A two loop induced neutrino mass model with modular $A_4$ symmetry

We propose a model with radiatively induced neutrino mass at two-loop level, applying modular $A_4$ symmetry. The neutrino mass matrix is formulated where the structure of associated couplings are restricted by the symmetry. Then we show several predictions in the lepton sector, satisfying lepton flavor violations as well as neutrino oscillation data. We also discuss muon anomalous magnetic moment and briefly comment on dark matter candidate.


I. INTRODUCTION
Understanding flavor structure for quark and lepton sectors is still important even after the discovery of Higgs boson in the standard model (SM). Along the idea of thought, non-Abelian discrete groups have been vastly adopted as flavor symmetries in order to predict and/or reproduce current experimental data of the mixings and masses of quarks and leptons [1][2][3][4][5][6][7][8]. However, since there are too many possibilities of their applications, one cannot address something concrete models independently in this manner. Recently, modular originated flavor symmetries have been proposed by [9,10] that are more promising ideas to obtain predictions to the quark and lepton sector, since Yukawa couplings also have a representation of the flavor groups. Their typical groups are found in basis of the A 4 modular group [10][11][12][13][14][15][16][17], S 3 [18,19], S 4 [20,21], A 5 [22,23], larger groups [24], and multiple modular symmetries [25] that have been applied to studies of flavor structures of quarks and leptons and the feature of dark matter (DM) candidate [14]. Another advantage of this modular groups is that fields and couplings have to assign a modular weight originated from modular groups. This number can be identified to be a symmetry to stabilize DM candidate if DM is included in a model.
In this paper, we apply an A 4 modular symmetry to the lepton sector in a framework of modified Zee-Babu type model [26,27] generating non-zero masses of neutrinos whose masses are generated at two-loop level. In the model, we introduce exotic vector-like charged leptons in addition to the field contents in original Zee-Babu model [26] which propagate inside a loop diagram generating neutrino mass. In our analysis, we show several predictions to the lepton sector, satisfying constraints of lepton flavor violations (LFVs) as well as neutrino oscillation data and discussing muon anomalous magnetic moment (muon g − 2). Finally, we briefly comment on our DM candidate in the conclusion. 1 This paper is organized as follows. In Sec. II, we give our model set up under the modular A 4 symmetry, writing down relevant fields and couplings and their assignments. Then, we formulate the valid Lagrangians, Higgs potential, exotic field mass matrix, LFVs, muon g − 2, neutrinos mass matrix, and numerical analysis in which we show several predictions to satisfy all the data that we will discuss. Finally we conclude and discuss in Sec. III.

Fermions
Bosons  Lepton couplings Higgs terms

II. MODEL
Here we explain our model with modular A 4 symmetry in which some fields have nonzero modular weight and couplings with non-zero modular weight are modular forms. In the fermion sector, we introduce three exotic singly-charged leptons as a triplet under A 4 with modular weight −1, while all the SM leptons L L , e R have zero modular weight assigned three kinds of singlet 1, 1 ′ , 1 ′′ for each flavor under A 4 . In the scalar sector, we introduce an isospin doublet field η and three singlet fields (ϕ, s − , k −− ) having non-zero modular weight (−1, −2, −1, −2) and hypercharge (1/2, 0, −1, −2) respectively, where all the scalar fields are true singlet under A 4 . We expect that η is an inert boson to induce nonzero neutrino mass at two-loop level, and its neutral component can be a DM candidate whose stability is assured by nonzero modular weight; this is due to the fact that all couplings should have even modular weight and fields with odd modular weight cannot singly appear in interactions.
Vacuum expectation value (VEV) of H and ϕ is respectively denoted by where H is identified as SM-like Higgs field. We summarize field assignments in table I and couplings in table II. Under these symmetries, one writes renormalizable Lagrangian as follows: where ω = e i 2 3 π and the charged-lepton mass eigenstate is directly given by the first term above. Thus, the observed mixing matrix for lepton sector is found in the neutrino sector only. The modular forms of weight 2, (f 1 , f 2 , f 3 ), transforming as a triplet of A 4 is written in terms of Dedekind eta-function η(τ ) and its derivative [10]: The overall coefficient in Eq. (II.2) is one possible choice; it cannot be uniquely determined.
Thus we just impose the purtabative limit f 1,2,3 √ 4π in the numerical analysis.
In the similar way as Yukawa couplings, M E with modular weight 2 is also written by where M 0 can be taken as a free parameter determining scale of vector-like charged lepton mass. Thus mass hierarchy among three vector-like charged leptons are given, once we fix modulus τ . While A 4 triplet g i with modular weight 4 is written by [16] 2 The A 4 singlets couplings g ′ i with modular weight 4 are also written by These structures are also determined by modulus τ .
A 4 modular invariant Higgs potential is given by where µ H,1,2 , λ H do not have modular weight and just real values. Here ϕ plays a role in inducing the mass of S ± and η after ϕ develops VEV. In the singly-charged bosons in basis 2 We use a different basis of A 4 group under which its triplet representations are constructed to be symmetric and anti-symmetric.
of S ± , η ± , which mix each other through λ, we define the mixing and its mass eigenvalue as follows: where s α (c α ) is the short-hand symbol of sin α(cos α).
After the electroweak spontaneous symmetry breaking, the charged-lepton mass matrix is given by while the E mass matrix is given by . LFVs, Muon g-2, Neutrino masses; First of all, let us rewrite the term of f as where E ′ is mass eigenstate and This interaction induces LFV decay process ℓ i → ℓ j γ at one-loop level, and the branching ratios are given by , (II.14) where G 21 = 1, G 31 = 0.1784, G 32 = 0.1736, α em is the electromagnetic fine structure constant, and G F is the Fermi constant. In addition, the muon g − 2 is given by The experimental upper bounds for LFVs are given by [31][32][33]  Neutrino mass matrix is given at two-loop level by the diagram in Fig.1, and its formula is found as , and here we assume to be g ≡ g L = g R for simplicity. Then g is given by given by the recent cosmological data [37]. Since the charged-lepton is mass eigenstate from the beginning, one identifies U ν as U M N S . Each of mixing is given in terms of the component of U M N S as follows: Also, the effective mass for the neutrinoless double beta decay is given by m ee = |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. 22) where its observed value could be measured by KamLAND-Zen in future [38].

A. Numerical analysis
Here, we show numerical analysis to satisfy all of the constraints that we discussed above, where we restrict ourselves the neutrino mass ordering is normal hierarchy. First of all, we provide the allowed ranges for neutrino mixings and mass difference squares at 3σ range [39] as follows: 2. α 21 is found to be zero.
3. The first generation of neutrino mass is at most 7 × 10 −5 eV.
4. Allowed region of sin 2 θ 13 lies in the whole region of experimental result.
5. The typical size of muon g − 2 is of the order 10 −13 , and its maximal value is about 6 × 10 −14 eV.

III. CONCLUSION AND DISCUSSION
We have proposed a two-loop induced neutrino mass model with a modular A 4 symmetry, and discussed predictions of neutrino oscillation data as well as LFVs, muon anomalous magnetic moment related to interactions generating neutrino mass. We have found several δ ℓ CP = [95 • −120 • ], α 21 = 0. Furthermore, the first generation of neutrino mass is tiny whose typical order is 10 −4 eV, and muon g − 2 is at most 6 × 10 −13 , which is smaller than the observed value by 10 4 order of magnitude.
Before closing this section, it would be worthwhile to mention dark matter candidate. In our model, neutral component of η can be the one, if there is mass difference between the real part and imaginary part to evade the direct detection search. This can be achieved by introducing, e.g., SU(2) L triplet boson with (1, −2) for hypercharge and modular weight, respectively. The systematic analysis has already been done by ref. [40] that tells us the dark matter mass is at around 534 GeV when the mass is larger than the mass of W/Z mass. In lighter region, one also finds the dark matter mass is at around the half of Higgs mass; 63 GeV.