Two A5 modular symmetries for Golden Ratio 2 mixing

We present a model of leptonic mixing based on two $A_5$ modular symmetries using the Weinberg operator. The two modular symmetries are broken to the respective diagonal $A_5$ subgroup. At the effective level, the model behaves as a model with a single $A_5$ modular flavour symmetry, but with two moduli. Using both as stabilisers, different residual symmetries are preserved, leading to golden ratio mixing that is perturbed by a rotation.

Models employing multiple modular symmetries [33,34], [30] have some advantages. As described in [33], introducing the generic mechanism for using multiple modular symmetries, it allows models to be built based on residual symmetries, left unbroken by distinct moduli stabilisers. The preserved residual symmetries then lead to the realisation of different mass textures in the charged lepton and neutrino sectors in modular flavour models without flavons.
As an example, a S 4 flavour model featuring TM1 mixing [42] is constructed in an elegant manner from multiple S 4 modular symmetries in [33,34], S 4 flavour models arise featuring e.g. TM1 mixing [42], from multiple S 4 modular symmetries, whereas in [30], multiple A 4 modular symmetries result in an A 4 model leading to TM2 mixing [43].
In this paper, we construct a model that uses two A 5 modular symmetries in order to obtain the golden ratio mixing plus a rotation between the first and the third columns, using the Weinberg operator to generate the neutrino masses. This is akin to the models with multiple S 4 modular symmetries in [33,34]. We refer also to [40], where models with a single A 5 modular symmetry but with two moduli (using the Weinberg operator) generate the neutrino masses. The model that uses some fixed points of the modular fields lead to the same mixing we are going to discuss here, although that is not explicit in [40]. At high energies, the model is based in two modular symmetries, A l 5 and A ν 5 , with modulus fields denoted by τ l and τ ν , respectively. After the modulus fields acquire different VEV's, different mass textures are realised in the charged lepton and neutrino sectors.
We will start by introducing some properties of the A 5 modular symmetry group. Subsequently, the various possibilities of a golden ratio mixing and a rotation among two of its columns are investigated and concluded that only a rotation between the first and third columns is compatible with the 3σ confidence interval from NuFit. Only then the explicit model will be introduced.
In Section II we briefly review the framework of multiple modular symmetries. In Section III we describe the A 5 modular symmetry and respective stabilisers associated to residual symmetries. In Section IV, the golden ratio mixing and variations are introduced and their compatibility with NuFit data verified. In Section V we present the model for GR2 mixing. We conclude in Section VI.
A. Modular group and modular forms Γ, the modular group, consists in all linear fractional transformations γ acting on the complex modulus τ (τ in the upper-half complex plane, i.e. Im(τ ) > 0): a, b, c, d are integers with ad − bc = 1. Often, 2 × 2 matrices represent these transformations: As γ and −γ correspond to the same element, the group Γ is isomorphic to P SL(2, Z) = SL(2, Z)/Z 2 , with SL(2, Z) being the group of 2 × 2 matrices with integer entries and unit determinant.
The set of generators, S τ and T τ , with S 2 τ = (S τ T τ ) 3 = 1 generates the modular group. We take: which are represented, in matrix form We continue now to the subgroups Γ(N ) of Γ. These are obtained by taking the integer entries of the matrices modulo N : These subgroups are infinite, even though they are discrete for a given N . In turn, the quotient groups obtained by taking Γ N = Γ/Γ(N ) are finite. They are referred to as the finite modular groups. For N ≤ 5, these groups are isomorphic to the popular flavour symmetry groups mentioned already: Γ 2 S 3 , Γ 3 A 4 , Γ 4 S 4 , Γ 5 A 5 . The finite modular groups can be obtained by imposing T N τ = 1, meaning that τ = τ + N . The Modular forms of a given modular weight 2k and for a fixed N (the level) are holomorphic functions of τ that transform in a well-defined way under action by elements of Γ(N ): k is here a non-negative integer, N defines the underlying group (due to the modulo N ), and we note that we will only consider even weights. The modular forms are invariant under transformations by Γ(N ), up to the complex factor of (cτ + d) 2k , but they do transform under the quotient group Γ N . Having fixed the level N = 5, modular forms of a given weight 2k span a linear space of finite dimension M 2k (Γ(5)). The dimension of these linear spaces for A 5 is given by 10k+1. Without loss of generality we select a basis in the linear space M 2k (Γ(N )) where the modular forms transform under Γ N with unitary representations ρ of Γ N : We assume here the minimal form of the Kähler potential. The effects of considering non-minimal forms of the Kähler potential may be relevant and are discussed in [44,45]. The superpotential is in general a function of the modulus τ i and superfields φ i and the expansion in powers of the superfields takes the form For the superpotential to be invariant under any modular transformation γ 1 , . . . , γ M in Γ 1 , Γ 2 , . . ., Γ M , the couplings Y (I Y,1 ,...,I Y,M ) must be multiplet modular forms, and the superfields φ i must transform as where J transforms and ρ I i,J (γ) and ρ I Y,J (γ) are the unitary representation matrices of γ J with γ J ∈ Γ J N J . Naturally, the superpotential can only be invariant when k Y,J = k i1,J + . . . + k in,J , and when there is a trivial singlet in I Y,J × I i1,J × . . . × I in,J for all J = 1, . . . , M .

