#  Quark-lepton mass relations from modular flavor symmetry

Mu-Chun Chen<EMAIL_ADDRESS>Department of Physics and Astronomy, University
of California,
Irvine, CA 92697-4575 USA  Stephen F. King<EMAIL_ADDRESS>School of
Physics and Astronomy, University of Southampton,
Southampton SO17 1BJ, United Kingdom  Omar Medina<EMAIL_ADDRESS>Instituto de Física Corpuscular (IFIC), Universidad de Valencia-CSIC,
Paterna (Valencia) E-46980, Spain  José W. F. Valle<EMAIL_ADDRESS>Instituto de Física Corpuscular (IFIC), Universidad de Valencia-CSIC,
Paterna (Valencia) E-46980, Spain

###### Abstract

The so-called Golden Mass Relation provides a testable correlation between
charged-lepton and down-type quark masses, that arises in certain flavor
models that do not rely on Grand Unification. Such models typically involve
broken family symmetries. In this work, we demonstrate that realistic fermion
mass relations can emerge naturally in modular invariant models, without
relying on ad hoc flavon alignments. We provide a model-independent derivation
of a class of mass relations that are experimentally testable. These relations
are determined by both the Clebsch-Gordan coefficients of the specific finite
modular group and the expansion coefficients of its modular forms, thus
offering potential probes of modular invariant models. As a detailed example,
we present a set of viable mass relations based on the $\Gamma_{4}\cong S_{4}$
symmetry, which have calculable deviations from the usual Golden Mass
Relation.

##  I Introduction

In the Standard Model (SM), fermion masses and mixings arise from the Yukawa
interaction of quarks and leptons with the Higgs field. Although the fields of
the three families have identical SM gauge group quantum numbers, they exhibit
largely distinct masses. Such hierarchical structure of masses across the
three families appears rather enigmatic [1, 2, 3]. Moreover, the mixing
pattern of quarks and leptons encoded in the CKM and lepton mixing matrices is
quite different, and unexplained from first principles in the SM.

Understanding the pattern of fermion masses and mixings presents a two-fold
puzzle for particle physics. While some success has been achieved towards
predicting fermion mixings through the imposition of family symmetries [4, 5,
6, 7, 8], less progress has been made concerning the formulation of a fully
convincing theory of fermion mass hierarchies, though there have been many
proposals in this direction [9, 10, 11, 12, 13, 14, 15, 16, 17, 18].

The idea of relating quark and lepton masses has a long history. Since the
$SU(5)$ model proposed by Georgi and Glashow [19], that places quarks and
leptons within a common representation, it has become usual to expect quark
and lepton mass relations to emerge from gauge unification [20, 21, 22, 23,
24, 25, 26, 27].

However, despite many efforts that started rather early on [28], no truly
definitive theory relating quarks and leptons has ever been devised. The so-
called flavor puzzle became more acute after the discovery of neutrino
oscillations [29, 30] and the need to account for neutrino masses and mixings
as well.

Interestingly, viable relations between quark and lepton masses can also
emerge in flavor symmetry models, even in the absence of genuine gauge
unification. This is the case for the so-called approximate golden quark-
lepton mass relation [31, 32, 33, 34, 35]

$\frac{m_{b}}{\sqrt{m_{s}m_{d}}}\approx\frac{m_{\tau}}{\sqrt{m_{\mu}m_{e}}}~{},$
(1)

that has been obtained both with discrete as well as continuous family
symmetry groups. This relation has also been obtained in the context of
orbifold extensions of the SM [36, 37, 38, 39, 40]. Given the experimental
uncertainty of down-quark mass measurements,

$\frac{m_{d}}{m_{s}}\approx\frac{1}{20}\,,\qquad\frac{m_{s}}{m_{b}}\approx\frac{1}{50}\,,\qquad\frac{m_{e}}{m_{\mu}}\approx\frac{1}{200}\,,\qquad\text{and}\qquad\frac{m_{\mu}}{m_{\tau}}\approx\frac{1}{17}\,,$
(2)

one can readily verify that this mass relation is consistent with experimental
data. Cleary this is just one relation and, by itself, does not exhaust the
complexity of the flavor problem. However, given its success and simplicity,
one may argue that it could constitute part of the ultimate theory of flavor.
Note also that it is consistent with the Georgi-Jarlskog mass relations [28],

$\frac{m_{e}}{m_{d}}\approx\frac{1}{3}\,,\qquad\frac{m_{\mu}}{m_{s}}\approx
3\,,\qquad\frac{m_{\tau}}{m_{b}}\approx 1\,,$ (3)

which was predicted to hold at the GUT scale, with quark masses increased by a
factor of about 3 at low energies, due to renormalization group (RG) running
(with the largest contribution coming from QCD). We emphasize that the
combination in Eq. (1) (satisfied by the Georgi-Jarlskog relations) is rather
stable under renormalization group evolution. As a consequence the Golden mass
relation, which holds at the electroweak scale could potentially hold also at
high energy scales, even all the way up to the gauge unification scale
$M_{\text{GUT}}\sim 10^{16}$ GeV (See discussion in Subsection III.2).

A common drawback of flavor symmetry model predictions, such as Eq. (1), is
that they usually rely on ad hoc flavor symmetry breaking and vacuum alignment
assumptions, for example through flavons in the scalar sector [41, 42, 43, 44,
45, 46, 47, 48, 49, 50, 51, 45, 4, 52]. By contrast, in modular invariant
models where the flavor symmetry is nonlinearly realized [53] (for a recent
review see [54]), minimal, realistic, and uniquely defined symmetry breaking
patterns can be obtained without the need for flavons. Moreover, these
symmetries could unveil a possible connection between the SM and strings or
extra dimensional field theories [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65].

In this work we point out that modular flavor symmetries can naturally yield
viable correlations between the SM fermion masses without invoking flavons nor
Grand Unification. This illustrates the potential of these symmetries towards
the formulation of a successful and experimentally testable flavor theory of
fermion masses. Mass relations can emerge from the symmetry structure of the
vector-valued modular forms that uniquely parametrize the breaking of modular
invariance [53, 66, 67, 54]. In particular, using modular invariance we obtain
analytically a generalization of the golden quark-lepton mass relation in Eq.
(1).

This work is structured as follows: In Section II we present the general
method, i.e. a model-independent derivation of fermion mass relations in
modular invariant models that contain few parameters. In Section III we
present an explicit example for the $\Gamma\cong S_{4}$ modular group, in
which we derive viable mass relations of down-quarks and charged leptons. We
also contrast these to experimental data. In Section IV we argue that, though
the chosen example is not intended to be a complete flavor model, the general
method may be useful to build more comprehensive modular symmetry models of
flavor, with significantly fewer free parameters than the SM.

##  II Mass relations from modular symmetry

To begin with, let us assume an $\mathcal{N}=1$ supersymmetric theory, whose
action is given as

$\mathcal{S}=\int d^{4}xd^{2}\theta
d^{2}\bar{\theta}\,\mathcal{K}(\bar{\tau},\bar{\psi},\tau,\psi)+\int
d^{4}xd^{2}\theta\mathcal{W}(\tau,\psi)+h.c\,,$ (4)

where $\mathcal{K}$ is the Kähler potential and $\mathcal{W}$ is the
superpotential. These are functions of the chiral superfields $\tau$ and
$\psi$.

###  II.1 Modular invariance

Besides the SM gauge symmetry, we require the action ${\mathcal{S}}$ to be
invariant under the modular group $SL(2,\mathbb{Z})\equiv\Gamma$, so it
remains unchanged under the modular transformation [53, 55, 56, 48]

$\tau\xrightarrow{\gamma}\frac{a\tau+b}{c\tau+d}\qquad\text{where}\qquad\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\,\quad\text{with}\quad ad-cb=1$ (5)

where the matrix $\gamma$ has integer entries and belongs to the modular group
$\Gamma$, generated by the elements

$S=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}\,,\quad T=\begin{pmatrix}1&1\\\
0&1\end{pmatrix}\,,\quad R=\begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix}\,,$ (6)

that obey the relations $S^{2}=R$, $(ST)^{3}=R^{2}=\mathbf{1}$, and $RT=TR$.

The set of matter chiral superfields $\psi$ of the Minimal Supersymmetric
Standard Model (MSSM) present in the action in Eq. (4) transform as weighted
representations $\psi\sim\bigoplus_{\alpha}(\mathbf{r}_{\alpha},-k_{\alpha})$
under the action of $\gamma\in\Gamma$.

$\psi_{\alpha}\,\xrightarrow{\gamma}\,(c\tau+d)^{-k_{\alpha}}\,\rho_{\mathbf{r}_{\alpha}}(\gamma)\,\psi_{\alpha}\,.$
(7)

Here $\alpha$ labels different irreducible representations
$\rho_{\mathbf{r}}(\gamma)$ of $\Gamma_{\text{F}}$, a finite subgroup of
$\Gamma$ that plays the role of the flavor symmetry. The corresponding
automorphic factor $(c\tau+d)^{-k_{\alpha}}$ depends on the weight
$k_{\alpha}$.

The superpotential $\mathcal{W}$, which is assumed to be invariant under a
modular transformation, can written as a power series in the superfields
$\psi_{\alpha}$

$\mathcal{W}=\sum_{n}\left(Y_{\alpha_{1}\dots\alpha_{n}}(\tau)\psi_{\alpha_{1}}\dots\psi_{\alpha_{n}}\right)_{\mathbf{1}}.$
(8)

Note that given the transformation properties of $\tau$ and $\psi_{\alpha}$
under the modular group in Eqs. (5) and (7) the Yukawa couplings
$Y_{\alpha_{1}\dots\alpha_{n}}(\tau)$ are requited to be the vector-valued
modular forms [66, 67, 68] of the finite modular group $\Gamma_{\text{F}}$

$Y_{\alpha_{1}\dots\alpha_{n}}(\gamma\tau)=(c\tau+d)^{k_{Y_{n}}}\rho_{Y}(\gamma)Y_{\alpha_{1}\dots\alpha_{n}}(\tau),$
(9)

with weight $k_{Y_{n}}$ and representation $\rho_{Y}(\gamma)$ of
$\Gamma_{\text{F}}$, such that each term of the superpotential in Eq. (8) is
invariant under a modular transformation.

