Predictions for the neutrino parameters in the minimal gauged U(1)$_{L_\mu-L_\tau}$ model

We study the structure of the neutrino mass matrix in the minimal gauged U(1)$_{L_\mu-L_\tau}$ model, where three right-handed neutrinos are added to the Standard Model in order to obtain non-zero masses for active neutrinos. Because of the U(1)$_{L_\mu-L_\tau}$ gauge symmetry, the structure of both Dirac and Majorana mass terms of neutrinos is tightly restricted. In particular, the inverse of the neutrino mass matrix has zeros in the $(\mu,\mu)$ and $(\tau,\tau)$ components, namely, this model offers a symmetric realization of the so-called two-zero-minor structure in the neutrino mass matrix. Due to these constraints, all the CP phases-the Dirac CP phase $\delta$ and the Majorana CP phases $\alpha_2$ and $\alpha_3$-as well as the mass eigenvalues of the light neutrinos $m_i$ are uniquely determined as functions of the neutrino mixing angles $\theta_{12}$, $\theta_{23}$, and $\theta_{13}$, and the squared mass differences $\Delta m_{21}^2$ and $\Delta m_{31}^2$. We find that this model predicts the Dirac CP phase $\delta$ to be $\delta\simeq 1.59\pi$-$1.70\pi$ ($1.54\pi$-$1.78\pi$), the sum of the neutrino masses to be $\sum_{i}m_i \simeq 0.14$-0.22 eV ($0.12$-0.40 eV), and the effective mass for the neutrinoless double beta decay to be $\langle m_{\beta \beta }\rangle \simeq 0.024$-0.055 eV ($0.017$-0.12 eV) at $1\sigma$ ($2\sigma$) level, which are totally consistent with the current experimental limits. These predictions can soon be tested in future neutrino experiments. Implications for leptogenesis are also discussed.


Introduction
scale, and thus can be regarded as a generic prediction in the minimal gauged U(1) Lµ−Lτ model. We find that the predicted values of the neutrino parameters are consistent with the present neutrino data and can be tested in future neutrino experiments. We also discuss the implications of our results for leptogenesis. This paper is organized as follows. In Sec. 2, we introduce the minimal gauged U(1) Lµ−Lτ model and examine the neutrino mass structure in this model. We then show in Sec. 3 the predicted values of the Dirac CP phase δ, the sum of the neutrino masses i m i , and the effective mass for the neutrinoless double beta decay m ββ using the neutrino mixing angles and the squared mass differences obtained in neutrino oscillation experiments. In Sec. 4, we discuss the implications for the leptogenesis. Finally, our conclusions are summarized in Sec. 5. This paper ends with two Appendices which give further details of our analysis. In Appendix A, we present some miscellaneous formulae which are useful to study the neutrino structure in the minimal gauged U(1) Lµ−Lτ model. In Appendix B, we perform similar analyses for the minimal gauged U(1) Le−Lµ and U(1) Le−Lτ models, and show that these minimal models fail to explain the observed neutrino oscillation data.
2 Neutrino mass structure in the minimal gauged U(1) L µ −L τ model To begin with, we describe the minimal U(1) Lµ−Lτ model which we discuss in this paper. The model possesses a new U(1) gauge symmetry U(1) Lµ−Lτ . Under this gauge symmetry, µ L,R and ν µ have the U(1) Lµ−Lτ charge +1, τ L,R and ν τ have the U(1) Lµ−Lτ charge −1, and the other SM fields have the zero U(1) Lµ−Lτ charge. We also introduce three right-handed neutrinos N e , N µ , and N τ to obtain non-zero neutrino masses. The U(1) Lµ−Lτ charges of these fields are 0, +1, and −1, respectively. 1 In the (e, µ, τ ) basis, the U(1) Lµ−Lτ charges of the Dirac Yukawa terms are Q Lµ−Lτ (Dirac) : where the (α, β) entry in the above matrix represents the U(1) Lµ−Lτ charge of the fermion bilinear term N c α L β , with α, β the flavor indices. For the Majorana mass terms of the righthanded neutrinos, on the other hand, we have 1 Generically speaking, the U(1)L µ −Lτ charges of the right-handed neutrinos can be (0, a, −a) without spoiling anomaly cancellation conditions. We however find that only the |a| = 1 case gives a neutrino mass structure that can explain the neutrino oscillations. Then we can define the right-handed neutrinos with the U(1)L µ−Lτ charge 0, +1, and −1, as Ne, Nµ, and Nτ , respectively, without loss of generality. We also note that the introduction of two right-handed neutrinos is insufficient since the observed neutrino mixing angles cannot be reproduced in this case.
where the (α, β) component indicates the U(1) Lµ−Lτ charge of the fermion bilinear term N α N β . From Eq. (1), we find that the Dirac Yukawa matrix is always diagonal in the gauged U(1) Lµ−Lτ models-for the same reason, the charged lepton Yukawa matrix is also diagonal. As long as renormalizable interactions are considered, this structure is not violated even if we introduce a U(1) Lµ−Lτ -breaking scalar field. On the other hand, the charges of the Majorana mass matrix in Eq. (2) show that only the (e, e), (µ, τ ), and (τ, µ) components can have non-zero values in the U(1) Lµ−Lτ -symmetric limit. As we mentioned above, with this simple structure, we cannot explain the required values of the neutrino mixing angles. We therefore introduce a scalar boson σ which has a non-zero U(1) Lµ−Lτ charge, and couple it to right-handed neutrinos. After this scalar field develops a VEV, these couplings lead to Majorana mass terms of the right-handed neutrinos. If the scalar field has the U(1) Lµ−Lτ charge ±1, then the (e, µ), (e, τ ), (µ, e), and (τ, e) components in Eq. (2) can be induced after the scalar field acquires a VEV, while the (µ, µ) and (τ, τ ) can be generated if the scalar has the U(1) Lµ−Lτ charge ±2. In the latter case, however, the Majorana mass matrix becomes block-diagonal, which makes it unable to explain the observed neutrino mixing angles. We are thus left with the case where the scalar field has the U(1) Lµ−Lτ charge ±1, and we take it to be +1 in the following discussion. We refer to this model as the minimal gauged U(1) Lµ−Lτ model.
The interaction terms relevant to neutrino masses are then given by where the dots indicate the contraction of the SU(2) L indices. After the Higgs field H and the singlet scalar σ acquire VEVs H = v/ √ 2 and σ , 2 respectively, these interaction terms lead to the neutrino mass terms: where The mass matrix for the light neutrinos is then given by [36][37][38][39] An explicit expression for M ν L can be found in Appendix A.1. We can diagonalize this mass matrix by using a unitary matrix U (PMNS matrix [48]): which can be parametrized as where c ij ≡ cos θ ij and s ij ≡ sin θ ij for θ ij = [0, π/2], δ = [0, 2π], and we have ordered m 1 < m 2 without loss of generality. We follow the convention of the Particle Data Group [48], where If m i = 0 (i = 1 or 3), then det(M ν L ) = 0. As shown in Appendix A.1, in this case we cannot have desired mixing angles since M ν L becomes block-diagonal. Thus, we focus on the m i = 0 case, where we obtain from Eqs. (6) and (7) We then notice that the (µ, µ) and (τ, τ ) components of these terms vanish since M D is diagonal and M R has zeros in these components. This structure is sometimes called two-zero minor [46,47]. For other previous studies of the two-zero minor structure, see Refs. [49,50]. These two vanishing conditions then lead to where the matrix V is defined by U = V · diag(1, e iα 2 /2 , e iα 3 /2 ). Notice that neither the U(1) Lµ−Lτ -breaking singlet VEV σ nor Majorana masses M ee and M µτ appear in these conditions explicitly. For this reason, the following discussions based on these equations have little dependence on the U(1) Lµ−Lτ -symmetry breaking scale; it may be around the electroweak scale, or as large as 10 (13−15) GeV, which is a prime scale for the masses of right-handed neutrinos since small neutrino masses are explained with O(1) Yukawa couplings via the seesaw mechanism [36][37][38][39]. It follows from Eqs. (10) and (11) that where we have used V T = V −1 and detV = 1 with V being the cofactor matrix of V . 4 In Appendix A.2, we give explicit expressions for R 2 and R 3 in terms of neutrino oscillation parameters. By taking the absolute values of the equations in (12), we find Therefore, these mass ratios are given as functions of the Dirac CP phase δ. Notice that R * 2,3 (−δ) = R 2,3 since R 2,3 contains a single CP phase δ. As a consequence, m 2,3 /m 1 are symmetric under the reflection δ → −δ (and π + δ → π − δ) as we see below.
In Fig. 1, we plot the mass ratios m 2 /m 1 and m 3 /m 1 against the Dirac CP phase δ using Eq. (15). The bands show uncertainty coming from the 1σ error in the neutrino mixing parameters. For input parameters of the neutrino mixing angles, we use sin 2 θ 12 = 2.97 +0. 17 −0.16 × 10 −1 , sin 2 θ 13 = 2.15 +0.07 −0.07 × 10 −2 , and sin 2 θ 23 = 4.25 +0.21 −0.15 × 10 −1 [51] (see Table 1). 5 It is found that the resultant uncertainty mainly comes from the error in θ 23 . From this figure, we find that when δ 0, 2π the observed neutrino mixing angles are incompatible with the condition m 1 < m 2 , and thus these regions are excluded. Moreover, around δ π we have m 1 < m 3 < m 2 , which disagrees with the possible neutrino mass ordering: either m 1 < m 2 < m 3 or m 3 < m 1 < m 2 [48]. As a consequence, in the region where a consistent neutrino mass ordering is obtained, the neutrino mass ordering is always Quasi-Degenerate Normal Ordering with m 1 m 2 m 3 , which is realized around δ ∼ π/2 and 3π/2.
According to the recent global-fit analysis in Ref. [51], the NO is somewhat favored over IO at ∼ 2σ level. There are quite a few proposed experiments that may determine the neutrino mass ordering at more than 3σ level within a decade [54,55], such as PINGU [56], ORCA [57], and JUNO [58,59]. To confirm NO in these future experiments would be a first consistency check of our model. Now that the mass ordering has been fixed, we determine m 1 and δ by using the above results. From the neutrino oscillation experiments, we can measure 6 These quantities are related to the neutrino mass ratios as By solving these equations, we can determine m 1 and δ. The observed values of δm 2 and ∆m 2 are given by δm 2 7.37 × 10 −5 eV 2 and ∆m 2 2.525 × 10 −3 eV 2 , respectively [51]. This means that the right-hand side of Eq. (19) is much smaller than that in Eq. (20). From Fig. 1, we see that such a hierarchy can be realized only when m 2 2 /m 2 When the best-fit values of the mixing angles θ ij are used, this leads to cos δ 0.46, which corresponds to δ 0.35π or 1.65π. In Eq. (A.6) in Appendix A.3, we give a cubic equation whose solution gives an exact value of cos δ as a function of the mixing angles θ ij and the squared mass differences δm 2 and ∆m 2 . As discussed there, the solution (21) approximates the real solution of the cubic equation (A.6) at O(δm 2 /∆m 2 ) level. By solving Eq. (A.6) numerically, we find cos δ = 0.445 for the best-fit values of θ ij , δm 2 , and ∆m 2 , which means δ = 0.353π or 1.647π, and justifies the expected accuracy of the approximated formula (21). 7 From Eqs. (13) and (19), we obtain m 1 as a function of δ. An explicit expression for m 1 is given in Appendix A.2. As noted above, |R 2,3 (δ)| are symmetric with respect to the reflection δ → −δ. This implies that m 1 is also symmetric under this reflection and thus depends only on cos δ (not sin δ). As a result, even though there are two solutions for δ, m 1 (and thus m 2,3 as well) is uniquely determined. Finally, by substituting the above results into Eq. (12), we determine the Majorana CP phases α 2 and α 3 . Again, since R * 2,3 (−δ) = R 2,3 (δ), we have α 2,3 (−δ) = −α 2,3 (δ), as seen in Fig. 7 in Appendix A.2.
In the next section, we discuss the predictions of the minimal gauged U(1) Lµ−Lτ model with the recent oscillation data. Before closing this section, let us give some general remarks. 6 In terms of the squared mass differences ∆m 2 21 ≡ m 2 2 −m 2 1 and ∆m 2 31 ≡ m 2 3 −m 2 1 , δm 2 and ∆m 2 are expressed as 7 This result disagrees with the observation in Ref. [21], where it was observed that δ 0 is predicted in the minimal gauged Lµ − Lτ model. In the analysis of Ref. [21], the Dirac Yukawa terms of neutrinos are supposed to be real in the basis where MR has only one CP phase, though it is not the generic case. In addition, we find disagreement between the neutrino mass matrix shown in Ref. [21] and ours, as pointed out in Appendix A.1.
• If the U(1) Lµ−Lτ symmetry breaking scale is much higher than the electroweak scale, we expect sizable quantum corrections to the neutrino mass matrix. Such quantum corrections can be taken into account by using renormalization group equations. Remarkably, it is found that the two-zero minor structure of M ν L is preserved throughout the renormalization group flow [46,47]. To see this, we first note that below the right-handed neutrino mass scale, which is around the U(1) Lµ−Lτ symmetry breaking scale in this model, righthanded neutrinos are integrated out to give the following dimension-five effective operator: where C αβ has the two-zero minor structure at the right-handed neutrino mass scale. The renormalization group equation of the Wilson coefficient C αβ at one-loop level is [60] with where Y u , Y d , and Y e denote the up-type, down-type, and charged-lepton Yukawa matrices, respectively, g 2 is the SU(2) L gauge coupling, and λ is the Higgs quartic coupling: 2 . Now recall that the charged lepton Yukawa matrix is diagonal in our model. In this case, the above equation can readily be solved as follows [61]: where t ≡ ln(µ/µ 0 ) with µ 0 being the initial scale, and Note that τ τ (t) = 0, which proves that the two-zero minor structure of the Wilson coefficient C remains at low energies. As a result, the two-zero minor neutrino-mass structure in our model is robust against quantum corrections, even if the U(1) Lµ−Lτ symmetry breaking scale is much higher than the electroweak scale.
• By performing a similar analysis, we can study the minimal gauged U(1) Le−Lµ and U(1) Le−Lτ models. We again obtain two conditions similar to Eqs. (10) and (11), which follow from the zero components in the inverse of the neutrino mass matrix in each model. We however find that in these cases it is unable to find parameter space consistent with the observed values of the neutrino oscillation parameters, as demonstrated in Appendix B. We thus conclude that within the minimal approach discussed in this paper, the U(1) Lµ−Lτ case is the only possibility that offers a desirable neutrino mass structure.
• If we go beyond the minimal model, we may obtain a different neutrino-mass structure in the presence of the U(1) Lµ−Lτ gauge symmetry. For example, Refs. [25,33] discuss gauged U(1) Lµ−Lτ models where neutrino masses are radiatively induced at one-loop level. It turns out that the neutrino mass matrices in these models have the form of the two-zero texture [62][63][64][65][66], which predicts IO for neutrino mass spectrum. Thus, the determination of neutrino mass hierarchy in future neutrino experiments allows us to distinguish this class of models from the minimal model, where NO is predicted as we have seen above.

Predictions for the neutrino parameters
Using the results obtained above, we now compute quantities relevant to neutrino experiments with the errors in the neutrino oscillation parameters taken into account. For input values, we use the values given in Ref. [51], which are summarized in Table 1. In particular, we take the three mixing angles and the two mass squared differences , as input parameters, and evaluate the predicted values of the other parameters, including Dirac CP phase δ, the absolute masses m i , their sum i m i , and the effective Majorana neutrino mass m ββ . The prediction for the Majorana phases α 2 and α 3 is also presented in Appendix A.2. In Fig. 2, we plot the Dirac CP phase δ as functions of θ 23 in the red lines. We vary θ 23 in the 2σ renege, where the 1σ range is in between the vertical thin dotted lines. The dark (light) red bands show the uncertainty coming from the 1σ (2σ) errors in the other parameters θ 12 , θ 13 , δm 2 , and ∆m 2 . We find that this uncertainty is dominated by the error in θ 12 . We also show the 1σ (2σ) favored region of δ in the dark (light) horizontal green bands. As we discussed in the previous section, there are two solutions for δ for each value of θ 23 . Intriguingly, the upper line is right in the middle of the favored range of δ; in particular, θ 23 41.5 • gives δ 1.6π, both of which are within the 1σ allowed region. Consequently, this model predicts δ 1.59π-1.70π (1.54π-1.78π) within 1σ (2σ). Future neutrino experiments can test this prediction through precision measurements of θ 23 and δ [67].
Next, we evaluate the neutrino masses m i , which are shown in Fig. 3a as functions of θ 23 . Here, the other parameters are fixed to be their best-fit values. We see that all of these masses are predicted to be √ ∆m 2 5 × 10 −2 eV. We also plot the sum of these neutrino masses as a function of θ 23 in Fig. 3b, where the dark (light) red band shows the uncertainty coming from the 1σ (2σ) errors in the parameters other than θ 23 . In this case, it turns out that the dominant contribution to the uncertainty (except for that from the error in θ 23 ) comes from the error in θ 13 , though the error in ∆m 2 also gives a sizable contribution. We also show in the black dashed line the present limit imposed by the Planck experiment: i m i < 0.23 eV (Planck TT+lowP+lensing+ext) [68]. 8  parameter region predicts a value of i m i which is below the present limit, though a part of the parameter region has already been disfavored by the Planck limit. We however note that this bound relies on the standard cosmological history, and thus if some new physics effects modify the cosmological evolution, then this bound may significantly be relaxed. In any case, our model predicts a rather large value of the sum of the neutrino masses, i m i 0.14-0.22 eV (0.12-0.40 eV) at 1σ (2σ), which may be probed in future cosmological observations. These relatively large values of m i open up a possibility of testing this model in neutrinoless double-beta decay experiments. The rate of neutrinoless double-beta decay is proportional to the square of the effective Majorana neutrino mass m ββ , which is given by It should be emphasized that, in the minimal gauged U(1) Lµ−Lτ model, not only the neutrino masses m i but also the Majorana phases α 2,3 are uniquely determined as functions of the other neutrino oscillation parameters. Thus, the value of the effective mass m ββ is also predicted unambiguously. Note also that this quantity has reflection symmetry with respect to δ → −δ and thus depends only on cos δ. In Fig. 4, we show m ββ as a function of θ 23 , where the dark (light) red band shows the uncertainty coming from the 1σ (2σ) errors in the parameters other than θ 23 . Currently, the KamLAND-Zen experiment gives the strongest bound on m ββ : m ββ < 0.061-0.165 eV [69] where the uncertainty stems from the estimation of the nuclear The dark (light) red band shows the uncertainty coming from the 1σ (2σ) errors in the parameters θ 12 , θ 13 , δm 2 , and ∆m 2 . The entire region is within the 2σ range of θ 23 , while its 1σ range is between the thin vertical dotted lines. We also show in the black dashed line the present limit imposed by the Planck experiment: i m i < 0.23 eV (Planck TT+lowP+lensing+ext) [68]. matrix element for 136 Xe. We also show in Fig. 4 the most severe bound from KamLAND-Zen, m ββ < 0.061 eV, in the black dashed line [69]. As can be seen, most parameter region predicts a value of m ββ lower than the strongest bound. At 1σ (2σ) level, this model predicts m ββ 0.024 eV (0.017 eV)-this can be within the reach of future neutrinoless double-beta decay experiments [70].

Implications for Leptogenesis
In this section, we discuss the implications of our results for the leptogenesis scenario [71], which is one of the most attractive mechanisms to explain the origin of the baryon asymmetry of the Universe. The minimal gauged U(1) Lµ−Lτ model has three right-handed neutrinos coupled to the Standard Model leptons, and therefore it contains enough ingredients for the leptogenesis: the CP-violating decay of the right-handed neutrino in the early universe can generate the lepton asymmetry, which is converted to the baryon asymmetry via the sphaleron process [72].
As we have seen in the previous sections, the light neutrino mass matrix M ν L U * diag(m 1 , m 2 , m 3 )U † in the minimal gauged U(1) Lµ−Lτ model is uniquely determined for a given set of neutrino oscillation parameters θ 12 , θ 23 , θ 13 , δm 2 , ∆m 2 , and sign(sin δ). Therefore, the mass matrix of the right-handed neutrino, M R −M D M −1 ν L M D , is also tightly constrained, having only three additional free parameters, λ e , λ µ and λ τ . Note that these neutrino Yukawa couplings can be taken to be real and positive by field redefinitions. Therefore, there is no additional phase pa- The entire region is within the 2σ range of θ 23 , while its 1σ range is between the thin vertical dotted lines. We also show the strongest limit from KamLAND-Zen, m ββ < 0.061 eV, in the black dashed line [69]. rameter in the model. By diagonalizing the masses of the right-handed neutrinos, we can rewrite the Lagrangian (3) as where N c i are the right-handed neutrino fields in the basis where the masses are diagonalized, λ iα are the neutrino Yukawa couplings in that basis, and M i are the masses which are taken to be real and positive. Explicitly, they are given by We emphasize again that both masses M i and the couplings λ iα are completely determined by the three real parameters λ e,µ,τ and the oscillation parameters θ 12 , θ 23 , θ 13 , δm 2 , ∆m 2 , sign(sin δ).
neutrino. At the leading order, it is given by [73][74][75] Here, one can see that there is a correlation between the sign of the Dirac phase δ and the baryon asymmetry of the Universe. Suppose that the sign of the Dirac phase δ is flipped, δ → −δ, while the other input oscillation parameters θ 12 , θ 23 , θ 13 , δm 2 , ∆m 2 as well as the neutrino Yukawa couplings λ e,µ,τ are fixed. As discussed in the previous sections, the Majorana phases then flip the sign, α 2,3 → −α 2,3 , while the absolute masses of light neutrinos do not change, m i → m i . Thus, the PMNS matrix transforms as U → U * . This then results in M ν L → M * ν L , leading to M R → M * R , Ω → Ω * , λ → λ * , and eventually 1 → − 1 . This means that, for a given input oscillation parameters θ 12 , θ 23 , θ 13 , δm 2 , ∆m 2 and the neutrino Yukawa couplings λ e,µ,τ , there is one-to-one correspondence between the sign of the Dirac phase δ and the sign of the baryon asymmetry of the Universe.
parameters θ 12 , θ 23 , θ 13 , δm 2 , ∆m 2 are taken to be their best-fit values in Table 1. The sign of the Dirac phase δ is taken to be negative, δ < 0 (or δ > π), as it is favored at 2σ level. Note that the observed baryon asymmetry of the Universe requires 1 < 0, because the sphaleron process predicts n B /n L < 0 [76]. Surprisingly, negative 1 is realized only in the limited regions of the parameter space. This is more clearly seen in Figs. 6, where we show the regions of 1 < 0 in the (θ, φ) plane as red shaded areas. We also show the contours of the right-handed neutrino mass ratios M 2 /M 1 and M 3 /M 1 in the left and right panels, respectively. As we see, the asymmetry parameter 1 can be negative when some of the right-handed neutrinos are degenerate in mass-M 1 M 2 in the negative 1 region around φ π/4 while M 2 M 3 for θ π/2 and φ 0, π/2. In the latter regions, θ π/2 and φ 0, π/2, the lightest right-handed neutrino mass is much smaller than the other ones, and λ e,τ λ µ (λ e,µ λ τ ) for φ 0 (φ π/2). In these cases, the absolute value of the asymmetry parameter | 1 | is found to be quite suppressed, and thus it is rather difficult to obtain a sizable value, say | 1 | 10 −6 . 10 Similarly, | 1 | gets small for φ π/4 and θ π/4. These observations further restrict the promising parameter region for leptogenesis to be (θ, φ) (π/4, π/4).
The final baryon asymmetry depends on the production mechanism of the right-handed neutrino. In the case of thermal leptogenesis, the predicted range of the lightest neutrino mass in the present model, m 1 0.03 eV, corresponds to the so-called strong wash-out region (see, e.g., [77]). Moreover, as we see from Fig. 6, around (θ, φ) (π/4, π/4), right-handed neutrino masses are close to each other. In this case, to properly estimate the net baryon asymmetry generated via the decay of right-handed neutrinos, we may need to include not only the contribution of the lightest right-handed neutrino but those of the other ones. Flavor effects [78][79][80] may also become important. A more detailed study will be given elsewhere [81].

Conclusions
We have studied the structure of the neutrino mass matrix in the minimal gauged U(1) Lµ−Lτ model. Because of the U(1) Lµ−Lτ gauge symmetry, the structure of both Dirac and Majorana mass terms of neutrinos is tightly restricted, which results in a two-zero minor structure of the neutrino mass matrix. Because of this restriction, all the CP phases and the neutrino masses are uniquely determined. We find that this model gives the following prediction at 1σ (2σ) level: • Dirac CP phase: δ 1.59π-1.70π (1.54π-1.78π), • the sum of the neutrino masses: i m i 0.14-0.22 eV (0.12-0.40 eV), • the effective mass for the neutrinoless double beta decay: m ββ 0.024-0.055 eV (0.017-0.12 eV).
They are totally consistent with the current experimental limits, and hold independently of the U(1) Lµ−Lτ breaking scale and the Majorana mass scale. In this sense, the above predictions are the generic features of the minimal gauged U(1) Lµ−Lτ model. Remarkably, these predictions can be tested in various neutrino experiments in the near future-regardless of the scale of the U(1) Lµ−Lτ symmetry breaking-which we believe sheds light on the gauge structure of physics beyond the Standard Model.
We have also discussed the implications of the minimal gauged U(1) Lµ−Lτ model for the leptogenesis scenario, and found that the correct sign of the baryon asymmetry of the Universe can be obtained only in the limited regions of the parameter space, as the right-handed neutrino mass structure is also severely restricted in this model. In particular, the observed value of baryon asymmetry can be realized when right-handed neutrinos are degenerate in mass, which requires a further detailed study to assess the viability of leptogenesis in this model [81].
The determinant of this mass matrix is given by .
We find that this determinant vanishes if and only if λ ν = 0 (ν = e, µ, or τ ). In this case, the mass matrix M ν L is block-diagonal, and thus cannot reproduce the required neutrino mixing angles.

B.1 U(1) Le−Lµ
Following an analysis similar to that in Sec. 2, we find that the (e, e) and (µ, µ) components in the inverse of the neutrino mass matrix vanish in the minimal gauged U(1) Le−Lµ model. The corresponding two vanishing conditions are Solving these equations, we have In Fig. 8a, we plot the mass ratios m 2 /m 1 and m 3 /m 1 as functions of δ. As we see from this figure, m 2 < m 1 is predicted for any value of δ, and thus there is no solution which gives an allowed pattern of neutrino mass spectrum.

B.2 U(1) Le−Lτ
In this case, the (e, e) and (τ, τ ) components in M −1 ν L are zero, which leads to These equations read (B.10) Using these equations, we plot the mass ratios m 2 /m 1 and m 3 /m 1 as functions of δ in Fig. 8b. Again, m 2 < m 1 over the whole range of δ, and thus this model cannot provide a desirable neutrino mass ordering.