Neutrino mass in a gauged $L_\mu - L_\tau$ model

We study the origin of neutrino mass through lepton-number violation and spontaneous $U(1)_{L_\mu-L_\tau}$ symmetry breaking. To accomplish the purpose, we include one Higgs triplet, two singlet scalars, and two vector-like doublet leptons in the $U(1)_{L_\mu-L_\tau}$ gauge extension of the standard model. To completely determine the free parameters, we employ the Frampton-Glashow-Marfatia (FGM) two-zero texture neutrino mass matrix as a theoretical input. It is found that when some particular Yukawa couplings vanish, an FGM pattern can be achieved in the model. Besides the explanation of neutrino data, we find that the absolute value of neutrino mass $m_j$ can be obtained in the model, and their sum can satisfy the upper bound of the cosmological measurement with $\sum_j |m_j|<0.12$ eV. The effective Majorana neutrino mass for neutrinoless double-beta decay is below the current upper limit and is obtained as $\langle m_{\beta \beta} \rangle =(0.34,\, 2.3)\times 10^{-2}$ eV. In addition, the doubly charged Higgs $H^{\pm\pm}$ decaying to $\mu^\pm \tau^\pm$ final states can be induced from a dimension-6 operator and is not suppressed, and its branching ratio is compatible with the $H^{\pm \pm}\to W^\pm W^\pm$ decay when the vacuum expectation value of Higgs triplet is $O(0.01)$ GeV.

In spite of the mass hierarchy among the quarks and charged leptons, the particle masses, except the neutrinos, in the standard model (SM) can be attributed to the Brout-Englert-Higgs (BEH) mechanism [1, 2], where the predicted Higgs boson was observed by ATLAS [3] and CMS [4] at a mass of 125 GeV. Based on the neutrino oscillation experiments, it was found that the neutrinos are also massive particles; however, the definite origin of their masses so far is unknown.
Moreover, although nonzero neutrino masses have been determined by the experiments, we still cannot tell their mass ordering, i.e., m 1 < m 2 < m 3 or m 3 < m 1 < m 2 is possible, where the former and latter are the mass spectrum with normal ordering (NO) and inverted ordering (IO), respectively. Hence, the current neutrino data can be shown in terms of the different mass ordering as [5]: where m 2 21 ≡ m 2 2 − m 2 1 , m 2 23 denotes m 2 3 − m 2 2 for NO or m 2 2 − m 2 3 for IO, and θ ij are the mixing angles of Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [6,7]. From the results, it is clearly seen that the PMNS matrix pattern is different from the Cabibbo-Kobayashi-Maskawa (CKM) for the quark-flavor mixing. In this work, we plan to study a model, where based on a flavor symmetry, the neutrino masses are dynamically generated without introducing singlet right-handed neutrinos, and all neutrino data can be explained.
In addition, the model can also have interesting phenomenological implications on flavor and collider physics.
In order to dynamically generate the neutrino masses, we require that each Majorana matrix entry has to be related to the effects of which arises from the lepton-number violation and spontaneous U(1) Lµ−Lτ symmetry breaking. To achieve the lepton-number violation, we introduce a Higgs triplet, which carries hypercharge Y = 1 and has no U(1) Lµ−Lτ charge.
Like the type-II seesaw [39,40], the vacuum expectation value (VEV) of this triplet can dictate the lepton-number violating effects. It is found that due to the protection of U(1) Lµ−Lτ gauge symmetry, we cannot obtain a realistic Majorana neutrino mass matrix without further discussing the U(1) Lµ−Lτ breaking. Therefore, to break the gauge symmetry, we employ two singlet scalars, which carry different U(1) Lµ−Lτ charges. Due to the chirality, the SM leptons cannot couple to the singlet scalars; therefore, we need to introduce proper exotic heavy leptons as the media. To avoid the anomaly cancellation, we employ two vector-like doublet leptons as the candidates. Based on the U(1) Lµ−Lτ gauge symmetry, the number of singlet scalars and vector-like leptons (VLLs) in this approach is the minimal requirement to obtain a proper Majorana neutrino mass matrix.
It will be demonstrated later that not all Yukawa couplings appearing in the neutrino mass matrix are small. Therefore, in addition to the neutrino issue, the model can also provide interesting phenomena on flavor and collider physics. For instance, the lepton-flavor violating h → µτ decay can be as large as the current measurements; the excess of muon g−2 can be resolved by the mediation of Z ′ gauge boson and new light scalars; and the doubly charged Higgs decaying to µτ and W W can be compatible each other without requiring the VEV of Higgs triplet to be the eV.
In the following, we start to introduce the model under the SU(2) L × U(1) Y × U(1) Lµ−Lτ local gauge symmetry. In order to dynamically generate the neutrino mass in the U(1) Lµ−Lτ extension of the SM, in addition to the SM particles, we include one Higgs triplet (∆), two vector-like doublet leptons (L 4 , L 5 ), and two singlet scalars (S, S ′ ). Their U(1) Lµ−Lτ charges are given in Table I, where the SM particles not shown in the table carry no such U(1) charges. Accordingly, the Yukawa couplings to the Higgs triplet are written as: From above equation, if the Higgs triplet ∆ carry two units of lepton number, the Yukawa where the pattern of mass matrix leads to m 2 = m 3 , θ 13 = θ 12 = 0, and θ 23 = π/4 [8,16,17].
Obviously, the results cannot explain the current neutrino data [5]. We clearly demonstrate that the neutrino mass matrix, which is arisen from the electroweak symmetry breaking and lepton-number violation, cannot explain the neutrino data due to the protection of U(1) Lµ−Lτ gauge invariance. In order to obtain a realistic neutrino mass matrix, we need to rely on other pieces of Yukawa interactions, which can break the U(1) symmetry. Concerning the magnitude of v ∆ , according to the electroweak symmetry breaking, the electroweak ρparameter at the tree-level can be written as [41]: Taking the current precision measurement for ρ-parameter within 2σ errors, the VEV of ∆ has to be less than 3.4 GeV.
(2), the gauge invariant Yukawa couplings to the Higgs and S (′) are given by: where H is the SM Higgs doublet, only the first term is from the SM, and the other Yukawa interactions arise from the new particles. Although Eq. (5) can cause rich interesting phenomena on the lepton-flavor physics, since we focus on the neutrino physics in this work, the detailed study on the flavor physics can be referred to [30]. Based on the Yukawa interactions in Eq. (5), it is found that the new entries of the Majorana mass matrix can be induced from higher dimensional operators, where the Feynman diagrams are sketched in Fig. 1, and the associated gauge invariant dimension-5 and -6 operators can be formulated as: with∆ = iτ 2 ∆. From the effective Lagrangian, when the U(1) Lµ−Lτ gauge symmetry is can be generated from Eq. (6) with ∆ = v ∆ / √ 2. We note that the dimension-6 operator

FIG. 1: Sketched Feynman diagrams for the Majorana neutrino mass matrix elements.
Since the neutrino masses are generated by the spontaneous U(1) Lµ−Lτ symmetry breaking, we need to find the conditions for the vacuum stability. We thus write the gauge invariant scalar potential in this model as: The VEVs of scalar fields are obtained by the minimal conditions ∂ V /∂v H,S,S ′ ,∆ = 0, and each condition can be expressed as: where we have ignored the v ∆ terms in the first three equations and the v 3 ∆ terms in the last equation due to v ∆ ≪ v H,S,S ′ . In order to avoid the precision Higgs measurements, we can adopt the mixing between H and S(S ′ ) to be small; then, the VEV of H can be simplified If we further take λ 13 and µ S to be small, the VEVs of S and S ′ can be found as The v S and v S ′ are free parameters and their relation to the Z ′ -boson mass is given by m 2 ; hence, their magnitudes can be taken as the value of the electroweak scale. From Eq. (11), the VEV of Higgs triplet can be determined as: Because of v ∆ < 3.4 GeV, in order to obtain the heavy Higgs triplet bosons, unlike Higgs doublet and S(S ′ ), m 2 ∆ has to be positive and dictates the masses of Higgs triplet bosons. From Eq. (12), it can be seen that similar to the type-II seesaw model [39,40], the Higgs triplet VEV is directly related to the lepton-number soft breaking term.
If we write the symmetric Majorana neutrino mass matrix as: from the Yukawa couplings in Eqs.
(2) and (6), each matrix element can then be expressed as: Although the neutrino mass matrix comes from the dimension-4, -5, and -6 operators, since the involving free parameters are different, the matrix entries in Eq. (14) can be taken as the same in the order of magnitude and have no particular hierarchy among them, unless there is a further indication. Due to the U(1) Lµ−Lτ gauge symmetry, the light charged-lepton mass matrix in the first term of Eq. (5) has been diagonal. Although the other Yukawa interactions can induce the off-diagonal elements, these induced terms indeed are suppressed [30]. If we neglect these small off-diagonal effects with s ij ≡ sin θ ij , c ij ≡ cos θ ij , and δ being the Dirac CP violating phase.
From Eq. (13), there are six different complex matrix elements. After rotating three unphysical phases, we have nine independent parameters. Since neutrino oscillation experiments cannot observe the two Majorana CP phases, even α 21 = α 31 = 0, we still have seven free parameters. However, we only have six observables: ∆m 2 21,31 , sin 2 θ 12,13,23 , and Dirac CP phase δ; that is, we cannot determine all free parameters without further theoretical or experimental inputs. It has been investigated that a class of neutrino mass matrices may suffice to explain all neutrino experiments if the matrix textures have two independent zeroes [42]. The seven possible Frampton-Glashow-Marfatia (FGM) matrix patterns are classified as: where the symbol X denotes a nonzero texture. The detailed study with two-zero textures can be found in [43][44][45]. In order to simplifying the analysis, it is a good approach to employ the FGM patterns as theoretical inputs.
As mentioned earlier that the neutrino mass ordering is still uncertain, i.e. m 1 < m 2 < m 3 or m 3 < m 1 < m 2 is allowed. With a FGM pattern, it helps understand what form of a neutrino mass matrix can lead to a certain mass ordering. According to the study in [46], it was concluded that by taking the neutrino data with 1σ errors, the NO spectrum can be achieved by the patterns A 1,2 and B 1,2,3,4 , while the patterns B 1,3 and C can conduct to IO spectrum. Accordingly, it is of interest to see how the matrix elements of Eq. (14) in our model realize each FGM matrix. It is found that when some Yukawa couplings are required to vanish, a definite FGM matrix pattern can then be accomplished. We show the vanishing Yukawa couplings for the corresponding FGM matrix in Table II. It is worth mentioning that the powerful FGM matrix pattern can also predict the absolute values of neutrino masses, where they so far have not yet observed in experiments.
Since our purpose is not to examine the all FGM patterns, for numerical analysis, we take the patterns A 1 and C as the representatives of NO and IO mass spectra, respectively.
To determine the non-vanishing entries of the neutrino mass matrix and |m i |, we use the neutrino data at the 1σ level as shown in Eq. (1). Due to the large experimental uncertainty, the values of Dirac CP phase are taken from a global data analysis with a χ 2 method [48], in which the results in the 1σ region are δ/π = (1.18, 1.61) for NO and δ/π = (1.12, 1.62) for IO. Combining the experimental inputs with two independent zero textures, we basically have eight known inputs; thus, we can completely constrain the four non-vanishing complex entries of the patterns A 1 and C.
According to the relation M ν = U * M ν dia U † and the zero textures in the M ν , the mass relations in the pattern A 1 can be expressed as: while in the pattern C, they are: where m k s in general are complex; however, there are only two independent phases among m 1,2,3 . If we take the central values of measured θ 12,13 in Eq. (1), sin 2 θ 23 ≈ 0.47, and δ ≈ 1.3π, we can easily obtain: However, it is found that the pattern C is very sensitive to the values of mixing angles and CP phase δ when ∆m 2 21 and ∆m 2 32 are required to fit the data within 1σ errors. If we take sin 2 θ 23 ≈ 0.4515 and δ ≈ 1.59205π, the results are: When we further take ∆m 2 21 ≈ 7.53 × 10 −5 eV 2 , the values of |m i | and ∆m 2 23 can be determined as: C : |m 1 | ≈ 9.07 × 10 −2 eV , |m 2 | ≈ 9.11 × 10 −2 eV , From the analysis, the pattern A 1 and C can fit the neutrino data for NO and IO mass spectra at the 1σ level, respectively. However, if we compare the results with the cosmological limit on the sum of neutrino masses, which is given as [47]: it can be found that the resulting j |m j | in the pattern A 1 can satisfy the upper bound while that in the pattern C is higher than the limit. In order to understand whether the tension with the cosmological neutrino mass bound can be relaxed when the ranges of experimental measurements are extended, we adopt neutrino data at the 2σ level instead of 1σ level for the pattern C. For numerical analysis, we generate 5 × In the end, the number of output points, which can fit the ∆m 2 21,23 data in 1(2)σ range, is 552(3004). The resulting Dirac CP phase δ and j |m j | are shown in Fig. 2, where the dots in black and red denote the results with 1σ and 2σ errors, respectively. From the figure, it can be clearly seen that j |m j | in the pattern C can still satisfy the bound from the cosmological measurements when the all neutrino data are taken at the 2σ level. Since the uncertainties of sin 2 θ 23 and ∆m 2 32 in Eq. (1) correspond to a 68% confidence level (CL), and the measurement of sin 2 θ 13 is associated with the value of m 2 32 [5], in our following analysis we only use the pattern A 1 , which can fit the neutrino data within 1σ errors, to show the constraints of the involving Yukawa couplings. From the mass diagonal relation M ν ℓℓ ′ = (U ℓk U ℓ ′ k ) * m k , when the PMNS matrix entries and m k are known, M ν ℓℓ ′ can then be determined. Following earlier discussions, where the FGM pattern A 1 can predict the value of each m j , we therefore show the correlation between δ and |m j | in Fig. 3(a), where the neutrino data within 1σ error have been satisfied. From the plot, it can be seen that each |m i | is located at around the value of Eq. (21) with a very narrow range. In the plot, we also show the effective Majorana neutrino mass m ββ , which is related to the neutrinoless double-beta (0νββ) decay rate, and is defined by [18]: allowed ranges for m ij as a correlation of |m τ τ |, where FGM pattern A 1 is applied and neutrino data within 1σ errors are taken.
We now discuss the limits on the Yukawa couplings in Eqs.
(2) and (5). To simplify the analysis, we take m 4L ≈ m 5L ≡ m L and v S ≈ v S ′ ≡ v X , and define the parameters as: where a R,L can lead to the Higgs lepton-flavor violating h → µτ decay, and its branching ratio (BR) is given by [30]: With m h ≈ 125 GeV and Γ h ≈ 4.21 MeV, the limit on a L,R can be obtained as where BR(h → µτ ) can be taken from the experimental data, and the current upper limits from ATLAS and CMS are 1.43% [50] and 1.26% [51,52], respectively. Taking BR(h → µτ ) ∼ 10 −3 and |a L | ∼ |a R |, we obtain |a L,R | ∼ 6.5 × 10 −4 . Based on the new parameters, the neutrino mass matrix entries in Eq. (14) are expressed as: According to Eq. (24), if we take |m µµ | ≈ |m τ τ | ∼ 2.7 × 10 −2 eV, |a L,R | ∼ 6.5 × 10 −4 , the magnitudes of parameters are obtained as: Assuming ξ 45 ≈ −ξ ′ 45 and taking |y τ | ∼ 0.1, v X ∼ 100 GeV, and m L ∼ 1000 GeV, which can lead to a sizable BR(h → µτ ), the magnitudes of ξ (′) ab can be shown as: |y * e ξ ′ 45 | ∼ 1.5 × 10 −8 GeV , |ξ µ4 | ≈ |ξ τ 5 | ∼ 2.9 × 10 −9 GeV , |ξ µτ | ∼ 2.3 × 10 −11 GeV , (30) where |m eτ | = 10 −2 eV and |m µτ | = 2.3 × 10 −2 eV are used, and the second and third terms in m µτ have been neglected due to y S , y ′ S ≪ 1. To avoid the strict constraint from µ → eγ and µ → 3e rare decays, we can take y e ≪ 1; thus, the Yukawa couplings can have the After determining the magnitudes of the Higgs-triplet Yukawa couplings, which are used to explain the neutrino data, we make some remarks on the implications of this model in flavor and collider physics. If the new Z ′ gauge boson is in the MeV to GeV range, in addition to explaining the excess of muon g − 2 and the large BR for h → µτ decay, the sizable Yukawa couplings y (′) τ and y (′) µ can lead to the τ → µZ ′ Z ′ decay through the mediation of light scalar S [30]. Unlike the type-II seesaw model, the doubly charged Higgs (H ±± ) can decay to the right-handed µ ± τ ± via the induced dimension-6 operator, expressed as where the corresponding H ±± Yukawa coupling to µ ± τ ± is Y H ±± = Y 45 y τ y µ v 2 H /(2m 2 L ). From Eq. (30), Y 45 can in principle be O(1), depending on the y e value. Thus, with m L ∼ 1000 GeV, v H ∼ 246 GeV, m H ±± ∼ 400 GeV, and y τ ∼ y µ ∼ 0.1, the decay rate ratio of H ±± → µ ± τ ± to H ±± → W ± W ± can be estimated as [53]: It can be seen that with |Y 45 | ∼ O(1) and v ∆ ∼ 0.1 GeV, the BR for H ±± → µ ± τ ± can be compatible with that for H ±± → W ± W ± .
In summary, we studied the origin of neutrino mass in the gauged L µ − L τ model. We learnt that although one Higgs triplet can violate the lepton number, it does not suffice to explain the neutrino data due to the U(1) Lµ−Lτ gauge invariance. It was found that a proper symmetric Majorana mass matrix can be obtained when a pair of vector-like leptons and two singlet scalars carrying the L µ − L τ charges are introduced. In this model, a certain Frampton-Glashow-Marfatia matrix pattern can be realized when some Yukawa couplings are set to vanish. Using pattern A 1 , we showed that when the neutrino data with 1σ errors and cosmological neutrino bound are satisfied, the involving Higgs-triplet Yukawa couplings have a hierarchy, i.e., Y µτ ≪ Y µ4,τ 5 ≪ Y 45 , Y ′ 45 , and Y (′) 45 can be O(1). As a result, the effective Majorana neutrino mass is below the current experimental upper limit. Moreover, the model can have interesting phenomena in flavor and collider physics, such as muon g − 2, h → µτ , τ → µZ ′ Z ′ , and H ±± → (W ± W ± , µ ± τ ± ) decays.