#  SISSA 14/2023/FISI On the Normalisation of the Modular Forms in Modular
Invariant Theories of Flavour

S. T. Petcov a,b111Also at: Institute of Nuclear Research and Nuclear Energy,
Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria.

aINFN/SISSA, Via Bonomea 265, 34136 Trieste, Italy bKavli IPMU(WPI), UTIAS,
The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

( Abstract
The problem of normalisation of the modular forms in modular invariant lepton
and quark flavour models is discussed. Modular invariant normalisations of the
modular forms are proposed. )

##  1 Introduction

The main new feature of the modular invariance approach to the fundamental
flavour problem in particle physics proposed in [1] 222For a review of the
flavour problem see, e.g., Ref. [2]. A brief discussion of the earlier
proposed “solutions” of the flavour problem and their drawbacks is given,
e.g., in Ref. [3]. , is that the elements of the Yukawa coupling and fermion
mass matrices in the Lagrangian of the theory are modular forms of a certain
level $N$, which are functions of a single complex scalar field – the modulus
$\tau$ – and, as like the fermion (matter) fields, have specific
transformation properties under the action of the inhomogeneous (homogeneous)
modular group $\Gamma\equiv PSL(2,\mathbb{Z})$ ($\Gamma^{\prime}\equiv
SL(2,\mathbb{Z})$). The Yukawa couplings and the fermion mass matrices depend
also on limited number of constant parameters (see further). The theory is
assumed to be invariant under the whole modular group. In addition, both the
the fermion (matter) fields and the modular forms, present in Yukawa couplings
and the fermion mass matrices, are assumed to transform in representations of
an inhomogeneous (homogeneous) finite modular group of level $N$,
$\Gamma^{(\prime)}_{N}$, $N=1,2,3,...$, which plays the role of a flavour
symmetry. For $N\leq 5$, the finite modular groups $\Gamma_{N}$ are isomorphic
to the permutation groups $S_{3}$, $A_{4}$, $S_{4}$ and $A_{5}$ (see, e.g.,
[4]), while the groups $\Gamma^{\prime}_{N}$ are isomorphic to the double
covers of the indicated permutation groups, $S^{\prime}_{3}\equiv S_{3}$,
$A^{\prime}_{4}\equiv T^{\prime}$, $S^{\prime}_{4}$ and $A^{\prime}_{5}$.

In the modular flavour models, the VEV of the modulus $\tau$, $\tau_{v}$, can
be the only source of flavour symmetry breaking, such that no flavons are
needed. The VEV of $\tau$ can also be the only source of breaking of the
generalised CP (gCP) symmetry, which can be consistently combined with the
modular symmetry [5] (see also [6]). When the flavour symmetry is broken, the
modular forms (which are holomorphic functions of $\tau_{v}$), and thus the
elements of the Yukawa coupling and fermion mass matrices get fixed, and a
certain flavour structure arises. As a consequence of the modular symmetry in
the lepton sector, for example, the charged-lepton masses, the two neutrino
mass squared differences, the three neutrino mixing angles (in the reference
3-neutrino mixing scheme, see, e.g., [7]) and the not yet known absolute
neutrino mass scale, neutrino mass ordering and the leptonic Dirac and
Majorana CP-violation phases, are simultaneously determined in terms of a
limited number of coupling constant parameters 333 This together with the fact
that they are, via the corresponding modular forms, also functions of a single
complex VEV – that of the modulus $\tau$ – leads to experimentally testable
correlations between, e.g., the neutrino mass and mixing observables (see,
e.g., [8]). Models of flavour based on modular invariance have then an
increased predictive power. . A unique characteristic of the modular framework
is the fact that fermion mass hierarchies may follow from the properties of
the modular forms, without fine-tuning [3].

The modular symmetry approach to the flavour problem has been widely
implemented so far primarily in theories with global (rigid) supersymmetry
(SUSY). Within this SUSY framework, modular invariance is assumed to be a
feature of the Kähler potential and the superpotential of the theory.

Bottom-up modular invariance approach to the lepton flavour problem has been
exploited first using the groups $\Gamma_{3}\simeq A_{4}$ [1],
$\Gamma_{2}\simeq S_{3}$ [9], $\Gamma_{4}\simeq S_{4}$ [10] and
$\Gamma_{3}\simeq A_{4}$ again [11] 444In the “bottom-up” constructions,
typically the minimal form of the Kähler potential is being used. Possible
non-minimal additions to the Kähler potential, compatible with the modular
symmetry, may affect the predictive power of the approach [12]. The indicated
problem is the subject of ongoing research. . After these first studies, the
interest in the approach grew significantly and various aspects of this
approach were and continue to be extensively studied 555A rather extensive
list of publications on the modular invariance approach to the flavour problem
can be found, e.g., in [13]. . Recently, for example, a phenomenologically
viable anomaly-free modular flavour model, which features also an “axion-less”
solution of the strong CP problem, was constructed in Ref. [14].

It should be clear from the preceding discussion that the modular forms play a
crucial role in the modular invariance approach to the flavour puzzle. As we
have already mentioned, the elements of the fermion mass matrices, on which
the successful description of the lepton and quark flavours depend, are
modular forms of some level $N$ furnishing irreducible representations of a
finite modular group $\Gamma^{(\prime)}_{N}$. It is a well known fact that the
modular forms furnishing irreducible representations of the finite modular
groups are determined up to a constant. Usually these normalisation constants
are absorbed in the constant parameters which multiply each modular form
present in the fermion mass matrices. Since the normalisation of the modular
forms is arbitrary, this makes the specific values of the constant parameters,
obtained in a given model by statistical analysis of the description of the
relevant experimental data by the model, of not much physical meaning.
Moreover, it also makes the comparison of models which use different
normalisations of the modular forms ambiguous. The problem of normalisation of
the modular forms can be particularly acute in modular flavour models in which
the charged lepton and quark mass hierarchies are obtained without fine-tuning
of the constant parameters present in the respective fermion mass matrices,
where these constant parameters have to be of the same order in magnitude.

In the present article we discuss the problem of normalisation of modular
forms in the modular invariant theories of flavour and consider two possible
modular-invariant solutions to this problem.

##  2 The Problem

Consider a modular-invariant bilinear

$\tilde{W}=v_{h}\psi^{c}\,M(\tau)\,\psi\,,$ (2.1)

where $v_{h}$ is a constant and the matter superfields $\psi$ and $\psi^{c}$
transform under the action of the inhomogeneous
$PSL(2,\mathbb{Z})\equiv\Gamma$ (homogeneous
$SL(2,\mathbb{Z})\equiv\Gamma^{\prime}$) modular group as

$\begin{split}\psi\,&\xrightarrow{\gamma}\,(c\tau+d)^{-k_{\psi}}\rho(\tilde{\gamma})\,\psi\,,\\\
\psi^{c}\,&\xrightarrow{\gamma}\,(c\tau+d)^{-k^{c}_{\psi}}\rho^{c}(\tilde{\gamma})\,\psi^{c}\,.\end{split}$
(2.2)

Here $\gamma$ is an element of $\Gamma^{(\prime)}$, $(-\,k^{(c)})$ is the
weight of the field $\psi^{(c)}$ and $\rho^{(c)}$ is an irreducible
representation of a finite inhomogeneous (homogeneous) modular group of level
$N$, $\Gamma^{(\prime)}_{\rm N}$. We recall that for $N\leq 5$, the groups
$\Gamma_{2,3,4,5}$ are isomorphic to the non-Abelian discrete symmetry groups
$S_{3}$, $A_{4}$, $S_{4}$ and $A_{5}$, while the groups
$\Gamma^{\prime}_{2,3,4,5}$ are isomorphic to their respective double covers.

In flavour models based on modular invariance, terms of the type in Eq. (2.1),
each multiplied by a Higgs doublet or a flavon fields, are present in the
superpotential of the theory (see, e.g., [1, 10, 9, 11]). These terms lead to
quark and charged lepton Yukawa couplings, and, if neutrino masses are
generated by the type I seesaw mechanism, they lead also to the neutrino
Yukawa coupling and the Majorana mass term for the right-handed (RH) neutrino
fields. Alternatively, they can give also the Weinberg dimension five
operator.

We will consider the simple case when the bilinear in Eq. (2.1) is generated
by couplings to Higgs fields, which are singlets under the action of the
finite modular group. Although we discuss further the case of only the two
Higgs fields present in the SUSY extension of the Standard Model, the
discussion can be easily applied to the case of models with flavons.

The term of interest in Eq. (2.1) appears when the corresponding Higgs field
obtains a non-zero vacuum expectation value (VEV), $v_{h}$. Then the Yukawa
couplings give rise to the quark, charged lepton and neutrino Dirac mass
terms, or to the neutrino Majorana mass term associated with the Weinberg
operator (in the latter case $v^{2}_{h}$ multiplies the mass matrix). The
quantity $M$ in Eq. (2.1) is a generic notation for the mass matrices in these
mass terms as well as for the mass matrix in the RH neutrino Majorana mass
term, which can originate from a term in the super potential than includes
neither flavon nor Higgs fields (with $v_{h}$ replaced by a generic constant).

The modular invariance of the term in Eq. (2.1) implies that $M$, and thus the
fermion mass matrices in the modular flavour models, is a modular form
$Y^{K}_{\rho_{Y}}(\tau)$ of level $N$ and weight $K$ furnishing a
representation $\rho_{Y}$ of $\Gamma^{(\prime)}_{\rm N}$, such that, if the
Higgs field leading to the bilinear (2.1) is a $\Gamma^{(\prime)}_{\rm N}$
singlet with modular weigh $(-\,k_{H})$, we have $K\equiv
k_{\psi}+k^{c}_{\psi}+k_{H}$ and
$\rho_{Y}\otimes\rho^{c}\otimes\rho\supset\mathbf{1}$, $\mathbf{1}$ being the
singlet representation of $\Gamma^{(\prime)}_{\rm N}$. Thus,
$M=v_{h}\,\alpha\,Y^{K}_{\rho_{Y}}(\tau)$, where $\alpha$ is a constant.

In realistic modular flavour models $M$ is a matrix involving more than one
modular forms and thus more than one constant. If, for example, $\psi$
furnishes a $\mathbf{3}$ representation of $S^{\prime}_{4}$ and $\psi^{c}$
represent three singlet representations of $S^{\prime}_{4}$ having different
weights, $M$ is a $3\times 3$ matrix involving at least three different
modular forms, each transforming as triplet representation of
$S^{\prime}_{4}$, and thus involves at least three different constant
parameters.

To be more specific, consider the case of $\psi$ being the three quark doublet
superfields, $\psi=Q=(Q_{1},Q_{2},Q_{3})$, and $\psi^{c}$ being the three
singlet up-type or down-type quark superfields,
$\psi^{c}=q^{c}=(q^{c}_{1},q^{c}_{2},q^{c}_{3})$. Let us assume further that
$Q$ furnishes the triplet representation of $S^{\prime}_{4}$, ${\bf 3}$, and
carry weight $k_{Q}$, $q^{c}_{1,2,3}$ transform as the “hatted” singlet
${\bf\hat{1}}$ of $S^{\prime}_{4}$ 666See Ref. [15] for the technical details
related to this discussion.  but have different weights
$k_{1}\neq\kappa_{2}\neq k_{3}$, and the Higgs fields $H_{q}$, $q=u,d$,
present in the theory are singlets with respect to $S^{\prime}_{4}$, but may
carry non-zero weights $k_{H_{q}}$. The modular invariance implies that the
quark mass matrix $M_{q}$ should be formed by modular forms which furnish the
${\bf\hat{3}}^{\prime}$ representation of $S^{\prime}_{4}$. These are: the
weigh 3 and 5 modular forms $Y^{3}_{\hat{3}^{\prime}}(\tau)$ and
$Y^{5}_{\hat{3}^{\prime}}(\tau)$, two weight 7 and two weigh 9 modular forms,
$Y^{7(9)}_{\hat{3}^{\prime},1}(\tau)$, $Y^{7(9)}_{\hat{3}^{\prime},2}(\tau)$,
the three weight 11 modular forms $Y^{11}_{\hat{3}^{\prime},i}$, $i=1,2,3$,
etc. [15]. Each modular form in $M_{q}$ is accompanied by a constant. Consider
the “minimal” case of four modular forms present in $M_{q}$ 777 Quark (charged
lepton) mass matrices which include only three different modular forms in the
considered case of doublets superfields forming a triplet representation of
$S^{\prime}_{4}$ and the three quark (charged lepton) singlet superfields
furnishing singlet representations of $S^{\prime}_{4}$, have one zero
eigenvalue and thus are not phenomenologically viable without some additional
input. . Choosing these to be the weight 3, 5 and 9 modular forms leads to
${\rm Det}(M_{q})=0$ and thus to one massless quark, which is incompatible
with the existing data on, and lattice calculations of, the light quark
masses. In this case one needs additional operators to generate non-zero mass
for the lightest quark (see, e.g., [16]). Thus, in the considered case the
only minimal and phenomenologically viable possibility involves the modular
forms of weights 3, 5 and 7 and four constants. The quark mass matrix $M_{q}$
has the following form in the R-L convention:

$\displaystyle
M_{q}=v_{q}\frac{1}{\sqrt{3}}\,\begin{pmatrix}\alpha^{\prime}_{q}&0&0\\\
0&\beta_{q}&0\\\ 0&0&\gamma_{q}\\\
\end{pmatrix}\begin{pmatrix}\tilde{Y}_{\hat{3}^{\prime};1}^{(7)}&\tilde{Y}_{\hat{3}^{\prime};3}^{(7)}&\tilde{Y}_{\hat{3}^{\prime}2}^{(7)}\\\
Y_{\hat{3}^{\prime};1}^{(5)}&Y_{\hat{3}^{\prime};3}^{(5)}&Y_{\hat{3}^{\prime};2}^{(5)}\\\
Y^{(3)}_{\hat{3}^{\prime};1}&Y^{(3)}_{\hat{3}^{\prime};3}&Y^{(3)}_{\hat{3}^{\prime};2}\end{pmatrix}\,,$
(2.3)

where $v_{q}$ is the Higgs doublet VEV,
$\tilde{Y}_{\hat{3}^{\prime};i}^{(7)}=Y_{\hat{3}^{\prime},2;i}^{(7)}+g_{q}Y_{\hat{3}^{\prime},1;i}^{(7)}$,
$g_{q}=\alpha_{q}/\alpha^{\prime}_{q}$, $\alpha_{q}$, $\alpha^{\prime}_{q}$,
$\beta_{q}$ and $\gamma_{q}$ are the four constants, and
$Y_{\hat{3}^{\prime},2;i}^{(7)}$, $Y_{\hat{3}^{\prime},1;i}^{(7)}$,
$Y_{\hat{3}^{\prime};i}^{(5)}$ and $Y^{(3)}_{\hat{3}^{\prime};i}$, $i=1,2,3$,
are the three components of the triplet modular forms. The mass matrix in Eq.
(2.3) leads in the “vicinity” of the fixed point $\tau_{\rm T}=i\infty$ to the
following hierarchy between the three quark masses
$1:\epsilon^{2}:\epsilon^{3}$, where $\epsilon\cong{\rm exp}(-\pi{\rm
Im}\tau/2)\ll 1$ is a measure of the “deviation” of the VEV $\tau_{v}$ of
$\tau$ from $i\infty$ [3]. In the “bottom-up” modular flavour models the value
of ${\rm Im}\tau_{v}$ is obtained from fits of the relevant data and typically
one finds ${\rm Im}\tau_{v}\sim 2.5-3.0$ (see, e.g., [3, 17, 16]) 888 We note
also that $\psi$ can be chosen to be the three lepton doublet superfields,
$\psi=L=(L_{1},L_{2},L_{3})$, and $\psi^{c}$ to be the three singlet charged
lepton fields $\psi=E^{c}=(E^{c}_{1},E^{c}_{2},E^{c}_{3})$. In this case the
mass matrix in Eq. (2.3) will lead to the hierarchy
$1:\epsilon^{2}:\epsilon^{3}$ of the charged lepton masses. .

If one uses the “minimal” Kähler potential,

$K(\tau,\overline{\tau},\psi_{I},\overline{\psi_{I}})=-\,\Lambda_{0}^{2}\,\log(-i\tau+i\overline{\tau})+\sum_{I}\frac{|\psi_{I}|^{2}}{(-i\tau+i\overline{\tau})^{k_{I}}}\,,$
(2.4)

$\Lambda_{0}$ being a constant of mass dimension one, which is typically done
within the bottom-up approach in the modular flavour model building, after the
modulus $\tau$ gets a VEV $\tau_{v}$, the fermion fields have to be
renormalised as follows in order to have a canonical kinetic term:

$\psi_{I}\rightarrow\sqrt{(2Im(\tau_{v}))^{k_{I}}}\psi_{I}\,.$ (2.5)

Correspondingly, the bilinear term in eq. (2.1) changes as:

$\psi^{c}\,Y(\tau_{v})\,\psi\rightarrow\sqrt{(2Im(\tau_{v}))^{k_{\psi}+k^{c}_{\psi}}}\,\psi^{c}\,Y(\tau_{v})\,\psi$
(2.6)

This implies that in the considered example of a quark mass matrix in Eq.
(2.3), each constant gets an additional factor:

$\displaystyle\alpha_{q}\rightarrow\hat{\alpha}_{q}=\alpha_{q}\,\sqrt{(2{\rm
Im}\tau_{v})^{7}}\,,$ (2.7)
$\displaystyle\alpha^{\prime}_{q}\rightarrow\hat{\alpha}^{\prime}_{q}=\alpha^{\prime}_{q}\,\sqrt{(2{\rm
Im}\tau_{v})^{7}}\,,$ (2.8)
$\displaystyle\beta_{q}\rightarrow\hat{\beta}_{q}=\beta_{q}\,\sqrt{(2{\rm
Im}\tau_{v})^{5}}\,,$ (2.9)
$\displaystyle\gamma_{q}\rightarrow\hat{\gamma}_{q}=\gamma_{q}\sqrt{(2{\rm
Im}\tau_{v})^{3}}\,.$ (2.10)

We note that since in certain cases of models, in which the fermion (charged
lepton and quark) mass hierarchies are obtained without fine tuning of the
constants present in the fermion mass matrices using the formalism developed
in [3] (see also [19, 18]), $2{\rm Im}\tau_{v}$ can have a relatively large
value. In the cases of models constructed in the “vicinity” of the fixed point
$\tau_{\rm T}=i\infty$ (see [8, 20]), for example, $2{\rm Im}\tau_{v}\sim 5$,
as we have already indicated (see, e.g., [3, 17, 16]). This can lead to
significant modifications of the coupling constants: if initially they were of
the same order in magnitude, they can become hierarchical, which in turn was
used, e.g., in [17, 16], to help explain the up-type quark mass hierarchies in
quark flavour models containing one modulus (for problems in constructing
viable non-fine-tuned quark flavour models see, e.g., [21]).

As we have already remarked, it is well known that the modular forms
furnishing irreducible representations of the finite modular groups are
determined up to a constant In addition, since for a given level N, the higher
weight modular forms are obtained as tensor products of the lowest weight
modular form – they are homogeneous polynomials of the lowest weight modular
form – the normalisation constant of the lowest weight modular form propagates
in the higher weight modular forms. Usually, as we have indicated, these
normalisation constants are absorbed in the constant parameters which multiply
each modular form present in the fermion mass matrices. Since the
normalisation of the modular forms is arbitrary, this makes the specific
values of the constant parameters, obtained in a given model by statistical
analysis of the description of the relevant experimental data by the model,
physically irrelevant.

Another problems is that some of the modular forms used in the “non-fine-tuned
models” of flavour which are studied in the vicinity of one of the three known
fixed points of the modular group [8, 20], namely, $\tau_{C}=i$,
$\tau_{L}=\omega\equiv{\rm exp}(i2\pi/3)$ (the left cusp) and
$\tau_{T}=i\infty$, vanish at the corresponding fixed point. In the case of
models based on $S^{\prime}_{4}$ and $\tau=i\infty$, for example, we have [15,
3] $Y_{\hat{1}^{\prime}}^{(3)}=0$, $Y_{3}^{(4)}=0$, $Y_{\hat{3},2}^{(5)}=0$,
$Y_{1^{\prime}}^{(6)}=0$, $Y_{3}^{(6)}=0$, $Y_{3^{\prime},2}^{(6)}=0$,
$Y_{\hat{1}^{\prime}}^{(7)}=0$, $Y_{\hat{2}}^{(7)}=0$,
$Y_{\hat{3},2}^{(7)}=0$, $Y_{\hat{3}^{\prime},2}^{(7)}=0$, etc. However, these
modular forms can be normalised in the vicinity of the fixed point (as well as
at the fixed point) in such a way that with the new normalisation the singlet
modular forms have a value of order 1,
$|\tilde{Y}_{\hat{1}^{\prime}}^{(3)}|\sim 1$,
$|\tilde{Y}_{1^{\prime}}^{(6)}|\sim 1$,
$|\tilde{Y}_{\hat{1}^{\prime}}^{(7)}|\sim 1$, while the triplet modular forms
listed above have at least one component whose value becomes much larger
(e.g., by one or two orders of magnitude, see further) or even of order 1. If
any of these modular forms plays an important role in making a given model
phenomenologically viable, it should be obvious that this conclusion would
depend on the normalisation employed of the modular form. For this reason, in
our opinion, modular forms which by changing their normalisation can change
significantly their values (e.g., from a value close to zero or zero to a
value $\sim 1$ ), which in turn can have phenomenological consequences, should
be avoided in constructing non-fine-tuned models of flavour.

In order to have a sensible comparison of the values of the constants obtained
in different models when they are confronted with the data, we need to adopt a
certain convention about the normalisation of the modular forms.

##  3 Modular Invariant Normalisations

###  3.1 “Local” Normalisation at $\tau=\tau_{v}$

Suppose we normalise the modular form of weight $K$, $Y^{(K)}(\tau)$, by a
factor $N^{(K)}_{Y}$, where

$N^{(K)}_{Y}=\left(\sum_{i}|Y^{(K)}_{i}(\tau)|^{2}(2Im(\tau))^{K}\right)^{\frac{1}{2}}\,,$
(3.1)

where $Y^{(K)}_{i}$ are the components of $Y^{(K)}$. Assuming that $Y^{(K)}$
furnishes a unitary representation r of the finite modular group of level N,
$Y^{(K)}=Y^{(K)}_{{\bf r}}$, this normalisation is modular invariant. Indeed,
under the modular transformation,

$\displaystyle\sum_{i}|Y^{(K)}_{{\bf
r}i}|^{2}\rightarrow\sum_{i}|Y^{(K)}_{{\bf r}i}|^{2}|(c\tau+d)|^{2K}\,,$ (3.2)
$\displaystyle(2Im(\tau))^{K}\rightarrow(2Im(\tau))^{K}|(c\tau+d)|^{-2K}\,.$

In a specific flavour model, the modulus $\tau$ in the expression of
$N^{(K)}_{Y}$ should be replaced by its VEV $\tau_{v}$ in the model, so the
normalisation should read $N^{(K)}_{Y}(\tau_{v})$. The quantity $N^{(K)}_{Y}$
determined in Eq. (3.1) seems an “appropriate” normalisation of $Y^{(K)}_{{\bf
r}}(\tau)$. Using $N^{(K)}_{Y}(\tau_{v}))$ as a normalisation, has an
additional important consequence. Indeed, employing the described
normalisation of $Y^{(K)}_{{\bf r}}(\tau)$ and taking into account the
normalisation factor coming from the Kähler potential, we get for the bilinear
of interest:

$\psi^{c}\,Y^{(K)}_{{\bf
r}}(\tau)\,\psi\rightarrow\sqrt{(2Im(\tau_{v}))^{k_{\psi}+k^{c}_{\psi}}}\,\psi^{c}\,Y^{(K)}_{{\bf
r}}(\tau_{v})\,\psi/N^{(K)}_{Y}(\tau_{v})=\psi^{c}\,R_{\rm
am}\,\frac{Y^{(K)}_{{\bf
r}}(\tau)}{\sqrt{\sum_{i}|Y^{(K)}_{{\bf}i}|^{2}}}\,\psi\,,$ (3.3)

where $R_{\rm am}$ is the ratio of the automorphy factors due to the re-
normalisation of $\psi$ and $\psi^{c}$ and the normalisation (3.1) of the
modular form:

$R_{\rm
am}=\dfrac{\sqrt{(2Im(\tau_{v}))^{k_{\psi}+k^{c}_{\psi}}}}{\sqrt{(2Im(\tau_{v}))^{K}}}=\dfrac{1}{\sqrt{(2Im(\tau_{v}))^{k_{H}}}}\,.$
(3.4)

We see that the automorphy factors coming from the re-normalisation of $\psi$
and $\psi^{c}$ cancel in the ratio $R_{\rm am}$ as a consequence of
$K=k_{\psi}+k^{c}_{\psi}+k_{H}$, which follows from the modular invariance.
This procedure (and cancellation) holds only after the modulus $\tau$ takes a
VEV. If $k_{H}=0$, as is assumed typically (but not universally, see, e.g.,
[18, 16]) in bottom-up modular flavour models, we have $R_{\rm am}=1$. We will
consider this case in what follows.

Under the assumed conditions, normalising the modular forms to their Euclidean
norms at $\tau=\tau_{v}$ seems to be a “natural” normalisation. In that case
the re-normalisation of the fermion fields in the Yukawa couplings, associated
with obtaining their canonical kinetic terms, is effectively taken into
account and does not need to be introduced in the fermion mass matrices.
Moreover, one can use flavour-dependent weights of the matter fields in the
non-fine-tuned approach to the fermion mass hierarchies [3]. The Euclidean
norms of the modular forms also do not depend on the basis one uses for the
generators of the respective finite modular group.

Let us consider a few examples of $N^{(K)}_{Y}(\tau_{v})$ in the case of level
$N=4$ modular forms. The modular forms of interest are expressed in terms of
two holomorphic functions of $\tau$, $\theta(\tau)$ and $\varepsilon(\tau)$
[15]:

$\theta(\tau)\,\equiv\,\frac{\eta^{5}(2\tau)}{\eta^{2}(\tau)\eta^{2}(4\tau)}\,=\,\Theta_{3}(2\tau)\,,\quad\,\,\varepsilon(\tau)\,\equiv\,\frac{2\,\eta^{2}(4\tau)}{\eta(2\tau)}\,=\,\Theta_{2}(2\tau)\,.$
(3.5)

Here $\eta(\tau)$ is the Dedekind eta-function and $\Theta_{2}(\tau)$ and
$\Theta_{3}(\tau)$ are the Jacobi theta constants. The functions
$\theta(\tau)$ and $\varepsilon(\tau)$ admit the following $q$-expansions,
i.e. power series expansions in $q_{4}\equiv\exp(i\pi\tau/2)$ 999See Ref. [15]
for technical details related to the properties of the functions
$\theta(\tau)$ and $\varepsilon(\tau)$. :

$\displaystyle\theta(\tau)$
$\displaystyle=\,1+2\sum_{k=1}^{\infty}q_{4}^{(2k)^{2}}\,=\,1+2\,q_{4}^{4}+2\,q_{4}^{16}+\ldots\,,$
(3.6) $\displaystyle\varepsilon(\tau)$
$\displaystyle=\,2\sum_{k=1}^{\infty}q_{4}^{(2k-1)^{2}}\,=\,2\,q_{4}^{\phantom{1}}+2\,q_{4}^{9}+2\,q_{4}^{25}+\ldots\,,$

so that $\theta\to 1$, $\varepsilon\to 0$ in the “large volume” limit
$\operatorname{Im}\tau\to\infty$. We give below the values of $\theta(\tau)$
and $\varepsilon(\tau)$ at the three fixed point values of $\tau$, namely,
$\tau_{C}\equiv i$, $\tau_{L}\equiv\omega={\rm
exp}(i2\pi/3)=-\,1/2+i\sqrt{3}/2$, and $\tau_{T}\equiv i\infty$, at which
there exist residual symmetries used in the construction of non-fine-tuned
modular flavour models:

$\displaystyle\theta(\tau_{C})$
$\displaystyle=1+2\,e^{-2\pi}+O(10^{-11})\,\simeq\,1.00373\,,$
$\displaystyle\varepsilon(\tau_{C})$
$\displaystyle=2\,e^{-\pi/2}+O(10^{-6})\,\simeq\,0.415761\,;$
$\displaystyle\theta(\tau_{L})$
$\displaystyle=1-2\,e^{-\sqrt{3}\,\pi}+O(10^{-9})\,\simeq\,0.991333\,,$ (3.7)
$\displaystyle\varepsilon(\tau_{L})$
$\displaystyle=2\,e^{-i\,\pi/4}\left[e^{-\sqrt{3}\,\pi/4}+O(10^{-5})\right]\,\simeq\,0.512152\;e^{-i\,\pi/4}\,;$
$\displaystyle\theta(\tau_{T})$ $\displaystyle=1\,,$
$\displaystyle\varepsilon(\tau_{T})$ $\displaystyle=0\,.$

We further give the exact relations between $\theta(\tau)$ and
$\varepsilon(\tau)$ at the symmetric points $\theta(\tau_{C})$ and $\tau_{L}$
:

$\frac{\varepsilon(\tau_{C})}{\theta(\tau_{C})}\,=\,\frac{1}{1+\sqrt{2}}\,,\qquad\frac{\varepsilon(\tau_{L})}{\theta(\tau_{L})}\,=\,\frac{1-i}{1+\sqrt{3}}\,.$
(3.8)

The level $N=4$ lowest weight 1 modular form furnishes a ${\bf\hat{3}}$
representation of $S^{\prime}_{4}$ [15]. In the group representation basis
used in [15] this modular form is given by:

$Y_{\mathbf{\hat{3}}}^{(1)}(\tau)=\begin{pmatrix}\sqrt{2}\,\varepsilon\,\theta\\\\[2.84526pt]
\varepsilon^{2}\\\\[2.84526pt] -\theta^{2}\end{pmatrix}\,.$ (3.9)

Further tensor products with $Y_{\mathbf{\hat{3}}}^{(1)}$ produce new modular
multiplets of even and odd weights 101010In what follows we use the
expressions for the modular form multiplets derived in [15], where the
multiplets with weights $k_{Y}\leq 10$ have been constructed. . We give below
the expressions of selected level 4 weight 3, 4, 5 and 7 modular forms which
we use for illustrative purposes in our discussion. At weight $k=3$ there are
three multiplets - a non-trivial singlet and two triplets - exclusive to
$S_{4}^{\prime}$:

$\displaystyle Y_{\mathbf{\hat{1}^{\prime}}}^{(3)}(\tau)$
$\displaystyle=\sqrt{3}\left(\varepsilon\,\theta^{5}-\varepsilon^{5}\,\theta\right)\,,$
(3.10) $\displaystyle Y_{\mathbf{\hat{3}}}^{(3)}(\tau)$
$\displaystyle=\begin{pmatrix}\varepsilon^{5}\,\theta+\varepsilon\,\theta^{5}\\\\[2.84526pt]
\frac{1}{2\sqrt{2}}\left(5\,\varepsilon^{2}\,\theta^{4}-\varepsilon^{6}\right)\\\\[2.84526pt]
\frac{1}{2\sqrt{2}}\left(\theta^{6}-5\,\varepsilon^{4}\,\theta^{2}\right)\end{pmatrix}\,,\quad
Y_{\mathbf{\hat{3}^{\prime}}}^{(3)}(\tau)=\frac{1}{2}\begin{pmatrix}-4\sqrt{2}\,\varepsilon^{3}\,\theta^{3}\\\\[2.84526pt]
\theta^{6}+3\,\varepsilon^{4}\,\theta^{2}\\\\[2.84526pt]
-3\,\varepsilon^{2}\,\theta^{4}-\varepsilon^{6}\end{pmatrix}\,.$

The set of four weight $k=4$ modular forms includes: [10, 15]:

$\displaystyle Y_{\mathbf{3}}^{(4)}(\tau)$
$\displaystyle=\frac{3}{2\sqrt{2}}\begin{pmatrix}\sqrt{2}\left(\varepsilon^{2}\,\theta^{6}-\varepsilon^{6}\,\theta^{2}\right)\\\\[2.84526pt]
\varepsilon^{3}\,\theta^{5}-\varepsilon^{7}\,\theta\\\\[2.84526pt]
-\varepsilon\,\theta^{7}+\varepsilon^{5}\,\theta^{3}\end{pmatrix}\,,\quad
Y_{\mathbf{3^{\prime}}}^{(4)}(\tau)=\begin{pmatrix}\frac{1}{4}\left(\theta^{8}-\varepsilon^{8}\right)\\\\[2.84526pt]
\frac{1}{2\sqrt{2}}\left(\varepsilon\,\theta^{7}+7\,\varepsilon^{5}\,\theta^{3}\right)\\\\[2.84526pt]
\frac{1}{2\sqrt{2}}\left(7\,\varepsilon^{3}\,\theta^{5}+\varepsilon^{7}\,\theta\right)\end{pmatrix}\,,$
(3.11)

The weight $k=5$ forms of interest are:

$\displaystyle Y_{\mathbf{\hat{2}}}^{(5)}(\tau)$
$\displaystyle=\begin{pmatrix}\frac{3}{2}\left(\varepsilon^{3}\,\theta^{7}-\varepsilon^{7}\,\theta^{3}\right)\\\\[2.84526pt]
\frac{\sqrt{3}}{4}\left(\varepsilon\,\theta^{9}-\varepsilon^{9}\,\theta\right)\end{pmatrix}\,,$
$\displaystyle Y_{\mathbf{\hat{3}},1}^{(5)}(\tau)$
$\displaystyle=\begin{pmatrix}\frac{6\sqrt{2}}{\sqrt{5}}\,\varepsilon^{5}\,\theta^{5}\\\\[2.84526pt]
\>\>\>\frac{3}{8\sqrt{5}}\left(5\,\varepsilon^{2}\,\theta^{8}+10\,\varepsilon^{6}\,\theta^{4}+\varepsilon^{10}\right)\\\\[2.84526pt]
-\frac{3}{8\sqrt{5}}\left(\theta^{10}+10\,\varepsilon^{4}\,\theta^{6}+5\,\varepsilon^{8}\,\theta^{2}\right)\end{pmatrix}\,,$
$\displaystyle Y_{\mathbf{\hat{3}},2}^{(5)}(\tau)$
$\displaystyle=\begin{pmatrix}\frac{3}{4}\left(\varepsilon\,\theta^{9}-2\,\varepsilon^{5}\,\theta^{5}+\varepsilon^{9}\,\theta\right)\\\\[2.84526pt]
\frac{3}{\sqrt{2}}\left(-\varepsilon^{2}\,\theta^{8}+\varepsilon^{6}\,\theta^{4}\right)\\\\[2.84526pt]
\frac{3}{\sqrt{2}}\left(-\varepsilon^{4}\,\theta^{6}+\varepsilon^{8}\,\theta^{2}\right)\end{pmatrix}\,,$
(3.12) $\displaystyle Y_{\mathbf{\hat{3}^{\prime}}}^{(5)}(\tau)$
$\displaystyle=\begin{pmatrix}2\left(\varepsilon^{3}\,\theta^{7}+\varepsilon^{7}\,\theta^{3}\right)\\\\[2.84526pt]
\frac{1}{4\sqrt{2}}\left(\theta^{10}-14\,\varepsilon^{4}\,\theta^{6}-3\,\varepsilon^{8}\,\theta^{2}\right)\\\\[2.84526pt]
\frac{1}{4\sqrt{2}}\left(3\,\varepsilon^{2}\,\theta^{8}+14\,\varepsilon^{6}\,\theta^{4}-\varepsilon^{10}\right)\end{pmatrix}\,.$

In the case of weight $k=7$ we have:

$\displaystyle Y_{\mathbf{\hat{1}^{\prime}}}^{(7)}(\tau)$
$\displaystyle=\frac{1}{4}\sqrt{\frac{3}{2}}\left(-\varepsilon^{13}\,\theta-13\,\varepsilon^{9}\,\theta^{5}+13\,\varepsilon^{5}\,\theta^{9}+\varepsilon\,\theta^{13}\right)\,,$
(3.13) $\displaystyle Y_{\mathbf{\hat{2}}}^{(7)}(\tau)$
$\displaystyle=\begin{pmatrix}\frac{3}{2}\left(\varepsilon^{3}\,\theta^{11}-\varepsilon^{11}\,\theta^{3}\right)\\\\[2.84526pt]
-\frac{\sqrt{3}}{8}\left(\varepsilon\,\theta^{13}-11\,\varepsilon^{5}\,\theta^{9}+11\,\varepsilon^{9}\,\theta^{5}-\varepsilon^{13}\,\theta\right)\end{pmatrix}\,,$
$\displaystyle Y_{\mathbf{\hat{3}},1}^{(7)}(\tau)$
$\displaystyle=\begin{pmatrix}\frac{12}{\sqrt{13}}\left(\varepsilon^{5}\,\theta^{9}+\varepsilon^{9}\,\theta^{5}\right)\\\\[2.84526pt]
\frac{3}{8\sqrt{26}}\left(\varepsilon^{2}\,\theta^{12}+45\,\varepsilon^{6}\,\theta^{8}+19\,\varepsilon^{10}\,\theta^{4}-\varepsilon^{14}\right)\\\\[2.84526pt]
\frac{3}{8\sqrt{26}}\left(\theta^{14}-19\,\varepsilon^{4}\,\theta^{10}-45\,\varepsilon^{8}\,\theta^{6}-\varepsilon^{12}\,\theta^{2}\right)\end{pmatrix}\,,$
$\displaystyle Y_{\mathbf{\hat{3}},2}^{(7)}(\tau)$
$\displaystyle=\begin{pmatrix}\frac{3}{8}\left(\varepsilon\,\theta^{13}-\varepsilon^{5}\,\theta^{9}-\varepsilon^{9}\,\theta^{5}+\varepsilon^{13}\,\theta\right)\\\\[2.84526pt]
\frac{3}{4\sqrt{2}}\left(\varepsilon^{2}\,\theta^{12}+6\,\varepsilon^{6}\,\theta^{8}-7\,\varepsilon^{10}\,\theta^{4}\right)\\\\[2.84526pt]
\frac{3}{4\sqrt{2}}\left(7\,\varepsilon^{4}\,\theta^{10}-6\,\varepsilon^{8}\,\theta^{6}-\varepsilon^{12}\,\theta^{2}\right)\end{pmatrix}\,,$
$\displaystyle Y_{\mathbf{\hat{3}^{\prime}},1}^{(7)}(\tau)$
$\displaystyle=\begin{pmatrix}\frac{3}{4\sqrt{37}}\left(7\,\varepsilon^{3}\,\theta^{11}+50\,\varepsilon^{7}\,\theta^{7}+7\,\varepsilon^{11}\,\theta^{3}\right)\\\\[2.84526pt]
-\frac{3}{4\sqrt{74}}\left(\theta^{14}+14\,\varepsilon^{4}\,\theta^{10}+49\,\varepsilon^{8}\,\theta^{6}\right)\\\\[2.84526pt]
\>\>\>\frac{3}{4\sqrt{74}}\left(49\,\varepsilon^{6}\,\theta^{8}+14\,\varepsilon^{10}\,\theta^{4}+\varepsilon^{14}\right)\end{pmatrix}\,,$
$\displaystyle Y_{\mathbf{\hat{3}^{\prime}},2}^{(7)}(\tau)$
$\displaystyle=\begin{pmatrix}\frac{9}{4}\left(\varepsilon^{3}\,\theta^{11}-2\,\varepsilon^{7}\,\theta^{7}+\varepsilon^{11}\,\theta^{3}\right)\\\\[2.84526pt]
\>\>\>\frac{9}{4\sqrt{2}}\left(\varepsilon^{4}\,\theta^{10}-2\,\varepsilon^{8}\,\theta^{6}+\varepsilon^{12}\,\theta^{2}\right)\\\\[2.84526pt]
-\frac{9}{4\sqrt{2}}\left(\varepsilon^{2}\,\theta^{12}-2\,\varepsilon^{6}\,\theta^{8}+\varepsilon^{10}\,\theta^{4}\right)\end{pmatrix}\,.$

