Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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4
6
4
S
eesaw Mechanisms o
f
Neutrino Masse
s
×
××
×
×
H
H
N
R
NN
N
R
NN
ν
L
νν
ν
L
νν
S
1
R
S
1R
××
×
×
Φ
1
Φ
1
×
×
S
2
R
S
2
R
×
×
×
×
×
H
H
N
R
NN
N
R
NN
ν
L
νν
ν
L
νν
S
1
R
S
1
R
××
×
×
Φ
1
1
Φ
1
1
××
×
×
Φ
2
Φ
2
S
2
R
S
2R
×
×
3
3
S
3R
S
S
3R
S
(
a
)
(
b
)
Fi
g
.4.3 Theori
g
in o
f
li
g
ht Majorana neutrino masses in multiple seesaw mecha-
nisms: (a) the minimal extension of the canonical seesaw mechanism (wit
h
n =2);
and
(
b
)
the minimal extension of the inverse seesaw mechanism
(
with
n
=3
)
N
R
N
N
an
d
M
μ
M
=
M
R
MM
)
.If
M
S
M
2
i
∼
M
D
MM
∼
O
(
Λ
E
W
)
an
d
M
S
M
2
i
−
1
∼
M
μ
M
∼O
(
Λ
SS
)
h
old
(
fo
r
i
=
1
,
2
,
···
,k
)
,thenEq.
(
4.51
)
leads t
o
M
ν
MM
∼
Λ
2
(
k
+1
)
E
W
/
Λ
2
k
+1
SS
,
which can effectivel
y
lower the conventional seesaw scale
Λ
SS
∼
10
1
4
G
e
V
d
own to t
h
e TeV sca
l
easi
ll
ustrate
d
in Fig. 4.2.
T
akin
g
n
=2
(
i.e.,
k
=1
)
for example, we simply arrive at
M
ν
MM
=
−
M
D
MM
M
T
S
M
1
−
1
M
S
M
2
M
−
1
μ
M
M
T
S
M
2
M
S
M
1
−
1
M
T
D
MM
.
(
4.52
)
This effective multiple seesaw mass term is illustrated in Fig. 4.3
(
a
)
.TheNN
I
pattern o
f
M
w
i
th
n
=
2can
b
eo
b
taine
db
yimposin
g
t
h
e
gl
o
b
a
l
U
(
1
)
×
Z
6
sy
mme
t
r
y
on
L
ν
,
where the proper charge assignment is listed in Table 4.1.
Table 4.1 The charges of relevant fermion and scalar fields under th
e
U
(
1
)
×
Z
6
s
y
mmetr
y
in t
h
emu
l
tip
l
e seesaw mec
h
anism wit
h
n
=
2
L
HE
R
N
R
NN
S
1R
S
2R
Φ
1
Φ
2
L
+
10
+
1
+
1
−
1
+
10 0
q
+
10
+
1
+
1
+
2
+
3
+
3
+1
C
lassBo
f
multiple seesaw mechanism
s
—
t
h
e
y
contain an o
dd
num
b
e
r
of gauge-singlet fermions
S
i
n
R
a
nd scalar
s
Φ
n
(
i.e.
,
n
=2
k
+
1wit
h
k
=
4.4 Multiple
S
eesaw Mechanisms 147
0
,
1
,
2
,
···
)
and correspond to an interesting extension of the inverse seesaw
picture. The effective mass matrix of three li
g
ht Majorana neutrinos read
s
M
ν
MM
=
M
D
MM
k
i
=
1
M
T
S
M
2
i
−
1
−
1
M
S
M
2
i
M
T
S
M
2
k
+1
−
1
×
M
μ
M
M
S
M
2
k
+1
−
1
k
i
=
1
M
T
S
M
2
i
−
1
−
1
M
S
M
2
i
T
M
T
D
MM
(
4.53
)
i
nt
h
e
l
ea
d
in
g
-or
d
er approximation. T
he
k
=
0 case
j
ust correspon
d
sto
the inverse seesaw scenario with a rather low mass scale of
M
μ
M
:
M
ν
MM
=
M
D
MM
(
M
T
S
M
1
)
−
1
M
μ
M
(
M
S
M
1
)
−
1
M
T
D
MM
. Note that the NNI pattern o
f
M
i
nt
h
e
d
ou
bl
e
s
eesaw mechanism is guaranteed by an implementation of the global
U
(
1
)
×
Z
4
s
ymmetry with the
f
ollowin
g
char
g
e assi
g
nment
:
L
(
L
)=
L
(
E
R
)=
L
(
N
R
NN
)=
+1
,
L
(
S
1
R
)=
−
1
,
L
(
H
)=
L
(
Φ
1
)
=0,
q
(
L
)
=
q
(
E
R
)=
q
(
N
R
NN
)=
q
(
Φ
1
)
=
+
1
,
q
(
H
)
=0an
d
q
(
S
1R
)
=+2.
If
M
S
M
2
i
∼
M
D
MM
∼O
(
Λ
EW
)
and
M
S
M
2
i
−
1
∼
O
(
Λ
SS
)
hold
(
fo
r
i
=1
,
2
,
···
,k
),
themassscaleo
f
M
μ
M
is in general unnecessary to be as small as that given
by the inverse seesaw mechanism. To be more speci
fi
c, let us consider the
m
inimal extension of the inverse seesaw picture by takin
g
n
=
3. In this case
,
we impose t
h
e
U
(
1
)
×
Z
10
symme
t
ry on
L
ν
w
it
h
aproperc
h
arge assignmen
t
l
i
sted
in T
ab
l
e
4.2
to assu
r
et
h
e
NNI
fo
rm
of
M
. The correspondin
gf
ormula
of
M
ν
MM
i
s
M
ν
MM
=
M
D
MM
M
T
S
M
1
−
1
M
S
M
2
M
T
S
M
3
−
1
M
μ
M
M
S
M
3
−
1
M
T
S
M
2
M
S
M
1
−
1
M
T
D
MM
.
(
4.54
)
This effective multiple seesaw mass term is illustrated in Fig. 4.3
(
b
)
.Itbe-
c
omes obvious that the proportionality of
M
ν
MM
to
M
μ
M
in Eq.
(
4.54
)
is doubly
s
uppresse
d
not on
l
y
b
yt
h
eratio
M
D
MM
/M
S
M
1
∼
Λ
E
W
/Λ
SS
b
ut a
l
so
b
yt
h
era
-
tio
M
S
M
2
/M
S
M
3
∼
Λ
E
W
/
Λ
SS
,an
d
t
h
u
s
M
ν
MM
∼
0
.
1
eV can naturall
y
result
f
rom
Y
ν
Y
Y
∼
Y
S
YY
1
∼
Y
S
YY
2
∼
Y
S
YY
3
∼O
(
1
)
and
M
μ
M
∼
1
k
eV at
Λ
SS
∼
1
TeV
.
Table 4.2 The charges of relevant fermion and scalar fields under the U (1
)
×
Z
1
0
s
y
mmetr
y
in t
h
emu
l
tip
l
e seesaw mec
h
anism wit
h
n
=
3
L
HE
R
N
R
NN
S
1R
S
2
R
S
3R
Φ
1
Φ
2
Φ
3
L
+1 0
+
1
+
1
−
1
+
1
−
10 0 0
q
+
10
+
1
+
1
+
2
+
3
+
5
+
7
+
5
+
2
4
.4.2
C
harged-current Interactions
The stren
g
th o
f
char
g
ed-current interactions is important
f
or both production
a
nd detection of light and heavy Majorana neutrinos in a realistic experiment
1
4
8
4
S
eesaw Mechanisms o
f
Neutrino Masse
s
(
e.g., at the LHC and in neutrino oscillations
)
. To define the neutrino mas
s
ei
g
enstates, we dia
g
onalize the overall mass matri
x
M
in Eq.
(
4.50
)
by mean
s
of the following unitary transformation:
V
˜
R
˜
S
˜
U
†
0
˜
M
D
MM
˜
M
T
D
MM
˜
M
μ
M
V
˜
R
˜
S
˜
U
∗
=
$
M
ν
M
M
0
0
$
M
N
M
+
S
,
(
4.55
)
whe
r
e
$
M
ν
M
M
≡
D
iag
{
m
1
,m
2
,
m
3
}
contains the masses of three light Majorana
n
eutrinos
(
ν
1
,
ν
2
,
ν
3
)
,an
d
$
M
N
M
+
S
denotes a dia
g
onal matrix whose ei
g
enval
-
u
es are the masses of 3
(
n
+
1
)
heavy Majorana neutrinos
(
ˆ
N
,
ˆ
S
1
,
·
·
·
,
ˆ
S
n
;
a
nd each of them consists of three components
)
. The SM charged-curren
t
i
nteractions o
f
ν
e
,
ν
μ
a
nd
ν
τ
can therefore be ex
p
ressed as
L
cc
=
g
√
2
eμτ
L
γ
μ
⎡
⎢
⎡
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎢
⎢
V
⎛
⎝
⎛
⎛
ν
1
ν
2
ν
3
⎞
⎠
⎞
⎞
L
+
˜
R
⎛
⎜
⎛
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎜⎜
ˆ
N
ˆ
S
1
.
.
.
ˆ
S
n
⎞
⎟
⎞⎞
⎟
⎟⎟
⎟
⎟⎟
⎠
⎟⎟
L
⎤
⎥
⎤
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎥⎥
W
−
μ
W
+
h
.
c
.
(
4.56
)
i
n the basis where the mass ei
g
enstates of three char
g
ed leptons are identifie
d
with their flavor eigenstates. Note tha
t
V
is t
h
e
3
×
3 neutrino mixing matri
x
responsible
f
or neutrino oscillations, while the
3
×
3(
n
+1
)
matri
x
˜
R
g
overns
the strength of charged-current interactions of heavy Majorana neutrinos.
N
ote a
l
so t
h
at
b
ot
h
VV
†
+
˜
R
˜
R
†
=
1
a
n
d
V
$
M
ν
MM
V
T
+
˜
R
$
M
N
M
+
S
˜
R
T
=
0
h
o
ld,
a
n
d
t
h
us
V
must
b
e non-unitary. It is
˜
R
t
hat measures the deviation of
V
f
rom unitarity in neutrino oscillations and determines the collider si
g
nature
s
of heav
y
Ma
j
orana neutrinos at the LHC.
Th
emu
l
tip
l
e seesaw i
d
ea may
l
ea
d
to ric
h
p
h
enomeno
l
ogy at
b
ot
h
t
h
eTe
V
s
cale and lower ener
g
ies. For simplicity, here we only mention a
f
ew aspect
s
of the phenomenological consequences of multiple seesaw mechanisms.
•
Non-unitary neutrino mixing and
C
Pviolation
.S
inc
e
V
is non-unitary
,
it
g
enerally involves a number o
f
new
fl
avor mixin
g
an
g
les and new
C
P
-
v
iolating phases
(
Xing, 2008b
)
. Novel CP-violating effects in the medium
-
basel
in
e
ν
μ
→
ν
τ
a
n
d
ν
μ
→
ν
τ
o
scillations ma
y
there
f
ore show up an
d
p
rovide a promisin
g
si
g
nature of the unitarity violation o
f
V
,
which coul
d
b
e measured at a neutrino factory
(
Fernandez-Martinez
e
ta
l.
,
2007
).
•
C
ollider si
g
natures o
f
heav
y
Ma
j
orana neutrinos
.G
ive
n
˜
M
D
MM
∼O
(
Λ
EW
)
an
d
˜
M
μ
M
∼
O
(
Λ
SS
)
∼O
(
1
)
TeV, it is easy to obtai
n
˜
R
≈
˜
M
D
MM
˜
M
−
1
μ
M
˜
U
∼
O
(
0
.
1)
, which actually saturates the present experimental upper boun
d
on
|
˜
R
|
(
Antusc
h
et al
., 2006
)
.Fo
r
Class
A
of
multiple seesaw mecha
-
nisms, their clear LHC signatures are expected to be the like-sign dilepton
e
vents arising from the lepton-number-violating processe
s
pp
→
l
±
α
l
±
β
X
(
fo
r
α,
β
=
e
,
μ
,
τ
)
mediated by heavy Majorana neutrinos
(
Atre
et al
.
,
2009
)
.ForClass B
o
fmulti
p
le seesaw mechanisms with
M
μ
M
Λ
EW
,
the
4.4 Multiple
S
eesaw Mechanisms 149
mass spectrum o
f
heavy Majorana neutrinos
g
enerally exhibits a pair
-
in
g
phenomenon in which the nearest-nei
g
hbor Majorana neutrinos have
nearly degenerate masses and can be combined to form pseudo-Dirac par
-
t
icles. This
f
eature has alread
y
been observed in the inverse seesaw model
(
Mohapatra and Valle, 1986
)
. Therefore, the discriminating collider signa-
t
ures at the LHC are expected to be the trilepton events coming from th
e
pp
→
l
±
α
l
±
β
l
∓
γ
X
processes
(
fo
r
α
,
β
,
γ
=
e
,
μ
,τ
)(
del Aguila and Aguilar-
S
aavedra, 2009b
).
• Possible candidates for dark matte
r
.
OneormoreoftheheavyMajo-
r
ana neutrinos an
dg
au
g
e-sin
gl
et sca
l
ars in our mu
l
tip
l
e seesaw mec
h
a
-
nisms could be arranged to have a sufficiently long lifetime. Such weakly
-
interactin
g
and massive particles mi
g
ht there
f
ore be a plausible candidat
e
for cold dark matter
(
Ponton and Randall, 2009
).
One may explore more low-energy effects of multiple seesaw mechanisms, such
a
s their contributions to the lepton-
fl
avor-violatin
g
processes
μ
→
eγ
a
n
dso
on. It should also be interesting to explore possible baryogenesis via leptoge
-
n
esis
(
Fukugita and Yanagida, 1986
)
, based on a multiple seesaw picture, t
o
i
nterpret t
h
ecosmo
l
o
g
ica
l
matter-antimatter asymmetry.
A
s a flexible and testable TeV seesaw scheme, the multi
p
le seesaw mech-
a
nisms can also provide us with plenty o
f
room
f
or model buildin
g
.Butthe
l
atter re
q
uires
f
urther in
p
uts or assum
p
tions. Here we
p
resent a sim
p
le bu
t
i
nstructive exam
p
le, in which all the textures of 3
×
3
sub-matrices in the over
-
all
n
eut
rin
o
m
ass
m
at
rix
M
are symmetric an
dh
ave t
h
ewe
ll
-
k
nown Fritzsc
h
pattern
(
Fritzsch, 1978
),
M
a
M
=
⎛
⎝
⎛
⎛
0
x
a
0
x
a
0
y
a
0
y
a
z
a
⎞
⎠
⎞
⎞
(
4.57
)
w
i
th
a
=
D
,S
1
,
···
,S
n
o
r
μ
,
f
or illustration.
C
hoosin
g
the Fritzsch textur
e
m
akes sense because it coincides with the NNI form o
f
M
i
tself. We make an
add
itiona
l
assumption t
h
at t
h
eratio
x
a
/y
a
is a constant independent of th
e
s
ubscri
p
t
a
.Thenitiseas
y
to show that the e
ff
ective mass matrix o
f
thre
e
l
ight Majorana neutrinos
M
ν
MM
h
as the same Fritzsch texture in the leading
-
or
d
er approximation
:
(
M
ν
MM
)
12
=
−
x
2
D
x
μ
k
i
=
1
x
2
S
2
i
x
2
S
2
i
−
1
,
(
M
ν
M
M
)
23
=
−
y
2
D
y
μ
k
i
=
1
y
2
S
2
i
y
2
S
2
i
−
1
,
(
M
ν
M
M
)
33
=
−
z
2
D
z
μ
k
i
=
1
z
2
S
2
i
z
2
S
2
i
−
1
,
(
4.58
)
150
4
S
eesaw Mechanisms o
f
Neutrino Masse
s
a
nd
(
M
ν
MM
)
1
1
=
(
M
ν
MM
)
1
3
=(
M
ν
MM
)
22
=
0 derived from Eq.
