Lepton Masses and Mixing from Modular $S_4$ Symmetry

We study models of lepton masses and mixing based on broken modular invariance. We consider invariance under the finite modular group $\Gamma_4 \simeq S_4$ and focus on the minimal scenario where the expectation value of the modulus is the only source of symmetry breaking, such that no flavons need to be introduced. After constructing a basis for the lowest weight modular forms, we build two minimal models, one of which successfully accommodates charged lepton masses and neutrino oscillation data, while predicting the values of the Dirac and Majorana CPV phases.


Introduction
It is a challenge to understand the origin of the flavour structure of quarks and leptons. Data obtained in neutrino oscillation experiments (see, e.g., [1]) has revealed a pattern of large mixing in the lepton sector, which is being probed with increasing precision. Additionally, a preference for both a neutrino mass spectrum with normal ordering (NO) and for a Dirac CP violation (CPV) phase close to maximal have been found in recent global fits (see, e.g., [2]).
A popular approach to explaining the observed mixing patterns consists in assuming the presence of a spontaneously broken discrete flavour symmetry. Typically, however, such breaking requires the introduction of a plethora of scalar fields (flavons) with a set of particularly aligned vacuum expectation values (VEVs). Arranging for such an alignment usually calls for the construction of rather "baroque" scalar potentials.
An interesting framework for the construction of flavour models has recently been put forward in Ref. [3] and further explored in Ref. [4]. In this setup, invariance under finite subgroups of the modular group shapes the relevant mass matrices. These models generalise existing approaches to the flavour puzzle, which can be seen as the particular cases of setting modular weigths to zero. In the most economical versions of these models, the VEV of the modulus τ is the only source of symmetry breaking, bypassing the need for flavon fields. Leading-order predictions may be derived by treating τ as a spurion.
In the present paper we investigate the consequences of enforcing modular invariance in the lepton sector. We focus on the action of the finite modular group Γ 4 , which is isomorphic to the group of permutations of four objects S 4 . After reviewing the necessary formalism, we explicitly construct the generators of modular forms of level N = 4. We then investigate two minimal models where neutrino masses arise from the Weinberg operator and where no flavons are introduced.

