Lepton Family Symmetry and Neutrino Mass Matrix

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by  Ernest Ma

UCRHEP-T368
January 2004
Lepton Family Symmetry
4 and Neutrino Mass Matrix
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Ernest Ma
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Physics Department, University of California, Riverside, California 92521
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Abstract
The standard model of leptons is extended to accommodate a discrete Z Z
3 2
×
family symmetry. After rotating the charged-lepton mass matrix to its diagonal form,
the neutrino mass matrix reveals itself as very suitable for explaining atmospheric
and solar neutrino oscillation data. A generic requirement of this approach is the
appearance of three Higgs doublets at the electroweak scale, with observable flavor
violating decays.
In the standard model of particle interactions, the fermion mass matrices are a priori
completely arbitrary, and yet data seem to indicate specific patterns which are not yet un-
derstood theoretically. Recent experimental advances in measuring the neutrino-oscillation
parameters in atmospheric [1] and solar [2] data have now fixed the 3 3 lepton mixing
×
matrix to a large extent. This information is not sufficient to fix all the elements of the
neutrino mass matrix (assumed here to be Majorana at the outset), but is enough to fix
ν
M
its approximate form in terms of a small number of parameters [3]. However, this works only
in the basis (ν ,ν ,ν ), i.e. where the charged-lepton mass matrix has been assumed
e µ τ l
M
diagonal.
If a symmetry is behind the observed pattern of , then it must also apply to . The
ν l
M M
realization of this in a complete theory of leptons has only been done in a few cases. Some
recent examples are those based on the symmetries of geometric objects, i.e. S (equilateral
3
triangle) [4], D (square) [5], and A (tetrahedron) [6, 7]. In this paper, a much simpler and
4 4
more flexible model is proposed, based on the discrete symmetry Z Z . [Because of the
3 2
×
choice of representations, this turns out to be equivalent to a specific realization of S .]
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The key idea which allows to be more restricted than is that they come from
ν l
M M
different mechanisms. Whereas the former comes from the naturally small vacuum expecta-
tion value (vev) of a single heavy Higgs triplet ξ = (ξ++,ξ+,ξ0) [8], thus dispensing with the
usual three heavy singlet neutrinos, the latter comes from the vev’s of three Higgs doublets
[9, 10]. As shown below, the use of Z Z as a lepton family symmetry results in 2 pa-
3 2
×
rameters for and 5 parameters for †. After rotating † to its diagonal form,
Mν MlMl MlMl
i.e. (m2,m2,m2), then under a condition to be derived below, the neutrino mass matrix will
e µ τ
become [10]
A B 0 0
−
ν =  0 A B, (1)
M −
 0 B A 
 − 
 
2
which is very suitable as a starting point for explaining atmospheric and solar neutrino
oscillations. The addition of a charged Higgs singlet χ+ and the soft breaking of Z Z
3 2
×
will then lead to a complete satisfactory description of all data, including the possibility of
CP violation, i.e a nonzero complex U .
e3
The representations of Z are denoted by 1, ω, ω2, where
3
1 3
ω = e2πi/3 = +i , (2)
−2 s2
with 1 + ω + ω2 = 0. Let the standard model be augmented with a Higgs triplet ξ =
(ξ++,ξ+,ξ0), three doublets φ = (φ0,φ−) and a charged singlet χ+, in addition to the usual
i i i
three lepton doublets (ν ,l ) and three singlets lc. Under Z , let
i i i 3
l , lc, φ , ξ, χ 1, (3)
1 1 1
∼
l , lc, φ ω, (4)
2 2 2
∼
l , lc, φ ω2. (5)
3 3 3
∼
Under Z , let
2
l l , lc lc, φ φ , χ χ. (6)
2 3 2 3 2 3
↔ ↔ ↔ ↔ −
Then the Yukawa couplings of
ν l +l ν
0 i j i j + ++
ν ν ξ ξ +l l ξ (7)
i j i j
− √2 !
imply that is of the form [11]
ν
M
a 0 0
ν = 0 0 b. (8)
M
0 b 0
 
