University of Zagreb
Faculty of Science
Lepton flavor violation
low-scale seesaw models
Doctoral Thesis submitted to the Physics Department
Faculty of Science, University of Zagreb
for the academic degree of
Doctor of Natural Sciences (Physics)
This thesis was made under the mentorship of Prof Amon Ilakovac, within University doctoral study at Physics Department of Faculty of Science of University of Zagreb.
Ova disertacija izrad–ena je pod vodstvom prof. dr. sc. Amona Ilakovca, u sklopu Sveučilišnog doktrorskog studija pri Fizičkom odsjeku Prirodoslovno-matematičkoga fakulteta Sveučilišta u Zagrebu.
A. M. D. G.
First and above all, I wish to thank my beloved wife Mirela, for illuminating our home with the warmth of her heart.
I would also like to thank my parents Danka and Bojan who never ceased to give me moral support throughout my work and studies.
My sincere gratitude goes to my colleagues Branimir, Mirko, Ivica and Sanjin for all the discussions regarding physics and beyond. I was honoured with your friendship.
I use this opportunity to say thanks to all my professors for generously sharing their knowledge with us.
Special gratitude goes to Mrs Marina Kavur from students’ office, for always being on our side during the never-ending bureaucratic battles.
I also which to express my gratitude to our collaborator Prof Apostolos Pilaftsis from University of Manchester for his participation on this project.
Last but not least, I thank my mentor Prof Amon Ilakovac, without whose patient guidance and support this thesis would never see the light of day.
Basic documentation card
University of Zagreb Doctoral Thesis
Faculty of Science
Lepton flavor violation in supersymmetric low-scale seesaw models
Faculty of Science, Zagreb
The minimal supersymmetric standard model with a low scale see-saw mechanism is presented. Within this framework, the lepton flavour violation in the charged lepton sector is thoroughly studied. Special attention is paid to the individual loop contributions due to the heavy neutrinos , sneutrinos and soft SUSY-breaking terms. For the first time, the complete set of box diagrams is included, in addition to the photon and -boson mediated interactions. The complete set of chiral amplitudes and their associate form-factors related to the neutrinoless three-body charged lepton flavor violating decays of the muon and tau, such as , , and , as well as the coherent conversion in nuclei, were derived. The obtained analytical results are general and can be applied to most of the New Physics models with charged lepton flavor violation. This systematic analysis has revealed the existence of two new box form factors, which have not been considered before in the existing literature in this area of physics.
In the same model, the systematic study of one-loop contributions to the muon anomalous magnetic dipole moment and the electron electric dipole moment is performed. Special attention is paid to the effect of the sneutrino soft SUSY-breaking parameters, and , and their universal CP phases ( and ) on and .
(135 pages, 169 references, original in English)
Keywords: Lepton Flavor Violation, Supersymmetry, MSSM, Seesaw mechanism, Low-scale seesaw, Lepton Dipole Moments
|Supervisor:||Prof Amon Ilakovac, University of Zagreb|
|Committee:||1. Prof Krešimir Kumerički, University of Zagreb|
|2. Dr Vuko Brigljević, Senior scientist, IRB Zagreb|
|3. Prof Amon Ilakovac, University of Zagreb|
|4. Prof Apostolos Pilaftsis, University of Machester|
|5. Prof Mirko Planinić, University of Zagreb|
|Replacements:||1. Prof Dubravko Klabučar, University of Zagreb|
|2. Dr Krešo Kadija, Senior scientist, IRB Zagreb|
- 1 Experimental survey
- 2 Theoretical framework
- 3 CLFV observables
- 4 Lepton Dipole Moments
- 5 Conclusions
- A Interaction vertices
- B Loop functions
- C One–loop form factors
- D Form factor analysis
- Prošireni sažetak
- Curriculum Vitæ
In the first part of this thesis the study of charged lepton flavor violation (CLFV) is performed in low-scale seesaw model of minimal supersymmetric standard model (MSSM) within the framework of minimal supergravity (mSUGRA). There are two dominant sources of CLFV: one originating from the usual soft supersymmetry-breaking sector, and other entirely supersymmetric coming from the supersymmetric neutrino Yukawa sector. Both sources are taken into account within this framework, and number of possible lepton-flavor-violating transitions are calculated. Supersymmetric low-scale seesaw models offer distinct correlated predictions for lepton flavor violating signatures, which might be discovered in current and projected experiments.
