PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

triplet representation of A 4 ,
l = (L e ,L µ ,L τ ) T ,
where L α denotes the standard model lepton doublets. The right handed charged leptons
e R , µ R and τ R are assumed to belong to the 1, 1 ′ and 1 ′′ representation respectively. The
standard Higgs doublets H u and H d remain invariant under A 4 . The form of the A 4
invariant Yukawa part of the Lagrangian is
L Y = y e e R (φ T l)+y µ µ R (φ T l) ′ +y τ τ R (φ T l) ′′ +x a ξ(ll)+x ′ a ξ′ (ll) ′′ +x ′′
a ξ′′ (ll) ′
+x b (φ S ll)+h.c.+... (5.3)
where, following [14] we have used the compact notation, y e e R (φ T l) ≡ y e e R (φ T l)H d /Λ,
x a ξ(ll) ≡ x a ξ(lH u lH u )/Λ 2 and so on, and Λ is the cut-off scale of the theory. Here we
adopt an effective field theory approach. We assume that φ S does not couple to charged
leptons and φ T does not contribute to the Majorana mass matrix. These two additional
features can be obtained by introducing extra abelian symmetries for example Z 3 [14].
After the spontaneous breaking of A 4 followed by SU(2) L
×U(1) Y
, we get the mass terms
for the charged leptons and neutrinos. Assuming the vacuum alignment
the charged lepton mass matrix is given as
⎛
m cl = v dv T
Λ
〈φ T 〉 = (v T ,0,0) , (5.4)
⎝
⎞
y e 0 0
0 y µ 0 ⎠ , (5.5)
0 0 y τ
Note that we could also obtain a diagonal charged lepton mass matrix even if we assume
that e R , µ R and τ R transform as 1 ′′ , 1 ′ and 1, and 〈φ T 〉 = (0,v T ,0) with appropriate
change in the Yukawa Lagrangian. Similarly, e R , µ R and τ R transforming 1 ′ , 1 and 1 ′′ ,
and 〈φ T 〉 = (0,0,v T ) could give us the same m cl .
In the most general case, where all three one dimensional A 4 Higgs as well as φ S
are present and we do not assume any particular vacuum alignment, the neutrino mass
matrix looks like
⎛
m ν = m 0
⎝ a+2b ⎞
1/3 c−b 3 /3 d−b 2 /3
c−b 3 /3 d+2b 2 /3 a−b 1 /3 ⎠ , (5.6)
d−b 2 /3 a−b 1 /3 c+2b 3 /3
where m 0 = v2 v
u
Λ
b i = 2x
Si
b , a = 2x Λ a u, c = Λ 2x′′ a u′′ and d = Λ 2x′ a u′
Λ
VEVs as
and we have written the
〈φ S 〉 = (v S1 ,v S2 ,v S3 ), 〈ξ〉 = u, 〈ξ ′ 〉 = u ′ , 〈ξ ′′ 〉 = u ′′ , 〈H u,d 〉 = v u,d . (5.7)
For simplicity, in this work we have considered all the Yukawa couplings as well as the
parameters a,b,c and d as real.
122