Modular $A_5$ Symmetry for Flavour Model Building

In the framework of the modular symmetry approach to lepton flavour, we consider a class of theories where matter superfields transform in representations of the finite modular group $\Gamma_5 \simeq A_5$. We explicitly construct a basis for the 11 modular forms of weight 2 and level 5. We show how these forms arrange themselves into two triplets and a quintet of $A_5$. We also present multiplets of modular forms of higher weight. Finally, we provide an example of application of our results, constructing two models of neutrino masses and mixing based on the supersymmetric Weinberg operator.


Introduction
Understanding the origins of flavour remains one of the major problems in particle physics. The power of symmetries in governing laws of particle interactions does not need to be advocated. In this regard, it is rather natural to expect that symmetry(ies) also hold the key to the solution of the flavour problem.
The fact that two out of three neutrino mixing angles are large [1][2][3] suggests the presence of a new flavour symmetry (at least in the lepton sector) described by a non-Abelian discrete (finite) group (see, e.g., [4][5][6][7]). While unifying the three known flavours at high energies, this symmetry may be broken at lower energies to residual symmetries of the charged lepton and neutrino mass terms, which correspond to Abelian subgroups of the original flavour symmetry group. In the bottom-up approach, starting from residual symmetries, one can successfully explain the observed pattern of neutrino mixing and, in addition, predict the value of the Dirac CP violation phase [8][9][10]. 1 However, predicting neutrino masses calls for the construction of specific models, in which the flavour symmetry is typically spontaneously broken by vacuum expectations values (VEVs) of flavons -scalar gauge singlets charged non-trivially under the flavour symmetry group. Usually, a numerous set of these fields is needed. Moreover, one may have to construct rather complicated flavon potentials in order to achieve vacuum alignments leading to viable phenomenology.
A very interesting generalisation of the discrete symmetry approach to lepton flavour has been recently proposed in Ref. [14]. In this proposal, modular invariance plays the role of flavour symmetry, and couplings of a theory are modular forms of a certain level N . In addition, both the couplings and matter supermultiplets are assumed to transform in representations of a finite modular group Γ N . In the simplest class of such models, the VEV of a complex field τ (the modulus) is the only source of flavour symmetry breaking, such that no flavons are needed. Another appealing feature of the proposed framework is that charged lepton and neutrino masses, neutrino mixing and CPV phases are simultaneously determined by the modular symmetry typically in terms of limited number of constant parameters. This leads to experimentally testable correlations between, e.g., the neutrino mass and mixing observables.
The cornerstone of the new approach is the modular forms of weight 2 and level N , and their arrangements into multiplets of Γ N . Modular forms of higher weights can be constructed from these building blocks. Remarkably, for N ≤ 5, the finite modular groups are isomorphic to well-known permutation groups. In Ref. [14], the group Γ 3 A 4 has been considered, and the three generating modular forms of weight 2 have been explicitly constructed and shown to furnish a 3-dimensional irreducible representation (irrep) of A 4 . Further, the group Γ 2 S 3 has been considered in [15], and the two forms shaping a doublet of S 3 have been identified. The five generating modular forms in the case of N = 4 have been found to organise themselves into a doublet and a triplet of Γ 4 S 4 in Ref. [16], where the first realistic model of lepton masses and mixing without flavons has also been constructed. Very recently, by studying Yukawa couplings in magnetised D-brane models, the authors of Ref. [17] have found multiplets of weight 2 modular forms corresponding to a triplet and a sextet of ∆(96), and a triplet of ∆(384). They have also reported an S 3 doublet and an S 4 triplet. Note that ∆(96) is isomorphic to a subgroup of Γ 8 , while ∆(384) is isomorphic to a subgroup of Γ 16 (see, e.g., [18]).
Lepton flavour models based on Γ 3 A 4 have been studied in more detail in Refs. [19,20], where several viable examples have been presented. In Ref. [21], we have constructed in a systematic way models based on Γ 4 S 4 , in which light neutrino masses are generated via the type I seesaw mechanism and where no flavons are introduced. We have shown that models with a relatively small number of free parameters can successfully describe data on the charged lepton masses, neutrino mass-squared differences and mixing angles. Furthermore, we have obtained predictions for the neutrino masses and the Dirac and Majorana CPV phases in the neutrino mixing matrix. In these models, the value of atmospheric mixing angle θ 23 is correlated with i) the Dirac phase δ, ii) the sum of neutrino masses, and iii) the effective Majorana mass in neutrinoless double beta decay.
In the present article, for the first time in this context, we consider the finite modular group Γ 5 A 5 . Our main focus is on constructing the 11 generating modular forms of weight 2 and demonstrating how they can be arranged into multiplets of A 5 , namely two triplets and a quintet. The group A 5 has been investigated in the context of the conventional discrete symmetry approach in Refs. [22][23][24][25][26][27] as well as in combination with so-called generalised CP symmetry in Refs. [28][29][30][31][32]. The characteristic phenomenological feature of the models based on the A 5 flavour symmetry is the golden ratio prediction for the solar mixing angle, θ 12 = arctan(1/ϕ) ≈ 32°, with ϕ = (1 + √ 5)/2 being the golden ratio, which is inside the experimentally allowed 3σ range [2,3]. An interesting theoretical feature of A 5 is that it is anomaly-free [5].
The article is organised as follows. In Section 2, we first summarise the modular symmetry approach to lepton masses and mixing proposed in Ref. [14], and then explicitly construct the two A 5 triplets and the A 5 quintet of modular forms of weight 2. Next, in Section 3, we give an example of application of the obtained results constructing a phenomenologically viable model of neutrino masses and mixing based on the Weinberg operator. Finally, in Section 4, we draw our conclusions.

