### Overview of the theoretical status

### of cLFV

A B S T R A C T

The goal of this thesis is to provide an overview of current influen- tial theories describing charged Lepton Flavour Violating Processes in order to be used as a field guide and a starting point for future study and research in the field of charged Lepton Flavour Violation.

Four topics concerning cLFV are discussed, i.e. Seesaw, Supersymme- try, leptoquarks, and leptogenesis. Finally it is concluded that seesaw is an important part in multiple theories to explain light neutrino masses, and that experimental results from a variety of experiments will have to be combined to decide which model will succeed the Standard Model.

2

C O N T E N T S

i p r e a m b l e 4 1 i n t r o d u c t i o n 5 2 s ta n d a r d m o d e l 7

2.1 Fermions 8

2.1.1 Neutrinos 9

2.2 Flavour changing currents 10 2.2.1 GIM mechanism 10 2.2.2 cLFV in SM 11 3 r e s e a r c h m e t h o d 12

ii l e p t o n f l av o u r v i o l at i n g m o d e l s 16 4 s e e s aw 17

4.1 Seesaw Models 19

4.1.1 Type I - Seesaw 19 4.1.2 Type II - Seesaw 20 4.1.3 Type III - Seesaw 22 4.1.4 Inverse - Seesaw 23 4.1.5 Linear - Seesaw 26 4.2 LRSM-Model 27

4.3 Summary 31 5 s u p e r-symmetry 32

5.1 *ν*_{R}MSSM 35

5.2 MSSM without R-parity 36 5.3 Compactified M-theory 38 5.4 ISS-MSSM 40

5.5 Non-Abelian RVV 41 5.6 AKM 41

5.7 Summary 42
6 l e p t o q ua r k s 43
6.1 *ω*^{2}^{3} *and χ*^{1}^{3} 45
6.2 Summary 47
7 l e p t o g e n e s i s 48

7.1 Resonant Leptogenesis 49
7.1.1 *RµL* 50

7.1.2 ReL 51
7.1.3 *RτL* 51
7.2 Summary 52

iii c o n c l u s i o n/bibliography 54 8 c o n c l u s i o n 55

Part I P R E A M B L E

## 1

I N T R O D U C T I O N

*CHAPTER 1. introduction*

The Standard Model is the best theory of elementary particles physics today [1]. It is a gauged quantum field theory which aims to explain all phenomena of particle physics with a small number of fundamen- tal particles, and has been very successful at providing these explana- tions while even making predictions. However, some questions still remain unresolved. Some of these unresolved problems by the Stan- dard Model are: the hierarchy problem, dark matter, neutrino oscilla- tions and masses, baryo- and leptogenesis, and the flavour structure.

To provide solutions to these problems a large number of new the- ories beyond the Standard Model are provided by theoretical physi- cists. Because charged Lepton Flavour Violating processes are negli- gibly small within the Standard Model, observing them would evi- dently point towards new physics beyond the Standard Model. For this reason Lepton Flavour Violation is a good way to distinguish dif- ferent theories and check them for validity. The goal of this thesis is to provide an overview of current influential theories describing charged Lepton Flavour Violating Processes in order to be used as a field guide and a starting point for future study and research in the field of charged Lepton Flavour Violation.

First a brief introduction to particle physics and Lepton Flavour Vi- olation within standard model (SM) will be given. Then a summary of how the research has been done will be given. Subsequently multi- ple theories and models which predict branching fractions exceeding the branching fractions predicted in the SM for various charged Lep- ton Flavour violating processes will be discussed. These are the see- saw mechanisms in chapter 4, supersymmetric models in chapter 5, leptoquarks in chapter 6 and a Leptogenesis model in chapter 7. Fi- nally an attempt will be made to give a decisive answer on which model and what area of research may provide the best clues on some of the problems stated before.

6

## 2

S TA N D A R D M O D E L

*2.1. FERMIONS* *CHAPTER 2. standard model*

The Standard Model is a gauge quantum field theory which aims to explain all phenomena of particle physics with a small number of fundamental ingredients. It contains two classes of particles, i.e.

fermions with half-integer spin, and bosons with integer spin. The
Bosons are also known as the ”force carriers” and are the Higgs
bosons, gluons, Z^{0}boson, W^{±}*bosons, and the photon γ. The fermions*
will be discussed in more detail in the next section.

2.1 f e r m i o n s

The Standard Model contains 3 generations of elementary Fermions,
all with spin |^{1}_{2}|. Each generation consist of a lepton doublet contain-
ing a charged lepton and neutrino [2] and a quark doublet containing
two quarks [3].

The three quark doublets are shown below. Where the three quarks
in the first column(up, charm, and top) have charge +^{2}_{3}e and their
partners(down, strange and, bottom respectively) have charge−^{1}_{3}^{e .}

u d

, c

s

, t

b

, (1)

In the remainder of this thesis we concentrate on the leptons. The quarks are therefore not elaborated on any further.

