(7.3.8) $\displaystyle W_{\nu}$
$\displaystyle=-\kappa\,\left(\left[\left(N^{c}L\right)_{\boldsymbol{3}_{S}}+g\left(N^{c}L\right)_{\boldsymbol{3}_{A}}\right]Y^{(2)}\right)_{\boldsymbol{1}}H_{u}-\Lambda\left(N^{c}N^{c}Y^{(2)}\right)_{\boldsymbol{1}}\,.$
(7.3.9)

The parameters $\kappa$ and $\Lambda$ can be made real without loss of
generality, whereas $g$ is complex in general. In the real basis for
$\boldsymbol{3}$ of $\Gamma_{3}$, this superpotential leads to the following
matrices $\mathcal{Y}^{e}(\tau)$, $\mathcal{Y}^{\nu}(\tau)$ and
$\mathcal{M}(\tau)$:

$\displaystyle\mathcal{Y}^{e}(\tau)$ $\displaystyle=2\begin{pmatrix}\alpha
Y_{2}Y_{3}&\alpha Y_{1}Y_{3}&\alpha Y_{1}Y_{2}\\\ \beta Y_{2}Y_{3}&\beta
Y_{1}Y_{3}&\beta Y_{1}Y_{2}\\\ \gamma Y_{2}Y_{3}&\gamma\omega
Y_{1}Y_{3}&\gamma\omega^{2}Y_{1}Y_{2}\end{pmatrix},$ (7.3.10)
$\displaystyle\mathcal{Y}^{\nu}(\tau)$
$\displaystyle=\kappa\left[\begin{pmatrix}0&Y_{3}&Y_{2}\\\ Y_{3}&0&Y_{1}\\\
Y_{2}&Y_{1}&0\end{pmatrix}+g\begin{pmatrix}0&Y_{3}&-Y_{2}\\\ -Y_{3}&0&Y_{1}\\\
Y_{2}&-Y_{1}&0\end{pmatrix}\right],$ (7.3.11) $\displaystyle\mathcal{M}(\tau)$
$\displaystyle=2\Lambda\begin{pmatrix}0&Y_{3}&Y_{2}\\\ Y_{3}&0&Y_{1}\\\
Y_{2}&Y_{1}&0\end{pmatrix}.$ (7.3.12)

The matrix $\mathcal{C}^{\nu}(\tau)$ of eq. (7.1.8) is now given by the seesaw
formula of eq. (7.1.10).

Some analytical considerations easily follow from the previous equations for
the “bare” quantities, i.e. those corresponding to the minimal Kähler
potential. We make use of the following $\epsilon$-expansion of
$Y^{(2)}\leavevmode\nobreak\ $101010One can prove, in general, that
$\left.\frac{\text{d}}{\text{d}\tau}\right|_{i}Y_{2,3}=i\,Y_{2,3}(i)$.
Moreover, we can rephase $Y^{(2)}$ in such a way that
$(Y_{3}(i))^{*}=Y_{2}(i)\equiv y$. In this basis, we find that
$\left.\frac{\text{d}}{\text{d}\tau}\right|_{i}Y_{1}\in-i\,\mathbb{R}^{+}$.:

$Y_{1}=-ix\epsilon\,,\qquad Y_{2}=y\left(1+i\epsilon\right)\,,\qquad
Y_{3}=y^{\ast}\left(1+i\epsilon\right),$ (7.3.13)

where, up to an overall constant, $x\approx 1.49087$ and
$y=\sqrt{3}/2+i(3/2-\sqrt{3})$. To first order in $\epsilon$ we obtain:

$\frac{\mathcal{Y}_{\nu}(\tau)}{\kappa}=\begin{pmatrix}0&\left(1+g\right)y^{\ast}&\left(1-g\right)y\\\
\left(1-g\right)y^{\ast}&0&0\\\
\left(1+g\right)y&0&0\end{pmatrix}+i\,\epsilon\begin{pmatrix}0&\left(1+g\right)y^{\ast}&\left(1-g\right)y\\\
\left(1-g\right)y^{\ast}&0&-\left(1+g\right)x\\\
\left(1+g\right)y&-\left(1-g\right)x&0\end{pmatrix},$ (7.3.14)
$\frac{\mathcal{M}(\tau)}{2\Lambda}=\begin{pmatrix}0&y^{\ast}&y\\\
y^{\ast}&0&0\\\ y&0&0\end{pmatrix}+i\,\epsilon\begin{pmatrix}0&y^{\ast}&y\\\
y^{\ast}&0&-x\\\ y&-x&0\end{pmatrix}.$ (7.3.15)

Notice that the bare Majorana mass matrix has one eigenvalue proportional to
$\epsilon$, thus vanishing in the limit $\tau\to i$. This corresponds to the
case of single right-handed neutrino dominance, in which one of the
electroweak singlet neutrinos is massless in the symmetric limit 111111As
already observed, in order for the standard seesaw analysis to be valid, we
must require the product $\left|\epsilon\right|\Lambda$ (that is the order of
magnitude of the lightest right-handed neutrino mass) to be large with respect
to the electroweak scale. For the present model this does not pose any
practical restriction, since the best-fit region (see below) is achieved for
values of $|\epsilon|\sim 10^{-2}\div 10^{-1}$.. Inverting the matrix in eq.
(7.3.15) and using the seesaw relation for the bare light neutrino mass matrix
$m_{\nu}^{(0)}$ we find to $\mathcal{O}(\epsilon)$:

$m_{\nu}^{(0)}=\begin{pmatrix}\frac{2ig^{2}|y|^{2}}{x}\frac{1}{\epsilon}-\frac{4g^{2}|y|^{2}}{x}&-\left(1+g^{2}\right)y^{\ast}&-\left(1+g^{2}\right)y\\\
-\left(1+g^{2}\right)y^{\ast}&0&0\\\
-\left(1+g^{2}\right)y&0&0\end{pmatrix}+\mathcal{O}(\epsilon)\,.$ (7.3.16)

The leading order form of the charged lepton mass matrix is as in eq. (7.3.1).