Modular symmetry and modular-invariant theories
The modular group Γ is the group of linear fractional transformations γ acting on the complex variable τ belonging to the upper-half complex plane as follows: The modular group is isomorphic to the projective special linear group P SL(2, Z), and it is generated by two elements S and T satisfying

2)
I being the identity element of a group. Representing S and T as The group SL(2, Z) = Γ(1) ≡ Γ contains a series of infinite normal subgroups Γ(N ), N = 1, 2, 3, . . . : called the principal congruence subgroups. For N = 1 and 2, we introduce the groups Γ(N ) ≡ Γ(N )/{I, −I} (note that Γ(1) ≡ Γ), and for N > 2, Γ(N ) ≡ Γ(N ). For each N , the associated linear fractional transformations of the form in eq. (2.1) are in a one-to-one correspondence with the elements of Γ(N ). The quotient groups Γ N ≡ Γ/Γ(N ) are called finite modular groups. For N ≤ 5, these groups are isomorphic to permutation groups widely used to build flavour models (see, e.g., [18]). Namely, Modular forms of weight k and level N are holomorphic functions f (τ ) transforming under the action of Γ(N ) in the following way: Here k is even and non-negative, and N is natural. Modular forms of weight k and level N span a linear space of finite dimension. There exists a basis in this space such that a multiplet of modular forms f i (τ ) transforms according to a unitary representation ρ of the finite group Γ N : In the case of N = 2, the modular forms of weight 2 span a two-dimensional linear space. In a certain basis the two generating modular forms are transformed in the 2-dimensional irreducible representation (irrep) of S 3 [15]. For level N = 3, weight 2 modular forms arrange themselves in a triplet of A 4 [14]. In the case of N = 4, the corresponding linear space has dimension 5, and weight 2 modular forms group in a doublet and a triplet of S 4 [16]. For N = 5, there are 11 modular forms of weight 2. They are organised in two triplets and a quintet of A 5 . In the next subsection, we will explicitly derive them, but before that, let us briefly recall how to construct supersymmetric modular-invariant theories.
In the case of N = 1 rigid supersymmetry, the matter action S reads where K is the Kähler potential, W is the superpotential and χ denotes a set of chiral supermultiplets contained in the theory apart from the modulus τ . The θ and θ denote Graßmann variables. The modular group acts on τ and supermultiplets χ I of a sector I of a theory in a certain way [33,34]. Assuming, in addition, that the supermultiplets χ I transform according to a representation ρ I of Γ N , we have (2.9) Note that χ I are not modular forms, and the weight (−k I ) can be odd and/or negative. Requiring invariance of S under eq. (2.9) leads to where the second line represents a Kähler transformation. The superpotential can be expanded in powers of χ I : where 1 stands for an invariant singlet of Γ N . To ensure invariance of W under the transformations specified in eq. (2.9), the functions Y I 1 ... In (τ ) must transform as follows: where ρ Y is a representation of Γ N , and k Y and ρ Y are such that (2.14) Thereby, the functions Y I 1 ... In (τ ) form a multiplet of weight k Y and level N modular forms transforming in the representation ρ Y of Γ N (cf. eq. (2.7)).

