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Linear seesaw model with T7 symmetry for neutrino mass and mixing

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1. Wu, Y.-L.. Gravidynamics, spinodynamics and electrodynamics within the framework of gravitational quantum field theory[J]. Science China: Physics, Mechanics and Astronomy, 2023, 66(6): 260411. doi: 10.1007/s11433-022-2052-6
2. Wu, Y.-L.. The foundation of the hyperunified field theory I-Fundamental building block and symmetry[J]. International Journal of Modern Physics A, 2021, 36(28): 21430016. doi: 10.1142/S0217751X21430016
3. Wu, Y.-L.. The foundation of the hyperunified field theory II - Fundamental interaction and evolving universe[J]. International Journal of Modern Physics A, 2021, 36(28): 2143002. doi: 10.1142/S0217751X21430028
4. Wu, Y.-L.. Hyperunified field theory and Taiji program in space for GWD[J]. International Journal of Modern Physics A, 2018, 33(31): 1844014. doi: 10.1142/S0217751X18440141
5. Wu, Y.-L.. Hyperunified field theory and gravitational gauge–geometry duality[J]. European Physical Journal C, 2018, 78(1): 28. doi: 10.1140/epjc/s10052-017-5504-3

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V. V. Vien. A linear seesaw model with T7 symmetry for neutrino mass and mixing[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac57b7
V. V. Vien. A linear seesaw model with T7 symmetry for neutrino mass and mixing[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac57b7 shu
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Linear seesaw model with T7 symmetry for neutrino mass and mixing

    Corresponding author: V. V. Vien, vovanvien@tdmu.edu.vn
  • 1. Institute of Applied Technology, Thu Dau Mot University, Binh Duong Province, Vietnam
  • 2. Department of Physics, Tay Nguyen University, Daklak province, Vietnam

Abstract: We propose a low-scale Standard Model extension with T7×Z4×Z3×Z2 symmetry that can successfully explain observed neutrino oscillation results within the 3σ range. Small neutrino masses are obtained via the linear seesaw mechanism. Normal and inverted neutrino mass orderings are considered with three lepton mixing angles in their experimentally allowed 3σ ranges. The model provides a suitable correlation between the solar and reactor neutrino mixing angles, which is consistent with the TM2 pattern. The prediction for the Dirac phase is δCP(295.80,330.0) for both normal and inverted orderings, including its experimentally maximum value, while those for the two Majorana phases are η1(349.60,356.60),η2=0 for normal ordering and η1(3.44,10.37),η2=0 for inverted ordering. In addition, the predictions for the effective neutrino masses are consistent with the present experimental bounds.

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    I.   INTRODUCTION
    • Despite its great success, the Standard Model (SM) is incomplete and must be improved to accommodate recent neutrino oscillation data. Except for the Dirac CP phase, neutrino oscillation parameters have been measured with high precision in the allowed ranges (taken from Ref. [1]), as shown in Table 1.

      Normalordering(NO) Invertedordering(IO)
      bfp±1σ(3σrange) bfp±1σ(3σrange)
      sin2θ12 0.318±0.016(0.271÷0.369) 0.318±0.016(0.271÷0.369)
      sin2θ23 0.574±0.014(0.434÷0.610) 0.578+0.0100.017(0.433÷0.608)
      sin2θ13102 2.200+0.0690.062(2.00÷2.405) 2.225+0.0640.070(2.018÷2.424)
      δ/π 1.08+0.130.12(0.71÷1.99) 1.58+0.150.16(1.11÷1.96)
      Δm221(meV2) 75.0+2.22.0(69.4÷81.4) 75.0+2.22.0(69.4÷81.4)
      |Δm231|(meV2)103 2.55+0.020.03(2.47÷2.63) 2.45+0.020.03(2.37÷2.53)

      Table 1.  Neutrino oscillation data determined from a global analysis taken from [1].

      The seesaw mechanism [2] is probably the most popular method of producing small neutrino masses; however, owing to the extremely high mass scale of right-handed neutrinos, they have not yet been observed experimentally. This problem can be solved with new physics on the TeV scale using the linear seesaw mechanism [37], which may be achieved by LHC experiments.

      While explaining the observed pattern of neutrino mixing, discrete symmetries have revealed many advantages. One of these symmetries, T7, has been widely used in various studies [814]. The linear seesaw mechanism with non-Abelian discrete symmetries has been investigated in [1523]; however, there are substantial differences between those studies and our present work. In those studies, the lepton and/or quark masses and mixings were obtained by i) many doublets [1523], ii) other non-Abelian discrete symmetries [16, 18, 2123], iii) other Abelian discrete symmetries [1523], and iv) other gauge symmetries [1623]. Thus, it is necessary to find a more optimal proposal to explain the observed neutrino oscillation data. In the present study, we propose a SM extension with T7×Z4×Z3×Z2 discrete symmetry, in which three left-handed leptons, three right-handed neutrinos, and extra neutral leptons lie in 3 of T7 while three right-handed charged leptons l1R,l2R,l3R lie in the singlets 10,11, and 12 of the T7 symmetry, respectively. As a result, the tiny neutrino masses and recently observed lepton mixing pattern are explained by the Yukawa terms up to five dimensions. An interesting feature of T7 is that its 3×ˉ3 tensor product contains three singlets 10,1,2 and two triplets that are complex conjugate to each other 3,ˉ3, which makes it convenient for constructing the desired mass matrices.

      The remainder of this paper is structured as follows. In Sec. II, we present the particle content of the model. The lepton mass and mixing are described in Sec. IV. Secs. V and VI are devoted to the numerical analysis and effective neutrino masses, respectively. Finally, the conclusion is presented in Sec. VII. The appendix provides the irreducible representations and tensor products of the T7 group.

    II.   THE MODEL
    • The full symmetry of the model is G=GSM×T7×Z4×Z3×Z2, in which the gauge symmetry of the SM, SU(3)C×SU(2)L×U(1)YG, is supplemented by one non-Abelian symmetry T7 and three Abelian discrete symmetries Z4,Z3, and Z2. Furthermore, three right-handed neutrinos (νR), two different types of neutral singlet leptons with both helicities (NL,R,SL,R), and four SU(2)L singlet scalars are added. The particle and scalar assignments of the model under G symmetry is summarized in Table 2.

      ψLl1R,l2R,l3RHνRNLNRSLSRφϕρη
      SU(2)L212111111111
      U(1)Y12−112000000000
      U(1)L110111110000
      T7310, 11, 121033333331010
      Z4ii11iiii−1−1i−1
      Z31ω21ω11111ωω21
      Z2+++++

      Table 2.  Assignment under G symmetry for leptons and scalars.

      The lepton Yukawa terms invariant under G (up to five dimensions) are

      Ll=h1Λ(ˉψLϕ)10(Hl1R)10+h2Λ(ˉψLϕ)12(Hl2R)11+h3Λ(ˉψLϕ)11(Hl3R)12+x1(ˉψLNR)10˜H+x2(ˉψLSR)10˜H+x3(¯NLνR)10ρ+x4(¯SLνR)10ρ+y1(¯SLNR)10η+y2(¯SLNR)ˉ3φ+y3(¯SLNR)3φ+z1(¯NLSR)10η+z2(¯NLSR)ˉ3φ+z3(¯NLSR)3φ+t1(¯NLNR)10η+t2(¯NLNR)ˉ3φ+t3(¯NLNR)3φ+u1(¯SLSR)10η+u2(¯SLSR)ˉ3φ+u3(¯SLSR)3φ+H.c.,

      (1)

      where hi,xi,4,yi,zi,ti and τi(i=1÷3) are dimensionless couplings, and Λ is the cut-off scale. All other Yukawa terms, listed in Table F1 of Appendix F, are prevented by one or some of the model's symmetries; thus, they are not included in the expression of the Lagrangian Ll in Eq. (1).