The structure of the Kähler potential $\mathcal{K}$, and its covariance under
a modular transformation is of great importance once a model is specified [69,
70]. Nonetheless, since our aim in this paper is not to build a particular
modular invariant model but rather to demonstrate a proof of principle that
the mass relations can be a result of modular invariance, we will not include
further discussion around it.

The superpotential terms in $\mathcal{W}$ involving the SM fermions can be
written compactly as modular invariant fermion bilinears,

$\mathcal{W}\,\supset\,\psi_{i}\,M(\tau)_{ij}\,\psi^{c}_{j}\,,$ (10)

with $i,j=1,\ldots,3$ denoting the three families of the SM. Here it is
understood that $M({\tau})$ includes the Higgs doublet chiral superfields
$\Phi_{u,d}$ of the MSSM. Their vacuum expectation values (VEVs) induce the
spontaneous breaking of the SM electroweak symmetry, while the VEV of the
scalar component of the field $\tau$, the modulus, characterizes the breaking
of modular invariance. Altogether, the $\Phi_{u,d}$ and $\tau$ VEVs give rise
to the mass matrices of the SM fermions.

Figure 1: This figure shows the fundamental domain for the action of the
modular group in cyan. We chose to include the solid line border. The three
symmetry points, $\tau_{\text{sym}}=i\infty,\,i,\,\omega$, are marked in red.
An arbitrary value of $\tau$ in the complex plane can be mapped to a point
within this domain via a modular transformation in Eq. (5).

We now turn to the issue of modular symmetry breaking and residual symmetries.
Note that, since modular invariance is nonlinearly realized, any value of
$\tau$ will break this symmetry. In Fig. 1 we display the fundamental domain
of the modular group action on $\tau$. Furthermore, there are three special
values of the modulus $\tau$ at which the modular group $\Gamma$ breaks down
into different preserved residual discrete symmetries [71, 53, 72, 73, 74]. We
generically denote these values as $\tau_{\text{sym}}$ and refer to them as
symmetry points. These are associated to some unbroken combination of the
generators of the modular group in Eq. (6). As seen from Eqs. (5) and (6) the
$R$ generator is unbroken for any value of $\tau$, so that a
$\mathbb{Z}_{2}^{R}$ symmetry is always preserved. Therefore the three
symmetry points are

  * •

$\tau_{\text{sym}}=i\infty$, invariant under $T$, preserving
$\mathbb{Z}_{N}^{T}\otimes\mathbb{Z}_{2}^{R}$.

  * •

$\tau_{\text{sym}}=i$, invariant under $S$, preserving $\mathbb{Z}_{4}^{S}$,
with $\mathbb{Z}_{2}^{R}$ as a subgroup.

  * •

$\tau_{\text{sym}}=\omega\equiv\exp(2\pi i/3)$, invariant under $ST$,
preserving $\mathbb{Z}_{3}^{ST}\otimes\mathbb{Z}_{2}^{R}$.

Notice that for $\tau_{\text{sym}}=i\infty$ the preserved symmetry
$\mathbb{Z}_{N}^{T}\otimes\mathbb{Z}_{2}^{R}$ is determined by the order $N$
of the $T$ generator ($T^{N}=\mathbb{I}$) of the corresponding finite modular
group.

When the modulus $\tau$ is close to any of the symmetry points
$\tau_{\text{sym}}$ the appealing properties of the modular forms in Eq. (9)
become manifest [75, 74, 76, 77]. Our main result, outlined below in this
section, emerges from these residual symmetries in a model-independent manner.
Our derivation holds as long as we are close to any of the three symmetry
points $\tau_{\text{sym}}=\,i\infty,\,i,\,\,\omega$. It is convenient to
define deviation parameters from each of the three symmetry points

$\epsilon_{i\infty}(\tau)=e^{i\frac{2\pi\tau}{N}},\qquad\quad\epsilon_{i}(\tau)=\frac{\tau-i}{\tau+i},\qquad\quad\text{and}\qquad\quad\epsilon_{\omega}(\tau)=\frac{\tau-\omega}{\tau-\omega^{2}}.$
(11)

Note that one can write the modular form multiplets in Eq. (9) in terms of
these variables. This is useful since the expansion the vector-valued modular
forms in terms of $\epsilon_{i\infty}$, $\epsilon_{i}$ or $\epsilon_{\omega}$
is related to their “charge” under the respective residual symmetry group, as
stressed in [74].

We now turn to the main result of our proposal, which is a model-independent
derivation of how correlations between fermion masses can emerge naturally
from modular flavor symmetries. We denote a generic pair of matter superfield
multiplets $\psi$ and $\psi^{c}$ resembling the three SM fermion families as:

$\psi=\begin{pmatrix}\psi_{1}\\\ \psi_{2}\\\
\psi_{3}\end{pmatrix},\quad\text{and}\quad\psi^{c}=\begin{pmatrix}\psi^{c}_{1}\\\
\psi^{c}_{2}\\\ \psi^{c}_{3}\end{pmatrix}.$ (12)

In general, one can write a general fermion mass matrix $M_{\psi}$ as a
function of $n$ parameters $a_{1},\dots,a_{n}$ present in the superpotential

$\mathcal{W}\,\supset\,\psi M_{\psi}(a_{1},\dots,a_{n})\psi^{c}\hskip
2.0pt\hskip 1.0pt,$ (13)

so that the three fermion masses are functions of these parameters:

$m_{1}(a_{1},\dots,a_{n}),\quad m_{2}(a_{1},\dots,a_{n}),\quad\text{and}\quad
m_{3}(a_{1},\dots,a_{n}).$ (14)

Given the chiral structure of the MSSM gauge interactions, it is convenient to
use the Hermitian and positive semi-definite matrix $H_{\psi}$ instead of the
original mass matrix $M_{\psi}$ in Eq. (13),

$H_{\psi}\equiv M_{\psi}M^{\dagger}_{\psi},$ (15)

this eliminates unphysical mixing parameters associated with the
$\mathrm{SU(2)_{L}}$ singlet superfields $\psi^{c}$. The squared masses of the
three fermions are obtained by performing a unitary basis tranformation
$U_{\psi}$ over the $\psi$ superfields:

$D^{2}_{\psi}=U^{\dagger}_{\psi}H_{\psi}U_{\psi},\qquad\text{where}\qquad
D^{2}_{\psi}\equiv\text{Diag}(m^{2}_{1},m^{2}_{2},m^{2}_{3}).$ (16)

Without loss of generality we assume the following ordering of the masses,

$m_{1}\leq m_{2}\leq m_{3}.$ (17)

Solving for the individual masses in Eq. (14) is, in general, a highly non-
trivial model-dependent task. In contrast, we can use a set of basis
invariants of $H_{\psi}(a_{1},\dots,a_{n})$ to extract useful information
concerning the SM fermion masses in terms of the $a_{1},\dots,a_{n}$
parameters (see for example [78]), i.e.

$\displaystyle\text{Tr}[H_{\psi}]$
$\displaystyle=m^{2}_{1}+m^{2}_{2}+m^{2}_{3},$ (18)
$\displaystyle\frac{1}{2}\left\\{(\text{Tr}[H_{\psi}])^{2}-\text{Tr}[H_{\psi}H_{\psi}]\right\\}$
$\displaystyle=m^{2}_{1}\,m^{2}_{3}+m^{2}_{2}\,m^{2}_{3}+m^{2}_{1}\,m^{2}_{2},$
(19) $\displaystyle\text{Det}[H_{\psi}]$
$\displaystyle=m^{2}_{1}\,m^{2}_{2}\,m^{2}_{3}.$ (20)

In these equations, the left-hand side are polynomials in the underlying
parameters $a_{1},\dots,a_{n}$, with mass-dimension of $2$, $4$, and $6$,
respectively.

###  II.2 Mass matrices at the symmetry points

Close to the three symmetry points $\tau_{\text{sym}}=i\infty,\,i,\,\omega$
one can conveniently write $H_{\psi}(a_{1},\dots,\epsilon(\tau))$ as a
function of the dimensionless parameter $\epsilon(\tau)$ [75, 74, 76, 77]. In
a given model the definition of $\epsilon$ will depend on the corresponding
symmetry point $\tau_{\text{sym}}$ according to the definitions in Eq. (11).
We will write the expressions in terms of $\epsilon(\tau)$ instead of $\tau$.

In a modular invariant model, the number of independent parameters of the
superpotential is reduced, since each term must be invariant under a modular
transformation, i.e.

$\mathcal{W}\supset\sum_{i}\alpha_{i}\left(Y^{(k)}_{\mathbf{r}}(\epsilon)\psi^{\dagger}\Phi_{u,d}\psi^{c}\right)_{\mathbf{1}}\,,$
(21)

where $\Phi_{u,d}$ denote the two MSSM Higgs doublets, while
$Y^{(k)}_{\mathbf{r}}(\epsilon)$ represents the vector-valued modular forms of
weight $k$. The index $i$ labels all the modular invariant contractions of
$Y^{(k)}_{\mathbf{r}}(\epsilon)$ with the MSSM superfields and $\alpha_{i}$
are independent numerical coefficients. To match with our notation in Eq.
(13), we will consider the dimensionful quantities
$a_{i}\equiv\alpha_{i}v_{u,d}/\sqrt{2}$ as the mass matrix parameters, along
with $\epsilon$ describing the deviation from one of the three residual
symmetry points. Note that in top-down constructions the superpotential
coefficients $\alpha_{i}$ are expected to be correlated quantities [58, 79,
80, 81, 82, 83, 64, 63, 84, 85], nonetheless in bottom-up constructions they
are taken to be independent.

