-
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+0.010−0.017(0.433÷0.608) sin2θ1310−2 2.200+0.069−0.062(2.00÷2.405) 2.225+0.064−0.070(2.018÷2.424) δ/π 1.08+0.13−0.12(0.71÷1.99) 1.58+0.15−0.16(1.11÷1.96) Δm221(meV2) 75.0+2.2−2.0(69.4÷81.4) 75.0+2.2−2.0(69.4÷81.4) |Δm231|(meV2)103 2.55+0.02−0.03(2.47÷2.63) 2.45+0.02−0.03(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 [3–7], 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 [8–14]. The linear seesaw mechanism with non-Abelian discrete symmetries has been investigated in [15–23]; 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 [15–23], ii) other non-Abelian discrete symmetries [16, 18, 21–23], iii) other Abelian discrete symmetries [15–23], and iv) other gauge symmetries [16–23]. 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 withT7×Z4× Z3×Z2 discrete symmetry, in which three left-handed leptons, three right-handed neutrinos, and extra neutral leptons lie in 3 ofT7 while three right-handed charged leptonsl1R,l2R,l3R lie in the singlets10,11 , and12 of theT7 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 ofT7 is that its3×ˉ3 tensor product contains three singlets10,1,2 and two triplets that are complex conjugate to each other3,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. -
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)Y≡G , is supplemented by one non-Abelian symmetryT7 and three Abelian discrete symmetriesZ4,Z3 , andZ2 . Furthermore, three right-handed neutrinos (νR ), two different types of neutral singlet leptons with both helicities(NL,R,SL,R) , and fourSU(2)L singlet scalars are added. The particle and scalar assignments of the model underG symmetry is summarized in Table 2.ψL l1R,l2R,l3R H νR NL NR SL SR φ ϕ ρ η SU(2)L 2 1 2 1 1 1 1 1 1 1 1 1 U(1)Y −12 −1 12 0 0 0 0 0 0 0 0 0 U(1)L 1 1 0 1 1 1 1 1 0 0 0 0 T7 3 10 ,11 ,12 10 3 3 3 3 3 3 3 10 10 Z4 i −i 1 1 −i i −i i −1 −1 −i −1 Z3 1 ω2 1 ω 1 1 1 1 1 ω ω2 1 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 andτi(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 LagrangianLl in Eq. (1).Couplings ( i=1,2,3 )Forbidden by (¯ψLνR)(˜Hϕ∗),(¯NLνR)(ϕ∗φ),(¯NLνR)(ϕ∗φ∗),(¯NLνR)(ϕ∗η),(¯NLνR)(ϕ∗η∗), (¯SLνR)(ϕ∗φ),(¯SLνR)(ϕ∗φ∗),(¯SLνR)(ϕ∗η),(¯SLνR)(ϕ∗η∗). 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)(φ∗η),(¯NLνR)(φ∗η∗), ¯SLνRρ∗,(¯SLνR)(ϕφ),(¯SLνR)(ϕφ∗),(¯SLνR)(ϕη),(¯SLνR)(ϕη∗), (¯SLνR)(φη),(¯SLνR)(φη∗),(¯SLνR)(φ∗η),(¯SLνR)(φ∗η∗), (¯SLNR)(ϕρ),(¯SLNR)(ϕρ∗),(¯SLNR)(ϕ∗ρ),(¯SLNR)(ϕ∗ρ∗), (¯SLNR)(φρ),(¯SLNR)(φρ∗),(¯SLNR)(φ∗ρ),(¯SLNR)(φ∗ρ∗), (¯SLNR)(ηρ),(¯SLNR)(ηρ∗),(¯SLNR)(η∗ρ),(¯SLNR)(η∗ρ∗), (¯NLSR)(ϕρ),(¯NLSR)(ϕρ∗),(¯NLSR)(ϕ∗ρ),(¯NLSR)(ϕ∗ρ∗), (¯NLSR)(φρ),(¯NLSR)(φρ∗),(¯NLSR)(φ∗ρ),(¯NLSR)(φ∗ρ∗), (¯NLSR)(ηρ),(¯NLSR)(ηρ∗),(¯NLSR)(η∗ρ),(¯NLSR)(η∗ρ∗),(¯NLSR)(η∗ρ∗), (¯NLNR)(ϕρ),(¯NLNR)(ϕρ∗),(¯NLNR)(ϕ∗ρ),(¯NLNR)(ϕ∗ρ∗), (¯NLNR)(φρ),(¯NLNR)(φρ∗),(¯NLNR)(φ∗ρ),(¯NLNR)(φ∗ρ∗), (¯NLNR)(ηρ),(¯NLNR)(ηρ∗),(¯NLNR)(η∗ρ),(¯NLNR)(η∗ρ∗), (¯SLSR)(ϕρ),(¯SLSR)(ϕρ∗),(¯SLSR)(ϕ∗ρ),(¯SLSR)(ϕ∗ρ∗), (¯SLSR)(φρ),(¯SLSR)(φρ∗),(¯SLSR)(φ∗ρ),(¯SLSR)(φ∗ρ∗), (¯SLSR)(ηρ),(¯SLSR)(ηρ∗),(¯SLSR)(η∗ρ),(¯SLSR)(η∗ρ∗). (¯ψLνR)(˜Hρ),(¯ψLNR)(˜Hφ),(¯ψLNR)(˜Hφ∗),(¯ψLNR)(˜Hη), (¯ψLNR)(˜Hη∗),(¯ψLSR)(˜Hφ),(¯ψLSR)(˜Hφ∗),(¯ψLSR)(˜Hη), Z4×Z2 (¯ψLSR)(˜Hη∗),(¯NLνR)(φρ),(¯NLνR)(φ∗ρ∗),(¯NLνR)(ρη), (¯NLνR)(ρη∗),(¯SLνR)(φρ),(¯SLνR)(φ∗ρ∗),(¯SLνR)(ρη),(¯SLνR)(ρη∗), (¯SLNR)(φη),(¯SLNR)(φη∗),(¯SLNR)(φ∗η),(¯SLNR)(φ∗η∗), (¯NLSR)(φη),(¯NLSR)(φη∗),(¯NLSR)(φ∗η),(¯NLSR)(φ∗η∗), (¯NLNR)(φη),(¯NLNR)(φη∗),(¯NLNR)(φ∗η),(¯NLNR)(φ∗η∗), (¯SLSR)(φη),(¯SLSR)(φη∗),(¯SLSR)(φ∗η),(¯SLSR)(φ∗η∗). (¯ψLνR)(˜Hρ∗),(¯NLνR)(ϕ∗ρ∗),(¯NLνR)(φ∗ρ∗),(¯NLνR)(ρ∗η), (¯NLνR)(ρ∗η∗),(¯SLνR)(ϕ∗ρ∗),(¯SLνR)(φ∗ρ∗),(¯SLνR)(ρ∗η), (¯SLνR)(ρ∗η∗) .Z3×Z2 (¯ψLνR)˜H,(¯ψLNR)(˜Hϕ),(¯ψLNR)(˜Hϕ∗),(¯ψLSR)(˜Hϕ),(¯ψLSR)(˜Hϕ∗), ¯NLνRϕ,¯NLνRϕ∗,¯NLνRφ,¯NLνRφ∗,¯NLνRη,¯NLνRη∗,(¯NLνR)(ϕρ) ,Z4×Z3×Z2 (¯NLνR)(ϕ∗ρ),¯SLνRϕ,¯SLνRϕ∗,¯SLνRφ,¯SLνRφ∗,¯SLνRη,¯SLνRη∗, (¯SLνR)(ϕρ),(¯SLνR)(ϕ∗ρ),(¯SLNR)ρ,(¯SLNR)ρ∗,(¯SLNR)(ϕη), (¯SLNR)(ϕη∗),(¯SLNR)(ϕ∗η),(¯SLNR)(ϕ∗η∗),(¯NLSR)ρ,(¯NLSR)ρ∗, (¯NLSR)(ϕη),(¯NLSR)(ϕη∗),(¯NLSR)(ϕ∗η),(¯NLSR)(ϕ∗η∗), (¯NLNR)ρ,(¯NLNR)ρ∗,(¯NLNR)(ϕη),(¯NLNR)(ϕη∗),(¯NLNR)(ϕ∗η), (¯NLNR)(ϕ∗η∗),(¯SLSR)ρ,(¯SLSR)ρ∗,(¯SLSR)(ϕη),(¯SLSR)(ϕη∗), (¯SLSR)(ϕ∗η),(¯SLSR)(ϕ∗η∗). (¯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.vϕ∼vφ∼vη∼vρ∼1010GeV,Λ∼1013GeV.
(3) -
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 usev∗α=vϑ (ϑ=H,φ,ϕ,ρ,η) . Therefore, the minimization condition ofVtot becomes∂Vtot∂vϑ=0,∂2Vtot∂v2ϑ>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) ∂2Vtot∂v2H=2λHv2H>0,∂2Vtot∂v2φ=4λφv2φ>0,∂2Vtot∂v2ϕ=24λϕv2ϕ>0,
(11) ∂2Vtot∂v2ρ=2(λHη+λρ)v2ρ>0,∂2Vtot∂v2η=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
∂2Vtot∂v2ρ=2v2ρ(λρ−2λHv2H+μ2H2v2φ+v2η+6v2ϕ+v2ρ)>0,
(13) ∂2Vtot∂v2η=2v2η(λη−2λHv2H+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]①,μ2H∼μ2φ∼μ2ϕ∼μ2ρ∼μ2η=−102GeV.
(16) With the help of Eqs. (3) and (16),
∂2Vtot/∂v2H depends onλH ,∂2Vtot/∂v2φ , and∂2Vtot/∂v2ϕ depend onλφ , and∂2Vtot/∂v2ρ and∂2Vtot/∂v2η depend onλH andλρ , 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). -
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=1√3(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=x1vHI3×3≡x1I3×3,MνS=x2vHI3×3≡x2I3×3,
(19) m′νN=x3vρI3×3≡x3I3×3,M′νS=x4vρI3×3≡x4I3×3,
(20) M′NS=(y1vη0y3v∗φ0y1vη0y2vφ0y1vη)≡(y10y30y10y20y1),
(21) MNS=(z1vη0z3v∗φ0z1vη0z2vφ0z1vη)≡(z10z30z10z20z1),
(22) MNN=(t1vη0t3v∗φ0t1vη0t2vφ0t1vη)≡(t10t30t10t20t1),
(23) MSS=(g1vη0g3v∗φ0g1vη0g2vφ0g1vη)≡(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 formMeff=(0mνNMνSm′νNMNNMNSM′νSM′NSMSS)≡(0MDMTDMR),
(25) where
MD=(mνNMνS),MTD=(m′νNM′νS),MR=(MNNMNSM′NSMSS),
and all the entries for
Meff are defined in Eqs. (19)–(24).The light Dirac neutrino mass matrix then gets the form
Mν=−MDM−1RMTD=−mνNM′−1NSM′νS−MνSM−1NSm′νN−MνSM−1SSM′νS+mνNM−1NNMNSM−1SSM′νS+MνSM−1SSM′NSM−1NNm′νN+MνSM−1NSMNNM′−1NSM′νS−mνNM−1NNMNSM−1SSM′NSM−1NNm′νN.
(26) Combining Eqs. (19)–(24) and (26) yields
Mν=(α0δ0β0κ0γ),
(27) where
α=x1x3α13+x1x4α14+x2x3α23+x2x4α24≡α0eiψα,
(28) β=(t1x2−x1z1)(x3y1−t1x4)g1t21+t1x2x4−x1x4z1−x2x3y1y1z1≡β0eiψβ,
(29) γ=x1x3γ13+x1x4γ14+x2x3γ23+x2x4γ24≡γ0eiψγ,
(30) δ=x1x3δ13+x1x4δ14+x2x3δ23+x2x4δ24≡δ0eiψδ,
(31) κ=x1x3κ13+x1x4κ14+x2x3κ23+x2x4κ24≡κ0eiψκ,
(32) with
αmn,γmn,δmn , andκmn(mn=13,14,23,24) given in Appendix D.First, we define the Hermitian matrix
M2ν given byM2ν=MνM+ν=(a200d0eiΨ0β200d0e−iΨ0c20),
(33) where
a20=α20+δ20,c20=γ20+κ20,
(34) d0eiψ=γ0δ0e−i(ψγ−ψδ)+α0κ0ei(ψα−ψκ).
(35) The mass matrix
M2ν is diagonalized byUν satisfyingU+νM2νUν={(m21000m22000m23),Uν=(cosθ0−sinθ.eiΨ010sinθ.e−iΨ0cosθ)forNO, (m23000m22000m21),Uν=(sinθ0cosθeiΨ010−cosθe−iΨ0sinθ)forIO, (36) where
m21=β20−Δm221,m22=β20,m23=β20+Δm231−Δm221.
(37) The sum of neutrino masses is given by
∑mν=β0+√β20−Δm221+√β20−Δm221+Δm231.
(38) Using the expressions for
ULep andUν in Eqs. (18) and (36), the leptonic mixing matrix,ULep=U+LUν , is given byULep={1√3(cosθ+sinθ.e−iΨ1cosθ−sinθ.eiΨcosθ+ω2sinθ.e−iΨωω2cosθ−sinθ.eiΨcosθ+ωsinθ.e−iΨω2ωcosθ−sinθ.eiΨ)forNO,1√3(sinθ−cosθ.e−iΨ1sinθ+cosθ.eiΨsinθ−ω2cosθ.e−iΨωω2sinθ+cosθ.eiΨsinθ−ωcosθ.e−iΨω2ωsinθ+cosθ.eiΨ)forIO.
(39) Expression (39) implies that
ULep possesses theTM2 form because(ULep)i2=1√3(i=1,2,3) .Now, comparing Eq. (39) with the standard parameterization of
UPMNS [24], we can parameterize the solar neutrino mixing angleθ12 , Dirac CP phaseδCP , and model parametersθ,Ψ,η1 in terms of the other two neutrino mixing anglesθ13 andθ23 , as follows②:s212=13c213forbothNOandIO,
(40) sθ={√12+√32√c413s′223−c′213forNO,√12−√32√c413s′223−c′213forIO,
(41) sinδCP=−√c413s′23−c′213s′23s13√2−3s213forbothNOandIO,
(42) Ψ={−sec−1(√4−3(s′213+4c413s′223)1−3s213)forNO,−sec−1(√4−3(s′213+4c413s′223)3s213−1)forIO,
(43) η1={−ilog(cθ+sθe−iΨ√3c12c13)forNO,i2log(3)−ilog(sθ−cθe−iΨc12c13)forIO,
(44) η2=0forbothNOandIO.
(45) -
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,Δm221 andΔm231 , for NO (Δm232 for IO), three mixing angles (θ12,θ13,θ23 ), one CP phase (δCP ), and two Majorana phases (η1,η2 ) in whichη1,η2 are currently undetermined while the others can be exactly observed. As shown in Table 1, five observable quantitiesθ12,θ13,θ23 ,Δm221 , andΔm231 for NO (Δm232 for IO) are more precisely determined, whereasδCP is less precise.First, Eq. (37) shows that
m1,2,3 depend on two observed parameters,Δm221 andΔm231 , which have been determined with high accuracy, and one model parameter,β0 , which is of the same order of magnitude as the second neutrino massm2 . The neutrino mass scale is unknown; however, from Eq. (38), we can estimate the allowed regions forβ0 using the upper limits of∑mν with∑mν<120meV for NO,∑mν<150meV for IO [1], and the best fit values ofΔm221 andΔm231 taken from [1] withΔm221=75meV2 andΔm231=2.55×103meV2 for NO andΔm231=−2.45×103meV2 for IO. From this, we obtain the allowed range ofβ0 .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 ofβ0 , which are estimated to beFigure 3. (color online)
m1 andm3 versusβ0≡m2 withβ0∈(10.0,30.0)meV for NO (left panel) andβ0∈(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 ofs212 .s212∈(0.3402,0.3416),i.e.,θ12(∘)∈(35.677,35.762).
(48) Using Eq. (4) and
3σ ranges forsinθ13 andθ23 , we plot in Fig. 4 the relationship betweensinθ andθ13,θ23 and obtain the allowed regions ofsinθ .Figure 4. (color online)
sθ versuss213 ands223 withs213∈(2.000,2.405)×10−2 ands223∈(0.434,0.610) for NH (left panel) whiles213∈(2.018,2.424)×10−2 ands223∈(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 andθ13,θ23 based on Eq. (42) and predict the allowed regions ofsinδCP as follows:Figure 5. (color online)
sinδCP versuss213 ands223 withs213∈ (2.000,2.405)×10−2 ands223∈(0.434,0.610) .sinδCP∈(−0.90,−0.50),i.e.,δCP(∘)∈(295.80,330.0).
(50) This value of
δCP lies in the3σ 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 thatFigure 6. (color online)
Ψ(rad) versuss213 ands223 withs213∈(2.000,2.405)×10−2 ands223∈(0.434,0.610) for NH (left panel) whiles213∈(2.018,2.424)×10−2 ands223∈(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 beη1(∘)∈{(3.44,10.37)forNO,(349.60,356.60)forIO,
(52) as shown in Fig. 7.
Figure 7. (color online)
η1(rad) versuss213 ands223 withs213∈(2.000,2.405)×10−2 ands223∈(0.434,0.610) for NH (left panel) whiles213∈(2.018,2.424)×10−2 ands223∈(0.433,0.608) for IH (right panel).As an example, at the best-fit points of
s213 ands223 given in Table 1,s213=2.20×10−2,s223=0.574 for NO ands213=2.225×10−2,s223=0.578 for IO; hence, we obtains212={0.3408forNO,0.3409forIO,θ12={35.72∘forNO,35.72∘forIO,
(53) sθ={0.792forNO,0.6151forIO,θ={52.37∘forNO,37.96∘forIO,
(54) sinδCP={−0.7204forNO,−0.686forIO,δCP={313.90∘forNO,316.70∘forIO,
(55) Ψ={345.00∘forNO,195.80∘forIO,η1={8.49∘forNO,351.10∘forIO.
(56) The unitary lepton mixing matrix becomes
ULep={(0.7941+0.1185i0.5774−0.08915+0.1185i0.2343−0.4417i−0.2887+0.5000i−0.6179−0.1867i0.0290+0.3232i−0.2887−0.5000i−0.6179+0.4238i)forNO,(0.7931−0.1240i0.5774−0.08292−0.1240i0.0287−0.3173i−0.2887+0.5000i−0.6156−0.4315i0.2435+0.4413i−0.2887−0.5000i−0.6156+0.1835i)forIO,
(57) which are all consistent with the entry constraints given in Ref. [26].
-
We now deal with the effective neutrino masses related to beta decay and neutrinoless double beta decay, which have the following respective forms [27–29]:
mβ=√3∑k=1|U1k|2m2k=√β20−23Δm221+Δm231s213forbothorderings,
(58) ⟨mee⟩=|3∑k=1U21kmk|={√βN3forNO,√βI3forIO,
(59) where
βN andβI are given in Appendix E. Expressions (58), (59), and (E1)–(E3) reveal that⟨mee⟩ depends on five parametersθ12,θ23 ,Δm221 ,Δm231 , andβ0 , whereasmβ depends on four parametersθ12 ,Δm221 ,Δm231 , andβ0 . At the best-fit points ofΔm221 andΔm231 taken from Table 1,β0 is fixed atβ0=20.0meV for NO andβ0=55.0meV for IO, and⟨mee⟩ depends ons213 ands223 , whereasmβ depends only ons213 , which are plotted in Figs. 8 and 9, respectively. These imply thatFigure 8. (color online)
⟨mee⟩ (in meV) versuss213 ands223 withs213∈(2.000,2.405)×10−2 ands223∈(0.434,0.610) for NH (left panel) whiles213∈(2.018,2.424)×10−2 ands223∈(0.433,0.608) for IH (right panel).Figure 9. (color online)
mβ (in meV) versuss213 withs213∈(2.000,2.405)×10−2 for NH (left panel) ands213∈(2.018,2.424)×10−2 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 ands223 taken from Table 1,s213=2.20×10−2 ands223=0.574 for NO whiles213= 2.225×10−2,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 andβ0=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 recent0νββ 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 . -
The Frobenius group
T7 is isomorphic toZ7⋊ and 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 ofT_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. -
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 forV_{\mathrm{tot}} in (B1). -
\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}
-
\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}
-
\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}
Linear seesaw model with T7 symmetry for neutrino mass and mixing
- Received Date: 2021-12-31
- Available Online: 2022-06-15
Abstract: We propose a low-scale Standard Model extension with