]>NUPHB14556S0550-3213(19)30024-010.1016/j.nuclphysb.2019.01.016The AuthorsHigh Energy Physics – PhenomenologyFig. 1Sketched Feynman diagrams for the Majorana neutrino mass matrix elements.Fig. 1Fig. 2Left: δμγγNP as a function of λ10 with ms = 200 GeV (dashed) and ms = 20 GeV (solid), where the taken values of the other parameters are shown on the plot. Right: BRs for the h → ss (dotted), h → Z′Z′ (dashed), and h → ss + Z′Z′ (solid) decays. The horizontal line denotes the experimental upper bound.Fig. 2Fig. 3Scatter plot for the Dirac CP phase and ∑j|mj|, where the dots in black, red, and blue denote the neutrino data with 1σ, 2σ, and 3σ errors, respectively. The dashed (dotted) line denote the cosmological neutrino mass bound 0.12(0.17) eV.Fig. 3Fig. 4(a) Predicted |mj| and effective Majorana neutrino mass for the 0νββ decay; (b) allowed ranges for |mij| as a correlation of |mττ|; (c)[(d)] correlation between Dirac phase δ and Majorana phase α13[α23], where FGM pattern A1 is applied, and neutrino data within 1σ errors are taken.Fig. 4Fig. 5Sketched Feynman diagram for τ → μγ.Fig. 5Fig. 6Correlation between BR(h → μτ) (in units of 10−3) and BR(τ → μZ′Z′) (in units of 10−9, where the horizontal dashed line is the upper bound of h → μτ, and we have fixed mS = 10 GeV and mZ′=0.2 GeV.Fig. 6Table 1U(1)Lμ−Lτ charges of involving leptons, S, and S′.Table 1eμτL4L5ΔS′S

U(1)01−1−11012

Table 2Vanishing Yukawa (VY) couplings to determine the FGM two-zero textures in the model.Table 2PatternA1A2B1B2

VY(Yee,Y45′,ye)≈0(Yee,Y45′,ye′)≈0(ye′,Y45′,yμ′)≈0(ye,Y45′,Yτ5)≈0

PatternB3B4C

VY(ye′,ye,Yμ4)≈0(ye, Yτ5, m4τ)≈0(Yμ4, Yτ5)≈0

Table 3Mass ratios and Majorana CP phases of each FGM pattern with some benchmark inputs, where in addition to the taken values of sin2θ12=0.304 and sin2θ13=0.0219, the values inside brackets correspond to two different inputs: for the left value, we fix δ = 1.5π and sin2θ23=0.5; for the right, δ = 1.59205π and sin2θ23=0.4515 are taken.Table 3mass relationCP-violating phases

A1|m1||m3|≃(0.10,0.087), |m2||m3|≃(0.23,0.22)α13 ≃ (0.43π, 0.33π), α23 ≃ (−0.47π, −0.56π)

A2|m1||m3|≃(0.10,0.12), |m2||m3|≃(0.23,0.25)α13 ≃ (−0.43π, −0.53π), α23 ≃ (0.47π, 0.38π)

B1|m1||m3|≃(1.0,0.95), |m2||m3|≃(1.0,0.74)α13 ≃ (1.0π,−0.98π), α23 ≃ (−1.0π, −0.99π)

B2|m1||m3|≃(1.0,1.1), |m2||m3|≃(1.0,1.3)α13 ≃ (−1.0π, −0.98π), α23 ≃ (1.0π, −0.99π)

B3|m1||m3|≃(1.0,0.73), |m2||m3|≃(1.0,0.87)α13 ≃ (−1.0π, 0.98π), α23 ≃ (−1.0π, −1.0π)

B4|m1||m3|≃(1.0,1.4), |m2||m3|≃(1.0,1.1)α13 ≃ (1.0π, 0.98π), α23 ≃ (1.0π, −1.0π)

C|m1||m3|≃(1.0,1.19), |m2||m3|≃(1.0,1.2)α13 ≃ (1.0π, 0.70π), α23 ≃ (1.0π, −0.89π)

Neutrino mass in a gauged Lμ − Lτ modelChuan-HungChenaphyschen@mail.ncku.edu.twTakaakiNomurab⁎nomura@kias.re.kraDepartment of Physics, National Cheng-Kung University, Tainan 70101, TaiwanDepartment of PhysicsNational Cheng-Kung UniversityTainan70101TaiwanbSchool of Physics, KIAS, Seoul 02455, Republic of KoreaSchool of PhysicsKIASSeoul02455Republic of Korea⁎Corresponding author.Editor: Hong-Jian HeAbstractWe study the origin of neutrino mass through lepton-number violation and spontaneous U(1)Lμ−Lτ symmetry breaking. To accomplish the purpose, we include one Higgs triplet, two singlet scalars, and two vector-like doublet leptons in the U(1)Lμ−Lτ 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 mj can be obtained in the model, and their sum can satisfy the upper bound of the cosmological measurement with ∑j|mj|<0.12 eV. The effective Majorana neutrino mass for neutrinoless double-beta decay is below the current upper limit and is obtained as 〈mββ〉=(0.34,2.3)×10−2 eV. In addition, the doubly charged Higgs H±± decaying to μ±τ± final states can be induced from a dimension-6 operator and is not suppressed, and its branching ratio is compatible with the H±±→W±W± decay when the vacuum expectation value of Higgs triplet is O(0.01) GeV.1IntroductionIn spite of the mass hierarchy among the quarks and charged leptons, the particle masses, with the exception of the neutrinos, in the standard model (SM) can be attributed to the Brout–Englert–Higgs (BEH) mechanism [1,2], where the predicted Higgs boson is observed using ATLAS [3] and CMS [4] at a mass of 125 GeV. Based on the neutrino oscillation experiments, it is 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 order, i.e., |m1|<|m2|<|m3| or |m3|<|m1|<|m2| 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]:(1)Δm212=(7.53±0.18)×10−5eV2,sin2θ12=0.304±0.014,Δm322=(2.44±0.06,2.51±0.06)×10−3eV2(NO,IO),sin2θ23=(0.51±0.05,0.50±0.05)(NO,IO),sin2θ13=(2.19±0.12)×10−2, where m212≡m22−m12, m232 denotes m32−m22 for NO or m22−m32 for IO, and θij are the mixing angles of the 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) matrix [8,9], which dictates 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 [10], and all neutrino data can be explained. Although it is inevitable to fine-tune the Yukawa couplings to fit the neutrino masses, the model can provide interesting phenomenological implications in flavor and collider physics.Inspired by the experimental indication of maximal θ23, large θ12, and small θ13, various Abelian flavor-symmetry based models have been proposed to understand the neutrino properties [11–21]. Among these flavor symmetries, we investigate the neutrino problems in an U(1)Lμ−Lτ gauge symmetry. We focus on such gauge symmetry based on some phenomenological considerations: (i) gauge anomaly-free conditions are automatically satisfied [22,23]; (ii) excess of muon anomalous magnetic dipole moment (muon g−2) can be resolved [24–26]; (iii) excesses in semileptonic B-meson decays can be explained [27–31]; (iv) potential signals for the processes e+e−→γZ′ [32,33] and τ→μZ′Z′ [34] can be observed at Belle II. Other interesting studies can be found in [35–43].In order to dynamically generate the neutrino masses, we require that each Majorana matrix entry is related to the lepton-number violating effect and the breaking of spontaneous U(1)Lμ−Lτ symmetry. To achieve the lepton-number violation, like type-II seesaw model [44,45], we introduce a Higgs triplet, which carries a hypercharge Y=1 and has no U(1)Lμ−Lτ charge. We find that due to the protection of U(1)Lμ−Lτ gauge symmetry, we cannot obtain a realistic Majorana neutrino mass matrix without further introducing the breaking of U(1)Lμ−Lτ. Therefore, to break the gauge symmetry, we employ two singlet scalars, which carry different U(1)Lμ−Lτ charges. Because the lepton chirality cannot be matched, the SM leptons cannot couple to the singlet scalars; therefore, we must introduce proper exotic heavy leptons as the media. To avoid gauge anomalies, 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 by which to obtain a proper Majorana neutrino mass matrix.Since the number of free parameters in the Majorana neutrino mass matrix is more than that of the neutrino data, not all free parameters can be determined. In order to completely determine the free parameters, we employ the Frampton–Glashow–Marfatia (FGM) matrix pattern [55], which has two independent zeros, as a theoretical input.It is 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 related to flavor and collider physics. For instance, the lepton-flavor violating h→μτ decay can be as large as the current measurements [56,57]; excess of muon g−2 can be resolved by the mediation of the Z′ gauge boson, and the doubly charged Higgs decaying to μτ and WW can be compatible each other without requiring the VEV of the Higgs triplet to be the eV.This paper is organized as follows. In Sec. 2, we introduce the model under the SU(2)L×U(1)Y×U(1)Lμ−Lτ local gauge symmetry. In Sec. 3, we generate the Majorana neutrino mass matrix without right-handed neutrinos in the model and discuss the relation to the FGM matrix pattern. The numerical analysis on neutrino physics and implications of the model on other phenomena are shown in Sec. 4. A summary is given in Sec. 4.2ModelIn this section, we 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 (Δ), two vector-like doublet leptons (L4,L5), and two singlet scalars (S,S′). Their U(1)Lμ−Lτ charges are given in Table 1, 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:(2)−LYΔ=12YeeLeTCiτ2ΔLe+YμτLμTCiτ2ΔLτ+Yμ4LμTCiτ2ΔL4L+Yτ5LτTCiτ2ΔL5L+Y45LT4LCiτ2ΔL5L+Y45′L4RTCiτ2ΔL5R+H.c. From the above equation, if the Higgs triplet Δ carries two lepton-number units, the Yukawa interactions are lepton-number conserved. However, when the Higgs triplet obtains a VEV, i.e. 〈Δ〉=vΔ/2, the lepton-number violating Majorana neutrino mass matrix for three light neutrinos is induced and expressed as:(3)Mν=(YeevΔ20000YμτvΔ20YμτvΔ20), where the pattern of mass matrix leads to m2=m3, θ13=θ12=0, and θ23=π/4 [11,19,20]. Obviously, the results cannot explain the current neutrino data [5]. We clearly demonstrate that the neutrino mass matrix, which arises from the breaking of the electroweak symmetry and lepton-number violations, 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 have to rely on other pieces of Yukawa interactions, which can break the U(1) symmetry. Concerning the magnitude of vΔ, according to the electroweak symmetry breaking, the electroweak ρ-parameter at the tree-level can be written as [45]:(4)ρ=mW2mZ2cθW2=1+2v2Δ/vH21+4vΔ2/vH2. Taking the current precision measurement for ρ-parameter within 2σ errors, the VEV of Δ has to be less than 3.4 GeV.In addition to Eq. (2), the gauge invariant Yukawa couplings to the Higgs and S(′) are given by:(5)−LY=YℓL¯ℓHℓR+yμL¯5LHμR+yτL¯4LHτR+yμ′L¯μL4RS+yτ′L¯τL5RS†+yeL¯eL4RS′+ye′L¯eL5RS′†+ySL¯5LL4RS+yS′L¯4LL5RS†+m4LL¯4LL4R+m5LL¯5LL5R+m4τL¯4RLτ+m5μL¯5RLμ+H.c., where H is the SM Higgs doublet; only the first term is from the SM, and the other terms are the new Yukawa interactions. Although Eq. (5) can cause rich interesting phenomena for lepton-flavor physics, we only focus on neutrino physics in this work, and a detailed study on the flavor physics can be found in [34]. 