At the symmetry point (in the $\epsilon\to 0$ limit) a discrete residual
symmetry is preserved. Given a specific finite modular flavor group for some
weighted representation assignments for $\psi$ and $\psi^{c}$ under this
group, the rank of $H_{\psi}$ will be reduced in the symmetric limit [74, 75,
76, 77],

$\lim_{\epsilon\to
0}\text{Det}[H_{\psi}(a_{1},\dots,\epsilon)]=m^{2}_{1}\,m^{2}_{2}\,m^{2}_{3}=0.$
(22)

This condition implies that at least the first family is massless in the
symmetric limit. Thus, the small mass $m_{1}$ would result from the deviation
of the modulus $\tau$ from the residual symmetry point $\tau_{\text{sym}}$,
providing a symmetry-based explanation for the lightness of the first-family
fermion. As detailed in [74], certain weighted representation assignments can
render $m_{1}$, $m_{2}$, and even $m_{3}$ massless at the symmetry point. In
our derivation, we focus on the minimal scenario where only $m_{1}$ is
massless in this limit, though the approach is directly applicable to the
other cases.

###  II.3 Conditions for mass relations

If the number of coefficients $\alpha_{i}$ is either one ($\alpha_{1}$) or two
($\alpha_{1},\alpha_{2}$) in Eq. (21), the mass matrix resulting from the
superpotential will depend on at most three independent parameters, one of
which is dimensionless ($\epsilon$). In this case the system in Eqs. (18)-(20)
is a solvable polynomial system of three variables.

This implies that in a given model, the $H_{\psi}$ matrix will lead to a
correlation among the three fermion masses, provided it satisfies the
following two conditions:

  1. 1.

The superpotential (21) must contain (at most) two independent coefficients,
$\alpha_{1}$ and $\alpha_{2}$, so that $H_{\psi}$ depends only on two
dimensionful parameters: $a_{1}=(\alpha_{1}v_{u,d}/\sqrt{2})$ and
$a_{2}=(\alpha_{2}v_{u,d}/\sqrt{2})$, in addition to $\epsilon$. This
condition is necessary, otherwise, Eqs. (18)-(20) form an unconstrained
polynomial system.

  2. 2.

The three-dimensional matrix $H_{\psi}$ reduces its rank at the symmetry
point, as indicated by Eq. (22), i.e.

$\text{ rank}\left[\lim_{\epsilon\to
0}H_{\psi}(a_{1},a_{2},\epsilon)\right]<\text{rank}[H_{\psi}(a_{1},a_{2},\epsilon)].$
(23)

In general, since $H_{\psi}$ is positive semi-definite, one can write Eq. (20)
as a power expansion in the dimensionless parameter $|\epsilon|$ around the
symmetry point $\tau_{\text{sym}}$, leading to

$\text{Det}[H_{\psi}(a_{1},\dots,\epsilon)]=\sum^{\infty}_{m=0}f_{m}(a_{1},\dots,a_{n})|\epsilon|^{2m}.$
(24)

If the second condition above is satisfied by $H_{\psi}$ then
$f_{0}(a_{1},\dots,a_{n})=0$ in the expasion. Therefore, when both conditions
hold we can write the last equation as

$\qquad\qquad\qquad\qquad\text{Det}[H_{\psi}(a_{1},a_{2},\epsilon)]=m^{2}_{1}\,m^{2}_{2}\,m^{2}_{3}\equiv
f_{\psi}(a_{1},a_{2})\hskip
2.0pt|\epsilon|^{\eta}+\mathcal{O}\left(|\epsilon|^{2\eta}\right)+\dots\hskip
2.0pt,\qquad\text{where}\quad\eta>0.$ (25)

where we defined $f_{\psi}(a_{1},a_{2})\equiv f_{\eta}(a_{1},a_{2})$, which is
the coefficient of the leading term in the expansion in Eq. (24). From
dimensional analysis, we know that $f_{\psi}$ is a homogeneous order-six
polynomial in the parameters $a_{1}$ and $a_{2}$. However, since $\epsilon$ is
dimensionless, the value of $\eta$ can not be inferred solely from dimensional
analysis.

Using Eqs. (17) and (22), we can write (18) and (19) in the symmetric limit

$\displaystyle\lim_{\epsilon\to 0}\text{Tr}[H_{\psi}]$ $\displaystyle\equiv
h_{\psi}(a_{1},a_{2})=m^{2}_{2}+m^{2}_{3},$ (26)
$\displaystyle\lim_{\epsilon\to
0}\frac{1}{2}\left\\{(\text{Tr}[H_{\psi}])^{2}-\text{Tr}[H_{\psi}H_{\psi}]\right\\}$
$\displaystyle\equiv g_{\psi}(a_{1},a_{2})=m^{2}_{2}\,m^{2}_{3}.$ (27)

The two homogeneous polynomials $h_{\psi}(a_{1},a_{2})$ and
$g_{\psi}(a_{1},a_{2})$ are of order $2$ and $4$ respectively. This is a two-
equation system of two variables $a_{1}$ and $a_{2}$. At the exact symmetry
point these polynomials give solutions of the form

$\displaystyle\tilde{a}_{1}(m_{2},m_{3})\,,\quad\text{and}\quad\tilde{a}_{2}(m_{2},m_{3})\,,$
(28)

which relate the model parameters $a_{1}$, $a_{2}$ with the fermion masses
$m_{3}$ and $m_{2}$. See Appendix B for further discussion about the
polynomial system solutions.

The solutions in the last equation, that hold at the symmetry point
($\epsilon\to 0$), together with Eq. (25) lead us to define the polynomial

$f(m_{2},m_{3})\equiv
f_{\psi}(\tilde{a}_{1}(m_{2},m_{3}),\tilde{a}_{2}(m_{2},m_{3})),$ (29)

Close to the symmetry point we can approximate Eq. (25) to its leading term in
$\lvert\epsilon\rvert\ll 1$. This yields an expression relating the three
fermion masses to $\lvert\epsilon\rvert$

$\frac{m^{2}_{1}\,m^{2}_{2}\,m^{2}_{3}}{f(m_{2},m_{3})}\approx|\epsilon|^{\eta}\,.$
(30)

This is our central result, it allows us to identify potentially viable
correlations between the masses of the SM fermions in modular invariant
models. In a specific model, the polynomial $f(m_{2},m_{3})$ is determined by
both the Clebsch-Gordan coefficients of the finite modular group
$\Gamma_{\text{F}}$ and the vector-valued modular forms in the superpotential
terms of Eq. (21). It is important to note that Eq. (30) serves as a valid
approximation near the symmetry points
$\tau_{\text{sym}}=i\infty,\,i,\,\omega$; otherwise, additional terms of the
expansion in Eq. (25) must be considered. Notice that Eq. (30) is fully
independent of the phase of $\epsilon(\tau)$.

##  III Example of viable mass relations

To illustrate the general derivation in the previous section and emphasize key
points, we now turn to a specific example. Let’s consider $\Gamma_{4}\cong
S_{4}$ as the finite modular group. There are several flavor models based on
this group or its double cover (see e.g. [86, 87, 88, 89, 90, 91, 72]).
Detailed information about the group properties, the chosen basis for the
representations, and the expansion of the relevant modular forms are given in
Appendix A.

The $\Gamma_{4}$ weighted-representation assignments of the down-sector
superfields that result in a quark-lepton mass relation are given in Table 1.
For simplicity we omit transformation properties under the
$\mathrm{SU(3)_{c}\otimes SU(2)_{L}\otimes U(1)_{Y}}$ gauge symmetry, as they
are the standard ones of the MSSM [92]. As mentioned in the previous section,
the first necessary condition for predicting a fermion mass relation is
related to the number of invariant contractions in the superpotential (see Eq.
(21)). In the case of $\Gamma_{4}$ at weight $2$, there are five modular forms
arranged in two multiplets:

$Y^{(2)}_{\mathbf{2}}(\tau)=\begin{pmatrix}Y_{1}(\tau)\\\
Y_{2}(\tau)\end{pmatrix},\hskip
10.0ptY^{(2)}_{\mathbf{3}}(\tau)=\begin{pmatrix}Y_{3}(\tau)\\\ Y_{4}(\tau)\\\
Y_{5}(\tau)\end{pmatrix}.$ (31)

Hence there are only two independent contractions of the superfields with the
modular forms,

| $Q$ | $D^{c}$ | $L$ | $E^{c}$ | $\Phi_{d}$ | $Y^{(2)}_{\mathbf{2}}$ | $Y^{(2)}_{\mathbf{3}}$
---|---|---|---|---|---|---|---
$\Gamma_{4}$ | $\mathbf{3}$ | $\mathbf{3^{\prime}}$ | $\mathbf{3}$ | $\mathbf{3^{\prime}}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{3}$
$k$ | $-1$ | $-1$ | $-2$ | $0$ | $0$ | $2$ | $2$
Table 1: This table contains the $\Gamma_{4}$ weighted-representation
assignments that render a quark-lepton mass relation. MSSM gauge symmetry
transformations are omitted in this table.