(
4.51
)
for
C
lass
A
of
m
ultiple seesaw mechanisms
(
wit
h
n
=
2
k for
k
=
0
,
1
,
2
,
···
)
;an
d
(
M
ν
MM
)
1
2
=
x
2
D
x
2
S
2
k
+1
k
i
=
1
x
2
S
2
i
x
2
S
2
i
−
1
x
μ
,
(
M
ν
MM
)
2
3
=
y
2
D
y
2
S
2
k
+
1
k
i
=
1
y
2
S
2
i
y
2
S
2
i
−
1
y
μ
,
(
M
ν
MM
)
33
=
z
2
D
z
2
S
2
k
+
1
k
i
=
1
z
2
S
2
i
z
2
S
2
i
−
1
z
μ
,
(
4.59
)
a
nd
(
M
ν
MM
)
11
=(
M
ν
M
M
)
13
=(
M
ν
MM
)
22
= 0 derived from Eq.
(
4.53
)
fo
r
C
lass B
of multiple seesaw mechanisms
(
wit
h
n
=
2
k
+1
f
o
r
k
=0
,
1
,
2
,
···
)
.Thi
s
s
eesaw-invariant propert
y
o
f
M
ν
MM
is interestin
g
since it exactly re
fl
ects how tw
o
c
lasses of multiple seesaw mechanisms work for every element of
M
ν
MM
.N
ot
e
t
h
at it is possi
bl
e to interpret current experimenta
ld
ata on sma
ll
neutrino
m
asses and lar
g
e flavor mixin
g
an
g
les by takin
g
both the texture of the li
g
ht
n
eutrino mass matrix
M
ν
MM
and that of the charged-lepton mass matrix
M
l
M
t
o
be of the Fritzsch form
(
Xing, 2002; Xing and Zhou, 2004; Zhou and Xing,
2
005
)
. Hence the above examples are phenomenologically viable. Once th
e
te
x
tu
r
eo
f
M
ν
MM
is fully reconstructed from more accurate neutrino oscillatio
n
d
ata, one ma
y
then consider to quantitativel
y
explore the textures o
f
those
3
×
3 sub-matrices o
f
M in such a multi
p
le seesaw model
.
T
o conclude, new ideas are eagerly wanted in the LHC era to achieve
a
proper
b
a
l
ance
b
etween t
h
eoretica
l
natura
l
ness an
d
experimenta
l
testa
b
i
l
it
y
of the elegant seesaw pictures, which ascribe the small masses of three known
n
eutrinos to the existence o
f
some heavy de
g
rees o
ff
reedom.Wehaveex-
tended the canonical and inverse seesaw scenarios and
p
ro
p
osed two classe
s
of multiple seesaw mechanisms at the TeV scale by introducing an arbitrar
y
n
umber o
fg
au
g
e-sin
g
let
f
ermions and scalars into the
S
M and by imple-
m
entin
g
the
g
lobal
U
(
1
)
×
Z
2
N
sy
mmetr
y
in the neutrino sector. These ne
w
TeV-scale seesaw mechanisms are expected to lead to rich phenomenology at
l
ow ener
g
ies and open some new prospects
f
or understandin
g
the ori
g
in o
f
tin
y
neutrino masses and lepton number violation.
4.5 Non-unitary Neutrino Mixing and CP Violatio
n
It is worth remarking that the charged-current interactions of light and heavy
Ma
j
orana neutrinos are not completel
y
independent o
f
each other in t
y
pe-I,
type-
(
I+II
)
, type-III or multiple seesaw mechanisms. This point can clearly
be seen from Eq.
(
4.56
)
,inwhic
h
VV
†
+
˜
R
˜
R
†
=
1
h
olds. Re
g
ardless o
f
the
d
etails o
f
each seesaw scheme, one ma
y
approximatel
y
parametrize the 3
×
3
m
ixing matrix of three light Majorana neutrinos a
s
4.5 Non-unitary Neutrino Mixing and
C
P Violation 151
V
≈
(
1
−
ξ
)
V
0
VV
w
i
t
h
ξ
=
1
2
˜
R
˜
R
†
,
(
4.60
)
where
V
0
V
V
i
s a unitar
y
matrix an
d
ξ
measures the deviation of
V
from
V
0
V
V
.In
view of current experimental data on the invisible width of the
Z
0
b
oson
,
u
niversalit
y
tests o
f
electroweak interactions, rare leptonic deca
y
s and neu
-
trino oscillations, one may find the followin
g
constraints on the elements o
f
ξ
a
tthe90
%
confidence level
(
Antusch et a
l
.
,
2006
)
:
|
ξ
|
=
⎡
⎣
|
ξ
ee
|
<
5
.
5
×
10
−
3
|
ξ
eμ
|
<
3
.
5
×
1
0
−
5
|
ξ
eτ
|
<
8
.
0
×
1
0
−
3
|
ξ
μe
|
<
3
.
5
×
10
−
5
|
ξ
μμ
|
<
5
.
0
×
10
−
3
|
ξ
μτ
|
<
5
.
0
×
1
0
−
3
|
ξ
τ
e
|
<
8
.
0
×
10
−
3
|
ξ
τμ
|
<
5
.
0
×
10
−
3
|
ξ
ττ
|
<
5
.
0
×
10
−
3
⎤
⎦
.
(
4.61
)
This result implies that the unitarity of
V
h
as
b
een teste
d
at t
h
e percent
l
eve
l
a
nd possible non-unitarit
y
o
f
V
ca
n
at
m
ost
r
each the sa
m
e level
1
.
N
ote that
the
p
hases o
f
ξ
a
re entirely unrestricted, and thus new CP-violatin
g
effect
s
of
O
(
1
0
−
2
)
or smaller may be induced by the non-unitarity of
V
a
n
d
ar
e
l
ikely to show up in neutrino oscillations
(
Fernandez-Martinez
et al.
,
200
7
;
X
ing, 2008b; Luo, 2008; Goswami and Ota, 2008;
A
ltarelli and Meloni, 2009
;
A
n
tusch
et al
., 2009; Ma
l
ins
ky
et al
.
,
2009
)
.
4
.5.1 Jarlsko
g
Invariants o
f
CP Violatio
n
The stren
g
th o
fC
P violation in neutrino oscillations is characterized by the
J
arlskog invariants
(
Jarlskog, 1985; Wu, 1986
)
J
ij
αβ
J
≡
Im
V
αi
VV
V
βj
VV
V
∗
αj
VV
V
∗
βi
VV
,
(
4.62
)
where Greek and Latin indices run over
(
e
,
μ
,
τ
)
and
(1
,
2
,
3
)
, respectively.
Without loss o
fg
enerality, the unitary matrix
V
0
V
V
i
nEq.
(
4.60
)
can b
e
p
arametrized as a
p
roduct of three rotation matrices in the com
p
lex
p
lane
:
V
0
VV
=
⎛
⎝
⎛⎛
10 0
0
c
2
3
ˆ
s
∗
23
0
−
ˆ
s
23
c
23
⎞
⎠
⎞⎞
⎛
⎝
⎛⎛
c
1
3
0ˆ
∗
1
3
010
−
ˆ
s
13
0
c
13
⎞
⎠
⎞⎞
⎛
⎝
⎛⎛
c
12
ˆ
s
∗
1
2
0
−
ˆ
s
12
c
1
2
0
001
⎞
⎠
⎞⎞
=
⎛
⎝
⎛⎛
c
12
c
13
ˆs
∗
12
c
13
ˆ
s
∗
13
−
ˆ
s
12
c
2
3
−
c
12
ˆ
s
1
3
ˆ
s
∗
23
c
12
c
23
−
ˆ
s
∗
1
2
ˆ
s
13
ˆ
s
∗
2
3
c
1
3
ˆ
s
∗
2
3
ˆ
s
12
ˆ
s
23
−
c
12
ˆ
s
13
c
23
−
c
12
ˆ
s
23
−
ˆ
s
∗
12
ˆ
s
13
c
23
c
13
c
23
⎞
⎠
⎞⎞
,
(
4.63
)
whe
r
e
c
ij
≡
cos
θ
i
j
,
s
i
j
≡
sin
θ
ij
and ˆ
i
j
≡
s
ij
e
i
δ
i
j
w
i
th
θ
i
j
a
n
d
δ
i
j
b
ein
g
t
h
e
rotation and phase an
g
les, respectively.
A
fter a proper phase rearran
g
ement,
1
Wh
en t
h
et
y
pe-III seesaw mec
h
anism is concerne
d
,t
h
e experimenta
lb
oun
d
s
on the elements of ξ
a
re slightly stronger than those given above for type-I, type
-
(
I+II
)
or multiple seesaw mechanisms
(
Abad
a
et al.
,
2007
)
. The reason is simpl
y
that in the type-III seesaw case the
fl
avor-chan
g
in
g
processes with char
g
ed leptons
are allowed at the tree level and can get stringent experimental constraints
.
15
24
S
eesaw Mechanisms o
f
Neutrino Masse
s
E
q.
(
4.63
)
is actually the standard parametrization advocated by the Particle
D
ata Group
(
Fritzsch and Xing, 2001; Nakamura
e
tal
.
,
2010
)
. The universal
J
arlskog invariant o
f
V
0
VV
r
ea
d
s
J
0
JJ
=
c
1
2
s
12
c
2
1
3
s
13
c
2
3
s
23
s
in
(
δ
1
3
−
δ
1
2
−
δ
23
)
in the
a
bove parametrization. Combining Eqs.
(
4.60
)
and
(
4.63
)
,onemaycalculate
J
ij
αβ
J
d
efined in Eq.
(
4.62
)
.Sincebot
h
s
13
an
d
ξ
α
β
a
re small
q
uantities, we
n
eglect the terms of
O
(
s
2
13
)
,
O
(
|
ξ
α
β
|
2
)
and
O
(
s
13
|
ξ
αβ
|
)
in the calculation
.
Then it is strai
g
ht
f
orward to obtain
J
12
eμ
JJ
=
J
0
J
J
−
c
12
s
12
c
23
1+
c
2
23
|
ξ
e
μ
|
s
in
(
δ
12
+
δ
eμ
)
+
c
12
s
1
2
c
2
2
3
s
23
|
ξ
e
τ
|
sin
(
δ
1
2
+
δ
23
+
δ
e
τ
)
,
J
23
eμ
J
J
=
J
0
J
J
+
c
12
s
1
2
c
23
s
2
2
3
|
ξ
eμ
|
sin
(
δ
1
2
+
δ
eμ
)
+
c
1
2
s
1
2
c
2
2
3
s
23
|
ξ
e
τ
|
sin
(
δ
1
2
+
δ
23
+
δ
e
τ
)
,
J
12
μτ
J
=
J
0
J
J
−
c
12
s
1
2
c
23
s
2
2
3
|
ξ
eμ
|
sin
(
δ
1
2
+
δ
eμ
)
−
c
1
2
s
1
2
c
2
2
3
s
23
|
ξ
e
τ
|
sin
(
δ
1
2
+
δ
2
3
+
δ
e
τ
)
,
J
23
μτ
J
=
J
12
μτ
J
−
2
c
2
1
2
c
2
3
s
23
|
ξ
μτ
|
s
in
(
δ
2
3
+
δ
μτ
)
,
J
31
μτ
J
=
J
12
μτ
J
+2
s
2
1
2
c
2
3
s
23
|
ξ
μτ
|
sin
(
δ
2
3
+
δ
μ
τ
)
,
J
12
τe
JJ
=
J
0
J
J
+
c
12
s
12
c
23
s
2
23
|
ξ
eμ
|
sin
(
δ
12
+
δ
eμ
)
−
c
1
2
s
12
s
2
3
1+
s
2
2
3
|
ξ
e
τ
|
sin
(
δ
1
2
+
δ
2
3
+
δ
e
τ
)
,
(
4.64
)
to
g
et
h
er wit
h
t
h
eequa
l
it
y
J
23
τe
J
J
=
J
31
τe
JJ
=
J
31
eμ
JJ
=
J
23
eμ
J
a
s
g
oo
d
approximations.
This equality holds simply because the relevant Jarlsko
g
invariants involve
t
h
esma
ll
est matrix e
l
emen
t
V
e
V
V
3
∼
s
1
3
(
Malinsky
e
ta
l.
,
2009
)
.
Eq.
(
4.64
)
shows that the deviation o
f
J
ij
αβ
J
f
r
o
m
J
0
JJ
is
g
overne
db
yt
he
m
a
g
nitude o
f
|
ξ
α
β
|
and ma
y
reach th
e
O
(
1
0
−
3
)
level. Hence the non-unitar
y
CP-violating effects are possible to be comparable with or even larger tha
n
the unitary
C
P-violatin
g
e
ff
ect characterized b
y
J
0
JJ
itsel
f
, in particular when
θ
13
i
sver
y
small. In th
e
θ
13
→
0
limit we hav
e
J
0
JJ
→
0bu
t
J
ij
αβ
J
=0.Itis
therefore important to detect CP violation in the future precision neutrino
oscillation experiments, so as to probe possible non-unitarit
y
o
f
V
as o
n
e
s
triking consequence of the TeV-scale seesaw mechanisms.
4
.5.2 Mixin
gA
n
g
les and CP-violatin
g
Phase
s
A
lthough the parametrization o
f
V
i
nEq.
(
4.60
)
is valid for an arbitrar
y
n
umber of heavy Majorana neutrinos, its unitarity-conserving part
(
V
0
VV
)
an
d
u
nitarity-violating part
(
ξ
)
consist of two sets of different parameters
(
i.e.
,
θ
ij
a
n
d
δ
ij
ve
r
sus
ξ
α
β
)
. Here we concentrate on the type-I and type-
(
I+II
)
seesaw
m
echanisms with three heav
y
Ma
j
orana neutrinos and propose a “standard
”
p
arametrization o
f
V
in terms of only rotation angles and phase angles
(
Xing,
2
008b
)
. This parametrization is phenomenologically useful because it ma
y
reveal the correlation between the char
g
ed-current interactions o
f
li
g
ht and
h
eavy Majorana neutrinos in a much more transparent manner.
4.5 Non-unitary Neutrino Mixing and
C
P Violation 153
Wh
en on
ly
t
h
ree
h
eav
y
Ma
j
orana neutrinos are concerne
d
,t
h
estan
d
ar
d
c
har
g
ed-current interactions o
f
ν
i
an
d
N
i
NN
i
nEq.
(
4.56
)
are simplified t
o
L
cc
=
g
√
2
(
eμ
τ
)
L
γ
μ
⎡
⎣
V
⎛
⎝
⎛⎛
ν
1
ν
2
ν
3
⎞
⎠
⎞⎞
L
+
R
⎛
⎝
⎛⎛
N
1
NN
N
2
N
N
N
3
NN
⎞
⎠
⎞⎞
L
⎤
⎦
W
−
μ
W
+h
.
c
.,
(
4.65
)
whe
r
e
VV
†
+
RR
†
=
1
holds
.N
ote that
V
a
n
d
R
b
e
l
ongtot
h
e same
6
×
6
u
nitary matrix, which can be parametrized in terms o
f
15 rotation an
g
les
θ
ij
a
nd 15 phase angle
s
δ
i
j
(
for
1
i
<
j
6
)
. Then the common parameters o
f
V
a
n
d
R
c
haracterize their correlation. Let us define the 2-dimensional
(
1,2
),
(
1,3
)
and
(
2,3
)
rotation matrices in a 6-dimensional complex space
:
O
12
=
⎛
⎜
⎛⎛
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎝
⎜⎜
c
1
2
ˆ
s
∗
1
2
0000
−
ˆ
s
12
c
12
0000
00
1
000
0 00100
00001
0
0 00001
⎞
⎟
⎞
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎟⎟
,
O
1
3
=
⎛
⎜
⎛⎛
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎝
⎜⎜
c
1
3
0ˆ
∗
1
3
000
0
10 00
0
−
ˆ
s
13
0
c
13
000
0 00 100
0
00 01
0
00000
1
⎞
⎟
⎞
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎟⎟
,
O
2
3
=
⎛
⎜
⎛⎛
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎝
⎜⎜
1 0 0 000
0
c
2
3
ˆ
s
∗
2
3
0
0
0
0
−
ˆs
23
c
23
000
0 0 0 100
00 001
0
00 000
1
⎞
⎟
⎞
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎟⎟
,
(
4.66
)
where
c
ij
,
s
ij
and ˆ
ij
aredefinedinthesamewayasdoneinEq.
(
4.63
)
.Othe
r
2
-
d
im
e
n
s
i
o
n
al
r
otat
i
o
nm
at
ri
ces
O
i
j
(
for 1
i
<j
6
)
can analogously be
d
efined
(
Harari and Leuer, 1980; Fritzsch and Plankl, 1987
)
. We parametrize
the
6
×
6
unitar
y
transformation as
AR
BU
⊗
V
0
V
V
0
01
=[
O
56
O
46
O
36
O
26
O
16
O
45
O
35
O
25
O
15
O
34
O
2
4
O
1
4
]
⊗
[
O
23
O
13
O
12
]
,
(
4.67
)
where the ex
p
licit
f
orm o
f
V
0
VV
h
as been presented in Eq.
(
4.63
)
. Comparin
g
E
q.
(
4.67
)
with Eq.
(
4.38
)
or Eq.
(
4.55
)
,weimmediatelyget
V
=
AV
0
V
V
.I
t
is obv
i
ous that
V
→
V
0
V
V
in
t
h
e
limi
tof
A
→
1
(
or equivalently,
R
→
0
)
.
Th
us
A
s
i
g
ni
fi
es the non-unitarity o
f
V
.A
fter a len
g
thy but strai
g
htforward
c
alculation, we obtain the ex
p
ressions o
f
A
and
R
as follows:
15
44
S
eesaw Mechanisms o
f
Neutrino Masse
s
A
=
⎛
⎜
⎛⎛
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎝
⎜⎜
c
1
4
c
1
5
c
1
6
00
−
c
14
c
1
5
ˆ
s
1
6
ˆ
s
∗
2
6
−
c
1
4
ˆ
s
1
5
ˆ
s
∗
2
5
c
2
6
−
ˆ
s
14
ˆs
∗
24
c
25
c
26
c
2
4
c
2
5
c
2
6
0
−
c
14
c
15
ˆ
s
16
c
26
ˆ
s
∗
36
+
c
14
ˆ
s
15
ˆ
s
∗
25
ˆ
s
26
ˆs
∗
36
−
c
14
ˆ
s
15
c
25
ˆ
s
∗
35
c
36
+ˆ
14
ˆ
s
∗
2
4
c
25
ˆ
s
26
ˆ
s
∗
36
+ˆ
14
ˆ
s
∗
24
ˆ
s
25
ˆs
∗
35
c
36
−
ˆ
s
14
c
2
4
ˆ
s
∗
3
4
c
35
c
36
−
c
24
c
25
ˆ
s
26
ˆ
s
∗
36
−
c
2
4
ˆ
s
2
5
ˆ
s
∗
35
c
36
−
ˆ
s
24
ˆ
s
∗
3
4
c
35
c
36
c
34
c
35
c
36
⎞
⎟
⎞⎞
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎠
⎟⎟
,
R
=
⎛
⎜
⎛⎛
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜
⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜
⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎝
⎜⎜
ˆ
s
∗
1
4
c
15
c
16
ˆ
s
∗
1
5
c
16
ˆ
s
∗
16
−
ˆ
s
∗
14
c
1
5
ˆ
s
16
ˆ
s
∗
26
−
ˆ
s
∗
1
4
ˆ
s
1
5
ˆ
s
∗
2
5
c
26
+
c
14
ˆs
∗
24
c
25
c
26
−
ˆ
s
∗
1
5
ˆ
s
16
ˆ
s
∗
2
6
+
c
15
ˆ
s
∗
25
c
26
c
16
ˆ
s
∗
26
−ˆ
s
∗
14
c
15
ˆ
s
16
c
26
ˆ
s
∗
36
+ˆ
∗
14
ˆ
s
15
ˆ
s
∗
25
ˆs
26
ˆ
s
∗
36
−
ˆ
s
∗
14
ˆ
s
15
c
25
ˆ
s
∗
35
c
36
−
c
14
ˆ
s
∗
24
c
25
ˆ
s
26
ˆ
s
∗
36
−
c
14
ˆ
s
∗
24
ˆ
s
25
ˆ
s
∗
35
c
36
+
c
14
c
2
4
ˆ
s
∗
34
c
35
c
36
−
ˆ
s
∗
15
ˆ
s
1
6
c
26
ˆ
s
∗
36
−
c
15
ˆ
s
∗
2
5
ˆ
s
26
ˆ
s
∗
36
+
c
15
c
25
ˆ
s
∗
35
c
36
c
1
6
c
2
6
ˆ
s
∗
36
⎞
⎟
⎞⎞
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟
⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟
⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎠
⎟⎟
.
(
4.68
)
O
ne can see that
A
a
n
d
R involve the same parameters: 9 rotation an
g
les an
d
9
phase angles, which signify non-unitary neutrino mixing and CP violation.
No
n
t
ri
v
i
al
A
is there
f
ore the brid
g
e between
V
=
AV
0
VV
a
n
d
R
.
Note that the trian
g
ular form o
f
A
is a salient feature of our
p
arametriza
-
tion. Some interesting consequences o
f
V
=
AV
0
V
V
c
an imme
d
iate
l
y
b
e
d
rawn
f
rom Eqs.
(
4.63
)
and
(
4.68
).
•
V
e
VV
3
=
c
14
c
1
5
c
1
6
ˆ
s
∗
1
3
holds.
G
ive
n
θ
1
3
=0f
or
V
0
VV
,whichmi
g
ht result
f
rom
certain
fl
avor s
y
mmetries
,
V
e
VV
3
tu
rn
souttova
ni
s
h.
•
T
he rati
o
|
V
e
VV
2
/V
e
VV
1
|
=t
a
n
θ
12
is completely irrelevant to the parameters of
A
or
R
. This result implies that the extraction o
f
θ
12
f
r
o
m
so
l
a
rn
eut
rin
o
o
scillation data should be independent o
f
the non-unitarit
y
o
f
V
.
•
|
V
e
VV
1
|
2
+
|
V
e
V
V
2
|
2
+
|
V
e
VV
3
|
2
=
c
2
14
c
2
15
c
2
16
h
o
ld
s. Hence non-vanis
h
in
g
θ
14
,
θ
15
a
n
d
θ
1
6
v
i
o
l
ate t
h
e
n
o
rm
a
liz
at
i
o
n
co
n
d
i
t
i
o
n
of t
hr
ee
m
at
rix
e
l
e
m
e
n
ts
in
t
he first row of V
.
•
m
ee
=
c
2
14
c
2
1
5
c
2
1
6
|
m
1
(
c
12
c
1
3
)
2
+
m
2
(ˆ
∗
12
c
1
3
)
2
+
m
3
(ˆ
∗
13
)
2
|
h
olds for the
eff
ective mass term o
f
the neutrinoless double-beta deca
y.
•
m
e
=
c
14
c
15
c
16
m
2
1
c
2
12
c
2
1
3
+
m
2
2
s
2
12
c
2
13
+
m
2
3
s
2
13
holds for the effective
mass term of the tritium beta decay
.
I
f
θ
14
,
θ
15
an
d
θ
16
a
re small
,
their effects o
n
m
e
an
d
m
ee
will be negligible
.
R
e
f
erences 15
5
I
n view o
f
the
f
act that the non-unitarit
y
o
f
V
m
ust be s
m
a
ll
effects
(
at most at the percent level
)
,weexpec
t
s
ij
O
(
0
.
1) (
fo
r
i
=1
,
2
,
3
and
j
=4
,
5
,
6)
to hold. Then we obtain
(
Xing, 2008b
)
A
=
1
−
⎛
⎜
⎛⎛
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎝
⎜⎜
1
2
6
i
=
4
s
2
1
i
00
6
i
=
4
ˆs
1
i
ˆ
s
∗
2
i
1
2
6
i
=4
s
2
2
i
0
6
i=
4
ˆ
s
1
i
ˆ
s
∗
3
i
6
i
=4
ˆ
s
2
i
ˆ
s
∗
3
i
1
2
6
i=
4
s
2
3
i
⎞
⎟
⎞
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎟
⎟
+
O
(
s
4
ij
)
,
R
=
0
+
⎛
⎝
⎛
⎛
ˆ
s
∗
14
ˆ
s
∗
15
ˆ
s
∗
16
ˆ
s
∗
2
4
ˆ
s
∗
25
ˆ
s
∗
26
ˆ
s
∗
34
ˆ
s
∗
35
ˆ
s
∗
36
⎞
⎠
⎞⎞
+
O
(
s
3
ij
)
(
4.69
)
a
s two excellent approximations. A strikin
g
consequence of the non-unitarity
of
V
is the loss o
f
universality
f
or the Jarlsko
g
invariants o
fC
P violation, as
we have shown in Section 4.5.1. Taking account of
ξ
=
RR
†
/
2
,
we arrive at
ξ
ee
=
1
2
6
i
=
4
s
2
1
i
,
ξ
μμ
=
1
2
6
i=
4
s
2
2
i
,ξ
ττ
=
1
2
6
i
=4
s
2
3
i
;
ξ
eμ
=
1
2
6
i
=
4
s
1
i
s
2
i
s
in
(
δ
2i
−
δ
1
i
)
,
ξ
e
τ
=
1
2
6
i
=
4
s
1
i
s
3
i
s
in
(
δ
3
i
−
δ
1
i
)
,
ξ
μ
τ
=
1
2
6
i
=
4
s
2
i
s
3
i
s
in
(
δ
3
i
−
δ
2
i
)
.
(
4.70
)
The deviation of the Jarlskog invariants
J
i
j
αβ
J
fr
o
m
J
0
JJ
c
an a
l
so
b
e given in
terms o
f
those non-unitary neutrino mixin
g
an
g
les and
C
P-violatin
g
phases.
Finally, it worth mentioning that one may simplify the above results to
s
ome extent w
h
en
d
iscussin
g
eit
h
er t
he
minimal
type-I seesaw mec
h
anism
with two heavy Majorana neutrinos
(
Frampton
et al.
, 2002;
G
uo
et al.
,
2007
)
or the
m
inimal type-
(
I+II
)
seesaw mechanism with only one heavy Majorana
n
eutrino
(
Gu
et al
.
,
2006;
C
han
et al
.
,
2007
)
.
R
e
f
erence
s
A
bada
,A
.
,
et al
.
,
2007
,JH
E
P
07
1
2
, 061
.
A
ltarelli, G., and Meloni, D., 2009, Nucl. Phys. B
8
0
9
,
158
.
A
ntusch, S.,
et al.
, 2006, JHEP
0610
,
084
.