PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

• The LL∆ term in Eq. (2.16) is generated from the interaction 16 F 16 F 126 H . The
SU(2) Higgs triplet field ∆ is a component of the 126 representation of SO(10)
and transforms as the (1,3,1,2) representation under the left right symmetry gauge
group SU(3) C
×SU(2) L
×SU(2) R
×U(1) B−L
.
Another very well-explored unified gauge group is the SU(5), originally proposed by
Georgi-Glashow [32]. For the type-III seesaw [33] one requires SU(2) triplet Y = 0
fermion field Σ R , which can be part of a fermionic 24 F representation under the gauge
group SU(5) [34]. The 24 F adjoint representation of SU(5) has the following decomposition
under the standard model gauge group,
24 F = (1,3,0)⊕(8,1,0)⊕(3,2,−5/6)⊕(3,2,5/6)⊕(1,1,0) (2.29)
Apart from the (1,3,0) field component, the 24 F field also contains the (1,1,0) field
component. Hence, adding 24 F field to the basic SU(5) field contents 5 F , 10 F , 5 H , 24 H
and 24 V one will obtain both type-I as well as type-III seesaw.
2.3 Flavor Symmetry
While the extremely small neutrino masses can be explained by the seesaw mechanism,
the non-trivial mixing in the leptonic sector still remains a puzzle. The aesthetic believe
of unification suggests that the standard model should be embedded in a higher ranked
gauge group, for example SU(5) or SO(10). Although the quarks and leptons in a higher
ranked gauge group belong to same representation, however the mixing in the leptonic
sector is drastically different than the mixings in the quark sector. Unlike the quark
mixing matrix V CKM , the PMNS mixing matrix in the lepton sector contains two large
mixing angles θ 12 and θ 23 and one small mixing angle θ 13 . These particular mixings have
been a puzzle to the physicists and without any symmetry it is difficult to understand the
origin of these mixings. The flavor symmetry [35,36] gives natural explanation in favor of
two large mixing angles and one small mixing angles. There are many symmetry groups
like A 4 , S 3 , S 4 , T ′ which can give explanation for the PMNS mixings as well as the mass
hierarchy in the leptonic sector. Below, we discuss the group theoretical properties of two
discrete symmetry groups A 4 and S 3 which have been extensively used in the literature
in justifying the mass and mixing of the leptons. Details of the different flavor symmetry
groups and its implication in the leptonic as well as in the quark sector can be found
in [35,36].
2.3.1 The Flavor Symmetry Group A 4
Alternating group A n is a group of even permutations of n objects. It is a subgroup of
the permutation group S n and has n!
2 elements. The non-Abelian group A 4 is the first
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