series expansion in powers of
$(\tau-\tau_{0})$. Details on the group theory of the residual symmetry
$G_{0}$ can be found in the Appendices.

The result is remarkably simple. Apart from the unrealistic case of a neutrino
mass matrix vanishing to all orders of the $(\tau-\tau_{0})$ expansion, only
five patterns are found. Four of them coincide with those arising in
$SL(2,\mathbbm{Z})$-invariant single modulus theories in the vicinity of the
fixed points $\tau_{0}=i$ and $\tau_{0}=-1/2+i~{}\sqrt{3}/2$. In particular,
one of these four patterns is especially effective in accommodating the
existing data, with no required tuning of the free parameters. The last
pattern predicts neutrino masses of the same order of magnitude and mixing
angles of approximately the same size and does not provide any explanation for
the smallness of $\Delta m^{2}_{sol}/\Delta m^{2}_{atm}$ and
$\sin^{2}\theta_{13}$. It is intriguing that, not only the universal behavior
of the theory near the fixed points is confirmed, but essentially no realistic
patterns of neutrino mass matrices different from those found in the single
modulus case are exhibited in this class of multi-moduli theories.

##  2 Siegel modular group

One of the most natural generalization of theories invariant under the modular
group $SL(2,\mathbbm{Z})$, where masses and mixing angles depend on a single
complex modulus $\tau$, is the class of theories invariant under the Siegel
(or symplectic) modular group $Sp(4,\mathbbm{Z})$ [72], where moduli are
described by a two-by-two complex symmetric matrix $\tau$ belonging to the
Siegel upper-half plane ${\cal H}_{2}$:

$\tau=\left(\begin{array}[]{cc}\tau_{1}&\tau_{3}\\\
\tau_{3}&\tau_{2}\end{array}\right)\,,~{}~{}~{}~{}~{}\det({\tt
Im}(\tau))>0\,,~{}~{}~{}~{}~{}{\rm Tr}({\tt Im}(\tau))>0\,.$ (1)

Assuming CP and ${\cal N}=1$ supersymmetric invariance, and neglecting gauge
interactions, these theories involve a set of chiral supermultiplets,
$\Phi=(\tau,\varphi^{(I)})$, where $\tau$ is dimensionless and gauge-
invariant. The Siegel modular group $Sp(4,\mathbbm{Z})$, adopted as flavour
symmetry, acts on $\Phi=(\tau,\varphi^{(I)})$ as [72]:

$\left\\{\begin{array}[]{l}\tau\to\gamma\tau=(A\tau+B)(C\tau+D)^{-1}\,,\\\\[5.69046pt]
\varphi^{(I)}\to[\det(C\tau+D)]^{-k_{I}}\rho_{I}(\gamma)\varphi^{(I)}~{}\,,\end{array}\right.~{}~{}~{}~{}~{}~{}~{}\gamma=\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\in Sp(4,\mathbbm{Z})\,.$ (2)

with suitable conditions on the submatrices $A$, $B$, $C$ and $D$. The weight
$k_{I}$ is assumed to be an integer and $\rho_{I}(\gamma)$ denotes a unitary
representation of a finite copy $\Gamma_{n}$ of $Sp(4,\mathbbm{Z})$ 111For a
generic genus $g$, the Siegel modular group is $Sp(2g,\mathbbm{Z})$ and the
finite modular groups are denoted by $\Gamma_{g,n}$. Here we use the concise
notation $\Gamma_{n}=\Gamma_{2,n}$..

Such a finite copy, known as finite Siegel modular group, is defined as the
quotient group $\Gamma_{n}=Sp(4,\mathbbm{Z})/\Gamma(n)$ where $\Gamma(n)$ is
the principal congruence subgroup $\Gamma(n)$ of level $n$:

$\Gamma(n)=\Big{\\{}\gamma\in
Sp(4,\mathbbm{Z})~{}\Big{|}~{}\gamma\equiv\mathbbm{1}\,\texttt{mod}\,n\Big{\\}}\,,$
(3)

$n$ being a generic positive integer. The finite Siegel modular group has
finite order [80, 81]:

$\displaystyle|\Gamma_{n}|=n^{10}\prod_{p|n}(1-\dfrac{1}{p^{2}})(1-\dfrac{1}{p^{4}})\,,$
(4)

where the product is over the prime divisors $p$ of $n$. $|\Gamma_{n}|$
rapidly grows with $n$: for example $|\Gamma_{2}|=720$, $|\Gamma_{3}|=51840$.
The group $\Gamma_{2}$ is isomorphic to $S_{6}$, and $\Gamma_{3}$ is
$Sp(4,F_{3})$, the double covering of Burkhardt group. These finite groups do
not possess three-dimensional irreducible representations suitable to
accommodate three fermion generations. The smallest irreducible representation
with degree bigger than one is 5 for $\Gamma_{2}$ and 4 for $\Gamma_{3}$. If
$n$ is a power of an odd prime, the next lowest dimensional (complex,
irreducible) representations of $\Gamma_{n}$, after the trivial
representation, are two of dimension $(n^{2}-1)/2$ and two of dimension
$(n^{2}+1)/2$ [82]. In this series we do not find any three-dimensional
irreducible representation. It is quite possible that also when $n$ is a power
of an even prime, the smallest nontrivial irreducible representation of
$\Gamma_{n}$ are quite large. Though we do not have a formal proof, we suspect
that none of the finite Siegel modular groups $\Gamma_{n}$ possesses three-
dimensional irreducible representations. The requirement of three-dimensional
irreducible representations is not mandatory, but is a very convenient one
since it reduces the number of independent parameters needed to describe the
fermion mass spectrum of the theory.

###  2.1 Invariant regions

In ref. [72] we have shown how to overcome this difficulty, by restricting
$\tau$ to a convenient region $\Sigma$ of the Siegel upper-half plane ${\cal
H}_{2}$, invariant under a subgroup of $Sp(4,\mathbbm{Z})$. Individual points
in $\Sigma$ are left invariant by a common subgroup $H$ of
$Sp(4,\mathbbm{Z})$, the stabilizer

$H~{}\tau=\tau\,.$ (5)

The region $\Sigma$, as a whole, is left invariant by the normalizer

$N(H)=\left\\{\gamma\in Sp(4,\mathbbm{Z})|\gamma^{-1}H\gamma=H\right\\}\,.$
(6)

In general, $H$ is a proper subgroup of $N(H)$. If $\Sigma$ consists of a
single point, the stabilizer and the normalizer coincide. In our theory, we
can consistently restrict the domain of moduli to a region $\Sigma$ of this
type, and replace $Sp(4,\mathbbm{Z})$ with $N(H)$ and $\Gamma_{n}$ with
$N_{n}(H)$, where the integer $n$ is the level of the representation and
$N_{n}(H)$ is a finite copy of the normalizer, obtained through the same steps
leading to the Siegel finite modular groups $\Gamma_{n}$. The group $N(H)$
acts on $\Phi=(\tau,\varphi^{(I)})$ as in eq. (2), where $\gamma$ belongs to
$N(H)$ and $\rho_{I}(\gamma)$ is a unitary representation of $N_{n}(H)$. Since
in general $N_{n}(H)$ is smaller than $\Gamma_{n}$, there is a chance that it
possesses three-dimensional irreducible representations.

Finally, up to Sigel modular transformations, CP transformations on the chiral
multiplets read (we use a bar to denote conjugation of fields) [32, 83, 84,
69, 70, 73]:

$\displaystyle\tau\xrightarrow{\mathcal{CP}}-\bar{\tau}\,,~{}~{}~{}~{}~{}~{}\varphi^{(I)}\xrightarrow{\mathcal{CP}}X_{I}\;\bar{\varphi}^{(I)}\,,$
(7)

where the CP transformation matrix $X_{I}$ is fixed by the following
consistency conditions [73]

$X_{I}\,\rho^{*}_{I}(S)X^{-1}_{I}=\rho_{I}(S^{-1}),~{}~{}~{}~{}X_{I}\,\rho^{*}_{I}(T_{i})X^{-1}_{I}=\rho_{I}(T_{i}^{-1})\,,$
(8)

up to an overall irrelevant phase. Here $S$ and $T_{i}\;(i=1,2,3)$ denote the
generators of $Sp(4,\mathbbm{Z})$:

$S=\begin{pmatrix}0&\mathbbm{1}_{2}\\\
-\mathbbm{1}_{2}&0\end{pmatrix}\,,~{}~{}~{}~{}T_{i}=\begin{pmatrix}\mathbbm{1}_{2}&B_{i}\\\
0&\mathbbm{1}_{2}\end{pmatrix}$ (9)

with

$B_{1}=\left(\begin{array}[]{cc}1&0\\\ 0&0\\\
\end{array}\right)\,,~{}~{}~{}B_{2}=\left(\begin{array}[]{cc}0&0\\\ 0&1\\\
\end{array}\right)\,,~{}~{}~{}B_{3}=\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right)\,.$ (10)

In the basis where the unitary representation matrices $\rho_{I}(S)$ and
$\rho_{I}(T_{i})$ are symmetric, the consistency conditions of eq. (8) is
solved by $X_{I}=\mathbbm{1}$ which is the canonical CP transformation. In our
analysis, it will be convenient to make use of generalized CP transformations,
which combine eq. (7) with a modular transformation. They will be discussed in
Section 3.1.

###  2.2 Two-dimensional invariant regions

Having to abandon the full Siegel upper-half plane ${\cal H}_{2}$, the widest
regions $\Sigma$ have complex dimension two. In the Siegel upper-half plane
${\cal H}_{2}$, there are two such regions, left invariant by the action of a
subgroup $N(H)$ of $Sp(4,\mathbbm{Z})$ [74, 75, 76].

Invariant region $\tau$ | Stabilizer $H$ | $\begin{array}[]{l}{\rm generators~{}of~{}the}\\\ {\rm Normalizer}~{}N(H)\end{array}$
---|---|---
$\Sigma_{1}=\begin{pmatrix}\tau_{1}&0\\\ 0&\tau_{2}\end{pmatrix}$ | $Z_{2}\times Z_{2}$ | eq. (13)
$\Sigma_{2}=\begin{pmatrix}\tau_{1}&\tau_{3}\\\ \tau_{3}&\tau_{1}\end{pmatrix}$ | $Z_{2}\times Z_{2}$ | eq. (18)
Table 1:  Invariant regions $\Sigma_{1,2}$ of complex dimension two in the
Siegel upper half plane $\mathcal{H}_{2}$. The generators of the corresponding
normalizers are shown in eqs. (13) and (18).

####  2.2.1 $\Sigma_{1}$

One of them is

$\Sigma_{1}=\left\\{\begin{pmatrix}\tau_{1}&0\\\
0&\tau_{2}\end{pmatrix}\in{\cal H}_{2}\right\\}\,.$ (11)

The stabilizer $H$ is the $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ group whose
generators are:

(12)

The normalizer $N(H)$ is generated by the elements:

$\displaystyle G_{1}=\begin{pmatrix}1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\
0&-1&0&0\end{pmatrix}\,,~{}~{}~{}~{}G_{2}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&1\\\
0&0&1&0\\\
0&0&0&1\end{pmatrix}\,,~{}~{}~{}~{}G^{\prime}_{1}=\begin{pmatrix}0&0&1&0\\\
0&1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\end{pmatrix},$ $\displaystyle
G^{\prime}_{2}=\begin{pmatrix}1&0&1&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{pmatrix}\,,~{}~{}~{}~{}G_{3}=\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\
0&0&0&1\\\ 0&0&1&0\end{pmatrix}\,.$ (13)

The action of the group generators on the moduli $\tau_{1}$ and $\tau_{2}$ is

$\displaystyle
G_{1}:\tau_{1}\rightarrow\tau_{1},\qquad\tau_{2}\rightarrow-\frac{1}{\tau_{2}}\,,$
$\displaystyle
G_{2}:\tau_{1}\rightarrow\tau_{1},\qquad\tau_{2}\rightarrow\tau_{2}+1\,,$
$\displaystyle
G^{\prime}_{1}:\tau_{1}\rightarrow-\frac{1}{\tau_{1}},\qquad\tau_{2}\rightarrow\tau_{2}\,,$
$\displaystyle
G^{\prime}_{2}:\tau_{1}\rightarrow\tau_{1}+1,\qquad\tau_{2}\rightarrow\tau_{2}\,,$
$\displaystyle
G_{3}:\tau_{1}\rightarrow\tau_{2}\qquad\tau_{2}\rightarrow\tau_{1}.$ (14)

The generators $G_{1,2}$, $G^{\prime}_{1,2}$ and $G_{3}$ obey the following
relations

$\displaystyle
G^{4}_{1}=(G_{1}G_{2})^{3}=1,~{}~{}~{}G^{2}_{1}G_{2}=G_{2}G^{2}_{1},~{}~{}~{}G^{\prime
4}_{1}=(G^{\prime}_{1}G^{\prime}_{2})^{3}=1,~{}~{}~{}G^{\prime
2}_{1}G^{\prime}_{2}=G^{\prime}_{2}G^{\prime 2}_{1}\,,$ $\displaystyle
G_{1}G^{\prime}_{1}=G^{\prime}_{1}G_{1},~{}G_{1}G^{\prime}_{2}=G^{\prime}_{2}G_{1},~{}G_{2}G^{\prime}_{1}=G^{\prime}_{1}G_{2},~{}G_{2}G^{\prime}_{2}=G^{\prime}_{2}G_{2}\,,$
$\displaystyle
G^{2}_{3}=1,~{}~{}~{}G_{3}G_{1}=G^{\prime}_{1}G_{3},~{}~{}~{}G_{3}G_{2}=G^{\prime}_{2}G_{3}\,.$
(15)

Therefore the normalizer $N(H)$ is isomorphic to $\left(SL(2,\mathbb{Z})\times
SL(2,\mathbb{Z})\right)\rtimes(\mathbbm{Z}_{2})_{M}$, where the last factor,
generated by $G_{3}$ that exchanges $\tau_{1}$ and $\tau_{2}$, represents the
so-called mirror symmetry in the string theory context. We find that the
normalizer $N_{n}(H)$ corresponding to region $\Sigma_{1}$ has no three-
dimensional irreducible representations. The triplet representations of the
group $SL(2,\mathbb{Z}_{n})\times SL(2,\mathbb{Z}_{n})$ can be obtained from
the direct product of $SL(2,\mathbb{Z}_{n})$ singlets with the irreducible
triplets of another $SL(2,\mathbb{Z}_{n})$. The two triplet representations
$SL(2,\mathbb{Z}_{n})\times SL(2,\mathbb{Z}_{n})$ related by the mirror
symmetry $(\mathbbm{Z}_{2})_{M}$ would form a six dimensional representation
of the normalizer $N_{n}(H)$. Hence $N_{n}(H)$ has no three-dimensional
irreducible representations. We tested this argument by varying $n$ from 2 to
18, and indeed we found no three-dimensional irreducible representations for
$N_{n}(H)$, but only irreducible representations of one, two, four, six,
eight, nine, and higher dimension. Thus the region $\Sigma_{1}$ is only
suitable to describe electroweak lepton doublets transforming in a reducible
representation of the finite normalizer. We do not consider such a case here
and we proceed by examining the region $\Sigma_{2}$.

