Radiative generation of neutrino masses in a 3-3-1 type model

Arindam Das, ∗ Kazuki Enomoto, † Shinya Kanemura, ‡ and Kei Yagyu § Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan Abstract A new model for tiny neutrino masses is proposed in the gauge theory of SU(3)C ⊗ SU(3)L ⊗ U(1)X , where neutrino masses are generated via the quantum effect of new particles. In this model, the fermion content is taken to be minimal to realize the gauge anomaly cancellation, while the scalar sector is extended from the minimal 3-3-1 model to have an additional SU(3)L triplet field. After SU(3)L⊗U(1)X is broken into SU(2)L⊗U(1)Y , the “Zee model” like diagrams are naturally induced, which contain sufficient lepton flavor violating interactions to reproduce current neutrino oscillation data. Furthermore, the remnant Z2 symmetry appears after the electroweak symmetry breaking, which guarantees the stability of dark matter. It is confirmed that this model can satisfy current dark matter data. As an important prediction to test this model, productions and decays of doubly-charged scalar bosons at collider experiments are discussed in successful benchmark scenarios.


I. INTRODUCTION
The structure of the electroweak gauge symmetry SU(2) L ⊗ U(1) Y has been well established by various experiments. The spontaneous breaking SU(2) L ⊗ U(1) Y → U(1) em by the Higgs mechanism has been confirmed by the discovery of the Higgs boson at the LHC. However, the possibility of larger gauge groups including the SU(2) L ⊗ U(1) Y symmetry can also be considered at higher energies than the electroweak scale. In fact based on various physics motivations, a plethora of models with extended gauge symmetries has been proposed.
One of the simple but well-motivated extensions is the SU(3) C ⊗ SU(3) L ⊗ U(1) X (3-3-1) gauge symmetry [1,2]. This extension can naturally give the answer to the question in the Standard Model (SM) "Why are there three generations of chiral fermions?" by the gauge anomaly cancellation. Namely, the number of generation has to be the same as the fundamental color degrees of freedom or its multiples [3]. There are many new particles in models with the 3-3-1 gauge symmetry. Clearly, these models necessarily introduce new gauge bosons. Cancellation of the gauge anomaly requires additional chiral fermions. In order to break the gauge symmetry into the SM one, additional scalar fields have also to be introduced. A next question is whether these new particles can play a role to explain phenomena which cannot be explained in the SM, such as the neutrino oscillation, the existence of dark matter and the baryon asymmetry of the Universe.
Apart from the 3-3-1 models, the radiative seesaw mechanism has been known as the attractive scenario to naturally explain tiny neutrino masses due to loop suppression factors without introducing super heavy new particles. The original model was proposed by Zee [4], in which neutrino masses are generated at 1-loop level. The model proposed by Zee and Babu [5,6] generates neutrino masses at 2-loop level. After these models appeared, radiative seesaw models with dark matter particles have also been proposed, in which the stability of dark matter is guaranteed by a discrete symmetry which is simultaneously forbid tree level diagrams for neutrino masses. For instance, the model proposed by Krauss-Nasri-Trodden [7] and that by Ma [8] correspond to those along this line. Therefore, if we can construct models for generating tiny neutrino masses radiatively based on the 3-3-1 scenario, we can explain origins of tiny neutrino masses, dark matter and generation of chiral fermions simultaneously. This is the subject of the present paper.
Several models with radiative generation of neutrino masses in the 3-3-1 scenario have been proposed in Refs. [9][10][11][12]. These models, however, do not contain dark matter candidates. Recently, in Refs. [13,14] one-loop neutrino mass models including a dark matter candidate have been constructed by the extension of the minimal model based on the SU(3) C ⊗ SU(3) L ⊗ U(1) X ⊗ U(1) N gauge symmetry, so-called 3-3-1-1 models. In the minimal version of 3-3-1-1 models [15][16][17], three right-handed neutrinos are introduced in order to realize the anomaly cancellation, and tiny masses of active neutrinos are obtained by the seesaw mechanism at tree level. In addition, dark matter candidates are naturally obtained by the remnant discrete symmetry after the spontaneous breaking of the 3-3-1-1 gauge symmetry [15,17]. Radiative seesaw scenarios can also be considered in the framework of 3-3-1-1 models [13,14] by changing U(1) N charges for three right-handed neutrinos to avoid the tree level mass term while keeping the dark matter candidates. However, this model is a bit complicated, because many additional particles are further required for the anomaly cancellation and for making all sterile neutrinos massive [14].