The Setup
Following Ref. [3], we consider the infinite groups Γ(N ), where N is a positive integer. The group Γ(1) SL(2, Z) is the modular group and Γ(N > 1) are normal subgroups of Γ (1). Taking the quotient of Γ(1) and Γ(2) by {1, −1} we obtain the groups of linear fractional transformations, Γ(N ) ≡ Γ(N )/{1, −1} for N = 1, 2, and Γ(N > 2) ≡ Γ(N ). Elements of Γ(N ) act on a complex variable τ as: and it can be shown that the upper half-plane {τ ∈ C | Im(τ ) > 0} is mapped to itself under this action. The complex variable τ is henceforth restricted to have positive imaginary part. We are interested in studying physical actions which are invariant under transformations of the finite modular groups Γ N . These discrete groups are obtained from the quotient of two of the aforementioned infinite groups, namely Γ N ≡ Γ(1)/Γ(N ). The group Γ(1) is generated by two elements S and T acting on τ as and satisfying The presentation of the groups Γ N can be obtained from that of Γ(1) by extending it with the condition T N = 1, cf. Eq. (1). Thus, the generators S and T of Γ N obey We consider modular-invariant N = 1 global supersymmetric (SUSY) actions [5,6], where χ i denotes the set of matter chiral superfields of the theory 1 . The physical action S is required to be invariant under the action of Γ N . Under an element of this group, one has the following transformations of τ and of the fields χ i : where ρ i are unitary representation matrices and the k i are integers. The fields χ i are said to carry weigth −k i . We require that the superpotential W remains invariant under Γ N and that the Kähler potential is changed at most by a Kähler transformation. To satisfy this last condition, we work with the Kähler: with h > 0 and Λ 0 a mass parameter. After τ develops a VEV, it gives rise to kinetic terms for the matter fields, These terms can be made canonical by rescaling the fields χ i , which in practice amounts to a redefinition of superpotential parameters. The superpotential reads and should remain unchanged under Γ N . Given the field transformations in Eq. (8), the functions Y {i 1 , ..., in} (τ ) in (11) must transform under Γ N as (we omit the indices i 1 , . . . , i n ): where the unitary matrix ρ Y (γ) and the non-negative integer k Y , as will shortly be shown, must satisfy specific conditions. Holomorphic functions f (τ ) satisfying f (γτ ) = (cτ + d) 2k f (τ ) with γ ∈ Γ(N ) and k ∈ N 0 are said to be modular forms of weight 2k and level N . For k = 0, the modular forms are constant functions, while for k < 0 modular forms do not exist. Modular forms are important objects in the present construction since, under Γ N , modular forms of weight 2k Y and level N transform in the way we require Y (τ ) in Eq. (12) to transform. The requirement of invariance of the superpotential then implies that the functions Y (τ ) are modular forms of level N . Their weights must cancel those of the fields in Eq. (11), 2k Y − k i 1 − . . . − k in = 0. Additionally, the tensor product ρ Y ⊗ ρ i 1 ⊗ . . . ⊗ ρ i 1 should contain at least one singlet. The subscript 1 in Eq. (11) indicates a sum (with independent coefficients) of all possible singlets one can extract from this product.
The effects of enforcing invariance under Γ N for N = 2 and N = 3 have been studied in Refs. [3,4]. The group Γ 2 is isomorphic to In what follows we consider invariance under the group Γ 4 S 4 , whose generators satisfy the following presentation rules: For a generic value of τ , Γ 4 will be fully broken. It is important to remark that the leading-order results obtained with this approach are susceptible to corrections from a small number of sources, namely from SUSY breaking and corrections to the Kähler potential.
In generalisations of our approach where flavons are introduced, additional corrections are expected from vacuum (mis)alignment.