The three lepton doublets can be seen in eq. 2 and eq. 3 for lepton and anti-lepton respectively.

e^{−}
*ν*_{e}

,

*µ*^{−}
*ν*_{µ}

,

*τ*^{−}
*ν*_{τ}

(2)

e^{+}

*¯ν*_{e}

,

*µ*^{+}

*¯ν*_{µ}

,

*τ*^{+}

*¯ν*_{τ}

(3)
The charged lepton and anti-lepton have a −^{1e and} +1e charge
respectively, while the neutrino are chargeless. Charged leptons in-
teract with both the electromagnetic and weak-force. Neutrinos only
interact with the weak-force, due to their chargeless nature. The Stan-
dard Model postulates that lepton flavour (also known as lepton fam-
ily) is conserved in every reaction. This results in the conservation of
lepton flavour number L_{i}. Where:

L_{i} =N_{l}_{i} −^{N}¯l_{i}+N*ν*_{i}−^{N}*¯ν*_{i} (4)

8

*2.1. FCC* *CHAPTER 2. standard model*

2.1.1 Neutrinos

The standard model includes three neutrino flavours ^{1}. The three
flavours, electron neutrino, muon neutrino, and tauon neutrino, are
created in conjunction with their charged lepton partner. Experi-
ments [5] provide compelling evidence that neutrino flavours do os-
cillate. Which means a neutrino with flavour i can change into a
neutrino with flavour j during propagation. This change in flavour
or neutrino oscillation can be explained by assuming the mass eigen-
states are not equal to the flavour eigenstates. This is realized by
connecting the neutrino flavour matrix with the neutrino mass ma-
trix using a unitary matrix U as seen in eq. 5. This matrix is
also known as the Pontecorvo-Maki-NakagawaSakata matrix or the
PMNS-matrix[6].

*ν*_{e}
*ν*_{µ}*ν**τ*

=U_{PMNS}

*ν*_{1}
*ν*_{2}
*ν*3

(5)

with

U=

c_{12}c_{13} s_{12}c_{13} s_{13}e^{iδ}

−^{s}12c23−^{c}12s23s_{13}e* ^{iδ}* c

_{12}c23−

^{c}12s23s

_{13}e

*s23c13 s*

^{iδ}_{12}s

_{23}−

^{c}12c

_{23}s

_{13}e

*c*

^{iδ}_{12}s

_{23}−

^{s}12c

_{23}s

_{13}e

*s*

^{iδ}_{23}c13

(6)

where s_{ij} =sinΩij, c_{ij} =cosΩij. While in the past many calculations
have been done with a tri-bimaximal( seen in eq. 7) PMNS neutrino
mass matrix[7, 8]. Experiments at Daya Bay in 2012 have shown that
unlike predicted |^{U}e3|^{2} was not equal to zero and it turned out be

|^{U}e3|^{2} = sin^{2}(_{2θ}_{13}) = _{0.092} 6= 0 [9]. This resulted in a none tri-
bimaximal PMNS-Matrix.

|^{U}e1|^{2} |^{U}e2|^{2} |^{U}e3|^{2}

|^{U}*µ1*|^{2} |^{U}*µ2*|^{2} |^{U}*µ3*|^{2}

|^{U}*τ1*|^{2} |^{U}*τ2*|^{2} |^{U}*τ3*|^{2}

=

2 3

1

3 0

1

6 1

3 1 1 2

6 1

3 1 2

(7)

From the neutrino oscillation data it can be concluded that lepton
flavour is not always conserved and that the neutrino masses are
quasi-degenerate. This quasi-degenerateness suppresses neutrino os-
cillations as a result of the GIM-mechanism. A brief summary of the
GIM mechanism will be given in the next section. Because neutrinos
are massive particles and have zero charge, they can either have Dirac
masses or Majorana masses. If neutrinos have Majorana masses, they
become their own antiparticle. This way neutrinos of the same gener-
ation can annihilate with each other. This could result in observable
*neutrinoless double beta decay(0νββ) as shown in figure 1.*

1 The number of light neutrinos N*ν* has been determined by[4] using the invisible
decay width of the Z BosonΛ and the charged leptonic partial widthΛ. Which

*2.2. FLAVOUR CHANGING CURRENTSCHAPTER 2. standard model*

*n*

*n*

*p*

*p*
*W*

*W*

*L*

*e*
ν

*L*

*L*

ν*L*
*L*

*e**L*

*Figure 1: Neutrinoless double beta decay with two Majorana neutrinos ν*L annihi-
lating with each other.

2.2 f l av o u r c h a n g i n g c u r r e n t s

Within the standard model flavour is conserved at the tree level with neutral currents and may be violated in charged current interactions involving quarks. These charged currents, also known as flavour changing charged currents(FCCC), can produce large rates for flavour changing processes induced at the loop level, but are highly sup- pressed by the GIM-mechanism.

2.2.1 GIM mechanism

The GIM mechanism was introduced in the 1970’s when only three
quark species were known. Measurements of the flavour oscillation
K^{0} to ¯K^{0} had been done by Joffe and Shabalin [10]. The measured
oscillation rate was far lower than theories predicted at that time.

Glashow, Iliopoulos, and Maiani invented a mechanism [11] which
could be the solution to this problem. Whereas previously only the
up-quark Feynman Diagrams depicted in figure 2a and 2b contributed
to this process, Glashow, Iliopoulos, and Maiani postulated a fourth
quarks(c) which gave rise to a third and fourth Feynman Diagram (fig-
ure 2c and 2d) that contributed to K^{0} →^{K}^{¯}^{0}. The new diagrams includ-
ing the c-quark would have the opposite sign. Due to the superposi-
tion of these diagrams the current becomes proportional to m^{2}_{c}−^{m}^{2}u.
If the c-quark mass mc and u-quark mass mu would have been de-
generate, u-quark, and c-quark diagrams would have cancelled and
consequently no K^{0} →^{K}^{¯}^{0}would have been observed. [12]

10

*2.2. FCC* *CHAPTER 2. standard model*

*d* *s*

*s**−* *w* *w* *d**−*

*u*

*u**−*

(a) u-quark

*d* *s*

*s**−* *w* *d**−*

*w*
*u*
*u*

(b) u-quark

*d* *s*

*s**−* *w* *w* *d**−*

*c*

*c**−*

(c) c-quark

*d* *s*

*s**−* *w* *d**−*

*w*
*c*
*c*

(d) c-quark

*Figure 2: Feynman diagrams contributing to K*^{0} → ^{K}^{¯}^{0} which is supressed by the
GIM-mechanism.

2.2.2 cLFV in SM

When charged lepton flavour is not conserved, flavour changing pro-
*cesses like: µ* → * ^{eγ, τ}* →

*→*

^{eγ, τN}*→ eee can take place.*

^{µN, and µ}In the current standard model the only source of Lepton Flavor Vio-
lation is: neutrino oscillations within loop diagrams. In this process
*a charged lepton l decays into a neutrino of the same flavour ν*_{l} while
sending out a W^{±}*. The ν*_{l} may oscillate into a neutrino of another
*flavour ν*_{l}0*. After this the ν*_{l}0 reacts with a W^{±}and becomes a charged
lepton l^{0}. The W^{±} can be the same as the one sent out by the first
process as can be seen in figure3^{2}.

*l* *l’*

*W*

*ν’*

ν
*γ ,Z*

*Figure 3: Lepton Flavor Violation in the standard model through neutrino oscilla-*
tion.

Because of the small differences in neutrino masses the oscillation
length is enormous. Branching fractions ^{3} of these cLFV have been
calculated to be 10^{−}^{54} as seen in eq. 8. With branching fractions this
small, observing cLFV processes would unambiguously hint to new
physics beyond the standard model.

BSM(*µ*→* ^{eγ}*) =

^{3α}*32π*

### ∑

i

U_{µi}^{∗}U_{ei}m^{2}_{/u}_{i}m^{2}_{W}

2

≈^{10}^{−}^{54} ^{(8)}

2 All Feynman diagrams included in this thesis have a horizontal time axis.

3 Branching fraction B_{i} is the decay rate of a certain channelΓidivided by the total
decay rate of the particleΓ e.g. B = ^{Γ}^{i}.Γ is summation over all Feynman diagrams

## 3

R E S E A R C H M E T H O D

12

*CHAPTER 3. research method*

A large number of rather different new physics models exist that introduce cLFV. In this thesis we create a comprehensive overview of the most discussed models, based on a survey of the pertinent literature. To analyse the large number of articles on Lepton Flavour Violation a meta-analyses was done in following fashion:

First a list of around two thousand articles was collected from the APS journals [13] and Web of Science[14] database. Only articles written after the year 2000 were used and selected by searching their databases for the following queries in article titles and abstracts:

• Lepton flavor violation

• Lepton flavour violation

• Lepton family violation

• Lepton universality

To get an overview on topics that are frequently assessed in corre- lation with cLFV, and on models that contribute to cLFV outside of the SM, these articles were analysed by looking for key-words that were frequently used in their titles and abstract. We concentrate on recently published articles, and frequently cited articles. The results can be seen in figure 4. It must be stressed that although theories or models like Seesaw and Supersymmetry get a lot of attention, these theories are still speculative. Current experimental data have not yet proven these theories to be reality or not.

*CHAPTER 3. research method*

0 50 100 150 200 250 300 350

supersymmetric Higgs seesaw cp violation neutrino oscillation leptogenesis heavy neutrino majorana susy Dirac bimaximal mssm tri-bimaximal dark matter r-parity SO(10) texture magnetic moment SU(5) cosmology two higgs doublet edm Chiral leptoquark Neutralino

Counts

*Figure 4: Histogram showing the frequency of the 25 most used keywords found in*
*2278 analyzed scientific articles*

As can be seen in fig 4 key words most frequently associated with Lepton Flavour Violation are supersymmetry, CP-violation, leptogen- esis, the Higgs particle, Seesaw and neutrinos.

14

*CHAPTER 3. research method*

Because seesaw is such a prominent topic within Lepton Flavour Violation, it has been subdivided in the following types:

0 5 10 15 20 25 30 35

inverse seesaw seesaw typ I seesaw type II seesaw type III seesaw type I+II seesaw type I+III

Counts

*Figure 5: Histogram showing the use various seesaw models in 2278 analyzed sci-*
entific articles

Based on these finding, the most prominent topics will be dis- cussed, specifically: Seesaw, Supersymmetry, Leptoquarks, and Lep- togenesis.

Part II

L E P T O N F L AV O U R V I O L AT I N G M O D E L S

## 4

S E E S AW

*CHAPTER 4. seesaw*

One of the problems the Standard Model cannot account for, is the small but non-zero mass of neutrinos. In the framework of the Standard Model, neutrinos would have zero mass. To explain the small but nonzero mass of the neutrinos, the Standard Model is often extended with a the Seesaw mechanism. Multiple seesaw models have been considered in the last decades. In this chapter five of these models will be discussed. After discussing types I, II, and III, the inverse and linear seesaw will be examined.

The seesaw mechanism is not just a stand alone model but is also used in other more encompassing theories. In section 4.2 of this chap- ter one such a theory, viz. the LRSM-model, will be treated and the subsequent chapter on supersymmetry will also contain models which make use of the seesaw mechanics.

18

*4.1. SEESAW MODELS* *CHAPTER 4. seesaw*

4.1 s e e s aw m o d e l s 4.1.1 Type I - Seesaw

The Type-I seesaw model expands the SM with three fermions which
are singlet under the SU(2)_{L} gauge group. These singlets are bet-
*ter known as the right-handed neutrinos (ν*_{R})[15, 16]. Adding these
particles results in a neutrino mass matrix9

M*ν*=

0 **m**D

**m**^{T}_{D} **m**^{∗}_{M}

(9)
**Where m**_{D} **and m**_{M} are 3×3 matrices relating to the Dirac and Majo-
rana masses respectively. Just like in the SM model cLFV only arises
trough neutrino oscillations at loop level in the Type-I seesaw model.