Generators of modular forms of level N = 5
The space of modular forms of level N = 5 and lowest nontrivial weight 2 is 11. Expansions for a standard basis {b 1 (τ ), . . . , b 11 (τ )} of this space of functions are given in Appendix A.1. Modular forms of higher weight can be constructed from homogeneous polynomials in these eleven modular forms. The action of the discrete quotient group Γ 5 divides the space of lowest weight modular functions into two triplets transforming in irreps 3 and 3 and a quintet transforming in 5 of Γ 5 A 5 (see also Section 4.4 of Ref. [35]). As in the cases of Γ 3 A 4 [14], Γ 2 S 3 [15] and Γ 4 S 4 [16], the lowest weight modular functions correspond to linear combinations of logarithmic derivatives of some "seed" functions α i,j (τ ). These functions form a set which is in a certain sense closed under the action of A 5 . As can be inferred from the results in Ref. [36], a convenient choice for α i,j (τ ) is given by the Jacobi theta functions θ 3 (z(τ ), t(τ )), and they explicitly read: 2 (2.15) 2 For properties of these special functions, see, e.g., Refs. [37,38]. In the notations of Ref. [37], θ3 ≡ θ 0 0 . Under the action of the generators S and T of Γ 5 (see Appendix B.1), each of these functions is mapped to another, up to (possibly τ -dependent) multiplicative factors. A diagram of said map is given in Fig. 1, and one can check that the actions of S 2 , (ST ) 3 and T 5 applied to each element correspond to the identity. Taking logarithmic derivatives, one obtains: where α S i,j and α T i,j are the images of α i,j under the S and T maps of Fig. 1, respectively. It then follows that the functions Then, as anticipated, the space in question is divided into the following multiplets of A 5 : where ζ = e 2πi/5 . The q-expansions for these modular forms are given in Appendix A.2. In Appendix B.1, we specify our basis choice for the representation matrices ρ(γ) of A 5 , and we list the Clebsch-Gordan coefficients for this basis in Appendix B.2.
Multiplets transforming in the other representations of A 5 can be obtained from tensor products of the lowest weight multiplets Y 5 , Y 3 and Y 3 . The missing 1 and 4 representations arise at weight 4. Even though one can form 66 products Y i Y j , the dimension of the space of weight k = 4 (and level 5) forms is 5k + 1 = 21. Therefore, there are 45 constraints between the Y i Y j , which we list in Appendix C, and which reduce the 66 potentially independent combinations to 21 truly independent ones. These last combinations arrange themselves into the following multiplets of A 5 : (2.24)

Phenomenology
To illustrate the use of the constructed modular multiplets for model building, we consider a minimal example where the neutrino masses originate from the Weinberg operator. We assume that the charged lepton mass matrix is diagonal, so it does not contribute to the mixing. We will show later an explicit example with residual symmetry where this possibility is realised. In this set-up, the only superpotential term relevant for the mixing is the Weinberg operator: where Y is a modular multiplet of weight k Y . We assume that the lepton SU(2) L doublets transform as an A 5 triplet (ρ L ∼ 3 or 3 ) of weight −k L , while the Higgs multiplet H u is an A 5 singlet (ρ u ∼ 1) of zero weight (k u = 0). After the breaking of the modular symmetry, we obtain: which leads to the Lagrangian term T , with C being the charge conjugation matrix. Given the above conditions, one needs to have k Y = 2k L to compensate the overall weight of the Weinberg operator term. Since k Y is a non-negative integer, we can systematically explore the possible neutrino mass matrices going from k Y = 0 to more and more positive integer k Y . In the case of k Y = 0 there are no modular forms in the Weinberg operator and the only possible 2), which leads to the following neutrino mass matrix: The case k Y = 0 is then excluded, since it leads to degenerate neutrino masses, which are ruled out by the neutrino oscillation data [1]. In the following subsections we consider the cases k Y = 2 and k Y = 4 (corresponding to k L = 1 and k L = 2, respectively).

