Neutrino phenomenology and keV dark matter in 2HDM with $A_4$ symmetry

We propose a minimal extended seesaw scheme based on the discrete symmetry $A_4\times Z_4\times Z_2\times Z_8$ which can successfully address neutrino phenomenology and keV sterile neutrino dark matter. The lepton mass hierarchy is naturally achieved. Active neutrino mixing angles can reached the best-fit points with the predictive Dirac CP violation phase. The active-sterile mixing matrix elements are small enough to access the observed cosmological dark matter abundance constraint with keV sterile neutrino dark matter. The effective neutrino masses are predicted to be in the ranges of the recent experimental limits.


I. INTRODUCTION
Although the mass hierarchy, the absolute neutrino mass and the Dirac CP-violating phase are still unknown, two squared neutrino mass differences and neutrino mixing angles have been measured with high accuracy [1].However, lepton mass hierarchy problem is one of the key subject in the elementary particle physics which cannot be explained within the framework of the standard model (SM).A possible solution to the fermion mass hierarchy problem is to introduce a new family symmetry acting between generations [2][3][4][5][6].Furthermore, one interesting anomaly derived from the experimental data of LSND [7] and MiniBooNE [8] that cannot be explained by the three-neutrino scenario but could be solved by adding at least one neutrino, motivating to extended scenarios with the existence of additional neutrinos which called sterile neutrinos since they mix with active neutrinos but do not participate in the SM gauge interactions.Besides, even though the strong evidences for the existence of dark matter (DM) [9][10][11], its properties is still an open question.It has been proved that [12][13][14][15][16] a keV-scale sterile neutrino is viable candidate for warm DM.Planck data [17] implies that 26.8% of the total energy density of the Universe and the dark matter abundance is given by Ω D h 2 ∈ (0.117, 0.120). ( The most recent result on the dark matter abundance from [18] sets a new limit Recently remarkable result by the STEREO experiment strongly rejected the hypothesis of the existence of a sterile neutrino with a few-eV-mass [19].However, sterile neutrinos with heavier masses such as keV scale and mixing very weakly with the active ones could exist but STEREO has not been sensitive to them or the mixing would be too weak to explain the reactor anomaly.
The outstanding feature of discrete symmetries is that they can give a satisfactorily interpretation of the neutrino oscillation data.The neutrino phenomenology and sterile neutrino dark matter with A 4 symmetry have been considered in Refs.[20][21][22][23] which is different from our current work with significantly different scalars and for the following basic properties: • Refs.[20] based on symmetry SU (2) L × U (1) Y × A 4 × Z 2 × Z 3 with five additional fermion singlets, four Higgs doublets, four scalar singlets and one scalar triplet in the frame work of inverse and type II seesaw mechanism with GeV-scale dark matter in which only inverted neutrino mass ordering is satisfied while the charged lepton mass hierarchy is not natural and active-sterile mixing angles are not mentioned.
• Ref. [21] based on symmetry SU (2) L ×U (1) Y ×U (1) ′ Y ′ ×A 4 ×Z 4 ×Z 3 with two Higgs doublets and up to fifteen scalar singlets for NO (sixteen scalar singlets for IO) in the frame work of MES framework with keV-scale dark matter in which the charged lepton mass hierarchy is not natural and the active-sterile mixing angles are not explicitly mentioned.
• Ref. [22] based on symmetry SU (2 one Higgs doublet and ten scalar singlets in 3 + 1 mixing scheme with GeV-scale dark matter in which the charged lepton mass hierarchy is not natural1 and the neutrino masses are generated by the help of up to seven-dimension term and the active-sterile mixing elementes are predicted to be U 14 = U 24 = U 34 which don't seem natural. • Ref. [23] based on symmetry SU (2 In the present study, we propose a minimal extended seesaw (MES) [24][25][26] with the discrete symmetry A 4 × Z 4 × Z 2 × Z 8 in the framework of the two Higgs doublets model with two A 4 triplet and two A 4 singlet flavon fields to explain neutrino phenomenology and keV sterile neutrino dark matter in both normal and inverted hierarchies.
The remaining part of this study is as follows.The description of the model is given in section II.Its implications in neutrino phenomenologyis given in III.Section IV is intended for the dark matter phenomenology.The numerical analysis is presented in section V. Section VI contains some conclusions.

