Up to the present day, we are yet to understand the data accumulated for lepton, quark masses and mixing. There still remains several unanswered questions within the frame- work of the SM, few of these are listed below:
1. Why there are three generations of quarks and leptons?
2. What is the origin of fermion mass hierarchies and why neutrinos are much lighter compared to the other SM fermions?
3. Why quark mixing pattern is quite different from the lepton mixing pattern?
All these questions together is known as flavor problem in particle physics. To address the issues related to the flavor problem we must go beyond SM. For this SM symmetries
and particle content are often extended. For example, the mass hierarchies of quarks (both up and down type) and leptons can be explained once we impose an Abelian flavor symmetry Gf = U(1)FN, as suggested by Froggatt and Nielsen (FN) [82]. Unlike SM symmetries, under this symmetry, different generations of SM particles transform differ- ently. Therefore, by introducing an additional scalar (also known as ‘flavon’, which is also charged under this additional symmetry) and considering spontaneous breaking of this symmetry, mass of the heaviest particle is generated. But masses of the lighter particles are generated from higher dimensional operators proportional to hθi/Λ≡ <<1, where θ is the additional scalar introduced and Λ is the cut off scale of the theory. This process of generation of fermion mass hierarchy is called Froggatt-Nielsen mechanism. However Abelian symmetry like this suffers from few shortcomings. Firstly, they can not explain why there exits three generations. Secondly, certain mixing patterns like TBM mixing can’t be predicted.
On the other hand a non-Abelian groupGf is usually much more predictive and there exits a wide range of such symmetries. The choice of a non-Abelian group can either be continuous or a discrete one. These exits large varieties of these groups. Example for non-Abelian continuous groups are: SU(2), SU(3), SO(3) etc. whereas example for non-Abelian discrete groups are:
• Alternating groups An : A4, A5
• Symmetric groups Sn: S3, S4
• Dihedral groups Dn : D4, D5
• Binary Dihedral groups Qn : Q4, Q6
• ∆(3n2) and ∆(6n2) : ∆(27) and ∆(54)
• T0 : double covering group of A4 etc.
For a brief review and breaking patterns of all these non-Abelain discrete groups, see [83].
Compared to Abelian groups, model building with non-Abelain discrete groups are much more interesting for two reasons: (a) non-Abelain discrete groups are well equipped in dealing with the flavor structures, (b) special attention is needed in constructing the scalar potential and hence realization of proper symmetry breaking pattern (as field content is
often larger in the latter case) can be successful. Furthermore, non-Abelian discrete symmetries can be originated from profound fundamental theory such as string theory and extra dimension [84–91]. Therefore in the study of the patterns of lepton, quark mixing and masses, non-Abelian discrete flavor symmetries can play an important role in providing possible explanation for the experimental observations.
Out of all the non-Abelian discrete flavor symmetries found in the literature, due to its minimality, the groupA4 turns out to be most popular one in explaining fermion masses and mixing, particularly in lepton sector. The main advantage of usingA4 is that one can fully unify the three flavors of leptons `α = (`e, `µ, `τ)T ∼ 3, as it is the smallest group which contains 3-dimensional representation. In this thesis, we have frequently used this symmetry to explain the observed lepton mixing. In the next subsection we now present a discussion on its representations and multiplication rules.
