Low scale seesaw models for low scale $U(1)_{L_\mu-L_\tau}$ symmetry

We propose models for neutrino masses and mixing in the framework of low scale $U(1)_{L_\mu-L_\tau}$ gauge extension of the standard model. The models are designed to spontaneously break $U(1)_{L_\mu-L_\tau}$ so that the $U(1)_{L_\mu-L_\tau}$ gauge boson acquires an MeV scale mass, which is required to solve the long-standing problem of muon anomalous magnetic moment. Tiny neutrino masses are obtained by simultaneously invoking the linear and the inverse seesaw mechanism, and we succeed in realizing two types of one-zero textures in the active neutrino mass matrix. Both of the obtained textures favor inverted neutrino mass ordering and are testable in next generation experiments of neutrinoless double beta decay. We also show that some of extra scalar bosons can have MeV scale masses and would have significant impacts on observations of high energy cosmic neutrinos.


I. INTRODUCTION
The extension of the standard model (SM) is one of the highest priority issues in modern particle physics; various types of new physics models have been proposed in the literature.
In most cases, new physics is anticipated to exist at an energy scale much higher than the electroweak scale, because it is strictly constrained by low energy experiments. Nevertheless, it may also be possible to consider new physics at lower energy, if its interactions are sufficiently weak so that the experimental constraints can be evaded. One interesting possibility in that direction is gauging the muon number minus the tau number, that is, the so-called U(1) L µ −L τ gauge extension [1][2][3]. This could be one of the most natural extensions of the SM since it is gauge anomaly free within the SM particle contents. Furthermore, it was recently found, in Ref. [4] (see also Refs. [5,6]), that the U(1) L µ −L τ gauge boson, Z µτ , with an MeV scale mass can settle the long-standing discrepancy of muon anomalous magnetic moment (g µ − 2) [7][8][9][10][11][12] without conflicting other experimental constraints. Then, this result motivated many authors to study low scale U(1) L µ −L τ extension of the SM in various contexts: for instance, it was found that an MeV scale Z µτ can relax the tension between the late time and the early time determination of the Hubble constant [13], implications for the dark matter problem were studied in Refs. [14][15][16][17], and the detectability of such a light Z µτ was discussed in Refs. [18][19][20][21][22][23][24][25]. Moreover, it was pointed out in Refs. [26][27][28][29] that an MeV scale Z µτ causes significant attenuation in the flux of high energy cosmic neutrinos and can explain the unexpected dip in the energy spectrum of high energy cosmic neutrinos reported by the IceCube Collaboration.
From a viewpoint of the lepton mixing, the U(1) L µ −L τ symmetry is well known to naturally explain the observed large atmospheric mixing angle [30][31][32]. However, it is also well known that the exact U(1) L µ −L τ symmetry forbids many of entries in the neutrino mass matrix, and as a result the solar and the reactor mixing angle are forced to be zero. In order to remedy such a situation, extra scalars are often introduced to spontaneously break U(1) L µ −L τ and to revive some of the entries forbidden by U(1) L µ −L τ , which provides us with an opportunity to realize zero textures in the neutrino mass matrix and testability of the model. Especially, the type-C [33] two-zero texture or two-zero-minor structure is frequently obtained [34][35][36][37][38][39][40][41][42], and these structures were consistent with experiments at that time. However, in Ref. [42], it has been pointed out that the type-C two-zero-minor structure is now driven into a corner by the combined upper bound on the sum of neutrino masses placed by the Planck Collaboration: i m i < 0.12 eV [43]. Similarly, it can be checked that the type-C two-zero texture is ruled out by the Planck bound 1 .