II. THE MODEL
We construct a MES model in which the SM model is supplemented by one non-Abelian discrete symmetry A 4 and three Abelian symmetries Z 2 , Z 4 and Z 8 .Three right-handed neutrinos ν 1,2,3R and one sterile neutrino ν s are added to the SM.On the other hand, one doublet H ′ and eight singlets (φ l , φ ν , χ, ρ) are added to the SM, i.e., the considered model contains two SU (2) L doublets2 .In this model, three left-handed leptons (ψ L ≡ ψ 1,2,3L ) are assigned in 3 under A 4 symmetry while the first right-handed charged lepton l 1R , three right-handed neutrinos, and the sterile neutrino are assigned in 1 and the last two right-handed charged leptons (l 2R , l 3R ) are assigned as 1 ′′ under A 4 .
The particle content of the model, under the symmetry are summarized in Table I.
The particle content in Table I yields the following up to 6D Yukawa interactions invariant under all the model symmetries: where, with H = iσ 2 H * , Λ being the cut-off scale, h i , y j (i = 1 ÷ 4; j = 1 ÷ 6) and z are the Yukawa-like couplings, and λ 1,2 are the Majorana mass scales.Each of additional Abelian symmetry Z 3 , Z 4 and Z 8 takes a crucial role in preventing the unwanted mass terms in the Lagrangian to obtain the desired lepton mass matrices which are listed Appendix A.
We consider the following vacuum expectation value (VEV) alignments for scalar fields, The Higgs doublet VEVs v and v ′ obey the total electroweak VEV, v 2 + v ′2 = v 2 w = (246 GeV) 2 ; thus, they can be conveniently parametrized as Furthermore, in order to have heavy right handed Majorana neutrino masses, and hence allowing the implementation of the type I seesaw mechanism that generates the small masses of the active neutrinos, the VEVs of flavons φ l , φ ν , ρ, χ and the cut-off scale Λ are at a very high scale: In models with more than one SU (2) L doublet as in this study, the FCNC processes such as b → sγ exist in the Higgs sector.However, they are suppressed by non-Abelian discrete symmetries [29,30].To make such processes below the experimental bounds, some restrictions on the model 3 For shortly, we employ the following notations: 12,23,13).
parameters such as the large masses for non SM scalars and Yukawa couplings need to be imposed.
This kind of the model contains many free parameters which allows us freedom to assume that the remaining scalars are sufficiently heavy to fullfil the current experimental bounds.Furthermore, the off-diagonal Yukawa couplings in the charged-lepton sector, Eq. ( 4), are proportional to v l Λ ∼ 10 −2 .Therefore, the LFV processes, such as l j → l i γ, are suppressed by the tiny factor v l Λ 1 m 2 H associated with the small Yukawa couplings and the large mass scale of the heavy scalars m H [31][32][33][34].

III. NEUTRINO MASS AND MIXING
A. Lepton mass and mixing in the three neutrino scheme From the Yukawa terms in Eq. ( 4), by using the multiplication rules of the A 4 group in the T -diagonal basis [35], when the scalar fields H, H ′ , φ l , χ and ρ get the VEVs in Eqs. ( 8) and ( 9), we obtain the following charged lepton mass matrix with In general, a l , b l , c l and d l are complex parameters, thus, M L is a complex matrix.For simplicity, we consider the case of arg c l = arg b l .The Hermitian matrix m l = M L M + L , whose real and positive eigenvalues, given by where α = arg b l − arg d l while a 0 , b 0 , c 0 , d 0 and α a , α b , α c , α d are4 respectively the magnitudes and the arguments of a l , b l , c l , d l .The matrix m l is diagonalised by the matrices U l,r satisfying where In the charged-lepton sector, there exist six parameters including h 1,2,3,4 , s β and c α .Since c α is determined in neutrino sector, Eq. ( 34); thus, leaving five parameters h 1,2,3,4 and s β corresponding to the three observed experimental parameters m e , m µ and m τ .
Expressions ( 12) and ( 15) yields the following relations: where Equation ( 16) tells us that U l is non trivial and hence it will affect on the lepton mixing matrix.
Next, we consider the neutrino sector.When the scalars get the VEVs in Eqs. ( 8) and ( 9), we get the following Dirac, Majorana and sterile mass matrices: where In the (ν c L , ν R , ν s ) basis, the (7 × 7) neutrino mass matrix takes the form: Expressions ( 9), ( 10), ( 21) and ( 22 By applying the type-I seesaw mechanism, the 3 × 3 active neutrino mass matrix is obtained as [24] Substituting Eq. ( 20) into Eq.( 25) yields: where The expressions ( 21), ( 22) and (27) show that, in general the 3×3 active neutrino mass matrix m ν in Eq. ( 25) is a complex matrix.Considering the case of real VEVs for the scalar fields H, H ′ , φ l , φ ν , χ and ρ, and the phase redefinition of lepton fields allows to rotate away the phases of three Yukawa couplings x 1÷6 , y and z that make the mass matrix m ν real and can be diagonalized by the unitary matrix U ν .The mass matrix m ν in Eq.( 26) has three eigenvalues and corresponding eigenvectors as follows where l = 1, 2, 3 and the explicit expressions of κ 1,2 , k 1,2,3 and n 1,2,3 are given in Appendix B, which obey the following relations It is noted that the eigenvalue m 1 = 0 corresponds to the first neutrino eigenvector ϕ 1 .Thus, the neutrino mass should be either (0, m 2 , m 3 ) or (m 2 , m 3 , 0).The considered model can accommodate both inverted ordering (IO) and normal ordering (NO) being consistent with the experimental data and different from that of Refs.[36,37] whereby only the NO is allowed.The eigenvalues and eigenvectors of m ν in Eq. ( 26) corresponding two mass hierarchies are given by6 : From Eqs. ( 28) and ( 31), we can express the model parameters κ 1 and κ 2 in terms of two observed parameters ∆m 2 21 and ∆m 2 31 as follows: for NO, The 3 × 3 leptonic mixing matrix, Comparing Eq. ( 34) with the standard parameterization of the lepton mixing matrix U PMNS , with the help of Eq. ( 30), yields the relations between the model parameters and three observed mixing angles7 s 12 , s 23 and s 13 : for NO, for NO, for NO, for IO, where k 1,2,3 and n 1,2,3 are defined in Appendix B. Therefore, we can express the model parameters k 1,2,3 and n 1,2,3 in terms of three observed parameters s 12 , s 13 , s 23 and two constrained parameters ψ, α as follows: • For NO: • For IO: The Jarlskog invariant parameter in the active sector [38], , is given by: for both NO and IO.(40) Combining expression (40) with that of the standard parameterization in the three neutrino scenario, J = −c 12 c 2 13 c 23 s 12 s 13 s 23 sin δ CP , we obtain: The effective neutrino masses in three neutrino scheme, m for NO, for IO, ( 42) for NO, √