 
Note that is proportional to ξ0 which can be naturally small if m2 is positive and
Mν h i ξ
large. On the other hand, the Yukawa couplings of
lc(l φ0 ν φ−) (9)
i j k − j k
3
imply that the mass matrix linking l to lc is given by
c f f
l = g d e, (10)
M
g e d
 
 
where v = v has been assumed for φ0 . [This assumption will be relaxed later to accom-
2 3 h ki
modate the soft breaking of Z . The role of χ+ will also be explained.]
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To rotate to its diagonal form, i.e.
l
M
m 0 0
e
VLMlVR† =  0 mµ 0 , (11)
 0 0 mτ 
 
 
consider
C F F
MlM†l = F∗ D E , (12)
F∗ E D
 
 
where
2 2
C = c +2 f , (13)
| | | |
2 2 2
D = d + e + g , (14)
| | | | | |
E = de∗ +ed∗ + g 2, (15)
| |
F = cg∗+f(d∗ +e∗). (16)
Then
m2 0 0
e
VLMlM†lVL† =  0 m2µ 0  (17)
 0 0 m2
 τ 
 
and the neutrino mass matrix in the basis (ν ,ν ,ν ) is given by
e µ τ
a 0 0
Mν = VL0 0 bVLT. (18)
0 b 0
 
 
As a trial, let
0 i/√2 i/√2
−
VL =  1/√2 1/2 1/2 , (19)
−
 1/√2 1/2 1/2 
 
 
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then V †V† becomes
LMlMl L
D E 0 0
−
 0 (C +D +E)/2 √2ReF (D +E C)/2 i√2ImF . (20)
− − −
 0 (D +E C)/2+i√2ImF (C +D +E)/2+√2ReF 
 − 
 
Comparing this with Eq. (17), it is clear that under the condition
C = D +E, ImF = 0, (21)
the charged-lepton matrix is diagonalized by the V of Eq. (19) with
L
2
m = D E, (22)
e −
m2 = (C +D +E)/2 √2ReF, (23)
µ −
m2 = (C +D +E)/2+√2ReF, (24)
τ
and in this basis, the neutrino mass matrix of Eq. (18) is given by
b 0 0
ν = 0 (b+a)/2 (b a)/2, (25)
M −
0 (b a)/2 (b+a)/2
 − 
 
which is identical to Eq. (1) with the substitution b = A B and a = A+B. The eigenvalues
−
and eigenvectors of are then
ν
M
b : ν , (26)
e
b : (ν +ν )/√2, (27)
µ τ
a : ( ν +ν )/√2. (28)
µ τ
−
This explains atmospheric neutrino oscillations with
2 2 2 2
∆m = a b , sin 2θ = 1, (29)
atm − atm
but ∆m2 = 0, which is nevertheless a goodfirst approximation. To split the two degenerate
sol
neutrino masses responsible for solar neutrino oscillations, consider the Yukawa coupling
(ν l l ν )χ+ = i[ν (µ+τ)/√2 e(ν +ν )/√2], (30)
2 3 2 3 e µ τ
− − −
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which is the only one allowed under Z Z . In addition, the most general trilinear scalar
3 2
×
couplings
χ+(φ0φ− φ−φ0) (31)
i j − i j
are assumed, thus breaking Z Z but only softly. As a result, there are one-loop contri-
3 2
×
butions to as shown in Fig. 1, and three nonzero parameters will emerge, i.e. ∆m2 ,
Mν sol
tan2θ , and U .
sol e3
×
χ+ φ−
ν l × lc ν
Figure 1: One-loop contributions to the neutrino mass matrix.
In the original ν basis, i.e. that of Eq. (8), the radiative corrections of Fig. 1 have the
1,2,3
form
0 r s
rad = r p t. (32)
M
s t q
 