The leading order predictions for neutrino masses and mixing angles strongly
depend upon the parameter $g$.

  * •

A neutrino mass spectrum with IO can be realised when $|g|^{2}\ll|\epsilon|$.
Then, $m_{\nu}^{(0)}$ has approximately the same form as in the model with the
Weinberg operator considered in subsection 7.3.1. We get the neutrino mass
spectrum with IO:

$m_{1}\,\colon\,m_{2}\,\colon\,m_{3}=1\,\colon\approx
1\colon\,\mathcal{O}(\epsilon)$ (7.3.17)

and the predictions for the mixing angles reported in eq. (7.3.6), in
particular, $\sin^{2}\theta_{23}=1$.

  * •

A neutrino mass spectrum with NO can be realised when $|\epsilon|\ll|g|^{2}$.
In this case,

$m_{\nu}^{(0)}=\begin{pmatrix}c(g^{2}/\epsilon)&a&b\\\ a&0&0\\\
b&0&0\end{pmatrix}+\mathcal{O}(\epsilon)\,,$ (7.3.18)

with $|a|$, $|b|$, $|c|$ being $\mathcal{O}(1)$ numbers. Therefore, we have
the neutrino mass spectrum with NO:

$m_{1}\,\colon\,m_{2}\,\colon\,m_{3}=\mathcal{O}\left(\epsilon^{2}/g^{2}\right)\,\colon\,\mathcal{O}\left(\epsilon^{2}/g^{4}\right)\,\colon\,\mathcal{O}(1)$
(7.3.19)

implying $r=O(\epsilon^{4}/g^{4})$. The mixing angles are:

$\sin^{2}\theta_{12}=\mathcal{O}(1)\,,\qquad\sin^{2}\theta_{23}\approx\sin^{2}\theta_{13}\approx
r^{1/2}\,.$ (7.3.20)

Again, the leading order prediction for $\sin^{2}\theta_{23}$ is far away from
its measured value and requires significant corrections from the Kähler
potential.

To verify the viability of the model we have performed a full numerical study,
also allowing for a non-minimal form of the Kähler potential for the matter
fields. In general, modular invariance allows many terms in the Kähler
potential [43, 258]. In the considered bottom-up approach, there seems to be
no way of reducing the number of these terms. However, this may change if
modular symmetry is augmented by a traditional finite flavour symmetry [266]
or, perhaps, if some other top-down principle is in action. In what follows,
to be concrete, we adopt three simplifying assumptions:

  * •

The new terms in $K$ are quadratic in $Y^{(2)}$. This is sufficient to
illustrate our results.

  * •

The minimal form (up to overall normalisation) is restored at
$\text{Im}\tau\to\infty$. This assumption is inspired by the minimal form of
the Kähler potential arising in certain string theory compactifications in the
large volume limit, corresponding to $\text{Im}\tau\to\infty$ (see, e.g. [258,
259] and references therein).

  * •

The diagonal entries, already controlled by the minimal Kähler potential, are
not affected by the new terms.

Under these assumptions and with the assignment of representations and weights
given in Table 7.2, we find 121212We present the full expressions for $K_{L}$
and $K_{E^{c}}$ quadratic in $Y^{(2)}$ in Appendix 7.5.:

$K=L^{\dagger}K_{L}L+E^{c\dagger}K_{E^{c}}E^{c}\,,$ (7.3.21)

where

$K_{L}=\frac{1}{2\text{Im}\tau}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}+2\text{Im}\tau\begin{pmatrix}0&\left(\alpha_{5}+i\alpha_{6}\right)X_{12}&\left(\alpha_{5}-i\alpha_{6}\right)X_{13}\\\
\left(\alpha_{5}-i\alpha_{6}\right)X_{12}^{\ast}&0&\left(\alpha_{5}+i\alpha_{6}\right)X_{23}\\\
\left(\alpha_{5}+i\alpha_{6}\right)X_{13}^{\ast}&\left(\alpha_{5}-i\alpha_{6}\right)X_{23}^{\ast}&0\end{pmatrix}.$
(7.3.22)

Here $\alpha_{5}$ and $\alpha_{6}$ are real coefficients and

$X_{12}=Y_{1}^{\ast}Y_{2}-Y_{1}Y_{2}^{\ast}\,,\qquad
X_{13}=Y_{1}^{\ast}Y_{3}-Y_{1}Y_{3}^{\ast}\,,\qquad
X_{23}=Y_{2}^{\ast}Y_{3}-Y_{2}Y_{3}^{\ast}\,.$ (7.3.23)

In the $E^{c}$ sector, we obtain:

$K_{E^{c}}=\frac{1}{8\left(\text{Im}\tau\right)^{3}}\begin{pmatrix}1&0&0\\\
0&1&0\\\
0&0&1\end{pmatrix}+\frac{1}{2\text{Im}\tau}\begin{pmatrix}0&0&c_{13}X\\\
0&0&c_{23}X\\\ c_{13}^{\ast}X^{\ast}&c_{23}^{\ast}X^{\ast}&0\end{pmatrix},$
(7.3.24)

where $c_{13}$ and $c_{23}$ are complex coefficients and

$X=Y_{1}^{\ast}Y_{1}+\omega
Y_{2}^{\ast}Y_{2}+\omega^{2}Y_{3}^{\ast}Y_{3}\sim\boldsymbol{1}^{\prime\prime}\,.$
(7.3.25)

Noteworthy, the seesaw formula (7.1.10) does not depend on the renormalisation
factor $z_{N^{c}}^{-1/2}$ of the heavy fields $N^{c}$, so that we will not
need to specify the Kähler metric of $N^{c}$ in what follows.