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:(6)−LY⊃Yμ4yμ′⁎m4LLμTCΔ¯LμS†+Yτ5y′⁎τm5LLτTCΔ¯LτS+(Yμ4ye⁎m4L+ye⁎Y45′m5μm4Lm5L)×LeTCΔ¯LμS′†+(Yτ5ye′⁎m5L+ye′⁎Y45′m4τm4Lm5L)LeTCΔ¯LτS′+Y45′(yeye′)⁎m4Lm5LLTeCΔ¯LeS′S′†+ye′⁎Y45′yμ′⁎m4Lm5LLeTCΔ¯LμS†S′+ye⁎Y45′y′⁎τm4Lm5LLeTCΔ¯LτSS′†+Yμ4yS′yτ′⁎m4Lm5LLμTCΔ¯LτSS†+Yτ5ySyμ′⁎m4Lm5LLτTCΔ¯LμSS†+(Y45+Y45′)y′⁎τyμ′⁎m4Lm5LLμTCΔ¯LτSS†+Yμ4m4τm4LLμTCΔ¯Lτ+Yτ5m5μm5LLτTCΔ¯Lμ+H.c. with Δ¯=iτ2Δ. From the effective Lagrangian, when the U(1)Lμ−Lτ gauge symmetry is spontaneously broken by 〈S〉=vS/2 and 〈S′〉=vS′/2, the vanishing elements in Eq. (3) can be generated from Eq. (6) with 〈Δ〉=vΔ/2. We note that the dimension-6 operator LμTCΔ¯Δ¯†Δ¯Lτ has been dropped due to vΔ≪vS,S′. From Eq. (6), it can be seen that after electroweak symmetry breaking, the m4μ and m5τ effects can be combined with other terms as:(7)Yμ4+m5μm5LY45′→Y˜μ4,Yτ5+m4τm4LY45′→Y˜τ5,vS22m4Lm5LyS′yτ′⁎+m4τm4L→vS22m4Lm5Ly˜S′yτ′⁎,vS22m4Lm5LySyμ′⁎+m5μm5L→vS22m4Lm5Ly˜Syμ′⁎. Thus, to fit the neutrino masses, we need to take m4τ,5μ≪m4L,5L.Since the neutrino masses are generated by the spontaneous U(1)Lμ−Lτ symmetry breaking, we need to find the necessary conditions for vacuum stability. We thus write the gauge invariant scalar potential in this model as:(8)V=mH2H†H+mΔ2Tr[Δ†Δ]+mS′2S′†S′+mS2S†S+μΔ[HT(iτ2)Δ†H+h.c.]+μS[S′S′S†+h.c.]+λ1|H†H|2+λ2(Tr[Δ†Δ])2+λ3Tr[(Δ†Δ)2]+λ4|S′†S′|2+λ5|S†S|2+λ6(H†H)Tr[Δ†Δ]+H†(λ7ΔΔ†+λ8Δ†Δ)H+λ9(S′†S′)(H†H)+λ10(S†S)(H†H)+λ11(S′†S′)Tr[Δ†Δ]+λ12(S†S)Tr[Δ†Δ]+λ13(S′†S′)(S†S). The VEVs of scalar fields are obtained by the minimal conditions ∂〈V〉/∂vH,S,S′,Δ=0, and each condition can be expressed as:(9)∂〈V〉∂vH≃mH2vH+λ1vH3+12λ9vS′2vH+12λ10vS2vH≃0,(10)∂〈V〉∂vS≃mS2vS+12μSvS′2+λ5vS3+12λ10vH2vS+12λ13vS′2vS≃0,(11)∂〈V〉∂vS′≃mS′2vS′+2μSvS′vS+λ4vS′3+12λ9vH2vS′+12λ13vS2vS′≃0,(12)∂〈V〉∂vΔ≃mΔ2vΔ+12μΔvH2+12(λ6+λ7)vH2vΔ+12λ11vS′2vΔ+12λ12vS2vΔ≃0, where we have ignored the vΔ terms in the first three equations and the vΔ3 terms in the last equation due to vΔ≪vH,S,S′. In order to avoid the precision Higgs measurements, we can assume the mixing between H and S(S′) to be small, where the scalar mixing is discussed below; then, the VEV of H can be simplified as vH≈−mH2/λ1. If we further assume λ13 and μS to be small, the VEVs of S and S′ can be found as vS≈−mS2/λ5 and vS′≈−mS′2/λ4 with mS,S′2<0. The vS and vS′ are free parameters and their relation to the Z′-boson mass is given by mZ′2=gZ′2(4vS2+vS′2); hence, their magnitudes can be taken as the electroweak scale. From Eq. (12), the VEV of Higgs triplet can be determined as [54]:(13)vΔ≃−12μΔvH2mΔ2+(λ6+λ7)vH2/2+λ11vS′2/2+λ12vS2/2. Because vΔ<3.4 GeV, in order to obtain the heavy Higgs triplet bosons, unlike the Higgs doublet and S(S′), mΔ2 has to be positive and must also dictate the masses of the Higgs triplet bosons. From Eq. (13), it can be seen that similar to the type-II seesaw model [44,45], the Higgs triplet VEV is directly related to the lepton-number soft breaking term.We make a remark on the oblique parameter constraint. For the Higgs triplet, the mass difference between the Higgs triplet components is predominantly dictated by the oblique T-parameter, where the mass splitting between singly and doubly charged Higgs mass is bounded as |mH±±−mH±(H0)|≲50 GeV [46–48]. Since our study does not directly relate to the mass splitting of the Higgs triplet, we can take mH±±≈mH±(H0) to satisfy the constraint. Similarly, because the particle masses in L4(5) are taken to be the same, the vector-like leptons contributing to the T-parameter are small and can be neglected.Although the involved new scalars do not directly affect the neutrino physics in this study, it is of interest to understand the limit from the current SM Higgs precision measurements. Thus, we briefly discuss the mixings among the SM Higgs and new scalar bosons in the following analysis. Since the mixing between the SM Higgs and the Higgs triplet is suppressed by the small VEV of the Higgs triplet field, therefore, we consider the situations in the SM Higgs and singlet scalars. Moreover, if we take λ9,13≪1, the mixing between the SM Higgs and S′ is suppressed and can be neglected. Thus, in order to show the constraint of the Higgs