Linear Seesaw for Dirac Neutrinos with $A_4$ Flavour Symmetry

We propose a linear seesaw model to realise light Dirac neutrinos within the framework of $A_4$ discrete flavour symmetry. The additional fields and their transformations under the flavour symmetries are chosen in such a way that naturally predicts the hierarchies of different elements of the seesaw mass matrix and also keeps the unwanted terms away. For generic choices of flavon alignments, the model predicts normal hierarchical light neutrino masses with the atmospheric mixing angle in the lower octant. Apart from predicting interesting correlations between different neutrino parameters as well as between neutrino and model parameters, the model also predicts the leptonic Dirac CP phase to lie in a specific range $-\pi/2\lesssim \delta \lesssim -\pi/5$ and $\pi/5\lesssim \delta \lesssim \pi/2$ that includes the currently preferred maximal value. The predictions for the absolute neutrino masses in one specific version of the model can also saturate the cosmological upper bound on sum of absolute neutrino masses.


I. INTRODUCTION
The fact that neutrinos have non-zero but tiny masses, several order of magnitudes smaller compared to the electroweak scale and large mixing [1] has been verified again and again in the last two decades. The present status of neutrino oscillation data can be found in the recent global fit analysis [2][3][4], which clearly indicate that we do not yet know some of the neutrino parameters namely, the mass hierarchy of neutrinos: normal (m 3 > m 2 > m 1 ) or inverted (m 2 > m 1 > m 3 ), leptonic CP violation as well as the octant of atmospheric mixing angle θ 23 . While the next generation neutrino experiments will be able to settle these issues,  [5]. Although such null results only disfavour the quasi-degenerate regime of light Majorana neutrinos and can never rule out Majorana nature of neutrinos, this has recently motivated the particle physics community to study the scenario of Dirac neutrinos with similar interest as given to Majorana neutrinos in the last few decades. The conventional seesaw mechanism for the origin of neutrino masses [6][7][8][9] and its many descendants predict light Majorana neutrinos. On the contrary, there were fewer proposals to generate light Dirac neutrino masses initially [10,11] but it has recently gained momentum with several new proposals to realise sub-eV scale Dirac neutrino masses . Since the coupling of left and right handed neutrinos to the standard model (SM) Higgs field will require fine tuning of Yukawa coupling to the level of 10 −12 or even less, it is important to forbid such couplings at tree level by introducing some additional symmetries such as U (1) B−L , Z N , A 4 which also make sure that the right handed singlet neutrinos do not acquire any Majorana mass terms.
There have been several discussions on other conventional seesaw mechanisms in the context of Dirac neutrinos for example, type I seesaw [32,35], type II seesaw [33], inverse seesaw [35] and so on. Here show how light Dirac neutrinos can be realised within another seesaw scenario, known as linear seesaw mechanism. We consider the presence of A 4 flavour symmetry augmented by additional discrete Z N and global lepton number symmetries which not only dictate the neutrino mixing patterns but also keep the unwanted terms away from the seesaw mass matrix, in order to realise linear seesaw. Linear seesaw for Majorana neutrino was proposed in earlier works [40,41] and further extended to radiative seesaw models in [42,43] and hidden gauge sector models in [44]. We extend it to Dirac neutrino scenarios in a minimal way incorporating the above-mentioned flavour symmetries. Apart from retaining the usual attractive feature of linear seesaw, like the viability of seesaw scale at TeV naturally without much fine-tuning, the model also predicts several other aspects of neutrinos that can be tested at upcoming experiments. Among them, the preference for normal hierarchy, specific range of Dirac CP phase that includes the maximal value, atmospheric mixing angle in lower octant are the ones which address the present puzzles in neutrino physics.
Rest of the paper is organised as follows. In section II, we discuss the conventional and Dirac linear seesaw model and its predictions for sub-eV Dirac neutrinos in details. Finally, we conclude in section III.

II. THE LINEAR SEESAW MODEL
In the conventional linear seesaw model for Majorana neutrinos [40,41], the standard model fermion content is effectively extended by two different types of neutral singlet fermions (N, S) per generation and the complete neutral fermion mass matrix (9 × 9) in the basis (ν L , N, S) assumes the form due to the chosen symmetries or scalar content of the model. The light neutrino mass matrix can be derived from this as which, being linear in Dirac neutrino mass matrix m D is known as the linear seesaw [40,41].
