Type II seesaw models with modular $A_4$ symmetry

We discuss type-II seesaw models adopting modular $A_4$ symmetry in supersymmetric framework. In our approach, the models are classified by the assignment of $A_4$ representations and modular weights for leptons and triplet Higgs fields. Then neutrino mass matrix is characterized by modulus $\tau$ and two free parameters. Carrying out numerical analysis, we find allowed parameter sets which can fit the neutrino oscillation data. For the allowed parameter sets, we obtain the predictions in neutrino sector such as CP violating phases and the lightest neutrino mass. Finally we also show the predictions for the branching ratios of doubly charged scalar boson focusing on the case where the doubly charged scalar boson dominantly decays into charged leptons.


I. INTRODUCTION
Understanding of the flavor structure of leptons and quarks is one of the well motivated issues to construct a model of new physics beyond the standard model (SM). In describing new physics, a new symmetry can play an important role to organize flavor structure.
The modular symmetry is a geometrical symmetry of torus and orbifold compactification, and very interesting, because it includes finite subgroups such as S 3 , A 4 , S 4 , and A 5 [1].
In realizing small neutrino masses, the so-called type-II seesaw mechanism is one of the interesting ideas in which an SU(2) triplet Higgs field is introduced [52][53][54]. The neutrino masses are generated through Yukawa interactions among the triplet and lepton doublets after the triplet developing a vacuum expectation value (VEV). In this scenario, we have a doubly charged scalar boson from the triplet which couples to charged leptons. The doubly charged scalar boson dominantly decays into the same sign charged lepton pair when the triplet VEV is less than around 10 −4 GeV, and it can give clear signals at the collider experiments such as the LHC. Importantly the branching ratios (BRs) of such decays are given by Yukawa couplings associated with neutrino mass generation and we can obtain 1 See also Refs. [8][9][10]. 2 Several reviews are helpful to understand the non-Abelian group and its applications to flavor structure [41][42][43][44][45][46][47][48].

Lepton
Higgs L (e R , µ R , τ R ) (1)  some correlations among the BRs and neutrino parameters.
In this paper, we apply the modular A 4 symmetry to the type-II seesaw mechanism in supersymmetric framework. Then some possible models are classified by the assignments of A 4 representations and modular weights to the leptons and the Higgs triplet. We then scan free parameters in these models and search for the region in which the neutrino oscillation data can be fitted. For the allowed parameter sets, we show the predictions of observables in the neutrino sector. Finally we show our predictions for the branching ratios of the doubly charged scalar boson applying the allowed parameter sets.
The paper is organized as follows. In section 2, we introduce our models. In section 3, we perform parameter scan to fit neutrino oscillation data and provide some predictions in observables in the neutrino sector. Also, we show the branching ratios of the doubly charged scalar boson applying the parameter sets accommodating with the neutrino oscillation data.
Section 4 is our conclusion and discussions. In Appendix, we summarize formulae to fix the coupling coefficients for the Yukawa interactions associated with charged lepton masses.