The case of k Y = 2
In this case, the available modular form multiplets are Y 3 , Y 3 and Y 5 . Note that L 2 decomposes as 3 ( ) ⊗ 3 ( ) = 1 ⊕ 3 ( ) ⊕ 5, but the 3 ( ) component vanishes due to antisymmetry (see Appendix B.2). Therefore the only way to form a singlet is by combining the quintets, L 2 Y 5 1 . If ρ L ∼ 3, one obtains: The difference between eq. (3.5) and eq. (3.6) resides in the cyclic exchange of Y 5 , Y 4 , Y 2 and Y 3 (in this order).
In both cases, τ determines neutrino masses up to the overall mass scale. Furthermore, given our assumption of a diagonal charged lepton mass matrix, after employing the permutation ordering the charged lepton masses, τ additionally determines the mixing parameters.
Through numerical search, we find that the agreement with data is optimised by choosing ρ L = 3 and τ = 0.48 + 0.873 i, giving rise to the following values of observables, for a spectrum with normal ordering: given an overall factor v 2 u g/Λ 0.006339 eV, and assuming the charged lepton sector induces a permutation of the first and third rows of the PMNS mixing matrix. While one obtains a good agreement with data for the mass-squared differences (and hence for the ratio r ≡ ∆m 2 21 /∆m 2 31 ), as well as for the values of sin 2 θ 12 and of δ, the value of sin 2 θ 23 is slightly outside its 3σ range and, more importantly, sin 2 θ 13 is many standard deviations away from its experimentally allowed range [2,3]. Nevertheless, it is encouraging to find that the predictions for the mixing angles are in qualitative agreement with the observed pattern, namely, sin 2 θ 13 < sin 2 θ 12 < sin 2 θ 23 . Note that the indicated value of τ is close to the "right cusp" τ R = 1/2 + i √ 3/2, which preserves a residual Z T S 3 symmetry (see, e.g., [21]).

The case of k Y = 4
In this case, the available modular form multiplets are those given in eq. (2.24). Again, since L 2 decomposes as 1 ⊕ 5, one can form singlets by using Y 5,1 or Y 5,2 . All three contributions should enter W with independent complex coefficients. If ρ L ∼ 3, one obtains: where through asterisks we (here and henceforth) omit some entries of symmetric matrices. If instead ρ L ∼ 3 , it follows that

The case of residual symmetries
Let us assume that the modular symmetry is broken to the residual Z S 2 = {I, S} symmetry of the neutrino mass matrix. This can be achieved by fixing τ = i in the neutrino sector, since this value of τ is invariant under the action of S, τ → −1/τ .
For the symmetric value of the VEV of the modulus τ = i, the lowest weight modular forms Y i take the values: It follows that in both cases ρ L ∼ 3, 3 for k Y = 4, aside from the permutation which orders charged lepton masses, 5 real parameters -namely, 2v 2 u g 1 /Λ, Re(g 2 /g 1 ), Im(g 2 /g 1 ), Re(g 3 /g 1 ) and Im(g 3 /g 1 ) -determine the neutrino masses and mixing. Through numerical search, we find for ρ L ∼ 3 a point given by:  The indicated value of the effective Majorana mass | m | which controls the rate of neutrinoless double beta decay may be probed in future experiments aiming to test values down to the | m | ∼ 10 −2 eV frontier.
In the vicinity of the point described by eq. (3.11), keeping τ = i, we find strong correlations between sin 2 θ 12 and sin 2 θ 13 , and between sin 2 θ 23 and δ. These correlations are shown in Fig. 2.
One possible way to force the charged lepton mass matrix to be diagonal in this set-up is to assume that it originates from a different modulus τ l which develops a VEV τ l = i ∞ breaking the modular symmetry to the residual Z symmetry generated by T , τ → τ + 1. The corresponding residual symmetry of the charged lepton mass matrix is Z 5 generated by the T generator of Γ 5 , which is diagonal for ρ L ∼ 3 ( ) . One can show that in the case ρ E c = ρ L the charged lepton Yukawa interaction terms (E c LY ) 1 H d with the multiplets of weight 4 modular forms lead to the following mass matrix (written in the left-right convention): 3 where the three matrix terms correspond to contributions from Y where α 2 /α 1 and α 3 /α 1 are complex parameters, and v d α 1 is the overall mass scale factor. To reproduce the charged lepton masses with these parameters one can choose, e.g., v d α 1 660 MeV, α 2 /α 1 = 1.34 and α 3 /α 1 = −0.500385. It is also possible to obtain the same matrix without an additional modulus τ l . Instead, let us assume that the charged lepton mass matrix originates from the term (E c L) 1 H d and Yukawa couplings to two flavons, an A 5 triplet ϕ 3 ( ) and an A 5 quintet ϕ 5 , each of which develops a VEV breaking A 5 to Z T 5 : ϕ 3 ( ) ∝ (1, 0, 0) and ϕ 5 ∝ (1, 0, 0, 0, 0). (3.15) In this case the three terms (E c L) 1 H d , (E c Lϕ 3 ( ) ) 1 H d and (E c Lϕ 5 ) 1 H d lead to the same mass matrix as in eq. (3.13). This is related to the fact that modular form multiplets of weight 4 take the following values at the symmetric point τ = i ∞: 0, 0, 0, 0). (3.16) Hence, they can be thought of as flavon multiplets developing the corresponding VEVs.