Input parameters
---
$\text{Re}\tau$ | $\pm 0.0235$
$\text{Im}\tau$ | $1.080$
$\beta/\alpha$ | $0.1459$
$\gamma/\alpha$ | $5.955$
$\text{Re}g$ | $-0.1494$
$\text{Im}g$ | $\mp 0.3169$
$\alpha_{5}$ | $-0.2071$
$\alpha_{6}$ | $-0.1437$
$c_{13}$ | $-0.2656$
$c_{23}$ | $0.0145$
$v_{u}^{2}\kappa^{2}/\Lambda$ [eV] | $0.0189$
$|\epsilon|\approx|\tau-i|$ | $0.0830$

Observables
---
$m_{e}/m_{\mu}$ | $0$
$m_{\mu}/m_{\tau}$ | $0.0565$
$r$ | $0.0299$
$\sin^{2}\theta_{12}$ | $0.304$
$\sin^{2}\theta_{13}$ | $0.02219$
$\sin^{2}\theta_{23}$ | $0.573$
$\delta m^{2}\leavevmode\nobreak\ [10^{-5}\leavevmode\nobreak\ \text{eV}^{2}]$ | $7.42$
$\Delta m^{2}\leavevmode\nobreak\ [10^{-3}\leavevmode\nobreak\ \text{eV}^{2}]$ | $2.480$

Predictions
---
$m_{1}$ [eV] | $0.0062$
$m_{2}$ [eV] | $0.0106$
$m_{3}$ [eV] | $0.0506$
$\delta/\pi$ | $\pm 0.92$
$\alpha_{21}/\pi$ | $\pm 0.97$
$\alpha_{31}/\pi$ | $\pm 0.93$
$|m_{ee}|$ [eV] | 0
$\sum_{i}m_{i}$ [eV] | 0.0673
Ordering | NO
$M_{1}/\Lambda$ | 0.225
$M_{2}/\Lambda$ | 2.298
$M_{3}/\Lambda$ | 2.524

Table 7.4. First pair of best-fit points in a vicinity of $\tau=i$ found
considering the Kähler potential in eqs. (7.3.22)–(7.3.25).

Input parameters
---
$\text{Re}\tau$ | $\pm 0.0328$
$\text{Im}\tau$ | $1.137$
$\beta/\alpha$ | $0.2388$
$\gamma/\alpha$ | $7.854$
$\text{Re}g$ | $-0.2234$
$\text{Im}g$ | $\pm 0.4469$
$\alpha_{5}$ | $-0.1865$
$\alpha_{6}$ | $-0.1116$
$c_{13}$ | $-0.2405$
$c_{23}$ | $-0.0959$
$v_{u}^{2}\kappa^{2}/\Lambda$ [eV] | $0.0191$
$|\epsilon|\approx|\tau-i|$ | $0.1408$

Observables
---
$m_{e}/m_{\mu}$ | $0$
$m_{\mu}/m_{\tau}$ | $0.0565$
$r$ | $0.0299$
$\sin^{2}\theta_{12}$ | $0.304$
$\sin^{2}\theta_{13}$ | $0.02219$
$\sin^{2}\theta_{23}$ | $0.573$
$\delta m^{2}\leavevmode\nobreak\ [10^{-5}\leavevmode\nobreak\ \text{eV}^{2}]$ | $7.42$
$\Delta m^{2}\leavevmode\nobreak\ [10^{-3}\leavevmode\nobreak\ \text{eV}^{2}]$ | $2.480$

Predictions
---
$m_{1}$ [eV] | $0.0063$
$m_{2}$ [eV] | $0.0107$
$m_{3}$ [eV] | $0.0506$
$\delta/\pi$ | $\pm 0.91$
$\alpha_{21}/\pi$ | $\pm 0.98$
$\alpha_{31}/\pi$ | $\pm 0.88$
$|m_{ee}|$ [eV] | 0
$\sum_{i}m_{i}$ [eV] | 0.0675
Ordering | NO
$M_{1}/\Lambda$ | 0.353
$M_{2}/\Lambda$ | 2.130
$M_{3}/\Lambda$ | 2.483

Table 7.5. Second pair of best-fit points in a vicinity of $\tau=i$ found
considering the Kähler potential in eqs. (7.3.22)–(7.3.25).