B. 3+1 sterile-active neutrino mixing
The fact that the neutrino mass spectrum can be NO (m depending on the sign of ∆m 2 31 [1,40].In MES scenario, the light neutrino masses are given in terms of three neutrino mass-squared differences as for NO, where 21,31,41).The mass of sterile neutrino [24], m s ≃ −M S M −1 R M T S , obtained from Eqs. ( 20) and ( 44) as follows: The 3 × 1 matrix R which control the strength of active-sterile mixing angles [24], , with the aid of Eqs. ( 20) and ( 44), reads: where The strength of the active-sterile mixing is determined by [24] (U e4 U µ4 U τ 4 ) T = U † L R. Combining Eqs. ( 16) and ( 46) yields: Three active-sterile neutrino mixing angles are given by: Equations ( 47), ( 21), ( 22) and (49) show that the active-sterile neutrino mixing angles depend on the VEVs of scalar fields H, H ′ , χ, ρ, φ ν , the cut-off scale Λ, Majorana masses λ 1,2 , and the Yukawa like coupling constants in the neutrino sectors.
The effective neutrino masses in 3+1 scheme, and m ee = 4 i=1 U 2 ei m i , read [39]: for NO, IV.

DARK MATTER PHENOMENOLOGY
If sterile neutrino mass m s is in keV-scale and the active-sterile neutrino mixing angles are tiny, it can be a warm DM candidate [12][13][14][15][16]. Thus, it is important to calculate the sterile-active mixing to investigate the DM phenomenology.The resulting relic abundance is proportional to the sterile neutrino mass m s and the sterile-active mixing as follows [41]: where s 2 2θ is the sum of the active-sterile mixing angles, with U i4 are determined in Eq. ( 48).
The sterile neutrino is not totally stable and may decay into an active neutrino and a photon γ via the process ν s −→ ν + γ.However, the decay rate Γ is negligible because of the tiny of sterile mixing angles [13].The decay rate is given by [42], Equations ( 52) and (54) show that the relic abundance Ω DM h 2 and the decay rate Γ depend on mass and the mixings of sterile neutrino.

V. NUMERICAL ANALYSIS AND DISCUSSION
We first start with the charged-lepton sector.Expressions ( 18) and (19) show that at the best-fit values [43], m e = 0.51099 MeV, m µ = 105.65837MeV, m τ = 1776.86MeV, and the VEV of the scalars in Eq. ( 10), h 1 depends on one parameter s β while h 2 and h 3 depend on two parameters h 4 and s β .In order to satisfy the experimental constraint on the dark matter abundance [17,18], Eqs.
Furthermore, Eqs.( 12) and (17) show that at the best-fit values of m e,µ,τ taken from Ref. [43] and the VEV of the scalars in Eq. ( 10), s ψ depends on two parameters s β and h 4 .With the help of Eq.
Similarly, at the best-fit points of the observed parameters and with the help of Eqs. ( 9) and (10), we can estimate the values of the model parameters k i , n i (i = 1, 2, 3) and the ranges of the magnitudes of the elements of the lepton mixing matrix |(U L ) ij | as follows: The effective neutrino masses in three neutrino scheme reach the following ranges: (65) 9 The analytical expressions in Eq. ( 41) is for both NH and IH, however, the slight difference values of the best-fit points of s13 and s23 for NO and IO [1] lead to the slight difference of sin δCP for the numerical analysis result, with sin δCP ∈ (−0.8749, −0.3641) , i.e., δ CP ∈ (299.000,338.700) for IO.
With the help of Eqs. ( 10) and (65), the sterile neutrino mass m s depends on two Yukawa like coupling x and y, m s = 9375z 2 /(2704y 2 − 6); thus, we find the possible range of m s as follows m s ∈ (6.783, 6.791) keV, (66) provided that y and z are given in Eq. (55).
) inly that5 the Dirac and sterile neutrino masses are much smaller than the right-handed neutrino (M D < M S ≪ M R ); thus, one can block-diagonalise the 7 × 7 matrix by using the seesaw formula and get the effective 4 × 4 light neutrino mass matrix in the (ν c L , ν s ) basis[24]