$\mathbf{3}\otimes\mathbf{3^{\prime}}\otimes
Y^{(2)}_{\mathbf{2}}\Longrightarrow\mathbf{1}\qquad\text{and}\qquad\mathbf{3}\otimes\mathbf{3^{\prime}}\otimes
Y^{(2)}_{\mathbf{3}}\Longrightarrow\mathbf{1}\hskip 3.0pt.$ (32)

Hence, the relevant superpotential terms in Eq. (21) for down-quarks and
charged-leptons have the following structure:

$\displaystyle\mathcal{W}^{\Gamma_{4}}_{\Phi_{d}}$ $\displaystyle\supset\hskip
3.0pt\alpha^{d}_{1}\left(Q\Phi_{d}D^{c}Y^{(2)}_{\mathbf{2}}(\tau)\right)_{\mathbf{1}}+\hskip
3.0pt\alpha^{d}_{2}\left(Q\Phi_{d}D^{c}Y^{(2)}_{\mathbf{3}}(\tau)\right)_{\mathbf{1}}$
$\displaystyle\hskip
3.0pt+\alpha^{e}_{1}\left(L\Phi_{d}E^{c}Y^{(2)}_{\mathbf{2}}(\tau)\right)_{\mathbf{1}}+\hskip
3.0pt\alpha^{e}_{2}\left(L\Phi_{d}E^{c}Y^{(2)}_{\mathbf{3}}(\tau)\right)_{\mathbf{1}}.$
(33)

There are four coefficients in Eq. (33): $\alpha^{d}_{1}$, $\alpha^{d}_{2}$,
$\alpha^{e}_{1}$, and $\alpha^{e}_{2}$. In our bottom-up setup, these are
independent parameters. Although they are potentially complex, we assume the
preservation of CP symmetry so these coefficients are real [93]. We define the
following set of dimensionful parameters

$a^{d}_{1}\equiv\frac{\alpha^{d}_{1}v_{d}}{\sqrt{2}},\quad
a^{d}_{2}\equiv\frac{\alpha^{d}_{2}v_{d}}{\sqrt{2}},\quad
a^{e}_{1}\equiv\frac{\alpha^{e}_{1}v_{d}}{\sqrt{2}},\quad\text{and}\quad
a^{e}_{2}\equiv\frac{\alpha^{e}_{2}v_{d}}{\sqrt{2}}.$ (34)

in terms of the standard MSSM VEVs $\langle\Phi_{u,d}\rangle\equiv
v_{u,d}/\sqrt{2}$ .

The mass matrices for down-type quarks and charged-leptons, denoted as $M_{d}$
and $M_{e}$ respectively, are given as:

$M_{f}=\begin{pmatrix}a^{f}_{1}Y_{2}&-a^{f}_{2}Y_{4}&a^{f}_{2}Y_{5}\\\
-a^{f}_{2}Y_{4}&\frac{1}{\sqrt{2}}\left(\sqrt{3}a^{f}_{1}Y_{1}-2a^{f}_{2}Y_{3}\right)&\frac{1}{2}a^{f}_{1}Y_{2}\\\
a^{f}_{2}Y_{5}&\frac{1}{2}a^{f}_{1}Y_{2}&\frac{1}{\sqrt{2}}\left(\sqrt{3}a^{f}_{1}Y_{1}+2a^{f}_{2}Y_{3}\right)\\\
\end{pmatrix},\qquad\text{for}\quad f=\hskip 1.0ptd,\hskip 1.0pte\hskip
1.0pt.$ (35)

For the sake of compactness, we have omitted the explicit $\tau$ dependence of
the modular forms $Y_{i}(\tau)$, with $i=1,..,5$. Notice that $M_{d}$ and
$M_{e}$, as described in Eq. (35), have a common structure arising from their
shared covariance properties under the gauge as well as the $\Gamma_{4}$
symmetries. Both mass matrices satisfy our first condition in Section II.3.

As prescribed in the previous section we define the Hermitian matrices

$H_{d}\equiv M_{d}M^{\dagger}_{d}\hskip 2.0pt,\qquad\text{and}\qquad
H_{e}\equiv M_{e}M^{\dagger}_{e}\hskip 2.0pt.$ (36)

For $\tau$ close to the $T$-symmetric point $\tau_{T}\equiv i\infty$ these
matrices fulfill the second condition outlined in Eq. (23). Consequently, they
will lead to mass correlations that we will now discuss. Following Eq. (11),
throughout this section we define the $\epsilon(\tau)$ parameter:

$\epsilon(\tau)\equiv q^{\frac{1}{4}}=e^{\frac{2\pi
i\tau}{4}},\qquad\text{thus}\qquad\lim_{\tau\to i\infty}\epsilon(\tau)=0.$
(37)

In this limit the superpotential preserves a residual $\mathbb{Z}^{T}_{4}$
symmetry, generated by $T$ in Table 2 of Appendix A.

Given that our general derivation applies near the residual symmetry points,
we can truncate the expansion of the modular forms given in Appendix A to
their leading order in $\epsilon$:

$Y^{(2)}_{\mathbf{2}}(\epsilon)\approx\begin{pmatrix}1+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}24}\epsilon^{4}\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-8\sqrt{3}}\epsilon^{2}\end{pmatrix},\hskip
10.0ptY^{(2)}_{\mathbf{3}}(\epsilon)\approx\begin{pmatrix}1{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-8}\epsilon^{4}\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-4\sqrt{2}}\epsilon\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-16\sqrt{2}}\epsilon^{3}\end{pmatrix}.$
(38)

It is manifest that the $T$-symmetric limit $\epsilon\to 0$ corresponds to the
following alignment of the vector-valued modular forms

$\lim_{\epsilon\to 0}Y^{(2)}_{\mathbf{2}}(\epsilon)=\begin{pmatrix}1\\\
0\end{pmatrix},\hskip 30.0pt\lim_{\epsilon\to
0}Y^{(2)}_{\mathbf{3}}(\epsilon)=\begin{pmatrix}1\\\ 0\\\ 0\end{pmatrix}.$
(39)

At this symmetry point we obtain the following mass spectrum for charged-
leptons and down-quarks

$\lim_{\epsilon\to 0}\begin{pmatrix}m_{\tau}\\\ m_{\mu}\\\
m_{e}\end{pmatrix}=\begin{pmatrix}m_{\tau}\left(a^{e}_{1},a^{e}_{2}\right)\\\
m_{\mu}\left(a^{e}_{1},a^{e}_{2}\right)\\\
0\end{pmatrix},\qquad\quad\lim_{\epsilon\to 0}\begin{pmatrix}m_{b}\\\ m_{s}\\\
m_{d}\end{pmatrix}=\begin{pmatrix}m_{b}\left(a^{d}_{1},a^{d}_{2}\right)\\\
m_{s}\left(a^{d}_{1},a^{d}_{2}\right)\\\ 0\end{pmatrix},$ (40)

meaning that in the limit $\tau\to i\infty$ the down-quark and the electron
become massless, hence their mass is only generated through a deviation from
this symmetry point. In this limit, the masses of the the second and third
family fermions are functions of the parameters defined in Eq. (34).

Following the general derivation in Section II we determine the polynomials
defined in Eqs. (26), and (27) for this example:

$h_{d}(a^{d}_{1},a^{d}_{2})=\frac{3}{2}{a^{d}_{1}}^{2}+2{a^{d}_{2}}^{2}=m^{2}_{s}+m^{2}_{b},\qquad
g_{d}(a^{d}_{1},a^{d}_{2})=\frac{1}{16}\left(3{a^{d}_{1}}^{2}-4{a^{d}_{2}}^{2}\right)^{2}=m^{2}_{s}\,m^{2}_{b},$
(41)

and analogously for charged-leptons $h_{e}(a^{e}_{1},a^{e}_{2})$ and
$g_{e}(a^{e}_{1},a^{e}_{2})$ which in this case have the equivalent structure.

We can obtain solutions for these two polynomial systems as described in Eq.
(28). For a detailed list of all the different solutions to this polynomial
system please refer to Appendix B. Notably, the following set of solutions
(obtained for $\epsilon\to 0$) produces viable mass relations:

$\displaystyle\tilde{a}^{d}_{1,\pm}(m_{s},m_{b})$
$\displaystyle=\frac{m_{b}\pm
m_{s}}{\sqrt{3}},\quad\tilde{a}^{d}_{2,\pm}(m_{s},m_{b})=\frac{m_{b}\mp
m_{s}}{2},$ (42) $\displaystyle\tilde{a}^{e}_{1,\pm}(m_{\mu},m_{\tau})$
$\displaystyle=\frac{m_{\tau}\pm
m_{\mu}}{\sqrt{3}},\quad\tilde{a}^{e}_{2,\pm}(m_{\mu},m_{\tau})=\frac{m_{\tau}\mp
m_{\mu}}{2}.$ (43)

When the modulus value $\tau$ departs from a given symmetry point, in this
case $\tau_{T}=i\infty$, there is a common scaling dependence on the parameter
$\epsilon(\tau)$ that generates both $m_{d}$ and $m_{e}$. Referring to Eq.
(25) we can express the equations for the determinants of $H_{e}$ and $H_{d}$
as an expansion in the $\epsilon$ parameter:

$\displaystyle\operatorname{Det}\left[H_{d}\right]$
$\displaystyle=m^{2}_{b}\hskip 1.0ptm^{2}_{s}\hskip
1.0ptm^{2}_{d}\approx\frac{4}{27}\left(3a^{d}_{1}+2\sqrt{3}a^{d}_{2}\right)^{4}\left(3a^{d}_{1}-4\sqrt{3}a^{d}_{2}\right)^{2}|\epsilon|^{4}+\mathcal{O}\left(|\epsilon|^{8}\right)\hskip
1.0pt.$ (44) $\displaystyle\operatorname{Det}\left[H_{e}\right]$
$\displaystyle=m^{2}_{\tau}\hskip 1.0ptm^{2}_{\mu}\hskip
1.0ptm^{2}_{e}\approx\frac{4}{27}\left(3a^{e}_{1}+2\sqrt{3}a^{e}_{2}\right)^{4}\left(3a^{e}_{1}-4\sqrt{3}a^{e}_{2}\right)^{2}|\epsilon|^{4}+\mathcal{O}\left(|\epsilon|^{8}\right)\hskip
1.0pt.$ (45)

Therefore close to the symmetry point ($|\epsilon|\ll 1$) we can plug in Eqs.
(42) and (43) into Eqs. (44), and (45) to obtain

$m^{2}_{b}\hskip 1.0ptm^{2}_{s}\hskip 1.0ptm^{2}_{d}\approx
f_{\pm}(m_{b},m_{s})\hskip 2.0pt\lvert\epsilon\lvert^{4}\hskip 2.0pt,\qquad
m^{2}_{\tau}\hskip 1.0ptm^{2}_{\mu}\hskip 1.0ptm^{2}_{e}\approx
f_{\pm}(m_{\tau},m_{\mu})\hskip 2.0pt\lvert\epsilon\lvert^{4}\hskip 2.0pt,$
(46)

where $f_{\pm}(m_{3},m_{2})$ are two polynomials of order 6 that depend on
masses of the third and second generation fermions. The viable polynomial
obtained $f_{\pm}(m_{3},m_{2})$ is given by

$f_{\pm}(m_{3},m_{2})={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}64}m^{4}_{3}\left(m_{3}{\mp\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}3}m_{2}\right)^{2}.$
(47)

We stress that the coefficients (marked in red in the last equation) in the
polynomials are determined not only by the scaling properties of the modular
forms in Eq. (38) with respect to $\lvert\epsilon\rvert$ (away from a symmetry
point), but they also involve Clebsch-Gordan coefficients of the $S_{4}$ group
and the coefficients of the Fourier expansion of the modular forms (also
marked in red in Eq. (38)). The expansion coefficients have been proven to
contain mathematical information (see e.g. [94]) whose significance in models
with modular flavor symmetry remains somewhat underexplored, as the focus so
far has primarily centered on the $\epsilon$ scaling properties [74, 75, 76,
77, 95].