In the second part, the same model is used to study the anomalous magnetic and electric dipole moments of charged leptons. The numerical estimates of the muon anomalous magnetic moment and the electron electric dipole moment will be given as a function of key parameters. The electron electric dipole moment is found to be naturally small in this model, and can be probed in the present and future experiments.
The thesis is organized as follows:
The first chapter gives a brief experimental survey, the current and projected experiments regarding the detection of charged lepton flavor violation and anomalous dipole moments of charged leptons.
The second chapter presents the theoretical framework which underlines the study of lepton flavor violation and anomalous dipole moments given in the thesis.
The third chapter exposes the analytic and numerical results for various lepton flavor violating transitions, as well as some important physical implications which follow.
The fourth chapter gives the analysis of the muon anomalous magnetic moment and the electron electric dipole moment in supersymmetric low-scale seesaw models with right-handed neutrino superfields.
Concluding remarks are given in the fifth chapter.
The appendices contain technical details regarding the relevant interaction vertices, loop functions and formfactors.
The main results of the thesis are the following:
The soft SUSY-breaking effects in the -boson-mediated graphs dominate the CLFV observables for appreciable regions of the MSSM parameter space in mSUGRA. But for the box diagrams involving heavy neutrinos in the loop can be comparable to, or even greater than the corresponding -boson-mediated diagrams in and conversion in nuclei. Therefore, the usual paradigm with the photon dipole-moment operators dominating the CLFV observables in high-scale seesaw models have to be radically modified.
Heavy singlet neutrino and sneutrino contributions to anomalous magnetic dipole moment of the muon are small, typically one to two orders of magnitude below the muon anomaly . The largest effect on instead comes from left-handed sneutrinos and sleptons, exactly as is the case in the MSSM without right-handed neutrinos. Heavy singlet neutrinos do not contribute to the electric dipole moment (EDM) of the electron either. The main contribution to EDM comes from SUSY-breaking terms, but only if one of the CP phases ( and/or ) introduced to SUSY-breaking sector is nonvanishing.
Chapter 1 Experimental survey
Neutrino oscillation experiments have provided undisputed evidence of lepton flavor violation (LFV) in the neutrino sector, pointing towards physics beyond the Standard Model (SM). Nevertheless, no evidence of LFV has been found in the charged lepton sector of SM, implying conservation of the individual lepton number associated with the electron , the muon and the tau lepton . All past and current experiments were only able to report upper limits on observables of charged lepton flavor violation (CLFV). The experimental detection of CLFV would certenaly pave the way to the New Physics.
Measurements of the anomalous magnetic dipole moment of the muon (i.e. its deviation form the SM prediction, ) can give an important constraint on model-building, since any New Physics contribution must remain within limit. Study of the electric dipole moment of the electron is even more compelling, since the observation of non-zero (i.e. ) value for would signify the existence of CP-violating physics beyond the Standard Model.
1.1 Neutrino oscillations
When a neutrino is produced in some weak interaction process, and it propagates through some finite distance, there is a non-zero probability that it will change its flavor. This well established and observed fact is known as neutrino oscillation [1, 2, 3], due to the oscillatory dependence of the flavor change probability with respect to the neutrino energy and the distance of the propagation.
There are numerous neutrino experiments which report the lepton flavor violation in the neutrino sector, by observing the disappearances or the appearances of a particular neutrino flavor.
In solar neutrino experiments, first by Homestake  and later confirmed by others [5, 6, 7, 8, 9, 10, 11, 12], the disappearance of the solar electron neutrino is observed. Atmospheric muon neutrinos and antineutrinos disappeared in Super-Kamiokande experiment [13, 14]. The disappearance of reactor electron antineutrinos is observed in Kam-LAND reactor [15, 16] and in DOUBLE-CHOOZ experiment . Muon neutrinos disappeared in the long-baseline accelerator neutrino experiments MINOS [18, 19] and K2K . Short-baseline reactor experiments Daya Bay [21, 22] and RENO  report the disappearance of the reactor electron antineutrinos .
