Explicit Perturbations to the Stabilizer $\tau={\rm i}$ of Modular
$A^{\prime}_{5}$ Symmetry and Leptonic CP Violation

Xin Wang ${}^{a,~{}b}$ ***E-mail<EMAIL_ADDRESS>Shun Zhou ${}^{a,~{}b}$
†††E-mail<EMAIL_ADDRESS>(corresponding author)

aInstitute of High Energy Physics, Chinese Academy of Sciences, Beijing
100049, China
bSchool of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China

###### Abstract

In a class of neutrino mass models with modular flavor symmetries, it has been
observed that CP symmetry is preserved at the fixed point (or stabilizer) of
the modulus parameter $\tau={\rm i}$, whereas significant CP violation emerges
within the neighbourhood of this stabilizer. In this paper, we first construct
a viable model with the modular $A^{\prime}_{5}$ symmetry, and explore the
phenomenological implications for lepton masses and flavor mixing. Then, we
introduce explicit perturbations to the stabilizer at $\tau={\rm i}$, and
present both numerical and analytical results to understand why a small
deviation from the stabilizer leads to large CP violation. As low-energy
observables are very sensitive to the perturbations to model parameters, we
further demonstrate that the renormalization-group running effects play an
important role in confronting theoretical predictions at the high-energy scale
with experimental measurements at the low-energy scale.

##  1 Introduction

The experimental discovery of neutrino oscillations indicates that neutrinos
are actually massive and leptonic flavor mixing is significant, but both the
origin of neutrino masses and the leptonic flavor structure are largely
unknown at present [1, 2]. One of the simplest ways to accommodate tiny
neutrino masses is to extend the standard model (SM) by introducing three
right-handed neutrino singlets $N_{i{\rm R}}$ (for $i=1,2,3$) and implementing
the canonical seesaw mechanism [3, 4, 5, 6, 7], which attributes the smallness
of three ordinary neutrino masses to the largeness of three heavy Majorana
neutrino masses. On the other hand, the leptonic flavor mixing can be
accounted for by imposing some discrete flavor symmetries on the seesaw model.
In these models, a number of scalar fields transforming nontrivially under the
flavor symmetry group have to be introduced, and the flavor structures of
lepton mass matrices depend much on the assignments of the relevant fields
into the representations of the symmetry group and the choices of the vacuum
expectation values (vev’s) of the scalar fields [8, 9, 10, 11].

Recently, the modular symmetry has been suggested as a possible solution to
the flavor mixing puzzle [12]. In this framework, the Yukawa couplings in the
leptonic sector turn out to be modular forms, which are holomorphic functions
of the complex modulus $\tau$ and transform as irreducible representations of
finite modular groups $\Gamma_{N}$ (with $N\geq 2$ being positive integers).
Once the modulus $\tau$ acquires its vev, the Yukawa couplings are determined
and thus leptonic flavor mixing pattern is obtained. Finite modular groups,
such as $\Gamma_{2}\simeq S_{3}$ [13, 14, 15], $\Gamma_{3}\simeq A_{4}$ [12,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], $\Gamma_{4}\simeq S_{4}$ [28,
29, 30, 31, 32, 33], $\Gamma_{5}\simeq A_{5}$ [34, 35, 36],
$\Gamma_{7}\simeq{\rm PSL}(2,Z_{7})$ [37], and some of their double coverings
$\Gamma^{\prime}_{3}\simeq A^{\prime}_{4}$ [38, 39],
$\Gamma^{\prime}_{4}\simeq S^{\prime}_{4}$ [40, 41] and
$\Gamma^{\prime}_{5}\simeq A^{\prime}_{5}$ [42, 43], have been extensively
investigated in the previous literature.

In the bottom-up approach to the modular-invariant flavor models, the modulus
$\tau$ is usually treated as a free parameter, which can be pinned down by
fitting it together with other model parameters to experimental observations.
In contrast, in the top-down approach, it can be fixed by the modulus
stabilization via the minimum of the supergravity scalar potential [44] or the
flux compactifications [45]. Interestingly, residual symmetries have been
found after the spontaneous breaking of the global modular symmetry at some
special values of $\tau$, which are called fixed points or stabilizers [29,
46, 47]. If $\tau$ is exactly located at one of the stabilizers, it seems
unlikely to realize viable lepton flavor models by using only one modular
symmetry [48, 49] or a common value of $\tau$ in both the charged-lepton and
neutrino sectors [46, 50]. One can alternatively construct modular-invariant
models with the modulus close to the stabilizers, with which the strong
hierarchy of charged-lepton masses can be successfully realized [51, 52].

As pointed out in Ref. [53], if the generalized CP (gCP) symmetry is further
imposed on the modular-invariant model, then both CP and modular symmetries
are spontaneously broken by the vev of $\tau$. Moreover, all the stabilizers
are CP conserving, while a small deviation of the modulus from them may result
in large CP violation [53]. Motivated by these observations, we propose in
this paper a feasible lepton flavor model with the gCP symmetry and the
$\Gamma^{\prime}_{5}\simeq A^{\prime}_{5}$ symmetry. A salient feature of our
model is the prediction for a nearly-maximal CP violation when
$\tau\approx{\rm i}$ holds, whereas CP symmetry is preserved at $\tau={\rm
i}$. It is worth mentioning that a modular-invariant model with large CP
violation around the stabilizer $\tau={\rm i}$ of the modular $A_{4}$ group
has been constructed in Ref. [54]. Nevertheless, the question why the large CP
violation can be generated with a modulus close to $\tau={\rm i}$ remains to
be answered. To this end, starting with the model at the stabilizer $\tau={\rm
i}$, we introduce explicit perturbations to the modulus and other model
parameters, and derive the approximate analytical formulas of neutrino masses,
mixing angles and CP-violating phases. Through these analytical calculations,
we can see that the nearly-maximal CP violation mainly comes from the large
ratio between the real and imaginary parts of $\tau$. In addition, theoretical
predictions for some low-energy observables are very sensitive to the
perturbations to model parameters when $\tau$ is approaching ${\rm i}$. Such
findings also imply that the renormalization-group (RG) running effects may
have remarkable impact on the flavor parameters. Therefore, we also take into
account the RG running effects and verify that this is indeed the case.

The remaining part of this paper is structured as follows. In Sec. 2 we give a
brief introduction to the modular symmetry and the modular
$\Gamma^{\prime}_{5}\simeq A^{\prime}_{5}$ group in order to establish our
notations. Then the concrete model for lepton masses and flavor mixing based
on the modular $A^{\prime}_{5}$ symmetry is constructed in Sec. 3. The
phenomenological implications for low-energy observables are studied in Sec. 4
in a completely numerical way, while the analytical analysis of the
perturbations to the model with the stabilizer $\tau={\rm i}$ is performed in
Sec. 5. Radiative corrections to the mixing parameters via the RG running are
discussed in Sec. 6. We summarize our main results in Sec. 7. Finally, the
basic properties of the finite modular group $\Gamma^{\prime}_{5}\simeq
A^{\prime}_{5}$ are presented in Appendix A.

##  2 Modular Symmetry

In this section, we briefly review some basic knowledge about modular
symmetries and the modular $\Gamma^{\prime}_{5}\simeq A^{\prime}_{5}$ group in
order to establish our notations and set up the framework for later
discussions. The modular group is isomorphic to the special linear group ${\rm
SL}(2,\mathbb{Z})$, which is defined as [12]

$\displaystyle\Gamma\equiv\left\\{\left(\begin{matrix}a&&b\\\
c&&d\end{matrix}\right)\bigg{|}a,b,c,d\in\mathbb{Z}\;,~{}ad-bc=1\right\\}\;.$
(2.1)

There are in total three generators $S$, $T$ and $R$ for the modular group
$\Gamma$, the matrix representations of which are given by

$\displaystyle S=\left(\begin{matrix}0&&1\\\ -1&&0\end{matrix}\right)\;,\quad
T=\left(\begin{matrix}1&&1\\\ 0&&1\end{matrix}\right)\;,\quad
R=\left(\begin{matrix}-1&&0\\\ 0&&-1\end{matrix}\right)\;.$ (2.2)

The modular transformations on the modulus $\tau$ and chiral supermultiplets
$\chi^{(I)}$ are defined as

$\displaystyle\gamma:\tau\rightarrow\dfrac{a\tau+b}{c\tau+d}\;,\quad\chi^{(I)}\rightarrow(c\tau+d)^{-k_{I}}\rho^{(I)}(\gamma)\chi^{(I)}\;,$
(2.3)

where $\gamma$ is an element of the modular group $\Gamma$, $k_{I}$ is the
weight of the chiral supermultiplet and $\rho^{(I)}(\gamma)$ denotes the
unitary representation matrix of $\Gamma$. For the modular-invariant
supersymmetric theories, the action ${\cal S}$ should be unchanged under the
modular transformations given in Eq. (2.3). As a consequence, the
superpotential ${\cal W}$ has to be modular-invariant and the Kähler potential
${\cal K}$ should also remain unchanged up to the Kähler transformation. For a
phenomenological purpose, in the present paper we consider only the minimal
form of the Kähler potential, namely,

$\displaystyle{\cal K}(\tau,\bar{\tau},\chi,\bar{\chi})=-h\log(-{\rm
i}\tau+{\rm i}\bar{\tau})+\sum_{I}\frac{\left|\chi^{(I)}\right|^{2}}{(-{\rm
i}\tau+{\rm i}\bar{\tau})^{k_{I}}}\;,$

where $h$ is a positive constant. From Eq. (2.3) one can easily observe that
the transformations on $\tau$ induced by $\gamma$ and $-\gamma$ are actually
the same, while the matter fields $\chi^{(I)}$ are generally allowed to
transform nontrivially under $R$. Therefore, one should consider $\Gamma$
rather than $\overline{\Gamma}\equiv\ \Gamma/\\{\mathbb{I},-\mathbb{I}\\}$ as
the symmetry group in such theories. In this case, we can introduce the double
covering of finite modular groups $\Gamma^{\prime}_{N}\equiv\Gamma/\Gamma(N)$
with $\Gamma(N)$ being the principal congruence subgroups of the modular group
$\Gamma$, e.g., $\Gamma^{\prime}_{3}\simeq A^{\prime}_{4}$,
$\Gamma^{\prime}_{4}\simeq S^{\prime}_{4}$ and $\Gamma^{\prime}_{5}\simeq
A^{\prime}_{5}$.

The modular form $f(\tau)$ of level $N$ and weight $k$ is a holomorphic
function of $\tau$, which transforms under $\Gamma(N)$ as

$\displaystyle
f\left(\gamma\tau\right)=(c\tau+d)^{k}f(\tau)\;,\quad\gamma\in\Gamma(N)\;,$
(2.4)

where $k\geq 0$ is an integer. As has been proved in Ref. [38], for a given
modular space ${\cal M}_{k}\left[\Gamma(N)\right]$, the modular forms can
always be decomposed into several multiplets that transform as irreducible
unitary representations of $\Gamma^{\prime}_{N}$. To be more precise, we can
always find a proper basis of the modular space ${\cal
M}_{k}\left[\Gamma(N)\right]$ such that a modular multiplet $Y^{(k)}_{\bf
r}=(f_{1}(\tau),f_{2}(\tau),\cdots)^{\rm T}$ in the representation ${\bf r}$
satisfies the following equation

$\displaystyle Y^{(k)}_{\bf r}(\gamma\tau)=(c\tau+d)^{k}\rho_{\bf
r}(\gamma)Y^{(k)}_{\bf r}(\tau)\;,\quad\gamma\in\Gamma\;,$ (2.5)

where $\rho_{\bf r}(\gamma)$ is the identity matrix for $\gamma\in\Gamma(N)$,
therefore it is essentially the representation matrix of the quotient group
$\Gamma^{\prime}_{N}\equiv\Gamma/\Gamma(N)$. For $\Gamma^{\prime}_{5}\simeq
A^{\prime}_{5}$ in question, the group structure and modular forms with
weights from one to six have been analyzed in detail in Ref. [42]. Hence we
just recapitulate the key points relevant for the following discussions. The
$A^{\prime}_{5}$ group has 120 elements, which can be produced by three
generators $S$, $T$ and $R$ satisfying the identities

$\displaystyle S^{2}=R\;,\quad(ST)^{3}=\mathbb{I}\;,\quad
T^{5}=\mathbb{I}\;,\quad R^{2}=\mathbb{I}\;,\quad RT=TR\;.$ (2.6)

There are nine distinct irreducible representations for $A^{\prime}_{5}$,
where the representations ${\bf 1}$, ${\bf 3}$, ${\bf 3}^{\prime}$, ${\bf 4}$
and ${\bf 5}$ with $R$ represented by the unit matrix coincide with those for
$A_{5}$, whereas $\widehat{\bf 2}$, $\widehat{\bf 2}^{\prime}$, $\widehat{\bf
4}$ and $\widehat{\bf 6}$ are unique for $A^{\prime}_{5}$ with $R$ represented
by the minus unit matrix. The irreducible representation matrices of $S$, $T$
and $R$, as well as the decomposition rules of the Kronecker products relevant
for the present work, can be found in Appendix A.

The modular forms $Y^{(1)}_{\widehat{\bf 6}}(\tau)$ with the lowest nontrivial
weight $k=1$ can be expressed as the linear combinations of six basis vectors
$\widehat{e}_{i}$ (for $i=1,\cdots,6$) in the modular space ${\cal
M}_{1}[\Gamma(5)]$, whose explicit forms as well as Fourier expansions are
given in Appendix A. Six components of $Y^{(1)}_{\widehat{\bf 6}}(\tau)$ will
be denoted as $Y_{i}(\tau)$ (for $i=1,2,\cdots,6$), i.e.,

$\displaystyle Y^{(1)}_{\widehat{\bf 6}}=\left(\begin{matrix}Y_{1}\\\ Y_{2}\\\
Y_{3}\\\ Y_{4}\\\ Y_{5}\\\ Y_{6}\\\
\end{matrix}\right)=\left(\begin{matrix}\widehat{e}_{1}-3\,\widehat{e}_{6}\\\
5\sqrt{2}\,\widehat{e}_{2}\\\ 10\,\widehat{e}_{3}\\\ 10\,\widehat{e}_{4}\\\
5\sqrt{2}\,\widehat{e}_{5}\\\ -3\,\widehat{e}_{1}-\widehat{e}_{6}\\\
\end{matrix}\right)\;,$ (2.7)

where the argument $\tau$ of all relevant functions has been suppressed. In
addition, we write down the modular forms that will be used for the model
building in Sec. 3. For weight two, we have

$\displaystyle Y^{(2)}_{{\bf
3}^{\prime}}=\dfrac{1}{2}\left(\begin{array}[]{c}\sqrt{6}\left(Y^{2}_{1}-2Y_{1}Y_{6}-Y^{2}_{6}\right)\\\
-\sqrt{3}Y_{3}(Y_{1}+Y_{6})\\\ \sqrt{3}Y_{4}(Y_{1}-Y_{6})\\\
\end{array}\right)\;,\quad Y^{(2)}_{\bf
5}=\dfrac{1}{2}\left(\begin{array}[]{c}\sqrt{2}\left(Y^{2}_{1}+Y^{2}_{6}\right)\\\
2\sqrt{6}Y_{2}\left(2Y_{1}+Y_{6}\right)\\\
\sqrt{3}Y_{3}\left(Y_{6}-3Y_{1}\right)\\\
\sqrt{3}Y_{4}\left(Y_{1}+3Y_{6}\right)\\\
2\sqrt{6}Y_{5}\left(2Y_{6}-Y_{1}\right)\\\ \end{array}\right)\;.$ (2.16)

For weight three, we will use

$\displaystyle Y^{(3)}_{\widehat{\bf 6},1}$ $\displaystyle=$
$\displaystyle-\dfrac{\sqrt{3}}{2}\left(\begin{array}[]{c}5Y_{1}^{3}-12Y_{1}^{2}Y_{6}-11Y_{1}Y_{6}^{2}-2Y_{6}^{3}\\\
-2Y_{2}\left(Y_{1}^{2}-5Y_{1}Y_{6}-2Y_{6}^{2}\right)\\\
Y_{3}\left(Y_{1}+Y_{6}\right)\left(Y_{1}+2Y_{6}\right)\\\
Y_{4}\left(Y_{1}-Y_{6}\right)\left(2Y_{1}-Y_{6}\right)\\\
2Y_{5}\left(2Y_{1}^{2}-5Y_{1}Y_{6}-Y_{6}^{2}\right)\\\
2Y_{1}^{3}-11Y_{1}^{2}Y_{6}+12Y_{1}Y_{6}^{2}+5Y_{6}^{3}\\\
\end{array}\right)\;,$ (2.23) $\displaystyle Y^{(3)}_{\widehat{\bf 6},2}$
$\displaystyle=$
$\displaystyle-\dfrac{\sqrt{3}}{2}\left(\begin{array}[]{c}3Y_{1}^{3}-9Y_{1}^{2}Y_{6}-Y_{1}Y_{6}^{2}+Y_{6}^{3}\\\
-2Y_{2}\left(2Y_{1}^{2}-2Y_{1}Y_{6}-Y_{6}^{2}\right)\\\ 2Y_{1}^{2}Y_{3}\\\
2Y_{4}Y_{6}^{2}\\\ 2Y_{5}\left(Y_{1}^{2}-2Y_{1}Y_{6}-2Y_{6}^{2}\right)\\\
-Y_{1}^{3}-Y_{1}^{2}Y_{6}+9Y_{1}Y_{6}^{2}+3Y_{6}^{3}\\\ \end{array}\right)\;.$
(2.30)

For weight four, two modular forms $Y^{(4)}_{\bf 1}$ and $Y^{(4)}_{\bf 3}$ are
involved, namely,

$\displaystyle Y^{(4)}_{\bf 1}$ $\displaystyle=$
$\displaystyle-\sqrt{2}\left(Y^{4}_{1}-3Y^{3}_{1}Y_{6}-Y^{2}_{1}Y^{2}_{6}+3Y_{1}Y^{3}_{6}+Y^{4}_{6}\right)\;,$
$\displaystyle Y^{(4)}_{\bf 3}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{4}\left(\begin{array}[]{c}\left(Y_{1}^{2}+Y_{6}^{2}\right)\left(7Y_{1}^{2}-18Y_{1}Y_{6}-7Y_{6}^{2}\right)\\\
Y_{2}\left(13Y_{1}^{3}-3Y_{1}^{2}Y_{6}-29Y_{1}Y_{6}^{2}-9Y_{6}^{3}\right)\\\
-Y_{5}\left(9Y_{1}^{3}-29Y_{1}^{2}Y_{6}+3Y_{1}Y_{6}^{2}+13Y_{6}^{3}\right)\\\
\end{array}\right)\;.$ (2.34)

As we have mentioned, once the modulus $\tau$ acquires its vev, the modular
symmetry will be spontaneously broken down. However, there are some special
values of $\tau$, which are stabilizers [29] and keep unchanged under the
transformations induced by one or more generators of the modular group.
Consequently, the global modular symmetry is only partially broken, and some
residual symmetries are left in the theory. Consider the fundamental domain
${\cal G}$ of the modular group $\Gamma$, which is defined as

Table 1: The charge assignment of the chiral superfields under the ${\rm SU(2)_{\rm L}}$ gauge symmetry and the modular $A^{\prime}_{5}$ symmetry in our model, with the corresponding weights listed in the last row. | $\widehat{L}_{e}$ | $\widehat{L}_{\mu\tau}$ | $\widehat{E}^{\rm C}_{1}$ | $\widehat{E}^{\rm C}_{23}$ | $\widehat{N}^{\rm C}$ | $\widehat{H}_{\rm u},\widehat{H}_{\rm d}$
---|---|---|---|---|---|---
SU(2) | 2 | 2 | 1 | 1 | 1 | 2
$A^{\prime}_{5}$ | 1 | $\widehat{\bf 2}$ | 1 | $\widehat{\bf 2}$ | ${\bf 3}^{\prime}$ | 1
$-k_{I}$ | 1 | 2 | $-1$ | 2 | 1 | 0

$\displaystyle{\cal G}=\left\\{\tau\in\mathbb{C}:\quad{\rm
Im}\,\tau>0,\;-0.5\leq{\rm Re}\,\tau\leq 0.5,\;|\tau|\geq 1\right\\}\;.$
(2.35)

There are three kinds of stabilizers which are not related to each other by
modular transformations in this fundamental domain, which are

  * •

$\tau_{\rm C}={\rm i}$, which is invariant under $S$;

  * •

$\tau_{\rm L}=-1/2+{\rm i}\sqrt{3}/2$, invariant under $ST$;

  * •

$\tau_{\rm T}={\rm i}\infty$, invariant under $T$.

Actually, there is an additional stabilizer $\tau_{\rm R}=1/2+{\rm
i}\sqrt{3}/2$, which, however, can be obtained from $\tau_{\rm L}$ by the $T$
transformation. In the present paper, we concentrate on the stabilizer
$\tau_{\rm C}={\rm i}$, which is invariant under the transformation
corresponding to $S$, i.e., $\tau\rightarrow-1/\tau$. Furthermore, it is
straightforward to verify that the residual symmetry at $\tau={\rm i}$ is
$Z^{S}_{4}=\\{\mathbb{I},S,R,RS\\}$, where $\mathbb{I}$ stands for the
identity element.

##  3 Modular $A^{\prime}_{5}$ Model

Now we are ready to construct a concrete model based on the modular
$A^{\prime}_{5}$ symmetry. To begin with, we need to assign properly the
weights and irreducible representations to all the superfields under the
modular group $A^{\prime}_{5}$. Different from most of the previous models,
the superfields of left-handed lepton doublets $\widehat{L}$ will not be
arranged as a triplet of $A^{\prime}_{5}$ in our model. Instead, we take
$\widehat{L}_{e}\sim{\bf 1}$ and
$\widehat{L}_{\mu\tau}\equiv(\widehat{L}_{\mu},\widehat{L}_{\tau})^{\rm
T}\sim\widehat{\bf 2}$. Correspondingly, the superfields of right-handed
charged leptons are also put into a singlet and a doublet of $A^{\prime}_{5}$,
i.e., $\widehat{E}^{\rm C}_{1}\sim{\bf 1}$ and $\widehat{E}^{\rm
C}_{23}\equiv(\widehat{E}^{\rm C}_{2},\widehat{E}^{\rm C}_{3})^{\rm
T}\sim\widehat{\bf 2}$. Since the two-dimensional modular forms exist only
with the odd weights, the flavor structure of the charged-lepton mass matrix
$M_{l}$ could be highly constrained. As we shall show soon, $M_{l}$ in our
model is restricted to be block-diagonal. In the neutrino sector, the
superfields of three right-handed neutrinos $N^{\rm C}$ are set to be ${\bf
3}^{\prime}$ of $A^{\prime}_{5}$. In addition, two Higgs doublets
$\widehat{H}_{\rm u}$ and $\widehat{H}_{\rm d}$ are both assumed to be in the
trivial one-dimensional irreducible representation. All these charge
assignments of the superfields in our model are listed in Table 1.

With the representations and weights of the supermultiplets in Table 1, one
can write down the superpotentials relevant for lepton masses

$\displaystyle{\cal W}_{l}$ $\displaystyle=$
$\displaystyle\xi_{1}\left[\widehat{L}_{e}\widehat{E}^{\rm C}_{1}\right]_{\bf
1}\widehat{H}_{\rm d}+\xi_{2}\left[\left(\widehat{L}_{\mu\tau}\widehat{E}^{\rm
C}_{23}\right)_{\bf 3}Y^{(4)}_{\bf 3}\right]_{\bf 1}\widehat{H}_{\rm
d}+\xi_{3}\left[\left(\widehat{L}_{\mu\tau}\widehat{E}^{\rm
C}_{23}\right)_{\bf 1}Y^{(4)}_{\bf 1}\right]_{\bf 1}\widehat{H}_{\rm d}\;,$
$\displaystyle{\cal W}_{\rm D}$ $\displaystyle=$ $\displaystyle
g_{1}\left[\left(\widehat{L}_{e}\widehat{N}^{\rm C}\right)_{{\bf
3}^{\prime}}Y^{(2)}_{{\bf 3}^{\prime}}\right]_{\bf 1}\widehat{H}_{\rm
u}+g_{2}\left[\left(\widehat{L}_{\mu\tau}\widehat{N}^{\rm
C}\right)_{\widehat{\bf 6}}Y^{(3)}_{\widehat{\bf 6},1}\right]_{\bf
1}\widehat{H}_{\rm u}+g_{3}\left[\left(\widehat{L}_{\mu\tau}\widehat{N}^{\rm
C}\right)_{\widehat{\bf 6}}Y^{(3)}_{\widehat{\bf 6},2}\right]_{\bf
1}\widehat{H}_{\rm u}\;,$ $\displaystyle{\cal W}_{\rm R}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Lambda\left[\left(\widehat{N}^{\rm
C}\widehat{N}^{\rm C}\right)_{\bf 5}Y^{(2)}_{\bf 5}\right]_{\bf 1}\;.$ (3.1)

Implementing the Kronecker product rules for $A^{\prime}_{5}$ collected in
Appendix A, we arrive at the explicit form of the charged-lepton mass matrix

$\displaystyle M_{l}$ $\displaystyle=$ $\displaystyle\dfrac{v_{\rm
d}}{\sqrt{2}}\left(\begin{matrix}\xi_{1}&&0&&0\\\
0&&\dfrac{\sqrt{3}}{3}\xi_{2}\left(Y^{(4)}_{\bf
3}\right)_{2}&&\xi_{2}\left[\dfrac{\sqrt{6}}{6}\left(Y^{(4)}_{\bf
3}\right)_{1}-\dfrac{\sqrt{2}}{2}\widetilde{\xi}Y^{(4)}_{\bf 1}\right]\\\
0&&\xi_{2}\left[\dfrac{\sqrt{6}}{6}\left(Y^{(4)}_{\bf
3}\right)_{1}+\dfrac{\sqrt{2}}{2}\widetilde{\xi}Y^{(4)}_{\bf
1}\right]&&-\dfrac{\sqrt{3}}{3}\xi_{2}\left(Y^{(4)}_{\bf
3}\right)_{3}\end{matrix}\right)^{\ast}\;,$ (3.2)

with $\xi_{3}/\xi_{2}\equiv\widetilde{\xi}$, the Dirac neutrino mass matrix

$\displaystyle M_{\rm D}$ $\displaystyle=$
$\displaystyle\dfrac{\sqrt{6}g_{1}v_{\rm
u}}{12}\left[\rule{0.0pt}{36.98866pt}\right.\left(\begin{matrix}2\left(Y^{(2)}_{\bf
3^{\prime}}\right)_{1}&&2\left(Y^{(2)}_{\bf
3^{\prime}}\right)_{3}&&2\left(Y^{(2)}_{\bf 3^{\prime}}\right)_{2}\\\
0&&0&&0\\\ 0&&0&&0\\\ \end{matrix}\right)$ (3.3)
$\displaystyle+\widetilde{g}_{2}\left(\begin{matrix}0&&0&&0\\\
-\sqrt{2}\left(Y^{(3)}_{\widehat{\bf
6},1}\right)_{4}&&-\sqrt{2}\left(Y^{(3)}_{\widehat{\bf
6},1}\right)_{2}&&\left(Y^{(3)}_{\widehat{\bf
6},1}\right)_{1}-\left(Y^{(3)}_{\widehat{\bf 6},1}\right)_{6}\\\
-\sqrt{2}\left(Y^{(3)}_{\widehat{\bf
6},1}\right)_{3}&&\left(Y^{(3)}_{\widehat{\bf
6},1}\right)_{1}+\left(Y^{(3)}_{\widehat{\bf
6},1}\right)_{6}&&-\sqrt{2}\left(Y^{(3)}_{\widehat{\bf 6},1}\right)_{5}\\\
\end{matrix}\right)$
$\displaystyle+\widetilde{g}_{3}\left(\begin{matrix}0&&0&&0\\\
-\sqrt{2}\left(Y^{(3)}_{\widehat{\bf
6},2}\right)_{4}&&-\sqrt{2}\left(Y^{(3)}_{\widehat{\bf
6},2}\right)_{2}&&\left(Y^{(3)}_{\widehat{\bf
6},2}\right)_{1}-\left(Y^{(3)}_{\widehat{\bf 6},2}\right)_{6}\\\
-\sqrt{2}\left(Y^{(3)}_{\widehat{\bf
6},2}\right)_{3}&&\left(Y^{(3)}_{\widehat{\bf
6},2}\right)_{1}+\left(Y^{(3)}_{\widehat{\bf
6},2}\right)_{6}&&-\sqrt{2}\left(Y^{(3)}_{\widehat{\bf 6},2}\right)_{5}\\\
\end{matrix}\right)\left.\rule{0.0pt}{36.98866pt}\right]^{\ast}\;,$

with $g_{2}/g_{1}\equiv\widetilde{g}_{2}$ and
$g_{3}/g_{1}\equiv\widetilde{g}_{3}$, and the Majorana mass matrix of right-
handed neutrinos

$\displaystyle M_{\rm R}$ $\displaystyle=$
$\displaystyle\dfrac{\Lambda}{4}\left(\begin{matrix}2\left(Y^{(2)}_{\bf
5}\right)_{1}&&-\sqrt{3}\left(Y^{(2)}_{\bf
5}\right)_{4}&&-\sqrt{3}\left(Y^{(2)}_{\bf 5}\right)_{3}\\\
-\sqrt{3}\left(Y^{(2)}_{\bf 5}\right)_{4}&&\sqrt{6}\left(Y^{(2)}_{\bf
5}\right)_{2}&&-\left(Y^{(2)}_{\bf 5}\right)_{1}\\\
-\sqrt{3}\left(Y^{(2)}_{\bf 5}\right)_{3}&&-\left(Y^{(2)}_{\bf
5}\right)_{1}&&\sqrt{6}\left(Y^{(2)}_{\bf 5}\right)_{5}\\\
\end{matrix}\right)^{\ast}\;,$ (3.4)

where $(Y^{(k)}_{\bf r})_{i}$ denotes the $i$-th element in the multiplet
$Y^{(k)}_{\bf r}$. Given $M_{\rm D}$ and $M_{\rm R}$, we can immediately
derive the effective neutrino mass matrix from the seesaw formula, i.e.,
$M_{\nu}\approx-M_{\rm D}M^{-1}_{\rm R}M^{\rm T}_{\rm D}$. Hence the lepton
mass spectra and flavor mixing parameters can be extracted from the charged-
lepton mass matrix $M_{l}$ and the effective neutrino mass matrix $M_{\nu}$.

Apart from the modulus $\tau$, the lepton mass matrices involve three extra
parameters $\widetilde{\xi}$, $\widetilde{g}_{2}$ and $\widetilde{g}_{3}$,
which are in general complex. All these complex parameters may contribute to
leptonic CP violation. Following Ref. [53], we further impose gCP symmetry on
our modular $A^{\prime}_{5}$ model such that the modulus $\tau$ becomes the
only source of CP violation. More explicitly, the gCP transformation of the
superfield $\chi^{(I)}(x)$ is

$\displaystyle\chi^{(I)}(x)\stackrel{{\scriptstyle\rm
CP}}{{\longrightarrow}}X_{\bf r}\overline{\chi}^{(I)}(x_{\rm P})\;,$ (3.5)

where $\overline{\chi}^{(I)}(x_{\rm P})$ denotes the conjugate superfield with
$x_{\rm P}=(t,-\vec{x})$ and $X_{\bf r}$ represents a unitary matrix acting on
the flavor space. For the gCP symmetry to be consistent with the modular
symmetry, the following condition should be satisfied

$\displaystyle
X_{\bf{r}}\rho_{\bf{r}}^{*}(\gamma)X_{\bf{r}}^{-1}=\rho_{\bf{r}}(u(\gamma))\;,$
(3.6)

where $u(\gamma)$ is an outer automorphism of the modular group.111It has been
mentioned in Ref. [40] that there are two distinct kinds of outer
automorphisms $u(\gamma)$ for the double covering group, corresponding to two
different gCP symmetries denoted as ${\rm CP_{1}}$ and ${\rm CP_{2}}$,
respectively. However, in the basis where the representation matrices of $S$
and $T$ are both symmetric, $X_{\bf r}$ will always be the trivial identity
matrix, i.e., $X_{\bf r}=\mathbb{I}$, regardless of whether ${\rm CP}_{1}$ or
${\rm CP}_{2}$ is combined with the $\Gamma^{\prime}_{5}$ group. The
consistency condition given in Eq. (3.6) indicates that the modulus $\tau$
transforms under CP as

$\displaystyle\tau\stackrel{{\scriptstyle\rm
CP}}{{\longrightarrow}}-\tau^{\ast}\;.$ (3.7)

On the other hand, Eq. (3.6) should be satisfied for all the elements in the
finite modular group, implying $X_{\bf r}=\mathbb{I}$ with $\mathbb{I}$ being
the identity element if the representation matrices of both $S$ and $T$ are
symmetric [53]. Furthermore, as can be seen in Appendix A, all the Clebsch-
Gordan coefficients are real, so the modular forms $Y^{(k)}_{\bf r}$ will
transform under CP as

$\displaystyle
Y_{\bf{r}}^{(k)}(\tau)\stackrel{{\scriptstyle\mathrm{CP}}}{{\longrightarrow}}Y_{\bf{r}}^{(k)}\left(-\tau^{*}\right)=\left[Y_{\bf{r}}^{(k)}(\tau)\right]^{*}\;.$
(3.8)

Therefore, to render the superpotentials invariant under the gCP
transformation, we require all the coupling constants in our model to be real.
As a result, the whole symmetry of modular and gCP transformations is
spontaneously broken down after the modulus $\tau$ gets its vev. However,
there are some special values of $\tau$, for which CP symmetry is conserved
while the modular symmetry is broken. As pointed out in Ref. [53], these
values are located along the imaginary axis (i.e., ${\rm Re}\,\tau=0$) and the
boundary of the fundamental domain ${\cal G}$.

##  4 Low-energy Phenomenology

Table 2: The best-fit values, the 1$\sigma$ and 3$\sigma$ intervals, together with the values of $\sigma_{i}$ being the symmetrized $1\sigma$ uncertainties, for three neutrino mixing angles $\\{\theta_{12},\theta_{13},\theta_{23}\\}$, two neutrino mass-squared differences $\\{\Delta m^{2}_{21},\Delta m^{2}_{31}~{}{\rm or}~{}\Delta m^{2}_{32}\\}$ and the Dirac CP-violating phase $\delta$ from a global-fit analysis of current experimental data [56]. Parameter | Best fit | 1$\sigma$ range | 3$\sigma$ range | $\sigma_{i}$
---|---|---|---|---
Normal neutrino mass ordering $(m_{1}<m_{2}<m_{3})$
$\sin^{2}\theta_{12}$ | $0.304$ | 0.292 — 0.317 | 0.269 — 0.343 | 0.0125
$\sin^{2}\theta_{13}$ | $0.02221$ | 0.02159 — 0.02289 | 0.02034 — 0.02430 | 0.00065
$\sin^{2}\theta_{23}$ | $0.570$ | 0.546 — 0.588 | 0.407 — 0.618 | 0.021
$\delta/{}^{\circ}$ | $195$ | 170 — 246 | 107 — 403 | 38
$\Delta m^{2}_{21}/(10^{-5}~{}{\rm eV}^{2})$ | $7.42$ | 7.22 — 7.63 | 6.82 — 8.04 | 0.205
$\Delta m^{2}_{31}/(10^{-3}~{}{\rm eV}^{2})$ | $+2.514$ | +2.487 — +2.542 | +2.431 — +2.598 | 0.0275
Inverted neutrino mass ordering $(m_{3}<m_{1}<m_{2})$
$\sin^{2}\theta_{12}$ | $0.304$ | 0.292 — 0.317 | 0.269 — 0.343 | 0.0125
$\sin^{2}\theta_{13}$ | $0.02240$ | 0.02178 — 0.02302 | 0.02053 — 0.02436 | 0.00062
$\sin^{2}\theta_{23}$ | $0.575$ | 0.554 — 0.592 | 0.411 — 0.621 | 0.019
$\delta/{}^{\circ}$ | $286$ | 254 — 313 | 192 — 360 | 29.5
$\Delta m^{2}_{21}/(10^{-5}~{}{\rm eV}^{2})$ | $7.42$ | 7.22 — 7.63 | 6.82 — 8.04 | 0.205
$\Delta m^{2}_{32}/(10^{-3}~{}{\rm eV}^{2})$ | $-2.497$ |  $-2.525$ — $-2.469$ |  $-2.583$ — $-2.412$ | $0.028$

Thanks to the gCP symmetry, only eight real parameters $\\{{\rm Re}\,\tau,{\rm
Im}\,\tau,\xi_{1}v_{\rm d},\xi_{2}v_{\rm
d},\widetilde{\xi},g^{2}_{1}v^{2}_{\rm
u}/\Lambda,\widetilde{g}_{2},\widetilde{g}_{3}\\}$ are involved in our model.
The allowed regions of model parameters can be found by following the
numerical analysis adopted in Ref. [42], and the basic strategy is summarized
as below.

