On the Normalisation of the Modular Forms in Modular Invariant Theories of Flavour

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.


Introduction
The main new feature of the modular invariance approach to the fundamental flavour problem in particle physics proposed in [1] 1 , 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 τ -and, as like the fermion (matter) fields, have specific transformation properties under the action of the inhomogeneous (homogeneous) modular group Γ ≡ P SL(2, Z) (Γ ′ ≡ SL(2, 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, Γ (′) N , N = 1, 2, 3, ..., which plays the role of a flavour symmetry.For N ≤ 5, the finite modular groups Γ N are isomorphic to the permutation groups S 3 , A 4 , S 4 and A 5 (see, e.g., [4]), while the groups Γ ′ N are isomorphic to the double covers of the indicated permutation groups, S ′ 3 ≡ S 3 , A ′ 4 ≡ T ′ , S ′ 4 and A ′ 5 .In the modular flavour models, the VEV of the modulus τ , τ v , can be the only source of flavour symmetry breaking, such that no flavons are needed.The VEV of τ 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 τ 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 2 .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 Γ 3 ≃ A 4 [1], Γ 2 ≃ S 3 [9], Γ 4 ≃ S 4 [10] and Γ 3 ≃ A 4 again [11] 3 .After these first studies, the interest in the approach grew significantly and various aspects of this approach were and continue to be extensively studied 4 .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 Γ (′) 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 modularinvariant solutions to this problem.

The Problem
Consider a modular-invariant bilinear where v h is a constant and the matter superfields ψ and ψ c transform under the action of the inhomogeneous P SL(2, Z) ≡ Γ (homogeneous SL(2, Z) ≡ Γ ′ ) modular group as Here γ is an element of Γ (′) , (− k (c) ) is the weight of the field ψ (c) and ρ (c) is an irreducible representation of a finite inhomogeneous (homogeneous) modular group of level N, Γ N .We recall that for N ≤ 5, the groups Γ 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 Γ ′ 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,[9][10][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 , where α 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, ψ furnishes a 3 representation of S ′ 4 and ψ c represent three singlet representations of S ′ 4 having different weights, M is a 3 × 3 matrix involving at least three different modular forms, each transforming as triplet representation of S ′ 4 , and thus involves at least three different constant parameters.
To be more specific, consider the case of ψ being the three quark doublet superfields, , and ψ c being the three singlet up-type or down-type quark superfields, ψ c = q c = (q c 1 , q c 2 , q c 3 ).Let us assume further that Q furnishes the triplet representation of S ′ 4 , 3, and carry weight k Q , q c 1,2,3 transform as the "hatted" singlet 1 of S ′ 4 5 but have different weights k 1 = κ 2 = k 3 , and the Higgs fields H q , q = u, d, present in the theory are singlets with respect to S ′ 4 , but may carry non-zero weights k Hq .The modular invariance implies that the quark mass matrix M q should be formed by modular forms which furnish the 3′ representation of S ′ 4 .These are: the weigh 3 and 5 modular forms Y 3 3′ (τ ) and Y 5 3′ (τ ), two weight 7 and two weigh 9 modular forms, Y 3′ ,2 (τ ), the three weight 11 modular forms Y 11 3′ ,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 6 .Choosing these to be the weight 3, 5 and 9 modular forms leads to 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: where v q is the Higgs doublet VEV, Ỹ (7) 3′ ;i = Y 3′ ,2;i + g q Y (7) 3′ ,1;i , g q = α q /α ′ q , α q , α ′ q , β q and γ q are the four constants, and Y 3′ ;i and Y (3) 3′ ;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 τ T = i∞ to the following hierarchy between the three quark masses 1 : ǫ 2 : ǫ 3 , where ǫ ∼ = exp(−πImτ /2) ≪ 1 is a measure of the "deviation" of the VEV τ v of τ from i∞ [3].In the "bottom-up" modular flavour models the value of Imτ v is obtained from fits of the relevant data and typically one finds Imτ v ∼ 2.5 − 3.0 (see, e.g., [3,16,17]) 7 .
If one uses the "minimal" Kähler potential, Λ 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 τ gets a VEV τ v , the fermion fields have to be renormalised as follows in order to have a canonical kinetic term: Correspondingly, the bilinear term in eq. ( 2.1) changes as: This implies that in the considered example of a quark mass matrix in Eq. (2.3), each constant gets an additional factor: 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 [18,19]), 2Imτ v can have a relatively large value.In the cases of models constructed in the "vicinity" of the fixed point τ T = i∞ (see [8,20]), for example, 2Imτ v ∼ 5, as we have already indicated (see, e.g., [3,16,17]).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 [16,17], 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-finetuned models" of flavour which are studied in the vicinity of one of the three known fixed points of the modular group [8,20], namely, τ C = i, τ L = ω ≡ exp(i2π/3) (the left cusp) and τ T = i∞, vanish at the corresponding fixed point.In the case of models based on S ′ 4 and τ = i∞, for example, we have [3,15] = 0, Y 3,2 = 0, Y = 0, Y 3 ′ ,2 = 0, Y 1′ = 0, Y = 0, Y 3,2 = 0, Y 3′ ,2 = 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, | 1′ | ∼ 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 ∼ 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.Y , where