II. MODELS
In this section we show type-II seesaw models with modular A 4 symmetry in supersymmetric framework under which superfields of leptons are non-trivially transformed by the modular symmetry. In the type-II seesaw mechanism, we introduce two SU(2) triplet su-perfields T 1 and T 2 which have hypercharge Y = 1 and −1 respectively; here we need two triplet superfields for gauge anomaly cancellation. We then obtain superpotential of the form where L is superfield for lepton doublet, and H u and H d are superfields for the Higgs doublets with hypercharge 1 2 and − 1 2 respectively. From the superpotential, we obtain the VEV of the neutral component of the T 1 scalar, denoted by where v u is that of the H u scalar. The VEV provides neutrino mass term [54]. The superpotential terms relevant to the charged lepton masses are written by where superfileds {e R , µ R , τ R } correspond to right-handed charged leptons. These superpotential terms are required to be invariant under A 4 symmetry with vanishing modular weight where the couplings can be modular forms associated with non-trivial A 4 representations and having non-zero modular weights. Then models are distinguished by the assignments of A 4 representations and modular weights for the leptons and scalar fields. In Table. I, we summarize the assignment of A 4 representations and modular weights to the fields in our models. With these representations and weights of the fields, those of the Yukawa couplings are fixed. Then, the structure of superpotential are determined. The other sectors are assumed to be equivalent to the supersymmetric type II seesaw model [54] and we do not discuss in this paper 3 .
The Yukawa coupling constants can have the modular weights under the modular symmetry. The modular form of A 4 triplet with weight 2, Y 3 (τ ), is given by where τ is a complex number. More precisely, the above modular forms can be written in terms of Dedekind eta-function η(τ ) and its derivative: where w = e 2πi/3 . Equation (4) is their q-expansions. The modular forms with higher weights can be constructed by products of Y where the modular form of 1 ′′ representation with weight 4 does not exist due to the relation 3 , is constructed as A. Model (1) In this model we can write the superpotential term relevant to the neutrino masses as and the superpotential term relevant to the charged lepton masses, The mass matrix for the charged leptons is given by where ℓ denotes three generations of charged leptons,γ ≡ v d γ/ √ 2,α ≡ α/γ,β = β/γ and Here v d is the VEV of H d . As in the SM, we can diagonalize the mass matrix by transforming lepton fields, ℓ L(R) → V e L(R) ℓ L(R) , providing diag(m e , m µ , m τ ) = (V e R ) † M e V e L . The parametersα andβ are determined to provide charged lepton mass eigenvalues as given in Appendix.
After the neutral component of T 1 developing its VEV, v T 1 , we obtain Majorana neutrino mass terms such as where ν ′ L i=1,2,3 denotes the neutral fermion component of L. Note that ν ′ L i s are not identified with ν e,µ,τ , the partners of the charged leptons in weak interaction, since they are in the basis where charged lepton mass matrix is not diagonalized. Then we find the lepton flavor basis by Thus the neutrino mass matrix in the flavor basis is given by Notice that the mixing matrix V e L is involved in the neutrino mass matrix.
B. Model (2) In this model, we take the A 4 representations of T 1 , T 2 and H u as 1 ′′ , 1 ′ and 1 ′ while the other setting is the same as model (1). Then the superpotential term relevant to the neutrino masses is Note that we do not have λ 1 H d T 1 H d term compared to Eq. (9) where the term is irrelevant in realizing the type-II seesaw mechanism and absence of the term does not affect our analysis.
For charged lepton mass term, the superpotential is the same as Eq. (10).
The neutrino mass matrix in this case is where the structure is different from model (1).

C. Model (3)
In this model, we take the modular weight −2 for leptons and assignment under the A 4 representation is the same as model (1). Then the superpotential term relevant to the neutrino masses is In this case, we have additional terms with free parameters since A 4 singlet modular forms are also available when couplings should have the modular weight 4.
For charged lepton mass term, we consider two cases depending on modular weight assignment for right-handed charged leptons. In cases A and B, the modular weights of ℓ R are assigned to 0 and −2, respectively. Then case A has the same superpotential as Eq. (10).
On the other hand, for case B we obtain the corresponding superpotential as In this case, the charged lepton mass matrix is 3,3 . We separately analyze cases A and B since the charged lepton mass matrix affects the neutrino mass matrix through V e L as we discussed above. The neutrino mass matrix in this case is 3,1 2Y whereŷ 2,3 ≡ y 2,3 /y 1 .

D. Model (4)
In this model, we chose A 4 singlet 1 ′′ for triplet T and the other assignments are the same as model (3). Then the superpotential term relevant to the neutrino masses is and it is the same as model (3) except for the A 4 structure. The superpotential term relevant to the charged lepton masses is the same as model (3), and we also analyze cases A and B separately.

III. NUMERICAL ANALYSIS
In this section, we carry out numerical analysis. Firstly the free parameters in each model are scanned to search for regions which accommodate with neutrino oscillation data. Here we parameterize the PMNS matrix U P M N S , diagonalizing the neutrino mass matrix m ν , in terms of three mixing angles θ ij (i, j = 1, 2, 3; i < j), one CP violating Dirac phase δ CP , and two Majorana phases {α 21 , α 32 } as follows: where c ij and s ij denote cos θ ij and sin θ ij respectively. Then we estimate the branching ratios of the doubly charged scalar boson focusing on the decays into the same sign charged lepton pairs using the allowed parameters explaining neutrino data.

A. Fitting neutrino oscillation data and relevant predictions
Here we scan the free parameters in the models and try to fit the neutrino oscillation data. In our analysis, we adopt experimentally allowed ranges for neutrino mixing and mass squared differences at 3σ range taken from ref. [55] as follows: where NO(IO) stands for normal(inverted) ordering for neutrino masses. Then the free parameters are scanned in the following range: (3) and (4).
The values of √ 2yv T 1 /3 and √ 2y 1 v T 1 are fixed to provide allowed range of |∆m 2 atm | taking the value of |∆m 2 atm | as input parameter. For parameters accommodating with neutrino data, we compute the Jarlskog invariant, δ CP given by PMNS matrix elements U αi : The Majorana phases are also estimated via other invariants I 1 and I 2 : We also calculate the effective mass for 0νββ decay given by m ee = |m 1 c 2 12 c 2 13 + m 2 s 2 12 c 2 13 e iα 21 + m 3 s 2 13 e i(α 31 −2δ CP ) |.
1. Model (1) In this model, the modulus τ is the only free parameter in the neutrino mass matrix

