Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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96 3 Neutrinos beyond the
S
tandard Mode
l
a
n
d
t
h
e present Borexino experiment
y
ie
ld
s
μ
ν
<
5
.
4
×
10
−
11
μ
B
(
Arpesell
a
et al., 2008b
)
. More stringent bounds on
μ
ν
can hopefull
y
be achieved fro
m
the future experiments of these types
(
Giunti and Studenikin, 2009
)
.
I
n view o
f
current experimental data on neutrino oscillations, we know
that neutrinos are actuall
y
massive. Hence the effects of finite neutrino masses
a
nd flavor mixing should be taken into account in calculating the cross sectio
n
of elastic neutrino-electron or antineutrino-electron scattering
(
Grimus an
d
Stockinger, 1998; Beacom and Vogel, 1999
)
. Here let us illustrate how th
e
n
eutrino oscillation may affect the weak and electromagnetic terms of elastic
ν
e
e
−
s
catterin
g
in a reactor experiment, w
h
ere t
h
ee
l
ectron antineutrinos ar
e
produced from the beta deca
y
of fission products and detected b
y
their elasti
c
s
catterin
g
wit
h
e
l
ectrons in a
d
etector. T
h
e antineutrino state create
d
in t
h
is
beta decay
(
vi
a
W
−
→
e
−
+
ν
e
)
at the reactor is a superposition of thre
e
a
ntineutrino mass eigenstates
:
|
ν
e
(
0
)
=
3
j
=
1
V
ej
VV
|
ν
j
,
(
3.97
)
where
V
i
sthe
3
×
3
neutrino mixing matrix. Such a
ν
e
b
eam propagates
ove
r
the d
i
sta
n
ce
L
t
ot
h
e
d
etector
,
|
ν
e
(
L
)
=
3
j
=
1
e
i
q
j
L
V
ej
VV
|
ν
j
,
(
3.98
)
i
n
wh
i
ch
q
j
q
=
E
2
ν
−
m
2
j
i
st
h
e
m
o
m
e
n
tu
m
of
ν
j
ν
w
i
th
E
ν
b
ein
g
t
h
e
b
ea
m
ener
g
yand
m
j
bein
g
the mass o
f
ν
j
ν
.A
fter takin
g
account of the effec
t
of neutrino oscillations
,
one obtains the differential cross section of elasti
c
a
ntineutrino-electron scattering as follows
(
Grimus and Stockinger, 1998
)
:
d
σ
d
T
=
d
σ
0
d
T
+
d
σ
μ
d
T
,
(
3.99
)
whe
r
e
d
σ
0
d
T
=
G
2
F
m
e
2
π
2
g
2
−
+
g
−
−
1
2
1
−
T
E
ν
2
−
g
−
g
−
−
1
m
e
T
E
2
ν
+2
g
−
3
j
=
1
e
i
q
j
L
|
V
ej
VV
|
2
2
2
1
−
T
E
ν
2
−
m
e
T
E
2
ν
⎫
⎪
⎫⎫
⎬
⎪⎪
⎪
⎬⎬
⎭
⎪⎪
(
3.100
)
wit
h
g
−
=2s
i
n
2
θ
w
for
ν
e
,
an
d
d
σ
μ
dT
=
α
2
em
π
m
2
e
3
k
=
1
3
j
=1
e
i
q
j
L
V
ej
VV
i
μ
jk
μ
B
+
jk
μ
B
2
1
T
−
1
E
ν
(
3.101
)
3.5 Lepton Flavor Mixing and
C
P Violation 97
w
i
th
μ
jk
a
n
d
jk
b
ein
g
t
h
ema
g
netic an
d
e
l
ectric transition
d
ipo
l
emoments
betwee
n
ν
j
ν
an
d
ν
k
n
eutrinos as defined in Eq.
(
3.85
)
. Because different neu
-
trino mass eigenstates are in princip
l
e
d
istinguis
h
a
bl
eint
h
ee
l
ectromagneti
c
ν
e
e
−
scatterin
g
,t
h
eir contri
b
utions to t
h
e tota
l
cross section are inco
h
erent
.
E
q.
(
3.101
)
shows that it is in general difficult to determine or constrain the
m
agnitudes o
f
μ
jk
a
n
d
jk
(
fo
r
j
,
k
=
1
,
2
,
3
)
from a single measurement. Bu
t
i
t is possible to simplify the above expression for
d
σ
μ
d
T
i
n a few special case
s
(
Beacom and Vogel, 1999
)
.