III. MODULAR A5 SYMMETRY AND RESIDUAL SYMMETRIES
In the following subsection the A 5 symmetry group is introduced, including some of its main properties as the modular forms of level 5 and its stabilisers which apply for the specific case of A 5 modular symmetries. The stabilisers for the modular groups from N = 2 to 5, can be found in [46].
A. Modular A5 symmetry and modular forms of level 5 The group A 5 is the group of even permutations of 5 objects and has 60 elements. It is generated by two operators S τ and T τ obeying This group has one singlet 1, two triplets 3 and 3 , one quadruplet 4 and one quintuplet 5 as its irreducible representations. The irreducible representations of the generators and the multiplication rules for the irreducible representations can be found in Appendix A.
The Yukawa couplings in a theory that is invariant under a Γ 5 ∼ A 5 symmetry are going to be modular forms of level 5. The eleven linearly independent weight 2 modular forms of level 5 form a quintuplet Y . These modular functions can be expressed in terms of the third theta function (see Appendix B for more details). The modular forms of higher weight are generated starting from these eleven modular forms of weight 2.
The space of the weight 4 modular forms of level 5 has dimension 21 and decomposes into a singlet 1, one triplet 3, one triplet 3 , a quadruplet 4 and two quintuplets 5. Using the weight 2 modular forms, one obtains the following expressions for the weight 4 modular forms [40]: Furthermore, the modular forms of weight 6, whose linear space has dimension 31 and decomposes into one singlet 1, two triplets 3, two triplets 3 , two quadruplet 4 and two quintuplets 5, are the following according to [40]: B. Stabilisers and residual symmetries of modular A5 As explained in [30], stabilisers of the symmetry play a crucial role in preserving residual symmetries. Given an element γ in the modular group A 5 , a stabiliser of γ corresponds to a fixed point in the upper half complex plane that transforms as γτ γ = τ γ . Once the modular field acquires a VEV at this special point, τ = τ γ , the modular symmetry is broken but an Abelian residual modular symmetry generated by γ is preserved. Obviously, acting γ on the modular form at its stabiliser leaves the modular form invariant, which implies that when τ is at a stabiliser, the respective modular form becomes an eigenvector of ρ I (γ) with (cτ γ + d) −2k as the eigenvalue. Making use of this characteristic, the alignment of modular forms at stabilisers are simple to obtain.
The stabilisers for the A 5 modular group are shown in TABLE I and can be found in [46].
The alignments of the modular forms of weight 2k = 2 and 4 for the τ that stabilise the generators S and T are shown in TABLE II. We present also the associated factors in terms of Y , defined as the first component Y 1 of the modular form Y (2) 5 . We used the definitions for the modular forms of weight 2 present in Appendix B. We include also the value of the singlet modular form of weight 4.

IV. GOLDEN RATIO MIXING AND RELATED MIXINGS
The golden ratio (GR) mixing is a mixing associated in previous works with models based in the A 5 symmetry, and this is not different for models using multiple modular A 5 . The mixing matrix that we will use is where φ = 1+ √ 5 2 . This mixing has the same problem as the TBM mixing: it is incompatible with the experimental results for θ 13 , and thus we want to work with models that preserve only the first or the second columns of the GR mixing matrix, that can be written as the GR matrix times a rotation between the other two columns.