In this work, we improve the previous studies mentioned above and propose experimentally consistent models for U(1) L µ −L τ extension of the SM. For this purpose, we combine the inverse and the linear seesaw mechanism and succeed in realizing two types of one-zero textures [45][46][47][48][49][50]. Both of the obtained textures prefer inverted neutrino mass ordering and are consistent with the Planck bound as well as the bounds from neutrino oscillation experiments. We calculate the effective mass of neutrinoless double beta decay as our prediction and find that it is testable in next generation experiments. In the models, extra scalars are introduced to spontaneously break U(1) L µ −L τ and to give an MeV scale mass to Z µτ in order that the g µ − 2 problem and the dip in the IceCube data can simultaneously be solved. We show that some of the extra scalar bosons can have MeV scale masses and can attenuate the flux of high energy cosmic neutrinos, just as the Z µτ does.
The paper is organized as follows. In Sec. II, we describe the particle contents and the charge assignments of the models. Then, in Sec. III, we show that tiny neutrino masses are obtained by combining the linear and the inverse seesaw mechanism. The numerical calculations of neutrino masses and mixing are given in Sec. IV. The scalar sector is studied in Sec. V, in which we also show that some scalar bosons can have MeV scale masses. In Sec. VI, we briefly discuss whether the MeV scale scalar bosons can attenuate the flux of high energy cosmic neutrinos. Finally, we summarize our results in Sec. VII. 1 See Fig. 9 in Ref. [35] and Fig. 4 in Ref. [44]. We have done the same calculation and obtained a similar conclusion that the sum of neutrino masses is constrained to be larger than 0.14 eV within 3σ error ranges of the neutrino oscillation parameters.
We begin with U(1) L µ −L τ extension of the SM and introduce left-and right-handed SM gauge singlet fermions, N L and N R . Although it may be natural to introduce three generations of N R and N L and assign them U(1) Lµ−Lτ charges as (N R(L)e , N R(L)µ , N R(L)τ ) = (0, 1, −1). Nevertheless, in this work, we introduce only two generations and aim at building a minimal model. In this case, the following two possibilities can be considered, and we refer to them as Model-A and Model-B: Note that we omit the case of (N R(L)µ , N R(L)τ ) = (1, −1) because it results in a neutrino mass matrix of which predicts unrealistic neutrino mass spectrum and mixing. Note also that the other combinations give rise to gauge anomalies, so we do not consider them in what follows.
In addition to the SM Higgs doublet, H, the scalar sector is also augmented with a new SU(2) W doublet scalar having hypercharge 1/2 (Y = 1/2), Φ, and two SM gauge singlet scalars, S L and S µτ . Here, S µτ breaks the U(1) L µ −L τ symmetry and gives an MeV scale mass to the U(1) L µ −L τ gauge boson, Z µτ , after developing a vacuum expectation value (VEV).
As studied in Refs. [26,29], the g µ − 2 problem and the dip in the IceCube data can simultaneously be solved with M Z µτ = 11 MeV and g µτ = 5 × 10 −4 , where M Z µτ is a mass of Z µτ and g µτ is the gauge coupling constant of U(1) L µ −L τ . We refer to these values in this work and require S µτ to develop Furthermore, we introduce a global lepton number symmetry U(1) L , which is explicitly broken in the scalar potential, see Sec. V. The gauge singlet scalar S L and the SU(2) W doublet scalar Φ are introduced to spontaneously break U(1) L and to generate tiny masses for neutrinos. In Table I, we summarize the particle contents and the charge assignments of the models.
The Lagrangian relevant to lepton masses is given by where ℓ L stands for the SU(2) W doublet left-handed leptons, ℓ R is the SU(2) W singlet right- and they take the forms of in the case of Model-A, or in the case of Model-B. The differences between Model-A and Model-B will be studied in more detail in Sec. IV.