  * •

First, we explain the experimental results that are used to constrain the
parameter space. For the charged-lepton masses, we take the best-fit values
$m_{e}=0.510~{}{\rm MeV}$, $m_{\mu}=107.8~{}{\rm MeV}$ and
$m_{\tau}=1.840~{}{\rm GeV}$, which are evaluated at the scale $\Lambda_{\rm
GUT}\approx 2\times 10^{16}~{}{\rm GeV}$ of grand unified theories (GUT) with
$\tan\beta\equiv v_{\rm u}/v_{\rm d}=10$ and the supersymmetry breaking scale
$m_{\rm SUSY}=10~{}{\rm TeV}$ in Ref. [55]. With the help of these charged-
lepton masses, one can determine the model parameters $\xi_{1}v_{\rm d}$,
$\xi_{2}v_{\rm d}$ and $\widetilde{\xi}$ if the complex modulus $\tau$ is
given. For the neutrino sector, we take the best-fit values, as well as
$1\sigma$ and $3\sigma$ ranges, of two neutrino mass-squared differences
$\Delta m^{2}_{21}\equiv m^{2}_{2}-m^{2}_{1}$ and $\Delta m^{2}_{31}\equiv
m^{2}_{3}-m^{2}_{1}$ in the normal mass ordering (NO) with $m_{1}<m_{2}<m_{3}$
or $\Delta m^{2}_{21}$ and $\Delta m^{2}_{32}\equiv m^{2}_{3}-m^{2}_{2}$ in
the inverted mass ordering (IO) with $m_{3}<m_{1}<m_{2}$, three flavor mixing
angles $\\{\theta_{12},\theta_{13},\theta_{23}\\}$, and the Dirac CP-violating
phase $\delta$, from the global-fit analysis by NuFIT 5.0 [56, 57] without
including the atmospheric neutrino data from Super-Kamiokande. All these
values are summarized in Table 2.

  * •

The model predictions for low-energy observables, including charged-lepton and
neutrino masses, flavor mixing angles, and CP-violating phases, can be
obtained by diagonalizing lepton mass matrices $M_{l}$ and $M_{\nu}$. Then,
the compatibility between the model predictions and the experimental
observations is measured by the $\chi^{2}$-function, which is constructed as
the sum of several one-dimensional functions $\chi^{2}_{j}$, namely,

$\displaystyle\chi^{2}(p_{i})=\sum_{j}\chi^{2}_{j}(p_{i})\;,$ (4.1)

where $p_{i}\in\\{{\rm Re}\,\tau,{\rm
Im}\,\tau,\widetilde{g}_{2},\widetilde{g}_{3},g^{2}_{1}v^{2}_{\rm
u}/\Lambda\\}$ stand for the model parameters, and $j$ is summed over the
observables
$\\{\sin^{2}\theta_{12},\sin^{2}\theta_{13},\sin^{2}\theta_{23},\Delta
m^{2}_{21},\Delta m^{2}_{31}(\Delta m^{2}_{32})\\}$ in the NO (IO) case. Here
we do not include the information of $\delta$ in the $\chi^{2}$-function due
to the weak constraints on $\delta$ from the global-fit results. For
$\sin^{2}\theta_{12}$, $\sin^{2}\theta_{13}$, $\Delta m^{2}_{21}$ and $\Delta
m^{2}_{31}$ ($\Delta m^{2}_{32}$), we make use of the Gaussian approximations

$\displaystyle\chi^{2}_{j}(p_{i})=\left(\frac{q_{j}(p_{i})-q^{\rm
bf}_{j}}{\sigma_{j}}\right)^{2}\;,$ (4.2)

where $q_{j}(p_{i})$ denote the model predictions for these observables and
$q^{\rm bf}_{j}$ are their best-fit values from the global analysis in Ref.
[56]. The associated uncertainties $\sigma_{j}$ are derived by symmetrizing
$1\sigma$ uncertainties from the global-fit analysis, as given in Table 2. For
$\sin^{2}\theta_{23}$, we utilize the one-dimensional projection of the
$\chi^{2}$-function provided by Refs. [56, 57]. By minimizing the overall
$\chi^{2}$-function in Eq. (4.1), we can determine the best-fit values of the
model parameters $\\{{\rm Re}\,\tau,{\rm
Im}\,\tau,\widetilde{g}_{2},\widetilde{g}_{3},g^{2}_{1}v^{2}_{\rm
u}/\Lambda\\}$.

Figure 1: The allowed parameter space of the model parameters $\\{{\rm
Re}\,\tau,{\rm Im}\,\tau,\widetilde{g}_{2},\widetilde{g}_{3}\\}$ and low-
energy observables at the $3\sigma$ level in the NO case, where the cyan stars
denote the best-fit values from the $\chi^{2}$-fit analysis. The gray shaded
region in the bottom-right panel represents the upper bound on
$m_{\beta\beta}$ from the KamLAND-Zen experiment [58], and the blue boundary
is obtained by using the $3\sigma$ ranges of $\\{\theta_{12},\theta_{13}\\}$
and $\\{\Delta m^{2}_{21},\Delta m^{2}_{31}\\}$ from the global-fit analysis.

After carrying out the numerical analysis, we find that our model can be
compatible with the experimental data at the $3\sigma$ level only in the NO
case. In Fig. 1, the $3\sigma$ allowed parameter space of model parameters as
well as the predictions for low-energy observables are shown. Some comments on
the numerical results are in order.

  1. 1.

In the top-left panel of Fig. 1, we can observe that the $3\sigma$ allowed
value of ${\rm Re}\,\tau$ is larger than $-0.16$, while the upper bound of
${\rm Im}\,\tau$ is around $1.03$. In fact, there should be a duplicate region
of $\\{{\rm Re}\,\tau,{\rm Im}\,\tau\\}$ in the right-half part of the
fundamental domain ${\cal G}$ with $0\leq{\rm Re}\,\tau\leq 0.5$, where the
signs of all CP-violating phases are reversed. However, the predicted value of
the Dirac CP-violating phase $\delta$ in this region is constrained to be
about $90^{\circ}$, which is lying outside the $3\sigma$ allowed range of
$\delta$ from the global-fit analysis. The allowed regions of the other two
parameters $\\{\widetilde{g}_{2},\widetilde{g}_{3}\\}$ in the neutrino sector
are shown in the top-right panel of Fig. 1, where it is worthwhile to mention
that the relation $\widetilde{g}_{3}=-\widetilde{g}_{2}/4$ holds
approximately, especially for large values of $\widetilde{g}_{2}$. Some of the
low-energy observables are strongly correlated with ${\rm Re}\,\tau$ (or ${\rm
Im}\,\tau$). For example, the middle-left panel of Fig. 1 reveals that the
value of $\theta_{23}$ increases as ${\rm Re}\,\tau$ decreases. When the value
of ${\rm Re}\,\tau$ is approaching zero, $\theta_{23}$ becomes very close to
$45^{\circ}$. Interestingly, the Dirac CP-violating phase $\delta$ shows a
similar behavior, i.e., $\delta$ tends to $270^{\circ}$ when ${\rm
Re}\,\tau\rightarrow 0$, implying a nearly-maximal CP violation in the
vicinity of $\tau={\rm i}$. This result seems quite confusing, since the
stabilizer $\tau={\rm i}$ should lead to CP conservation. Why does a small
deviation from $\tau={\rm i}$ (i.e., ${\rm Im}\,\tau-1<0.5\%$) give rise to a
large CP-violating phase? In the next section, we shall come back to answer
this question by performing an analytical analysis with some reasonable
approximations. The other mixing angles $\theta_{12}$ and $\theta_{13}$ and
two neutrino mass-squared differences are loosely constrained and thus not
presented here.

  2. 2.

The sum of three light neutrino masses $\sum m_{\nu}=m_{1}+m_{2}+m_{3}$ also
depends crucially on the value of ${\rm Re}\,\tau$. As indicated in the
bottom-left panel of Fig. 1, the minimal value of $\sum m_{\nu}$ predicted in
our model is $0.15~{}{\rm eV}$, which runs into contradiction with the most
stringent bound $\sum m_{\nu}<0.12~{}{\rm eV}$ from the Planck observations of
cosmic microwave background [59]. Notice that the limit $\sum
m_{\nu}<0.12~{}{\rm eV}$ has also been gained earlier in Ref. [60, 61].
However, this upper bound is dependent on the choices of observational data
sets and cosmological models. The latest global-fit analysis of absolute
neutrino masses yields $\sum m_{\nu}<(0.12\cdots 0.69)~{}{\rm eV}$ at the
$2\sigma$ level [62], where the “aggressive” and “conservative” combinations
of different observational data sets are considered. Therefore, our model is
still consistent with the “conservative” cosmological bound on neutrino
masses.

  3. 3.

Two Majorana CP-violating phases $\rho$ and $\sigma$ can also be determined,
so we can calculate the effective mass for neutrinoless double-beta decays

$\displaystyle
m_{\beta\beta}\equiv|m_{1}\cos^{2}\theta_{13}\cos^{2}\theta_{12}e^{2{\rm
i}\rho}+m_{2}\cos^{2}\theta_{13}\sin^{2}\theta_{12}e^{2{\rm
i}\sigma}+m_{3}\sin^{2}\theta_{13}e^{-2{\rm i}\delta}|\;,$ (4.3)

where the standard parametrization of leptonic flavor mixing matrix has been
taken [63]. The numerical result of $m_{\beta\beta}$ is presented in the
bottom-right panel of Fig. 1, where we can find that the predicted range of
$m_{\beta\beta}$ is overlapping with the upper bound $m^{\rm
upper}_{\beta\beta}=(61\cdots 165)~{}{\rm meV}$ from the KamLAND-Zen
experiment [58]. Hence the next-generation neutrinoless double-beta decay
experiments with a sensitivity of $m_{\beta\beta}\sim 15~{}{\rm meV}$ [64]
will be able to make a final verdict on whether our model is ruled out or not.

Based on the $\chi^{2}$-fit analysis, we find that the minimum $\chi^{2}_{\rm
min}=0.735$ is reached in the NO case with the following best-fit values of
the model parameters

$\displaystyle{\rm Re}\,\tau=-0.0735\;,\quad{\rm
Im}\,\tau=1.00897\;,\quad\widetilde{g}_{2}=1.0199\;,\quad\widetilde{g}_{3}=-0.286\;,\quad
g^{2}_{1}v^{2}_{\rm u}/\Lambda=62.35~{}{\rm meV}\;.$

Combining the above values with charged-lepton masses, we obtain
$\xi_{1}v_{\rm d}=0.721~{}{\rm MeV}$, $\xi_{2}v_{\rm d}=0.246~{}{\rm GeV}$ and
$\widetilde{\xi}=3.981$. These best-fit values of model parameters lead to a
nearly-degenerate spectrum of three neutrino masses $m_{1}=72.58~{}{\rm meV}$,
$m_{2}=73.08~{}{\rm meV}$, $m_{3}=88.21~{}{\rm meV}$, and three mixing angles
$\theta_{12}=33.58^{\circ}$, $\theta_{13}=8.58^{\circ}$ and
$\theta_{23}=42.51^{\circ}$. Meanwhile, the predictions for three CP-violating
phases are $\delta=271.3^{\circ}$, $\rho=88.74^{\circ}$ and
$\sigma=90.53^{\circ}$. In addition, the effective mass for neutrinoless
double-beta decays is $m_{\beta\beta}=71.11~{}{\rm meV}$. All these
predictions are readily to be tested in future neutrino oscillation and
neutrinoless double-beta decay experiments.

##  5 Explicit Perturbations and Analytical Results

As we have seen in Sec. 4, our model predicts a nearly-maximal CP-violating
phase $\delta=271.4^{\circ}$, given ${\rm Re}\,\tau=-0.0736$ and ${\rm
Im}\,\tau=1.00897$, which is lying in the vicinity of $\tau={\rm i}$. Since CP
violation is completely absent for $\tau={\rm i}$, it deserves a further
investigation to clarify how such significant CP violation is generated. In
this section, we attempt to explore the analytical properties of our model in
the parameter space, where $\tau$ is close to the stabilizer $\tau={\rm i}$,
by introducing explicit perturbations to the stabilizer and calculating lepton
mass spectra and flavor mixing parameters.

Let us first consider the NO case. If $\tau={\rm i}$ exactly holds, with the
help of the explicit expressions of basis vectors $\widehat{e}_{i}$ (for
$i=1,\cdots,6$) shown in Eq. (A.122), we can find

$\displaystyle\frac{\widehat{e}_{2}(\rm i)}{\widehat{e}_{1}(\rm
i)}=\frac{\widehat{e}_{3}(\rm i)}{\widehat{e}_{2}(\rm
i)}=\frac{\widehat{e}_{4}(\rm i)}{\widehat{e}_{3}(\rm
i)}=\frac{\widehat{e}_{5}(\rm i)}{\widehat{e}_{4}(\rm
i)}=\frac{\widehat{e}_{6}(\rm i)}{\widehat{e}_{5}(\rm i)}=A_{0}\;,$ (5.1)

where $A_{0}=\sqrt{\sqrt{5}\phi}-\phi$ with $\phi\equiv(\sqrt{5}+1)/2$. The
elements of $Y^{(1)}_{\widehat{\bf 6}}(\rm i)$ can be explicitly written as

$\displaystyle\begin{array}[]{lllll}Y_{1}({\rm i})=\widehat{e}_{1}({\rm
i})(1-3\,A^{5}_{0})\;,&&Y_{2}({\rm i})=5\sqrt{2}\,\widehat{e}_{1}({\rm
i})A_{0}\;,&&Y_{3}({\rm i})=10\,\widehat{e}_{1}({\rm i})A^{2}_{0}\;,\\\
Y_{4}({\rm i})=10\,\widehat{e}_{1}({\rm i})A^{3}_{0}\;,&&Y_{5}({\rm
i})=5\sqrt{2}\,\widehat{e}_{1}({\rm i})A^{4}_{0}\;,&&Y_{6}({\rm
i})=-\widehat{e}_{1}({\rm i})(3+A^{5}_{0})\;.\end{array}$ (5.4)

Then the modular forms at $\tau={\rm i}$ with higher weights can be
constructed by using Eq. (5.4). Since the stabilizer $\tau={\rm i}$ keeps
unchanged under the transformation of $S$, $H_{l}\equiv
M_{l}M^{{\dagger}}_{l}$ from the charged-lepton sector is invariant under $S$,
i.e.,

$\displaystyle H_{l}=\rho^{{\dagger}}_{l}(S)H_{l}\rho_{l}(S)\;,$ (5.5)

where $\rho_{l}(S)\equiv{\rm Diag}\\{1,\rho_{\bf 2}(S)\\}$ with $\rho_{\bf
2}(S)$ being the representation matrix of the two-dimensional irreducible
representation 2 of $S$ in the $A^{\prime}_{5}$ group. The identity in Eq.
(5.5) tells us that a unitary matrix converting $\rho_{l}(S)$ into its
diagonal form can also diagonalize the matrix $H_{l}$. It is easy to verify
that both $\rho_{l}(S)$ and $H_{l}$ can be diagonalized by the real orthogonal
matrix

$\displaystyle U_{l}=\left(\begin{matrix}1&0&0\\\
0&\cos\theta_{l}&-\sin\theta_{l}\\\ 0&\sin\theta_{l}&\cos\theta_{l}\\\
\end{matrix}\right)\;,$ (5.6)

where $\tan 2\theta_{l}=\sqrt{(\phi-1)/\phi}$. Two comments on $U_{l}$ in Eq.
(5.6) are helpful. First, substituting $\phi=(\sqrt{5}+1)/2\approx 1.618$ into
Eq. (5.6), we arrive at $\cos\theta_{l}\approx 0.962$ and
$\sin\theta_{l}\approx 0.273$. Therefore, $U_{l}$ is roughly an identity
matrix, and its contribution to the lepton flavor mixing is insignificant.
Second, we have checked that the form of $U_{l}$ given in Eq. (5.6) holds as a
good approximation to the unitary matrix that diagonalizes $H_{l}$, even if
$\tau$ slightly deviates from the stabilizer $\tau={\rm i}$. It is then safe
to assume $U_{l}$ in Eq. (5.6) to be valid in the vicinity of $\tau={\rm i}$.

We turn to the neutrino sector. Without loss of generality, one can work in
the flavor basis where the charged-lepton mass matrix is diagonal, and
redefine the effective neutrino mass matrix as
$\widetilde{M}_{\nu}=U^{\dagger}_{l}M_{\nu}U_{l}$. For $\tau={\rm i}$,
$\widetilde{M}_{\nu}$ is simply written as

$\displaystyle\widetilde{M}_{\nu}=\frac{\widehat{g}^{2}_{1}v^{2}_{\rm u}}{{\rm
Det}(M_{\rm
R})}\left(\begin{matrix}50.67&&0&&-17.19(\widehat{g}_{2}+4\widehat{g}_{3})\\\
0&&0&&0\\\
-17.19(\widehat{g}_{2}+4\widehat{g}_{3})&&0&&5.834(\widehat{g}_{2}+4\widehat{g}_{3})^{2}\\\
\end{matrix}\right)\;,$ (5.7)

with $\widehat{g}_{1}=g_{1}|\widehat{e}_{1}(\tau)|^{2}$ and
$\widehat{g}_{2,3}=\widetilde{g}_{2,3}|\widehat{e}_{1}(\tau)|$. It is obvious
that the right-handed neutrino mass matrix $M_{\rm R}$ has one zero eigenvalue
for $\tau={\rm i}$, i.e., ${\rm Det}(M_{\rm R})=0$ in Eq. (5.7). The seesaw
formula $M_{\nu}=-M_{\rm D}M^{-1}_{\rm R}M^{\rm T}_{\rm D}$ is not directly
applicable for $\tau={\rm i}$, but anyway we are interested in the region
where $\tau={\rm i}+\epsilon$ with $|\epsilon|\ll 1$. In this case, the
determinant of $M_{\rm R}$ is found to be ${\rm Det}(M_{\rm R})\propto{\rm
i}\Lambda\epsilon$ to the first order of $\epsilon$. Although $\epsilon$ is a
small parameter, the overall factor $\Lambda$ corresponding to the mass scale
of right-handed neutrinos can be quite large, giving rise to a sizable value
of ${\rm Det}(M_{\rm R})$. On the other hand, Eq. (5.7) indicates that if
$\widehat{g}_{3}=-\widehat{g}_{2}/4$, only the (1,1)-element of
$\widetilde{M}_{\nu}$ survives. As can be seen from the top-right panel of
Fig. 1, the ratio
$\widehat{g}_{2}/\widehat{g}_{3}=\widetilde{g}_{2}/\widetilde{g}_{3}$ is
indeed around $-4$, corresponding to
$\widetilde{g}_{3}/\widetilde{g}_{2}=-0.25$ that is not far from the best-fit
value $\widetilde{g}_{3}/\widetilde{g}_{2}\approx-0.28$. Inspired by this
observation, we further assume $\widehat{g}_{3}=-\widehat{g}_{2}/4$ to hold
for the stabilizer $\tau={\rm i}$, and introduce perturbations to this
identity when $\tau$ deviates from ${\rm i}$.