####  2.2.2 $\Sigma_{2}$

The other two-dimensional region is

$\Sigma_{2}=\left\\{\begin{pmatrix}\tau_{1}&\tau_{3}\\\
\tau_{3}&\tau_{1}\end{pmatrix}\in{\cal H}_{2}\right\\}\,.$ (16)

The stabilizer $H$ is the $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ group
generated by:

(17)

The normalizer $N(H)$ is generated by:

$\displaystyle G_{1}=\begin{pmatrix}1&0&1&0\\\ 0&1&0&1\\\ 0&0&1&0\\\
0&0&0&1\end{pmatrix},~{}~{}~{}~{}~{}G_{2}=\begin{pmatrix}1&0&0&1\\\ 0&1&1&0\\\
0&0&1&0\\\ 0&0&0&1\end{pmatrix},$ $\displaystyle
G_{3}=\begin{pmatrix}0&0&1&0\\\ 0&0&0&1\\\ -1&0&0&0\\\
0&-1&0&0\end{pmatrix},~{}~{}~{}~{}~{}G_{4}=\begin{pmatrix}1&0&0&0\\\
0&-1&0&0\\\ 0&0&1&0\\\ 0&0&0&-1\end{pmatrix}\,.$ (18)

The action of the group generators on the moduli $\tau_{1}$ and $\tau_{2}$ is

$\displaystyle
G_{1}:\tau_{1}\rightarrow\tau_{1}+1,\qquad\tau_{3}\rightarrow\tau_{3}\,,$
$\displaystyle
G_{2}:\tau_{1}\rightarrow\tau_{1},\qquad\tau_{3}\rightarrow\tau_{3}+1\,,$
$\displaystyle
G_{3}:\tau_{1}\rightarrow\frac{\tau_{1}}{\tau^{2}_{3}-\tau^{2}_{1}},\qquad\tau_{3}\rightarrow-\frac{\tau_{3}}{\tau^{2}_{3}-\tau^{2}_{1}}\,,$
$\displaystyle
G_{4}:\tau_{1}\rightarrow\tau_{1},\qquad\tau_{3}\rightarrow-\tau_{3}\,.$ (19)

The generators $G_{1}$, $G_{2}$, $G_{3}$ and $G_{4}$ fulfill the following
relations

$\displaystyle
G^{2}_{3}=R,~{}~{}~{}R^{2}=G^{2}_{4}=(G_{1}G_{3})^{3}=(G_{2}G_{3})^{6}=(G_{2}G_{4})^{2}=1\,,$
$\displaystyle
G_{2}G_{3}G^{2}_{2}G_{3}G_{2}=G^{3}_{3}G^{-2}_{2}G_{3},~{}~{}~{}G_{1}G_{2}=G_{2}G_{1},~{}~{}~{}G_{1}G_{4}=G_{4}G_{1},~{}~{}~{}G_{3}G_{4}=G_{4}G_{3}\,,$
$\displaystyle
G_{1}R=RG_{1},~{}~{}~{}G_{2}R=RG_{2},~{}~{}~{}G_{3}R=RG_{3},~{}~{}~{}G_{4}R=RG_{4}\,,$
(20)

where $R=-\mathbbm{1}$. In table 2, for $n$ small and equal to a power of two
or an odd prime, we show the number of irreducible triplets of $N_{n}(H)$. We
see that when the level $n$ is a prime (or a power of primes) different from
two, there are no three-dimensional irreducible representations, at least for
the first few values of $n$. When $n$ is a power of two, there are several
three-dimensional irreducible representations and their number grows with $n$,
until $n$ is equal to 8. We find that for $n=2,4,8$ there are $4,24,56$ three-
dimensional irreducible representations, and for $n=16$ there are still only
56 three-dimensional irreducible representations. Moreover, there are 24
independent singlet representations in all $N_{n}(H)$.

level $n$ | order | n. of conjugacy classes | n. of irreducible triplets
---|---|---|---
2 | 48 | 10 | 4
4 | 6144 |  109 |  24
8 | 393216 |  938 |  56
16 | 25165824 |  5572 |  56
32 | 1610612736 |  ? |  ?
… | … | … | …
3 | 1152 | 35 | 0
5 | 28800 | 54 | 0
7 | 225792 | 77 | 0
11 | 3484800 | 135 | 0
13 | 9539712 | 170 | 0
… | … | … | …
Table 2: Orders, number of conjugacy classes and of irreducible triplets of
the finite normalizers $N_{n}(H)$ in the region $\Sigma_{2}$. The results have
been obtained with the software GAP.

We conjecture that there are no new three-dimensional irreducible
representations for $n>8$. Moreover, the 56 three-dimensional irreducible
representations at level 8 include those al the lower levels $n<8$. If our
conjecture is correct, all irreducible triplets of $N_{n}(H)$ can be obtained
from the product of the 24 singlets and 56 triplets of $\Gamma_{2,4,8}$. Out
of the resulting 1344 representations, we find that the inequivalent ones are
168. In the rest of this paper, we assume that these 168 representations
exhaust the number of inequivalent irreducible triplets.

###  2.3 Fixed points

We analyze a class of theories, where the matrix $\tau$ is restricted to
$\Sigma_{2}$ and the flavor group is the corresponding normalizer $N(H)$
defined in eq. (18). In a generic point of $\Sigma_{2}$, the flavour symmetry
is broken down to the stabilizer $H$, eq. (17). An exception is provided by
the fixed points $\tau_{0}$ of $\Sigma_{2}$, where the residual symmetry group
$G_{0}$, a subgroup of $N(H)$, is bigger than $H$. The inequivalent fixed
points of $Sp(4,\mathbbm{Z})$ in the upper half-plane are displayed in table
3. Fixed points 2 and 5 belong to $\Sigma_{2}$. Fixed points 3 and 4 belong to
both $\Sigma_{1}$ and $\Sigma_{2}$. Here we will only consider the latter
option. The fixed point 6 belongs to the region $\Sigma_{1}$ and will not be
discussed. Finally, the fixed point 1, where CP is preserved, does not belong
to either $\Sigma_{1}$ or $\Sigma_{2}$. We have also checked that it is not
related by any $Sp(4,\mathbbm{Z})$ transformation to one point in either
$\Sigma_{1}$ or $\Sigma_{2}$. This point will be dismissed in our analysis,
too.

| Fixed points $\tau$ | Residual symmetry in $Sp(4,\mathbbm{Z})$ |  $G_{0}=$ Residual symmetry in $N(H)$ | CP
---|---|---|---|---
1. | $\begin{pmatrix}\zeta&\zeta+\zeta^{-2}\\\ \zeta+\zeta^{-2}&-\zeta^{-1}\end{pmatrix}$ | $Z_{10}$ | — | $+$
2. | $\begin{pmatrix}\eta&\frac{1}{2}(\eta-1)\\\ \frac{1}{2}(\eta-1)&\eta\end{pmatrix}$ | $GL(2,3)$ |  $D_{4}$ $(\tau\in\Sigma_{2}$) | $+$
3. | $\begin{pmatrix}i&0\\\ 0&i\end{pmatrix}$ | $(Z_{4}\times Z_{4})\rtimes Z_{2}$ | $\left\\{\begin{array}[]{cc}(Z_{4}\times Z_{4})\rtimes Z_{2}&(\tau\in\Sigma_{1})\\\ D_{4}\circ Z_{4}&(\tau\in\Sigma_{2})\end{array}\right.$ | $+$
4. | $\begin{pmatrix}\omega&0\\\ 0&\omega\end{pmatrix}$ | $[72,30]\cong(Z_{6}\times Z_{6})\rtimes Z_{2}$ | $\left\\{\begin{array}[]{cc}(Z_{6}\times Z_{6})\rtimes Z_{2}&(\tau\in\Sigma_{1})\\\ D_{4}\times Z_{3}&(\tau\in\Sigma_{2})\end{array}\right.$ | $+$
5. | $\dfrac{i\sqrt{3}}{3}\begin{pmatrix}2&1\\\ 1&2\end{pmatrix}$ | $(Z_{6}\times Z_{2})\rtimes Z_{2}$ |  $D_{4}$ $(\tau\in\Sigma_{2}$) | $+$
6. | $\begin{pmatrix}\omega&0\\\ 0&i\end{pmatrix}$ | $Z_{12}\times Z_{2}$ |  $Z_{12}\times Z_{2}$ $(\tau\in\Sigma_{1})$ | $+$

Table 3:  Inequivalent fixed points of $Sp(4,\mathbb{Z})$ in the Siegel upper
half plane $\mathcal{H}_{2}$. We set $\zeta=e^{2\pi
i/5},~{}\eta=\frac{1}{3}(1+i2\sqrt{2}),~{}\omega=-1/2+i\sqrt{3}/2$. All other
fixed points are related to one of these through a $Sp(4,\mathbbm{Z})$
transformation. The group with GAP id [72,30] is isomorphic to $(Z_{6}\times
Z_{6})\rtimes Z_{2}$.

In modular invariant theories depending on a single modulus $\tau$ where
lepton doublets are assigned to irreducible triplets of the finite modular
group, it has been shown that in the vicinity of the fixed points the
predictions are universal: they are independent of the finite modular group
acting on the lepton multiplets, the weights of the matter multiplets and even
the form of the kinetic terms. We are led to study the behavior of Siegel
modular invariant theories when the matrix $\tau$, conveniently restricted to
$\Sigma_{2}$, falls in the vicinity of one of fixed points 2, 3, 4 and 5.

##  3 Modular invariant models of lepton masses

In the rest of this paper, we analyze CP-invariant and ${\cal N}=1$ locally
supersymmetric models of lepton masses and mixing angles. Neglecting gauge
interactions that are not of interest in our analysis, they are described by
an action ${\mathscr{S}}$ fully specified by a gauge-invariant real function
${\cal G}$

${\cal G}=\frac{K}{M_{Pl}^{2}}+\log\left|\frac{w}{M_{Pl}^{3}}\right|^{2},$
(21)

where restriction to the scalar components of the supermultiplets is
understood, and $M_{Pl}$ is the Planck mass. Here $K$ is the Kähler potential,
a real gauge-invariant function of the chiral multiplets and their conjugate
of dimensionality (mass)2, and $w$ is the superpotential, an analytic gauge-
invariant function of the chiral multiplets of dimensionality (mass)3. We do
not discuss supersymmetry breaking effects, which in theories with a single
modulus are negligible in a large portion of the parameter space [4]. The
chiral superfield content includes the matrix $\tau$ of eq. (1), restricted to
the region $\Sigma_{2}$, and a set of matter fields $\varphi^{(I)}$. Siegel
modular invariance requires ${\mathscr{S}}$ to remain unchanged under the
transformations of eq. (2), where $\gamma$ is restricted to the normalizer
$N(H)$ of $\Sigma_{2}$. The transformation law of $\varphi^{(I)}$ is specified
by a unitary representation $\rho_{I}(\gamma)$ of the finite modular group
$N_{n}(H)$ and by the integer weight $k_{I}$ (irreducible components of
$\varphi^{(I)}$ admitting independent weights). Invariance under CP requires
${\mathscr{S}}$ to remain unmodified under the transformations of eq. (7).

Notice that for the action ${\mathscr{S}}$ to be invariant under the
transformations of eq. (2), the Kähler potential $K$ and the superpotential
$w$ do not need to be separately invariant. If modular transformations of eq.
(2), restricted to the scalar components $z$ of the supermultiplets
$(\tau,\varphi^{(I)})$, induce a Kähler transformation:

$\displaystyle K\xrightarrow{\gamma}$
$\displaystyle~{}K+M_{Pl}^{2}F_{\gamma}(z)+M_{Pl}^{2}\bar{F}_{\gamma}(\bar{z})\,,$
$\displaystyle w\xrightarrow{\gamma}$ $\displaystyle~{}e^{\displaystyle-
F_{\gamma}(z)}w\,,$ (22)

${\cal G}$ does not change and the theory is invariant, provided the fermionic
partners $\psi$ of the scalar multiplets $z$ undergo an extra chiral rotation
of the type 222Gauginos are required to transform with an opposite phase.:

$\psi\xrightarrow{\gamma}e^{\frac{F_{\gamma}(z)-\bar{F}_{\gamma}(z)}{4}}\psi\,.$
(23)

If $F_{\gamma}(z)$ has a nontrivial field dependence, eq. (23) represents a
local chiral rotation on the fermion components and anomaly cancellation is
required to guarantee invariance. Thus, up to potential anomalies requiring a
cancellation mechanism, a Kähler transformation is always a symmetry of the
theory. As a particular case, we can consider $F=-2i\alpha_{\gamma}$, with
$\alpha_{\gamma}$ a field-independent real constant and eq. (3) becomes

$\displaystyle
K\xrightarrow{\gamma}~{}K\,,~{}~{}~{}~{}~{}~{}w\xrightarrow{\gamma}~{}e^{\displaystyle
2i\alpha_{\gamma}}w\,.$ (24)

For example, consider the choice:

$K=-3M_{Pl}^{2}\log(-i\tau+i\bar{\tau})\,,~{}~{}~{}~{}~{}~{}w=c\frac{M_{Pl}^{3}}{\eta(\tau)^{6}}\,,$
(25)

where $\eta(\tau)$ is the Dedekind eta function and $c$ a constant. When
performing the transformation $T:\tau\to\tau+1$, we reproduce eq. (24) with
$\alpha_{T}=-\pi/4$, thanks to the property

$\eta(\tau+1)=e^{\displaystyle i\frac{\pi}{12}}~{}\eta(\tau).$ (26)

This theory is invariant under $T$ 333Indeed, it is invariant under the full
$SL(2,\mathbbm{Z})$., despite the presence of nontrivial phases in the
transformation of the superpotential. We conclude that in the local case, the
covariance of $w$ is only required up to a phase $2\alpha_{\gamma}$. Since
this phase depends on the group element $\gamma$, consistency with the group
law requires 444We can replace this condition with a less restrictive one,
where $v(\gamma)=e^{\displaystyle 2i\alpha_{\gamma}}$ is a multiplier system
obeying:
$v(\gamma_{1})v(\gamma_{2})j(\gamma_{1},\gamma_{2}\tau)^{-k_{w}}j(\gamma_{2},\tau)^{-k_{w}}=v(\gamma_{1}\gamma_{2})j(\gamma_{1}\gamma_{2},\tau)^{-k_{w}}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}j(\gamma,\tau)=\det(C\tau+D),$
(27) where $k_{w}$ is the weight of $w$. When $k$ is an integer, this
condition becomes equivalent to the one in eq. (28).

$\alpha_{\gamma_{1}}+\alpha_{\gamma_{2}}=\alpha_{\gamma_{1}\gamma_{2}}\,.$
(28)

Here we will make use of the possibility in eq. (24), which is rarely
exploited in the bottom-up approach. Working in the context of local
supersymmetry, we consider the general case where $K$ and $w$ satisfy

$\displaystyle K\xrightarrow{\gamma}$
$\displaystyle~{}K+k_{w}M^{2}_{Pl}\log\det(C\tau+D)+M^{2}_{Pl}k_{w}\log\det(C\bar{\tau}+D)\,,$
$\displaystyle w\xrightarrow{\gamma}$
$\displaystyle~{}\det(C\tau+D)^{-k_{w}}\bm{r}_{s}(\gamma)w,$ (29)

where $\bm{r}_{s}$ is a singlet representation of $N_{n}(H)$, which can differ
from the trivial one by a phase factor. A suitable mechanism of anomaly
cancellation, not affecting the analysis of the light fermion masses, is
understood in the present analysis.

As a side remark, we stress that, in general, requiring $w$ to transform as a
nontrivial singlet $\bm{r}_{s}$ is not possible in rigid supersymmetry, where
the theory depends separately on $K$ and $w$. Indeed, if $w$ acquires a phase,
the invariance of the theory can be guaranteed by an $R$-symmetry. The
transformation in eq. (24) should read:

$w(\theta)\xrightarrow{\gamma}e^{\displaystyle
2i\alpha_{\gamma}}w(e^{\displaystyle-i\alpha_{\gamma}}\theta),$ (30)

which can be absorbed by a redefinition of the Grassmann measure

$\displaystyle\theta\xrightarrow{\gamma}$
$\displaystyle~{}e^{\displaystyle+i\alpha_{\gamma}}~{}\theta\,,$
$\displaystyle d^{2}\theta\xrightarrow{\gamma}$
$\displaystyle~{}e^{\displaystyle-2i\alpha_{\gamma}}~{}d^{2}\theta\,.$ (31)

When $w$ is a polynomial, the required $R$-invariance can be achieved if the
$R$-charges of the chiral multiplets add up to $+2$ in each term of $w$. In a
modular invariant theory, $w$ is not necessarily polynomial since one of these
fields is typically a modular form $Y(\tau)$. Due to the non-homogeneous
dependence of $Y(\tau)$ on $\tau$, if we assign a nonvanishing $R$-charge to
$\tau$, $Y(\tau)$ will not possess a definite $R$-charge, in general. Thus,
the only possible $R$-charge assignment of both $\tau$ and $Y(\tau)$ is $R=0$.
But this amounts to saying that the scalar component of $\tau$ does not
transform under the considered symmetry. This is what happens when dealing
with a traditional flavour symmetry. Specific combinations of modular
transformations behave as traditional flavour symmetries, such as $S^{2}$ in
$SL(2,\mathbbm{Z})$-invariant theories or transformations belonging to the
center of the metaplectic modular group, arising in the context of fractional
weight modular forms [85, 86, 87, 88]. In the following, we will not consider
these special cases, and we focus on nontrivial transformations of the moduli
$\tau$.

