An Inverse Seesaw model with $A_4$-modular symmetry

We discuss an inverse seesaw model based on right-handed fermion specific $U(1)$ gauge symmetry and $A_4$-modular symmetry. These symmetries forbid unnecessary terms and restrict structures of Yukawa interactions which are relevant to inverse seesaw mechanism. Then we can obtain some predictions in neutrino sector such as Dirac-CP phase and sum of neutrino mass, which are shown by our numerical analysis. Besides the relation among masses of heavy pseudo-Dirac neutrino can be obtained since it is also restricted by the modular symmetry. We also discuss implications to lepton flavor violation and collider physics in our model.


I. INTRODUCTION
One of the challenging issue in particle physics is the understanding of flavor structure of fermions in the standard model (SM). In the SM, we do not have any principle to determine the structure and we expect it can be explained in a framework of new physics beyond the SM. In constructing a new physics model to describe the flavor structure, a new symmetry can play an important role to control the structure of flavors.
Recently the framework of modular flavor symmetries have been proposed by [1,2] to realize more predictable flavor structures. In this framework, a coupling can be transformed under a non-trivial representation of a non-Abelian discrete group and many scalar fields such as flavons are not necessary to realize flavor structure. Then some typical groups are found to be available in basis of the A 4 modular group [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], S 3 [18][19][20][21], S 4 [22][23][24][25][26][27][28], A 5 [27,29,30], larger groups [31], multiple modular symmetries [32], and double covering of A 4 [33] in which masses, mixing, and CP phases for quark and/or lepton are predicted. 1 A possible correction from Kähler potential is also discussed in Ref. [42]. Furthermore, a systematic approach to understand the origin of CP transformations has been recently discussed in ref. [43], and CP violation in models with modular symmetry is discussed in Ref. [44]. In particular, it is interesting to apply a modular symmetry in constructing a new physics model for neutrino mass generation since we would obtain prediction for signals of new physics correlated with observables in neutrino sector.
In this paper, we apply modular A 4 symmetry to the inverse seesaw mechanism which is realized introducing new local Abelian gauge symmetry, denoted as U (1) R , which is righthanded fermion specific [45]. The inverse seesaw mechanism requires a left-handed neutral fermions S L in addition to the right-handed ones N R , and provides us more complicated neutrino mass matrix which can make mass hierarchies softer than the other models such as canonical seesaw [46][47][48][49] and provide rich phenomenologies such as unitarity constraints [50,51]. The U (1) R symmetry requires three SM singlet fermions with non-zero U (1) R charge to cancel gauge anomaly [52][53][54][55][56] and forbid unnecessary Yukawa interactions to obtain inverse seesaw mechanism. We then assign A 4 triplet representation to S L and N R , and some relevant Yukawa couplings are written in terms of modular form providing a constrained 1 Some reviews are useful to understand the non-Abelian group and its applications to flavor structure [34][35][36][37][38][39][40][41]. structure of the neutrino mass matrix. Then we perform numerical analysis scanning free parameters in the model and search for the region which can fit neutrino data. For the allowed parameter sets, we show predictions in observables in the neutrino sector. This paper is structured as follows. In Sec.II we briefly revisit the well-known inverse seesaw mechanism with discrete A 4 -modular flavor symmetry and its appealing feature resulting in simple mass structures for charged leptons and neutral leptons including light active neutrinos and other two types of sterile neutrinos. We provide also the analytic formula for light neutrino masses and mixing along with the discussion on the non-unitarity effect. In Sec.III we study numerically the correlations between observables in the neutrino sector along with the input model parameters arising in A 4 -modular symmetry and its predictions to lepton flavor violation. We briefly comment on collider aspects of TeV scale pseudo-Dirac neutrinos in Sec.IV and conclude our results in Sec.V.