In the expressions for the modular forms in Eqs. (3.10) - (3.13) we have not
included the overall arbitrary constant factors, which we assume to be real.
They will be present in the respective modular form normalisation constants
and therefore will be canceled in the expressions for the normalised modular
forms. Thus, the normalised modular forms will not depend on these arbitrary
constants.

Consider the effect of the normalisation by $N^{(K)}_{Y}(\tau_{v})$ of level
$N=4$ modular forms in the vicinity of the fixed point $\tau_{T}=i\infty$. For
concreteness we fix ${\rm Im}\tau_{v}=2.5$, as suggested by flavour models
considered in the literature (see, e.g., [3, 17, 16])), while ${\rm
Re}\tau_{v}$ is left arbitrary. In this case $|q_{4v}|={\rm exp(-\pi
Im\tau_{v}/2)}\cong 0.019703$, and $|q_{4v}|^{4}\cong 1.507\times 10^{-7}$.
Therefore, to an excellent approximation $\theta(\tau_{v})\cong 1$ and
$\varepsilon(\tau_{v})\cong 2q_{4v}=2\,e^{i\,\pi{\rm
Re}\tau_{v}/2}\,e^{-\,\pi{\rm Im}\tau_{v}/2}$. In the considered case, for the
normalisation factor $N^{(K)}_{Y}(\tau_{v})$ of the modular forms
$Y_{\mathbf{\hat{3}}}^{(1)}(\tau)$,
$Y_{\mathbf{\hat{1}^{\prime}}}^{(3)}(\tau)$, $Y_{\mathbf{3}}^{(4)}(\tau)$,
$Y_{\mathbf{\hat{3}},2}^{(5)}(\tau)$ and
$Y_{\mathbf{\hat{3}^{\prime}},2}^{(7)}(\tau)$, for example, we get:

$\displaystyle|Y_{\mathbf{\hat{3}}}^{(1)}(\tau_{v})|\cong|\theta(\tau_{v})|^{2}(1+2|\varepsilon(\tau_{v})|^{2}/|\theta(\tau_{v})|^{2})^{\frac{1}{2}}=1+O(10^{-3})\,,$
(3.14)
$\displaystyle|Y_{\mathbf{\hat{1}^{\prime}}}^{(3)}(\tau_{v})|\cong\sqrt{3}|\varepsilon(\tau_{v})||\theta(\tau_{v})|^{5}\,,$
$\displaystyle|Y_{\mathbf{3}}^{(4)}(\tau_{v})|\cong\frac{3}{2\sqrt{2}}\,|\varepsilon(\tau_{v})||\theta(\tau_{v})|^{7}\,,$
$\displaystyle|Y_{\mathbf{\hat{3}},2}^{(5)}(\tau_{v})|\cong\frac{3}{4}\,|\varepsilon(\tau_{v})||\theta(\tau_{v})|^{9}\,,$
$\displaystyle|Y_{\mathbf{\hat{3}^{\prime}},2}^{(7)}(\tau_{v})|\cong\frac{9}{4\sqrt{2}}|\varepsilon(\tau_{v})|^{2}|\theta(\tau_{v})|^{12}\,,$

where the notation used for $N^{(K)}_{Y}(\tau_{v})$ is self explanatory.

Since in the considered case the normalisation of the lowest weight modular
form $|Y_{\mathbf{\hat{3}}}^{(1)}(\tau_{v})|=1+O(10^{-4})$, the effect of
propagation of this normalisation to higher weight modular form is practically
negligible. The normalised $Y_{\mathbf{\hat{3}}}^{(1)}(\tau_{v})$ has the
form:

$\dfrac{Y_{\mathbf{\hat{3}}}^{(1)}(\tau_{v})}{|Y_{\mathbf{\hat{3}}}^{(1)}(\tau_{v})|}=\begin{pmatrix}\sqrt{2}\,\frac{\varepsilon_{v}\,\theta_{v}}{|\theta_{v}|^{2}}\\\\[2.84526pt]
\frac{\varepsilon^{2}_{v}}{|\theta_{v}|^{2}}\\\\[2.84526pt]
-\,\frac{\theta^{2}_{v}}{|\theta_{v}|^{2}}\end{pmatrix}\cong\begin{pmatrix}\sqrt{2}\,\varepsilon_{v}\\\\[2.84526pt]
\varepsilon^{2}_{v}\\\\[2.84526pt] -\,1\end{pmatrix}\,,$ (3.15)

where we have used $\theta_{v}\cong 1$ (see Eq. (3.6) and the discussion after
Eq. (3.13)).

If we normalise the singlet modular form
$Y_{\mathbf{\hat{1}^{\prime}}}^{(3)}(\tau)$ at $\tau=\tau_{v}$, we get:

$\dfrac{Y_{\mathbf{\hat{1}^{\prime}}}^{(3)}(\tau_{v})}{|Y_{\mathbf{\hat{1}^{\prime}}}^{(3)}(\tau_{v})|}=\dfrac{\varepsilon_{v}\,\theta^{5}_{v}}{|\varepsilon_{v}\,\theta^{5}_{v}|}-\dfrac{\varepsilon^{5}_{v}}{|\varepsilon_{v}|}\,\dfrac{\theta_{v}}{|\theta^{5}_{v}|}\cong
e^{i\,\pi\,Re(\tau_{v})/2}\left(1-\varepsilon^{4}_{v})\right)\,.$ (3.16)

We see that the used normalisation changes drastically the magnitude of this
singlet modular form from being of order $|\varepsilon|\ll 1$ to being of
order 1.

Consider next the effect of the normalisation on the triplet modular forms
$Y_{\mathbf{3}}^{(4)}(\tau)$, $Y_{\mathbf{\hat{3}},2}^{(5)}(\tau)$ and
$Y_{\mathbf{\hat{3}^{\prime}},2}^{(7)}(\tau)$:

$\displaystyle\dfrac{Y_{\mathbf{3}}^{(4)}(\tau_{v})}{|Y_{\mathbf{3}}^{(4)}(\tau_{v})|}$
$\displaystyle=\frac{1}{|\varepsilon(\tau_{v})||\theta(\tau_{v})|^{7}}\begin{pmatrix}\sqrt{2}\left(\varepsilon^{2}_{v}\,\theta^{6}_{v}-\varepsilon^{6}_{v}\,\theta^{2}\right)\\\\[2.84526pt]
\varepsilon^{3}_{v}\,\theta^{5}_{v}-\varepsilon^{7}_{v}\,\theta\\\\[2.84526pt]
-\varepsilon_{v}\,\theta^{7}_{v}+\varepsilon^{5}_{v}\,\theta^{3}_{v}\end{pmatrix}\cong
e^{i\,\pi{\rm Re}\tau/2}\begin{pmatrix}\sqrt{2}\varepsilon_{v}\\\\[2.84526pt]
\varepsilon^{2}_{v}\\\\[2.84526pt] -\,1\end{pmatrix}\,,$
$\displaystyle\dfrac{Y_{\mathbf{\hat{3}},2}^{(5)}(\tau_{v})}{|Y_{\mathbf{\hat{3}},2}^{(5)}(\tau_{v})|}$
$\displaystyle=\frac{4}{3\,|\varepsilon(\tau_{v})||\theta(\tau_{v})|^{9}}\begin{pmatrix}\frac{3}{4}\left(\varepsilon_{v}\,\theta^{9}_{v}-\,2\,\varepsilon^{5}_{v}\,\theta^{5}_{v}+\varepsilon^{9}_{v}\,\theta_{v}\right)\\\\[2.84526pt]
\frac{3}{\sqrt{2}}\left(-\varepsilon^{2}_{v}\,\theta^{8}_{v}+\varepsilon^{6}_{v}\,\theta^{4}_{v}\right)\\\\[2.84526pt]
\frac{3}{\sqrt{2}}\left(-\varepsilon^{4}_{v}\,\theta^{6}_{v}+\varepsilon^{8}_{v}\,\theta^{2}_{v}\right)\end{pmatrix}\cong
e^{i\,\pi{\rm Re}\tau/2}\begin{pmatrix}1\\\\[2.84526pt]
-\,\frac{4}{\sqrt{2}}\,\varepsilon_{v}\\\\[2.84526pt]
-\,\frac{4}{\sqrt{2}}\,\varepsilon^{3}_{v}\end{pmatrix}\,,$ (3.17)
$\displaystyle\dfrac{Y_{\mathbf{\hat{3}^{\prime}},2}^{(7)}(\tau_{v})}{|Y_{\mathbf{\hat{3}^{\prime}},2}^{(7)}(\tau_{v})|}$
$\displaystyle=\frac{4\sqrt{2}}{9|\varepsilon(\tau_{v})|^{2}|\theta(\tau_{v})|^{12}}\,\begin{pmatrix}\frac{9}{4}\left(\varepsilon^{3}_{v}\,\theta^{11}_{v}-\,2\,\varepsilon^{7}_{v}\,\theta^{7}_{v}+\varepsilon^{11}_{v}\,\theta^{3}_{v}\right)\\\\[2.84526pt]
\>\>\>\frac{9}{4\sqrt{2}}\left(\varepsilon^{4}_{v}\,\theta^{10}_{v}-\,2\,\varepsilon^{8}_{v}\,\theta^{6}_{v}+\varepsilon^{12}_{v}\,\theta^{2}_{v}\right)\\\\[2.84526pt]
-\,\frac{9}{4\sqrt{2}}\left(\varepsilon^{2}_{v}\,\theta^{12}_{v}-\,2\,\varepsilon^{6}_{v}\,\theta^{8}_{v}+\varepsilon^{10}_{v}\,\theta^{4}_{v}\right)\end{pmatrix}\cong
e^{i\,\pi{\rm Re}\tau}\begin{pmatrix}\sqrt{2}\,\varepsilon_{v}\\\\[2.84526pt]
\>\>\>\varepsilon^{2}_{v}\\\\[2.84526pt] -\,1\end{pmatrix}\,.$

We see that in the normalised weight 4 and 5 triplet modular forms all
components are increased by the factor $|\varepsilon(\tau_{v})|^{-1}>>1$,
while in the weight 7 modular form the components are increased by
$|\varepsilon(\tau_{v})|^{-2}>>1$. As a consequence, instead of being of order
$|\varepsilon(\tau_{v})|\ll 1$ ($|\varepsilon(\tau_{v})|^{2}\ll 1$), the 3rd
and the 1st (the 3rd) components respectively of the weight 4 and 5 (weight 7)
normalised triplet modular forms are of order 1. If any of these three triplet
modular forms is present in a given fermion mass matrix, the results obtained
by diagonalising the mass matrix, i.e., the mass eigenvalues and the values of
the elements of the diagonalising unitary or orthogonal matrix, will depend on
the chosen normalisation of this modular form.

Indeed, consider our example of fermion mass matrix given in Eq. (2.3). It
contains the modular form $Y_{\mathbf{\hat{3}^{\prime}},2}^{(7)}(\tau)$ in
addition to the modular forms $Y_{\mathbf{\hat{3}^{\prime}},1}^{(7)}(\tau)$,
$Y_{\mathbf{\hat{3}^{\prime}}}^{(5)}(\tau)$ and
$Y_{\mathbf{\hat{3}^{\prime}}}^{(3)}(\tau)$. If we use the expressions for
these four modular forms given in Eqs. (3.10), (3.12) and (3.10), we find that
in the limit of $|\varepsilon(\tau_{v})|\rightarrow 0$ (${\rm
Im}\tau_{v}\rightarrow\infty$),
$Y_{\mathbf{\hat{3}^{\prime}},2}^{(7)}(\tau_{v})=0$ and that only the second
components of $Y_{\mathbf{\hat{3}^{\prime}},1}^{(7)}(\tau)$,
$Y_{\mathbf{\hat{3}^{\prime}}}^{(5)}(\tau)$ and
$Y_{\mathbf{\hat{3}^{\prime}}}^{(3)}(\tau)$ are non-zero and are of order
$3/(4\sqrt{74})$, $1/(4\sqrt{2})$ and 1/2, respectively. As a consequence, in
the considered limit only the [3,3] element of the matrix
$M^{\dagger}_{q}M_{q}$, whose diagonalising matrix contributes to the CKM
quark mixing matrix, is non-zero. This implies that $M_{q}$ has only one
“large” eigenvalue, $m_{3}$, which is given by

$m_{3}=\dfrac{v_{q}}{4\sqrt{6}}\left(\frac{9}{37}\,|\alpha_{q}|^{2}+|\beta_{q}|^{2}+8\,|\gamma_{q}|^{2}\right)^{\frac{1}{2}}\,.$
(3.18)

The other two eigenvalues, $m_{1,2}$, as can be shown, are by the factors
$|\varepsilon(\tau_{v})|^{2}$ and $|\varepsilon(\tau_{v})|^{3}$ smaller [3],
which, in principle, may allow to explain the hierarchy of quark (or charged
lepton) masses without fine-tuning of the constants $\alpha_{q}$,
$\alpha^{\prime}_{q}$, $\beta_{q}$ and $\gamma_{q}$ present in $M_{q}$.

We get a completely different result if we use the expressions for the
normalised modular forms
$Y_{\mathbf{\hat{3}^{\prime}}}^{(3)}(\tau_{v})/|Y_{\mathbf{\hat{3}^{\prime}}}^{(3)}(\tau_{v})|$,
$Y_{\mathbf{\hat{3}^{\prime}}}^{(5)}(\tau_{v})/|Y_{\mathbf{\hat{3}^{\prime}}}^{(5)}(\tau_{v})|$,
$Y_{\mathbf{\hat{3}^{\prime}},1}^{(7)}(\tau_{v})/|Y_{\mathbf{\hat{3}^{\prime}},1}^{(7)}(\tau_{v})|$
and
$Y_{\mathbf{\hat{3}^{\prime}},2}^{(7)}(\tau_{v})/|Y_{\mathbf{\hat{3}^{\prime}},2}^{(7)}(\tau_{v})|$
in $M_{q}$. In the limit $|\varepsilon(\tau_{v})|\rightarrow 0$, again only
the 2nd components of
$Y_{\mathbf{\hat{3}^{\prime}}}^{(3)}(\tau_{v})/|Y_{\mathbf{\hat{3}^{\prime}}}^{(3)}(\tau_{v})|$,
$Y_{\mathbf{\hat{3}^{\prime}}}^{(5)}(\tau_{v})/|Y_{\mathbf{\hat{3}^{\prime}}}^{(5)}(\tau_{v})|$,
$Y_{\mathbf{\hat{3}^{\prime}},1}^{(7)}(\tau_{v})/|Y_{\mathbf{\hat{3}^{\prime}},1}^{(7)}(\tau_{v})|$
are non-zero. Now they are of order 1, 1 and (-1), respectively. The third
component of
$Y_{\mathbf{\hat{3}^{\prime}},2}^{(7)}(\tau_{v})/|Y_{\mathbf{\hat{3}^{\prime}},2}^{(7)}(\tau_{v})|$
is also non-zero in this case and is given by $(-\,{\rm exp}(i\pi{\rm
Re}\tau_{v}))$. The matrix of interest $M^{\dagger}_{q}M_{q}$ can be cast in
the form:

$\displaystyle M^{\dagger}_{q}M_{q}=v_{q}^{2}\,\begin{pmatrix}0&0&0\\\
0&|\alpha^{\prime}_{q}|^{2}&|\alpha_{q}\alpha^{\prime}_{q}|\,e^{-\,i(\kappa_{v}+\kappa_{g})}\\\
0&|\alpha_{q}\alpha^{\prime}_{q}|\,e^{i(\kappa_{v}+\kappa_{g})}&|\alpha_{q}|^{2}+|\beta_{q}|^{2}+|\gamma_{q}|^{2}\end{pmatrix}\,,$
(3.19)

where $\kappa_{v}=\pi{\rm Re}\tau_{v}$ and $\kappa_{g}=-\,{\rm
arg}(g_{q})=-\,{\rm arg}(\alpha_{q}/\alpha^{\prime}_{q})$.
$M^{\dagger}_{q}M_{q}$ in Eq. (3.19) (and thus $M_{q}$) has two “large” mass
eigenvalues $m^{2}_{2,3}$ ($m_{2,3}$). The hierarchy $m_{2}/m_{3}\ll 1$ can be
obtained in the considered case only by fine-tuning the values of the
constants $|\alpha_{q}|$, $|\alpha^{\prime}_{q}|$ and
$|\beta_{q}|^{2}+|\gamma_{q}|^{2}$. One possibility is, e.g.,
$|\alpha_{q}|\cong|\alpha^{\prime}_{q}|\ll(|\beta_{q}|^{2}+|\gamma_{q}|^{2})^{\frac{1}{2}}$
111111 In order for the (2-3) mixing to be sufficiently small in the quark
case, we should have also
$|\alpha_{q}\alpha^{\prime}_{q}|\ll(|\alpha_{q}|^{2}+|\beta_{q}|^{2}+|\gamma_{q}|^{2}-|\alpha^{\prime}_{q}|^{2})$.
. In this case to a good approximation
$m_{2}/m_{3}\cong(|\alpha_{q}|^{2}+|\alpha^{\prime}_{q}|^{2})^{\frac{1}{2}}/(\sqrt{2}(|\beta_{q}|^{2}+|\gamma_{q}|^{2})^{\frac{1}{2}})$.

###  3.2 “Global” (or “Integral”) Normalisation

In the proposed “local” solution to the modular form normalisation problem
within the non-fine-tuned approach to the fermon mass hierarchies in the
modular flavour models, the normalisation factors $N^{(K)}_{Y}$ depend on the
VEV of the modulus $\tau$, $N^{(K)}_{Y}=N^{(K)}_{Y}(\tau_{v})$. If the model
is constructed in the vicinity of a given fixed point, $N^{(K)}_{Y}(\tau_{v})$
will depend on the chosen fixed point. That will allow meaningful comparison
of different models constructed in the vicinity of the same fixed point, but
somewhat ambiguous the comparison of models constructed at different fixed
points.