For a model where the second column is preserved, the matrix that diagonalizes M ν is U = U GR U r , where U r is a rotation between the first and third columns. Using the parametrisation we are then able to diagonalize M ν . Here, θ is the angle that governs the rotation and the three α i are introduced such that m i are purely real values. The angles and phases from the standard parametrisation of the PMNS matrix in [47] can be expressed in terms of the model parameters θ, α 1 and α 2 using the expressions between the parameters and the PMNS matrix elements: cos θ 12 sin θ 13 cos 2 θ 13 cos θ 23 + cos θ 12 sin θ 13 cos θ 23 Using the 3σ C.L. range of sin 2 θ 13 for NO(IO), 0.02034(0.02053) → 0.02430(0.02434) [48], we obtain the allowed range for sin θ: which implies also ranges for the other mixing angles (using that −1 ≤ cos(α 1 − α 2 ) ≤ 1): The 1σ NuFit region is within the interval found for sin 2 θ 23 , which overlaps with the 3σ region for this parameter, with our result extending below 0.405(0.410) for NO(IO) and not reaching its upper limit. The range of allowed values for sin 2 θ 12 is near the lowest limit of the 1σ region although outside. For a model where the first column is preserved instead, the rotation matrix U r between the second and third columns can be parametrised as: Again, θ is the angle that governs the rotation and the three α i are introduced such that the three neutrino masses m i have purely real values. For this model, the expressions for the angles and phases from the standard parametrisation of the PMNS matrix in [47] in terms of the model parameters θ, α 1 and α 2 are cos θ 12 sin θ 13 cos 2 θ 13 cos θ 23 + cos θ 12 sin θ 13 cos θ 23 Using the 3σ C.L. range of sin 2 θ 13 for NO(IO), 0.02034(0.02053) → 0.02430(0.02434) [48], we obtain the allowed range for sin θ: which implies also ranges for the other mixing angles (using that −1 ≤ cos(α 1 − α 2 ) ≤ 1): We conclude that the range of allowed values for sin 2 θ 12 is outside the 3σ region and thus the class of models that preserve the first column of the golden ratio mixing matrix, which we call GR 1 mixing, are disfavoured by experiment.
Consequently, in the following we are only interested in models that preserve the second column of the golden ratio mixing, which we call GR 2 , although, as pointed out previously, even for these models sin 2 θ 12 is outside the experimental 1σ interval.

V. MODEL WITH TWO MODULAR A5 SYMMETRIES USING THE WEINBERG OPERATOR
Now that the A 5 modular symmetry and the mixing derived from the GR mixing were introduced, the model that uses this symmetry in order to get what we called the GR 2 mixing can now be described, assuming that neutrinos get their mass through the Weinberg operator. At high energies, these models are based in two modular symmetries, A l 5 and A ν 5 , with modulus fields denoted by τ l and τ ν , respectively. After the modulus fields acquire different VEV's, different mass textures are realised in the charged lepton and neutrino sectors, in such a way that the GR 2 mixing is recovered for the PMNS.
We consider then that neutrinos get their mass through an effective term of the type 1 Λ Y L 2 H 2 u . The transformation properties of fields and Yukawa couplings can be found in TABLE III. All the Yukawa coefficients Y l and Y ν are modular forms of weight 4. The right-handed lepton fields E c are arranged as a triplet 3 or 3 of A l 5 and singlets 1 of A ν 5 , with weights 2k l = +4 and 2k ν = −2. Similarly, the lepton doublets L transform as a 3 ( ) of A l 5 and a 1 of A ν 5 , with weights 2k l = 0 and 2k ν = +2. These are the correct choices for the weights such that the modular forms and fields in each term sum up to zero since the weight for the fields is not 2k, which are the values that were introduced in this section, but −2k instead. H d and H u are the usual Higgs and an additional Higgs doublet as required in supersymmetric models. A bi-quintuplet Φ, which is a quintuplet under both A l 5 and A ν 5 , is introduced. The multiplication of two triplets has the decomposition 3 ( ) ⊗ 3 ( ) = 1 ⊕ 3 ( ) ⊕ 5, where the 3 ( ) component is antisymmetric. This means that L 2 only decomposes as 1 ⊗ 5, and so it must combine with a singlet or quintuplet. This implies that we have only to consider the contributions from Y ν 1 , Y ν 51 and Y ν 52 , each associated with a different complex constant g i . For Y ν , we only consider the contribution from 5 1 since the other weight 4 5 2 will vanish at the chosen stabiliser for τ ν as is shown below.