In this paper, we construct a new model with the 3-3-1 gauge symmetry in order to explain tiny neutrino masses, dark matter and generation of chiral fermions simultaneously. We take the minimal content of fermions required for the gauge anomaly cancellation, while we introduce an additional scalar SU(3) L triplet field to break the lepton number by the scalar self-interactions. Our model then induces "Zee model" like diagrams after the spontaneous breaking of SU(3) L ⊗ U(1) X → SU(2) L ⊗ U(1) Y . It has been known that the neutrino oscillation data cannot be reproduced by the Zee model due to a too restricted structure of lepton flavor violating (LFV) interactions, see e.g., [18,19]. On the contrary, our model includes additional sources of LFV interactions, so that we can explain current neutrino data.
Another interesting feature of our model is that the appearance of an unbroken discrete Z 2 parity, which guarantees the stability of dark matter candidate, i.e., the lightest neutral particle with a Z 2 -odd charge. This discrete symmetry arises as the remnant symmetry of the 3-3-1 gauge symmetry and a global U(1) ′ symmetry 1 , where the latter is softly-broken and is introduced in order to avoid dangerous flavor changing neutral interactions between a SM fermion and an extra fermion. We find that the Z 2 -odd scalar dark matter can explain the thermal relic abundance satisfying the current direct search results. This paper is organized as follows. In Sec. II, we show the particle content in our model, and give the Higgs potential, kinetic terms for scalar fields and Yukawa interaction terms. In Sec. III, we discuss the generation mechanism for neutrino masses and their mixings. Constraints from LFV decays of charged leptons are considered in Sec. IV. We show numerical results for the correlations between branching ratios of the LFV decays in the parameter sets satisfying the current neutrino data. Sec. V is devoted for the discussion of the dark matter and the collider phenomenology. We conclude the article in Sec. VI. In appendices, we present the mass formulae of physical scalar bosons (App. A) and those for the decay branching ratios for the LFV decays (App. B).

II. MODEL
In this section, we define our model based on the 3-3-1 gauge symmetry. We first present the particle content and give expressions for component fields of SU(3) L (anti-)triplet fields.
We then discuss the Higgs potential, kinetic terms for scalar fields and Yukawa interaction terms in the following subsections in order.

A. Particle content
Fermion Scalar The U (1) ′ symmetry is a global, which is softly-broken by scalar interactions. The flavor indices i and a run over 1-3 and 1-2, respectively.
The particle content and the charge assignment under the 3-3-1 gauge symmetry are shown in Table I, in which the fermion content is taken to be minimal based on the requirement of the gauge anomaly cancellation. From the cancelation of the pure SU(3) L gauge anomaly, generation must be 3, number of the color, which can be regarded as the origin of the three-generation structure for leptons and quarks [3]. In addition to the gauge symmetry, we introduce a global U(1) ′ symmetry, which is softly-broken. This U(1) ′ symmetry is imposed to avoid the dangerous flavor changing neutral current, while it maintains necessary scalar interaction terms in the potential to generate one-loop induced neutrino masses.
The embedding scheme of leptons and quarks into 3-3-1 multiplets is the same as that given in Ref. [22], in which the electric charge Q is defined by where T 3 , T 8 and Q X are the third and the eighth components of the SU(3) L generator and the U(1) X charge, respectively. After the 3-3-1 symmetry is broken into the SU(2) L ⊗U(1) Y symmetry, Y in Eq. (1) is identified with the weak hypercharge.
The component fields of left-handed leptons and quarks are then determined as follows: where the flavor indices i and a run over 1-3 and 1-2, respectively. In Eq. (2), component fields denoted as lowercase (uppercase) letters can be identified as SM (extra) fermions.
We have three, two and one extra charged leptons, up-type quarks and down-type quark, respectively, whose electric charges are the same as the corresponding charged leptons and quarks in the SM. All these extra fermions have Dirac mass terms proportional to the Vacuum Expectation Value (VEV) which breaks the SU(3) L ⊗ U(1) X symmetry into the The scalar triplet fields are represented as In 3-3-1 gauge theories the Higgs sector contains at least three SU(3) L -triplet scalar fields [1,2,22], which give masses to the fermions except for the neutrinos after the electroweak symmetry breaking. In our model, we further introduce an additional triplet field Φ ℓ for the neutrino mass generation, which will be discussed in Sec. III.