All these experiments have provided undisputed evidence for neutrino oscillations caused by finite (non-zero) neutrino masses and, consequently, neutrino mixing parameters. Since neutrinos are massive, the transition from the neutrino flavor eigenstate fields (, , ) which makes the lepton charged current in weak interactions to the neutrino mass eigenstate fields (, , ) is non-trivial:
where , and . denotes solar mixing angle, atmospheric mixing angle and reactor mixing angle. Phases , and stand for Dirac CP violating phase and two Majorana CP violating phases, respectively.
1.2 Searching for CLFV
The existence of lepton flavor violation (LFV) in the neutrino sector implies the possibility of LFV in the charged sector as well. However, in spite of intense experimental searches [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] no evidence of LFV in the charged lepton sector of the Standard Model (SM) has yet been found.
All past and current experiments searching for the charged lepton flavor violation (CLFV) were only able to report upper limits on the observables associated with CLFV. Recently, the MEG collaboration  has announced an improved upper limit on the branching ratio of the CLFV decay , with at the 90% confidence level (CL). As also shown in Table 1.1, future experiments searching for the CLFV processes, , , coherent conversion in nuclei, , and , are expected to reach branching-ratio sensitivities to the level of [38, 39] ( ),  ( ), [42, 43, 44, 45] ( [40, 46, 47]), [48, 49],  and , respectively. The values in parentheses indicate the sensitivities that are expected to be achieved by the new generation CLFV experiments in the next decade. Most interestingly, the projected sensitivity for and conversion in nuclei is expected to increase by five and six orders of magnitude, respectively. The history and current status of the experimental search for CLFV is very nicely exposed in Ref , which is highly recommended for further reading.
|No.||Observable||Upper Limit||Future Sensitivity|
|1.||||– [38, 39], |
|3.||||– [42, 43, 44, 45], [40, 46, 47]|
|4.||||– [42, 43, 44, 45], [40, 46, 47]|
|5.||[30, 31, 32, 33, 34, 35, 36, 37]||– [49, 48]|
|6.||[30, 31, 32, 33, 34, 35, 36, 37]||[49, 48]|
|7.||[30, 31, 32, 33, 34, 35, 36, 37]||[49, 48]|
|8.||[30, 31, 32, 33, 34, 35, 36, 37]|||
|9.||[30, 31, 32, 33, 34, 35, 36, 37]||[49, 48]|
|10.||[30, 31, 32, 33, 34, 35, 36, 37]|||
Given that CLFV is forbidden in the SM, its observation would constitute a clear signature for New Physics, which makes this field of investigation ever more exciting.
1.3 Measuring lepton dipole moments
The anomalous magnetic dipole moment (MDM) of the muon, is a high precision observable extremely sensitive to physics beyond the Standard Model. Its current experimental value, according to PDG , is
The Standard Model prediction of this observable reads
The difference between measured and predicted value,
is at the confidence level (CL) and has therefore been called the muon anomaly. This value limits the allowed contributions of New Physics to MDM and consequently can be used as a strong constraint on model-building, or even eliminate some of the proposed New Physics models.
Likewise, the electric dipole moment (EDM) of the electron, , constitutes a very sensitive probe for CP violation induced by new CP phases present in the physics beyond the Standard Model. The present upper limit on is reported to be [37, 51, 52]
Future projected experiments utilizing paramagnetic systems, such as Cesium, Rubidium and Francium, may extend the current sensitivity to the level [52, 53, 54, 55, 56, 57, 58, 59]. In the Standard Model, the predictions for range from to depending on whether the Dirac CP phase in light neutrino mixing is zero or not (for detalis see Ref ). Therefore, any observation of non-zero value of , i.e. value larger than , would signify the existence of CP-violating physics beyond the Standard Model.
For that reason, these observables are of great interest for the investigation of possible scenarios for the New Physics. The announced higher-precision measurement of by a factor of 4 in the future Fermilab experiment E989 [61, 62, 63, 64, 65] as well as the expected future sensitivities of the electron EDM down to the level of , renders the study of the dipole moments even more actual and interesting.
Chapter 2 Theoretical framework
In this chapter we will expose some basic features of the theoretical framework which underlines the study of lepton flavor violation and anomalous dipole moments given in the thesis.