With the fields assigned in this manner, the superpotential for this model, which can be separated into one part containing the mass terms for the charged leptons and the other the neutrino mass terms, has the following form: Considering the multiplication rules for two quintuplets to get a trivial singlet, the term 1 Λ 2 L 2 ΦY ν H 2 u can be explicitly expanded as: where P π is the matrix that describes the permutation π = 1 2 3 4 5 which is explicitly If Φ acquires the VEV Φ = v Φ P π (see Appendix C for more details), we get for the term in Eq.
which implies that w ν gets the form (the w e terms remain exactly the same): This means that the symmetry A l 5 × A ν 5 is broken but given that the same transformation γ can be performed in A l 5 and A ν 5 simultaneously and being the terms in the superpotential above all left invariant by such a transformation, there is still a single modular symmetry A D 5 , the diagonal subgroup, that is conserved. The superpotential above implies a neutrino mass matrix. Expanding Y ν 51 and Y ν 52 in terms of the weight 2 modular functions gives the results already derived in [40]. If the triplets L, E c and ν c are triplets 3, which we will simply write as ρ L ∼ 3, the neutrino mass matrix after the Higgs field acquires the VEV H u = (0, v u ) gets the form: where asterisks were used to omit the off diagonal entries of symmetric matrices and g 1 , g 51 and g 52 are arbitrary complex constants associated with the respective modular form contribution. The factors 2v 2 u /Λ and 2v 2 u v Φ /Λ 2 are included inside these constants.
If the triplets L, E c and ν c are triplets 3 instead, which can be equivalently expressed as ρ L ∼ 3 , one obtains: where again g 1 , g 51 and g 52 are arbitrary complex constants associated with the respective modular form contribution that absorbed the factors 2v 2 u /Λ and 2v 2 u v Φ /Λ 2 .

B. A D 5 breaking
The flavour structure after A D 5 symmetry breaking will now be covered. We assume that the charged lepton modular field τ l acquires the VEV τ l = τ T = i∞. This is a stabiliser of T τ which means that a residual modular Z T 5 symmetry is preserved in the charged lepton sector. The directions the modular forms take at this stabiliser are in TABLE II. These directions lead to an almost diagonal charged lepton mass matrix when the Higgs field H d acquires a VEV H d = (0, v d ): The masses for the charged leptons can be reproduced by adjusting the parameters α i . These constants were redefined to include the constant associated with Y l (τ l ). This matrix can be diagonalized by multiplying on the left and right by P L and P R (P T L m e P R = m e d ) by taking P L as the identity matrix and P R = P 23 . Consequently, the PMNS matrix is simply the matrix that diagonalizes the mass matrix for the neutrinos. These considerations are valid whether we choose the triplets in the model to be 3 or 3 .
For the other modular field τ ν , we want to find a VEV that leads to a mixing that preserves the second column of the GR mixing matrix. This occurs for τ ν = τ S = i and for Y ν with weight 4 (see TABLE II for the directions the modular forms get at this stabiliser). In this case, a residual modular Z S 2 symmetry is preserved in the neutrino sector.
If ρ L ∼ 3, this implies the following structure for the neutrino mass matrix: where g 1 , g 51 and g 52 were redefined to include factors coming from the modular forms Y ν 1 , Y ν 51 and Y ν 52 . We want now to diagonalize M ν , such that U T M ν U = M ν d = diag(m 1 , m 2 , m 3 ), where m i are the neutrino masses and U is an unitary matrix. When we apply the golden ratio mixing matrix Eq.(32) to the neutrino mass matrix for triplets 3 one obtains: g 51 + 55−24 √ 5 2 g 52 and c = 3 √ 5 + 5 g 51 + 3(7 √ 5−15) 2 g 52 . This implies that the PMNS is simply the Golden Ratio matrix times a rotation among the second and third columns, conserving only its first column. We have already discussed the compatibility of the GR 1 mixing and experimental values in Section IV, where it has already been seen that this mixing is incompatible with the 3σ confidence interval for θ 12 . For this reason, we will not further develop the case ρ L ∼ 3.
We now turn our attention to M 3 ν . For ρ L ∼ 3 , we have the following structure for the neutrino mass matrix: where once again g 1 , g 51 and g 52 were redefined to include the factors coming from the modular forms Y ν 1 , Y ν 51 and Y ν 52 . When we apply the golden ratio mixing matrix Eq. (32) to the neutrino mass matrix for triplets 3 we obtain: where a = g 1 − 5+ This matrix has only an element on the second row and second column and four elements on the corners that form a 2 × 2 symmetric matrix and so it can be fully diagonalized by introducing a matrix U r that describes a rotation among the first and third columns. The matrix that diagonalizes M ν is then U = U GR U r , where U r is given by Eq. (33). We are then able to diagonalize M ν and the lepton mixing obeys a GR 2 mixing.