      Couplings (i=1,2,3)Forbidden by
      (¯ψLνR)(˜Hϕ),(¯NLνR)(ϕφ),(¯NLνR)(ϕφ),(¯NLνR)(ϕη),(¯NLνR)(ϕη),(¯SLνR)(ϕφ),(¯SLνR)(ϕφ),(¯SLνR)(ϕη),(¯SLνR)(ϕη). Z4
      (¯ψLliR)(Hφ),(¯SLNR)ϕ,(¯SLNR)ϕ,(¯NLSR)ϕ,(¯NLSR)ϕ,(¯NLNR)ϕ,(¯NLNR)ϕ,(¯SLSR)ϕ,(¯SLSR)ϕ. Z3
      (¯NLνR)(ϕρ),(¯NLνR)(φρ),</td>(¯SLνR)(ϕρ),(¯SLνR)(φρ).Z2
      (¯ψLliR)(Hϕ),(¯ψLliR)(Hφ),(¯ψLliR)(Hη),(¯ψLliR)(Hη).T7×Z3
      (¯ψLνR)(˜Hϕ),(¯ψLνR)(˜Hφ),(¯ψLνR)(˜Hφ),(¯ψLνR)(˜Hη),(¯ψLνR)(˜Hη),(¯ψLNR)(˜Hρ),(¯ψLNR)(˜Hρ),(¯ψLSR)(˜Hρ),Z4×Z3
      (¯ψLSR)(˜Hρ),¯NLνRρ,(¯NLνR)(ϕφ),(¯NLνR)(ϕφ),(¯NLνR)(ϕη),(¯NLνR)(ϕη),(¯NLνR)(φη),(¯NLνR)(φη),(¯NLνR)(φη),(¯NLνR)(φη),
      ¯SLνRρ,(¯SLνR)(ϕφ),(¯SLνR)(ϕφ),(¯SLνR)(ϕη),(¯SLνR)(ϕη),(¯SLνR)(φη),(¯SLνR)(φη),(¯SLνR)(φη),(¯SLνR)(φη),
      (¯SLNR)(ϕρ),(¯SLNR)(ϕρ),(¯SLNR)(ϕρ),(¯SLNR)(ϕρ),(¯SLNR)(φρ),(¯SLNR)(φρ),(¯SLNR)(φρ),(¯SLNR)(φρ),
      (¯SLNR)(ηρ),(¯SLNR)(ηρ),(¯SLNR)(ηρ),(¯SLNR)(ηρ),(¯NLSR)(ϕρ),(¯NLSR)(ϕρ),(¯NLSR)(ϕρ),(¯NLSR)(ϕρ),
      (¯NLSR)(φρ),(¯NLSR)(φρ),(¯NLSR)(φρ),(¯NLSR)(φρ),(¯NLSR)(ηρ),(¯NLSR)(ηρ),(¯NLSR)(ηρ),(¯NLSR)(ηρ),(¯NLSR)(ηρ),
      (¯NLNR)(ϕρ),(¯NLNR)(ϕρ),(¯NLNR)(ϕρ),(¯NLNR)(ϕρ),(¯NLNR)(φρ),(¯NLNR)(φρ),(¯NLNR)(φρ),(¯NLNR)(φρ),
      (¯NLNR)(ηρ),(¯NLNR)(ηρ),(¯NLNR)(ηρ),(¯NLNR)(ηρ),(¯SLSR)(ϕρ),(¯SLSR)(ϕρ),(¯SLSR)(ϕρ),(¯SLSR)(ϕρ),
      (¯SLSR)(φρ),(¯SLSR)(φρ),(¯SLSR)(φρ),(¯SLSR)(φρ),(¯SLSR)(ηρ),(¯SLSR)(ηρ),(¯SLSR)(ηρ),(¯SLSR)(ηρ).
      (¯ψLνR)(˜Hρ),(¯ψLNR)(˜Hφ),(¯ψLNR)(˜Hφ),(¯ψLNR)(˜Hη),(¯ψLNR)(˜Hη),(¯ψLSR)(˜Hφ),(¯ψLSR)(˜Hφ),(¯ψLSR)(˜Hη),Z4×Z2
      (¯ψLSR)(˜Hη),(¯NLνR)(φρ),(¯NLνR)(φρ),(¯NLνR)(ρη),(¯NLνR)(ρη),(¯SLνR)(φρ),(¯SLνR)(φρ),(¯SLνR)(ρη),(¯SLνR)(ρη),
      (¯SLNR)(φη),(¯SLNR)(φη),(¯SLNR)(φη),(¯SLNR)(φη),(¯NLSR)(φη),(¯NLSR)(φη),(¯NLSR)(φη),(¯NLSR)(φη),
      (¯NLNR)(φη),(¯NLNR)(φη),(¯NLNR)(φη),(¯NLNR)(φη),(¯SLSR)(φη),(¯SLSR)(φη),(¯SLSR)(φη),(¯SLSR)(φη).
      (¯ψLνR)(˜Hρ),(¯NLνR)(ϕρ),(¯NLνR)(φρ),(¯NLνR)(ρη),(¯NLνR)(ρη),(¯SLνR)(ϕρ),(¯SLνR)(φρ),(¯SLνR)(ρη),(¯SLνR)(ρη).Z3×Z2
      (¯ψLνR)˜H,(¯ψLNR)(˜Hϕ),(¯ψLNR)(˜Hϕ),(¯ψLSR)(˜Hϕ),(¯ψLSR)(˜Hϕ),¯NLνRϕ,¯NLνRϕ,¯NLνRφ,¯NLνRφ,¯NLνRη,¯NLνRη,(¯NLνR)(ϕρ),Z4×Z3×Z2
      (¯NLνR)(ϕρ),¯SLνRϕ,¯SLνRϕ,¯SLνRφ,¯SLνRφ,¯SLνRη,¯SLνRη,(¯SLνR)(ϕρ),(¯SLνR)(ϕρ),(¯SLNR)ρ,(¯SLNR)ρ,(¯SLNR)(ϕη),
      (¯SLNR)(ϕη),(¯SLNR)(ϕη),(¯SLNR)(ϕη),(¯NLSR)ρ,(¯NLSR)ρ,(¯NLSR)(ϕη),(¯NLSR)(ϕη),(¯NLSR)(ϕη),(¯NLSR)(ϕη),
      (¯NLNR)ρ,(¯NLNR)ρ,(¯NLNR)(ϕη),(¯NLNR)(ϕη),(¯NLNR)(ϕη),(¯NLNR)(ϕη),(¯SLSR)ρ,(¯SLSR)ρ,(¯SLSR)(ϕη),(¯SLSR)(ϕη),
      (¯SLSR)(ϕη),(¯SLSR)(ϕη).
      (¯ψLliR)H,(¯ψLliR)(Hρ),(¯ψLliR)(Hρ).T7×Z4×Z3×Z2

      Table F1.  Forbidden interactions.

      To reproduce the recently observed neutrino oscillation data from the scalar potential minimum condition, as will be presented in Sec. III, the following VEV configurations for the scalar fields are obtained:

      H=(0vH)T,φ=(0,0,φ3),φ3=vφ,ρ=vρ,η=vη,ϕ=(ϕ1,ϕ2,ϕ3),ϕ1=ϕ2=ϕ3=vϕ.

      (2)

      In this study, we assume that the VEV of SU(2) singlet scalars and the cut-off scale are at extremely high scales.

      vϕvφvηvρ1010GeV,Λ1013GeV.

      (3)
    III.   SCALAR POTENTIAL MINIMUM CONDITION
    • The scalar potential invariant under G is explicitly given in Appendix B. To show that the scalar VEVs in Eq. (2) is a natural solution of the minimum condition of the total Higgs potential, we use vα=vϑ(ϑ=H,φ,ϕ,ρ,η). Therefore, the minimization condition of Vtot becomes

      Vtotvϑ=0,2Vtotv2ϑ>0.

      (4)

      Using the benchmark points

      λHφ1=λHφ2=λHϕ1=λHϕ2=λHρ1=λHρ2=λHη1=λHη2=λHη,λϕρ1=λϕρ2=λϕη1=λϕη2=λϕη,λρη1=λρη2=λρη,λφη1=λφη2=λφρ1=λφρ2=λφρ,λϕ1=λϕ3=λφ1=λφ2=λφ,λφϕ1=λφϕ6=λφϕ7=λφϕ8=λφϕ9=λφϕ10=λφϕ,

      (5)

      the expressions in (4) reduce to

      μ2H+2λHv2H+2λHϕ(v2φ+3v2ϕ)+λHη(v2η+v2ρ)=0,

      (6)

      μ2φ+2λHϕv2H+4λφv2φ+2λφϕ(v2η+4v2ϕ+v2ρ)=0,

      (7)

      3μ2ϕ+6λHϕv2H+8λφϕv2φ+24λϕv2ϕ+6λφϕ(v2η+v2ρ)=0,

      (8)

      μ2ρ+λHηv2H+2λφϕ(v2φ+v2η+3v2ϕ)+2(λHη+λρ)v2ρ=0,

      (9)

      μ2η+λHηv2H+2λφϕv2φ+2(λη+λHη)v2η+6λφϕv2ϕ+2λφϕv2ρ=0,

      (10)

      2Vtotv2H=2λHv2H>0,2Vtotv2φ=4λφv2φ>0,2Vtotv2ϕ=24λϕv2ϕ>0,

      (11)

      2Vtotv2ρ=2(λHη+λρ)v2ρ>0,2Vtotv2η=2(λη+λHη)v2η>0.

      (12)

      The system of Eqs. (6)–(10) always possesses the solution defined in Appendix C. With the aid of (C1)–(C4), the expressions in (11) remain unchanged, while those in (12) become

      2Vtotv2ρ=2v2ρ(λρ2λHv2H+μ2H2v2φ+v2η+6v2ϕ+v2ρ)>0,

      (13)

      2Vtotv2η=2v2η(λη2λHv2H+μ2H2v2φ+v2η+6v2ϕ+v2ρ)>0.

      (14)

      The conditions in the expressions (11), (13), and (14) are satisfied only if the following conditions are simultaneously held:

      λH,λφ,λϕ>0;λρ,λη>2λHv2H+μ2H2v2φ+v2η+6v2ϕ+v2ρ.

      (15)

      As a concrete example, assuming that μ2ϑ are negative and of the same order of magnitude as that of the SM [24],

      μ2Hμ2φμ2ϕμ2ρμ2η=102GeV.

      (16)

      With the help of Eqs. (3) and (16), 2Vtot/v2H depends on λH, 2Vtot/v2φ, and 2Vtot/v2ϕ depend on λφ, and 2Vtot/v2ρ and 2Vtot/v2η depend on λH and λρ, which are plotted in Figs. 1 and 2, respectively. These show that the inequalities in (4) are always satisfied by the VEV alignments in (2).

      Figure 1.  (color online) (2Vtot/v2H)×1021 versus λH with λH(103,101) (upper left), (2Vtot/v2φ)×1037 (bottom left), and (2Vtot/v2ϕ)×1037 (bottom right) versus λφ with λφ(103,101)

      Figure 2.  (color online) (2Vtot/v2ρ)×1036 versus λH and λρ (left panel) and (2Vtot/v2η)×1036 versus λH and λη (right panel) with λH(103,101), λρ(103,101) and λη(103,101).

    IV.   LEPTON MASSES AND MIXINGS
    • Using the tensor product of the T7 group [13, 25], after symmetry breaking, the charged lepton masses and corresponding mixing matrix obtained from the first line of Eq. (1) have the following forms:

      Mcl=vHvϕΛ(h1h2h3h1ω2h2ωh3h1ωh2ω2h3)(ω=ei2π/3),

      (17)

      U+L=13(1111ωω21ω2ω),Ur=I3×3.

      (18)

      Turning now to the neutrino sector. From Eq. (1), when the scalar fields obtain their VEVs, we obtain the following neutrino mass matrices:

      mνN=x1vHI3×3x1I3×3,MνS=x2vHI3×3x2I3×3,

      (19)

      mνN=x3vρI3×3x3I3×3,MνS=x4vρI3×3x4I3×3,

      (20)

      MNS=(y1vη0y3vφ0y1vη0y2vφ0y1vη)(y10y30y10y20y1),

      (21)

      MNS=(z1vη0z3vφ0z1vη0z2vφ0z1vη)(z10z30z10z20z1),

      (22)

      MNN=(t1vη0t3vφ0t1vη0t2vφ0t1vη)(t10t30t10t20t1),

      (23)

      MSS=(g1vη0g3vφ0g1vη0g2vφ0g1vη)(g10g30g10g20g1).

      (24)

      In contrast to the charged lepton sector, all the mass matrices in the neutrino sector are generated from the renormalizable Yukawa terms. The neutrino mass matrix for the linear seesaw, in the (νL,NL,SL), (νR,NR,SR) basis, takes the form

      Meff=(0mνNMνSmνNMNNMNSMνSMNSMSS)(0MDMTDMR),

      (25)

      where

      MD=(mνNMνS),MTD=(mνNMνS),MR=(MNNMNSMNSMSS),

      and all the entries for Meff are defined in Eqs. (19)–(24).

      The light Dirac neutrino mass matrix then gets the form

      Mν=MDM1RMTD=mνNM1NSMνSMνSM1NSmνNMνSM1SSMνS+mνNM1NNMNSM1SSMνS+MνSM1SSMNSM1NNmνN+MνSM1NSMNNM1NSMνSmνNM1NNMNSM1SSMNSM1NNmνN.

      (26)

      Combining Eqs. (19)–(24) and (26) yields

      Mν=(α0δ0β0κ0γ),

      (27)

      where

      α=x1x3α13+x1x4α14+x2x3α23+x2x4α24α0eiψα,

      (28)

      β=(t1x2x1z1)(x3y1t1x4)g1t21+t1x2x4x1x4z1x2x3y1y1z1β0eiψβ,

      (29)

      γ=x1x3γ13+x1x4γ14+x2x3γ23+x2x4γ24γ0eiψγ,

      (30)

      δ=x1x3δ13+x1x4δ14+x2x3δ23+x2x4δ24δ0eiψδ,

      (31)

      κ=x1x3κ13+x1x4κ14+x2x3κ23+x2x4κ24κ0eiψκ,

      (32)

      with αmn,γmn,δmn, and κmn(mn=13,14,23,24) given in Appendix D.

      First, we define the Hermitian matrix M2ν given by

      M2ν=MνM+ν=(a200d0eiΨ0β200d0eiΨ0c20),

      (33)

      where

      a20=α20+δ20,c20=γ20+κ20,

      (34)

      d0eiψ=γ0δ0ei(ψγψδ)+α0κ0ei(ψαψκ).

      (35)

      The mass matrix M2ν is diagonalized by Uν satisfying

      U+νM2νUν={(m21000m22000m23),Uν=(cosθ0sinθ.eiΨ010sinθ.eiΨ0cosθ)forNO,  (m23000m22000m21),Uν=(sinθ0cosθeiΨ010cosθeiΨ0sinθ)forIO,

      (36)

      where

      m21=β20Δm221,m22=β20,m23=β20+Δm231Δm221.

      (37)

      The sum of neutrino masses is given by

      mν=β0+β20Δm221+β20Δm221+Δm231.

      (38)

      Using the expressions for ULep and Uν in Eqs. (18) and (36), the leptonic mixing matrix, ULep=U+LUν, is given by

      ULep={13(cosθ+sinθ.eiΨ1cosθsinθ.eiΨcosθ+ω2sinθ.eiΨωω2cosθsinθ.eiΨcosθ+ωsinθ.eiΨω2ωcosθsinθ.eiΨ)forNO,13(sinθcosθ.eiΨ1sinθ+cosθ.eiΨsinθω2cosθ.eiΨωω2sinθ+cosθ.eiΨsinθωcosθ.eiΨω2ωsinθ+cosθ.eiΨ)forIO.

      (39)

      Expression (39) implies that ULep possesses the TM2 form because (ULep)i2=13(i=1,2,3).

      Now, comparing Eq. (39) with the standard parameterization of UPMNS [24], we can parameterize the solar neutrino mixing angle θ12, Dirac CP phase δCP, and model parameters θ,Ψ,η1 in terms of the other two neutrino mixing angles θ13 and θ23, as follows:

      s212=13c213forbothNOandIO,

      (40)

      sθ={12+32c413s223c213forNO,1232c413s223c213forIO,

      (41)

      sinδCP=c413s23c213s23s1323s213forbothNOandIO,

      (42)

      Ψ={sec1(43(s213+4c413s223)13s213)forNO,sec1(43(s213+4c413s223)3s2131)forIO,

      (43)

      η1={ilog(cθ+sθeiΨ3c12c13)forNO,i2log(3)ilog(sθcθeiΨc12c13)forIO,

      (44)

      η2=0forbothNOandIO.

      (45)
    V.   NUMERICAL ANALYSIS
    • The considered model contains three degrees of freedom in the neutrino sector corresponding to three free parameters, including β0,θ, and Ψ. In the three neutrino framework, there are eight independent parameters in total, including two neutrino squared mass splittings, Δm221and Δm231, for NO (Δm232 for IO), three mixing angles (θ12,θ13,θ23), one CP phase (δCP), and two Majorana phases (η1,η2) in which η1,η2 are currently undetermined while the others can be exactly observed. As shown in Table 1, five observable quantities θ12,θ13,θ23, Δm221, and Δm231 for NO (Δm232 for IO) are more precisely determined, whereas δCP is less precise.

      First, Eq. (37) shows that m1,2,3 depend on two observed parameters, Δm221 and Δm231, which have been determined with high accuracy, and one model parameter, β0, which is of the same order of magnitude as the second neutrino mass m2. The neutrino mass scale is unknown; however, from Eq. (38), we can estimate the allowed regions for β0 using the upper limits of mν with mν<120meV for NO,mν<150meV for IO [1], and the best fit values of Δm221 and Δm231 taken from [1] with Δm221=75meV2 and Δm231=2.55×103meV2 for NO and Δm231=2.45×103meV2 for IO. From this, we obtain the allowed range of β0.

      β0{(10,30)meVforNO,(51,60)meVforIH.

      (46)

      The estimation of β0 allows us to predict neutrino masses. Figure 3 shows the allowed neutrino mass regions as a function of β0, which are estimated to be

      Figure 3.  (color online) m1 and m3 versus β0m2 with β0(10.0,30.0)meV for NO (left panel) and β0(50.0,60.0)meV for IO (right panel).

      {m1(5.0,28.0)meV,m3=(51.0,58.0)meVforNO,m1(50.5,59.3)meV,m3=(8.7,32.8)meVforIO.

      (47)

      We now return to the lepton mixing sector. The global reassessment of neutrino oscillation [1] reveals that in the 3σ range of the best-fit value, s213(0.02,0.02405); thus, using Eq. (40), we can estimate the allowed regions of s212.

      s212(0.3402,0.3416),i.e.,θ12()(35.677,35.762).

      (48)

      Using Eq. (4) and 3σ ranges for sinθ13 and θ23, we plot in Fig. 4 the relationship between sinθ and θ13,θ23 and obtain the allowed regions of sinθ.

      Figure 4.  (color online) sθ versus s213 and s223 with s213(2.000,2.405)×102 and s223(0.434,0.610) for NH (left panel) while s213(2.018,2.424)×102 and s223(0.433,0.608) for IH (right panel).

      sθ{(0.77,0.82)forNO,(0.58,0.68)forIO,i.e.,θ(){(50.35,55.08)forNO,(35.45,42.84)forIO.

      (49)

      Next, in Fig. 5, we plot the relationship between sinδCP and θ13,θ23 based on Eq. (42) and predict the allowed regions of sinδCP as follows:

      Figure 5.  (color online) sinδCP versus s213 and s223 with s213(2.000,2.405)×102 and s223(0.434,0.610).

      sinδCP(0.90,0.50),i.e.,δCP()(295.80,330.0).

      (50)

      This value of δCP lies in the 3σ ranges taken from Ref. [1] for both NO and IO.

      Similarly, from Eq. (43), we plot the correlation between the model parameter Ψ and θ13,θ23, as in Fig. 6. This figure indicates that

      Figure 6.  (color online) Ψ(rad) versus s213 and s223 with s213(2.000,2.405)×102 and s223(0.434,0.610) for NH (left panel) while s213(2.018,2.424)×102 and s223(0.433,0.608) for IH (right panel).

      Ψ(){(5.73,20.05)forNO,(185.20,199.60)forIO.

      (51)

      In our model, one Majorana phase is predicted to be zero (η2=0) for both mass orderings, and the other (η1) is predicted to be

      η1(){(3.44,10.37)forNO,(349.60,356.60)forIO,

      (52)

      as shown in Fig. 7.

      Figure 7.  (color online) η1(rad) versus s213 and s223 with s213(2.000,2.405)×102 and s223(0.434,0.610) for NH (left panel) while s213(2.018,2.424)×102 and s223(0.433,0.608) for IH (right panel).

      As an example, at the best-fit points of s213 and s223 given in Table 1, s213=2.20×102,s223=0.574 for NO and s213=2.225×102,s223=0.578 for IO; hence, we obtain

      s212={0.3408forNO,0.3409forIO,θ12={35.72forNO,35.72forIO,

      (53)

      sθ={0.792forNO,0.6151forIO,θ={52.37forNO,37.96forIO,

      (54)

      sinδCP={0.7204forNO,0.686forIO,δCP={313.90forNO,316.70forIO,

      (55)

      Ψ={345.00forNO,195.80forIO,η1={8.49forNO,351.10forIO.

      (56)

      The unitary lepton mixing matrix becomes

      ULep={(0.7941+0.1185i0.57740.08915+0.1185i0.23430.4417i0.2887+0.5000i0.61790.1867i0.0290+0.3232i0.28870.5000i0.6179+0.4238i)forNO,(0.79310.1240i0.57740.082920.1240i0.02870.3173i0.2887+0.5000i0.61560.4315i0.2435+0.4413i0.28870.5000i0.6156+0.1835i)forIO,

      (57)

      which are all consistent with the entry constraints given in Ref. [26].

    VI.   EFFECTIVE NEUTRINO MASS PARAMETERS
    • We now deal with the effective neutrino masses related to beta decay and neutrinoless double beta decay, which have the following respective forms [2729]:

      mβ=3k=1|U1k|2m2k=β2023Δm221+Δm231s213forbothorderings,

      (58)

      mee=|3k=1U21kmk|={βN3forNO,βI3forIO,

      (59)

      where βN and βI are given in Appendix E. Expressions (58), (59), and (E1)–(E3) reveal that mee depends on five parameters θ12,θ23, Δm221, Δm231, and β0, whereas mβ depends on four parameters θ12, Δm221, Δm231, and β0. At the best-fit points of Δm221 and Δm231 taken from Table 1, β0 is fixed at β0=20.0meV for NO and β0=55.0meV for IO, and mee depends on s213 and s223, whereas mβ depends only on s213, which are plotted in Figs. 8 and 9, respectively. These imply that

      Figure 8.  (color online) mee (in meV) versus s213 and s223 with s213(2.000,2.405)×102 and s223(0.434,0.610) for NH (left panel) while s213(2.018,2.424)×102 and s223(0.433,0.608) for IH (right panel).

      Figure 9.  (color online) mβ (in meV) versus s213 with s213(2.000,2.405)×102 for NH (left panel) and s213(2.018,2.424)×102 for IH (right panel).

      mee{(47.00,50.50)meV2forNO,(48.40,49.40)meVforIO,

      (60)

      mβ{(51.02,51.10)meVforNO,(49.92,50.02)meVforIO.

      (61)

      At the best-fit points of s213 and s223 taken from Table 1, s213=2.20×102 and s223=0.574 for NO while s213=2.225×102,s223=0.578 for NO; hence, we obtain

      mee={17.60meVforNO,48.74meVforIO,

      (62)

      mβ={20.15meVforNO,49.96meVforIO,

      (63)

      provided that β0=20.0meV for NO and β0=55.0meV for IO.

      The predicted ranges of mβ and mee in Eqs. (60) and (61) for both orderings satisfy all the upper bounds taken from recent 0νββ decay experiments, such as the PLANCK Collaboration mee<80÷180meV [30, 31], CUORE Collaboration [32] mee<75÷350meV, and GERDA Collaboration [33] mee<70÷160meV.

    VII.   CONCLUSIONS
    • We propose a low-scale Standard Model extension with T7×Z4×Z3×Z2 symmetry that can successfully explain the observed neutrino oscillation results within the 3σ range. The small neutrino masses are obtained via the linear seesaw mechanism. Normal and inverted neutrino mass orderings are considered with three lepton mixing angles in their allowed 3σ ranges. The prediction for the Dirac phase is δCP(295.80,330.0) for both normal and inverted orderings, including its experimentally maximum value, while those for the two Majorana phases are η1(349.60,356.60),η2=0 for normal ordering and η1(3.44,10.37),η2=0 for inverted ordering. In addition, the predictions for the effective neutrino masses are mee(47.00,50.50)meV,mβ(51.02,51.10)meV for normal ordering and mee(48.40,49.40)meV,mβ(49.92,50.02)meV for inverted ordering, which are consistent with the present experimental bounds.

    APPENDIX A: IRREDUCIBLE REPRESENTATIONS AND TENSOR PRODUCTS OF T7 GROUP
    • The Frobenius group T7 is isomorphic to Z7and has 21 elements divided into five conjugacy classes corresponding to its five irreducible representations, including three singlets, {\bf{1}}_{0},\, {\bf{1}}_{1}, \, {\bf{1}}_{2} , and two triplets, {\bf{3}}, \, {\bar{\bf{3}}} . All the group multiplication rules of T_7 as given below.

      The tensor products between singlets of T_7 are [13, 25]

      \begin{aligned}[b]\\[-7pt] &{\bf{1}}_0 (a) \otimes {\bf{1}}_k (b) = {\bf{1}}_k(ab) (k = 1, 2), \\ &{\bf{1}}_0 (a) \otimes {\bf{1}}_0 (b) = {\bf{1}}_1 (a) \otimes {\bf{1}}_2 (b) = {\bf{1}}_2 (a) \otimes {\bf{1}}_1 (b) = {\bf{1}}_0 (ab), \\ &{\bf{1}}_1 (a) \otimes {\bf{1}}_1 (b) = {\bf{1}}_2(ab), \,\, {\bf{1}}_2 (a) \otimes {\bf{1}}_2 (b) = {\bf{1}}_1(ab). \end{aligned}\tag{A1}

      The tensor products between singlets and triplets of T_7 are [13, 25]

      \begin{aligned}[b] {\bf{1}}_0 (a) \otimes {\bf{3}} (b_1, b_2, b_3) =& {\bf{3}} (ab_1, ab_2, ab_3), \,\, {\bf{1}}_1 (a)\otimes {\bf{3}} (b_1, b_2, b_3) = {\bf{3}}(ab_1, \omega ab_2, \omega^2 ab_3), \\ {\bf{1}}_2 (a)\otimes {\bf{3}} (b_1, b_2, b_3) =& {\bf{3}}(ab_1, \omega^2 ab_2, \omega ab_3),\,\, {\bf{1}}_0 (a)\otimes {\bar{\bf{3}}} (b_1, b_2, b_3) = {\bar{\bf{3}}}(ab_1, ab_2, ab_3), \\ {\bf{1}}_1 (a) \otimes {\bar{\bf{3}}} (b_1, b_2, b_3) =& {\bar{\bf{3}}}(ab_1, \omega ab_2, \omega^2 ab_3),\,\, {\bf{1}}_2 (a)\otimes {\bar{\bf{3}}} (b_1, b_2, b_3) = {\bar{\bf{3}}}(ab_1, \omega^2 ab_2, \omega ab_3). \,\, \end{aligned}\tag{A2}

      The tensor products between triplets of T_7 are [13, 25]

      \begin{aligned}[b]{\bf{3}} (a_1, a_2, a_3)& \otimes {\bf{3}}(b_1, b_2, b_3) = {\bf{3}} (a_3b_3, a_1b_1, a_2b_2)\oplus {\bar{\bf{3}}} (a_2b_3,a_3b_1,a_1b_2) \oplus {\bar{\bf{3}}} (a_3b_2,a_1b_3,a_2b_1) , \\ {\bar{\bf{3}}} (a_1, a_2, a_3)&\otimes {\bar{\bf{3}}}(b_1, b_2, b_3) = {\bar{\bf{3}}} (33,11,22)\oplus {\bf{3}} (23,31,12) \oplus {\bf{3}}(32,13,21), \\ {\bf{3}} (a_1, a_2, a_3)& \otimes {\bar{\bf{3}}} (b_1, b_2, b_3) = {\bf{1}}_0 (a_1 b_1 + a_2 b_2 + a_3 b_3)\oplus {\bf{1}}_1 (a_1 b_1 + \omega a_2 b_2 + \omega^2 a_3 b_3) \\ &\oplus\, {\bf{1}}_2 (a_1 b_1 + \omega^2 a_2 b_2 + \omega a_3 b_3)\oplus {\bf{3}} (a_2 b_1, a_3 b_2, a_1 b_3) \oplus {\bar{\bf{3}}} (a_1 b_2, a_2 b_3, a_3 b_1). \end{aligned}\tag{A3}

      Note that {\bf{3}} \times {\bf{3}} \times {\bf{3}} or {\bar{\bf{3}}} \times {\bar{\bf{3}}} \times {\bar{\bf{3}}} contains two invariants, whereas {\bf{3}} \times {\bf{3}} \times {\bar{\bf{3}}} or {\bf{3}} \times {\bar{\bf{3}}} \times {\bar{\bf{3}}} contains one invariant.

    APPENDIX B: SCALAR POTENTIAL
    • The scalar potential invariant under {\cal{G}} takes the form

      \begin{aligned}[b]\\[-5pt] V_{\mathrm{tot}} = & V(H)+ V(\varphi)+V(\phi)+ V(\rho)+ V(\eta) +V(H, \varphi) +V(H,\phi) + V(H, \rho) + V(H, \eta) + V(\varphi, \phi)+ V(\varphi, \rho) \\ &+V(\varphi, \eta) + V(\phi, \rho) + V(\phi, \eta) + V(\rho, \eta), \end{aligned}\tag{B1}

      where

      V(H) = \mu_{H}^2 H^{\dagger} H +\lambda^{H}({H}^{\dagger}H)^2,\tag{B2}

      V(\varphi) = \mu_{\varphi}^2 (\varphi^* \varphi)_{{\bf{1}}_0} +\lambda^{\varphi}_1(\varphi^*\varphi)_{{\bf{1}}_0} (\varphi^*\varphi)_{{\bf{1}}_0} + \lambda^{\varphi}_2(\varphi^*\varphi)_{{\bf{1}}_1} (\varphi^*\varphi)_{{\bf{1}}_2} +\lambda^{\varphi}_3(\varphi^*\varphi)_{\bf{3}} (\varphi^*\varphi)_{\overline{\bf{3}}} \tag{B3}

      V(\phi) = V(\varphi\rightarrow \phi), V(\rho) = \mu_{\rho}^2 (\rho^* \rho)_{{\bf{1}}_0} +\lambda^{\rho}_1(\rho^* \rho)_{{\bf{1}}_0} (\rho^* \rho)_{{\bf{1}}_0}, V(\eta) = V(\rho\rightarrow \eta), \tag{B4}

      V(H, \varphi) = \lambda^{H \varphi}_1(H^{\dagger}H)_{{\bf{1}}_0} (\varphi^* \varphi)_{{\bf{1}}_0}+\lambda^{H \varphi}_2({H}^{\dagger}\varphi)_{\bf{3}} (\varphi^* H)_{\overline{\bf{3}}}, \,\, V(H, \phi) = V(H, \varphi\rightarrow \phi), \tag{B5}

      V(H, \rho) = \lambda^{H \rho}_1(H^{\dagger}H)_{{\bf{1}}_0} (\rho^* \rho)_{{\bf{1}}_0}+\lambda^{H \rho}_2({H}^{\dagger}\rho)_{{\bf{1}}_0} (\rho^* H)_{{\bf{1}}_0}, \,\, V(H, \eta) = V(H, \rho\rightarrow \eta), \tag{B6}

      \begin{aligned}[b] V(\varphi, \phi) =& \lambda^{\varphi \phi}_1(\varphi^*\varphi)_{{\bf{1}}_0} (\phi^* \phi)_{{\bf{1}}_0} +\lambda^{\varphi \phi}_2(\varphi^*\varphi)_{{\bf{1}}_1} (\phi^* \phi)_{{\bf{1}}_2} +\lambda^{\varphi \phi}_3(\varphi^*\varphi)_{{\bf{1}}_2} (\phi^* \phi)_{{\bf{1}}_1} +\, \lambda^{\varphi \phi}_4(\varphi^*\varphi)_{{\bf{3}}} (\phi^* \phi)_{{\overline{\bf{3}}}} +\lambda^{\varphi \phi}_5(\varphi^*\varphi)_{{\overline{\bf{3}}}} (\phi^* \phi)_{{\bf{3}}} \\ &+\, \lambda^{\varphi \phi}_6(\varphi^*\phi)_{{\bf{1}}_0} (\phi^* \varphi)_{{\bf{1}}_0} +\lambda^{\varphi \phi}_7(\varphi^*\phi)_{{\bf{1}}_1} (\phi^* \varphi)_{{\bf{1}}_2} +\lambda^{\varphi \phi}_8(\varphi^*\phi)_{{\bf{1}}_2} (\phi^* \varphi)_{{\bf{1}}_1} +\, \lambda^{\varphi \phi}_9(\varphi^*\phi)_{{\bf{3}}} (\phi^* \varphi)_{{\overline{\bf{3}}}} +\lambda^{\varphi \phi}_{10}(\varphi^* \phi)_{{\overline{\bf{3}}}} (\phi^* \varphi)_{{\bf{3}}}, \end{aligned}\tag{B7}

      V(\varphi, \rho) = \lambda^{\varphi \rho}_1(\varphi^*\varphi)_{{\bf{1}}_0} (\rho^* \rho)_{{\bf{1}}_0} +\lambda^{\varphi \rho}_2(\varphi^*\rho)_{{\overline{\bf{3}}}}(\rho^* \varphi)_{{\bf{3}}},\,\, V(\varphi, \eta) = V(\varphi, \rho\rightarrow \eta), \tag{B8}

      V(\phi, \rho) = \lambda^{\phi \rho}_1(\phi^*\phi)_{{\bf{1}}_0} (\rho^* \rho)_{{\bf{1}}_0} +\lambda^{\phi \rho}_2(\phi^*\rho)_{{\overline{\bf{3}}}}(\rho^* \phi)_{{\bf{3}}}, \,\, V(\phi, \eta) = V(\phi, \rho\rightarrow \eta), \tag{B9}

      V(\rho, \eta) = \lambda^{\rho \eta}_1(\rho^*\rho)_{{\bf{1}}_0} (\eta^* \eta)_{{\bf{1}}_0} +\lambda^{\rho \eta}_2(\rho^* \eta)_{{\bf{1}}_0} (\eta^* \rho)_{{\bf{1}}_0}. \tag{B10}

      Here, we have used the notation V(x\rightarrow x', y\rightarrow y') = V(x, y)|_{x = x',\; y = y'} . Other interaction terms contain three and four different scalar fields. For example, V(H, \,\varphi,\,\phi), \, V(H,\, \varphi,\,\rho), V(H, \,\varphi,\,\eta),\; V(H, \,\phi,\,\rho), \; V(H,\, \phi,\,\eta), \; V(H,\, \rho,\,\eta), \; V(\varphi, \,\phi, \,\rho), V(\varphi, \,\phi, \,\eta), \; V(\varphi,\, \rho,\,\eta), \; V(\phi,\, \rho,\,\eta), \; V(H, \,\varphi,\, \phi, \,\rho), V(H, \,\varphi,\, \phi,\, \eta), V(H, \,\varphi, \,\rho,\,\eta), \; V(H,\, \phi,\, \rho,\,\eta), \; V(\varphi, \,\phi,\, \rho,\,\eta) are not invariant under {\cal{G}} and thus are not included in the expression for V_{\mathrm{tot}} in (B1).

    APPENDIX C: EXPLICIT EXPRESSIONS FOR \boldsymbol \lambda^{\boldsymbol H\eta}, \boldsymbol\lambda^{\boldsymbol\rho\eta}, \boldsymbol\lambda^{\boldsymbol\phi\eta}, \boldsymbol\lambda^{\boldsymbol\varphi\rho} , AND \boldsymbol\lambda^{\boldsymbol\varphi\phi}
    • \begin{aligned}[b]\\[-5pt]\lambda^{H\eta} = -\frac{2\lambda^{H} v^2_H+\mu_H^2}{v_+},\quad \lambda^{\rho\eta} = \lambda^{H\eta}+\frac{2 \lambda^{\eta} v_\eta^2+\mu_\eta^2-\mu_\rho^2}{2 \delta_-}-\frac{\lambda^{\rho} v_\rho^2}{\delta_-}, \end{aligned}\tag{C1}

      \begin{aligned}[b]\lambda^{\phi\eta} = &\left\{v_+ \left[\left( \mu_\rho^2 v_\eta^2-\mu_\eta^2 v_\eta^2- 2 \lambda^{\eta} v_\eta^4 + 3 \mu_\phi^2 v_\phi^2\right) v_\rho^2+ (\mu_\rho^2 + 2 \lambda^{\rho} v_\eta^2) v_\rho^4\right.\right. -\, v_\eta^2 (\mu_\eta^2 v_\eta^2 + 2 \lambda^{\eta} v_\eta^4 + 3 \mu_\phi^2 v_\phi^2) \\& \left.\left.+ 2 \lambda^{\rho} v_\rho^6 + \mu_\varphi^2 v_\varphi^2 \delta_- + 4 \lambda^{\varphi} (v_\varphi^4 - 6 v_\phi^4) \delta_-\right] \right. \left. +\, \mu_H^2 \left(2 \delta_+^2 + v^2_H v_-\right)\delta_- + 2 \lambda^{H} v^2_H \left(2 \delta_+^2 + v^2_H v_-\right)\delta_- \right\}/\left(12 v_\phi^2 \delta_+\delta_- v_+\right), \end{aligned}\tag{C2}

      \begin{aligned}[b] \lambda^{\varphi\rho} =& \left\{-v_+ \left[\mu_\eta^2 v_\eta^4 + 2 \lambda^{\eta} v_\eta^6 - 3 \mu_\phi^2 v_\eta^2 v_\phi^2 + \left(\mu_\eta^2 v_\eta^2 - \mu_\rho^2 v_\eta^2 + 2 \lambda^{\eta} v_\eta^4 + 3 \mu_\phi^2 v_\phi^2\right) v_\rho^2\right.\right. -\, (\mu_\rho^2 +2 \lambda^{\rho} v_\eta^2) v_\rho^4 - 2 \lambda^{\rho} v_\rho^6 \\ &\left. \left.+ \mu_\varphi^2 v_\varphi^2 \delta_- + 4 \lambda^{\varphi} (v_\varphi^4 - 6 v_\phi^4) \delta_-\right] \right. \left. +\, \mu_H^2 \delta_- (2 \delta_+^2 +v^2_H V_+) + 2 \lambda^{H} v^2_H (2 \delta_+^2 +v^2_H V_+) \delta_- \right\}/\left\{4 v_\varphi^2 \delta_+\delta_- v_+\right\}, \end{aligned}\tag{C3}

      \begin{aligned}[b] \lambda^{\varphi\phi} =& -\left\{v_+ \left[3 \mu_\phi^2 v_\eta^2 v_\phi^2-\mu_\eta^2 v_\eta^4 - 2 \lambda^{\eta} v_\eta^6 - (\mu_\eta^2 v_\eta^2- \mu_\rho^2 v_\eta^2+ 2 \lambda^{\eta} v_\eta^4 + 3 \mu_\phi^2 v_\phi^2) v_\rho^2 \right.\right. +\, (\mu_\rho^2 + 2 \lambda^{\rho} v_\eta^2) v_\rho^4 + 2 \lambda^{\rho} v_\rho^6 \\ &\left. \left.+ \mu_\varphi^2 v_\varphi^2 \delta_- + 4 \lambda^{\varphi} (v_\varphi^4 + 6 v_\phi^4) \delta_-\right] \right. \left. +\, \mu_H^2 \delta_- (2 \delta_+^2 + v^2_H V_-) + 2 \lambda^{H} v^2_H (2 \delta_+^2 +v^2_H V_-) \delta_-\right\}/\left\{16 v_\varphi^2 v_\phi^2 \delta_- v_+\right\}, \end{aligned}\tag{C4}

      v_{\pm} = 6 v_\phi^2\pm 2 v_\varphi^2 + v_\eta^2 + v_\rho^2, V_{\pm} = v_{\pm}-12 v_\phi^2, \delta_{\pm} = v_\eta^2 \pm v_\rho^2. \tag{C5}

    APPENDIX D: EXPLICIT EXPRESSIONS FOR \boldsymbol \alpha_{\boldsymbol {mn}}, \boldsymbol\gamma_{\boldsymbol {mn}}, \boldsymbol\delta_{\boldsymbol {mn}} , AND \boldsymbol \kappa_{\boldsymbol {mn}}
    • \begin{aligned}[b] \alpha_{13} = &\left\{{\boldsymbol{g}}_{1} \left[{\boldsymbol{t}}_{1} ({\boldsymbol{t}}_{2} {\boldsymbol{y}}_{1} {\boldsymbol{z}}_{3} +{\boldsymbol{t}}_{2} {\boldsymbol{y}}_{3} {\boldsymbol{z}}_{1} +{\boldsymbol{t}}_{3} {\boldsymbol{y}}_{1} {\boldsymbol{z}}_{2} +{\boldsymbol{t}}_{3} {\boldsymbol{y}}_{2} {\boldsymbol{z}}_{1})-{\boldsymbol{t}}_{1}^2 ({\boldsymbol{y}}_{1} {\boldsymbol{z}}_{1}+{\boldsymbol{y}}_{2} {\boldsymbol{z}}_{3})\right.\right. -\left.\left. {\boldsymbol{t}}_{2} {\boldsymbol{t}}_{3} ({\boldsymbol{y}}_{1} {\boldsymbol{z}}_{1}+{\boldsymbol{y}}_{3}{\boldsymbol{z}}_{2})\right] +{\boldsymbol{g}}_{2} ({\boldsymbol{t}}_{1} {\boldsymbol{y}}_{1}-{\boldsymbol{t}}_{2} {\boldsymbol{y}}_{3}) ({\boldsymbol{t}}_{1} {\boldsymbol{z}}_{3}-{\boldsymbol{t}}_{3} {\boldsymbol{z}}_{1}) \right. \\ &+\left. {\boldsymbol{g}}_{3} ({\boldsymbol{t}}_{1} {\boldsymbol{y}}_{2}-{\boldsymbol{t}}_{2} {\boldsymbol{y}}_{1}) ({\boldsymbol{t}}_{1}{\boldsymbol{z}}_{1}-{\boldsymbol{t}}_{3} {\boldsymbol{z}}_{2}) \right\}/\left({\boldsymbol{g}}_{1}^2-{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}\right) \left({\boldsymbol{t}}_{1}^2-{\boldsymbol{t}}_{2} {\boldsymbol{t}}_{3}\right)^2, \end{aligned}\tag{D1}

      \begin{eqnarray} \alpha_{14}& = &\frac{{\boldsymbol{g}}_{1} {\boldsymbol{t}}_{1} {\boldsymbol{z}}_{1}-{\boldsymbol{g}}_{2} {\boldsymbol{t}}_{1} {\boldsymbol{z}}_{3}+{\boldsymbol{g}}_{2} {\boldsymbol{t}}_{3} {\boldsymbol{z}}_{1}-{\boldsymbol{g}}_{1} {\boldsymbol{t}}_{3} {\boldsymbol{z}}_{2}}{\left({\boldsymbol{g}}_{1}^2-{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}\right) \left({\boldsymbol{t}}_{1}^2-{\boldsymbol{t}}_{2} {\boldsymbol{t}}_{3}\right)} +\frac{ {\boldsymbol{y}}_{1}}{{\boldsymbol{y}}_{2} {\boldsymbol{y}}_{3}-{\boldsymbol{y}}_{1}^2}, \end{eqnarray} \tag{D2}

      \begin{eqnarray} \alpha_{23}& = &\frac{{\boldsymbol{g}}_{1} {\boldsymbol{t}}_{1} {\boldsymbol{y}}_{1}-{\boldsymbol{g}}_{3} {\boldsymbol{t}}_{1}{\boldsymbol{y}}_{2}+{\boldsymbol{g}}_{3} {\boldsymbol{t}}_{2} {\boldsymbol{y}}_{1}-{\boldsymbol{g}}_{1} {\boldsymbol{t}}_{2} {\boldsymbol{y}}_{3}}{\left({\boldsymbol{g}}_{1}^2-{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}\right) \left({\boldsymbol{t}}_{1}^2-{\boldsymbol{t}}_{2} {\boldsymbol{t}}_{3}\right)} + \frac{{\boldsymbol{z}}_{1}}{{\boldsymbol{z}}_{2} {\boldsymbol{z}}_{3}-{\boldsymbol{z}}_{1}^2}, \end{eqnarray}\tag{D3}

      \begin{eqnarray} \alpha_{24}& = &\frac{{\boldsymbol{t}}_{1} {\boldsymbol{y}}_{1} {\boldsymbol{z}}_{1}+{\boldsymbol{t}}_{1} {\boldsymbol{y}}_{2} {\boldsymbol{z}}_{3}-{\boldsymbol{t}}_{2} {\boldsymbol{y}}_{1} {\boldsymbol{z}}_{3}-{\boldsymbol{t}}_{3} {\boldsymbol{y}}_{2} {\boldsymbol{z}}_{1}}{\left({\boldsymbol{y}}_{1}^2-{\bf{y}}_{2} {\boldsymbol{y}}_{3}\right) \left({\boldsymbol{z}}_{1}^2-{\boldsymbol{z}}_{2} {\boldsymbol{z}}_{3}\right)}+\frac{{\boldsymbol{g}}_{1}}{{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}-{\boldsymbol{g}}_{1}^2}, \end{eqnarray} \tag{D4}

      \begin{aligned}[b] \gamma_{13} = &\left\{{\boldsymbol{g}}_{1} \left[{\boldsymbol{t}}_{1} ({\boldsymbol{t}}_{2} {\boldsymbol{y}}_{1} {\boldsymbol{z}}_{3}+{\boldsymbol{t}}_{2} {\boldsymbol{y}}_{3} {\boldsymbol{z}}_{1}+{\boldsymbol{t}}_{3} {\boldsymbol{y}}_{1} {\boldsymbol{z}}_{2}+{\boldsymbol{t}}_{3} {\boldsymbol{y}}_{2} {\boldsymbol{z}}_{1})- {\boldsymbol{t}}_{1}^2 ({\boldsymbol{y}}_{1} {\boldsymbol{z}}_{1}+{\boldsymbol{y}}_{3} {\boldsymbol{z}}_{2})\right.\right. \left.\left. -\, {\boldsymbol{t}}_{2} {\boldsymbol{t}}_{3} ({\boldsymbol{y}}_{1} {\boldsymbol{z}}_{1}+{\boldsymbol{y}}_{2} {\boldsymbol{z}}_{3})\right]+{\boldsymbol{g}}_{2} ({\boldsymbol{t}}_{1} {\boldsymbol{z}}_{1}-{\boldsymbol{t}}_{2} {\boldsymbol{z}}_{3}) ({\boldsymbol{t}}_{1} {\boldsymbol{y}}_{3}-{\boldsymbol{t}}_{3} {\boldsymbol{y}}_{1}) \right. \\ &\left. +\,{\boldsymbol{g}}_{3} ({\boldsymbol{t}}_{1} {\boldsymbol{z}}_{2}-{\boldsymbol{t}}_{2} {\boldsymbol{z}}_{1}) ({\boldsymbol{t}}_{1} {\boldsymbol{y}}_{1}-{\boldsymbol{t}}_{3} {\boldsymbol{y}}_{2})\right\}/\left({\boldsymbol{g}}_{1}^2-{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}\right) \left({\boldsymbol{t}}_{1}^2-{\boldsymbol{t}}_{2} {\boldsymbol{t}}_{3}\right)^2, \end{aligned}\tag{D5}

      \begin{eqnarray} \gamma_{14}& = &\frac{ {\boldsymbol{g}}_{1} {\boldsymbol{t}}_{1} {\boldsymbol{z}}_{1}-{\boldsymbol{g}}_{3} {\boldsymbol{t}}_{1} {\boldsymbol{z}}_{2}+{\boldsymbol{g}}_{3} {\boldsymbol{t}}_{2} {\boldsymbol{z}}_{1}-{\boldsymbol{g}}_{1} {\boldsymbol{t}}_{2} {\boldsymbol{z}}_{3}}{\left({\boldsymbol{g}}_{1}^2-{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}\right) \left({\boldsymbol{t}}_{1}^2-{\boldsymbol{t}}_{2} {\boldsymbol{t}}_{3}\right)}+\frac{ {\boldsymbol{y}}_{1}}{{\boldsymbol{y}}_{2} {\boldsymbol{y}}_{3}-{\boldsymbol{y}}_{1}^2}, \end{eqnarray}\tag{D6}

      \begin{eqnarray} \gamma_{23}& = &\frac{{\boldsymbol{g}}_{1} {\boldsymbol{t}}_{1} {\boldsymbol{y}}_{1}-{\boldsymbol{g}}_{2} {\boldsymbol{t}}_{1} {\boldsymbol{y}}_{3}+{\boldsymbol{g}}_{2} {\boldsymbol{t}}_{3} {\boldsymbol{y}}_{1}-{\boldsymbol{g}}_{1} {\boldsymbol{t}}_{3} {\boldsymbol{y}}_{2}}{\left({\boldsymbol{g}}_{1}^2-{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}\right) \left({\boldsymbol{t}}_{1}^2-{\boldsymbol{t}}_{2} {\boldsymbol{t}}_{3}\right)}+\frac{{\boldsymbol{z}}_{1}}{{\boldsymbol{z}}_{2} {\boldsymbol{z}}_{3}-{\boldsymbol{z}}_{1}^2}, \end{eqnarray}\tag{D7}

      \begin{eqnarray} \gamma_{24}& = & \frac{{\boldsymbol{t}}_{1} {\boldsymbol{y}}_{1} {\boldsymbol{z}}_{1}+{\boldsymbol{t}}_{1} {\boldsymbol{y}}_{3} {\boldsymbol{z}}_{2}-{\boldsymbol{t}}_{2} {\boldsymbol{y}}_{3} {\boldsymbol{z}}_{1}-{\boldsymbol{t}}_{3} {\boldsymbol{y}}_{1} {\boldsymbol{z}}_{2}}{\left({\boldsymbol{y}}_{1}^2-{\boldsymbol{y}}_{2} {\boldsymbol{y}}_{3}\right) \left({\boldsymbol{z}}_{1}^2-{\boldsymbol{z}}_{2} {\boldsymbol{z}}_{3}\right)}+\frac{{\boldsymbol{g}}_{1}}{{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}-{\boldsymbol{g}}_{1}^2}, \end{eqnarray} \tag{D8}

      \begin{aligned}[b] \delta_{13} = & \left\{{\boldsymbol{g}}_{1} \left[{\boldsymbol{t}}_{1} {\boldsymbol{t}}_{2} (2 {\boldsymbol{y}}_{1}{\boldsymbol{z}}_{1}+{\boldsymbol{y}}_{2} {\boldsymbol{z}}_{3}+{\boldsymbol{y}}_{3} {\boldsymbol{z}}_{2}) - {\boldsymbol{t}}_{1}^2 ({\boldsymbol{y}}_{1} {\boldsymbol{z}}_{2}+{\boldsymbol{y}}_{2} {\boldsymbol{z}}_{1}) \right.\right. -\left.\left. {\boldsymbol{t}}_{2}^2 ({\boldsymbol{y}}_{1} {\boldsymbol{z}}_{3}+{\boldsymbol{y}}_{3} {\boldsymbol{z}}_{1})\right] +{\boldsymbol{g}}_{2} ({\boldsymbol{t}}_{1} {\boldsymbol{y}}_{1}-{\boldsymbol{t}}_{2} {\boldsymbol{y}}_{3}) ({\boldsymbol{t}}_{1} {\boldsymbol{z}}_{1}-{\boldsymbol{t}}_{2} {\boldsymbol{z}}_{3}) \right. \\ &+\left. {\boldsymbol{g}}_{3} ({\boldsymbol{t}}_{2} {\boldsymbol{y}}_{1}-{\boldsymbol{t}}_{1} {\boldsymbol{y}}_{2}) ({\boldsymbol{t}}_{2} {\boldsymbol{z}}_{1}-{\boldsymbol{t}}_{1} {\boldsymbol{z}}_{2})\right\}/\left({\boldsymbol{g}}_{1}^2-{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}\right) \left({\boldsymbol{t}}_{1}^2-{\boldsymbol{t}}_{2} {\boldsymbol{t}}_{3}\right)^2, \end{aligned}\tag{D9}

      \begin{eqnarray} \delta_{14}& = &\frac{{\boldsymbol{g}}_{1} {\boldsymbol{t}}_{1} {\boldsymbol{z}}_{2}-{\boldsymbol{g}}_{2} {\boldsymbol{t}}_{1} {\boldsymbol{z}}_{1}+{\boldsymbol{g}}_{2} {\boldsymbol{t}}_{2} {\boldsymbol{z}}_{3}-{\boldsymbol{g}}_{1} {\boldsymbol{t}}_{2} {\boldsymbol{z}}_{1}}{\left({\boldsymbol{g}}_{1}^2-{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}\right) \left({\boldsymbol{t}}_{1}^2-{\boldsymbol{t}}_{2} {\boldsymbol{t}}_{3}\right)}+\frac{{\boldsymbol{y}}_{2}}{{\boldsymbol{y}}_{1}^2-{\boldsymbol{y}}_{2} {\boldsymbol{y}}_{3}}, \end{eqnarray}\tag{D10}

      \begin{eqnarray} \delta_{23}& = &\frac{{\boldsymbol{g}}_{1} {\boldsymbol{t}}_{1} {\boldsymbol{y}}_{2}-{\boldsymbol{g}}_{2} {\boldsymbol{t}}_{1} {\boldsymbol{y}}_{1}+{\boldsymbol{g}}_{2} {\boldsymbol{t}}_{2} {\boldsymbol{y}}_{3}-{\boldsymbol{g}}_{1} {\boldsymbol{t}}_{2} {\boldsymbol{y}}_{1}}{\left({\boldsymbol{g}}_{1}^2-{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}\right) \left({\bf{t}}_{1}^2-{\boldsymbol{t}}_{2} {\boldsymbol{t}}_{3}\right)}+\frac{ {\boldsymbol{z}}_{2}}{{\boldsymbol{z}}_{1}^2-{\boldsymbol{z}}_{2} {\boldsymbol{z}}_{3}}, \end{eqnarray} \tag{D11}

      \begin{eqnarray} \delta_{24}& = &\frac{{\boldsymbol{t}}_{2} {\boldsymbol{y}}_{1} {\boldsymbol{z}}_{1}+{\boldsymbol{t}}_{3} {\boldsymbol{y}}_{2} {\boldsymbol{z}}_{2}-{\boldsymbol{t}}_{1} ({\boldsymbol{y}}_{1} {\boldsymbol{z}}_{2}+{\boldsymbol{y}}_{2} {\boldsymbol{z}}_{1})}{\left({\boldsymbol{y}}_{1}^2-{\boldsymbol{y}}_{2} {\boldsymbol{y}}_{3}\right) \left({\boldsymbol{z}}_{1}^2-{\boldsymbol{z}}_{2} {\boldsymbol{z}}_{3}\right)}+\frac{{\boldsymbol{g}}_{2} }{{\boldsymbol{g}}_{1}^2-{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}}, \end{eqnarray} \tag{D12}

      \begin{aligned}[b] \kappa_{13} = & \left\{{\boldsymbol{g}}_1 \left[{\boldsymbol{t}}_1 {\boldsymbol{t}}_3 (2 {\boldsymbol{y}}_1 {\boldsymbol{z}}_1 + {\boldsymbol{y}}_3 {\boldsymbol{z}}_2 + {\boldsymbol{y}}_2 {\boldsymbol{z}}_3)- {\boldsymbol{t}}^2_1 ({\boldsymbol{y}}_3 {\boldsymbol{z}}_1 + {\boldsymbol{y}}_1 {\boldsymbol{z}}_3)\right.\right. \left.\left. -\, {\boldsymbol{t}}^2_3 ({\boldsymbol{y}}_2 {\boldsymbol{z}}_1 + {\boldsymbol{y}}_1 {\boldsymbol{z}}_2) \right] + {\boldsymbol{g}}_2 ({\boldsymbol{t}}_3 {\boldsymbol{y}}_1 -{\boldsymbol{t}}_1 {\boldsymbol{y}}_3) ({\boldsymbol{t}}_3 {\boldsymbol{z}}_1 -{\boldsymbol{t}}_1 {\boldsymbol{z}}_3)\right. \\ &\left. +\, {\boldsymbol{g}}_3 ({\boldsymbol{t}}_1 {\boldsymbol{y}}_1 - {\boldsymbol{t}}_3 {\boldsymbol{y}}_2) ({\boldsymbol{t}}_1 {\boldsymbol{z}}_1 -{\boldsymbol{t}}_3 {\boldsymbol{z}}_2) \right\}/\left({\boldsymbol{g}}_{1}^2-{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}\right) \left({\boldsymbol{t}}_{1}^2-{\boldsymbol{t}}_{2} {\boldsymbol{t}}_{3}\right)^2, \end{aligned}\tag{D13}

      \begin{eqnarray} \kappa_{14}& = &\frac{{\boldsymbol{g}}_{1} {\boldsymbol{t}}_{1} {\boldsymbol{z}}_{3}-{\boldsymbol{g}}_{3} {\boldsymbol{t}}_{1} {\boldsymbol{z}}_{1}+{\boldsymbol{g}}_{3} {\boldsymbol{t}}_{3} {\boldsymbol{z}}_{2}-{\boldsymbol{g}}_{1} {\boldsymbol{t}}_{3} {\boldsymbol{z}}_{1}}{\left({\boldsymbol{g}}_{1}^2-{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}\right) \left({\boldsymbol{t}}_{1}^2-{\boldsymbol{t}}_{2} {\boldsymbol{t}}_{3}\right)}+\frac{{\boldsymbol{y}}_{3}}{{\boldsymbol{y}}_{1}^2-{\boldsymbol{y}}_{2} {\boldsymbol{y}}_{3}}, \end{eqnarray} \tag{D14}

      \begin{eqnarray} \kappa_{23}& = &\frac{{\boldsymbol{g}}_{1} {\boldsymbol{t}}_{1} \text{y3}-{\boldsymbol{g}}_{3} {\boldsymbol{t}}_{1}{\boldsymbol{y}}_{1}+{\boldsymbol{g}}_{3} {\boldsymbol{t}}_{3} {\boldsymbol{y}}_{2}-{\boldsymbol{g}}_{1} {\boldsymbol{t}}_{3} {\boldsymbol{y}}_{1}}{\left({\boldsymbol{g}}_{1}^2-{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}\right) \left({\boldsymbol{t}}_{1}^2-{\boldsymbol{t}}_{2} {\boldsymbol{t}}_{3}\right)}+\frac{{\boldsymbol{z}}_{3}}{{\boldsymbol{z}}_{1}^2-{\boldsymbol{z}}_{2} {\boldsymbol{z}}_{3}}, \end{eqnarray}\tag{D15}

      \begin{eqnarray} \kappa_{24}& = &\frac{{\boldsymbol{t}}_{2} {\boldsymbol{y}}_{3} {\boldsymbol{z}}_{3}+{\boldsymbol{t}}_{3} {\boldsymbol{y}}_{1} {\boldsymbol{z}}_{1}-{\boldsymbol{t}}_{1} ({\boldsymbol{y}}_{1} {\boldsymbol{z}}_{3}+{\boldsymbol{y}}_{3} {\boldsymbol{z}}_{1})}{\left({\boldsymbol{y}}_{1}^2-{\boldsymbol{y}}_{2} {\boldsymbol{y}}_{3}\right) \left({\boldsymbol{z}}_{1}^2-{\boldsymbol{z}}_{2} {\boldsymbol{z}}_{3}\right)}+\frac{{\boldsymbol{g}}_{3}}{{\boldsymbol{g}}_{1}^2-{\boldsymbol{g}}_{2} {\boldsymbol{g}}_{3}}. \end{eqnarray} \tag{D16}

    APPENDIX E: EXPLICIT EXPRESSIONS FOR \boldsymbol\beta_{\boldsymbol N} AND \boldsymbol\beta_{\boldsymbol I}
    • \begin{aligned}[b] \beta_N = & 2\beta_0 \sqrt{\beta_1} \left[c^2_\theta\cos \left(\arg \beta_1/2 \right)+ s'_\theta \cos \left(\Psi-\arg \beta_1/2\right) + s^2_\theta \cos \left(2 \Psi -\arg \beta_1/2 \right) \right] +2\beta_0 \sqrt{\beta_2} \Big[c^2_\theta\cos \left(\arg \beta_2/2 \right)\\ &-s'_\theta \cos \left(\Psi+\arg \beta_2/2\right) + s^2_\theta \cos \left(2 \Psi + \arg \beta_2/2\right) \Big] + 2 \sqrt{\beta_1 \beta_2}\Big[c^4_\theta \cos \left(\Delta \beta_{12}/2\right)-\left(s'^2_\theta/2\right) \cos \left(2 \Psi-\Delta \beta_{12}/2\right) \\ &+s^4_\theta \cos \left(4\Psi -\Delta \beta_{12}/2\right)\Big] + \left(\beta_1+\beta_2\right)\left(1+s'^2_\theta/2\right) +\left(\beta_1+\beta_2\right) s'^2_\theta c'_\Psi + \beta_0^2 + 2\left(\beta_1-\beta_2\right) s'_\theta c_\Psi , \end{aligned}\tag{E1}

      \begin{aligned}[b] \beta_I = &2\beta_0 \sqrt{\beta_1} \left[c^2_\theta\cos \left(2 \Psi-\arg \beta_1/2 \right)-s'_\theta \cos \left(\Psi-\arg \beta_1/2 \right) + s^2_\theta \cos \left(\arg \beta_1/2 \right)\right] +2\beta_0 \sqrt{\beta_2} \Big[c_\theta^2 \cos \left(2 \Psi + \arg \beta_2/2\right) \\ &+s'_\theta \cos \left(\Psi + \arg \beta_2/2\right)+s^2_\theta \cos \left(\arg \beta_2/2 \right) \Big] + 2\sqrt{\beta_1 \beta_2} \Big[c_\theta^4 \cos \left(4\Psi \Delta \beta_{12}/2\right)-\left(s'^2_\theta/2\right) \cos \left(2\Psi -\Delta \beta_{12}/2\right)\\ &+ s^4_\theta \cos \left(\Delta \beta_{12}/2 \right)\Big] +\left(1+s'^2_\theta/2\right) \left(\beta_1+\beta_2\right)+ \left(\beta_1+\beta_2\right) s'^2_\theta c'_\Psi +\beta_0^2 -2\left(\beta_1-\beta_2\right) s'_\theta c_\Psi, \end{aligned}\tag{E2}

      where

      \beta_1 = \beta_0^2-\Delta m^2_{21}, \;\;\beta_2 = \beta_0^2-\Delta m^2_{21}+\Delta m^2_{31}, \;\;\Delta \beta_{12} = \arg \beta_1-\arg \beta_2. \tag{E3}

    APPENDIX F: FORBIDDEN INTERACTIONS
    Reference (33)

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