Generators of Modular Forms of Level N = 4
The functions Y (τ ) are modular forms of level N = 4 and weight 2k Y . The dimension of the space of modular forms of level 4 and weight 2k is 4k + 1. Thus, the space of (level 4) forms which carry the lowest nontrivial weight, 2k = 2, has dimension 4k + 1 = 5. It proves useful to explicitly find a basis {Y 1 (τ ), . . . , Y 5 (τ )} of this lowest weight space, since modular forms of higher weights can be constructed from homogeneous polynomials in these five modular forms Y i (i = 1, . . . , 5). A starting point in this search is the recognition of certain properties of the Dedekind eta function η(z ∈ C), defined as: The η function satisfies η(z +1) = e iπ/12 η(z) and η(−1/z) = √ −iz η(z), as well as the identity (see, e.g., [7]): One sees that the set respects a certain notion of closure under the action of Γ 4 generators S and T , since under their action one has: S : and T : The transformations under S of the elements η(τ + 1/2) and η((τ + 2)/4) can be derived by making use of the relation (15). Up to multiplicative factors, this set is closed under S and T . Furthermore, each element is taken into itself (up to a factor) by the (left-)actions of S 2 , (ST ) 3 and T 4 . The above suggests that the desired k = 1 modular forms can be written as linear combinations of the logarithmic derivatives of the elements of the set {η i }. We define: The use of logarithmic derivatives allows one to eliminate the multiplicative factors in the transformations (17) and (18) by requiring i a i = 0. We are thus left with five independent linear combinations of the η i /η i , as expected. Under the action of the generators S and T this function transforms as: We find a basis for lowest weight modular forms, where ω ≡ e 2πi/3 . These five linearly independent forms Y i (τ ) arrange themselves into two irreducible representations of Γ 4 S 4 , a doublet 2 and a triplet 3 , The multiplets Y 2 and Y 3 transform under Γ 4 as indicated in Eq. (12). In Appendix A.1 we specify our basis choice for the representation matrices ρ(γ) of S 4 and we list the Clebsch-Gordan coefficients for this basis in Appendix A.2. In Appendix B we give the q-expansions of (combinations of) the five functions defined in Eq. (22). Multiplets transforming in the other representations of S 4 can be obtained from tensor products of Y 2 and Y 3 . The representations 1 and 3 arise at weight 2k = 4, while the 1 representation first arises at weight 6. Since we can form 15 combinations Y i Y j , one may expect 15 independent (level 4) forms at weight 2k = 4. However, the dimension of the space of these forms is 4k + 1 = 9. In fact, we find 6 constraints between the several Y i Y j , which we list in Appendix C. These constraints reduce the 15 potentially independent combinations to 9 truly independent ones, which are organised in the following representations of S 4 : 3 Phenomenology To understand how invariance under the subgroup Γ 4 of the modular group may play a role in determining lepton masses and mixing, one needs to specify the S 4 representations ρ i and the modular weights −k i of the relevant fields χ i , which transform as indicated in Eq. (8).
In what follows, we search for choices of representations and weights which are in line with a certain notion of minimality.
In a minimal approach, the superpotential W includes only the Yukawa interactions of the Minimal Supersymmetric Standard Model (MSSM) and the SUSY Weinberg operator as the origin of (Majorana) neutrino masses: where, as mentioned before, different coefficients are implied for different singlet combinations.
No flavons are present in the above superpotential. Recall also that the functions Y 2 = Y 2 (τ ) and Y 3 = Y 3 (τ ) have definite transformation properties under Γ 4 . After the breaking of modular symmetry we obtain: which leads to the Lagrangian terms written in terms of four-spinors, The generic assignments of representations and weights to the MSSM fields present in Eq. (25) are defined in Table 1. We will keep the Higgs sector assignments trivial for simplicity. We will also take lepton doublets (singlets) to transform as three (one) dimensional representations of S 4 , as is customary. Minimal models are then built by adhering to the following guidelines: • No flavons are introduced, • Neutrino masses arise from the Weinberg operator, • Higgs multiplets transform trivially ρ u,d ∼ 1 under Γ 4 , with k u,d = 0, • Lepton SU(2) L doublets transform as a triplet ρ L ∼ 3, 3 under Γ 4 , • Lepton SU(2) L singlets transform as singlets ρ 1,2,3 ∼ 1, 1 under Γ 4 , and • Lowest possible weights are chosen such that a rank 3 charged-lepton mass matrix M e is possible without imposing additional "shaping" symmetries.
Given the above conditions, we further expand the superpotential as: where the integer, non-negative exponents of the modular forms satisfy 2(a i + b i ) = k L + k i + k d = k L + k i and 2(c + d) = k L + k u = k L . In order to obtain some nontrivial structure in the Weinberg operator, we assume that k L > 0. We explore in what follows two minimal choices of weights and representations, corresponding to k L = 1 (model I) and k L = 2 (model II).

