PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

where
Σ ± Ri = Σ1 Ri ∓iΣ2 Ri
√ and Σ 0 Ri = 2 Σ3 Ri
(3.5)
are the components of the triplet in the charge eigenbasis. The corresponding chargeconjugated
multiplet will then be
Σ R
C
(
T Σ
0 C
i = CΣ R = Ri / √ 2
C
Σ − Ri
Σ + C
Ri
Σ + C
Ri
−Σ 0 RiC / √ 2
−Σ 0 RiC / √ 2
)
. (3.6)
The object which transforms as the left-handed component of the Σ multiplet can then
be written as
(
˜Σ C C Σ
0 C
R i
= iσ 2 Σ Ri iσ 2 = Ri / √ 2 Σ − C )
Ri
, (3.7)
such that Σ i = Σ Ri + ˜Σ C R i
, where
(
Σ
Σ =
0 / √ )
2 Σ +
Σ − −Σ 0 / √ . (3.8)
2
For notational convenience, in this chapter we have changed our notations and we denote
the standard model Higgs as Φ 1 . In addition to the usual standard model doublet Φ 1 ,
in our model we include a new SU(2) scalar doublet Φ 2 . This new doublet couples only
to the triplet fermions introduced above. The triplet fermions on the other hand are
restricted to couple with only the new Φ 2 doublet and not with Φ 1 . This can be ensured
very easily by giving Z 2 charge of −1 to the triplet fermions Σ i and the scalar doublet Φ 2 ,
and Z 2 charge +1 to all standard model particles. In this model we work with a mildly
broken Z 2 symmetry. We will break this Z 2 symmetry mildly in the scalar potential. We
discuss the phenomenological consequences of this Z 2 symmetry and its breaking when we
introduce the scalar potential and present the Higgs mass spectrum in Appendix A. The
part of the Lagrangian responsible for the lepton masses which respects the Z 2 symmetry
can be written as
−L Y =
[
]
Y lij l Ri Φ † 1 L j +Y Σij˜Φ† 2 Σ R i
L j +h.c.
+ 1 [ ]
2 M ijTr Σ Ri˜ΣC Rj
+h.c. , (3.9)
where L and l R are the usual left-handed lepton doublet and right-handed charged leptons
respectively, Y l and Y Σ are the 3×3 Yukawa coupling matrices, and ˜Φ 2 = iσ 2 Φ ∗ 2. Once
the Higgs doublets Φ 1 and Φ 2 take Vacuum Expectation Value (VEV)
( ( )
0 0
〈Φ 1 〉 = , 〈Φ
v)
2 〉 =
v ′ , (3.10)
38