We will exploit the above framework to describe masses, mixing angles, and
phases in the lepton sector. The matter multiplets $\varphi^{(I)}$ include two
Higgs doublets $H_{u,d}$, electroweak doublets $L$, singlets $E^{c}$ and, when
neutrino masses arise from the seesaw mechanism, heavy singlets $N^{c}$.
Assuming Majorana neutrino masses, the low-energy superpotential $w$ of the
lepton sector reads:

$\displaystyle w=-\displaystyle\frac{1}{2\Lambda_{L}}(H_{u}L)^{T}{\cal
Y}(\tau)(H_{u}L)-E^{cT}{\cal Y}_{e}(\tau)(H_{d}L),$ (32)

where the first term can arise from the seesaw mechanism when heavy singlets
$N^{c}$ are integrated out. To minimize the number of free parameters, $L$
will be assigned to an irreducible triplet of the finite modular group
$N_{n}(H)$. In the class of theories described by the action ${\mathscr{S}}$,
the lepton mass matrices are obtained by combining the holomorphic
contribution arising from the superpotential $w$ with the non-holomorphic data
coming from the Kähler potential. In general, there is a large freedom related
to both the allowed transformation properties of the matter fields (level $n$,
weight $k_{I}$ and representation $\rho_{I}$) and the inefficiency of modular
invariance in constraining the Kähler potential. Moreover, the holomorphic
contribution is expressed in terms of Siegel modular forms, whose knowledge
for generic levels and weights $(n,k)$ is limited. These obstacles can be
largely overcome if $\tau$ falls in the vicinity of a fixed point $\tau_{0}$.
In this case, even giving up the full power of modular invariance, a
considerable amount of information about lepton masses and mixing angles
survives from the approximate invariance under CP and the stability group
$G_{0}$.

###  3.1 A field redefinition

At the fixed points $\tau_{0}$, both CP and the stability group $G_{0}$ are
preserved. To establish CP conservation, it might be convenient to use a
nonstandard definition of CP transformation
$g\mathcal{CP}\equiv\gamma^{-1}\circ\mathcal{CP}$ on $\tau$ and the matter
fields:

$\displaystyle\tau\xrightarrow{g\mathcal{CP}}$
$\displaystyle~{}\tau_{CP}=\gamma^{-1}(-\bar{\tau})\,,~{}~{}~{}~{}~{}~{}~{}~{}\gamma=\begin{pmatrix}A_{0}&B_{0}\\\
C_{0}&D_{0}\end{pmatrix}\,,$
$\displaystyle\varphi^{(I)}\xrightarrow{g\mathcal{CP}}$
$\displaystyle~{}\det(C^{t}_{0}\bar{\tau}+A_{0}^{t})^{-k_{I}}\rho_{I}(\gamma^{-1})X_{I}\;\bar{\varphi}^{(I)}~{}\,,$
(33)

where $\gamma$ is an element of $Sp(4,\mathbbm{Z})$ fulfilling
$\gamma\tau_{0}=-\bar{\tau}_{0}$, so that the fixed point $\tau_{0}$ is
invariant under $g\mathcal{CP}$. In general both CP and the stability group
$G_{0}$ are nonlinearly realized on $(\tau,\varphi^{(I)})$ and it is
preferable to move to a field basis $(u,\Phi^{(I)})$ where CP and $G_{0}$ act
linearly. We will choose a basis where $u(\tau_{0})=0$, so that the group
$G_{0}$ and CP are unbroken at the origin of the field space, $u=0$. This
allows us to adopt $u$ as an order parameter for the breaking of $G_{0}$ and
CP. The new basis can be defined by the transformation:

$\displaystyle u=$
$\displaystyle~{}e^{\displaystyle-i\alpha}(\tau-\tau_{0})(\tau-\bar{\tau}_{0})^{-1}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}\tau=(1-e^{\displaystyle
i\alpha}u)^{-1}(\tau_{0}-e^{\displaystyle i\alpha}u\bar{\tau}_{0})\,,$
$\displaystyle\Phi^{(I)}=$ $\displaystyle~{}[\det(1-e^{\displaystyle
i\alpha}u)]^{k_{I}}\varphi^{(I)}\,,$ (34)

where the phase $\alpha$ can be conveniently adjusted to simplify the action
of CP. Denoting by

$h_{i}=\left(\begin{array}[]{cc}A_{i}&B_{i}\\\ C_{i}&D_{i}\end{array}\right)$
(35)

the generators of the stability group $G_{0}$, $h_{i}\tau_{0}=\tau_{0}$,we
find 555We used the identities
$\det(C_{i}\tau+D_{i})=\det(C_{i}\tau_{0}+D_{i})\frac{\det(1-e^{i\alpha}h_{i}u)}{\det(1-e^{i\alpha}u)}$
and $A_{i}-\tau_{0}C_{i}=(\tau_{0}C_{i}^{t}+D_{i}^{t})^{-1}$.

$\displaystyle u\xrightarrow{h_{i}}$
$\displaystyle~{}h_{i}u=(A_{i}-\tau_{0}C_{i})~{}u~{}(A_{i}-\bar{\tau}_{0}C_{i})^{-1}\,,$
$\displaystyle\Phi^{(I)}\xrightarrow{h_{i}}$
$\displaystyle~{}\Omega_{I}(h_{i})\Phi^{(I)}\,,~{}~{}~{}~{}\Omega_{I}(h_{i})=\det(C_{i}\tau_{0}+D_{i})^{-k_{I}}\rho_{I}(h_{i})\,.$
(36)

It is always possible to parametrize $u$ in such a way that this linear
transformation is unitary. Moreover, if $\rho_{I}(h_{i})$ is unitary, so is
$\Omega_{I}(h_{i})$, since $[\det(C_{i}\tau_{0}+D_{i})]^{q}=1$, where $q$ is
the order of $h_{i}$. The explicit parametrization of $u$ and $\Phi^{(I)}$
will be specified in the next Sections. Moreover, under the general CP
symmetry $g\mathcal{CP}$, the new fields $u$ and $\Phi^{(I)}$ transform as
follows

$\displaystyle u$ $\displaystyle\xrightarrow{g\mathcal{CP}}$ $\displaystyle
e^{\displaystyle-2i\alpha}\left(\bar{\tau}_{0}C_{0}+A_{0}\right)^{-1}\bar{u}\left(\tau_{0}C_{0}+A_{0}\right)\,,$
$\displaystyle\Phi^{(I)}$ $\displaystyle\xrightarrow{g\mathcal{CP}}$
$\displaystyle\det(\tau_{0}C_{0}+A_{0})^{k_{I}}X^{\prime}_{I}\bar{\Phi}^{(I)},~{}~{}~{}X^{\prime}_{I}=\rho_{I}(\gamma^{-1})X_{I}\,.$
(37)

The phase $\det(\tau_{0}C_{0}+A_{0})^{k_{I}}$ can be absorbed by field
redefinition further, thus the CP transformation matrix is essentially
$X^{\prime}_{I}$. The explicit form of $X^{\prime}_{I}$ can be determined by
solving a set of consistency conditions, as shown in the following Sections.
In the basis $(u,\Phi^{(I)})$, CP and $G_{0}$ are linearly realized in the
field space, and are broken by the VEV of the flavon $u(\tau)$, which remains
small if $\tau$ is near $\tau_{0}$. Assuming that the lepton doublets $L$
transform as an irreducible triplet of $N_{n}(H)$, we can build the most
general set of lepton mass matrices with the correct transformation properties
under $G_{0}$ and CP from the decomposition of such a triplet under $G_{0}$.
As we will see, both the moduli $u$ and the irreducible triplets of $N_{n}(H)$
decompose into the direct sum of $G_{0}$-singlets. Thus, a suitable choice of
the phase $\alpha$ allows us to cast the CP transformations of the relevant
fields in the simple form:

$\displaystyle\Phi^{(I)}\xrightarrow{g\mathcal{CP}}~{}\bar{\Phi}^{(I)},~{}~{}~{}~{}u_{i}\xrightarrow{g\mathcal{CP}}~{}\bar{u}_{i}\,.$
(38)

Working in the vicinity of the fixed point, we only need a few terms of the
expansion of such matrices in powers of the symmetry-breaking order parameter
$u(\tau)$. In general, the kinetic terms originating from the most general
Kähler potential allowed by CP and Siegel modular invariance are not
canonical. After moving to the basis where such terms are canonical, it is not
difficult to prove that, when the theory is invariant under $G_{0}$ and CP,
charged lepton and neutrino mass matrices, $m_{e}^{2}(u,\bar{u})\equiv
m_{e}(u,\bar{u})^{\dagger}m_{e}(u,\bar{u})$ and $m_{\nu}(u,\bar{u})$, should
transform as shown in table 4.

| $G_{0}$ | CP
---|---|---
$m_{\nu}(u,\bar{u})$ | $\bm{r}_{s}\Omega^{*}~{}m_{\nu}(u,\bar{u})~{}\Omega^{\dagger}$ | $m_{\nu}(u,\bar{u})^{*}$
$m_{\nu}(u,\bar{u})^{-1}$ | $\bm{r}_{s}^{-1}\Omega~{}m_{\nu}(u,\bar{u})^{-1}~{}\Omega^{T}$ | $m_{\nu}(u,\bar{u})^{-1*}$
$m_{e}(u,\bar{u})$ | $\bm{r}_{s}\Omega_{c}^{*}~{}m_{e}(u,\bar{u})~{}\Omega^{\dagger}$ | $m_{e}(u,\bar{u})^{*}$
$m_{\bar{e}e}(u,\bar{u})$ | $\Omega~{}m_{\bar{e}e}(u,\bar{u})~{}\Omega^{\dagger}$ | $[m_{\bar{e}e}(u,\bar{u})]^{*}$
Table 4: Transformation properties of the lepton mass matrices under the group
$G_{0}$ and CP. We have defined $m_{\bar{e}e}(u,\bar{u})\equiv
m_{e}(u,\bar{u})^{\dagger}m_{e}(u,\bar{u})$. In the column ”$G_{0}$”, we show
the various mass matrices evaluated at $(h_{i}u,h_{i}\bar{u})$, while in the
column ”CP” they are evaluated at $(\bar{u}_{i},u_{i})$. We allow for the
possibility that the superpotential $w$ transforms as a generic singlet
$\bm{r}_{s}$ of $N_{n}(H)$.

The unitary matrices in table 4 read:

$\displaystyle\Omega\equiv\Omega_{H_{u}}\Omega_{L}\,,~{}~{}~{}~{}~{}~{}\Omega_{c}\equiv\Omega_{H_{u}}^{*}\Omega_{H_{d}}\Omega_{E^{c}}\,.$
(39)

If the neutrino mass matrix arises from the seesaw mechanism, it may occur
that $m_{\nu}(0,0)$ is singular 666That is the limit of $m_{\nu}(u,\bar{u})$
when $u$ goes to zero does not exist or is infinite.. In such a case it is
convenient to enforce the transformations on the inverse
$[m_{\nu}(u,\bar{u})]^{-1}$, also reported in table 4. Once $\Omega$ and
$\Omega_{c}$ are specified, table 4 can be used to get the most general
parametrization of $m_{\nu}(u,\bar{u})$ and $m_{\bar{e}e}(u,\bar{u})$ in the
vicinity of $\tau_{0}$.

##  4 Siegel modular invariant models near the fixed points

In this Section we analyze the fixed points 2, 3, 4 and 5, embedded in the
region $\Sigma_{2}$, whose normalizer $N(H)$ is generated by the elements
$G_{i}$ $(i=1,...,4)$ of eq. (18). We show that CP is conserved at each of
these fixed points. We find out the stability group $G_{0}$ and its
generators. Moving to the basis where $G_{0}$ acts linearly, we determine the
transformation properties of moduli and fields under $G_{0}$ and CP. Finally,
we find the decomposition of any irreducible triplet of the finite Siegel
modular group $N_{n}(H)$ under the subgroup $G_{0}$.

###  4.1 Fixed point 2

We start by analyzing the fixed point

$\tau_{0}=\begin{pmatrix}\eta&\frac{1}{2}(\eta-1)\\\
\frac{1}{2}(\eta-1)&\eta\end{pmatrix}\,,~{}~{}~{}~{}~{}~{}~{}\eta=\frac{1}{3}(1+i2\sqrt{2})\,.$
(40)

In $\tau_{0}$ CP is unbroken, since 777The matrix $\gamma$ is not unique: both
$\gamma$ and $\gamma^{\prime}=\gamma h$, with $h\tau_{0}=\tau_{0}$, satisfy
eq. (41).

$-\bar{\tau}_{0}=\gamma\tau_{0}\,,~{}~{}~{}~{}~{}~{}~{}\gamma=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\\\ -1&1&1&0\\\ 1&-1&0&1\end{array}\right)\,.$ (41)

It is convenient to define the CP action on $\tau$ as in eq. (3.1), with the
matrix $\gamma$ of eq. (41), such that $\tau_{CP}$ belongs to the region
conjugate to $\Sigma_{2}$ and $\tau_{0CP}=\tau_{0}$.

####  4.1.1 Stability group

The stability group $G_{0}$ of $\tau_{0}$, subgroup of the flavour group
$N(H)$ defined in eq. (18), is isomorphic to $D_{4}$, generated by the
elements $a$ and $b$, satisfying $a^{4}=b^{2}=(ab)^{2}=1$. In terms of the
generators $G_{i}$ $(i=1,2,3,4)$ of $N(H)$, $a$ and $b$ read:

$a=G_{4}G_{2}G_{3}G_{2}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}b=(G_{3}G_{2})^{3}\,.$
(42)

The irreducible representations of $D_{4}$ are four singlets $\bm{1}_{+}$,
$\bm{1}_{-}$, $\bm{1}^{\prime}_{+}$, $\bm{1}^{\prime}_{-}$ and one doublet
$\bm{2}$. Further details about $D_{4}$ are given in Appendix A.

####  4.1.2 New basis

To realize a linear and unitary action of the stability group $D_{4}$ on
$\tau$ and an antiunitary action of CP, we perform the field redefinition of
eq. (3.1), choosing

$e^{2i\alpha}=-\frac{1}{3}-i\frac{2\sqrt{2}}{3}\,.$ (43)

Moreover, we use the parametrization:

$u=\left(\begin{array}[]{cc}\frac{1}{\sqrt{3}}u_{1}+i\sqrt{\frac{2}{3}}u_{3}&i\sqrt{\frac{2}{3}}u_{1}+\frac{1}{\sqrt{3}}u_{3}\\\
i\sqrt{\frac{2}{3}}u_{1}+\frac{1}{\sqrt{3}}u_{3}&\frac{1}{\sqrt{3}}u_{1}+i\sqrt{\frac{2}{3}}u_{3}\end{array}\right)\,.$
(44)

The phase $\alpha$ reveals useful in simplifying the action of CP on
$u_{1,3}$. Under the stability group $D_{4}$, the new fields $u$ split into
the sum of an invariant singlet, $u_{1}$, and a component $u_{3}$ transforming
as $\bm{1}^{\prime}_{-}$, see Appendix A:

$\displaystyle u_{1}\xrightarrow{a}+u_{1},\qquad
u_{1}\xrightarrow{b}+u_{1},\qquad u_{1}\sim\bm{1}_{+}$ $\displaystyle
u_{3}\xrightarrow{a}-u_{3},\qquad u_{3}\xrightarrow{b}+u_{3},\qquad
u_{3}\sim\bm{1}^{\prime}_{-}\,.$ (45)

Under CP, we get

$u_{1}\xrightarrow{g\mathcal{CP}}\bar{u}_{1}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}u_{3}\xrightarrow{g\mathcal{CP}}\bar{u}_{3}\,.$
(46)

The action of $D_{4}$ on the new field $\Phi^{(I)}$ of eq. (3.1) reads

$\displaystyle\Phi^{(I)}\xrightarrow{\gamma}\Omega_{I}(\gamma)\Phi^{(I)}\,,~{}~{}~{}~{}~{}~{}~{}\Omega_{I}(\gamma)=j(\gamma,\tau_{0})^{-k_{I}}\rho_{I}(\gamma)\,.$
(47)

where $j(\gamma,\tau)=\det(C\tau+D)$. In particular, we have:

$j(a,\tau_{0})^{-1}=+1\,,~{}~{}~{}~{}~{}~{}j(b,\tau_{0})^{-1}=-1,$ (48)

which reproduces the representation $\bm{1}_{-}$ of the stability group. We
see that $\Omega_{I}(\gamma)=j(\gamma,\tau_{0})^{-k_{I}}\rho_{I}(\gamma)$ is
also a representation of the stability group, the direct product of
$(\bm{1}_{-})^{k_{I}}=(\bm{1}_{-},\bm{1}_{+})$ and $\rho_{I}(\gamma)$. In a
bottom-up approach, by varying all possible representations of the matter
multiplets, we can absorb the factor $(\bm{1}_{-})^{k_{I}}$ into the choice of
$\rho_{I}(\gamma)$. From Eq. (8) and the expression of generators in eq. (42),
we know the CP transformation $X^{\prime}_{I}$ satisfies the following
consistency conditions on the stability group $D_{4}$:

$X^{\prime}_{I}\,\rho^{*}_{I}(a)X^{\prime-1}_{I}=\rho_{I}(a),~{}~{}~{}~{}X^{\prime}_{I}\,\rho^{*}_{I}(b)X^{\prime-1}_{I}=\rho_{I}(b^{-1})\,,$
(49)

which lead to

$X^{\prime}_{I}=\mathbbm{1}$ (50)

in the basis of Appendix A, no matter whether $\Phi^{(I)}$ transforms as
singlets or doublet of $D_{4}$. Hence the CP transformation simply maps
$\Phi^{(I)}$ to its conjugate.

####  4.1.3 Triplet decomposition

From the 168 triplet representation of the finite Siegel modular groups
$N_{n}(H)$, the three-dimensional representation matrices of the $D_{4}$
generators $a$ and $b$ can be obtained via the relations in Eq. (42). This
allows the decomposition of the three-dimensional irreducible representations
of $N_{n}(H)$ under the stability group $D_{4}$. We find that each irreducible
triplet $\rho_{L}$ decomposes into one of the following sum of three singlets
of $D_{4}$

$\rho_{L}\sim\begin{cases}\mathbf{1}_{-}\oplus\mathbf{1}^{\prime}_{+}\oplus\mathbf{1}^{\prime}_{+}\\\
\mathbf{1}_{+}\oplus\mathbf{1}^{\prime}_{-}\oplus\mathbf{1}^{\prime}_{-}\\\
\mathbf{1}^{\prime}_{+}\oplus\mathbf{1}_{-}\oplus\mathbf{1}_{-}\\\
\mathbf{1}^{\prime}_{-}\oplus\mathbf{1}_{+}\oplus\mathbf{1}_{+}\\\
\end{cases}\,.$ (51)

We end up with

$\Omega_{L}(\gamma)=j(\gamma,\tau_{0})^{-k_{I}}\rho_{L}(\gamma)\,,$ (52)

where $j(\gamma,\tau_{0})^{-1}\sim\bm{1}_{-}$. Finally, we get
$\Omega\equiv\Omega_{H_{u}}\Omega_{L}$ by multiplying $\Omega_{L}$ and the
generic singlet associated with $\Omega_{H_{u}}$.

###  4.2 Fixed point 3

The fixed point

$\tau_{0}=\left(\begin{array}[]{cc}i&0\\\ 0&i\end{array}\right)$ (53)

belongs to both $\Sigma_{1}$ and $\Sigma_{2}$. It preserves CP using the
standard definition of eq. (3.1), where $\gamma=\mathbbm{1}$. We assume that
in the theory under examination, the matrix $\tau$ is restricted to
$\Sigma_{2}$ and the flavor group is the corresponding normalizer $N(H)$
defined in eq. (18).

####  4.2.1 Stability group

The stability group $G_{0}$ of $\tau_{0}$, subgroup of the flavour group
$N(H)$ defined in eq. (18), is isomorphic to the Pauli group, namely the
central product of $D_{4}$ and $Z_{4}$: $D_{4}\circ Z_{4}$. $G_{0}$ is
generated by the elements $a$ and $b$ and $c$, satisfying
$a^{4}=c^{4}=b^{2}=(ab)^{2}=1$, $a^{2}=c^{2}$, $ac=ca$, $bc=cb$. The elements
$(a,b)$ generate $D_{4}$ and $c$ generates $Z_{4}$. The elements of $D_{4}$
commute with those of $Z_{4}$. In terms of the generators $G_{i}$
$(i=1,2,3,4)$ of $N(H)$, $a$, $b$ and $c$ read:

$a=G_{3}G_{4}\,,~{}~{}~{}~{}~{}~{}b=(G_{3}G_{2})^{3}\,,~{}~{}~{}~{}~{}~{}c=G_{3}\,.$
(54)

The irreducible representations of $D_{4}\circ Z_{4}$ are eight singlets and
two doublets. Further details about $D_{4}\circ Z_{4}$ are given in Appendix
B.

####  4.2.2 New basis

We move to the new basis of eq. (3.1), choosing $\alpha=0$. Using the
parametrization:

$u=\left(\begin{array}[]{cc}u_{1}{}&~{}u_{3}\\\
u_{3}{}&~{}u_{1}\end{array}\right)\,,$ (55)

we find that, under the stability group $D_{4}\circ Z_{4}$, the new multiplet
$u$ splits into the sum of two singlets: $\bm{1}_{-+-}\oplus\bm{1}_{++-}$, see
Appendix B:

$\displaystyle u_{1}\xrightarrow{a}-u_{1}\,,\qquad
u_{1}\xrightarrow{b}+u_{1}\,,\qquad u_{1}\xrightarrow{c}-u_{1}\,,\qquad
u_{1}\sim\bm{1}_{-+-}$ $\displaystyle u_{3}\xrightarrow{a}+u_{3}\,,\qquad
u_{3}\xrightarrow{b}+u_{3}\,,\qquad u_{3}\xrightarrow{c}-u_{3}\,,\qquad
u_{3}\sim\bm{1}_{++-}~{}\,.$ (56)

Under CP, with $\gamma=\mathbbm{1}$ in eq. (3.1), we get

$u_{1}\xrightarrow{\mathcal{CP}}\bar{u}_{1}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}u_{3}\xrightarrow{\mathcal{CP}}\bar{u}_{3}\,.$
(57)

The action of $D_{4}\circ Z_{4}$ on the new matter fields $\Phi^{(I)}$ of eq.
(3.1) is

$\displaystyle\Phi^{(I)}\xrightarrow{\gamma}\Omega_{I}(\gamma)\Phi^{(I)}\,,~{}~{}~{}~{}~{}~{}\Omega_{I}(\gamma)=j(\gamma,\tau_{0})^{-k_{I}}\rho_{I}(\gamma)\,.$
(58)

where $j(\gamma,\tau)=\det(C\tau+D)$. In particular, we have:

$j(a,\tau_{0})^{-1}=+1\,,~{}~{}~{}~{}~{}~{}j(b,\tau_{0})^{-1}=-1\,,~{}~{}~{}~{}~{}~{}j(c,\tau_{0})^{-1}=-1\,,$
(59)

which reproduces the representation $\bm{1}_{+--}$ of the stability group. We
see that $\Omega_{I}(\gamma)=j(\gamma,\tau_{0})^{-k_{I}}\rho_{I}(\gamma)$ is
also a representation of the stability group, the direct product of
$(\bm{1}_{+--})^{k_{I}}$ and $\rho_{I}(\gamma)$. In a bottom-up approach, by
varying all possible representations of the matter multiplets, we can absorb
the factor $(\bm{1}_{+--})^{k_{I}}$ into the choice of $\rho_{I}(\gamma)$. The
consistency conditions of CP transformation $X^{\prime}_{I}$ on the stability
group $D_{4}\circ Z_{4}$ is given by

$X^{\prime}_{I}\,\rho^{*}_{I}(a)X^{\prime-1}_{I}=\rho_{I}(a^{-1}),~{}~{}~{}~{}X^{\prime}_{I}\,\rho^{*}_{I}(b)X^{\prime-1}_{I}=\rho_{I}(b^{-1}),~{}~{}~{}~{}X^{\prime}_{I}\,\rho^{*}_{I}(c)X^{\prime-1}_{I}=\rho_{I}(c^{-1})\,.$
(60)

In the basis of Appendix B, we have $X^{\prime}_{I}=\mathbbm{1}$ regardless of
the transformation of $\Phi^{(I)}$ under $D_{4}\circ Z_{4}$. As a result, the
CP transformation of the matter field is
$\Phi^{(I)}\xrightarrow{\mathcal{CP}}\bar{\Phi}^{(I)}$.

####  4.2.3 Triplet decomposition

We find that each irreducible triplet of $N_{n}(H)$ decomposes under
$D_{4}\circ Z_{4}$ into a direct sum of three singlets, wherein two of them
are identical. There are only eight distinct cases as follows,

$\displaystyle\rho_{L}\sim\begin{cases}\bm{1}_{+++}\oplus\bm{1}_{-+-}\oplus\bm{1}_{-+-}\\\
\bm{1}_{-+-}\oplus\bm{1}_{+++}\oplus\bm{1}_{+++}\\\
\bm{1}_{+--}\oplus\bm{1}_{--+}\oplus\bm{1}_{--+}\\\
\bm{1}_{--+}\oplus\bm{1}_{+--}\oplus\bm{1}_{+--}\\\
\bm{1}_{+-+}\oplus\bm{1}_{---}\oplus\bm{1}_{---}\\\
\bm{1}_{---}\oplus\bm{1}_{+-+}\oplus\bm{1}_{+-+}\\\
\bm{1}_{-++}\oplus\bm{1}_{++-}\oplus\bm{1}_{++-}\\\
\bm{1}_{++-}\oplus\bm{1}_{-++}\oplus\bm{1}_{-++}\end{cases}\,.$ (61)

We end up with

$\Omega_{L}(\gamma)=j(\gamma,\tau_{0})^{-k_{I}}\rho_{L}(\gamma)\,,$ (62)

where $j(\gamma,\tau_{0})^{-1}\sim\bm{1}_{+--}$. Finally, we get
$\Omega\equiv\Omega_{H_{u}}\Omega_{L}$ by multiplying $\Omega_{L}$ and the
generic singlet associated with $\Omega_{H_{u}}$.

###  4.3 Fixed point 4

The fixed point

$\tau_{0}=\left(\begin{array}[]{cc}\omega&0\\\ 0&\omega\end{array}\right)$
(63)

belongs to both $\Sigma_{1}$ and $\Sigma_{2}$. It preserves CP using the
definition of eq. (3.1), with

$\gamma=\left(\begin{array}[]{cccc}1&0&1&0\\\ 0&1&0&1\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right).$ (64)

We assume that in the theory under examination the matrix $\tau$ is restricted
to $\Sigma_{2}$ and the flavor group is the corresponding normalizer $N(H)$
defined in eq. (18).

####  4.3.1 Stability group

The stability group $G_{0}$ of $\tau_{0}$, subgroup of the flavour group
$N(H)$ defined in eq. (18), is isomorphic to $D_{4}\times Z_{3}$ generated by
the elements $a$ and $b$ and $c$, satisfying $a^{4}=b^{2}=(ab)^{2}=c^{3}=1$,
$ca=ac$, $cb=bc$. The elements $(a,b)$ generate $D_{4}$ and $c$ generates
$Z_{3}$. In terms of the generators $G_{i}$, $(i=1,2,3,4)$ of $N(H)$, eq.
(18), $a$, $b$ and $c$ read:

$a=(G_{2}G_{3})^{3}G_{4}\,,~{}~{}~{}~{}~{}~{}~{}b=G_{3}^{2}G_{4}\,,~{}~{}~{}~{}~{}~{}~{}c=(G_{3}G_{1})^{2}.$
(65)

The irreducible representations of $D_{4}\times Z_{3}$ are 12 singlets and 3
doublets. Further details about $D_{4}\times Z_{3}$ are given in Appendix C.

####  4.3.2 New basis

We move to the new basis of eq. (3.1), choosing $\alpha=0$. Using the
parametrization:

$u=\left(\begin{array}[]{cc}u_{1}&u_{3}\\\ u_{3}&u_{1}\end{array}\right)\,,$
(66)

we find that, under the stability group $G_{0}=D_{4}\times Z_{3}$, the new
multiplet $u$ transforms as the sum of two singlets: $u_{1}\sim\bm{1}_{++1}$
and $u_{3}\sim\bm{1}_{--1}$, see Appendix C.

$\displaystyle u_{1}\xrightarrow{a}+u_{1}\,,\qquad
u_{1}\xrightarrow{b}+u_{1}\,,\qquad u_{1}\xrightarrow{c}\omega u_{1}$
$\displaystyle u_{3}\xrightarrow{a}-u_{3}\,,\qquad
u_{3}\xrightarrow{b}-u_{3}\,,\qquad u_{3}\xrightarrow{c}\omega u_{3}\,,$ (67)

where $\omega=-1/2+i\sqrt{3}/2$. Under CP, with $\gamma$ in eq. (64), we get

$u_{1}\xrightarrow{g\mathcal{CP}}\bar{u}_{1}\,,~{}~{}~{}~{}~{}~{}~{}u_{2}\xrightarrow{g\mathcal{CP}}\bar{u}_{2}\,.$
(68)

The action of $D_{4}\times Z_{3}$ on the new matter fields $\Phi^{(I)}$ of eq.
(3.1) is

$\displaystyle\Phi^{(I)}\xrightarrow{\gamma}\Omega_{I}(\gamma)\Phi^{(I)}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}\Omega_{I}(\gamma)=j(\gamma,\tau_{0})^{-k_{I}}\rho_{I}(\gamma)\,.$
(69)

where $j(\gamma,\tau)=\det(C\tau+D)$. In particular, we have:

$j(a,\tau_{0})^{-1}=1\,,~{}~{}~{}~{}~{}~{}j(b,\tau_{0})^{-1}=-1\,,~{}~{}~{}~{}~{}~{}j(c,\tau_{0})^{-1}=\omega\,,$
(70)

which reproduces the representation $\bm{1}_{+-1}$ of the stability group. We
see that $\Omega_{I}(\gamma)=j(\gamma,\tau_{0})^{-k_{I}}\rho_{I}(\gamma)$ is
also a representation of the stability group, the direct product of
$(\bm{1}_{+-1})^{k_{I}}$ and $\rho_{I}(\gamma)$. In a bottom-up approach, by
varying all possible representations of the matter multiplets, we can absorb
the factor $(\bm{1}_{+-1})^{k_{I}}$ into the choice of $\rho_{I}(\gamma)$. The
CP transformation $X^{\prime}_{I}$ fulfills the following consistency
conditions on the stability group $D_{4}\times Z_{3}$:

$X^{\prime}_{I}\,\rho^{*}_{I}(a)X^{\prime-1}_{I}=\rho_{I}(a)\,,~{}~{}~{}~{}X^{\prime}_{I}\,\rho^{*}_{I}(b)X^{\prime-1}_{I}=\rho_{I}(b^{-1})\,,~{}~{}~{}~{}X^{\prime}_{I}\,\rho^{*}_{I}(c)X^{\prime-1}_{I}=\rho_{I}(c^{-1})\,.$
(71)

A basis for the $D_{4}\times Z_{3}$ generators exists, where the consistency
conditions of CP transformation restricted to the stability group $D_{4}\times
Z_{3}$ are solved by $X_{I}^{\prime}=\mathbbm{1}$, see comment at the end of
Appendix C.

####  4.3.3 Triplet decomposition

We find that each irreducible triplet of $N_{n}(H)$ decomposes under
$D_{4}\times Z_{3}$ into a direct sum of three singlets which transform in the
same way under the $D_{4}\times Z_{3}$ subgroup. There are only four distinct
cases as follows,

$\displaystyle\rho_{L}\sim\begin{cases}\bm{1}_{++0}\oplus\bm{1}_{++1}\oplus\bm{1}_{++2}\\\
\bm{1}_{+-0}\oplus\bm{1}_{+-1}\oplus\bm{1}_{+-2}\\\
\bm{1}_{-+0}\oplus\bm{1}_{-+1}\oplus\bm{1}_{-+2}\\\
\bm{1}_{--0}\oplus\bm{1}_{--1}\oplus\bm{1}_{--2}\end{cases}\,.$ (72)

We end up with

$\Omega_{L}(\gamma)=j(\gamma,\tau_{0})^{-k_{I}}\rho_{L}(\gamma)\,,$ (73)

where $j(\gamma,\tau_{0})^{-1}\sim\bm{1}_{+-1}$. Finally, we get
$\Omega\equiv\Omega_{H_{u}}\Omega_{L}$ by multiplying $\Omega_{L}$ and the
generic singlet associated with $\Omega_{H_{u}}$. Since each triplet of
$N_{n}(H)$ is decomposed into three singlets of $D_{4}\times Z_{3}$, the CP
symmetry transforms the lepton fields into their conjugate.

###  4.4 Fixed point 5

The last case to be examined is the fixed point

$\tau_{0}=\frac{i}{\sqrt{3}}\left(\begin{array}[]{cc}2&1\\\
1&2\end{array}\right).$ (74)

It preserves CP using the standard definition of eq. (3.1), where
$\gamma=\mathbbm{1}$.

####  4.4.1 Stability group

The stability group $G_{0}$ of $\tau_{0}$, subgroup of the flavour group
$N(H)$ defined in eq. (18), is isomorphic to $D_{4}$, generated by the
elements $a$ and $b$, satisfying $a^{4}=b^{2}=(ab)^{2}=1$. In terms of the
generators $G_{i}$ $(i=1,2,3,4)$ of $N(H)$, $a$ and $b$ read:

$a=G_{3}G_{4}\,,~{}~{}~{}~{}~{}~{}~{}b=(G_{3}G_{2})^{3}\,.$ (75)

The irreducible representations of $D_{4}$ are four singlets $\bm{1}_{+}$,
$\bm{1}_{-}$, $\bm{1}^{\prime}_{+}$, $\bm{1}^{\prime}_{-}$ and one doublet
$\bm{2}$. Further details about $D_{4}$ are given in Appendix A.

####  4.4.2 New basis

We move to the new basis of eq. (3.1), choosing $\alpha=0$. We define:

$u=\left(\begin{array}[]{cc}u_{1}&u_{3}\\\ u_{3}&u_{1}\end{array}\right)\,,$
(76)

and we find that, under the stability group $D_{4}$, the new fields $u$ split
into the sum of an invariant singlet, $u_{3}$, and a component $u_{1}$
transforming as $\bm{1}^{\prime}_{-}$, see Appendix A:

$\displaystyle u_{1}\xrightarrow{a}-u_{1}\,,\qquad
u_{1}\xrightarrow{b}+u_{1}\,,\qquad u_{1}\sim\bm{1}^{\prime}_{-}$
$\displaystyle u_{3}\xrightarrow{a}+u_{3}\,,\qquad
u_{3}\xrightarrow{b}+u_{3}\,,\qquad u_{3}\sim\bm{1}_{+}\,.$ (77)

Under CP, we get

$u_{1}\xrightarrow{\mathcal{CP}}\bar{u}_{1}\,,~{}~{}~{}~{}~{}~{}~{}~{}u_{3}\xrightarrow{\mathcal{CP}}\bar{u}_{3}\,.$
(78)

The action of $D_{4}$ on the new matter fields $\Phi^{(I)}$ of eq. (3.1) is

$\displaystyle\Phi^{(I)}\xrightarrow{\gamma}\Omega_{I}(\gamma)\Phi^{(I)}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}\Omega_{I}(\gamma)=j(\gamma,\tau_{0})^{-k_{I}}\rho_{I}(\gamma).$
(79)

where $j(\gamma,\tau)=\det(C\tau+D)$. In particular, we have:

$j(a,\tau_{0})^{-1}=+1\,,~{}~{}~{}~{}~{}~{}j(b,\tau_{0})^{-1}=-1\,,$ (80)

which reproduces the representation $\bm{1}_{-}$ of the stability group. We
see that $\Omega_{I}(\gamma)=j(\gamma,\tau_{0})^{-k_{I}}\rho_{I}(\gamma)$ is
also a representation of the stability group, the direct product of
$(\bm{1}_{-})^{k_{I}}=(\bm{1}_{-},\bm{1}_{+})$ and $\rho_{I}(\gamma)$. In a
bottom-up approach, by varying all possible representations of the matter
multiplets, we can absorb the factor $(\bm{1}_{-})^{k_{I}}$ into the choice of
$\rho_{I}(\gamma)$. The consistency conditions of CP transformation
$X^{\prime}_{I}$ on the stability group $D_{4}$ is given by

$X^{\prime}_{I}\,\rho^{*}_{I}(a)X^{\prime-1}_{I}=\rho_{I}(a^{-1}),~{}~{}~{}~{}X^{\prime}_{I}\,\rho^{*}_{I}(b)X^{\prime-1}_{I}=\rho_{I}(b^{-1})\,.$
(81)

A basis for the $D_{4}$ generators exists, where the consistency conditions of
CP transformation restricted to the stability group $D_{4}$ are solved by
$X_{I}^{\prime}=\mathbbm{1}$, see comment at the end of Appendix A. Notice
that this basis does not coincide with the one allowing to choose
$X_{I}^{\prime}=\mathbbm{1}$ at the fixed point 2, despite the stability group
for the fixed points 2 and 5 is the same.

####  4.4.3 Triplet decomposition

We need the decomposition of the irreducible triplets $\rho_{L}$ of
$N_{n}(H)$, to which lepton doublets $L$ are assigned, within the stability
group $D_{4}$. With the help of Appendix A, by considering the representation
matrices of the two generators, $a$ and $b$, we find that each irreducible
triplet of $N_{n}(H)$ decomposes under $D_{4}$ into a direct sum of three
singlets, wherein two of them are identical. There are only four distinct
cases, illustrated in eq. (82).

$\displaystyle\rho_{L}\sim\begin{cases}\mathbf{1}_{-}\oplus\mathbf{1}^{\prime}_{+}\oplus\mathbf{1}^{\prime}_{+}\\\
\mathbf{1}_{+}\oplus\mathbf{1}^{\prime}_{-}\oplus\mathbf{1}^{\prime}_{-}\\\
\mathbf{1}^{\prime}_{+}\oplus\mathbf{1}_{-}\oplus\mathbf{1}_{-}\\\
\mathbf{1}^{\prime}_{-}\oplus\mathbf{1}_{+}\oplus\mathbf{1}_{+}\\\
\end{cases}\,,$ (82)

We end up with

$\Omega_{L}(\gamma)=j(\gamma,\tau_{0})^{-k_{I}}\rho_{L}(\gamma)\,,$ (83)

where $j(\gamma,\tau_{0})^{-1}\sim\bm{1}_{-}$. Finally, we get
$\Omega\equiv\Omega_{H_{u}}\Omega_{L}$ by multiplying $\Omega_{L}$ and the
generic singlet associated with $\Omega_{H_{u}}$.

##  5 Neutrino Mass Matrix

In this Section, we discuss the patterns of the lepton mass matrices of a
generic theory whose moduli, belonging to the region $\Sigma_{2}$
parameterized by

$\tau=\begin{pmatrix}\tau_{1}&\tau_{3}\\\ \tau_{3}&\tau_{1}\end{pmatrix}$ (84)

are close to one of the fixed points examined previously. We assume the most
general CP-invariant ${\cal N}=1$ local supersymmetric action and enforce
Siegel modular invariance by asking that the superpotential $w$ transforms as
in eq. (3) with $\bm{r}_{s}$ either trivial or non-trivial singlet of the
finite modular group $N_{n}(H)$, and thus also a singlet of the stability
group $G_{0}$. In such a theory the level $n$, the weights $k_{I}$ and the
representations $\rho_{I}$ of the matter multiplets, and even the specific
form of the Kähler potential are the most general ones consistent with the
requirement of CP and modular invariance. In this completely general
framework, we make a single assumption: the lepton doublets $L$ are assigned
to an irreducible triplet $\rho_{L}$ of the finite modular group $N_{n}(H)$.
In a bottom-up approach this assumption usually minimizes the number of free
parameters of the theory.

Working with the local coordinates $(u,\Phi^{(I)})$ of eq. (3.1), on which the
stability group $G_{0}$ has a linear action, the lepton mass matrices
$m_{\bar{e}e}$ and $m_{\nu}$ must transform as required by table 4. Making use
of the decomposition of each irreducible triplet of $N_{n}(H)$ and the
transformation laws of $u$ under $G_{0}$, we can easily fulfill this
requisite, which can be implemented order by order in the expansion in powers
of $|u|$, a small quantity when $\tau$ is close to the fixed point. We provide
the general expression of $m_{\bar{e}e}$ and $m_{\nu}$ at the first nontrivial
order in $|u|$.

As a general result, we find that in each fixed point, the mass matrices
$m_{\bar{e}e}$ and $m_{\nu}$ are sensitive only to $\bm{r}_{s}$ and not to the
decomposition of $\rho_{L}$ under $G_{0}$. Moreover, the mass matrices
$m_{\bar{e}e}$ are the same for all $\bm{r}_{s}$, while the neutrino mass
matrices $m_{\nu}$(or $m_{\nu}^{-1}$) depend on $\bm{r}_{s}$. The examined
fixed points fall into two classes. The fixed points 2, 3 and 5 give rise to
similar patterns, which are different from those arising around the fixed
point 4.

###  5.1 Fixed points 2, 3, and 5

From table 4 and the analysis of the previous Section, we find the following
pattern for $m_{\bar{e}e}(u,\bar{u})$:

$\displaystyle
m_{\bar{e}e}(u,\bar{u})=m_{0e}^{2}~{}\left(\begin{array}[]{ccc}y^{0}_{11}&y_{12}~{}x&y_{13}~{}x\\\
y_{12}^{*}~{}x&y^{0}_{22}&y^{0}_{23}\\\
y_{13}^{*}~{}x&y^{0}_{23}&y^{0}_{33}\\\ \end{array}\right)+\ldots\,,$ (88)

where $x=|u_{3}|$ for the fixed point 2 and $x=|u_{1}|$ for the fixed points 3
and 5. The overall real coefficient $m_{0e}^{2}$ has the dimension of (mass)2.
Dots stand for higher orders in the $x$ expansion. The coefficients
$y^{0}_{ij}(y_{ij})$ are real (complex) numbers, independent of the moduli.
They are not constrained in the present analysis, though they are expected to
be of the same order. To discuss the lepton mixing matrix, we diagonalize
$m_{\bar{e}e}(u,\bar{u})$:

$\displaystyle U_{e}^{\dagger}m_{\bar{e}e}(u,\bar{u})U_{e}={\tt
diag}[m_{\bar{e}e}(u,\bar{u})]\,.$ (89)

Up to a permutation matrix $P$ related to the ordering of the charged lepton
masses, $U_{e}$ has the pattern:

$\displaystyle U_{e}=\left(\begin{array}[]{ccc}{\cal O}(1)&{\cal O}(x)&{\cal
O}(x)\\\ {\cal O}(x)&{\cal O}(1)&{\cal O}(1)\\\ {\cal O}(x)&{\cal O}(1)&{\cal
O}(1)\end{array}\right)\,.$ (93)

Next, we examine the neutrino mass matrices. We find that they fall into three
classes, depending on the singlet representation $\bm{r}_{s}$ under which $w$
transforms, see table 5.

  * $\bullet$ A

The following pattern is obtained when $\bm{r}_{s}$ embeds one of the $G_{0}$
singlets listed in table 5, first line. The overall factor $m_{0\nu}$ is a
real mass parameter.

$\displaystyle
m_{\nu}(u,\bar{u})=m_{0\nu}~{}\left(\begin{array}[]{ccc}x^{0}_{11}&x_{12}~{}x&x_{13}~{}x\\\
\cdot&x^{0}_{22}&x^{0}_{23}\\\ \cdot&\cdot&x^{0}_{33}\\\
\end{array}\right)+\ldots\,.$ (97)

The coefficients $x^{0}_{ij}(x_{ij})$ are real (complex) numbers, independent
of the moduli $u_{1,3}$, except for the choice $\bm{1}_{++-}$ at the fixed
point 3, where both $x^{0}_{ij}$ and $x_{ij}$ are proportional to a linear
combination of $u_{3}$ and $\bar{u}_{3}$. In any case, $x^{0}_{ij}(x_{ij})$
are expected to be of the same order, and we do not need to treat separately
this special case.

  * $\bullet$ B

The following pattern is obtained when $\bm{r}_{s}$ embeds one of the $G_{0}$
singlets listed in table 5, second line. The overall factor $m_{0\nu}$ is a
real mass parameter.

$\displaystyle
m_{\nu}(u,\bar{u})=m_{0\nu}~{}\left(\begin{array}[]{ccc}x_{11}~{}x&x^{0}_{12}&x^{0}_{13}\\\
\cdot&x_{22}~{}x&x_{23}~{}x\\\ \cdot&\cdot&x_{33}~{}x\\\
\end{array}\right)+\ldots\,.$ (101)

The coefficients $x^{0}_{ij}(x_{ij})$ are real (complex) numbers, independent
of the moduli $u_{1,3}$, except for the choice $\bm{1}_{-++}$ at the fixed
point 3, where both $x^{0}_{ij}$ and $x_{ij}$ are proportional to a linear
combination of $u_{3}$ and $\bar{u}_{3}$. In any case, $x^{0}_{ij}(x_{ij})$
are expected to be of the same order, and we do not need to discuss separately
this particular case.

Since $m_{\nu}(u,\bar{u})$ has rank two at $x=0$, we also consider the
expansion of $[m_{\nu}(u,\bar{u})]^{-1}$, which is identical to the one in eq.
(101), with the replacement $m_{\nu}(u,\bar{u})\to[m_{\nu}(u,\bar{u})]^{-1}$
and $m_{0\nu}\to m_{0\nu}^{-1}$. The two possibilities arising from the
analysis of $m_{\nu}(u,\bar{u})$ and $[m_{\nu}(u,\bar{u})]^{-1}$ are
physically distinct. At $x=0$, the pattern of $[m_{\nu}(u,\bar{u})]$ predicts
a vanishing mass, while that of $[m_{\nu}(u,\bar{u})]^{-1}$ predicts a
divergent neutrino mass. We understand such an infinite mass in terms of an
extra degree of freedom of the full theory becoming massless at the fixed
point. It is natural to interpret such a degree of freedom as a right-handed
neutrino. Thus, the pattern of $[m_{\nu}(u,\bar{u})]^{-1}$ is expected to
arise in the context of the seesaw mechanism when a right-handed neutrino,
whose mass depends on the moduli, becomes massless at the fixed point.

  * $\bullet$ C

In this case, the only solution to the constraint specified in table 4, is a
vanishing neutrino mass matrix $m_{\nu}(u,\bar{u})=0$, to any order in the
expansion in powers of $u_{1}$ and $u_{3}$. We dismiss this unphysical
possibility.

pattern | $\bm{r}_{s}({\tt FP2})$ | $\bm{r}_{s}({\tt FP3})$ | $\bm{r}_{s}({\tt FP5})$
---|---|---|---
A \- eq. (97) | $\bm{1}_{+}$ |  $\bm{1}_{+++}$, $\bm{1}_{++-}$ | $\bm{1}_{+}$
B \- eq. (101) | $\bm{1}^{\prime}_{-}$ |  $\bm{1}_{-+-}$, $\bm{1}_{-++}$ | $\bm{1}^{\prime}_{-}$
C |  $\bm{1}_{-}$, |  $\bm{1}_{+--}$, $\bm{1}_{+-+}$, |  $\bm{1}_{-}$,
| $\bm{1}^{\prime}_{+}$ |  $\bm{1}_{---}$, $\bm{1}_{--+}$ | $\bm{1}^{\prime}_{+}$
Table 5:  Patterns of neutrino mass matrices $m_{\nu}(u,\bar{u})$, or their
inverse $m_{\nu}(u,\bar{u})^{-1}$, and their dependence on the singlet
representation of the superpotential $w$, for the fixed points $i=2,3,5$. We
have displayed the singlet representations $\bm{r}_{s}(\texttt{FPi})$ of the
stability group $G_{0}$. Each representation $\bm{r}_{s}(\texttt{FPi})$ can be
embedded in one (or more) representations $\bm{r}_{s}$ of $N_{n}(H)$.

From eq. (93) we see that moving to the basis where $m_{\bar{e}e}(u,\bar{u})$
is diagonal, up to a common permutation matrix of rows and columns and up to
higher-order terms in the expansion, the neutrino mass matrix maintains the
same pattern shown in eqs. (97) and (101). To first order in $x$, the effect
of the basis change can be absorbed in the coefficients $x_{ij}^{0}$,
$x_{ij}$. The same conclusion holds for the inverse $m_{\nu}(u,\bar{u})^{-1}$
and, without losing generality, we can discuss the neutrino mass spectrum,
mixing angles and phases by directly analyzing the matrices (97) and (101).
These two patterns coincide with those occurring in single-modulus
$SL(2,\mathbbm{Z})$-invariant theories close to the fixed point $\tau_{0}=i$
[43, 44]. We briefly recapitulate the results here and we refer the reader to
the literature for more details 888In particular, the predictions for the
lepton masses and lepton mixing matrix as well as mixing parameters have been
presented in Appendices D.1.1 and D.1.2 and Section 6 of ref. [44]..