It is also possible to start from the diagonal matrix M ν d and get U T GR M ν U GR . We have that: and comparing with (63) we obtain that and, more importantly, we get a mass sum rule for m i : The sum rule (65) and (34)(35)(36)(37) give us relations between the six observables (the three mixing angles, the atmospheric and solar neutrino squared mass differences and the Dirac neutrino CP violation phase) and the five parameters of the GR 2 mixing (θ, α 1 , α 2 , m 1 and m 2 ), and hence we can do a numerical minimisation using the χ 2 function: P i are the model values, BF are the best fit values from NuFit [48], σ i is obtained by averaging the upper and lower σ provided by NuFit. The fit parameters obtained for normal ordering (NO) and inverted ordering (IO) of neutrino masses can be found in TABLE IV. The best fit values lie inside the 1σ range for all the observables except θ 12 , for both orderings near the lower limit of the 1σ range, and θ 23 for IO. Nonetheless, all the observables are within their 3σ intervals. The best-fit occurs for NO with a χ 2 = 3.22.
It is also possible to obtain the expected m ββ for neutrinoless beta decay using the formula where m 2 is given by Eq.(65). Doing a numerical computation, the allowed regions of m lightest vs m ββ of FIG.1 (for NO, m lightest = m 1 and for IO, m lightest = m 3 ) were obtained, using again as constraints the data from [48]. In both figures it is also shown the current upper limit provided by KamLAND-Zen, m ββ < 61 − 165 meV [49]. Results from PLANCK 2018 also constrain the sum of neutrino masses, although different constrains can be obtained depending on the data considered (for more details, see [50]). In the figures are plotted two shadowed regions, a very disfavoured region m i > 0.60 eV (considering the limit 95%C.L.,Planck lensing+BAO+θ M C ) and a disfavoured region m i > 0.12 eV (considering the only, except θ 12 ) and 3σ data from [48]. In both figures there were also included the current upper limit from KamLAND-Zen m ββ < 61 − 165 meV [49] and cosmological constraints from PLANCK 2018 (disfavoured region 0.12 eV < m i < 0.60 eV and very disfavoured region m i > 0.60 eV) [50]. In both figures there were also included the current upper limit from KamLAND-Zen m ββ < 61 − 165 meV [49] and cosmological constraints from PLANCK 2018 (disfavoured region 0.12 eV < m i < 0.60 eV and very disfavoured region m i > 0.60 eV) [50].
limit 95%C.L.,Planck TT,TE,EE+lowE+lensing+BAO+θ M C ). These constraints on m i can be expressed as constraints on m lightest using the best fit value for the squared mass differences: m lightest > 0.198 eV and m lightest > 0.030 eV for NO and m lightest > 0.196 eV and m lightest > 0.016 eV for IO, for the very disfavoured and the disfavoured regions respectively. We conclude then that both fits in TABLE IV are in the disfavoured region.
For NO, the points that were compatible with the 1σ regions for the observables other than θ 12 were plotted with a darker red colour. Only for normal mass orderings do we have points outside the disfavoured region.
Taking these considerations into account, we conclude that NO is the preferred mass ordering.

VI. CONCLUSION
In this paper we make use of the framework of multiple modular symmetries and build a model with the viable Golden Ratio 2 mixing. Relying on a minimal field content, we describe how to break the multiple A 5 to a single modular symmetry. We present choices of representations and weights for the fields that produce the model, which leads to leptonic mixing matrix that fits well with the observed values of the angles in a predictive manner. We predict neutrinoless double beta decay. Inverted ordering is disfavoured by cosmological observations and is also disfavoured by the fits to the experimental leptonic mixing observables, with the normal ordering leading to a better fit to the experimental observables. and ζ = e 2πi/5 .
The factors considered for the representation in TABLE V lead to the following decomposition, with the Clebsch-Gordan coefficients in [40]: Appendix B: Modular forms of weight 2 for A5 The linear space of modular forms of level N = 5 and weight 2 has dimension 11. These modular functions are arranged into two triplets 3 and 3 and a quintuplet 5 of Γ 5 . Modular forms of higher weight can be constructed from polynomials of these eleven modular functions.

Appendix C: Vacuum alignments for bi-quintuplet Φ in A5
In this Appendix we consider how to align the VEV of the bi-quintuplet Φ. Following from [33] and [30] where an alignment was obtained in the context of S 4 and A 4 respectively, we add two driving fields, with the properties present in TABLE VI.  The superpotential responsible for the vacuum alignment that will be minimised with relation to the driving fields is w = ΦΦχ lν + M Φχ lν + ΦΦχ l .
With this field content, we are only interested in contractions of quintuplets to give quintuplets or singlets. Minimising this superpotential in order to the driving fields leads us to the constraints: (C7)