The inclusion of a non-minimal Kähler potential, even within the above
restrictive assumptions, brings in several additional free parameters:
$\alpha_{5,6}$, ${\tt Re}(c_{13,23})$ and ${\tt Im}(c_{13,23})$ 131313In our
numerical analysis, we have set ${\tt Im}(c_{13,23})=0$.. Adding them to
$\beta/\alpha$, $\gamma/\alpha$, ${\tt Re}(g)$, ${\tt Im}(g)$, ${\tt
Re}(\tau)$ and ${\tt Im}(\tau)$, we have a total of 12 dimensionless input
parameters, more than the number of observables. Thus the focus of our
analysis cannot be on predictability. Rather, we are interested in accounting
for the mass hierarchies in terms of the $Z_{4}$ parameter $\epsilon$, in the
context of a model reproducing all lepton masses and mixings. While the mass
hierarchies alone can be easily accommodated without the need of hierarchical
Lagrangian parameters, some degree of hierarchy turns out to be required by
the need to fix the mixing parameters. Useful parameters to estimate such
hierarchies in the Kähler potential are the condition numbers of eq. (7.3.7).
To establish the possibility to reproduce all the relevant observables, and
the role of $Z_{4}$ breaking in setting the mass hierarchies, we have selected
several benchmark points with slightly different features. We show the results
of two (pairs) of the benchmark points in Tables 7.4 and 7.5. In all such
benchmark points, all five dimensionless observables take exactly their
experimental best-fit values (for the time being we set $m_{e}=0$). In
addition, the model predicts a normal ordered neutrino mass spectrum and the
values of the CPV phases. Interestingly, for both pairs of the benchmark
points, the predicted value of $\delta$ (the one with minus sign) matches its
experimental best-fit value. Notice also the interesting result $|m_{ee}|=0$
which at the leading order can be seen as a simple consequence of the matrix
patterns (7.3.16) and (7.3.1) 141414Given the column ordering in eq. (7.3.1),
$|m_{ee}|$ is given at the leading order by
$\big{(}m_{\nu}^{(0)}\big{)}_{33}=0+O(\epsilon)$.. Finally, we report in the
last column the masses $M_{i}$, $i=1,2,3$, of the heavy neutrinos in the units
of $\Lambda$. Although the value of $\Lambda$ cannot be uniquely fixed, it can
be estimated as (see the first column of the tables) $\Lambda\approx
v_{u}^{2}\kappa^{2}/(0.02\leavevmode\nobreak\ \text{eV})\approx
10^{15}\sin^{2}\beta$ GeV, where we have used $v_{u}=v\sin\beta$, with $v=174$
GeV, and $\kappa\sim\mathcal{O}(1)$. This implies that for $\tan\beta\gtrsim
1$, the scale $\Lambda\gtrsim 5\times 10^{14}$ GeV. Let us stress once again
that in the considered model, $M_{1}\sim\left|\epsilon\right|\Lambda$, and
thus, it is generated by a small departure of $\tau$ from $i$.

Our analysis shows that the mass hierarchies are indeed governed by $Z_{4}$
breaking, whereas, in general, Kähler corrections reflect on the lepton mass
spectrum through $\mathcal{O}(1)$ changes. For example, in the first pair of
benchmark points (see Table 7.4), we verified numerically that Kähler
corrections only affect the mass ratios by about a factor of 2 151515For the
minimal Kähler potential, i.e. setting
$\alpha_{5}=\alpha_{6}=c_{13}=c_{23}=0$, we find $m_{\mu}/m_{\tau}=0.0520$ and
$r=0.0637$, whereas the angles $\sin^{2}\theta_{12}=0.228$,
$\sin^{2}\theta_{13}=0.03751$ and $\sin^{2}\theta_{23}=0.256$ are far away
from their experimental values.; on the other hand, at these points, the
Kähler metrics are by themselves somewhat hierarchical, as shown by the
condition numbers $\kappa(K_{L})\approx 12$ and $\kappa(K_{E^{c}})\approx 16$.
In the second pair of benchmark points shown in Table 7.5, the hierarchies in
the Kähler metrics are both reduced (the condition numbers are
$\kappa(K_{L})\approx 6$ and $\kappa(K_{E^{c}})\approx 12$), and points with
even milder hierarchies may potentially be found. These observations lead us
to conclude that the deviations from the canonical Kähler metric present in
the best-fit points, have little to do with the mass spectrum hierarchies;
rather, they are necessary in order to reproduce the correct PMNS mixing
angles.

Figure 7.3.1. Dependence of the mixing angles and two mass ratios on
$\alpha_{5}$ (left) and $\alpha_{6}$ (right), fixing all other input
parameters to their best-fit values from Table 7.4. The horizontal dashed
lines indicate the boundaries of the respective $1\sigma$ ranges from Table
7.1. The vertical dashed line in the left (right) panels stands for the best-
fit value of $\alpha_{5}$ ($\alpha_{6}$) from Table 7.4.

Figure 7.3.2. Dependence of the mixing angles and two mass ratios on $c_{13}$
(left) and $c_{23}$ (right), fixing all other input parameters to their best-
fit values from Table 7.4. The horizontal dashed lines indicate the boundaries
of the respective $1\sigma$ ranges from Table 7.1. The vertical dashed line in
the left (right) panels stands for the best-fit value of $c_{13}$ ($c_{23}$)
from Table 7.4.

We have analysed more in detail the dependence of the fitted observables on
the parameters of the Kähler potential. In Figs. 7.3.1 and 7.3.2, we plot the
values of the five dimensionless observables versus $\alpha_{5,6}$ and
$c_{13,23}$, respectively 161616For the values of $\alpha_{5,6}$ ($c_{13,23}$)
beyond the range displayed in the $x$-axis, the matrix $K_{L}$ ($K_{E^{c}}$)
is not positive definite, and thus, the corresponding Kähler metric is well-
defined only for the displayed range of $\alpha_{5,6}$ ($c_{13,23}$).. All
other input parameters are fixed to their best-fit values as in Table 7.4. We
see that the parameters $\alpha_{5}$ and $\alpha_{6}$ strongly impact the
predictions for the mixing angles and the two mass ratios, $r$ and
$m_{\mu}/m_{\tau}$, whereas $c_{13}$ and $c_{23}$ in $K_{E^{c}}$ mainly affect
the predictions for $\sin^{2}\theta_{23}$ and $m_{\mu}/m_{\tau}$.

In conclusion, we see that a mass matrix of reduced rank at the self-dual
point $\tau=i$ can explain the observed mass hierarchies in terms of
$\mathcal{O}(1)$ Lagrangian parameters. At the same time, at least in the
model considered here, moderately hierarchical Kähler and superpotential
parameters are needed to fix the mixing angle predictions. Whether or not this
is a general feature of this class of models is a question which definitely
requires further investigation, but is beyond the scope of the present
Chapter. On the other hand, in order to get a fully realistic description of
lepton masses we should still generate a non-vanishing electron mass, without
perturbing too much the results achieved so far. We discuss this point in the
next subsection.

#####  7.3.3. Generating $m_{e}\neq 0$

Both the models discussed above yield, by construction, $m_{e}=0$. One can
easily concoct mechanisms to generate the small electron mass without spoiling
the other predictions. We give below two examples, where $m_{e}$ is generated
by supersymmetry breaking and by dimension six operators, respectively.

If supersymmetry is broken by some $F$-term, fermion masses get corrected by
the second term of eq. (7.1.6), which, as discussed below the same equation,
scales as $m_{\mathrm{SUSY}}\,v/M$ for SM fermion masses (where $M$ is the
SUSY breaking messenger scale). For instance, a Kähler interaction of the
form:

$K\supset\frac{1}{\Lambda^{2}}\chi^{\dagger}E_{i}^{c}\left[a_{i}(\tau)H_{d}+b_{i}(\tau)\widetilde{H}_{u}\right]L+\text{h.c.},$

where the superfield $\chi$ gets a supersymmetry breaking expectation value
$\langle\chi\rangle=F\theta^{2}$, gives a contribution to the charged lepton
Yukawa matrix proportional to $F/\Lambda^{2}$, which in turn generically
induces an electron Yukawa coupling of the same order.

As a second possibility, one may generate $m_{e}\neq 0$ through the dimension
six operator:

$(E_{i}^{c}LH_{d})(H_{u}H_{d}),$ (7.3.26)

whose Wilson coefficient should be a modular form of the appropriate weight.
In order for this mechanism to work, we need to generalise the weight
assignments in Table 7.2. We make the following requirements:

$\displaystyle k_{L}$ $\displaystyle=1-k_{u}\,,$ (7.3.27) $\displaystyle
k_{E^{c}}$ $\displaystyle=3+k_{u}-k_{d}\,,$ (7.3.28) $\displaystyle
k_{u}+k_{d}$ $\displaystyle\neq 0\,.$ (7.3.29)

The first two conditions ensure that the superpotentials discussed in the
previous subsections have weight zero; the last condition implies that the
operator (7.3.26) has different weight from the corresponding renormalisable
Yukawa term 171717Curiously, the same condition can be exploited to make the
Higgs $\mu$-term vanish at $\tau=i$. The Higgs $\mu$-term, being a
$\Gamma_{3}$ singlet modular form of weight $k_{u}+k_{d}$, vanishes by eq.
(7.2.8) at $\tau=i$ if $k_{u}+k_{d}\neq 0\leavevmode\nobreak\
(\texttt{mod}\leavevmode\nobreak\ 4)$, since all $\Gamma_{3}$ singlets have
$\rho(\widetilde{S})=1$., so that it couples to a functionally independent
modular form multiplet (making the resulting charged lepton mass matrix of
rank three). Such a mechanism thus generates $m_{e}\sim
v_{u}v_{d}^{2}/\Lambda^{2}$, where $\Lambda$ is the scale at which the
operator in eq. (7.3.26) is generated.

While in some flavour models, the ratios $m_{e}/m_{\tau}$ and
$m_{\mu}/m_{\tau}$ are associated to different powers of the same expansion
parameter, we note that here the two ratios are associated to independent
parameters.

####  7.4. Conclusions

Supersymmetric modular invariant theories offer an attractive framework to
address the flavour puzzle. The role of flavour symmetry is played by modular
invariance, regarded as a discrete gauge symmetry, thus circumventing the
obstruction concerning fundamental global symmetries. The arbitrary symmetry
breaking sector of the conventional models based on flavour symmetries is
replaced by the moduli space. Yukawa couplings become modular forms, severely
restricted by the matter transformation properties. So far this framework has
delivered interesting preliminary results especially in the lepton sector,
where neutrino masses and lepton mixing parameters can be efficiently
described in terms of a limited number of input parameters.

Weak points in most of the existing constructions are the need of independent
hierarchical parameters to describe charged lepton masses, the reduced
predictability caused by a general form of the Kähler potential, and the
absence of a reliable dynamical mechanism to determine the value of $\tau$ in
the vacuum. As a matter of fact, in several models reproducing lepton masses
and mixing parameters, the required value of $\tau$ is close to $i$, the self-
dual point where the generator $S$ of the modular group and CP (if the
Lagrangian is CP invariant) are unbroken. A small departure of $\tau$ from $i$
suffices to generate sizeable CP-violating effects in the lepton sector.

For these reasons, we were led to analyse more in detail the vicinity of
$\tau=i$. Our goal was to show that a small deviation from the self-dual point
can be responsible for the observed mass hierarchy $m_{e}\ll m_{\mu}\ll
m_{\tau}$ and $\delta m^{2}\ll\Delta m^{2}$. At $\tau=i$, the theory has an
exact $Z_{4}$ symmetry, generated by the element $S$ of the modular group. In
the neighbourhood of $\tau=i$, the breaking of $Z_{4}$ can be fully described
by the (small) spurion $\epsilon\approx\tau-i$, that flips its sign under
$Z_{4}$. We explained how to exploit this residual $Z_{4}$ symmetry in order
to obtain lepton mass matrices having reduced rank at $\tau=i$. This can be
easily done with a suitable assignment of modular weights and representations
for matter fields. There is a twofold advantage in this strategy. First, mass
ratios that are forced to vanish at $\tau=i$ by the $Z_{4}$ symmetry are
expected to acquire small values $\propto|\epsilon|^{n}$ $(n>0)$ near the
self-dual point. Second, the reduced rank of the mass matrices can tame the
contribution from a non-minimal Kähler potential, provided the metrics of the
matter fields do not display large hierarchies.

To see whether this strategy can be successfully realised or not, we built a
concrete model at level 3, where neutrinos get masses through the type I
seesaw mechanism. The model predicts a normal mass ordering. The number of
parameters exceeds the number of fitted observables and we cannot claim
predictability. However, with $\tau$ being near $i$, mass ratios and mixing
angles are reproduced with input parameters nearly of the same order of
magnitude and matter kinetic terms display only a moderate hierarchy. We saw
that the main contribution to the mass hierarchy can be induced by the
singular mass matrix at the $Z_{4}$ symmetric point. In the model we
considered, the Kähler potential and the other Lagrangian parameters are
crucial in order to correctly reproduce the values of the mixing angles. While
the $Z_{4}$ symmetry plays a fundamental role in all our discussion, we notice
that our model could not have been realised in the context of a $Z_{4}$
flavour symmetry alone. In particular, the electron mass vanishes in the
models we have considered due to the correlations among generic
$Z_{4}$-invariant operators provided by the underlying modular invariance.
Also the leading order values of the mixing angles are dictated by $Z_{4}$. We
have discussed possible sources of a non-vanishing electron mass. While the
models we formulated have clearly room for improvement, we consider them as a
good starting point to naturally accommodate the observed fermion mass
hierarchies within a modular invariant framework.

####  7.5. Appendix: Kähler potential quadratic in $Y^{(2)}$

In the real basis for the $\Gamma_{3}$ generators $S$ and $T$ in the
3-dimensional representation we employ in this Chapter, $L^{\ast}$ and
$Y^{(2)\ast}$ transform as triplets, i.e.
$L^{\ast}\to\rho_{\boldsymbol{3}}(\tilde{\gamma})L^{\ast}$ and
$Y^{(2)\ast}\to\rho_{\boldsymbol{3}}(\tilde{\gamma})Y^{(2)\ast}$. Thus, we can
contract first $L^{\ast}$ with $L$ and $Y^{(2)\ast}$ with $Y^{(2)}$, and after
that perform contractions of the obtained multiplets. Proceeding in this way,
we obtain:

$\displaystyle\left(L^{\ast}L\right)_{\boldsymbol{1}}$
$\displaystyle=L^{\dagger}L\,,\qquad\left(L^{\ast}L\right)_{\boldsymbol{1}^{\prime}}=L^{\dagger}M_{\boldsymbol{1}^{\prime}}L\,,\qquad\left(L^{\ast}L\right)_{\boldsymbol{1}^{\prime\prime}}=L^{\dagger}M_{\boldsymbol{1}^{\prime\prime}}L\,,$
(7.5.1) $\displaystyle\left(L^{\ast}L\right)_{\boldsymbol{3}_{S}}$
$\displaystyle=\begin{pmatrix}L^{\dagger}M_{\boldsymbol{3}_{S}}^{(1)}L\\\
L^{\dagger}M_{\boldsymbol{3}_{S}}^{(2)}L\\\
L^{\dagger}M_{\boldsymbol{3}_{S}}^{(3)}L\end{pmatrix},\qquad\left(L^{\ast}L\right)_{\boldsymbol{3}_{A}}=\begin{pmatrix}L^{\dagger}M_{\boldsymbol{3}_{A}}^{(1)}L\\\
L^{\dagger}M_{\boldsymbol{3}_{A}}^{(2)}L\\\
L^{\dagger}M_{\boldsymbol{3}_{A}}^{(3)}L\end{pmatrix},$ (7.5.2)

with the matrices $M_{\bullet}$ being:

$\displaystyle M_{\boldsymbol{1}^{\prime}}$
$\displaystyle=\begin{pmatrix}1&0&0\\\ 0&\omega^{2}&0\\\
0&0&\omega\end{pmatrix},\qquad
M_{\boldsymbol{1}^{\prime\prime}}=\begin{pmatrix}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{2}\end{pmatrix},$ (7.5.3) $\displaystyle
M_{\boldsymbol{3}_{S}}^{(1)}$ $\displaystyle=\begin{pmatrix}0&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix},\qquad M_{\boldsymbol{3}_{S}}^{(2)}=\begin{pmatrix}0&0&1\\\
0&0&0\\\ 1&0&0\end{pmatrix},\qquad
M_{\boldsymbol{3}_{S}}^{(3)}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix},$ (7.5.4) $\displaystyle M_{\boldsymbol{3}_{A}}^{(1)}$
$\displaystyle=\begin{pmatrix}0&0&0\\\ 0&0&1\\\ 0&-1&0\end{pmatrix},\qquad
M_{\boldsymbol{3}_{A}}^{(2)}=\begin{pmatrix}0&0&-1\\\ 0&0&0\\\
1&0&0\end{pmatrix},\qquad M_{\boldsymbol{3}_{A}}^{(3)}=\begin{pmatrix}0&1&0\\\
-1&0&0\\\ 0&0&0\end{pmatrix},$ (7.5.5)

and $\omega=e^{2\pi i/3}$. The same equations hold for
$\left(Y^{(2)\ast}Y^{(2)}\right)_{\bullet}$. Taking further invariant
contractions of the obtained multiplets, we find