3
.5 Lepton Flavor Mixing and CP Violatio
n
R
e
g
ardless o
f
the ori
g
in o
f
tiny neutrino masses
8
,wema
y
discuss lepton
fl
avor
m
ixing by taking account of the effective mass terms of charged leptons an
d
Majorana neutrinos at
l
ow energie
s
9
,
−L
l
epton
=
(
eμ
τ
)
L
M
l
M
⎛
⎝
⎛⎛
e
μ
τ
⎞
⎠
⎞⎞
R
+
1
2
ν
e
ν
μ
ν
τ
L
M
ν
MM
⎛
⎝
⎛⎛
ν
c
e
ν
c
μ
ν
c
τ
⎞
⎠
⎞⎞
R
+h
.
c
..
(
3.102
)
The phenomenon of lepton flavor mixing arises from a mismatch betwee
n
the dia
g
onalizations o
f
M
l
M
a
n
d
M
ν
MM
i
n an arbitrar
yfl
avor basis:
V
†
l
VV
M
l
M
U
l
U
=
D
ia
g
{
m
e
,μ
μ
,m
τ
}
a
n
d
V
†
ν
VV
M
ν
MM
V
∗
ν
VV
=
Dia
g
{
m
1
,m
2
,
m
3
}
,
wher
e
V
l
VV
,
U
l
U
a
n
d
V
ν
VV
a
re t
h
e
3
×
3 unitary matrices. In the basis of mass eigenstates, it is the unitar
y
mat
r
ix
V
=
V
†
l
VV
V
ν
VV
t
h
at wi
ll
appear in t
h
ewea
k
c
h
ar
g
e
d
-current interactions
i
nEq.
(
3.9
)
. Although the basis o
f
M
l
M
=Diag
{
m
e
,
m
μ
,m
τ
}
with
V
l
VV
=
1
a
nd
V
=
V
ν
VV
is often chosen in neutrino phenomenology, one should keep in min
d
t
h
at
b
ot
h
t
h
ec
h
ar
g
e
d
-
l
epton an
d
neutrino sectors may in
g
enera
l
contri
b
ute
to lepton flavor mixing. In other words, bot
h
V
l
V
V
and
V
ν
V
V
are not full
y
ph
y
sical,
a
n
d
on
l
yt
h
eir pro
d
uc
t
V
=
V
†
l
VV
V
ν
VV
i
s a physical description of lepton flavo
r
m
ixin
g
and
C
P violation at low ener
g
ies.
3
.
5
.1
C
lassification of Different Parametrizations
F
lavor mixing among
n
d
ifferent lepton families can be described b
y
an
n
×
n
u
nitary matrix
V
,
whose number of independent parameters relies on the
n
ature o
f
neutrinos. I
f
neutrinos are Dirac particles, one ma
y
make use o
f
n
(
n
−
1
)
/
2
rotation angles and
(
n
−
1)(
n
−
2
)
/
2
phase angles to parametrize
8
F
or simplicity, here we do not consider possible non-unitarity of the
3
×
3
neutrino mixin
g
matrix because its e
ff
ects are either absent or very small
.
9
A
s for Dirac neutrinos, the corresponding mass term is the same as that given in
Eq.
(
3.11
)
. In this case the neutrino mass matri
x
M
ν
M
M
is in genera
l
not symmetric an
d
can be dia
g
onalized by means o
f
the trans
f
ormation
V
†
ν
V
V
M
ν
M
M
U
ν
UU
=
D
ia
g
{
m
1
,m
2
,m
3
}
,
where bot
h
V
ν
VV
a
nd
U
ν
UU
a
re unitary.
98 3 Neutrinos beyond the
S
tandard Mode
l
V
.I
f
neutrinos are Ma
j
orana particles, however, a
f
ull parametrization o
f
V
n
eeds
n
(
n
−
1
)
/
2
rotation an
g
les and the same number of phase an
g
les
10
.
The flavor mixing between charged leptons and Dirac neutrinos is completel
y
a
nalo
g
ous to that o
f
quarks,
f
or which a number o
f
di
ff
erent parametrization
s
h
ave been proposed and classified in the literature
(
Fritzsch and Xing, 1998a
)
.
H
ere we classify all possible parametrizations for the flavor mixing betwee
n
ch
ar
g
e
dl
eptons an
d
Majorana neutrinos wit
h
n
=
3. Re
g
ardless o
f
the
f
ree
-
d
om of phase reassignments, we find that there are nine structurally different
parametrizations for the 3 ×
3
lepton flavor mixing matri
x
V
.
Th
e3
×
3
lepton
fl
avor mixin
g
matri
x
V
,whichiso
f
ten called the Maki-
N
akagawa-Sakata
(
MNS
)
matrix
(
Mak
i
e
tal
.