Looking at Eq. (46) it is straightforward to obtain the predicted mass
relations for this example

$\frac{m^{2}_{b}\hskip 1.0ptm^{2}_{s}\hskip
1.0ptm^{2}_{d}}{f_{\pm}(m_{b},m_{s})}\approx\frac{m^{2}_{\tau}\hskip
1.0ptm^{2}_{\mu}\hskip 1.0ptm^{2}_{e}}{f_{\pm}(m_{\tau},m_{\mu})}$ (48)

which can be simplified to obtain

$\displaystyle\frac{m_{s}\hskip 1.0ptm_{d}}{m_{b}\left(m_{b}\pm
3m_{s}\right)}\approx\frac{m_{\mu}\hskip 1.0ptm_{e}}{m_{\tau}\left(m_{\tau}\pm
3m_{\mu}\right)}.$ (49)

Different possible signs in the denominators correspond to different solutions
in Eqs. (42) and (43). There are four distinct mass relations emerging from
Eq. (49), which we label as follows:

$\displaystyle\text{R1:}\quad\frac{m_{s}\hskip
1.0ptm_{d}}{m_{b}\left(m_{b}-3m_{s}\right)}\approx\frac{m_{\mu}\hskip
1.0ptm_{e}}{m_{\tau}\left(m_{\tau}-3m_{\mu}\right)}.$ (50)
$\displaystyle\text{R2:}\quad\frac{m_{s}\hskip
1.0ptm_{d}}{m_{b}\left(m_{b}-3m_{s}\right)}\approx\frac{m_{\mu}\hskip
1.0ptm_{e}}{m_{\tau}\left(m_{\tau}+3m_{\mu}\right)}.$ (51)
$\displaystyle\text{R3:}\quad\frac{m_{s}\hskip
1.0ptm_{d}}{m_{b}\left(m_{b}+3m_{s}\right)}\approx\frac{m_{\mu}\hskip
1.0ptm_{e}}{m_{\tau}\left(m_{\tau}-3m_{\mu}\right)}.$ (52)
$\displaystyle\text{R4:}\quad\frac{m_{s}\hskip
1.0ptm_{d}}{m_{b}\left(m_{b}+3m_{s}\right)}\approx\frac{m_{\mu}\hskip
1.0ptm_{e}}{m_{\tau}\left(m_{\tau}+3m_{\mu}\right)}.$ (53)

To scrutinize the viability of these mass relations we performed a scan over
the following set of parameters determining the masses of down quarks and
charged leptons

$\\{a^{d,e}_{1,2},\lvert\epsilon\rvert\\}\Longrightarrow\\{m_{d,s,b,e,\mu,\tau}\\}.$
(54)

Notice that the light down-quark masses $m_{d}$ and $m_{s}$ are determined
through lattice QCD computations, and exhibit the largest uncertainties
amongst all fermion masses in Eqs. (50)-(53). In Figs. 2 and 3 we display the
results of our parameter scan, each point plotted falls within the
experimental $3$-$\sigma$ region for $m_{b}$, $m_{e}$, $m_{\mu}$, and
$m_{\tau}$.

Figure 2: This plot displays the results of our parameter scan
$\\{a^{d,e}_{1,2},\lvert\epsilon\rvert\\}\Longrightarrow\\{m_{d,s,b,e,\mu,\tau}\\}$
in the $m_{d}-m_{s}$ plane. Each point lies inside the experimental
$3$-$\sigma$ range for $m_{b}$, $m_{e}$, $m_{\mu}$, and $m_{\tau}$. The
allowed 1-$\sigma$ and 2-$\sigma$ regions for $m_{d}$ and $m_{s}$ are shown as
a dark- and light-cyan rectangle respectively [96]. We include the plots of
the mass relations R1 and R2 from Eqs. (50) and (53). There can be sizable
differences between these mass relations. For comparison we also plot (dashed)
the golden quark-lepton mass relation in Eq. (55) obtained in "non-modular"
flavor models [31, 32, 33, 34, 35].

We include the plots of R1 and R2 in Fig. 2 illustrating how the mass
relations differing only on signs in the denominators can be substantially
different. In Fig. 3 we show a close-up plot of R1 and R3 which are the most
favored experimentally. These curves use the experimental central values for
$m_{b}$, $m_{e}$, $m_{\mu}$, and $m_{\tau}$, allowing us to compare the
analytical mass relations with the numerical results from the parameter scan.
The scatter plot and the plotted lines show excellent agreement, indicating
that the approximations we used to derive the mass relations are accurate
enough.

Figure 3: This is a close-up of Fig. 2 where we plot R1 and R3 from Eqs. (50)
and (52), which are the experimentally favored ones. The dashed curve in the
plot shows the golden quark-lepton mass prediction in Eq. (55) obtained in
"non-modular" flavor models [31, 32, 33, 34, 35]. An improved determination of
$m_{s}$ and $m_{d}$ could probe our mass relations.

###  III.1 Zooming in the golden quark-lepton mass relation

We will now take a closer look at the derived mass relations in Eqs. (50)-(53)
in comparison to the golden quark-lepton mass relation obtained from
traditional flavor symmetries [31, 32, 33, 34, 35]. In the usual derivation of
the latter, using flavons for example, there are small corrections which are
usually neglected, i.e.

$\displaystyle\frac{m_{b}\hskip 1.0ptm_{s}\hskip
1.0ptm_{d}}{m_{b}^{3}+\mathcal{O}(m_{s}^{3})}\approx\frac{m_{\tau}\hskip
1.0ptm_{\mu}\hskip 1.0ptm_{e}}{m^{3}_{\tau}+\mathcal{O}(m^{3}_{\mu})}.$ (55)

These corrections in the denominators of Eq. (55) are very small. In contrast,
the corrections in Eq. (49) arising from Clebsch-Gordan coefficients of the
$\Gamma_{4}$ symmetry and those of the modular forms expansion around the
symmetry point, are typically much larger. They can yield substantial
differences, as shown in Figs. 2 and 3, where we also display the prediction
from Eq. (55).

Indeed, the mass relations in Eqs. (50)-(53) have an interesting property.
They involve polynomials $f_{\pm}(m_{3},m_{2})$, Eq. (47), which include a
term proportional to $m^{6}_{3}$. Given the strong hierarchy between the
masses of fermions in the second and third families ($m_{3}\gg m_{2}$), these
polynomials $f_{\pm}(m_{3},m_{2})$ can be approximated using their leading
term:

$\qquad\frac{m_{2}}{m_{3}}\ll 1\quad\Longrightarrow\quad
f_{\pm}(m_{3},m_{2})\approx 64\hskip 1.5ptm^{6}_{3}.$ (56)

This illustrates that there is a class of polynomials $f(m_{3},m_{2})$,
defined in Eq. (29), which have a leading term proportional to $m^{6}_{3}$.
And when used to relate charged-leptons and down-type quarks, this class of
polynomials will approximate the golden quark-lepton mass relation in Eq. (55)
for largely hierarchical fermion masses $m_{1}\ll m_{2}\ll m_{3}$. In fact, by
using this insight, one can infer that the polynomials in Eq. (47) will be
roughly consistent with experimental data even without performing a
quantitative parameter scan.

What is important to note is that the mass relations in Eqs. (50)-(53) are
indeed valid for the values of quark and lepton masses (up to theoretical or
experimental uncertainty). Our general derivation in Section II and the above
example demonstrate that a class of viable mass predictions for down-quarks
and charged-leptons can arise from modular invariant models near the symmetry
points. Although the obtained mass relation can be similar to the non-modular
golden mass relation, these predictions can yield testable deviations from it.
As a result, improving determinations of light-quark masses could help
constrain modular invariant flavor models, where the mass hierarchies of
quarks and leptons emerge from deviations from residual symmetry points.

###  III.2 Stability under renormalization group evolution

Mass relations are generally very sensitive to renormalization group (RG)
running. For instance, the Georgi-Jarlskog mass relations in Eq. (3), derived
at the GUT scale [28], undergo significant evolution to fit experimental quark
and lepton masses at lower scales, such as $M_{Z}$. In contrast, the mass
relations in Eqs. (50)-(53), derived from Eq. (30), involve ratios of down-
quark to charged-lepton masses. Thus, they are expected to be fairly stable
under RG running. This follows from the fact that the dominant contribution to
the Yukawa couplings’ $\beta$-functions comes from gauge couplings, which
effectively cancel in the mass ratios. However, for the bottom quark mass
$m_{b}$ there is a contribution involving the top quark Yukawa coupling
$y_{t}$, making it necessary to analyze more closely the stability of our
fermion mass relations.