B. Higgs potential
The most general Higgs potential is given by where Φ 4 = Φ ℓ . The m 2 13 and µ terms softly break the U(1) ′ symmetry, by which the appearance of an additional Nambu Goldstone (NG) boson is avoided. The parameters m 2 13 , µ, ξ 1 and ξ 2 are complex in general, and these complex phases cannot be simultaneously taken to be zero by phase redefinitions of the scalar fields. For simplicity, we take these parameters to real.
The VEVs of the Higgs triplet fields can generally be taken as The VEV of the third component of Φ 1 can be taken to zero without any loss of generality by using the field rotation of Φ 1 and Φ 3 . The VEV v ′ causes phenomenologically dangerous mixing between the SM fermions and the extra ones. Therefore, we arrange µ = 0 and m 2 13 = 0 so as to have a remnant Z 2 symmetry, by which v ′ = 0 is guaranteed and such a dangerous mixing can be avoided. We call this Z 2 symmetry asZ 2 . The charge of theZ 2 symmetry can be defined as (−1) |Q ′ |/q with Q ′ being the U(1) ′ charge, by which Φ 1,2 and all the SM right-handed fermions are assigned to be odd, while the other fields are even. Thẽ Z 2 symmetry is spontaneously broken by the VEVs v 1 and v 2 , so that domain walls would appear in the early Universe [23,24]. We will briefly comment on this issue at the end of this subsection.
In the following, we assume that V ≫ v 1 , v 2 . Under the setup with v ′ = 0, SU(3) L ⊗U(1) X is spontaneously broken into SU(2) L ⊗ U(1) Y by the VEV V at higher energy scales than the electroweak scale. Then, the SU(2) L ⊗ U(1) Y symmetry is broken down to U(1) em by v 1 and v 2 at the electroweak scale. The Fermi constant G F is reproduced by For the later convenience, we introduce tan β = v 2 /v 1 as the analogue of two Higgs doublet models (THDMs).
After the spontaneous breakdown of the 3-3-1 gauge symmetry and theZ 2 symmetry, another remnant Z 2 symmetry, let us denote it as Z rem 2 , appears, whose charge can be defined as (−1) 2s+2 √ 3T 8 +|Q ′ |/q with s being the spin of the particle. From this definition, all the extra fermions and scalar bosons denoted by η in Eq. (3) are assigned to be odd under Z rem 2 . Because the Z rem 2 symmetry is unbroken, the lightest neutral Z rem 2 -odd particle can be a candidate of dark matter. We will discuss dark matter physics in Sec. V.
Because of the Z rem 2 symmetry, we can classify the physical scalar fields into the Z rem 2 -even and Z rem 2 -odd ones as follows. In the Z rem 2 -even sector, we have two-pairs of singly-charged scalar bosons H ± andĤ ± , one CP-odd Higgs boson A and three CP-even Higgs bosons H i (i = 1, 2, 3). The discovered Higgs boson with a mass of about 125 GeV can be identified with the H 1 state. On the other hand in the Z rem 2 -odd sector, we have one-pair of doublycharged scalar bosons η ±± ℓ , two-pairs of singly-charged scalar bosons η ± andη ± , and one complex neutral scalar boson η 0 . The other eight scalar states are the NG bosons which are absorbed into the longitudinal components of the massive gauge bosons (W µ , W ′ µ , Y µ , Z µ and Z ′ µ ), see Sec. II C. In Appendix A, we explicitly show the relation between the mass eigenstates and the weak eigenbasis of the scalar states and their mass formulae.
Let us discuss the effective theory of our 3-3-1 model in the large VEV limit V ≫ v with V µ ≡ M 2 . In this case, the masses of H ± , A and H 2 are determined by the M parameter 2 , while that of H 1 is determined by v. On the other hand, all the other physical Higgs bosons are decoupled from the theory, as their masses are determined by V . Therefore, the scalar sector effectively coincides with a THDM with a special flavor structure which cannot be realized in THDMs with a softly-broken Z 2 symmetry, see Sec. II D. Similar to the usual THDMs, we can define the decoupling limit by M ≫ v, where only the SM-like Higgs boson H 1 remains at the scale v. We can also define the so-called alignment limit, where the SMlike Higgs boson couplings with the SM gauge bosons and fermions become the same values as those of the SM Higgs boson at tree level. This alignment limit can be taken by choosing potential parameters such that the (1,2) element of the mass matrix of the CP-even Higgs bosons given in Eq. (A14) is zero. Therefore, our 3-3-1 model provides another important example that predicts the THDM as the low energy effective theory other than the minimal supersymmetric extension of the SM [25] and composite Higgs models [26][27][28].