,
1962
)
, can be expressed as a
product o
f
three unitary matrice
s
O
1
,
O
2
a
n
d
O
3
.T
h
ey correspon
d
to simp
l
e
rotations in the complex
(
1,2
)
,
(
2,3
)
and
(
3,1
)
planes
:
O
1
(
θ
1
,
α
1
,
β
1
,γ
1
)=
⎛
⎝
⎛
⎛
c
1
e
i
α
1
s
1
e
−
i
β
1
0
−
s
1
e
i
β
1
c
1
e
−
i
α
1
0
00e
i
γ
1
⎞
⎠
⎞
⎞
,
O
2
(
θ
2
,α
2
,β
2
,γ
2
)=
⎛
⎝
⎛⎛
e
i
γ
2
0
0
0
c
2
e
i
α
2
s
2
e
−
i
β
2
0
−
s
2
e
i
β
2
c
2
e
−
i
α
2
⎞
⎠
⎞
⎞
,
O
3
(
θ
3
,α
3
,
β
3
,γ
3
)=
⎛
⎝
⎛⎛
c
3
e
i
α
3
0
s
3
e
−
i
β
3
0
e
i
γ
3
0
−
s
3
e
i
β
3
0
c
3
e
−
i
α
3
⎞
⎠
⎞
⎞
,
(
3.103
)
whe
r
e
s
i
≡
s
in
θ
i
a
n
d
c
i
≡
cos
θ
i
(
for
i
=1
,
2
,
3)
. Obviously
O
i
O
†
i
=
O
†
i
O
i
=
1
h
olds, and an
y
two rotation matrices do not commute with each other. W
e
find twelve different ways to arrange the product of
O
1
,
O
2
an
d
O
3
,
whic
h
c
an cover t
h
ew
h
o
l
e
3
×
3 space and provide a
f
ull description o
f
V
(
Fritzsch
a
nd Xing, 2001
)
. Explicitly, six of the twelve different combinations o
f
O
i
b
e
l
ong to t
h
etyp
e
V
=
O
i
(
θ
i
,α
i
,β
i
,
γ
i
)
⊗
O
j
(
θ
j
,α
j
,β
j
β
,γ
j
γ
)
⊗
O
i
(
θ
i
,
α
i
,β
i
,γ
i
)(
3.104
)
w
i
th
i
=
j
,wherethecom
p
lex rotation matri
x
O
i
occurs twice; and the other
s
ix belong to the typ
e
V
=
O
i
(
θ
i
,α
i
,
β
i
,γ
i
)
⊗
O
j
(
θ
j
,
α
j
,β
j
β
,γ
j
γ
)
⊗
O
k
(
θ
k
,
α
k
,β
k
,γ
k
)(
3.105
)
w
i
th
i
=
j
=
k
, in which the rotations take
p
lace in three di
ff
erent com
p
lex
p
lanes. Note that the
p
roduct
s
O
i
O
j
O
i
a
nd
O
i
O
k
O
i
(
for
j
=
k
)
in Eq.
(
3.104
)
a
re correlated with each other, i
f
the relevant phase parameters are switche
d
1
0
N
o matter w
h
et
h
er neutrinos are Dirac or Majorana partic
l
es, t
he
n
×
n
un
i
tar
y
flavor mixing matrix has
(
n
−
1)
2
(
n
−
2)
2
/
4Jarlsko
g
invariants o
fC
P violatio
n
defined as
J
i
j
αβ
J
J
≡
Im
V
αi
V
V
V
βj
V
V
V
∗
αj
V
V
V
∗
βi
V
V
(Botella and Chau, 1986)
.
3.5 Lepton Flavor Mixing and
C
P Violation 99
off. Hence only nine of the twelve parametrizations, three from Eq.
(
3.104
)
a
nd six from Eq.
(
3.105
)
, are structurally different
(
Xing, 2004
)
.
I
n each parametrization of
V
,
t
h
ere apparent
l
y exist nine p
h
ase parame
-
ters.