To illustrate this point with an example we define the quantity
$F_{\text{R1}}(\mu_{E})$ and rewrite the mass relation R1 in Eq. (50) with the
down-quark and lepton masses given as functions of the scale $\mu_{\text{E}}$,

$F_{\text{R1}}(\mu_{E})\equiv\frac{m_{s}m_{d}m_{\tau}(m_{\tau}-3m_{\mu})}{m_{\mu}m_{e}m_{b}(m_{b}-3m_{s})}\approx
1.$ (57)

For definiteness we perform the RG running of the mass relation R1 in the
constrained MSSM (CMSSM) [97, 98, 99], using the Mathematica Package REAP
[100] with the extension SusyTC [24] so as to account for SUSY threshold
corrections [101, 102, 96].

Figure 4: $F_{\text{R1}}(\mu_{E})$ from Eq. (57) is plotted in dark cyan, from
the electroweak scale $M_{Z}$ to the GUT scale $M_{\text{GUT}}=2\times
10^{16}$ GeV, using the REAP and SusyTC packages [100, 24]. We take central
values of quark and lepton masses [96]. The dark green solid line
$F_{\text{R1}}(\mu_{E})=1$ is taken as reference and corresponds to the R1
mass relation. The plot shows how its changes remain within the light-green
band corresponding to $10\%$ deviation, showing the validity of R1 from the
electroweak to the GUT scale, given the uncertainty in the light quark masses
$m_{d}$ and $m_{s}$.

To quantify the RG running of the mass relation R1 we choose the mass of the
$Z$-boson ($M_{Z}$) as the initial scale [96], where it holds quite well, as
seen from Figs. 2 and 3. For definiteness we choose as benchmark the following
values for the SUSY breaking scale $M_{\text{SUSY}}=10$ TeV and
$\tan{\beta}=\nicefrac{{v_{u}}}{{v_{d}}}=50$. Concerning the four CMSSM soft
SUSY-breaking parameters, we take the common scalar mass $m_{0}=12$ TeV,
fermion mass $m_{1/2}=10$ TeV, and the trilinear coupling $A_{0}=-15$ TeV.

With these assumptions, we plot in Fig. 4 the evolution of
$F_{\text{R1}}(\mu_{E})$ in Eq. (57) from the electroweak scale $M_{Z}$ to the
GUT scale $M_{\text{GUT}}=2\times 10^{16}$ GeV, using the central values of
the experimental quark and lepton masses as input. We also display as a
reference the case $F_{\text{R1}}(\mu_{E})=1$, and the $\mathit{10}\%$
deviation band. It is clear from this figure that the mass relation is quite
stable under RG running, and holds all the way from the electroweak scale to
the GUT scale, given the large uncertainty of the light quark masses $m_{d}$
and $m_{s}$.

##  IV Discussion and outlook

We have given a model-independent method to derive mass relations for the SM
fermions in modular flavor symmetry models containing few parameters. They are
determined by the Clebsch-Gordan coefficients of the specific modular flavor
group as well as the expansion coefficients of the corresponding vector-valued
modular forms around the symmetry points.

We illustrated our results with a detailed example in Section III, where we
obtained viable mass relations for charged leptons and down-type quarks from
the $\Gamma_{4}\cong S_{4}$ flavor symmetry. We showed that these mass
relations are experimentally testable, and distinguishable from their non-
modular counterpart the so-called golden mass relation given by Eq. (55) [31,
32, 33, 34, 35]. Since the mass relations can have sizable differences from
one modular flavor group to another, they could be used to experimentally
probe different models.

In Appendix C we point out the existence of a second class of mass relations
that could yield viable predictions regarding neutrino masses and the up-quark
sector. Nonetheless, these are model-dependent. We presented a particular
example to highlight the differences with respect to our general derivation in
Section III. This example is very suggestive, as it relates the smallness of
the solar squared mass splitting to the lightness of the up quark.

We want to stress that the examples presented in Section III and Appendix C
are not meant to be understood as complete flavor models, rather as
illustrations of the use of our general method for deriving mass relations.
Note that, as discussed in the Introduction (Section I), the main significance
of the mass relations in Eqs. (50)-(53) and Eq. (55) is that they are
compatible with experimental data of the SM fermion masses, which is a non-
trivial fact. Our two examples appear to be complementary, since in Section
III we only deal with the down sector of the MSSM, while in Appendix C we
discuss the up sector and neutrinos. Since we present them independently, we
can not make any meaningful statement concerning CKM mixing, which by
definition involves both up and down quark sectors simultaneously. Although
one can complete each of the models trivially, we can not preserve both
predictions simultaneously in a simple manner. In other words, it is
straightforward to extend the model of the down sector in Section III to
include also the up-type quarks and neutrinos, while preserving the mass
relations in Eqs. (50)-(53), at the expense of losing the prediction for the
up sector in Appendix C.

It has been proven challenging to build experimentally viable quark and lepton
modular flavor models while keeping the number of input parameters to a
minimum. We expect that the results obtained in this work, largely model-
independent, may prove useful also for more comprehensive modular invariant
models and for top-down constructions. We demonstrated that viable mass
relations for the SM fermions can emerge in modular flavor symmetry in a
general manner relying only on the modular flavor group and its vector-valued
modular forms, rather than ad hoc flavon alignments. The mass relations can
differentiate amongst models. This was illustrated with one explicit example
where the mass relations predicted are distinguishable from the one obtained
in a traditional flavor symmetry model with flavons. The differences may be
resolved experimentally and this may, perhaps, shed light on the symmetry
underlying the flavor problem.

###### Acknowledgements.

We would like to thank Stefan Antusch, Salvador Centelles Chuliá, Xue-Qi Li,
Xiang-Gan Liu, and Michael Ratz for insightful discussions. OM thanks the
Department of Physics and Astronomy at UC, Irvine, for their hospitality
during his visit. SFK thanks IFIC, Valencia for their hospitality during his
visit where this work began. Work supported by the Spanish grants
PID2020-113775GB-I00 (AEI/10.13039/501100011033) and Prometeo CIPROM/2021/054
(Generalitat Valenciana). OM acknowledges financial support from the
Generalitat Valenciana through Programa Santiago Grisolía (No.
GRISOLIA/2020/025) and the research visit grant CIBEFP/2022/63. The research
of MCC was supported, in part, by the U.S. National Science Foundation under
Grant No. PHY-1915005. SFK acknowledges the STFC Consolidated Grant
ST/L000296/1 and the European Union’s Horizon 2020 Research and Innovation
programme under Marie Sklodowska-Curie grant agreement HIDDeN European ITN
project (H2020-MSCA-ITN-2019//860881-HIDDeN).

##  Appendix A The $\Gamma_{4}$ Group - Basis and Modular Forms

The $\Gamma_{4}\simeq S_{4}$ group is of order $24$, all its elements can be
written in terms of two generators $S$ and $T$ following its presentation
equation:

$S_{4}\simeq\\{S,T|S^{2}=T^{4}=(ST)^{3}=e\\}.$ (58)

The $S_{4}$ group has five irreducible representations: two singlets $\bf{1}$,
$\bf{1^{\prime}}$, one doublet $\bf{2}$, and two triplets $\bf{3}$,
$\bf{3}^{\prime}$. In this work we use the basis displayed in Table 2. In this
basis the $T$ generator is diagonal for all irreducible representations, and
both generators are symmetric matrices.

| $S$ | $T$
---|---|---
$\mathbf{1},\mathbf{1^{\prime}}$ | $\pm 1$ | $\pm 1$
$\mathbf{2}$ | $\dfrac{1}{2}\begin{pmatrix}-1&\sqrt{3}\\\ \sqrt{3}&1\\\ \end{pmatrix}$ | $\begin{pmatrix}1&0\\\ 0&-1\\\ \end{pmatrix}$
$\mathbf{3},\mathbf{3^{\prime}}$ | $\pm\dfrac{1}{2}\begin{pmatrix}0&\sqrt{2}&\sqrt{2}\\\ \sqrt{2}&-1&1\\\ \sqrt{2}&1&-1\\\ \end{pmatrix}$ | $\pm\begin{pmatrix}1&0&0\\\ 0&i&0\\\ 0&0&-i\\\ \end{pmatrix}$
Table 2: Generators of $S_{4}$ in the symmetric $T$-diagonal basis.

The $S_{4}$ representation products relevant for the examples outlined in
Section III and Appendix C are given by