precision measurements, we only focus on the H–S mixing. Using the scalar potential in Eq. (8) and HT=(G+,(v+h˜+iG0)/2) and S=(vS+s˜+iηS)/2, where G+ and G0 are the Nambu–Goldstone bosons in the SM, the squared mass matrix for the h˜ and s˜ scalar bosons can be obtained as:(14)L⊃12(h˜s˜)T(λ1v2λ102vvSλ102vvSλ5vS2)(h˜s˜). Using the 2×2 orthogonal matrix, written as:(15)(hs)=(cosαsinα−sinαcosα)(h˜s˜), the eigenvalues of the mass-square matrix and the mixing angle α can be obtained as:(16)mh,s2=λ1v2+λ5vS22±12(λ1v2−λ5vS2)2+λ102v2vS2,sin2α=λ10vvSmh2−mS2, where α is the mixing angle, and h is identified as the SM-like Higgs boson. It is clearly seen that in addition to the VEV of the Higgs field, the mixing effect of h and s is associated with the λ10 parameter and the VEV of S field.Although there are several channels for the SM-like Higgs production and decays, the most accurate measurement in the LHC is the gluon-gluon fusion (ggF) Higgs production and the Higgs diphoton decay, i.e. pp(gg)→h→γγ. Thus, we only concentrate on the h→γγ mode. For illustrating the influence of the new physics effects, we use the signal strength for pp→h→γγ, defined as:(17)μγγ=σ(pp→h)SM+NPσ(pp→h)SMBR(h→γγ)SM+NPBR(h→γγ)SM, where the ATLAS and CMS results using luminosities of 80 fb−1 and 35 fb−1 at s=13 TeV are given by μggF=0.97+0.15−0.14 [49] and μggF=1.02−0.18+0.19 [50], respectively. According to the current data, we can take δμγγNP=μγγ−1=±15% to constrain the new physics effect.From Eq. (16), the SM Higgs couplings are modified by a factor of cosα; thus, we obtain σ(pp→h)SM+NP≃cos2α×σ(pp→h)SM. For the h decays, in addition to the SM channels, the h can also decay into the ss and Z′Z′ final states when kinematically allowed in this model. In order to include these two decay modes, we write the relevant interactions as:(18)L⊃4gZ′2vSsinαhZμ′Z′μ−12ghsshss, where with λ1≃(mh/v)2 and λ5≃(ms/vS)2, the effective coupling ghss from the scalar potential can be obtained as:(19)ghϕϕ≃6sinαcosα(mh2vsinα+ms2vScosα)+λ10(vcos3α+vSsin3α−2vSsinαcos2α−2vsin2αcosα). Accordingly, the partial decay rates for the h→ss and h→Z′Z′ processes can be formulated as:(20)Γh→Z′Z′=2g′4vS2sin2απmh1−4mZ′2mh2(2+mh44mZ′4(1−2mZ′2mh2)2),Γh→ss=ghss232πmh1−(2msmh)2. As a result, the μγγ signal strength in Eq. (17) can be obtained as:(21)μγγ=cos4αΓhSMcos2αΓhSM+Γh→ss+Γh→Z′Z′, where ΓhSM≃4.07 MeV is the decay width of the SM Higgs [51]. Using Eq. (16) and vS=mZ′/(5gZ′), which arises from vS=vS′, we show δμγγNP as a function of λ10 in the left panel of Fig. 2, where mZ′=0.2 GeV and gZ′=10−3 motivated from the muon g−2 are used. With ms=10(200) GeV, the upper limit of λ10 can be ∼0.01(0.05), whereas the corresponding value of sinα is ∼0.004(0.01). Since we focus on a light S-boson in the phenomenological analysis, the effects of the small mixing α angle can be neglected. In the considered parameter region, s and Z′ mainly decay into Z′Z′ and ν¯ν, respectively, it is of interest to see the constraint from the invisible Higgs decays, where the current upper limit of branching ratio (BR) is BR(h→invisible)<0.24 [52,53]. Thus, we show BR(h→ss) (dotted), BR(h→Z′Z′) (dashed), and BR(h→ss+Z′Z′) (solid) as a function of λ10 in the right panel of Fig. 2. It can be clearly seen that the constraint from δμγγNP is stricter than that from the invisible Higgs decays.3Charged lepton flavor mixing matrixSince the PMNS matrix is related to the neutrino and charged-lepton flavor mixing matrices, before discussing the neutrino mass generation in this model, we first analyze the possibly sizable charged-lepton flavor mixing. As mentioned before, the active neutrino mass matrix is dictated by the Yukawa couplings in Eqs. (2) and (5), therefore, to explain the neutrino masses below the eV scale, most parameters have to be many orders of magnitude smaller than one. On the other hand, in order to have implications on the flavor physics, such as h→μτ and H−−→μτ, we need some parameters to be of O(10−2−10−1). In order to simplify the analysis on the charged-lepton flavor mixing, we thus ignore the small parameters, which are dictated by the neutrino masses, select the potentially sizable parameters, such as yμ,τ, yμ,τ′, and Y45, and use these parameters to formulate the flavor mixing matrix. The reason to select these parameters will be clear in the later analysis.The SM charged leptons and the introduced heavy leptons form a multiplet state in flavor space, denoted by ℓ′T=(ℓ,Ψℓ) with ℓ=(e,μ,τ) and ΨTℓ=(L4,L5). From Eq. (5), the 5×5 lepton mass matrix can be written as:(22)ℓ¯L′Mℓ′ℓR′=(ℓ¯L,Ψ¯ℓL)(mℓ3×3δm1δm2TmL)5×5(ℓRΨℓR), where diagmℓ=(me,mμ,mτ), mf=vHYf/2, diagmL=(m4L,m5L), and δm1,2 are given