S
ome o
f
them or their combinations can be absorbed by rede
fi
nin
g
the
relevant phases of charged-lepton and neutrino fields. For example,
⎛
⎝
⎛
⎛
e
i
γ
2
0
0
0
c
2
e
i
α
2
s
2
e
−
i
β
2
0
−
s
2
e
i
β
2
c
2
e
−
i
α
2
⎞
⎠
⎞
⎞
⎛
⎝
⎛
⎛
c
3
e
i
α
3
0
s
3
e
−
i
β
3
0
e
i
γ
3
0
−
s
3
e
i
β
3
0
c
3
e
−
i
α
3
⎞
⎠
⎞
⎞
⎛
⎝
⎛
⎛
c
1
e
i
α
1
s
1
e
−
i
β
1
0
−
s
1
e
i
β
1
c
1
e
−
i
α
1
0
00
e
i
γ
1
⎞
⎠
⎞
⎞
=
P
⎛
⎝
⎛
⎛
c
1
c
3
s
1
c
3
s
3
e
−
i
δ
−
s
1
c
2
−
c
1
s
2
s
3
e
i
δ
c
1
c
2
−
s
1
s
2
s
3
e
i
δ
s
2
c
3
s
1
s
2
−
c
1
c
2
s
3
e
i
δ
−
c
1
s
2
−
s
1
c
2
s
3
e
i
δ
c
2
c
3
⎞
⎠
⎞⎞
P
,
(
3.106
)
whe
r
e
δ
=(
α
1
−
β
1
)
−
(
α
2
+
β
2
)
+
(
β
3
−
γ
3
)
,
P
=Dia
g
{
e
i
a
,
e
i
b
,
e
i
c
}
w
i
th
a
=
(
α
1
−
β
1
)
−
(
α
2
+
β
2
−
γ
2
)
−
γ
3
,
b
=
−
(
β
2
+
α
3
)
and
c
=
−
(
α
2
+
α
3
),
a
n
d
P
=D
iag
{
e
i
x
,
e
i
y
,
e
i
z
}
w
i
th
x
=
β
1
+(
α
2
+
β
2
)
+
(
α
3
+
γ
3
)
,
y
=
−
α
1
+
(
α
2
+
β
2
)
+
(
α
3
+
γ
3
)
an
d
z
=
γ
1
.I
f
neutrinos are Dirac particles, one ca
n
a
lwaysremovebot
h
P
an
d
P
such that only
δ
i
s left as the single nontrivia
l
C
P-violatin
g
phase. I
f
neutrinos are Majorana particles, however, there is no
f
reedom to rearran
g
e the relative phases o
f
three Majorana neutrino
fi
elds
.
H
ence onl
y
P
itself and one common
p
hase o
f
P
are removable; namely, w
e
a
re left with three nontrivial CP-violating phases in the Majorana case
(
δ
a
n
d
two Ma
j
orana phases in
P
)
. Both CP- and T-violating effects in neutrin
o
osci
ll
ations
d
epen
d
on
l
yupont
h
e Dirac-
l
i
k
ep
h
ase
δ
.
Di
ff
erent parametrizations o
f
V
are mat
h
ematica
lly
equiva
l
ent, so a
d
opt
-
i
n
g
any of them does not directly point to physical si
g
nificance. But it is very
l
ikely that one particular parametrization is more useful and transparent than
the others in studying the neutrino phenomenology and
(
or
)
exploring the
u
nderlyin
g
dynamics responsible for lepton mass
g
eneration and CP viola-
tion. Here we highlight two particular parametrizations of the MNS matrix
V .The
fi
rst one is the so-called “standard” parametrization advocated b
y
the Particle Data Group
(
Nakamura
e
tal
.
,
2010
):
V
=
⎛
⎝
⎛
⎛
10 0
0
c
23
s
23
0
−
s
2
3
c
2
3
⎞
⎠
⎞
⎞
⎛
⎝
⎛
⎛
c
1
3
0
s
1
3
e
−
i
δ
010
−
s
1
3
e
i
δ
0
c
1
3
⎞
⎠
⎞
⎞
⎛
⎝
⎛
⎛
c
12
s
1
2
0
−
s
12
c
12
0
001
⎞
⎠
⎞
⎞
P
=
⎛
⎝
⎛
⎛
c
1
2
c
13
s
12
c
1
3
s
1
3
e
−
i
δ
−
s
1
2
c
23
−
c
12
s
2
3
s
13
e
i
δ
c
1
2
c
23
−
s
12
s
2
3
s
1
3
e
i
δ
s
2
3
c
13
s
12
s
23
−
c
12
c
23
s
13
e
i
δ
−
c
12
s
23
−
s
12
c
23
s
13
e
i
δ
c
23
c
13
⎞
⎠
⎞
⎞
P
,
(
3.107
)
whe
r
e
c
ij
≡
cos
θ
ij
a
n
d
s
ij
≡
sin
θ
ij
(
fo
r
ij
=12
,
13
,
2
3
)
together with the
Majorana phase matri
x
P
=
D
ia
g
{
e
i
ρ
,
e
i
σ
,
1
}
.
Without loss of generality, the
three mixing angles
(
θ
12
,θ
13
,
θ
2
3
)
can all be arranged to lie in the first quad-
rant.
A
rbitrar
y
values between 0 and 2
π
a
re allowed
f
or three
C
P-violatin
g
phases
(
δ
,ρ,
σ
)
. A remarkable merit of this parametrization is that its three
1
00 3 Neutrinos beyond the
S
tandard Mode
l
m
ixing angles are approximately equivalent to the mixing angles of solar
(
θ
1
2
),
a
tmospheric
(
θ
23
)
and CHOOZ reactor
(
θ
13
)
neutrino oscillation experiments
.
A
nother useful parametrization is the Fritzsch-Xing
(
FX
)
parametrizatio
n
proposed originally for quark mixing
(
Fritzsch and Xing, 1997
)
and later fo
r
l
epton mixing
(
Fritzsch and Xing, 1998b
):
V
=
⎛
⎝
⎛⎛
c
l
s
l
0
−
s
l
c
l
0
001
⎞
⎠
⎞⎞
⎛
⎝
⎛⎛
e
−
i
φ
00
0
cs
0
−
sc
⎞
⎠
⎞⎞
⎛
⎝
⎛⎛
c
ν
−
s
ν
0
s
ν
c
ν
0
00
1
⎞
⎠
⎞⎞
P
=
⎛
⎝
⎛
⎛
s
l
s
ν
c
+
c
l
c
ν
e
−
i
φ
s
l
c
ν
c
−
c
l
s
ν
e
−
i
φ
s
l
s
c
l
s
ν
c
−
s
l
c
ν
e
−
i
φ
c
l
c
ν
c
+
s
l
s
ν
e
−
i
φ
c
l
s
−
s
ν
s
−
c
ν
sc
⎞
⎠
⎞
⎞
P
,
(
3.108
)
where c
l
,ν
≡
cos
θ
l
,ν
,
s
l
,ν
≡
s
i
n
θ
l,
ν
,
c
≡
cos
θ
,
s
≡
sin
θ
,
an
d
P
i
s a diagonal
phase matrix containing two nontrivial CP-violating phases. Although the
fo
rm
of
V
i
nEq.
(
3.108
)
is apparently different from that in Eq.
(
3.107
),
their corresponding flavor mixing angles
(
θ
l
,θ
ν
,
θ
)
and
(
θ
12
,
θ
13
,
θ
23
)
have
q
uite simi
l
ar meanin
g
s in interpretin
g
t
h
e experimenta
ld
ata on neutrino
osc
ill
at
i
o
n
s
.In
t
h
e
limi
t
θ
l
=
θ
13
=
0, one easil
y
arrives at
θ
ν
=
θ
12
a
n
d
θ
=
θ
23
. As a natural consequence of very smal
l
θ
l
,
three mixing angles of th
e
F
X parametrization can also be related to those of solar
(
θ
ν
)
, atmospheric
(
θ
)
a
nd CHOOZ reactor
(
θ
l
si
n
θ
)
neutrino oscillation experiments in the leading-
order approximation
(
Xing, 2009a
)
. A striking merit of this parametrization
i
st
h
at its six parameters
h
ave very simp
l
erenorma
l
ization-
g
roup equations
when they run from a superhi
g
h-ener
g
y scale to the electroweak scale or vic
e
versa
(
Xing, 2006
)
, as one will see in Section 3.6.
3.5.2 Democrat
i
candTr
i
-b
i
max
i
mal M
i
x
i
n
g
Patterns
Tables 3.1 and 3.2 indicate the essential
f
eature o
f
lepton
fl
avor mixin
g
:two
m
ixing angles are quite large
(
θ
12
∼
3
4
◦
a
nd θ
23
∼
45
◦
)
while the third one
i
sverysmall
(
θ
13
<
10
◦
)
. Such a flavor mixing pattern is far beyond the
ori
g
inal ima
g
ination o
f
most people because it is rather di
ff
erent
f
rom th
e
well-known quark mixing pattern
(
ϑ
12
≈
1
3
◦
,
ϑ
23
≈
2
.
3
◦
,
ϑ
13
≈
0
.
2
2
◦
a
nd
δ
≈
69
◦
)
described by the same parametrization of the Cabibbo-Kobayashi-
Maskawa
(
CKM
)
matrix
(
Cabibbo, 1963; Kobayashi and Maskawa, 1973
)
.To
u
nderstand this difference, a number of constant lepton mixing patterns have
been proposed as the starting point of model building
(
Xing, 2004
)
. Possible
flavor symmetries and their spontaneous or explicit breakin
g
mechanisms
h
idden in those constant patterns might finally help us pin down the dynamic
s
responsible
f
or lepton mass
g
eneration and
fl
avor mixin
g
. To illustrate, let
u
s first comment on the “democratic” neutrino mixing pattern and then pay
m
ore attention to t
h
e“tri-
b
imaxima
l”
neutrino mixing pattern.