 
Rotating to the basis ν ,(ν + ν )/√2,( ν + ν )/√2 and using Eqs. (26) to (28), the
e µ τ µ τ
{ − }
neutrino mass matrix is then given by
b+t (p+q)/2 i(q p)/2 i(s r)/√2
− − −
ν =  i(q p)/2 b+t+(p+q)/2 (r +s)/√2 . (33)
M −
 i(s r)/√2 (r +s)/√2 a 
 − 
 
The two-fold degeneracy of the solar neutrino doublet is now lifted with
2
∆m = 4(b+t)√pq, (34)
sol
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and
1 x 2 q
2
tan θ = − , x = . (35)
sol
1+x p
(cid:18) (cid:19) s
For x 0.22, tan2θ 0.41, as indicated [12] by most recent data. A nonzero
sol
≃ ≃
i(r s)
U − (36)
e3
≃ √2(a b t)
− −
is also obtained. Note that the phase of U is undetermined because r s is in general
e3
−
complex. ThesofttermsofEq.(31)breakZ ,hencetheconditionv = v assumedpreviously
2 2 3
should also berelaxed, in which case the zero entries ofEq. (20)would become nonzero. This
means another contribution to U . The condition of Eq. (21) may also be relaxed to account
e3
for any possible deviation from maximum mixing in atmospheric neutrino data.
Therepresentationcontent ofleptonfamiliesandHiggsmultipletsunderZ Z proposed
3 2
×
here is such that the resulting model is equivalent to the following specific realization of
S . There are 3 irreducible representations of S : 1+, 1−, 2. If 1+ and 2 are chosen as
3 3
representations here with 1+ as 1 and 2 as (ω,ω2) under Z , together with Z of Eq. (6),
3 2
then the group multiplication rules of the two are the same. For example, the S invariant
3
2 2 2 1 is given by (1,2) (1,2) (1,2) = 111+222, whereas the Z Z analog is
3 2
× × → × × ×
(ω,ω2) (ω,ω2) (ω,ω2) = ω3 +ω6. It can also be checked easily that of Eq. (8) has
ν
× × M
2 invariants and of Eq. (10) has 5 invariants in both cases as expected.
l
M
Diagonalizing Eq. (12) with V of Eq. (19) under the condition of Eq. (21) leads to the
L
neutrino mass matrix of Eq. (25) which yields the eigenvalues and eigenvectors of Eqs. (26)
to (28). Adding the one-loop radiative corrections induced by the interactions of Eqs. (30)
and (31), the full of Eq. (33) is then obtained. It is very suitable for explaining present
ν
M
data on atmospheric and solar neutrino oscillations, with sin22θ 1, tan2θ 0.4, and
atm sol
≃ ≃
U small but nonzero. It also has the flexibility to allow for either the normal hierarchy
e3
m < m < m or the inverted hierarchy m < m < m of neutrino masses.
1 2 3 3 1 2
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In this model, lepton masses come from three different sources in the Higgs sector:
charged-lepton Dirac masses come from three Higgs doublets, neutrino Majorana masses
come from a Higgs triplet at tree level, and the interactions of a charged Higgs singlet with
the already present Higgs doublets at the one-loop level. Singlet (right-handed) neutrinos
are not needed.
To have a naturally small ξ0 , m2 should be positive and large [8], which means that
h i ξ
the interactions of ξ are not observable at the electroweak scale. On the other hand, the
three Higgs doublets (φ0,φ−) and the one Higgs singlet χ+ should have masses below the
i i
TeV scale, resulting in observable phenomena at future colliders. The typical decay of any
one of the three physical charged scalars is into a charged lepton and a neutrino, with large
violations of lepton family universality. There are also five neutral scalars, each decaying
into a pair of charged leptons l−l+. Since i = j is allowed, there should be many distinct
i j 6
observable experimental signatures.
I thank M. Frigerio for discussions. This work was supported in part by the U. S. De-
partment of Energy under Grant No. DE-FG03-94ER40837.
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