As mentioned in the above, our model potentially has the domain wall problem. It has been known that the energy density of domain walls is only suppressed by the inverse of the radius of the Universe, which is much slower than the dilution of the energy density for ordinary matter and radiation. Therefore, the existence of domain walls could significantly change the history of the Universe. In Refs. [29][30][31][32], solutions for the domain wall problem have been discussed. According to Ref. [29], if a discrete symmetry which is spontaneously broken by Higgs VEVs (in our model, this corresponds to the electroweak symmetry breaking VEV v) is not restored at high temperature, the domain wall problem might not arise. Such situation can happen if finite temperature effects which are proportional to T 2 on a negative mass squared term are also negative [33]. In the SM, this does not happen, because there is only one scalar quartic coupling. Such quartic coupling gives a positive effect of finite temperature on the negative mass squared term, so that the broken symmetry at zero temperature is restored at high temperature as it is seen in the usual thermal history of the Universe. On the other hand, if we consider models with multi-scalar fields as in our model, this is not always the case, because some combinations of scalar quartic parameters can be taken to negative so as to realize the symmetry non-restoration scenario. Thus, we might be able to avoid the domain wall problem. Clearly, more dedicated discussions for this solution have to be done in order to ensure its justification, which is beyond the scope of this paper.

C. Kinetic terms for scalar fields
Kinetic terms for the scalar triplet fields are expressed as where Φ 4 = Φ ℓ . The covariant derivative D µ for SU(3) L triplet fields is given by with g and g X being the SU(3) L and U(1) X gauge couplings, respectively. The SU(3) L gauge boson A µ is expressed by the 3 × 3 matrix form as: (4,5) and (6,7). We can identify W µ ≡ A µ 12 and W ′µ ≡ A µ 45 with the SM W boson and the additional charged gauge boson, respectively, while Y µ ≡ A µ 67 with a neutral complex gauge boson. Their masses are given by the VEVs of the Higgs triplet fields as follows: where s θ = sin θ and c θ = cos θ. In addition to these complex states, there are three real neutral gauge bosons, where one of them can be identified with the massless photon γ µ . We can define the basis where γ µ is separated from the other two massive states (Z µ andZ ′µ ) as follows: TheZ andZ ′ states can be mixed. Their mass matrix is given by In the large V limit, the mass eigenvalues are expressed as From the expression of m 2 Z , we see that the weak mixing angle θ W can be identified with We then have the same expression of m Z as that of the mass of the SM Z boson. The U(1) em coupling e is consistently given by e = g sin θ W as that in the SM.
The electroweak rho parameter can be expressed at tree level by using the definition of θ W given in Eq. (13), In order to satisfy |ρ − 1| ≤ 10 −3 , V has to be taken to be larger than around 8 TeV. We note that the calculation of one-loop corrections to the rho parameter is different from that in models with ρ = 1 at tree level, e.g., the SM. In models with ρ = 1 at tree level, the electroweak sector is described by four input parameters which can be chosen to be α em , m Z , G F and δρ with the last one being the deviation of the rho parameter from unity.
This means that the radiative correction to the rho parameter cannot be a prediction, because the additional parameter δρ provides an additional counterterm by which loop corrections to the rho parameter can be absorbed by imposing a renormalization condition.
The similar situation can also happen in models with higher isospin scalar multiplets with a non-vanishing VEV such as SU(2) L triplet scalar fields [34,35].

D. Yukawa interactions
Thanks to the global U(1) ′ symmetry, Yukawa interaction terms for the SM right-handed fermions and those for the extra right-handed fermions are separately given as follows: where ǫ αβγ is the complete antisymmetric tensor with α, β and γ being the indices for the SU(3) L triplet. Because of ǫ αβγ , the complex 3 × 3 matrix f ij has also to be antisymmetric.
From the structure of the VEVs and the Yukawa interactions given in Eqs. where We note that M E , M U and M D are respectively the 3 × 3, the 2 × 2 and the 1 × 1 matrices.
They are diagonalized by bi-unitary transformations: As we can see in Eq. (17), the mass matrix for the SM charged leptons has the same form as in the SM; i.e., only one of the Higgs fields gives their masses. On the other hand, the mass matrices for the up-type and the down-type SM quarks are given by two VEVs of the Higgs fields; i.e., 3 × 3 matrices are composed of the two independent Yukawa matrices.
This structure predicts characteristic flavor-dependent Higgs-boson couplings to quarks [36] in the THDM which is effectively deduced after the SU(3) L ⊗ U(1) X symmetry breaking as discussed in Sec. II B.
The f ij terms in Eq. (15) do not contribute to the masses of charged leptons, but they play an important role for the neutrino mass generation. From these interaction terms, we can assign two units of the lepton number to the Higgs triplet field Φ ℓ . This lepton number is explicitly broken by the ξ 1 and ξ 2 terms in the Higgs potential given in Eq. (4), and they turn out to be the source of the Majorana masses for the active neutrinos. We will discuss the neutrino mass generation in Sec. III.

III. NEUTRINO MASSES
Majorana neutrino masses are generated at one-loop level as shown in Fig. 1, in which the Z rem 2 -even and Z rem 2 -odd particles run in the loop in the left and right diagram, respectively. We here give relevant interaction terms among physical charged scalar bosons and fermions in their mass eigenbases for the calculation of one-loop induced neutrino masses; where The total contribution to the Majorana neutrino masses is then expressed as where M e ν and M E ν represent the contribution from the left and right diagram, respectively. They are calculated as The flavor independent coefficients C e and C E are given by The mass matrix given in Eq. (20) is diagonalized by introducing the Pontecorvo-Maki-Nakagawa-Sakata matrix U PMNS as; where m i ν (i = 1, 2, 3) are mass eigenvalues for neutrinos, and with s ij = sin θ ij , c ij = cos θ ij and δ CP being the CP phase. We consider both the cases for the orders of the neutrino masses; i.e., the normal hierarchy (|m 1 ν | < |m 2 ν | < |m 3 ν |) and the inverted hierarchy (|m 3 ν | < |m 1 ν | < |m 2 ν |). The flavor structure of M e ν is the same as that of the Zee model. It has been known that the Zee model cannot explain the current neutrino oscillation data because of the too restricted structure of flavor violating couplings which only arise from the antisymmetric 3 × 3 F matrix, see e.g., [18,19] 3 . Although the flavor structure of M E ν also takes the similar form to that of M e ν , another flavor violating source in the matrix W is inserted into the mass matrix. Consequently, M E ν has a different flavor mixing pattern. Hence, it can explain the current neutrino data. We note that such an additional flavor violating source vanishes if we take the masses of the extra leptons degenerate; i.e., M 1 = M 2 = M 3 , by which M 2 k and G k in Eq. (21) commute with W † and then the effect of the W matrix disappears by the unitarity. We also note that the contribution from M E ν is typically much larger than that from M e ν , because the latter is proportional to the mass squared of the SM charged leptons. Therefore, the neutrino masses and the mixings are determined essentially only by the contribution from M E ν . We thus switch off the contribution from M e ν hereafter for simplicity, which can be realized by taking C e = 0 or equivalently θ H = 0.

IV. LEPTON FLAVOR VIOLATION
In our model, new particles can contribute to LFV decays of the charged leptons. In this section, we discuss the constraints from ℓ → ℓ ′ γ and ℓ → ℓ ′ ℓ ′′ ℓ ′′′ types of LFV decays in the parameter sets which satisfy the current neutrino oscillation data. 3 If we take the general Yukawa interactions for leptons, then there is a corner of parameter space which can satisfy the current neutrino results [37,38].
Diagrams of the ℓ → ℓ ′ γ processes are shown in Fig. 2. The branching ratios of these processes are calculated as where (ℓ 1 , ℓ 2 , ℓ 3 ) = (e, µ, τ ) and (C 21 , C 31 , C 32 ) = (1, 0.1784, 0.1736). In the above expression, a ϕ R and a ϕ L represent ϕ-loop (ϕ = η 0 , η ±± ℓ , H ± andĤ ± ) contributions to the amplitude for the decays with the right-and left-handed initial charged lepton ℓ i , respectively. Their analytic expressions are presented in Appendix B. From the first diagram in Fig. 2, the structure of the W matrix is constrained. Since W is a unitary matrix, we cannot simply take small values for each component. We can instead take the small mass difference among E i in order to suppress the contribution from the first diagram. On the other hand, the magnitude of the contributions from the second and the third diagrams can be easily suppressed by taking small values for the F matrix elements. Typically, |F ij | 10 −3 is required from the constraint by the µ → eγ data. Fig. 3 shows one-loop box diagrams contributing to the ℓ → ℓ ′ ℓ ′′ ℓ ′′′ type of the LFV decay process. Among the six possible processes, the branching ratio of µ → 3e is most strongly constrained by the data. In our model, the branching ratio of µ → 3e is calculated as where a ijkl (i, j, k, l = L or R) denote contributions from the diagrams with the i-handed muon and the j-handed electron in the µ-e current, and the k-handed positron and the l-handed electron in the e-e current. Detailed expressions for each contribution are given in Appendix B. We note that the µ → 3e data typically do not further constrain the parameter region allowed by the ℓ → ℓ ′ γ data.
Now, let us numerically show the prediction of the branching ratios of ℓ → ℓ ′ γ and µ → 3e decays in the parameter sets which satisfy the current neutrino data. As aforementioned in Sec. III, we take C e = 0 (or θ H = 0) in the numerical evaluation. We assume the CPconservation in the Yukawa interaction terms. The W matrix given in Eq.
We then take the following parameters as inputs; The parameters in the first line are required for the neutrino mass calculation. For the calculation of the LFV decays, we also need to specify the parameters in the second line.
We refer to the neutrino oscillation data given in Ref. [42], and we apply the 3σ allowed ranges of two squared mass differences and three mixing angles to our analysis.
In Fig. 4, we show various correlations between the branching ratios of the LFV decays.
In these plots, we scan six parameters F ij and w ij , and fix other parameters as written in the caption of Fig. 4. The coefficient C E is determined so as to reproduce the mass squared difference of atmospheric neutrinos, i.e., ∆m 2 atm = |(m 3 ν ) 2 − (m 1 ν ) 2 |. All these points satisfy  the current neutrino oscillation data assuming the normal hierarchy for neutrino masses, where the black (red) points show the case with tan β = 30 (100). We see that the red points are given in the lower-left region of these planes as compared with the black points, because values of the branching ratios of ℓ → ℓ ′ γ are dominantly determined by the second term of Eq. (B1), which is proportional to c β . We note that the loop contributions of η ±± ℓ andĤ ± are unimportant as long as |F ij | become larger, whose magnitude is typically taken to be smaller than O(10 −3 ). We see that the µ → eγ data give the most sever constraint on the parameter space, because of its strongest upper bound on the branching ratio. For the other two modes τ → µγ and τ → eγ, our predictions are typically smaller than the current limit by two or more orders of magnitude, because the branching ratio has already been highly suppressed by the µ → eγ data. It is also seen that the branching ratio of µ → 3e is significantly lower than the current upper limit. We observe an anticorrelation between the branching ratios of µ → eγ and τ → eγ, see the most upper-left panel, which is predicted by our characteristic flavor structure of the Yukawa interactions. In addition, we find a very strong correlation between the branching ratios of µ → eγ and µ → 3e. This can be understood from the fact that the largest factor f L or f R given in Eq. (B4) mainly determines the size of these branching ratios.
Similar plots but for the inverted hierarchy case are shown in Fig. 5. We see that B(τ → µγ) tends to have similar values with the order of 10 −11 (10 −12 ) for tan β = 30 (100) as a function of the other branching ratios, which cannot be seen in the normal hierarchy case.
The other behavior is quite similar to the normal hierarchy case.
Let us give a comment on cases for the other sets of the fixed parameters in the above analysis. Among the fixed parameters, the mass differences between the extra leptons can significantly affect the results of the LFV branching ratios. For larger values of the mass difference, these branching ratios tend to become larger, because the suppression by the unitarity of the W matrix becomes weaker. Therefore, larger values of tan β or V are required to avoid the constraint from the µ → eγ data. Varying the other mass parameters such as m η ± does not change the above results significantly.

V. PHENOMENOLOGY
In this section, we discuss phenomenological consequences of our model.

A. Dark matter physics
As we discussed in Sec. II, our model has a dark matter candidate η 0 , which is the Z rem 2 -odd complex neutral scalar boson. This scalar boson has trilinear interaction terms with neutral Z rem 2 -and CP-even scalar bosons H i (i = 1, 2, 3), among which H 1 can be identified with the discovered Higgs boson (h) with the mass of 125 GeV. Therefore, the phenomenology of dark matter is similar to the Higgs portal scenario in which annihilation processes occur via the s-channel Higgs-boson mediations.
If the dark matter mass m η 0 is smaller than 2m W and if the additional Higgs bosons H 2 and H 3 are much heavier than 2m W , the dominant annihilation process is η 0 η 0 * → h ( * ) → ff with f = t, whose thermal averaged cross section is evaluated at the leading order as where Γ h is the width of h (∼ 4 MeV) and λ hηη is the dimensionful η 0 η 0 * h coupling. From Eq. (31), the annihilation cross section becomes significant when m η 0 is getting close to m h /2 due to the resonant effect of h, so that smaller values of λ hηη are required to keep the observed relic abundance of dark matter, Ω DM h 2 ≃ 0.12 [43].
On the other hand, constraints from dark matter direct detections have to be taken into account. In our scenario, the dark matter scattering with a nucleon N through the t-channel Higgs mediation becomes to be most important. Using the effective vertex, which is given by with g N ≃ 1.1 × 10 −3 [44], the scattering cross section is expressed as where m N is the mass of the nucleon. We here neglect the 3-momentum of the dark matter. The black curve shows the required value of λ hηη /v satisfying Ω DM h 2 = 0.12 as a function of the dark matter mass.
In Fig. 6, we show the combined results for the calculations of the relic abundance and the bound from the direct search experiment (XENON1T) [45]. The red curve represents the upper limit on the normalized hη 0 η 0 * coupling by the VEV λ hηη /v as a function of the dark matter mass m η 0 . The required value of λ hηη /v to satisfy Ω DM h 2 = 0.12 is shown as the black curve. As already mentioned, smaller values of λ hηη are required to keep the observed value of the abundance when the dark matter mass is around the resonance region ∼ m h /2. Our dark matter candidate can simultaneously satisfy both the relic abundance and the direct detection bounds at m η 0 ∼ m h /2 as it has been known in Higgs portal models, see e.g., [46][47][48]. Another scenario with a much larger mass ( a few TeV) may also be considered for the dark matter to satisfy both the dark matter data. However, we do not discuss details of this case because such a scenario strongly depends on the parameters of extra fields. Instead, we only have shown that there is at least a solution in our model to satisfy the dark matter data in addition to the neutrino data. Because of the special Yukawa interaction, they can decay into quarks with different flavors such as A/H 2 → tc and H ± → ts [36]. Dedicated simulation studies for these flavor violating decays of the extra Higgs bosons at the LHC have been performed in Ref. [49]. In addition, there are the other extra particles whose masses are proportional to V such as η ±± ℓ , η ± ,η ± , η 0Ĥ ± and H 3 . They can also be detected at the LHC if the associated coupling constants are small enough.
One of the most interesting signatures in our model arises from the doubly-charged scalar bosons η ±± ℓ . At the LHC, they can be created in pair via the Drell-Yan process pp → γ * /Z * → η ++ ℓ η −− ℓ and in associated with the singly-charged scalar bosons pp → W ± * → η ±± ℓη ∓ /η ±± ℓ η ∓ [50]. In Fig. 7, we show the cross sections for these production processes at the pp colliders with the collision energy of 13 TeV (left) and 27 TeV (right). We use Using the mass spectrum assumed in the analyses for the neutrino masses and LFV decays, the decay pattern of η ±± where η 0 is the dark matter candidate. The intermediate state E ± k can be on-shell in this case. The flavor of the same-sign dilepton in the final state is determined by the F and W matrices, which are constrained by the neutrino oscillation data and the LFV data. Therefore, by measuring the flavor of the same-sign dilepton system, we can indirectly test the mechanism for the neutrino mass generation. The partial decay rates of E ± i and η ±± ℓ are calculated as where f L and f R are given in Eq. (B4), and the mass of the SM charged leptons is neglected.  All these benchmark points satisfy the neutrino oscillation data and the LFV data assuming the normal hierarchy case for the neutrino masses.

Inputs
Outputs In these expressions, small differences of the masses and the widths of E i 4 are ignored; i.e., . The total width Γ E is typically of order 0.1 GeV, so that the narrow width approximation is valid for the calculation of the decay rate of η ±± ℓ . In Table II and Table III,  contains missing energies which are carried by the dark matter η 0 , the invariant mass distribution of the same-sign dilepton system does not have a peak at around the mass of η ±± ℓ . This property is different from that of doubly-charged Higgs bosons from an SU(2) L triplet or a singlet field, which can decay into the same-sign dilepton without missing energy. Therefore, the current bounds on the mass of such doublycharged Higgs bosons, around 800 GeV depending on the flavor of the final state leptons at the LHC [52], cannot be applied to that on η ±± ℓ . In order to extract the bound on the mass of η ±± ℓ from current experiments and its discovery potential at future experiments, dedicated simulation studies are needed, which are beyond the scope of this paper.
For the sake of completeness, let us discuss the phenomenology for the singly-charged scalar bosons η ± ,η ± andĤ ± . For simplicity, we neglect the effect of the small mixing angle θ η (θ H ) between η ± andη ± (H ± andĤ ± ). These bosons can be produced in pair via the Drell-Yan process at collider experiments. In addition,η ± can also be produced in association with η ±± ℓ as already discussed in the above text, see also Fig. 7 for its production cross section. Their decay processes can be η ± /η ± → ν i E ± k → ν i ℓ ± j η 0 5 andĤ ± → ℓ ± i ν j . Because of the Z rem 2 symmetry, the decays of η ± andη ± include the dark matter. The decay of η ± (η ± andĤ ± ) occurs via the Yukawa coupling y E (F ), so that the flavor dependence of the charged lepton in the final state could be different among the decaying particles. We note that the singly-charged scalar bosons in the inert doublet model can decay into the W boson and a lighter Z 2 -odd scalar particle. Therefore, the signatures from the decays of η ± andη ± can be different from those in the inert doublet model [53,54]. On the other hand, the decay property ofĤ ± is quite similar to that of singly-charged scalar bosons in the Zee 5 If η ±± ℓ are lighter thanη ± , theη ± → η ±± ℓ W ∓( * ) processes are also possible.

VI. CONCLUSIONS
We have proposed a new model for the generation of tiny neutrino masses based on the 3-3-1 gauge symmetry within the minimal fermion content required for the gauge anomaly cancellation. In this model, the source for lepton number violation is obtained by extending the minimal Higgs sector of 3-3-1 models with an additional SU(3) L triplet scalar field.
Majorana masses for the active neutrinos are generated at one-loop level. We have found the parameter sets which satisfy the current neutrino data under the constraint from the LFV decays of the charged leptons such as µ → eγ and µ → 3e.
In our model, Z rem 2 appears as a remnant symmetry after the breaking of the electroweak symmetry and that of the global U(1) ′ symmetry, where the latter symmetry is introduced to avoid the dangerous flavor changing neutral current. The symmetry Z rem 2 guarantees the stability of the dark matter candidate which is the lightest neutral Z rem 2 -odd scalar particle η 0 . We have confirmed that the dark matter candidate can satisfy the relic abundance and the direct search results when the dark matter mass is taken to be at around the half of the discovered Higgs boson mass.
We then have discussed the collider phenomenology of our model. One of the most interesting signatures arise from productions and decays of the doubly-charged scalar bosons  We first give the mass formulae for the Z rem 2 -even scalar bosons. The neutral components of the scalar triplet fields can be expressed as with v 3 = V . There are 3 pairs of singly-charged, 3 CP-odd and 3 CP-even scalar states in the Z rem 2 -even sector. Their mass eigenstates are obtained by introducing the following orthogonal transformations: where G ± , G 0 and G ′0 are the NG bosons which are absorbed into the longitudinal component of W ± , Z and Z ′ , respectively. In Eq. (A4), R H is the 3 × 3 orthogonal matrix which can be expressed by three independent mixing angles. These mixing angles are determined from the mass matrix for the CP-even Higgs bosons given in Eq. (A14). In Eq. (A3), x ± 12 ≡ (x 1 ± x 2 )/|x ± 12 | and x 3 are three component vectors defined as x T 2 = (c β , −s β , 0) , (A14) In the above expressions, the lower-left elements are the same as the corresponding transposed elements.
Next, we present the mass formulae for the Z rem 2 -odd scalar states, in which there are one pair of doubly-charged, 3 pairs of singly-charged and one neutral complex scalar states.
The squared mass of the doubly-charged scalar bosons η ±± ℓ is given by The mass eigenstates of the singly-charged and neutral states are defined as follows: where G ′± and G 0 Y are the NG bosons which are absorbed into the longitudinal component of W ′ and Y , respectively. The squared masses of the physical scalar bosons and the mixing angle θ η are given by The mass matrix for the singly-charged state M 2 η ± is calculated as The amplitudes a ϕ L are obtained by taking f L ↔ f R in Eq. (B1) and m i ↔ m j in Eqs. (B2)-(B3).
For the µ → 3e process, we have with F ij box (m) ≡