In the first section, we will give the basic structure of the Minimal Supersymmetric Standard Model (MSSM), as well as some main features regarding the Soft Supersymmetry Breaking in the MSSM. The notation used when discussing the Supersymmetry (SUSY) will correspond to the one used in Drees et al. , adapted to Petcov et al. . For further reading regarding SUSY in general and MSSM in particular, the reader is encouraged to consult Refs [68, 70, 71, 72, 73].
Second section is dedicated to the seesaw mechanisms, with the main focus on the low-scale version of the seesaw mechanism type I.
Finally, the the MSSM extended by low-scale right handed neutrinos (or MSSM) is introduced.
2.1 Basic features of the MSSM
The basic idea behind all supersymmetric models is that there is a symmetry (conveniently called supersymmetry) which transforms a fermion into the boson and vice versa. The Minimal Supersymmetric Standard Model supersymmetrizes the SM with minimal extension of the SM particle spectrum: every SM particle is accompanied by one superparticle or a superpartner. The superpartners of matter fermions are spin zero particles, called sfermions. They can be further classified into the scalar leptons or sleptons and scalar quarks or squarks. Matter fermions and their superpartners are described by chiral superfields. The superpartners of SM gauge bosons are spin one-half particles called gauginos. They can be further classified into the stronlgy interacting gluinos and electroweak zino and winos (superpartners of and bosons, respectively). Together with SM gauge bosons, they are described by vector superfields. Superpartners of Higgs bosons are spin one-half particles called higgsinos and, along with the latter, are described by chiral superfields. The electroweak symmetry breaking mixes the electroweak gauginos with higgsinos resulting in physical particles referred to as charginos and neutralinos. Table 2.1 displays full filed contents of the MSSM, with the corresponding quantum numbers.
|Field contents of the MSSM|
As can be seen from Table 2.1, there are two Higgs superfields in the MSSM. These can be written as
field is sometimes referred to as the down type Higgs (), superfield containing and , while is referred to as the up type Higgs superfield containing and . The component fields denoted by lower case letters are given by
After the spontaneous breakdown of electroweak symmetry, the Higgs vacuum expectation values (VEVs) are given by real, positive quantities and ,
which arise from the minimization of the Higgs potential. The ratio of these values,
is considered to be a free parameter of the theory, at least regarding the fermion masses.
Let us proceed to the interaction and mass terms in the Lagrangian density which partly comes from the exact supersymmetrization of the SM. Full MSSM Lagrangian can be written as the sum of two parts,
While is fully supersymmetric, the contains terms which explicitly break the supersymmetry (acronym SSB stands for SuperSymmetry Breakdown).
Let’s first take a look to the contents of . The supersymmetric part of the MSSM Lagrangian can be further decomposed as
where , and are pure gauge, matter and Higgs-Yukawa parts, respectively. Detailed expressions for these terms can be found in the literature [68, pp 171-172]. The part which is most interesting for the purposes of this thesis is the superpotential, which constitutes important part of , and reads
where matrices are given by
Here, , and represent lepton, down-quark and up-quark mass matrices, respectively. The dot products are defined in two-component notation [72, 74] as (). Second, third and fourth terms in right-hand side of Eq (2.8) are just supersymmetric generalization of the Yukawa couplings in the Standard Model Lagrangian (for this and other aspects of the SM see Ref ). The first term is however new, and can be thought of as a supersymmetric generalization of a higgsino mass term. It can be shown that the consistent incorporation of spontaneus electroweak symmetry breakdown requires to be of the order of the weak scale.
One more thing needs to be adressed at this point, and that is the implicit assumption of the conservation of -parity defined by a quantum number given by
where , and stand for barion number, lepton number and spin of the particle, respectively. The conservation of in the MSSM may be posited as a natural assumption in a minimal supersymmetric extensions of the SM, due to the barion and lepton number conservations in the SM Lagrangian.