Summary and Conclusions
In the framework of modular invariance approach to lepton flavour proposed in Ref. [14], we have considered a class of theories in which couplings and matter superfields transform in irreps of the finite modular group Γ 5 A 5 . The building blocks needed to construct such theories are modular forms of weight 2 and level 5. We have explicitly constructed the 11 generating modular forms of weight 2, using the Jacobi theta function and its properties, which lead to closure of the set of 12 seed functions (see eq. (2.15)) under the action of Γ 5 , as shown in Fig. 1. Further, we have demonstrated how these 11 modular forms arrange themselves into multiplets of A 5 . Namely, we have found two triplets transforming in the irreps 3 and 3 of A 5 , and a quintet transforming in the irrep 5 of A 5 . They are given in eqs. (2.21) -(2.23), and their explicit q-expansions are listed in Appendix A.2. From these triplets and quintet we have constructed multiplets of modular forms of weight 4 (see eq. (2.24) and Appendix C).
While thorough analysis of modular-invariant theories with the Γ 5 symmetry is left for future work, we have presented two examples of application of the obtained results to neutrino masses and mixing. In both of them, we have assumed that neutrino masses are generated via the Weinberg operator, and considered the charged lepton mass matrix to be diagonal. The first model involving the quintet of weight 2 modular forms leads to the neutrino mass matrix containing three real parameters -complex VEV of the modulus τ and a real overall scale. We have found that the value of τ lying very close to the "right cusp" τ R = 1/2 + i √ 3/2 (τ R preserves a residual Z T S 3 symmetry) leads to a good agreement with neutrino oscillation data except for sin 2 θ 13 , which falls many standard deviations away from its experimentally allowed region. The second model contains an A 5 singlet and two A 5 quintets of modular forms of weight 4. The neutrino mass matrix in this case depends on five real parameters (three real constants and two phases) apart from τ . Assuming that τ = i -a self-dual point which preserves a residual Z S 2 symmetry -we have obtained a viable benchmark point compatible with the data at 1.7σ confidence level. In this case the neutrino mass matrix depends on three real parameters and two phases. Varying these free parameters we found strong correlations between the values of sin 2 θ 12 and sin 2 θ 13 , and the values of sin 2 θ 23 and the Dirac CPV phase δ (Fig. 2).
In conclusion, the results obtained in the present study can be used to build in a systematic way modular-invariant flavour models with the Γ 5 A 5 symmetry. In this regard, this article is expected to serve as a useful handbook for future studies.
(A.4) B A 5 Group Theory

B.1 Basis
A 5 is the group of even permutations of five objects. It contains 5!/2 = 60 elements and admits five irreducible representations, namely 1, 3, 3 , 4 and 5 (see, e.g., [5]). It can be generated by two elements S and T satisfying We will employ the group theoretical results of Ref. [24], using in particular the following explicit basis for the irreducible representations of A 5 : where ζ = e 2πi/5 and ϕ = (1 + √ 5)/2.

B.2 Clebsch-Gordan coefficients
For completeness, we reproduce here the nontrivial Clebsch-Gordan coefficients of Ref. [24], given in the above basis. Entries of each multiplet entering the tensor product are denoted by α i and β i .

C Higher weight Forms and Constraints
Through tensor products of Y 5 , Y 3 and Y 3 , one can find, at weight 4, the multiplets: Not all of the above multiplets are expected to be independent. Indeed, from the qexpansions of the Y i (τ ) given in Appendix A.2 we find 45 constraints between the 66 different Y i (τ )Y j (τ ) products, namely: