Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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136
4
S
eesaw Mechanisms o
f
Neutrino Masse
s
m
ore clearl
y
in
S
ection 4.5, and its deviation
f
rom unitarit
y
is o
f
O
(
M
2
D
MM
/M
2
R
MM
)
.
We consider two interestin
g
possibilities:
•
M
D
MM
∼
O
(
10
2
)
GeV an
d
M
R
MM
∼
O
(
10
1
5
)
GeVtoget
M
ν
MM
∼
O
(
1
0
−
2
)
eV. In
th
i
sco
n
ve
n
t
i
o
n
al a
n
d
natural
case,
M
D
MM
/M
R
MM
∼O
(
1
0
−
13
)
holds. Hence the
non-unitarit
y
o
f
V
i
son
ly
at t
he
O
(
1
0
−
26
)
level, too small to be observed
.
•
M
D
MM
∼O
(
1
0
2
)
GeV and
M
R
MM
∼O
(
1
0
3
)
GeVtoget
M
ν
MM
∼O
(
1
0
−
2
)
eV.
I
n
th
i
s
unnatural
c
ase, a si
g
ni
fi
cant “structural cancellation” has to b
e
im
p
osed on the textures o
f
M
D
MM
a
n
d
M
R
MM
.
B
ecause of
M
D
MM
/M
R
MM
∼
O
(
0
.
1),
t
he non-unitarity o
f
V
c
an reach the percent level and may lead to som
e
o
bservable e
ff
ects at low ener
g
ies.
Now we d
i
scuss
h
ow to
r
ea
liz
et
h
eabove
“
st
r
uctu
r
a
l
ca
n
ce
ll
at
i
o
n”
fo
r
t
h
e
t
y
pe-I seesaw mechanism at the TeV scale. For the sake of simplicit
y
, we take
t
h
ebas
i
so
f
M
R
MM
=
D
iag
{
M
1
,M
2
MM
,M
3
MM
}
f
or three heavy Majorana neutrinos
(
N
1
NN
,N
2
NN
,N
3
N
N
)
.Itiswellknowntha
t
M
ν
MM
va
ni
s
h
es
i
f
M
D
M
M
=
m
⎛
⎝
⎛⎛
y
1
y
2
y
3
α
y
1
α
y
2
αy
3
βy
1
β
y
2
β
y
3
⎞
⎠
⎞⎞
an
d
3
i
=
1
y
2
i
M
i
MM
=0
(
4.33
)
s
imultaneously hold
(
Kersten and Smirnov, 2007
)
. Tiny neutrino masses ca
n
be generated from tiny corrections to the texture o
f
M
D
MM
i
nEq.
(
4.33
)
.For
examp
l
e
,
M
D
MM
=
M
D
MM
−
X
D
w
i
th
M
D
MM
g
iven a
b
ove an
d
b
ein
g
asma
ll d
imen-
s
ionless parameter
(
i.e.
,
|
|
1)
yield
s
M
ν
M
M
=
−
M
D
MM
M
−
1
R
M
M
M
T
D
MM
M
D
MM
M
−
1
R
MM
X
T
D
+
X
D
M
−
1
R
M
M
M
T
D
MM
,
(
4.34
)
f
r
o
m
w
hi
c
h
M
ν
MM
∼
O
(
1
0
−
2
)
eV can be obtained by adjusting the size of
.
A
lot of attention has recentl
y
been paid to a viable t
y
pe-I seesaw model
a
nd its collider signatures at the TeV scale
(
Xing, 2008a; 2009
)
. At least th
e
f
ollowin
g
lessons can be learnt:
•
T
wo necessar
y
conditions must be satisfied in order to test a t
y
pe-I seesaw
model at the LHC:
(
a
)
M
i
M
M
are o
f
O
(
1
)
TeV or smaller; and
(
b
)
the strengt
h
o
f light-heavy neutrino mixing
(
i.e.,
M
D
MM
/M
R
MM
)
is large enough. Otherwise,
it would be im
p
ossible to
p
roduce and detec
t
N
i
NN
a
t the LHC
.
•
T
he collider signatures o
f
N
i
N
N
a
re essentially decoupled from the mass and
mixin
g
parameters o
f
three li
g
ht neutrino
s
ν
i
. For instance, t
h
esma
ll
p
arame
t
er
i
nEq.
(
4.34
)
has nothing to do with the ratio
M
D
MM
/M
R
MM
.
•
T
he non-unitarity o
f
V
m
ight lead to some observable effects in long
-
b
aseline neutrino oscillations and some other lepton-
fl
avor-violatin
g
or
lepton-number-violating processes, i
f
M
D
MM
/M
R
MM
i
so
f
O
(0
.1
)
. More discus-
sions will be given in Section 4.5
.
•
T
he clean LH
C
si
g
natures o
f
heavy Majorana neutrinos are th
e
Δ
L
=
2
like-sign dilepton events
(
Keung and Senjanovic, 1983
)
,suchas
pp
→
W
∗
±
W
∗
±
→
μ
±
μ
±
j
j
(
a collider analogue to the neutrinoless double-
b
eta decay
)
and
pp
→
W
∗±
→
μ
±
N
i
NN
→
μ
±
μ
±
jj
(
a dominant channe
l
due to the resonant
p
roduction of
N
i
NN
)
.
4
.
3S
eesaw Mechanisms at the Te
VS
cale 1
37
S
ome instructive and comprehensive anal
y
ses o
f
possible LH
C
events
f
or
a
s
ingle heavy Majorana neutrino have recently been done
(
Han and Zhang,
2
006; del Aguila and Aguilar-Saavedra, 2009a; Atr
e
et a
l.
, 2009
)
, but the
y
onl
y
serve
f
or illustration because such a simpli
fi
ed t
y
pe-I seesaw scenario i
s
a
ctuall
y
unrealistic
.
4
.3.2 T
y
pe-II Seesaw Mechanis
m
The type-II seesaw
f
ormula
M
ν
M
M
=
Y
Δ
Y
v
Δ
=
λ
Δ
Y
Δ
Y
v
2
/M
Δ
M
h
as a
l
rea
d
y
b
een
given in Eq.
(
4.13
)
. Note that the third term in Eq.
(
4.7
)
violates both
L
a
nd
B
−
L
,
so the smallness of
λ
Δ
i
s naturally allowed according to ’t Hooft’
s
n
atura
l
ness criterion; i.e., settin
g
λ
Δ
=
0 will increase the s
y
mmetr
y
o
f
L
l
epton
(
’t Hooft, 1980
)
.Give
n
M
Δ
M
∼
O
(
1
)
TeV, for example, this seesaw mechanis
m
wor
k
s to generate
M
ν
MM
∼O
(
10
−
2
)
eV provide
d
λ
Δ
Y
Δ
Y
∼O
(
1
0
−
12
)
holds. Not
e
al
so t
h
at t
h
e neutrino mixin
g
matrix
V
i
sexact
ly
unitar
y
in t
h
et
y
pe-II seesa
w
m
echanism, simply because the heavy de
g
rees of freedom do not mix with
t
h
e
l
ig
h
tones
.
Th
ere are tota
ll
y seven p
h
ysica
l
Hi
gg
s
b
osons in t
h
e type-II seesa
w
s
cheme: doubly-charged
H
++
an
d
H
−−
,
singly-charged
H
+
and
H
−
,
neutral
A
0
(
CP-odd
)
, and neutra
l
h
a
n
d
H
0
(
CP-even
)
,where
h
i
s the SM-like Higg
s
boson. Except
f
or
M
2
h
M
,
we
g
et a quasi-de
g
enerate mass spectrum
f
or othe
r
s
calars
(
Fileviez P´erez et al
.
, 2008; del Aguila and Aguilar-Saavedra, 2009a
)
:
M
2
H
M
±±
=
M
2
Δ
M
≈
M
2
H
M
0
≈
M
2
H
M
±
≈
M
2
A
M
0
. As a consequence, the decay channel
s
H
±±
→
W
±
H
±
a
n
d
H
±±
→
H
±
H
±
a
re kinematicall
yf
orbidden. The pro-
d
uction o
f
H
±±
at the LHC is mainly throug
h
q
¯
q
→
γ
∗
,Z
∗
→
H
++
H
−−
a
nd
q
¯
q
→
W
∗
→
H
±±
H
∓
processes, w
h
ic
hd
onot
d
epen
d
on t
h
esma
ll
Yu
k
awa
c
ouplin
g
s and thus may not be suppressed.
T
he typical collider signatures in this seesaw scenario are the lepton
-
n
um
b
er-vio
l
atin
g
H
±±
→
l
±
α
l
±
β
d
eca
y
saswe
ll
as
H
+
→
l
+
α
ν
a
n
d
H
−
→
l
−
α
ν
d
ecays
(
fo
r
α
,
β
=
e,
μ
,
τ
)
. Their branching ratio
s
B
(
H
±±
→
l
±
α
l
±
β
)=
|
(
M
ν
MM
)
αβ
|
2
ρ
,
σ
|
(
M
ν
MM
)
ρ
σ
|
2
2
−
δ
αβ
,
B
(
H
+
→
l
+
α
ν
)=
β
|
(
M
ν
M
M
)
αβ
|
2
ρ,
σ
|
(
M
ν
MM
)
ρσ
|
2
(
4.35
)
are closely related to the masses, flavor mixing angles and CP-violating phases
of three light neutrinos, becaus
e
M
ν
M
M
=
V
$
M
ν
M
M
V
T
w
i
th
$
M
ν
M
M
=D
ia
g
{
m
1
,m
2
,
m
3
}
h
olds.
S
ome detailed analyses o
f
such decay modes to
g
ether with the LH
C
s
ignatures o
f
H
±±
a
n
d
H
±
b
osons have been done in the literature
(
see, for
example, Fileviez P´erez
et al.
,
2008
).
138
4
S
eesaw Mechanisms o
f
Neutrino Masse
s
I
t is worth pointin
g
out that the
f
ollowin
g
dimension-6 operator can easily
be derived from the t
y
pe-II seesaw mechanism:
L
d=
6
Λ
2
=
−
(
Y
Δ
Y
)
αβ
(
Y
Δ
Y
)
†
ρ
σ
4
M
2
Δ
M
α
L
γ
μ
σ
L
β
L
γ
μ
ρ
L
,
(
4.36
)
which has two immediate low-ener
g
ye
ff
ects: the non-standard interactions o
f
n
eutrinos and the lepton-flavor-violating interactions of charged leptons.
A
n
a
nalysis of such effects provides us with some preliminary information
(
Malin
-
s
k
y
et al.
, 2009
)
:
(
a
)
the magnitudes of non-standard interactions of neutrinos
a
nd the widths of lepton-flavor-violating tree-level decays of charged leptons
a
re
d
epen
d
ent on t
h
e neutrino masses
m
i
a
nd
fl
avor mixin
g
parameters o
f
V ;
(
b
)
for a long-baseline neutrino oscillation experiment, the neutrino bea
m
encounters the Earth matter and the electron-type non-standard interaction
c
ontributes to the matter potential; and
(
c
)
at a neutrino factory, the lepton-
flavor-violatin
g
processes
μ
−
→
e
−
ν
e
ν
μ
a
nd
μ
+
→
e
+
ν
e
ν
μ
c
ould cause som
e
wrong-sign muons at a near detector. Current experimental constraints tell us
that such low-ener
g
ye
ff
ects are very small, but they mi
g
ht be experimentall
y
a
ccessible in the future
p
recision measurements.
4
.3.3 Type-
(
I+II
)
Seesaw Mechanis
m
The type-
(
I+II
)
seesaw mechanism can be achieved by combining the neu-
trino mass terms in Eqs.
(
4.6
)
and
(
4.7
)
. After spontaneous gauge symmetr
y
breakin
g
,wearele
f
t with the overall neutrino mass ter
m
−
L
mass
=
1
2
(
ν
L
N
c
R
NN
)
M
L
M
M
M
D
M
M
M
T
D
MM
M
R
MM
ν
c
L
N
R
NN
+
h
.
c
.
,
(
4.37
)
where
M
D
M
M
=
Y
ν
YY
v
/
√
2
an
d
M
L
M
M
=
Y
Δ
Y
v
Δ
w
ith
H
≡
v/
√
2an
d
Δ
≡
v
Δ
c
orrespondin
g
to the vacuum expectation values o
f
the neutral component
s
o
f
the Hi
gg
s doublet
H
and the Hi
gg
striple
t
Δ
.
This mass term is e
q
uivalen
t
to the one given in Eq.
(
3.30
)
.The6
×
6 neutrino mass matrix in Eq.
(
4.37
)
i
s symmetric and can be dia
g
onalized by the unitary trans
f
ormation
VR
SU
†
M
L
MM
M
D
MM
M
T
D
MM
M
R
MM
VR
SU
∗
=
$
M
ν
MM
0
0
$
M
N
M
,
(
4.38
)
whe
r
e
$
M
ν
MM
=
Dia
g
{
m
1
,
m
2
,m
3
}
a
n
d
$
M
N
M
=
Dia
g
{
M
1
,M
2
MM
,M
3
M
M
}
.
N
eedless to
s
a
y,
V
†
V
+
S
†
S
=
VV
†
+
RR
†
=
1
h
olds as a consequence of the unitarit
y
o
f
t
h
eabovet
r
a
n
s
f
o
rm
at
i
o
n. H
e
n
ce
V
,
the flavor mixing matrix of three ligh
t
Ma
j
orana neutrinos, must be non-unitar
y
i
f
R
a
n
d
S
a
r
e
n
o
nz
e
r
o.
I
n the leading-order approximation, the type-
(
I+II
)
seesaw formula reads
M
ν
M
M
=
M
L
M
M
−
M
D
M
M
M
−
1
R
M
M
M
T
D
M
M
.
(
4.39
)
4
.
3S
eesaw Mechanisms at the Te
VS
cale 1
39
H
ence type-I and type-II seesaws can be re
g
arded as two extreme cases o
f
th
e
type-
(
I+II
)
seesaw. Note that the two mass terms in Eq.
(
4.39
)
are possibly
c
omparable in magnitude. If both of them are small, their contributions to
M
ν
MM
m
ay have si
g
ni
fi
cant inter
f
erence e
ff
ects which make it practically impossibl
e
to distinguish between type-II and type-
(
I+II
)
seesaws
(
Ren and Xing, 2008
);
but if both of them are large, their contributions t
o
M
ν
MM
must be dest
r
uct
i
ve.
The latter case unnaturally requires a si
g
ni
fi
cant cancellation between two
big quantities in order to obtain a small quantity
(
Chao
e
tal
.
,
2008a
)
, but it
i
s interesting in t
h
e sense t
h
at it mig
h
t give rise to some o
b
serva
bl
eco
ll
i
d
e
r
s
i
g
natures o
f
heavy Majorana neutrinos
.
Let us briefly describe a particular type-
(
I+II
)
seesaw model and commen
t
onitspossibleLH
C
si
g
natures. First, we assume that both
M
i
MM
a
n
d
M
Δ
M
a
r
e
of
O
(
1
)
TeV. Then the production o
f
H
±±
a
n
d
H
±
bosons at the LH
C
i
s guaranteed, and their lepton-number-violating signatures will probe the
H
iggs triplet sector of the type-
(
I+II
)
seesaw mechanism. On the other hand
,
O
(
M
D
M
M
/M
R
MM
)
O
(0
.
1)
is possible as a result of
O
(
M
R
MM
)
∼O
(
1
)
TeV an
d
O
(
M
D
M
M
)
O
(
v
)
, such that appreciable signatures of
N
i
NN
can
b
eac
h
ieve
d
at t
he
L
H
C
.
S
econd, the small mass scale o
f
M
ν
MM
i
mp
l
ies t
h
at t
h
ere
l
atio
n
O
(
M
L
MM
)
∼
O
(
M
D
M
M
M
−
1
R
MM
M
T
D
M
M
)
must hold. In other words, it is the significant but incomplet
e
c
ance
ll
ation
b
etwee
n
M
L
MM
a
n
d
M
D
MM
M
−
1
R
M
M
M
T
D
MM
t
erms t
h
at resu
l
ts in t
h
enon
-
vanishin
g
but tiny masses
f
or three li
g
ht neutrinos. We admit that dan
g
erou
s
radiative corrections to two mass terms of
M
ν
MM
require a delicate fine-tunin
g
of the cancellation at the loop level
(
Cha
o
et al
.
,
2008b
)
. But this scenari
o
a
ll
owsusto
r
eco
n
st
r
uct
M
L
MM
via the excellent a
pp
roximatio
n
M
L
MM
=
V
$
M
ν
MM
V
T
+
R
$
M
N
M
R
T
≈
R
$
M
N
M
R
T
, such that the elements o
f
Y
Δ
Y
read
(
Y
Δ
Y
)
αβ
=
(
M
L
MM
)
α
β
v
Δ
≈
3
i
=
1
R
α
i
R
β
i
M
i
MM
v
Δ
,
(
4.40
)
whe
r
e
α
a
n
d
β
ru
n
ove
r
e
,
μ
a
n
d
τ
.
T
h
is resu
l
timp
l
ies t
h
at t
h
e
l
eptoni
c
d
eca
y
so
f
H
±±
a
n
d
H
±
b
osons de
p
end on both R
a
n
d
M
i
M
M
,
which determine
the production and decays of
N
i
NN
. Thus we have established an interestin
g
c
orre
l
ation
b
etween t
h
esin
gl
y- or
d
ou
bl
y-c
h
ar
g
e
d
Hi
gg
s
b
osons an
d
t
h
e
h
eavy
Majorana neutrinos. If the correlative si
g
natures of
H
±
,
H
±±
a
n
d
N
i
NN
can be
observed at the LHC, they will serve for a direct experimental test of this
type-
(
I+II
)
seesaw model.
T
o illustrate
,
here we focus on th
e
m
inima
l
type-
(
I+II
)
seesaw model wit
h
a
sing
l
e
h
eavy Majorana neutrino, w
h
ere
R
c
an be parametrized in terms o
f
t
h
ree rotation an
gl
e
s
θ
i
4
an
d
t
h
ree p
h
ase an
gl
e
s
δ
i
4
(
fo
r
i
=
1
,
2
,
3
)(
Xing
,
2
008b
)
.Inthiscasewehav
e
ω
1
≡
σ
(
pp
→
μ
+
μ
+
W
−
X
)
|
N
1
σ
(
pp
→
μ
+
μ
+
H
−
X
)
|
H
++
≈
σ
N
σ
H
·
s
2
14
+
s
2
24
+
s
2
3
4
4
,
ω
2
≡
σ
(
pp
→
μ
+
μ
+
W
−
X
)
|
N
1
σ
(
pp
→
μ
+
μ
+
H
−−
X
)
|
H
+
+
≈
σ
N
σ
pair
·
s
2
14
+
s
2
24
+
s
2
34
4
(
4.41
)
1
4
0
4
S
eesaw Mechanisms o
f
Neutrino Masse
s
10
-
3
10
-
2
10
-
1
1
10
10
2
10
3
ω
i
ω
2
ω
1
M
Δ
M
=1Te
V
ω
2
ω
1
M
Δ
M
= 500 Ge
V
M
1
[
GeV
]
s
3
4
=
0
.
1
s
24
=
0
.
1
v
Δ
=1
G
e
V
s
1
4
=
0
Fi
g
. 4.1 An illustration of the correlative si
g
natures o
f
N
1
an
d
H
±±
at the LHC
with an integrated luminosity of 300 f
b
−
1
(
Chao
et al.
, 2008b. With permission
from the Elsevier
)
f
o
r
s
i
4
≡
s
i
n
θ
i
4
O
(
0.1
)
,wher
e
σ
N
≡
σ
(
pp
→
l
+
α
N
1
N
N
X
)
/
|
R
α
1
|
2
,
σ
H
≡
σ
(
pp
→
H
++
H
−
X
)
an
d
σ
pair
≡
σ
(
pp
→
H
++
H
−−
X
)
are three reduced
c
ross sections
(
Chao
et al
.
,
2008b
)
. Fig. 4.1 shows a numerical illustration of
ω
1
a
n
d
ω
2
c
h
anging wit
h
M
1
at the LHC with an integrated luminosity of
300 f
b
−
1
,anditmay
g
ive one a ball-park
f
eelin
g
o
f
possible collider si
g
natures
of
N
1
N
N
an
d
H
±±
and their correlation
.
I
f an additional heavy Majorana neutrino with a nearly equal mass i
s
i
ntro
d
uce
d
into t
h
ea
b
ove scenario, t
h
eco
ll
i
d
er si
g
natures can
b
e resonant
l
y
enhanced due to the constructive interference between the contribution
s
f
rom two heavy Majorana neutrinos
(
Cha
o
et al.
,
2010
)
.IftwoheavyMajo-
rana neutrinos with nearly de
g
enerate masses
f
orm a pseudo-Dirac particle
,
h
owever, the lepton-number-violating signatures will be highly suppressed
a
nd the trilepton signatures arising from the processe
s
pp
→
l
+
α
l
+
β
l
−
γ
ν
a
n
d
pp
→
l
−
α
l
−
β
l
+
γ
ν
(
for α, β,
γ
=
e
,μ,
τ
)
should be most promising. This cas
e
h
as been discussed in the minimal type-I seesaw model with a softly-broken
gl
o
b
a
l
U
(
1
)
symmetry
(
Zhang and Zhou, 2010
)
.
4
.3.4 T
y
pe-III Seesaw Mechanism
The lepton mass terms in the type-III seesaw scheme have already been
g
iven
i
nEq.
(
4.9
)
. After spontaneous gauge symmetry breaking, we are left wit
h
−
L
mass
=
1
2
(
ν
L
Σ
0
)
0
M
D
MM
M
T
D
M
M
M
Σ
M
ν
c
L
Σ
0
c
+
h
.
c
.,
4
.
3S
eesaw Mechanisms at the Te
VS
cale 14
1
−L
mass
=
(
e
L
Ψ
L
ΨΨ
)
M
l
M
√
2
M
D
MM
0
M
Σ
M
E
R
Ψ
R
ΨΨ
+
h
.
c
.
,
(
4.42
)
respectively,
f
or neutral and char
g
ed
f
ermions, wher
e
M
l
M
=
Y
l
YY
v/
√
2,
M
D
MM
=
Y
Σ
Y
Y
v/
√
2
,
an
d
Ψ
=
Σ
−
+
Σ
+
c
.Thes
y
mmetric 6
×
6neutr
i
no mass matr
i
x
c
an be diagonalized by the following unitary transformation:
VR
SU
†
0
M
D
MM
M
T
D
MM
M
Σ
M
VR
SU
∗
=
$
M
ν
MM
0
0
$
M
Σ
M
,
(
4.43
)
i
n
w
hi
c
h
$
M
ν
MM
=Dia
g
{
m
1
,
m
2
,
m
3
}
a
n
d
$
M
Σ
M
=Dia
g
{
M
1
,M
2
MM
,M
3
M
M
}
.In
t
h
e
l
eading-order approximation, this diagonalization yields the type-III seesaw
fo
rm
u
l
a
M
ν
MM
=
−
M
D
MM
M
−
1
Σ
M
M
T
D
MM
,
which is equivalent to the one derived
f
ro
m
the effective dimension-5 operator in Eq.
(
4.13
)
. Let us use one sentence t
o
c
omment on the similarities and differences between type-I and type-III see-
s
aw mechanisms
(
Abad
a
et al
.
,
2007
)
: the non-unitarity of the 3
×
3
neutr
i
n
o
mi
x
i
n
g
matr
i
x V
h
as appeared in both cases, althou
g
h the modified couplin
gs
b
etween t
h
e
Z
0
b
oson and three light neutrinos differ and the non-unitary
fl
avor mixin
g
is also present in the couplin
g
s between th
e
Z
0
boso
n
a
n
dth
r
ee
c
harged leptons in the type-III seesaw scenario
.
At the LHC, the typical lepton-number-violating signatures of the type-
III
seesaw
m
echa
ni
s
m
ca
n
be
pp
→
Σ
+
Σ
0
→
l
+
α
l
+
β
+
Z
0
W
−
(
→
4
j
)
and
pp
→
Σ
−
Σ
0
→
l
−
α
l
−
β
+
Z
0
W
+
(
→
4
j
)
processes. A detailed analysis of such
c
ollider signatures have been done in the literature
(
Franceschin
i
et al.
,
2008;
d
el Aguila and Aguilar-Saavedra, 2009a
)
. As for the low-energy phenomenol-
ogy, a consequence of this seesaw scenario is the non-unitarity of the 3
×
3
fl
avor mixin
g
matri
x
˜
V
(
≈
V
)
in both charged- and neutral-current inter-
a
ctions
(
Abada
e
tal
.
, 2007
)
. Current experimental bounds on the deviation
o
f
˜
V
˜
V
†
from the identity matrix are at the 0
.
1
%
level, much stronger than
those obtained in the type-I seesaw scheme, just because the
fl
avor-chan
g
in
g
processes with charged leptons are allowed at the tree level in the type-II
I
seesaw
m
echa
ni
s
m
.
I
t is worth mentioning that an interesting type-
(
I+III
)
seesaw model has
recently been proposed
(
Bajc and Senjanovic, 2007
)
, and its phenomenolog-
i
ca
l
an
d
cosmo
l
o
g
ica
l
consequences to
g
et
h
er wit
h
its possi
bl
eco
ll
i
d
er si
g
na
-
tures have also been explored
(
Bajc
et al
.
,
2007;
A
rhri
b
et al
., 2010
)
.
4
.3.5 Inverse
S
eesaw Mechanism
Given the naturalness and testability as two prerequisites, the so-called in
-
verse or double seesaw mechanism
(
Wyler and Wolfenstein, 1983; Mohapatra
a
nd Valle, 1986; Ma, 1987
)
is another interesting possibility of generating tiny
n
eutrino masses at the TeV scale. The idea of this seesaw picture is to add
t
h
ree
h
eavy ri
gh
t-
h
an
d
e
d
neutrino
s
N
R
N
N
,three
S
M
g
au
g
e-sin
g
let neutrino
s
S
R
1
42 4
S
eesaw Mechanisms o
f
Neutrino Masse
s
a
n
d
one Hi
gg
ssin
gl
et
Φ
into the
S
M, such that the
g
au
g
e-invariant lepto
n
m
ass terms can be written a
s
−L
l
e
p
ton
=
L
Y
l
YY
HE
R
+
L
Y
ν
Y
Y
˜
HN
R
N
N
+
N
c
R
NN
Y
S
YY
Φ
S
R
+
1
2
S
c
R
M
μ
M
S
R
+h
.
c
.,
(
4.44
)
where
M
μ
M
i
s naturally small according to ’t Hooft’s naturalness criterion
(
’t Hooft, 1980
)
, because its corresponding mass term violates the lepto
n
n
umber.
A
fter spontaneous
g
au
g
e symmetry breakin
g
, the overall neutrino
m
ass term turns out to be
−
L
m
ass
=
1
2
(
ν
L
N
c
R
NN
S
c
R
)
⎛
⎝
⎛⎛
0
M
D
MM
0
M
T
D
MM
0
M
S
M
0
M
T
S
M
M
μ
M
⎞
⎠
⎞⎞
⎛
⎝
⎛⎛
ν
c
L
N
R
NN
S
R
⎞
⎠
⎞⎞
,
(
4.45
)
whe
r
e
M
D
MM
=
Y
ν
YY
H
a
n
d
M
S
M
=
Y
S
Y
Y
Φ
.A
dia
g
onalization of the symmetric
9
×
9matrixinEq.
(
4.45
)
leads us to the effective light neutrino mass matri
x
M
ν
MM
=
M
D
MM
M
T
S
M
−
1
M
μ
M
(
M
S
M
)
−
1
M
T
D
MM
(
4.46
)
i
n the leading-order approximation
(
Chan
et al.
,
2009
)
. Hence the smallness o
f
M
ν
MM
i
s attributed to both the smallness o
f
M
μ
M
i
tself and the doubl
y
-suppressed
M
D
MM
/M
S
M
term fo
r
M
D
MM
M
S
M
.
For examp
l
e
,
M
μ
M
∼O
(
1
)
keV an
d
M
D
MM
/M
S
M
∼
O
(
10
−
2
)
naturally yield a sub-e
V
M
ν
MM
.O
ne ha
s
M
ν
MM
=
0
in
the l
imi
t
M
μ
M
→
0
,
reflectin
g
the restoration of the sli
g
htly-broken lepton number. The heavy
s
ector consists of three pairs of pseudo-Dirac neutrinos, whose CP-conjugate
d
Majorana components
h
ave a tiny mass sp
l
ittin
g
c
h
aracterize
db
y
O
(
M
μ
M
).
4.4 Multiple Seesaw Mechanisms
We have mentioned that a test of the t
y
pe-I seesaw mechanism at the TeV
s
cale necessitates an appreciable magnitude of
M
D
MM
/M
R
MM
s
oastoma
k
eitpos
-
s
ible to produce and detect heav
y
Ma
j
orana neutrinos at the LH
C
.Thi
s
prerequisite unavoidably requires a terrible fine-tuning o
f
M
D
MM
a
n
d
M
R
MM
,
be
-
c
ause one
h
as to impose
M
R
MM
∼
1Te
V,
M
D
MM
/M
R
MM
∼
10
−
3
to
10
−
1
a
n
d
M
ν
MM
∼
0
.
1
eV simu
l
taneous
ly
on t
h
e seesaw re
l
ation
M
ν
MM
=
−
M
D
M
M
M
−
1
R
MM
M
T
D
MM
(
Xing, 2009
)
.
It is therefore desirable to invoke new ideas to resolve this
u
nnaturalness
pro
bl
em
b
ui
l
tint
h
e TeV seesaw mec
h
anism.
A
m
ulti
p
le seesaw mechanism at the TeV scale ma
y
satis
fy
both
natu
-
ralness an
d
t
estabilit
y
r
equirements
(
Xing and Zhou, 2009; Liao, 2010a
)
.To
i
llustrate, we assume that the small mass scale o
f
three li
g
ht neutrinos arise
s
f
r
o
m
a
n
a
i
ve seesaw
r
e
l
at
i
o
n
m
∼
(
λ
Λ
EW
)
n
+1
/
Λ
n
SS
,
where
λ
i
sad
im
e
n
s
i
o
n
-
l
ess Yukawa coupling coefficient an
d
n
is an ar
b
itrary integer
l
arger t
h
an one.
Without any terrible
fi
ne-tunin
g
, the seesaw scale can be estimated
f
rom
Λ
SS
∼
λ
n
+1
n
Λ
EW
100 G
e
V
n
+
1
n
0
.
1
e
V
m
1
n
10
2(
n
+
6
)
n
G
e
V
.
(
4.47
)
4.4 Multiple
S
eesaw Mechanisms 143
1
0
0
10
1
10
2
10
−
5
10
0
1
0
5
1
0
10
1
0
15
Λ
S
S
[
GeV
]
m
=0
.
1e
V
Λ
EW
=
100
G
eV
n
λ
=
1
λ
=1
0
−
2
λ
=
10
−
4
Fig. 4.2 A numerical illustration of the seesaw scale Λ
SS
c
hanging wit
h
n
a
nd
λ
as specified in Eq.
(
4.47
)
. Here the horizontal line stands for the TeV scal
e
A
numerical chan
g
eof
Λ
SS
w
i
th
n
a
n
d
λ
is s
h
own in Fi
g
.4.2,w
h
er
e
Λ
SS
∼
1
TeV ma
y
naturall
y
result fro
m
n
2
and
λ
10
−
3
. Hence the multi
p
le
s
eesaw i
d
ea is expecte
d
to wor
k
at t
h
e TeV sca
l
ean
d
provi
d
euswit
h
anove
l
a
pproac
h
to
b
ri
dg
et
h
e
g
ap
b
etween t
h
eoretica
l
natura
l
ness an
d
experimenta
l
testabilit
y
of the canonical seesaw mechanism
.
Th
esimp
l
est way to
b
ui
ld
amu
l
tip
l
e seesaw mo
d
e
l
at t
h
e TeV sca
l
eisto
extend the canonical seesaw mechanism by introducin
g
anumbero
fg
au
g
e
-
s
inglet fermions S
i
n
R
and scalar
s
Φ
n
(
for
i
=1
,
2
,
3
an
d
n
=1
,
2
,
···
)
.W
e
fi
nd that a proper implementation o
f
the
g
lobal
U
(
1
)
×
Z
2
N
s
ymmetry
l
ea
d
s
u
s to two classes o
f
multiple seesaw mechanisms with the nearest-nei
g
hbor-
i
nteraction pattern — an interesting form of the overall 3
(
n
+2
)
×
3
(
n
+
2
)
n
eutrino mass matrix in w
h
ic
h
every
3
×
3
su
b
-matrix on
l
y interacts wit
h
its
n
earest nei
g
hbor. The
fi
rst class contains an even number o
f
S
i
n
R
a
n
d
Φ
n
a
n
d
c
orresponds to an appealing extension of the canonical seesaw mechanism
,
w
hil
et
h
e seco
n
dc
l
ass
h
as a
n
odd
n
u
m
be
r
of
S
i
n
R
a
n
d
Φ
n
an
d
is actua
lly a
s
trai
g
htforward extension of the inverse seesaw mechanism.
4
.4.1 Two Classes of Multiple Seesaw Mechanism
s
The spirit o
f
multiple seesaw mechanisms is to make a harmless extension
o
f
the
S
M by addin
g
three ri
g
ht-handed neutrino
s
N
i
R
NN
t
o
g
ether with some
gauge-singlet fermions
S
i
n
R
a
nd scalars
Φ
n
(
for
i
=
1
,
2
,
3
and
n
=
1
,
2
,
·
·
·
)
.
1
44 4
S
eesaw Mechanisms o
f
Neutrino Masse
s
A
llowin
g
for lepton number violation to a certain extent, we can write the
gauge-invariant Lagrangian for neutrino masses as
(
Xing and Zhou, 2009
)
−
L
ν
=
L
Y
ν
YY
˜
HN
R
NN
+
N
c
R
NN
Y
S
YY
1
S
1R
Φ
1
+
n
i=2
S
c
(
i
−
1)R
Y
S
YY
i
S
i
R
Φ
i
+
1
2
S
c
n
R
M
μ
M
S
n
R
+
h
.
c
.
,
(
4.48
)
where
Y
ν
Y
Y
an
d
Y
S
YY
i
(
fo
r
i
=
1
,
2
,
···
,
n
)
are the 3
×
3 Yukawa coupling matrices
,
a
n
d
M
μ
M
is a symmetric Majorana mass matrix. After spontaneous
g
au
ge
s
ymmetry breaking, we arrive at the overall 3
(
n
+2
)
×
3
(
n
+
2
)
neutrino
m
ass matri
x
M
i
n the flavor basis defined by
(
ν
L
,N
c
R
NN
,
S
c
1R
,
·
·
·
,S
c
n
R
)
an
d
t
h
eir c
h
ar
g
e-conju
g
ate states:
M
=
⎛
⎜
⎛⎛
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜
⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜
⎜
⎜
⎜⎜
⎝
⎜⎜
0
M
D
MM
00 0
···
0
M
T
D
MM
0
M
S
M
1
0
0
···
0
0
M
T
S
M
1
0
M
S
M
2
0
···
0
00
M
T
S
M
2
0
.
.
.
.
.
.
.
.
.
00
0
.
.
.
.
.
.
M
S
M
n
−
1
0
.
.
.
.
.
.
.
.
.
.
.
.
M
T
S
M
n
−
1
0
M
S
M
n
000
·
·
·
0
M
T
S
M
n
M
μ
M
⎞
⎟
⎞
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎟⎟
,
(
4.49
)
whe
r
e
M
D
MM
≡
Y
ν
YY
H
a
n
d
M
S
M
i
=
Y
S
YY
i
Φ
i
(
fo
r
i
=
1
,
2
,
···
,
n
)
are the
3
×
3
m
ass matrices. Settin
g
N
R
NN
=
S
0R
for simplicity, one can observe that the
ii
Yu
k
awa interactions
b
etween
S
i
R
an
d
S
j
S
R
e
xist if and only if their subscript
s
s
atis
fy
the selection rul
e
|
i
−
j
|
=
1
(
for
i
,
j
=
0
,
1
,
2
,
···
,
n
)
. Note tha
t
M
m
anifests a very suggestive nearest-neighbor-interaction
(
NNI
)
pattern,
which has attracted a lot of attention in the quark sector to understand the
observed hierarchies of quark masses and flavor mixing angles
(
Fritzsch, 1978
,
1
979
;
Branco et al
.
,
1989
)
. Such a special structure of M ma
y
arise fro
m
a
proper implementation of the global
U
(
1
)
×
Z
2
N
s
ymmetry. T
h
e unique
g
enerator o
f
the cyclic
g
roup
Z
2
N
is
"
=e
i
π
/N
,
w
h
ic
h
pro
d
uces a
ll
t
h
e
g
roup
element
s
Z
2
N
=
{
1
,
"
,"
2
,"
3
,
···
,
"
2
N
−
1
}
.
B
y
definition, a fiel
d
Ψ
w
ith th
e
ch
arge
q
t
r
a
n
s
f
o
rm
sas
Ψ
→
e
i
π
q
/
N
Ψ
u
n
der
Z
2
N
(
fo
r
q
=0
,
1
,
2
,
·
·
·
,
2
N
−
1)
.
H
ence we assi
g
nt
he
U
(
1
)
an
d
Z
2
N
c
har
g
es o
f
the relevant
fi
elds in
L
ν
as
below
:
(
1
)
Th
e
gl
o
b
a
l
U
(
1
)
symmetry can be identified with the lepton number
L
,namel
y
L
(
L
)
=
L
(
E
R
)
= +1, where
E
R
r
epresents the char
g
ed-
lepton singlets in the SM. We arrange the lepton numbers of gauge-single
t
fe
rmi
o
n
sa
n
d sca
l
a
r
stobe
L
(
N
R
NN
)
=+1,
L
(
S
k
R
)
=
(
−
1)
k
a
n
d
L
(
Φ
k
)
=0
(
fo
r
k
=1
,
2
,
···
,n
)
. It turns out that only the Majorana mass term
S
c
n
R
M
μ
M
S
n
R
in
L
ν
exp
l
icit
l
y vio
l
ates t
h
e
U
(
1
)
symmetry. After this as-
signment, other lepton-number-violating mass terms
(
e.g.
,
N
c
R
N
N
M
R
M
M
N
R
N
N
in
4.4 Multiple
S
eesaw Mechanisms 145
t
he canonical seesaw mechanism
)
may also appear in the Lagrangian, bu
t
t
hey can be eliminated by invokin
g
the discrete
Z
2
N
s
y
mme
t
r
y.
(
2
)
W
e assign t
h
e
Z
2
N
c
harge o
f
S
n
R
as
q
(
S
n
R
)=
N
.T
h
en it is easy
t
overi
fy
that the Ma
j
orana mass ter
m
S
c
n
R
M
μ
M
S
n
R
is
in
va
ri
a
n
tu
n
der
t
h
e
Z
2
N
transformation. If all the other
g
au
g
e-sin
g
let fermions S
k
R
(
for
k
=
1
,
2
,
·
·
·
,
n
−
1)
take any charges in
{
1
,
2
,
·
·
·
,
2
N
−
1
}
othe
r
tha
n
N
,
their correspondin
g
Majorana mass terms are accordin
g
ly
f
orbid-
den. Given
q
(
L
)=
q
(
E
R
)
=
q
(
N
R
NN
)
= 1, both the charges o
f
S
k
R
(
for
k
=1
,
2
,
·
·
·
,
n
−
1)
and those of
Φ
i
(
for
i
=
1
,
2
,
···
,n
)
can be properl
y
c
h
ose
n
so as to ac
hi
eve t
h
e
NNI
fo
rm
of
L
ν
as shown in Eq.
(
4.48
)
.Bu
t
t
he solution to this kind of charge assignment may not be unique, be-
cause
f
or a
g
iven value o
f
n
o
ne can a
l
ways ta
k
e
N
n
to fu
l
fi
ll
a
ll
t
h
e
above-mentioned requirements. Simple examples
(
with
n
=1
,
2
,
3)
will b
e
p
resented below
.
This multi
p
le seesaw
p
icture should be the sim
p
lest extension of the canoni
-
c
al seesaw mechanism, since it does not invoke the help of either additional
SU
(
2
)
L
f
ermion doublets
(
Dudas and Savoy, 2002
)
or a new isospin 3
/
2Higg
s
m
ultiplet
(
Babu
e
tal.
,
2009; Liao, 2010b
)
. In addition, the inverse seesaw
s
cenario is merely the simplest example in one class o
f
the multiple seesaw
m
echanisms under discussion
(
with an odd number o
f
S
i
n
R
or
Φ
n
).
Now let us diagonalize
M
i
nEq.
(
4.49
)
to achieve the effective mass
m
atrix o
f
three li
g
ht neutrino
s
M
ν
MM
.N
ote that
M
ca
n
be
r
ew
ri
tte
n
as
M
=
0
˜
M
D
MM
˜
M
T
D
MM
˜
M
μ
M
,
(
4.50
)
whe
r
e
˜
M
D
MM
=
(
M
D
MM
0
)
denotes a 3
×
3(
n
+1
)
matrix an
d
˜
M
μ
M
i
s a symmetric
3(
n
+1
)
×
3(
n
+1
)
matrix. Takin
g
O
(
˜
M
μ
M
)
O
(
˜
M
D
MM
)
, one can easily obtain
M
ν
MM
=
−
˜
M
D
M
M
˜
M
−
1
μ
M
˜
M
T
D
MM
f
or three li
g
ht Majorana neutrinos in the leadin
g
-order
app
roximation.
A
s the elements in the fourth to
3
n
-t
h
co
l
u
mn
sof
˜
M
D
M
M
a
r
e
exactly zero, only the 3
×
3
to
p
left block of
˜
M
−
1
μ
M
is relevant to the calculation
o
f
M
ν
MM
.
Without loss of generality, the inverse of
˜
M
μ
M
c
an be figured out b
y
a
ssumin
g
a
ll
t
h
e non-zero
3
×
3
sub-matrices o
f
M
t
obeo
f
rank three. We
find two types of solutions, depending on whethe
r
n
is even or odd
,
and thu
s
a
rrive at two classes of multiple seesaw mechanisms:
C
lass
A
o
f
multiple seesaw mechanism
s
—the
y
containanevennumbero
f
g
au
g
e-sin
g
let
f
ermion
s
S
i
n
R
a
n
d scala
r
s
Φ
n
(
i.e.
,
n
=2
k
w
i
th
k
=
0
,
1
,
2
,
···
)
a
nd correspond to a novel extension of the canonical seesaw picture. Th
e
effective mass matrix of three light Majorana neutrinos is given by
M
ν
M
M
=
−
M
D
MM
k
i
=
1
M
T
S
M
2
i
−
1
−
1
M
S
M
2
i
M
−
1
μ
M
k
i
=1
M
T
S
M
2
i
−
1
−
1
M
S
M
2
i
T
M
T
D
MM
(
4.51
)
i
n the leadin
g
-order approximation. The
k
= 0 case is obviousl
y
equivalent to
the canonical seesaw mechanism
(
i.e.,
M
ν
M
M
=
−
M
D
M
M
M
−
1
R
M
M
M
T
D
M
M
b
y setting
S
0R
=