Model I (k L = 1)
The choices k L = 1 and k u = 0 along with the fact that ρ L ∼ 3 or ρ L ∼ 3 are enough to determine the structure of the Weinberg operator and hence of the neutrino mass matrix M ν . On the charged lepton side, the cancellation-of-weights condition 2(a i +b i ) = 1+k i implies that the k i are odd, k i ∈ {−1, 1, 3, . . .}. To avoid having rank(λ) < 3, one must make sure that different singlets can be extracted from each term in W . If two lepton singlet superfields share the same weight and representation under Γ 4 , the rank of the charged-lepton mass matrix is lower than its possible maximum. If some k i = −1, then a i = b i = 0 and no singlet can be formed for that generation. If instead k i = 1, a singlet can only be formed if ρ i ∼ 1 (1) when ρ L ∼ 3 (3 ). We summarise in Table 2 the weight and representation assignments of the minimal model allowing for rank(λ) = 3.
With the weight assignments of Table 2, the superpotential reads: Table 2: Transformation properties of chiral superfields under Γ 4 , for the minimal model with k L = 1 (model I). Both lines of lepton assignments lead to the same results.
Making use of the Clebsch-Gordan coefficients given in Appendix A.2 and of the Γ 4 S 4 representation choices in Table 2 we find that only some terms in Eq. (29) contain non-zero singlets. We are left with where α ≡ α 2 , β ≡ β 2 + β 3 and γ ≡ γ 2 . It is interesting to note that the constraints listed in Appendix C imply that the singlets of the β 2 and β 3 terms coincide. This superpotential gives rise to the mass matrices: and Specifying values for the parameters α, β, γ, and τ determines both mass matrices up to global factors 3 , and hence determines mass ratios as well as lepton mixing. After performing a numerical search, it seems this minimal model does not allow to reproduce known data. As a benchmark, we find the point α = 1, β = 1.7 × 10 −4 , γ = 0.025, and τ = 0.19 + 0.99i, which accommodates a neutrino mass spectrum with inverted ordering, but does not provide acceptable values for either sin 2 θ 12 or sin 2 θ 13 . For this point, we have The obtained predictions are to be compared with neutrino oscillation data and information on charged-lepton mass ratios, which we collect in Table 3.  Table 3: Best-fit values and 1σ ranges for neutrino oscillation parameters, obtained from the global analysis of Ref. [2], and for charged-lepton mass ratios, given at the scale 2 × 10 16 GeV with the tan β averaging described in [3], obtained from Ref. [8]. The parameters entering the definition of r are δm 2 ≡ m 2 2 − m 2 1 and ∆m 2 ≡ m 2 3 − (m 2 1 + m 2 3 )/2. The best-fit value and 1σ range of δ did not drive the numerical searches here reported.

Model II (k L = 2)
As before, the choices k L = 2 and k u = 0 along with the fact that ρ L ∼ 3 or ρ L ∼ 3 are enough to determine the structure of M ν .
On the charged lepton side, the cancellation-of-weights condition 2(a i +b i ) = 2+k i implies that the k i are even, k i ∈ {−2, 0, 2, . . .}. If some k i = −2, then again a i = b i = 0 and no singlet can be formed for that generation. If instead k i = 0, a singlet can only be formed if ρ i ∼ 1 (1) when ρ L ∼ 3 (3 ). We summarise in Table 4 the weight and representation assignments of the minimal model allowing for rank(λ) = 3. Table 4: Transformation properties of chiral superfields under Γ 4 , for the minimal model with k L = 2 (model II). Both lines of lepton assignments lead to the same results.
With the weight assignments of Table 4, the charged-lepton Yukawa part of the superpotential matches that of the case k L = 1, leading to the same charged-lepton mass matrix as the one given in Eq. (32), i.e. M II e = M I e . The Weinberg operator part of the superpotential reads instead: The first term in Eq. (34) contributes with two different non-zero singlets, since L 2 ∼ 1 ⊕ 2 ⊕ 3 ⊕ 0, Y 2 2 ∼ 1 ⊕ 0 ⊕ 2, and singlets may be obtained from both the 1 ⊗ 1 and the 2 ⊗ 2 combinations. The second term contributes with a third singlet, as Y 2 Y 3 ∼ 3 ⊕ 3 and thus a singlet is obtained from the 3 ⊗ 3 combination. The third term contains all the three singlets: 1 ⊗ 1, 2 ⊗ 2, and 3 ⊗ 3. Due to the constraints in Appendix C, there are only three independent singlets which enter the Weinberg operator part of W II . Explicitly, using the Clebsch-Gordan coefficients of Appendix A.2: which gives rise to the mass matrix: Specifying values for the parameters α, β, γ, g/g , g /g and τ determines both mass matrices up to global factors 4 , and hence determines mass ratios as well as lepton mixing.
Through numerical search, we find that this minimal model is successful in accommodating the data. We find a first benchmark, α = 0.12, β = 4.2 × 10 −4 , γ = 1, g/g = 0.34 − 0.94i, g /g = 1.93 − 0.53i, and τ = −0.1 + 1.2i, which admits a neutrino mass spectrum with normal ordering, with These results are in good agreement with the values of Table 3, except in what regards the experimental hint of δ ∼ 3π/2. We find a second benchmark, α = 0.12, β = 1, γ = 4.5 × 10 −4 , g/g = 15.1 + 7.7i, g /g = −0.40 − 1.08i, and τ = 0.35 + 0.85i, also admitting a neutrino mass spectrum with normal ordering, for which the values of δ and of other measured parameters are less than 2σ away from the best-fit values of For this second benchmark, in order to fit the individual mass-squared differences δm 2 and ∆m 2 , we set the global factor 2g v 2 u /Λ 0.0024 eV. In this case, the neutrino masses read m 1 0.046 eV, m 2 0.047 eV, and m 3 0.068 eV. A distinctive feature of this framework is the prediction of the Dirac and Majorana CPV phases. One is then in a position to extract a prediction for the effective Majorana mass | m | which controls the rate of neutrinoless double beta decay. Using the values in (38) we find | m | 0.043 eV. This value can be probed by new-generation experiments which are working towards the | m | ∼ 10 −2 eV frontier.
In this setup, the correlations between pairs of mixing angles, phases and mass ratios are non-trivial. The existence and the success of the above benchmarks warrant further exploration of the present framework.