Let’s now turn back to (2.6) and analyse the contents of the . There are several constraints which need to be put upon the supersymmetry breaking terms. First, they need to be “small” compared to the fully supersymmetric part . Second, and most important, they must obey certain mass dimensional constrains in order to preserve the desired convergent behavior of the supersymmetric theory at high energies as well as the nonrenormalization of its superpotential couplings. According to the Symanzik’s rule [76, pp 107-8] this turns out to be possible in all orders in perturbation theory only if the explicit supersymmetry breaking terms are soft [77, 78, 79, 80], i.e. that every field operator occuring in has mass dimension strictly less then four. The Eq (2.6) is therefore usually written as
Taking all this into account, one can write down the expression for , by collecting all allowed soft SUSY-breaking terms [68, p 185],
Practical calculations within the MSSM usually include several simplifying assumptions in order to drastically reduce the number of additional parameters in the model. Different assumptions result in different versions of the Constrained Minimal Supersymmetric Standard Model or CMSSM.
In this thesis, we will adopt the framework of Minimal Super Gravity (mSUGRA) model. Since MSSM fields alone cannot break supersymmetry spontaneously at the weak scale [68, pp 183-5], spontaneous supersymmetry breakdown needs to be effected in a sector of fields which are singlets with respect to the SM gauge group. This sector is known as the hidden or secluded sector. SUSY breaking is then transmitted to the gauge nonsinglet observable or visible sector by a messenger sector associated by a typical mass scale . Unlike the details of the spontaneous SUSY-breaking in the hidden sector, the mechanism of its transmission from hidden sector to the MSSM fields does have an immediate impact on the observable sparticle spectrum and then also on the SUSY phenomenology. The most economical mechanism of this kind uses gravitational strength interactions based on local supersymmetry also known as supergravity [70, 81].
The great benefit in using the mSUGRA model is the fact that it reduces the extra one hundred and five parameters (compared to the nineteen parameters of the SM) to the set of just five parameters,
where stands for the sign of the parameter in superpotential (2.8), constitute masses of the scalars (), is common mass of all three MSSM gauginos, is common trilinear coupling constant (higgs-sfermion-sfermion) and is ratio of VEVs defined by Eq (2.5). These parameters are also referred to as the supersymmetry breaking parameters. Their values are usually imposed on the scale of Grand Unification (GUT), and then via Renormalization Group Equations (RGE)  transmitted down to the weak scale.
There are quite a few reasons to work in the framework of the MSSM with -parity conserved. The MSSM provides a quantum-mechanically stable solution to the guage hierarchy problem and predicts rather accurate unification of the SM gauge couplings close to the grand unified theory (GUT) scale. The lightest supersymmetric particle (LSP) is stable and, if neutral, such as the neutralino, could represent a good candidate for the dark matter in the Universe. Besides that, the MSSM typically predicts a SM-like Higgs boson lighter than 135 GeV, in agreement with the recent observations for a Higgs boson, made by ATLAS  and CMS [83, 84] Collaborations.
2.2 Seesaw mechanism
Neutrino oscillation experiments (see Chapter 1) have indisputably shown that neutrinos are not massless, as was once believed to be. This imposes the necessity to extend the Standard Model (as well as the MSSM) in a way that will consistently allow the existence of massive neutrinos. One of the most interesting extensions in that sense is provided by so-called seesaw mechanism. There are three realizations of the seesaw mechanism: the seesaw type one [85, 86, 87, 88, 89, 90], the seesaw type two [91, 90, 92, 93, 94, 95] and the seesaw type three . These three scenarios differ by the nature of their seesaw messengers needed to explain the small neutrino masses. For the purpose of this thesis, we will explain and adopt a low-scale variant of the seesaw type-I realization, whose messengers are three singlet neutrinos . But first let us examine the usual, high-scale variant, seesaw type-I mechanism in order to detect its weaknesses and to demonstrate how low-scale variant can overcome them.
The leptonic Yukawa sector of the SM with massless neutrinos is described by
Here, the primes indicate that the fields are not written in the mass basis (so-called physical states), but rather in the interaction basis. and are lepton and neutrino Yukawa matrices, respectively.
The consistent and straightforward extension of this sector by a right-handed neutrinos includes both the extra Yukawa neutrino term and the mass term which is singlet under the SM gauge group ,
After the spontaneous breakdowns of the electroweak symmetry,
one ends with the well-known expression for lepton masses,
Here represents masses of the charged leptons, stands for the Dirac mass matrix, and is the Majorana mass matrix. The former two make the mass term for neutrinos,
In order to get from the interaction to mass basis, i.e. to write the Lagrangian in terms of physical states, one needs to diagonalize the matrix. This is performed with unitary matrix ,
Matrix transforms fields written in the interaction basis to the one written in the mass basis,
Finally one can re-write the Lagrangian (2.20) in the mass basis,
If we allow the Yukawa matrices to be of arbitrary form, we have to face two unpleasant consequences:
From Eq (2.22) we see that mass of light neutrinos is roughly given by . Since the light neutrino masses are of the order , and if we assume that Yukawa couplings are of order , it follows that the heavy singlet neutrinos must assume masses of order . That is inconvenient by itself, since its direct detection is way beyond the reach of experiments in high energy physics.
From Eq (2.23) we see that the mixing between light and heavy neutrinos is of the order , for light neutrino masses . That means that the heavy neutrinos decouple form low-energy processes of CLFV in the SM with right-handed neutrinos, giving rise to extremely suppressed and unobservable rates.
One way to overcome these difficulties is to impose the presence of the approximate lepton flavor symmetries [99, 100, 101, 102, 103, 104, 105] in the theory. These symmetries result in a specific structure of Yukawa matrices which, if exact, can provide massless light neutrinos regardless of the masses of heavy neutrinos, so that
Small neutrino masses can then be reproduced by breaking the imposed symmetry by just the right amount. This scenario allows the heavy neutrino mass scale to be as low as . Unlike in the usual seesaw scenario, the light-to-heavy neutrino mixings are not correlated to the light neutrino masses . Instead, are free parameters, constrained by experimental limits on the deviations of the and -boson couplings to leptons with respect to their SM values [106, 107, 108, 109].
Approximate lepton flavor symmetries do not restrict the size of the LFV, and so potentially large phenomena of CLFV may be predicted. This feature is quite generic both in the SM  and in the MSSM [111, 112] extended with low-scale right-handed neutrinos. This new source of LFV, in addition to the one resulting from the frequently considered soft SUSY breaking sector [113, 114, 115, 116, 117, 118, 119], will be in particular interest in the study provided in this thesis.
2.3 MSSM extended with right-handed neutrinos
The SM and the MSSM extended by low-scale right-handed neutrinos in the presence of the approximate lepton-number symmetries will be denoted by SM and MSSM, respectively. Although some of the results displayed in this thesis may be applicable to the more general soft SUSY breaking scenarios, this study will be performed within the mSUGRA framework.
The MSSM has some interesting features compared with the MSSM. In particular, the heavy singlet sneutrinos may emerge as a new viable candidates of cold dark matter [120, 121, 122, 123, 124]. In addition, the mechanism of low-scale resonant leptogenesis [125, 126, 127, 128, 129] could provide a possible explanation for the observed baryon asymmetry in the Universe, as the parameter space for successful electroweak baryogenesis gets squeezed by the current LHC data [130, 131].
Given the multitude of quantum states mediating LFV in the MSSM, the predicted values for observables of CLFV in this model turn out to be generically larger than the corresponding ones in the SM, except possibly for [111, 112], where . The origin of suppression for the latter branching ratios may partially be attributed to the SUSY no-go theorem due to Ferrara and Remiddi , which states that the magnetic dipole moment operator necessarily violates SUSY and it must therefore vanish in the supersymmetric limit of the theory.
In this section, we will describe the leptonic sector of the MSSM and introduce the neutrino Yukawa structure of two baseline scenarios based on approximate lepton-number symmetries and universal Majorana masses at the GUT scale. These scenarios will be used to present generic predictions of the CLFV within the framework of mSUGRA, and to analyze the anomalous magnetic and electric dipole moments within the same framework.
The leptonic superpotential part of the MSSM reads:
where , , and denote the two Higgs-doublet superfields, the three left- and right-handed charged-lepton superfields and the three right-handed neutrino superfields, respectively. The Yukawa couplings and the Majorana mass parameters form complex matrices. Here, the Majorana mass matrix is taken to be SO(3)-symmetric at the scale, i.e. .