Here the effective light neutrino mass is roughly given by ∼ m D /M where is originated from a small lepton number violating term in M L , the (13) entry of the neutral fermion mass matrix given in Eq. (1). This is a simple alternative to the usual inverse seesaw model [45][46][47][48] where we also introduce two sets of gauge singlet Majorana neutrinos at the TeV scale to obtain light neutrino mass in sub-eV range. We now consider an extension of this simple linear seesaw model to generate sub-eV Dirac neutrino masses following a similar roadmap that was used to accommodate light Dirac neutrinos in type I seesaw [32,35], type II seesaw [33], inverse seesaw [35] etc. Apart from introducing the right handed counterpart of the usual left handed neutrinos, the other heavy fermions introduced for seesaw purpose are of Dirac nature, having both helicities: (N L , N R ) and (S L , S R ). In such a case the complete linear seesaw mass matrix can be written in (ν L , N L , S L ) T , (ν R , N R , S R ) basis as The corresponding formula for light Dirac neutrinos can be written as To obtain the desired structure of the seesaw mass matrix given in Eq. (3) and to obtain the required hierarchy among its elements, we consider Z 4 × Z 3 symmetry and a global lepton number U (1) L symmetry in addition to A 4 flavour symmetry which plays a crucial role in realising flavour structures of the corresponding mass matrices. In table I we elaborate the where Λ is the cut-off scale of the theory and both y's are the respective dimensionless Yukawa coupling constants. Here the leading order contribution to the charged leptons viā where v is the vev of the SM Higgs doublet H and ω = e i2π/3 is the cube root of unity. This charged lepton mass matrix now can be diagonalised by using a matrix U ω (also known as the magic matrix), given by Now, for neutrino sector the relevant Yukawa Lagrangian is given by Here bothLHN R andLHS R terms, involving SM lepton doublet are generated at tree level.
As both SM lepton doublets (L) and gauge singlet Dirac fermions (N, S) are A 4 triplets (the SM Higgs H being a singlet under the same), following the A 4 multiplication rules given in appendix A, we find the associated mass matrices to be diagonal. These mass matrices can be written as where I is a 3 × 3 identity matrix. On the other hand, owing to the specific discrete Z 4 × Z 3 symmetry, S L -ν R and ν R -N L couplings are generated at dimension five level, ensuring the smallness of these couplings. These contributions come via involvement of the A 4 singlet flavons ξ, η and ρ as well as the triplet flavon φ S . Unlike in the conventional linear seesaw mechanism for Majorana neutrinos, here we do not have any approximate global symmetry to make certain terms of the mass matrix small, from naturalness arguments. Therefore, we need to assign these additional discrete symmetries so that at least one of the mass matrices contributing to the light neutrino mass formula in Eq. (4) arises at next to leading order. In this set-up φ S and ξ share same discrete charges like η, hence all of them therefore contribute to the S L -ν R and ν R -N L couplings. This essentially leads to same non-diagonal contributions in these couplings. Now, with the vev alignment for the flavons φ S , ξ, η and ρ as, φ S = (0, v S , 0), ξ = v ξ , η = v η and ρ = v ρ respectively, the most general mass matrices corresponding to these two couplings can be written as where where Here also s i and a i are the symmetric and anti-symmetric contributions originated from A 4 multiplication. This unique contribution (a i or a i ) is a specific feature of A 4 flavour models for Dirac neutrinos and usually do not appear for Majorana neutrinos due to symmetry property of the Majorana mass matrix. It is worth mentioning that, these anti-symmetric parts, originated due to the Dirac nature of neutrinos, significantly dictate the pattern of neutrino mixing and can explain non-zero θ 13 in a very minimal scenario [34] compared to what is usually done with Majorana neutrinos [49]. Such anti-symmetric contribution from A 4 triplet products can also play a non-trivial role in generating nonzero θ 13 in Majorana neutrino scenarios (through Dirac Yukawa coupling appearing in type I seesaw) [50]. Now, substituting these mass matrices obtained in Eq. (9)-(11) in the linear seesaw formula given in Eq. (4) one can obtain the effective light neutrino mass matrix as Here all the elements appearing in the effective neutrino mass matrix are complex in general.