by:(23)δm1T=(0,vSyμ′2,00,0,vSyτ′2),δm2T=(0,0,vHyτ20,vHyμ2,0). The mass matrix Mℓ′ in Eq. (22) can be diagonalized by the unitary matrices UℓR,L through Mdiaℓ′=UℓLMℓ′UℓR†. Due to vH,S≪m4L,5L, we can expand the flavor mixing effects in terms of vH,S/m4L,5L; therefore, the 5×5 flvaor mixing matrices can be simplified as:(24)Uℓχ≈(13×3−ϵχϵχ†12×2)5×5, where we only retain the leading contributions, and the effects, which are smaller than ϵχ with χ=R,L, have been dropped, such as ϵχ†ϵχ, mℓδm1,2/mL2, etc. The explicit expressions of ϵχ are given as:(25)ϵL†=(0,vSyμ′2m4L,00,0,vSyτ′2m5L),ϵR†=(0,0,vHyτ2m4L0,vHyμ′2m5L,0), where the Yukawa couplings yμ,τ and yμ,τ′ are taken as real numbers. If we use vS≈100 GeV, m4L≈m5L≈1000 GeV, and yμ(τ)(′)∼0.1, the off-diagonal mixing matrix elements of Uℓχ are of O(10−2). Comparing with the PMNS matrix, where the minimal element is U13∼0.14 and is one oder of magnitude larger than (Uℓχ)ij with i≠j, we can approximate the PMNS matrix to be U≡UνLUℓL†≈UνL. That is, in this leading order approximation, we can use the PMNS matrix to diagonalize the induced neutrino mass matrix.After rotating the lepton weak states to physical states based on the UℓR and UℓL, the Yukawa couplings of the SM Higgs to the charged leptons are expressed as:(26)−Lh=(ℓ¯L,Ψ¯ℓL)UℓL(mℓ3×30δm2T0)UℓR†(ℓRΨℓR)hvH, where we still use ℓ and ΨℓT to represent the charged leptons. As a result, the SM Higgs Yukawa couplings to the light charged leptons can be found as:(27)−Lh⊃mℓvHℓ¯LℓRh−vSyμ′yτ2m4Lμ¯LτRh−vSyτ′yμ2m5Lτ¯LμRh+H.c. The second and third terms can lead to the h→μτ decay.4Majorana neutrino mass matrix and FGM patternsIn this section, we discuss the neutrino mass matrix and some phenomenology in our model. When we write the symmetric Majorana neutrino mass matrix as:(28)Mν=(meemeμmeτmeμmμμmμτmeτmμτmττ), from the Yukawa couplings in Eqs. (2) and (6), each matrix element can then be expressed as:(29)mee=YeevΔ2+Y45′y⁎eye′⁎vS′2vΔ22m4Lm5L,meμ=ye⁎Yμ4vS′vΔ2m4L+Y45′ye′⁎yμ′⁎22vSvS′vΔm4Lm5L+Y45′ye⁎m5μ2vS′vΔm4Lm5L,meτ=ye′Yτ5vS′vΔ2m5L+Y45′ye⁎yτ′⁎22vSvS′vΔm4Lm5L+Y45′ye′⁎m4τ2vS′vΔm4Lm5L,mμμ=Yμ4yμ′⁎vSvΔ2m4L,mμτ=YμτvΔ2+Yμ4m4τvΔ2m4L+Yτ5m5μvΔ2m5L+η22vS2vΔm4Lm5L,mττ=Yτ5yτ′⁎vSvΔ2m5L with η=Yμ4yS′yτ′⁎+Yτ5ySy′⁎μ+(Y45+Y45′)yτ′⁎yμ′⁎. Although the neutrino mass matrix comes from the dimension-4, -5, and -6 operators, since the involved free parameters are different, the matrix entries in Eq. (29) can be taken as the same order of magnitude with no particular hierarchy, 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) is diagonal. Although the other Yukawa interactions can induce off-diagonal elements, as shown earlier, these induced terms indeed are suppressed. If we neglect these small off-diagonal effects as a leading approximation, the Majorana neutrino mass matrix can be diagonalized by the PMNS matrix as Mdiaν=diag(m1,m2,m3)=diag(|m1|e−iα13,|m2|e−iα23,|m3|)=UTMνU, where α13 and α23 are the Majorana CP violating phases, and the standard parametrization of PMNS matrix is given as [5]:(30)U=(c12c13s12c13s13e−iδ−s12c23−c12s23s13eiδc12c23−s12s23s13eiδs23c13s12s23−c12c23s13eiδ−c12s23−s12c23s13eiδc23c13) with sij≡sinθij, cij≡cosθij, and δ being the Dirac CP violating phase.From Eq. (28), 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 we assume α13=α23=0, there are seven free parameters. However, we only have six observables: Δm21,312, sin2θ12,13,23, and Dirac CP phase δ; that is, we cannot determine all free parameters without further theoretical or experimental inputs. It has been suggested that a class of neutrino mass matrices may suffice to explain all neutrino experiments if the matrix textures have two independent zeroes [55]. The seven possible Frampton–Glashow–Marfatia (FGM) matrix patterns are classified as:(31)A1:(00X0XXXXX),A2:(0X0XXX0XX),B1:(XX0X0X0XX),B2:(X0X0XXXX0),B3:(X0X00XXXX),B4:(XX0XXX0X0),C:(XXXX0XXX0), where the symbol X denotes a nonzero texture. A detailed study with two-zero textures can be found in [58–60]. In order to simplify the analysis, we thus employ the FGM patterns as the theoretical inputs.As mentioned earlier, the neutrino mass order is still uncertain, i.e. |m1|<|m2|<|m3| or |m3|<|m1|<|m2| is allowed. With an FGM pattern, it helps understand what form of a neutrino mass matrix can lead to a specific mass order. According to the study referenced in [61], by taking the neutrino data with 1σ errors, the NO spectrum could be achieved by the patterns A1,2 and B1,2,3,4, while the IO could be achieved by the patterns B1,3 and C. Accordingly, it is of interest to see how the matrix elements of Eq. (29) in our model connect to those of a specific FGM matrix. It is found that when some Yukawa couplings are required to vanish, a definite FGM matrix