*But due to the larger mass of ν*_{R}, branching fractions of cLFV pro-
cesses are increased. As an example, we consider radiative muon
*decay, µ*→*eγ. The branching fraction for this decay is given by:*

B(*µ*→* ^{eγ}*) =

^{3α}*8π*

### ∑

i

|^{Y}NieY_{N}^{†}* _{iµ}*|

^{2}

^{v}

^{4}

^{(10)}

*where α is the fine structure constant, Y the neutrino Yukawa cou-*plings matrix, and v = 246GeV. As can be seen branching fractions depend on the unknown Yukawa couplings matrix Y. To be able to get numerical results, one can consider the ratios of two cLFV branch- ing fractions involving the same l →

^{l}

^{0}transition resulting in the can- cellation of the Yukawa couplings. from Branching fraction ratios for B(l →

^{l}

^{0}

*)/B(l →*

^{γ}^{3l}

^{0}) and B(lN →

^{l}

^{0}

^{N})/B(l →

^{3l}

^{0})for this model can be found in fig 6.

10^{2} 10^{3} 10^{4} 10^{5} 10^{6} 10^{7}

0 10 20 30 40

*m**N*HGeVL
Br HΜ ® eΓL Br HΜ ® eeeL
Br HΤ ® eΓL Br HΤ ® eeeL
Br HΤ ® ΜΓL Br HΤ ® ΜΜΜL

(a) Branching fraction ratios for
B(l →^{l}^{0}* ^{γ}*)/B(l→

^{3l}

^{0})expected

*in a seesaw type I model [17, 18].*

0 5000 10 000 15 000 20 000
10^{-4}

0.001 0.01 0.1 1 10

*m**N*@GeVD
*R*Ð®eeeÐ®*e*

*Au*
*Pb*
*Ti*
*Al*

(b) Branching fraction ratios
B(lN → ^{l}^{0}^{N})/B(l → ^{3l}^{0})
expected in a seesaw type I model.

*Figure 6: Branching fraction ratios for a type I seesaw model*

*4.1. SEESAW MODELS* *CHAPTER 4. seesaw*

4.1.2 Type II - Seesaw

The type II seesaw mechanism adds a SO(2)_{L} scalar triplet ∆ with
hypercharge Y =2 to the SM. This triplet contributes to the neutrino
mass by generating a Majorana mass as a consequence of adding
Yukawa termLSSI I to the Lagrangian [19].

LSSI I = ^{M}

ij
*ν*

v_{∆} L^{T}_{i} *Cıσ*2∆Lj (11)
Where L_{i} is the left handed lepton doublet, C the charge conju-
gation operator, M*ν* the neutrino mass matrix in the charged lepton
basis and v_{∆} the∆^{0} vev^{1}, and with scalar triplet∆ in matrix form as
seen in eq. 12.

∆=

*δ*^{+}

√2 *δ*^{++}

*δ*^{+} −^{√}^{δ}^{+}_{2}

!

(12) The matrix 12 can result in two different mass scenarios:

A : m* _{δ}*0 ≥

^{m}

*δ*

^{+}≥

^{m}

*δ*

^{++}

B : m_{δ}^{++} > m_{δ}^{+} >m* _{δ}*0

(13) The size of the mass splittings influences the main decay channel.

In figure 8 these influences are shown for ∆^{++} ^{2} decay in the B sce-
nario. It can be seen that for small v_{∆} the main decay channel is
expected to be the same sign charged lepton decay channel 7a, which
is Lepton Flavour Violating. The Cascade decay 7c is most dominant
for large mass differences.

∆

*l*
*l*

*i*

*j*

*++*

(a) Same sign lepton de- cay

∆^{++}

*W*^{+}

*W*^{+}

(b) Same sign gauge bo- son decay

*W*^{+}

*W*^{+}

∆^{+}

∆^{++}

∆^{0}

(c) Cascade decay

*Figure 7: The different decay channels for*∆^{++}decay.

This mechanism has four lepton interactions (l → ^{3l}^{0}) at the tree
level due to the exchange of the scalar triplet ∆ and other charged
Lepton Flavour interactions at the loop level. This results in rather
large branching fraction for l → lll compared to the other Seesaw
models according to [16]. Branching fractions l →lll for look like the
equation below:

B(*µ*→^{eee}) = |^{Y}*∆eµ*|^{2}||^{Y}∆ee|^{2}

4m^{4}_{∆}G^{2}_{F} (14)

1 The vacuum expectation value (vev) is the average expected value in the vacuum of an operator

2 Not to be confused with the∆^{++}baryon which consists of three u quarks

20

*4.1. SEESAW MODELS* *CHAPTER 4. seesaw*

Cascade Decay

Leptonic Decay

Gauge
Boson Decay
10^{-}^{10} 10^{-}^{8} 10^{-}^{6} 10^{-}^{4} 0.01 1
0.5

1.0 5.0 10.0 50.0 100.0

*v*DHGeVL

DMHGeVL

*Figure 8: Decay phase diagram for*∆ for m_{∆}^{++} = 150 GeV and ∆M = m_{∆}^{++}−
m_{∆}^{+}*. Dashed, thin and solid line correspond to 99, 90 and 50% of the*
*branching ratios respectively from ref [19].*

with Y_{∆ei} as the Yukawa coupling of a delta with an electron, and
a lepton ”i”, m_{∆} being the triplet mass, and G_{F} being the Fermi con-
stant. Because of the dependence on multiple Yukawa couplings Y,
which do not cancel upon taking the ratios with other lepton flavour
violating process, no ratios have been calculated for l → lll by [16].

By only using dipole operators, a branching fraction has been approx-
imated by [15]. This resulted in a B(*µ*→ ^{eee}) = 3.3×^{10}^{−}^{15}^{. Which}
is the same as branching fractions calculated in a seesaw type I or
III with just dipole operators. The ratio R^{B}_{B}^{(}_{(}_{µ}^{µN}_{→}^{→}_{eγ}^{eN}_{)} ^{)} = ^{B}_{B}^{(}^{µN}_{(} ^{→}^{eN}^{)}

*µ*→* ^{eγ}*) only
depends on the triplet mass m

_{∆}and can be seen in in fig 9.

*Figure 9:* ^{B(µZ}_{B(µ}_{→eγ)}^{→eZ)} versus m_{∆}in a Seesaw type II model for various nuclei Z.

*4.1. SEESAW MODELS* *CHAPTER 4. seesaw*

4.1.3 Type III - Seesaw

Type-III seesaw model introduces fermionic SU(2)_{L} triplets Σi with
hypercharge Y = 0 [15](displayed in matrix form in eq. 15). At
least two of those triplets have to be introduced to generate non-zero
neutrino mass [20].

Σi =

Σ^{0}_{ki}

√2 Σ^{+}_{k}

i

Σ^{−}_{k}

i −^{Σ}

0ki

√2

(15)

Couplings generated in a type III seesaw framework can generate
cLFV directly at the tree level. The only cLFV process that cannot hap-
pen at tree level is: l_{1} → ^{l}2*γ. This process still happens only on the*
loop level as a result of the QED coupling being flavour diagonal [16].

[20] have shown that branching fractions take the form:

B(l_{2} →^{l}1*γ*) = ^{3a}
*32π*

(^{13}

3 +C)*e*_{l}_{1}_{l}_{2}−

### ∑

i

(^{M}

2vi

M^{2}_{W})(U_{PMNS})_{l}_{1}_{i}(U_{PMNS}^{†} )_{il}_{2}

2

(16)
with C=−^{6.56 and}

*e*_{l}_{1}_{l}_{2} = ^{v}

2

2 |^{Y}_{Σ}^{†} ^{1}

M_{Σ}^{†}M_{Σ}Y_{Σ}|l_{2}l_{1} (17)
*The e term generates the contribution from the triplets, and oscil-*
lations of the light neutrinos result in the summation contribution.

Because branching fractions of l_{1} →^{l}2*γ*and l_{1}→^{3l}2obtained by [21]

*depend on the same parameter e the following branching fraction*
ratios can be obtained:

B(*µ*→* ^{eγ}*) =1.3×

^{10}

^{−}

^{3}×

^{B}(

*µ*→

^{eee}) B(

*τ*→

*) =1.3×*

^{µγ}^{10}

^{−}

^{3}×

^{B}(

*τ*→

*) =2.1×*

^{µµµ}^{10}

^{−}

^{3}×

^{B}(

*τ*

^{−}→

^{e}

^{−}

^{e}

^{+}

^{µ}^{−}) B(

*τ*→

*) =1.3×*

^{eγ}^{10}

^{−}

^{3}×

^{B}(

*τ*→

^{eee}) =2.1×

^{10}

^{−}

^{3}×

^{B}(

*τ*

^{−}→

^{µ}^{−}

^{µ}^{+}

^{e}

^{−})

(18)
Type I and III seesaw mechanism both generate ratios for B(*µ* →
*eγ*)and B(*τ*→* ^{eγ}*). But in type III l →

^{3l}

^{0}is possible at the tree level, while l →

^{l}

^{0}

*happens at the one-loop level. Therefore the ratios*

^{γ}*shown in eq. 18 are smaller than one. The same µ*−

^{e}−Z vertex that generates l →

^{3l}

^{0}

*at tree level also enables µ*→e conversion in nuclei at the tree level. Calculating ratios for this conversion in

^{48}

_{22}Ti results in:

B(*µN* →^{eN}) =1.4×^{10}^{1}× |^{e}*eµ*|^{2} ^{(19)}

B(*µ*→^{eee}) =2.4×^{10}^{−}^{1}×^{B}(*µN* →^{eN})

B(*µ*→* ^{eγ}*) =3.1×

^{10}

^{−}

^{4}×

^{B}(

*µN*→

^{eN})

^{(20)}

22

*4.1. SEESAW MODELS* *CHAPTER 4. seesaw*

µ *e*

*γ , Z*
Φ*−*

*ν, Σ** ^{0}*
Φ

*−*

µ *e*

*γ , Z*
Φ*−*

*ν, Σ*^{0}*W**−*

µ *e*

*γ , Z*
Φ*−*
*ν, Σ*^{0}*W**−*

µ *e*

*γ , Z*

*ν, Σ*^{0}*W**−*

*W**−*

µ *e*

*γ , Z*
*Z*

*l, ψ* µ *e*

*γ , Z*
*H, η*

*l, ψ*

*Figure 10: Feynman diagrams contributing to µ* → *eγ in Seesaw type III. Where*
*φ*^{±} *and η are the three Goldstone bosons associated with W*^{−} and Z
bosons. H is a Higgs boson.