To diagonalise this general complex matrix let us first define a Hermitian matrix M, given This structure of the Hermitian matrix M suggests that it can be diagonalised by a rotation and m 2 1,2,3 are the light neutrino mass eigenvalues. Here the rotation angle θ and phase ψ can be evaluated using the complex parameters in Eq. (13). From Eqs. (13)- (19) it is clear that there exists several parameters in M (obtained from the effective light neutrino matrix) to constrain θ and ψ satisfying correct neutrino oscillation data. Therefore due to presence of several non-trivial matrices having many complex parameters in the effective light neutrino mass matrix, it does not lead to very specific constraints on the parameters appearing in the neutrino linear seesaw mass matrix.
It turns out, there is a way to have a more constrained scenario. Now along with the symmetry mentioned in Table I, for simplicity one can introduce an additional Z 2 symmetry under which both η and ρ are odd (with all other particles are even under this symmetry). Therefore this two flavons will always appear together and under this additional symmetry the Lagrangian presented in Eq. (8) can be re-written in a simplified form as Subsequently, in the present set-up we work with this Z 2 symmetry to keep the analysis minimal and more predictive. Clearly, the Y νN and Y νS couplings remain unchanged and hence corresponding mass are given by Eq. (9). As the triplet flavon φ s (and singlet ξ) do not share same Z 2 symmetry with η, the mass matrices involved in S L -ν R and ν R -N L couplings now can be written in much simpler way as In this simplified scenario, the mixing between the heavy neutrinos S L − N R and S R − N L now takes the form where x 1 = y ξ 1 v ξ , s 1 = y s 1 v S , a 1 = y a 1 v S , x 2 = y ξ 2 v ξ , s 2 = y s 2 v S and a 2 = y a 2 v S . Clearly, presence of the same Z 2 symmetry forbids any contribution from the singlet flavon η in these matrices as evident from Eq. (21). Now, in this simplified scenario, substituting these mass matrices given in Eqs. (9), (22) and (23) in the linear seesaw formula given in Eq. (4) one can obtain the effective light neutrino mass matrix as where λ 1 = to fit neutrino oscillation data. We discuss these two cases below.

A. Case A:
In this simplest scenario, we first consider λ 1 = λ 2 = λ, a 1 = a 2 = a, s 1 = s 2 = s and Hence the general structure for the effective light neutrino matrix as given in Eq. (24) reduces to Here s and a take care of the symmetric and anti-symmetric contributions respectively originating from the two terms in the linear seesaw formula. In order to diagonalise this mass matrix, let us first define a Hermitian matrix as Here we have defined κ 2 = 4|λ| 2 |x| 2 , α = |a|/|x|, β = |s|/|x|, with s = |s|e iφs , a = |a|e iφa and x = |x|e iφx respectively. For notational convenience, the relative phases φ sx and φ ax will be denoted just as φ s and φ a respectively from here onwards. It can be clearly seen from the expressions for mass eigenvalues that m 2 3 > m 2 1 implying the preference for normal hierarchical light neutrino masses. From these definitions it is clear that α is associated with the anti-symmetric contribution whereas β is related to the symmetric contribution in the Dirac neutrino mass matrix. Using Eq. (7), (20), the final lepton mixing matrix in our framework is given by Now, using Eq. (26) and Eq. (30), one can obtain the correlation between the rotation angle θ and phase ψ as tan 2θ = β sin φ s cos ψ − α cos φ a sin ψ) αβ cos(φ s − φ a ) and tan ψ = − α sin φ a β cos φ s .
Such correlation between the model parameters and neutrino mixing angles θ 13 , θ 12 , θ 23 , Dirac CP phase δ can also be found in [34,35,[51][52][53][54]. Therefore from Eq. (31) and Eq. (34) it is clear that the neutrino mixing angles are functions of four model parameters namely, α, β, φ s and φ a . These are the parameters associated with symmetric and anti-symmetric part of the effective light neutrino mass matrix and corresponding relative phases. These parameters then can be constrained using the current data on neutrino mixing angles [2][3][4].
In addition to the bounds obtained from the mixing angles, the parameter space can be further constrained in oder to satisfy correct value for mass squared differences. Here one can define a ratio for the solar to atmospheric mass squared difference as From Eq. (27)- (29), it is evident that this ratio r is a function of the model parameters α, β, φ s and φ a . In order to satisfy correct neutrino oscillation data, we use the 3σ allowed range of the neutrino mixing angles and mass squared differences given in global fit analysis [2,4] to constrain these model parameters. Here in Fig. 1 we have shown the allowed regions for parameters α, β, φ s and φ a satisfying 3σ ranges for neutrino mixing angles (θ 13 , θ 12 , θ 23 ) and ratio of the mass squared differences r. In the left panel of Fig. 1 we show the allowed Jarlskog invariant J CP as a function of φ a (right panel). Here each points in both panels also satisfy 3σ allowed ranges for θ 13 , θ 12 , θ 23 and the ratio (r) of solar to atmospheric mass squared differences [2,4].
points in α-φ a plane whereas in the right panel we have plotted the same in β-φ s plane.
Here we find that the parameter β, associated with the symmetric part of the neutrino mass matrix ranges between 0.7-1. On the other hand, in the right panel of Fig. 2, we have plotted the allowed regions for the as a function of the relative phase φ a associated with the anti-symmetric contribution of the neutrino mass matrix and estimated to be within the range |J CP | ∼ 0 − 0.024. In Fig. 3, we show the most important among such correlations namely, the one between the Dirac CP phase δ and atmospheric mixing angle θ 23 . Interestingly, here we find that, the model predicts the CP phase δ to be in the range −π/2 δ −π/5 and π/5 δ π/2 whereas sin 2 θ 23 lies in the lower octant. This value of δ falls in the current preferred ballpark suggested by experiments [57] as well as global fit analysis [2,4], predicting atmospheric mixing angle θ 23 to be in the lower octant.

B.
Case B: In this subsection, we analyse the effective light neutrino mass matrix given in Eq. (12) in a more general canvas to illustrate the effects of contributions coming from symmetric and anti-symmetric parts appearing in the two different terms of the linear seesaw formula, without assuming any equality between two symmetric (and anti-symmetric) terms. Considering the most general structure for the light neutrino mass matrix as given in Eq. (12), we can define a Hermitian matrix as, where X 1 = 4|x| 2 + |(a 1 + a 2 ) + (s 1 + s 2 )| 2 , X 2 = 2x{(a 1 + a 2 ) * − (s 1 + s 2 ) * } − 2x * {(a 1 + a 2 ) + (s 1 + s 2 )}, Here for simplicity, we have considered x 1 = x 2 = x and λ 1 = λ 2 = λ while keeping the other terms distinct. This Hermitian matrix M now can also be diagonalised by a similar rotation matrix (in the 13 plane) given in Eq. (20) with rotation angle θ and phase factor ψ. These parameters can therefore be expressed as with where we have defined the parameters as α j = |a j |/|x|, β j = |s j |/|x|, φ a j x = φ a j − φ x , φ s j x = φ s j − φ x , s j = |s|e jφs j , a j = |a|e iφa j and x = |x|e iφx with j = 1, 2. For notational compactness we have written the relative phases φ a j x , φ s j x in equation (39)(40)(41) as φ a j , φ s j with j = 1, 2. Hence, diagonalising the Hermitian matrix via U † 13 MU 13 = diag(m 2 1 , m 2 2 , m 2 3 ), we obtain the light neutrino masses as where C 5 = 2{α 1 α 2 cos(φ a 1 − φ a 2 ) + β 1 β 2 cos(φ s 1 − φ s 2 )}, C 6 = 4(α 2 1 cos 2φ a 1 + α 2 2 cos 2φ a 2 ) − 4(β 2 1 cos 2φ s 1 + β 2 2 cos 2φ s 2 ) Considering the contributions from both charged lepton and neutrino sectors, the complete lepton mixing matrix in this general case also is given by allowed ranges of θ 13 , θ 12 , θ 23 and the ratio (r) of solar to atmospheric mass squared differences [2,4]. These points additionally also satisfy the upper limit for sum of the three absolute neutrino mass m i ≤ 0.17 eV [57].
Comparing this mixing matrix with U PMNS as given in Eq. (32), one can obtain the correlations between the the mixing angles (θ 13 , θ 12 and θ 23 ) and Dirac CP phase δ as previously given in Eq. (34). Further using Eq. (38) we find the correspondence between neutrino mixing angles and the relevant model parameters. Here the parameters α j , β j , φ a j and φ s j with j = 1, 2 essentially dictate the neutrino mixing patterns. Obviously, the number of parameters controlling neutrino mixing in this general case is more than what it was in the simple scenario described earlier. Jarlskog invariant J CP as a function of φ a 1 (right panel). Here each points in both panels also satisfy 3σ allowed values of θ 13 , θ 12 , θ 23 and the ratio (r) of solar to atmospheric mass squared differences [2,4].