pattern can then be achieved. We show the vanishing Yukawa couplings for the corresponding FGM matrix in Table 2. It is worth mentioning that a powerful FGM matrix pattern can also predict the absolute values of neutrino masses and Majorana CP-phases, which so far have not yet been observed in experiments. From the zero textures Mijν=Mklν=0 (ij≠kl), the neutrino mass ratios and Majorana CP phases can be obtained as [58]:(32)|m1||m3|=|Ui3Uj3Uk2Ul2−Ui2Uj2Uk3Ul3Ui2Uj2Uk1Ul1−Ui1Uj1Uk2Ul2|,|m2||m3|=|Ui1Uj1Uk3Ul3−Ui3Uj3Uk1Ul1Ui2Uj2Uk1Ul1−Ui1Uj1Uk2Ul2|α13=arg[Ui3Uj3Uk2Ul2−Ui2Uj2Uk3Ul3Ui2Uj2Uk1Ul1−Ui1Uj1Uk2Ul2],α23=arg[Ui1Uj1Uk3Ul3−Ui3Uj3Uk1Ul1Ui2Uj2Uk1Ul1−Ui1Uj1Uk2Ul2]. The values of the neutrino mass ratios and CP phases for each pattern with two benchmark inputs are shown in Table 3, where in addition to the taken values of sin2θ12=0.304 and sin2θ13=0.0219, the values inside brackets correspond to two different inputs: for the left value, we fix δ=1.5π and sin2θ23=0.5; for the right, δ=1.59205π and sin2θ23=0.4515 are used. From the results, it can be seen that the patterns A1 and A2 prefer the normal hierarchy, and the patter C shows the inverted hierarchy and degenerate case. The mass ordering in patterns B1−4 depends on the taken parameters. For illustration, in the following analysis, we focus the detailed analysis on the patterns A1 and C.5Numerical analysis and other phenomena of interest5.1Explain neutrino data and predict the absolute neutrino massesSince our purpose is not to examine all FGM patterns, in the following numerical analysis, we take A1 and C as the representatives of the NO and IO mass spectra, respectively. To determine the non-vanishing entries of the neutrino mass matrix and |mi|, we scan the parameters with the neutrino data at the 1σ level. Due to large experimental uncertainty, the Dirac CP phase is taken from a global data analysis using an χ2 method [64], in which the result in the 1σ region is 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 A1 and C.Using the relation Mν=U⁎MdiaνU† and the zero textures in Mν, the mass relations in A1 can be expressed as:(33)m1⁎=U13U11(U12U23−U13U22U11U22−U12U21)m3⁎,m2⁎=−U13U12(U11U23−U13U21U11U22−U12U21)m3⁎, while in C, they are:(34)m1⁎=U222U332−U232U322U212U322−U222U312m3⁎,m2⁎=−U212U332−U232U312U212U322−U222U312m3⁎, where the mks values in general are complex; however, there are only two independent phases among m1,2,3. With the chosen Majorana phases, such as m1=|m1|e−iα13 and m2=|m2|e−iα23, we obtain the relations(35)α13=arg[U13U11(U12U23−U13U22U11U22−U12U21)],α23=arg[−U13U12(U11U23−U13U21U11U22−U12U21)] for the A1 case, and(36)α13=arg[U222U332−U232U322U212U322−U222U312],α23=arg[−U221U332−U232U312U212U322−U222U312] for the C case. If we take the central values of measured θ12,13 in Eq. (1), sin2θ23≈0.50, and δ≈1.5π, we can easily obtain:(37)A1:{|m1|/|m3|≈0.230,|m2|/|m3|≈0.102,|m2|2−|m1|2≈0.029|m3|2,α13≈0.430π,α23≈−0.469π. However, it is found that the pattern C is very sensitive to the values of the mixing angles and CP phase δ when Δm212 and Δm322 are required to fit the data within 1σ errors. If sin2θ23≈0.4515 and δ≈1.59205π are taken, we obtain:(38)C:{|m1|/|m3|≈1.19,|m2|/|m3|≈1.20,|m2|2−|m1|2≈0.0130|m3|2,α13≈−0.705π,α23≈0.887π. Accordingly, if we further take Δm212≈7.53×10−5 eV2, the values of |mi| and Δm223 can be determined as:(39)A1:{|m1|≈5.5×10−3eV,|m2|≈1.03×10−2eV,|m3|≈5.06×10−2eV,Δm322≈2.45×10−3eV2;C:{|m3|≈7.60×10−2eV,|m1|≈9.07×10−2eV,|m2|≈9.11×10−2eV,Δm232≈2.53×10−3eV2.From above analysis, A1 and C can fit the neutrino data for the 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:(40)∑νmν<(0.12,0.17)eV, ([62], [63]) it can be found that the resulting ∑j|mj| in A1 can satisfy the upper bound while that in C is higher than the limit. In order to determine whether the tension with the cosmological neutrino mass bound can be relaxed when the ranges of the experimental measurements are extended, we adopt neutrino data up to the 3σ level instead of those at the 1σ level for C. In the numerical analysis, we generate 5×108 sampling points by randomly selecting the experimental values of s12,23,13 and δ within {1σ,2σ,3σ} errors and the values of m1 in the range of [0.01,0.17] eV; then, m2 and m3 are obtained via Eq. (34). In the end, the number of output points, which can fit the Δm21,232 data in the {1σ,2σ,3σ} range, is {552,3004,3467}. The obtained Dirac CP phase δ and ∑j|mj| are shown in Fig. 3, where the dots in black, red and blue denote the results with 1σ, 2σ and 3σ errors, respectively. From the figure, it can be clearly seen that ∑j|mj| in C is excluded even at the 3σ level if we adopt the bound from the cosmological measurements ∑νmν<0.12 eV while it can still satisfy the bound at the 2σ level if we adopt the upper limit of 0.17 eV.Since the uncertainties of sin2θ23 and Δm232 in Eq. (1) correspond to a 68% confidence level (CL), and the pattern C cannot fit the data within 1σ errors, in the remaining part of the paper, we only use the pattern A1 to show the constraints for the relevant Yukawa couplings. From the mass diagonal relation Mℓℓ′ν=(UℓkUℓ′k)⁎mk, when the PMNS matrix entries and mk are known, Mℓℓ′ν can then be determined. Thus, the correlation between δ and |mj| in A1 is shown in Fig. 4(a), where the neutrino data within 1σ error have been included. From the plot, it can be seen that each |mi| narrowly spreads around the value of Eq. (39). In the plot, we also show the effective Majorana neutrino mass 〈mββ〉, which is related to the neutrinoless double-beta (0νββ) decay rate and is defined by [21]:(41)〈mββ〉=|∑kUek2mk|, where a 90% CL upper limit of 〈mββ〉<0.061−0.165 eV was obtained by the KamLAND-Zen collaboration [65]. Our result of 〈mββ〉≈(0.34,2.3)×10−2 eV clearly satisfies the bound. According the results, the allowed ranges of |mij| as a correlation of |mττ| are shown in Fig. 4(b), where we scan the parameters using 107 sampling points to fit the neutrino data, and |m1|∈[0.001,0.1] eV is taken. As a result, the obtained ranges of mij in A1 are given as:(42)meτ=(0.99,1.11)×10−2eV,mμμ=(2.5,3.0)×10−2eV,mμτ=(2.2,2.4)×10−2eV,mττ=(2.4,2.8)×10−2eV, where mee and meμ are zero in neutrino mass pattern A1. In addition, the correlation between the Dirac phase δ and Majorana phase α13[α23] are shown in Fig. 4(c)[(d)].5.2Limits of Yukawa couplings and the h→μτ decayBased on the results obtained above, we now discuss the limits on the introduced Yukawa couplings shown in Eqs. (2) and (5). To simplify the analysis, we take m4L≈m5L≡mL and vS≈vS′≡vX, and define the parameters as:(43)aL=yτ′⁎yμ⁎vX2mL,aR=yμ′yτvX2mL,ξab(′)=Yab(′)vΔ2. The parameters aR,L can lead to the Higgs lepton-flavor violating h→μτ decay, where the associated interactions from Eq. (27) are expressed as [56,57]:(44)Lhτμ=hμ¯(aRPR+aLPL)τ+H.c. The BR for h→τμ can be obtained as:(45)BR(h→μτ)=|aL|2+|aR|28πΓhmh. With mh≈125 GeV and Γh≈4.21 MeV, the limit on aL,R can be obtained as(46)|aL|2+|aR|2≈1.56×10−3BR(h→τμ)2.5×10−3, where BR(h→μτ) can be taken from the experimental data, and the current upper limits from ATLAS and CMS are 1.43% [66] and 0.25% [67,68], respectively.Using Yee≈Y45′≈ye≈0 in A1, the neutrino mass matrix entries in Eq. (29) are formed as:(47)meτ=ye′vXmLξτ5,mμμ=2yτ⁎aR⁎ξμ4,mττ=2yμ⁎aLξτ5,mμτ=ξμτ+(m4τmL+yS′yμ⁎vXmLaL)ξμ4+(m5μmL+ySyτ⁎vXmLaR⁎)ξτ5+2yτ⁎yμ⁎aLaR⁎(ξ45+ξ45′). In order to get sizable BR(h→μτ) and ξ45, we find that |aL|≪|aR| or |aR|≪|aL| has to be satisfied. According to Eq. (42), if we take |mμμ|≈|mττ|≈2.7×10−2 eV, |meτ|≈10−2 eV, |mμτ|≈2.3×10−2 eV, |aR(L)|≈10−3(10−8), vX≈100 GeV, and mL≈1000 GeV, we can obtain BR(h→μτ)≈1.2×10−3, and the magnitudes of parameters are obtained as:(48)|ye′ξτ5|≈1.0×10−10GeV,|ξμ4yτ|≈1.9×10−8GeV,|ξτ5yμ|≈1.9×10−3GeV,|ξμτ|≈|2.3−2ξ45yμ⁎yτ⁎|×10−11GeV, where the second and third terms in mμτ have been ignored due to yS,yS′,m4τ,5μ/mL≪1. With yμ≈yτ≈0.1, the Higgs triplet Yukawa couplings then have the hierarchy Yμτ≪Yμ4≪Yτ5≪Y45; that is, we cannot avoid fine-tuning the Yukawa couplings to explain the neutrino data in this model.According to above analysis, we see that the Yukawa couplings, which are not highly suppressed by the neutrino masses, are only yμ, yτ, yμ′, and Y45. We need to examine if they will be further constrained by other rare decays. Since the new physics effects occur in the lepton sector, the strict constraints may come from the lepton-flavor violating processes, such as τ→3μ, τ→(e,μ)γ, and μ→eγ. From Eq. (44), it is known that τ→3μ can be induced through off-shell h decay into the muon pair. The BR for this three-body decay can be expressed as:(49)BR(τ→3μ)=ττmτ53⋅29π3mh4|yμμaR|2≈1.2×10−7|aR|2, where yμμ=mμ/vH is the Higgs coupling to the muon in the SM, and the small aL has been ignored. Taking aR≈10−3, the h-mediated BR(τ→3μ) is much less than the current upper bound of 2.1×10−8 [5]. To induce the rare μ→eγ process, the new Yukawa couplings have to couple to the electron. From Eqs. (2) and (5), the relevant couplings are Yee, ye, and ye′, however, Yee≈ye≈0 and ye′≪1 have been used to fit the neutrino masses. Thus, the rare μ→eγ process is suppressed in our model.Similarly, τ→(e,μ)γ are suppressed by most Yukawa couplings with the exception of yτ and yμ′, where the associated Feynman diagram is shown in Fig. 5. It can be seen that in addition to yτ and yμ′, the quartic scalar coupling λ10 involves in the τ→μγ process. As a result, the interaction of