$\displaystyle\begin{array}[]{@{}cll@{}}\mathbf{3}\,\otimes\,\mathbf{3^{\prime}}\,\Longrightarrow&\left\\{\begin{array}[]{@{}l@{\hspace{3pt}\sim\hspace{3pt}}l@{}}\hskip
3.5pt\mathbf{1^{\prime}}\hfil\hskip 3.0pt\sim\hskip
3.0&\alpha_{1}\beta_{1}+\alpha_{2}\beta_{3}+\alpha_{3}\beta_{2}\\\\[5.69054pt]
\hskip 3.5pt\mathbf{2}\hfil\hskip 3.0pt\sim\hskip
3.0&\begin{pmatrix}\frac{\sqrt{3}}{2}\left(\alpha_{2}\beta_{2}+\alpha_{3}\beta_{3}\right)\\\
-\alpha_{1}\beta_{1}+\frac{1}{2}\left(\alpha_{2}\beta_{3}+\alpha_{3}\beta_{2}\right)\end{pmatrix}\\\\[11.38109pt]
\hskip 3.5pt\mathbf{3}\hfil\hskip 3.0pt\sim\hskip
3.0&\begin{pmatrix}\alpha_{3}\beta_{2}-\alpha_{2}\beta_{3}\\\
\alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}\\\
\alpha_{1}\beta_{3}-\alpha_{3}\beta_{1}\end{pmatrix}\\\\[17.07164pt] \hskip
3.5pt\mathbf{3^{\prime}}\hfil\hskip 3.0pt\sim\hskip
3.0&\begin{pmatrix}\alpha_{3}\beta_{3}-\alpha_{2}\beta_{2}\\\
\alpha_{1}\beta_{3}+\alpha_{3}\beta_{1}\\\
-\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\end{pmatrix}\end{array}\right.\end{array}\quad\begin{array}[]{@{}cll@{}}\mathbf{3}\,\otimes\,\mathbf{3}\Longrightarrow&\left\\{\begin{array}[]{@{}l@{\hspace{3.5pt}\sim\hspace{3.5pt}}l@{}}\hskip
3.5pt\mathbf{1}\hfil\hskip
3.5pt\sim\hskip&\alpha_{1}\beta_{1}+\alpha_{2}\beta_{3}+\alpha_{3}\beta_{2}\\\\[5.69054pt]
\hskip 3.5pt\mathbf{2}\hfil\hskip
3.5pt\sim\hskip&\begin{pmatrix}\alpha_{1}\beta_{1}-\frac{1}{2}\left(\alpha_{2}\beta_{3}+\alpha_{3}\beta_{2}\right)\\\
\frac{\sqrt{3}}{2}\left(\alpha_{2}\beta_{2}+\alpha_{3}\beta_{3}\right)\end{pmatrix}\\\\[11.38109pt]
\hskip 3.5pt\mathbf{3}\hfil\hskip
3.5pt\sim\hskip&\begin{pmatrix}\alpha_{3}\beta_{3}-\alpha_{2}\beta_{2}\\\
\alpha_{1}\beta_{3}+\alpha_{3}\beta_{1}\\\
-\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\end{pmatrix}\\\\[17.07164pt] \hskip
3.5pt\mathbf{3^{\prime}}\hfil\hskip
3.5pt\sim\hskip&\begin{pmatrix}\alpha_{3}\beta_{2}-\alpha_{2}\beta_{3}\\\
\alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}\\\
\alpha_{1}\beta_{3}-\alpha_{3}\beta_{1}\end{pmatrix}\end{array}\right.\end{array}$
(69)
$\displaystyle\begin{array}[]{@{}c@{{}\,\otimes\,{}}c@{{}\,\,\,{}}ll@{}}\mathbf{2}&\mathbf{2}&\Longrightarrow&\left\\{\begin{array}[]{@{}l@{\hspace{3.5pt}\sim\hspace{3.5pt}}l@{}}\hskip
3.5pt\mathbf{1}\hfil\hskip
3.5pt\sim\hskip&\alpha_{1}\beta_{1}+\alpha_{2}\beta_{2}\\\\[5.69054pt] \hskip
3.5pt\mathbf{1^{\prime}}\hfil\hskip
3.5pt\sim\hskip&\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\\\\[5.69054pt] \hskip
3.5pt\mathbf{2}\hfil\hskip
3.5pt\sim\hskip&\begin{pmatrix}\alpha_{2}\,\beta_{2}-\alpha_{1}\,\beta_{1}\\\
\alpha_{1}\,\beta_{2}+\alpha_{2}\,\beta_{1}\end{pmatrix}\end{array}\right.\end{array}$
(74)

The vector-valued modular forms of the $\Gamma_{4}$ group at weight $2$ are

$Y^{(2)}_{\mathbf{2}}(\tau)=\begin{pmatrix}Y_{1}(\tau)\\\
Y_{2}(\tau)\end{pmatrix},\hskip
10.0ptY^{(2)}_{\mathbf{3}}(\tau)=\begin{pmatrix}Y_{3}(\tau)\\\ Y_{4}(\tau)\\\
Y_{5}(\tau)\end{pmatrix}.$ (75)

these can be written as power expansions

$Y_{i}(\tau)=\sum^{\infty}_{n=0}c_{i,n}q^{\frac{n}{4}}\hskip
2.0pt,\qquad\text{with}\qquad q\equiv e^{2\pi i\tau},$ (76)

where $c_{i,n}$ are constant coefficients. We obtained the explicit expansion
following the derivation in [103].

$\displaystyle Y_{1}(\tau)$
$\displaystyle=1+24q+24q^{2}+96q^{3}+24q^{4}+144q^{5}+96q^{6}+192q^{7}+...\hskip
2.0pt,$ $\displaystyle Y_{2}(\tau)$
$\displaystyle=q^{\frac{1}{2}}(-8\sqrt{3}-32\sqrt{3}q-48\sqrt{3}q^{2}-64\sqrt{3}q^{3}-104\sqrt{3}q^{4}-96\sqrt{3}q^{5}+...\hskip
2.0pt),$ $\displaystyle Y_{3}(\tau)$
$\displaystyle=1-8q+24q^{2}-32q^{3}+24q^{4}-48q^{5}+96q^{6}-64q^{7}+...\hskip
2.0pt,$ (77) $\displaystyle Y_{4}(\tau)$
$\displaystyle=q^{\frac{1}{4}}(-4\sqrt{2}-24\sqrt{2}q-52\sqrt{2}q^{2}-56\sqrt{2}q^{3}-72\sqrt{2}q^{4}-128\sqrt{2}q^{5})...\hskip
2.0pt,$ $\displaystyle Y_{5}(\tau)$
$\displaystyle=q^{\frac{3}{4}}(-16\sqrt{2}-32\sqrt{2}q^{2}-48\sqrt{2}q^{3}-96\sqrt{2}q^{4}-80\sqrt{2}q^{5}+...)\hskip
2.0pt.$

##  Appendix B Polynomial system solutions

In Eqs. (26) and (27) of Section II we defined the polynomials
$h_{\psi}(a_{1},a_{2})$ and $g_{\psi}(a_{1},a_{2})$, these are homogeneous
polynomials of order $2$ and $4$ respectively. Ultimately the solutions of the
polynomial system

$\displaystyle\begin{split}h_{\psi}(a_{1},a_{2})&=m^{2}_{2}+m^{2}_{3},\\\
g_{\psi}(a_{1},a_{2})&=m^{2}_{2}\,m^{2}_{3},\end{split}$ (78)

determine $f(m_{3},m_{2})$ in Eq. (30), which is the main result of this work.

By naive counting one could infer that this polynomial system has eight
solutions, the product of the orders of the two polynomials. Nonetheless the
actual number of independent and physically inequivalent solutions of the
polynomial system in (78) cannot be determined generically without the
explicit form of $h_{\psi}(a_{1},a_{2})$ and $g_{\psi}(a_{1},a_{2})$.

To illustrate this, consider the mass matrix $M_{f}$ in Eq. (35), for this
particular case the polynomials take the form

$h_{\psi}(a_{1},a_{2})=\frac{3}{2}{a_{1}}^{2}+2{a_{2}}^{2}=m^{2}_{2}+m^{2}_{3},\qquad
g_{\psi}(a_{1},a_{2})=\frac{1}{16}\left(3{a_{1}}^{2}-4{a_{2}}^{2}\right)^{2}=m^{2}_{2}\,m^{2}_{3}.$
(79)

This polynomial system has in fact only three independent and physically
inequivalent solutions

$\displaystyle\textbf{a)}\quad\qquad\tilde{a}_{1,+}(m_{2},m_{3})$
$\displaystyle=\frac{m_{3}+m_{2}}{\sqrt{3}},\qquad\tilde{a}_{2,-}(m_{2},m_{3})=\frac{m_{3}-m_{2}}{2},$
(80) $\displaystyle\textbf{b)}\quad\qquad\tilde{a}_{1,-}(m_{2},m_{3})$
$\displaystyle=\frac{m_{3}-m_{2}}{\sqrt{3}},\qquad\tilde{a}_{2,+}(m_{2},m_{3})=\frac{m_{3}+m_{2}}{2},$
(81) $\displaystyle\textbf{c)}\quad\qquad\hat{a}_{1,+}(m_{2},m_{3})$
$\displaystyle=\frac{m_{3}+m_{2}}{\sqrt{3}},\qquad\hat{a}_{2,-}(m_{2},m_{3})=-\frac{m_{3}-m_{2}}{2},$
(82)

These solutions, following Eqs. (29) and (30) lead to the following three mass
relations around the $\tau_{\text{sym}}=i\infty$, in terms of $\epsilon$
defined in Eq. (37)

$\textbf{a)}\quad\frac{m_{2}m_{1}}{8m_{3}(m_{3}+3m_{2})}\approx\lvert\epsilon\rvert^{2}\hskip
2.0pt,\qquad\quad\textbf{b)}\quad\frac{m_{2}m_{1}}{8m_{3}(m_{3}-3m_{2})}\approx\lvert\epsilon\rvert^{2}\hskip
2.0pt,\qquad\quad\textbf{c)}\quad\frac{m_{3}m_{1}}{8m_{2}(m_{2}+3m_{3})}\approx\lvert\epsilon\rvert^{2}.\hskip
2.0pt$ (83)

##  Appendix C Another class of mass relations

Our derivation in Section II is general, and model-independent. We now
consider a second class of mass relations that can emerge in modular invariant
models. This happens when an $H_{\psi}$ matrix, defined in Eq. (15), satisfies
the first condition listed in Subsection II.3, while it does not satisfy the
second. Meaning that at the symmetric point $\epsilon\to 0$ the rank of the
matrix is not reduced.

$\text{ rank}\left[\lim_{\epsilon\to
0}H_{\psi}(a_{1},a_{2},\epsilon)\right]=\text{rank}[H_{\psi}(a_{1},a_{2},\epsilon)].$
(84)

This implies that, in the limit, the three non-vanishing masses $m_{1}$,
$m_{2}$ and $m_{3}$ can be written as functions of only two parameters
$a_{1}$, and $a_{2}$. This yields algebraic relations amongst the three masses
at the symmetry point. This relation can be obtained by solving the polynomial
system in Eqs. (26), and (27). However, it cannot be derived in a model
independent way, as it does not only depend on the representation and weight
of the fields, but also on the specific symmetry point.