In the following discussions, we consider explicit perturbations to both
$\tau={\rm i}$ and $\widehat{g}_{3}=-\widehat{g}_{2}/4$, and explore their
implications for lepton masses, flavor mixing angles and CP-violating phases
with some reasonable approximations.

  * •

First, we introduce the perturbation to the stabilizer, i.e., $\tau={\rm
i}+\epsilon$, where $\epsilon\equiv\epsilon_{\rm R}+{\rm i}\epsilon_{\rm I}$
is a complex parameter with $|\epsilon|=\sqrt{\epsilon^{2}_{\rm
R}+\epsilon^{2}_{\rm I}}\ll 1$ and $\epsilon_{\rm R}$ ($\epsilon_{\rm I}$)
being the real (imaginary) part. In the presence of this perturbation, the
ratios of two adjacent basis vectors of ${\cal M}_{1}[\Gamma(5)]$ read

$\displaystyle\frac{\widehat{e}_{2}(\rm i+\epsilon)}{\widehat{e}_{1}(\rm
i+\epsilon)}=\frac{\widehat{e}_{3}(\rm i+\epsilon)}{\widehat{e}_{2}(\rm
i+\epsilon)}=\frac{\widehat{e}_{4}(\rm i+\epsilon)}{\widehat{e}_{3}(\rm
i+\epsilon)}=\frac{\widehat{e}_{5}(\rm i+\epsilon)}{\widehat{e}_{4}(\rm
i+\epsilon)}=\frac{\widehat{e}_{6}(\rm i+\epsilon)}{\widehat{e}_{5}(\rm
i+\epsilon)}\approx A_{0}(1+1.245\,{\rm i}\epsilon)\;,$ (5.8)

to the first order of $\epsilon$. With Eq. (5.8), we can also write down the
approximate expressions of all modular forms for $\tau={\rm i}+\epsilon$. Then
the neutrino mass matrix $\widetilde{M}_{\nu}$ up to ${\cal O}(\epsilon^{2})$
is

$\displaystyle\widetilde{M}_{\nu}\approx-\frac{\widehat{g}^{2}_{1}v^{2}_{\rm
u}}{\Lambda\epsilon}\left(\begin{matrix}0.089\,{\rm i}-0.331\epsilon&&0&&0\\\
0&&-0.823\,{\rm
i}\widehat{g}^{2}_{2}\epsilon^{2}&&\widehat{g}^{2}_{2}\epsilon(1.258+5.731\,{\rm
i}\epsilon)\\\ 0&&\widehat{g}^{2}_{2}\epsilon(1.258+5.731\,{\rm
i}\epsilon)&&2.332\,{\rm i}\widehat{g}^{2}_{2}\epsilon^{2}\\\
\end{matrix}\right)^{\ast}\;,$ (5.9)

where the identity $\widehat{g}_{3}=-\widehat{g}_{2}/4$ is assumed. In the
present paper, we focus on the case where $\epsilon_{\rm R}<0$ while
$\epsilon_{\rm I}>0$. It is then evident that
$H_{\nu}\equiv\widetilde{M}_{\nu}\widetilde{M}^{{\dagger}}_{\nu}$ can be
diagonalized via $U^{{\dagger}}_{23}H_{\nu}U_{23}={\rm
Diag}\,\\{m^{2}_{1,0},m^{2}_{2,0},m^{2}_{3,0}\\}$ by the following unitary
matrix

$\displaystyle U_{23}=\left(\begin{matrix}1&&0&&0\\\ 0&&1&&0\\\ 0&&0&&-{\rm
i}\\\ \end{matrix}\right)\cdot\left(\begin{matrix}1&&0&&0\\\
0&&\cos\theta_{23}&&-\sin\theta_{23}\\\
0&&\sin\theta_{23}&&\cos\theta_{23}\end{matrix}\right)\;,$ (5.10)

where $\sin\theta_{23}\approx\sqrt{2}/2+0.212\epsilon_{\rm R}$. This simple
formula implies that $\theta_{23}$ is approximately $45^{\circ}$ when ${\rm
Re}\,\tau=\epsilon_{\rm R}$ is small in magnitude, which is in good agreement
with numerical results in the middle-left panel of Fig. 1. In addition, three
eigenvalues of $H_{\nu}$ can be expressed as

$\displaystyle m^{2}_{1,0}$ $\displaystyle\approx$
$\displaystyle(0.00793-0.0590\epsilon_{\rm I}-0.0934\epsilon^{2}_{\rm
R})\mu^{2}_{0}\;,$ $\displaystyle m^{2}_{2,0}$ $\displaystyle\approx$
$\displaystyle(1.583+3.971\epsilon_{\rm R}-14.42\epsilon_{\rm
I}+35.91\epsilon^{2}_{\rm R})\widehat{g}^{4}_{2}\epsilon^{2}_{\rm
R}\mu^{2}_{0}\;,$ $\displaystyle m^{2}_{3,0}$ $\displaystyle\approx$
$\displaystyle(1.583-3.971\epsilon_{\rm R}-14.42\epsilon_{\rm
I}+35.91\epsilon^{2}_{\rm R})\widehat{g}^{4}_{2}\epsilon^{2}_{\rm
R}\mu^{2}_{0}\;,$ (5.11)

where $\mu_{0}\equiv(\widehat{g}^{2}_{1}v^{2}_{\rm u})/(\Lambda|\epsilon|)$.
Note that we have retained the terms up to next-to-leading order of
$\epsilon_{\rm R}$ but only the leading order terms of $\epsilon_{\rm I}$.
This treatment has been guided by the numerical results, which demonstrate
that $|\epsilon_{\rm R}|$ is about ten times larger than $\epsilon_{\rm I}$ in
the allowed parameter space. Eq. (5.11) indicates that the values of
$m^{2}_{2,0}$ and $m^{2}_{3,0}$ are proportional to
$\widehat{g}^{2}_{2}\epsilon^{2}_{\rm R}$, which should be highly suppressed
because of a small value of $\epsilon_{\rm R}$. However, the coefficients in
the parentheses on the right-hand side of Eq. (5.11) for $m^{2}_{2,0}$ and
$m^{2}_{3,0}$ can be much larger than that for $m^{2}_{1,0}$. Therefore, it is
possible to have the feasible parameter space of $\widehat{g}_{2}$ where the
normal mass ordering is allowed.

  * •

From previous discussions, we have observed that the perturbation to
$\tau={\rm i}$ gives $\theta_{23}\approx 45^{\circ}$, and the identity
$\widehat{g}_{3}=-\widehat{g}_{2}/4$ leads to zero values of $\theta_{12}$ and
$\theta_{13}$. In order to accommodate realistic mixing angles, we have to
break the identity by assuming $\widehat{g}_{3}=-\widehat{g}_{2}/4+\kappa$,
where $\kappa$ is another perturbative parameter. To the first order of
$\kappa$, the matrix $H_{\nu}$ after the $(2,3)$-rotation is modified to be
$H^{\prime}_{\nu}=\widehat{H}_{\nu}+\Delta H_{\nu}$ with
$\widehat{H}_{\nu}={\rm Diag}\,\\{m^{2}_{1,0},m^{2}_{2,0},m^{2}_{3,0}\\}$,
where the leading-order mass eigenvalues have been given in Eq. (5.11), and
the perturbation matrix $\Delta H_{\nu}$ is given by

$\displaystyle\Delta
H_{\nu}\approx\mu^{2}_{0}\kappa\left[0.00761\left(\begin{matrix}0&&-{\rm
i}&&+{\rm i}\\\ +{\rm i}&&0&&0\\\ -{\rm i}&&0&&0\\\
\end{matrix}\right)+0.108\,\widehat{g}^{2}_{2}\left(\begin{matrix}0&&-{\rm
i}\epsilon^{*}&&-{\rm i}\epsilon\\\ +{\rm i}\epsilon&&0&&0\\\ +{\rm
i}\epsilon^{\ast}&&0&&0\end{matrix}\right)\right]\;.$ (5.12)

One can numerically check that the $(1,3)$-element of $\Delta H_{\nu}$ is
about ten times larger than its $(1,2)$-element if both $\epsilon$ and
$\widehat{g}_{2}$ are within their individual $3\sigma$ allowed range.
Therefore, we implement a sequence of rotations, namely, the $(1,3)$-rotation
followed by the $(1,2)$-rotation, on $H^{\prime}_{\nu}$ to diagonalize it. For
the $(1,3)$-rotation, the unitary matrix $U_{13}$ approximates to

$\displaystyle U_{13}=\left(\begin{matrix}e^{{\rm i}\varphi}&&0&&0\\\
0&&1&&0\\\ 0&&0&&1\\\
\end{matrix}\right)\cdot\left(\begin{matrix}\cos\theta_{13}&&0&&-\sin\theta_{13}\\\
0&&1&&0\\\ \sin\theta_{13}&&0&&\cos\theta_{13}\end{matrix}\right)\;,$ (5.13)

where

$\displaystyle\sin\theta_{13}\approx\frac{\mu^{2}_{0}|\kappa|\left(0.00761-0.108\,\widehat{g}^{2}_{2}\epsilon_{\rm
R}\right)}{m^{2}_{3,0}-m^{2}_{1,0}}\;,\quad\varphi\approx\arctan\left(\frac{0.0705-\widehat{g}^{2}_{2}\epsilon_{\rm
R}}{\widehat{g}^{2}_{2}\epsilon_{\rm I}}\right)\;.$ (5.14)

From Eq. (5.14) we see that $\theta_{13}$ is proportional to the new
perturbative parameter $\kappa$, so it is always possible to get
$\sin\theta_{13}\sim 0.15$ by adjusting properly the value of $\kappa$. It is
apparent that $\Delta H_{\nu}$ also induces some corrections to the
eigenvalues $m^{2}_{1,0}$ and $m^{2}_{3,0}$, which turn out to be proportional
to $\kappa^{2}$ and thus can be neglected. After the above $(1,3)$-rotation is
applied to $H^{\prime}_{\nu}$, we have

$\displaystyle
H^{\prime\prime}_{\nu}=U^{\dagger}_{13}H^{\prime}_{\nu}U_{13}\approx\left(\begin{matrix}m^{2}_{1,0}&&-0.108\,\mu^{2}_{0}\widehat{g}^{2}_{2}\epsilon_{\rm
I}\kappa e^{-{\rm i}\varphi}&&0\\\
-0.108\,\mu^{2}_{0}\widehat{g}^{2}_{2}\epsilon_{\rm I}\kappa e^{{\rm
i}\varphi}&&m^{2}_{2,0}&&0\\\ 0&&0&&m^{2}_{3,0}\\\ \end{matrix}\right)\;,$
(5.15)

where the approximation $\cos\theta_{13}\approx 1$ has been made. Note that
the modulus of the off-diagonal element
$0.108\,\mu^{2}_{0}\widehat{g}^{2}_{2}\epsilon_{\rm I}\kappa$ is much smaller
than $m^{2}_{1,0}$ and $m^{2}_{2,0}$ in magnitude, but the high degeneracy
between $m^{2}_{1,0}$ and $m^{2}_{2,0}$ ensures a relatively large mixing
angle $\theta_{12}$. By using the degenerate perturbation theory, we obtain

$\displaystyle m^{2}_{1}$ $\displaystyle\approx$ $\displaystyle
m^{2}_{1,0}-0.108\,\mu^{2}_{0}\widehat{g}^{2}_{2}\epsilon_{\rm I}|\kappa|\;,$
$\displaystyle m^{2}_{2}$ $\displaystyle\approx$ $\displaystyle
m^{2}_{2,0}+0.108\,\mu^{2}_{0}\widehat{g}^{2}_{2}\epsilon_{\rm I}|\kappa|\;,$
$\displaystyle m^{2}_{3}$ $\displaystyle\approx$ $\displaystyle
m^{2}_{3,0}\;,$ (5.16)

which are the final results for three light neutrino masses. Meanwhile, the
unitary matrix $U_{12}$ to diagonalize $H^{\prime\prime}_{\nu}$ is found to be

Figure 2: The $3\sigma$ allowed regions of $\\{{\rm Re}\,\tau,{\rm
Im}\,\tau\\}$ and $\\{\widetilde{g}_{2},\widetilde{g}_{3}\\}$ obtained by
inputting the approximate analytical formulas of three neutrino masses and
mixing angles are represented by blue shaded areas. For comparison, the
allowed regions obtained by exact numerical calculations are plotted in red.

$\displaystyle U_{12}=\left(\begin{matrix}e^{-{\rm i}\varphi}&&0&&0\\\
0&&1&&0\\\ 0&&0&&1\\\
\end{matrix}\right)\cdot\left(\begin{matrix}\cos\theta_{12}&&-\sin\theta_{12}&&0\\\
\sin\theta_{12}&&\cos\theta_{12}&&0\\\ 0&&0&&1\end{matrix}\right)\;,$ (5.17)

with

$\displaystyle\sin^{2}\theta_{12}\approx\dfrac{0.108\,\mu^{2}_{0}\widehat{g}^{2}_{2}\epsilon_{\rm
I}|\kappa|}{m^{2}_{2,0}-m^{2}_{1,0}+0.216\,\mu^{2}_{0}\widehat{g}^{2}_{2}\epsilon_{\rm
I}|\kappa|}\;,$ (5.18)

implying that the value of $\sin^{2}\theta_{12}$ can be greatly enhanced if
$m^{2}_{2,0}$ is very close to $m^{2}_{1,0}$. For example, if
$m^{2}_{2,0}-m^{2}_{1,0}\approx
0.108\,\mu^{2}_{0}\widehat{g}^{2}_{2}\epsilon_{\rm I}|\kappa|$ or equivalently
$\Delta m^{2}_{21}\approx 0.324\,\mu^{2}_{0}\widehat{g}^{2}_{2}\epsilon_{\rm
I}|\kappa|$, we will obtain $\sin^{2}\theta_{12}\approx 1/3$, which agrees
well with the experimental observation. To illustrate how well the analytical
approximations are in agreement with exact numerical results, we have also
scanned over the model parameters by using the approximate expressions of
three neutrino masses and mixing angles. The $3\sigma$ allowed regions of
$\\{{\rm Re}\,\tau,{\rm Im}\,\tau,\widetilde{g}_{2},\widetilde{g}_{3}\\}$ are
shown in Fig. 2, where an excellent agreement with the exact numerical results
can be found for $|{\rm Re}\,\tau|\lesssim 0.1$ and $\widetilde{g}_{2}\gtrsim
1.0$.

Figure 3: The distributions of
$\\{\sin^{2}\delta,\sin^{2}\rho,\sin^{2}\sigma\\}$ in the vicinity of
$\tau={\rm i}$, where $\widetilde{g}_{2}=1.456$ and $\widetilde{g}_{3}=-0.350$
are fixed. The vertical dotted line corresponds to ${\rm Re}\,\tau=-0.0343$,
which is the typical value of ${\rm Re}\,\tau$ in its $3\sigma$ allowed
parameter space.

Combining three unitary matrices $U_{23}$, $U_{13}$ and $U_{12}$ together, we
finally obtain the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [65, 66]

$\displaystyle
U=U_{23}U_{13}U_{12}=\left(\begin{matrix}c_{12}c_{13}&&s_{12}c_{13}&&s_{13}e^{{\rm
i}\varphi}\\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{-{\rm
i}\varphi}&&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{-{\rm
i}\varphi}&&s_{23}c_{13}\\\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{-{\rm
i}\varphi}&&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{-{\rm
i}\varphi}&&c_{23}c_{13}\end{matrix}\right)\;,$ (5.19)

where $s_{ij}\equiv\sin\theta_{ij}$ and $c_{ij}\equiv\cos\theta_{ij}$ (for
$ij=12,13,23$) have been defined, and the unphysical phases have been
eliminated by redefining the phases of the charged-lepton fields. Comparing
Eq. (5.19) with the standard parametrization of $U$ [63], we can extract the
Dirac CP-violating phase $\delta$ as

$\displaystyle\delta=2\pi-\varphi\approx
2\pi-\arctan\left(\frac{0.0705-\widehat{g}^{2}_{2}\epsilon_{\rm
R}}{\widehat{g}^{2}_{2}\epsilon_{\rm I}}\right)\;.$ (5.20)

Since the value of $|\epsilon_{\rm R}|$ is much larger than $\epsilon_{\rm I}$
in the allowed parameter space, we obtain a nearly-maximal Dirac CP-violating
phase, i.e., $\delta\approx 270^{\circ}$. Two Majorana CP-violating phases can
be figured out by diagonalizing the neutrino mass matrix via
$U^{\dagger}\widetilde{M}_{\nu}U^{*}={\rm Diag}\\{m_{1}e^{{\rm
i}\rho},m_{2}e^{{\rm i}\sigma},m_{3}\\}$. It turns out that

$\displaystyle\rho\approx\frac{\pi}{2}+\arctan\left(\frac{\epsilon_{\rm
I}}{\epsilon_{\rm
R}}\right)\;,\quad\sigma\approx\frac{\pi}{2}-\arctan\left(\frac{\epsilon_{\rm
I}}{\epsilon_{\rm R}}\right)\;,$ (5.21)

which are both close to $90^{\circ}$ since $\epsilon_{\rm I}\ll|\epsilon_{\rm
R}|$. In the left panel of Fig. 3, we present the distribution of
$\sin^{2}\delta$ in the vicinity of $\tau={\rm i}$ where
$\widetilde{g}_{2}=1.456$ and $\widetilde{g}_{3}=-0.350$ are fixed for
illustration. For the chosen values of $\widetilde{g}_{2}$ and
$\widetilde{g}_{3}$, the $3\sigma$ allowed value of ${\rm Re}\,\tau$ is
tightly restricted to be around $-0.0343$, which is denoted by the vertical
dotted line in Fig. 3. Along this dotted line, the value of $\sin^{2}\delta$
goes down as ${\rm Im}\,\tau$ increases, which can be well described by Eq.
(5.20). For completeness, we also show the distributions of $\sin^{2}\rho$ and
$\sin^{2}\sigma$ around $\tau={\rm i}$ in the right two panels of Fig. 3. Note
that the values of $\sin^{2}\rho$ and $\sin^{2}\sigma$ along some parts of the
imaginary axis as well as the lower boundary of the fundamental domain ${\cal
G}$ can be maximal, corresponding to $\rho=\sigma=90^{\circ}$. However, CP
violation associated with Majorana phases should be always proportional to
$\sin 2\rho$ or $\sin 2\sigma$, implying CP conservation along the imaginary
axis and the lower boundary of ${\cal G}$.