In table 6 we summarize the predictions of Siegel modular invariant models for
lepton masses in the vicinity of one of the fixed points, up to possible
permutations affecting the mixing matrix. When the pattern in eq. (97) is
realized, both normal (NO) and inverted ordering (IO) of the neutrino mass
spectrum can be realized. However, $\Delta m^{2}_{sol}/\Delta m^{2}_{atm}$ is
generically expected to be of order one. Moreover $\sin^{2}\theta_{12}$ and
$\sin^{2}\theta_{13}$ are expected to be of the same order, contrary to
observation. Therefore this pattern can only be reconciled with the data at
the price of tuning the coefficients $x_{ij}^{0}$ and $x_{ij}$.

| $\begin{array}[]{c}{\rm mass}\\\ {\rm ordering}\end{array}$ | $\displaystyle\frac{\Delta m^{2}_{sol}}{\Delta m^{2}_{atm}}$ | $\sin^{2}\theta_{12}$ | $\sin^{2}\theta_{13}$ | $\sin^{2}\theta_{23}$
---|---|---|---|---|---
A \- eq. (97) $m_{\nu}(0,0)$ regular | NO/IO | ${\mathcal{O}}(1)$ | ${\mathcal{O}}(x^{2})$ | ${\mathcal{O}}(x^{2})$ | ${\mathcal{O}}(1)$
B \- eq. (101) $m_{\nu}(0,0)$ regular | IO | ${\mathcal{O}}(x)$ | $\frac{1}{2}(1+{\mathcal{O}}(x))$ | ${\mathcal{O}}(x^{2})$ | ${\mathcal{O}}(1)$
B \- eq. (101) $m_{\nu}(0,0)$ singular | NO | ${\mathcal{O}}(x^{3})$ | $\frac{1}{2}(1+{\mathcal{O}}(x))$ | ${\mathcal{O}}(x^{2})$ | ${\mathcal{O}}(1)$
D \- eq. (114) $m_{\nu}(0,0)$ regular | NO/IO | ${\mathcal{O}}(x)$ | $\frac{1}{2}(1+{\mathcal{O}}(x))$ | ${\mathcal{O}}(x^{2})$ | ${\mathcal{O}}(x^{2})$
E \- eq. (118) $m_{\nu}(0,0)$ regular | NO/IO | ${\mathcal{O}}(1)$ | ${\mathcal{O}}(1)$ | ${\mathcal{O}}(1)$ | ${\mathcal{O}}(1)$
Table 6:  Synopsis of predictions in modular invariant flavor models of
leptons, when the modulus $\tau$ falls in the vicinity of the fixed points and
$\rho_{L}$ is an irreducible triplet. For the fixed point 4, $x=|u_{1}|$.

When the pattern in eq. (101) is realized, and $m_{\nu}(0,0)$ has a vanishing
eigenvalue, an inverted ordering of neutrino masses is predicted. To reproduce
the observed values of $\sin^{2}\theta_{13}$ and $\sin^{2}\theta_{12}$, $x$
should be close to 0.15. This is in tension with the value of $x$ required by
$r=\Delta m^{2}_{sol}/\Delta m^{2}_{atm}={\cal O}(x)$, experimentally close to
0.03. This pattern can match the data too, at the price of tuning the
coefficients $x_{ij}^{0}$ and $x_{ij}$.

When the outcome is the pattern in eq. (101), a particularly appealing
scenario occurs when $[m_{\nu}(0,0)]^{-1}$ has a vanishing eigenvalue, which
can occur within the seesaw mechanism. In this case, all the data can be
reproduced by parameters $x^{0}_{ij}$ and $x_{ij}$ of the same order of
magnitude, by adjusting the overall scale $m_{0\nu}$ and choosing $x$ close to
$0.1$. No tuning of the unknown order-one parameters is needed in this case.
At the fixed point, $\Delta m^{2}_{sol}/\Delta
m^{2}_{atm}=\sin\theta_{13}=\sin^{2}\theta_{12}-1/2=0$ and CP is conserved.
Nonvanishing values of these three quantities and CP-violating effects all
originate from a small departure of $\tau$ from the fixed point. This is the
most successful pattern among all those discussed in this paper.

###  5.2 Fixed point 4

From table 4 and the analysis of the previous Section, we find the following
pattern for $m_{\bar{e}e}(u,\bar{u})$:

$\displaystyle
m_{\bar{e}e}=m_{0e}^{2}~{}\left(\begin{array}[]{ccc}y^{0}_{11}&y^{01}_{12}~{}\bar{u}_{1}&y^{10}_{13}~{}u_{1}\\\
y^{01}_{12}~{}u_{1}&y^{0}_{22}&y^{01}_{23}\bar{u}_{1}\\\
y^{10}_{13}~{}\bar{u}_{1}&y^{01}_{23}~{}u_{1}&y^{0}_{33}\end{array}\right)+\ldots\,.$
(105)

The overall real coefficient $m_{0e}^{2}$ has the dimension of (mass)2. Dots
stand for higher orders in the moduli expansion. The parameters
${y^{(0,01,10)}}_{ij}$ are real, independent of the moduli and expected to be
of the same order. To discuss the lepton mixing matrix, we diagonalize
$m_{\bar{e}e}(u,\bar{u})$:

$\displaystyle U_{e}^{\dagger}m_{\bar{e}e}(u,\bar{u})U_{e}={\tt
diag}[m_{\bar{e}e}(u,\bar{u})]\,.$ (106)

where $U_{e}$ is a nearly diagonal matrix:

$\displaystyle U_{e}=\left(\begin{array}[]{ccc}{\cal O}(1)&{\cal
O}(1)~{}\bar{u}_{1}&{\cal O}(1)~{}u_{1}\\\ {\cal O}(1)~{}u_{1}&{\cal
O}(1)&{\cal O}(1)~{}\bar{u}_{1}\\\ {\cal O}(1)~{}\bar{u}_{1}&{\cal
O}(1)~{}u_{1}&{\cal O}(1)\end{array}\right).$ (110)

Next, we examine the neutrino mass matrices. We find that they fall into three
classes, depending on the singlet representation $\bm{r}_{s}$ under which $w$
transforms.

  * $\bullet$ D

Up to cyclic permutations of rows and columns, when $\bm{r}_{s}$ embeds one of
the $G_{0}$ singlets $\bm{1}_{++n}$ $(n=0,1,2)$, we get the following pattern,
$m_{0\nu}$ denoting a real mass parameter:

$\displaystyle
m_{\nu}(u,\bar{u})=m_{0\nu}~{}\left(\begin{array}[]{ccc}x^{0}_{11}&x^{01}_{12}~{}\bar{u}_{1}&x^{10}_{13}~{}u_{1}\\\
\cdot&x^{10}_{22}~{}u_{1}&x^{0}_{23}\\\
\cdot&\cdot&x^{01}_{33}~{}\bar{u}_{1}\\\ \end{array}\right)+\ldots\,.$ (114)

The coefficients ${x^{(0,01,10)}}_{ij}$ are real and independent of the moduli
$u_{1,3}$.

  * $\bullet$ E

Up to cyclic permutations of rows and columns, when $\bm{r}_{s}$ embeds one of
the $G_{0}$ singlets $\bm{1}_{--n}$ $(n=0,1,2)$, we get the following pattern,
$m_{0\nu}$ denoting a real mass parameter:

$\displaystyle
m_{\nu}(u,\bar{u})=m_{0\nu}~{}\left(\begin{array}[]{ccc}x^{01}_{11}~{}\bar{u}_{3}&x^{10}_{12}~{}u_{3}&x^{11}_{13}~{}u_{1}\bar{u}_{3}+x^{{}^{\prime}11}_{13}~{}\bar{u}_{1}u_{3}\\\
\cdot&x^{11}_{22}~{}u_{1}\bar{u}_{3}+x^{{}^{\prime}11}_{22}~{}\bar{u}_{1}u_{3}&x^{01}_{23}~{}\bar{u}_{3}\\\
\cdot&\cdot&x^{10}_{33}~{}u_{3}\\\ \end{array}\right)+\ldots\,.$ (118)

The coefficients ${x^{(01,10,11)}}_{ij}$ and $x^{{}^{\prime}11}_{ij}$ are real
and independent of the moduli $u_{1,3}$.

  * $\bullet$ F

When $\bm{r}_{s}$ embeds one of the $G_{0}$ singlets $\bm{1}_{+-n}$ or
$\bm{1}_{-+n}$ $(n=0,1,2)$, to all orders in the moduli expansion the neutrino
mass matrix vanishes, an option we discard as unphysical.

Given the form of $U_{e}$ in eq. (110), in the basis where
$m_{\bar{e}e}(u,\bar{u})$ is diagonal, up to a common permutation matrix of
rows and columns and up to higher-order terms in the expansion, the neutrino
mass matrix has always the same pattern shown in eqs. (114) and (118). To
first order in $u_{1}$ and $\bar{u}_{1}$, the effect of the basis change can
be absorbed in the coefficients ${x^{(01,10,11)}}_{ij}$ and
$x^{{}^{\prime}11}_{ij}$. Thus, without losing generality, we can discuss the
neutrino mass spectrum, mixing angles and phases by directly analyzing the
matrices (114) and (118).

The pattern in eq. (114) coincides with the one occurring in single-modulus
$SL(2,\mathbbm{Z})$-invariant theories close to the fixed point
$\tau_{0}=\omega$ [44]. We refer the reader to the literature for more details
999In particular, the predictions for the lepton masses and lepton mixing
matrix as well as mixing parameters have been presented in Appendix D.2 and
Section 6 of ref. [44].. In table 6 we summarize the predictions of Siegel
modular invariant models for lepton masses in the vicinity of the fixed point
4, up to possible permutations affecting the mixing matrix. Within the pattern
in eq. (114), both normal and inverted ordering of the neutrino mass spectrum
can be realized. However, $\Delta m^{2}_{sol}/\Delta m^{2}_{atm}={\cal O}(x)$
would demand $x\approx 0.03$, which is inadequate to describe
$\sin^{2}\theta_{13}$ and $\sin^{2}\theta_{23}$, both expected of ${\cal
O}(x^{2})$. A considerable tuning of the order-one coefficients is required to
reconcile this pattern with the data. Finally, the pattern in eq. (118)
predicts neutrino masses of the same order of magnitude and mixing angle of
approximately the same size, as a numerical simulation shows. While data can
be reproduced by fitting the unknown order-one coefficients, this pattern does
not suggest any explanation for the smallness of $\Delta m^{2}_{sol}/\Delta
m^{2}_{atm}$ and/or $\sin^{2}\theta_{13}$.

###  5.3 Summary

Excluding the unphysical case of a vanishing neutrino mass matrix, the four
patterns of eqs. (97,101,114,118) exhaust all possible cases that can arise
from a CP and Siegel modular invariant locally supersymmetric theory when the
moduli are close to a fixed point. The predictions are summarized in table 6,
where the scaling properties of the leptonic mixing angles and the ratio
between solar and atmospheric squared mass differences are shown. The distance
from the fixed point is parametrized by a small variable $x$. The only pattern
that accommodates all the data without tuning of the unknown parameters is B
of eq. (101), when $m_{\nu}(0,0)$ is singular. It can be realized by the
seesaw mechanism when one of the right-handed neutrinos happens to be massless
at the fixed point. All the other patterns require an adjustment of the
parameters to overcome the wrong scaling of one or several observable
quantities. An explicit model belonging to the class analyzed in this Section
has been discussed in ref. [73] and, for completeness, is illustrated in the
Appendix D, where we show that it matches pattern A of eq. (97).

Many properties of the mass spectrum and the mixing matrix follow mainly from
the decomposition of the representation $\Omega$ into irreducible components.
This idea was developed in ref. [89], where the decompositions of $\Omega$,
and its charged lepton counterpart $\Omega_{c}$, compatible with a realistic
leading-order pattern of lepton masses and mixing angles have been classified.
While our results apply to a specific context and are not aimed to cover the
case of the most general flavour group acting linearly on matter fields, they
have been obtained under less restrictive assumptions. First, we go beyond the
zeroth order approximation by including the correction linear in the moduli.
Second, we also consider the case where the neutrino mass matrix develops
diverging eigenvalues at the fixed point, as happens in the seesaw mechanism
when a right-handed neutrino mass vanishes at the symmetric point. Third,
consistently with the freedom permitted by supergravity, we allow an overall
phase factor in the transformation of neutrino mass matrices.

##  6 Conclusion and outlook

To explore theories depending on more than one modulus, we have analyzed a
class of locally supersymmetric models of lepton masses invariant under CP and
under a subgroup $N(H)$ of the Siegel modular group $Sp(4,\mathbbm{Z})$. We
have concentrated on the subgroup $N(H)$ whose finite copies $N_{n}(H)$
contain three-dimensional irreducible representations, to which we assign
lepton electroweak doublets. For consistency, the moduli space is restricted
to a subset $\Sigma_{2}$ of the Siegel upper half plane, spanned by two
complex moduli and invariant under $N(H)$. We have identified 168 irreducible
triplets of $N_{n}(H)$, which we conjecture to exhaust the number of
inequivalent three-dimensional irreducible representations. There are four
inequivalent fixed points in $\Sigma_{2}$, each left invariant by a specific
finite group $G_{0}$. By exploiting the decomposition of each irreducible
triplet of $N_{n}(H)$ under $G_{0}$, and a convenient basis for moduli and
matter fields, we have built all possible patterns of neutrino mass matrices,
consisting of a series expansion around each fixed point, of which we keep the
first nontrivial term. The leading-order contribution is invariant under both
$G_{0}$ and CP, which are spontaneously broken by the vacuum expectation value
of an order parameter, measuring the distance from the fixed point. After
moving to the basis where kinetic terms are canonical and the charged lepton
mass matrix is diagonal, we can read neutrino masses and lepton mixing angles.

Apart from the unrealistic case of a vanishing neutrino mass matrix, only five
patterns are found. Four of them coincide with those arising in
$SL(2,\mathbbm{Z})$-invariant single-modulus theories in the vicinity of the
fixed points $\tau_{0}=i$ and $\tau_{0}=-1/2+i~{}\sqrt{3}/2$. In each pattern,
all physical quantities scale with the distance of the moduli from the fixed
point in a way that is largely independent of the details of the theory. All
the patterns but a single one require tuning the free parameters to match such
a scaling with the smallness of $\Delta m^{2}_{sol}/\Delta m^{2}_{atm}$ and
$\sin^{2}\theta_{13}$ and the largeness of $\sin^{2}\theta_{12}$ and
$\sin^{2}\theta_{23}$. The matrix that best describes the data and needs no
tuning is pattern B of eq. (101), when $m_{\nu}(0,0)$ is singular. This is the
same pattern preferred by the majority of single-modulus models when $\tau$ is
close to the self-dual point $\tau_{0}=i$.

We stress the generality of this result. Except for the assumption that the
lepton doublets transform under $N_{n}(H)$ as any of the 168 irreducible
triplets, our finding is independent of the level, of the weights of matter
multiplets, and the form of the Kähler potential. In particular, we are not
forced to assume a minimal or flavor universal Kähler potential: our
conclusion holds for the most general Kähler potential compatible with Siegel
modular invariance. It would be very difficult and time-consuming to reproduce
our results by inspecting one-by-one all models compatible with our
assumptions. Today very few models of this type exist in the literature [72,
73, 90, 91]. When moduli fall in the vicinity of a fixed point, we find
agreement with the present analysis.

On the other hand, our analysis has some limitations. First, the dependence on
the chosen level, weights, and Kähler potential affect the unknown
coefficients of the patterns we found. We have assumed that such dependence
does not conspire to produce hierarchies among these coefficients, but we
cannot provide a mathematical proof of this statement, which seems to work
well in the case of single-modulus theories. Second, we have no dynamical
justification for working close to a fixed point. Here again, we rely on the
statistics accumulated in the single modulus theories and on the few
semianalytical results derived from the minimization of modular invariant
energy densities. Finally, new patterns may arise if we relax the assumption
that the lepton doublets transform as irreducible triplets of the finite
modular group. This is a realistic possibility, especially in the context of
string theory compactifications where typically the modular group is part of a
bigger eclectic group [32, 33, 34, 35, 36, 37, 38]. Allowing for reducible
representations would presumably open many new unexplored possibilities. Both
the two-dimensional region $\Sigma_{1}$ or the entire Siegel upper half plane
might be relevant in a more general analysis. A complete classification of all
representations of dimensionality one and two would be required to undertake
such a demanding task, which we leave for future work.

## Acknowledgements

We thank warmly Gianguido Dall’Agata and Davide Cassani, for a stimulating
correspondence. This work is supported by the INFN. GJD is supported by the
National Natural Science Foundation of China under Grant Nos. 12375104 and
11975224. XGL is supported by the National Science Foundation, under Grant No.
PHY-1915005.