1.6.1 The A4 Group
Non-Abelian discrete group A4 is a group of even permutation of four objects. There are 4!/2 = 12 elements in this group. It is symmetry group of a tetrahedron. Geometri- cally, these twelve elements are: one unit operator, three rotations by 180◦, four clockwise rotations by 120◦ and four anti-clockwise rotations by 120◦. The four irreducible repre- sentations of A4 are 1 (one trivial singlet) ,10,100 (two non-trivial one-dimensional repre- sentations are conjugate to each other) and 3 (triplet representation). Two generators, known as S and T can form all the 12 elements (through multiplications in all possible way) which satisfy
S2=T3 = (ST)2 = 1. (1.54)
This relation dictates the ‘presentation’ of the group. Therefore, the three one dimensional representations are given by
1 → (S = 1, T = 1), (1.55)
10 → (S = 1, T =ω), (1.56)
100 → (S = 1, T =ω2), (1.57)
where ω = e2πi/3 is cubic root of unity. Now the multiplication rule for the singlets can be written as
1×1 = 1,1×10 = 10,1×100 = 100, (1.58) 10×10= 100,10×100= 1 and 100×100= 10. (1.59) Now the basis for three dimensional representation can be written as
S= 1 3
−1 2 2 2 −1 2
2 2 −1
, T =
1 0 0
0 ω 0
0 0 ω2
, (1.60)
from which again all 12 matrices of the three dimensional representation of A4 can be obtained and the product rule involving singlet and triplets are
1×3 = 3,10×3 = 3,100×3 = 3, (1.61) 3×3 = 1 + 10+ 100+ 3S+ 3A, (1.62) where subscripts S and A stands for ‘symmetric’ and ‘asymmetric’ combinations respec- tively. If we have two triplets having elements a = (a1, a2, a3) and b = (b1, b2, b3), their products are given by
(ab)1 = a1b1+a2b3+a3b2, (1.63) (ab)10 = a3b3+a1b2+a2b1, (1.64) (ab)100 = a2b2+a3b1+a1b3, (1.65) (ab)3S = 1
3
2a1b1−a2b3−a3b2
2a3b3−a1b2−a2b1 2a2b2−a1b3−a3b1
, (1.66)
(ab)3A = 1 2
a2b3−a3b2
a1b2−a2b1 a3b1−a1b3
. (1.67)
Lastly, if c,dand eare three one dimensional representation transforming as 1,10 and 100 respectively, their products with a triplet a= (a1, a2, a3) are given by
(ca)3 =
ca1 ca2
ca3
, (da)3=
da3 da1
da2
(ea)3 =
ea2 ea3
ea1
. (1.68)
Now A4 has two subgroups GS and GT. GS is the reflection subgroup generated by S whereas GT (isomorphic to Z3) is generated by T. Now when the A4 flavor symmetry is broken by vev of a triplet scalar field, φ = (φ1, φ2, φ3) one can obtain two important breaking pattern hφi = (vs, vs, vs) (when A4 breaks down to GS) and hφi = (vT,0,0) (when A4 breaks down to GT). This symmetry breaking patterns plays a crucial role in obtaining fermion mass matrices.
As an illustrative purpose, here we show howA4can be responsible for flavor structures of the mass matrices. Let us consider the charged lepton mass matrix is being generated from a higher dimensional operator involving a scalar fieldφT (triplet underA4but singlet under SM gauge symmetry) having vev alignment hφTi = (vT,0,0). Now the three SM lepton doublets (Le, Lµ, Lτ) can form a triplet under A4 as L = (Le, Lµ, Lτ) and we consider RH charged leptons eR, µR, τR are to be 1,100 and 10 under A4. Hence relevant Lagrangian can be written as
Ll= ye
Λ( ¯LφT)HeR+ yµ
Λ( ¯LφT)0HµR+yτ
Λ( ¯LφT)00HτR, (1.69) where we consider the SM Higgs doublet is a trivial singlet under A4. Here Λ is the cutoff scale and ye, yµ and yτ are the respective coupling constants. Terms in the first parenthesis represent products of twoA4triplets, which further contracts withA4 singlets 1, 100 and 10 corresponding to eR, µR and τR respectively to make a true singlet underA4. After breaking of A4 symmetry in the direction hφTi = (vT,0,0), the lagrangian can be written as
L0l= vT
Λ yeL¯eHeR+yµL¯µHµR+yτL¯τHτR
. (1.70)
Finally after breaking of the electoweak symmetry we obtain the charged lepton mass matrix as
Ml =vvT Λ
ye 0 0 0 yµ 0 0 0 yτ
. (1.71)