ABSTRACT
Title of dissertation: LEFT-RIGHT SYMMETRIC MODEL
AND ITS TEV-SCALE PHENOMENOLOGY
Chang-Hun Lee, Doctor of Philosophy, 2017
Dissertation directed by: Professor Rabindra N. Mohapatra
Department of Physics
The Standard Model of particle physics is a chiral theory with a broken parity
symmetry, and the left-right symmetric model is an extension of the SM with the
parity symmetry restored at high energies. Its extended particle content allows us
not only to find the solution to the parity problem of the SM but also to solve the
problem of understanding the neutrino masses via the seesaw mechanism. If the scale
of parity restoration is in the few TeV range, we can expect new physics signals that
are not present in the Standard Model in planned future experiments. We investigate
the TeV-scale phenomenology of the various classes of left-right symmetric models,
focusing on the charged lepton flavour violation, neutrinoless double beta decay,
electric dipole moments of charged leptons, and leptogenesis.
LEFT-RIGHT SYMMETRIC MODEL
AND ITS TEV-SCALE PHENOMENOLOGY
by
Chang-Hun Lee
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2017
Advisory Committee:
Professor Rabindra N. Mohapatra, Chair/Advisor
Professor Kaustubh Agashe
Professor Sarah Eno
Professor Niranjan Ramachandran
Professor Raman Sundum
c© Copyright by
Chang-Hun Lee
2017
Table of Contents
List of Abbreviations iv
1 Introduction 1
2 Minimal left-right symmetric model 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Review of the minimal left-right symmetric model . . . . . . . . . . . 8
2.3 Construction of lepton mass matrices . . . . . . . . . . . . . . . . . . 13
2.4 Conditions for the TeV-scale minimal left-right symmetric model . . . 16
2.5 Numerical procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Natural TeV-scale left-right symmetric model 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Outline of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Numerical procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 TeV-scale resonant leptogenesis 52
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 One-loop resummed effective Yukawa couplings and decay rates . . . 56
4.3 Boltzmann equations and the lepton asymmetry . . . . . . . . . . . . 57
4.4 Numerical procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5 Conclusion 66
ii
A Derivation of various expressions in the minimal left-right symmetric model 67
A.1 Gauge group and fields . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A.2 Current and generators . . . . . . . . . . . . . . . . . . . . . . . . . . 68
A.3 Yukawa interaction Lagrangian . . . . . . . . . . . . . . . . . . . . . 69
A.4 Spontaneous symmetry breaking and fermion masses . . . . . . . . . 69
A.5 Gauge bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
B Expressions of observables 85
C Parametrization of the Dirac neutrino mass matrix 105
D Boltzmann equation 108
E Lepton asymmetry 124
Bibliography 129
iii
List of Abbreviations
CLFV Charged lepton flavour violation
EDM Electric dipole moment
LH Left-handed
LHC Large Hadron Collider
LRSM Left-right symmetric model
MLRSM Minimal left-right symmetric model
PMNS Pontecorvo-Maki-Nakagawa-Sakata
RH Right-handed
SM Standard Model
TeV Teraelectron-Volts
0νββ Neutrinoless double beta decay
iv
Chapter 1: Introduction
The Standard Model (SM) of particle physics is the theoretical framework to ex-
plain the fundamental principles of nature. The gauge group of the SM before the
spontaneous symmetry breaking is
SU(2)L ⊗ U(1)Y . (1.1)
The representations of the leptons are
Li =
 νLi
`Li
 ∼ (2,−1), `Ri ∼ (1,−2), (1.2)
and for quarks, we have
Qi =
 uLi
dLi
 ∼ (2, 1/3), uRi ∼ (1, 4/3), dRi ∼ (1,−2/3) (1.3)
where i is the generation index. In addition, the scalar doublet field is given by
Φ =
 φ+
φ0
 ∼ (2, 1). (1.4)
The Yukawa interaction Lagrangian is written as
LY = −f `ijLiΦ`Rj − fuijQiΦ˜uRj − fdijQiΦdRj (1.5)
1
where Φ˜ ≡ iσ2Φ∗. After spontaneous symmetry breaking of the electroweak gauge
group SU(2)L⊗ U(1)Y to U(1)em via the vacuum expectation value (VEV) of Higgs
〈Φ〉 =
 0
vEW/
√
2
 (1.6)
where vEW = 246 GeV, the Yukawa interaction Lagrangian can be written as
〈LY 〉 = − 1√
2
f `ijvEW`Li`Rj −
1√
2
fuijvEWuLiuRj −
1√
2
fdijvEWdLidRj. (1.7)
In other words, the charged leptons and quarks acquire masses, and neutrinos remain
massless in the SM.
The observation of nonzero neutrino masses and mixing has provided the first
experimental evidence for physics beyond the SM. Since the origin of mass for all
charged fermions in the SM appears to have been clarified by the discovery of the
Higgs boson with mass of 125 GeV at the LHC [1, 2], an important question is
whether the same Higgs field is also responsible for neutrino masses. If we simply
add three right-handed (RH) neutrinos νR to the SM, Yukawa coupling terms of the
form
L`Y = −
1√
2
f `ijLiΦνRj + H.c. (1.8)
can be written in the lepton sector. After spontaneous symmetry breaking, this
Yukawa term gives masses of the form f `vEW/
√
2 to the neutrinos. However, to get
sub-eV neutrino masses as observed, it requires f ` . 10−12 which is an unnaturally
small number. This provides sufficient reason to believe that there is new physics
behind neutrino masses beyond adding just three RH neutrinos to the SM, thereby
providing the first clue to the nature of physics beyond the SM.
2
A simple paradigm for understanding the small neutrino masses is the type-I
seesaw mechanism [3–6] where the RH neutrinos alluded to above have a Majorana
mass of the form mNν
T
RνR, in addition to having Dirac masses like all charged
fermions in the SM. Neutrinos being electrically neutral allow for this possibility,
distinguishing them from the charged fermions, and this feature might be at the
heart of such diverse mass and mixing patterns for leptons in contrast with the
quark sector. The seesaw mechanism leads to the generic 6 × 6 neutrino mass
matrix
MνN =
 0 MD
MTD MR
 (1.9)
where the 3×3 Dirac mass matrix MD mixes the νL and νR states and is generated by
the SM Higgs field, while MR is the Majorana mass for νR which embodies the new
neutrino mass physics. In the usual seesaw approximation where |(MDM−1N )ij|  1,
the light neutrino mass matrix is given by the seesaw formula
Mν ≈ −MDM−1N MTD. (1.10)
Seesaw mechanism provides a very simple way to understand the smallness of
neutrino mases. Two main ingredients of this mechanism are: (i) the introduction
of RH neutrinos νR to the SM, and (ii) endowing the νR’s with a Majorana mass
which breaks the accidental B − L symmetry of the SM. In the context of the SM
gauge group, these two features do not follow from any underlying principle, but
are rather put in by hand. There is, however, a class of theories where both these
ingredients of seesaw emerge in a natural manner: the left-right symmetric theories
3
of weak interactions [7–9] based on the gauge group SU(2)L⊗ SU(2)R⊗ U(1)B−L.
The existence of the RH neutrinos is guaranteed by the gauge symmetry in both
cases and their Majorana masses are connected to the breaking scale of local B−L
symmetry, which is a subgroup of the above gauge groups. Furthermore they also
predict the number of νR’s to be three. Thus, the essential ingredients of seesaw are
no more adhoc but are rather connected to symmetries of the extended theory. It
is then important to explore how new features of these symmetries can be probed
in laboratory experiments. Our focus is on the low-scale left-right symmetric model
(LRSM) where the seesaw scale can be in the few TeV range and be accessible to
the LHC, while satisfying the observed charged lepton and neutrino mass spectra.
The first question for such models is how the small neutrino masses can be un-
derstood if the seesaw scale is indeed in the TeV range, since by naive expectations,
the Dirac masses are expected to be similar to the charged lepton masses, which
after seesaw would give rise to too large tau neutrino mass. In the context of the
minimal LRSM, this question becomes specially important since the Higgs sector
relates the neutrino Yukawa couplings with charged lepton ones. There are three
ways to fit both charged lepton and neutrino masses in such TeV scale LRSM: (i)
by choosing one set of the Yukawa couplings to be . 10−5.5 for a particular VEV
assignment for the SM-doublet Higgs fields; (ii) by choosing larger Yukawa couplings
and invoking cancellations between Yukawa couplings in the Dirac neutrino mass
matrix to get smaller Dirac masses for neutrinos to get seesaw to work and (iii) by
choosing particular textures for the Yukawa couplings that guarantees the leading
order seesaw contribution to neutrino masses to vanish. We call these models Class
4
I, II, and III models respectively.
5
Chapter 2: Minimal left-right symmetric model
2.1 Introduction
In the lepton sector of the minimal left-right symmetric model (MLRSM), we have
four mass matrices: the charged lepton mass matrix M`, the Dirac neutrino mass
matrix MD, and the left-handed and RH Majorana neutrino mass matrices ML and
MR. The light neutrino mass matrix Mν is determined by MD, ML, and MR through
the seesaw mechanism Mν ≈ML−MDM−1R MTD. Since we have experimental data on
the masses of charged leptons and the squared-mass differences of neutrinos as well
as their mixing angles, M` is completely known in the charged lepton mass basis and
Mν is also partially determined in its own mass basis and in the charged lepton mass
basis. The neutrino mass matrices MD, ML, and MR are nonetheless completely
unknown, and constructing those matrices compatible with experimental data is a
nontrivial problem, not only because M` and MD in the MLRSM are determined
from common Yukawa couplings and electroweak VEV’s, but also because those
Yukawa coupling matrices have a specific structure (i.e. Hermitian or symmetric)
in a specific basis (i.e. symmetry basis) due to the discrete symmetry (i.e. parity
or charge conjugation symmetry) of the model that realizes the manifest left-right
symmetry at high energies.
6
For simplicity, we may assume that the electroweak VEV’s are all real, in
which case M` and MD have the same structure (i.e. Hermitian or symmetric) as
the Yukawa coupling matrices. Since M` in that case is diagonalized by a similarity
transformation (i.e. V `R = V
`
L for Hermitian M`, and V
`
R = V
`∗
L for symmetric
M`), the mass matrices in the charged lepton mass basis maintain that structure.
Hence, we can work in that basis where M` is completely determined so that we
can practically forget about it while keeping the structure of mass matrices. Now
using that structure itself, we can find MR from known MD [10] or alternatively find
MD from known MR [11]. Without loss of generality, however, we can make only
one of two electroweak VEV’s real by gauge transformation. Furthermore, for the
TeV-scale MLRSM, MD assumed or constructed in such ways usually requires fine-
tuning of Yukawa couplings and VEV’s, and it would be rather difficult to make
natural predictions for the TeV-scale phenomenology of the MLRSM using those
mass matrices.
Here, we develop a different approach appropriate for the case of type-I dom-
inance (i.e. ML = 0) with complex electroweak VEV’s: (i) the Yukawa coupling
matrices with a desired structure are constructed from M` in the symmetry ba-
sis; (ii) MD is determined from those Yukawa couplings as well as the electroweak
VEV’s, and MR is calculated from MD we have found. Since Yukawa couplings are
explicitly constructed and MD is calculated from them, fine-tuned MD can only ap-
pear rarely. With this method, we collect a huge amount of data points that satisfy
all the major experimental constraints, and conduct a comprehensive study of the
TeV-scale phenomenology of the model, focusing on the CLFV, 0νββ, and EDM’s
7
of charged leptons.
There are several works which studied CLFV and 0νββ in the MLRSM: in
reference [12], those effects were discussed in the type-I or type-II seesaw dominance,
and several processes of 0νββ were examined in detail; in reference [13], CLFV and
0νββ processes were investigated also in type-I or type-II dominance with emphasis
on the allowed masses of doubly charged scalar fields; in reference [14], the type-I+II
seesaw contributions were simultaneously considered as in references [10] and [11],
but with richer results on the phenomenology; in reference [15], the CLFV effects
were studied in detail also in the type-I+II seesaw cases by a slightly different method
from the one originally proposed by reference [10]. However, the common features
of those works are: (i) real electroweak VEV’s were explicitly or implicitly assumed,
and (ii) MD or MR was chosen for numerical analysis without considering the issue of
fine-tuning. Even though we can still obtain meaningful results focusing on specific
regions of parameter space with rich phenomenologies, it is important to investigate
the predictions of the model in a more natural situation. Furthermore, some works
assumed that the tree-level contribution to µ → eee is always dominant over the
type-I contribution in their analyses. We will also see that this is an inadequate
assumption.
2.2 Review of the minimal left-right symmetric model
In this section, we briefly review the MLRSM. The gauge group of the MLRSM is
SU(2)L ⊗ SU(2)R ⊗ U(1)B−L, (2.1)
8
and the representations of the leptons are
L′Li =
 ν ′Li
`′Li
 ∼ (2,1,−1), L′Ri =
 ν ′Ri
`′Ri
 ∼ (1,2,−1) (2.2)
where i is the flavour index. The bi-doublet scalar field is given by
Φ =
 φ01 φ+2
φ−1 φ
0
2
 ∼ (2,2, 0), (2.3)
and the triplet scalar fields are
∆L =
 δ+L /
√
2 δ++L
δ0L −δ+L /
√
2
 ∼ (3,1, 2), ∆R =
 δ+R/
√
2 δ++R
δ0R −δ+R/
√
2
 ∼ (1,3, 2).
(2.4)
The Lagrangian terms of Yukawa interactions are written as
L`Y = −L′Li(fijΦ + f˜ijΦ˜)L′Rj − hLijL′cLiiσ2∆LL′Lj − hRijL′cRiiσ2∆RL′Rj + H.c. (2.5)
where
Φ˜ ≡ σ2Φ∗σ2 =
 φ0∗2 −φ+1
−φ−2 φ0∗1
 . (2.6)
Here, ψc ≡ Cψ∗, and thus ψc = −ψTC where C = iγ2γ0 is the charge conjugation
operator in the Dirac-Pauli representation. Note that hL and hR are symmetric
matrices. Without loss of generality, we can write the VEV’s of scalar fields as
Φ =
 κ1/
√
2 0
0 κ2e
iα/
√
2
 , ∆L =
 0 0
vLe
iθL/
√
2 0
 , ∆R =
 0 0
vR/
√
2 0
 .
(2.7)
9
After spontaneous symmetry breaking, the mass matrix of charged leptons is written
as
M` =
1√
2
(fκ2e
iα + f˜κ1), (2.8)
and the neutrino mass term is given by
Lmassν = −
1
2
(ν ′L ν
′c
R)
 ML MD
MTD MR

 ν ′cL
ν ′R
+ H.c. (2.9)
where
MD =
1√
2
(fκ1 + f˜κ2e
−iα), ML =
√
2h∗LvLe
−iθL , MR =
√
2hRvR. (2.10)
When vL  κ1, κ2  vR, the light neutrino mass matrix is given by the seesaw
mechanism
Mν ≈ML −MDM−1R MTD. (2.11)
In this paper, we only consider the case of type-I dominance by assuming vL = 0,
and the light neutrino mass matrix is given by the type-I seesaw formula
Mν ≈ −MDM−1R MTD. (2.12)
We denote the mass eigenstates of the light and heavy neutrinos as νi and Ni (i =
1, 2, 3), respectively. The charged gauge bosons W−L , W
−
R in the gauge basis can be
written in terms of the mass eigenstates W−1 , W
−
2 as W−L
W−R
 =
 cos ξ sin ξeiα
− sin ξe−iα cos ξ

 W−1
W−2
 (2.13)
10
where ξ is the WL-WR mixing parameter given by
ξ ≈ −κ1κ2
v2R
. (2.14)
The masses of charged gauge bosons are
m2W1 ≈
1
4
g2v2EW, m
2
W2
≈ 1
2
g2v2R (2.15)
where vEW =
√
κ21 + κ
2
2 = 246 GeV is the VEV of the SM. In addition, the masses
of neutral gauge bosons Z1, Z2, A are given by
m2Z1 ≈
g2v2EW
4 cos2 θW
, m2Z2 ≈
g2 cos2 θWv
2
R
cos 2θW
, m2A = 0 (2.16)
where θW is the Weinberg angle. We can identify W1, Z1, A as W , Z, the photon
of the SM, respectively. The neutral gauge bosons W 3L, W
3
R, B in the gauge basis
are expressed in terms of the mass eigenstates as
W 3L
W 3R
B
 =

1 0 0
0 cos ζ1 sin ζ1
0 − sin ζ1 cos ζ1


cos ζ2 0 sin ζ2
0 1 0
− sin ζ2 0 cos ζ2


cos ζ3 sin ζ3 0
− sin ζ3 cos ζ3 0
0 0 1


Z1
Z2
A

(2.17)
where
ζ1 = sin
−1 (tan θW ), ζ2 ≈ θW , ζ3 ≈ −g
2
√
cos 2θWv
2
EW
4 cos2 θWm2Z2
. (2.18)
For the MLRSM with a manifest left-right symmetry before spontaneous symmetry
breaking, we need a discrete symmetry which could be either the parity symmetry
or the charge conjugation symmetry. In case of the parity symmetry, we have the
11
relationships of fields and Yukawa couplings given by
L′Li ↔ L′Ri, ∆L ↔ ∆R, Φ↔ Φ†, f = f †, f˜ = f˜ †, hL = hR,
(2.19)
and in case of the charge conjugation symmetry
L′Li ↔ L′cRi, ∆L ↔ ∆∗R, Φ↔ ΦT, f = fT, f˜ = f˜T, hL = h∗R.
(2.20)
We consider only the parity symmetry here. This symmetry is manifest in a specific
basis in the flavour space, which we call the symmetry basis. The scalar potential
invariant under the parity symmetry is written as
V = −µ21Tr
[
Φ†Φ
]− µ22 (Tr[Φ†Φ˜]+ Tr[Φ˜†Φ])− µ23 (Tr[∆†L∆L]+ Tr[∆†R∆R])
+ λ1Tr
[
Φ†Φ
]2
+ λ2
(
Tr
[
Φ†Φ˜
]2
+ Tr
[
Φ˜†Φ
]2)
+ λ3Tr
[
Φ†Φ˜
]
Tr
[
Φ˜†Φ
]
+ λ4Tr
[
Φ†Φ
] (
Tr
[
Φ†Φ˜
]
+ Tr
[
Φ˜†Φ
])
+ ρ1
(
Tr
[
∆†L∆L
]2
+ Tr
[
∆†R∆R
]2)
+ ρ2
(
Tr
[
∆†L∆
†
L
]
Tr
[
∆L∆L
]
+ Tr
[
∆†R∆
†
R
]
Tr
[
∆R∆R
])
+ ρ3Tr
[
∆†L∆L
]
Tr
[
∆†R∆R
]
+ ρ4
(
Tr
[
∆†L∆
†
L
]
Tr
[
∆R∆R
]
+ Tr
[
∆L∆L
]
Tr
[
∆†R∆
†
R
])
+ α1Tr
[
Φ†Φ
] (
Tr
[
∆†L∆L
]
+ Tr
[
∆†R∆R
])
+
{
α2e
iδ2
(
Tr
[
Φ†Φ˜
]
Tr
[
∆†L∆L
]
+ Tr
[
Φ˜†Φ
]
Tr
[
∆†R∆R
])
+ H.c.
}
+ α3
(
Tr
[
ΦΦ†∆L∆
†
L
]
+ Tr
[
Φ†Φ∆R∆
†
R
])
+ β1
(
Tr
[
Φ†∆†LΦ∆R
]
+ Tr
[
Φ†∆LΦ∆
†
R
])
+ β2
(
Tr
[
Φ†∆†LΦ˜∆R
]
+ Tr
[
Φ˜†∆LΦ∆
†
R
])
+ β3
(
Tr
[
Φ˜†∆†LΦ∆R
]
+ Tr
[
Φ†∆LΦ˜∆
†
R
])
.
(2.21)
In this paper, we study the TeV-scale MLRSM without fine-tuning, for which κ1 
κ2 is one of the sufficient conditions, as we will see in section 2.4. The physical
scalar fields and their masses when vL = 0 and vR  κ1  κ2 are summarized in
table 2.1 [16].
12
2.3 Construction of lepton mass matrices
Now, we discuss the procedure to construct lepton mass matrices that satisfy the
experimental constraints in the light lepton sector (i.e. light neutrino masses and
mixing angles) in case of type-I dominance. The Yukawa coupling matrices f , f˜ in
the symmetry basis are Hermitian due to the parity symmetry before spontaneous
symmetry breaking. However, the mass matrices M` and MD in the same basis
do not have such structures when the electroweak VEV’s are complex, and it is
therefore a non-trivial problem to construct mass matrices that would give Yukawa
couplings with the right structure in the symmetry basis and simultaneously satisfy
all the constraints in the light lepton sector.
The procedure to construct such lepton mass matrices is as follows: (i) first,
we find M` in the symmetry basis that gives the right masses of charged leptons,
and build up f , f˜ , and VEV’s out of it. The solutions are not unique; (ii) MD is
constructed in the straightforward way from the Yukawa couplings and VEV’s we
have obtained, and MR can also be easily calculated from this MD and the type-I
seesaw formula of equation 2.12.
Since the masses of charged leptons are already known, M` in the symmetry
basis can be easily constructed from
M` = V
`
LM
c
`V
`†
R (2.22)
where V `L and V
`
R are arbitrary unitary matrices and M
c
` is the diagonal matrix which
has charged lepton masses as its entries. The superscript c denotes mass matrices
13
in the charged lepton mass basis, and we always assume that matrices without any
superscript are in the symmetry basis. Note that V `L and V
`
R are totally different
matrices in general even with a manifest discrete symmetry when the electroweak
VEV’s are complex. With the parity symmetry, we have M` = Ae
iα + B (A ≡
fκ2/
√
2, B ≡ f˜κ1/
√
2) where A, B are Hermitian matrices. Therefore, for the rest
of step (i), we claim that, for an arbitrary matrix M , it is always possible to find
Hermitian matrices A, B such that M = Aeiα +B.
In order to prove it, we explicitly construct Hermitian matrices A, B that
satisfy M = Aeiα + B. First, we write Aij = |Aij|eiθij and Bij = |Bij|eiφij where
θji = −θij and φji = −φij. Then, we have Mij = |Aij|ei(α+θij) + |Bij|eiφij and
Mji = |Aij|ei(α−θij) + |Bij|e−iφij . From these expressions, it is straightforward to
derive
2|Aij| sinα = ±
√
Re[Mji −Mij]2 + Im[Mji +Mij]2 (2.23)
and
tan θij =
Re[Mji −Mij]
Im[Mji +Mij]
. (2.24)
Note that two different values of θij are allowed in the range −pi < θij < pi for each
pair of i, j. In addition, since | sinα| ≤ 1, we must have
|Aij| ≥ 1
2
√
Re[Mji −Mij]2 + Im[Mji +Mij]2 (2.25)
which sets the lower bound of |Aij| for given M . If |Aij| 6= 0, we can write
sinα = ± 1
2|Aij|
√
Re[Mji −Mij]2 + Im[Mji +Mij]2. (2.26)
14
Now we choose an arbitrary real number |A11| that satisfies
|A11| >
∣∣Im[M11]∣∣, (2.27)
and determine α from
sinα = ±
∣∣Im[M11]∣∣
|A11| . (2.28)
Note that four different values of α are allowed in the range −pi < α < pi. We can
find all the other |Aij| from
|Aij| = 1
2| sinα|
√
Re[Mji −Mij]2 + Im[Mji +Mij]2 (2.29)
=
|A11|
2
∣∣Im[M11]∣∣
√
Re[Mji −Mij]2 + Im[Mji +Mij]2. (2.30)
By equations 2.30 and 2.24, A is completely determined. Alternatively we can write
Aij = ± 1
2| sinα|
(
Im[Mji +Mij] + iRe[Mji −Mij]
)
(2.31)
= ± |A11|
2
∣∣Im[M11]∣∣(Im[Mji +Mij] + iRe[Mji −Mij]). (2.32)
It is now trivial to find B from B = M − Aeiα, and explicitly
Re[Bij] =
1
2
Re[Mji +Mij]− Re[Aij] cosα,
Im[Bij] = −1
2
Im[Mji −Mij]− Im[Aij] cosα, (2.33)
or
Bij =
1
2
(
Re[Mji +Mij]− iIm[Mji −Mij]
)− Aij cosα. (2.34)
Note that A and B are indeed Hermitian matrices. Since we have two choices of Aij
for each pair of i, j as well as each choice of α and |A11|, there are 26 choices of A
15
for each α and |A11| as we have three diagonal and three off-diagonal independent
components in A. Moreover, since we have four choices of α for each |A11|, there
are total 26 · 4 = 256 different choices of A, B, and α for each choice of |A11|. We
use this method to construct lepton mass matrices in the TeV-scale MLRSM.
2.4 Conditions for the TeV-scale minimal left-right symmetric model
In the MLRSM, M` and MD are determined from common Yukawa couplings and
VEV’s: f , f˜ , κ1, and κ2e
iα. Hence, it would be natural if the largest component
of MD is O(1) GeV, since the largest component of M` should be comparable to
mτ ∼ O(1) GeV. However, this implies that the smallest heavy neutrino mass should
be larger than O(1010) GeV, since Mν is determined from the seesaw formula of
equation 2.12 and the present upper bound of the light neutrino mass is mν . O(0.1)
eV [17].
For the TeV-scale MLRSM, i.e. 0.1 TeV . mN . 100 TeV, we need |MDij| .
10−3 GeV. Since MD = (fκ1 + f˜κ2e−iα)/
√
2 in the MLRSM, its largest component
could be as small as 10−3 GeV when the corresponding components of fκ1 and
f˜κ2e
−iα almost cancel each other, which is however unnatural. One solution to avoid
such cancellation is that either fκ2 or f˜κ1 is dominant in M` while f˜κ2 and fκ1 are
both small and comparable to each other in MD. Note that we need hierarchies in
both Yukawa couplings and VEV’s to satisfy this condition. Even though it is good
enough if only a few components of either fκ2 or f˜κ1 that correspond to mτ and
mµ are dominant in M`, we assume that all the components of either fκ2 or f˜κ1 are
16
dominant over the others for simplicity.
Now we write A ≡ fκ2/
√
2 and B ≡ f˜κ1/
√
2, and thus M` = Ae
iα + B,
as before. When |Aij|  |Bij|, M` must be close to a Hermitian matrix, which is
equivalent to V `†L V
`
R ≈ 1. When |Aij|  |Bij|, we have M` ≈ Aeiα, which implies
that M`e
−iα is approximately Hermitian, i.e. V `†L V
`
R ≈ eiα. Note that we need the
condition on mixing matrices in addition to the conditions on the Yukawa couplings
and VEV’s since constructing M` from mixing matrices is one of the first steps to
construct all the mass matrices.
In this paper, we only consider the first case, i.e. |Aij|  |Bij|. For simplicity,
we could assume A = 0, for which we need either f = 0 or κ2 = 0. In these cases,
the mass matrices are rather simple: M` = f˜κ1/
√
2, MD = f˜κ2e
−iα/
√
2 if f = 0,
and M` = f˜κ1/
√
2, MD = fκ1/
√
2 if κ2 = 0. However, f = 0 is the limiting case
of an extreme hierarchy between two Yukawa coupling matrices f and f˜ , which
is rather unnatural. Furthermore, we must have M` ∝ MD ∝ f˜ , and thus MD is
diagonal in the mass basis of charged leptons, which means that we have to resort to
only restrictive structures of mass matrices. On the other hand, with the condition
κ2 = 0, the WL-WR mixing parameter ξ ≈ −κ1κ2/v2R vanishes, and we have to lose
the rich phenomenology dependent upon ξ, especially the EDM’s of charged leptons.
Therefore, we do not introduce these extreme conditions.
In summary, for the TeV-scale MLRSM without fine-tuning in MD, we can
assume the conditions either that (i) fij  f˜ij and κ1  κ2, when M` is approxi-
mately Hermitian, i.e. V `L ≈ V `R, or that (ii) fij  f˜ij and κ1  κ2, when M`e−iα is
approximately Hermitian, i.e. V `L ≈ V `Re−iα. We study the first case here.
17
2.5 Numerical procedure
In this paper, we only consider the normal hierarchy in light neutrino masses. The
procedure to calculate all the model parameters that determine the phenomenology
of the MLRSM in type-I dominance is as follows:
1. Randomly generate the lightest light neutrino mass mν1 , and calculate mν2 =√
m2ν1 + ∆m
2
21 and mν3 =
√
m2ν1 + ∆m
2
31.
2. Calculate M cν from M
c
ν = UPMNSM
diag
ν U
T
PMNS where M
c
ν and M
diag
ν are the light
neutrino mass matrices in the charged lepton and light neutrino mass bases,
respectively. The mixing matrix UPMNS is the Pontecorvo-Maki-Nakagawa-
Sakata (PMNS) matrix whose CP phases are also randomly generated.
3. Randomly generate V `L, V
`
R, and calculate M` = V
`
LM
c
`V
`†
R where M` and M
c
`
are charged lepton mass matrices in the symmetry and charged lepton mass
bases, repectively.
4. Find A ≡ fκ2/
√
2, B ≡ f˜κ1/
√
2 from M` = Ae
iα + B using the method
discussed in section 2.3. Randomly generate κ2, and calculate f , f˜ from A, B.
5. Calculate MD = (fκ1 + f˜κ2e
−iα)/
√
2 from f , f˜ , α, κ2, κ1 =
√
v2EW − κ22, and
find M cD = V
`†
L MDV
`
R where M
c
D is the Dirac neutrino mass matrix in the
charged lepton mass basis.
6. Calculate M cR from the type-I seesaw formula M
c
ν = −M cDM c−1R M cTD where
M cR is the RH neutrino mass matrix in the charged lepton mass basis.
18
7. Construct the 6 × 6 neutrino mass matrix M cνN from M cD and M cR, and find
the 6× 6 mixing matrix VνN that diagonalizes M cνN .
Here, the 6×6 neutrino mass matrix M cνN in the charged lepton mass basis is written
as
M cνN =
 0 M cD
M cTD M
c
R
 , (2.35)
and this matrix is diagonalized by the 6× 6 unitary matrix VνN :
MdiagνN = V
T
νNM
c
νNVνN (2.36)
where MdiagνN is the diagonal matrix with positive entries. Following the convention
of reference [12], we write
V ∗νN =
 U S
T V
 (2.37)
where U , S, T , and V are 3 × 3 mixing matrices. Note that U = UPMNS. The
straightforward numerical diagonalization might not work appropriately because of
the hierarchy in the components of M cνN . Instead, VνN is calculated in two steps:
VνN = VνN1VνN2 (2.38)
where
VνN1 =
 1 −M cDM c−1R
−M c−1R M cTD −1
 , VνN2 =
 U∗ 0
0 −V ∗
 . (2.39)
Here, VνN1 transforms MνN into the block-diagonal matrix
MBDνN =
 M cν 0
0 M cR +M
c−1
R M
cT
D M
c
D +M
cT
D M
c
DM
c−1
R
 , (2.40)
19
and VνN2 is the matrix that diagonalizes M
BD
νN . In addition, we use the standard
parametrization of the PMNS matrix:
UPMNS =

1 0 0
0 cos θ23 sin θ23
0 − sin θ23 cos θ23


cos θ13 0 sin θ13e
−iδD
0 1 0
− sin θ13eiδD 0 cos θ13


cos θ12 sin θ12 0
− sin θ12 cos θ12 0
0 0 1

×

1 0 0
0 e−iδM1 0
0 0 e−iδM2
 (2.41)
where δD and δMi are Dirac and Majorana CP phases, respectively. On the other
hand, we parametrize V `L and V
`
R as
V = V1V2V3 (2.42)
where
V1 =

1 0 0
0 e−iδ2 0
0 0 e−iδ3
 , (2.43)
V2 =

1 0 0
0 cos θ23 sin θ23
0 − sin θ23 cos θ23


cos θ13 0 sin θ13e
−iδ1
0 1 0
− sin θ13eiδ1 0 cos θ13


cos θ12 sin θ12 0
− sin θ12 cos θ12 0
0 0 1
 ,
(2.44)
V3 =

e−iδ4 0 0
0 e−iδ5 0
0 0 e−iδ6
 . (2.45)
20
Note that it is always possible to absorb V `R3 into V
`
L3 since M` = V
`
LM
c
`V
`†
R where
M c` is a diagonal matrix. We can therefore write
V `L = V
`
L1V
`
L2V
`
L3, V
`
R = V
`
R1V
`
R2. (2.46)
In addition, the Hermitian matrix A (≡ fκ2/
√
2) is parametrized as
A =

A11 |A12|eiθA12 |A13|eiθA13
|A12|e−iθA12 A22 |A23|eiθA23
|A13|e−iθA13 |A23|e−iθA23 A33
 (2.47)
where Aii are real numbers. The list of model parameters and the ranges where they
are randomly generated are summarized in table 2.2. Several appropriate constraints
are imposed on some model parameters, and they are presented in table 2.3.
2.6 Numerical results
The present and future experimental bounds on CLFV, 0νββ, and EDM’s of charged
leptons are summarized in table 2.4. The upper bound of light neutrino masses
from the Planck observation is also considered. The experimental bounds on the
dimensionless parameters associated with the various processes of 0νββ are given
in table 2.5. The numerical results are presented in figures 2.1−2.7. The plots
on the various branching ratios and conversion rates of CLFV in the MLRSM for
2 TeV < mWR < 30 TeV are given in figure 2.1. The results on the dimensionless
parameters of 0νββ for the same range of mWR are presented in figure 2.2. The
plots on the EDM’s of charged leptons are presented in figure 2.3. The effect of
experimental constraints on the masses of the RH gauge boson, neutrinos, and
21
scalar fields are shown in figures 2.4−2.7. The benchmark model parameters and
their predicitons are given in appendix B.
The most notable result is that the regions of parameter space that allow
small light neutrino masses are largely constrained by the experimental bounds
from CLFV as well as the constraints from the light neutrino mass and mixing an-
gles. Since the type-I seesaw formula implies det(Mν) ≈ det(MD)2/det(MR), we
need a hierarchy in the eigenvalues of MD or MR when light neutrino masses have
a hierarchy. However, MD is determined from Yukawa couplings and VEV’s, and
it generally does not have the appropriate hierarchy in its eigenvalues to give hier-
achical light neutrino masses for most of the available parameter space. In other
words, we generally need a hierarchy in the eigenvalues of MR, i.e. in the heavy
neutrino masses as well, in order to obtain hierachical light neutrino masses. Since
we are considering a range of mN , i.e. 0.1 TeV . mN . 100 TeV, the cases of
large hierarchies in light neutrino masses are supposed to get constrained accord-
ingly. Furthermore, since the regions of parameter space with large mN are largely
affected by the experimental constraints from CLFV, small light neutrino masses
are disfavored by all those experimental constraints. These results are all clearly
presented in several plots in figures 2.4, 2.6, and 2.7. For example, the 99 % contour
in figure 2.7a shows that mν1 ∼ 0.1 eV for mWR = 5 TeV and mν1 & 6 · 10−3 eV for
mWR = 10 TeV. Note that this does not necessarily mean that there exists a strict
lower bound of the light neutrino mass for given mWR , since the results of this paper
are based on the naturalness argument such as no fine-tuning in MD. Note also that
we can observe similar patterns in neutrino mass correlations in any type-I seesaw
22
(a) BRτ→µγ vs. BRµ→eγ (b) BRτ→eγ vs. BRµ→eγ (c) BRtype-Iµ→eee vs. BR
tree
µ→eee
(d) BRµ→eee vs. BRµ→eγ (e) BRµ→eee vs. RTiµ→e (f) R
Ti
µ→e vs. BRµ→eγ
(g) RAlµ→e vs. R
Ti
µ→e (h) R
Au
µ→e vs. R
Ti
µ→e (i) R
Pb
µ→e vs. R
Ti
µ→e
Figure 2.1: CLFV in the MLRSM for 2 TeV < mWR < 30 TeV. The green dots
are data points that satisfy only the experimental constraints from the light lepton
masses and PMNS matrix. The red dots are data points that also satisfy present
bounds from the CLFV, 0νββ, EDM’s of charged leptons, and Planck observation.
The purple dots are those that satisfy the strongest bounds from future experiments.
The shaded regions are regions of parameter space excluded by present experimental
bounds. Figures 2.1a and 2.1b show that there exist only small chances that τ → µγ
or τ → eγ could be detected in near-future experiments. In figure 2.1c, the tree-
level and 1-loop contributions to µ → eee are compared, and it shows that we
should consider both when calculating BRµ→eee. Figures 2.1d−2.1f show the linear
correlations among various CLFV effects. Note that the strongest future bounds on
CLFV come from PRISM/PRIME and PSI, as clearly shown in figure 2.1e. Figures
2.1g−2.1i show that the µ→ e conversion rates for various nuclei have very strong
linear correlations with each other. The total number of data points is 83724 (total)
= 81132 (green) + 2573 (red) + 19 (purple).
23
(a) T 0ν1/2
∣∣max
Ge
vs. |ην | (b) T 0ν1/2
∣∣max
Te
vs. |ην | (c) T 0ν1/2
∣∣max
Xe
vs. |ην |
(d) |ην | vs. |ηRNR | (e) |ην | vs. |ηδR | (f) |ηδR | vs. |ηRNR |
(g) |ηLNR | vs. |ηRNR | (h) |ηη| vs. |ηλ| (i) |ην | vs. mν1
Figure 2.2: Parameters of 0νββ in the MLRSM for 2 TeV < mWR < 30 TeV.
Figures 2.2a−2.2c show that only cases where ην dominantly determines T 0ν1/2
∣∣max
are allowed with a few exceptions by the present and future experimental bounds.
Even though the contributions of ηRNR and ηδR could be comparable to that of ην
in principle, such cases have been actually almost excluded by the constraints from
CLFV, as shown in figures 2.2d−2.2f. The contributions from ηη or ηλ are too small
compared with experimental bounds, as shown in figure 2.2h. Figure 2.2i shows
that the present upper bound of the light Majorana neutrino mass from Planck
is already below the bounds from KamLAND-Zen and CUORE, which means that
0νββ processes are difficult to be detected in near-future experiments since the light
neutrino exchange diagrams are dominant for most of the parameter space due to
the CLFV constraints.
24
(a) |dµ| vs. |de| (b) |dτ | vs. |de| (c) |de| vs. RTiµ→e
Figure 2.3: EDM’s of charged leptons in the MLRSM for 2 TeV < mWR < 30 TeV.
The predicted values are found to be too small compared with the present and future
bounds, since large EDM’s require small mWR whose regions of parameter space
have been largely constrained as shown in figure 2.4a. Even though the correlations
between EDM’s and CLFV are rather weak, as shown in figure 2.3c, the larger
EDM’s generally require the larger CLFV effects since mWR affects both CLFV and
EDM’s.
models, even in the simple extension of the SM only with gauge singlet neutrinos.
The difference in the MLRSM, or in a more general class of the left-right symmetric
model, is that we can have large CLFV effects and thus the experimental bounds
on CLFV are constraining the light neutrino masses. Moreover, since the largest
possible hierarchy in heavy neutrino masses is directly associated with mWR and
the regions of parameter space with smaller mWR are more constrained by CLFV
bounds, we can expect that the discovery of light WR as well as any improved ex-
perimental bounds on CLFV would largely constrain the regions of parameter space
of the normal hierarchy.
Another interesting result is that the mass of the lightest heavy neutrino mN1
25
has been also notably constrained by the present experimental constraints, which is,
of course, associated with the result on light neutrino masses just mentioned. This
is shown in figures 2.5a, 2.5b, 2.6a, and 2.7b. For example, the 99 % density contour
of figure 2.7b shows that mN1 . 200 GeV for mWR = 5 TeV and mN1 . 2 TeV for
mWR = 10 TeV. Due to the mass insertion in the Dirac propagators of heavy neu-
trinos in some CLFV processes, large heavy neutrino masses generally induce large
CLFV effects. Figure 2.4b explicitly shows how the CLFV bound is constraining
mN1 . The heaviest heavy neutrino mass is also affected by the experimental bounds,
although its effect is rather small, as shown in figures 2.5c, 2.6b, and 2.7c.
While the CLFV effects of muons could be large enough for the associated
processes to be detected in near-future experiments, the branching ratios of tau
decays are either too small or just around the sensitivities of future experiments,
as shown figure 2.1. The experimental bounds of CLFV are also constraining small
masses of charged scalar fields as well as the RH gauge boson, as shown in figure 2.7.
As a result, the 0νββ processes through the heavy neutrinos as well as RH gauge
boson (denoted by ηRNR) and also processes through δ
++
R as well as the RH gauge
boson (denoted by ηδR) are both suppressed. Hence, for most data points that satisfy
the present experimental constraints, the dominant contribution to 0νββ comes
from the process of the light neutrino exchange (denoted by ην), as shown in figures
2.2a−2.2c. However, since the upper bound of the light neutrino mass by Planck is
already below the bounds of future experiments as shown in figure 2.2i, i.e. the light
neutrino exchange channel has been largely constrained by the Planck observation,
the possibility to detect 0νββ processes in near-future experiments is small. As for
26
the EDM’s of electrons, there seems to be also only small chances that they could
be detected in near-future experiments as shown in figure 2.3, since the largest
possible EDM’s of electrons are well below the future sensitivities of the planned
experiement. In addition, the EDM’s of muons and taus are too small compared with
the present upper bounds. Note that the EDM’s of charged leptons has been also
constrained by the experimental bounds from CLFV, since large EDM’s generally
require small mWR and large mN and such regions of parameter space are largely
affected by those experimental constraints. Note also that, even with the relatively
small values of the RH scale, i.e. vR < 65 TeV corresponding to mWR < 30 TeV,
the observables of CLFV, 0νββ, and EDM’s cover very wide ranges, e.g. roughly
10−20 . BRµ→eγ . 10−3 and 10−35 e · cm . |de| . 10−29 e · cm. Hence, neither a
success nor a failure in detecting one of these effects rules out even the TeV-scale
MLRSM, unless any other experimental results are simultaneously considered.
2.7 Conclusion
The procedure to construct lepton mass matrices has been presented in the MLRSM
of type-I dominance with the parity symmetry, and the conditions for the TeV-
scale MLRSM without fine-tuning have also been discussed, i.e. either (i) κ1  κ2
and fij  f˜ij, which implies V `L ≈ V `R, or (ii) κ1  κ2 and fij  f˜ij, which
implies V `L ≈ V `Re−iα. Based on these results, the phenomenology of the TeV-scale
MLRSM has been numerically investigated when the masses of light neutrinos are
in the normal hierarchy, and the numerical results on how the present and future
27
experimental bounds from the CLFV, 0νββ, EDM’s of charged leptons, and Planck
observation constrain the parameter space of the MLRSM have been presented.
According to the numerical results, the regions of parameter space of small
light neutrino masses have been constrained by the experimental bounds on CLFV
effects, although it does not necessarily mean there exists a strict lower bound
of light neutrino masses. The lightest heavy neutrino mass is also found to have
been notably constrained by the present experimental bounds especially for small
mWR . In addition, it has been shown that all the 0νββ processes and the EDM’s of
charged leptons have been suppressed by the experimental constraints from CLFV,
and we have at best only small chances to detect any of these effects in near-future
experiments.
Note that the results here are based on several nontrivial assumptions such as
(i) type-I seesaw dominance, (ii) the parity symmetry, and (iii) the normal hierarchy
in light neutrino masses. Furthermore, it should be emphasized that this paper is
considering the TeV-scale phenomenology of the MLRSM without fine-tuning of
model parameters. If fine-tuning is allowed, significantly different predictions could
be made.
28
Physical scalar fields Mass-squared
h0 =
√
2Re[φ0∗1 + 2e
−iαφ02]
1
2
(4λ1 − α21/ρ1)κ21 + 12α3v2R22
H01 =
√
2Re[−2eiαφ0∗1 + φ02] 12α3v2R
H02 =
√
2Re[δ0R] 2ρ1v
2
R
H03 =
√
2Re[δ0L]
1
2
(ρ3 − 2ρ1)v2R
A01 =
√
2Im[−2eiαφ0∗1 + φ02] 12α3v2R
A02 =
√
2Im[δ0L]
1
2
(ρ3 − 2ρ1)v2R
H+1 = δ
+
L
1
2
(ρ3 − 2ρ1)v2R + 14α3κ21
H+2 = φ
+
2 + 2e
iαφ+1 +
1√
2
1δ
+
R
1
2
α3
(
v2R +
1
2
κ21
)
δ++R 2ρ2v
2
R +
1
2
α3κ
2
1
δ++L
1
2
(ρ3 − 2ρ1)v2R + 12α3κ21
Table 2.1: Physical scalar fields and their masses in the MLRSM when vL = 0 and
vR  κ1  κ2. Here, 1 ≡ κ1/vR and 2 ≡ κ2/κ1. The SM Higgs field is identified
as h0. Note that mH+1 ≈ mδ++L for vR  vEW. The mixing between δ
++
L and δ
++
R
is assumed to be small, although it could be large in principle for relatively small
values of ρ3− 2ρ1 and vR [15]. It is, however, a good assumption even for such cases
if we introduce an additional assumption β1, β3 . O(10−1).
29
Parameter Range
log10 (mν1/eV) −4− log10 2
mWR 2− 35 TeV
log10 (κ2/GeV) −4− 1
δD, δM1, δM2,
θL12, θL13, θL23,
δL1, δL2, δL3
−pi − pi rad
δL4 (−1− 1)·10−3 rad
log10 (|A11|/GeV) log10
∣∣Im[M`11]∣∣− log10 (5√2pivEW)
log10 α3, log10 ρ2 log10 (1000 GeV
2/v2R)− log10 (5
√
4pi)
log10 (ρ3 − 2ρ1) log10 (1000 GeV2/v2R)− log10 (15
√
4pi)
Table 2.2: List of parameters and the ranges where those parameters are randomly
generated. It is also assumed that δL5 = δL6 = 0, θRij = θLij, and δRi = δLi
(i, j = 1, 2, 3). Here, A is defined as A ≡ fκ2/
√
2, and M` = V
`
LM
c
`V
`†
R is the
charged lepton mass matrix in the symmetry basis. The electroweak VEV is vEW =√
κ21 + κ
2
2 = 246 GeV, and vR = mWR
√
2/g (g = 0.65) is the VEV of the SU(2)R
triplet. Since Yukawa coupling matrices f , f˜ are constructed from given M` by the
method presented in section 2.3, we explicitly consider only the condition κ1  κ2
for the TeV-scale MLRSM. Any Yukawa couplings that do not satisfy fij  f˜ij can
be excluded by filteringMR with large entries, which is one of the constraints given in
table 2.3. The ranges and values of δL4, δL5, δL6, θRij, and δRi are chosen to guarantee
V `R ≈ V `L for TeV-scale mN . In principle, we only need δL4 ≈ 0, δL5 ≈ 0, δL6 ≈ 0,
θRij ≈ θLij, and δRi ≈ δLi for V `R ≈ V `L. However, for the parameters other than
δL4, it turned out that only extremely small deviations (. 10−6) from the values
assumed above are allowed to obtain TeV-scale mN . Therefore, for convenience,
only δL4 is varied around 0 while all the other parameters are set to the fixed values
mentioned above. The coupling constants α3, ρ2, ρ3−2ρ1 are assumed to be positive,
which is a sufficient condition to have real masses of charged scalar fields. Note that
slightly broader ranges than necessary are chosen for several parameters, in order
to generate contour plots less distorted around the borders.
Parameter Constraint
mH+1 , mH
+
2
, mδ++L
, mδ++R
> 500 GeV
|Eigenvalues of f , f˜ , h|, α3, ρ2 <
√
4pi
ρ3 − 2ρ1 < 3
√
4pi
|Eigenvalues of MD| > 1 keV
|Eigenvalues of MR| 100 GeV−
√
8pivR
Table 2.3: List of constraints imposed on several model parameters. The lower limits
of scalar field masses are set to 500 GeV to safely neglect many loop diagrams by
those charged scalar fields. Note that the upper limits of all the coupling constants
are set to
√
4pi. The lower limit of the eigenvalues of MD is appropriately chosen to
avoid singularity in calculating M−1D . The constraint from the absence of the flavour
changing neutral current in the quark sector requires mH01 ,mH+2 & 10 TeV [16, 18],
which is not considered in this paper because the contribution of H+2 to CLFV is
almost negligible, as shown in figures 2.7e. The constraint from the SM Higgs mass
mh0 = 125 GeV is not explicitly considered as well, because we can always find λ1, α1
that would give the correct Higgs mass for given ρ1, α3 if 2 . 0.01 and mWR < 30
TeV. The condition 2 . 0.01 is found to be satisfied for all the data points due to
the perturbativity constraint, as shown in figure 2.4f.
30
Present bound Future sensitivity
BRµ→eγ < 4.2 · 10−13 (MEG) [19] < 5.0 · 10−14 (Upgraded MEG) [20]
BRτ→µγ < 4.4 · 10−8 (BaBar) [21] < 1.0 · 10−9 (Super B factory) [22]
BRτ→eγ < 3.3 · 10−8 (BaBar) [21] < 3.0 · 10−9 (Super B factory) [22]
BRµ→eee < 1.0 · 10−12 (SINDRUM) [23] < 1.0 · 10−16 (PSI) [24]
RAlµ→e · < 3.0 · 10−17 (COMET) [25]
RTiµ→e < 6.1 · 10−13 (SINDRUM II) [26] < 1.0 · 10−18 (PRISM/PRIME) [27]
RAuµ→e < 6.0 · 10−13 (SINDRUM II) [25] ·
RPbµ→e < 4.6 · 10−11 (SINDRUM II) [28] ·
T 0ν1/2
∣∣
Ge
> 2.1 · 1025 yrs. (GERDA) [29] > 1.35 · 1026 yrs. (GERDA II) [29]
T 0ν1/2
∣∣
Te · > 2.1 · 1026 yrs. (CUORE) [29]
T 0ν1/2
∣∣
Xe
> 1.9 · 1025 yrs. (KamLAND-Zen) [29] ·
|de| < 8.7 · 10−29 e·cm (ACME) [53] < 5.0 · 10−30 e·cm (PSU) [54]
|dµ| < 1.9 · 10−19 e·cm (Muon (g − 2)) [55] ·
|dτ | . 5.0 · 10−17 e·cm (Belle) [56] ·∑3
i mνi < 0.23 eV (Planck) [17] ·
Table 2.4: Experimental bounds on CLFV, 0νββ, EDM’s of charged leptons, and
light neutrino masses. The actual present bounds on dτ reported by Belle Col-
laboration are −2.2 · 10−17e·cm < Re[dτ ] < 4.5 · 10−17e·cm and −2.5 · 10−17e·cm
< Im[dτ ] < 0.8 · 10−17e·cm. For the normal hierarchy, the constraint from the
Planck observation corresponds to the upper bound of the lightest neutrino mass
mν1 < 0.071 eV.
31
Present bound (KamLAND-Zen) Future sensitivity (CUORE)
|ην | < 7.1 · 10−7 < 1.4 · 10−7
|ηLNR | < 6.8 · 10−9 < 1.4 · 10−9
|ηRNR | < 6.8 · 10−9 < 1.4 · 10−9
|ηδR | < 6.8 · 10−9 < 1.4 · 10−9
|ηλ| < 5.7 · 10−7 < 1.2 · 10−7
|ηη| < 3.0 · 10−9 < 8.2 · 10−10
Table 2.5: Experimental bounds on the dimensionless parameters associated with
the various processes of 0νββ. The present bounds come from KamLAND-Zen,
and the strongest future bounds are from CUORE [29]. To obtain each bound, the
associated decay channel is assumed to be dominant over the others. Even though
there exist regions of parameter space where contributions from ην , η
R
NR
, and ηδR
are comparable to each other, it does not invalidate the assumption at least for the
data points of interest around the present and future bounds, since larger values of
|ηRNR | and |ηδR | are rarely allowed by the constraints from CLFV, as shown in figures
2.2d−2.2f.
32
(a) RTiµ→e vs. mWR (b) R
Ti
µ→e vs. mN1 (c) R
Ti
µ→e vs. mN3
(d) RTiµ→e vs. mν1 (e) BRµ→eγ vs. mν1 (f) |ην | vs. 2 (≡ κ2/κ1)
Figure 2.4: Figures 2.4a−2.4e show the effect of CLFV constraints on the masses
of neutrinos and the RH gauge boson. Here, RTiµ→e is chosen since it most clearly
divides the colors of data points through its experimental bounds. The smaller
values of the lightest light neutrino mass mν1 produce the larger CLFV effects, as
in figures 2.4d and 2.4e, since they require the larger values of the heaviest heavy
neutrino mass mN3 in most of the parameter space, as shown in figure 2.6f. As a
result, the regions of parameter space of small light neutrino masses get constrained
by the experimental bounds on CLFV. In figure 2.4f, additional data points (yellow
dots) are also presented in order to show the effects of the perturvativity constraints,
and all the data points generated in the ranges of parameters given in table 2.2 are
shown in this plot. For those yellow points, at least one of the coupling constants
are larger than
√
4pi while the experimental constraints in the light neutrino sector
are still satisfied. This figure shows that 2 ≡ κ2/κ1 . 0.01 is satisfied for all the
data points due to the perturvativity constraints as well as the condition κ2 < 10
GeV, and thus the Higgs mass constraint can be easily satisfied, as mentioned in
table 2.3.
33
(a) mWR vs. mN1 (b) mWR vs. mN1 (c) mWR vs. mN3
Figure 2.5: Masses of heavy neutrinos in the TeV-scale MLRSM for 2 TeV < mWR <
30 TeV. For figure 2.5a, the same data set as in the previous plots are used to show
the effect of the consraints from CLFV, 0νββ, EDM’s, and Planck on the parameter
space. The non-perturbative regions are where at least one coupling constant is
larger than
√
4pi. Note that green dots in figure 2.5a do not completely fill the
available parameter space because of the constraints on masses and angles in the
light lepton sector. For figures 2.5b and 2.5c, much more amount of data points was
used to show how the present and future bounds constrain the parameter space.
Figures 2.5a and 2.5b show that the lightest heavy neutrino mass mN1 has been
notably constrained by the experimental bounds, especially for smaller mWR . Figure
2.5c is the plot on the heaviest heavy neutrino mass mN3 , and it shows that only
a small region of parameter space with small mWR seems to have been excluded.
Even though these plots in the linear scale are better in presenting the effect of
experimental constraints on largest possible masses of heavy neutrinos, they do not
correctly show the density distributions since the matrix A (≡ fκ2/
√
2) is generated
in the logarithmic scale. Plots of mN in the logarithmic scale are presented in figure
2.7. For figures 2.5b and 2.5c, the data sets for figures 2.7b and 2.7c are used,
respectively.
34
(a) mWR vs. mN1 (b) mWR vs. mN3 (c) mN3 vs. mN1
(d) mWR vs. mν1 (e) mN1 vs. mν1 (f) mN3 vs. mν1
Figure 2.6: Figures 2.6a−2.6d show the effect of experimental bounds on the masses
of neutrinos and the RH gauge boson. Figures 2.6a and 2.6b show that the regions
with smaller mWR and larger mN are more affected by the present bounds on CLFV,
0νββ, and EDM’s. Figures 2.6e and 2.6f show that, for smaller mν1 , i.e. for the light
neutrino masses with a larger hierarchy, the heavy neutrino masses also generally
need to have a larger hierarchy accordingly since MD itself does not have the struc-
ture that would give hierarchical light neutrino masses. Due to this effect, only
larger mWR is generally allowed for smaller mν1 , as shown in figure 2.7a, since large
mN3 requires large vR.
35
(a) mWR vs. mν1 (b) mWR vs. mN1 (c) mWR vs. mN3
(d) mWR vs. mH+1
( ≈ mδ++L ) (e) mWR vs. mH+2 (f) mWR vs. mδ++R
Figure 2.7: Masses of neutrinos and charged scalar fields in the MLRSM for mWR <
30 TeV. The contours of 90 % and 99 % densities are also presented for illustration
purposes. According to the 99 % contour in figure 2.7a, mν1 ∼ 0.1 eV for mWR = 5
TeV and mν1 & 6 · 10−3 eV for mWR = 10 TeV. In addition, the 99 % contour in
figure 2.7b shows that mN1 . 200 GeV for mWR < 5 TeV and mN1 . 2 TeV for
mWR < 10 TeV. While the masses of H
+
1 , δ
++
L , and δ
++
R have been also constrained
by the experimental bounds, the mass of H+2 which appears only in the Z1-exchange
diagrams of CLFV processes has been barely constrained, as shown in figure 2.7e.
Hence, the constraint of mH+2 & 10 TeV from the absence of flavour changing neutral
current in the quark sector is not considered in this paper. The total number of
data points is 51971 = 51561 (red) + 410 (purple).
36
Chapter 3: Natural TeV-scale left-right symmetric model
3.1 Introduction
Our goal here is to explore whether the two key aspects of the seesaw physics, i.e. (i)
the Majorana character of heavy and light neutrino masses, and (ii) the heavy-light
neutrino mixing, can be tested at the LHC as well as in complementary experiments
at low energies, e.g. in planned high sensitivity searches for CLFV, 0νββ, etc. A
necessary requirement for this synergic exploration to have any chance of success is
that the seesaw scale be in the TeV range as well as the heavy-light mixing being
relatively large. With this in mind, we discuss a class of the LRSM where both the
above ingredients of type-I seesaw, i.e. TeV seesaw scale and observable heavy-light
neutrino mixing emerge in a natural manner.
A simple candidate seesaw model is based on the left-right symmetric theory
of weak interactions where the key ingredients of seesaw, i.e. the RH neutrino and its
Majorana mass, appear naturally. The RH neutrino field νR arises as the necessary
parity gauge partner of the left-handed (LH) neutrino field νL and is also required by
anomaly cancellation, whereas the seesaw scale is identified as the one at which the
RH counterpart of the SM SU(2)L gauge symmetry, namely the SU(2)R symmetry,
is broken. The RH neutrinos are therefore a necessary part of the model and do not
37
have to be added just to implement the seesaw mechanism. An important point is
that the RH neutrinos acquire a Majorana mass as soon as the SU(2)R symmetry
is broken at a scale vR. This is quite analogous to the way the charged fermions
get mass as soon as the SM gauge symmetry SU(2)L is broken at the electroweak
scale v. The Higgs field that gives mass to the RH neutrinos becomes the analog
of the 125 GeV Higgs boson discovered at the LHC. Clearly, the seesaw scale is not
added in an adhoc manner but rather becomes intimately connected to the SU(2)R⊗
U(1)B−L symmetry breaking scale.
In generic TeV-scale seesaw models without any special structures for MD and
MN , in order to get small neutrino masses, we must fine-tune the magnitude of the
elements of MD to be very small (of order MeV for MN ∼ TeV), as is evident from
the seesaw formula in equation 1.10. As a result, the heavy-light neutrino mixing
ξ ∼ MDM−1N ' (MνM−1N )1/2 . 10−6. This suppresses all heavy-light mixing effects
to an unobservable level which keeps this key aspect of seesaw shielded from being
tested experimentally. To overcome this shortcoming, some special textures for MD
and MN have been studied in the literature [30–39] for which even with TeV-scale
seesaw, the mixing parameter ξ can be significantly enhanced whereas the neutrino
masses still remain small, thereby enriching the seesaw phenomenology. Here, we
present an LRSM embedding of one such special texture using an appropriate family
symmetry. This is a highly non-trivial result since in the LRSM the charged lepton
mass matrix and the Dirac neutrino mass matrix are related, especially when there
are additional discrete symmetries to guarantee a specific form of the Dirac mass
matrix MD.
38
When we have the mass matrices of
MD =

m1 0 0
m2 0 0
m3 0 0
 , MR =

0 M1 0
M1 0 0
0 0 M2
 , (3.1)
then the type-I seesaw formula gives
Mν = −MDM−1R MTD = 0. (3.2)
By introducting small values in the zero entries, we can generate small light neutrino
masses. We want the mass matrices in the symmetry basis to be
M` =

δa11 a12 a13
δa21 a22 a23
δa31 a32 a33
 , MD =

m11 δm12 δm13
m21 δm22 δm23
m31 δm32 δm33
 , MR =

0 M1 0
M1 0 0
0 0 M2

(3.3)
where |δaij|  |akl| and |δmij|  |mkl|  |Mn| (i, j, k, l = 1, 2, 3 and n = 1, 2). If
the symmetry basis is close to the charged lepton mass basis, then we can expect
the followings:
1. Explanation of the small mass of an electron.
2. Large CLFV and EDM of an electron.
As for the second point, note that both CLFV and EDM’s of charged leptons have
a contribution of the form
∑3
i=1 SαiVαimNi . Since S ≈ (M cDM c−1R )∗V , we can write
3∑
i=1
SαiVαimNi ≈
3∑
i,j=1
(M cDM
c−1
R )
∗
αjVjiVαimNi =
3∑
j=1
(M cDM
c−1
R )
∗
αjM
c∗
Rjα = M
c∗
Dαα.
(3.4)
39
Since M cD11 is large in this model, we can expect the large CLFV and EDM of an
electron.
3.2 Outline of the model
When we have multiple bi-doublet and RH triplet scalar fields, the Lagrangian terms
with Yukawa couplings are given by
L`Y = −L′Li(f `aijΦa + f˜ `aijΦ˜a)L′Rj − hLaijL′cLiiσ2∆LaL′Lj − hRaijL′cRiiσ2∆RaL′Rj + H.c..
(3.5)
Now we introduce a discrete symmetry Z4⊗Z4⊗Z4, and define the transformation
rule of the fermions and scalar fields as in table 3.1. The Yukawa interaction terms
Field Z4 ⊗ Z4 ⊗ Z4
LLa (1, 1, 1)
LR1 (−i, 1, 1)
LR2 (1,−i, 1)
LR3 (1, 1,−i)
Φ1 (−i, 1, 1)
Φ2 (1, i, 1)
Φ3 (1, 1, i)
∆R1 (i, i, 1)
∆R2 (1, 1,−1)
Table 3.1: Transformation property of leptons and scalar fields under Z4⊗Z4⊗Z4.
40
under this symmetry are written as
L`Y = −fi1L′LiΦ˜1L′R1 − fi2L′LiΦ2L′R2 − fi3L′LiΦ3L′R3
− h12L′cR1iσ2∆R1L′R2 − h12L′cR2iσ2∆R1L′R1 − h33L′cR3iσ2∆R2L′R3 + H.c..
(3.6)
The scalar potential is given by
V = −µ21atr
[
Φ†aΦa
]− µ22atr[∆†Ra∆Ra]
+ λ1abtr
[
Φ†aΦa
]
tr
[
Φ†bΦb
]
+
(
λ2ae
iδatr
[
Φ˜†aΦa
]2
+ H.c.
)
+ λ3abtr
[
Φ˜†aΦb
]
tr
[
Φ†aΦ˜b
]
+ λ4abtr
[
Φ†aΦb
]
tr
[
Φ˜†aΦ˜b
]
+ α1abtr
[
Φ†aΦa
]
tr
[
∆†Rb∆Rb
]
+ α2abtr
[
Φ†aΦa∆Rb∆
†
Rb
]
+ α3abtr
[
∆†Ra∆Ra
]
tr
[
∆†Rb∆Rb
]
+ α′3tr
[
∆†R1∆R2
]
tr
[
∆†R2∆R1
]
+ α4abtr
[
∆†Ra∆
†
Rb
]
tr
[
∆Ra∆Rb
]
. (3.7)
Note that the potential terms tr
[
Φ˜†aΦb
]
are not allowed due to the discrete symmetry.
Without loss of generality, the VEV’s of scalar fields are written as
〈Φa〉 =
 κaeiαa 0
0 κ′aeiα
′
a/
√
2
 , 〈∆R1〉 =
 0 0
vR1/
√
2 0
 , 〈∆R〉 =
 0 0
vR2e
iθR/
√
2 0
 ,
(3.8)
41
where we can choose α1 = 0 by gauge transformation. Some of the minimazation
conditions of the scalar potential are written as
∂〈V 〉
∂κ1
= κ1
[
3∑
a=1
(λ′a1κ
′2
a + λa1κ
2
a) +
2∑
a=1
αa1v
2
Ra + 2µ
2
1
]
+ λ′′12κ
′
1κ
′
2κ2 + λ
′′
13κ
′
1κ
′
3κ3 = 0,
(3.9)
∂〈V 〉
∂κ′1
= κ′1
[
3∑
a=1
(λ′a1κ
2
a + λa1κ
′2
a ) +
2∑
a=1
α′a1v
2
Ra + 2µ
2
1
]
+ λ′′12κ1κ2κ
′
2 + λ
′′
13κ1κ3κ
′
3 = 0
(3.10)
where the coefficients are appropriately defined from the coefficients of the potential
and the phases of VEV’s. We can write similar equations for κ2, κ
′
2, κ3, and κ
′
3.
Now we assume that vRa are determined from the other minimization conditions.
Further assuming κa  κ′a and there exists no large hierarchy among the same type
of coupling constants, we can obtain from the equations of type 3.10.
3∑
a=1
λabκ
′2
a +
2∑
a=1
α′abv
2
Ra + 2µ
2
b ≈ 0 (3.11)
where b = 1, 2, 3. These are coupled linear equations, from which κ′a is easily
determined. Now, equation 3.9 can be written as
κ1
[
3∑
a=1
λ′a1κ
′2
a +
2∑
a=1
αa1v
2
Ra + 2µ
2
1
]
+ λ′′12κ
′
1κ
′
2κ2 + λ
′′
13κ
′
1κ
′
3κ3 ≈ 0, (3.12)
and we can write similar equations from the derivatives with respective to κ2 and κ3.
They are also coupled linear equations whose solution is clearly κ1 = κ2 = κ3 = 0.
Note that this derivation of VEV’s is possible due to the absence of the term tr
[
Φ˜†aΦb
]
which is prohibited by the discrete symmetry, although its absence is not a sufficient
condition for κ1 = κ2 = κ3 = 0. Therefore, when there exists the potential terms
tr
[
Φ˜†aΦb
]
with very small coefficients, we can have very small nonzero κa.
42
Now we introduce that potential as a soft symmetry-breaking term
VSB = −
3∑
a,b=1
µ2SBabtr
[
Φ˜†aΦb
]
+ H.c. (3.13)
which would change the minimization conditions above into
∂〈V 〉
∂κ1
= κ1
∑
a
(λ′a1κ
′2
a + λa1κ
2
a + αa1v
2
Ra + 2µ
2
1) + λ
′′
12κ
′
1κ
′
2κ2 + λ
′′
13κ
′
1κ
′
3κ3 +
∑
a
µ′2SBa1κ
′
a
≈ κ1
∑
a
(λ′a1κ
′2
a + αa1v
2
Ra + 2µ
2
1) + λ
′′
12κ
′
1κ
′
2κ2 + λ
′′
13κ
′
1κ
′
3κ3 +
∑
a
µ′2SBa1κ
′
a ≈ 0,
(3.14)
∂〈V 〉
∂κ′1
= κ′1
∑
a
(λ′a1κ
2
a + λa1κ
′2
a + α
′
a1v
2
Ra + 2µ
2
1) + λ
′′
12κ1κ2κ
′
2 + λ
′′
13κ1κ3κ
′
3 +
∑
a
µ′2SBa1κa
≈ κ′1
∑
a
(λa1κ
′2
a + α
′
a1v
2
Ra + 2µ
2
1) + λ
′′
12κ1κ2κ
′
2 + λ
′′
13κ1κ3κ
′
3 ≈ 0. (3.15)
The second equation and its companions from κ′2 and κ
′
3 would give (approximately)
the same expressions of κ′a, and the first equation and its companions from κ2 and
κ3 are now coupled linear equations with solutions κa = δκa.
With those VEV’s, we can write the lepton mass matrices in the symmetry
43
basis as
M` =
1√
2

f11δκ1 f12κ
′
2e
iα′2 f13κ
′
3e
iα′3
f21δκ1 f22κ
′
2e
iα′2 f23κ
′
3e
iα′3
f31δκ1 f32κ
′
2e
iα′2 f33κ
′
3e
iα′3
 , (3.16)
MD =
1√
2

f11κ
′
1e
−iα′1 f12δκ2eiα2 f13δκ3eiα3
f21κ
′
1e
−iα′1 f22δκ2eiα2 f23δκ3eiα3
f31κ
′
1e
−iα′1 f32δκ2eiα2 f33δκ3eiα3

= M`D where D ≡

κ′1
δκ1
e−iα
′
1 0 0
0 δκ2
κ′2
e−i(α
′
2−α2) 0
0 0 δκ3
κ′3
e−i(α
′
3−α3)
 , (3.17)
MR =
√
2

0 h12vR1 0
h21vR1 0 0
0 0 h33vR2
 (3.18)
where we have redefined the phase of LR3 to absorb θR into α3 and α
′
3, i.e. α3 −
θR/2→ α3 and α′3 − θR/2→ α′3.
The motivation for the discrete symmetry is now clear:
1. No scalar potential terms of the form tr
[
Φ˜aΦ
†
b
]
.
2. No fine-tuning in MD for the TeV-scale phenomenology.
3. Explanation of the small mass of an electron.
4. Large branching ratios of various muon decay processes and a large EDM of
44
an electron.
The mass term for charged weak gauge bosons is
Lmassg = (W+µL W+µR )
 14g2L
∑3
a=0(δκ
2
a + κ
′2
a ) − 12gLgR
∑3
a=0 δκaκ
′
ae
i(α′a−αa)
− 12gLgR
∑3
a=0 δκaκ
′
ae
−i(α′a−αa) 1
4g
2
R
∑3
a(δκ
2
a + κ
′2
a ) + 2
∑2
a=1 v
2
Ra

 W−Lµ
W−Rµ
 .
(3.19)
Their masses can still be written as
m2W1 ≈
1
4
g2Lv
2
EW, m
2
W2
≈ 1
2
g2Rv
2
R (3.20)
where vEW =
√∑3
a=0(δκ
2
a + κ
′2
a ) = 246 GeV and vR ≡
√
v2R1 + v
2
R2, and the WL-WR
mixing parameter is given by
ξeiα ≈ −gL
∑3
a=0 δκaκ
′
ae
i(α′a−αa)
gRv2R
(3.21)
where α is defined as the complex phase of the mixing parameter in this model, not
the phase of the electroweak VEV as in the MLRSM.
3.3 Numerical procedure
In the symmetry basis, we assume that MR has the form
MR =

0 M1 0
M1 M3 0
0 0 M2
 , (3.22)
where M3 is not necessarily small yet. In the same basis, we have MD = M`D,
and thus Mν = −MDM−1R MTD = −Ma`Ma−1R MaT` ≡ Maν where Ma` ≡ M` and
45
MaR ≡ D−1MRD−1. Note that MaR has the same structure as MR since D is diagonal.
While Ma` and M
a
ν (i.e. M` and Mν) can be easily constructed by M
a
` = V
`
LM
c
`V
`†
R
and Maν = V
`
LM
c
νV
`T
L for arbitrary unitary matrices V
`
L and V
`
R, the matrix M
a
R
calculated from MaR = −MaT` Ma−1ν Ma` does not have the structure we want for
those arbitary mixing matrices in general.
In order to have MaR with the desired structure, we generate arbitrary V
`b
L and
V `bR (instead of V
`
L and V
`
R), and calculate M
b
` ≡ V `bL M c`V `b†R , M bν ≡ V `bL M cνV `bTL , and
M bR ≡ −M bT` (M bν)−1M b` . Note that there always exists a unitary matrix VR that
transforms M bR into M
a
R = V
T
RM
b
RVR where M
a
R is in the form of 3.22. Defining
Ma` ≡ M b`VR and Maν ≡ M bν , we obtain MaR = −MaT` Ma−1ν Ma` . Further defining
M` ≡ Ma` (= M b`VR), MR ≡ DMaRD (= DV TRM bRVRD), MD ≡ Ma`D (= M b`VRD),
andMν ≡Maν (= M bν), we can finally obtainMR = −MTDM−1ν MD whereMD = M`D
and MR is in the form of 3.22. For M
a
R and D given by
MaR =