Figure 4: The distribution of the absolute value of Jarlskog invariant $|{\cal
J}|$ in the vicinity of $\tau={\rm i}$, where $\widetilde{g}_{2}=1.456$ and
$\widetilde{g}_{3}=-0.350$ are fixed. The black star corresponds to ${\rm
Re}\,\tau=-0.0343$ and ${\rm Im}\,\tau=1.00466$, which are the typical values
of $\\{{\rm Re}\,\tau,{\rm Im}\,\tau\\}$ in their $3\sigma$ allowed parameter
space.

One can also adopt a parametrization-independent approach to measure the
magnitude of CP violation. For instance, the Jarlskog invariant ${\cal
J}\equiv{\rm Im}\left[U_{e1}U_{e2}^{*}U_{\mu 1}^{*}U_{\mu 2}\right]$ for CP
violation in leptonic sector [67, 68] can be calculated via

$\displaystyle\mathcal{I}_{1}=-6\mathrm{i}\Delta m^{2}_{21}\Delta
m^{2}_{31}\Delta m^{2}_{32}\Delta m^{2}_{e\mu}\Delta m^{2}_{\mu\tau}\Delta
m^{2}_{\tau e}\mathcal{J}\;,$ (5.22)

where $\Delta m^{2}_{\alpha\beta}\equiv m_{\alpha}^{2}-m_{\beta}^{2}$ (for
$\alpha,\beta=e,\mu,\tau$) denote the mass-squared differences of charged
leptons and
$\mathcal{I}_{1}\equiv\operatorname{Tr}\left\\{\left[H_{\nu},H_{l}\right]^{3}\right\\}$
is one of the CP-odd weak-basis invariants [69, 70, 71, 72]. Substituting the
approximate expressions of $H_{\nu}$ and $H_{l}$ into ${\cal I}_{1}$ and using
Eq. (5.22), we obtain

$\displaystyle{\cal
J}\approx\frac{0.00963\,\mu^{6}_{0}\kappa^{2}\widehat{g}^{6}_{2}\epsilon^{3}_{\rm
R}(\epsilon_{\rm I}-1.474\,\epsilon^{2}_{\rm R})}{\Delta m^{2}_{21}\Delta
m^{2}_{31}\Delta m^{2}_{32}}\;,$ (5.23)

where one can observe that ${\cal J}$ may be highly suppressed due to a tiny
numerator on the right-hand side. Therefore, a sizable value of ${\cal J}$ can
be achieved if $\Delta m^{2}_{21}$ is small enough. For example, if we take
$\Delta m^{2}_{21}=0.324\,\mu^{2}_{0}\widehat{g}^{2}_{2}\epsilon_{\rm
I}|\kappa|$, corresponding to $\sin^{2}\theta_{12}=1/3$ as discussed below Eq.
(5.18), then Eq. (5.23) can be simplified to

$\displaystyle{\cal J}\approx\dfrac{0.283\,r^{2}\epsilon^{3}_{\rm
R}(\epsilon_{\rm I}-1.474\,\epsilon^{2}_{\rm R})}{|\kappa|\epsilon^{3}_{\rm
I}(1-r)}\;,$ (5.24)

where the definition $r\equiv\Delta m^{2}_{21}/\Delta m^{2}_{31}$ in the NO
case has been used. The maximal CP violation with $\delta\approx 270^{\circ}$
corresponds to ${\cal
J}=s_{12}c_{12}s_{23}c_{23}s_{13}c_{13}^{2}\sin\delta\approx-0.03$, if all the
mixing angles take their individual best-fit values from the global-fit
analysis as shown in Table 2. Assuming $\epsilon_{\rm R}\approx-0.03$ and
$\epsilon_{\rm I}\approx 0.003$ in Eq. (5.24), we can find that ${\cal
J}\approx-0.03$ requires $\kappa\approx 15r^{2}\approx 0.015$. The
distribution of $|{\cal J}|$ in the vicinity of $\tau={\rm i}$ with
$\widetilde{g}_{2}=1.456$ and $\widetilde{g}_{3}=-0.350$ is shown in Fig. 4,
where one can clearly see that the red ring of radius $\sqrt{\epsilon^{2}_{\rm
R}+\epsilon^{2}_{\rm I}}\sim 0.035$ corresponds to the relatively large values
of ${\cal J}$. The $3\sigma$ allowed values of $\\{{\rm Re}\,\tau,{\rm
Im}\,\tau\\}$ denoted by the black star are exactly located at this ring,
indicating the nearly-maximal CP violation.

Before closing this section, let us briefly discuss why the IO case is not
permitted in our model. For the IO case, we can also introduce explicit
perturbations to $\tau={\rm i}$ in a similar way. After the first
$(2,3)$-rotation, the $(1,2)$-element of $H^{\prime}_{\nu}$ becomes much
larger than the counterpart in the NO case. As a consequence, if we require
$\sin^{2}\theta_{12}\sim 1/3$, $r\equiv\Delta m^{2}_{21}/|\Delta m^{2}_{32}|$
will be close to one, which is apparently incompatible with the experimental
observation.

##  6 Radiative Corrections

The modular symmetry and seesaw mechanism are supposed to work at a very high
energy scale $\Lambda_{\rm SS}$, whereas the leptonic flavor mixing parameters
are measured in the neutrino oscillation experiments whose energy scales are
much lower than the electroweak scale characterized by the mass of the $Z$
gauge boson $m_{Z}\approx 91.2~{}{\rm GeV}$, varying from several MeV to GeV.
In order to confront theoretical predictions with experimental observations,
one has to take into account the radiative corrections to physical parameters
via their RG equations. It is well known that the RG running effects can be
significant, especially for large values of $\tan\beta$ and (or) nearly-
degenerate neutrino masses [73]. Similar conclusions have also been reached in
the modular-invariant flavor models [32, 74]. As indicated in Eq. (5.18), the
value of $\sin^{2}\theta_{12}$ in our model is very sensitive to small changes
of model parameters, particularly in the region close to $\tau={\rm i}$. Hence
it is reasonable to expect large corrections from RG running even if the value
of $\tan\beta$ is relatively small.

Figure 5: Correlations between low-energy observables
$\\{r,\theta_{12},\theta_{13},\delta\\}$ and the model parameter ${\rm
Re}\,\tau$, where the red-shaded areas are gained by inputting the $3\sigma$
allowed parameter space in Sec. 4 without the RG running effects, while the
blue-shaded areas are obtained with same input but by taking account of the RG
running effects from the GUT scale to the electroweak scale.

For definiteness, we first carry out a quantitative analysis of RG running
effects on the mixing parameters from the GUT scale $\Lambda_{\rm
SS}\sim\Lambda_{\rm GUT}$ to the electroweak scale, where $\tan\beta=10$ will
be assumed. The RG running effects from the electroweak scale to the energy
scale of oscillation experiments are not significant and thus can be
neglected. In the basis where the charged-lepton Yukawa coupling matrix
$\widetilde{Y}_{l}\equiv{\rm Diag}\\{y_{e},y_{\mu},y_{\tau}\\}$ is diagonal
with $y_{\alpha}=\sqrt{2}m_{\alpha}/v_{\rm d}$ (for $\alpha=e,\mu,\tau$), the
Dirac neutrino Yukawa coupling matrix becomes
$\widetilde{Y}_{\nu}=U^{\dagger}_{l}Y_{\nu}$ with $Y_{\nu}=\sqrt{2}M_{\rm
D}/v_{\rm u}$. After integrating out heavy Majorana neutrinos below the scale
$\Lambda_{\rm SS}$, we obtain the effective neutrino mass parameter ${\cal
M}\equiv-U^{{\dagger}}_{l}Y_{\nu}M^{-1}_{\rm R}Y^{\rm T}_{\nu}U^{\ast}_{l}$.
In the minimal supersymmetric SM, the radiative corrections to
$\widetilde{Y}_{l}$ and ${\cal M}$ can be described by the one-loop RG
equations [75, 76, 77, 78, 79, 80]

$\displaystyle 16\pi^{2}\frac{{\rm d}\widetilde{Y}_{l}}{{\rm d}t}$
$\displaystyle=$
$\displaystyle\left[\alpha_{l}+3\left(\widetilde{Y}_{l}\widetilde{Y}^{{\dagger}}_{l}\right)\right]\widetilde{Y}_{l}\;,$
(6.1) $\displaystyle 16\pi^{2}\frac{{\rm d}{\cal M}}{{\rm d}t}$
$\displaystyle=$ $\displaystyle\alpha_{\nu}{\cal
M}+\left[\left(\widetilde{Y}_{l}\widetilde{Y}^{{\dagger}}_{l}\right){\cal
M}+{\cal M}\left(\widetilde{Y}_{l}\widetilde{Y}^{{\dagger}}_{l}\right)^{\rm
T}\right]\;,$ (6.2)

where $t\equiv\ln(\mu/\Lambda_{\rm SS})$ with $\mu$ being the renormalization
scale, $\alpha_{l}=-1.8\mathfrak{g}^{2}_{1}-3\mathfrak{g}^{2}_{2}+3{\rm
Tr}(Y_{\rm d}Y^{{\dagger}}_{\rm d})+{\rm
Tr}(\widetilde{Y}_{l}\widetilde{Y}^{{\dagger}}_{l})$ and
$\alpha_{\nu}=-1.2\mathfrak{g}^{2}_{1}-6\mathfrak{g}^{2}_{2}+6{\rm Tr}(Y_{\rm
u}Y^{{\dagger}}_{\rm u})$. Notice that $\mathfrak{g}_{2}$ and
$\mathfrak{g}_{1}$ represent respectively the ${\rm SU}(2)_{\rm L}$ and ${\rm
U}(1)_{\rm Y}$ gauge couplings, $Y_{\rm u}$ and $Y_{\rm d}$ denote the up- and
down-type quark Yukawa coupling matrices. In the chosen flavor basis where
$\widetilde{Y}_{l}$ is diagonal, only the diagonal elements on both sides of
Eq. (6.1) survive. Therefore, the radiative corrections to neutrino masses and
lepton flavor mixing parameters are determined by the RG equation of ${\cal
M}$ [81]. The solution to ${\cal M}$ in Eq. (6.2) can be formally written as

$\displaystyle{\cal M}(m_{Z})=I_{\nu}\left(\begin{matrix}I_{e}&&&0&&&0\\\
0&&&I_{\mu}&&&0\\\ 0&&&0&&&I_{\tau}\end{matrix}\right){\cal M}(\Lambda_{\rm
SS})\left(\begin{matrix}I_{e}&&&0&&&0\\\ 0&&&I_{\mu}&&&0\\\
0&&&0&&&I_{\tau}\end{matrix}\right)\;,$ (6.3)

where $I_{\nu}$ and $I_{\alpha}$ (for $\alpha=e,\mu,\tau$) are the evolution
functions [81]

$\displaystyle I_{\nu}$ $\displaystyle=$ $\displaystyle{\rm
exp}\left[-\frac{1}{16\pi^{2}}\int^{\ln(\Lambda_{\rm
SS}/m_{Z})}_{0}\alpha_{\nu}(t){\rm d}t\right]\;,$ $\displaystyle I_{\alpha}$
$\displaystyle=$ $\displaystyle{\rm
exp}\left[-\frac{1}{16\pi^{2}}\int^{\ln(\Lambda_{\rm
SS}/m_{Z})}_{0}y^{2}_{\alpha}(t){\rm d}t\right]\;.$ (6.4)

While $I_{\nu}$ in Eq. (6.3) affects only the absolute scale of light neutrino
masses, $I_{\alpha}$ could modify both neutrino masses and flavor mixing
parameters. By using Eq. (6.3), one can establish the direct connection
between the effective neutrino mass parameter at the high-energy scale
$\Lambda_{\rm SS}$ and that at the electroweak scale $m_{Z}$.

To illustrate radiative corrections to flavor mixing parameters, we
numerically solve the RG equations and search for the allowed parameter space
by re-scanning over the model parameters. It turns out that the $3\sigma$
allowed regions of $\\{{\rm Re}\,\tau,{\rm
Im}\,\tau,\widetilde{g}_{2},\widetilde{g}_{3}\\}$ obtained with or without
radiative corrections are hardly distinguishable, implying that the RG running
effects will almost not change the allowed parameter space. However, they can
greatly modify the predictions for flavor mixing parameters if we fix the
values of $\\{{\rm Re}\,\tau,{\rm
Im}\,\tau,\widetilde{g}_{2},\widetilde{g}_{3}\\}$. To make this point clearer,
we identify the $3\sigma$ allowed parameter space obtained without considering
the RG running effects in Sec. 4 as the input at the GUT scale, and calculate
the predictions for neutrino masses and mixing parameters at the electroweak
scale with radiative corrections. The correlations between low-energy
observables $\\{r,\theta_{12},\theta_{13},\delta\\}$ and the model parameter
${\rm Re}\,\tau$ are shown in Fig. 5, where both the red- and blue-shaded
areas are gained by inputting the $3\sigma$ allowed parameter space in Sec. 4,
while only the blue-shaded areas correspond to the results with the radiative
corrections. Notice that the correction to $\theta_{23}$ is negligibly small,
so we do not show the result of $\theta_{23}$. As one can see from Fig. 5, the
RG running effects can modify the predicted values of oscillation parameters
to a large extent especially when $|{\rm Re}\,\tau|$ is small. For instance,
if $|{\rm Re}\,\tau|$ is as small as $0.03$, the mixing angle $\theta_{12}$
can reach $55^{\circ}$, which is about $60\%$ larger than that without the RG
running effects.

To better understand why radiative corrections to $\theta_{12}$ are so large,
we consider the RG running effects on the model parameters. Since the electron
and muon Yukawa couplings are extremely small, $I_{e}\approx I_{\mu}\approx 1$
holds as an excellent approximation and $I_{\tau}$ contributes dominantly to
the running effects on ${\cal M}$ in Eq. (6.3). With the help of Eq. (6.3), we
can derive the approximate formula of $\widetilde{M}_{\nu}(m_{Z})$ around the
stabilizer $\tau={\rm i}$ after including the RG running. Then, three light
neutrino masses at the electroweak scale are

$\displaystyle m^{2}_{1}(m_{Z})$ $\displaystyle\approx$ $\displaystyle
m^{2}_{1,0}-0.108\,I_{\tau}\mu^{2}_{0}\widehat{g}^{2}_{2}\epsilon_{\rm
I}|\kappa|\;,$ $\displaystyle m^{2}_{2}(m_{Z})$ $\displaystyle\approx$
$\displaystyle
I^{2}_{\tau}m^{2}_{2,0}+0.108\,I_{\tau}\mu^{2}_{0}\widehat{g}^{2}_{2}\epsilon_{\rm
I}|\kappa|\;,$ $\displaystyle m^{2}_{3}(m_{Z})$ $\displaystyle\approx$
$\displaystyle I^{2}_{\tau}m^{2}_{3,0}\;,$ (6.5)

and the expression of $\sin^{2}\theta_{12}(m_{Z})$ is approximately given by

$\displaystyle\sin^{2}\theta_{12}(m_{Z})\approx\dfrac{0.108\,I_{\tau}\mu^{2}_{0}\widehat{g}^{2}_{2}\epsilon_{\rm
I}|\kappa|}{I^{2}_{\tau}m^{2}_{2,0}-m^{2}_{1,0}+0.216\,I_{\tau}\mu^{2}_{0}\widehat{g}^{2}_{2}\epsilon_{\rm
I}|\kappa|}\;,$ (6.6)

which will be reduced to Eq. (5.18) for $I_{\tau}=1$. As pointed out in the
last section, $m^{2}_{2,0}$ can be very close to $m^{2}_{1,0}$ when ${\rm
Re}\,\tau$ approaches zero. Therefore, although $I_{\tau}-1\approx-0.0015$ is
extremely small in our case, $I^{2}_{\tau}m^{2}_{2,0}-m^{2}_{1,0}$ in the
denominator on the right-hand side of Eq. (6.6) leads to large corrections to
$\sin^{2}\theta_{12}$. For instance, let us set the values of model parameters
to be $\epsilon=-0.0301+0.00359\,{\rm i}$, $\widehat{g}_{2}=1.555$ and
$\kappa=0.0125$, then Eq. (5.18) gives $\theta_{12}\approx 31.5^{\circ}$. On
the other hand, if the radiative corrections are taken into consideration, Eq.
(6.6) will give $\theta_{12}(m_{Z})\approx 50.1^{\circ}$, where one can
observe a remarkable modification of $\theta_{12}$ due to the RG running
effects. In conclusion, despite the fact that the RG running effects on the
model parameters are inconspicuous, the low-energy observables are sensitive
to small perturbations to free parameters in the vicinity of $\tau={\rm i}$
and radiative corrections are actually playing a significant role.

Although our analytical calculations are performed in a specific model, they
are indeed helpful in understanding sizable radiative corrections to the
mixing angle $\theta_{12}$ for nearly-degenerate neutrino masses in the most
general case. The key point is that the large mixing angle $\theta_{12}$ is
essentially determined by the ratio of two small model parameters, to which
radiative corrections themselves are tiny but change considerably the ratio.

##  7 Summary

Modular flavor symmetries provide us with an attractive way to account for
lepton flavor mixing and CP violation. Instead of just fitting the modulus
$\tau$ to the experimental data, one can also start with one of some special
values $\tau$ (i.e., the fixed points or stabilizers), at which residual
symmetries are retained after the breaking of the global modular symmetry. Any
realistic models for lepton masses, flavor mixing and CP violation require the
modulus $\tau$ to slightly deviate from the stabilizers. Therefore, it is
interesting to explore the basic properties of modular-symmetry models with a
modulus $\tau$ in the vicinity of the stabilizer. Such an exploration is
useful for understanding the phenomenological implications of the modular-
symmetry models, and gives a clue to possible residual symmetries and the
modular symmetry itself.