II. MODEL
We briefly discuss here the model framework for inverse seesaw mechanism introducing right-handed fermion under specific local Abelian symmetry U (1) R and modular A 4 symmetry. In the model, we introduce three families of right(left)-handed SU (2) singlet fermions N R (S L ) with 1(0) charge under the U (1) R gauge symmetry, and an isospin singlet boson ϕ with 1 charge under the same U (1) symmetry. Furthermore, the SM Higgs boson H also has charge 1 under U (1) R to induce the masses of SM fermions from the Yukawa Lagrangian after the spontaneous symmetry breaking. 2 Here we denote each of vacuum expectation value to be H ≡ v/ √ 2, and ϕ ≡ v / √ 2. The scalar and gauge sector are the same as in Ref. [55] where U (1) R gauge boson mass is given by VEV of ϕ and new gauge coupling In this paper, we omit the details of the scalar sector and focus on the neutrino sector.
Using the particle content and symmetries mentioned in Table I, the relevant Yukawa Lagrangian for leptons-including charged leptons and neutral leptons-can be written as, where L M is interaction Lagrangian for charge leptons, L M D is for Dirac neutrino mass term connecting active light neutrinos ν L and other sterile neutrino N R , L M is for mixing term between two types of sterile neutrinos N R and S L and L µ is for Majorana mass term for sterile neutrino S L . The Majorana mass terms for the sterile neutrinos N R and another term L L HS c L connecting ν L and S L are absent in the present framework due to appropriate U (1) R charge assignments. These Lagrangians should be invariant under A 4 symmetry and sum of modular weight should be zero for each term.

Fermions
Bosons   After spontaneous symmetry breaking, the charged lepton mass matrix is found to be diagonal, Then it is found that the generic Dirac Yukawa term L LH N R is protected by A 4 modular symmetry. The advantage of A 4 modular symmetry here is to allow this term without introducing additional fields while allowing the corresponding Yukawa coupling transforming under A 4 modular group as triplets shown in Table II. We use the modular forms of weight 2, Y (τ ) = (y 1 (τ ), y 2 (τ ), y 3 (τ )), transforming as a triplet of A 4 which is given in terms of Dedekind eta-function η(τ ) and its derivative [2] which is given as Eq. (V.8) in the Appendix.
As a result of this, the relevant term for Dirac neutrino mass connecting light neutrinos ν L and sterile neutrinos N R is given by where subscript for the operator Y N R indicates A 4 representation constructed by the product and {α D , β D , γ D } are free parameters. Using H = v/ √ 2, the resulting Dirac neutrino mass matrix is found to be, Mixing term connecting N R and S L (M): We chose both types of sterile neutrinos N R and S L transforming as triplets 3 under A 4 modular group. However the mixing term S L N R is forbidden due to U (1) R charge assignment. Then this term is obtained from a Yukawa interaction with scalar singlet ϕ which has non-zero U (1) R charge and singlet under modular A 4 symmetry. The allowed mixing term for N R and S L is given by where first and second term in the first line correspond to symmetric and anti-symmetric product forS L N R making 3 representation of A 4 . Using ϕ = v / √ 2, the resulting mass matrix is found to be, Majorana mass term for S L (µ): Since the sterile neutrino S L transforming as triplet 3 under A 4 modular group with zero modular weight, the Majorana mass term can be written as, A. Inverse Seesaw mechanism for light neutrino Masses Within the present model invoked with A 4 modular symmetry the complete 9 × 9 neutral fermion mass matrix for inverse seesaw mechanism in the flavor basis of (ν L , N c R , S L ) is given by (II.10) Using the appropriate mass hierarchy among mass matrices as given below 3 , the inverse seesaw mass formula for light neutrinos is given by (II.12) The above relation can be read as, Since mass parameters for M D , µ, M N S are overall factors, we can define a dimensionless neutrino mass matrixm ν as follows: The hierarchies among mass parameters could be explained by several mechanisms such as radiative models [57][58][59] and effective models with higher order terms [60].
. In this case, κ is determined by where ∆m 2 atm is atmospheric neutrino mass difference squares, and NO and IO stand for normal and inverted ordering respectively. Subsequently, the solar mass difference squares can be written in terms of κ as follows: which can be compared to the observed value. For heavy sterile neutrino, we obtain pseudo Dirac mass for µ 0 M N S and mass eigenvalues are obtained by diagonalizing M N S . We write these eigenvalues as M 1,2,3 which will be numerically estimated.
In our model, one finds U P M N S = V ν since the charged-lepton is diagonal basis, and it is parametrized by three mixing angle θ ij (i, j = 1, 2, 3; i < j), one CP violating Dirac phase where c ij and s ij stands for cos θ ij and sin θ ij respectively. Then, each of mixing is given in terms of the component of U P M N S as follows: (II.17) Also we compute the Jarlskog invariant and δ CP derived from PMNS matrix elements U αi : (II. 19) In addition, the effective mass for the neutrinoless double beta decay is given by m ee = κ|D ν 1 cos 2 θ 12 cos 2 θ 13 +D ν 2 sin 2 θ 12 cos 2 θ 13 e iα 21 +D ν 3 sin 2 θ 13 e i(α 31 −2δ CP ) |, (II. 20) where its observed value could be measured by KamLAND-Zen in future [61]. We will adopt the neutrino experimental data at 3σ interval [62]  We apply these ranges in searching for allowed parameter space in our numerical analysis.

