Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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2
6
4
S
eesaw Mechanisms o
f
Neutrino Masse
s
if
and onl
y
i
f
neutrinos are massive and lepton
fl
avors are mixed, and thus it i
s
a
kind of new physics beyond the SM. To
g
enerate nonzero but tiny neutrino
m
asses
,
one or more of the above-mentioned constraints on the SM must b
e
ab
an
d
one
d
or re
l
axe
d
. It is certain
l
yinto
l
era
bl
etoa
b
an
d
on t
h
e
g
au
g
esym-
m
etr
y
and Lorentz invariance; otherwise, one would be led astra
y
.Giventh
e
f
ramework of the SM as a consistent field theory, its particle content can b
e
m
odified and
(
or
)
its renormalizability can be abandoned to accommodate
m
assive neutrinos. There are several ways to this goal. For simplicity, w
e
roughly classify the viable ideas about neutrino mass generation into non-
seesaw
m
echa
ni
s
m
sa
n
d seesaw
m
echa
ni
s
m
s
.
4
.1.1 N
o
n-
seesa
wM
ec
h
a
ni
s
m
s
A
mong many non-seesaw mechanisms of neutrino masses proposed in the
l
iterature, three o
f
them are particularly interestin
g
and have attracted a lot
of attention in recent
y
ears
.
(
1
)
Non-renorma
l
iza
bl
e neutrino mass operators
.
In 1979
,
Steven Wein
-
ber
g
extended the
S
M by introducin
g
some hi
g
her-dimension operators in
terms of the fields of the SM itself
(
Weinberg, 1979
)
:
L
eff
=
L
S
M
+
L
d=
5
Λ
+
L
d
=
6
Λ
2
+
···
,
(
4.1
)
whe
r
e
Λ
d
enotes the cutoff scale of this effective theory. Within such a frame
-
work, the lowest-dimension operator that violates the lepton number
(
L
)
i
s
the uni
q
ue dimension-5 o
p
erato
r
H
HLL
/Λ
.A
fter spontaneous gauge sym
-
m
etry
b
rea
k
in
g
,t
h
is Wein
b
er
g
operator yie
lds
m
i
∼
H
2
/
Λ
fo
rn
eut
rin
o
m
asses, which can be sufficiently small
(
1
eV
)
i
f
Λ
i
snot
f
ar awa
yf
rom th
e
s
cales of possible grand unified theories
(
i.e.
,
Λ
1
0
13
G
e
V
fo
r
H
∼
10
2
GeV
)
. In this sense, people argue that neutrino masses can serve as a low
-
ener
g
y window onto new physics at superhi
g
h-ener
g
y scales.
(
2
)
Pure Dirac neutrino masses wit
h
B
−
L
symme
t
ry
.
Given three right
-
h
an
d
e
d
neutrinos, t
h
e
g
au
g
e-invariant an
dl
epton-num
b
er-conservin
g
mas
s
terms of char
g
ed leptons and neutrinos are
−
L
l
epto
n
=
L
Y
l
YY
H
E
R
+
L
Y
ν
YY
˜
HN
R
NN
+h
.
c
.
,
(
4.2
)
w
h
ere
˜
H
≡
i
σ
2
H
∗
is defined and
L
denotes the left-handed lepton doublet
.
A
fter spontaneous
g
au
g
e symmetry breakin
g
, we arrive at the char
g
ed-lepto
n
m
ass matr
i
x
M
l
M
=
Y
l
YY
v/
√
2 and the Dirac neutrino mass matri
x
M
ν
MM
=
Y
ν
YY
v
/
√
2
w
i
th
v
246 GeV. In this case, the smallness of three neutrino masse
s
m
i
(
for
i
=
1
,
2
,
3)
is attributed to the smallness of three eigenvalues of
Y
ν
YY
(
denote
d
as
y
i
fo
r
i
=1
,
2
,
3
)
. Then we encounter a transparent hierarchy problem
:
y
i
/y
e
=
m
i
/
m
e
0
.
5
e
V
/
0
.
5M
e
V
∼
10
−
6
.W
h
yis
y
i
so s
m
a
ll? Th
e
r
e
i
s
n
o
ex
p
lanation at all in this Dirac-mass
p
icture.
4.1 How to
G
enerate Tiny Neutrino Masses 12
7
A
speculative wa
y
out is to invoke extra dimensions; namel
y
, the smallness
of Dirac neutrino masses is ascribed to the assumption that three ri
g
ht
-
h
anded neutrinos have access to one or more extra spatial dimensions
(
Dienes
et al
.
,
1999;
A
rkani-Hame
d
et al.
, 2002
)
. The idea is simply to confine the S
M
p
articles onto a brane and to allo
w
N
R
NN
to travel in the bulk. For exam
p
le
,
t
h
ewave
f
u
n
ct
i
o
n
o
f
N
R
NN
s
prea
d
soutovert
h
eextra
d
imensio
n
y
,
giving rise
to a suppresse
d
Yu
k
awa interaction at
y
=0
(
i.e., the location of the brane
)
:
L
Y
ν
Y
Y
˜
HN
R
NN
y
=0
∼
1
√
L
L
Y
ν
Y
Y
˜
HN
R
NN
y
=
L
.
(
4.3
)
The magnitude of 1
/
√
L
is measure
db
y
Λ/Λ
P
lanck
a
n
d
can natura
ll
y
be
s
mall, because the
f
undamental scal
e
Λ
of
a
f
ull theor
y
with extra dimension
s
m
a
y
be much lower than the Planck mass scale
Λ
P
lanck
o
f the effective four
-
d
imensional theory
(
e.g.
,
Λ
∼
1
Te
V
an
d
Λ
Pla
n
ck
∼
10
19
G
eV
)
.
(
3
)
R
adiative
g
eneration o
f
tin
y
neutrino masse
s
. In a semina
l
paper pu
b
-
l
ished in 1972, Weinber
g
pointed out that “in theories with spontaneously
broken gauge symmetries, various masses or mass differences may vanish in
zeroth order as a consequence o
f
the representation content o
f
the
fi
elds ap
-
pearing in the Lagrangian. These masses or mass differences can then be
c
alculated as finite higher-order effects.”
(
Weinberg, 1972
)
. Such a mecha
-
n
ism may allow us to sli
g
htly
g
obeyondthe
S
M and radiatively
g
enerate
tiny neutrino masses. A typical example is the well-known Zee model
(
Zee
,
1
980
)
,
−
L
le
p
ton
=
L
Y
l
YY
HE
R
+
L
Y
S
YY
S
−
i
σ
2
c
L
+
˜
Φ
T
F
S
+
i
σ
2
˜
H
+h
.
c
.,
(
4.4
)
where S
+
an
d
S
−
a
re charged
SU
(
2
)
L
s
inglet scalars,
Φ
d
enotes a ne
w
S
U
(
2
)
L
d
oublet scalar which has the same quantum number as the SM Higgs double
t
H
,
Y
S
YY
i
sanantis
y
mmetric matrix, an
d
F
r
epresents a coup
l
in
g
constant an
d
h
as the dimension of mass. Without loss of generality, we choose the basis
of
M
l
M
=
Y
l
Y
Y
H
=D
ia
g
{
m
e
,m
μ
,
m
τ
}
.
In
th
i
s
m
odel
n
eut
rin
os a
r
e
m
assless at
the tree level, but their masses can be
g
enerated via the one-loop corrections.
G
ive
n
M
S
M
M
h
M
∼
M
Φ
M
M
∼
F
a
nd
Φ
∼
H
,
the elements of the effectiv
e
m
ass matrix o
f
three li
g
ht Majorana neutrinos turns out to b
e
(
M
ν
M
M
)
α
β
∼
M
h
M
16
π
2
·
m
2
α
−
m
2
β
M
2
S
M
(
Y
S
YY
)
α
β
,
(
4.5
)
whe
r
e
α
a
n
d
β
r
u
n
over
e
,
μ
a
n
d
τ
.Th
es
m
a
lln
ess o
f
M
ν
MM
is t
h
e
r
e
f
o
r
easc
ri
bed
to t
h
es
m
a
lln
ess of
Y
S
YY
and
(
m
2
α
−
m
2
β
)
/M
2
S
M
.
A
lthou
g
h the ori
g
inal version
oftheZeemodelisdisfavoredb
y
current experimental data on neutrino
osci
ll
ations, its extensions or variations at t
h
eone-
l
oop or two-
l
oop
l
eve
l
ca
n
s
urvive
(
Babu, 1988; Ma, 1998
)
.
1
2
8
4
S
eesaw Mechanisms o
f
Neutrino Masse
s
4
.1.2
S
eesaw Mechanism
s
Without loss of the
g
au
g
e symmetry and Lorentz invariance, the essentia
l
s
pirit of seesaw mechanisms is to add a few new particles into the SM an
d
allow
B
−
L
v
iolation — tin
y
masses o
f
three known neutrinos are attributed to
the existence of heavy de
g
rees of freedom and lepton number violation. Ther
e
a
re t
h
ree typica
l
seesaw mec
h
anisms on t
h
emar
k
et, an
d
t
h
eir properties
h
av
e
b
een extensive
ly
stu
d
ie
d.
(
1
)
Type-I seesaw — three heavy right-handed neutrinos are added int
o
the SM and the lepton number is violated by their Majorana mass term
(
Fritzsch
et al
., 1975; Minkowski, 1977; Yana
g
ida, 1979;
G
ell-Mann
et al
.
,
1
979; Glashow, 1980; Mohapatra and Senjanovic, 1980
)
:
−L
l
e
p
ton
=
L
Y
l
YY
HE
R
+
L
Y
ν
YY
˜
HN
R
NN
+
1
2
N
c
R
NN
M
R
M
M
N
R
NN
+h
.
c
.,
(
4.6
)
whe
r
e
M
R
MM
i
st
h
e symmetric Majorana mass matrix
.
(
2
)
Type-II seesaw — one heavy Higgs triplet is added into the SM an
d
the lepton number is violated by its interactions with both the lepton doublet
a
nd the Higgs doublet
(
Konetschny and Kummer, 1977, Magg and Wetterich,
1
980;
S
chechter and Valle, 1980;
C
hen
g
and Li, 1980; Lazaride
s
et al
.
,
1981;
Mohapatra and Senjanovic, 1981
)
:
−
L
le
p
to
n
=
L
Y
l
YY
HE
R
+
1
2
L
Y
Δ
Y
Δ
i
σ
2
c
L
− λ
Δ
M
Δ
M
H
T
i
σ
2
ΔH
+
h
.
c
.,
(
4.7
)
whe
r
e
Δ
≡
Δ
−
−
√
2
Δ
0
√
2
Δ
−−
−
Δ
−
(
4.8
)
de
n
otes t
h
e
SU
(
2
)
L
H
i
gg
striplet
.
(
3
)
Type-III seesaw — three heavy triplet fermions are added into th
e
SM and the lepton number is violated by their Majorana mass term
(
Foo
t
et
al.
,
1989; Ma, 1998
):
−L
l
epto
n
=
L
Y
l
YY
HE
R
+
L
√
2
Y
Σ
YY
Σ
c
˜
H
+
1
2
Tr
ΣM
Σ
M
Σ
c
+h
.
c
.,
(
4.9
)
where
Σ
=
Σ
0
/
√
2
Σ
+
Σ
−
−
Σ
0
/
√
2
(
4.10
)
de
n
otes the
SU
(
2
)
L
f
ermion triplet
.
Of
course, there are a number o
f
variations or combinations o
f
these thre
e
typical seesaw mechanisms in the literature. Some of them will be introduce
d
i
n
S
ections 4.3 and 4.4. For each o
f
the above seesaw pictures, one may
g
et th
e
u
nique dimension-5 Weinber
g
operator of neutrino masses after inte
g
ratin
g
out the corresponding heavy degrees of freedom
:
4.1 How to
G
enerate Tiny Neutrino Masses 12
9
L
d
=
5
Λ
=
⎧
⎪
⎧⎧
⎪
⎪⎪
⎪
⎪⎪
⎪
⎪⎪
⎪
⎪⎪
⎪
⎪⎪
⎨
⎪⎪
⎪
⎨⎨
⎪
⎪⎪
⎪
⎪⎪
⎪
⎪⎪
⎪
⎪⎪
⎪
⎪⎪
⎩
⎪⎪
1
2
Y
ν
YY
M
−
1
R
MM
Y
T
ν
YY
α
β
α
L
˜
H
˜
H
T
c
β
L
+h
.
c
.
(
Type I
)
,
−
λ
Δ
M
Δ
M
(
Y
Δ
Y
)
αβ
α
L
˜
H
˜
H
T
c
β
L
+h.
c
.
(
Type II
)
,
1
2
Y
Σ
YY
M
−
1
Σ
MM
Y
T
Σ
YY
α
β
α
L
˜
H
˜
H
T
c
β
L
+h
.
c
.
(
Type III
)
.
(
4.11
)
A
fter spontaneous
g
au
g
e symmetry breakin
g
,
˜
H
a
c
h
ieves its vacuum expecta
-
tion valu
e
˜
H
=
v
/
√
2
with
v
2
46 GeV. Then we are left with the effective
Majorana neutrino mass term for three light neutrinos
,
−L
m
ass
=
1
2
ν
L
M
ν
M
M
ν
c
L
+
h
.
c
.,
(
4.12
)
where the symmetric Majorana mass matri
x
M
ν
MM
i
sgivenb
y
M
ν
M
M
=
⎧
⎪
⎧⎧
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪⎪
⎪
⎪⎪
⎪
⎪⎪
⎨
⎪⎪
⎪
⎨⎨
⎪
⎪⎪
⎪
⎪⎪
⎪
⎪⎪
⎪
⎪⎪
⎪
⎪⎪
⎪
⎪⎪
⎩
⎪⎪
−
1
2
Y
ν
Y
Y
v
2
M
R
MM
Y
T
ν
Y
Y
(
Type I
)
,
λ
Δ
Y
Δ
Y
v
2
M
Δ
M
(
Type II
)
,
−
1
2
Y
Σ
YY
v
2
M
Σ
M
Y
T
Σ
YY
(
Type III
)
.
(
4.13
)
I
t beco
m
es obv
i
ous t
h
at t
h
es
m
a
lln
ess of
M
ν
M
M
can
b
e attri
b
ute
d
to t
h
e
l
arge
-
ness of
M
R
M
M
,
M
Δ
M
o
r
M
Σ
M
i
n
the seesaw
m
echa
ni
s
m
.
4
.1.3 The Weinberg
O
perator
N
ow let us derive the dimension-5 Weinberg operator of neutrino masses
L
d
=
5
/
Λ given in Eq.
(
4.11
)
. The cutoff scale
Λ
i
srou
ghl
yc
h
aracterize
dby
the masses of heav
y
particles in a seesaw mechanism, and thus it is naturall
y
expected to be much higher than the electroweak scale. At low energies one
m
ay inte
g
rate out the heavy de
g
rees o
ff
reedom and obtain an e
ff
ective
fi
el
d
theory, which only contains the li
g
ht de
g
rees of freedom, to describe the finit
e
masses o
f
t
hr
ee
kn
ow
nn
eut
rin
os.
W
ithout loss o
fg
enerality, we illustrate the approach by considerin
g
a
field theory consisting of two real scalars
φ
(
x
)
an
d
Φ
(
x
)
, whose masses are
d
enote
d
respective
l
y
b
y
m
φ
a
n
d
M
Φ
M
M
.
W
e assum
e
m
φ
M
Φ
MM
in t
h
is t
h
eory
.
Its La
g
ran
g
ian can
b
esc
h
ematica
ll
y written a
s
L
f
ul
l
[
φ
;
Φ
]=
L
f
ree
[
φ
]+
L
f
ree
[
Φ
]
+
L
i
n
t
[
φ
,
Φ
]
,
(
4.14
)
whe
r
e
L
f
r
ee
d
enotes t
h
e
k
inetic an
d
mass terms, an
d
L
i
n
t
r
epresents t
h
ein
-
teraction term.
A
tanener
g
y scale far below
M
Φ
MM
,
it s
h
ou
ld b
e
g
oo
d
enou
gh
to describe ph
y
sical phenomena b
y
meansofaneffectivetheor
y
which onl
y
130
4
S
eesaw Mechanisms o
f
Neutrino Masse
s
i
n
vo
l
ves t
h
efie
l
d
φ
.
The action o
f
this e
ff
ective theory can accordin
g
ly be
d
erived as follows
(
Weinberg, 1980; Bilenky and Santamaria, 1994
)
:
e
x
p
{
i
S
eff
S
}
=e
x
p
*
i
L
eff
[
φ
(
x
)]d
4
x
)
=
[D
Φ
][
D
Φ
†
]
exp
*
i
L
full
[
φ
(
x
)
;
Φ
(
x
)]
)
=e
x
p
*
i
[
L
fu
l
l
[
φ
(
x
)
;
Φ
(
x
)
=0
]+
O
[
φ
(
x
)]]
d
4
x
)
,
(
4.15
)
i
n which the effective operator arisin
g
from the path-inte
g
ration is denote
d
as
O
[
φ
(
x
)]
. This operator only depends on the fiel
d
φ
(
x
)
and may not nec
-
essarily be o
f
dimension-
f
our. The e
ff
ective La
g
ran
g
ian at low ener
g
ies is
therefore a sum of the La
g
ran
g
ian without the heavy de
g
rees of freedom an
d
the operator obtained from integrating out the heavy degrees of freedom;
i
.e.,
L
eff
[
φ
(
x
)] =
L
fu
l
l
[
φ
(
x
);
Φ
(
x
)
=0
]
+
O
[
φ
(
x
)]
. Taking the type-I seesa
w
m
echanism for example, we shall subsequently derive the Weinberg operato
r
b
yintegratingoutt
h
e
h
eavy Majorana neutrinos
.
T
he t
y
pe-I seesaw model is a simple extension o
f
the
S
M with three heav
y
Majorana neutrinos. The full Lagrangian of this model read
s
L
=
L
SM
+
N
R
NN
i
/
∂N
//
R
NN
−
1
2
N
c
R
NN
M
R
MM
N
R
NN
+
L
Y
ν
YY
˜
HN
R
NN
+
h
.
c
.
,
(
4.16
)
whe
r
e
L
SM
stands
f
or the
S
MLa
g
ran
g
ian. The Majorana mass matrix
M
R
MM
is symmetric and can be diagonalized by a unitary matrix
U
v
i
at
h
et
r
a
n
s-
f
ormatio
n
U
†
M
R
M
M
U
∗
=
$
M
=
D
iag
{
M
1
,M
2
MM
,M
3
MM
}
.
In the mass basis of heav
y
Majorana neutrinos, the La
g
ran
g
ian involvin
g
the heavy de
g
rees o
ff
reedom
c
an be rewritten a
s
L
N
=
1
2
N
i
NN
i
/
∂N
//
i
N
N
−
1
2
M
i
M
M
N
i
NN
N
i
N
N
−
α
L
(
˜
Y
ν
YY
)
αi
˜
HN
i
N
N
+
h
.
c
.
,
(
4.17
)
whe
r
e
3
N
R
NN
≡
U
T
N
R
NN
,
˜
Y
ν
YY
≡
Y
ν
YY
U
∗
a
n
d
N
≡
3
N
R
NN
+
3
N
c
R
NN
h
ave been de
fi
ned, an
d
the relevant
f
amil
y
indices have been omitted. Becaus
e
L
˜
Y
ν
YY
˜
HN
T
=
N
˜
Y
T
ν
YY
˜
H
T
c
L
,
N
˜
Y
†
ν
YY
˜
H
†
L
T
=
c
L
˜
Y
∗
ν
YY
˜
H
∗
N,
(
4.18
)
where
c
L
≡C
L
T
similar to the definitions made in Eq.
(
3.6
)
,wehav
e
α
L
(
˜
Y
ν
Y
Y
)
αi
˜
HN
i
NN
+
h
.
c
.
=
1
2
N
˜
Y
T
ν
YY
˜
H
T
c
L
+
˜
Y
†
ν
YY
˜
H
†
L
+
L
˜
Y
ν
YY
˜
H
+
c
L
˜
Y
∗
ν
YY
˜
H
∗
N
,
(
4.19
)
which will be use
f
ul in the inte
g
ration over the heavy Majorana
fi
elds.
S
ub-
s
tituting Eq.
(
4.19
)
into Eq.
(
4.17
)
,weimmediatelyarrivea
t
4.1 How to
G
enerate Tiny Neutrino Masses 13
1
L
N
=
1
2
NKN
−
1
2
N
χ
−
1
2
χN ,
(
4.20
)
w
h
e
r
e
K
≡
i
/
∂
//
−
$
M
,
χ
≡
χ
†
γ
0
a
n
d
χ
≡
˜
Y
T
ν
YY
˜
H
T
c
L
+
˜
Y
†
ν
YY
˜
H
†
L
have bee
n
d
efined. We proceed to apply the formulism in Eqs.
(
4.14
)
and
(
4.15
)
to the
f
ull Lagrangian of the type-I seesaw model in Eq.
(
4.16
)
,soastoobtainth
e
effective La
g
ran
g
ian of three li
g
ht neutrinos at low ener
g
ies. First of all, w
e
i
ntegrate
L
N
in Eq.
(
4.20
)
over the heavy Majorana neutrino fields; viz.
,
[
D
N
][
D
N
]
exp
*
i
1
2
NKN
−
N
χ
−
χN
d
4
x
)
,
(
4.21
)
which is typically Gaussian. In
g
eneral, the Gaussian inte
g
rals can be ex
-
pressed in the following way
(
Weinberg, 1995
)
:
I
=
+
∞
−∞
r
d
ξ
r
e
−
Q
(
ξ
)
,
(
4.22
)
i
n
wh
i
ch
Q
(
ξ
)
≡
1
2
r,s
K
rs
K
ξ
r
ξ
s
+
r
L
r
ξ
r
+
M.
(
4.23
)
N
ote that
K
is a nonsingular, positive-definite and symmetric matrix, an
d
the va
ri
ables
ξ
r
a
r
e
r
eal
.H
e
n
ce
K
c
an
b
e
d
ia
g
ona
l
ize
d
via t
h
eort
h
o
g
o
-
na
l
t
r
a
n
sfo
rm
at
i
on
S
T
K
S
=
κ
, or equivalentl
y
S
rp
SS
S
sq
SS
K
rs
K
=
κ
p
δ
pq
δ
w
i
th
S
T
S
=
SS
T
=
1
.
G
iven the definitio
n
˜
ξ
r
≡
S
sr
SS
ξ
s
,
Eq.
(
4.23
)
becomes
Q
(
˜
ξ
)=
1
2
r
κ
r
˜
ξ
2
r
+
r
(
L
S
)
r
˜
ξ
r
+
M
,
(
4.24
)
a
nd the integration in Eq.
(
4.22
)
turns out to b
e
I
=e
−
M
r
+
∞
−∞
d
˜
ξ
r
ex
p
*
−
1
2
κ
r
˜
ξ
2
r
−
(
L
S
)
r
˜
ξ
r
)
=e
−
M
r
1
2
π
κ
r
1
/
2
e
x
p
*
1
2
κ
−
1
r
(
L
S
)
2
r
)
=(
2
π
D
et
K
)
−
1
/
2
e
x
p
*
1
2
LK
−
1
L
T
−
M
)
,
(
4.25
)
w
h
ere Det
K
i
s the determinant o
f
K
. It is instructive to rewrite t
h
e resu
lt
i
nEq.
(
4.25
)
by evaluating the extremum o
f
Q
(
ξ
)
throug
h
∂Q
(
ξ
)
∂ξ
r
=
0.
A
f
-
ter a strai
g
ht
f
orward calculation, we
fi
nd the extremum at
ˆ
ξ
=
−
K
−
1
L
T
.
Therefore
,
13
24
S
eesaw Mechanisms o
f
Neutrino Masse
s
I
=(2
π
Det
K
)
−
1
/
2
ex
p
*
1
2
LK
−
1
L
T
−
M
)
=(2
π
D
et
K
)
−
1
/
2
e
x
p
"
−
Q
(
ˆ
ξ
)
#
.
(
4.26
)
This result can be
g
eneralized to the complex variables and to a
fi
eld theor
y
with infinite dimensions. The important point indicated by Eq.
(
4.26
)
is that
the integral can be obtained by solving the equation of motion, if the variable
s
ξ
r
a
re replaced b
y
the relevant
fi
elds. For the complex scalar
fi
elds, we hav
e
the functional integration
[D
φ
][
D
φ
†
]
ex
p
*
i
φ
†
(
x
)
K
(
x
,
y
)
φ
(
y
)
−
i
J
†
(
z
)
φ
(
z
)
+h
.
c
.
)
=(
De
t
K
)
−
1
e
x
p
*
−
i
d
4
x
d
4
yJ
†
(
x
)
K
−
1
(
x,
y
)
J
(
y
)
)
;
(
4.27
)
a
nd
f
or the
f
ermion
fi
elds, we hav
e
[
D
ψ
][
D
ψ
]
ex
p
*
i
ψ
(
x
)
K
(
x,
y
)
ψ
(
y
)
−
i
J
(
z
)
ψ
(
z
)
+h
.
c
.
)
=
D
et
K
e
x
p
*
−
i
d
4
x
d
4
y
J
(
x
)
K
−
1
(
x
,
y
)
J
(
y
)
)
.
(
4.28
)
N
ote that the integration over the spacetime arguments in the first line of
E
q.
(
4.27
)
or Eq.
(
4.28
)
is implied. Note also that the factors appearing in the
bosonic and fermionic integrations
(
i.e.,
(
De
t
K
)
−
1
a
n
d
D
et
K
)
are different
.
N
ow we apply the result in Eq.
(
4.28
)
to Eq.
(
4.21
)
:
[D
N
][D
N
]
exp
*
i
1
2
NKN
−
Nχ
−
χN
d
4
x
)
=
D
et
K
e
x
p
*
−
i
2
χK
−
1
χ
d
4
x
)
=e
xp
*
i
2
d
4
x
L
˜
Y
ν
YY
˜
H
+
c
L
˜
Y
∗
ν
YY
˜
H
∗
$
M
−
1
˜
Y
T
ν
YY
˜
H
T
c
L
+
˜
Y
†
ν
Y
Y
˜
H
†
L
)
,
(
4.29
)
where the
fi
eld-independent
f
actor Det
K
co
n
t
ri
butes a co
n
sta
n
t to the act
i
on
a
nd thus can be omitted, and the operator
(i
/
∂
//
−
$
M
)
−
1
has been ex
p
anded in
powers o
f
$
M
−
1
. In the leadin
g
-order approximation we can there
f
ore arrive
a
ttheWeinber
g
operator in the type-I seesaw model:
L
d=
5
Λ
=
1
2
L
˜
H
˜
Y
ν
YY
$
M
−
1
˜
Y
T
ν
YY
˜
H
T
c
L
+h
.
c
.
=
1
2
L
˜
HY
ν
YY
U
∗
$
M
−
1
U
†
Y
T
ν
YY
˜
H
T
c
L
+h
.
c
.
=
1
2
Y
ν
YY
M
−
1
R
M
M
Y
T
ν
Y
Y
αβ
α
L
˜
H
˜
H
T
c
β
L
+
h
.
c
.,
(
4.30
)
4.2
O
nthe
S
cales o
fS
eesaw Mechanisms 1
33
whe
r
e
˜
Y
ν
YY
=
Y
ν
YY
U
∗
a
n
d
M
R
MM
=
U
$
MU
T
have bee
n
used
.T
h
i
s
r
esult
i
sval
i
d
i
n any flavor basis of heavy Majorana neutrinos.
A
fter spontaneous gauge
s
ymmetry breaking, the Weinberg operator gives rise to the effective neutrin
o
m
ass term in Eq.
(
4.12
)
with
M
ν
MM
=
−
M
D
MM
M
−
1
R
MM
M
T
D
MM
a
n
d
M
D
MM
=
Y
ν
YY
v
/
√
2
.T
he
d
imension-5 operators in the t
y
pe-II and t
y
pe-III seesaw models, which hav
e
been summarized in Eq.
(
4.11
)
, can be derived in an analogous way
.
4.2
O
nthe
S
cales o
fS
eesaw Mechanisms
A
s we have seen, the key point of a seesaw mechanism is to ascribe the
s
mallness o
f
neutrino masses to the existence o
f
some new de
g
rees o
ff
reedom
h
eavier than the Fermi scal
e
v
246 GeV. The energy scale where a see-
s
aw mec
h
anism wor
k
siscrucia
l
,asitisre
l
evant to w
h
et
h
er t
h
is mec
h
anis
m
i
s theoretically natural and experimentally testable
(
Xing, 2009
)
.Betwee
n
F
ermi and Planck scales, there might exist two other fundamental scales: on
e
i
s the scale of a grand unified theory
(
GUT
)
at which strong, weak and elec
-
troma
g
netic forces can be unified, and the other is the TeV scale at whic
h
the unnatural gauge hierarchy problem of the SM can be solved or at least
s
o
f
tened b
y
a kind o
f
new ph
y
sics
.
4
.2.1
S
eesaw-induced Hierarchy Problem
Many theorists ar
g
ue that the conventional seesaw scenarios are natural be
-
c
ause their scales
(
i.e., the masses of heavy degrees of freedom
)
are close to
the
G
UT scale. This ar
g
ument is reasonable on the one hand, but it re
fl
ects
the drawbacks of the conventional seesaw models on the other hand. In othe
r
wor
d
s, t
h
econventiona
l
seesaw mo
d
e
l
s
h
ave no
d
irect experimenta
l
testa
b
i
l-
i
t
y
an
d
invo
l
ve a potentia
lh
ierarc
hy
pro
bl
em. T
h
e
l
atter is usua
lly
spo
k
e
of when two largely different energy scales exist in a model, but there is n
o
s
ymmetry to stabilize the low-scale physics suffering from large correction
s
c
omin
gf
rom the hi
g
h-scale physics.
S
uch a seesaw-induced fine-tuning problem means that the SM Higgs
mass
M
h
M
is very sensitive to quantum corrections from the heavy degrees o
f
f
reedom in a seesaw mechanism. For example
(
Vissani, 1998; Casa
s
et al
.
,
2
004; Abad
a
et al
., 2007
)
,
δM
2
h
M
=
⎧
⎪
⎧
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪⎪
⎪
⎨
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
−
y
2
i
8
π
2
Λ
2
+
M
2
i
MM
l
n
M
2
i
MM
Λ
2
(
Type I
)
3
16
π
2
λ
3
Λ
2
+
M
2
Δ
M
l
n
M
2
Δ
M
Λ
2
+4
λ
2
Δ
M
2
Δ
M
l
n
M
2
Δ
M
Λ
2
(Type II
)
−
3
y
2
i
8
π
2
Λ
2
+
M
2
i
MM
l
n
M
2
i
MM
Λ
2
(
Type III
)
(
4.31
)
i
nthreet
y
pical seesaw scenarios, where
Λ
i
sthere
g
ulator cuto
ff,
y
i
a
n
d
M
i
MM
(
fo
r
i
=1
,
2
,
3)
stand respectively for the eigenvalues of
Y
ν
Y
Y
(
o
r
Y
Σ
Y
Y
)
an
d
13
44
S
eesaw Mechanisms o
f
Neutrino Masse
s
M
R
MM
(
or
M
Σ
M
)
, and the contributions proportional to
v
2
a
n
d
M
2
h
M
have bee
n
omitted. Eq.
(
4.31
)
shows a quadratic sensitivity to the new scale whic
h
i
s characteristic of the seesaw model, implying that a high degree of fine
-
tunin
g
wou
ld b
e necessary to accommo
d
ate t
h
e experimenta
ld
ata o
n
M
h
M
i
f the seesaw scale is much larger than
v
(
or the Yukawa couplings are not
extremely fine-tuned in type-I and type-III seesaws
)(
Abad
a
et al
., 2007
).
T
akin
g
the type-I seesaw scenario
f
or illustration, we assum
e
Λ
∼
M
i
MM
a
n
d
re
q
u
i
re
|
δM
2
h
M
|
0
.
1
Te
V
2
.
Then Eq.
(
4.31
)
leads us to the rough estimat
e
M
i
M
M
∼
(2
π
v
)
2
|
δM
2
h
M
|
m
i
1
/
3
10
7
G
e
V
0
.
2eV
m
i
1
/
3
|
δM
2
h
M
|
0
.
1
Te
V
2
1
/
3
.
(
4.32
)
This naive result indicates that a hierarchy problem will arise i
f
M
i
MM
10
7
G
eV in the t
y
pe-I seesaw scheme. Because o
f
m
i
∼
y
2
i
v
2
/
(2
M
i
M
M
)
, the boun
d
M
i
MM
1
0
7
GeV im
p
lie
s
y
i
∼
2
m
i
M
i
M
/
v
2
.
6
×
1
0
−
4
for
m
i
∼
0
.
2
eV. Such
a
small magnitude of y
i
seems to
b
ea
b
it unnatura
l
in t
h
e sense t
h
at t
h
e
co
n
ve
n
t
i
o
n
a
l
seesaw
i
dea att
ri
butes t
h
es
m
a
lln
ess of
m
i
to the lar
g
eness o
f
M
i
MM
o
ther than the smallness o
f
y
i
.
T
here are two possible ways out of this impasse: one is to appeal for th
e
s
upersymmetry, and the other is to lower the seesaw scale. In the
f
ollowin
g
d
iscussions we shall focus on whether a seesaw mechanism at low energies i
s
t
h
eoretica
ll
y natura
l
an
d
interesting.
4
.2.2
S
eesaw-induced Naturalness Proble
m
In reality, there is no direct evidence
f
or a hi
g
h or extremely hi
g
h seesa
w
s
cale.
S
o eV-, keV-, MeV- and
G
eV-scale seesaw mechanisms are all
p
ossible,
a
t least in principle; and they are technically natural in the sense that their
l
epton-number-violatin
g
mass terms are naturally small accordin
g
to
G
erar
-
d
us ’t Hooft’s naturalness criterion
(
’t Hooft, 1980
)
— “At any energy scale
μ
, a set of parameter
s
α
i
(
μ
)
describing a system can be small, if and onl
y
if
, in the limi
t
α
i
(
μ
)
→
0f
or each o
f
these parameters, the s
y
stem exhibit
s
a
n enhanced s
y
mmetr
y
.” But there are several potential problems associate
d
with the seesaw ideas at a very low energy scale
(
de Gouvea et a
l
., 2007
)
:
(
a
)
a low-scale seesaw picture does not give any obvious connection to a the
-
oretically well-justified fundamental physical scale
(
such as the Fermi scale
,
the TeV scale, the GUT scale or the Planck scale
)
;
(
b
)
the neutrino Yukawa
c
oup
l
in
g
sina
l
ow-sca
l
e seesaw mo
d
e
l
turn out to
b
etiny,
g
ivin
g
no actua
l
explanation of why the masses of three known neutrinos are so small; and
(
c
)
i
n genera
l
,avery
l
ow seesaw sca
l
e
d
oes not a
ll
ow t
h
e “canonica
l”
t
h
erma
l
l
eptogenesis mechanism
(
Fukugita and Yanagida, 1986
)
to work, althoug
h
there might be an exotic way out.
I
tisthere
f
ore more natural to consider the possibility o
f
realizin
g
a seesa
w
mec
h
a
ni
s
m
above t
h
ee
l
ect
r
owea
k
sca
l
e but fa
r
be
l
ow
Λ
∼
1
0
7
G
eV. That
i
s just the TeV energy region which is being explored by the Large Hadro
n
4
.
3S
eesaw Mechanisms at the Te
VS
cale 1
35
Collider
(
LHC
)
. There are several reasons for possible existence of new physic
s
a
t the TeV scale. This kind of new ph
y
sics should be able to stabilize th
e
H
iggs-
b
oson mass an
dh
ence t
h
ee
l
ectrowea
k
sca
l
e; in ot
h
er wor
d
s, it s
h
ou
ld
be able to solve or so
f
ten the unnatural
g
au
g
e hierarchy problem. It has also
been argued that the weakly-interacting particle candidates for dark matter
s
hould weigh about one TeV or less
(
Dimopoulos, 1990
)
. If the TeV scal
e
i
sreall
y
a
f
undamental scale, we shall be reasonabl
y
motivated to speculat
e
that possible new physics existing at the TeV scale and responsible for th
e
electroweak symmetry breaking might also be responsible for the origin of
n
eutrino masses
(
Xing, 2008a
)
. It is interesting and meaningful in this sens
e
to investigate and balance the naturalness and testability of TeV seesa
w
m
echanisms at the ener
g
y
f
rontier set by the LH
C
.
As a big bonus of the conventional
(
type-I
)
seesaw mechanism, the ther
-
m
al leptogenesis mechanism
(
Fukugita and Yanagida, 1986
)
provides us wit
h
a
ne
l
e
g
ant
d
ynamic picture to interpret t
h
ecosmo
l
o
g
ica
l
matter-antimatter
a
s
y
mmetr
y
characterized b
y
the observed ratio o
f
the bar
y
on number den
-
s
ity to t
h
ep
h
oton num
b
er
d
ensity,
η
B
≡
n
B
/n
γ
=(6
.
1
±
0
.2
)
×
10
10
.
Wh
en
h
eav
y
Ma
j
orana neutrino masses are
d
own to t
h
e TeV sca
l
e, t
h
eYu
k
awa
c
ouplin
g
s should be reduced by more than six orders of ma
g
nitude so as to
generate tiny neutrino masses via the type-I seesaw mechanism and satisfy
the out-o
f
-equilibrium condition, but the
C
P-violatin
g
asymmetries o
f
heavy
Ma
j
orana neutrino deca
y
s can still be enhanced b
y
the resonant effects i
n
o
r
de
r
to accou
n
t
f
or
η
B
.T
h
is “resonant
l
eptogenesis
”
scenario mig
h
twor
k
i
n a specific TeV seesaw model
(
Pilaftsis and Underwood, 2004, 2005; Xing
a
nd Zhou, 2007
)
, but its naturalness is certainly questionable. Other see
-
s
aw mec
h
anisms mig
h
t
h
ave simi
l
ar pro
bl
ems at t
h
e TeV sca
l
etorea
l
ize t
h
e
s
uccess
f
ul baryo
g
enesis via lepto
g
enesis.
I
s there a TeV Noah’s Ark which can naturally and simultaneously accom
-
m
odate the seesaw idea, the lepto
g
enesis picture and the collider si
g
natures
?
We are most likel
y
not so luck
y
and should not be too ambitious at present
.
4.3
S
eesaw Mechanisms at the TeV
S
cal
e
The neutrino mass terms in three typical seesaw mechanisms have been
g
iven
i
n Section 4.1.2. Without loss of generality, we choose the basis in whic
h
the mass ei
g
enstates o
f
three char
g
ed leptons are identi
fi
ed with their
fl
avor
eigenstates. Let us discuss a few typical TeV seesaw mechanisms
.
4
.3.1 T
y
pe-I Seesaw Mechanis
m
G
ive
n
M
D
M
M
=
Y
ν
YY
v
/
√
2
, the approximate type-I seesaw formula in Eq.
(
4.13
)
ca
n
be
r
ew
ri
tte
n
as
M
ν
MM
=
−
M
D
MM
M
−
1
R
MM
M
T
D
MM
. Note that the 3
×
3
li
g
ht neutrin
o
m
ixing matri
x
V
i
s not exactly unitary in this seesaw scheme, as one can se
e