On the basis of the expression for $N^{(K)}_{Y}$, Eq. (3.1), one can form a
modular invariant normalisation of the modular forms which does not depend on
the VEV of the modulus $\tau$ and thus on the fixed point in the vicinity of
which the model is constructed. This can be done by integrating
$(N^{(K)}_{Y})^{2}$ over the fundamental domain $D$ of the modular group using
the modular invariant hyperbolic measure (or volume form)
$d\mu(\tau)=dx\,dy/y^{2}$, where $x\equiv{\rm Re}\tau$ and $y\equiv{\rm
Im}\tau$:

$(\overline{{\rm N}}^{(K)}_{\rm Y})^{2}=\int_{D}\int(N^{(K)}_{\rm
Y}(\tau,\tau^{*}))^{2}\,\frac{dxdy}{y^{2}}\equiv\int_{D}\int\sum_{i}|Y^{(K)}_{{\bf
r}i}(\tau)|^{2}(2\,y)^{K}\frac{dx\,dy}{y^{2}}\,.$ (3.20)

The integration should be understood as integration over all possible VEVs of
$\tau$ in the fundamental domain of the modular group 121212A normalisation
with integration over the fundamental domain ${\rm D}$ with the modular
invariant hyperbolic measure $d\mu(\tau)=dx\,dy/y^{2}$ and integrand
$N^{(K)}_{Y}$ given in Eq. (3.1) was suggested to the author by P. Novichkov
after the author proposed in July of 2022 and discussed with P. Novichkov and
J. Penedo the normalisation defined in Eq. (3.1). . The normalisation squared
$(\overline{{\rm N}}^{(K)}_{\rm Y})^{2}$ is a particular case of what is known
in mathematical literature as “Petersson inner product of two modular forms”
[22] 131313The author learned this fact from Dr. A. M. Kidambi during the 2023
spring visit of Kavli IPMU, University of Tokyo. . The integral in Eq. (3.20)
is independent of the choice of the fundamental domain since both
$\sum_{i}|Y_{{\bf r}i}(\tau)|^{2}(2\,y)^{K}$ and the hyperbolic measure
$d\mu=dx\,dy/(y^{2})$ are modular invariant.

We show next that the hyperbolic measure $d\mu=dxdy/y^{2}$ is indeed modular
invariant. Consider two moduli $z=u+i\,v$ and $\tau=x+iy$ related by a modular
transformation:

$z=\frac{a\tau+b}{c\tau+d}\,,~{}~{}{\rm with~{}integer~{}a,b,c,d}\,~{}{\rm
and~{}ad-bc=1}\,.$ (3.21)

In the context of the problem we are considering Eq. (3.21) describes a change
of variables $(u,v)$ to $(x,y)$:

$u=\frac{ac(x^{2}+y^{2})+x(ad+bc)+bd}{|c\tau+d|^{2}}\,,~{}~{}v=\frac{y}{|c\tau+d|^{2}}\,.$
(3.22)

For the measure of interest we get:

$\mu=\frac{du\,dv}{v^{2}}={\rm
J(x,y)}\,|c\tau+d|^{4}\,\frac{dx\,dy}{y^{2}}\,,$ (3.23)

where ${\rm J(x,y)}$ is the Jacobian of the change of variables of
integration:

${\rm
J(x,y)}=\frac{\partial{u}}{\partial{x}}\,\frac{\partial{v}}{\partial{y}}-\frac{\partial{u}}{\partial{y}}\,\frac{\partial{v}}{\partial{x}}\,.$
(3.24)

Using the expressions for $u$ and $v$ in Eq. (3.22) we find:

$\frac{\partial{u}}{\partial{x}}=\frac{\partial{v}}{\partial{y}}=\frac{(cx+d)^{2}-c^{2}\,y^{2}}{|c\tau+d|^{4}}\,,~{}~{}\frac{\partial{u}}{\partial{y}}=-\,\frac{\partial{v}}{\partial{x}}=\frac{2c\,(cx+d)\,y}{|c\tau+d|^{4}}\,.$
(3.25)

Thus,

${\rm J(x,y)}=|c\tau+d|^{-4}\,.$ (3.26)

Inserting the expression for ${\rm J(x,y)}$ in Eq. (3.23) we obtain:

$d\mu=\frac{du\,dv}{v^{2}}=\frac{dx\,dy}{y^{2}}\,.$ (3.27)

####  3.2.1 Cusp Modular Forms

The integral in Eq. (3.20) is convergent and the normalisation $\overline{{\rm
N}}^{(K)}_{\rm Y}$ is well-defined and positive for “cusp” modular forms,
i.e., for the modular forms such that $|Y^{(K)}(\tau)|\sim O(e^{-2\pi\,y})$
when $y\equiv{\rm Im\tau}\rightarrow\infty$ 141414The $q$-expansions of the
holomorphic cusp modular forms do not contain constant $q$-independent terms.
. Among the considered modular forms these are
$Y_{\mathbf{\hat{1}^{\prime}}}^{(3)}(\tau)$, $Y_{\mathbf{3}}^{(4)}(\tau)$,
$Y_{\mathbf{\hat{3}},2}^{(5)}(\tau)$ and
$Y_{\mathbf{\hat{3}^{\prime}},2}^{(7)}(\tau)$. We will discuss later a
modification of the “integral” normalisation $\overline{N}^{(K)}_{\rm Y}$ for
the modular forms which do not satisfy the “cusp” condition. These are the
majority of modular forms in Eqs. (3.9) - (3.13).

As an example, consider the normalisation of the cusp modular form
$Y_{\mathbf{\hat{1}^{\prime}}}^{(3)}(\tau)=\sqrt{3}(\varepsilon\,\theta^{5}-\varepsilon^{5}\,\theta)$.
We will use the $q-$expansions of $\varepsilon(\tau)$ and $\theta(\tau)$ given
in Eq. (3.6). In these expansions $|q_{4}|={\rm exp}(-\pi y/2)\leq 0.257$
since $y\geq\sqrt{3}/2$. Correspondingly, to a very good approximation
$\theta\cong 1+2q^{4}_{4}$ and $\varepsilon\cong 2q_{4}$. We can safely
neglect the terms $\propto|\varepsilon|^{10}$ and $\propto{\rm
Re(\varepsilon\,2q^{4}_{4}\,(\varepsilon^{5})^{*}}$ in the expression for
$|Y_{\mathbf{\hat{1}^{\prime}}}^{(3)}(\tau)|^{2}$ since their contributions to
$\overline{{\rm N}}^{(3)}_{\rm\hat{1}^{\prime}}$ is negligible. For the
normalisation squared of the considered modular form we get up to corrections
$O(10^{-3})$

$(\overline{N}^{(3)}_{\rm\hat{1}^{\prime}})^{2}\cong\int_{-\,\frac{1}{2}}^{\frac{1}{2}}dx\int_{\sqrt{1-x^{2}}}^{\infty}dy\,\,8\,y\left[12\,e^{-\,\pi
y}-144\,e^{-\,3\pi y}\cos(2\pi x)\right]\,.$ (3.28)

The lower limit of the integration over $y$, $\sqrt{1-x^{2}}$, corresponds to
the arc bounding the fundamental domain from below, on which $x^{2}+y^{2}=1$.
Performing the integration we find:

$\overline{{\rm N}}^{(3)}_{\rm\hat{1}^{\prime}}\cong 1.38\,.$ (3.29)

It follows form this result that the introduced “global” normalisation of
$Y_{\mathbf{\hat{1}^{\prime}}}^{(3)}(\tau)$ is not precisely 1, but is close
to 1 and thus cannot alter significantly the value of
$Y_{\mathbf{\hat{1}^{\prime}}}^{(3)}(\tau)$ at $\tau=\tau_{v}$. As can be
shown, the same conclusion holds for the “global” normalisations of the other
cusp modular forms $Y_{\mathbf{3}}^{(4)}(\tau)$,
$Y_{\mathbf{\hat{3}},2}^{(5)}(\tau)$ and
$Y_{\mathbf{\hat{3}^{\prime}},2}^{(7)}(\tau)$ considered by us.

####  3.2.2 Non-Cusp Modular Forms

A method of normalisation of the holomorphic modular forms such that
$|Y^{(K)}_{{\bf r}}(\tau)|$ does not decrease as $O(e^{-2\pi\,y})$ when
$y\equiv{\rm Im\tau}\rightarrow\infty$ was proposed by D. Zagier [23]. It
represents the following modification of the “global” normalisation defined in
Eq. (3.20):

$\displaystyle(\overline{{\rm N}}^{(K)}_{\rm YR})^{2}$
$\displaystyle=\lim_{T\to\infty}\left(\int_{D_{T}}\int(N^{(K)}_{\rm
Y}(\tau,\tau^{*}))^{2}\,\frac{dxdy}{y^{2}}-|a_{0}|^{2}{\rm\frac{2^{K}}{K-1}\,T^{K-1}}\right)$
(3.30)
$\displaystyle\equiv\lim_{T\to\infty}\left(\int_{D_{T}}\int\sum_{i}|Y^{(K)}_{{\bf
r}i}(\tau)|^{2}(2\,y)^{\rm
K}\frac{dx\,dy}{y^{2}}-|a_{0}|^{2}{\rm\frac{2^{K}}{K-1}\,T^{K-1}}\right)\,.$

Here $T$ is a real constant, the region $D_{T}$ is the lower part of the
fundamental domain limited from above by the line $y=T$, $T>1$, and
$|a_{0}|^{2}$ is the constant $q$-independent term in the $q$-expansion of
$\sum_{i}|Y^{(K)}_{{\bf r}i}(\tau)|^{2}$. The mathematical justification of
the normalisation (3.30) and the theorem on which it is based can be found in
Ref. [23] and we are not going to reproduce them here. The integration in
(3.30) is in the intervals $-\,0.5\leq x\leq 0.5$ and $\sqrt{1-x^{2}}\leq
y\leq T$. The limit $T\rightarrow i\infty$ is taken after the integration over
the variables $x$ and $y$ is performed 151515The procedure of normalisation of
the “non-cusp” modular forms proposed in [23] formally resembles the procedure
of renormalisation of the amplitudes of processes in quantum field theory by
which the infinite terms are removed. Actually, considering the inner product
of two modular forms $f(z)$ and $g(z)$ of weight $k$ neither of which is a
cusp form, the author of [23] writes that the modular invariant function
$F(z)=y^{k}f(z)\overline{g}(z)$ is “renormalisable” and that one can define
the inner product of $f(z)$ and $g(z)$ as its “renormalisable integral” over
the fundamental domain $D$. As should be clear, we are considering the case of
$g(z)\equiv f(z)$. .