##  Appendix A Group theory of $D_{4}$

The dihedral group $D_{4}$ is the symmetry group of a square, generated by the
two elements $a$ and $b$, satisfying $a^{4}=b^{2}=(ab)^{2}=1$. The element $a$
represents a rotation of $\pi/2$ of the square around its center, while the
element $b$ is a reflection around a symmetry axis. The group has 5 conjugacy
classes and 5 irreducible representations: four singlets $\bm{1}_{+}$,
$\bm{1}_{-}$, $\bm{1}^{\prime}_{+}$, $\bm{1}^{\prime}_{-}$ and 1 doublet
$\bm{2}$. The representation matrices of the generators $a$ and $b$ are

$\displaystyle\begin{array}[]{ccc}\bm{1}_{+}:{}{}&~{}a=+1,{}&~{}b=+1\,,\\\
\bm{1}_{-}:{}{}&~{}a=+1,{}&~{}b=-1\,,\\\
\bm{1}^{\prime}_{-}:{}{}&~{}a=-1,{}&~{}b=+1\,,\\\
\bm{1}^{\prime}_{+}:{}{}&~{}a=-1,{}&~{}b=-1\,,\\\
\bm{2}:{}{}&~{}a=\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right),{}&~{}b=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\,.\end{array}$ (A.10)

| $1C_{1}$ | $1C_{2}$ | $2C_{2}$ | $2C^{\prime}_{2}$ | $2C_{4}$
---|---|---|---|---|---
| $\\{b^{2}\\}$ | $\\{a^{2}\\}$ | $\\{b,a^{2}b\\}$ | $\\{ab,a^{3}b\\}$ | $\\{a,a^{3}\\}$
$\bm{1}_{+}$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$
$\bm{1}_{-}$ | $+1$ | $+1$ | $-1$ | $-1$ | $+1$
$\bm{1}^{\prime}_{-}$ | $+1$ | $+1$ | $+1$ | $-1$ | $-1$
$\bm{1}^{\prime}_{+}$ | $+1$ | $+1$ | $-1$ | $+1$ | $-1$
$\bm{2}$ | $+2$ | $-2$ | $0$ | 0 | $0$
Table 7: Character table of $D_{4}$, where $mC_{n}$ denotes a conjugacy class
with $m$ elements of order $n$.

The character table is shown in table 7. The product of representations
decomposes as follows:

$\displaystyle\bm{1}_{m}\otimes\bm{1}_{n}=\bm{1}_{mn},\qquad\bm{1}_{m}\otimes\bm{1^{\prime}}_{n}=\bm{1^{\prime}}_{mn},\qquad\bm{1^{\prime}}_{m}\otimes\bm{1^{\prime}}_{n}=\bm{1}_{mn}$
$\displaystyle\bm{1}_{m}\otimes\bm{2}=\bm{2},\qquad\bm{1}^{\prime}_{m}\otimes\bm{2}=\bm{2}\,,\qquad\bm{2}\otimes\bm{2}=\bm{1}_{+}\oplus\bm{1}_{-}\oplus\bm{1^{\prime}}_{+}\oplus\bm{1^{\prime}}_{-}\,.$
(A.11)

We define:

$\displaystyle\bm{1}_{\pm}=\beta_{\pm}\quad\quad\quad\bm{1^{\prime}}_{\pm}=\beta^{\prime}_{\pm}\quad\quad\quad\bm{2}=\left(\begin{array}[]{c}\alpha_{1}\\\
\alpha_{2}\end{array}\right)$ (A.14)

and we get:

$\displaystyle\bm{1}_{+}\otimes\bm{2}=\bm{2}=\left(\begin{array}[]{c}\beta_{+}\alpha_{1}\\\
\beta_{+}\alpha_{2}\end{array}\right)\,,\qquad$
$\displaystyle\bm{1}_{-}\otimes\bm{2}=\bm{2}=\left(\begin{array}[]{c}\beta_{-}\alpha_{2}\\\
-\beta_{-}\alpha_{1}\end{array}\right)\,.$ (A.19)
$\displaystyle\bm{1^{\prime}}_{+}\otimes\bm{2}=\bm{2}=\left(\begin{array}[]{c}\beta^{\prime}_{+}\alpha_{2}\\\
\beta^{\prime}_{+}\alpha_{1}\end{array}\right)\,,\qquad$
$\displaystyle\bm{1^{\prime}}_{-}\otimes\bm{2}=\bm{2}=\left(\begin{array}[]{c}\beta^{\prime}_{-}\alpha_{1}\\\
-\beta^{\prime}_{-}\alpha_{2}\end{array}\right)\,.$ (A.24)

$\displaystyle\bm{2}\otimes\bm{2}=\bm{1}_{+}\oplus\bm{1}_{-}\oplus\bm{1^{\prime}}_{+}\oplus\bm{1^{\prime}}_{-},~{}~{}~{}~{}~{}~{}~{}~{}\left\\{\begin{array}[]{c}\bm{1}_{+}=~{}\alpha_{1}\beta_{1}+\alpha_{2}\beta_{2}\\\
\bm{1}_{-}=~{}\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\\\
\bm{1}^{\prime}_{-}=~{}\alpha_{1}\beta_{1}-\alpha_{2}\beta_{2}\\\
\bm{1}^{\prime}_{+}=~{}\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1}\end{array}\right.\,.$
(A.29)

An equivalent basis $(a^{\prime},b^{\prime})$ for the $D_{4}$ generators is
obtained by the similarity transformation acting on doublets:

$a^{\prime}=U^{-1}a~{}U=\left(\begin{array}[]{cc}i&0\\\
0&-i\end{array}\right)\,,~{}~{}~{}~{}~{}~{}~{}~{}b^{\prime}=U^{-1}b~{}U=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),$ (A.30)

where $U$ is the matrix

$U=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}i&i\\\ 1&-1\end{array}\right).$
(A.31)

Singlet representations are unchanged in this new basis, where the consistency
conditions of CP transformation in eq. (81) are solved by
$X_{I}^{\prime}=\mathbbm{1}$, both for singlets and doublets when working at
the fixed point 5. Of course, this new basis is inconvenient when dealing with
the fixed point 2, since a nontrivial matrix $X_{I}^{\prime}$ would be
required in the CP transformation law of doublets.

##  Appendix B Group theory of $D_{4}\circ Z_{4}$

The Pauli group is the central product of $D_{4}$ and $Z_{4}$: $D_{4}\circ
Z_{4}$, which has the GAP id [16,13] and is isomorphic to $(Z_{2}\times
Z_{4})\rtimes Z_{2}$. It is generated by the elements $a$, $b$ and $c$,
satisfying $a^{4}=c^{4}=b^{2}=(ab)^{2}=1$, $a^{2}=c^{2}$, $ac=ca$, $bc=cb$.
The elements $(a,b)$ generate $D_{4}$ and $c$ generates $Z_{4}$. The elements
of $D_{4}$ commute with those of $Z_{4}$. They have in common the element
$a^{2}=c^{2}$ and for this reason the central product does not coincide with
the direct product. The group has ten irreducible representations: eight
singlets and two doublets. The generators $a$, $b$ and $c$ in each irreducible
representation are represented by

$\displaystyle\begin{array}[]{cccc}\bm{1}_{+++}:{}{}&~{}a=+1,{}&~{}b=+1,{}&~{}c=+1\,,\\\
\bm{1}_{+--}:{}{}&~{}a=+1,{}&~{}b=-1,{}&~{}c=-1\,,\\\
\bm{1}_{+-+}:{}{}&~{}a=+1,{}&~{}b=-1,{}&~{}c=+1\,,\\\
\bm{1}_{---}:{}{}&~{}a=-1,{}&~{}b=-1,{}&~{}c=-1\,,\\\
\bm{1}_{--+}:{}{}&~{}a=-1,{}&~{}b=-1,{}&~{}c=+1\,,\\\
\bm{1}_{-+-}:{}{}&~{}a=-1,{}&~{}b=+1,{}&~{}c=-1\,,\\\
\bm{1}_{-++}:{}{}&~{}a=-1,{}&~{}b=+1,{}&~{}c=+1\,,\\\
\bm{1}_{++-}:{}{}&~{}a=+1,{}&~{}b=+1,{}&~{}c=-1\,,\\\
\bm{2}:{}{}&~{}a=\left(\begin{array}[]{cc}i&0\\\
0&-i\end{array}\right),{}&~{}b=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),{}&~{}c=\left(\begin{array}[]{cc}-i&0\\\
0&-i\end{array}\right)\,,\\\\[21.68121pt]
\bm{2}^{\prime}:{}{}&~{}a=\left(\begin{array}[]{cc}i&0\\\
0&-i\end{array}\right),{}&~{}b=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),{}&~{}c=\left(\begin{array}[]{cc}i&0\\\
0&i\end{array}\right)\,.\end{array}$ (B.23)

| $1C_{1}$ | $1C_{2}$ | $1C_{4}$ | $1C^{\prime}_{4}$ | $2C_{2}$ | $2C^{\prime}_{2}$ | $2C^{\prime\prime}_{2}$ | $2C_{4}$ | $2C^{\prime}_{4}$ | $2C^{\prime\prime}_{4}$
---|---|---|---|---|---|---|---|---|---|---
| $\\{b^{2}\\}$ | $\\{c^{2}\\}$ | $\\{c\\}$ | $\\{a^{2}c\\}$ | $\\{b,a^{2}b\\}$ | $\\{ab,abc^{2}\\}$ | $\\{ac,ac^{3}\\}$ | $\\{a,ac^{2}\\}$ | $\\{bc,a^{2}bc\\}$ | $\\{abc,abc^{3}\\}$
$\bm{1}_{+++}$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$
$\bm{1}_{+--}$ | $+1$ | $+1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $+1$ | $+1$ | $+1$
$\bm{1}_{+-+}$ | $+1$ | $+1$ | $+1$ | $+1$ | $-1$ | $-1$ | $+1$ | $+1$ | $-1$ | $-1$
$\bm{1}_{---}$ | $+1$ | $+1$ | $-1$ | $-1$ | $-1$ | $+1$ | $+1$ | $-1$ | $+1$ | $-1$
$\bm{1}_{--+}$ | $+1$ | $+1$ | $+1$ | $+1$ | $-1$ | $+1$ | $-1$ | $-1$ | $-1$ | $+1$
$\bm{1}_{-+-}$ | $+1$ | $+1$ | $-1$ | $-1$ | $+1$ | $-1$ | $+1$ | $-1$ | $-1$ | $+1$
$\bm{1}_{-++}$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$ | $-1$ | $-1$ | $-1$ | $+1$ | $-1$
$\bm{1}_{++-}$ | $+1$ | $+1$ | $-1$ | $-1$ | $+1$ | $+1$ | $-1$ | $+1$ | $-1$ | $-1$
$\bm{2}$ | $+2$ | $-2$ | $-2i$ | $+2i$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\bm{2}^{\prime}$ | $+2$ | $-2$ | $+2i$ | $-2i$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$

Table 8: Character table of $D_{4}\circ Z_{4}$.

The character table is shown in table 8. The product of representations
decomposes as follows:

$\displaystyle\bm{1}_{mnp}\otimes\bm{1}_{m^{\prime}n^{\prime}p^{\prime}}=$
$\displaystyle\bm{1}_{(mm^{\prime})(nn^{\prime})(pp^{\prime})}\,,$
$\displaystyle\bm{2}\otimes\bm{2}=\bm{2}^{\prime}\otimes\bm{2}^{\prime}=$
$\displaystyle~{}\bm{1}_{---}\oplus\bm{1}_{+--}\oplus\bm{1}_{-+-}\oplus\bm{1}_{++-}\,,$
$\displaystyle\bm{2}\otimes\bm{2}^{\prime}=$
$\displaystyle~{}\bm{1}_{+++}\oplus\bm{1}_{-++}\oplus\bm{1}_{+-+}\oplus\bm{1}_{--+}\,,$
$\displaystyle\hskip-86.72377pt\left(\begin{array}[]{c}\bm{2}\\\
\bm{2}^{\prime}\end{array}\right)\otimes\left(\bm{1}_{+++},\bm{1}_{-++},\bm{1}_{+-+},\bm{1}_{--+}\right)=~{}\left(\begin{array}[]{c}\bm{2}\\\
\bm{2}^{\prime}\end{array}\right)\,,$ (B.28)
$\displaystyle\hskip-86.72377pt\left(\begin{array}[]{c}\bm{2}\\\
\bm{2}^{\prime}\end{array}\right)\otimes\left(\bm{1}_{---},\bm{1}_{+--},\bm{1}_{-+-},\bm{1}_{++-}\right)=~{}\left(\begin{array}[]{c}\bm{2}^{\prime}\\\
\bm{2}\end{array}\right)\,.$ (B.33)

The tensor products between singlets and doublets are given by

$\displaystyle\bm{1}_{+++}\otimes\bm{2}=\bm{2}=\alpha\begin{pmatrix}\beta_{1}\\\
\beta_{2}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{+++}\otimes\bm{2}^{\prime}=\bm{2}^{\prime}=\alpha\begin{pmatrix}\beta_{1}\\\
\beta_{2}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{++-}\otimes\bm{2}=\bm{2}^{\prime}=\alpha\begin{pmatrix}\beta_{1}\\\
\beta_{2}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{++-}\otimes\bm{2}^{\prime}=\bm{2}=\alpha\begin{pmatrix}\beta_{1}\\\
\beta_{2}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{+--}\otimes\bm{2}=\bm{2}^{\prime}=\alpha\begin{pmatrix}\beta_{1}\\\
-\beta_{2}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{+--}\otimes\bm{2}^{\prime}=\bm{2}=\alpha\begin{pmatrix}\beta_{1}\\\
-\beta_{2}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{+-+}\otimes\bm{2}=\bm{2}=\alpha\begin{pmatrix}\beta_{1}\\\
-\beta_{2}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{+-+}\otimes\bm{2}^{\prime}=\bm{2}^{\prime}=\alpha\begin{pmatrix}\beta_{1}\\\
-\beta_{2}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{---}\otimes\bm{2}=\bm{2}^{\prime}=\alpha\begin{pmatrix}\beta_{2}\\\
-\beta_{1}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{---}\otimes\bm{2}^{\prime}=\bm{2}=\alpha\begin{pmatrix}\beta_{2}\\\
-\beta_{1}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{--+}\otimes\bm{2}=\bm{2}=\alpha\begin{pmatrix}\beta_{2}\\\
-\beta_{1}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{--+}\otimes\bm{2}^{\prime}=\bm{2}^{\prime}=\alpha\begin{pmatrix}\beta_{2}\\\
-\beta_{1}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{-+-}\otimes\bm{2}=\bm{2}^{\prime}=\alpha\begin{pmatrix}\beta_{2}\\\
\beta_{1}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{-+-}\otimes\bm{2}^{\prime}=\bm{2}=\alpha\begin{pmatrix}\beta_{2}\\\
\beta_{1}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{-++}\otimes\bm{2}=\bm{2}=\alpha\begin{pmatrix}\beta_{2}\\\
\beta_{1}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{-++}\otimes\bm{2}^{\prime}=\bm{2}^{\prime}=\alpha\begin{pmatrix}\beta_{2}\\\
\beta_{1}\end{pmatrix}\,.$ (B.34)

The tensor products between two doublets are

$\displaystyle\bm{2}\otimes\bm{2}=\bm{1}_{+--}\oplus\bm{1}_{---}\oplus\bm{1}_{-+-}\oplus\bm{1}_{++-},~{}~{}~{}~{}~{}~{}~{}~{}\left\\{\begin{array}[]{c}\bm{1}_{+--}=~{}\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\\\
\bm{1}_{---}=~{}\alpha_{1}\beta_{1}-\alpha_{2}\beta_{2}\\\
\bm{1}_{-+-}=~{}\alpha_{1}\beta_{1}+\alpha_{2}\beta_{2}\\\
\bm{1}_{++-}=~{}\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1}\end{array}\right.\,,$
(B.39)
$\displaystyle\bm{2}\otimes\bm{2}^{\prime}=\bm{1}_{+++}\oplus\bm{1}_{+-+}\oplus\bm{1}_{--+}\oplus\bm{1}_{-++},~{}~{}~{}~{}~{}~{}~{}~{}\left\\{\begin{array}[]{c}\bm{1}_{+++}=~{}\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1}\\\
\bm{1}_{+-+}=~{}\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\\\
\bm{1}_{--+}=~{}\alpha_{1}\beta_{1}-\alpha_{2}\beta_{2}\\\
\bm{1}_{-++}=~{}\alpha_{1}\beta_{1}+\alpha_{2}\beta_{2}\end{array}\right.\,,$
(B.44)
$\displaystyle\bm{2}^{\prime}\otimes\bm{2}^{\prime}=\bm{1}_{+--}\oplus\bm{1}_{---}\oplus\bm{1}_{-+-}\oplus\bm{1}_{++-},~{}~{}~{}~{}~{}~{}~{}~{}\left\\{\begin{array}[]{c}\bm{1}_{+--}=~{}\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\\\
\bm{1}_{---}=~{}\alpha_{1}\beta_{1}-\alpha_{2}\beta_{2}\\\
\bm{1}_{-+-}=~{}\alpha_{1}\beta_{1}+\alpha_{2}\beta_{2}\\\
\bm{1}_{++-}=~{}\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1}\end{array}\right.\,,$
(B.49)

##  Appendix C Group theory of $D_{4}\times Z_{3}$

The group $D_{4}\times Z_{3}$ is generated by the elements $a$ and $b$ and
$c$, satisfying $a^{4}=b^{2}=(ab)^{2}=c^{3}=1$, $ca=ac$, $cb=bc$. The elements
$(a,b)$ generate $D_{4}$ and $c$ generates $Z_{3}$. This group has 15
conjugacy classes as follows,

$\displaystyle 1C_{1}=\left\\{1\right\\}\,,$ $\displaystyle
1C_{2}=\left\\{a^{2}\right\\}\,,$ $\displaystyle 1C_{3}=\left\\{c\right\\}\,,$
$\displaystyle 1C^{\prime}_{3}=\left\\{c^{2}\right\\}\,,$ $\displaystyle
1C_{6}=\left\\{a^{2}c\right\\}\,,$ $\displaystyle
1C^{\prime}_{6}=\left\\{a^{2}c^{2}\right\\}\,,$ $\displaystyle
2C_{2}=\left\\{b,a^{2}b\right\\}\,,$ $\displaystyle
2C^{\prime}_{2}=\left\\{ab,a^{3}b\right\\}\,,$ $\displaystyle
2C_{6}=\left\\{bc,a^{2}bc\right\\}\,,$ $\displaystyle
2C^{\prime}_{6}=\left\\{bc^{2},a^{2}bc^{2}\right\\}\,,$ $\displaystyle
2C^{\prime\prime}_{6}=\left\\{abc,a^{3}bc\right\\}\,,$ $\displaystyle
2C^{(3)}_{6}=\left\\{abc^{2},a^{3}bc^{2}\right\\}\,,$ $\displaystyle
2C_{4}=\left\\{a,a^{3}\right\\}\,,$ $\displaystyle
2C_{12}=\left\\{ac,a^{3}c\right\\}\,,$ $\displaystyle
2C^{\prime}_{12}=\left\\{ac^{2},a^{3}c^{2}\right\\}\,.$ (C.1)

The irreducible representations of $D_{4}\times Z_{3}$ are 12 singlets and 3
doublets, the representation matrices of the generators are

$\displaystyle\begin{array}[]{cccc}\bm{1}_{++n}:{}{}&~{}a=+1,{}&~{}b=+1,{}&~{}c=\omega^{n}\,,\\\
\bm{1}_{+-n}:{}{}&~{}a=+1,{}&~{}b=-1,{}&~{}c=\omega^{n}\,,\\\
\bm{1}_{-+n}:{}{}&~{}a=-1,{}&~{}b=+1,{}&~{}c=\omega^{n}\,,\\\
\bm{1}_{--n}:{}{}&~{}a=-1,{}&~{}b=-1,{}&~{}c=\omega^{n}\,,\\\
\bm{2}_{n}:{}{}&~{}a=\left(\begin{array}[]{cc}+i&0\\\
0&-i\end{array}\right),{}&~{}b=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),{}&~{}c=\left(\begin{array}[]{cc}\omega^{n}&0\\\
0&\omega^{n}\end{array}\right)\,,\end{array}$ (C.13)

with $n=0,1,2$ and $\omega=e^{2\pi i/3}$. The character table is listed in
table 9.

| $1C_{1}$ | $1C_{2}$ | $1C_{3}$ | $1C^{\prime}_{3}$ | $1C_{6}$ | $1C^{\prime}_{6}$ | $2C_{2}$ | $2C^{\prime}_{2}$ | $2C_{6}$ | $2C^{\prime}_{6}$ | $2C^{\prime\prime}_{6}$ | $2C^{(3)}_{6}$ | $2C_{4}$ | $2C_{12}$ | $2C^{\prime}_{12}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\bm{1}_{++0}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
$\bm{1}_{++1}$ | 1 | 1 | $\omega$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | 1 | 1 | $\omega$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | 1 | $\omega$ | $\omega^{2}$
$\bm{1}_{++2}$ | 1 | 1 | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $\omega$ | 1 | 1 | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $\omega$ | 1 | $\omega^{2}$ | $\omega$
$\bm{1}_{+-0}$ | 1 | 1 | 1 | 1 | 1 | 1 | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | 1 | 1 | 1
$\bm{1}_{+-1}$ | 1 | 1 | $\omega$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $-1$ | $-1$ | $-\omega$ | $-\omega^{2}$ | $-\omega$ | $-\omega^{2}$ | 1 | $\omega$ | $\omega^{2}$
$\bm{1}_{+-2}$ | 1 | 1 | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $\omega$ | $-1$ | $-1$ | $-\omega^{2}$ | $-\omega$ | $-\omega^{2}$ | $-\omega$ | 1 | $\omega^{2}$ | $\omega$
$\bm{1}_{-+0}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | $-1$ | 1 | 1 | $-1$ | $-1$ | $-1$ | $-1$ | $-1$
$\bm{1}_{-+1}$ | 1 | 1 | $\omega$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | 1 | $-1$ | $\omega$ | $\omega^{2}$ | $-\omega$ | $-\omega^{2}$ | $-1$ | $-\omega$ | $-\omega^{2}$
$\bm{1}_{-+2}$ | 1 | 1 | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $\omega$ | 1 | $-1$ | $\omega^{2}$ | $\omega$ | $-\omega^{2}$ | $-\omega$ | $-1$ | $-\omega^{2}$ | $-\omega$
$\bm{1}_{--0}$ | 1 | 1 | 1 | 1 | 1 | 1 | $-1$ | 1 | $-1$ | $-1$ | 1 | 1 | $-1$ | $-1$ | $-1$
$\bm{1}_{--1}$ | 1 | 1 | $\omega$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $-1$ | 1 | $-\omega$ | $-\omega^{2}$ | $\omega$ | $\omega^{2}$ | $-1$ | $-\omega$ | $-\omega^{2}$
$\bm{1}_{--2}$ | 1 | 1 | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $\omega$ | $-1$ | 1 | $-\omega^{2}$ | $-\omega$ | $\omega^{2}$ | $\omega$ | $-1$ | $-\omega^{2}$ | $-\omega$
$\bm{2}_{0}$ | 2 | $-2$ | 2 | 2 | $-2$ | $-2$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\bm{2}_{1}$ | 2 | $-2$ | $2\omega$ | $2\omega^{2}$ | $-2\omega$ | $-2\omega^{2}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\bm{2}_{2}$ | 2 | $-2$ | $2\omega^{2}$ | $2\omega$ | $-2\omega^{2}$ | $-2\omega$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0

Table 9: Character table of $D_{4}\times Z_{3}$.

The Kronecker products of different representations are as follows:

$\displaystyle\bm{1}_{mnp}\otimes\bm{1}_{m^{\prime}n^{\prime}p^{\prime}}=\bm{1}_{(mm^{\prime})(nn^{\prime})[p+p^{\prime}]}\,,~{}~{}~{}\bm{1}_{mnp}\otimes\bm{2}_{p^{\prime}}=\bm{2}_{[p+p^{\prime}]}\,,$
$\displaystyle\bm{2}_{p}\otimes\bm{2}_{p^{\prime}}=\bm{1}_{++[p+p^{\prime}]}\oplus\bm{1}_{+-[p+p^{\prime}]}\oplus\bm{1}_{-+[p+p^{\prime}]}\oplus\bm{1}_{--[p+p^{\prime}]}\,.$
(C.14)

where the integer $[n]\equiv n~{}(\text{mod}~{}3)$. The tensor products
between singlets and doublets are given by

$\displaystyle\bm{1}_{++p}\otimes\bm{2}_{p^{\prime}}=\bm{2}_{[p+p^{\prime}]}=\alpha\begin{pmatrix}\beta_{1}\\\
\beta_{2}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{+-p}\otimes\bm{2}_{p^{\prime}}=\bm{2}_{[p+p^{\prime}]}=\alpha\begin{pmatrix}\beta_{1}\\\
-\beta_{2}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{-+p}\otimes\bm{2}_{p^{\prime}}=\bm{2}_{[p+p^{\prime}]}=\alpha\begin{pmatrix}\beta_{2}\\\
\beta_{1}\end{pmatrix}\,,$
$\displaystyle\bm{1}_{--p}\otimes\bm{2}_{p^{\prime}}=\bm{2}_{[p+p^{\prime}]}=\alpha\begin{pmatrix}\beta_{2}\\\
-\beta_{1}\end{pmatrix}\,.$ (C.15)

The tensor products between the doublets are given by

$\displaystyle\bm{2}_{p}\otimes\bm{2}_{p^{\prime}}=\bm{1}_{++[p+p^{\prime}]}\oplus\bm{1}_{+-[p+p^{\prime}]}\oplus\bm{1}_{-+[p+p^{\prime}]}\oplus\bm{1}_{--[p+p^{\prime}]},~{}~{}~{}~{}\left\\{\begin{array}[]{c}\bm{1}_{++[p+p^{\prime}]}=~{}\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1}\\\
\bm{1}_{+-[p+p^{\prime}]}=~{}\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\\\
\bm{1}_{-+[p+p^{\prime}]}=~{}\alpha_{1}\beta_{1}+\alpha_{2}\beta_{2}\\\
\bm{1}_{--[p+p^{\prime}]}=~{}\alpha_{1}\beta_{1}-\alpha_{2}\beta_{2}\end{array}\right.\,.$
(C.20)

An equivalent basis $(a^{\prime},b^{\prime},c^{\prime})$ for the $D_{4}\times
Z_{3}$ generators is obtained by the similarity transformation acting on
doublets:

$a^{\prime}=U^{-1}a~{}U=\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right)~{}~{}~{}~{}~{}~{}~{}b^{\prime}=U^{-1}b~{}U=\left(\begin{array}[]{cc}-1&0\\\
0&1\end{array}\right)~{}~{}~{}~{}~{}~{}~{}c^{\prime}=U^{-1}c~{}U=c,$ (C.21)

where $U$ is the matrix

$U=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}-i&1\\\ i&1\end{array}\right).$
(C.22)

Singlet representations are unchanged in this new basis, where the consistency
conditions of CP transformation are solved by $X_{I}^{\prime}=\mathbbm{1}$,
both for singlets and doublets.

##  Appendix D  Lepton model of ref. [73]

To verify the above general analysis, we take the lepton model in ref. [73] as
an example,

$\displaystyle\rho_{E^{c}}=\mathbf{2}\oplus\mathbf{1},~{}~{}~{}\rho_{L}=\mathbf{3^{\prime}},~{}~{}~{}\rho_{H_{u}}=\rho_{H_{d}}=\mathbf{1}\,,$
$\displaystyle
k_{H_{u}}=k_{H_{d}}=0,~{}~{}~{}k_{E_{D}^{c}}=-3,~{}~{}k_{E_{3}^{c}}=k_{L}=-1\,.$
(D.1)

In this model, the superpotential $w$ are required to be modular invariant,
i.e. $\bm{r}_{s}=\bm{1}$. The weighted representation matrices can be obtained
from the appendix of ref [73] , which happen to have the following diagonal
form:

$\Omega_{L}(a)=\begin{pmatrix}-1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}\,,\qquad\Omega_{L}(b)=\begin{pmatrix}-1&0&0\\\ 0&-1&0\\\
0&0&-1\end{pmatrix}\,,$ (D.2) $\Omega_{E^{c}}(a)=\begin{pmatrix}-1&0&0\\\
0&1&0\\\ 0&0&1\end{pmatrix}\,,\qquad\Omega_{E^{c}}(b)=\begin{pmatrix}-1&0&0\\\
0&-1&0\\\ 0&0&-1\end{pmatrix}\,,$ (D.3)

Namely, they are decomposed under $D_{4}$ into

$\Omega_{L}\sim\mathbf{1}^{\prime}_{+}\oplus\mathbf{1}_{-}\oplus\mathbf{1}_{-}\,,\qquad\Omega_{E^{c}}\sim\mathbf{1}^{\prime}_{+}\oplus\mathbf{1}_{-}\oplus\mathbf{1}_{-}\,,$
(D.4)

they indeed belong to one of the four cases in eq. (82). The linearized moduli
$u_{1,3}$ are transformed as $\bm{1}^{\prime}_{-}\oplus\bm{1}_{+}$. As we
analyzed in Section 5, the charged lepton mass matrix $m_{e}$ and neutrino
mass matrix $m_{\nu}$ have the pattern A of eq. (97):

$\displaystyle m_{\bar{e}e}(u_{1},u_{3})$
$\displaystyle\simeq\begin{pmatrix}1+u_{3}+u_{3}^{2}+u_{1}^{2}&u_{1}+u_{1}u_{3}&u_{1}+u_{1}u_{3}\\\
u_{1}+u_{1}u_{3}&1+u_{3}+u_{3}^{2}+u_{1}^{2}&1+u_{3}+u_{3}^{2}+u_{1}^{2}\\\
u_{1}+u_{1}u_{3}&1+u_{3}+u_{3}^{2}+u_{1}^{2}&1+u_{3}+u_{3}^{2}+u_{1}^{2}\end{pmatrix}+\mathcal{O}(u_{i}^{3})$
$\displaystyle\simeq\begin{pmatrix}1&u_{1}&u_{1}\\\ u_{1}&1&1\\\
u_{1}&1&1\end{pmatrix}+\dots\,.$ (D.5) $\displaystyle m_{\nu}(u_{1},u_{3})$
$\displaystyle\simeq\begin{pmatrix}1+u_{3}+u_{3}^{2}+u_{1}^{2}&u_{1}+u_{1}u_{3}&u_{1}+u_{1}u_{3}\\\
u_{1}+u_{1}u_{3}&1+u_{3}+u_{3}^{2}+u_{1}^{2}&1+u_{3}+u_{3}^{2}+u_{1}^{2}\\\
u_{1}+u_{1}u_{3}&1+u_{3}+u_{3}^{2}+u_{1}^{2}&1+u_{3}+u_{3}^{2}+u_{1}^{2}\end{pmatrix}+\mathcal{O}(u_{i}^{3})$
$\displaystyle\simeq\begin{pmatrix}1&u_{1}&u_{1}\\\ u_{1}&1&1\\\
u_{1}&1&1\end{pmatrix}+\dots\,.$ (D.6)

Here we omit the unknown constant coefficients. Through the $u$-expansion of
the Siegel modular forms:

$\displaystyle((1-u_{1})^{2}-u_{3}^{2})^{-2}Y^{(2)}_{\mathbf{3},1}(u)=15.68u_{1}+\mathcal{O}(u_{i}^{3})\,,$
$\displaystyle((1-u_{1})^{2}-u_{3}^{2})^{-2}Y^{(2)}_{\mathbf{3},2}(u)=1.21+25.49u_{1}^{2}-6.37u_{3}^{2}+\mathcal{O}(u_{i}^{3})\,,$
$\displaystyle((1-u_{1})^{2}-u_{3}^{2})^{-2}Y^{(2)}_{\mathbf{3},3}(u)=-2.56+5.54u_{3}-27.04u_{1}^{2}+\mathcal{O}(u_{i}^{3})\,,$
$\displaystyle((1-u_{1})^{2}-u_{3}^{2})^{-2}Y^{(2)}_{\mathbf{1}}(u)=1.81+3.92u_{3}+19.12u_{3}^{2}+19.12u_{1}^{2}+\mathcal{O}(u_{i}^{3})\,.$
(D.7)

We can obtain the $u$-expansion form of mass matrices in ref. [73],

$\displaystyle m_{e}(u_{1},u_{3})$
$\displaystyle\simeq\begin{pmatrix}2.57&71.84u_{1}&-101.60u_{1}\\\
-71.84u_{1}&23.12&18.17\\\ 0.18u_{1}&0.014&-0.03\end{pmatrix}v_{d}+\dots\,.$
$\displaystyle m_{\nu}(u_{1},u_{3})$
$\displaystyle\simeq\begin{pmatrix}2.01&22.17u_{1}&15.68u_{1}\\\
22.17u_{1}&-1.40&1.21\\\
15.68u_{1}&1.21&7.97\end{pmatrix}\frac{v_{u}^{2}}{\Lambda}+\dots\,.$ (D.8)

where the free Lagrangian parameters are fixed at their best-fit values.
101010The best-fit values for Lagrangian parameters are
$\alpha=1,~{}\beta=-0.83991,~{}\gamma=0.01176,~{}g_{1}=1,~{}g_{2}=1.58030$.
And the best-fit values of the moduli vacua are
$\tau_{1}=\tau_{2}=-0.03376+1.11329i,\tau_{3}=-0.02376+0.50670i$, i.e.
$u_{1}=u_{2}=-0.00418+0.01298i,u_{3}=-0.02895+0.00474i$. This result is
consistent with our above general analysis. Since this pattern is not optimal
for reproducing the data, the coefficients multiplying $u_{1}$ are not of the
same order, but appear to be tuned to overcome the bad scaling shown in table
6.

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