T
he “democratic” lepton
fl
avor mixin
g
patter
n
3.5 Lepton Flavor Mixing and
C
P Violation 10
1
U
0
UU
=
⎛
⎜
⎛⎛
⎝
⎜⎜
1
√
2
1
√
2
0
−
1
√
6
1
√
6
√
2
√
3
1
√
√
√
3
−
1
√
3
1
√√
√
3
⎞
⎟
⎞
⎞
⎠
⎟⎟
(
3.109
)
was ori
g
inally obtained by Fritzsch and Xin
g
as the leadin
g
term o
f
the 3
×
3
l
epton mixin
g
matrix from the breakin
g
of flavor democracy o
r
S
(
3
)
L
×
S
(
3
)
R
s
ymmetry of the charged-lepton mass matrix in the basis where the Majorana
n
eutrino mass matrix is
d
ia
g
ona
l
an
d
possesses t
he
S
(
3
)
symmetry
(
Fritzsc
h
a
nd Xing, 1996
)
. Its naive predictions
θ
12
=
4
5
◦
a
nd
θ
23
≈
54
.
7
◦
a
r
e
n
o
m
o
r
e
f
avored today, but they may receive proper corrections from the symmetry
-
breaking perturbations so as to fit current neutrino oscillation data
(
See, e.g.,
F
ritzsch and Xing, 1998b, 2000a, 2004; Xing et al., 2010
)
.
T
oday’s most popular constant pattern of neutrino mixing is the “tri-
bimaximal” mixing matrix
(
Harriso
n
et al
., 2002; Xin
g
, 2002a; He and Zee
,
2
003
):
V
0
VV
=
⎛
⎜
⎛⎛
⎝
⎜⎜
√
2
√
3
1
√
3
0
−
1
√
6
1
√
√
√
3
1
√
2
1
√
√
√
6
−
1
√
3
1
√√
√
2
⎞
⎟
⎞⎞
⎠
⎟⎟
(
3.110
)
which looks like a twisted
f
orm o
f
the democratic mixin
g
pattern with th
e
s
ame entries. Its strange name comes from the fact that this flavor mixing
pattern is actually a product of the “tri-maximal” mixing matrix
(
Cabibbo,
1
978
)
and a “bi-maximal” mixing matrix
:
V
0
V
V
=
⎛
⎜
⎛⎛
⎝
⎜
⎜
1
√
3
1
√
3
1
√
3
1
√
3
ω
√
3
ω
2
√
√
3
1
√
3
ω
2
√
√
3
ω
√
3
⎞
⎟
⎞⎞
⎠
⎟
⎟
⎛
⎝
⎛
⎛
1
√
2
0
−
1
√
2
010
1
√
2
0
1
√
2
⎞
⎠
⎞
⎞
=
PV
0
VV
P
,
(
3.111
)
whe
r
e
ω
=e
i2
π
/
3
denotes the complex cube-root of unity
(
i.e.
,
ω
3
=1
)
,and
P
=
Diag
{
1
,
ω
,
ω
2
}
an
d
P
=
Diag
{
1
,
1
,i
}
a
re two diagonal phase matrices
.
V
0
VV
o
r
V
0
VV
pre
d
ict
s
θ
1
2
=
arctan
(
1
/
√
2)
≈
35
.
3
◦
,
θ
1
3
=0
◦
a
n
d
θ
2
3
=4
5
◦
,
c
onsistent quite we
ll
wit
h
current neutrino osci
ll
ation
d
ata. Because t
h
een
-
tries o
f
U
0
UU
or
V
0
VV
a
re all formed from small integers
(
0, 1, 2 and 3
)
and their
s
quare roots, it is often suggestive of certain discrete flavor symmetries i
n
the lan
g
ua
g
eo
fg
roup theories. That is why the democratic or tri-bimaxima
l
n
eutrino mixing pattern can serve as a good starting point of model buildin
g
based on a variety o
ffl
avor symmetries, such a
s
Z
2
,
Z
3
Z
,
S
3
,
S
4
S
,
A
4
,
D
4
,
D
5
,
Q
4
,
Q
6
,
Δ
(
27
)
an
d
Σ
(
81
)(
Ma, 2007; King and Luhn, 2009
)
. In particular, a
l
ot of interest has been
p
aid to the derivation o
f
V
0
VV
with the hel
p
of the non
-
Abe
li
a
n
d
i
sc
r
ete
A
4
symmetry
(
see, e.g., Ma, 2004; Altarelli and Feruglio
,
2
005
;
Zee
,
2006
;
H
e
e
tal.
,
2006
)
or the Friedberg-Lee symmetry
(
Friedberg
a
n
d
Lee, 2006, 2008, 2010; Xin
g
et a
l.
, 2006; Xing, 2007; Chan
e
ta
l.
,
2009
)
.
Note t
h
at t
h
e
d
emocratic mixin
g
matrix
U
0
U
U
a
n
d
t
h
etri-
b
imaxima
l
mixin
g
m
atr
ix
V
0
V
V
a
re related with each other via the followin
g
transformation
:
1
02 3 Neutrinos beyond the
S
tandard Mode
l
V
0
VV
=
⎛
⎝
⎛⎛
10 0
0
co
s
θ
0
−
sin
θ
0
0
sin
θ
0
cos
θ
0
⎞
⎠
⎞⎞
U
0
UU
⎛
⎝
⎛⎛
cos
θ
0
−
s
i
n
θ
0
0
sin
θ
0
cos
θ
0
0
001
⎞
⎠
⎞⎞
,
(
3.112
)
where
θ
0
=arctan
(
√
2
−
1)
2
≈
9
.
7
◦
.
This an
g
le is actually a measure of the
d
ifference between the mixing angles of
U
0
U
U
an
d
V
0
VV
(
namely, 4
5
◦
−
35
.
3
◦
=
54
.
7
◦
−
45
◦
=9
.
7
◦
)
. In this sense, we argue that it is worthwhile to explore
possible flavor s
y
mmetries behind bot
h
V
0
V
V
an
d
U
0
U
U
so as to build realisti
c
m
odels for neutrino mass generation and lepton flavor mixing
.
Note also that it is possible to construct di
ff
erent but viable neutrin
o
m
ixing patterns with a few small integers and their square roots. Here we giv
e
a
n interesting example in terms of only two small integers 1 and 2 togethe
r
with their square roots and the imaginary unit i
(
Xing, 2008c
):
V
=
1
2
⎛
⎜
⎛⎛
⎜
⎜⎜
⎝
⎜⎜
1+
1
√
2
11
−
1
√
2
−
1
√
2
1
−
i
1
−
1
√
2
1
−
i
1
√
2
1
√
2
1
+i
1+
1
√
2
−
1
√
2
1
+i
1
−
1
√
2
1
+i
1
√
2
1
√
2
1
−
i
1+
1
√
2
⎞
⎟
⎞⎞
⎟
⎟⎟
⎠
⎟⎟
.
(
3.113
)
We re
f
er to this ansatz as the “tetra-maximal” mixin
g
pattern because i
t
c
an be expressed as a product of four rotation matrices whose mixing angles
a
r
eall
π
/
4
in t
h
e comp
l
ex p
l
ane. It pre
d
ict
s
θ
12
=
arctan
(
2
−
√
2)
≈
30
.
4
◦
,
θ
1
3
= arcsin
[(
√
2
−
1)
/
(
2
√
2)]
≈
8
.
4
◦
,
θ
23
=
45
◦
a
n
d
δ
=
90
◦
i
n
the sta
n
da
r
d
parametrization. These results are certainl
y
compatible with current neutrin
o
osci
ll
ation
d
ata an
d
wi
ll
soon
b
e teste
d
in t
h
e upcoming an
d
more accurat
e
n
eutrino oscillation ex
p
eriments.
Let us remark that a specific constant mixing pattern should be regarde
d
a
s the leadin
g
-order approximation o
f
the “true” lepton
fl
avor mixin
g
ma-
trix, whose mixin
g
an
g
les should in
g
eneral depend on both the ratios o
f
c
harged-lepton masses and those of neutrino masses. We may at least make
the
f
ollowin
g
naive speculation about how to phenomenolo
g
ically understand
the observed pattern of lepton flavor mixing
(
Xing, 2004
)
:
•
Lar
g
evalueso
f
θ
12
an
d
θ
23
could arise from a weak hierarch
y
or a near
degeneracy of the neutrino mass spectrum, because the strong hierarch
y
of
char
g
ed-lepton masses implies tha
t
m
e
/
m
μ
≈
4
.
7
×
10
−
3
a
n
d
m
μ
/
m
τ
≈
5.9
×
1
0
−
2
a
t the electroweak scale are unlikel
y
to contribute to
θ
12
a
nd
θ
2
3
i
na
d
ominant way.
•
S
pecial values o
f
θ
12
a
n
d
θ
2
3
mi
g
ht stem
f
rom an underlyin
gfl
avor sym
-
metry of the charged-lepton mass matrix or the neutrino mass matrix
.
T
hen the contributions of lepton mass ratios to flavor mixing angles, due
t
o
fl
avor symmetry breakin
g
, are expected to serve as the perturbativ
e
corrections to
U
0
UU
or
V
0
VV
, or another constant flavor mixing pattern.