In this paper, we construct a feasible lepton flavor model based on the
modular $A^{\prime}_{5}$ group combined with the gCP symmetry, and focus on
the fixed point $\tau={\rm i}$ where CP symmetry is preserved. A seemingly
puzzling feature is that it predicts a nearly-maximal CP-violating phase in
the region where $\tau={\rm i}+\epsilon$ with $\epsilon$ being a small
parameter. By treating $\epsilon$ as a perturbation to the stabilizer
$\tau={\rm i}$, we derive the approximate analytical expressions of light
neutrino masses, flavor mixing angles and the CP-violating phases. We find
that $\sin^{2}\theta_{12}$ depends on the mass-squared difference $\Delta
m^{2}_{21}$, and the enhancement of $\sin^{2}\theta_{12}$ can be attributed to
the high degeneracy between two neutrino masses $m_{1}$ and $m_{2}$. The
approximate expression of $\tan\delta$ is found to be
$-(0.0705-\widehat{g}^{2}_{2}\epsilon_{\rm
R})/(\widehat{g}^{2}_{2}\epsilon_{\rm I})$, where $\epsilon_{\rm R}\equiv{\rm
Re}\,\epsilon$ is about ten times larger than $\epsilon_{\rm I}\equiv{\rm
Im}\,\epsilon$ in the $3\sigma$ allowed parameter space. This analytical
result shows that a nearly-maximal CP-violating phase can be obtained near
$\tau={\rm i}$. We find that $\epsilon\approx-0.03+0.003\,{\rm i}$ is
sufficient to give ${\cal J}\approx-0.03$, corresponding to the nearly-maximal
CP-violating phase $\delta\approx 270^{\circ}$ when all the mixing angles are
consistent with their observed values.

Since $\sin^{2}\theta_{12}$ is very sensitive to small perturbations to model
parameters in the region around $\tau={\rm i}$, radiative corrections via RG
running to the model parameters should be important. Given the same input
parameters and ${\rm Re}\,\tau\sim-0.03$, the mixing angle $\theta_{12}$ would
be changed from $32^{\circ}$ to $50^{\circ}$ after the radiative corrections
are taken into consideration. The main reason for such a significant RG
running effect is that the mass degeneracy between $m_{1}$ and $m_{2}$ is
governed by small perturbation parameters while $\theta_{12}$ is determined by
the ratio of two small perturbation parameters. Although the details of such
an analysis in this paper are quite model-dependent, it does shed some light
on the common features of the modular-symmetry models that seem to be
“unstable” around the stabilizers. For instance, neutrino masses are usually
predicted to be nearly degenerate in such kinds of models.

In the near future, it is necessary and interesting to further investigate the
properties of modular-symmetry models around all the stabilizers in a
systematic and model-independent way. As such scenarios are strictly
constrained by the modular symmetry and the residual symmetries, their
predictions for neutrino masses, mixing angles and the Dirac CP-violating
phase are readily to be tested in next-generation neutrino oscillation
experiments, neutrinoless double-beta decays and cosmological observations.

## Acknowledgements

This work was supported in part by the National Natural Science Foundation of
China under grant No. 11775232 and No. 11835013, by the Key Research Program
of the Chinese Academy of Sciences under grant No. XDPB15, and by the CAS
Center for Excellence in Particle Physics.

##  Appendix A Modular $A^{\prime}_{5}$ group and the modular space ${\cal
M}_{1}\left[\Gamma(5)\right]$

The $A^{\prime}_{5}$ group has 120 elements, which can be divided into nine
conjugacy classes, indicating that $A^{\prime}_{5}$ has nine distinct
irreducible representations that are normally denoted as ${\bf 1}$,
$\widehat{\bf 2}$, $\widehat{\bf 2}^{\prime}$, ${\bf 3}$, ${\bf 3}^{\prime}$,
${\bf 4}$, $\widehat{\bf 4}$, ${\bf 5}$ and $\widehat{\bf 6}$ by their
dimensions. The representation matrices of all three generators $S$, $T$ and
$R$ in the irreducible representations are summarized as below

$\displaystyle{\bf 1}$ $\displaystyle:$
$\displaystyle\rho(S)=+1\;,\quad\rho(T)=+1\;,\quad\rho(R)=+1\;,$
$\displaystyle\widehat{\bf 2}$ $\displaystyle:$
$\displaystyle\rho(S)=\dfrac{\rm
i}{\sqrt[4]{5}}\left(\begin{matrix}\sqrt{\phi}&\sqrt{\phi-1}\\\
\sqrt{\phi-1}&-\sqrt{\phi}\\\
\end{matrix}\right)\;,\quad\rho(T)=\left(\begin{matrix}\omega^{2}&0\\\
0&\omega^{3}\\\ \end{matrix}\right)\;,\quad\rho(R)=-\mathbb{I}_{2\times 2}\;,$
$\displaystyle\widehat{\bf 2}^{\prime}$ $\displaystyle:$
$\displaystyle\rho(S)=\dfrac{\rm
i}{\sqrt[4]{5}}\left(\begin{matrix}\sqrt{\phi-1}&\sqrt{\phi}\\\
\sqrt{\phi}&-\sqrt{\phi-1}\\\
\end{matrix}\right)\;,\quad\rho(T)=\left(\begin{matrix}\omega&0\\\
0&\omega^{4}\\\ \end{matrix}\right)\;,\quad\rho(R)=-\mathbb{I}_{2\times 2}\;,$
$\displaystyle{\bf 3}$ $\displaystyle:$
$\displaystyle\rho(S)=\dfrac{1}{\sqrt{5}}\left(\begin{matrix}1&-\sqrt{2}&-\sqrt{2}\\\
-\sqrt{2}&-\phi&\phi-1\\\ -\sqrt{2}&\phi-1&-\phi\\\
\end{matrix}\right)\;,\quad\rho(T)=\left(\begin{matrix}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{4}\end{matrix}\right)\;,\quad\rho(R)=+\mathbb{I}_{3\times 3}\;,$
$\displaystyle{\bf 3}^{\prime}$ $\displaystyle:$
$\displaystyle\rho(S)=\dfrac{1}{\sqrt{5}}\left(\begin{matrix}-1&\sqrt{2}&\sqrt{2}\\\
\sqrt{2}&1-\phi&\phi\\\ \sqrt{2}&\phi&1-\phi\\\
\end{matrix}\right)\;,\quad\rho(T)=\left(\begin{matrix}1&0&0\\\
0&\omega^{2}&0\\\ 0&0&\omega^{3}\\\
\end{matrix}\right)\;,\quad\rho(R)=+\mathbb{I}_{3\times 3}\;,$
$\displaystyle{\bf 4}$ $\displaystyle:$
$\displaystyle\rho(S)=\dfrac{1}{\sqrt{5}}\left(\begin{matrix}1&\phi-1&\phi&-1\\\
\phi-1&-1&1&\phi\\\ \phi&1&-1&\phi-1\\\ -1&\phi&\phi-1&1\\\
\end{matrix}\right)\;,\quad\rho(T)=\left(\begin{matrix}\omega&0&0&0\\\
0&\omega^{2}&0&0\\\ 0&0&\omega^{3}&0\\\ 0&0&0&\omega^{4}\\\
\end{matrix}\right)\;,\quad\rho(R)=+\mathbb{I}_{4\times 4}\;,$
$\displaystyle\widehat{\bf 4}$ $\displaystyle:$
$\displaystyle\rho(S)=\dfrac{\rm
i}{5^{\frac{3}{4}}}\left(\begin{matrix}-\sqrt{2\phi+1}&\sqrt{3\phi}&\sqrt{3(\phi-1)}&\sqrt{2\phi-3}\\\
\sqrt{3\phi}&\sqrt{2\phi-3}&\sqrt{2\phi+1}&\sqrt{3(\phi-1)}\\\
\sqrt{3(\phi-1)}&\sqrt{2\phi+1}&-\sqrt{2\phi-3}&-\sqrt{3\phi}\\\
\sqrt{2\phi-3}&\sqrt{3(\phi-1)}&-\sqrt{3\phi}&\sqrt{2\phi+1}\\\
\end{matrix}\right)\;,$
$\displaystyle\rho(T)=\left(\begin{matrix}\omega&0&0&0\\\ 0&\omega^{2}&0&0\\\
0&0&\omega^{3}&0\\\ 0&0&0&\omega^{4}\\\
\end{matrix}\right)\;,\quad\rho(R)=-\mathbb{I}_{4\times 4}\;,$
$\displaystyle{\bf 5}$ $\displaystyle:$
$\displaystyle\rho(S)=\dfrac{1}{5}\left(\begin{matrix}-1&\sqrt{6}&\sqrt{6}&\sqrt{6}&\sqrt{6}\\\
\sqrt{6}&(\phi-1)^{2}&-2\phi&2(\phi-1)&\phi^{2}\\\
\sqrt{6}&-2\phi&\phi^{2}&(\phi-1)^{2}&2(\phi-1)\\\
\sqrt{6}&2(\phi-1)&(\phi-1)^{2}&\phi^{2}&-2\phi\\\
\sqrt{6}&\phi^{2}&2(\phi-1)&-2\phi&(\phi-1)^{2}\end{matrix}\right)\;,$
$\displaystyle\rho(T)=\left(\begin{matrix}1&0&0&0&0\\\ 0&\omega&0&0&0\\\
0&0&\omega^{2}&0&0\\\ 0&0&0&\omega^{3}&0\\\ 0&0&0&0&\omega^{4}\\\
\end{matrix}\right)\;,\quad\rho(R)=+\mathbb{I}_{5\times 5}\;,$
$\displaystyle\widehat{\bf 6}$ $\displaystyle:$
$\displaystyle\rho(S)=\dfrac{-\rm
i}{5^{\frac{3}{4}}}\left(\begin{matrix}\sqrt{\phi}&-\sqrt{2(\phi-1)}&\sqrt{2\phi-3}&-\sqrt{2\phi+1}&\sqrt{2\phi}&\sqrt{\phi-1}\\\
-\sqrt{2(\phi-1)}&-\sqrt{\phi-1}&\sqrt{2(\phi-1)}&\sqrt{2\phi}&\sqrt{\phi}&\sqrt{2\phi}\\\
\sqrt{2\phi-3}&\sqrt{2(\phi-1)}&\sqrt{\phi}&-\sqrt{\phi-1}&-\sqrt{2\phi}&\sqrt{2\phi+1}\\\
-\sqrt{2\phi+1}&\sqrt{2\phi}&-\sqrt{\phi-1}&-\sqrt{\phi}&\sqrt{2(\phi-1)}&\sqrt{2\phi-3}\\\
\sqrt{2\phi}&\sqrt{\phi}&-\sqrt{2\phi}&\sqrt{2(\phi-1)}&\sqrt{\phi-1}&\sqrt{2(\phi-1)}\\\
\sqrt{\phi-1}&\sqrt{2\phi}&\sqrt{2\phi+1}&\sqrt{2\phi-3}&\sqrt{2(\phi-1)}&-\sqrt{\phi}\\\
\end{matrix}\right)\;,$
$\displaystyle\rho(T)=\left(\begin{matrix}1&0&0&0&0&0\\\ 0&\omega&0&0&0&0\\\
0&0&\omega^{2}&0&0&0\\\ 0&0&0&\omega^{3}&0&0\\\ 0&0&0&0&\omega^{4}&0\\\
0&0&0&0&0&1\\\ \end{matrix}\right)\;,\quad\rho(R)=-\mathbb{I}_{6\times 6}\;,$

where $\omega\equiv e^{2{\rm i}\pi/5}$ and $\mathbb{I}_{n\times n}$ denotes
the $n$-dimensional identity matrix. Notice that the representations ${\bf
1}$, ${\bf 3}$, ${\bf 3}^{\prime}$, ${\bf 4}$ and ${\bf 5}$ with
$R=\mathbb{I}$ coincide with those for $A_{5}$, whereas $\widehat{\bf 2}$,
$\widehat{\bf 2}^{\prime}$, $\widehat{\bf 4}$ and $\widehat{\bf 6}$ are unique
for $A^{\prime}_{5}$ with $R=-\mathbb{I}$.

The whole list of decomposition rules for the Kronecker products of any two
nontrivial irreducible representations of $A^{\prime}_{5}$ can be found in
Ref. [42]. Here we only summarize the decomposition rules relevant to this
paper.

  * •

$\widehat{\bf 2}\otimes\widehat{\bf 2}={\bf 1}_{\rm a}\oplus{\bf 3}_{\rm s}$

$\displaystyle{\bf 1}_{\rm
a}:-\dfrac{\sqrt{2}}{2}\left(\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\right)\;,\quad{\bf
3}_{\rm
s}:\dfrac{\sqrt{2}}{2}\left(\begin{array}[]{c}\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1}\\\
-\sqrt{2}\alpha_{2}\beta_{2}\\\ \sqrt{2}\alpha_{1}\beta_{1}\\\
\end{array}\right)\;.$ (A.4)

  * •

$\widehat{\bf 2}\otimes{\bf 3}^{\prime}=\widehat{\bf 6}$

$\displaystyle\begin{array}[]{c}\widehat{\bf
6}:-\dfrac{\sqrt{2}}{2}\left(\begin{array}[]{c}\alpha_{1}\beta_{3}-\alpha_{2}\beta_{2}\\\
\sqrt{2}\alpha_{2}\beta_{3}\\\ -\sqrt{2}\alpha_{1}\beta_{1}\\\
\sqrt{2}\alpha_{2}\beta_{1}\\\ -\sqrt{2}\alpha_{1}\beta_{2}\\\
\alpha_{1}\beta_{3}+\alpha_{2}\beta_{2}\\\ \end{array}\right)\;.\end{array}$
(A.12)

  * •

$\mathbf{3}\otimes\bm{3}=\mathbf{1}_{\rm s}\oplus\bm{3}_{\rm
a}\oplus\bm{5}_{\rm s}$

$\displaystyle\begin{array}[]{c}\mathbf{1}_{\rm
s}:\dfrac{\sqrt{3}}{3}(\alpha_{1}\beta_{1}+\alpha_{2}\beta_{3}+\alpha_{3}\beta_{2})\;,\\\
{\bf 3}_{\rm
a}:\dfrac{\sqrt{2}}{2}\left(\begin{array}[]{c}\alpha_{2}\beta_{3}-\alpha_{3}\beta_{2}\\\
\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\\\
\alpha_{3}\beta_{1}-\alpha_{1}\beta_{3}\end{array}\right)\;,\quad{\bf 5}_{\rm
s}:\dfrac{\sqrt{6}}{6}\left(\begin{array}[]{c}2\alpha_{1}\beta_{1}-\alpha_{2}\beta_{3}-\alpha_{3}\beta_{2}\\\
-\sqrt{3}\alpha_{1}\beta_{2}-\sqrt{3}\alpha_{2}\beta_{1}\\\
\sqrt{6}\alpha_{2}\beta_{2}\\\ \sqrt{6}\alpha_{3}\beta_{3}\\\
-\sqrt{3}\left(\alpha_{1}\beta_{3}+\alpha_{3}\beta_{1}\right)\end{array}\right)\;.\end{array}$
(A.23)

  * •

${\bf 3}^{\prime}\otimes{\bf 3}^{\prime}={\bf 1}_{\rm s}\oplus{\bf
3}^{\prime}_{\rm a}\oplus{\bf 5}_{\rm s}$

$\displaystyle\begin{array}[]{c}{\bf 1}_{\rm
s}:\dfrac{\sqrt{3}}{3}\left(\alpha_{1}\beta_{1}+\alpha_{2}\beta_{3}+\alpha_{3}\beta_{2}\right)\;,\\\
{\bf 3}^{\prime}_{\rm
a}:\dfrac{\sqrt{2}}{2}\left(\begin{array}[]{c}\alpha_{2}\beta_{3}-\alpha_{3}\beta_{2}\\\
\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\\\
\alpha_{3}\beta_{1}-\alpha_{1}\beta_{3}\end{array}\right)\;,\quad{\bf 5}_{\rm
s}:\dfrac{\sqrt{6}}{6}\left(\begin{array}[]{c}2\alpha_{1}\beta_{1}-\alpha_{2}\beta_{3}-\alpha_{3}\beta_{2}\\\
\sqrt{6}\alpha_{3}\beta_{3}\\\
-\sqrt{3}\left(\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1}\right)\\\
-\sqrt{3}\left(\alpha_{1}\beta_{3}+\alpha_{3}\beta_{1}\right)\\\
\sqrt{6}\alpha_{2}\beta_{2}\end{array}\right)\;.\end{array}$ (A.34)

  * •

${\bf 5}\otimes{\bf 5}={\bf 1}_{\rm s}\oplus{\bf 3}_{\rm a}\oplus{\bf
3}^{\prime}_{\rm a}\oplus{\bf 4}_{\rm s}\oplus{\bf 4}_{\rm a}\oplus{\bf
5}_{\rm s,1}\oplus{\bf 5}_{\rm s,2}$