In the case of the cusp modular forms $a_{0}=0$, and since $D_{T}\rightarrow
D$ when $T\rightarrow i\infty$, the normalisation defined in Eq. (3.30)
reduces to that defined in Eq. (3.20).

We note also that the method of normalisation of “non-cusp” modular forms was
derived in [23] under the assumption that the modular form weights satisfy
$K\geq 2$. We assume that in the case of modular forms with weight $K=1$ the
term $|a_{0}|^{2}{\rm 2^{K}T^{K-1}/(K-1)}$ in Eq. (3.30) should be replaced by
$|a_{0}|^{2}\,{\rm 2^{K}\,lnT}$.

Finally, it follows from Eq. (3.30) that $(\overline{{\rm N}}^{(K)}_{\rm
YR})^{2}$ is a real number but is not necessarily positive-definite [23]. In
the cases when $(\overline{{\rm N}}^{(K)}_{\rm YR})^{2}<0$ we propose to use
the absolute value $|\overline{{\rm N}}^{(K)}_{\rm YR}|$ as a “global”
normalisation of $Y^{(K)}_{\bf r}(\tau)$.

As an example of the application of the method defined in Eq. (3.30) we show
next the normalisation of the level 4 weight 3 “non-cusp” modular form
$Y^{(3)}_{\hat{3}^{\prime}}(\tau)$ whose expression in terms of the functions
$\theta(\tau)$ and $\varepsilon(\tau)$ is given in Eq. (3.10). We recall that
to a very good approximation $\theta\cong 1+2q^{4}_{4}$ and $\varepsilon\cong
2q_{4}$ as $|q_{4}|={\rm exp}(-\pi y/2)\leq 0.257$ since $y\geq\sqrt{3}/2$. Up
to corrections of $O(10^{-2})$, we find:

$\sum_{i}|Y^{(3)}_{\hat{3}^{\prime}i}(\tau)|^{2}\cong\frac{1}{4}\left[1+2048\,e^{-3\pi
y}+\left(144+120\,\cos(2x\pi)\right)\,e^{-2\pi y}\right]\,.$ (3.31)

Inserting the derived expression for
$\sum_{i}|Y^{(3)}_{\hat{3}^{\prime}i}(\tau)|^{2}$ in Eq. (3.30) and performing
the integration we find for the “global” normalisation factor for
$Y^{(3)}_{\hat{3}^{\prime}}(\tau)$ a value close to 1:

$|\overline{{\rm N}}^{(3)}_{\rm\hat{3}^{\prime}R}|\cong 0.9\,.$ (3.32)

As in the case of the example of normalisation of the cusp form
$Y_{\mathbf{\hat{1}^{\prime}}}^{(3)}(\tau)$ we have considered, the global
normalisation factor of the “non-cusp” form $Y^{(3)}_{\hat{3}^{\prime}}(\tau)$
is close to 1 and therefore, when applied, it cannot change significantly the
magnitude of the components of $Y_{\mathbf{\hat{1}^{\prime}}}^{(3)}(\tau)$ at
$\tau=\tau_{v}$.

The “global” normalisations of other modular forms, which are holomorphic
functions of the modulus $\tau$ and do not satisfy the cusp condition, can be
calculated in a similar way.

##  4 Summary

We have discussed the normalisation of the modular forms which play a crucial
role in the modular invariance approach to the flavour problem. In this
approach the elements of the fermion Yukawa couplings and mass matrices in the
Lagrangian of the theory are modular forms of a certain level $N$ and weights
$K$. As like the fermion (matter) fields, the modular forms of level $N$ have
specific transformation properties under the action of the inhomogeneous
(homogeneous) modular symmetry group $\Gamma\equiv PSL(2,\mathbb{Z})$
($\Gamma^{\prime}\equiv SL(2,\mathbb{Z})$). They also furnish irreducible
representations of the the inhomogeneous (homogeneous) finite modular group
$\Gamma^{(\prime)}_{\rm N}$, which describes the flavour symmetry. The modular
forms are holomorphic functions of a single complex scalar field – the modulus
$\tau$. When $\tau$ develops a VEV, $\tau_{v}$, the modular forms in the
fermion mass matrices get fixed and certain flavour structure arises.

It is a well known fact that the modular forms furnishing irreducible
representations of the finite modular groups are determined up to a constant.
Usually these normalisation constants are absorbed in the constant parameters
which multiply each modular form present in the fermion mass matrices. Since
the normalisation of the modular forms is arbitrary, this makes the specific
values of the constant parameters, obtained in a given model by statistical
analysis of the description of the relevant experimental data by the model, of
not much physical meaning. Moreover, it also makes the comparison of models
which use different normalisations of the modular forms ambiguous. The problem
of normalisation of the modular forms can be particularly acute in modular
flavour models in which the charged lepton and quark mass hierarchies are
obtained without fine-tuning the constant parameters present in the respective
fermion mass matrices, where the constants have to be of the same order in
magnitude.

We have discussed two possible modular invariant normalisations of the modular
forms $Y^{(K)}$ of interest – a “local” one at $\tau=\tau_{v}$ (Eq. (3.1)),
$N^{(K)}_{Y}(\tau_{v})$, and a “global” one, $\overline{{\rm N}}^{(K)}_{\rm
Y}$, involving an integration of $(N^{(K)}_{Y}(\tau))^{2}$ over the
fundamental domain $D$ of the modular group over the modular invariant
hyperbolic measure (or volume form) $d\mu(\tau)=dx\,dy/y^{2}$ (Eq. (3.20)).
Both normalisations do not depend on the basis employed for the generators of
the finite modular group $\Gamma^{(\prime)}_{\rm N}$ describing the flavour
symmetry. We have considered the simple case when the Higgs fields present in
the Yukawa couplings are singlets with zero weights of the considered finite
modular group describing the flavour symmetry. However, our results can be
easily generalised to more sophisticated cases including also flavon fields.
In the considered case of singlet Higgs fields, the local normalisation of a
given modular form $Y^{(K)}_{\bf r}$, present in a modular invariant fermion
mass term (2.1), r being the representation of the respective finite modular
(flavour) group, effectively coincides with the Euclidean norm of the form at
$\tau=\tau_{v}$, $(\sum_{i}|Y^{(K)}_{{\bf r}i}(\tau_{v})|^{2})^{\frac{1}{2}}$,
after one performs the necessary renormalisation of the fermion fields in
order to get canonical fermion kinetic terms in the Kahler potential of the
theory. In this case the fermion normalisation factors do not appear in the
fermion mass matrices.

For the cusp modular forms, the global normalisation $\overline{{\rm
N}}^{(K)}_{\rm Y}$ (as we have learned in 2023) is based on the Petersson
inner product of two modular forms [22], while in the case of “non-cusp”
(holomorphic) modular forms the global normalisation $\overline{{\rm
N}}^{(K)}_{\rm YR}$ (Eq. (3.30)) is based on the Zagier inner product of two
modular forms [23], which represents a “renormalised” modification of the
Petersson inner product. The square of $\overline{{\rm N}}^{(K)}_{\rm YR}$
calculated following the Zagier recipe is real, but is not necessarily
positive-definite. In the cases of $(\overline{{\rm N}}^{(K)}_{\rm
YR})^{2}<0$, we have proposed to use $|\overline{{\rm N}}^{(K)}_{\rm YR}|$ as
a global normalisation of the non-cusp holomorphic modular forms.

We find that the local normalisation of the cusp modular forms at
$\tau=\tau_{v}$, present in the mass term (2.1), which coincides with the
Euclidean norms of the forms at $\tau=\tau_{v}$, can change drastically their
magnitude if the Euclidean norm is much smaller than 1. As we have shown, this
can have important implications for the description of observables such as the
fermion masses, if cusp forms are present in the fermion (quark, charged
lepton) mass matrices in the modular flavour models. In the case of the non-
cusp holomorphic modular forms, their local normalisations are $\sim 1$ and
essentially have no effect on the magnitude of the forms at $\tau=\tau_{v}$.

In what concerns the global normalisation of cusp and non-cusp holomorphic
modular forms, we have found that their respective normalisation factors are
close to 1 and can change somewhat, but not in a significant way, the
magnitude of the forms. In contrast to the case of the local normalisation,
the fermion normalisation factors have to be taken into account in the fermion
mass matrices if one uses the global normalisation of the modular forms.

Our results indicate that the modular invariant global normalisation of the
holomorphic modular forms present in the fermion Yuakawa couplings and mass
matrices in modular flavour models seems better suited for the formalism of
the modular invariance approach to the flavour problem.

## Acknowledgements

The author would like to thank P. Novichkov, J. Penedo, A. Titov and A. M.
Kidambi for very useful discussions, and J. Penedo for checking the results
for the global normalisations of the modular forms
$Y_{\mathbf{\hat{1}^{\prime}}}^{(3)}(\tau)$ and
$Y^{(3)}_{\hat{3}^{\prime}}(\tau)$ and correcting my result for that of
$Y_{\mathbf{\hat{1}^{\prime}}}^{(3)}(\tau)$. This work was supported in part
by the European Union’s Horizon 2020 research and innovation programme under
the Marie Sklodowska-Curie grant agreement No. 860881-HIDDeN, by the Italian
INFN program on Theoretical Astroparticle Physics and by the World Premier
International Research Center Initiative (WPI Initiative, MEXT), Japan.

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