Summary and Conclusions
In the present article we have explored the consequences of the presence of modular invariance in the lepton sector via the action of the finite modular group Γ 4 S 4 which is a subgroup of SL(2, Z).
Fields carrying a non-trivial modular weight transform with a scale factor in addition to the usual unitary rotation. To build an invariant theory under such transformations, special functions with the appropriate scaling properties need to be present in order to provide compensating factors. These functions are modular forms. For a fixed scaling (modular) weight, they make up a finite-dimensional space. In section 2.1, we have detailed the construction of a basis {Y 1 , . . . , Y 5 } for the lowest weight modular forms of level 4 (corresponding to Γ 4 S 4 ), necessary to the generation of higher weight forms. We have additionally shown how the weight 2 and 4 forms organise themselves into different multiplets of S 4 , and that they satisfy non-trivial constraints which guarantee the correct dimensionality of higher-weight spaces (see Appendix C).
We have then studied supersymmetric models of lepton masses and mixing based on the breaking of the Γ 4 S 4 modular invariance. Focusing on minimality as a guiding principle, we considered models where neutrino Majorana masses have their origin in the Weinberg operator and where no flavons are introduced. The expectation value of the modulus τ is the only source of symmetry breaking.
We study two minimal models, differing in the weight −k L of the lepton doublet under modular transformations. While the first model (model I, k L = 1) does not seem to be able to reproduce the data, the second model (model II, k L = 2) successfully accommodates chargedlepton masses and neutrino oscillation data. Deviations of individual parameters from best-fit values are found to be below the 2σ level for the benchmark of Eq. (38).
Since in the present scheme the mass matrices are fully determined by the VEV τ and by superpotential parameters -see Eqs. (31), (32) and (36) -the values of the Dirac and Majorana CPV phases can be predicted. For the benchmark of Eq. (38), which corresponds to a neutrino mass spectrum with normal ordering, a prediction for the effective Majorana mass | m | 0.043 eV is possible. This value can be probed by new-generation experiments working towards the | m | ∼ 10 −2 eV frontier.
The predictions here derived are expected to hold at leading order. There is a limited number of sources for deviations, namely SUSY breaking and Kähler corrections. The predictive power of the present approach and the existence of successful benchmarks make this model building avenue worthy of future study.

A.2 Clebsch-Gordan Coefficients
After establishing a dictionary between presentations (see previous subsection), we can directly use the Clebsch-Gordan coefficients from Ref. [10], since no change of basis on the representation matrices has been performed. We reproduce the coefficients here for completeness. Entries of each multiplet entering the tensor product are denoted by α i and β i . where y ≡ i q/3, z ≡ e iπ/4 (q/4) 1/4 , and as usual q = e 2πi τ .

C Forms of higher weight and constraints
Through tensor products of Y 2 and Y 3 , one can find the multiplets: