1. Introduction

In particle physics, the hierarchy of fermion masses is one of the most exciting issues [¹,²]. The basically experimental results related to flavour problem including the origin of charged-lepton mass hierarchy [³], me:mμ:mτ∼10-4:10-1:1, and the origins of two large and one small mixing angles of neutrinos, and of two neutrino squared mass differences [⁴],7.50×10-5 eV2∼Δm212bf≪Δm312bf∼2.50×10-3 eV2.

A remarkable feature of discrete symmetries is that they can be combined with the Standard Model (SM) extensions to explain the neutrino mass and mixing data (see, for example, Refs. [⁵,⁸] and the references therein). D5 symmetry [⁹] has been exploited in previous works [¹⁰-¹²] which differ from our current workⁱ with the following basic properties:

Thence, it is necessary to suggest another D ₅-based model with fewer scalar fields and a simpler scalar potential. With such a motivation, we suggest a B - L model with D ₅ symmetryⁱⁱ in which the first two generations of the left- and right-handed charged leptons (ψ1,2L,l1,2R) are assigned in 2₁ while the others (ψ3L,l3R) are in 1₁ of D5. Furthermore, the first and second generations of the right-handed neutrinos are put in 1₁ while the third one is in 1₂ of D5. As a consequence, only one SU(2)L doublets and six singlet scalars are needed to obtain the current neutrino mass and mixing data. As will be shown below, the considered model can also naturally explain the SM charged-lepton mass hierarchy problem.

The rest of this paper is as follows. In Sec. 2 we present the particle content of the model. Section 3 is devoted to the lepton sector. The numerical analysis is presented in Sec. 4. The conclusions are drawn in Sec. 5.

2. The model

The total symmetry of the model is Γ=GB-L×D5×Z4 with GB-L is the gauge symmetry of the B - L model [¹³]. The particle content of the considered model under Γ symmetry is given in Tables I and II.

The Yukawa interactions which are invariant under Γ symmetry take the form:

All other Yukawa terms, up to five-dimension, listed in Table III of the Appendix E are prevented by one or some of the model’s symmetries; thus, they have not been included in the expression of LYlep in Eq. (1).

The minimization condition of the Higgs potential, explicitly presented in Appendix B, provides a vacuum expectation value (VEV):

We note that, for each D ₅ doublet (ϕ, η), there may be four following alignments: i) 0=〈Φ1〉≠〈Φ2〉≠0, D ₅ is broken into {identity}; ii) 0≠〈Φ1〉≠〈Φ2〉=0, D ₅ is broken into Z ₂ consisting of two elements {e, b} where a being the 2π/5 rotation and b is the reflection; iii) 〈Φ1〉=〈Φ2〉≠0, D ₅ is broken into {identity}; and iv) 0≠〈Φ1〉≠〈Φ2〉≠0, D ₅ is broken into {identity}. Therefore, the VEV of 𝜑 and η in Eq. (2) will break D ₅ into {identity} (i.e., discrete symmetry D ₅ is completely broken).

With the help of Eqs. (24)-(30), the expressions in (18)-(22) reduce to

Expression (4) implies the following conditions:

3. Lepton mass and mixing

The lepton Yukawa terms invariant under Γ symmetry take the form:

After symmetry breaking, the mass Lagrangian for the charged leptons can be rewritten in the form:

where

In general the Yukawa couplings xk (k=1,2,3) are complex, i.e., al,b1,2l,cl and therefore M _cl in Eq. (8) are complex. We construct a Hermitian matrix whose real and positive eigenvalues:

whereⁱⁱⁱ

The matrix Mcl2 in Eq. (9) can be diagonalized by biunitary transform VL+Mcl2VR=diag(me2,mμ2,mτ2), where^iv

with

The left-handed charged lepton mixing matrix V _L in Eq. (13) differs from unity; thus, it will contribute to the lepton mixing matrix in Eq. (31).

Combining Eqs. (8), (12), (14) and (15) yields^v:

where

Equations (18)-(19) yields a ratio between x ₀₁ and x ₀₂:

Furthermore, substituting Eq. (8) in Eq. (15), we get:

Regarding the neutrino sector, when the scalar fields get VEVs as in Eq. (2), from Eq. (6), we get the mass matrices for Dirac and Majorana neutrinos:

where

By using the type-I seesaw mechanism, the light effective neutrino mass matrix is given by

where Aν,Bν,Cν,Dν,Gν and H _v are given in Appendix C.

The mass matrix Meff in Eq. (26) possesses three eigenvalues and the corresponding mixing matrix as follows

where Ω1,2, κ1,2, τ1,2 and r1,2 are given in Appendix D. The neutrino mass spectrum can be normal (m1<m2<m3) or inverted (m3<m1<m2) hierarchy depends on the sign of Δm312 [³]. Note that the eigenvalue λ1=0 corresponds to the first neutrino eigenvector,

Thus, the neutrino mass hierarchy should be either m1=λ1=0, m2=λ2, m3=λ3 or (m1=λ2, m2=λ3, m3=λ1=0). As will see below, the model obtained results are in agreement with the current neutrino oscillation data for both normal and inverted orderings. The eigenvalues and corresponding vectors of Meff in Eq. (26), for the two mass hierarchies, are defined by:

It is easily to check that κ1,2,τ1,2 and r1,2 satisfy the following relations

The leptonic mixing matrix, ULep=VL†Uν, reads:

In the three neutrino framework, the leptonic mixing matrix can be parameterized as

where c12=cosθ12, c23=cosθ23, c13=cosθ13, s12=sinθ12, s23=sinθ23 and s13=sinθ13,δCP is the Dirac CP phase and η1,2 are two Majorana phases.

By comparing Eqs. (31) and (32), with the aid of Eq.(30), we obtain:

Comparing the expressions of the Jalskog’s CP-violation parameter, JCP=Im(U23U13*U12U22*), in term of the model parameters in Eq. (31) and the standard parameterization in Eq. (32), we get

Equations (34)-(35) yield a solution:

• For NH:

where

• For IH:

where

Expressions (30), (36)-(52) imply that k1,2, τ1,2, r1,2, sinδCP and η1,2 depend on s13, s12, s23, θ and ψ which will be presented in Sec. 4.