###  C.1 Example of a mass relation for the up-type quarks and neutrinos

We regard it illustrative to give an explicit example of how to obtain the
second kind of mass relations which could also potentially yield interesting
and viable predictions.

| $Q$ | $U^{c}$ | $L$ | $N^{c}$ | $\Phi_{u}$ | $Y^{(2)}_{\mathbf{2}}$ | $Y^{(2)}_{\mathbf{3}}$
---|---|---|---|---|---|---|---
$S_{4}$ | $\mathbf{3}$ | $\mathbf{3^{\prime}}$ | $\mathbf{3}$ | $\mathbf{3}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{3}$
$k$ | $-1$ | $-1$ | $-2$ | $0$ | $0$ | $2$ | $2$
Table 3:  $\Gamma_{4}$ weight and representation assignments that lead to a
correlation between neutrino and up-quark masses. Gauge MSSM transformation
properties are omitted.

For definiteness, we assume that neutrinos acquire a mass through a Type-I
seesaw mechanism, including thus the $N^{c}$ superfields for the three heavy
right-handed (RH) neutrinos. The representation and weight assignments are
given in Table 3, leading to the following up-sector MSSM superpotential terms

$\displaystyle\mathcal{W}^{S_{4}}_{\Phi_{u}}$ $\displaystyle\supset\hskip
3.0pt\alpha^{u}_{1}\left(Q\Phi_{u}U^{c}Y^{(2)}_{\mathbf{2}}(\tau)\right)_{\mathbf{1}}+\hskip
3.0pt\alpha^{u}_{2}\left(Q\Phi_{u}U^{c}Y^{(2)}_{\mathbf{3}}(\tau)\right)_{\mathbf{1}}$
$\displaystyle\hskip
4.0pt+\alpha^{\nu}_{1}\left(L\Phi_{u}N^{c}Y^{(2)}_{\mathbf{2}}(\tau)\right)_{\mathbf{1}}+\hskip
3.0pt\alpha^{\nu}_{2}\left(L\Phi_{u}N^{c}Y^{(2)}_{\mathbf{3}}(\tau)\right)_{\mathbf{1}}+\frac{m_{N}}{2}N^{c}N^{c}.$
(85)

Here the modular forms are the ones given in Eq. (31). The Dirac and RH
neutrino mass blocks are given by

$M^{D}_{\nu}=\begin{pmatrix}a^{\nu}_{1}Y_{1}&-a^{\nu}_{2}Y_{5}&a^{\nu}_{2}Y_{4}\\\
a^{\nu}_{2}Y_{5}&\frac{\sqrt{3}}{2}a^{f}_{1}Y_{2}&-\frac{1}{2}\left(a^{\nu}_{1}Y_{1}+2a^{\nu}_{2}Y_{3}\right)\\\
-a^{f}_{2}Y_{4}&-\frac{1}{2}\left(a^{\nu}_{1}Y_{1}-2a^{\nu}_{2}Y_{3}\right)&\frac{\sqrt{3}}{2}a^{\nu}_{1}Y_{2}\\\
\end{pmatrix},\qquad\text{and},\qquad M_{N}=m_{N}\begin{pmatrix}1&0&0\\\
0&0&1\\\ 0&1&0\end{pmatrix}.$ (86)

The effective mass of the light neutrinos is given by the standard seesaw
formula

$M_{\nu}\approx-M^{D}_{\nu}M^{-1}_{N}({M^{D}_{\nu}})^{\top}.$ (87)

We choose these particular representations to illustrate a relevant point. The
Hermitian matrix $H_{\nu}\equiv M_{\nu}M^{\dagger}_{\nu}$ fulfills the first
property in subsection II.3, with only two dimensionful parameters;
$a^{\nu}_{1}$, and $a^{\nu}_{2}$. However, at the symmetry point
$\tau_{\text{sym}}=i\infty$ its rank is not reduced, Eq. (84).

In this example, for simplicity we choose the up-quark mass matrix $M_{u}$ to
have the same structure that $M_{f}$ in Eq. (35) thus around
$\tau_{\text{sym}}=i\infty$ we have the following expression relating up-quark
masses

$\frac{m^{2}_{t}m^{2}_{c}m^{2}_{u}}{f(m_{t},m_{c})}\approx\lvert\epsilon\rvert^{4}$
(88)

where the polynomial can be either one of the three solutions listed in Eq.
(83) in Appendix B. The last equation indicates that at this symmetric point
the masses of the three families satisfy

$\lim_{\epsilon\to 0}\begin{pmatrix}m_{t}\\\ m_{c}\\\
m_{u}\end{pmatrix}=\begin{pmatrix}m_{t}\left(a^{u}_{1},a^{u}_{2}\right)\\\
m_{c}\left(a^{u}_{1},a^{u}_{2}\right)\\\
0\end{pmatrix},\qquad\quad\lim_{\epsilon\to 0}\begin{pmatrix}m^{\nu}_{3}\\\
m^{\nu}_{2}\\\
m^{\nu}_{1}\end{pmatrix}=\begin{pmatrix}m^{\nu}_{3}\left(a^{\nu}_{1},a^{\nu}_{2}\right)\\\
m^{\nu}_{2}\left(a^{\nu}_{1},a^{\nu}_{2}\right)\\\
m^{\nu}_{1}\left(a^{\nu}_{1},a^{\nu}_{2}\right)\end{pmatrix},$ (89)

Notice that $m_{u}$ vanishes at the symmetric limit as expected from the
relation in Eq. (88). However, since neither of the neutrino masses vanish,
the three neutrino masses are not independent. In fact at the symmetry point
two neutrino masses are degenerate:

$\lim_{\tau\to
i\infty}m^{\nu}_{2}=m^{\nu}_{1}\qquad\Longrightarrow\quad\lim_{\tau\to
i\infty}\Delta m^{2}_{21}=0.$ (90)

Therefore in this example the smallest of the required neutrino mass squared
splitting, the solar $\Delta m^{2}_{21}$, must result from a departure from
the symmetry point $\tau_{T}=i\infty$ yielding $\Delta
m^{2}_{21}\propto\lvert\epsilon\rvert^{2}$. For this illustrative example we
restrict ourselves to the region Im $\epsilon=0$, hence for values
$\lvert\epsilon\rvert\ll 1$ we obtain the simplified expression

$\Delta m^{2}_{21}\approx
f^{\nu}_{\pm}(m^{\nu}_{3},m^{\nu}_{1})\lvert\epsilon\rvert^{2},$ (91)

This expression is analogous to Eq. (30) but manifestly different. The
function $f^{\nu}_{\pm}(m^{\nu}_{3},m^{\nu}_{1})$ has mass-dimension $2$ and
is given by

$f^{\nu}_{\pm}(m^{\nu}_{3},m^{\nu}_{1})=\frac{8m^{\nu}_{1}}{\Delta
m^{2}_{31}}\left[8{m^{\nu}_{1}}^{3}+16{m^{\nu}_{1}}^{2}{m^{\nu}_{3}}+14{m^{\nu}_{1}}{m^{\nu}_{3}}^{2}-3{m^{\nu}_{3}}^{3}\pm
3\sqrt{4{m^{\nu}_{1}}+{m^{\nu}_{3}}}\left(4{m^{\nu}_{1}}{m^{\nu}_{3}}^{\frac{3}{2}}+{m^{\nu}_{3}}^{\frac{5}{2}}\right)\right],$
(92)

where $\Delta m^{2}_{31}$ is the atmospheric neutrino squared mass difference.
Notice that last equation is valid for both normal (NO) and inverted ordering
(IO) of neutrino masses since the polynomial form is common.

A viable correlation between the up-quark and neutrino masses can be obtained
by using Eqs. (88), (92), the plus sign function $f^{\nu}_{+}$ and the third
solution in Eq. (83), i.e.

$\frac{\Delta
m^{2}_{21}}{f^{\nu}_{+}(m^{\nu}_{3},m^{\nu}_{1})}\approx\frac{m_{t}m_{c}m_{u}}{8m^{2}_{c}(m_{c}+3m_{t})}.$
(93)

This correlation becomes manifest when we scan our parameter space
$\\{a^{u,\nu}_{1,2},\lvert\epsilon\rvert\\}\Longrightarrow\\{m_{u,c,t},m^{\nu}_{1,2,3}\\}$
as shown in Fig. 5.

Figure 5:  Predicted quark-lepton mass correlations. In the left panel we show
$m_{u}$ versus the lightest neutrino mass $m^{\nu}_{1}$, each point lies
inside the experimental $3$-$\sigma$ range for $m_{t}$, $m_{c}$, $\Delta
m^{2}_{21}$, and $\Delta m^{2}_{31}$ from PDG [104] and [105] respectively.
This correlation is the prediction in Eq. (93). The right panel assumes normal
ordering, fixing $m^{\nu}_{1}=35$ meV. It illustrates how both $m_{u}$ and
$\Delta m^{2}_{21}$ are generated simultaneously, when departing from the
symmetry point as predicted by Eqs. (88) and (91).

Indeed, this figure shows a correlation in the $m_{u}$ vs. squared solar mass
splitting plane that results by taking points $\\{m_{t},\hskip
1.0ptm_{c},\hskip 1.0pt\Delta m^{2}_{21},\hskip 1.0pt\Delta m^{2}_{31}\text{
(NO)}\\}$ within their $3\sigma$ ranges given the PDG [104] and Ref. [105],
respectively. This result is very suggestive, as it correlates the smallness
of the solar squared mass splitting to the lightness of $m_{u}$, which is the
quark mass with the largest uncertainty [96].

This example is relevant as it shows that there is a (second) kind of mass
relations that can emerge in modular symmetric models that is not included in
our general derivation in Section II. We want to highlight that this example,
and in particular Eqs. (90) and (91), illustrates how mass matrices satisfying
Eq. (84) could lead to viable predictions for the neutrino sector.

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