the loop induced τ→μγ can be written as:(50)Lτμγ=−emτ16π2CRμ¯σμνPRτFμν,CR=λ10vHaR2mτmL2I(zh,zS),I(zh,zS)=∫01dx1∫0x1dx2x22(zh−(zh−zS)x1+(1−zS)x2)2, with zh=mh2/mL2 and zS=mS2/mL2. The BR for τ→μγ can be expressed as:(51)BR(τ→μγ)BR(τ→μν¯μντ)=3αe4πGF|CR|2≈1.51×10−13(|aR|λ1010−3)2, where we have used aR≈10−3, mh≈125 GeV, mS≈10 GeV, mL≈1000 GeV, and I(zh,zS)≈0.46. Clearly, the BR for τ→μγ in our model is still below the current upper bound of 4.4×10−8 [5]. Note that we have ignored the aL effect due to the use of aL≪aR.5.3Phenomenological implications on the muon g−2, rare τ, and H−−→μτ decaysAfter determining the magnitudes of the Higgs-triplet Yukawa couplings, which are used to explain the neutrino data, we state some implications of this model in flavor and collider physics, which have been studied in the literature and are still interesting in this model. In addition to the large BR(h→μτ), if the new Z′ gauge boson is in the MeV to GeV range, the muon g−2 anomaly can be resolved by the intermediate Z′-gauge boson [26,27,33], depending on the magnitude of gZ′ gauge coupling. The contribution from the Z′-penguin diagram to the muon g−2 can be expressed as [26,33](52)ΔaμZ′=gZ′28π2∫01dx2mμ2x2(1−x)x2mμ2+(1−x)mZ′2. It is found that to explain the muon g−2 anomaly, Δaμ=aμexp−aμSM=(28.7±8.0)×10−10 [5], the allowed ranges of gZ′ and mZ′ are :(53)2×10−4≤gZ′≤2×10−3,(54)5≤mZ′≤210MeV, where other regions have been experimentally excluded, such as the neutrino trident production [70], BABAR collaboration [71], and Borexino experiment [72].With the value of aR∼10−3, the sizable Yukawa couplings yμ′ and yτ can induce the lepton-flavor violating interaction τ-μ-S through the mixing between vector-like lepton and τ(μ)-lepton. From the S-Z′-Z′ interaction, the τ→μZ′Z′ decay can be generated by the mediation of light scalar S, and its partial decay rate as a function of Z′-pair invariant can be derived as [34]:(55)dBR(τ→μZ′Z′)dq2≈mτ64π2mhΓhΓτBR(h→μτ)×(q2−2mZ′2)2+8mZ′4vS4mS2(1−q2mτ2)21−4mZ′2q2. It can be seen that τ→μZ′Z′ and h→μτ can be correlated in the model when mZ′ is below GeV. We show the BR(τ→μZ′Z′) (in units of 10−9) as a function of BR(h→μτ) (in units of 10−3) and vS in Fig. 6, where mS=10 GeV and mZ′=0.2 GeV are fixed. With 50 ab−1 of data accumulated at the Belle II, the sample of τ pairs can be increased up to around 5×1010, where the sensitivity necessary to observe the LFV τ decays can reach 10−10−10−9 [69]. Therefore, the BR(τ→μZ′Z′) of 10−9 allowed in the model could be tested at the Belle II.Moreover, we find that a sizable Y45 Yukawa coupling can change the decay property of doubly charged Higgs H±± in the Higgs triplet. In this model, H±± can decay to the μ±τ± final states via the dimension-4 and the induced dimension-6 operators, which are expressed as:(56)YμτLμTCiτ2ΔLτ+Y45yτyμmL2τRTHTiτ2ΔHμR, where the corresponding H±± Yukawa coupling to μ±τ± can be written as:(57)YH±±=Yμτ⁎+Y45yτyμvH22mL2. From Eq. (48), Y45 can in principle be O(0.1) when other neutrino mass related parameters are tuned to be small, e.g. YμτvΔ/2∼10−10. Thus, with mL≈1000 GeV, vH≈246 GeV, mH±±≈800 GeV, and yτ∼yμ∼0.1, the decay rate ratio of H±±→μ±τ± to H±±→W±W± can be estimated as [73]:(58)Γ(H±±→μ±τ±)Γ(H±±→W±W±)≈|YH±±|2vH22vΔ2vH2mH±±2≈2.6×10−4|Y45|2vΔ2, where the small Yμτ is neglected. With |Y45|∼0.05 and vΔ∼0.01 GeV, the ratio can be at the 10% level; that is, the BR for H±±→μ±τ± is not suppressed and can be a good channel to observe the doubly charged-Higgs. In addition, the τ→ℓiℓjℓ¯k decays can be induced by the H±± couplings shown in Eq. (56). Since we focus on the A1 pattern, the potential mode is τ→μμμ¯ and its BR can be estimated as [74]:(59)BR(τ→μμμ¯)BR(τ→μν¯ν)=14GF2mH±±4|YH±±|2(2mμμvΔ)2. This BR is tiny since it is suppressed by (mμμ/vΔ)2∼4×10−17 when vΔ∼0.01 GeV is used; therefore, this process cannot give a strict constraint on YH±±.6SummaryWe studied the origin of the neutrino mass in the gauged Lμ−Lτ model. We learned that although including one Higgs triplet can violate the lepton number, the effect is not sufficient 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, which carry the Lμ−Lτ charges, are introduced. In this model, a specific Frampton–Glashow–Marfatia matrix pattern can be achieved when some Yukawa couplings are set to vanish. Using the pattern A1, we showed that when the neutrino data within 1σ errors and cosmological neutrino bound are satisfied, the involving Higgs-triplet Yukawa couplings have a hierarchy, i.e., Yμτ≪Yμ4≪Yτ5≪Y45, and Y45 can be O(0.1). As a result, the effective Majorana neutrino mass is below the upper limit of neutrinoless double-beta decay experiment. 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