Modular Invariant Normalisations
where are the components of Y (K) .Assuming that Y (K) furnishes a unitary representation r of the finite modular group of level N, , this normalisation is modular invariant.Indeed, under the modular transformation, In a specific flavour model, the modulus τ in the expression of N (K) Y should be replaced by its VEV τ v in the model, so the normalisation should read )) as a normalisation, has an additional important consequence.Indeed, employing the described normalisation of Y (K) r (τ ) and taking into account the normalisation factor coming from the Kähler potential, we get for the bilinear of interest: ) where R am is the ratio of the automorphy factors due to the re-normalisation of ψ and ψ c and the normalisation (3.1) of the modular form: We see that the automorphy factors coming from the re-normalisation of ψ and ψ c cancel in the ratio R am as a consequence of K = k ψ + k c ψ + k H , which follows from the modular invariance.This procedure (and cancellation) holds only after the modulus τ takes a VEV.If k H = 0, as is assumed typically (but not universally, see, e.g., [16,18]) in bottom-up modular flavour models, we have R am = 1.We will consider this case in what follows.
Under the assumed conditions, normalising the modular forms to their Euclidean norms at τ = τ 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.
We further give the exact relations between θ(τ ) and ε(τ ) at the symmetric points θ(τ C ) and τ L : The level N = 4 lowest weight 1 modular form furnishes a 3 representation of S ′ 4 [15].In the group representation basis used in [15] this modular form is given by: Further tensor products with Y (1) 3 produce new modular multiplets of even and odd weights 9 .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 tripletsexclusive to S ′ 4 : The set of four weight k = 4 modular forms includes: [10,15]: The weight k = 5 forms of interest are: (3.12) In the case of weight k = 7 we have: 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.
If we normalise the singlet modular form Y 1′ (τ ) at τ = τ v , we get: We see that the used normalisation changes drastically the magnitude of this singlet modular form from being of order |ε| ≪ 1 to being of order 1.
Consider next the effect of the normalisation on the triplet modular forms Y (4) 3′ ,2 (τ v )| is also non-zero in this case and is given by (− exp(iπReτ v )).The matrix of interest M † q M q can be cast in the form: where κ v = πReτ v and κ g = − arg(g q ) = − arg(α q /α ′ q ).M † 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 ≪ 1 can be obtained in the considered case only by fine-tuning the values of the constants |α q |, |α ′ q | and |β q | 2 + |γ q | 2 .One possibility is, e.g., . In this case to a good approximation 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 depend on the VEV of the modulus τ , Y (τ 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 τ 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) dµ(τ ) = dx dy/y 2 , where x ≡ Reτ and y ≡ Imτ : The integration should be understood as integration over all possible VEVs of τ in the fundamental domain of the modular group 11 .The normalisation squared Y ) 2 is a particular case of what is known in mathematical literature as "Petersson inner product of two modular forms" [22] 12 .The integral in Eq. (3.20) is independent of the choice of the fundamental domain since both i |Y ri (τ )| 2 (2 y) K and the hyperbolic measure dµ = dx dy/(y 2 ) are modular invariant.
We show next that the hyperbolic measure dµ = dxdy/y 2 is indeed modular invariant.Consider two moduli z = u + i v and τ = x + iy related by a modular transformation: For the measure of interest we get: (N (3.30) 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 i |Y 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 ≤ x ≤ 0.5 and √ 1 − x 2 ≤ y ≤ T .The limit T → i∞ is taken after the integration over the variables x and y is performed 14 .
In the case of the cusp modular forms a 0 = 0, and since D T → D when T → i∞, 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 ≥ 2. We assume that in the case of modular forms with weight K = 1 the term Finally, it follows from Eq. (3.30) that (N YR ) 2 is a real number but is not necessarily positive-definite [23].In the cases when (N 3′ i (τ )| 2 in Eq. (3.30) and performing the integration we find for the "global" normalisation factor for Y (3) 3′ (τ ) a value close 14 The 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)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) ≡ f (z).
1′ (τ ) we have considered, the global normalisation factor of the "non-cusp" form Y  The "global" normalisations of other modular forms, which are holomorphic functions of the modulus τ and do not satisfy the cusp condition, can be calculated in a similar way.

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 Γ ≡ P SL(2, Z) (Γ ′ ≡ SL(2, Z)).They also furnish irreducible representations of the the inhomogeneous (homogeneous) finite modular group Γ (′) N , which describes the flavour symmetry.The modular forms are holomorphic functions of a single complex scalar field -the modulus τ .When τ develops a VEV, τ 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 τ = τ v (Eq.(3.1)), 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) 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 τ = τ v , ( i |Y 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 N (K) 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 N (K) 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 N (K) YR calculated following the Zagier recipe is real, but is not necessarily positive-definite.In the cases of (N We find that the local normalisation of the cusp modular forms at τ = τ v , present in the mass term (2.1), which coincides with the Euclidean norms of the forms at τ = τ 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 ∼ 1 and essentially have no effect on the magnitude of the forms at τ = τ 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.

Y
(τ v ) is self explanatory.Since in the considered case the normalisation of the lowest weight modular form |Y (1) 3 (τ v )| = 1 + O(10 −4 ), the effect of propagation of this normalisation to higher weight modular form is practically negligible.The normalised Y (1) 3 (τ v ) has the form: integer a, b, c, d 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): )where J(x, y) is the Jacobian of the change of variables of integration: for u and v in Eq. (3.22) we find: , y) = |cτ + d| −4 .(3.26)Inserting the expression for J(x, y) in Eq. (3.23) we obtain:

( 3 )
3′ (τ ) is close to 1 and therefore, when applied, it cannot change significantly the magnitude of the components of Y

Y
(τ )) 2 over the fundamental domain D of the modular group over the modular invariant hyperbolic measure (or volume form) dµ(τ ) = dx dy/y 2 (Eq.(3.20)).Both normalisations do not depend on the basis employed for the generators of the finite modular group Γ (′) N

YR ) 2
< 0, we have proposed to use |N (K) YR | as a global normalisation of the non-cusp holomorphic modular forms.
of level N and weight K furnishing a representation ρ Y of Γ