B. Non-unitarity
Here, let us briefly discuss non-unitarity matrix U M N S . This is typically parametrized by the form (II.24) free parameters inM N S andm D are taken to be the same order. Therefore, Non-unitarity can be controlled by v which is expected to be large mass scale.

III. NUMERICAL ANALYSIS
In this section, we carry out numerical analysis. Scanning free parameters in the model, we search for parameter sets satisfying neutrino data and obtain some predictions in neutrino sector.

Neutrino mass and mixing
Here we numerically analyze neutrino mass matrix applying the formulas in the previous section. To fit neutrino data, we consider the free input parameters in following ranges: where parameter µ 0 is determined by Eq. (II.13) and not a free parameter. Under these regions, we randomly scan the parameters and search for the allowed parameter sets satisfying all the neutrino oscillation data.
As a result, we find parameter sets which can fit the neutrino data for both NO and IO cases. The typical region of modulus τ is found in narrow space as -1.  In Fig. 2  mixing allowed in the present A 4 -modular inverse seesaw mechanism and heavy pseudo-Dirac neutrino around few hundreds GeV mass range, the branching ratio for popular lepton flavor violating decay µ → eγ is given by [65] BR(µ → eγ) = 3α 32π Here, M i represents the heavy pseudo-Dirac neutrinos and With M i = 1 TeV, the branching ratio for lepton flavor violating decay µ → eγ is recasted in following way [66] The current bound derived from MEG experiment is BR(µ → eγ) < 4.2 × 10 −13 [68,69].
The only way to distinguish between pseudo-Dirac from Majorana neutrinos at collider through careful analysis of their decay channels. In case of heavy Majorana neutrinos at TeV scale, like in type-I seesaw mechanism, the typical mixing between light-heavy neutrinos is θ νN m ν M −1 N ≤ 10 −6 (see ref [73] and references therein). In case of inverse seesaw mechanism with the introduction of small lepton number violating term µ, the seesaw scale can be naturally in the testable range leading to large light-heavy neutrino mixing. At first, we produce heavy neutrinos, if kinematically allowed, through qq → W + L → + N for heavy pseudo-Dirac neutrinos. After that heavy pseudo-Dirac neutrino decays to − + ν α which crucially depends on large light-heavy neutrino mixing.

V. CONCLUSION
We have studied an inverse seesaw model based on right-handed specific U (1) gauge symmetry and modular A 4 symmetry where these symmetries forbid unnecessary terms and restrict structures of relevant Yukawa interactions. Majorana neutrino mass matrix has been formulated and it is characterized by modulus τ and some free parameters.
We have then carried out a numerical analysis to search for parameter sets that can fit neutrino oscillation data in both normal and inverted ordering cases. For the allowed parameter sets, we find predictions such that; the Dirac CP phases to be ∼ The modular groupΓ is the group of linear fractional transformation γ acting on the modulus τ belonging to the upper-half complex plane and its transformation is given as where k is the so-called as the modular weight.
Here we discuss the modular symmetric theory without imposing supersymmetry explicitly. In this paper, we consider the where −k I is the modular weight and ρ (I) (γ) denotes an unitary representation matrix of γ ∈ Γ(2).
The kinetic terms of scalar fields can be written by which is invariant under the modular transformation and overall factor is eventually absorbed by a field redefinition. Then the Lagrangian should be invariant under the modular symmetry.
The modular forms with weight 2, Y = (y 1 , y 2 , y 3 ), transforming as a triplet of A 4 is written in terms of Dedekind eta-function η(τ ) and its derivative [2]: