Fully Constrained Majorana Neutrino Mass Matrices Using $Σ(72\times 3)$
R. Krishnan P. F. Harrison W. G. Scott
hep-ph




                     



University of Warwick, Coventry, CV4 7AL, United KingdomRutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, United Kingdom  




In 2002, two neutrino mixing ansatze having trimaximally-mixed middle (ν_2) columns, namely tri-chi-maximal mixing () and tri-phi-maximal mixing (), were proposed. In 2012, it was shown that  with χ=±π/16 as well as  with ϕ = ±π/16 leads to the solution, sin^2 θ_13 = 2/3sin^2 π/16, consistent with the latest measurements of the reactor mixing angle, θ_13. To obtain _(χ=±π/16) and _(ϕ=±π/16), the type I see-saw framework with fully constrained Majorana neutrino mass matrices was utilised. These mass matrices also resulted in the neutrino mass ratios, m_1:m_2:m_3=(2+√(2))/1+√(2(2+√(2))):1:(2+√(2))/-1+√(2(2+√(2))). In this paper we construct a flavour model based on the discrete group Σ(72×3) and obtain the aforementioned results. A Majorana neutrino mass matrix (a symmetric 3×3 matrix with 6 complex degrees of freedom) is conveniently mapped into a flavon field transforming as the complex 6 dimensional representation of Σ(72×3). Specific vacuum alignments of the flavons are used to arrive at the desired mass matrices.


      14.60.PqNeutrino mass and mixing   11.30.HvFlavor symmetries
      
 

Fully Constrained Majorana Neutrino Mass Matrices Using Σ(72×3)
    R. Krishnan1https://orcid.org/0000-0002-0707-3267orcid.org/0000-0002-0707-3267, P. F. Harrison1mailto:p.f.harrison@warwick.ac.uk p.f.harrison@warwick.ac.uk, W. G. Scott3mailto:w.g.scott@rl.ac.ukw.g.scott@rl.ac.uk



    Received: date / Revised version: date
===========================================================================================================================================================================================================================




§ INTRODUCTION


The neutrino mixing information is encapsulated in the unitary PMNS mixing matrix which, in the standard PDG parameterisation <cit.>, is given by

    U_PMNS=
       (
    c_12 c_13   s_12 c_13   s_13e^-iδ
    
    -s_12 c_23-c_12 s_23 s_13 e^iδ   c_12c_23 -s_12s_23 s_13 e^iδ   s_23 c_13
    
    s_12 s_23-c_12 c_23 s_13e^iδ   -c_12s_23-s_12c_23s_13e^iδ   c_23c_13)
                  ×diag(1,  e^iα_21/2,  e^iα_31/2)

where s_ij=sinθ_ij, c_ij=cosθ_ij. The three mixing angles θ_12 (solar angle), θ_23 (atmospheric angle) and θ_13 (reactor angle) along with the CP-violating complex phases (the Dirac phase, δ, and the two Majorana phases, α_21 and α_31) parameterise U_PMNS. In comparison to the small mixing angles observed in the quark sector, the neutrino mixing angles are found to be relatively large <cit.>:

    sin^2 θ_12   = 0.271→ 0.345  ,
    sin^2 θ_23   = 0.385→ 0.635  ,
    sin^2 θ_13   = 0.01934→ 0.02392 .

The values of the complex phases are unknown at present. Besides measuring the mixing angles, the neutrino oscillation experiments also proved that neutrinos are massive particles. These experiments measure the mass-squared differences of the neutrinos and currently their values are known to be  <cit.>,

    Δ m_21^2=70.3→ 80.9 meV^2,
    
    |Δ m_31^2|=2407 → 2643 meV^2.


Several mixing ansatze with a trimaximally-mixed second column for U_PMNS, i.e. |U_e2|=|U_μ2|=|U_τ2|=1/√(3), were proposed during the early 2000s <cit.>. Here we briefly revisit two of those, the tri-chi-maximal mixing () and the tri-phi-maximal mixing (), which are relevant to our model. They can be conveniently parameterised <cit.> as follows   

    U_   =(√(2/3)cosχ   1/√(3)   √(2/3)sinχ
    
    -cosχ/√(6)-isinχ/√(2)   1/√(3)    icosχ/√(2)-sinχ/√(6)
    
    -cosχ/√(6)+isinχ/√(2)   1/√(3)    -icosχ/√(2)-sinχ/√(6)),
    
    U_   =(√(2/3)cosϕ   1/√(3)   √(2/3)sinϕ
    
    -cosϕ/√(6)-sinϕ/√(2)   1/√(3)   cosϕ/√(2)-sinϕ/√(6)
    
    -cosϕ/√(6)+sinϕ/√(2)   1/√(3)    -cosϕ/√(2)-sinϕ/√(6)).

Both  and [TM_i (TM^i) has been proposed <cit.> as a nomenclature to denote the mixing matrices that preserve various columns (rows) of the tribimaximal mixing <cit.>. Under this notation, both  and  fall under the category of TM_2. To be more specific, TM_2 which breaks CP maximally is  and TM_2 which conserves CP is .]  have one free parameter each (χ and ϕ) which directly corresponds to the reactor mixing angle, θ_13, through the U_e3 elements of the mixing matrices. The three mixing angles and the Dirac CP phase obtained by relating Eq. (<ref>) with Eqs. (<ref>, <ref>) are shown in Table <ref>.


In , since δ = ±π/2, CP violation is maximal for a given set of mixing angles. The Jarlskog CP-violating invariant <cit.> in the context of  <cit.> is given by 

    J=sin 2χ/6√(3).
 
On the other hand,  is CP conserving, i.e. δ = 0, π, and thus J=0. 

Since the reactor angle was discovered to be non-zero at the Daya Bay reactor experiment in 2012 <cit.>, there has been a resurgence of interest <cit.> in  and  and their equivalent forms. For any CP-conserving (δ = 0, π) mixing matrix with non-zero θ_13 and trimaximally-mixed ν_2 column, we can have an equivalent parameterisation realised using the  matrix. Here the “equivalence” is with respect to the neutrino oscillation experiments. The oscillation scenario is completely determined by the three mixing angles and the Dirac phase (Majorana phases are not observable in neutrino oscillations), i.e. we have a total of four degrees of freedom in the mixing matrix. If we assume CP conservation and also assume that the ν_2 column is trimaximally mixed, then there is only one degree of freedom left. It is exactly this degree of freedom which is parameterised using ϕ in  mixing. Similarly any mixing matrix with δ = ±π/2, θ_13≠0 and trimaximal ν_2 column is equivalent to  mixing.

In 2012 <cit.>, shortly after the discovery of the non-zero reactor mixing angle, it was shown that _(χ=±π/16) as well as _(ϕ=±π/16) results in a reactor mixing angle, 

    sin^2 θ_13   = 2/3sin^2 π/16
       = 0.025,
 
consistent with the experimental data. The model was constructed in the Type-1 see-saw framework <cit.>. Four cases of Majorana mass matrices were discussed:

    M_Maj   ∝((2-√(2))     0    1/√(2)
    
    0     1     0
    1/√(2)    0     0
    ),      ∝(0     0    1/√(2)
    
    0     1     0
    1/√(2)    0     (2-√(2))
    ),
    
    M_Maj   ∝(i+1-i/√(2)    0     1-1/√(2)
    
    0     1     0
    
    1-1/√(2)    0     -i+1+i/√(2)),      ∝(-i+1+i/√(2)    0     1-1/√(2)
    
    0     1     0
    
    1-1/√(2)    0     i+1-i/√(2))

where M_Maj is the coupling among the right-handed neutrino fields, i.e. (ν_R)^cM_Majν_R. In Ref. <cit.>, the mixing matrix was modelled in the form

    U_PMNS = 1/√(3)(1     1     1
    
    1    ω   ω̅
    
    1    ω̅   ω)U_ν ,   with ω=e^i2π/3 and ω̅=e^-i2π/3,

in which the 3×3 trimaximal contribution came from the charged-lepton sector. U_ν, on the other hand, was the contribution from the neutrino sector. The four U_νs vis-a-vis the four Majorana neutrino mass matrices given in Eqs. (<ref>) and Eqs. (<ref>), gave rise to _(χ=±π/16) and _(ϕ=±π/16) respectively.  All the four mass matrices, Eqs. (<ref>, <ref>), have the eigenvalues 1+√(2(2+√(2)))/(2+√(2)), 1 and -1+√(2(2+√(2)))/(2+√(2)). Due to the see-saw mechanism, the neutrino masses become inversely proportional to the eigenvalues of the Majorana mass matrices, resulting in the neutrino mass ratios

    m_1:m_2:m_3=(2+√(2))/1+√(2(2+√(2))):1:(2+√(2))/-1+√(2(2+√(2)))  .

Using these ratios and the experimentally-measured mass-squared differences, the light neutrino mass was predicted to be around 25 meV.

In this paper we use the discrete group Σ(72×3) to construct a flavon model that essentially reproduces the above results. Unlike in Ref. <cit.> where the neutrino mass matrix was decomposed into a symmetric product of two matrices, here a single sextet representation of the flavour group is used to build the neutrino mass matrix. A brief discussion of the group Σ(72×3) and its representations is provided in Section 2 of this paper. Appendix A contains more details such as the tensor product expansions of its various irreducible representations (irreps) and the corresponding Clebsch-Gordan (C-G) coefficients. In Section 3, we describe the model with its fermion and flavon field content in relation to these irreps. Besides the aforementioned sextet flavon, we also introduce triplet flavons in the model to build the charged-lepton mass matrix. The flavons are assigned specific vacuum expectation values (VEVs) to obtain the required neutrino and charged-lepton mass matrices. A detailed description of how the charged-lepton mass matrix attains its hierarchical structure is deferred to Appendix B. In Section 4, we obtain the phenomenological predictions and compare them with the current experimental data along with the possibility of further validation from future experiments. Finally the results are summarised in Section 5. The construction of suitable flavon potentials which generate the set of VEVs used in our model is demonstrated in Appendix C. 



§ THE GROUP Σ(72×3) AND ITS REPRESENTATIONS


Discrete groups have been used extensively in the description of flavour symmetries. Historically, the study of discrete groups can be traced back to the study of symmetries of geometrical objects. Tetrahedron, cube, octahedron, dodecahedron and icosahedron, which are the famous Platonic solids, were known to the ancient Greeks. These objects are the only regular polyhedra with congruent regular polygonal faces. Interestingly, the symmetry groups of the platonic solids are the most studied in the context of flavour symmetries too - A_4 (tetrahedron), S_4 (cube and its dual octahedron) and A_5 (dodecahedron and its dual icosahedron). These polyhedra live in the three-dimensional Euclidean space. In the context of flavour physics, it might be rewarding to study similar polyhedra that live in three-dimensional complex Hilbert space. In fact, five such complex polyhedra that correspond to the five Platonic solids exist as shown by Coxeter <cit.>. They are 3{3}3{3}3, 2{3}2{4}p, p{4}2{3}2, 2{4}3{3}3, 3{3}3{4}2 where we have used the generalised schlafli symbols <cit.> to represent the polyhedra. The polyhedron 3{3}3{3}3 known as the Hessian polydehron can be thought of as the tetrahedron in the complex space. Its full symmetry group has 648 elements and is called Σ(216×3). Like the other discrete groups relevant in flavour symmetry, Σ(216×3) is also a subgroup of the continuous group U(3). 

The principal series of Σ(216×3) <cit.> is given by

    {e}◃ Z_3 ◃Δ(27) ◃Δ(54) ◃Σ(72×3) ◃Σ(216×3).

Our flavour symmetry group, Σ(72×3), is the maximal normal subgroup of Σ(216×3). So we get Σ(216×3)/Σ(72×3)=Z_3. Various details about the properties of the group Σ(72×3) and its representations can be found in Refs. <cit.>. Note that Σ(72×3) is quite distinct from Σ(216) which is defined using the relation Σ(216×3)/Z_3=Σ(216). In other words, Σ(216×3) forms the triple cover of Σ(216). Σ(216×3) as well as Σ(216) is sometimes referred to as the Hessian group. In terms of the GAP <cit.> nomenclature, we have Σ(216×3)≡SmallGroup(648,532), Σ(72×3)≡SmallGroup(216,88) and Σ(216)≡SmallGroup(216,153).

We find that, in the context of flavour physics and model building, Σ(72×3) has an appealing feature: it is the smallest group containing a complex three-dimensional representation whose tensor product with itself results in a complex six-dimensional representation [We studied the comprehensive list of finite subgroups of U(3) provided in Ref. <cit.> and determined that Σ(72×3) is the smallest group having this feature.], i.e.

    3⊗3=6⊕.

With a suitably chosen basis for 6 we get

    6≡(a_1 b_1 
    
    	a_2 b_2
    
    	a_3 b_3
    1/√(2)(a_2 b_3 + a_3 b_2)
    1/√(2)(a_3 b_1 + a_1 b_3)
    1/√(2)(a_1 b_2 + a_2 b_1)
    ),   3̅≡(1/√(2)(a_2 b_3 - a_3 b_2)
    1/√(2)(a_3 b_1 - a_1 b_3)
    1/√(2)(a_1 b_2 - a_2 b_1)
    )

where (a_1, a_2, a_3)^T and (b_1, b_2, b_3)^T represent the triplets appearing in the LHS of Eq. (<ref>). All the symmetric components of the tensor product together form the representation 6 and the antisymmetric components form . For the SU(3) group it is well known that the tensor product of two 3s gives rise to a symmetric 6 and an antisymmetric . Σ(72×3) being a subgroup of SU(3), of course, has its 6 and  embedded in the 6 and  of SU(3).  

Consider the complex conjugation of Eq. (<ref>), i.e. ⊗=⊕3. Let the right-handed neutrinos form a triplet, ν_R=(ν_R1,ν_R2,ν_R3)^T, which transforms as a . Symmetric (and also Lorentz invariant) combination of two such triplets leads to a conjugate sextet, S̅_ν, which transforms as a ,

    S̅_ν= (ν_R1.ν_R1
    ν_R2.ν_R2
    ν_R3.ν_R3
    1/√(2)(ν_R2.ν_R3 + ν_R3.ν_R2)
    1/√(2)(ν_R3.ν_R1 + ν_R1.ν_R3)
    1/√(2)(ν_R1.ν_R2 + ν_R2.ν_R1)
    )≡

where ν_Ri.ν_Rj is the Lorentz invariant product of the right-handed neutrino Weyl spinors. We may couple S̅_ν to a flavon field

    ξ=(ξ_1,ξ_2,ξ_3,ξ_4,ξ_5,ξ_6)^T

which transforms as a 6 to construct the invariant term

    S̅_ν^T ξ = (ν_R1
    ν_R2
    ν_R3)^T. (ξ_1    1/√(2)ξ_6    1/√(2)ξ_5
    1/√(2)ξ_6    ξ_2    1/√(2)ξ_4
    1/√(2)ξ_5    1/√(2)ξ_4    ξ_3
    ).(ν_R1
    ν_R2
    ν_R3).

In general, the 3×3 Majorana mass matrix is symmetric and has six complex degrees of freedom. Therefore, using Eq. (<ref>), any required mass matrix can be obtained through a suitably chosen vacuum expectation value (VEV) for the flavon field. 




 


To describe the representation theory of Σ(72×3) we largely follow Ref. <cit.>. Σ(72×3) can be constructed using four generators, namely C, E, V and X <cit.>. For the three-dimensional representation, we have

    C ≡(1     0     0
    
           0        0
    
           0     0    ),           E ≡(0     1     0
    
           0     0     1
    
           1     0     0
    ),
       V≡
    -i/√(3)(1     1     1
    
           1       
    
           1       ),    X≡
    -i/√(3)(1     1    
    
           1       
        1    ).

The characters of the irreducible representations of Σ(72×3) are given in Table <ref>. Tensor product expansions of various representations relevant to our model are given in Appendix A. There we also provide the corresponding C-G coefficients and the generator matrices.




§ THE MODEL

toc

In this paper we construct our model in the Standard Model framework with the addition of heavy right-handed neutrinos. Through the type I see-saw mechanism, light Majorana neutrinos are produced. The fermion and flavon content of the model, together with the representations to which they belong, are given in Table <ref>. In addition to Σ(72×3), we have introduced a flavour group C_4 = {1, -1, i, -i} for obtaining the observed mass hierarchy for the charged leptons. The Standard Model Higgs field is assigned to the trivial (singlet) representation of the flavour groups.





For the charged leptons, we obtain the mass term

    (y_τ L^†τ_R ϕ̅_β/Λ+y_μ L^†μ_R √(2)A̅_βα/Λ^2)H+  ℋ.𝒯.

where H is the Standard Model Higgs, Λ is the cut-off scale, y_τ and y_μ are the coupling constants for the τ-sector and the μ-sector respectively. A̅_βα is the conjugate triplet obtained from ϕ_β and ϕ_α, constructed in the same way as the second part of Eq. (<ref>),

    A̅_βα≡(1/√(2)(ϕ_β 2ϕ_α 3 - ϕ_β 3ϕ_α 2)
    1/√(2)(ϕ_β 3ϕ_α 1 - ϕ_β 1ϕ_α 3)
    1/√(2)(ϕ_β 1ϕ_α 2 - ϕ_β 2ϕ_α 1)
    )

where ϕ_α=(ϕ_α 1, ϕ_α 2, ϕ_α 3)^T and ϕ_β=(ϕ_β 1, ϕ_β 2, ϕ_β 3)^T. 

L^†τ_R transforms as 3× i under the flavour group, Σ(72×3)× C_4. The flavon ϕ̅_β transforms as × -i and hence it couples to L^†τ_R as shown in Eq. (<ref>). No other coupling involving τ_R, μ_R or e_R with either ϕ̅_β or ϕ̅_α is allowed, given the C_4 assignments in Table <ref>. However, L^†μ_R and A̅_βα, which transform as 3× 1 and × 1 respectively, can couple, Eq. (<ref>). Note that A̅_βα is a second order product of ϕ_β and ϕ_α and it is antisymmetric. No other second order product transforming as  exists, since the antisymmetric product of ϕ_β with itself or ϕ_α with itself vanishes. ℋ.𝒯. represents all the higher order terms, i.e.  the terms consisting of higher order products of the flavons, coupling to e_R, μ_R and τ_R. It can be shown that, for obtaining a flavon term coupling to the e_R, we require at least quartic order[Refer to Appendix B for an analysis of the higher order products of ϕ_α and ϕ_β.].

The VEV of the Higgs, (0, h_o), breaks the weak gauge symmetry. For the flavons ϕ̅_α and ϕ̅_β, we assign the vacuum alignments[Refer to Appendix C for the details of the flavon potential that leads to these VEVs.] footnote-2Hfootnote-2

    ⟨ϕ̅_α⟩=V^†(1,0,0)^Tm,  ⟨ϕ̅_β⟩=V^†(0,0,1)^Tm

where V is one of the generators of Σ(72×3) given in Eqs. (<ref>) and is proportional to the 3×3 trimaximal matrix. The constant m has dimensions of mass. Substituting these vacuum alignments in Eq. (<ref>) leads to the following charged-lepton mass term

    ( e_L
    μ_L
    τ_L
    )^† V^†(𝒪(ϵ^4)    𝒪(ϵ^4)     0
    
    0     y_μ h_oϵ^2+𝒪(ϵ^4)     0
    𝒪(ϵ^4)    𝒪(ϵ^4)     y_τ h_o ϵ+𝒪(ϵ^3)
    )( e_R
    μ_R
    τ_R
    )

where ϵ=m/Λ. The matrix elements, 𝒪(ϵ^3) and 𝒪(ϵ^4), are  of the order of ϵ^3 and ϵ^4 respectively. They are the result of the higher order terms in Eq. (<ref>) containing cubic and quartic flavon productsfootnote1Hfootnote1. The mass matrix shown in Eq. (<ref>) is approximately diagonalised [The effect of higher order elements on diagonalisation is discussed in Section 4.] by left multiplying it with V. It is apparent that the charged-lepton masses, i.e. the eigenvalues of the mass matrix, are in the ratio 𝒪(ϵ):𝒪(ϵ^2):𝒪(ϵ^4). This is consistent with the experimentally-observed mass hierarchy, (m_μ/m_e) ≈(m_τ/m_μ)^2.          

Now, we write the Dirac mass term for the neutrinos:

    2 y_w L^†ν_R H̃

where H̃ is the conjugate Higgs and y_w is the coupling constant. With the help of Eq. (<ref>), we also write the Majorana mass term for the neutrinos:

    y_m S̅_ν^Tξ

where y_m is the coupling constant. Let ⟨ξ⟩ be the VEV acquired by the sextet flavon ξ, and let ⟨ξ⟩ be the corresponding 3× 3 symmetric matrix of the form given in Eq. (<ref>). Combining the mass terms, Eq. (<ref>) and Eq. (<ref>), and using the VEVs of the Higgs and the flavon, we obtain the Dirac-Majorana mass matrix:

    M=(0     y_w h_o I
    
           y_w h_o I      y_m ⟨ξ⟩).

The 6×6 mass matrix M, forms the coupling

    M_ij ν_i.ν_j   with  ν=(ν_L^*
    ν_R)

where ν_L=(ν_e,ν_μ,ν_τ)^T are the left-handed neutrino flavour eigenstates.

Since y_w h_o is at the electroweak scale and y_m ⟨ξ⟩ is at the high energy flavon scale (>10^10 GeV), small neutrino masses are generated through the see-saw mechanism. The resulting effective see-saw mass matrix is of the form

    M_ss=-(y_w h_o)^2(y_m ⟨ξ⟩)^-1.

From Eq. (<ref>), it is clear that the see-saw mechanism makes the light neutrino masses inversely proportional to the eigenvalues of the matrix ⟨ξ⟩. We now proceed to construct the four cases of the mass matrices, Eqs. (<ref>, <ref>), all of which result in the neutrino mass ratios, Eq. (<ref>). To achieve this we choose suitable vacuum alignments[Refer to Appendix C for the details of the flavon potentials that lead to these VEVs.] for the sextet flavon ξ. 



 §.§ _(χ=+π/16)
 

Here we assign the vacuum alignment

    ⟨ξ⟩ = ((2-√(2)),1,0,0,1,0)^Tm.

Using the symmetric matrix form of the sextet given in Eq. (<ref>), we obtain

    ⟨ξ⟩ =((2-√(2))     0    1/√(2)
    
    0     1     0
    1/√(2)    0     0
    )m.

Diagonalising the corresponding effective see-saw mass matrix M_ss, Eq. (<ref>), we get

    U_ν^† M_ss U_ν^* = (y_w h_o)^2/y_m mDiag((2+√(2))/1+√(2(2+√(2))),1,(2+√(2))/-1+√(2(2+√(2))))

leading to the neutrino mass ratios, Eq. (<ref>). The unitary matrix U_ν is given by

    U_ν = i (cos(3π/16)     0     -isin(3π/16)
    
    0     1     0
    sin(3π/16)     0     icos(3π/16)
    ).

The product of the contribution from the charged-lepton sector i.e. V from Eqs. (<ref>, <ref>) and the contribution from the neutrino sector i.e. U_ν from Eqs. (<ref>, <ref>) results in the _(χ=+π/16) mixing:

    U_PMNS=VU_ν=
       (1     0     0
    
    0        0
    
    0     0    )(√(2/3)cosχ   1/√(3)   √(2/3)sinχ
    
    -cosχ/√(6)-isinχ/√(2)   1/√(3)    icosχ/√(2)-sinχ/√(6)
    
    -cosχ/√(6)+isinχ/√(2)   1/√(3)    -icosχ/√(2)-sinχ/√(6))(1     0     0
    
    0     1     0
    
    0     0     i
    )

with χ=+π/16.



 §.§ _(χ=-π/16)
 
Here we assign the vacuum alignment

    ⟨ξ⟩ = (0,1,(2-√(2)),0,1,0)^Tm

resulting in the symmetric matrix

    ⟨ξ⟩=(0     0    1/√(2)
    
    0     1     0
    1/√(2)    0     (2-√(2))
    )m.

In this case, the diagonalising matrix is

    U_ν =i (cos(5π/16)     0     isin(5π/16)
    
    0     1     0
    sin(5π/16)     0     -icos(5π/16)
    )

and the corresponding mixing matrix is

    U_PMNS=VU_ν=
       (1     0     0
    
    0        0
    
    0     0    )(√(2/3)cosχ   1/√(3)   √(2/3)sinχ
    
    -cosχ/√(6)-isinχ/√(2)   1/√(3)    icosχ/√(2)-sinχ/√(6)
    
    -cosχ/√(6)+isinχ/√(2)   1/√(3)    -icosχ/√(2)-sinχ/√(6))(1     0     0
    
    0     1     0
    
    0     0     -i
    )

with χ=-π/16.



 §.§ _(ϕ=+π/16)
 
Here we assign the vacuum alignment

    ⟨ξ⟩ = (i+1-i/√(2),1,-i+1+i/√(2),0, (√(2)-1), 0)^Tm

resulting in the symmetric matrix

    ⟨ξ⟩=(i+1-i/√(2)    0     1-1/√(2)
    
    0     1     0
    
    1-1/√(2)    0     -i+1+i/√(2))m.

In this case, the diagonalising matrix is

    U_ν =i (1/√(2) e^-iπ/16    0     -1/√(2) e^-iπ/16
    
    0     1     0
    1/√(2) e^iπ/16    0    1/√(2) e^iπ/16)

and the corresponding mixing matrix is

    U_PMNS=VU_ν=
       (1     0     0
    
    0        0
    
    0     0    )(√(2/3)cosϕ   1/√(3)   √(2/3)sinϕ
    
    -cosϕ/√(6)-sinϕ/√(2)   1/√(3)   cosϕ/√(2)-sinϕ/√(6)
    
    -cosϕ/√(6)+sinϕ/√(2)   1/√(3)    -cosϕ/√(2)-sinϕ/√(6))(1     0     0
    
    0     1     0
    
    0     0     i
    )

with ϕ=+π/16.



 §.§ _(ϕ=-π/16)
 
Here we assign the vacuum alignment

    ⟨ξ⟩ =  (-i+1+i/√(2),1,i+1-i/√(2),0, (√(2)-1), 0)^Tm

resulting in the symmetric matrix

    ⟨ξ⟩=(-i+1+i/√(2)    0     1-1/√(2)
    
    0     1     0
    
    1-1/√(2)    0     i+1-i/√(2))m.

In this case, the diagonalising matrix is

    U_ν =i (1/√(2) e^iπ/16    0    1/√(2) e^iπ/16
    
    0     1     0
    1/√(2) e^-iπ/16    0     -1/√(2) e^-iπ/16)

and the corresponding mixing matrix is

    U_PMNS=VU_ν=
       (1     0     0
    
    0        0
    
    0     0    )(√(2/3)cosϕ   1/√(3)   √(2/3)sinϕ
    
    -cosϕ/√(6)-sinϕ/√(2)   1/√(3)   cosϕ/√(2)-sinϕ/√(6)
    
    -cosϕ/√(6)+sinϕ/√(2)   1/√(3)    -cosϕ/√(2)-sinϕ/√(6))(1     0     0
    
    0     1     0
    
    0     0     -i
    )

with ϕ=-π/16.

As stated earlier, the four cases, Eqs. (<ref>, <ref>, <ref>, <ref>), result in the same neutrino mass ratios, Eq. (<ref>).



 §.§ Symmetries of the VEVs of the sextet flavons


A careful inspection of the Majorana matrices, Eqs. (<ref>, <ref>), reveals several symmetries which could be attributed to the underlying symmetries of the VEVs of the sextet flavons, Eqs. (<ref>, <ref>, <ref>, <ref>). The VEVs, Eqs. (<ref>, <ref>), (and thus the mass matrices, Eqs. (<ref>)) are composed of real numbers implying they remain invariant under complex conjugation. Therefore, they do not contribute to CP violation. In our model, U_PMNS=V U_ν where V originates from the charged-lepton mass matrix, Eq. (<ref>). Since V is maximally CP-violating (δ=π/2), the resulting leptonic mixing, V U_ν, is also maximally CP-violating (). Note that U_, Eq. (<ref>), is symmetric under the conjugation and the exchange of μ and τ rows. This generalised CP symmetry under the combined operations of μ-τ exchange and complex conjugation is referred to as μ-τ reflection symmetry in previous publications <cit.>. The conjugation symmetry in the neutrino VEVs together with maximal CP violation from the charged-lepton sector produces the μ-τ reflection symmetry of U_PMNS.

Consider the exchange of the first and the third rows as well as the columns of the mass matrix, Eq. (<ref>). This is equivalent to the exchange of the first and the third elements and the fourth and the sixth elements of the sextet flavon, Eq. (<ref>). In Σ(72×3), this exchange can be realised using the group transformation by the unitary matrix E.V.V,

    E.V.V≡(0     0     -1
    
           0     -1     0
    
           -1     0     0
    )≡(0     0     1     0     0     0
    
    	0     1     0     0     0     0
    
    	1     0     0     0     0     0
    
    	0     0     0     0     0     1
    
    	0     0     0     0     1     0
    
    	0     0     0     1     0     0
    ),

with E and V given in Eqs. (<ref>, <ref>). By the group transformation we imply left and right multiplication of the mass matrix using the 3× 3 unitary matrix and its transpose or equivalently left multiplication of the sextet flavon using the 6× 6 unitary matrix. The mass matrices, Eqs. (<ref>), and the corresponding flavon VEVs, Eqs. (<ref>, <ref>), are invariant under the transformation by E.V.V together with the conjugation. The VEVs break Σ(72×3) almost completely except for E.V.V with conjugation which remains as their residual symmetry[Here we apply E.V.V and complex conjugation together even though complex conjugation is not a part of Σ(72×3).]. The resulting mixing matrix, U_PMNS=V U_ν, is  which is real and CP conserving. E.V.V-conjugation symmetry in the neutrino VEVs together with maximal CP violation from the charged-lepton sector produces the CP symmetry of U_PMNS.


 In our model, the second column becomes trimaximal because of the vanishing of the forth and the sixth elements of the VEVs, Eqs. (<ref>, <ref>, <ref>, <ref>) which correspond to the off-diagonal zeros present in the mass matrices, Eqs. (<ref>, <ref>).



    E.E.A.E≡(-1     0     0
    
           0     1     0
    
           0     0     -1
    )

Another interesting transformation is the group action by the unitary matrix 

    E.E.A(16).E≡(-1     0     0
    
           0     1     0
    
           0     0     -1
    )

where A(16) is diag(1,e^2π i/16,e^-2π i/16) as defined in Ref. <cit.>


Both  and  have a trimaximal second column. This feature of the mixing matrix was linked to the "magic" symmetry of the mass matrix <cit.>. In our model, the charged-leptonic contribution, V, is trimaximal. Because of the vanishing of the forth and the sixth elements of the sextet VEVs, Eqs. (<ref>, <ref>, <ref>, <ref>), which correspond to the off-diagonal (1-2, 2-3) zeros present in the mass matrices, Eqs. (<ref>, <ref>), the trimaximality of V carries over to U_PMNS=V U_ν. Consider the unitary matrix,

    A≡diag(-1,1,-1).

Group transformation by A results in the multiplication of the off-diagonal (1-2, 2-3) elements of the Majorana matrix by -1. Invariance under A, implies these elements vanish and ensures trimaximality. In the literature, small groups like the Klein group <cit.> are often used to implement symmetries like the generalised CP and the trimaximality as the residual symmetries of the mass matrix. However, A is not a group member of Σ(72×3). In our model, the vanishing mass matrix elements arise as a consequence of the specific choice of the flavon potential, Eq. (<ref>), rather than the result of a residual symmetry under Σ(72×3). 

The presence of a simple set of numbers in the VEVs (and the mass matrices) is suggestive of additional symmetry transformations (like the one generated by A, Eq. (<ref>)) which are not a part of Σ(72×3). The present model only serves as a template for constructing any fully constrained Majorana mass matrix using Σ(72×3). We impose additional symmetries on the mass matrix by using flavon potentials with a carefully chosen set of parameters, Table <ref>. Realising these symmetries naturally by incorporating more group transformations along with Σ(72×3) in an expanded flavour group requires further investigation.




§ PREDICTED OBSERVABLES


For comparing our model with the neutrino oscillation experimental data, we use the global analysis done by the NuFIT group and their latest results reproduced in Eqs. (<ref>-<ref>). They are a leading group doing a comprehensive statistical data analysis based on essentially all currently available neutrino oscillation experiments. Their results are updated regularly and published on the NuFIT website <cit.>. The value sin^2 θ_13 = 2/3sin^2 π/16 = 0.02537,[Besides in Ref. <cit.>, this value was predicted in the context of Δ(6n^2) symmetry group in Ref. <cit.> and later obtained in Ref. <cit.>] is slightly more than the upper limit of the 3σ range, 0.02392. We provide a solution to this discrepancy in the following discussion.

In our previous analysis in Subsections 3.1-3.4, we used the relation U_PMNS=VU_ν where V is the left-diagonalising matrix for the charged-lepton mass matrix, Eq. (<ref>). However, the diagonalisation achieved by V is only an approximation. In Eq. (<ref>), the presence of the 𝒪(ϵ^4) element in the e_L-μ_R off-diagonal position in relation to the 𝒪(ϵ^4) electron mass and 𝒪(ϵ^2) muon mass produces an 𝒪(ϵ^2) correction to the diagonalisation, i.e. a more accurate left-diagonalisation matrix is

    (𝒪(1)    𝒪(ϵ^2)     0 
    𝒪(ϵ^2)    𝒪(1)     0
    
    0     0     1
    ).V.

The resulting correction in the e3 element of U_PMNS is

    (U_PMNS)_e3→ (U_PMNS)_e3+𝒪(ϵ^2) ( U_PMNS)_μ3.

Since (U_PMNS)_e3=sinθ_13 e^-iδ and ϵ≈m_μ/m_τ=𝒪(0.1), we obtain

    sinθ_13 e^-iδ→sinθ_13 e^-iδ + 𝒪(0.01).

The above correction is sufficient to reduce[Whether this correction has a reducing or enhancing effect on sin^2 θ_13, is determined by the relative phase between the (U_PMNS)_e3 and the correction, which in turn is determined by the phases of the elements in the mass matrix, Eq. (<ref>). For a range of values of the mass matrix elements, we have numerically verified that a reducing effect can be achieved.] our prediction for sin^2 θ_13 to within the 3σ range. 


For the solar angle, using the formula given in Table <ref>, we get

    sin^2 θ_12   = 1/3-2sin^2(π/16)
       = 0.342 .

This is within 3σ errors of the experimental values, although there is a small tension towards the upper limit. For the atmospheric angle,  predicts maximal mixing:

    sin^2 θ_23 = 1/2

which is also within 3σ errors. The NuFIT data as well as other global fits <cit.> are showing a preference for non-maximal atmospheric mixing. As a result there has been a lot of interest in the problem of octant degeneracy of θ_23 <cit.>.  predicts this non-maximal scenario of atmospheric mixing. _(ϕ=π/16) and _(ϕ=-π/16) correspond to the first and the second octant solutions respectively. Using the formula for θ_23 given in Table <ref>, we get 

    _(ϕ=+π/16):   sin^2 θ_23   = 2sin^2(2π/3+π/16)/3-2sin^2(π/16)
       = 0.387 ,
    
    
    _(ϕ=-π/16):   sin^2 θ_23   = 2sin^2(2π/3-π/16)/3-2sin^2(π/16)
       = 0.613 .

The Dirac CP phase, δ, has not been measured yet. The discovery that the reactor mixing angle is not very small has raised the possibility of a relatively earlier measurement of δ <cit.>.  having δ=±π/2 should lead to large observable CP-violating effects. Substituting χ=±π/16 in Eq. (<ref>), our model gives

    J   =±sinπ/8/6√(3)
       = ±0.0368
 
which is about 40% of the maximum value of the theoretical range, -1/6√(3)≤ J≤ +1/6√(3).
On the other hand, , with δ=0, π and J=0, is CP conserving. 

The neutrino mixing angles are fully determined by the model, Eqs. (<ref>, <ref>, <ref>, <ref>, <ref>). Hence, we simply compared the individual mixing angles with the experimental data in the earlier part of this section. Regarding the neutrino masses, the model predicts their ratios, Eq. (<ref>). To compare this result with the experimental data, which gives the mass-squared differences, Eqs. (<ref>, <ref>), we utilise a χ^2 analysis, 

    χ^2=∑_ x=Δ m^2_21, Δ m^2_31(x_model-x_expt/σ_x expt)^2.

We report that the predicted neutrino mass ratios are consistent with the experimental mass-squared differences. Using the χ^2 analysis we obtain,

    m_1=25.04^+0.17_-0.15 meV,
    
    m_2=26.50^+0.18_-0.16 meV,
    
    m_3=56.09^+0.37_-0.34 meV.

The best fit values correspond to χ^2_min=0.03 and the error ranges correspond to Δχ^2=1, where Δχ^2=χ^2-χ^2_min. The results from our analysis are also shown in Figure <ref>.

Note that the mass ratios Eq. (<ref>), are incompatible with the inverted mass hierarchy. Considerable experimental studies are being conducted to determine the mass hierarchy <cit.> and we may expect a resolution in the not-too-distant future. Observation of the inverted hierarchy will obviously rule out the model.



Cosmological observations can provide limits on the sum of the neutrino masses. The strongest such limit has been set recently by the data collected using the Planck satellite <cit.>:

    ∑_i m_i < 183 meV.

Our predictions Eqs. (<ref>), give a sum 

    ∑_i m_i = 107.6^+0.71_-0.65 meV

which is not far below the current cosmological limit. Improvements in the cosmological bounds from Planck data are expected. Future ground-based CMB polarisation experiments such as Polarbear-2 <cit.> and Square Kilometer Array-2 <cit.>, could lower the cosmological limit to below 100 meV and could also determine the mass hierarchy. Such results may support or rule out our model.

Neutrinoless double beta decay experiments seek to determine the nature of the neutrinos as Majorana or not. These experiments have so far set limits on the effective electron neutrino mass <cit.> |m_ββ|, where

    m_ββ   =m_1 U_e1^2+m_2 U_e2^2+m_3 U_e3^2
       =m_1 |U_e1|^2+m_2 |U_e2|^2 e^iα_21+m_3 |U_e3|^2 e^i(α_31-2δ)

with U representing U_PMNS. In all the four mixing scenarios predicted by the model, Eqs. (<ref>, <ref>, <ref>, <ref>), we have |U_e1|=√(2/3)cosπ/16, |U_e2|=1/√(3) and |U_e3|=√(2/3)sinπ/16. By comparing with the standard PDG parameterisation, Eq. (<ref>), we can also show that, all these scenarios lead to

    α_21=0,   α_31-2δ= π.

Therefore the model predicts

    m_ββ=2/3m_1cos^2 π/16+1/3m_2-2/3m_3sin^2 π/16.

Substituting the neutrino masses from Eqs. (<ref>) in Eq. (<ref>) we get 

    m_ββ=23.47^+0.16_-0.14 meV.

The most stringent upper bounds on the value of |m_ββ| have been set by Heidelberg-Moscow <cit.>, Cuoricino <cit.>, NEMO3 <cit.>, EXO200 <cit.> and GERDA <cit.> experiments. Combining their results leads to the bounds of the order of a few hundreds of meV <cit.>. New experiments such as CUORE <cit.>, SuperNEMO <cit.> and GERDA-2 <cit.> will improve the measurements on |m_ββ| to a few tens of meV and thus may support or rule out our model.



 §.§ Renormalisation Effects on the Observables


The see-saw mechanism requires the existence of a heavy Majorana mass term coupling the right-handed neuntrinos together. Our model, combined with the observed neutrino mass-squared differences, predicts that the neutrino masses are a few tens of meVs. This places the see-saw scale (also the flavon scale) at around 10^12 GeV. As such, this is the scale at which the fully constrained mass matrices, as proposed in our model, are generated. In order to accurately compare the the model with the observed masses and mixing parameters, it is necessary to calculate its renormalisation group (RG) evolution from the high energy scale down to the electroweak scale. 

We use the Mathematica package, REAP <cit.>, to numerically study the RG evolution of the masses and the mixing observables. The Mathematica code for calculating the RG evolution relevant to the model is given below:

Needs["REAP`RGESM`"];
RGEAdd["SM"];
RGESetInitial[10^12, 
 RGE
    Theta]12 -> 33.79 Degree, 
     RGE
        Theta]23 -> 45.00 Degree, 
         RGE
            Theta]13 -> 9.165 Degree, 
             RGE
                Delta] -> 90.00 Degree, 
                 RGEMlightest -> 0.03055, 
                 RGE
                    CapitalDelta]m2sol -> 0.0001119, 
                     RGE
                        CapitalDelta]m2atm -> 0.0037490, 
                         RGEY
                            Nu] -> .01, 0, 0, 0, .01, 0, 
                             0, 0, .01];
                            RGESolve[100, 10^12];
                            MNSParameters[RGEGetSolution[100,
                             RGEM
                                Nu]], RGEGetSolution[100,RGEYe]]
                                
                                
                                In the above code, the initial values of the mixing observables and the masses are set at 10^12 GeV. The mixing observables are chosen such that they correspond to _(χ=+π/16).  We set the masses to be 30.55 meV, 32.33 meV and 69.24 meV. These specific values are chosen such that they are consistent with Eq. (<ref>) (at 10^12 GeV) and give the best fit to the observed mass-squared differences when renormalised to the electroweak scale (100 GeV). MSNParameters in the code gives the renormalised parameters at 100 GeV as its output. Here we obtain θ_12=33.78^∘, θ_23=45.00^∘, θ_13=9.165^∘, δ=90.00^∘, m_1=24.63 meV, m_2=26.07 meV, m_3=56.16 meV [These masses correspond to χ^2_min=1.23 as calculated using Eq. (<ref>).] as the output. From these values we conclude that, under the conditions of our model, renormalisation has virtually no effect on the mixing parameters. On the other hand, it affects our predictions for the masses, Eqs. (<ref>, <ref>, <ref>), by a few percentage points.
                                
                                Analysis of RG equations <cit.> show that, even though the neutrino masses (the fermion masses in general) evolve appreciably, their ratios evolve slowly. This behaviour is sometimes referred to as "universal scaling". For our model, the light neutrino masses evolve by around 20%, while their ratios by less than 1%. This ensures that the mass ratios, Eq. (<ref>), theorised at the high energy scale remain practically valid at the electroweak scale as well. 
                                
                                
                                
                                § SUMMARY
                                
                                
                                In this paper we utilise the group Σ(72×3) to construct fully-constrained Majorana mass matrices for the neutrinos. These mass matrices reproduce the results obtained in Ref. <cit.> i.e. _(χ=±π/16) and _(ϕ=±π/16) mixings along with the neutrino mass ratios, Eq. (<ref>). The mixing observables as well as the neutrino mass ratios are shown to be consistent with the experimental data. _(χ=±π/16) and _(ϕ=±π/16) predict the Dirac CP-violating effect to be maximal (at fixed θ_13) and null respectively. Using the neutrino mass ratios in conjunction with the experimentally-observed neutrino mass-squared differences, we calculate the individual neutrino masses. We note that our predicted neutrino mass ratios are incompatible with the inverted mass hierarchy. We also predict the effective electron neutrino mass for the neutrinoless double beta decay, |m_ββ|. We briefly discuss the current status and future prospects of determining experimentally the neutrino observables leading to the confirmation or the falsification of our model. In the context of model-building, we carry out an in-depth analysis of the representations of Σ(72×3) and develop the necessary groundwork to construct the flavon potentials satisfying the Σ(72×3) flavour symmetry. In the charged-lepton sector, we use two triplet flavons with a suitably chosen set of VEVs which provide a 3× 3 trimaximal contribution towards the PMNS mixing matrix. It also explains the hierarchical structure of the charged lepton masses. In the neutrino sector, we discuss four cases of Majorana mass matrices. The Σ(72×3) sextet acts as the most general placeholder for a fully constrained Majorana mass matrix. The intended mass matrices are obtained by assigning appropriate VEVs to the sextet flavon. It should be noted that we need additional symmetries to `explain' any specific texture in the mass matrix.
                                
                                This work was supported by the UK Science and Technology Facilities Council (STFC). Two of us (RK and PFH) acknowledge the hospitality of the Centre for Fundamental Physics (CfFP) at the Rutherford Appleton Laboratory. RK acknowledges the support from the University of Warwick. RK thanks the management of the School of the Good Shepherd, Thiruvananthapuram, for providing a convenient and flexible working arrangement conducive to research.
                                
                                
                                
                                § APPENDIX A:   IRREPS OF Σ(72×3) AND THEIR TENSOR PRODUCT EXPANSIONS
                                   i)   3⊗3=6⊕   
                                The generator matrices for the triplet representation are provided in Eq. (<ref>). We define the basis for the sextet representation using Eqs. (<ref>). The resulting generator matrices are 
                                
                                    C ≡(1     0     0     0     0     0
                                    
                                    	0        0     0     0     0
                                    
                                    	0     0        0     0     0
                                    
                                    	0     0     0     1     0     0
                                    
                                    	0     0     0     0        0
                                    
                                    	0     0     0     0     0    ),    E ≡(0     1     0     0     0     0
                                    
                                    	0     0     1     0     0     0
                                    
                                    	1     0     0     0     0     0
                                    
                                    	0     0     0     0     1     0
                                    
                                    	0     0     0     0     0     1
                                    
                                    	0     0     0     1     0     0
                                    ),
                                       V≡
                                    -1/3(1     1     1    √(2)   √(2)   √(2)
                                    
                                    	1          √(2)   √(2)   √(2)
                                    
                                    	1          √(2)   √(2)   √(2)
                                    √(2)   √(2)   √(2)    -1     -1     -1
                                    √(2)   √(2)   √(2)    -1     -    -
                                    √(2)   √(2)   √(2)    -1     -    -),
                                       X≡
                                    -1/3(1     1       √(2)   √(2)   √(2)
                                    
                                    	1          √(2)   √(2)   √(2)
                                        1       √(2)   √(2)   √(2)
                                    √(2)   √(2)   √(2)    -1     -1     -
                                    √(2)   √(2)   √(2)    -1     -    -
                                    √(2)   √(2)   √(2)    -    -1     -).
                                   ii)   3⊗=1⊕8   
                                With (a_1,a_2,a_3)^T and (_1,_2,_3)^T transforming as 3 and , the tensor product expansion, Eq. (<ref>), is given by
                                
                                    1≡1/√(3)(a_1 _1 + a_2 _2+ a_3 _3),
                                       8≡(1/√(6)a_1 b̅_1 - √(2)/√(3)a_2 b̅_2+1/√(6)a_3 b̅_3
                                    1/√(2)(a_1 _1 - a_3 _3)
                                    1/√(2)(a_2 _3 + a_3 _2)
                                    1/√(2)(a_3 _1 + a_1 _3)
                                    1/√(2)(a_1 _2 + a_2 _1)
                                    i/√(2)(a_2 _3 - a_3 _2)
                                    i/√(2)(a_3 _1 - a_1 _3)
                                    i/√(2)(a_1 _2 - a_2 _1)
                                    ).
                                
                                In this basis, the generator matrices of the octet representation are 
                                
                                    C ≡1/2(2     0     0     0     0     0     0     0
                                    
                                    	0     2     0     0     0     0     0     0
                                    
                                    	0     0     -1     0     0     -√(3)    0     0
                                    
                                    	0     0     0     -1     0     0     -√(3)    0
                                    
                                    	0     0     0     0     -1     0     0     -√(3)
                                    
                                    	0     0    √(3)    0     0     -1     0     0
                                    
                                    	0     0     0    √(3)    0     0     -1     0
                                    
                                    	0     0     0     0    √(3)    0     0     -1
                                    ),
                                       E ≡1/2(-1    √(3)    0     0     0     0     0     0
                                    
                                    	-√(3)    -1     0     0     0     0     0     0
                                    
                                    	0     0     0     2     0     0     0     0
                                    
                                    	0     0     0     0     2     0     0     0
                                    
                                    	0     0     2     0     0     0     0     0
                                    
                                    	0     0     0     0     0     0     2     0
                                    
                                    	0     0     0     0     0     0     0     2
                                    
                                    	0     0     0     0     0     2     0     0
                                    ),
                                       V≡1/6(0     0    √(3)   √(3)   √(3)    3     3     3
                                    
                                    	0     0     3     3     3     - √(3)    - √(3)    - √(3)
                                    √(3)    3     4     -2     -2     0     0     0
                                    √(3)    3     -2     1     1     -2 √(3)   √(3)   √(3)
                                    √(3)    3     -2     1     1     2 √(3)    - √(3)    - √(3)
                                    
                                    	-3    √(3)    0     2 √(3)    -2 √(3)    0     0     0
                                    
                                    	-3    √(3)    0     - √(3)   √(3)    0     -3     3
                                    
                                    	-3    √(3)    0     - √(3)   √(3)    0     3     -3
                                    ),
                                       X≡1/6(0     0      -2√(3)   √(3)   √(3)    0     -3     3
                                    
                                    	0     0     0     -3     3     2√(3)    - √(3)    - √(3)
                                    √(3)    -3     1     1     4     -√(3)   √(3)    0
                                    
                                    	 -2√(3)    0     -2     -2     1     0      2√(3)   √(3)
                                    √(3)    3     -2     -2     1     -2√(3)    0     - √(3)
                                    
                                    	3    √(3)    -√(3)   √(3)    0     3     3     0
                                    
                                    	0      -2√(3)    -2√(3)    0      -√(3)    0     0     -3
                                    
                                    	-3    √(3)    0     2√(3)   √(3)    0     0     -3
                                    ).
                                
                                The octet is a real representation.
                                
                                   iii)   2⊗=6   
                                We define the basis for the doublet representation in such a way that 6 is simply the Kronecker product of 2 and , i.e.
                                
                                    6≡(a_1 _1 
                                    
                                    	a_1 _2
                                    
                                    	a_1 _3
                                    
                                    	a_2 _1
                                    
                                    	a_2 _2
                                    
                                            a_2 _3
                                    )
                                
                                where (a_1, a_2)^T and (_1, _2, _3)^T represent 2 and  respectively. In such a basis, the generator matrices for the doublet are
                                
                                    C ≡(1     0
                                    
                                    	0     1
                                    ),             E ≡(1     0
                                    
                                    	0     1
                                    ),
                                       V ≡i/√(3)(1    √(2)
                                    √(2)    -1
                                    ),    X ≡i/√(3)(1    √(2)
                                    √(2)    -1
                                    ).
                                   iv)   2⊗2=1⊕1^(0,1)⊕1^(1,0)⊕1^(1,1)   
                                The singlets 1^(p,q) transform as
                                
                                    C ≡ 1,    E ≡ 1,    V ≡ (-1)^p,    X ≡ (-1)^q.
                                
                                In terms of the tensor product expansion, Eq. (<ref>), these singlets are given by
                                
                                    1≡ a^T u   b,
                                       1^(0,1)≡ a^T u_1   b,    1^(1,0)≡ a^T u_  b,    1^(1,1)≡ a^T u_  b
                                
                                where a and b represent the doublets in Eq. (<ref>) and u, u_1, u_ and u_ are unitary matrices,
                                
                                    u=i σ_2
                                       u_1 = u V,    u_= u X,    u_= -u X̅,
                                
                                with σ_2 being the second Pauli matrix and V, X being the generators of the doublet representation, Eq (<ref>).
                                
                                    1≡1/√(2)(a_1 b_2 - a_2 b_1),
                                       1^(0,1)≡1/√(3) a_1 b_1 -1/√(6)a_2 b_1-1/√(6)a_1 b_2-1/√(3)a_2 b_2,
                                       1^(1,0)≡/√(3) a_1 b_1 -1/√(6)a_2 b_1-1/√(6)a_1 b_2-/√(3)a_2 b_2,
                                       1^(1,1)≡/√(3) a_1 b_1 -1/√(6)a_2 b_1-1/√(6)a_1 b_2-/√(3)a_2 b_2
                                
                                where (a_1, a_2)^T and (b_1, b_2)^T represent the doublets appearing in the LHS of Eq. (<ref>)
                                   v)   6⊗3=2⊕8⊕8   
                                The C-G coefficients for the above tensor product expansion are given by
                                   2≡(1/√(3)a_1 b_1 +1/√(3)a_2 b_2+1/√(3)a_3 b_3
                                1/√(3)a_4 b_1 +1/√(3)a_5 b_2+1/√(3)a_6 b_3
                                ),      8≡(
                                -1/√(2)a_1 b_1+1/√(2)a_3 b_3
                                1/√(6)a_1 b_1-√(2)/√(3)a_2 b_2+1/√(6)a_3 b_3
                                1/√(6)a_2 b_3-1/√(6)a_3 b_2+1/√(3)a_4 b_2-1/√(3)a_4 b_3
                                1/√(6)a_3 b_1-1/√(6)a_1 b_3+1/√(3)a_5 b_3-1/√(3)a_5 b_1
                                1/√(6)a_1 b_2-1/√(6)a_2 b_1+1/√(3)a_6 b_1-1/√(3)a_6 b_2
                                
                                -i/√(6)a_2 b_3-i/√(6)a_3 b_2-i/√(3)a_4 b_2-i/√(3)a_4 b_3
                                
                                -i/√(6)a_3 b_1-i/√(6)a_1 b_3-i/√(3)a_5 b_3-i/√(3)a_5 b_1
                                
                                -i/√(6)a_1 b_2-i/√(6)a_2 b_1-i/√(3)a_6 b_1-i/√(3)a_6 b_2
                                ),      8≡(
                                -1/√(2)a_4 b_1+1/√(2)a_6 b_3
                                1/√(6)a_4 b_1-√(2)/√(3)a_5 b_2+1/√(6)a_6 b_3
                                
                                -1/√(6)a_5 b_3+1/√(6)a_6 b_2-1/√(3)a_2 b_1+1/√(3)a_3 b_1
                                
                                -1/√(6)a_6 b_1+1/√(6)a_4 b_3-1/√(3)a_3 b_2+1/√(3)a_1 b_2
                                
                                -1/√(6)a_4 b_2+1/√(6)a_5 b_1-1/√(3)a_1 b_3+1/√(3)a_2 b_3
                                i/√(6)a_5 b_3+i/√(6)a_6 b_2-i/√(3)a_2 b_1-i/√(3)a_3 b_1
                                i/√(6)a_6 b_1+i/√(6)a_4 b_3-i/√(3)a_3 b_2-i/√(3)a_1 b_2
                                i/√(6)a_4 b_2+i/√(6)a_5 b_1-i/√(3)a_1 b_3-i/√(3)a_2 b_3
                                ),   
                                where (a_1, a_2, a_3, a_4, a_5, a_6)^T and (b_1, b_2, b_3)^T represent the sextet and the triplet appearing in the LHS of Eq. (<ref>).
                                
                                   vi)   6⊗=3⊕⊕3^(0,1)⊕3^(1,0)⊕3^(1,1)   
                                The representations 3^(0,1), 3^(1,0) and  3^(1,1) are simply the product of the triplet 3 and the singlets 1^(0,1), 1^(1,0) and 1^(1,1) respectively, 
                                
                                    3^(p,q)=3 1^(p,q).
                                
                                The C-G coefficients for the tensor product expansion, Eq. (<ref>), are given by
                                   3≡(1/√(2)a_1 _1 +1/2a_5 _3+1/2a_6 _2
                                1/√(2)a_2 _2 +1/2a_6 _1+1/2a_4 _3
                                1/√(2)a_3 _3 +1/2a_4 _2+1/2a_5 _1
                                ),      ≡(
                                -1/√(2)a_5 _3 +1/√(2)a_6 _2
                                
                                -1/√(2)a_6 _1 +1/√(2)a_4 _3
                                
                                -1/√(2)a_4 _2 +1/√(2)a_5 _1
                                1/√(2)a_2 _3 -1/√(2)a_3 _2
                                1/√(2)a_3 _1 -1/√(2)a_1 _3
                                1/√(2)a_1 _2 -1/√(2)a_2 _1
                                ),      3^(0,1)≡(1/√(6)a_1 _1+1/√(6)a_{2_3}+1/√(3)a_4 _1-1/2√(3)(a_5 _3+a_6 _2)
                                1/√(6)a_2 _2+1/√(6)a_{3_1}+1/√(3)a_5 _2-1/2√(3)(a_6 _1+a_4 _3)
                                1/√(6)a_3 _3+1/√(6)a_{1_2}+1/√(3)a_6 _3-1/2√(3)(a_4 _2+a_5 _1)
                                ),      3^(1,0)≡(1/√(6)a_1 _1+/√(6)a_{2_3}+/√(3)a_4 _1-1/2√(3)(a_5 _3+a_6 _2)
                                1/√(6)a_2 _2+/√(6)a_{3_1}+/√(3)a_5 _2-1/2√(3)(a_6 _1+a_4 _3)
                                1/√(6)a_3 _3+/√(6)a_{1_2}+/√(3)a_6 _3-1/2√(3)(a_4 _2+a_5 _1)
                                ),      3^(1,1)≡(1/√(6)a_1 _1+/√(6)a_{2_3}+/√(3)a_4 _1-1/2√(3)(a_5 _3+a_6 _2)
                                1/√(6)a_2 _2+/√(6)a_{3_1}+/√(3)a_5 _2-1/2√(3)(a_6 _1+a_4 _3)
                                1/√(6)a_3 _3+/√(6)a_{1_2}+/√(3)a_6 _3-1/2√(3)(a_4 _2+a_5 _1)
                                ),   
                                where (a_1, a_2, a_3, a_4, a_5, a_6)^T and (_1, _2, _3)^T represent the sextet and the conjugate triplet appearing in the LHS of Eq. (<ref>). In Eqs. (<ref>-<ref>) we have used the curly bracket to denote the symmetric sum, i.e. a_{i_j}=a_i _j + a_j _i. 
                                
                                   vii)   6⊗6=⊕⊕⊕3_symm⊕⊕3^(0,1)⊕3^(1,0)⊕3^(1,1)_antisymm   
                                Here the sextet, , appears more than once in the symmetric part. So there is no unique way of decomposing the product space into the sum of the irreducible sextets, i.e. the C-G coefficients are not uniquely defined. To solve this problem, we utilise the group Σ(216×3) which has Σ(72×3) as one of its subgroups. Σ(216×3) has three distinct types of sextets <cit.>, 6^0, 6^1, 6^2. The sextet of Σ(72×3) can be embedded in any of these three sextets of Σ(216×3). The tensor product expansion for two 6^0s of Σ(216×3) is given by 
                                
                                    6^0⊗6^0=^0⊕^1⊕^2⊕3^0_symm⊕^0⊕9_antisymm.
                                
                                In Eq. (<ref>), the decomposition of the symmetric part into the irreducible sextets is unique. Hence we embed the irreps of Σ(72×3) in the irreps of Σ(216×3),
                                
                                    [>=stealth,baseline=(current bounding box.center),anchor=base,inner sep=0pt]
                                     (foil) [matrix of math nodes,nodes=minimum height=1em] Σ(216×3)    :    6^0   ⊗   6^0   =   6̅^̅0̅   ⊕   6̅^̅1̅   ⊕   6̅^̅2̅   ⊕   3^0   ⊕   6̅^̅0̅   ⊕   9
                                     
                                    Σ(72×3)    :    6   ⊗   6   =      ⊕      ⊕      ⊕   3   ⊕      ⊕   3^(0,1)   ⊕   3^(1,0)   ⊕   3^(1,1),
                                    ;
                                    [->] ((foil-1-1.south))  edge[]     ((foil-3-1.north));
                                    [->] ((foil-1-3.south))  edge[]     ((foil-3-3.north));
                                    [->] ((foil-1-5.south))   edge[]    ((foil-3-5.north));
                                    [->] ((foil-1-7.south))  edge[]     ((foil-3-7.north));
                                    [->] ((foil-1-9.south))   edge[]    ((foil-3-9.north));
                                    [->] ((foil-1-11.south))  edge[]     ((foil-3-11.north));
                                    [->] ((foil-1-13.south))   edge[]    ((foil-3-13.north));
                                    [->] ((foil-1-15.south))  edge[]     ((foil-3-15.north));
                                    [->] ((foil-1-17.south))   edge[]    ((foil-3-17.north));
                                    [->] ((foil-1-17.south))  edge[]     ((foil-3-19.north));
                                    [->] ((foil-1-17.south))   edge[]    ((foil-3-21.north));
                                to obtain a unique decomposition for the case of Σ(72×3) as well. Thus the C-G coefficients for Eq. (<ref>) are given by
                                
                                   ≡(1/√(3)a_{2b_3}-1/√(3)a_4 b_4
                                1/√(3)a_{3b_1}-1/√(3)a_5 b_5
                                1/√(3)a_{1b_2}-1/√(3)a_6 b_6
                                1/√(6)a_{5b_6}-1/√(3)a_{1b_4}
                                1/√(6)a_{6b_4}-1/√(3)a_{2b_5}
                                1/√(6)a_{4b_5}-1/√(3)a_{3b_6}
                                ),      ≡(-1/√(3)a_1 b_1+1/√(6)a_{2b_6}+1/√(6)a_{3b_5}
                                
                                -1/√(3)a_2 b_2+1/√(6)a_{3b_4}+1/√(6)a_{1b_6}
                                
                                -1/√(3)a_3 b_3+1/√(6)a_{1b_5}+1/√(6)a_{2b_4}
                                √(2)/√(3)a_4 b_4+1/√(6)a_{2b_3}
                                √(2)/√(3)a_5 b_5+1/√(6)a_{3b_1}
                                √(2)/√(3)a_6 b_6+1/√(6)a_{1b_2}
                                ),      ≡(1/√(6)a_{1b_4}+1/√(3)a_{5b_6}
                                1/√(6)a_{2b_5}+1/√(3)a_{6b_4}
                                1/√(6)a_{3b_6}+1/√(3)a_{4b_5}
                                
                                -√(2)/√(3)a_1 b_1-1/2√(3)a_{2b_6}-1/2√(3)a_{3b_5}
                                
                                -√(2)/√(3)a_2 b_2-1/2√(3)a_{3b_4}-1/2√(3)a_{1b_6}
                                
                                -√(2)/√(3)a_3 b_3-1/2√(3)a_{1b_5}-1/2√(3)a_{2b_4}
                                ),      3≡(1/2a_{2b_6}-1/2a_{3b_5}
                                1/2a_{3b_4}-1/2a_{1b_6}
                                1/2a_{1b_5}-1/2a_{2b_4}
                                ),      ≡(1/√(2)a_[1b_4]
                                1/√(2)a_[2b_5]
                                1/√(2)a_[3b_6]
                                1/2a_[2b_6]+1/2a_[3b_5]
                                1/2a_[3b_4]+1/2a_[1b_6]
                                1/2a_[1b_5]+1/2a_[2b_4]
                                ),      3^(0,1)≡(
                                -1/√(6)a_[2b_3]+1/2√(3)a_[2b_6]-1/2√(3)a_[3b_5]+1/√(6)a_[5b_6]
                                
                                -1/√(6)a_[3b_1]+1/2√(3)a_[3b_4]-1/2√(3)a_[1b_6]+1/√(6)a_[6b_4]
                                
                                -1/√(6)a_[1b_2]+1/2√(3)a_[1b_5]-1/2√(3)a_[2b_4]+1/√(6)a_[4b_5]
                                ),      3^(1,0)≡(
                                -/√(6)a_[2b_3]+1/2√(3)a_[2b_6]-1/2√(3)a_[3b_5]+/√(6)a_[5b_6]
                                
                                -/√(6)a_[3b_1]+1/2√(3)a_[3b_4]-1/2√(3)a_[1b_6]+/√(6)a_[6b_4]
                                
                                -/√(6)a_[1b_2]+1/2√(3)a_[1b_5]-1/2√(3)a_[2b_4]+/√(6)a_[4b_5]
                                ),      3^(1,1)≡(
                                -/√(6)a_[2b_3]+1/2√(3)a_[2b_6]-1/2√(3)a_[3b_5]+/√(6)a_[5b_6]
                                
                                -/√(6)a_[3b_1]+1/2√(3)a_[3b_4]-1/2√(3)a_[1b_6]+/√(6)a_[6b_4]
                                
                                -/√(6)a_[1b_2]+1/2√(3)a_[1b_5]-1/2√(3)a_[2b_4]+/√(6)a_[4b_5]
                                )   
                                where (a_1, a_2, a_3, a_4, a_5, a_6)^T and (b_1, b_2, b_3, b_4, b_5, b_6)^T represent the sextets appearing in the LHS of Eq. (<ref>). In Eqs. (<ref>-<ref>) we have used the curly bracket and the square bracket to denote the symmetric sum and the antisymmetric sum respectively, i.e. a_{i b_j}=a_i b_j + a_j b_i and a_[i b_j]=a_i b_j - a_j b_i .
                                
                                   viii)   6⊗=1⊕8⊕1^(0,1)⊕1^(1,0)⊕1^(1,1)⊕8⊕8⊕8   
                                We are not listing the C-G coefficients for the above expansion, since they are not used in our model.
                                
                                
                                
                                
                                § APPENDIX B:    HIERARCHICAL STRUCTURE  OF  THE CHARGED-LEPTON MASS MATRIX
                                
                                
                                The triplet flavons, ϕ_α and ϕ_β, transform as 3×-i and 3× i under Σ(72×3)× C_4, Table <ref>. L^†τ_R, L^†μ_R, L^† e_R transform as 3× i, 3× 1, 3× -1 respectively. Therefore, the flavons and their tensor products which transform as  under Σ(72×3) and -i, 1, -1 under C_4 couple with L^†τ_R, L^†μ_R, L^† e_R respectively. C_4 is responsible for restricting the allowed couplings and produces the hierarchical structure of the mass matrix. In Section 3, we showed that ϕ̅_β and A̅_βα couple to the τ and μ sectors. After symmetry breaking, the flavons attain the VEVs ⟨ϕ_α⟩ =V^†(1,0,0)^Tm and ⟨ϕ_β⟩ =V^†(0,0,1)^Tm, Eq. (<ref>). The resulting tau and muon masses are of the order of ϵ and ϵ^2 respectively. The term ℋ. 𝒯. in Eq. (<ref>) contains all the higher order products of the flavons transforming as  and -i, 1, -1 coupling to τ, μ, e sectors. In this Appendix, we analyse the cubic and the quartic products which give rise 𝒪(ϵ^3) and 𝒪(ϵ^4) mass matrix elements respectively in Eq. (<ref>). We neglect the products beyond quartic order.
                                
                                
                                
                                
                                 §.§ Cubic Products
                                i)  3⊗3⊗3   =(6⊕)⊗3   
                                   =2⊕8⊕8⊕1⊕8   
                                ii)   ⊗⊗   =(⊕3)⊗   
                                   =2⊕8⊕8⊕1⊕8   
                                iii)  3⊗3⊗   =(6⊕)⊗   
                                   =3⊕⊕3^(0,1)⊕3^(1,0)⊕3^(1,1)⊕⊕3   
                                The above expansions, Eqs. (<ref>-<ref>), do not contribute to any coupling, since 3̅ does not appear in their RHS.
                                
                                iv)  ⊗⊗3   =(⊕3)⊗3   
                                   =⊕6⊕⊕⊕⊕6⊕   
                                In the above expansion,  appears twice in the RHS. In terms of the components of the triplets, these s are given by
                                
                                    1^st =   1/2√(2)(_1 (2_1 c_1+_2 c_2+_3 c_3)+_1(_2 c_2+_3 c_3),. 
                                       .       _2 (2_2 c_2+_3 c_3+_1 c_1)+_2(_3 c_3+_1 c_1),. 
                                       .       _3 (2_3 c_3+_1 c_1+_2 c_2)+_3( _1 c_1+_2 c_2))^T,
                                    
                                    
                                    2^nd =   1/2(_1 (_2 c_2+_3 c_3)-_1 (_2 c_2+_3 c_3),. 
                                       .     _2 (_3 c_3+_1 c_1)-_2 (_3 c_3+_1 c_1) ,. 
                                       .     _3 (_1 c_1+_2 c_2)-_3 (_1 c_1+_2 c_2))^T
                                
                                where (_1, _2, _3)^T, (_1, _2, _3)^T and (c_1, c_2, c_3)^T are the ,  and 3 appearing in the LHS of Eq. (<ref>). The product ⊗⊗3 can be obtained in terms of the flavons ϕ_α and ϕ_β in several different ways. These are listed in Table <ref>. For each combination of flavons, we provide the corresponding C_4 representation. Under ⟨ϕ_α⟩∝(1,0,0) and ⟨ϕ_β⟩∝(0,0,1)[For the sake of brevity, in this Appendix we omit V^†, m and the transposition when referring to the VEVs, i.e. (1,0,0)≡ V^†(1,0,0)^Tm], we calculate the vacuum alignments of the cubic 3̅s given in Eq. (<ref>) and Eq. (<ref>). These are listed in the last two columns of the table.
                                
                                
                                
                                The products transforming as i under C_4 can not couple to any of the right-handed charged leptons. On the other hand ϕ̅_αϕ̅_αϕ_β, ϕ̅_αϕ̅_βϕ_α and ϕ̅_βϕ̅_βϕ_β which transform as -i, couple with τ_R. From the table, it is clear that these products lead to non-vanishing elements in the third position only. The cubic products provide 𝒪(ϵ^3) contributions to the mass matrix. The aforementioned position corresponds to the position of the 𝒪(ϵ^3) element in the mass matrix, Eq. (<ref>).
                                
                                
                                
                                 §.§ Quartic Products
                                i)  3⊗   3⊗3⊗3   
                                   =(6⊕)⊗(6⊕)   
                                   =⊕⊕⊕3⊕⊕3^(0,1)⊕3^(1,0)⊕3^(1,1)   
                                       ⊕3⊕⊕3^(0,1)⊕3^(1,0)⊕3^(1,1)   
                                       ⊕3⊕⊕3^(0,1)⊕3^(1,0)⊕3^(1,1)   
                                       ⊕⊕3   
                                ii)  3⊗   3⊗⊗   
                                   =(6⊕)⊗(⊕3)   
                                   =1⊕8⊕1^(0,1)⊕1^(1,0)⊕1^(1,1)⊕8⊕8   
                                       ⊕2⊕8⊕8⊕2⊕8⊕8   
                                       ⊕1⊕8   
                                iii)  ⊗   ⊗3⊗   
                                   =(⊕3)⊗3⊗   
                                   =(⊕6⊕⊕⊕⊕6⊕)⊗   
                                   =⊕3⊕3⊕⊕3^(0,1)⊕3^(1,0)⊕3^(1,1)   
                                       ⊕⊕3^(0,1)⊕⊕3^(1,0)⊕⊕3^(1,1)   
                                       ⊕3⊕⊕3^(0,1)⊕3^(1,0)⊕3^(1,1)⊕⊕3   
                                The above expansions, Eqs. (<ref>-<ref>), do not contribute to any coupling, since 3̅ does not appear in their RHS.
                                
                                iv)  ⊗   ⊗⊗   
                                This tensor product corresponds to the conjugation of Eq. (<ref>). The conjugate expansion will have four s[For the sake of brevity, we do not provide the explicit expressions of these quartic s. However it is straightforward to obtain them, as was the case for the cubic s, Eqs. (<ref>, <ref>).] in the RHS. The product ⊗⊗⊗ can be obtained in terms of the flavons ϕ_α and ϕ_β in several different ways. All these are listed in Table <ref>.  For each combination of flavons, we provide the corresponding C_4 representation. Under ⟨ϕ_α⟩∝(1,0,0) and ⟨ϕ_β⟩∝(0,0,1), we calculate the vacuum alignments of the above mentioned four 3̅s and list them in the table.
                                
                                v)  3⊗   3⊗⊗3   
                                This tensor product corresponds to the conjugation of Eq. (<ref>). The conjugate expansion will have four s[We do not provide the expressions of these s also.] in the RHS. All the products of ϕ_α and ϕ_β in the form of 3⊗3⊗⊗3 are listed in Table <ref>, along with their respective C_4 representations. Under ⟨ϕ_α⟩∝(1,0,0) and ⟨ϕ_β⟩∝(0,0,1), the four s attain specific alignments which we have calculated and provided in the table.
                                
                                
                                
                                The quartic products in Tables (<ref>, <ref>) with the C_4 representations 1 and -1 couple to μ_R and e_R respectively. The VEVs with C_4≡ 1 have non-zero elements in the first, the second and the third positions while the VEVs with C_4≡ -1 have non-zero elements in the first and the third positions only. The quartic products provide 𝒪(ϵ^4) contributions to the mass matrix. The aforementioned positions correspond to the positions of the 𝒪(ϵ^4) elements in the mass matrix, Eq. (<ref>).
                                
                                
                                
                                
                                
                                § APPENDIX C:    FLAVON POTENTIALS
                                
                                
                                Here we discuss the flavon potentials that lead to the vacuum alignments assumed in our model. It should be noted that even though our construction results in the required VEVs, we are not doing an exhaustive analysis of the most general flavon potentials involving all the possible invariant terms. However, the content we include is sufficient to realise our VEVs.
                                
                                
                                
                                 §.§ The triplet flavons: ϕ_α, ϕ_β
                                
                                
                                First we consider the triplet flavons ϕ_α and ϕ_β. Our target is to obtain the VEVs, ⟨ϕ_α⟩=V(1,0,0)^Tm and ⟨ϕ_β⟩=V(0,0,1)^Tm, Eqs. (<ref>). The flavons ϕ_α and ϕ_β transform as 3, Table <ref>. The 3×3 maximal matrix V, Eqs. (<ref>), is one of the generators of 3. Therefore if the potentials of ϕ_α and ϕ_β have minima at (1,0,0)^Tm and (0,0,1)^Tm, then they have minima at V(1,0,0)^Tm and V(0,0,1)^Tm as well. The 3×3 cyclic matrix E, Eqs. (<ref>), is another generator of 3. Therefore, if the potential has a minimum at (1,0,0)^Tm, then it has a minimum at (0,0,1)^Tm also. So, for obtaining the target VEVs, all we need to do is to construct a potential with a minimum at (1,0,0)^Tm.
                                
                                An invariant term (singlet) can be constructed using the tensor product expansion of a 3 and a , Eq. (<ref>). This expansion is valid for both Σ(72×3) and SU(3).  It is well known that the singlet constructed from a triplet and its conjugate is the square of the norm of the triplet, e.g. for the flavon ϕ_α=(ϕ_α 1,ϕ_α 2,ϕ_α 3)^T, we have |ϕ_α|^2=ϕ̅_α 1ϕ_α 1+ϕ̅_α 2ϕ_α 2+ϕ̅_α 3ϕ_α 3. Next we take the symmetric part of the tensor product of two 3s to obtain a 6, similar to Eq. (<ref>),
                                
                                    S_α= (ϕ_α1^2
                                    ϕ_α2^2
                                    ϕ_α3^2
                                    √(2)ϕ_α2ϕ_α3
                                    √(2)ϕ_α3ϕ_α1
                                    √(2)ϕ_α1ϕ_α2).
                                
                                With this sextet, we may construct a singlet by combining it with its conjugate, i.e. S̅_α^T S_α. It can be shown that S̅_α^T S_α=|ϕ_α|^4. Therefore, without loss of generality we may choose,
                                
                                    T_ϕ_α=(|ϕ_α|^2-m^2)^2
                                
                                as the potential term having a minimum at (1,0,0)^Tm. T_ϕ_α is invariant not only under Σ(72×3) but also under the continuous symmetry, SU(3), and hence the minima of the potential are not discrete. This issue can be tackled in two different ways. We may add higher-order non-renormalisable terms which break SU(3) to Σ(72×3). Or we may introduce extra flavons whose sole purpose is to break SU(3) while maintaining renormalisability. In this paper we choose the later approach.
                                
                                To achieve SU(3) breaking we introduce a flavon η_α=(η_α 1,η_α 2)^T which transforms as a doublet under Σ(72×3), Eqs. (<ref>) . For η_α, we construct the following potential:
                                
                                    T_η_α=    (|η_α|^2-m^2)^2 
                                       + Re^2(η_α^Tu_1 η_α) +  Re^2( η_α^Tu_ η_α) + Re^2( η_α^Tu_ η_α),
                                
                                where Re^2 denotes the square of the real part. Since the terms  η_α^Tu_1 η_α,  η_α^Tu_ η_α  and η_α^Tu_ η_α  transform as 1^(0,1), 1^(1,0) and 1^(1,1) respectively, Eqs. (<ref>), their squares are invariants. Therefore it is evident that Re^2(η_α^Tu_1 η_α), Re^2( η_α^Tu_ η_α) and Re^2( η_α^Tu_ η_α) are also invariants. In terms of the components of η_α, the invariants in Eq. (<ref>) are given by
                                
                                    (|η_α|^2-m^2)^2   =(η̅_α 1η_α 1+η̅_α 2η_α 2-m^2)^2,
                                    Re^2(η_α^Tu_1 η_α)   =Re^2(i√(2)/√(3)(η_α1^2-√(2)η_α1η_α2-η_α2^2)),
                                    Re^2( η_α^Tu_ η_α)   =Re^2(i√(2)/√(3)(η_α1^2-√(2)η_α1η_α2-η_α2^2)),
                                    Re^2( η_α^Tu_ η_α)   =Re^2(i√(2)/√(3)(η_α1^2-√(2)η_α1η_α2-η_α2^2)).
                                
                                If we assign
                                
                                    ⟨η_α⟩ = (1,0)^Tm,
                                
                                it is evident that each of these invariants vanishes. Therefore the potential, Eq. (<ref>), attains its minimum value of zero at η_α = (1,0)^Tm (and also at the states generated by the discrete transformations on (1,0)^Tm). Note that the first term, (|η_α|^2-m^2)^2, is SU(2) invariant. The other three terms break the continuous SU(2) group and all its U(1) subgroups so that only discrete symmetries generated by Eqs. (<ref>) are present in the potential. 
                                
                                The Kronecker product of η_α (2) and ϕ̅_α (), calculated using Eq. (<ref>), leads to a sextet (6),
                                
                                    K_α= (η_α1ϕ̅_α1
                                    η_α1ϕ̅_α2
                                    η_α1ϕ̅_α3
                                    η_α2ϕ̅_α1
                                    η_α2ϕ̅_α2
                                    η_α2ϕ̅_α3).
                                
                                We utilise S_α, Eq. (<ref>), and K_α, Eq. (<ref>), to couple together the flavons η_α, ϕ_α and their conjugates and thus we construct
                                
                                    T_ϕ_αη_α=(S̅_α - K̅_α)^T(S_α - K_α)
                                
                                as an invariant[Under the group C_4, Table <ref>, ϕ_α belongs to -i. Hence η_α needs to transform as i to ensure the invariance of Eq. (<ref>)]. If we assign 
                                
                                    ⟨ϕ_α⟩ = (1,0,0)^Tm,
                                
                                its symmetric product, S_α,  becomes (1,0,0,0,0,0)^Tm^2. The Kronecker product, K_α, of ⟨η⟩=(1,0)^Tm and ⟨ϕ_α⟩=(1,0,0)^Tm also becomes (1,0,0,0,0,0)^Tm^2. Therefore, T_ϕ_αη_α vanishes (which is its minimum value) at assigned VEVs, Eqs. (<ref>, <ref>).
                                
                                Combining Eqs. (<ref>, <ref>, <ref>), we obtain the following renormalisable potential term for the flavon ϕ_α:
                                
                                    T_ϕ_α+ T_η_α+ T_ϕ_αη_α
                                
                                which is Σ(72×3) invariant and at the same time devoid of continuous symmetries. A similar potential can be constructed for the flavon ϕ_β also,
                                
                                    T_ϕ_β+ T_η_β+ T_ϕ_βη_β,
                                
                                by introducing a doublet η_β[Under the group C_4, Table <ref>, ϕ_β belongs to i. Hence η_β needs to transform as -i to ensure the invariance of T_ϕ_βη_β in Eq. (<ref>)]. The expressions for the three invariants in Eq. (<ref>) can be found by replacing α with β in Eqs. (<ref>-<ref>, <ref>, <ref>). We also write the term,
                                    T_ϕ_αϕ_β=|ϕ_α^†ϕ_β|^2,
                                
                                which couples ϕ_α and ϕ_β together and ensures that their VEVs are orthogonal to each other, Eqs. (<ref>). In conclusion, the potential,
                                
                                    T_ϕ_α+ T_ϕ_β+ T_η_α+ T_η_β+ T_ϕ_αη_α+ T_ϕ_βη_β+ T_ϕ_αϕ_β
                                
                                which is invariant under Σ(72×3), has a discrete set of minima. One among them corresponds to the required VEVs, Eqs. (<ref>). The flavons ϕ_α and ϕ_β attain these VEVs through the spontaneous symmetry breaking of Σ(72×3).
                                
                                
                                
                                 §.§ The sextet flavon: ξ
                                
                                
                                We studied the invariants that can be constructed using the sextet ξ up to the quartic order (renormalisable) and found that these terms are insufficient to obtain a potential devoid of continuous symmetries (SU(3) and its continuous subgroups). Therefore, as in Subsection <ref>, we introduce extra flavons to break the continuous symmetries and to ensure that the potential has a discrete set of minima. The extra flavons introduced here are a doublet η and two triplets ϕ_a, ϕ_b. The flavons used in the charged-lepton sector ( ϕ_α, ϕ_β, η_α, η_β in Subsection <ref>) are kept distinct from the flavons used in the neutrino sector (ξ, η, ϕ_a, ϕ_b in Subsection <ref>) in order to avoid unwanted couplings between the two sectors. Table <ref> provides the complete list of flavons in the model along with the fermions. 
                                
                                
                                Our first step is to write the potential terms for η, ϕ_a and ϕ_b, similar to Eq. (<ref>),
                                
                                    T_ϕ_a+ T_ϕ_b+ T_η+ T_ϕ_aη+ T_ϕ_bη+ T_ϕ_aϕ_b.
                                
                                The individual invariant terms in Eq. (<ref>) are
                                 T_ϕ_a   =(|ϕ_a|^2-m^2)^2,
                                 T_ϕ_b   =(|ϕ_b|^2-m^2)^2,
                                 T_η   =(|η|^2-m^2)^2 
                                     + Re^2(η^Tu_1 η) +  Re^2( η^Tu_ η) + Re^2( η^Tu_ η),
                                 T_ϕ_aη   =(S̅_a - K̅_a )^T(S_a - K_a ),
                                 T_ϕ_bη   =(S̅_b - K̅_b )^T(S_b - K_b ),
                                 T_ϕ_aϕ_b   =|ϕ_a^†ϕ_b|^2,
                                
                                where S_a, K_a and S_b, K_b are defined similar to S_α, K_α in Eqs. (<ref>, <ref>) having ϕ_α, η_α replaced with ϕ_a, η and ϕ_b, η respectively. As described earlier, it is straightforward to show that, each term in Eqs. (<ref>-<ref>) vanishes, if we assign the following VEVs:
                                
                                    ⟨η⟩ =    (1,0)^Tm,
                                    ⟨ϕ_a⟩ =    (1,0,0)^Tm,
                                    ⟨ϕ_b⟩ =    (0,0,1)^Tm.
                                S_a and S_b are the sextets constructed from ϕ_a and ϕ_b respectively. We may also construct a sextet combining ϕ_a and ϕ_b together, 
                                
                                    S_ab= (ϕ_a1ϕ_b1
                                    ϕ_a2ϕ_b2
                                    ϕ_a3ϕ_b3
                                    1/√(2)(ϕ_a2ϕ_b3 + ϕ_a3ϕ_b2)
                                    1/√(2)(ϕ_a3ϕ_b1 + ϕ_a1ϕ_b3)
                                    1/√(2)(ϕ_a1ϕ_b2 + ϕ_a2ϕ_b1)
                                    ).
                                
                                Under the VEVs, Eqs. (<ref>, <ref>), we obtain
                                ⟨ S_a⟩ =    (1,0,0,0,0,0)^Tm^2,
                                ⟨ S_b⟩ =    (0,0,1,0,0,0)^Tm^2,
                                ⟨ S_ab⟩ =    (0,0,0,0,1/√(2),0)^Tm^2.
                                
                                We use the sextet ξ=(ξ_1 ,ξ_2 ,ξ_3 ,ξ_4 ,ξ_5, ξ_6) to construct the Majorana neutrino mass matrix, Eq. (<ref>). In the VEV of ξ, if any two among the three elements ξ_4, ξ_5 and ξ_6 become zero, then two off-diagonal elements in the mass matrix vanishes. U_ν effectively becomes a 2×2 unitary matrix and U_PMNS=VU_ν attains one trimaximal column. In all the four VEVs, Eqs. (<ref>, <ref>, <ref>, <ref>), we can see that ξ_4=0 and ξ_6=0 leading to the trimaximal second column, i.e. |U_e2|=|U_μ2|=|U_τ2|=1/√(3). Both  and  belong to the larger class of mixing schemes in which one neutrino is trimaximally mixed<cit.>. As the first step in constructing the potential for ξ, we consider the tensor product of two ξs, Eq. (<ref>), and obtain a sextet,
                                
                                    X̅=(1/√(3)ξ_{2ξ_3}-1/√(3)ξ_4 ξ_4
                                    1/√(3)ξ_{3ξ_1}-1/√(3)ξ_5 ξ_5
                                    1/√(3)ξ_{1ξ_2}-1/√(3)ξ_6 b_6
                                    1/√(6)ξ_{5ξ_6}-1/√(3)ξ_{1ξ_4}
                                    1/√(6)ξ_{6ξ_4}-1/√(3)ξ_{2ξ_5}
                                    1/√(6)ξ_{4ξ_5}-1/√(3)ξ_{3ξ_6}
                                    ),
                                
                                as shown in Eq. (<ref>). X̅ transforms as a . Note that, when ξ_4=0 and ξ_6=0, the fourth and sixth elements of X̅ also vanishes.
                                
                                If the second column of U_PMNS is trimaximally mixed, then the VEV, ⟨ξ⟩, as well as the resulting ⟨X̅⟩ have non-zero elements only in the first, second, third and the fifth positions. As shown in Eqs. (<ref>, <ref>, <ref>), ⟨ S_a⟩, ⟨ S_b⟩ and ⟨ S_ab⟩ have non-zero elements only in the first, third and the fifth position respectively. Therefore, a linear combination of ⟨ξ⟩, ⟨ X_ξ⟩, ⟨ S_a⟩, ⟨ S_b⟩ and ⟨ S_ab⟩ can be constructed which fully vanishes. With this information in hand, we construct the potential term,
                                
                                    T_ξ=    (m ξ̅ + c̅_1 X̅ + c̅_2S̅_a + c̅_3S̅_b + c̅_4S̅_ab)^T 
                                               (m ξ + c_1 X + c_2 S_a + c_3 S_b + c_4 S_ab),
                                
                                where c_1, c_2, c_3 and c_4 are constants. T_ξ couples the sextet flavon, ξ with the triplet flavons, ϕ_a and ϕ_b. Any neutrino mass matrix which leads to a trimaximally-mixed column can be obtained using a potential of the form, Eq. (<ref>). The values of the constants resulting in the four VEVs, Eqs. (<ref>, <ref>, <ref>, <ref>), are given in Table <ref>. 
                                
                                Using an appropriate choice of the constants, c_1, c_2, c_3, and c_4, we may obtain any mixing scheme within the constraint of a trimaximal column. It can be shown that, having the symmetry of c_2 and c_3 being real (invariant under complex conjugation) leads to . In the case of , the symmetry is the simultaneous conjugation and interchange of c_2 and c_3. Additionally, the fact that c_1, c_2, c_3 and c_4 are related by simple ratios points to the presence of more symmetries, the study of which is beyond the scope of this paper. 
                                
                                
                                
                                With the help of first and second order partial derivatives of a given potential, its minima can be calculated, as was followed in previous works, e.g.  in Ref. <cit.>. Using such a procedure, along with numerical analysis, we have verified that every potential discussed here has a discrete set of minima and that the quoted VEVs are included among those minima in each case. 
                                
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