4.1.4 Inverse - Seesaw

The Inverse Seesaw model(ISS) introduces two SU(_{3})×^{SU}(_{2})×^{U}(_{1})
*singlets ν*^{C}, S. This model has a neutrino mass matrix[22, 23]:

M*ν* =

0 **m**_{D} 0

**m**^{T}_{D} 0 **m**_{M}
0 **m**^{T}_{M} *µ*

(21)

**with m**_{D}**, m**_{M}*, and µ being 3*×3 Dirac, lepton conserving, and Ma-
jorana mass matrices respectively. This M*ν* can be diagonalized to
obtain the light neutrino masses. If an U(1)_{L} global lepton number
*symmetry is assumed, µ will be equal to 0. Which would result in*
*mass-less light neutrinos. However if this symmetry is broken, µ*6=^{0,}
light neutrinos become massive particles. The mass matrix M*ν* can
be used to calculate branching fractions for Lepton Flavour Violating
processes as seen in eq. 22.

In this model the largest contributions to lepton flavor violation
*processes come from the heavy singlet neutrinos ν*^{C} and S in one-
loop diagrams. Branching fraction for these one-loop diagrams are
given by:

B(l_{i} →^{l}j*γ*) = ^{α}

3s^{2}_{W}
*256π*^{2}

m^{5}_{l}

i

M_{W}^{4} Γli

### ∑

9 k=4(^{M}

2
*ν*_{k}

M_{W}^{2} )K_{ik}^{∗}K_{jk}G_{γ}

2

(22) where

G* _{γ}* =−

^{2x}

^{2}+5x

^{2}−

^{x}

4(1−^{x})^{3} − ^{3x}^{3}

2(1−^{x})^{4}^{ln x} ^{(23)}
, Γ is the total decay width of the lepton, Kik the lepton mixing

*4.1. SEESAW MODELS* *CHAPTER 4. seesaw*

The difference between the branching fraction within the SM and the
*Inverse Seesaw, is the sum over nine neutrinos, three light(ν) and six*
*heavy(ν*^{C} and C) in ISS, instead of the sum over just the three light
neutrinos in the SM.

The remark has to be made that branching fractions given in the inverse and linear seesaw were calculated with a tri-bimaximal lepton mixing matrix. This does not include the Higgs decay.

10^{-2} 10^{-1} 10^{0} 10^{1} 10^{2}

10^{-16}
10^{-15}
10^{-14}
10^{-13}
10^{-12}
10^{-11}
10^{-10}

*v** _{L}*HblueL, Μ HredL HeVL

BrHΜ->eΓL

*Figure 11: B*(*µ* → * ^{eγ}*)

*vs µ for the inverse seesaw, and v*

_{L}for the linear seesaw, continuous line m

_{M}=100 GeV, dashed m

_{M}=200 GeV and dot-dashed mM=

*1000 GeV from ref [22].*

24

*4.1. SEESAW MODELS* *CHAPTER 4. seesaw*

-22

-20

-20

-18

-20 -19

-19

-17

-16

-17 -18

-17

-16

-14

-12

-11

-9 -8

-6

-4

**R****=****I**
**m**_{Ν}_{1}**=****0.1 eV**

-15

-13

-5

-7 -10

200 10^{3} 10^{4} 10^{5} 10^{6}

10^{-9}
10^{-8}
10^{-7}
10^{-6}
10^{-5}
10^{-4}

*M**R*HGeVL
Μ*X*HGeVL

Log10*BR HH ® ΜΤL*

(a) Contour lines for B(H → ^{µ ¯}* ^{τ}*).
With m

_{ν}_{1}being the light electron neutrino mass

-20

-19

-18 -18

-17

-17 -16

-16

-15

-15

-14

-13

-12

-11

-10

-9 -8

-7

-6

-5

-4

**R****=****I**
**m**_{Ν}_{1}**=****0.1 eV**
**M****R****1****=****900 GeV**
**M**_{R}

**2****=****1000 GeV**

10^{3} 10^{4} 10^{5} 10^{6}

10^{-9}
10^{-8}
10^{-7}
10^{-6}
10^{-5}
10^{-4}

*M**R*3HGeVL
Μ*X*HGeVL

Log10*BR HH ® ΜΤL*

(b) Contour lines for B(H→^{µ ¯}* ^{τ}*).

*Figure 12: Contour lines for B*(H → ^{µ ¯}* ^{τ}*) within a Inverse seesaw model. With
pink and blue areas excludedd due to upper bound on B(

*µ*→

*)and the pertubrativity requirement of the neutrino Yukawa couplings from*

^{eγ}*ref [24].*

Given the recent discovery of the Higgs particle, and the search
for rare Higgs decays at the LHC, lepton flavour violating Higgs
decays have been studied by [24]. To get to the results shown in
figure eq. 12a for degenerate heavy neutrino masses, interactions
including neutrino Yukawa coulings, gauge-couplings of W^{±}bosons
with neutrino-electron pairs, and G^{±}goldstone-boson couplings with
neutrino-electron pairs are used . The same coupling are used to ob-
tain results for hierarchical neutrino masses with m*ν*_{1}, M_{R}_{1} and M_{R}_{2}
for the light electron neutrino mass, and the first, and second genera-
tion heavy neutrino masses respectively as seen in fig 12b. Relevant
Feynman diagrams can be found in figure 13.

*H* *G*
ν_{i}

ν_{j}

*l**k*

*l**−*_{m}*W*

*H*
ν_{i}

ν*j*

*l*_{k}*l**−*_{m}

*H* *W* ν

*i*

*l**k*

*l**−**m*

*W*

*G*

*H* ν_{i}

*l*_{k}

*l**−*_{m}*W*

*G*

*H* ν_{i}

*l*_{k}

*l**−*_{m}*G*

*H* ν_{i}

*W*

*l**−**m*

*l**k*

*l**m*

*H* ν_{i}

*G*

*l**−*_{m}*l*_{k}*l*_{m}

*Figure 13: Feynman Diagrams contributing to H* → ^{l}k¯lm in a Inverse seesaw
model.

*4.1. SEESAW MODELS* *CHAPTER 4. seesaw*

4.1.5 Linear - Seesaw

The Linear Seesaw mechanism is similar to the inverse seesaw mech-
*anism. The Linear seesaw also adds a ν*^{C} and an S neutrino. But it’s
neutrino effective mass matrix is different.

M* _{ν}* =

0 **m**D **v**^{T}_{L}
**m**^{T}_{D} 0 **m**M

**v**^{T}_{L} **m**^{T}_{M} 0

(24)

As in the inverse seesaw light neutrinos become massive particles
when the symmetry is broken and v_{L} 6= 0, . To calculate branching
fractions within linear-seesaw framework, the same equation as in the
inverse seesaw(eq. 22) can be used. Which results in the branching
fractions for B(*µ* → * ^{eγ}*) shown in fig. 11, and the ratio obtained by
[22] seen below.

Br(*τ*→* ^{µγ}*) =4×

^{Br}(

*τ*→

*) (25)*

^{eγ}26

*4.2. LRSM-MODEL* *CHAPTER 4. seesaw*

4.2 l r s m-model

Next we will consider a more encompassing model, the left-right sym-
metric model(LRSM), which model makes use of the type I and II
seesaw mechanic, proposed by [25]. This particular model has been
chosen to show how Seesaw mechanics, multiple coupling constants,
and masses can emerge from the breaking of symmetries in a GUT
*theory. Additionally it has influences on cLFV- and 0νββ processes.*

The left-right symmetric model is based on the gauge group SU(2)_{L}×
SU(2)_{R}×^{U}(1)_{B}_{−}_{L} which is embedded in a SO(10) GUT with Pati-
Salam symmetry. In a Pati-Salam model the number of quark colors
is extended from three to four colours. Besides the usual red, green
and blue colour the fourth colour violet or lilac is added. This fourth
colour is better known as the leptons within our Standard Model.

The scale of the breaking of the SU(2)_{L}×^{SU}(2)_{R}×^{U}(1)_{B}_{−}_{L} sym-
metries govern the masses of neutrinos and scalar triplets in a LRSM-
model. If the symmetries break at the TeV scale, RH-neutrinos, scalar
triplets, and the RH gauge bosons W_{R} and Z_{R} acquire TeV scale
masses and type I and type II seesaw mechanisms contribute to the
neutrino mass. Introducing new particles results in new Feynman di-
*agrams as shown in fig. 14 for LFV processes and in fig. 17 for 0µββ*
decay.

*l* *l’*

*l**i*

∆^{+}^{_ +_}

(a) LRSM contribution
*to µ* → ^{eγ with a}

∆^{+}+_{SSI I}

*l* *l’*

*N*_{i}

*u* *d* *u*

*W* *W*

(b) LRSM contribution
*to µN* → eN with
heavy neutrinos

*l* *N*_{i}*l’*

*u* *d* *u*

*W* *W*

(c) LRSM contribution
*to µN* → ^{eN with}
heavy neutrinos

*l* *N**i* *l’*

*W* *W*

*N**i* *l’*

*l’*

(d) LRSM contribution
*to µ* → eee with
heavy neutrinos

*l* *N**i* *l’*

*W* *W*

*N**i*

*l’* *l’*

(e) LRSM contribution
*to µ* → ^{eee with}
heavy neutrinos

*l* *l’*

*l’* *l’*

∆
*++**−**−*

(f) LRSM contribution
*to µ* → eee with a

∆^{+}+_{SSI I}
*Figure 14: LRSM contribution to Lepton Flavour violating processes*

*4.2. LRSM-MODEL* *CHAPTER 4. seesaw*

The Feynman Diagrams shown in fig. 14 result in branching frac- tions:

B(*µ*→* ^{eγ}*) =

*384π*

^{2}e

^{2}(|

^{A}L|

^{2}+|

^{A}R|

^{2}) B(

*µN*→

^{eN}) =

^{2G}

2F

Γcapt

(|^{A}^{∗}RD+ ˜g^{(}_{LV}^{p}^{)}V^{(}^{p}^{)}+ ˜g^{(}_{LV}^{n}^{)}V^{(}^{n}^{)}|^{2}+A^{∗}_{L}D+ ˜g^{(}_{RV}^{p}^{)}V^{(}^{p}^{)}+ ˜g^{(}_{RV}^{n}^{)}V^{(}^{n}^{)}|^{2}

B(*µ*→^{eee}) =^{1}

2|^{h}*eµ*h^{∗}_{ee}|^{2}

M_{W}^{4}

L

M^{4}

*δ*^{−−}_{R}

+ ^{M}

4W_{L}

M^{4}

*δ*^{−−}_{L}

(26)
where A_{L} and |^{A}R| are amplitudes depending on the heavy neu-
tron flavour mixing matrix, a summation over the heavy neutrinos,
and on the ∆^{++} W bosons mass, gLV and gRV are couplings, V^{(}^{p,n}^{)}
and D are overlap integrals, and h_{ij} is the lepton-Higgs LFV cou-
pling^{3}.

As before, many of these parameters are unknown. But by assum-
ing double charged Higgs diagrams dominate, a proportionate mass
spectrum for the heavy particles, and ^{M}

WL4

M^{4}

*δ*−−

L

≈0, the ratio _{B}^{B}_{(}^{(}_{µ}^{µ}_{→}^{→}_{eγ}^{e}^{)}_{)} can
be approximated. This has been done by [26] for the LRSM-model
and can be seen in fig 15 for different values of rL= ^{M}_{M}^{δ}^{L}^{++}

W2 .

0 100 200 300 400 500

0.5 1 1.5 2 2.5 3

0 100 200 300 400 500

0.5 1 1.5 2 2.5 3

r = 0.2_{L}
r = 1_{L}
r = 5_{L}

M /Mw2 w1

B(μN eN)/B(μ eΥ)^{__}^{>} ^{__}^{>}

*Figure 15: The ratio* ^{B(µN}_{B(µ}_{→eγ)}^{→eN)} as a function of ^{M}_{M}^{W2}

W1 different values of rL.
While most LRSM models assume an explicit left-right symmetry(D-
parity) at the TeV scales, i.e. g_{L} = g_{R} some of the LRSMs assume a
spontaneous breaking of this D-parity at a higher scale as the SU(2)_{R}
breaking. The D-parity breaking model treated in [25] will be de-
scribed next.

Like most LRSM models this symmetric model is embedded in a SO(10) GUT with Pati-Salam symmetry. However D-parity breaks at

3 For a more elaborate description of these amplitudes, couplings, and overlap inte- grals we refer to [26], because that is beyond the scope of this theses.

28

*4.2. LRSM-MODEL* *CHAPTER 4. seesaw*

10^{9} GeV which causes an asymmetry in the Left- and Right-handed
Higgs fields. This ensures the SU(_{2})_{R} _{and SU}(_{2})_{L} evolve separately,
and the divergence of the left and right handed gauge couplings, as
seen in fig. 16 which illustrates the evolution of gauge couplings
within this model. As a consequence of the D-parity breaking at 10^{9}
GeV the ratio g_{R}/g_{L}becomes 0.6 approximately at the TeV scale. The
breaking also generates a large mass splitting between LH and RH
scalar particles.

**M**_{C}**M**_{P}**M**_{U}

**M**_{W}

**M**_{BL}**M**_{Z}

**2 L = 2 R**

**4 C****4 C****2 L****2 R**

**3 C****3 C**

**BL**
**2 L****2 L****Y**

**1 R****2 R**

0 2 4 6 8 10 12 14 16 18

0 10 20 30 40 50 60 70 80 90

Log@ΜGeVD
Α*i*-1

*Figure 16: The symmetry breaking energy levels versus their coupling constant for*
*a SO(10) GUT with Pati-Salam symmetry*

Due to the breaking of D-parity the branching fractions gain a(^{g}_{g}^{R}

L)^{4}
term as illustarted below.

B(*µ*→^{eee}) = ^{1}

2|^{h}*eµ*h^{∗}_{ee}|^{2}^{ g}_{g}^{R}

L

4M_{W}^{4}_{L}
M^{4}

*δ*^{−−}_{R}

+ ^{M}

W4L

M^{4}

*δ*^{−−}_{L}

(27)

In the LRSM an interesting connection appears between cLFV and
*0νββ. In fig. 17 additional diagrams for this decay are given.*

*n*

*n*

*p*

*p*
*W*

*W*

*L*

*e*
ν

*L*
*L*

ν*L*
*L*

*e**L*

(a) With left handed neu- trinos

*n*

*n*

*p*

*p*
*W*

*W*

*R*

*N* *e*

*R*
*R*

*N*_{R}

*R*

*e**R*

(b) With right handed neutrinos

*n*

*n*

*p*

*p*
*W*

*W*

*R* *e*

*R*

*R*

*e**R*

δ_{R}

(c) With RH triplet higgs scalar

*Figure 17: Feynman Diagrams for 0νββ decay in the LRSM-model*

*4.2. LRSM-MODEL* *CHAPTER 4. seesaw*

*The inverse half life of 0µββ-decay can be calculated using these*
*diagrams. With contribution only from ν, N and doubly charged RH*
Higgs(∆^{++}) this results in:

1
T^{0ν}_{1}

2

= G_{01}* ^{0ν}*|

^{M}

*ν*

*η*

*+M*

_{ν}_{N}(

*η*

_{N}

_{R}+

*η*

_{δ}_{R}|

^{2}

^{(28)}

Where G^{0ν}_{01} is the nuclear phase space factor, M*ν* and M_{N} are the
*respective nuclear matrix elements and η*_{i} are the dimensionless pa-
rameters shown below.

*η** _{ν}* =

^{1}

m_{e}

### ∑

^{U}

^{ei}

^{m}

^{i}

*η*_{N}_{R} =m_{p} g_{R}
gL

M_{W}^{4}_{L}
MR

4

### ∑

_{M}

^{V}

^{ei}

^{∗}

_{N}

^{2}

i

*η*_{δ}_{R} = ^{ g}^{R}
g_{L}

M_{W}^{4}

L

M_{R}

4

### ∑

^{V}

_{M}

^{ei}

^{∗}

^{2}2

^{M}

^{N}

^{i}

*δ*

^{−−}

_{R}

(29)

This results in graphs for the effective mass and half life time for
*0νββ-decay shown in fig18. With the red and blue areas showing the*
variation caused by the Majorana phases for normal and inverted hi-
erarchy respectively. And the green and yellow bands corresponding
*with a 3σ variation.*

30