0 Ma1 0
Ma1 M
a
3 0
0 0 Ma2
 , D =

d1 0 0
0 d2 0
0 0 d3
 , (3.23)
we have
MR = DM
a
RD =

0 Ma1 d1d2 0
Ma1 d1d2 M
a
3 d
2
2 0
0 0 Ma2 d
2
3
 . (3.24)
Hence, by choosing small |d2| and large |d1|, we can have MR with |M1|, |M2| 
|M3| ≈ 0, and also MD = M`D with the first column large and the second column
small, as desired. If |M3| is small enough, we can set it to zero to have the structure
46
allowed by the exact discrete symmetry while all the experimental constraints are
still satisfied within their uncertainties.
In order for this model to explain the small electron mass, the mixing matrix
V `R = V
†
RV
`b
R should not largely mix the first column of M` with the others. Note
that the stronger condition |V `Rij| ≈ δij is the general prediction of the model. By
construction, we have
MaR = V
T
RM
b
RVR = −V `∗R M cT` (M cν)−1M c`V `†R , (3.25)
i.e. V `R is the mixing matrix that transforms M
cT
` (M
c
ν)
−1M c` into the form of 3.22.
SinceM ≡M cT` (M cν)−1M c` has the structure ofM33 Mi3 (i = 1, 2) andM22,M12 
M11 due to the mass hierarchy in charged leptons as well as the large mixing in light
neutrinos, we always have |V `Rij| ≈ δij.
In summary, the numerical procedure to generate the model parameters is as follows:
1. Randomly generate mν1 , and calculate mν2 =
√
m2ν1 + ∆m
2
21 and mν3 =√
m2ν1 + ∆m
2
31.
2. Calculate M cν from M
c
ν = UPMNSM
d
νU
T
PMNS where the CP phases of UPMNS are
also randomly generated.
3. Randomly generate V `bL , V
`b
R , and calculateM
b
` = V
`b
L M
c
`V
`b†
R , M
b
ν = V
`b
L M
c
νV
`bT
L ,
and M bR = M
bT
` (M
b
ν)
−1M b` .
4. Find VR which transforms M
b
R into a matrix M
a
R in the form of 3.22 by M
a
R =
V TRM
b
RVR. In that basis, we have M
a
` = M
b
`VR and M
a
ν = M
b
ν .
47
5. Randomly generate D, and calculate the lepton mass matrices in the symmetry
basis by M` = M
a
` , MD = M
a
`D, and MR = DM
a
RD.
6. Randomly generate κ0, κ
′
0, κ
′
1, κ
′
2, κ
′
3 which satisfies
√
κ20 + κ
′2
0 + κ
′2
1 + κ
′2
2 + κ
′2
3 =
vEW, and calculate δκ1 = κ
′
1/D11, δκ2 = κ
′
2D22, δκ3 = κ
′
3D33. Calculate the
Yukawa coupling matrix f in the symmetry basis from M` and the electroweak
VEV’s.
7. Define V `L ≡ V `bL , V `R ≡ V †RV `bR , and calculate M cD = V `†L MDV `R and M cR =
V `TR MRV
`
R.
8. Construct the 6 × 6 neutrino mass matrix M cνN from M cD and M cR, and find
the 6× 6 mixing matrix VνN that diagonalizes M cνN .
The mixing matrices UPMNS, V
`
L, V
`
R, and VνN are parametrized in the same way as
in the MLRSM.
The ranges of model parameters where they are randomly generated are pre-
sented in table 3.2. The constraints imposed on model parameters are given in table
3.3. We assume that the contribution of charged scalar fields to CLFV and 0νββ
are negligible. It is usually a good assumption for all the CLFV and 0νββ processes
of our interest even when the masses of those charged scalar fileds are small, since
we have h11 = h13 = h23 = 0 and |V `Rij| ≈ δij. For example, the Feynman diagrams
of muon or tau decays in the symmetry basis always involve one of h11, h13, h23.
48
3.4 Numerical results
The numerical results for mWR = 3 TeV are presented in figure 3.1. The most
notable result is that a large EDM of an electron is allowed in spite of the CLFV
constraints, as expected. The prediction |V `Rij| ≈ δij has been also verified, as shown
in figure 3.1f.
(a) RTiµ→e vs. BRµ→eγ (b) T
Ge
1/2 vs. |ην | (c) |ην | vs. |ηRNR |
(d) |ηLNR | vs. |ηRNR | (e) |de| vs. RTiµ→e (f) |V `R11V `R22V `R33| vs. mN3
Figure 3.1: Predictions of the model for mWR = 3 TeV. Figures 3.1b−3.1d show
that other processes such as ηLNR can be dominant in this model. In addition, a
large EDM of e is allowed as shown in figure 3.1e. Figure 3.1f shows |V `Rij| ≈ δij.
3.5 Conclusion
We have presented a new TeV-scale seesaw model based on the left-right symmetric
gauge group but without parity symmetry where a particular texture for the Dirac
49
and Majorana masses guarantees that neutrino masses are naturally small while
keeping the heavy-light neutrino mixing in the LHC-observable range. A discrete
flavour symmetry has been shown to guarantee the stability of this texture, while
being consistent with the observed lepton masses and mixing. We then explored its
tests in the domain of the CLFV and EDM of an electron.
50
Parameter Range
log10 (mν1/eV) −4− log10 2
log10 (κ
′
0/GeV) log10 70− log10
√
v2EW − 4 · 102
log10 (κ0/GeV) −1− 1
log10 (κ
′
1/GeV) 1− log10
√
v2EW − κ′20 − κ20 − 2 · 102
log10 (κ
′
2/GeV) 1− log10
√
v2EW − κ′20 − κ20 − κ′21 − 102
log10 (κ
′
1/δκ1) 2− 5
log10 (κ
′
2/δκ2) 5− 8
log10 (κ
′
3/δκ3) 2− 5
δD, δM1, δM2, α
′
a − αa,
θLij, δLi, θRij, δRi
−pi − pi rad
Table 3.2: List of parameters and the ranges where those parameters are randomly
generated. We have set mWR = 3 TeV and gR = gL = 0.65. Here, α
′
a − αa is the
difference of the phases of κ′a and δκa, and we have used v
2
EW =
∑3
a=1(δκ
2
i + κ
′2
a ) ≈∑3
a=1 κ
′2
a . The angles δD, δM1, δM2 are the CP phases of the PMNS matrix, and
θLij, δLi and θRij, δRi are the parameters of V
`
L and V
`
R, respectively.
Parameter Constraint
|fij| <
√
4pi
|MR11|2 + |MR12|2 < 8piv2R
|M cR11/M cR12| < 0.1
Table 3.3: List of constraints imposed on the model parameters. Here, MR and
M cR are the Majorana mass matrices in the symmetry and charged lepton mass
bases, respectively. Since |MR11|2/|
√
2h11|2 + |MR12|2/|
√
2h12|2 = v2R1 + v2R2 = v2R,
we can always find vR1 and vR2 which satisfies |h12|, |h33| <
√
4pi if and only if
|MR11|2+|MR12|2 < 8piv2R. In addition, |M cR11/M cR12| is supposed to be small because
MR is of the form 3.22 and |V `Rij| ≈ δij. However, there can appear a small number
of data points with |M cR11/M cR12| ≈ 1 due to numerical errors, since κ′1/δκ1 can be
very large. The last condition has been imposed to remove those evidently erroneous
data points.
51
Chapter 4: TeV-scale resonant leptogenesis
4.1 Introduction
An attractive feature of the seesaw mechanism is that the same Yukawa couplings
that give rise to light neutrino masses, can also resolve one of the outstanding
puzzles of cosmology, namely, the origin of matter-antimatter asymmetry, via lepto-
genesis [40]. The key driver of leptogenesis are the out-of-equilibrium decays of the
RH Majorana neutrinos via the modes N → Liφ and N → Lciφ†, where Li = (νi, `i)T
(i = 1, 2, 3) are the SU(2)L lepton doublets, and φ are the Higgs doublets. In the
presence of CP violation in the Yukawa sector, these decays can lead to a dynamical
lepton asymmetry in the early Universe. This asymmetry will undergo thermody-
namic evolution as the universe expands and different reactions present in the model
have their impact on washing out part of the asymmetry. The remaining final lepton
asymmetry is converted to the baryon asymmetry via sphaleron transitions before
the electroweak phase transition. There is also a weak connection between the CP
violation in neutrino oscillations and the amount of lepton asymmetry.
For TeV-scale seesaw models, the generation of adequate lepton asymmetry
requires one to invoke resonant leptogenesis [41–43], where at least two of the heavy
neutrinos have a small mass difference comparable to their decay widths. In this
52
case, the heavy Majorana neutrino self-energy contributions [44] to the leptonic
CP asymmetry become dominant [?, 45] and get resonantly enhanced, even up to
order one [41,42]. In the context of an embedding of seesaw into TeV-scale LRSM,
there are additional complications due to the presence of RH gauge interactions
that contribute to the dilution and washout of the primordial lepton asymmetry
generated via resonant leptogenesis. This was explored in detail in [46] where it was
pointed out that there is significant dilution of the primordial lepton asymmetry
due to ∆L = 1 scattering processes such as N`R → udc mediated by WR. This
leads to an extra suppression of the final lepton symmetry, in addition to the usual
inverse decay Lφ→ N and ∆L = 0, 2 scattering processes Lφ↔ Lφ (Lcφ†) present
in generic SM seesaw scenarios. This additional dilution factor κ (also sometimes
called efficiency) in this case is of order
Y 2ν m
4
WR
g4Rm
4
N
, which formN ∼ TeV andmWR ∼ 3−4
TeV can be easily ≤ 10−7 or so for Y ' 10−5.5. Combined with entropy dilution
effect and the dilution from inverse decays, this implies that even for the maximal
CP asymmetry ε ∼ O(1), the observed baryon to photon ratio can be obtained only
if mWR ≥ 18 TeV. This result is very important because, as argued in [46], this can
provide a way to falsify leptogenesis if a WR with mass below this limit is observed
in colliders.
We investigate whether there are any allowed parameter space in the TeV-scale
LRSM where leptogenesis can work with a weakened lower bound on mWR , without
conflicting with observed neutrino data and charged lepton masses. We work in a
version of the model that is parity asymmetric at the TeV scale, which is anyway
necessary if we want type-I seesaw to be the only contribution to neutrino masses.
53
According to our classification above, the work of [46] falls into the class I models.
We explore whether the lower bound can be weakened in the other classes of models
discussed above. It could very well be that if other observations push the Yukawa
parameters to the range of class I models, the bound of [46] cannot be avoided,
thereby providing a way to disprove leptogenesis at the LHC. However, to see how
widely applicable the bound of [46] is, we consider in this paper an example of a
model which belongs to class II, i.e. neutrino fits are done by cancellation leading
to a specific texture for Dirac masses.
We implement the class II strategy for small neutrino masses in the minimal
LRSM with a single bi-doublet field in the lepton sector where all leptonic Yukawa
couplings are significantly larger than the canonical value of O(10−5.5) and the WR
mass is in the few TeV range. As noted above, to get small neutrino masses via
type-I seesaw, we invoke cancellation between two Yukawa couplings to generate
extra suppression and a particular resulting texture for the Dirac masses. We find
that due to enhanced Yukawa couplings, the dilution of lepton asymmetry due to
the WR mediated scatterings as well as due to 3-body decays of RH neutrinos such
as N → `Rudc become considerably less than the CP-violating 2-body decay modes
N → Lφ,Lcφ†, and as a result, the lower limit on WR mass can be brought within
the LHC reach for a range of Yukawa couplings for which the washout effect due to
inverse decay is in control. New aspects in our work that goes beyond that of [46] are
the following: (i) we give a realistic fit for all lepton masses and mixing with larger
Yukawa couplings (∼ 10−2 or so); (ii) reference [46] assumes that the CP asymmetry
ε ∼ 1 whereas we calculate the primordial CP asymmetry ε in our model using the
54
Yukawa couplings demanded by our specific neutrino fit. As a result, our ε is still
of order 10−1 (see text for precise numbers); (iii) finally, we take the flavour effects
into account in our washout and lepton asymmetry calculation. It is a consequence
of (i) and (iii), which leads us to lower the WR mass bound from leptogenesis.
When Tc < T < TR, we write the scalar bi-doublet as
Φ = (φ, φ′) (4.1)
where φ and φ′ are SU(2)L doublets. Then, we can write
Φ˜ ≡ σ2Φ∗σ2 =
(
φ˜′, φ˜
)
(4.2)
where φ˜ ≡ iσ2φ∗ and φ˜′ ≡ −iσ2φ′∗. For simplicity, we assume that φ′ acquires a
mass larger than mN through vR while φ remains massless. Then, φ is identified
as the Higgs doublet of the SM. The Yukawa interaction Lagrangian of the lepton
sector in the RH neutrino mass basis when Tc < T < TR is written as
L`Y = −LLi(fijΦ + f˜ijΦ˜)LRj − hRijLcRiiσ2∆RLRj + H.c. (4.3)
= −fijLLiφNRj − fijLLiφ′`Rj − f˜ijLLiφ˜′NRj − f˜ijLLiφ˜`Rj
− f ∗jiNRjφ†LLi − f ∗ji`Rjφ′†LLi − f˜ ∗jiNRjφ˜′
†
LLi − f˜ ∗ji`Rjφ˜†LLi
− 1√
2
hRijvRN cRiNRj − hRijδ0RN cRiNRj +
√
2hRijδ
+
R`
c
RiNRj + hRijδ
++
R `
c
Ri`Rj
− 1√
2
h∗RjivRNRjN
c
Ri − h∗Rjiδ0∗R NRjN cRi +
√
2h∗Rjiδ
−
RNRj`
c
Ri + h
∗
Rjiδ
−−
R `Rj`
c
Ri,
(4.4)
from which we can identify interactions that contribute to the lepton asymmetry.
55
4.2 One-loop resummed effective Yukawa couplings and decay rates
For simplicity, we write
Li ≡ LLi, `i ≡ `Ri, Q ≡ QL, u ≡ uR, d ≡ dR (4.5)
where i is the lepton flavour index, and u, d can be any pair in the three flavours. We
also use Greek and Roman indices for RH neutrino and LH lepton doublet flavours,
respectively. The partial decay rates Γ(Nα → Liφ) and Γ(Nα → Lciφ†) at T = 0 are
given by
Γ(Nα → Liφ) = mNαAiαα(f̂), Γ(Nα → Lciφ†) = mNαAiαα(f̂ c) (4.6)
where Aiαβ is the absorptive transition amplitude defined by
Aiαβ(f̂) =
1
16pi
f̂iαf̂
∗
iβ. (4.7)
Here, f̂iα is the one-loop resummed effective Yukawa couplings given by [50]
f̂iα = fiα − i
∑
β,γ
|αβγ |fiβ
× mα(mαAαβ +mβAβα)− iRαγ
[
mαAγβ(mαAαγ +mγAγα) +mβAβγ(mαAγα +mγAαγ)
]
m2α −m2β + 2im2αAββ + 2iIm[Rαγ ]
(
m2α|Aβγ |2 +mβmγRe[A2βγ ]
)
(4.8)
where αβγ is the Levi-Civita anti-symmetric tensor and
mα ≡ mNα , Aαβ ≡
3∑
i=1
Aiαβ(f̂), Rαβ =
m2α
m2α −m2β + 2im2αAββ
. (4.9)
The CP-conjugate effective Yukawa couplings f̂ ciα are obtained by replacing hiα with
h∗iα. The total two-body decay rate at T = 0 is written as
ΓNαLφ =
3∑
i=1
[
Γ(Nα → Liφ) + Γ(Nα → Lciφ†)
]
=
mNα
16pi
[(
f̂ †f̂)αα +
(
f̂ c†f̂ c
)
αα
]
(4.10)
56
and the total three-body decay rate at T = 0 as
ΓNα`αudc = Γ(Nα → `αudc) + Γ(Nα → `cαucd). (4.11)
Here,
Γ(Nα → `αudc) = Γ(Nα → `cαucd) =
3g4R
29pi3m3Nα
∫ m2Nα
0
ds
m6Nα − 3m2Nαs2 + 2s3
(s−m2WR)2 +m2WRΓ2WR
(4.12)
where ΓWR ≈ (g2R/4pi)mWR is the total decay rate of WR at T = 0 when mNα < WR,
and all three quark flavours and colors have been considered. Note that we can have
only one lepton flavour `α for each Nα.
4.3 Boltzmann equations and the lepton asymmetry
The generic Boltzmann equation is written as [48]
dna
dt
+ 3Hna = −
∑
aX↔Y
[
nanX
neqa n
eq
X
γ(aX → Y )− nY
neqY
γ(Y → aX)
]
(4.13)
where γ is the thermally averaged collision term. We define the CP-conserving
collision terms for various decay and scattering processes by
γaXY ≡ γ(aX → Y ) + γ(aX → Y ) (4.14)
where aX → Y is the CP-conjugate process of aX → Y . Note that the CPT
invariance implies
γaXY = γ
Y
aX . (4.15)
We introduce the dimensionless time variable defined by
z ≡ mN1
T
. (4.16)
57
Then, the thermally averaged decay rates of Nα are wirtten as
γNαLiφ ≈ neqN1
K1(z)
K2(z)
ΓNαLiφ =
m3N1
pi2z
K1(z)Γ
Nα
Liφ
, (4.17)
γNα`αudc ≈ neqN1
K1(z)
K2(z)
ΓNα`αudc =
m3N1
pi2z
K1(z)Γ
Nα
`αudc
(4.18)
where K1(z)/K2(z) is the thermally averaged time dilation factor. Defining the
leptonic CP-asymmetry by
δiNα =
Γ(Nα → Liφ)− Γ(Nα → Lciφ†)∑3
j=1
[
Γ(Nα → Ljφ) + Γ(Nα → Lcjφ†)
] , (4.19)
we can write the CP-violating decay term as
δγNαLiφ ≡ γ(Nα → Liφ)− γ(Nα → Lciφ†) = δiNαγNαLφ (4.20)
where
γNαLφ ≡
3∑
i=1
γNαLiφ. (4.21)
For 2→ 2 scattering processes, the thermally averaged collision term can be written
as
γXYAB =
T
64pi4
∫ ∞
smin
ds
√
s σˆXYAB (s) K1
(√
s
T
)
=
m4N1
64pi4z
∫ ∞
xmin
dx
√
xK1(z
√
x)σˆXYAB (x) (4.22)
where σˆXYAB is CP-conserving reduced cross section defined by
σˆXYAB ≡ σˆ(XY → AB) + σˆ(AB → XY ). (4.23)
58
The CP-conserving reduced cross sections for the dominant scattering processes are
derived in appendix D, and they are given by
σˆNαu
c
`αdc
(s) =
9g4R
4pis
∫ 0
m2N−s
dt
(s+ t)(s+ t−m2N)
(t−m2WR)2
(4.24)
σˆNαu
c
`αdc
(s) =
9g4R
4pis
∫ 0
m2N−s
dt
(s+ t)(s+ t−m2N)
(t−m2WR)2
(4.25)
σˆNαd`αu (s) =
9g4R
4pi
(m2N − s)2
m2WR(s+m
2
WR
−m2N)
. (4.26)
Following the steps in appendix D, we can write the Boltzmann equations for the
RH neutrino density and the LH lepton doublet density as
dnNα
dt
+ 3HnNα =
(
1− nNα
neqNα
)(
γNαLφ + γ
Nα
`αudc
+ γSRα
)− 3∑
j=1
n∆Lj
2neq`j
δγNαLjφ, (4.27)
dn∆Li
dt
+ 3Hn∆Li =
3∑
α=1
(
nNα
neqNα
− 1
)
δγNαLiφ −
n∆Li
2neq`i
3∑
α=1
γNαLiφ (4.28)
where
γSRα ≡ γNα`αucd + γNαu
c
`αdc
+ γNαd`αu . (4.29)
We can simplify the Boltzmann equations using the dimensionless time variable
z introduced above and also using normalized densities of RH neutrinos and lepton
asymmetry. First, we write the Hubble parameter at z = 1 as
HN ≡ H(z = 1) ≈
√
8pi3g∗
90
m2N1
MPl
= z2H (4.30)
where g∗ is the SM degree of freedom. The photon number density is given by
nγ =
2m3N1ζ(3)
pi2z3
. (4.31)
59
where ζ(x) =
∑∞
n=1 n
−x is the Riemann zeta function. Now we introduce the nor-
malized densities of RH neutrinos and lepton asymmetry defined by
ηNα ≡
nNα
nγ
, η∆Li ≡
n∆Li
nγ
, (4.32)
and the normalized RH neutrino density in equilibrium is
ηeqNα =
neqNα
nγ
≈ n
eq
N1
nγ
=
1
2ζ(3)
z2K2(z). (4.33)
As shown in appendix D, we can simplify the left-hand sides of the Boltzmann
equations 4.27 and 4.28 using the normalized densities, and they are written as
HNnγ
z
dηNα
dz
= −
(
ηNα
ηeqNα
− 1
)(
γNαLφ + γ
Nα
`αudc
+ γSRα
)− 2
3
3∑
j=1
η∆Ljδ
j
Nα
γNαLφ , (4.34)
HNnγ
z
dη∆Li
dz
=
3∑
α=1
δiNα
(
ηNα
ηeqNα
− 1
)
γNαLφ −
2
3
η∆Li
3∑
α=1
γNαLiφ (4.35)
where we have used neq`i = 3/4. When the lepton asymmetry satisfies |δiNα|  1, we
can safely neglect the second term in equation 4.34 to obtain
HNnγ
z
dηNα
dz
= −
(
ηNα
ηeqNα
− 1
)(
γNαLφ + γ
Nα
`αudc
+ γSRα
)
, (4.36)
HNnγ
z
dη∆Li
dz
=
3∑
α=1
δiNα
(
ηNα
ηeqNα
− 1
)
γNαLφ −
2
3
η∆Li
3∑
α=1
γNαLiφ. (4.37)
From equation 4.36, we can find the expression
ηNα
ηeqNα
− 1 = −HNnγ
z
dηNα
dz
1
γNαLφ + γ
Nα
`αudc
+ γSRα
, (4.38)
which we can substitute into equation 4.37 to obtain
dη∆Li
dz
= −
3∑
α=1
δiNα
dηNα
dz
D˜α
Dα + Sα
− 2
3
η∆LiWi (4.39)
60
where
D˜α ≡ z
HNnγ
γNαLφ , Dα ≡
z
HNnγ
(γNαLφ + γ
Nα
`αudc
), Sα ≡ z
HNnγ
γSRα , Wi =
z
HNnγ
3∑
α=1
γNαLiφ.
(4.40)
The differential equation 4.39 can be solved by the integrating factor method, as
shown in appendix E. Assuming the initial lepton asymmetry is negligible, we obtain
the expression
η∆Li(z) = −
3∑
α=1
δiNακ
i
Nα(z) (4.41)
where κiNα is the efficiency factor defined by
κiNα(z) ≡
∫ z
z0
dz′
dηNα
dz′
D˜α
Dα + Sα
[
−2
3
∫ z
z′
dz′′Wi(z′′)
]
. (4.42)
Due to the strong washout of RH neutrino densities in the TeV-scale leptogenesis,
we have |ηNα/ηeqNα − 1|  1. We may therefore approximately assume ηNα ≈ ηeqNα ,
and thus
dηNα
dz
≈ dη
eq
Nα
dz
≈ dη
eq
N1
dz
= − 1
2ζ(3)
z2K1(z) (4.43)
where z0 is the initial time with the initial lepton asymmetry. If the lepton washout
term satisfies Wi(zc) . 1, the lepton asymmetry freezes out at zB < zc where zB can
be found by the steepest descent method [51]. On the other hand, if Wi(zc)  1
as in the TeV-scale leptogenesis, we can find an approximate expression of the
lepton asymmetry from equations 4.41 and 4.42, as shown in appendix D [52]. The
approximate form of the asymmetry in the LH lepton doublet is now given by
η∆Li(z) ≈
3z2K1(z)
4Wi(z)
3∑
α=1
δiNα
D˜α
Dα + Sα
, (4.44)
61
and the total asymmetry is
η∆L(z) ≡
3∑
i=1
η∆Li(z). (4.45)
4.4 Numerical procedure
For the successful leptogenesis, we should be able to find the model parameters that
would give |η∆L(zc)| = (2.47 ± 0.03) · 10−8 which is the value consistent with the
observed baryon asymmetry. The following is the numerical procedure:
1. Randomly generate the lightest light neutrino mass mν1 , and calculate mν2 =√
m2ν1 + ∆m
2
21 and mν3 =
√
m2ν1 + ∆m
2
31.
2. Calculate M cν from M
c
ν = UPMNSM
d
νU
T
PMNS where M
c
ν and M
d
ν are the light
neutrino mass matrices in the charged lepton and light neutrino mass bases,
respectively. The mixing matrix UPMNS is the PMNS matrix whose CP phases
are also randomly generated.
3. Randomly generate mN2 , mN2 −mN1 , and mN3 −mN2 which determine MR in
the RH neutrino mass basis.
4. Randomly generate a complex orthogonal matrix O, and calculate MD =
−iUPMNS
√
MdνO
√
MdR [47] where MD is the Dirac mass matrix in the RH
neutrino mass basis.
5. Randomly generate V `R, and calculate M` = M
c
`V
`†
R where M` is the charged
lepton mass basis in the RH neutrino mass basis.
62
6. Randomly generate κ2, α, and calculate κ1 =
√
v2EW − κ22. Find the Yukawa
coupling matrices in RH neutrino mass basis from
f =
√
2
κ1MD − κ2e−iαM`
κ21 − κ22
, f˜ =
√
2
κ1M` − κ2eiαMD
κ21 − κ22
, (4.46)
where f is the Yukawa couplings associated with the decay and scattering
processes of our interest under the assumption we have introduced.
7. Calculate one-loop resummed effective Yukawa couplings f̂ , f̂ c from f , mNα ,
and calculate the CP asymmetry and collision terms.
8. Calculate η∆Li(zc), the normalized asymmetry in the LH lepton doublet at zc,
from the CP asymmetry and collision terms we have obtained.
9. Calculate M cD = MDV
`
R and M
c
R = V
`T
R MRV
`
R where M
c
D and M
c
R are the Dirac
and RH neutrino mass matrices in the charged lepton mass basis, respectively.
10. Construct the 6 × 6 neutrino mass matrix M cνN from M cD and M cR, and find
the 6× 6 mixing matrix VνN that diagonalizes M cνN .
The mixing matrices UPMNS, V
`
L, V
`
R, and VνN are parametrized in the same way as
in the MLRSM. The complex orthogonal matrix can be parametrized as O = eS
where S is a skew-symmetric complex matrix, i.e. ST = −S.
4.5 Numerical results
The lower bound of mWR compatible with leptogenesis is found to be 6.9 TeV,
which is beyond the upper limit observable at the LHC. The numerical results are
63
presented in figure 4.1. If we discover WR much lighter than this value, the idea of
leptogenesis can be falsified.
4.6 Conclusion
We have analyzed the leptogenesis constraints on the mass of the right-handed
gauge boson in TeV scale Left-Right Symmetric Models. While the existing bound
of mWR > 18 TeV applies for generic LRSM scenarios with small Yukawa couplings,
we have found a significantly weaker bound of mWR > 6.9 TeV in a new class of
LRSM scenarios with relatively larger Yukawa couplings, which is consistent with
charged lepton and neutrino mass data. The key factors responsible for our result
is the inclusion of flavour effects in the lepton asymmetry calculation. This lower
bound, mWR > 6.9 TeV is for the case gL = gR and will be proportionately weaker
for the case gR < gL.
64
Figure 4.1: Values of parameters and mass matrices that give the lower bound of
mWR = 6.9 TeV. The lepton asymmetry is slightly larger than 2.47 · 10−8, and thus
slightly smaller value of mWR is allowed.
65
Chapter 5: Conclusion
We have investigated the TeV-scale phenomenology of the LRSM. We have provided
a new method to construct lepton mass matrices in the MLRSM of type-I dominance
with the parity symmetry. Using this method, we have investigated the TeV-scale
phenomenology of the MLRSM in the normal hierarchy of light neutrino masses,
and explored the model predictions for the CLFV, 0νββ, EDM’s of charged leptons.
We have also presented a natual TeV-scale seesaw model which does not require fine-
tuning of model parameters for the TeV-scale phenomenology. A discrete flavour
symmetry is shown to guarantee a specific texture of lepton mass matrices. In ad-
dition, we have studied the leptogenesis with TeV-scale WR and mN , and presented
a lower bound of mWR which allows leptogenesis.
66
Appendix A: Derivation of various expressions in the minimal left-
right symmetric model
A.1 Gauge group and fields
The gauge group of the left-right symmetric model (LRSM) is given by
SU(2)L ⊗ SU(2)R ⊗ U(1)B−L. (A.1)
The representations of the leptons are
L′Li =
 ν ′Li
`′Li
 ∼ (2,1,−1), L′Ri =
 ν ′Ri
`′Ri
 ∼ (1,2,−1), (A.2)
and for quarks, we have
Q′Li =
 u′Li
d′Li
 ∼ (2,1, 1/3), Q′Ri =
 u′Ri
d′Ri
 ∼ (1,2, 1/3) (A.3)
where i is the generation index. In addition, the scalar bi-doublet field is given by
Φ =
 φ01 φ+2
φ−1 φ
0
2
 ∼ (2,2, 0), (A.4)
67
and the scalar triplet field is
∆L =
 δ+L /
√
2 δ++L
δ0L −δ+L /
√
2
 ∼ (3,1, 2), ∆R =
 δ+R/
√
2 δ++R
δ0R −δ+R/
√
2
 ∼ (1,3, 2).
(A.5)
A.2 Current and generators
The SU(2)L⊗ SU(2)R generators are
TL+ =
∫
d3x(ν′†L e
′
L + u
′†
Ld
′
L), TL− = (TL+)
†, TL3 =
1
2
∫
d3x(ν′†L ν
′
L − e′†Le′L + u′†Lu′L − d′†Ld′L),
TR+ =
∫
d3x(ν′†Re
′
R + u
′†
Rd
′
R), TR− = (TR+)
†, TR3 =
1
2
∫
d3x(ν′†Rν
′
R − e′†Re′R + u′†Ru′R − d′†Rd′R).
(A.6)
The electric charge generator is given by
Q =
∫
d3x
(
−e′†e′ + 2
3
u′†u′ − 1
3
d′†d′
)
. (A.7)
Now we can find the U(1)B−L generator given by
Q− TL3 − TR3 =
∫
d3x
[
−1
2
(ν ′†Lν
′
L + ν
′†
Rν
′
R + e
′†
Le
′
L + e
′†
Re
′
R) +
1
6
(u′†Lu
′
L + u
′†
Ru
′
R + d
′†
Ld
′
L + d
′†
Rd
′
R)
]
=
B − L
2
. (A.8)
Since we have Y = 2(Q− TL3), the generators satisfy
Y
2
= TR3 +
B − L
2
. (A.9)
68
A.3 Yukawa interaction Lagrangian
The Yukawa interaction Lagrangian is written as
L`Y = −L′Li(fijΦ + f˜ijΦ˜)L′Rj − hLijL′cLiiσ2∆LL′Lj − hRijL′cRiiσ2∆RL′Rj + H.c.
(A.10)
= −fijφ02`′Li`′Rj − fijφ01ν ′Liν ′Rj − fijφ−1 `′Liν ′Rj − fijφ+2 ν ′Li`′Rj
− f˜ijφ0∗1 `′Li`′Rj − f˜ijφ0∗2 ν ′Liν ′Rj + f˜ijφ−2 `′Liν ′Rj + f˜ijφ+1 ν ′Li`′Rj
− hLijδ0Lν ′cLiνLj +
1√
2
hLijδ
+
L `
′c
Liν
′
Lj +
1√
2
hLijδ
+
L ν
′c
Li`
′
Lj + hLijδ
++
L `
′c
Li`
′
Lj
− hRijδ0Rν ′cRiν ′Rj +
1√
2
hRijδ
+
R`
′c
Riν
′
Rj +
1√
2
hRijδ
+
Rν
′c
Ri`
′
Rj + hRijδ
++
R `
′c
Ri`
′
Rj
+ H.c. (A.11)
where
Φ˜ ≡ σ2Φ∗σ2 =
 φ0∗2 −φ+1
−φ−2 φ0∗1
 . (A.12)
We have also defined ψc ≡ Cψ∗ and ψc = −ψTC where C = iγ2γ0 is the charge
conjugation operator in the Dirac-Pauli representation.
A.4 Spontaneous symmetry breaking and fermion masses
Without loss of generality, the scalar fields after the spontaneous symmetry breaking
are written as
〈Φ〉 =
 κ1/
√
2 0
0 κ2e
iα/
√
2
 , 〈∆L〉 =
 0 0
vLe
iθL/
√
2 0
 , 〈∆R〉 =
 0 0
vR/
√
2 0
 .
(A.13)
69
After spontaneous symmetry breaking, the Yukawa coupling terms are written as
〈L`Y 〉 = −
1√
2
fijκ2e
iα`′Li`
′
Rj −
1√
2
fijκ1ν ′Liν
′
Rj −
1√
2
f˜ijκ1`′Li`
′
Rj −
1√
2
f˜ijκ2e
−iαν ′Liν
′
Rj
− 1√
2
hLijvLe
iθLν ′cLiν
′
Lj −
1√
2
hRijvRν ′cRiν
′
Rj + H.c.. (A.14)
The mass terms for leptons are written as
Lmass` = −
1√
2
(fijκ2e
iα + f˜ijκ1)`′Li`
′
Rj + H.c.. (A.15)
We therefore have
M` =
1√
2
(fκ2e
iα + f˜κ1). (A.16)
The neutrino mass terms are given by
Lmassν = −
1√
2
(fijκ1 + f˜ijκ2e
−iα)ν ′Liν
′
Rj −
1√
2
hLijvLe
iθLν ′cLiν
′
Lj −
1√
2
hRijvRν ′cRiν
′
Rj + H.c..
(A.17)
We have the identity
ν ′Lν
′
R = (ν
′
Lν
′
R)
T = −ν ′TR γT0 ν ′∗L = −ν ′TR C†CγT0 ν ′∗L = (Cν ′∗R )†γ0Cν ′∗L = ν ′cRν ′cL (A.18)
where we have used C−1γµC = −γTµ . Similarly,
ν ′cLiν
′
Lj = ν
′c
Ljν
′
Li, ν
′c
Riν
′
Rj = ν
′c
Rjν
′
Ri. (A.19)
Hence, we can write
Lmassν = −
1
2
(ν ′L ν
′c
R)
 ML MD
MTD MR

 ν ′cL
ν ′R
+ H.c. (A.20)
where
MD =
1√
2
(fκ1 + f˜κ2e
−iα), ML =
√
2h∗LvLe
−iθL , MR =
√
2hRvR. (A.21)
70
A.5 Gauge bosons
The covariant derivative is given by
Dµ = ∂µ − igLTL ·WLµ − igRTR ·WRµ − ig′B − L
2
Bµ. (A.22)
Now we define
W+µ ≡
1√
2
(W 1µ − iW 2µ), W−µ ≡
1√
2
(W 1µ + iW
2
µ). (A.23)
Kinetic terms
The kinetic terms for SU(2) gauge bosons are
−1
4
F µνa F
a
µν = −
1
4
(∂µW νa − ∂νW µa − gfabcW µb W νc )(∂µW aν − ∂νW aµ − gfabcW bµW cν )
= −1
4
(∂µW νa − ∂νW µa )(∂µW aν − ∂νW aµ ) +
1
2
gfabcW µb W
ν
c (∂µW
a
ν − ∂νW aµ )
− 1
4
g2fabcfadeW µb W
ν
cW
d
µW
e
ν . (A.24)
Lepton sector
For the LH leptons and neutrinos, we have
L′Liiγ
µDµL
′
Li = (ν
′
Li `
′
Li)iγ
µ∂µ
 ν′Li
`′Li
+ 12(ν′Li `′Li)γµ
 gLW 3Lµ − g′Bµ
√
2gLW
+
Lµ
√
2gLW
−
Lµ −gLW 3Lµ − g′Bµ

 ν′Li
`′Li

= ν′Liiγ
µ∂µν
′
Li + `
′
Liiγ
µ∂µ`
′
Li +
1
2
ν′Liγ
µν′Li(gLW
3
Lµ − g′Bµ)
− 1
2
`′Liγ
µ`′Li(gLW
3
Lµ + g
′Bµ) +
1√
2
gLν′Liγ
µ`′LiW
+
Lµ +
1√
2
gL`′Liγ
µν′LiW
−
Lµ
(A.25)
71
Similarly, the kinetic terms for the RH leptons and neutrinos are written as
L′Riiγ
µDµL
′
Ri = ν
′
Riiγ
µ∂µν
′
Ri + `
′
Riiγ
µ∂µ`
′
Ri +
1
2
ν ′Riγ
µν ′Ri(gRW
3
Rµ − g′Bµ)
− 1
2
`′Riγ
µ`′Ri(gRW
3
Rµ + g
′Bµ) +
1√
2
gRν ′Riγ
µ`′RiW
+
Rµ +
1√
2
gR`′Riγ
µν ′RiW
−
Rµ.
(A.26)
Quark sector
For quarks, we have
QLiiγ
µDµQLi = uLiiγ
µ∂µuLi + dLiiγ
µ∂µdLi +
1
2
uLiγ
µuLi
(
gLW
3
Lµ +
1
3
g′Bµ
)
− 1
2
dLiγ
µdLi
(
gLW
3
Lµ −
1
3
g′Bµ
)
+
1√
2
gLuLiγ
µdLiW
+
Lµ +
1√
2
gLdLiγ
µuLiW
−
Lµ
(A.27)
and
QRiiγ
µDµQRi = uRiiγ
µ∂µuRi + dRiiγ
µ∂µdRi +
1
2
uRiγ
µuRi
(
gLW
3
Rµ +
1
3
g′Bµ
)
− 1
2
dRiγ
µdRi
(
gLW
3
Rµ −
1
3
g′Bµ
)
+
1√
2
gRuRiγ
µdRiW
+
Rµ +
1√
2
gRdRiγ
µuRiW
−
Rµ.
(A.28)
Scalar field sector
For scalar fields, we have
Ls = tr[(DµΦ)†(DµΦ)] + tr[(Dµ∆L)†(Dµ∆L)] + tr[(Dµ∆R)†(Dµ∆R)]. (A.29)
Now we explicitly calculate the masses of gauge bosons.
72
Contribution from 〈Φ〉
We have
LΦ = tr[(DµΦ)†(DµΦ)]
= tr
[(
∂µΦ† + igLΦ†
σaL
2
W aµL − igR
σaR
2
W aµR Φ
†
)(
∂µΦ− igLσ
b
L
2
W bLµΦ + igRΦ
σbR
2
W bRµ
)]
= tr
[
∂µΦ†∂µΦ− i
2
gL∂
µΦ†σaLW
a
LµΦ +
i
2
gR∂
µΦ†ΦσaRW
a
Rµ +
i
2
gLΦ
†σaLW
aµ
L ∂µΦ−
i
2
gRσ
a
RW
aµ
R Φ
†∂µΦ
+
1
4
(
g2LΦ
†σaLσ
b
LΦW
aµ
L W
b
Lµ − gLgRΦ†σaLΦσbRW aµL W bRµ
−gLgRσaRΦ†σbLΦW aµR W bLµ + g2RσaRΦ†ΦσbRW aµR W bRµ
)]
. (A.30)
After Φ acquires the VEV, we can write
tr
[
〈Φ†〉σaLσbL〈Φ〉W aµL W bLµ
]
=
1
2
(κ21 + κ
2
2)W
3µ
L W
3
Lµ + (κ
2
1 + κ
2
2)W
+µ
L W
−
Lµ, (A.31)
tr
[
〈Φ†〉σaL〈Φ〉σbRW aµL W bRµ
]
=
1
2
(κ21 + κ
2
2)W
3µ
L W
3
Rµ + κ1κ2e
iαW+µL W
−
Rµ + κ1κ2e
−iαW−µL W
+
Rµ,
(A.32)
tr
[
〈Φ〉σaR〈Φ†〉σbLW aµL W bRµ
]
= tr
[
〈Φ†〉σaL〈Φ〉σbRW aµL W bRµ
]†
=
1
2
(κ21 + κ
′2)W 3µL W
3
Rµ + κ1κ2e
iαW+µL W
−
Rµ + κκ2e
−iαW−µL W
+
Rµ,
(A.33)
tr
[
〈Φ〉σbRσaR〈Φ†〉W aµR W bRµ
]
= tr
[
〈Φ†〉σaRσbR〈Φ〉W aµR W bRµ
]†
=
1
2
(κ21 + κ
2
2)W
3µ
R W
3
Rµ + (κ
2
1 + κ
2
2)W
+µ
R W
−
Rµ. (A.34)
We therefore have
〈LΦ〉 = 1
8
(κ21 + κ
2
2)
(
g2LW
3µ
L W
3
Lµ − 2gLgRW 3µL W 3Rµ + g2RW 3µR W 3Rµ
)
+
1
4
g2L(κ
2
1 + κ
2
2)W
+µ
L W
−
Lµ −
1
2
gLgRκ1κ2e
iαW+µL W
−
Rµ
− 1
2
gLgRκ1κ2e
−iαW−µL W
+
Rµ +
1
4
g2R(κ
2
1 + κ
2
2)W
+µ
R W
−
Rµ + · · · . (A.35)
73
Contribution from 〈∆〉
We can write the scalar triplet ∆ as
∆ =
1√
2
σaδa (A.36)
where
δ0 =
1√
2
(δ1 + iδ2), δ+ = δ3, δ++ =
1√
2
(δ1 − iδ2). (A.37)
The gauge invariant kinetic term for ∆ is given by
L∆ = tr[(Dµ∆)†(Dµ∆)] = 1
2
tr[{Dµ(σaδa)}†{Dµ(σbδb)}]
=
1
2
tr[σaσb](∂µδa∗ + igδc∗Tca ·Wµ + ig′Bµδa∗)(∂µδb − igTbd ·Wµδd − ig′Bµδb)
= ∂µδa∗∂µδa − ig∂µδa∗(T i)adδdW iµ − ig′∂µδa∗δaBµ
+ igδc∗(T i)ca∂µδaW iµ + g2δc∗(T i)ca(T j)adδdW iµW jµ + gg
′δc∗(T i)caδaW iµBµ
+ ig′δa∗∂µδaBµ + gg′δa∗(T j)adδdW jµB
µ + g′2δa∗δaBµBµ
= ∂µδa∗∂µδa − ig∂µδa∗(T i)adδdW iµ + igδc∗(T i)ca∂µδaW iµ + g2δc∗(T i)ca(T j)adδdW iµW jµ
+ 2gg′δc∗(T i)caδaW iµBµ − ig′∂µδa∗δaBµ + ig′δa∗∂µδaBµ + g′2δa∗δaBµBµ
(A.38)
where T i is the generator of the SU(2) adjoint representation. Since (T c)ab = −iabc,
we have
δc∗(T i)ca(T j)adδd = δc∗aciadjδd = δc∗(δcdδij − δcjδid)δd = δc∗δcδij − δj∗δi,
δc∗(T i)caδa = −iδc∗caiδa. (A.39)
74
Therefore, the kinetic terms can be written as
L∆ = ∂µδa∗∂µδa − gabc∂µδa∗δbW cµ + gabcδa∗∂µδbW cµ + g2δa∗δaW bµW bµ − g2δa∗δbW aµW bµ
− 2igg′abcδa∗δbW cµBµ − ig′∂µδa∗δaBµ + ig′δa∗∂µδaBµ + g′2δa∗δaBµBµ.
(A.40)
We also have
δa∗δa = δ0∗δ0 + δ−δ+ + δ−−δ++,
δa∗W aµ = δ
−−W+µ + δ
0∗W−µ + δ
−W 3µ ,
δaW aµ = δ
++W−µ + δ
0W+µ + δ
+W 3µ ,
abcδa∗δbW cµ = (δ
1∗δ2 − δ2∗δ1)W 3µ + (δ2∗δ3 − δ3∗δ2)W 1µ + (δ3∗δ1 − δ1∗δ3)W 2µ
= i(δ++δ−−W 3µ − δ0∗δ0W 3µ + δ0δ−W+µ − δ−−δ+W+µ + δ0∗δ+W−µ − δ++δ−W−µ ).
(A.41)
After ∆ acquires the VEV, the Lagrangian terms relevant to the masses of gauge
bosons can be written as
〈L∆〉 = 1
2
g2v2(2W+µW−µ +W
3µW 3µ)−
1
2
g2v2W+µW−µ − gg′v2W 3µBµ +
1
2
g′2v2BµBµ + · · ·
=
1
2
g2v2W+µW−µ +
1
2
g2v2W 3µW 3µ − gg′v2W 3µBµ +
1
2
g′2v2BµBµ + · · · .
(A.42)
75
Total contributions
Hence, we have
〈Ls〉 = 〈LΦ〉+ 〈L∆L〉+ 〈L∆R〉
=
1
8
(κ21 + κ
2
2)
(
g2LW
3µ
L W
3
Lµ − 2gLgRW 3µL W 3Rµ + g2RW 3µR W 3Rµ
)
+
1
4
g2L(κ
2
1 + κ
2
2)W
+µ
L W
−
Lµ −
1
2
gLgRκ1κ2e
iαW+µL W
−
Rµ
− 1
2
gLgRκ1κ2e
−iαW−µL W
+
Rµ +
1
4
g2R(κ
2
1 + κ
2
2)W
+µ
R W
−
Rµ
+
1
2
g2Lv
2
LW
+µ
L W
−
Lµ +
1
2
g2Lv
2
LW
3µ
L W
3
Lµ − gLg′v2LW 3µL Bµ +
1
2
g′2v2LB
µBµ
+
1
2
g2Rv
2
RW
+µ
R W
−
Rµ +
1
2
g2Rv
2
RW
3µ
R W
3
Rµ − gRg′v2RW 3µR Bµ +
1
2
g′2v2RB
µBµ + · · ·
=
1
8
g2L(κ
2
1 + κ
2
2 + 4v
2
L)W
3µ
L W
3
Lµ −
1
4
gLgR(κ
2
1 + κ
2
2)W
3µ
L W
3
Rµ +
1
8
g2R(κ
2
1 + κ
2
2 + 4v
2
R)W
3µ
R W
3
Rµ
− gLg′v2LW 3µL Bµ − gRg′v2RW 3µR Bµ +
1
2
g′2(v2L + v
2
R)B
µBµ
+
1
4
g2L(κ
2
1 + κ
2
2 + 2v
2
L)W
+µ
L W
−
Lµ −
1
2
gLgRκ1κ2e
iαW+µL W
−
Rµ −
1
2
gLgRκ1κ2e
−iαW−µL W
+
Rµ
+
1
2
g2R(κ
2
1 + κ
2
2 + 2v
2
R)W
+µ
R W
−
Rµ + · · · . (A.43)
We therefore can write the mass terms for gauge bosons as
Lmassg =
1
2
(W 3µL W
3µ
R B
µ)

1
4g
2
L(κ
2
1 + κ
2
2 + 4v
2
L) −14gLgR(κ21 + κ22) −gLg′v2L
−14gLgR(κ21 + κ22) 14g2R(κ21 + κ22 + 4v2R) −gRg′v2R
−gLg′v2L −gRg′v2R g′2(v2L + v2R)


W 3Lµ
W 3Rµ
Bµ

+ (W+µL W
+µ
R )
 14g2L(κ21 + κ22 + 2v2L) −12gLgRκ1κ2eiα
−12gLgRκ1κ2e−iα 14g2R(κ21 + κ22 + 2v2R)

 W−Lµ
W−Rµ
+ · · · .
(A.44)
(i) Charged gauge bosons
76
Without loss of generality, the general form of the change of basis for charged gauge
bosons can be written as W−L
W−R
 =
 cos ξ sin ξeiα
− sin ξe−iα cos ξ

 W−1
W−2
 (A.45)
where W−1 and W
−
2 are mass eigenstates. We can find
cos ξ =
b− a+√(b− a)2 + 4c2√
[b− a+√(b− a)2 + 4c2]2 + 4c2 , sin ξ = −
2c√
[b− a+√(b− a)2 + 4c2]2 + 4c2
(A.46)
where
a ≡ 1
4
g2L(κ
2
1 + κ
2
2 + 2v
2
L), b ≡
1
4
g2R(κ
2
1 + κ
2
2 + 2v
2
R), c ≡
1
2
gLgRκ1κ2. (A.47)
Note that we have
tan 2ξ = − 2c
b− a = −
4gLgRκ1κ2
(g2R − g2L)(κ21 + κ22) + 2(g2Rv2R − g2Lv2L)
. (A.48)
The masses of charged gauge bosons are found to be
m2W1 =
1
2
[b+ a−
√
(b− a)2 + 4c2], m2W2 =
1
2
[b+ a+
√
(b− a)2 + 4c2].
(A.49)
With the phenomenological assumption vL  κ1, κ2  vR, we have a, c b. Then,
we can approximately write
√
(b− a)2 + 4c2 ≈ b− a+ 2c2/b, which gives
cos ξ ≈ 1− c
2
2b2
= 1− 2g
2
Lκ
2
1κ
2
2
g2R(κ
2
1 + κ
2
2 + 2v
2
R)
2
≈ 1− g
2
Lκ
2
1κ
2
2
2g2Rv
4
R
, (A.50)
sin ξ ≈ −c
b
(
1 +
a
b
)
= − 2gLκ1κ2
gR(κ2 + κ22 + 2v
2
R)
[
1 +
g2L(κ
2
1 + κ
2
2 + 2v
2
L)
g2R(κ
2
1 + κ
2
2 + 2v
2
R)
]
≈ −gLκ1κ2
gRv2R
,
(A.51)
(A.52)
77
and
tan 2ξ ≈ −2gLκ1κ2
gRv2R
. (A.53)
Note that we have 0 < −ξ  1. The charged gauge boson masses can also be
written as
m2W1 ≈ a−
c2
b
=
1
4
g2L(κ
2
1 + κ
2
2 + 2v
2
L)−
2g2Lκ
2
1κ
2
2
κ21 + κ
2
2 + 2v
2
R
, (A.54)
m2W2 ≈ b+
c2
b
=
1
4
g2R(κ
2
1 + κ
2
2 + 2v
2
R) +
2g2Lκ
2
1κ
2
2
κ21 + κ
2
2 + 2v
2
R
, (A.55)
or simply as
m2W1 ≈
1
4
g2L(κ
2
1 + κ
2
2), m
2
W2
≈ 1
2
g2Rv
2
R. (A.56)
These approximate expressions are obtained by systematically expanding the trigono-
metric functions and gauge boson masses in terms of the small parameters a/b and
c/b up to the second order.
(ii) Neutral gauge bosons
Without loss of generality, the general form of the change of basis for neutral gauge
bosons can be written as
W 3L
W 3R
B
 =

1 0 0
0 cos ζ1 sin ζ1
0 − sin ζ1 cos ζ1


cos ζ2 0 sin ζ2
0 1 0
− sin ζ2 0 cos ζ2


cos ζ3 sin ζ3 0
− sin ζ3 cos ζ3 0
0 0 1


Z1
Z2
A

(A.57)
=

cos ζ2 cos ζ3 cos ζ2 sin ζ3 sin ζ2
− sin ζ1 sin ζ2 cos ζ3 − cos ζ1 sin ζ3 cos ζ1 cos ζ3 − sin ζ1 sin ζ2 sin ζ3 sin ζ1 cos ζ2
− cos ζ1 sin ζ2 cos ζ3 + sin ζ1 sin ζ3 − sin ζ1 cos ζ3 − cos ζ1 sin ζ2 sin ζ3 cos ζ1 cos ζ2


Z1
Z2
A

(A.58)
78
where Z1, Z2, and A are the mass eigenstates, and the mixing angles are given by
cos ζ1 =
gR√
g2R + g
′2 , sin ζ1 =
g′√
g2R + g
′2 , (A.59)
cos ζ2 =
gL
√
g2R + g
′2√
g2Lg
2
R + g
2
Lg
′2 + g2Rg′2
, sin ζ2 =
gRg
′√
g2Lg
2
R + g
2
Lg
′2 + g2Rg′2
, (A.60)
tan 2ζ3 =
2
√
g2Lg
2
R + g
2
Lg
′2 + g2Rg′2[4g
′2v2L − g2R(κ21 + κ22)]
(g4R − g2Lg2R − g2Lg′2 − g2Rg′2)(κ21 + κ22) + 4(g′4 − g2Lg2R − g2Lg′2 − g2Rg′2)v2L + 4(g2R + g′2)2v2R
.
(A.61)
Note that we have the identity
gLgRg
′√
g2Lg
2
R + g
2
Lg
′2 + g2Rg′2
= g′ cos ζ1 cos ζ2 = gL sin ζ2 (A.62)
or
g′ =
gL tan ζ2
cos ζ1
. (A.63)
The gauge field A corresponds to the photon with zero mass, and the masses of the
other neutral gauge bosons are
m2Z1 =
1
8
(g2L + g
2
R)(κ
2
1 + κ
2
2) +
1
2
(g2L + g
′2)v2L +
1
2
(g2R + g
′2)v2R
− 1
4(g2R + g
′2)
{
(g2Lg
2
R + g
2
Lg
′2 + g2Rg
′2)[4g′2v2L − g2R(κ21 + κ22)]2
+
[
1
2
(g4R − g2Lg2R − g2Lg′2 − g2Rg′2)(κ21 + κ22)
+ 2(g′4 − g2Lg2R − g2Lg′2 − g2Rg′2)v2L + 2(g2R + g′2)2v2R
]2}1/2
,
(A.64)
m2Z2 =
1
8
(g2L + g
2
R)(κ
2
1 + κ
2
2) +
1
2
(g2L + g
′2)v2L +
1
2
(g2R + g
′2)v2R
+
1
4(g2R + g
′2)
{
(g2Lg
2
R + g
2
Lg
′2 + g2Rg
′2)[4g′2v2L − g2R(κ21 + κ22)]2
+
[
1
2
(g4R − g2Lg2R − g2Lg′2 − g2Rg′2)(κ21 + κ22)
+ 2(g′4 − g2Lg2R − g2Lg′2 − g2Rg′2)v2L + 2(g2R + g′2)2v2R
]2}1/2
.
(A.65)
79
The neutral gauge bosons that couple to the LH fermions can be written as
gLW
3
Lµ − g′Bµ = gL(cos ζ2 cos ζ3Z1 + cos ζ2 sin ζ3Z2 + sin ζ2A)
− g′[(− cos ζ1 sin ζ2 cos ζ3 + sin ζ1 sin ζ3)Z1
+ (− sin ζ1 cos ζ3 − cos ζ1 sin ζ2 sin ζ3)Z2 + cos ζ1 cos ζ2A
]
=
[
gL cos ζ2 cos ζ3 − g′(− cos ζ1 sin ζ2 cos ζ3 + sin ζ1 sin ζ3)
]
Z1
+
[
gL cos ζ1 sin ζ2 − g′(− sin ζ1 cos ζ3 − cos ζ1 sin ζ2 sin ζ3)
]
Z2
+ (gL sin ζ2 − g′ cos ζ1 cos ζ2)A
= (gL cos ζ2 cos ζ3 + g
′ cos ζ1 sin ζ2 cos ζ3 − g′ sin ζ1 sin ζ3)Z1
+ (gL cos ζ2 sin ζ3 + g
′ sin ζ1 cos ζ3 + g′ cos ζ1 sin ζ2 sin ζ3)Z2
=
gL
cos ζ2
[
(cos ζ3 − tan ζ1 sin ζ2 sin ζ3)Z1 + (tan ζ1 sin ζ2 cos ζ3 + sin ζ3)Z2
]
(A.66)
and
gLW
3
Lµ + g
′Bµ = gL(cos ζ2 cos ζ3Z1 + cos ζ2 sin ζ3Z2 + sin ζ2A)
+ g′
[
(− cos ζ1 sin ζ2 cos ζ3 + sin ζ1 sin ζ3)Z1
+ (− sin ζ1 cos ζ3 − cos ζ1 sin ζ2 sin ζ3)Z2 + cos ζ1 cos ζ2A
]
=
[
gL cos ζ2 cos ζ3 + g
′(− cos ζ1 sin ζ2 cos ζ3 + sin ζ1 sin ζ3)
]
Z1
+
[
gL cos ζ2 sin ζ3 + g
′(− sin ζ1 cos ζ3 − cos ζ1 sin ζ2 sin ζ3)
]
Z2
+ (gL sin ζ2 + g
′ cos ζ1 cos ζ2)A
= (gL cos ζ2 cos ζ3 − g′ cos ζ1 sin ζ2 cos ζ3 + g′ sin ζ1 sin ζ3)Z1
+ (gL cos ζ2 sin ζ3 − g′ sin ζ1 cos ζ3 − g′ cos ζ1 sin ζ2 sin ζ3)Z2
+ 2gL sin ζ2A
=
gL
cos ζ2
[
(cos 2ζ2 cos ζ3 + tan ζ1 sin ζ2 sin ζ3)Z1
+ (− tan ζ1 sin ζ2 cos ζ3 + cos 2ζ2 sin ζ3)Z2
]
+ 2gL sin ζ2A. (A.67)
For the RH sector, we have
gRW
3
Rµ − g′Bµ = gR
[
(− sin ζ1 sin ζ2 cos ζ3 − cos ζ1 sin ζ3)Z1
80
+ (cos ζ1 cos ζ3 − sin ζ1 sin ζ2 sin ζ3)Z2 + sin ζ1 cos ζ2A
]
− g′[(− cos ζ1 sin ζ2 cos ζ3 + sin ζ1 sin ζ3)Z1
+ (− sin ζ1 cos ζ3 − cos ζ1 sin ζ2 sin ζ3)Z2 + cos ζ1 cos ζ2A
]
=
[
gR(− sin ζ1 sin ζ2 cos ζ3 − cos ζ1 sin ζ3)− g′(− cos ζ1 sin ζ2 cos ζ3 + sin ζ1 sin ζ3)
]
Z1
+
[
gR(cos ζ1 cos ζ3 − sin ζ1 sin ζ2 sin ζ3)− g′(− sin ζ1 cos ζ3 − cos ζ1 sin ζ2 sin ζ3)
]
Z2
+ (gR sin ζ1 cos ζ2 − g′ cos ζ1 cos ζ2)A
= (−gR sin ζ1 sin ζ2 cos ζ3 − gR cos ζ1 sin ζ3 + g′ cos ζ1 sin ζ2 cos ζ3 − g′ sin ζ1 sin ζ3)Z1
+ (gR cos ζ1 cos ζ3 − gR sin ζ1 sin ζ2 sin ζ3 + g′ sin ζ1 cos ζ3 + g′ cos ζ1 sin ζ2 sin ζ3)Z2
=
gR
cos ζ1
(− sin ζ3Z1 + cos ζ3Z2) (A.68)
and
gRW
3
Rµ + g
′Bµ = gR
[
(− sin ζ1 sin ζ2 cos ζ3 − cos ζ1 sin ζ3)Z1
+ (cos ζ1 cos ζ3 − sin ζ1 sin ζ2 sin ζ3)Z2 + (sin ζ1 cos ζ2)A
]
+ g′
[
(− cos ζ1 sin ζ2 cos ζ3 + sin ζ1 sin ζ3)Z1
+ (− sin ζ1 cos ζ3 − cos ζ1 sin ζ2 sin ζ3)Z2 + cos ζ1 cos ζ2A
]
=
[
gR(− sin ζ1 sin ζ2 cos ζ3 − cos ζ1 sin ζ3) + g′(− cos ζ1 sin ζ2 cos ζ3 + sin ζ1 sin ζ3)
]
Z1
+
[
gR(cos ζ1 cos ζ3 − sin ζ1 sin ζ2 sin ζ3) + g′(− sin ζ1 cos ζ3 − cos ζ1 sin ζ2 sin ζ3)
]
Z2
+ (gR sin ζ1 cos ζ2 + g
′ cos ζ1 cos ζ2)A
= (−gR sin ζ1 sin ζ2 cos ζ3 − gR cos ζ1 sin ζ3 − g′ cos ζ1 sin ζ2 cos ζ3 + g′ sin ζ1 sin ζ3)Z1
+ (gR cos ζ1 cos ζ3 − gR sin ζ1 sin ζ2 sin ζ3 − g′ sin ζ1 cos ζ3 − g′ cos ζ1 sin ζ2 sin ζ3)Z2
+ 2gR sin ζ1 cos ζ2A
= gR
[− 2 sin ζ1 sin ζ2 cos ζ3 − cos ζ1(1− tan2 ζ1) sin ζ3]Z1
+ gR
[
cos ζ1(1− tan2 ζ1) cos ζ3 − 2 sin ζ1 sin ζ2 sin ζ3
]
Z2
+ 2gR sin ζ1 cos ζ2A
=
gR
cos ζ1
[− (sin 2ζ1 sin ζ2 cos ζ3 + cos 2ζ1 sin ζ3)Z1
81
+ (cos 2ζ1 cos ζ3 − sin 2ζ1 sin ζ2 sin ζ3)Z2
]
+ 2gR sin ζ1 cos ζ2A. (A.69)
With the phenomenological assumption vL  κ1, κ2  vR, we can approximately
write
tan 2ζ3 ≈ −g
2
R
√
g2Lg
2
R + g
2
Lg
′2 + g2Rg′2(κ
2
1 + κ
2
2)
2(g2R + g
′2)2v2R
(A.70)
where 0 < −ζ3  1. The neutral gauge boson masses can be written as
m2Z1 ≈
g2Lg
2
R + g
2
Lg
′2 + g2Rg
′2
4(g2R + g
′2)
(κ21 + κ
2
2 + 4v
2
L) ≈
g2Lg
2
R + g
2
Lg
′2 + g2Rg
′2
4(g2R + g
′2)
(κ2 + κ′2),
(A.71)
m2Z2 ≈
g4R
4(g2R + g
′2)
(κ21 + κ
2
2) +
g′4
g2R + g
′2v
2
L + (g
2
R + g
′2)v2R ≈ (g2R + g′2)v2R. (A.72)
The first approximate approximate expressions are obtained by expanding the gauge
boson masses in terms of the small parameters (κ21 + κ
2
2)/v
2
R and v
2
L/v
2
R up to the
first order. From the second approximate expressions, we can identify the Weinberg
angle θW from its experimental definition
cos θW ≡ mW1
mZ1
≈ cos ζ2 (A.73)
and also the electric charge from
e = gL sin ζ2 = gR sin ζ1 cos ζ2
≈ gL sin θW = gR sin ζ1 cos θW (A.74)
where we have chosen e, gL, gR, θW > 0. Now we can rewrite
m2Z1 ≈
g2L(κ
2
1 + κ
2
2)
4 cos2 θW
, m2Z2 ≈
g2Rv
2
R
1− (g2L/g2R) tan2 θW
, (A.75)
82
and
ζ1 = sin
−1
(
gL
gR
tan θW
)
, ζ2 ≈ θW , ζ3 ≈ −gL
√
g2R − g2L tan2 θW (κ21 + κ22)
4 cos θWm2Z2
.
(A.76)
Since 0 < sin ζ1 ≤ 1, we must have
0 <
gL
gR
tan θW ≤ 1 (A.77)
where tan θW ≈ 0.548. In addition,
tan 2ζ3 ≈ − 2g
2
R√
g2Lg
2
R + g
2
Lg
′2 + g2Rg′2
m2Z1
m2Z2
= −2 cos θW
√
g2R/g
2
L − tan2 θW
m2Z1
m2Z2
.
(A.78)
Now we simply write ζ ≡ ζ3. Then, we have
gLW
3
Lµ − g′Bµ =
gL
cos ζ2
[
(cos ζ3 − tan ζ1 sin ζ2 sin ζ3)Z1 + (tan ζ1 sin ζ2 cos ζ3 + sin ζ3)Z2
]
≈ gL
cos θW
1− ζ gL sin2 θW√
g2R sin
2 θW − g2L cos2 θW
Z1 +
 gL sin2 θW√
g2R sin
2 θW − g2L cos2 θW
+ ζ
Z2

(A.79)
and
gLW
3
Lµ + g
′Bµ =
gL
cos ζ2
[
(cos 2ζ2 cos ζ3 + tan ζ1 sin ζ2 sin ζ3)Z1 + (− tan ζ1 sin ζ2 cos ζ3 + cos 2ζ2 sin ζ3)Z2
]
+ 2gL sin ζ2A
≈ gL
cos θW
cos 2θW + ζ gL sin2 θW√
g2R sin
2 θW − g2L cos2 θW
Z1
+
− gL sin2 θW√
g2R sin
2 θW − g2L cos2 θW
+ ζ cos 2θW
Z2
+ 2gL sin θWA.
(A.80)
For the RH sector,
gRW
3
Rµ − g′Bµ =
gR
cos ζ1
(− sin ζ3Z1 + cos ζ3Z2) ≈ g
2
R√
g2R − g2L tan2 θW
(−ζZ1 + Z2) (A.81)
83
and
gRW
3
Rµ + g
′Bµ =
gR
cos ζ1
[− (sin 2ζ1 sin ζ2 cos ζ3 + cos 2ζ1 sin ζ3)Z1 + (cos 2ζ1 cos ζ3 − sin 2ζ1 sin ζ2 sin ζ3)Z2]
+ 2gR sin ζ1 cos ζ2A
≈ gL√
g2R − g2L tan2 θW
[
−
(
2 sin θW tan θW
√
g2R/g
2
L − tan2 θW + ζ
[
g2R/g
2
L − 2 tan2 θW
])
Z1
+
(
g2R/g
2
L − 2 tan2 θW − 2ζ sin θW tan θW
√
g2R/g
2
L − tan2 θW
)
Z2
]
+ 2gL sin θWA
=
gL
cos θW
−
2 sin2 θW + ζ [g2R/g2L − 2 tan2 θW ] cos θW√
g2R/g
2
L − tan2 θW
Z1
+
[g2R/g2L − 2 tan2 θW ] cos θW√
g2R/g
2
L − tan2 θW
− 2ζ sin2 θW
Z2
+ 2gL sin θWA.
(A.82)
84
Appendix B: Expressions of observables
For the observables discussed here, the expressions presented in reference [12] are
mostly used. The exceptions are the form factors FZ1R and B
µeee
RR : for F
Z1
R , a mixed
expression from references [12] and [57] is used; for BµeeeRR , the suppression factor
(mWL/mWR)
2 is multiplied to the whole expression. The normalized Yukawa cou-
plings h˜L and h˜R are explicitly distinguished in this paper, since they are generally
different even with the manifest left-right symmetry.
Charged lepton flavour violation
The normalized Yukawa couplings h˜L, h˜R in the charged lepton mass basis are given
by [58]
h˜L ≡ 2
g
V `TL hLV
`
L =
2
g
V `TL
M∗Le
−iθL
√
2vL
V `L, (B.1)
h˜R ≡ 2
g
V `TR hRV
`
R =
2
g
V `TR
MR√
2vR
V `R = V
`T
R
MR
mWR
V `R. (B.2)
Note that h˜L 6= h˜R in general since V `L 6= V `R for nonzero α, although h ≡ hL = hR
with the parity symmetry. The loop functions of CLFV are given in appendix B.
85
`a → `bγ
For on-shell decay `a → `bγ, the branching ratio is given by
BR`a→`bγ =
α3W s
2
Wm
5
`a
256pi2m4WLΓ`a
(|GγL|2 + |GγR|2) (B.3)
where αW ≡ g2/(4pi), sW ≡ sin θW , and Γ`a is the decay rates of `a: Γµ = 2.996·10−19
GeV and Γτ = 2.267 · 10−12 GeV [59]. The form factors GγL, GγR are given by
GγL =
3∑
i=1
[
VµiV
∗
eiξ
2Gγ1(xi)− S∗µiV ∗eiξe−iαGγ2(xi)
mNi
m`a
+ VµiV
∗
ei
m2WL
m2WR
Gγ1(yi) + h˜Rµih˜
∗
Rei
2
3
m2WL
m2
δ++R
]
,
(B.4)
GγR =
3∑
i=1
[
S∗µiSeiG
γ
1(xi)− VµiSeiξeiαGγ2(xi)
mNi
m`a
+ h˜Lµih˜
∗
Lei
(
2
3
m2WL
m2
δ++L
+
1
12
m2WL
m2
H+1
)]
(B.5)
where xi = (mNi/mWL)
2 and yi = (mNi/mWR)
2. The initial and final charged
leptons have opposite chiralities, and L or R in GγL,R denotes the chirality of the
initial charged lepton. The Feynman diagrams of on-shell µ→ eγ are given in figure
B.1.
µ→ eee
The tree-level contribution to µ→ eee is
BRtreeµ→eee =
α4Wm
5
µ
24576pi3m4WLΓµ
(4pi)2
2α2W
(∣∣h˜Lµeh˜∗Lee∣∣2 m4WLm4
δ++L
+
∣∣h˜Rµeh˜∗Ree∣∣2 m4WLm4
δ++R
)
. (B.6)
The Feynman diagrams of the tree-level processes are given in figure B.2. The
86
one-loop type-I seesaw contribution is given by [60,61]
BRtype-Iµ→eee =
α4Wm
5
µ
24576pi3m4WLΓµ
[
2
{∣∣∣∣12BµeeeLL + FZ1L − 2s2W (FZ1L − F γL)
∣∣∣∣2 + ∣∣∣∣12BµeeeRR − 2s2W (FZ1R − F γR)
∣∣∣∣2}
+
∣∣∣∣2s2W (FZ1L − F γL)−BµeeeLR ∣∣∣∣2 + ∣∣∣∣2s2W (FZ1R − F γR)− (FZ1R +BµeeeRL )∣∣∣∣2
+ 8s2W
{
Re
[(
2FZ1L +B
µeee
LL +B
µeee
LR
)
Gγ∗R
]
+ Re
[(
FZ1R +B
µeee
RR +B
µeee
RL
)
Gγ∗L
]}
− 48s4W
{
Re
[(
FZ1L − F γL
)
Gγ∗R
]
+ Re
[(
FZ1R − F γR
)
Gγ∗L
]}
+ 32s4W
(|GγL|2 + |GγR|2){ ln(m2µm2e
)
− 11
4
}]
, (B.7)
and the interference terms are
BRtree+type-Iµ→eee =
α4Wm
5
µ
24576pi3m4WLΓµ
2(4pi)
αW
×[
m2WL
m2
δ++L
Re
[
h˜∗Lµeh˜Lee
{
2s2WF
γ
L + 4s
2
WG
γ
R +B
µeee
LL + F
Z1
L (1− 2s2W )
}]
+
m2WL
m2
δ++R
Re
[
h˜∗Rµeh˜Ree
{
2s2WF
γ
R + 4s
2
WG
γ
L +B
µeee
RR − 2s2WFZ1R
}]]
.
(B.8)
The form factors for the off-shell photon exchange are
F γL =
3∑
i=1
[
S∗µiSeiFγ(xi)− h˜Lµih˜∗Lei
(
2
3
m2WL
mδ++L
ln
m2µ
mδ++L
+
1
18
m2WL
mH+1
)]
, (B.9)
F γR =
3∑
i=1
[
VµiV
∗
ei
(
ξ2Fγ(xi) +
m2WL
m2WR
Fγ(yi)
)
− h˜Rµih˜∗Rei
2
3
m2WL
mδ++R
ln
m2µ
mδ++R
]
. (B.10)
87
For the Z1-exchange diagrams, the form factors are given by
FZ1L =
3∑
i,j=1
S∗µiSej
[
δij
{
FZ(xi) + 2GZ(0, xi)
}
+ (STS∗)ij
{
GZ(xi, xj)−GZ(0, xi)−GZ(0, xj)
}
+ (S†S)ijHZ(xi, xj)
]
,
(B.11)
FZ1R =
3∑
i=1
VµiV
∗
ei
[
8ζ3c
2
W√
1− 2s2W
{
FZ(yi) + 2GZ(0, yi)− yi
2
}
+ 2
(
κ1κ2
vEWvR
)2
DZ(yi, xi)
+
(
κ21 − κ22√
2vEWvR
)2
DZ(yi, zi)
]
(B.12)
where zi = (mNi/mH+2 )
2, cW ≡ cos θW , and ζ3 is the Z1-Z2 mixing parameter given
by equation 2.18. The Feynman diagrams that contribute to F γL,R and F
Z1
L,R are
presented in reference [58]. The form factors of the box diagrams are written as
BµeeeLL = −2
3∑
i=1
S∗µiSei
[
FXbox(0, xi)− FXbox(0, 0)
]
+
3∑
i,j=1
S∗µiSej
[
− 2S∗ejSei
{
FXbox(xi, xj)− FXbox(0, xj)− FXbox(0, xi) + FXbox(0, 0)
}
+ S∗eiSejGbox(xi, xj, 1)
]
, (B.13)
BµeeeRR = −2
m2WL
m2WR
3∑
i,j=1
VµiV
∗
ei
[
FXbox(0, yi)− FXbox(0, 0)
]
+
m2WL
m2WR
3∑
i,j=1
VµiV
∗
ej
[
− 2VejV ∗ei
{
FXbox(yi, yj)− FXbox(0, yj)− FXbox(0, yi) + FXbox(0, 0)
}
+ VeiV
∗
ejGbox(yi, yj, 1)
]
, (B.14)
BµeeeLR =
1
2
m2WL
m2WR
3∑
i,j=1
S∗µiSejVeiV
∗
ejGbox
(
xi, xj,
m2WL
m2WR
)
, (B.15)
BµeeeRL =
1
2
m2WL
m2WR
3∑
i,j=1
VµiV
∗
ejS
∗
eiSejGbox
(
xi, xj,
m2WL
m2WR
)
. (B.16)
88
Here, the masses of light neutrinos and the momenta of external fields are assumed
to be zero. The Feynman diagrams of the box diagrams are presented in figure B.3.
µ→ e
The µ→ e conversion rate is given by [58,61–63]
RA(N,Z)µ→e =
α3emα
4
Wm
5
µ
16pi2m4WLΓcapt
Z4eff
Z
∣∣Fp(−m2µ)∣∣2(∣∣QWL ∣∣2 + ∣∣QWR ∣∣2). (B.17)
Here, A, N , and Z are the mass, neutron, and atomic numbers of a nucleus, respec-
tively, and Zeff is the effective atomic number. The parameter Fp is the nuclear form
factor, Γcapt is the capture rate, and αem ≡ e2/(4pi). The values of Fp and Γcapt of
various nuclei are summarized in table B.1 [63]. The form factors in equation B.17
Nucleus AZN Zeff |Fp(−m2µ)| Γcapt (106 s−1)
27
13Al 11.5 0.64 0.7054
48
22Ti 17.6 0.54 2.59
197
79 Au 33.5 0.16 13.07
208
82 Pb 34.0 0.15 13.45
Table B.1: Form factors and capture rates of various nuclei associated with µ → e
conversion.
are given by
QWL,R = (2Z +N)
[
W uL,R −
2
3
s2WG
γ
R,L
]
+ (Z + 2N)
[
W dL,R +
1
3
s2WG
γ
R,L
]
(B.18)
89
and
W uL,R =
2
3
s2WF
γ
L,R +
(
− 1
4
+
2
3
s2W
)
FZ1L,R +
1
4
(
BµeuuLL,RR +B
µeuu
LR,RL
)
, (B.19)
W dL,R = −
1
3
s2WF
γ
L,R +
(
1
4
− 1
3
s2W
)
FZ1L,R +
1
4
(
BµeddLL,RR +B
µedd
LR,RL
)
. (B.20)
The box diagram form factors are
BµeuuLL =
3∑
i=1
S∗µiSei[Fbox(0, xi)− Fbox(0, 0)], (B.21)
BµeddLL =
3∑
i=1
S∗µiSei
[
FXbox(0, xi)− FXbox(0, 0)
+ |V qLtd|2{FXbox(xt, xi)− FXbox(0, xi)− FXbox(0, xt) + FXbox(0, 0)}
]
,
(B.22)
BµeuuRR =
3∑
i=1
VµiV
∗
ei[Fbox(0, xi)− Fbox(0, 0)], (B.23)
BµeddRR =
3∑
i=1
VµiV
∗
ei
[
FXbox(0, xi)− FXbox(0, 0)
+ |V qRtd|2{FXbox(xt, xi)− FXbox(0, xi)− FXbox(0, xt) + FXbox(0, 0)}
]
,
(B.24)
and BµeqqLR = B
µeqq
RL = 0 due to their chiral structures. Here, xt = m
2
t/m
2
WL
and
yt = m
2
t/m
2
WR
where mt is the mass of a top quark, and the masses of all the
other quarks as well as light neutrinos are assumed to be zero. The matrix V qL is
the Cabibbo-Kobayashi-Maskawa matrix, and V qR is its RH counterpart. Note that
V qL 6= V qR for nonzero α, although V qLtd = V qRtd is assumed for the numerical analysis
in this paper. The momenta of external fields are also assumed to be zero. The
Feynman diagrams of the box diagrams are given in figure B.4.
90
Loop functions
The loop functions of CLFV are
Fγ(x) =
7x3 − x2 − 12x
12(1− x)3 −
x4 − 10x3 + 12x2
6(1− x)4 lnx, (B.25)
Gγ1(x) = −
2x3 + 5x2 − x
4(1− x)3 −
3x3
2(1− x)4 lnx, (B.26)
Gγ2(x) =
x2 − 11x+ 4
2(1− x)2 −
3x2
(1− x)3 lnx, (B.27)
FZ(x) = − 5x
2(1− x) −
5x2
2(1− x)2 lnx, (B.28)
GZ(x, y) = − 1
2(1− x)
[
x2(1− y)
1− x lnx−
y2(1− x)
1− y ln y
]
, (B.29)
HZ(x, y) =
√
xy
4(x− y)
[
x(x− 4)
1− x lnx−
y(y − 4)
1− y ln y
]
, (B.30)
DZ(x, y) = x
(
2− ln y
x
)
+
x(−8 + 9x− x2)− x2(8− x) lnx
(1− x)2 +
xy(1− y + y ln y)
(1− y)2
+
2xy(4− x) lnx
(1− x)(1− y) +
2x(x− 4y) ln y
x
(1− y)(x− y) , (B.31)
Fbox(x, y) =
(
4 +
xy
4
)
I2(x, y, 1)− 2xyI1(x, y, 1), (B.32)
FXbox(x, y) = −
(
1 +
xy
4
)
I2(x, y, 1)− 2xyI1(x, y, 1), (B.33)
Gbox(x, y, η) = −√xy [(4 + xyη)I2(x, y, η)− (1 + η)I1(x, y, η)] (B.34)
91
where
I1(x, y, η) =
[
x lnx
(1− x)(1− ηx)(x− y) + (x↔ y)
]
− η ln η
(1− η)(1− ηx)(1− ηy) ,
(B.35)
I2(x, y, η) =
[
x2 lnx
(1− x)(1− ηx)(x− y) + (x↔ y)
]
− ln η
(1− η)(1− ηx)(1− ηy) ,
(B.36)
Ii(x, y, 1) ≡ lim
η→1
Ii(x, y, η). (B.37)
Neutrinoless double beta decay
The dimensionless parameter associated with the WL- and light neutrino exchange
is
ην =
∑3
i=1(Uei)
2mνi
me
. (B.38)
For the WL- and heavy neutrino exchange, we have
ηLNR = mp
3∑
i=1
(Sei)
2
mNi
(B.39)
where mp is the mass of a proton. For the WR- and heavy neutrino exchange, the
parameter is given by
ηRNR = mp
(
mWL
mWR
)4 3∑
i=1
(V ∗ei)
2
mNi
. (B.40)
For the δ++R -exchange, we have
ηδR =
∑3
i=1(Vei)
2mNi
m2
δ++R
m4WR
mp
G2F
. (B.41)
92
For the λ-diagram with final state electrons of different helicities, the parameter is
written as
ηλ =
(
mWL
mWR
)2 3∑
i=1
UeiT
∗
ei. (B.42)
For the η-diagram with WL-WR mixing,
ηη = −ξe−iα
3∑
i=1
UeiT
∗
ei. (B.43)
The Feynman diagrams corresponding to those parameters are given in figure B.5.
The phase space factors G0ν01 and matrix elements M0ν for various processes that
lead to 0νββ are summarized in table B.2 [12,64–71]. The inverse half-life is written
as
[T 0ν1/2]
−1 = G0ν01
(|M0νν |2|ην |2 + |M0νN |2|ηLNR |2 + |M0νN |2|ηRNR + ηδR |2 + |M0νλ |2|ηλ|2 + |M0νη |2|ηη|2)
+ interference terms. (B.44)
Isotope G0ν01 (10
−14 yrs.−1) M0νν M0νN M0νλ M0νη
76Ge 0.686 2.58− 6.64 233− 412 1.75− 3.76 235− 637
82Se 2.95 2.42− 5.92 226− 408 2.54− 3.69 209− 234
130Te 4.13 2.43− 5.04 234− 385 2.85− 3.67 414− 540
136Xe 4.24 1.57− 3.85 164− 172 1.96− 2.49 370− 419
Table B.2: Phase space factors and matrix elements associated with 0νββ.
93
Electric dipole moments of charged leptons
The EDM of the charged lepton `α (α = e, µ, τ) is given by [11,72]
dα =
eαW
8pim2WL
Im
[ 3∑
i=1
SαiVαiξe
iαGγ2(xi)mNi
]
. (B.45)
The Feynman diagrams that generate the EDM of an electron are given in figure
B.6.
Benchmark model parameters and their predictions
The benchmark model parameters and their predictions are summarized in tables
B.3 and B.4. These parameters are chosen to obtain BRµ→eγ, BRµ→eee, Rµ→e, and
T 0ν1/2 large enough to be observable in near-future experiments.
The Yukawa coupling matrices f , f˜ in the symmetry basis calculated from
these parameters are
f =

−0.117629 −0.0954074− 0.303042i −0.287722− 0.316317i
−0.0954074 + 0.303042i 0.858098 −0.581546− 0.997804i
−0.287722 + 0.316317i −0.581546 + 0.997804i 1.55438
 · 10
−6,
(B.46)
f˜ =

9.02581 0.362808− 3.15221i −0.217594 + 0.423914i
0.362808 + 3.15221i 1.53907 3.98014 · 10−4 − 0.328771i
−0.217594− 0.423914i 3.98014 · 10−4 + 0.328771i 0.260124
 · 10
−3.
(B.47)
94
Parameter Value Parameter Value
log10 (mν3/eV) −10.2 log10 (κ2/GeV) −1.12
mWR 3.60 TeV α 0.7843093682120977pi rad
δD −0.700pi rad log10 (|A11|/GeV) −8.20
δM1 −0.0640pi rad A11/|A11| 1
δM2 0.850pi rad A22/|A22| −1
θL12 0.287pi rad A33/|A33| −1
θL13 0.387pi rad θA12 −0.5970870460412485pi rad
θL23 0.546pi rad θA13 0.26505775139215687pi rad
δL1 −0.488pi rad θA23 −0.6679707059438431pi rad
δL2 −0.953pi rad log10 α3 0.520
δL3 −0.769pi rad log10 (ρ3 − 2ρ1) 0.328
δL4 −5.30 · 10−5pi rad log10 ρ2 0.450
Table B.3: Benchmark parameters for large CLFV and 0νββ. The predictions from
these parameters are given in table B.4.
95
Parameter Value
mWR 3.60 TeV
mν1 0.0631 eV
mν2 0.0637 eV
mν3 0.0807 eV
mN1 0.139 TeV
mN2 0.280 TeV
mN3 4.13 TeV
mH+1
8.08 TeV
mH+2
10.1 TeV
mδ++L
8.09 TeV
mδ++R
18.6 TeV
κ1 246 GeV
κ2e
iα 0.0759ei0.784pi GeV
α3 3.31
ρ3 − 2ρ1 2.13
ρ2 2.82
The charged lepton and Dirac neutrino mass matrices in the symmetry basis are
M` =
1√
2
(fκ2e
iα + f˜κ1)
=

1.57002− 3.95569 · 10−9i 0.0631099− 0.548321i −0.0378502 + 0.0737391i
0.0631098 + 0.548321i 0.267718 + 2.88565 · 10−8i 6.92918 · 10−5 − 0.0571891i
−0.0378501− 0.0737391i 6.92247 · 10−5 + 0.0571891i 0.0452481 + 5.22714 · 10−8i
 GeV,
(B.48)
96
MD =
1√
2
(fκ1 + f˜κ2e
−iα)
=

−3.97641− 3.03524i −1.37761 + 0.668135i 0.733973 + 0.446252i
0.742466− 0.912148i 0.849485− 0.517565i −1.12232− 1.59841i
0.448861− 0.299905i −0.901194 + 1.59814i 2.59511− 0.0874759i
 · 10
−4 GeV.
(B.49)
The mixing matrices that diagonalize M` are
V `L =

0.215620 + 3.59016 · 10−5i 0.272630 0.0353401 + 0.936980i
−0.174794− 0.555520i 0.00850025− 0.736518i −0.340224 + 0.0506041i
−0.527503 + 0.579736i 0.526439− 0.325580i 0.0374439− 0.0332209i
 ,
(B.50)
V `R =

0.215620 0.272630 0.0353401 + 0.936980i
−0.174886− 0.555491i 0.00850025− 0.736518i −0.340224 + 0.0506041i
−0.527407 + 0.579824i 0.526439− 0.325580i 0.0374439− 0.0332209i
 .
(B.51)
The neutrino mass matrices in the charged lepton mass basis are written as
M cν = UPMNSM
diag
ν U
T
PMNS
=

6.14141 + 0.604007i −0.641188 + 1.37500i −0.414134− 0.161926i
−0.641188 + 1.37500i 5.21993 + 3.90978i −0.721679 + 2.37952i
−0.414134− 0.161926i −0.721679 + 2.37952i 5.35910 + 4.32684i
 · 10
−11 GeV,
(B.52)
97
M cD = V
`†
L MDV
`
R
=

−0.887458− 0.00113569i −0.596983− 1.80367i −0.364728− 0.967911i
−0.596682 + 1.80377i 2.44772− 0.204264i 0.650485− 0.676299i
−0.364567 + 0.967972i 0.650486 + 0.676299i −3.86700− 3.43503i
 · 10
−4 GeV,
(B.53)
M cR = −M cTD (M cν)−1M cD
=

327.179− 124.513i −141.421− 201.931i 36.0396 + 816.162i
−141.421− 201.931i 56.2978 + 60.4971i 517.744− 74.6682i
36.0396 + 816.162i 517.744− 74.6682i −2486.91− 2973.37i
 GeV.
(B.54)
The neutrino mixing matrices are given by
U = UPMNS =

0.824240 0.535780 + 0.109200i 0.131084− 0.0667906i
−0.365548 + 0.0658493i 0.632967 + 0.173591i −0.585126− 0.298136i
0.420911 + 0.0741679i −0.516908 + 0.0551401i −0.659043− 0.335799i
 ,
(B.55)
S =

−0.492113− 0.340868i 0.999284 + 0.0561499i 0.239615 + 0.0281506i
−0.0475962 + 0.503081i −0.231028− 1.26661i −0.00795814− 0.320325i
0.232020− 0.00648341i −0.401571 + 0.125068i −0.175188 + 0.136668i
 · 10
−6,
(B.56)
T =

−6.53107− 6.47350i −8.46370 + 5.72968i −1.16360− 8.20634i
2.04202− 6.05309i −4.69170− 5.30774i 3.49735− 2.06263i
−1.83711− 0.641098i 0.0608069 + 1.46932i 0.103607− 0.502026i
 · 10
−7,
(B.57)
98
V =

−0.183724 + 0.375972i 0.879386 + 0.0740900i 0.195953− 0.0876577i
−0.881006 + 0.210057i −0.242230− 0.320460i −0.0720947− 0.114618i
−0.0677616 + 0.00212502i 0.177470− 0.168300i −0.408123 + 0.876937i
 .
(B.58)
The Yukawa coupling matrix h in the symmetry basis is
h =
1√
2vR
V `∗R M
c
RV
`†
R
=

0.206578 + 0.223735i 0.120506− 0.0241230i −0.0469350− 0.0641918i
0.120506− 0.0241230i 0.00351664− 0.0376782i −0.0257606 + 0.00173595i
−0.046935− 0.0641918i −0.0257606 + 0.00173595i −0.00335158 + 0.0385022i
 ,
(B.59)
and the normalized Yukawa couplings h˜L, h˜R in the charged lepton mass basis are
h˜L =
2
g
V `TL hV
`
L
=

0.0908945− 0.0345568i −0.0392741− 0.0560986i 0.00997325 + 0.226713i
−0.0392741− 0.0560986i 0.0156383 + 0.0168047i 0.143818− 0.0207412i
0.00997325 + 0.226713i 0.143818− 0.0207412i −0.690808− 0.825936i
 ,
(B.60)
h˜R =
2
g
V `TR hV
`
R
=

0.0908830− 0.0345871i −0.0392835− 0.0560921i 0.0100110 + 0.226712i
−0.0392835− 0.0560921i 0.0156383 + 0.0168047i 0.143818− 0.0207412i
0.0100110 + 0.226712i 0.143818− 0.0207412i −0.690808− 0.825936i
 .
(B.61)
Note that h˜L ≈ h˜R since we are considering the cases of V `L ≈ V `R for the TeV-scale
phenomenology.
99
µ−R NRi
W+1
µ−L e
−
R
mµ −Vµiξeiα −V ∗eiξe−iα
γ
(a) GγL
N cRi NRi
W+1
µ−L e
−
R
S∗µi mNi −V ∗eiξe−iα
γ
(b) GγL
µ−R NRi
W+2
µ−L e
−
R
mµ Vµi V ∗ei
γ
(c) GγL
µ−R `+Ri
δ++R
µ−L e
−
R
mµ h˜Rµi h˜
∗
Rei
γ
(d) GγL
µ−R δ++R
`+Ri
µ−L e
−
R
mµ h˜Rµi h˜
∗
Rei
γ
(e) GγL
µ−L N cRi
W+1
µ−R e
−
L
mµ S
∗
µi Sei
γ
(f) GγR
NRi N
c
Ri
W+1
µ−R e
−
L
−Vµiξeiα mNi Sei
γ
(g) GγR
µ−L `+Li
δ++L
µ−R e
−
L
mµ h˜Lµi h˜
∗
Lei
γ
(h) GγR
µ−L δ++L
`+Li
µ−R e
−
L
mµ h˜Lµi h˜
∗
Lei
γ
(i) GγR
µ−L νcLi
H+1
µ−R e
−
L
mµ
1√
2
h˜Lµi
1√
2
h˜∗Lei
γ
(j) GγR
Figure B.1: Feynman diagrams of on-shell µ→ eγ. Here, W+L ≈ W+1 +ξe−iαW+2 and
W+R ≈ −ξeiαW+1 +W+2 . Figures B.1a−B.1e contribute to GγL, and figures B.1f−B.1j
to GγR. The arrows in neutrino propagators denote the directions of the propagation
of Ni = NRi +N
c
Ri.
100
δ++L
µ−L
e−L
e−L
h˜Lµe
h˜∗Lee
e+L
(a)
δ++R
µ−R
e−R
e−R
h˜Rµe
h˜∗Ree
e+R
(b)
Figure B.2: Feynman diagrams of the tree-level processes of µ→ eee.
νLj , N
c
Rj
νLi, N
c
Ri
e−L
µ−L
e−L
e−L
U∗ej , S
∗
ej Uej , Sej
U∗µi, S
∗
µi Uei, Sei
W+1W
+
1
(a) BµeeeLL
N cRi
W+1
N cRi
W+1
N cRj
N cRj
e−L
µ−L
e−L
e−L
S∗ei Sej
S∗µi Sej
mNi mNj
(b) BµeeeLL
νcLj , NRj
νcLi, NRi
e−R
µ−R
e−R
e−R
Tej , Vej T
∗
ej , V
∗
ej
Tµi, Vµi T ∗ei, V
∗
ei
W+2W
+
2
(c) BµeeeRR
NRi
W+2
NRi
W+2
NRj
NRj
e−R
µ−R
e−R
e−R
Vei V
∗
ej
Vµi V
∗
ej
mNi mNj
(d) BµeeeRR
NRi
W+2
N cRi
W+1
NRj
N cRj
e−R
µ−L
e−R
e−L
Vei V
∗
ej
S∗µi Sej
(e) BµeeeLR
N cRi
W+1
NRi
W+2
N cRj
NRj
e−L
µ−R
e−L
e−R
S∗ei Sej
Vµi V
∗
ej
(f) BµeeeRL
Figure B.3: Feynman diagrams of Bµeee. Note that the arrows in neutrino propaga-
tors indicate the directions of the propagation of νi = νLi + ν
c
Li or Ni = NRi +N
c
Ri.
101
dLj
W+1
νLi, N
c
Ri
W+1
uL
µ−L
uL
e−L
V q∗Ludj V
q
Ludj
U∗µi, S
∗
µi Uei, Sei
(a) BµeuuLL
uLj
νLi, N
c
Ri
dL
µ−L
dL
e−L
V qLujd V
q∗
Lujd
U∗µi, S
∗
µi Uei, Sei
W+1W
+
1
(b) BµeddLL
dRj
W+2
νcLi, NRi
W+2
uR
µ−R
uR
e−R
V q∗Rudj V
q
Rudj
Tµi, Vµi T ∗ei, V
∗
ei
(c) BµeuuRR
uRj
νcLi, NRi
dR
µ−R
dR
e−R
V qRujd V
q∗
Rujd
Tµi, Vµi T ∗ei, V
∗
ei
W+2W
+
2
(d) BµeddRR
Figure B.4: Feynman diagrams of Bµeqq.
102
W+1
W+1
νLi
νLi
dL
dL
uL
e−L
e−L
uL
Uei
mνi
Uei
(a) ην
W+1
W+1
N cRi
N cRi
dL
dL
uL
e−L
e−L
uL
Sei
mNi
Sei
(b) ηLNR
W+2
W+2
NRi
NRi
dR
dR
uR
e−R
e−R
uR
V ∗ei
mNi
V ∗ei
(c) ηRNR
δ++R
dR
dR
uR
e−R
e−R
uR
W+2
√
2g2vR
hcRee
W+2
(d) ηδR
W+2
W+1
νcLi
νLi
dR
dL
uR
e−R
e−L
uL
T ∗ei
q
Uei
(e) ηλ
W+1
W+1
νcLi
νLi
dL
dL
uL
e−R
e−L
uL
−T ∗eiξe−iα
q
Uei
(f) ηη
Figure B.5: Feynman diagrams of 0νββ. Here, W+L ≈ W+1 + ξe−iαW+2 and W+R ≈
−ξeiαW+1 +W+2 . The coupling hcR ≡ V `TR hV `R = M cR/(
√
2vR) is the Yukawa coupling
matrix in the charged lepton mass basis. The typical momentum transfer of the
processes is q ≈ 100 MeV.
N cRi NRi
W+1
e−L e
−
R
S∗ei mNi −V ∗eiξe−iα
γ
(a)
NRi N
c
Ri
W+1
e−R e
−
L
−Veiξeiα mNi Sei
γ
(b)
Figure B.6: Feynman diagrams contributing to the EDM of e.
103
Prediction Near-future sensitivity
BRµ→eγ 5.98 · 10−14 < 5.0 · 10−14 (Upgraded MEG)
BRτ→µγ 1.94 · 10−13 ·
BRτ→eγ 4.85 · 10−13 ·
BRµ→eee 8.12 · 10−14 < 1.0 · 10−15 (PSI) [24]
RAlµ→e 2.17 · 10−13 < 3.0 · 10−17 (COMET)
RTiµ→e 4.13 · 10−13 < 1.0 · 10−18 (PRISM/PRIME)
RAuµ→e 3.98 · 10−13 ·
RPbµ→e 3.83 · 10−13 ·
|ην | 1.21 · 10−7 . 1.4 · 10−7 (CUORE)
|ηLNR | 4.97 · 10−15 ·
|ηRNR | 4.77 · 10−10 ·
|ηδR | 4.24 · 10−11 ·
|ηλ| 4.61 · 10−10 ·
|ηη| 2.81 · 10−13 ·
T 0ν1/2
∣∣
Ge
2.12 · 1026 − 1.31 · 1027 yrs. ·
T 0ν1/2
∣∣
Se
6.11 · 1025 − 3.43 · 1026 yrs. ·
T 0ν1/2
∣∣
Te
5.91 · 1025 − 2.41 · 1026 yrs. > 2.1 · 1026 yrs. (CUORE)
T 0ν1/2
∣∣
Xe
1.05 · 1026 − 5.48 · 1026 yrs. ·
|de| |−2.98 · 10−31| e·cm ·
|dµ| |1.99 · 10−31| e·cm ·
|dτ | |−3.13 · 10−31| e·cm ·
Table B.4: Predictions from the benchmark model parameters of table B.3. Only
near-future experiments that would detect the corresponding processes are presented
here.
104
Appendix C: Parametrization of the Dirac neutrino mass matrix
In this section, we show that the Casas-Ibarra parametrization [47] of the Dirac
neutrino mass matrix is the most general form of MD for given heavy neutrino
masses.
Standard Model with right-handed Majorana neutrinos
For a diagonal matrix D with positive entries, i.e.
D =

d1 0 0
0 d2 0
0 0 d3
 (C.1)
with di > 0, we define
√
D ≡

√
d1 0 0
0
√
d2 0
0 0
√
d3
 . (C.2)
We write C which satisfies CTC = CCT = D as C =
√
DB where B ≡ √D−1C.
Then, BBT = (
√
D−1C)(
√
D−1C)T =
√
D−1CCT
√
D−1 = I, i.e. B is an orthog-
onal matrix. In other words, any matrix C which satisfies CTC = CCT = D is
orthogonally equivalent to
√
D−1.
105
Now we go to the basis in the flavour space where the light and heavy neutrino
mass matrices are diagonal with positive entries. In that basis, we denote the charged
lepton mass matrix as M`, the Dirac neutrino mass matrix as MD, the right-handed
Majorana neutrino mass matrix as MdR, and light neutrino mass matrix as M
d
ν . We
assume that the neutrino mass matrices are invertible, which is trivially satisfied as
long as the lightest neutrino mass is nonzero. Then, for a matrix CR which satisfy
CRC
T
R = M
d
R, we can write CR =
√
MdROR for an orthogonal matrix OR. The
neutrino mass matrices satisfy the type-I seesaw formula, and thus
Mdν = −MD(MdR)−1MTD = −MD
(√
(MdR)
−1OR
)(√
(MdR)
−1OR
)T
MD
=
[
iMD
√
(MdR)
−1OR
] [
iMD
√
(MdR)
−1OR
]T
. (C.3)
We can therefore write
iMD
√
(MdR)
−1OR =
√
MdνOν (C.4)
for an orthogonal matrix Oν , and
MD = −i
√
MdνO
√
MdR (C.5)
where O ≡ OνOTR is also an orthogonal matrix.
In the charged lepton mass basis, we have
M c` = UM`V
`
R, M
c
ν = UM
d
νU
T, M cD = UMD, M
c
R = M
d
R (C.6)
where U and V `R are the unitary matrices which transform M` into the diagonal
matrix M c` with charged lepton masses as its entries. Note that U ≡ UPMNS is the
106
PMNS matrix. We can write
M cD = −iU
√
MdνO
√
MdR. (C.7)
Without loss of generality, the complex orthogonal matrix O can be parametrized
as O = eS where S is a skew-symmetric matrix, i.e. ST = −S, as the exponential
map is surjective.
Left-right symmetric model
We follow the same steps up to the proof of the generality of equations ?? and ??.
In the charged lepton mass basis, we have
M c` = UM`V
`
R, M
c
ν = UM
d
νU
T, M cD = UMDV
`
R, M
c
R = V
`T
R M
d
RV
`
R.
(C.8)
Hence,
M cD = −iU
√
MdνO
√
MdRV
`
R. (C.9)
107
Appendix D: Boltzmann equation
In this section, we explicitly derive the Boltzmann equations for the RH neutrino
density and LH lepton doublet asymmetry. Here, we consider the extension of the
SM only with three RH neutrinos for simplicity. Note that the relations of collision
terms and the correct forms of Boltzmann equations in any other models should be
carefully derived in a similar way.
The generic form of the Boltzmann equation is
dna
dt
+ 3Hna = −
∑
aX↔Y
[
nanX
neqa n
eq
X
γ(aX → Y )− nY
neqY
γ(Y → aX)
]
. (D.1)
Since φ is a massless scalar field, we have nφ = n
eq
φ . In addition, n
eq
Li
= neqLci =
neq`i where n
eq
`i
≡ neq`Li + neq`Ri is the total lepton number density of each flavour in
equilibrium. The CP-conserving decay term is defined by
γNαLiφ ≡ γ(Nα → Liφ) + γ(Nα → Lciφ†), (D.2)
and the CP-violating decay term by
δγNαLiφ ≡ γ(Nα → Liφ)− γ(Nα → Lciφ†). (D.3)
By CPT invariance, we have
γ(Liφ→ Nα) = γ(Nα → Lciφ†) =
1
2
(γNαLiφ − δγNαLiφ), (D.4)
γ(Lciφ
† → Nα) = γ(Nα → Liφ) = 1
2
(γNαLiφ + δγ
Nα
Liφ
). (D.5)
108
The Boltzmann equation for the RH neutrino density is written as
dnNα
dt
+ 3HnNα = −
3∑
j=1
[
nNα
neqNα
{
γ(Nα → Ljφ) + γ(Nα → Lcjφ†)
}
− nLjnφ
neqLjn
eq
φ
γ(Ljφ→ Nα)−
nLcjnφ
neqLcjn
eq
φ
γ(Lcjφ
† → Nα)
]
= −
3∑
j=1
[
nNα
neqNα
γNαLjφ −
nLj
2neq`j
(γNαLjφ − δγNαLjφ)−
nLcj
2neq`j
(γNαLjφ + δγ
Nα
Ljφ
)
]
= −
3∑
j=1
[
nNα
neqNα
γNαLjφ −
nLj + nLcj
2neq`j
γNαLjφ +
nLj − nLcj
2neq`j
δγNαLjφ
]
≈ −
(
nNα
neqNα
− 1
)
γNαLφ −
3∑
j=1
n∆Lj
2neq`j
δγNαLjφ. (D.6)
In addition, the RIS-subtracted CP-conserving scattering terms are defined by
γ′Liφ
Lcjφ
† ≡ γ′(Liφ→ Lcjφ†) + γ′(Lciφ† → Ljφ), (D.7)
γ′LiφLjφ ≡ γ′(Liφ→ Ljφ) + γ′(Lciφ† → Lcjφ†). (D.8)
The corresponding CP-violating terms can be written as [49]
γ′(Liφ→ Lcjφ†)− γ′(Lciφ† → Ljφ) =
1
2
3∑
α=1
(BiNαδ
j
Nα
+BjNαδ
i
Nα)γ
Nα
Lφ , (D.9)
γ′(Liφ→ Ljφ)− γ′(Lciφ† → Lcjφ†) = −
1
2
3∑
α=1
(BiNαδ
j
Nα
−BjNαδiNα)γNαLφ (D.10)
where
δiNα =
Γ(Nα → Liφ)− Γ(Nα → Lciφ†)∑3
j=1
[
Γ(Nα → Ljφ) + Γ(Nα → Lcjφ†)
] , (D.11)
BiNα =
Γ(Nα → Liφ) + Γ(Nα → Lciφ†)∑3
j=1
[
Γ(Nα → Ljφ) + Γ(Nα → Lcjφ†)
] . (D.12)
109
We therefore have
γ′(Liφ→ Lcjφ†) =
1
2
γ′Liφ
Lcjφ
† +
1
4
3∑
α=1
(BiNαδ
j
Nα
+BjNαδ
i
Nα)γ
Nα
Lφ , (D.13)
γ′(Lciφ
† → Ljφ) = 1
2
γ′Liφ
Lcjφ
† − 1
4
3∑
α=1
(BiNαδ
j
Nα
+BjNαδ
i
Nα)γ
Nα
Lφ , (D.14)
γ′(Liφ→ Ljφ) = 1
2
γ′LiφLjφ −
1
4
3∑
α=1
(BiNαδ
j
Nα
−BjNαδiNα)γNαLφ , (D.15)
γ′(Lciφ
† → Lcjφ†) =
1
2
γ′LiφLjφ +
1
4
3∑
α=1
(BiNαδ
j
Nα
−BjNαδiNα)γNαLφ . (D.16)
The Boltzmann equations for the LH lepton doublet number density are written as
dnLi
dt
+ 3HnLi = −
3∑
α=1
nLinφ
neqLin
eq
φ
γ(Liφ→ Nα)−
3∑
j=1
nLinφ
neqLin
eq
φ
γ′(Liφ→ Lcjφ†)
−
3∑
j=1
nLinφ
neqLin
eq
φ
γ′(Liφ→ Ljφ) +
3∑
α=1
nNα
neqNα
γ(Nα → Liφ)
+
3∑
j=1
nLcjnφ
neqLcj
neqφ
γ′(Lcjφ
† → Liφ) +
3∑
j=1
nLjnφ
neqLjn
eq
φ
γ′(Ljφ→ Liφ) + · · · ,
(D.17)
dnLci
dt
+ 3HnLci = −
3∑
α=1
nLcinφ
neqLci
neqφ
γ(Lciφ
† → Nα)−
3∑
j=1
nLcinφ
neqLci
neqφ
γ′(Lciφ
† → Ljφ)
−
3∑
j=1
nLcinφ
neqLci
neqφ
γ′(Lciφ
† → Lcjφ†) +
3∑
α=1
nNα
neqNα
γ(Nα → Lciφ)
+
3∑
j=1
nLjnφ
neqLjn
eq
φ
γ′(Ljφ→ Lciφ†) +
3∑
j=1
nLcjnφ
neqLcj
neqφ
γ′(Lcjφ
† → Lciφ†) + · · ·
(D.18)
where we have explicitly written only the terms that would contribute to δγNαLiφ. We
can thus write
dn∆Li
dt
+ 3Hn∆Li = −
3∑
α=1
nLinφ
neqLin
eq
φ
γ(Liφ→ Nα)−
3∑
j=1
nLinφ
neqLin
eq
φ
γ′(Liφ→ Lcjφ†)
−
3∑
j=1
nLinφ
neqLin
eq
φ
γ′(Liφ→ Ljφ) +
3∑
α=1
nNα
neqNα
γ(Nα → Liφ)
+
3∑
j=1
nLcjnφ
neqLcj
neqφ
γ′(Lcjφ
† → Liφ) +
3∑
j=1
nLjnφ
neqLjn
eq
φ
γ′(Ljφ→ Liφ),
110
+3∑
α=1
nLcinφ
neqLci
neqφ
γ(Lciφ
† → Nα) +
3∑
j=1
nLcinφ
neqLci
neqφ
γ′(Lciφ
† → Ljφ)
+
3∑
j=1
nLcinφ
neqLci
neqφ
γ′(Lciφ
† → Lcjφ†)−
3∑
α=1
nNα
neqNα
γ(Nα → Lciφ)
−
3∑
j=1
nLjnφ
neqLjn
eq
φ
γ′(Ljφ→ Lciφ†)−
3∑
j=1
nLcjnφ
neqLcj
neqφ
γ′(Lcjφ
† → Lciφ†) + · · ·
= −
3∑
α=1
nLi
2neq`i
(γNαLiφ − δγNαLiφ)−
3∑
j=1
nLi
2neq`i
[
γ′Liφ
Lcjφ
† +
1
2
3∑
α=1
(BiNαδ
j
Nα
+BjNαδ
i
Nα)γ
Nα
Lφ
]
−
3∑
j=1
nLi
2neq`i
[
γ′LiφLjφ −
1
2
3∑
α=1
(BiNαδ
j
Nα
−BjNαδiNα)γNαLφ
]
+
3∑
α=1
nNα
2neqNα
(γNαLiφ + δγ
Nα
Liφ
) +
3∑
j=1
nLcj
2neq`j
[
γ′Liφ
Lcjφ
† − 1
2
3∑
α=1
(BiNαδ
j
Nα
+BjNαδ
i
Nα)γ
Nα
Lφ
]
+
3∑
j=1
nLj
2neq`j
[
γ′LiφLjφ +
1
2
3∑
α=1
(BiNαδ
j
Nα
−BjNαδiNα)γNαLφ
]
,
+
3∑
α=1
nLci
2neq`i
(γNαLiφ + δγ
Nα
Liφ
) +
3∑
j=1
nLci
2neq`i
[
γ′Liφ
Lcjφ
† − 1
2
3∑
α=1
(BiNαδ
j
Nα
+BjNαδ
i
Nα)γ
Nα
Lφ
]
+
3∑
j=1
nLci
2neq`i
[
γ′LiφLjφ +
1
2
3∑
α=1
(BiNαδ
j
Nα
−BjNαδiNα)γNαLφ
]
−
3∑
α=1
nNα
2neqNα
(γNαLiφ − δγNαLiφ)−
3∑
j=1
nLj
2neq`j
[
γ′Liφ
Lcjφ
† +
1
2
3∑
α=1
(BiNαδ
j
Nα
+BjNαδ
i
Nα)γ
Nα
Lφ
]
−
3∑
j=1
nLcj
2neq`j
[
γ′LiφLjφ −
1
2
3∑
α=1
(BiNαδ
j
Nα
−BjNαδiNα)γNαLφ
]
+ · · ·
= −
3∑
α=1
nLi − nLci
2neq`i
δγNαLiφ +
3∑
α=1
nLi + nLci
2neq`i
δγNαLiφ +
3∑
α=1
nNα
neqNα
δγNαLiφ
−
3∑
j=1
nLi − nLci
2neq`i
γ′Liφ
Lcjφ
† −
3∑
j=1
nLi + nLci
4neq`i
3∑
α=1
(BiNαδ
j
Nα
+BjNαδ
i
Nα)γ
Nα
Lφ
−
3∑
j=1
nLi − nLci
2neq`i
γ′LiφLjφ +
3∑
j=1
nLi + nLci
4neq`i
3∑
α=1
(BiNαδ
j
Nα
−BjNαδiNα)γNαLφ
−
3∑
j=1
nLj − nLcj
2neq`j
γ′Liφ
Lcjφ
† −
3∑
j=1
nLj + nLcj
4neq`j
3∑
α=1
(BiNαδ
j
Nα
+BjNαδ
i
Nα)γ
Nα
Lφ
+
3∑
j=1
nLj − nLcj
2neq`j
γ′LiφLjφ +
3∑
j=1
nLj + nLcj
4neq`j
3∑
α=1
(BiNαδ
j
Nα
−BjNαδiNα)γNαLφ + · · ·
≈ −
3∑
α=1
n∆Li
2neq`i
δγNαLiφ +
3∑
α=1
δγNαLiφ +
3∑
α=1
nNα
neqNα
δγNαLiφ
−
3∑
j=1
n∆Li
2neq`i
γ′Liφ
Lcjφ
† − 1
2
3∑
j=1
3∑
α=1
(BiNαδ
j
Nα
+BjNαδ
i
Nα)γ
Nα
Lφ
111
−
3∑
j=1
n∆Li
2neq`i
γ′LiφLjφ +
1
2
3∑
j=1
3∑
α=1
(BiNαδ
j
Nα
−BjNαδiNα)γNαLφ
−
3∑
j=1
n∆Lj
2neq`j
γ′Liφ
Lcjφ
† − 1
2
3∑
j=1
3∑
α=1
(BiNαδ
j
Nα
+BjNαδ
i
Nα)γ
Nα
Lφ
+
3∑
j=1
n∆Lj
2neq`j
γ′LiφLjφ +
1
2
3∑
j=1
3∑
α=1
(BiNαδ
j
Nα
−BjNαδiNα)γNαLφ + · · ·
= −
3∑
α=1
n∆Li
2neq`i
δγNαLiφ +
3∑
α=1
δγNαLiφ +
3∑
α=1
nNα
neqNα
δγNαLiφ
−
3∑
j=1
n∆Li
2neq`i
γ′Liφ
Lcjφ
† − 1
2
3∑
α=1
(BiNαδγ
Nα
Lφ + δγ
Nα
Liφ
)
−
3∑
j=1
n∆Li
2neq`i
γ′LiφLjφ +
1
2
3∑
α=1
(BiNαδγ
Nα
Lφ − δγNαLiφ)
−
3∑
j=1
n∆Lj
2neq`j
γ′Liφ
Lcjφ
† − 1
2
3∑
α=1
(BiNαδγ
Nα
Lφ + δγ
Nα
Liφ
)
+
3∑
j=1
n∆Lj
2neq`j
γ′LiφLjφ +
1
2
3∑
α=1
(BiNαδγ
Nα
Lφ − δγNαLφ ) + · · ·
=
3∑
α=1
(
nNα
neqNα
− 1
)
δγNαLiφ −
3∑
α=1
n∆Li
2neq`i
δγNαLiφ
−
3∑
j=1
n∆Li
2neq`i
(γ′Liφ
Lcjφ
† + γ
′Liφ
Ljφ
)−
3∑
j=1
n∆Lj
2neq`j
(γ′Liφ
Lcjφ
† − γ′LiφLjφ ) + · · · . (D.19)
Now we simplify the left-hand side of the Boltzmann equation D.1. Since
T ∝ a where a is the scale factor in the Friedmann-Robertson-Walker metric, we
have
1
T
dT
dt
=
a˙
a
= −H, (D.20)
and thus
dz
dt
= − z
T
dT
dt
= zH. (D.21)
Hence,
dnX
dt
=
dz
dt
dnX
dz
= zH
dnX
dz
, (D.22)
112
and for ηX ≡ nX/nγ, we have
dηX
dz
=
1
nγ
dnX
dz
− nX
n2γ
dnγ
dz
=
1
nγ
(
dnX
dz
+
3
z
nX
)
=
1
zHnγ
(
dnX
dt
+ 3HnX
)
=
z
HNnγ
(
dnX
dt
+ 3HnX
)
. (D.23)
Therefore, we can write
dnX
dt
+ 3HnX =
HNnγ
z
dηX
dz
. (D.24)
Reduced scattering cross section
The thermally averaged scattering rate is given by
γ(ab→ 12) = neqa neqb 〈σ(ab→ 12)|v|〉
=
T
64pi4
∫ ∞
smin
ds
√
s σˆ(s) K1
(√
s
T
)
. (D.25)
Here, smin ≡ max[(ma +mb)2, (m1 +m2)2]. The Ka¨lle´n function is defined by
λ(a, b, c) = a2 + b2 + c2 − 2ab− 2bc− 2ca. (D.26)
The scattering cross section is given by
σ(ab→ Y ) = 1
4
√
λ
∫ (∏
c∈Y
d3pc
(2pi)32Ec
)
(2pi)4δ(4)(pa + pb − Y )
∑
spin
|M(ab→ Y )|2.
(D.27)
In reference [48], the phase space factors are defined as follows:
dpiX =
∏
b∈X
dpib, dpib = gb
d3pb
(2pi)3
1
2E(pb)
. (D.28)
113
Note that equation A7 in that paper has a typo in the expression of dpib: (2pi)
2 →
(2pi)3. The reduced cross section is defined as
σˆ(s) ≡ 8piΦ2(s)
∫
dpiY (2pi)
4δ4(pa + pb − pY )|A(ab→ Y )|2
= 8piΦ2(s)
∫ (∏
c∈Y
gc
d3pc
(2pi)3
1
2E(pc)
)
(2pi)4δ4(pa + pb − pY )|A(ab→ Y )|2.
(D.29)
Rewriting the multipicative degrees of freedom, ga, gb, and gc as spin sums, we
obtain
σˆ(s) = 8piΦ2(s)
∫ (∏
c∈Y
d3pc
(2pi)3
1
2E(pc)
)
(2pi)4δ4(pa + pb − pY ) 1
gagb
∑
spin
|M(ab→ Y )|2
= 8piΦ2(s)
4
√
λ
gagb
σ(ab→ Y ) (D.30)
The two-body phase space factor Φ2(s) is given by
Φ2(s) ≡
∫
dpiadpib(2pi)
4δ4(pa + pb − pY )
=
gagb
8pis
√
[s− (ma +mb)2][s− (ma −mb)2]
=
gagb
8pis
√
λ. (D.31)
Therefore, we can write
σˆ(s) =
4
s
λσ(ab→ Y ) (D.32)
which is twice the expression below equation 2.8 in reference [48]. The differential
cross section is given by
dσ
dt
=
1
16pis
√
λ
∑
spin
|M(ab→ Y )|2. (D.33)
114
In principle, the differential scattering cross section is given by
dσ
dt
=
1
4
√
λ
∫ (∏
c∈Y
d3pc
(2pi)32Ec
)
(2pi)4δ(4)(pa + pb − Y )
∑
spin
|M(ab→ Y )|2δ(t− (pa − p1)2).
(D.34)
According to reference [48], the differential reduced scattering cross section is given
by
dσˆ
dt
=
gagbgcgd
8pis
|A(ab→ Y )|2
→ 1
8pis
∑
spin
|M(ab→ Y )|2. (D.35)
Nα`Rα → ucRdR
The Feynman amplitude for this process is given by
iM =
(
i
gR√
2
)2
vN(pN)γ
µRu`(p`)
−i
(pN + p`)2 −m2WR + imWRΓWR
ud(pd)γµRvu(pu)
= i
g2R
2
vN(pN)γ
µRu`(p`)
1
(pN + p`)2 −m2WR + imWRΓWR
ud(pd)γµRvu(pu),
(D.36)
and thus
−iM† = −ig
2
R
2
v†u(pu)Rγ
†
νγ
0ud(pd)
1
(pN + p`)2 −m2WR − imWRΓWR
u†`(p`)Rγ
ν†γ0vN(pN)
= −ig
2
R
2
vu(pu)γνRud(pd)
1
(pN + p`)2 −m2WR − imWRΓWR
u`(p`)γ
νRvN(pN)
(D.37)
where we have used ucs = v−s. We therefore have∑
spin
|M|2 = g
4
R
4
1
[(pN + p`)2 −m2WR ]2 +m2WRΓ2WR
tr[γνRvNvNγ
µRu`u`]tr[γνRududγµRvuvu].
(D.38)
115
The trace part is calculated as follows:
tr[γνRvNvNγ
µRu`u`]tr[γνRududγµRvuvu]
= tr[γνR(/pN +mN )γ
µR/p`]tr[γνR/pdγµR/pu] = tr[γ
ν
/pNγ
µ
/p`L]tr[γν/pdγµ/puL]
= tr[γνγργµγσL]tr[γνγαγµγβL]pNρpeσpd
αpu
β
=
1
4
(
tr[γνγργµγσ]− tr[γνγργµγσγ5]) (tr[γνγαγµγβ ]− tr[γνγαγµγβγ5]) pNρp`σpdαpuβ
= 4(gνρgµσ − gνµgρσ + gνσgρµ + iνρµσ)(gναgµβ − gνµgαβ + gνβgαµ + iναµβ)pNρp`σpdαpuβ
= 4[pN
νp`
µ − gνµ(pN · p`) + p`νpNµ + iνρµσpNρp`σ][pdνpuµ − gνµ(pd · pu) + puνpdµ + iναµβpdαpuβ ]
= 4[(pN · pd)(p` · pu)− (pd · pu)(pN · p`) + (p` · pd)(pN · pu) + iνρµσpNρp`σpdνpuµ
− (pN · p`)(pd · pu) + 4(pN · p`)(pd · pu)− (p` · pN )(pd · pu)
+ (pN · pu)(p` · pd)− (pu · pd)(pN · p`) + (p` · pu)(pN · pd) + iνρµσpNρp`σpuνpdµ
+ iναµβpN
νp`
µpd
αpu
β + iναµβp`
νpN
µpd
αpu
β − νρµσναµβpNρpeσpdαpuβ ]
= 4[(pN · pd)(p` · pu)− (pd · pu)(pN · p`) + (p` · pd)(pN · pu) + 2(pN · p`)(pd · pu)
+ (pN · pu)(p` · pd)− (pd · pu)(pN · p`) + (p` · pu)(pN · pd) + 2(δραδσβ − δρβδσα)pNρpeσpdαpuβ ]
= 4[2(pN · pd)(p` · pu) + 2(p` · pd)(pN · pu) + 2(pN · pd)(p` · pu)− 2(pN · pu)(p` · pd)]
= 16(pN · pd)(p` · pu). (D.39)
Since we have
s = (pN + p`)
2 = (pd + pu)
2,
t = (pN − pu)2 = (pd − p`)2,
u = (pN − pd)2 = (pu − p`)2, (D.40)
116
we can write
2(pN · p`) = −(m2N − s), 2(pd · pu) = s,
2(pN · pu) = m2N − t, 2(pd · p`) = −t,
2(pN · pd) = m2N − u, 2(pu · p`) = −u. (D.41)
Hence, we have
tr[γνRvNvNγ
µRu`u`]tr[γνRududγµRvuvu] = 4(m
2
N − u)(−u) = 4(s+ t)(s+ t−m2N)
= 4(s2 + 2st+ t2 −m2Ns−m2N t) = 4[t2 − (m2N − 2s)t− s(m2N − s)], (D.42)
and thus
∑
spin
|M|2 = g4R
t2 − (m2N − 2s)t− s(m2N − s)
(s−m2WR)2 +m2WRΓ2WR
. (D.43)
Now the differential reduced scattering cross section is written as
dσˆ
dt
=
9
8pis
∑
spin
|M|2 = 9g
4
R
8pis
t2 − (m2N − 2s)t− s(m2N − s)
(s−m2WR)2 +m2WRΓ2WR
(D.44)
where the multiplicative factor 9 is from the numbers of quark flavours and color
factors. Note that this is the result for one flavour of RH neutrino. The Mandelstam
variable t is written as
t = (pN − pu)2 = E2N − 2ENEu + E2u − |pN |2 − |pu|2 + 2pN · pu
= m2N − 2(ENEu − |pN ||pu| cos θ). (D.45)
In the center-of-momentum (CM) frame, we have
|pN | =
1
2
√
s
√
s2 − 2m2Ns+m4N =
s−m2N
2
√
s
, |pu| =
√
s
2
, (D.46)
117
and
EN =
√
|pN |2 +m2N =
s+m2N
2
√
s
, Eu =
√
s
2
. (D.47)
Hence, we have
t = m2N −
1
2
[s+m2N − (s−m2N) cos θ], (D.48)
and thus
tmin = m
2
N − s, tmax = 0. (D.49)
Therefore, the reduced cross section is
σˆ(s) =
9g4R
8pis[(s−m2WR)2 +m2WRΓ2WR ]
∫ 0
m2N−s
dt [t2 − (m2N − 2s)t− s(m2N − s)]
=
9g4R
8pis[(s−m2WR)2 +m2WRΓ2WR ]
1
6
(m2N − s)2(m2N + 2s), (D.50)
which is the same as equation 2.15 in [46]. Hence, the CP-conserving reduced cross
section is
σˆNα`αucd (s) =
9g4R
4pis[(s−m2WR)2 +m2WRΓ2WR ]
1
6
(m2N − s)2(m2N + 2s). (D.51)
Nαu
c
R → `RαdcR
The Feynman amplitude is written as
iM = ig
2
R
2
u`(p`)γ
µRuN(pN)
1
(pN − p`)2 −m2WR
vu(pu)γµRvd(pd), (D.52)
and its Hermitian conjugate as
−iM† = −ig
2
R
2
v†d(pd)Rγ
†
νγ
0vu(pu)
1
(pN − p`)2 −m2WR
u†N(pN)Rγ
ν†γ0u`(p`)
= −ig
2
R
2
vd(pd)γνRvu(pu)
1
(pN − p`)2 −m2WR
uN(pN)γ
νRu`(p`). (D.53)
118
We thus have
∑
spin
|M|2 = g
4
R
4
1
[(pN − p`)2 −m2WR ]2
tr[γνRu`u`γ
µRuNuN ]tr[γνRvuvuγµRvdvd],
(D.54)
where
tr[γνRu`u`γ
µRuNuN ]tr[γνRvuvuγµRvdvd]
= tr[γνR/peγ
µR(/pN −mN)]tr[γνR/puγµR/pd] = tr[γν/peγµ/pNL]tr[γν/puγµ/pdL]
= 16(pN · pd)(p` · pu). (D.55)
Since we have
s = (pN + pu)
2 = (p` + pd)
2,
t = (pN − p`)2 = (pd − pu)2,
u = (pN − pd)2 = (p` − pu)2, (D.56)
we can write
2(pN · p`) = m2N − t, 2(pd · pu) = −t,
2(pN · pu) = −(m2N − s), 2(p` · pd) = s,
2(pN · pd) = m2N − u, 2(p` · pu) = −u. (D.57)
Therefore, we have
tr[γνRu`u`γ
µRuNuN ]tr[γνRvuvuγµRvdvd] = 4(m
2
N − u)(−u)
= 4(s+ t)(s+ t−m2N). (D.58)
119
Hence, we have
∑
spin
|M|2 = g4R
(s+ t)(s+ t−m2N)
(t−m2WR)2
, (D.59)
and the differential reduced scattering cross section is given by
dσˆ
dt
=
9
8pis
∑
spin
|M|2 = 9g
4
R
8pis
(s+ t)(s+ t−m2N)
(t−m2WR)2
(D.60)
where the multiplicative factor 9 is from the numbers of quark flavours and color
factors. We have
t = (pN − p`)2 = E2N − 2ENEe + E2e − |pN |2 − |pe|2 + 2pN · pe
= m2N − 2(ENEe − |pN ||pe| cos θ). (D.61)
In the CM frame, we can write
|pN | =
1
2
√
s
√
s2 − 2m2Ns+m4N =
s−m2N
2
√
s
, |pe| =
√
s
2
, (D.62)
and
EN =
√
|pN |2 +m2N =
s+m2N
2
√
s
, Ee =
√
s
2
. (D.63)
Hence, we obtain
t = m2N −
1
2
[s+m2N − (s−m2N) cos θ], (D.64)
and thus
tmin = m
2
N − s, tmax = 0. (D.65)
Therefore, the reduced cross section is
σˆ(s) =
9g4R
8pis
∫ 0
m2N−s
dt
(s+ t)(s+ t−m2N)
(t−m2WR)2
, (D.66)
120
which is the same as equation 2.16 in [46]. Hence, the CP-conserving reduced cross
section is
σˆNαu
c
`αdc
(s) =
9g4R
4pis
∫ 0
m2N−s
dt
(s+ t)(s+ t−m2N)
(t−m2WR)2
. (D.67)
NαdR → `RαuR
The Feynman amplitude is
iM = ig
2
R
2
u`(p`)γ
µRuN(pN)
1
(pN − p`)2 −m2WR
uu(pu)γµRud(pd), (D.68)
and
−iM† = −ig
2
R
2
ud(pd)γνRuu(pu)
1
(pN − p`)2 −m2WR
uN(pN)γ
νRu`(p`). (D.69)
We thus have
∑
spin
|M|2 = g
4
R
4
1
[(pN − p`)2 −m2WR ]2
tr[γνRu`u`γ
µRuNuN ]tr[γνRuuuuγµRudud],
(D.70)
where
tr[γνRu`u`γ
µRuNuN ]tr[γνRuuuuγµRudud]
= tr[γνR/peγ
µR(/pN −mN)]tr[γνR/puγµR/pd] = tr[γν/peγµ/pNL]tr[γν/puγµ/pdL]
= 16(pN · pd)(p` · pu). (D.71)
Since we have
s = (pN + pd)
2 = (p` + pu)
2,
t = (pN − p`)2 = (pu − pd)2,
u = (pN − pu)2 = (p` − pd)2, (D.72)
121
we can write
2(pN · p`) = m2N − t, 2(pd · pu) = −t,
2(pN · pu) = m2N − u, 2(p` · pd) = −u,
2(pN · pd) = −(m2N − s), 2(p` · pu) = s. (D.73)
We therefore have
tr[γνRu`u`γ
µRvNvN ]tr[γνRududγµRvuvu] = −4s(m2N − s). (D.74)
Hence, we have
∑
spin
|M|2 = −g4R
s(m2N − s)
(t−m2WR)2
, (D.75)
and the differential reduced scattering cross section is given by
dσˆ
dt
=
9
8pis
∑
spin
|M|2 = −9g
4
R
8pi
m2N − s
(t−m2WR)2
(D.76)
where the multiplicative factor 9 is from the numbers of quark flavours and color
factors. As in the previous case, we have
tmin = m
2
N − s, tmax = 0. (D.77)
Therefore, the reduced cross section is
σˆ(s) = −9g
4
R
8pi
∫ 0
m2N−s
dt
m2N − s
(t−m2WR)2
=
9g4R
8pi
(m2N − s)
[
1
t−m2WR
]0
m2N−s
=
9g4R
8pi
(m2N − s)
[
− 1
m2WR
− 1
m2N − s−m2WR
]
=
9g4R
8pi
(m2N − s)
[
1
s+m2WR −m2N
− 1
m2WR
]
=
9g4R
8pi
(m2N − s)2
m2WR(s+m
2
WR
−m2N)
, (D.78)
122
which is the same as equation 2.17 in equation [46]. Hence, the CP-conserving
reduced cross section is
σˆNαd`αu (s) =
9g4R
4pi
(m2N − s)2
m2WR(s+m
2
WR
−m2N)
. (D.79)
123
Appendix E: Lepton asymmetry
Exact solution
We can also derive the expression 4.41 by directly solving the differential equation
4.39. This equation is in the form
dy
dx
= Q(x)− P (x)y (E.1)
where
x ≡ z, y ≡ η∆Li , P (x) ≡
2
3
Wi(x), Q(x) ≡
3∑
α=1
δiNα
dηNα
dz
D˜α(x)
Dα(x) + Sα(x)
.
(E.2)
The differential equation E.1 can be rewritten as
0 =
[
Q(x)− P (x)y]dx− dy. (E.3)
In order to solve this differential equation, we need an integrating factor f(x, y):
dϕ = f(x, y)
[
Q(x)− P (x)y]dx− f(x, y)dy
=
∂ϕ
∂x
dx+
∂ϕ
∂y
dy
= 0. (E.4)
124
Then,
∂2ϕ
∂y∂x
=
∂
∂y
{
f(x, y)
[
Q(x)− P (x)y]} = ∂f(x, y)
∂y
[
Q(x)− P (x)y]− f(x, y)P (x)
∂2ϕ
∂x∂y
=
∂
∂x
[− f(x, y)] = −∂f(x, y)
∂x
, (E.5)
Thus, we need
∂f(x, y)
∂y
[
Q(x)− P (x)y]− f(x, y)P (x) = −∂f(x, y)
∂x
(E.6)
to have an exact differential dϕ. Now we assume f(x, y) = f(x). Then, the condition
E.6 is written as
f(x)P (x) =
df(x)
dx
, (E.7)
thus we can write
P (x)dx =
df(x)
f(x)
. (E.8)
The solution of this equation is given by
∫ x
x0
P (x′)dx′ = ln f(x)− ln f(x0), (E.9)
thus
f(x) = f(x0) exp
[∫ x
x0
P (x′)dx′
]
. (E.10)
The differential dϕ is given by
dϕ = f(x)[Q(x)− P (x)y]dx− f(x)dy = 0. (E.11)
125
We choose the integration path (x0, y0) −−−→
x=x0
(x0, y) −−−→
y=y0
(x, y). Then, we obtain
ϕ(x, y) = C
=
∫ x
x0
f(x′)[Q(x′)− P (x′)y]dx′ − f(x0)
∫ y
y0
dy′
=
∫ x
x0
f(x′)[Q(x′)− P (x′)y]dx′ − f(x0)(y − y0)
=
∫ x
x0
f(x′)Q(x′)dx′ −
[∫ x
x0
f(x′)P (x′)dx′ + f(x0)
]
y + f(x0)y0 (E.12)
where C = ϕ(x0, y0) = 0. Hence, we can write
y(x) =
f(x0)y0 +
∫ x
x0
f(x′)Q(x′)dx′
f(x0) +
∫ x
x0
f(x′)P (x′)dx′
=
y0 +
∫ x
x0
exp
[∫ x′
x0
P (x′′)dx′′
]
Q(x′)dx′
1 +
∫ x
x0
exp
[∫ x′
x0
P (x′′)dx′′
]
P (x′)dx′
=
y0 +
∫ x
x0
exp
[∫ x′
x0
P (x′′)dx′′
]
Q(x′)dx′
1 +
{
exp
[∫ x
x0
P (x′′)dx′′
]
− 1
}
= y0 exp
[
−
∫ x
x0
P (x′′)dx′′
]
+
∫ x
x0
exp
[
−
∫ x
x′
P (x′′)dx′′
]
Q(x′)dx′. (E.13)
At x = xc, we have
y(xc) = y0 exp
[
−
∫ xc
x0
P (x′′)dx′′
]
+
∫ xc
x0
exp
[
−
∫ xc
x′
P (x′′)dx′′
]
Q(x′)dx′. (E.14)
Using the definitions of variables E.2, we can rewrite this as
η∆Li(zc) = η∆Li(z0) exp
[
−2
3
∫ zc
z0
dz′′Wi(z′′)
]
+
1
2ζ(3)
∫ zc
z0
dz′z′2K1(z′)
∑
α
δiNα
D˜α(z
′)
Dα(z′) + Sα(z′)
exp
[
−2
3
∫ zc
z′
dz′′Wi(z′′)
]
= η∆Li(z0) exp
[
−2
3
∫ zc
z0
dz′′Wi(z′′)
]
−
∑
α
δiNακ
l
Nα(zc) (E.15)
where
κlNα(z) ≡
∫ zc
z0
dz′
dηNα
dz′
D˜α(z
′)
Dα(z′) + Sα(z′)
exp
[
−2
3
∫ zc
z′
dz′′Wi(z′′)
]
(E.16)
126
is the efficiency factor. Assuming the first term in equation E.15 is much smaller
than the second, (i.e. the initial lepton asymmetry is not so large as to be completely
washed out at the critical temperature), we can write
η∆Li(zc) = −
3∑
α=1
δiNακ
i
Nα(zc). (E.17)
Approximate solution
Now we derive the approximate solution 4.44 from equations E.16 and E.17. Note
that we do not follow mathematically rigorous steps in this derivation. We define
A(z) ≡ 2
3
Wi(z), (E.18)
B(z) ≡ −dη
eq
Nα
dz
D˜α(z)
Dα(z) + Sα(z)
=
1
2ζ(3)
z2K1(z)
D˜α(z)
Dα(z) + Sα(z)
. (E.19)
Note that we have A(z) 1 in the strong washout regime. We have
−κiNα(zc) =
∫ zc
z0
dz′B(z′) exp
[
−
∫ zc
z′
dz′′A(z′′)
]
≈
∫ zc
z1
dz′B(z′) exp
[
−
∫ zc
z′
dz′′A(z′′)
]
(E.20)
for some z1 which is very close to zc due to the large suppresion by the exponential
factors. Since z1 is close to zc, we can also approximately write
∫ zc
z1
dz′B(z′) exp
[
−
∫ zc
z′
dz′′A(z′′)
]
≈ B(zc)
∫ zc
z1
dz′ exp
[
−A(zc)
∫ zc
z′
dz′′
]
= B(zc)
∫ zc
z1
dz′ exp [−A(zc)(zc − z′)]
=
B(zc)
A(zc)
{1− exp [−A(zc)(zc − z1)]}
≈ B(zc)
A(zc)
. (E.21)
127
At the last step, we assumed
A(zc)(zc − z1) > 1, (E.22)
which can be satisfied when A(zc) 1, i.e. Wi(zc) 1, with appropriate z1. Then,
we obtain
−κiNα(zc) =
B(zc)
A(zc)
, (E.23)
which gives the expression 4.44. Note that this solution corresponds to
yc =
Q(xc)
P (xc)
. (E.24)
In other words, the expression 4.44 is approximately valid solution of the differential
equation E.1 when
∣∣∣∣dydx(xc)
∣∣∣∣ Q(xc) ≈ P (xc)yc, (E.25)
which can be satisfied if P (xc) 1, i.e. Wi(zc) 1.
128
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