Overview of the theoretical status of cLFV

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_RMSSM 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 ω²³ and χ¹³ 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, Z⁰boson, 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 |¹₂|. 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 +²₃e and their partners(down, strange and, bottom respectively) have charge−¹₃^{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_li −^N¯l_i+Nν_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 ντ



=U_PMNS



 ν₁ ν₂ ν3



 (5)

with

U=





c₁₂c₁₃ s₁₂c₁₃ s₁₃e^iδ

−^s12c23−^c12s23s₁₃e^iδ c₁₂c23−^c12s23s₁₃e^iδ s23c13 s₁₂s₂₃−^c12c₂₃s₁₃e^iδ c₁₂s₂₃−^s12c₂₃s₁₃e^iδ s₂₃c13



 (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 |^Ue3|² was not equal to zero and it turned out be

|^Ue3|² = sin²(_2θ13) = _0.092 6= 0 [9]. This resulted in a none tri- bimaximal PMNS-Matrix.





|^Ue1|² |^Ue2|² |^Ue3|²

|^Uµ1|² |^Uµ2|² |^Uµ3|²

|^Uτ1|² |^Uτ2|² |^Uτ3|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 Nν 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

νL L

eL

Figure 1: Neutrinoless double beta decay with two Majorana neutrinos νL 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 K⁰ to ¯K⁰ 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 K⁰ →^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 m²_c−^m2u. 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 K⁰ →^K¯0would 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 K⁰ → ^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} → ^{µN, and µ} → eee can take place.

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_l0. After this the ν_l0 reacts with a W^±and becomes a charged lepton l⁰. The W^± can be the same as the one sent out by the first process as can be seen in figure3².

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_eim²_/uim²_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 = ^Γi.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

Mν=

 0 mD

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

|^YNieY_N^†_iµ|^{2v4 (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 →^l0 transition resulting in the can- cellation of the Yukawa couplings. from Branching fraction ratios for B(l → ^l0γ)/B(l → ^3l0) and B(lN → ^l0N)/B(l → ^3l0)for this model can be found in fig 6.

10² 10³ 10⁴ 10⁵ 10⁶ 10⁷

0 10 20 30 40

mNHGeVL 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 →^l0γ)/B(l→^3l0)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

mN@GeVD RЮeeeЮe

Au Pb Ti Al

(b) Branching fraction ratios B(lN → ^l0N)/B(l → ^3l0) 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 Cıσ2∆Lj (11) Where L_i is the left handed lepton doublet, C the charge conju- gation operator, Mν the neutrino mass matrix in the charged lepton basis and v_∆ the∆⁰ vev¹, 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_δ0 ≥^mδ⁺ ≥^mδ⁺⁺

B : m_δ⁺⁺ > m_δ⁺ >m_δ0

(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 → ^3l0) 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

4m⁴_∆G²_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

vDHGeVL

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 =







Σ⁰_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: l₁ → ^l2γ. 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(l₂ →^l1γ) = ^3a 32π

(¹³

3 +C)e_l1l2−

∑

i

(^M

2vi

M²_W)(U_PMNS)_l1i(U_PMNS^† )_il2    

2

(16) with C=−^{6.56 and}

e_l1l2 = ^v

2

2 |^Y_Σ^{† 1}

M_Σ^†M_ΣY_Σ|l₂l₁ (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 l₁ →^l2γand l₁→^3l2obtained 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→^µγ) =1.3×¹⁰⁻³×^B(τ→^µµµ) =2.1×¹⁰⁻³×^B(τ⁻→^e−e+µ−) B(τ→^eγ) =1.3×¹⁰⁻³×^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 →^3l0 is possible at the tree level, while l → ^l0γ happens at the one-loop level. Therefore the ratios shown in eq. 18 are smaller than one. The same µ−^e−Z vertex that generates l →^3l0 at tree level also enables µ→e conversion in nuclei at the tree level. Calculating ratios for this conversion in⁴⁸₂₂Ti results in:

B(µN →^eN) =1.4×¹⁰¹× |^eeµ|^{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, Σ⁰ Φ−

µ e

γ , Z Φ−

ν, Σ⁰ W−

µ e

γ , Z Φ− ν, Σ⁰ W−

µ e

γ , Z

ν, Σ⁰ 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]:

Mν =





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 Mν 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, µ6=^0, light neutrinos become massive particles. The mass matrix Mν 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 →^ljγ) = ^α

3s²_W 256π²

m⁵_l

i

M_W⁴ Γli

∑

(^M

2 ν_k

M_W² )K_ik^∗K_jkG_γ    

2

(22) where

G_γ =−^2x2+5x²−^x

4(1−^x)³ − ^3x3

2(1−^x)^{4ln 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^-1 10⁰ 10¹ 10²

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

v_LHblueL, 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_Ν1=0.1 eV

-15

-13

-5

-7 -10

200 10³ 10⁴ 10⁵ 10⁶

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

MRHGeVL ΜXHGeVL

Log10BR HH ® ΜΤL

(a) Contour lines for B(H → ^{µ ¯τ}). With m_ν1 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_Ν1=0.1 eV MR1=900 GeV M_R

2=1000 GeV

10³ 10⁴ 10⁵ 10⁶

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

MR3HGeVL ΜXHGeVL

Log10BR 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→^eγ)and the pertubrativity requirement of the neutrino Yukawa couplings from 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 mν₁, M_R1 and M_R2 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

lk

l−_m W

H ν_i

νj

l_k l−_m

H W ν

i

lk

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

lk

lm

H ν_i

G

l−_m l_k l_m

Figure 13: Feynman Diagrams contributing to H → ^lk¯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 mD v^T_L m^T_D 0 mM

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→^eγ) (25)

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’

li

∆^{+_ +_}

(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 Ni l’

W W

Ni l’

l’

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

l Ni l’

W W

Ni

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) =384π²e²(|^AL|²+|^AR|²) B(µN →^eN) =^2G

2F

Γcapt

(|^A∗RD+ ˜g⁽_LV^p)V^(p)+ ˜g⁽_LVⁿ⁾V⁽ⁿ⁾|²+A^∗_LD+ ˜g⁽_RV^p)V^(p)+ ˜g⁽_RVⁿ⁾V⁽ⁿ⁾|²

B(µ→^eee) =¹

2|^heµh^∗_ee|²

M_W⁴

L

M⁴

δ⁻⁻_R

+ ^M

4W_L

M⁴

δ⁻⁻_L



(26) where A_L and |^AR| 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- pling³.

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

M⁴

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^δL++

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

10⁹ 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 10⁹ GeV the ratio g_R/g_Lbecomes 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@̐GeVD Αi-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|^heµh^∗_ee|^{2 g}_g^R

L

4M_W⁴_L M⁴

δ⁻⁻_R

+ ^M

W4L

M⁴

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

νL L

eL

(a) With left handed neu- trinos

n

n

p

p W

W

R

N e

R R

N_R

R

eR

(b) With right handed neutrinos

n

n

p

p W

W

R e

R

R

eR

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^0ν₁

2

= G₀₁^0ν|^Mνη_ν+M_N(η_NR+η_δR|^{2 (28)}

Where G^0ν₀₁ is the nuclear phase space factor, Mν and M_N are the respective nuclear matrix elements and η_i are the dimensionless pa- rameters shown below.

η_ν = ¹

m_e

∑

η_NR =m_p g_R gL

M_W⁴_L MR

4

∑

i

η_δR = ^{ gR} g_L

M_W⁴

L

M_R

4

∑

(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