III. NEUTRINO MASSES AND MIXING
The mass matrices in Eq. (4) compose the 7 × 7 neutrino mass matrix where In order to generate tiny neutrino masses, we invoke the seesaw mechanism by assuming the hierarchy of Then, one can approximately block-diagonalize M ν as with where the active neutrino mass matrix is given by the mixing parameter Θ is defined as 1 n×n stands for the n × n unit matrix, and V S is a diagonalizing matrix of m S . Here, the inverse matrix of M 4×4 is approximately given by which in turn leads to As a result, the active neutrino mass matrix is derived as . We emphasize that the obtained mass matrix is twofold: the first two terms stem from the so-called linear seesaw mechanism, while the last term from the inverse seesaw one. As will be discussed in Sec. IV, in order to reproduce experimentally allowed neutrino mass spectrum and mixing, both contributions are necessary. To this end, throughout this paper, we suppose the following parameter setting: which results in m ν = O(10 −10 ) GeV. We here note that H serves as the SM like Higgs doublet, and S µτ is determined to induce an MeV scale mass to Z µτ in Eq. (2). A more detailed discussion of the scalar potential is given in Sec. V.
Let us mention the active-sterile mixing and derive the leptonic mixing matrix in our framework. The mixing between the active neutrinos and the sterile ones is described in Eq. (11). In view of the hierarchy Eq. (9), we simplify Θ in Eq. (15) as leading to and the leptonic mixing matrix is obtained as Here, P = Diag(1, e i α 21 2 , e i α 31 2 ) includes the Majorana CP-violating phases, and V MNS is the MNS mixing matrix [51] parametrized as where c ij = cos θ ij , s ij = sin θ ij and δ denotes the Dirac CP-violating phase. In Eq.
for normal (inverted) mass ordering, and the 2σ upper bounds on η's derived in Ref. [53]: In terms of η, the active neutrino mass matrix in Eq. (16) is rewritten by and it takes the form of for Model-A, while it takes for Model-B. As can be seen, both the neutrino mass matrices contain one zero 2 , and thus one can predict two observables. In the following subsections, we calculate the effective mass of neutrinoless double beta decay, m ee , as a function of the sum of active neutrino masses, i m i , and check the consistency with the current bounds: from the KamLAND-Zen Collaboration [55], where the uncertainty comes from the nuclear matrix element calculation, and the combined upper bound from the Planck Collaboration [43].

A. Inverted ordering
Let us first investigate the inverted ordering case. In view of our parameter setting given in Eqs. (17) and (18), we vary the active neutrino mass matrix within the following ranges: while imposing the constraints in Eqs. (23) and (24). Here, we take a basis in which only η ee and η eµ (η eτ ) are complex and vary their phases within 0 − 2π. We vary these parameters 2 There is another contribution to the neutrino mass term from one-loop diagrams [54]. In our framework, the diagrams yield a new contribution, which is proportional to m D m RR m T D , to (m ν ) ee and do not break the obtained one-zero textures. Thus, by tuning y RR in Eq. (3), we simply assume that the contribution is negligibly small compared with Eqs. (26) and (27). For instance, in the areas where the density is low, it is difficult to find solutions because relatively strong parameter tuning is necessary to satisfy Eqs. (23) and (24), especially the small squared-mass-differences. As a reference, we also show the current 3σ upper and lower bounds (solid curves), which are derived by calculating with Eq. (23)

V. SCALAR SECTOR
The scalar potential invariant under G SM × U(1) L µ −L τ × U(1) L , where G SM denotes the SM gauge group, is given by Since the potential preserves the global U(1) L symmetry, a massless Nambu-Goldstone boson (NGB) appears after the spontaneous symmetry breaking. In addition, as will be seen later, an almost massless scalar boson appears in the CP-even sector for our parameter setting given in Eq. (17). The existence of the NGB and the almost massless scalar boson has interesting implications for observations of high energy cosmic neutrinos. In Sec. VI, we show that if these particles have MeV scale masses, they would significantly attenuate the flux of high energy cosmic neutrinos. In order to give finite masses to them, we here introduce the following U(1) L breaking term: This breaking term becomes important also when we solve stationary conditions of the potential: tiny VEVs can naturally be obtained for |m 2 Φ | ≫ |B 2 | [57,58]. Note that in the potential all the parameters are taken to be real.

We define components of the scalars and their VEVs as
Then, the stationary conditions are obtained as In the second and the third condition, for |m 2 Φ |, |m 2 L | ≫ v 2 ew , v 2 µτ , v 2 L , v 2 Φ , the conditions can be written as Therefore, v Φ and v L can simultaneously be small for |m 2 L | ≃ |µv ew | and |m 2 Φ | ≫ |B 2 |(or |µv ew | ≫ |B 2 |).

A. Masses of charged scalars
In a basis of (h ± , φ ± ), the squared mass matrix of the charged components is given by This mass matrix can easily be diagonalized, and one obtains an eigenvalue of and a zero eigenvalue corresponding to the NGB for the SM W ± gauge boson.
Charged scalar bosons are searched at the LHC experiments; they can be produced via top quark decays, t → H ± b, or in association with a top and a bottom quark, pp → H ± tb.
In our models, however, interactions between H ± and quarks are suppressed by the small v Φ (or large tan β = v ew /v Φ ), so the LHC constraints are irrelevant for our studies. In contrast, at the LEP experiments, charged scalar bosons can be produced via the Drell-Yan process e + e − → γ/Z → H + H − , which does not rely on interactions with quarks and may be applicable to our models. In Ref. [59], lower bounds on a charged scalar boson mass are derived for several decay channels, in the framework of two-Higgs-doublet-models. Since decays to quarks are strongly suppressed in our models, we refer to the bound for type-II two-Higgs-doublet-models with Br(H ± → τ ± ν) = 1, i.e., m H ± > 94 GeV, and that for type-I models with Br(H ± → W ± A) ≃ 1 (tan β = 100), i.e., m H ± > 84 − 89 GeV, where A denotes a lighter CP-odd neutral boson. Also, in order to prevent m H ± from being tachyonic, we (40). As a consequence, we place a constraint of in what follows.

B. Masses of CP-even neutral scalars
In a basis of (h, φ, s L , s µτ ), the squared mass matrix of the CP-even neutral components is given by Although the mass matrix is too complicated to diagonalize in a general way, one can see that the (φ, s L ) block part is dominated by the terms proportional to µ and an almost rank one matrix in case of B 2 = 0. Consequently, an almost massless scalar boson inevitably appears in the absence of explicit U(1) L symmetry breaking for our parameter setting proposed in Eq. (17).
In order to simplify the diagonalization, in this work, we restrict ourselves to the parameter region of | µ √ 2 v L + B 2 | < O(1) GeV 2 so that the mass matrix can be simplified to Note that µ and B 2 are also constrained by the lower bound on the charged scalar boson Then, the diagonalization can easily be done by rotating where c 1(2) = cos θ 1(2) and s 1(2) = sin θ 1 (2) . However, the exact expressions for the mixing angles and the eigenvalues are somewhat complicated. Thus, we refrain from showing them and, instead, derive approximate expressions by considering certain limits.
Let us first consider the (h, s µτ ) block part. As we will show later, the value of λ 6 is restricted to be λ 6 < O(0.01) to suppress invisible decays of the Higgs boson. Hence, given v ew = 246 GeV and v µτ = 20 GeV, one finds a hierarchy of (M 2 even ) 11 ≫ (M 2 even ) 44 ≫ (M 2 even ) 14 for λ 1,5 = O(1). In this case, the diagonalization can approximately be done by which lead to The mass of h 1 is proportional to v ew ≃ 246 GeV, and thus we identify h 1 as the SM Higgs boson, while the mass of h 4 is expected to be somewhat small, that is m h 4 = O(10) GeV.
Next, we tackle with the (φ, s L ) block part for the following two cases.

C. Masses of CP-odd neutral scalars
In a basis of (η, ξ, σ L , σ µτ ), the squared mass matrix of the CP-odd neutral components is written as Here, σ µτ neither mixes with the others nor acquires a non-zero mass; it is the NGB eaten by the Z µτ gauge boson. Omitting σ µτ , the diagonalization can approximately be done by where r = v Φ /v ew . The mixing r block-diagonalizes the squared mass matrix into a zero eigenvalue and a 2 × 2 block part Note that the mass of ζ 1 is exactly vanishing, and it is the NGB eaten by the SM Z gauge boson. The 2 × 2 block part M 2×2 has the same form as the (φ, s L ) block part of Eq. (43), and it can be diagonalized in the same way as demonstrated in the previous subsection.

D. Invisible decays of the Higgs boson
As we have seen in the previous subsections, there are three scalar bosons whose masses can be smaller than half of the Higgs boson mass, namely h 3 , h 4 , and ζ 3 ; these particles cause invisible decays of the Higgs boson h 1 . In addition, the Higgs boson can decay into a pair of Z µτ through the mixing between h 1 and h 4 . We here formulate those interactions as where ρ denotes the Lorentz index and In the above equations, we have neglected r in Eq. (52). Since G h 3 h 4 is suppressed by v Φ or v L in comparison with the others, we will ignore h 1 → h 3 h 4 decay in the following discussion.
The branching ratio of the Higgs boson invisible decays is constrained by the LHC experiments, and it is defined as where and Γ SM is the SM expectation. The partial decay widths of h 1 → XX and h 1 → Z µτ Z µτ are calculated to be  MeV.
Next, depending on µ and B 2 , the masses of h 3 and ζ 3 can also be smaller than half of the Higgs boson mass, so they cause the Higgs boson invisible decays, too. We are particularly interested in the case of | µ √ 2 v L | ≫ |B 2 | as will be discussed in the next section. In this case, by approximating the mixing with s 1 <  IceCube data, see Ref. [67] for the recent analysis. In our models, we have three candidates for such light bosons, that is, the CP-even neutral scalar h 3 and the CP-odd one ζ 3 , as well as Z µτ . The detailed analysis for Z µτ was already done in Refs. [26,29]. Thus, in this section, we focus only on h 3 and ζ 3 and check whether they can serve as the mediator of the resonant scattering, just as Z µτ does.
Suppose the light boson is scalar and has Yukawa interactions with active neutrinos, the interaction can be formulated as g νν ν c χ + h.c. .
According to Ref. [29], in order for the scalar χ to significantly attenuate the cosmic neutrino flux in the PeV region, its mass and the coupling constant should satisfy As shown in Sec. V, both h 3 and ζ 3 can have an MeV scale mass by suitably tuning the U(1) L breaking parameters B 2 . As for the Yukawa coupling g ν , in our models, it arises from the third term in Eq. (3), through the mixing in Eqs. (20) and (49). In the case of h 3 , it arises as while for ζ 3 , it is described as Given our parameter setting: η = O(10 −2 ), R = v Φ /v L = O(10 −1 ), and r = v Φ /v ew = O(10 −10 ), one obtains g (′) ν = O(10 −3 ), which is presumably large enough to attenuate the cosmic neutrino flux. A more detailed study will be done elsewhere.

VII. SUMMARY
In summary, we propose models for an MeV scale U(1) L µ −L τ gauge boson, Z µτ , which is anticipated to exist to resolve the g µ − 2 problem as well as the possible dip in the energy spectrum of high energy cosmic neutrinos. We introduce extra scalars to spontaneously break U(1) L µ −L τ and to generate an MeV scale mass to Z µτ . Tiny neutrino masses and mixing are obtained by simultaneously invoking the linear and the inverse seesaw mechanism.
Depending on the U(1) L µ −L τ charge assignment, the active neutrino mass matrix enjoys two types of one-zero textures, which endows the models with predictive power. Both of the textures prefer inverted neutrino mass ordering and would be tested in next generation experiments of neutrinoless double beta decay. Furthermore, we find that two of the extra scalars can acquire MeV scale masses while having interactions with active neutrinos. We briefly confirm that they can serve as the mediator of resonant scattering between high energy cosmic neutrinos and cosmic neutrino background, and that they could help us understand the existence of the unexpected dip in the IceCube data.
Finally, we comment on kinetic mixing between U(1) L µ −L τ and the SM electromagnetic U(1) em gauge symmetry. In our framework, we do not introduce the kinetic mixing term at the tree level, but it appears at the one loop level. In Refs. [20,29], we studied implications of the loop-induced kinetic mixing for solar neutrino measurements and the detectability of Z µτ at e + e − colliders. In contrast to the previous work, we here introduce extra fermions and scalars charged under U(1) L µ −L τ . Nevertheless, there are no mass eigenstates that are charged under both U(1) L µ −L τ and U(1) em , except for the SM mu and tau leptons. Hence, we expect that the same kinetic mixing as those in Refs. [20,29] will also be obtained for the framework proposed in this paper.