Prog. Theor. Exp. Phys. 2013, 063B04 (15 pages)
DOI: 10.1093/ptep/ptt042
Separate seesaw and its applications to dark matter
and baryogenesis
Ryo Takahashi∗
Department of Physics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan
∗
E-mail: [email protected]
Received March 08 2013; Accepted April 23 2013; Published June 1, 2013
...............................................................................
We propose a new seesaw model in an extra-dimensional setup where only right-handed neutrinos are bulk fields. In the model, the localizations of an extra-dimensional wave function and
brane Majorana mass of the right-handed neutrinos can be different among each generation of
the right-handed neutrinos. The setup can lead to different suppression factor dependences of
the effective right-handed neutrino masses and neutrino Yukawa couplings for each generation.
It is shown that the resultant mass spectra of the right-handed neutrinos and neutrino Yukawa
couplings are favored in models of neutrino dark matter with baryogenesis.
...............................................................................
Subject Index
1.
B40, B43, B70, B71, B73
Introduction
The results from neutrino experiments have established the phenomenon of neutrino oscillations. The
experimental data can be nicely understood by three flavor mixings of active neutrinos and described
by three mixing angles, one CP-phase, and two mass-squared differences of the active neutrinos.
The non-zero masses of the active neutrinos require a physics beyond the standard model (SM). In
addition, one has to explain the smallness of the active neutrino mass scales. One famous explanation
for realizing the small active neutrino masses is the seesaw mechanism [1–4]. There are several
types of seesaw mechanism. In the canonical type-I seesaw mechanism, heavy right-handed (sterile)
neutrinos are introduced into the SM as Majorana particles. After decoupling the heavy right-handed
neutrinos, the small active neutrino masses can be realized. Furthermore, it is well known that decays
of such heavy Majorana (right-handed) neutrinos can also generate the baryon asymmetry of the
universe (BAU), so-called leptogenesis [5].
In addition to leptogenesis, the right-handed (sterile) neutrinos can also play a cosmological role
as a dark matter (DM) [6] candidate. If the mass of the sterile neutrino is in the keV range, the
sterile neutrino can serve as DM [7,8] (see also Refs. [9–21]) and such a sterile neutrino DM
model is constrained by cosmological and astrophysical observations [9–11,22–55]. There are also
some investigations of astrophysical phenomena [55–63] and related considerations for the keV
sterile neutrino DM (e.g., see Refs. [66–83]). One interesting discussion about the keV sterile neutrino DM gives simultaneous explanations for problems in particle physics and cosmology such
as the smallness of the active neutrino masses, the origin of BAU, and the LSND/MiniBooNE
anomaly [84,85], in addition to the DM [64,65,80–83]. However, such models for simultaneous
explanations of several problems require strongly hierarchical mass spectra of the sterile neutrinos
© The Author(s) 2013. Published by Oxford University Press on behalf of the Physical Society of Japan.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),
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PTEP 2013, 063B04
R. Takahashi
and the additional suppression of left–right mixing of the keV sterile neutrino in the context of the
seesaw mechanism. A split seesaw [80]1 and the Froggatt–Nielsen mechanism [86] can naturally
realize strongly hierarchical mass spectra of the right-handed neutrinos [78,79] but do not give the
additional suppression of the left–right mixing without taking the corresponding neutrino Yukawa
coupling to be small enough to satisfy the cosmological bound. In the split seesaw mechanism, an
essential point to obtain hierarchical mass spectra comes from the properties of the wave function in
an extra dimension.
Extra-dimensional theory is one of the fascinating approaches to afford a compelling solution to
the gauge hierarchy problem [87–89]. In fact, the discovery of evidence of extra dimension(s) in addition to the Higgs particle and supersymmetry is one of the missions of the Large Hadron Collider
experiment. Further, extra-dimensional theories can also give rich phenomenologies such as alternatives to the electroweak (EW) symmetry breaking mechanisms in the SM (e.g., see Refs. [90–98]),
DM candidates [99–103], deviations of coupling constants from standard ones [96–98,104–107],
and realizations of fermion mass hierarchies [103,108–111] and the smallness of the active neutrino
mass [112–125], etc. In this work, we will propose a new model with right-handed neutrinos in an
extra-dimensional space and apply it to models of keV sterile neutrino DM with the baryogenesis
mechanism. Then, it will be shown that the new seesaw can be favored in the DM models with respect
to a realization of the additional suppression of left–right mixing.
The paper is organized as follows: In Sect. 2, brief reviews of the canonical type-I seesaw and
split seesaw mechanisms are given. In Sect. 3, we discuss all possible localizations of the righthanded neutrinos and Majorana masses in an extra-dimensional space-time, and show the results for
relevant parameters to the neutrino masses. In Sect. 4, a new seesaw model from the extra dimension
is proposed. In Sect. 5, we apply the new seesaw to models of the neutrino DM with the baryogenesis
mechanism. Section 6 is devoted to the summary.
2. Seesaw mechanisms
2.1. Type-I seesaw mechanism
The relevant Lagrangian for the type-I seesaw mechanism [1–4] reads
(M R )i j c
μ
L = iν Ri γ ∂μ ν Ri − (yν )iα ν Ri L α φ +
ν Ri ν R j + h.c. ,
2
(1)
where ν Ri (i = 1, 2, 3) are the right-handed neutrinos, L α (α = e, μ, τ ) are left-handed lepton doublets, φ is the SM Higgs, yν is the neutrino Yukawa coupling matrix, M R is the Majorana mass matrix
of the right-handed neutrinos, and μ = 0, 1, 2, 3, respectively. Here we have introduced three generations of right-handed neutrinos to the SM. After integrating out the right-handed neutrinos, the
mass matrix of light (active) neutrinos is given by
Mν yνT M R−1 yν v 2 ,
(2)
where v is the vacuum expectation value (VEV) of the Higgs. This type-I seesaw mechanism with
heavy Majorana masses of the right-handed neutrinos can lead to small active neutrino masses
without tiny neutrino Yukawa couplings.
1
See also Ref. [81] for a consideration of A4 flavor models in the split seesaw mechanism.
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R. Takahashi
Split seesaw mechanism
Next, let us show a seesaw model in an extra dimension. This seesaw model is known as the split
seesaw mechanism [80]. The split seesaw has been considered in a flat 5D space-time compactified
on an orbifold S 1 /Z 2 with the 5D coordinate, y ≡ x 5 . A fundamental region of the orbifold is given
by y ∈ [0, ]. The orbifold gives two fixed points and one can consider branes at the fixed points.
One of the branes (y = 0) is an SM brane where the SM particles reside, while the other (y = ) is
a hidden brane.
Then three generations of Dirac spinor, i (x, y) = (χi (x, y), ψ̄i (x, y))T , are introduced on the
bulk with bulk masses m i in this flat 5D setup. A fundamental action for the 5D Dirac spinor is
4
dy M(ii A ∂ A i + m i i i ), A = 0, 1, 2, 3, 5,
(3)
S= d x
0
where M is a 5D fundamental scale and we assume that the 5D Dirac mass matrix m is diagonal
(m i ≡ m ii ) for simplicity. The 5D gamma matrices A are defined by
0 σμ
1 0
μ
5
=
.
(4)
and = −i
0 −1
σ̄ μ 0
Note that the mass dimension of i (x, y) is 3/2. The zero modes of i (x, y) follow Dirac
equations as
(0)
(i 5 ∂5 + m i )i (x, y) = 0.
(5)
The solutions of the 5D Dirac equations determine the wave function profiles of the zero modes on
the bulk as e∓m i y for χi (x, y) and ψi (x, y). In order to obtain chiral fermions in 4D, the fields are
transformed under an orbifold parity as
Z 2 : i (x, y) → Pi (x, y) = +i (x, y),
(6)
where P = −i5 . Under this orbifold parity, only ψi (x, y) can have a zero mode. Note that the bulk
masses m i have to carry negative Z 2 parity to make the fundamental action invariant under the above
parity transformation. After canonical normalization of the Dirac fermions in 4D, the zero modes of
(0)
Ri (x, y) ≡ (0, ψi (x, y))T are represented by a normalized field ψ Ri (x) as
2m i
(0)
(0)
Ri (x, y) =
(7)
em i y ψ Ri (x),
(e2m i − 1)M
(0)
(0)
where ψ Ri (x) are identified with the right-handed neutrinos in 4D, ψ Ri = ν Ri . A point to note is that
(0)
the 5D wave function profiles of Ri (x, y) with real positive (negative) m i localize at the hidden
(SM) brane since the profile is described by the exponential function, em i y . Therefore, the right(0)
handed neutrinos, ψ Ri (= ν Ri ), have exponentially suppressed Yukawa couplings at the SM brane
when m i 1 with real positive m i .
The split seesaw mechanism utilizes the above property of the wave function profiles of the righthanded neutrinos. The relevant Lagrangian for the split seesaw mechanism is given by
(0) A
(0)
(0) (0)
4
S= d x
dy M i Ri ∂ A Ri + m i Ri Ri
0
− δ(y)
(0)
( ỹν )iα Ri L α φ
( M̃ R )i j (0)c (0)
+
Ri R j + h.c. .
2
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(8)
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R. Takahashi
Inserting (7) into (8), we can obtain the effective 4D Majorana mass matrix of the right-handed
neutrinos and the neutrino Yukawa coupling matrix as
(M R )i j = f i f j ( M̃ R )i j ,
(yν )iα = f i ( ỹν )iα ,
respectively, where
(9)
(10)
fi ≡
2m i
.
− 1)M
(e2m i (11)
It is convenient for the following discussion to define a diagonal matrix F as Fi j ≡ δi j f j . The effective 4D Majorana mass matrix of the right-handed neutrinos and the neutrino Yukawa coupling matrix
given in (9) and (10) are rewritten with the use of F,
M R = F M̃ R F,
yν = F ỹν .
(12)
(13)
The seesaw mechanism leads to
Mν = yνT M R−1 yν v 2 = ỹνT M̃ R−1 ỹν v 2
(14)
for the mass matrix of the light neutrinos. Note that the factors f i are completely canceled out in the
seesaw mechanism. This is one of the interesting features of the split seesaw mechanism.
The most important feature of the mechanism is as follows: By taking appropriate values for m i , we
can obtain strongly hierarchical mass spectra of the right-handed neutrinos and the neutrino Yukawa
matrix elements shown in (9) and (10) due to the exponential suppression in f i , without introducing
strongly hierarchical many-mass scales in the model. For instance, if one takes (m 1 , m 2 , m 3 ) (23.3, 3.64, 2.26), M = 5 × 1017 GeV, −1 = 1016 GeV, ( M̃ R )ii = 1015 GeV in a diagonal basis of
M̃ R , we can obtain a splitting mass spectrum of the right-handed neutrinos including both keV and
intermediate mass scales as
(M1 , M2 , M3 ) = (5 keV, 1011 GeV, 1012 GeV)
(15)
in the 4D effective theory. In this case, mass scales in the fundamental action are super heavy as
O(1015–17 ) GeV or on the EW scale O(102 ) GeV, and the tiny active neutrino mass scale can be
realized by the seesaw mechanism without tiny neutrino Yukawa couplings. Further, the lightest
sterile neutrino with a keV mass could be a candidate for the DM and heavier sterile neutrinos with
masses of O(1011–12 ) GeV could lead to the BAU via leptogenesis.
3. Localizations of right-handed neutrinos and Majorana masses
3.1. Localized right-handed neutrinos at the hidden brane and Majorana masses at the
SM brane
In the previous section, we have presented a brief review of the split seesaw mechanism. In this
mechanism, the 5D wave functions of the right-handed neutrinos exponentially localize at the hidden
brane (7), and the Majorana masses of the right-handed neutrinos are localized at the SM brane (8). As
a result, one can easily obtain strongly hierarchical mass spectra of the right-handed neutrinos and the
neutrino Yukawa couplings without introducing strongly hierarchical many-mass scales in the fundamental action. Regarding the resultant active neutrino mass after the seesaw mechanism, it is given
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by the same formula as in the type-I seesaw mechanism. The setup and results are summarized as
⎧
(0)
(0)
⎪
⎨ Ri (x, y) = f i em i y ψ Ri (x) for the localization of the 5D wave functions
,
(16)
⎪
⎩ S ⊃ −δ(y) ( M̃ R )i j (0)c (0) for the localization of the Majorana mass matrix
Ri
Rj
2
and
⎧
⎪
⎪
⎨(M R )i j = f i f j ( M̃ R )i j
(yν )iα = f i ( ỹν )iα
⎪
⎪
⎩ M = ỹ T M̃ −1 ỹ v 2
ν
ν
R ν
for the 4D Majorana mass matrix
for the 4D neutrino Yukawa coupling matrix,
(17)
for the light (active) neutrino mass matrix
(0)
with the suppression factors f i (if m i 1), respectively. We call this case ( R , M̃ R ) = (H, S)
(0)
where “H ” and “S” mean that the 5D wave function ( R ) or Majorana mass ( M̃ R ) of the righthanded neutrinos localize at the hidden and SM branes, respectively. Therefore, note that the other
(0)
three possibilities of localizations (( R , M̃ R ) = (S, H ), (H, H ), and (S, S)) can be generically
considered. In the following sections, we investigate the other three localizations and their results.
3.2.
Localized right-handed neutrinos at the SM brane and Majorana masses at the
hidden brane
The split seesaw mechanism assumes (H, S) localizations, namely, the 5D wave functions of the
right-handed neutrinos localize at the hidden brane while the Majorana mass matrix localizes at the
(0)
SM brane. Here we consider the reverse situation as ( R , M̃ R ) = (S, H ), i.e., the 5D wave functions
of the right-handed neutrinos localize at the SM brane while the Majorana mass matrix localizes at the
hidden brane. The localization of the 5D wave function of the right-handed neutrinos is determined
by the sign for the bulk mass in (3). Therefore, if one take a minus sign for the bulk mass, unlike the
case of the split seesaw mechanism (H, S), one can obtain a 5D wave function profile described by
(0)
e−m i y for the zero mode of the right-handed neutrinos. In this case, after normalization, Ri (x, y)
is written as
2m i
(0)
(0)
(0)
(18)
e−m i y ψ Ri (x) ≡ gi e−m i y ψ Ri (x).
Ri (x, y) =
−2m
i
(1 − e
)M
Note that the factor gi is not an exponential suppression one if m i 1, unlike the f i in the split
seesaw mechanism. Therefore, the right-handed neutrinos couple to the SM particles without any
exponential suppressions.
For the localization of the Majorana mass matrix at the hidden brane, the relevant action becomes
S=
d4x
−
0
(0)
(0)
(0)
(0)
dy M i Ri A ∂ A Ri − m i Ri i R
(0)
δ(y)( ỹν )iα Ri L α φ
( M̃ R )i j (0)c (0)
+ δ(y − )
Ri R j + h.c. .
2
(19)
After substituting (18) into (19), one can obtain the 4D Majorana mass matrix of the right-handed
neutrinos and the 4D neutrino Yukawa coupling matrix as
(M R )i j = gi g j e−(m i +m j ) ( M̃ R )i j = f i f j ( M̃ R )i j ,
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(yν )iα = gi ( ỹν )iα ,
(20)
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R. Takahashi
respectively, where the relation f i = gi e−m i is utilized. By taking the matrix forms as (9) and yν =
G ỹν with G i j ≡ δi j gi , the seesaw mechanism leads to Mν = ỹνT E M̃ R−1 E ỹν v 2 , where E ≡ G F −1 =
F −1 G and E i j = δi j em i .
We find that the 4D Majorana masses of the right-handed neutrinos are exponentially suppressed
from the fundamental scale M̃ R , like the split seesaw mechanism. Therefore, strongly hierarchical
mass spectra of the right-handed neutrinos can also be realized without introducing strongly hierarchical many-mass scales for the right-handed neutrinos. On the other hand, the neutrino Yukawa
couplings are not suppressed, unlike the split seesaw case. As a result, the fundamental mass scale of
the active neutrino mass matrix is enhanced compared with that in the type-I and split seesaw cases.
These are summarized as
⎧
(0)
(0)
⎪
for the localization of the 5D wave functions
⎨ Ri (x, y) = gi e−m i y ψ Ri (x)
, (21)
⎪
⎩ S ⊃ −δ(y − ) ( M̃ R )i j (0)c (0) for the localization of the Majorana mass matrix
Ri
Rj
2
and
⎧
⎪
⎪
⎨(M R )i j = f i f j ( M̃ R )i j
(yν )iα = gi ( ỹν )iα
⎪
⎪
⎩ M = ỹ T E M̃ −1 E ỹ v 2
ν
3.3.
ν
ν
R
for the 4D Majorana mass matrix
for the 4D neutrino Yukawa coupling matrix .
(22)
for the light (active) neutrino mass matrix
Localized right-handed neutrinos and Majorana masses at the hidden brane
Next, we investigate the (H, H ) case, i.e., both the wave functions and the Majorana masses of
the right-handed neutrinos are localized at the hidden brane. The 5D wave function profiles of the
right-handed neutrinos are given by (7). Then the relevant action is described by
S=
d4x
−
0
(0)
(0)
(0)
(0)
dy M i Ri A ∂ A Ri + m i Ri i R
(0)
δ(y)( ỹν )iα Ri L α φ
Inserting (7) into (23), we find
⎧
(0)
(0)
⎪
⎨ Ri (x, y) = f i em i y ψ Ri (x)
⎪
⎩ S ⊃ −δ(y − ) ( M̃ R )i j (0)c (0)
Ri
Rj
2
as the setup, and
⎧
⎪
⎪
⎨(M R )i j = gi g j ( M̃ R )i j
(yν )iα = f i ( ỹν )iα
⎪
⎪
⎩ M = ỹ T E −1 M̃ −1 E −1 ỹ v 2
ν
ν
ν
R
( M̃ R )i j (0)c (0)
+ δ(y − )
Ri R j + h.c. .
2
(23)
for the localization of the 5D wave functions
,
(24)
for the localization of the Majorana mass matrix
for the 4D Majorana mass matrix
for the 4D neutrino Yukawa coupling matrix,
(25)
for the light (active) neutrino mass matrix
as the results. It is seen that the 4D Majorana masses do not include exponential suppression factors,
and thus this (H, H ) case requires some mass scales for the right-handed neutrino sector in order
to realize strongly hierarchical mass spectra. For the 4D neutrino Yukawa coupling, it is suppressed
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by f i as well as the split seesaw case (H, S). The resultant active neutrino mass matrix includes a
suppression factor E −1 ((E −1 )i j = δi j e−m i ) compared to the type-I and split seesaw mechanisms.
3.4.
Localized right-handed neutrinos and Majorana masses at the SM brane
Finally, we show the (S, S) case, i.e., both the wave functions and the Majorana masses of the righthanded neutrinos are localized at the SM brane. The 5D wave function profiles of the right-handed
neutrinos are given by (18) and the relevant action is described by
(0) A
(0)
(0) (0)
4
S= d x
dy M i Ri ∂ A Ri − m i Ri Ri
0
− δ(y)
(0)
( ỹν )iα Ri L α φ
Substituting (18) into (26), we find
⎧
(0)
(0)
⎪
⎨ Ri (x, y) = gi e−m i y ψ Ri (x)
( M̃ R )i j (0)c (0)
+
Ri R j + h.c. .
2
(26)
for the localization of the 5D wave functions
⎪
⎩ S ⊃ −δ(y) ( M̃ R )i j (0)c (0) for the localization of the Majorana mass matrix
Ri
Rj
2
as the setup, and
⎧
⎪
⎪
⎨(M R )i j = gi g j ( M̃ R )i j for the 4D Majorana mass matrix
,
(yν )iα = gi ( ỹν )iα
for the 4D Yukawa coupling matrix
⎪
⎪
⎩ M = ỹ T M̃ −1 ỹ v 2
for the light (active) neutrino mass matrix
ν
ν
R ν
,
(27)
(28)
as the results. It can be seen that both the 4D Majorana masses of the right-handed neutrinos and the
neutrino Yukawa couplings are not suppressed because of the factor gi (if m i 1), and the active
neutrino mass matrix is described by the same formula as in the type-I and split seesaw mechanisms.
Therefore, this (S, S) case is similar to the type-I seesaw because of the localizations of the wave
functions and Majorana masses at the SM brane.
4.
Separate seesaw
We propose a new seesaw model using the above results. A fundamental assumption in this seesaw
model is that each generation of right-handed neutrinos can have different localizations of the
5D wave function and Majorana mass, e.g., one of the generations of the right-handed neutrinos
has (H, S) localization and the others have (S, H ) localization. This means that the localizations
(0)
of ( R , M̃ R ) are separate for each generation of right-handed neutrinos. Under this assumption,
when one introduces two generations of right-handed neutrinos into the SM, one can consider 10
(0)
(0)
possible combinations of the localizations,2 e.g., (( R1 , M̃ R1 ), ( R2 , M̃ R2 )) = ((H, S), (H, S)),
((H, S), (S, H )), ((H, S), (H, H )), ((H, S), (S, S)), and ((S, H ), (S, H )) etc., where M̃ Ri is the
fundamental mass scale of the ith generation of the right-handed neutrinos in a 5D action with a
diagonal basis for the 5D Dirac mass matrix. Therefore, in this seesaw model, M̃ R is not generically matrix. In this work, we investigate a combination and its results as an example, which will be
2
Here we do not consider the ordering of generations; for instance, we do not distinguish the
((H, S), (S, H )) case from the ((S, H ), (H, S)) one. There are 24 possible combinations of localizations in
the case of three generations of right-handed neutrinos.
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R. Takahashi
interestingly applied to DM and baryogenesis models with respect to a left–right mixing angle, as
we will see later.
(0)
(0)
We consider the case of (( R1 , M̃ R1 ), ( R2 , M̃ R2 )) = ((H, S), (S, H )) as an example with two
generations of right-handed neutrinos for simplicity.3 The relevant action for this case of the separate
seesaw is given by
S=
d4x
0
(0)
(0)
(0)
(0)
(0)
(0)
dy M i Ri A ∂ A Ri + m 1 R1 R1 − m 2 R2 R2
M̃
M̃
1
2
(0)
(0)c (0)
(0)c (0)
− δ(y) ( ỹν )iα Ri L α φ +
+ h.c. ,
+ δ(y − )
2 R1 R1
2 R2 R2
(29)
where i = 1, 2 and M̃i are the fundamental mass scales of each generation of right-handed neutrinos.
Note that the Majorana masses of the right-handed neutrinos ( M̃i ) in this action are not described by
a matrix form because of the separate localizations at different branes. The solutions of the 5D Dirac
equations from this action determine the extra-dimensional wave function profiles of the right-handed
neutrinos as
(0)
(0)
R1 (x, y) = f 1 em 1 y ψ R1 (x),
(0)
(0)
R2 (x, y) = g2 e−m 2 y ψ R2 (x).
(30)
Then the resultant 4D Majorana masses, 4D Yukawa coupling matrices, and the light neutrino mass
matrix after the separate seesaw are written down as
Mi = f i2 M̃i ,
(yν )1α = f 1 ( ỹν )1α ,
(31)
(yν )2α = g2 ( ỹν )2α ,
(Mν )αβ = (( ỹν )1α ( ỹν )1β M̃1−1 + e2m 2 ( ỹν )2α ( ỹν )2β M̃2−1 )v 2 ,
(32)
(33)
where α, β = e, μ, τ . After diagonalizing the light neutrino mass matrix Mν , one can obtain three
mass eigenvalues of the light neutrinos as
m ν1 = 0,
m ν2 O(|( ỹν )1α |2 v 2 M̃ R−1 ),
m ν3 O(e2m 2 |( ỹν )2α |2 v 2 M̃ R−1 ),
(34)
where we assume |( ỹν )iα | |( ỹν )iβ | (α = β) and take M̃ R , which is a fundamental mass scale of
the right-handed neutrino sector in the action, as M̃ R M̃1 M̃2 . Note that this kind of separate
seesaw can lead to hierarchical mass eigenvalues of the light neutrinos, which are determined by
exponential factors, due to the different localizations of the wave function and Majorana mass for
each generation of right-handed neutrinos, unlike the type-I and split seesaw mechanisms. This model
can also realize hierarchical mass spectra of the right-handed neutrinos and the neutrino Yukawa
couplings among each generation of right-handed neutrinos. Other combinations of localizations
lead to different patterns of mass spectra for the light neutrinos and right-handed neutrinos, and
neutrino Yukawa couplings for each generation. Such formalism for all cases and extensions to three
generations of right-handed neutrino models are straightforward. We now concentrate on this type of
3
Such a separate localization of the Majorana masses can be realized by introducing gauge singlet scalars
with lepton number and an additional discrete symmetry. We will show an example of a realistic case of three
generations in the next section.
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(0)
(0)
separate seesaw, (( R1 , M̃ R1 ), ( R2 , M̃ R2 )) = ((H, S), (S, H )), and apply it to realistic models of
neutrino DM and baryogenesis with three generations of right-handed neutrinos in the next section.
5.
Applications to models of dark matter and baryogenesis
In this section, we apply the separate seesaw to models of neutrino DM with baryogenesis. It is
known that a sterile neutrino with a keV scale mass can be a candidate for DM. Further, models with
three generations of right-handed neutrinos including the keV sterile neutrino can simultaneously
explain the BAU, the tiny active neutrino mass scale, and the LSND/MiniBooNE anomaly in addition
to the DM [64,65,80–83]. In such models, the tiny active neutrino mass is described by the type-I
seesaw formula, and the models can be embedded into the split seesaw mechanism in order to realize
split mass spectra without introducing strongly hierarchical many-mass scales. However, it is also
known that, in models with the keV sterile neutrino, the sterile neutrino (with keV mass) should not
contribute to the active neutrino masses to satisfy cosmological (X-ray) bounds on left–right mixing
(neutrino Yukawa couplings) for the keV sterile neutrino as
θ2 ≡
|(yν )1α |2 v 2
M12
O(10−11 − 10−9 ),
(35)
in a mass region of the sterile neutrino of O(1) keV Ms (≡ M1 ) O(10) keV (e.g., see Ref. [55]
and references therein). This cosmological bound means that the resultant neutrino mass, m ν1 , from
the keV sterile neutrino after the seesaw mechanism is much smaller than the solar and atmospheric neutrino mass scales as m ν1 m sol < m atm , where m sol 8.73 × 10−3 eV and m atm 5.05 × 10−2 eV [126]. In other words, once the keV sterile neutrino contributes to the active neutrino
mass through the seesaw mechanism, the model is in conflict with the cosmological bound because
of the large left–right mixing angle,
θ2 =
|(yν )1α |2 v 2
M12
m sol(atm)
O(10−6 − 10−5 ).
Ms
(36)
Therefore, in a realistic model with such keV sterile neutrino DM, an additional suppression for
θ (equivalent to |(yν )1α |) is needed to ensure consistency with the cosmological constraint. Further, at least two more generations of right-handed neutrinos are required to reproduce the solar
and atmospheric neutrino mass scales. Such a situation is realized by taking the Yukawa couplings of the keV sterile neutrino to be additionally small compared to those of the other two
sterile neutrinos, which give the solar and atmospheric neutrino mass scales, in some models with
keV sterile neutrino DM. The separate seesaw can lead to the additional suppression of the left–
right mixing angle (the Yukawa couplings) and give a typical mass spectrum of active neutrinos as
m ν1 m ν2 ( m sol ) < m ν3 ( m atm ) in the keV sterile neutrino DM models.
In order to see this, we extend the separate seesaw with two generations of right-handed neutrinos
given in the previous section to the three-generation case. The relevant Lagrangian is given by
S=
d4x
−
0
δ(y)
(0)
(0)
(0)
(0)
(0)
(0)
dy M i Ri A ∂ A Ri + m 1 R1 R1 − m j R j R j
(0)
( ỹν )iα Ri L α φ
M̃ j (0)c (0)
M̃1 (0)c (0)
+
+ h.c. ,
+ δ(y − )
2 R1 R1
2 Rj Rj
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R. Takahashi
where i = 1, 2, 3, j = 2, 3, and we take a 2 × 2 diagonal mass matrix, M̃ R = Diag{ M̃2 , M̃3 }.4 Note
(0)
(0)
that this is an extension of the (( R1 , M̃ R1 ), ( R2 , M̃ R2 )) = ((H, S), (S, H )) case to the three(0)
(0)
generation case, (( R1 , M̃ R1 ), ( R j , M̃ R j )) = ((H, S), (S, H )). In a similar manner to the twogeneration case, we obtain extra-dimensional wave function profiles of the right-handed neutrinos
as
(0)
(0)
(0)
(0)
R1 (x, y) = f 1 em 1 y ψ R1 (x), R j (x, y) = g j e−m j y ψ R j (x),
(38)
and read the resultant 4D Majorana masses, 4D neutrino Yukawa coupling matrices, light neutrino
mass matrix, and three mass eigenvalues of the light neutrinos as
Mi = f i2 M̃i ,
(yν )1α = f 1 ( ỹν )1α ,
⎛
(39)
(yν ) jα = g j ( ỹν ) jα ,
(Mν )αβ = ⎝( ỹν )1α ( ỹν )1β M̃1−1 +
⎞
(40)
⎠ v2,
e2m j ( ỹν ) jα ( ỹν ) jβ M̃ −1
j
(41)
m ν j O(e2m j |( ỹν ) jα |2 v 2 M̃ R−1 ),
(42)
j=2,3
m ν1 O(|( ỹν )1α |2 v 2 M̃ R−1 ),
respectively. This separate seesaw can realize strongly hierarchical mass spectra of the sterile neutrinos and neutrino Yukawa couplings among the first and other generations of right-handed neutrinos.
As a result, one of the three neutrino mass eigenvalues m ν1 is suppressed from the other two mass
eigenvalues m ν j . In a realistic model, these two mass eigenvalues are of the order of the solar and
atmospheric scales, m ν j ∼ m sol(atm) . However, in the expressions of the mass eigenvalues m ν j , an
exponential factor e2m j appears, while m ν1 does not include the factor. This affects the evaluation
of the left–right mixing as
|(yν )1α |2 v 2
m sol(atm)
θ2 =
2m ,
(43)
2
e j Ms
M1
where we assume that all |( ỹν )iα | are of the same order.
We apply this separate seesaw to two typical mass spectra of the sterile neutrinos, which can give
the sterile neutrino DM and generate the BAU in addition to the tiny active neutrino mass scales. The
first example of the mass spectra is
(M1 , M2 , M3 ) ∼ (O(keV), O(1011 GeV), O(1012 GeV)).
(44)
In this type of model, the lightest sterile neutrino with keV mass is the DM, and heavy sterile
neutrinos with an intermediate mass scale can lead to the BAU via leptogenesis, as mentioned in
4
Such a separate localization of the Majorana masses can be realized by introducing gauge singlet scalars
with an additional discrete symmetry. For instance, we introduce two gauge singlet scalars with lepton number, which are localized at the SM or the hidden brane, e.g., φ S is localized at the SM brane while φ H is
localized at the hidden brane. Further, we impose a Z 6 discrete symmetry and assign charges under Z 6 as
(0)
(0)
4
iπ/3
. The setup can
Q(φ S ) = 1, Q(φ H ) = ω64 , Q((0)
R1 ) = 1, Q( R2 ) = ω6 , and Q( R3 ) = ω6 , where ω6 ≡ e
induce Majorana masses in the second line of (37) after the lepton number violation by the VEVs of φ S and
φ H as δ(y) M̃1 = δ(y)λ1 φ S and δ(y − ) M̃ j = δ(y − )λ j φ H , where λi are the Yukawa couplings among
the right-handed neutrinos and new gauge singlets. For the neutrino Yukawa interactions, we have to assign
appropriate charges to L α with Q(φ) = 1 without reducing the rank of the neutrino Dirac mass matrix in order
to realize three mass eigenvalues of the active neutrinos. If one introduces more gauge singlets, which do not
have the lepton number but have Z 6 charges, one can obtain rich flavor structures in the neutrino sector to realize experimentally observed values of lepton mixing angles. This is one of the examples inducing the above
separate localization of the right-handed Majorana masses.
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PTEP 2013, 063B04
R. Takahashi
the split seesaw mechanism. This mass spectrum of the sterile neutrinos requires corresponding
neutrino Yukawa couplings |(yν ) jα | of order |(yν ) jα | m sol(atm) M j /v ∼ O(10−2 − 10−1 ), as
known in the canonical type-I seesaw mechanism. In our realization using the separate seesaw, when
we take (m 1 , m 2 , m 3 ) (23.3, 3.64, 2.26), M = 5 × 1017 GeV, −1 = 1016 GeV, M̃ R = M̃i =
1015 GeV, and |( ỹν )iα | (5 − 10) × 10−2 for all Yukawa couplings, the model leads to a mass spectrum (44), and the solar and atmospheric neutrino mass scales. Further, an additional suppression for
the left–right mixing angle can be naturally realized as shown in (43), θ 2 ∼ O(10−10 ), which is consistent with the cosmological bound. In this realization, the mass scales in the model are super-heavy
O(1015-17 ) GeV or on the EW scale.
The next application of the separate seesaw is to a model with a mass spectrum of the sterile
neutrinos as
(M1 , M2 , M3 ) ∼ (O(keV), O(10GeV), O(10GeV)).
(45)
This type of model can explain the BAU through the mechanism proposed in Ref. [127], i.e., an
oscillation between the second and third generations of right-handed neutrinos. The model described
by the mass spectrum (45) is known as the νMSM [64,65], which can also realize the tiny active
neutrino masses through the seesaw mechanism by utilizing two heavier sterile neutrinos. Since
this model also includes the keV sterile neutrino DM, the model should require an additional
suppression for the left–right mixing (neutrino Yukawa couplings) of the keV sterile neutrino to
satisfy the cosmological bound as well as the previous example. In the realization of the separate
seesaw, when we take (m 1 , m 2 , m 3 ) (23.3, 15.9, 15.9), M = 5 × 1017 GeV, −1 = 1016 GeV,
M̃ R = M̃i = 1015 GeV, and | ỹ jα | O(10−7 ), the model can lead to a mass spectrum (45) and
θ 2 < O(10−11 ) in addition to both the solar and atmospheric neutrino mass scales.5
6.
Summary
The introduction of right-handed neutrinos into the SM can solve several problems in particle physics
and cosmology, such as the DM, the generation of the BAU, the neutrino anomaly, and the realization
of the small active neutrino mass through the seesaw mechanism. Further, extra-dimensional theories
can also lead to various interesting results for phenomenology. In this work, we have focused on the
case in which only right-handed neutrinos live in flat 5D space. In particular, the localizations of
the extra-dimensional wave functions and Majorana masses of the right-handed neutrinos, and the
results for relevant parameters for the neutrino masses have been investigated. First, it has been shown
that different localizations of the wave functions and Majorana masses of the right-handed neutrinos
in the extra dimension lead to different suppression factor dependences of the effective Majorana
neutrino masses and the neutrino Yukawa couplings.
Next, a new extra-dimensional seesaw model has been proposed by combining the above results. A
fundamental assumption of the new seesaw is that each generation of right-handed neutrinos has different localizations for the wave function and Majorana masses in the extra dimension. As a result,
the Majorana neutrino masses and the neutrino Yukawa couplings for each generation can be different to those of the canonical type-I seesaw and split seesaw mechanisms. We have shown the
We have taken smaller values of neutrino Yukawa couplings in the action as | ỹ jα | O(10−7 ), which are
the same as those required in the νMSM for the Yukawa couplings of the second and third generations of righthanded neutrinos. The split seesaw does not need such tiny Yukawa couplings in the action for the heavier
right-handed neutrinos but this separate seesaw does require them. Therefore, an additional mechanism to give
such tiny Yukawa couplings, e.g., the Froggatt–Nielsen mechanism [86], should be introduced.
5
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R. Takahashi
(0)
(0)
results of the case, (( R1 , M̃ R1 ), ( R2 , M̃ R2 )) = ((H, S), (S, H )), as an example of the case of two
generations of right-handed neutrinos. It has been found that this separate seesaw model case can
lead to hierarchical mass eigenvalues of the light neutrinos, which are determined by exponential
factors, due to the different localizations of wave function and Majorana masses for each generation
of right-handed neutrinos, unlike the type-I and split seesaw mechanisms. Further, it has also been
shown that this model can realize hierarchical mass spectra of the right-handed neutrinos and the
neutrino Yukawa couplings among each generation of right-handed neutrinos.
Finally, we have applied the results of the separate seesaw to models of the keV sterile neutrino
DM with baryogenesis. It is known that an additional suppression for the left–right mixing angle of
the keV sterile neutrino is needed to be consistent with the cosmological constraint in the framework
of the seesaw mechanism. We have shown that the above separate seesaw extended to three generations of right-handed neutrinos can naturally realize the additional suppression for the mixing and
be favored in two typical models of the keV sterile neutrino DM with baryogenesis.
At the end of the paper, we comment on realizations of leptonic mixing angles determined by
neutrino oscillation experiments. The separate seesaw can be consistent with two possible discussions
to obtain experimentally observed values of leptonic mixing angles, i.e., a discussion of A4 flavor
models in the split seesaw mechanism [81] and a minimal neutrino texture analysis for the model
with the keV sterile neutrino DM [128]. The fundamental assumption of the separate seesaw might
also be interesting for other phenomenological applications and/or other types of seesaw mechanism,
such as a radiative seesaw model [129]. These applications will be presented in a different publication
(M. Aoki and R. Takahashi, manuscript in preparation).
Acknowledgements
This work is supported by Research Fellowships of the Japan Society for the Promotion of Science for Young
Scientists. The author thanks M. Aoki, T. Asaka, N. Haba, H. Nakano, and K. Yoshioka for fruitful discussions
at early stages of this work.
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