$\displaystyle\begin{array}[]{c}{\bf 1}_{\rm
s}:\dfrac{\sqrt{5}}{5}\left(\alpha_{1}\beta_{1}+\alpha_{2}\beta_{5}+\alpha_{3}\beta_{4}+\alpha_{4}\beta_{3}+\alpha_{5}\beta_{2}\right)\;,\\\
{\bf 3}_{\rm
a}:\dfrac{\sqrt{10}}{10}\left(\begin{array}[]{c}\alpha_{2}\beta_{5}+2\alpha_{3}\beta_{4}-2\alpha_{4}\beta_{3}-\alpha_{5}\beta_{2}\\\
-\sqrt{3}\alpha_{1}\beta_{2}+\sqrt{3}\alpha_{2}\beta_{1}+\sqrt{2}\alpha_{3}\beta_{5}-\sqrt{2}\alpha_{5}\beta_{3}\\\
\sqrt{3}\alpha_{1}\beta_{5}+\sqrt{2}\alpha_{2}\beta_{4}-\sqrt{2}\alpha_{4}\beta_{2}-\sqrt{3}\alpha_{5}\beta_{1}\end{array}\right)\;,\\\
{\bf 3}^{\prime}_{\rm
a}:\dfrac{\sqrt{10}}{10}\left(\begin{array}[]{c}2\alpha_{2}\beta_{5}-\alpha_{3}\beta_{4}+\alpha_{4}\beta_{3}-2\alpha_{5}\beta_{2}\\\
\sqrt{3}\alpha_{1}\beta_{3}-\sqrt{3}\alpha_{3}\beta_{1}+\sqrt{2}\alpha_{4}\beta_{5}-\sqrt{2}\alpha_{5}\beta_{4}\\\
-\sqrt{3}\alpha_{1}\beta_{4}+\sqrt{2}\alpha_{2}\beta_{3}-\sqrt{2}\alpha_{3}\beta_{2}+\sqrt{3}\alpha_{4}\beta_{1}\end{array}\right)\;,\\\
{\bf 4}_{\rm
s}:\dfrac{\sqrt{30}}{30}\left(\begin{array}[]{l}\sqrt{6}\alpha_{1}\beta_{2}+\sqrt{6}\alpha_{2}\beta_{1}-\alpha_{3}\beta_{5}+4\alpha_{4}\beta_{4}-\alpha_{5}\beta_{3}\\\
\sqrt{6}\alpha_{1}\beta_{3}+4\alpha_{2}\beta_{2}+\sqrt{6}\alpha_{3}\beta_{1}-\alpha_{4}\beta_{5}-\alpha_{5}\beta_{4}\\\
\sqrt{6}\alpha_{1}\beta_{4}-\alpha_{2}\beta_{3}-\alpha_{3}\beta_{2}+\sqrt{6}\alpha_{4}\beta_{1}+4\alpha_{5}\beta_{5}\\\
\sqrt{6}\alpha_{1}\beta_{5}-\alpha_{2}\beta_{4}+4\alpha_{3}\beta_{3}-\alpha_{4}\beta_{2}+\sqrt{6}\alpha_{5}\beta_{1}\end{array}\right)\;,\\\
{\bf 4}_{\rm
a}:\dfrac{\sqrt{10}}{10}\left(\begin{array}[]{c}\sqrt{2}\alpha_{1}\beta_{2}-\sqrt{2}\alpha_{2}\beta_{1}+\sqrt{3}\alpha_{3}\beta_{5}-\sqrt{3}\alpha_{5}\beta_{3}\\\
-\sqrt{2}\alpha_{1}\beta_{3}+\sqrt{2}\alpha_{3}\beta_{1}+\sqrt{3}\alpha_{4}\beta_{5}-\sqrt{3}\alpha_{5}\beta_{4}\\\
-\sqrt{2}\alpha_{1}\beta_{4}-\sqrt{3}\alpha_{2}\beta_{3}+\sqrt{3}\alpha_{3}\beta_{2}+\sqrt{2}\alpha_{4}\beta_{1}\\\
\sqrt{2}\alpha_{1}\beta_{5}-\sqrt{3}\alpha_{2}\beta_{4}+\sqrt{3}\alpha_{4}\beta_{2}-\sqrt{2}\alpha_{5}\beta_{1}\end{array}\right)\;,\\\
{\bf 5}_{\rm
s,1}:\dfrac{\sqrt{14}}{14}\left(\begin{array}[]{c}2\alpha_{1}\beta_{1}+\alpha_{2}\beta_{5}-2\alpha_{3}\beta_{4}-2\alpha_{4}\beta_{3}+\alpha_{5}\beta_{2}\\\
\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1}+\sqrt{6}\alpha_{3}\beta_{5}+\sqrt{6}\alpha_{5}\beta_{3}\\\
-2\alpha_{1}\beta_{3}+\sqrt{6}\alpha_{2}\beta_{2}-2\alpha_{3}\beta_{1}\\\
-2\alpha_{1}\beta_{4}-2\alpha_{4}\beta_{1}+\sqrt{6}\alpha_{5}\beta_{5}\\\
\alpha_{1}\beta_{5}+\sqrt{6}\alpha_{2}\beta_{4}+\sqrt{6}\alpha_{4}\beta_{2}+\alpha_{5}\beta_{1}\end{array}\right)\;,\\\
{\bf 5}_{\rm
s,2}:\dfrac{\sqrt{14}}{14}\left(\begin{array}[]{c}2\alpha_{1}\beta_{1}-2\alpha_{2}\beta_{5}+\alpha_{3}\beta_{4}+\alpha_{4}\beta_{3}-2\alpha_{5}\beta_{2}\\\
-2\alpha_{1}\beta_{2}-2\alpha_{2}\beta_{1}+\sqrt{6}\alpha_{4}\beta_{4}\\\
\alpha_{1}\beta_{3}+\alpha_{3}\beta_{1}+\sqrt{6}\alpha_{4}\beta_{5}+\sqrt{6}\alpha_{5}\beta_{4}\\\
\alpha_{1}\beta_{4}+\sqrt{6}\alpha_{2}\beta_{3}+\sqrt{6}\alpha_{3}\beta_{2}+\alpha_{4}\beta_{1}\\\
-2\alpha_{1}\beta_{5}+\sqrt{6}\alpha_{3}\beta_{3}-2\alpha_{5}\beta_{1}\end{array}\right)\;.\end{array}$
(A.66)

  * •

$\widehat{\bf 6}\otimes\widehat{\bf 6}={\bf 1}_{\rm a}\oplus{\bf 3}_{\rm
s,1}\oplus{\bf 3}_{\rm s,2}\oplus{\bf 3}^{\prime}_{\rm s,1}\oplus{\bf
3}^{\prime}_{\rm s,2}\oplus{\bf 4}_{\rm s}\oplus{\bf 4}_{\rm a}\oplus{\bf
5}_{\rm s}\oplus{\bf 5}_{\rm a,1}\oplus{\bf 5}_{\rm a,2}$

$\displaystyle\begin{array}[]{c}{\bf 1}_{\rm
a}:\dfrac{\sqrt{6}}{6}\left(\alpha_{1}\beta_{6}+\alpha_{2}\beta_{5}-\alpha_{3}\beta_{4}+\alpha_{4}\beta_{3}-\alpha_{5}\beta_{2}-\alpha_{6}\beta_{1}\right)\;,\\\
{\bf 3}_{\rm
s,1}:\dfrac{1}{2}\left(\begin{array}[]{l}\alpha_{1}\beta_{6}+\alpha_{3}\beta_{4}+\alpha_{4}\beta_{3}+\alpha_{6}\beta_{1}\\\
\alpha_{2}\beta_{6}+\alpha_{3}\beta_{5}+\alpha_{5}\beta_{3}+\alpha_{6}\beta_{2}\\\
\alpha_{1}\beta_{5}-\alpha_{2}\beta_{4}-\alpha_{4}\beta_{2}+\alpha_{5}\beta_{1}\end{array}\right)\;,\end{array}$
(A.72) $\displaystyle\begin{array}[]{c}{\bf 3}_{\rm
s,2}:\dfrac{\sqrt{6}}{6}\left(\begin{array}[]{c}-\alpha_{1}\beta_{1}+\alpha_{1}\beta_{6}-\alpha_{2}\beta_{5}-\alpha_{5}\beta_{2}+\alpha_{6}\beta_{1}+\alpha_{6}\beta_{6}\\\
\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1}+\alpha_{3}\beta_{5}-\sqrt{2}\alpha_{4}\beta_{4}+\alpha_{5}\beta_{3}\\\
-\alpha_{2}\beta_{4}+\sqrt{2}\alpha_{3}\beta_{3}-\alpha_{4}\beta_{2}-\alpha_{5}\beta_{6}-\alpha_{6}\beta_{5}\end{array}\right)\;,\\\
{\bf 3}^{\prime}_{\rm
s,1}:\dfrac{\sqrt{6}}{6}\left(\begin{array}[]{c}\alpha_{1}\beta_{6}+\alpha_{2}\beta_{5}-\alpha_{3}\beta_{4}-\alpha_{4}\beta_{3}+\alpha_{5}\beta_{2}+\alpha_{6}\beta_{1}\\\
-\sqrt{2}\left(\alpha_{1}\beta_{3}-\alpha_{2}\beta_{2}+\alpha_{3}\beta_{1}\right)\\\
-\sqrt{2}\left(\alpha_{4}\beta_{6}+\alpha_{5}\beta_{5}+\alpha_{6}\beta_{4}\right)\end{array}\right)\;,\\\
{\bf 3}^{\prime}_{\rm
s,2}:\dfrac{\sqrt{2}}{4}\left(\begin{array}[]{c}\sqrt{2}\left(\alpha_{1}\beta_{1}-\alpha_{2}\beta_{5}-\alpha_{5}\beta_{2}-\alpha_{6}\beta_{6}\right)\\\
\alpha_{1}\beta_{3}+\alpha_{3}\beta_{1}-\alpha_{3}\beta_{6}+\sqrt{2}\alpha_{4}\beta_{5}+\sqrt{2}\alpha_{5}\beta_{4}-\alpha_{6}\beta_{3}\\\
\alpha_{1}\beta_{4}+\sqrt{2}\alpha_{2}\beta_{3}+\sqrt{2}\alpha_{3}\beta_{2}+\alpha_{4}\beta_{1}+\alpha_{4}\beta_{6}+\alpha_{6}\beta_{4}\end{array}\right)\;,\\\
{\bf 4}_{\rm
s}:\dfrac{\sqrt{30}}{30}\left(\begin{array}[]{c}\begin{array}[]{c}-3\alpha_{1}\beta_{2}-3\alpha_{2}\beta_{1}-\alpha_{2}\beta_{6}+\alpha_{3}\beta_{5}-2\sqrt{2}\alpha_{4}\beta_{4}+\alpha_{5}\beta_{3}-\alpha_{6}\beta_{2}\\\
\sqrt{2}\alpha_{1}\beta_{3}+2\sqrt{2}\alpha_{2}\beta_{2}+\sqrt{2}\alpha_{3}\beta_{1}+2\sqrt{2}\alpha_{3}\beta_{6}+\alpha_{4}\beta_{5}+\alpha_{5}\beta_{4}+2\sqrt{2}\alpha_{6}\beta_{3}\\\
2\sqrt{2}\alpha_{1}\beta_{4}-\alpha_{2}\beta_{3}-\alpha_{3}\beta_{2}+2\sqrt{2}\alpha_{4}\beta_{1}-\sqrt{2}\alpha_{4}\beta_{6}+2\sqrt{2}\alpha_{5}\beta_{5}-\sqrt{2}\alpha_{6}\beta_{4}\\\
\alpha_{1}\beta_{5}+\alpha_{2}\beta_{4}-2\sqrt{2}\alpha_{3}\beta_{3}+\alpha_{4}\beta_{2}+\alpha_{5}\beta_{1}-3\alpha_{5}\beta_{6}-3\alpha_{6}\beta_{5}\end{array}\end{array}\right)\;,\\\
{\bf 4}_{\rm
a}:-\dfrac{\sqrt{6}}{6}\left(\begin{array}[]{c}\begin{array}[]{c}\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}+\alpha_{2}\beta_{6}+\alpha_{3}\beta_{5}-\alpha_{5}\beta_{3}-\alpha_{6}\beta_{2}\\\
\sqrt{2}\alpha_{1}\beta_{3}-\sqrt{2}\alpha_{3}\beta_{1}+\alpha_{4}\beta_{5}-\alpha_{5}\beta_{4}\\\
\alpha_{2}\beta_{3}-\alpha_{3}\beta_{2}+\sqrt{2}\alpha_{4}\beta_{6}-\sqrt{2}\alpha_{6}\beta_{4}\\\
\alpha_{1}\beta_{5}-\alpha_{2}\beta_{4}+\alpha_{4}\beta_{2}-\alpha_{5}\beta_{1}-\alpha_{5}\beta_{6}+\alpha_{6}\beta_{5}\end{array}\end{array}\right)\;,\\\
{\bf 5}_{\rm
s}:\dfrac{\sqrt{3}}{6}\left(\begin{array}[]{c}\sqrt{6}\left(\alpha_{1}\beta_{1}+\alpha_{6}\beta_{6}\right)\\\
\sqrt{2}\left(\alpha_{2}\beta_{6}-\alpha_{3}\beta_{5}-\sqrt{2}\alpha_{4}\beta_{4}-\alpha_{5}\beta_{3}+\alpha_{6}\beta_{2}\right)\\\
-\alpha_{1}\beta_{3}-2\alpha_{2}\beta_{2}-\alpha_{3}\beta_{1}+\alpha_{3}\beta_{6}+\sqrt{2}\alpha_{4}\beta_{5}+\sqrt{2}\alpha_{5}\beta_{4}+\alpha_{6}\beta_{3}\\\
\alpha_{1}\beta_{4}-\sqrt{2}\alpha_{2}\beta_{3}-\sqrt{2}\alpha_{3}\beta_{2}+\alpha_{4}\beta_{1}+\alpha_{4}\beta_{6}-2\alpha_{5}\beta_{5}+\alpha_{6}\beta_{4}\\\
-\sqrt{2}\left(\alpha_{1}\beta_{5}+\alpha_{2}\beta_{4}+\sqrt{2}\alpha_{3}\beta_{3}+\alpha_{4}\beta_{2}+\alpha_{5}\beta_{1}\right)\end{array}\right)\;.\\\
{\bf 5}_{\rm
a,1}:\dfrac{\sqrt{3}}{6}\left(\begin{array}[]{c}\alpha_{1}\beta_{6}-2\alpha_{2}\beta_{5}-\alpha_{3}\beta_{4}+\alpha_{4}\beta_{3}+2\alpha_{5}\beta_{2}-\alpha_{6}\beta_{1}\\\
-\sqrt{3}\left(\alpha_{2}\beta_{6}-\alpha_{3}\beta_{5}+\alpha_{5}\beta_{3}-\alpha_{6}\beta_{2}\right)\\\
\sqrt{6}\left(\alpha_{3}\beta_{6}-\alpha_{6}\beta_{3}\right)\\\
-\sqrt{6}\left(\alpha_{1}\beta_{4}-\alpha_{4}\beta_{1}\right)\\\
-\sqrt{3}\left(\alpha_{1}\beta_{5}+\alpha_{2}\beta_{4}-\alpha_{4}\beta_{2}-\alpha_{5}\beta_{1}\right)\end{array}\right)\;,\\\
{\bf 5}_{\rm
a,2}:\dfrac{\sqrt{3}}{6}\left(\begin{array}[]{c}\sqrt{3}\left(\alpha_{1}\beta_{6}+\alpha_{3}\beta_{4}-\alpha_{4}\beta_{3}-\alpha_{6}\beta_{1}\right)\\\
-2\alpha_{1}\beta_{2}+2\alpha_{2}\beta_{1}+\alpha_{2}\beta_{6}+\alpha_{3}\beta_{5}-\alpha_{5}\beta_{3}-\alpha_{6}\beta_{2}\\\
-\sqrt{2}\left(\alpha_{1}\beta_{3}-\alpha_{3}\beta_{1}-\sqrt{2}\alpha_{4}\beta_{5}+\sqrt{2}\alpha_{5}\beta_{4}\right)\\\
\sqrt{2}\left(\sqrt{2}\alpha_{2}\beta_{3}-\sqrt{2}\alpha_{3}\beta_{2}-\alpha_{4}\beta_{6}+\alpha_{6}\beta_{4}\right)\\\
\alpha_{1}\beta_{5}-\alpha_{2}\beta_{4}+\alpha_{4}\beta_{2}-\alpha_{5}\beta_{1}+2\alpha_{5}\beta_{6}-2\alpha_{6}\beta_{5}\end{array}\right)\;.\end{array}$
(A.115)

For a given non-negative integer $k$, the modular space ${\cal
M}_{k}\left[\Gamma(5)\right]$ of weight $k$ for $\Gamma(5)$ contains $5k+1$
linearly-independent modular forms, which can be regarded as the basis vectors
of the modular space. According to Ref. [82], we have

$\displaystyle{\cal
M}_{k}\left[\Gamma(5)\right]=\bigoplus_{\begin{subarray}{c}a+b=5k\\\ a,b\geq
0\end{subarray}}\mathbb{C}\,\frac{\eta(5\tau)^{15k}}{\eta(\tau)^{3k}}\,{\mathfrak{k}}^{a}_{\frac{1}{5},\frac{0}{5}}(5\tau)\,{\mathfrak{k}}^{b}_{\frac{2}{5},\frac{0}{5}}(5\tau)\;,$
(A.116)

where $\eta(\tau)$ is the Dedekind eta function

$\displaystyle\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}\left(1-q^{n}\right)\;,$
(A.117)

with $q\equiv e^{2{\rm i}\pi\tau}$, and ${\mathfrak{k}}_{r_{1},r_{2}}(\tau)$
is the Klein form

$\displaystyle\mathfrak{k}_{r_{1},r_{2}}(\tau)=q_{z}^{(r_{1}-1)/2}\left(1-q_{z}\right)\times\prod_{n=1}^{\infty}\left(1-q^{n}q_{z}\right)\left(1-q^{n}q_{z}^{-1}\right)\left(1-q^{n}\right)^{-2}\;,$
(A.118)

with $(r_{1},r_{2})$ being a pair of rational numbers in the domain of
${\mathbb{Q}}^{2}-{\mathbb{Z}}^{2}$, $z\equiv\tau r_{1}+r_{2}$ and
$q_{z}\equiv e^{2{\rm i}\pi z}$. Now we take $k=1$, then the basis vectors of
the modular space ${\cal M}_{1}\left[\Gamma(5)\right]$ can be expressed as

$\displaystyle\begin{array}[]{lcl}\widehat{e}_{1}(\tau)=\dfrac{\eta^{15}(5\tau)}{\eta^{3}(\tau)}\,\mathfrak{k}^{5}_{\frac{2}{5},\frac{0}{5}}(5\tau)\;,&{}\hfil&\widehat{e}_{2}(\tau)=\dfrac{\eta^{15}(5\tau)}{\eta^{3}(\tau)}\,\mathfrak{k}_{\frac{1}{5},\frac{0}{5}}(5\tau)\,\mathfrak{k}^{4}_{\frac{2}{5},\frac{0}{5}}(5\tau)\;,\\\
\widehat{e}_{3}(\tau)=\dfrac{\eta^{15}(5\tau)}{\eta^{3}(\tau)}\,\mathfrak{k}^{2}_{\frac{1}{5},\frac{0}{5}}(5\tau)\,\mathfrak{k}^{3}_{\frac{2}{5},\frac{0}{5}}(5\tau)\;,&{}\hfil&\widehat{e}_{4}(\tau)=\dfrac{\eta^{15}(5\tau)}{\eta^{3}(\tau)}\,\mathfrak{k}^{3}_{\frac{1}{5},\frac{0}{5}}(5\tau)\,\mathfrak{k}^{2}_{\frac{2}{5},\frac{0}{5}}(5\tau)\;,\\\
\widehat{e}_{5}(\tau)=\dfrac{\eta^{15}(5\tau)}{\eta^{3}(\tau)}\,\mathfrak{k}^{4}_{\frac{1}{5},\frac{0}{5}}(5\tau)\,\mathfrak{k}_{\frac{2}{5},\frac{0}{5}}(5\tau)\;,&{}\hfil&\widehat{e}_{6}(\tau)=\dfrac{\eta^{15}(5\tau)}{\eta^{3}(\tau)}\,\mathfrak{k}^{5}_{\frac{1}{5},\frac{0}{5}}(5\tau)\;.\end{array}$
(A.122)