Model (2)
This model is similar to model (1) except for the neutrino mass structure. For NO, it is found that we can fit the values of |∆m 2 atm |, |∆m 2 sol | and sin 2 θ 12 . However, the predicted values for the other mixing angles are sin 2 θ 23 ∼ 0.2 and sin 2 θ 13 ∼ 0.45 for both Eq. (36) and (37) solutions, and they cannot be fully fitted to the observed data. For IO, we find that only |∆m 2 atm | and |∆m 2 sol | can be consistent with the observed data as in the model (1).

Model (3)
In this model we can fit the neutrino oscillation data due to the additional free parameterŝ y 2,3 compared to model (1) and (2). For cases A and B, the results are summarized as follows.  in Fig. 2 Fig. 3. The Dirac CP phase is found to be around ±50 • in this case. We also find specific regions on α 21 and α 32 plane and the lightest neutrino mass is restricted to be

Model (4)
This model can be also accommodated with the neutrino oscillation data due to the additional parameters. The results for cases A and B are as follows. also found that IO case is disfavored where we cannot fit three mixing angles simultaneously when we fit the mass differences.

B. Branching ratio of doubly charged scalar boson
Here we calculate the branching ratios (BR) of the doubly charged scalar boson δ ±± . In the type-II seesaw model, δ ±± → ℓ ± ℓ ± decay modes are induced via Yukawa couplings where m ν is the neutrino mass matrix. The doubly charged scalar also decays into same sign W boson pair through the gauge interaction which is proportional to v T 1 . Then leptonic modes are dominant when v T 1 < 10 −4 GeV 4 . In our following analysis we focus on the 4 We can also have decay modes with other scalar bosons in triplet. They can be ignored when masses for components of triplet are degenerated. case where leptonic modes are dominant choosing small v T 1 value since we are interested in prediction for leptonic decay BRs in the model. In addition, we assume the doubly charged scalar mass to be around TeV scale to avoid collider constraints [57][58][59]. In this case the BRs for leptonic modes are simply given by where we ignored decay width for δ ±± → W ± W ± and δ ij is the Kronecker delta. We then estimate the BRs for the parameter sets which can accommodate with the neutrino oscillation data in models (3) and (4).

IV. CONCLUSION
We have discussed type-II seesaw models with modular A 4 symmetry in supersymmetric framework. In our approach, models are classified by the assignment of A 4 representations and modular weights for leptons and triplet Higgs field. The free parameters in models are scanned to fit the neutrino oscillation data and we find the minimal case including only weight 2 modular form is disfavored. We can fit the data for the models with weight 4 modular form applied to the neutrino mass matrix due to additional two free parameters.
Then we have shown the predictions in the neutrino parameters for the allowed parameter small triplet VEV. We can predict the branching ratios where these values are realized to be in some restricted regions. Therefore it can be clear indication of our models if we find the pattern of the branching ratios at the collider experiment. Furthermore we have the relations between predictions in the neutrino parameters and the branching ratios.
Importantly measurement of these branching ratios can test a flavor structure by modular symmetry comparing predictions in the neutrino sector.
Before closing the conclusion, we would like to comment on the other lepton flavor violating (LFV) processes such as µ → eγ, τ → µ(e)γ and µ → 3e, etc. As shown in the branching ratios of the doubly charged scalar boson, the elements of the neutrino mass matrix are constrained and related with each other due to modular A 4 symmetry. Therefore it is expected that the BRs of the LFV processes will also have correlations among them.
If it is the case, such correlations will provide another useful information which enable us to discriminate our model from others. The LFV BRs depend on the mass of the charged scalar bosons and need detailed analyses including the spectrum of the scalar bosons. Such analyses are beyond the scope of this paper and we will leave this for our future work. In this appendix we summarize determination of free parameters, {α, β, γ}, in charged lepton mass matrix in Eq. (11) following discussion in ref. [13]. We have three equations with charged lepton mass eigenvalues: where χ ≡ m 2 e m 2 µ + m 2 µ m 2 τ + m 2 τ m 2 e . The coefficients C 1 , C 2 and C 3 are given byŶ 2 ≡ Y e uφ Y , where Y is real positive and φ Y is a phase parameter, such that The values of these coefficients are determined when we fix the value of modulus τ . We then obtain the general equations to determineα andβ: where s ≡α 2 +β 2 and t ≡α 2β2 . We thus obtainα andβ by the relation: where we separately write the possible two solutions forα and β. Finallyγ is determined byα andβ via Eq. (33a).