r(= ∆m 2 /|∆m 2 A | = ∆m 2 21 /|∆m 2 32 |) in terms of very same parameters α j , β j , φ a j and φ s j . Here also to satisfy correct neutrino oscillation data, we use the 3σ range of the neutrino mixing angle and mass squared differences [2,4] to constrain these parameters and we find the correlations among them. In addition to the bounds from neutrino oscillation experiments, these parameters can also get constrained in order to satisfy the cosmological upper limit on sum of the three absolute neutrino mass, given by m i ≤ 0.17 eV [57]. Therefore, using all these constraints, in Fig. 4 we have all the allowed points in α 1 -α 2 (left panel) plane and β 1 - the allowed regions for α 1,2 , β 1,2 , φ a 1,2 and φ s 1,2 , one can again find out the common factor κ appearing in the light neutrino mass eigenvalues involved in this general case using Eq.  Fig. 7, we have now plotted the correlation between the Dirac CP phase δ and atmospheric mixing angle θ 23 . Similar to the previous case, here also we find that, the model predicts the Dirac CP phase δ to be in the range −π/2 δ −π/5 and π/5 δ π/2 whereas θ 23 lies in the lower octant.
In this two different limits of Dirac linear seesaw discussed above, we have observed that the allowed range of the light neutrino mass is different in the two cases. In Case A, due to much constrained scenario of the neutrino mass matrix the correlation between m 1 and m i is mainly concentrated within a narrow region. This is shown in the left panel of basically allowed, as shown in the right panel of Fig. 8. Finally, it is important to mention that, within these two different limits, inverted hierarchy of neutrino mass is not allowed, another important prediction of our model.

III. CONCLUSION
We have proposed a linear seesaw model for Dirac neutrinos within the framework of A 4 flavour symmetry, augmented by additional discrete and global lepton number symmetry in order to make sure that the correct hierarchy between different terms appearing in the complete neutral fermion mass matrix is naturally obtained without making any ad hoc assumptions. The interesting feature of the conventional linear seesaw framework where a small lepton number breaking term in seesaw formula, linear in Dirac neutrino mass, can give rise to correct neutrino mass with heavy neutrinos lying in TeV scale, is retained in the Dirac version of it by appropriately generating hierarchical terms at different orders (dimension four and dimension five). Since lepton number is present as a global unbroken symmetry in the model, all the mass matrices involved are of Dirac type and hence the A 4 triple products contain the anti-symmetric components which play a crucial role in generating the correct neutrino phenomenology. Since we use the S diagonal basis of A 4 for Dirac neutrino case, the charged lepton mass matrix is also non trivial in our scenarios and hence can contribute to the leptonic mixing matrix. For generic choices of A 4 flavon alignments, we find that the model remains very predictive in terms of neutrino mass hierarchy, leptonic CP phase, octant of atmospheric mixing angle as well as absolute neutrino masses. While the neutrino mass hierarchy is predicted to be the normal one, the Dirac CP phase δ is found to lie in the range −π/2 δ −π/5 and π/5 δ π/2 whereas the atmospheric mixing angle θ 23  Appendix A: A 4 Multiplication Rules A 4 , the symmetry group of a tetrahedron, is a discrete non-abelian group of even permutations of four objects. It has four irreducible representations: three one-dimensional and one three dimensional which are denoted by 1, 1 , 1 and 3 respectively, being consistent with the sum of square of the dimensions i n 2 i = 12. We denote a generic permutation (1, 2, 3, 4) → (n 1 , n 2 , n 3 , n 4 ) simply by (n 1 n 2 n 3 n 4 ). The group A 4 can be generated by two basic permutations S and T given by S = (4321), T = (2314). This satisfies S 2 = T 3 = (ST ) 3 = 1 which is called a presentation of the group. Their product rules of the irreducible representations are given as where a and s in the subscript corresponds to anti-symmetric and symmetric parts respectively. Denoting two triplets as (a 1 , b 1 , c 1 ) and (a 2 , b 2 , c 2 ) respectively, their direct product can be decomposed into the direct sum mentioned above. In the S diagonal basis, the products are given as In the T diagonal basis on the other hand, they can be written as 1 a 1 a 2 + b 1 c 2 + c 1 b 2 1 c 1 c 2 + a 1 b 2 + b 1 a 2