Furthermore, making use of Eqs. (A.117) and (A.118), we can derive the Fourier
expansions of the above six basis vectors

$\displaystyle\widehat{e}_{1}$ $\displaystyle=$ $\displaystyle
1+3q+4q^{2}+2q^{3}+q^{4}+3q^{5}+6q^{6}+4q^{7}-q^{9}+\cdots\;,$
$\displaystyle\widehat{e}_{2}$ $\displaystyle=$ $\displaystyle
q^{1/5}\left(1+2q+2q^{2}+q^{3}+2q^{4}+2q^{5}+2q^{6}+q^{7}+2q^{8}+2q^{9}+\cdots\right)\;,$
$\displaystyle\widehat{e}_{3}$ $\displaystyle=$ $\displaystyle
q^{2/5}\left(1+q+q^{2}+q^{3}+2q^{4}+q^{6}+q^{7}+2q^{8}+q^{9}+\cdots\right)\;,$
$\displaystyle\widehat{e}_{4}$ $\displaystyle=$ $\displaystyle
q^{3/5}\left(1+q^{2}+q^{3}+q^{4}-q^{5}+2q^{6}+2q^{8}+q^{9}+\cdots\right)\;,$
$\displaystyle\widehat{e}_{5}$ $\displaystyle=$ $\displaystyle
q^{4/5}\left(1-q+2q^{2}+2q^{6}-2q^{7}+2q^{8}+q^{9}+\cdots\right)\;,$
$\displaystyle\widehat{e}_{6}$ $\displaystyle=$ $\displaystyle
q\left(1-2q+4q^{2}-3q^{3}+q^{4}+2q^{5}-2q^{6}+3q^{8}-2q^{9}+\cdots\right)\;.$
(A.123)

## References

  * [1] Z. z. Xing and S. Zhou, “Neutrinos in particle physics, astronomy and cosmology,” Springer-Verlag, Berlin Heidelberg (2011).
  * [2] Z. z. Xing, “Flavor structures of charged fermions and massive neutrinos,” Phys. Rept. 854, 1-147 (2020) [arXiv:1909.09610].
  * [3] P. Minkowski, “$\mu\to e\gamma$ at a Rate of One Out of $10^{9}$ Muon Decays?,” Phys. Lett. 67B, 421 (1977).
  * [4] T. Yanagida, “Horizontal Symmetry And Masses Of Neutrinos,” Conf. Proc. C 7902131, 95 (1979).
  * [5] M. Gell-Mann, P. Ramond and R. Slansky, “Complex Spinors and Unified Theories,” Conf. Proc. C 790927, 315 (1979) [arXiv:1306.4669].
  * [6] S. L. Glashow, “The Future of Elementary Particle Physics,” NATO Sci. Ser. B 61, 687 (1980).
  * [7] R. N. Mohapatra and G. Senjanovic, “Neutrino Mass and Spontaneous Parity Violation,” Phys. Rev. Lett. 44, 912 (1980).
  * [8] G. Altarelli and F. Feruglio, “Discrete Flavor Symmetries and Models of Neutrino Mixing,” Rev. Mod. Phys. 82, 2701 (2010) [arXiv:1002.0211].
  * [9] H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada and M. Tanimoto, “Non-Abelian Discrete Symmetries in Particle Physics,” Prog. Theor. Phys. Suppl. 183, 1 (2010) [arXiv:1003.3552].
  * [10] S. F. King and C. Luhn, “Neutrino Mass and Mixing with Discrete Symmetry,” Rept. Prog. Phys. 76, 056201 (2013) [arXiv:1301.1340].
  * [11] S. F. King, A. Merle, S. Morisi, Y. Shimizu and M. Tanimoto, “Neutrino Mass and Mixing: from Theory to Experiment,” New J. Phys. 16, 045018 (2014) [arXiv:1402.4271].
  * [12] F. Feruglio, “Are neutrino masses modular forms?,” [arXiv:1706.08749].
  * [13] T. Kobayashi, K. Tanaka and T. H. Tatsuishi, “Neutrino mixing from finite modular groups,” Phys. Rev. D 98, no.1, 016004 (2018) [arXiv:1803.10391].
  * [14] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T. H. Tatsuishi and H. Uchida, “Finite modular subgroups for fermion mass matrices and baryon/lepton number violation,” Phys. Lett. B 794, 114-121 (2019) [arXiv:1812.11072].
  * [15] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T. H. Tatsuishi, “Modular $S_{3}$-invariant flavor model in SU(5) grand unified theory,” PTEP 2020, no.5, 053B05 (2020) [arXiv:1906.10341].
  * [16] T. Kobayashi, N. Omoto, Y. Shimizu, K. Takagi, M. Tanimoto and T. H. Tatsuishi, “Modular A4 invariance and neutrino mixing,” JHEP 11, 196 (2018) [arXiv:1808.03012].
  * [17] F. J. de Anda, S. F. King and E. Perdomo, “$SU(5)$ grand unified theory with $A_{4}$ modular symmetry,” Phys. Rev. D 101, no.1, 015028 (2020) [arXiv:1812.05620].
  * [18] H. Okada and M. Tanimoto, “CP violation of quarks in $A_{4}$ modular invariance,” Phys. Lett. B 791, 54-61 (2019) [arXiv:1812.09677].
  * [19] H. Okada and M. Tanimoto, “Towards unification of quark and lepton flavors in $A_{4}$ modular invariance,” Eur. Phys. J. C 81, no.1, 52 (2021) [arXiv:1905.13421].
  * [20] G. J. Ding, S. F. King and X. G. Liu, “Modular A4 symmetry models of neutrinos and charged leptons,” JHEP 09, 074 (2019) [arXiv:1907.11714].
  * [21] T. Asaka, Y. Heo, T. H. Tatsuishi and T. Yoshida, “Modular $A_{4}$ invariance and leptogenesis,” JHEP 01, 144 (2020) [arXiv:1909.06520].
  * [22] D. Zhang, “A modular $A_{4}$ symmetry realization of two-zero textures of the Majorana neutrino mass matrix,” Nucl. Phys. B 952, 114935 (2020) [arXiv:1910.07869].
  * [23] T. Kobayashi, T. Nomura and T. Shimomura, “Type II seesaw models with modular $A_{4}$ symmetry,” Phys. Rev. D 102, no.3, 035019 (2020) [arXiv:1912.00637].
  * [24] X. Wang, “Lepton flavor mixing and CP violation in the minimal type-(I+II) seesaw model with a modular $A_{4}$ symmetry,” Nucl. Phys. B 957, 115105 (2020) [arXiv:1912.13284].
  * [25] H. Okada and M. Tanimoto, “Quark and lepton flavors with common modulus $\tau$ in $A_{4}$ modular symmetry,” [arXiv:2005.00775].
  * [26] C. Y. Yao, J. N. Lu and G. J. Ding, “Modular Invariant $A_{4}$ Models for Quarks and Leptons with Generalized CP Symmetry,” [arXiv:2012.13390].
  * [27] P. Chen, G. J. Ding and S. F. King, “$SU(5)$ GUTs with $A_{4}$ modular symmetry,” [arXiv:2101.12724].
  * [28] J. T. Penedo and S. T. Petcov, “Lepton Masses and Mixing from Modular $S_{4}$ Symmetry,” Nucl. Phys. B 939, 292-307 (2019) [arXiv:1806.11040].
  * [29] P. P. Novichkov, J. T. Penedo, S. T. Petcov and A. V. Titov, “Modular S4 models of lepton masses and mixing,” JHEP 04, 005 (2019) [arXiv:1811.04933].
  * [30] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T. H. Tatsuishi, “New $A_{4}$ lepton flavor model from $S_{4}$ modular symmetry,” JHEP 02, 097 (2020) [arXiv:1907.09141].
  * [31] X. Wang and S. Zhou, “The minimal seesaw model with a modular $S_{4}$ symmetry,” JHEP 05, 017 (2020) [arXiv:1910.09473].
  * [32] X. Wang, “Dirac neutrino mass models with a modular $S_{4}$ symmetry,” Nucl. Phys. B 962, 115247 (2021) [arXiv:2007.05913].
  * [33] Y. Zhao and H. H. Zhang, “Adjoint SU(5) GUT model with Modular $S_{4}$ Symmetry,” [arXiv:2101.02266].
  * [34] P. P. Novichkov, J. T. Penedo, S. T. Petcov and A. V. Titov, “Modular A5 symmetry for flavour model building,” JHEP 04, 174 (2019) [arXiv:1812.02158].
  * [35] G. J. Ding, S. F. King and X. G. Liu, “Neutrino mass and mixing with $A_{5}$ modular symmetry,” Phys. Rev. D 100, no.11, 115005 (2019) [arXiv:1903.12588].
  * [36] J. C. Criado, F. Feruglio and S. J. D. King, “Modular Invariant Models of Lepton Masses at Levels 4 and 5,” JHEP 02, 001 (2020) [arXiv:1908.11867].
  * [37] G. J. Ding, S. F. King, C. C. Li and Y. L. Zhou, “Modular Invariant Models of Leptons at Level 7,” JHEP 08, 164 (2020) [arXiv:2004.12662].
  * [38] X. G. Liu and G. J. Ding, “Neutrino Masses and Mixing from Double Covering of Finite Modular Groups,” JHEP 08, 134 (2019) [arXiv:1907.01488].
  * [39] J. N. Lu, X. G. Liu and G. J. Ding, “Modular symmetry origin of texture zeros and quark lepton unification,” Phys. Rev. D 101, no.11, 115020 (2020) [arXiv:1912.07573].
  * [40] P. P. Novichkov, J. T. Penedo and S. T. Petcov, “Double cover of modular $S_{4}$ for flavour model building,” Nucl. Phys. B 963, 115301 (2021) [arXiv:2006.03058].
  * [41] X. G. Liu, C. Y. Yao and G. J. Ding, “Modular Invariant Quark and Lepton Models in Double Covering of $S_{4}$ Modular Group,” [arXiv:2006.10722].
  * [42] X. Wang, B. Yu and S. Zhou, “Double Covering of the Modular $A_{5}$ Group and Lepton Flavor Mixing in the Minimal Seesaw Model,” [arXiv:2010.10159].
  * [43] C. Y. Yao, X. G. Liu and G. J. Ding, “Fermion Masses and Mixing from Double Cover and Metaplectic Cover of $A_{5}$ Modular Group,” [arXiv:2011.03501].
  * [44] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T. H. Tatsuishi, “$A_{4}$ lepton flavor model and modulus stabilization from $S_{4}$ modular symmetry,” Phys. Rev. D 100, no.11, 115045 (2019) [erratum: Phys. Rev. D 101, no.3, 039904 (2020)] [arXiv:1909.05139].
  * [45] K. Ishiguro, T. Kobayashi and H. Otsuka, “Landscape of Modular Symmetric Flavor Models,” [arXiv:2011.09154].
  * [46] P. P. Novichkov, S. T. Petcov and M. Tanimoto, “Trimaximal Neutrino Mixing from Modular $A_{4}$ Invariance with Residual Symmetries,” Phys. Lett. B 793, 247-258 (2019) [arXiv:1812.11289].
  * [47] I. de Medeiros Varzielas, M. Levy and Y. L. Zhou, “Symmetries and stabilisers in modular invariant flavour models,” JHEP 11, 085 (2020) [arXiv:2008.05329].
  * [48] I. de Medeiros Varzielas, S. F. King and Y. L. Zhou, “Multiple modular symmetries as the origin of flavor,” Phys. Rev. D 101, no.5, 055033 (2020) [arXiv:1906.02208].
  * [49] S. F. King and Y. L. Zhou, “Trimaximal TM1 mixing with two modular $S_{4}$ groups,” Phys. Rev. D 101, no.1, 015001 (2020) [arXiv:1908.02770].
  * [50] G. J. Ding, S. F. King, X. G. Liu and J. N. Lu, “Modular S4 and A4 symmetries and their fixed points: new predictive examples of lepton mixing,” JHEP 12, 030 (2019) [arXiv:1910.03460].
  * [51] H. Okada and M. Tanimoto, “Modular invariant flavor model of $A_{4}$ and hierarchical structures at nearby fixed points,” Phys. Rev. D 103, no.1, 015005 (2021) [arXiv:2009.14242].
  * [52] F. Feruglio, V. Gherardi, A. Romanino and A. Titov, “Modular Invariant Dynamics and Fermion Mass Hierarchies around $\tau={\rm i}$,” [arXiv:2101.08718].
  * [53] P. P. Novichkov, J. T. Penedo, S. T. Petcov and A. V. Titov, “Generalised CP Symmetry in modular invariant Models of Flavour,” JHEP 1907 165 (2019) [arXiv:1905.11970].
  * [54] H. Okada and M. Tanimoto, “Spontaneous CP violation by modulus $\tau$ in $A_{4}$ model of lepton flavors,” [arXiv:2012.01688].
  * [55] S. Antusch and V. Maurer, “Running quark and lepton parameters at various scales,” JHEP 11, 115 (2013) [arXiv:1306.6879].
  * [56] I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz and A. Zhou, “The fate of hints: updated global analysis of three-flavor neutrino oscillations,” [arXiv:2007.14792].
  * [57] NuFIT 5.0 (2020), www.nu-fit.org.
  * [58] A. Gando et al. [KamLAND-Zen Collaboration], “Search for Majorana Neutrinos near the Inverted Mass Hierarchy Region with KamLAND-Zen,” Phys. Rev. Lett. 117, no.8, 082503 (2016) Addendum: [Phys. Rev. Lett. 117 no.10, 109903 (2016)] [arXiv:1605.02889].
  * [59] N. Aghanim et al. [Planck Collaboration], “Planck 2018 results. VI. Cosmological parameters,” arXiv:1807.06209.
  * [60] E. Giusarma, M. Gerbino, O. Mena, S. Vagnozzi, S. Ho and K. Freese, “Improvement of cosmological neutrino mass bounds,” Phys. Rev. D 94, no.8, 083522 (2016) [arXiv:1605.04320].
  * [61] S. Vagnozzi, E. Giusarma, O. Mena, K. Freese, M. Gerbino, S. Ho and M. Lattanzi, “Unveiling $\nu$ secrets with cosmological data: neutrino masses and mass hierarchy,” Phys. Rev. D 96, no.12, 123503 (2017) [arXiv:1701.08172].
  * [62] F. Capozzi, E. Di Valentino, E. Lisi, A. Marrone, A. Melchiorri and A. Palazzo, “Global constraints on absolute neutrino masses and their ordering,” Phys. Rev. D 95, no. 9, 096014 (2017) Addendum: [Phys. Rev. D 101, no. 11, 116013 (2020)] [arXiv:2003.08511, arXiv:1703.04471].
  * [63] P. A. Zyla et al. (Particle Data Group), “The Review of Particle Physics (2020),” Prog. Theor. Exp. Phys. 2020, 083C01 (2020).
  * [64] M. J. Dolinski, A. W. P. Poon and W. Rodejohann, “Neutrinoless Double-Beta Decay: Status and Prospects,” arXiv:1902.04097.
  * [65] B. Pontecorvo, “Mesonium and anti-mesonium,” Sov. Phys. JETP 6, 429 (1957) [Zh. Eksp. Teor. Fiz. 33, 549 (1957)].
  * [66] Z. Maki, M. Nakagawa and S. Sakata, “Remarks on the unified model of elementary particles,” Prog. Theor. Phys. 28, 870-880 (1962).
  * [67] C. Jarlskog, “Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Violation,” Phys. Rev. Lett. 55, 1039 (1985)
  * [68] D. d. Wu, “The Rephasing Invariants and CP,” Phys. Rev. D 33, 860 (1986)
  * [69] J. Bernabeu, G. C. Branco and M. Gronau, “CP Restrictions on Quark Mass Matrices,” Phys. Lett. B 169, 243-247 (1986)
  * [70] G. C. Branco, L. Lavoura and M. N. Rebelo, “Majorana Neutrinos and CP Violation in the Leptonic Sector,” Phys. Lett. B 180, 264 (1986).
  * [71] B. Yu and S. Zhou, “The number of sufficient and necessary conditions for CP conservation with Majorana neutrinos: three or four?,” Phys. Lett. B 800, 135085 (2020) [arXiv:1908.09306].
  * [72] B. Yu and S. Zhou, “Sufficient and Necessary Conditions for CP Conservation in the Case of Degenerate Majorana Neutrino Masses,” [arXiv:2009.12347].
  * [73] T. Ohlsson and S. Zhou, “Renormalization group running of neutrino parameters,” Nature Commun. 5, 5153 (2014) [arXiv:1311.3846].
  * [74] J. C. Criado and F. Feruglio, “Modular Invariance Faces Precision Neutrino Data,” SciPost Phys. 5, no.5, 042 (2018) [arXiv:1807.01125].
  * [75] M. E. Machacek and M. T. Vaughn, “Two Loop Renormalization Group Equations in a General Quantum Field Theory. 2. Yukawa Couplings,” Nucl. Phys. B 236 (1984) 221.
  * [76] H. Arason, D. J. Castano, B. Keszthelyi, S. Mikaelian, E. J. Piard, P. Ramond and B. D. Wright, “Renormalization group study of the standard model and its extensions. 1. The Standard model,” Phys. Rev. D 46 (1992) 3945.
  * [77] D. J. Castano, E. J. Piard and P. Ramond, “Renormalization group study of the Standard Model and its extensions. 2. The Minimal supersymmetric Standard Model,” Phys. Rev. D 49 (1994) 4882 [hep-ph/9308335].
  * [78] P. H. Chankowski and Z. Pluciennik, “Renormalization group equations for seesaw neutrino masses,” Phys. Lett. B 316 (1993) 312 [hep-ph/9306333].
  * [79] K. S. Babu, C. N. Leung and J. T. Pantaleone, “Renormalization of the neutrino mass operator,” Phys. Lett. B 319 (1993) 191 [hep-ph/9309223].
  * [80] S. Antusch, M. Drees, J. Kersten, M. Lindner and M. Ratz, “Neutrino mass operator renormalization revisited,” Phys. Lett. B 519 (2001) 238 [hep-ph/0108005].
  * [81] J. w. Mei and Z. z. Xing, “Radiative corrections to neutrino mixing and CP violation in the minimal seesaw model with leptogenesis,” Phys. Rev. D 69 (2004) 073003 [hep-ph/0312167].
  * [82] D. Schultz, “Notes on modular forms”,
https://faculty.math.illinois.edu/~schult25/ModFormNotes.pdf (2015).