Radiatively Induced Neutrino Mass Model with Flavor Dependent Gauge Symmetry

We study a radiative seesaw model at one-loop level with a flavor dependent gauge symmetry $U(1)_{\mu-\tau}$, in which we consider bosonic dark matter. We also analyze the constraints from lepton flavor violations, muon $g-2$, relic density of dark matter, and collider physics, and carry out numerical analysis to search for allowed parameter region which satisfy all the constraints and to investigate some predictions. Furthermore we find that a simple but adhoc hypothesis induces specific two zero texture with inverse mass matrix, which provides us several predictions such as a specific pattern of Dirac CP phase.


I. INTRODUCTION
The observation of neutrino oscillation confirms at least two non-zero masses of active neutrinos indicating physics beyond the standard model (SM) to generate the neutrino masses. Radiative seesaw models are one of the attractive candidate to generate the neutrino masses where a neutrino mass matrix is induced at loop level and a dark matter (DM) candidate can be included as a particle propagating inside a loop diagram for generating neutrino mass. It is also interesting to include flavor dependent gauge symmetry with which we can obtain predictive structure of neutrino mass matrix [1,2].
In this paper, we construct a radiative seesaw model with U(1) µ−τ gauge symmetry and Z 2 symmetry in which we introduce exotic SU(2) L doublet leptons with U(1) µ−τ , a Z 2 even singlet scalar field, and Z 2 odd triplet and singlet scalar fields. In the model, active neutrino mass matrix is generated at one loop level where Z 2 odd particles propagate inside a loop diagram. Furthermore we have DM candidate which is the lightest Z 2 odd neutral particle.
Then global numerical analysis is carried out to search for allowed parameter region and to investigate some predictions in the model, taking into account constraints from charged lepton flavor violation (cLFV), ∆a µ , and relic density of DM. In addition, we find that structure of the Dirac mass matrix of exotic lepton determines that of the active neutrino mass matrix when we apply assumptions i) degenerate masses for exotic leptons, or ii) some vanishing Yukawa couplings which are associated with interactions among SM leptons, exotic leptons and exotic scalars. In that case, we have two zero texture of the neutrino mass matrix which provides some predictions in neutrino oscillation experiments. This paper is organized as follows. In Sec. II, we introduce our model and discuss some  phenomenologies such as neutrino mass matrix, lepton flavor violation, and some processes induced by Z ′ interactions. The numerical analysis is carried out in Sec. III to search for parameter region satisfying experimental constraints and to obtain some prediction for neutrino mass matrix. Finally we summarize the results in Sec. IV.

II. MODEL, PARTICLE PROPERTIES AND PHENOMENOLOGY
In this section, we introduce our model and discuss some phenomenologies. As extra symmetries, local U(1) µ−τ and discrete Z 2 symmetries are added. In the fermion sector, we introduce SU(2) L doublet vector like fermions L ′ e,µ,τ ≡ [N, E] T e,µ,τ , and impose a flavor dependent gauge symmetry U(1) µ−τ as summarized in Table I. Also Z 2 odd parity is imposed for this new fermion in order to discriminate the SM model leptons with SU(2) L and forbid the mixing between them. 1 In the scalar sector, we add an SU(2) L triplet inert scalar ∆, real singlet inert scalar S, and singlet scalar ϕ to the SM Higgs Φ as summarized in Table II. Notice here the Higgs doublet Φ (that spontaneously breaks electroweak symmetry), the SU(2) singlet field ϕ (that spontaneously break U(1) µ−τ symmetry), have the vacuum expectation values (VEVs), which are respectively symbolized by v/ √ 2, v ′ / √ 2, and Z 2 odd parity is also imposed for the inert scalars ∆ and S to forbid the tree level neutrino masses through VEVs. Therefore the lightest neutral scalar boson with Z 2 odd parity can be a DM candidate.
Yukawa interactions and scalar potential: Under these fields and symmetries, the renormalizable Lagrangians for quark and lepton sector are given by where σ 2 is the second Pauli matrix, and again L ′ ≡ [N, E] T .
We parametrize the scalar fields as where v ≃ 246 GeV is VEV of the Higgs doublet, and w ± , z, and z ′ are respectively Nambu-

matrices as
where c(s) a(α) is the short-hand notation of cos(sin) a(α) . Notice here that we assmue small mixing case O a ≈ 1 in following analysis, which could however be an natural assumption because s a 0.4 is indicated from the data of LHC experiment [20][21][22][23]; therefore we take After the µ −τ gauge symmetry breaking, vector-like fermion mass matrix can be written in the basis [L ′ e , L ′ µ , L ′ τ ] T as follows: where we have simply assumed M L ′ to be a real symmetric matrix and define M eµ ≡ where M 1,2,3 is the mass eigenstate.

A. Active neutrino mass and lepton flavor violating processes
Our active neutrino mass matrix is given in general at one-loop level by the diagram shown in Fig. 1 which is calculated as [19] m th where , y S and y ∆ are diagonal Yukawa matrices respectively. Here is neutrino mass eigenvalues [24]. Therefore we have to satisfy the relation m th ν ≈ m exp ν . The smallness of neutrino masses ∼ 10 −12 GeV partly arises from loop suppression factor and small mixing of s α ∼0.1; s α c α /(4π) 2 ∼ 10 −3 , but the other factor is controlled by Yukawa couplings; y ∆ y S ∼ 10 −11 for D N ≈ O(100) GeV. Thus each of Yukawa couplings could typically be the same order of electron Yukawa coupling; y ∆ ∼ y S ∼ 10 −5 . On the other hand the mass hierarchy between M and m Ha does not severely affect the order of neutrino masses.
Lepton flavor violations(LFVs) arises from the term y ∆ and y S at one-loop level, and its form can be given by 1784, and C 32 ≈ 0.1736. Experimental upper bounds are respectively given by New contributions to the muon anomalous magnetic moment (muon g − 2: ∆a µ ) arises from Yukawa terms y ∆ with negative contribution and y S with positive contribution. Also another source via additional gauge sector can also be induced by where r ≡ (m µ /M Z ′ ) 2 , and Z ′ is the new gauge vector boson. Thus we could explain the [3], if we can satisfy the constraint of trident process. Notice here that g Z ′ 10 −3 [27] has to be satisfied due to the trident process.

B. Dark matter
Here we consider the lightest inert boson X ≡ H 1 ≈ S, assuming s α << 1 for simplicity.
As we commented in previous subsection, this small mixing plays a role of suppression factor in neutrino mass formula while mass relation does not give significant change in the neutrino mass. Then annihilation modes generally arise from interactions associated with coupling constants y ∆ and y S , and SM-Higgs portal. However we found that Yukawa modes cannot explain the sizable relic density; Its cross section is of the order 10 −10 GeV −2 at most, even when large coupling y S is favor of the muon g − 2. Thus we should rely on interactions in the scalar sector to explain thermal relic density of DM. Then we focus on interactions between h 1 and X since the SM Higgs portal interaction is highly constrained by the direct detection experiments. Before considering the relic density, we also have to discuss the direct detection bound for h 1 portal coupling. The stringent bound comes from spin independent nucleon-DM scattering via the h 1 scalar boson portal, 4 and its cross section is evaluated as where λ ϕS is coefficient of |ϕ| 2 S 2 , and m N ≈ 0.939 GeV is neutron mass. The recent experiment LUX [31] provides the bound on the scattering cross section as σ SI 2.
where s, t, u are Mandelstam valuables, c a ≃ 1 is taken, we have assumed narrow width of Then the relic density of DM is given by [33] Ωh 2 ≈ 1.07 × 10 9 , (II. 19) where g * (x f ≈ 25) ≈ 100 is the degrees of freedom for relativistic particles at the freeze- x 2 ) is given by [36] J Then one has to satisfy the current relic density of DM; Ωh 2 ≈ 0.12 [34]. In our numerical analysis below we focus on annihilation mode of 2X → {ZZ, W + W − , tt, 2h 2 } assuming m h 1 to be heavy. Also we have assumed other scalar contact interactions such as λ ΦS is small, and we have neglected mixing between Z − Z ′ , thus we do not consider the modes 2X → Z ′ Z ′ .