$\displaystyle K_{L}=(2\text{Im}\tau)^{-k_{L}}$ $\displaystyle+$
$\displaystyle+(2\text{Im}\tau)^{k_{Y}-k_{L}}\bigg{\\{}$
$\displaystyle\alpha_{1}Y^{(2)\dagger}Y^{(2)}\mathbb{1}+\alpha_{2}\left[\left(Y^{(2)\dagger}M_{\boldsymbol{1}^{\prime\prime}}Y^{(2)}\right)M_{\boldsymbol{1}^{\prime}}+(Y^{(2)\dagger}M_{\boldsymbol{1}^{\prime}}Y^{(2)})M_{\boldsymbol{1}^{\prime\prime}}\right]$
$\displaystyle+\alpha_{3}\,i\left[\left(Y^{(2)\dagger}M_{\boldsymbol{1}^{\prime\prime}}Y^{(2)}\right)M_{\boldsymbol{1}^{\prime}}-\left(Y^{(2)\dagger}M_{\boldsymbol{1}^{\prime}}Y^{(2)}\right)M_{\boldsymbol{1}^{\prime\prime}}\right]$
$\displaystyle+\alpha_{4}\sum_{n=1}^{3}\left(Y^{(2)\dagger}M_{\boldsymbol{3}_{S}}^{(n)}Y^{(2)}\right)M_{\boldsymbol{3}_{S}}^{(n)}+\alpha_{5}\sum_{n=1}^{3}\left(Y^{(2)\dagger}M_{\boldsymbol{3}_{A}}^{(n)}Y^{(2)}\right)M_{\boldsymbol{3}_{A}}^{(n)}$
$\displaystyle+\alpha_{6}\sum_{n=1}^{3}i\left(Y^{(2)\dagger}M_{\boldsymbol{3}_{A}}^{(n)}Y^{(2)}\right)M_{\boldsymbol{3}_{S}}^{(n)}+\alpha_{7}\sum_{n=1}^{3}i\left(Y^{(2)\dagger}M_{\boldsymbol{3}_{S}}^{(n)}Y^{(2)}\right)M_{\boldsymbol{3}_{A}}^{(n)}\bigg{\\}}\,,$
(7.5.6)

where $\alpha_{j}$, $j=1\,,\dots\,,7$, are real coefficients, which accompany
hermitian matrices. (We have used the fact that
$M_{\boldsymbol{1}^{\prime}}^{\dagger}=M_{\boldsymbol{1}^{\prime\prime}}$,
$M_{\boldsymbol{3}_{S}}^{(n)\dagger}=M_{\boldsymbol{3}_{S}}^{(n)}$, and
$M_{\boldsymbol{3}_{A}}^{(n)\dagger}=-M_{\boldsymbol{3}_{A}}^{(n)}$.)

One of our assumptions is that the canonical form of $K_{L}$ is restored at
$\text{Im}\tau\to\infty$. The $q$-expansions of $Y_{i}$ in the real basis
read:

$\displaystyle Y_{1}(\tau)$
$\displaystyle=\frac{1}{\sqrt{3}}\left(1-6q^{1/3}-18q^{2/3}+12q+\dots\right),$
(7.5.7) $\displaystyle Y_{2}(\tau)$
$\displaystyle=\frac{1}{\sqrt{3}}\left(1-6\omega
q^{1/3}-18\omega^{2}q^{2/3}+12q+\dots\right),$ (7.5.8) $\displaystyle
Y_{3}(\tau)$
$\displaystyle=\frac{1}{\sqrt{3}}\left(1-6\omega^{2}q^{1/3}-18\omega
q^{2/3}+12q+\dots\right),$ (7.5.9)

where $q=e^{2\pi i\tau}$. Thus, at $\text{Im}\tau\to\infty$

$Y_{1}=Y_{2}=Y_{3}\sim\frac{1}{\sqrt{3}}\,,$ (7.5.10)

and $K_{L}$ has the following form:

$K_{L}\sim\frac{1}{2\text{Im}\tau}+\frac{2}{3}\text{Im}\tau\begin{pmatrix}\frac{3}{2}\alpha_{1}&\alpha_{4}+i\alpha_{7}&\alpha_{4}-i\alpha_{7}\\\
\alpha_{4}-i\alpha_{7}&\frac{3}{2}\alpha_{1}&\alpha_{4}+i\alpha_{7}\\\
\alpha_{4}+i\alpha_{7}&\alpha_{4}-i\alpha_{7}&\frac{3}{2}\alpha_{1}\end{pmatrix},$
(7.5.11)

where we have used $k_{L}=1$ and $k_{Y}=2$. To satisfy our assumption of the
asymptotic behaviour of $K_{L}$, the coefficients
$\alpha_{1}=\alpha_{4}=\alpha_{7}=0$. Thus, the number of free parameters in
$K_{L}$ is reduced from seven to four. Then, the elements of $K_{L}$ from eq.
(7.5.6) read:

$\displaystyle\left(K_{L}\right)_{11}$
$\displaystyle=\frac{1}{2\text{Im}\tau}+2\text{Im}\tau\left[2\alpha_{2}\left|Y_{1}\right|^{2}-\left(\alpha_{2}+\sqrt{3}\alpha_{3}\right)\left|Y_{2}\right|^{2}-\left(\alpha_{2}-\sqrt{3}\alpha_{3}\right)\left|Y_{3}\right|^{2}\right],$
(7.5.12) $\displaystyle\left(K_{L}\right)_{22}$
$\displaystyle=\frac{1}{2\text{Im}\tau}+2\text{Im}\tau\left[-\left(\alpha_{2}-\sqrt{3}\alpha_{3}\right)\left|Y_{1}\right|^{2}+2\alpha_{2}\left|Y_{2}\right|^{2}-\left(\alpha_{2}+\sqrt{3}\alpha_{3}\right)\left|Y_{3}\right|^{2}\right],$
(7.5.13) $\displaystyle\left(K_{L}\right)_{33}$
$\displaystyle=\frac{1}{2\text{Im}\tau}+2\text{Im}\tau\left[-\left(\alpha_{2}+\sqrt{3}\alpha_{3}\right)\left|Y_{1}\right|^{2}-\left(\alpha_{2}-\sqrt{3}\alpha_{3}\right)\left|Y_{2}\right|^{2}+2\alpha_{2}\left|Y_{3}\right|^{2}\right],$
(7.5.14) $\displaystyle\left(K_{L}\right)_{12}$
$\displaystyle=2\text{Im}\tau\left(\alpha_{5}+i\alpha_{6}\right)\left[Y_{1}^{\ast}Y_{2}-Y_{1}Y_{2}^{\ast}\right],$
(7.5.15) $\displaystyle\left(K_{L}\right)_{13}$
$\displaystyle=2\text{Im}\tau\left(\alpha_{5}-i\alpha_{6}\right)\left[Y_{1}^{\ast}Y_{3}-Y_{1}Y_{3}^{\ast}\right],$
(7.5.16) $\displaystyle\left(K_{L}\right)_{23}$
$\displaystyle=2\text{Im}\tau\left(\alpha_{5}+i\alpha_{6}\right)\left[Y_{2}^{\ast}Y_{3}-Y_{2}Y_{3}^{\ast}\right].$
(7.5.17)

For the sake of simplicity, we set further $\alpha_{2}=\alpha_{3}=0$. In this
case, the diagonal entries of $K_{L}$ are not affected by the contributions
containing modular forms, on the contrary to the off-diagonal elements.
Thereby, we arrive at the form of $K_{L}$ in eqs. (7.3.22) and (7.3.23).

What concerns $K_{E^{c}}$, with the assignment of representations and weights
given in Table 7.2, the most general Kähler potential quadratic in $Y^{(2)}$
reads

$\displaystyle K_{E^{c}}$
$\displaystyle=\frac{1}{8\left(\text{Im}\tau\right)^{3}}\begin{pmatrix}c_{11}^{0}&c_{12}^{0}&0\\\
c_{12}^{0\ast}&c_{22}^{0}&0\\\ 0&0&c_{33}^{0}\end{pmatrix}$
$\displaystyle+\frac{1}{2\text{Im}\tau}\begin{pmatrix}c_{11}Y^{(2)\dagger}Y^{(2)}&c_{12}Y^{(2)\dagger}Y^{(2)}&c_{13}Y^{(2)\dagger}M_{\boldsymbol{1}^{\prime\prime}}Y^{(2)}\\\
c_{12}^{\ast}Y^{(2)\dagger}Y^{(2)}&c_{22}Y^{(2)\dagger}Y^{(2)}&c_{23}Y^{(2)\dagger}M_{\boldsymbol{1}^{\prime\prime}}Y^{(2)}\\\
c_{13}^{\ast}Y^{(2)\dagger}M_{\boldsymbol{1}^{\prime}}Y^{(2)}&c_{23}^{\ast}Y^{(2)\dagger}M_{\boldsymbol{1}^{\prime}}Y^{(2)}&c_{33}Y^{(2)\dagger}Y^{(2)}\end{pmatrix}\,,$
(7.5.18)

with $c_{ii}^{(0)}$ being real and $c_{ij}^{(0)}$, $i\neq j$, complex
coefficients.

Taking into account that at $\text{Im}\tau\to\infty$, the invariant
combination $Y^{(2)\dagger}Y^{(2)}\sim 1$, whereas $X\equiv
Y^{(2)\dagger}M_{\boldsymbol{1}^{\prime\prime}}Y^{(2)}$ and
$X^{\ast}=Y^{(2)\dagger}M_{\boldsymbol{1}^{\prime}}Y^{(2)}$ decay
exponentially, we find

$K_{E^{c}}\sim\frac{1}{8\left(\text{Im}\tau\right)^{3}}\begin{pmatrix}c_{11}^{0}&c_{12}^{0}&0\\\
c_{12}^{0\ast}&c_{22}^{0}&0\\\
0&0&c_{33}^{0}\end{pmatrix}+\frac{1}{2\text{Im}\tau}\begin{pmatrix}c_{11}&c_{12}&0\\\
c_{12}^{\ast}&c_{22}&0\\\ 0&0&c_{33}\end{pmatrix}.$ (7.5.19)

In order to restore the canonical form of $K_{E^{c}}$ in the considered limit,
$c_{12}^{0}=c_{12}=0$. Furthermore, we set $c_{ii}=0$ for simplicity. Finally,
we can always make $c_{ii}^{0}=1$ by independent rescalings of $E_{i}^{c}$,
$i=1,2,3$. Thus, we recover $K_{E^{c}}$ given by eqs. (7.3.24) and (7.3.25).

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