•
V
anis
h
in
g
or sma
ll
θ
1
3
c
ould be a natural consequence o
f
the explici
t
t
extures o
f
lepton mass matrices. It mi
g
ht also be related to the
fl
avo
r
symmetry which gives rise to sizabl
e
θ
12
a
n
d
θ
23
(
e.g., i
n
U
0
UU
o
r
V
0
VV
)
.
3.5 Lepton Flavor Mixing and
C
P Violation 103
•
S
mall corrections to a constant
fl
avor mixin
g
pattern may also result
f
ro
m
t
he renormalization-
g
roup runnin
g
effects of leptons and quarks, e.
g
., from
asuper
h
ig
h
-energy sca
l
eto
l
ow energies or vice versa.
We admit that there are too many possibilities of linking the observed patter
n
o
f
lepton
fl
avor mixin
g
to a certain
fl
avor symmetry, and none o
f
them i
s
u
nique from the theoretical point of view. In this sense flavor s
y
mmetries
m
i
g
ht not be a per
f
ect
g
uidin
g
principle o
f
model buildin
g
.
3.5.3 Rephas
i
ng Invar
i
ants and Un
i
tar
i
ty Tr
i
angles
In the basis where the flavor eigenstates of charged leptons are identifie
d
with their mass ei
g
enstates, the MN
S
matri
x
V
relates the
n
eut
rin
o
m
ass
eigenstates
(
ν
1
,ν
2
,
ν
3
)
to the neutrino flavor eigenstates
(
ν
e
,ν
μ
,
ν
τ
):
⎛
⎝
⎛⎛
ν
e
ν
μ
ν
τ
⎞
⎠
⎞
⎞
=
⎛
⎝
⎛⎛
V
e
VV
1
V
e
VV
2
V
e
VV
3
V
μ
V
V
1
V
μ
V
V
2
V
μ
V
V
3
V
τ
VV
1
V
τ
VV
2
V
τ
VV
3
⎞
⎠
⎞⎞
⎛
⎝
⎛⎛
ν
1
ν
2
ν
3
⎞
⎠
⎞⎞
.
(
3.114
)
The unitarity o
f
V
r
epresents two sets of normalization and orthogonalit
y
co
n
d
i
t
i
o
n
s:
i
V
αi
VV
V
∗
βi
VV
=
δ
αβ
,
α
V
αi
VV
V
∗
αj
VV
=
δ
ij
,
(
3.115
)
where Greek and Latin subscripts run over
(
e
,μ,τ
)
and
(1
,
2
,
3)
, respectively
.
In the complex plane the six orthogonality relations in Eq.
(
3.115
)
define six
triangles
(
e
,
μ
,
τ
)
and
(
1
,
2
,
3
)
shown in Fig. 3.10, the so-calle
d
u
nitarity triangles. These six triangles have eighteen different sides and nin
e
d
ifferent inner
(
or outer
)
angles. But the unitarity of
V
r
equires t
h
at a
ll
si
x
trian
g
les have the same area amountin
g
to
J
/
2, wher
e
J
i
stheJarlsko
g
i
nvariant of CP violation
(
Jarlskog, 1985; Wu, 1986
)
defined throug
h
Im
V
αi
VV
V
βj
VV
V
∗
αj
VV
V
∗
βi
VV
=
J
γ
α
β
γ
k
ijk
.
(
3.116
)
O
ne has
J
=
c
12
s
12
c
2
13
s
13
c
23
s
23
si
n
δ
i
n the standard
p
arametrization o
f
V
a
nd
J
=
c
l
s
l
c
ν
s
ν
c
s
2
s
i
n
φ
in the FX
p
arametrization of
V
.
No matter whethe
r
n
eutrinos are Dirac or Majorana particles, the stren
g
th o
fC
P or T violation
i
n neutrino oscillations depends onl
y
upo
n
J
.
T
o show why the areas of six unitarity triangles are identical with on
e
a
not
h
er,
l
et us ta
k
etrian
gl
es
τ
a
n
d
3
f
or example. The
y
correspond to
the ortho
g
onality relation
s
V
e
VV
1
V
∗
μ
VV
1
+
V
e
V
V
2
V
∗
μ
VV
2
+
V
e
VV
3
V
∗
μ
VV
3
=
0
,
V
e
V
V
1
V
∗
e
VV
2
+
V
μ
VV
1
V
∗
μ
VV
2
+
V
τ
VV
1
V
∗
τ
VV
2
=
0
.
(
3.117
)
Multiplyin
g
these two equations b
y
V
μ
V
V
2
V
∗
e
V
V
2
a
n
d
V
μ
V
V
2
V
∗
μ
V
V
1
respectivel
y
, we arrive
a
t two rescaled triangles which share the side
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