III. NUMERICAL ANALYSIS
In this section, we show a global analysis, where we have fixed some parameters for simplicity. At first, we fix m H 2 = m ∆ ± = m ∆ ±± in order to evade the constraints from oblique parameters in the triplet boson; the S, T, U-parameters are suppressed when the masses in the triplet are degenerated [35]. Also we numerically solve our parameters Y ≡ (y Se , y Sµ , y Sτ , y ∆ 2 , y ∆ 3 ), by using the relation m th ν = m exp ν , 5 where we impose the perturbative bounds on these output parameters; Y √ 4π. Thus we randomly select the following range of reduced input parameters as 100 GeV ≤ M X GeV, m H 2 ∈ [1.2M X , 2500] GeV, where we have used experimental neutrino oscillation data in ref. [25] with 3σ range. In In Fig. 4, we show the allowed scattering plots in terms of sum of neutrino masses and m ν 1 . It suggests that the lightest neutrino mass is of the order 10 −12 eV. 5 In principle, six parameters can numerically be solved, but it is technically difficult in our model.
In Fig. 5, we demonstrate Majorana phases; ρ(with red points) and σ(with blue points) in terms of Dirac phase δ, where the red/blue present the region in M X ∈ [100, 350] GeV. 6 It displays that δ runs over π ∼ 2π, whereas Majorana phases tend to be localized, depending on ρ and σ. Especially, both of these phases are in favor of being localized at around π/2 that could be one of the remarkable features of this model.
In fig. 6,   Comment on the specific case: It is worth mentioning the following two hypotheses that lead a predictive two-zero texture with (m ν ) 22 = (m ν ) 33 = 0: The case i) suggests that a fermion DM is in a coannihilation system to satisfy the correct relic density of Universe when M < m H 1,2 . Notice here that the lower mass bound on M is around 100 GeV from the LEP experiment. Transversely a bosonic DM candidate can simply satisfy the relic density.
The case ii) suggests that a fermion DM does not require a coannihilation process among neutral fermions, however it must still be considered between the exotic charged fermions E due to the constraint of oblique parameter.
In both of the cases, the situation could be more or less same if we identify DM as the bosonic DM candidate, and we adopt i) in our discussion below. Before starting the discussion of neutrinos, let us roughly estimate the degree of our predictability from µ − τ symmetry. Since we have eleven free parameters (three in y S , three in y ∆ , and five in M L ′ ) which contribute to form the texture, it still seems to remain nine free parameters even after imposing the above conditions i) or ii). Thus naive expectation gives no predictions while one finds the type-C of neutrino texture that has only seven parameters. It suggests that two more freedom in the parameter sets are reduced by our specific textures of y S , y ∆ , and M L ′ which are determined by the µ − τ symmetry. Thus our model still improve predictability by two degrees of freedom due to the symmetry. 7 Then m ν is simplified as Eq. (III.5) corresponds to the type C two zero texture that provides several predictions that only an inverted neutrino mass ordering is allowed and specific pattern of phases. In fig. 7, we show ρ(red) and σ(blue) in terms of δ, where we adapt the recent global neutrino oscillation data [25] up to 3σ confidence level and the same input value in the general analysis. It implies that the region of ρ is restricted to be 0 ∼ 3π/2, whereas σ be 3π/2 ∼ 2π, and these are overlapped at around δ = 3π/2 that is in good agreement with the current neutrino experiments as the best fit value. In this case, the dominant contribution of muon g − 2 arises from ∆a Z ′ µ , where g Z ′ 10 −3 [27] is satisfied due to the trident process. While the relic density of DM can be obtained by the Yukawa coupling y S that leads to the d-wave dominant. This result is opposite to the one of general feature, although we do not show the detailed analysis here because this is nothing but ad-hoc hypothesis.

IV. CONCLUSIONS AND DISCUSSIONS
We have proposed a radiative seesaw model at one-loop level with a flavor dependent gauge symmetry U(1) µ−τ , in which we have consider gauge singlet-like bosonic dark matter The allowed region between ρ(red) and σ(blue) in terms of δ, where we adapt the recent global neutrino oscillation data [25] up to 3σ confidential level and the same input value in the general analysis. It implies that the region of ρ is restricted to be 0 ∼ 3π/2, whereas σ 3π/2 ∼ 2π, and these are overlapped at around δ = 3π/2 that is the best fit value in the current neutrino experiments.
candidate and explained muon g − 2 without conflict of LFVs. In the numerical analysis, we have shown several features as follows: 1. Whole the DM mass region with λ ϕS s a ≈ 0.021 is obtained by the experimental bounds on spin independent scattering and relic density of DM. And this range is in good agreement with the current experimental data of muon g − 2 without conflict of LFVs as well as neutrino oscillation data.
2. The typical lightest neutrino mass is of the order 10 −12 eV.
3. There exist a mild correlation between the Dirac phase δ and Majorana phases ρ, σ. Therefore, δ runs over π ∼ 2π, whereas Majorana phases tend to be localized, depending on ρ and σ. Especially, both of these phases are in favor of being localized at around π/2. 4. As a specific case such as M ≡ M 1 ≈ M 2 ≈ M 3 , we have found the predictive two zero texture(type-C) and their features are clearer than the generic one. As an example, the region of ρ is restricted to be 0 ∼ 3π/2, whereas σ be 3π/2 ∼ 2π, and these are overlapped at around δ = 3π/2 that is in good agreement with the current neutrino experiments as the best fit value.
Finally, we have an inert doubly charged Higgs boson which decay into dark matter and SM fermions by cascade decay modes. It will be interesting to search for the signal of "missing E T + same sign leptons" as a signature of the inert Higgs triplet as well as our dark matter. The detailed analysis of the signal is beyond the scope of this paper and it will be studied elsewhere.
If global U(1) µ−τ symmetry is applied to our model, a few results could change. The first one is that the muon g − 2 due to the absence of Z ′ contribution. The second one is that a new annihilation mode of DM relic density has to be added; 2X → 2G, where G is a physical massless goldstone boson. As a result, the allowed range of DM mass is wider, since whole the cross section increases.