In the low-scale seesaw models models with the presence of approximate lepton symmetries, the neutrino induced LFV transitions from a charged lepton to another charged lepton are functions of the ratios [110, 133, 134, 135, 136]
and are not constrained by the usual seesaw factor , where is the vacuum expectation value (VEV) of the Higgs doublet , with . The mixing matrix that occurs in the interaction of the bosons with the charged leptons and the three heavy neutrinos is defined in Appendix A. It is important to note that the LFV parameters do not directly depend on the RGE evolution of the soft SUSY-breaking parameters, except through the VEV defined at the minimum of the Higgs potential.
In the electroweak interaction basis , the neutrino mass matrix in the MSSM takes on the standard seesaw type-I form:
where and are the Dirac- and Majorana-neutrino mass matrices, respectively. Complex conjugation of matrix is a consequence of the Majorana mass term in the superpotential (2.27). In this thesis, we consider two baseline scenarios of neutrino Yukawa couplings. The first one realizes a U(1) leptonic symmetry [125, 126, 127] and is given by
In the second scenario, the structure of the neutrino Yukawa matrix is motivated by the discrete symmetry group and has the following form :
In Eqs (2.30) and (2.31), the Yukawa parameters , and are assumed to be real. As was explained in the previous section, the small neutrino masses can be obtained by adding small symmetry-breaking terms into these matrices thus making the above mentioned symmetries approximate rather than exact. The predictions for CLFV observables, however, remain independent of the flavor structure of these small terms, needed to fit the low-energy neutrino data. For this reason, the particular symmetry breaking patterns of the above two baseline Yukawa scenarios will not be discussed in this thesis.
Another source of LFV in the models under consideration comes from sneutrino interactions. Specifically, the sneutrino mass Lagrangian in flavor and mass bases is given by
where is a Hermitian mass matrix in the flavor basis and is the corresponding diagonal mass matrix in the mass basis. More explicitly, in the flavor basis , the sneutrino mass matrix may be cast into the following form:
where the block entries are the matrices, namely
Here, , and are soft SUSY-breaking matrices associated with the left-handed slepton doublets, the right-handed sneutrinos and their trilinear couplings, respectively.
In the supersymmetric limit, all the soft SUSY-breaking matrices are equal to zero, and . As a consequence, the sneutrino mass matrix can be expressed in terms of the neutrino mass matrix in (2.29) as follows:
resulting with the expected equality between neutrino and sneutrino mixings. Sneutrino LFV mixings do depend on the RGE evolution of the MSSM parameters, but unlike the LFV mixings induced by soft SUSY-breaking terms, the sneutrino LFV mixings do not vanish at the GUT scale.
The sneutrino LFV mixings are obtained as combinations of unitary matrices which diagonalize the sneutrino, slepton and chargino mass matrices. It is interesting to notice that in the diagonalization of the sneutrino mass matrix in (2.34), the sneutrino fields , and their complex conjugates , are treated independently. As a result, the expressions for and , in terms of the real-valued mass eigenstates , are not manifestly complex conjugates to and , thus leading to a two-fold interpretation of the flavor basis fields,
where and , with and . For this reason, in Appendix A we include all equivalent forms in which Lagrangians, such as and , can be written down.
Finally, a third source of LFV in the MSSM comes from soft SUSY-breaking LFV terms [113, 115]. These LFV terms are induced by RGE running and, in the mSUGRA framework, vanish at the GUT scale. Their size strongly depends on the interval of the RGE evolution from the GUT scale to the universal heavy neutrino mass scale .
All the three different mechanisms of LFV, mediated by heavy neutrinos, heavy sneutrinos and soft SUSY-breaking terms, depend explicitly on the neutrino Yukawa matrix and vanish in the limit .
We will end this chapter with a technical remark. The diagonalization of sneutrino mass matrix and the resulting interaction vertices will be evaluated numerically, without approximations. To perform the diagonalization of numerically, the method developed in Ref  for the neutrino mass matrix will be used. This method becomes very efficient if one of the diagonal submatrices has eigenvalues larger than the entries in all other submatrices. It will therefore be assumed that the heavy neutrino mass scale is of the order of, or larger than the scale of the other mass parameters in the MSSM.
Chapter 3 Charged lepton flavor violation
In this chapter, the results and key details regarding the calculations for a number of CLFV observables in the MSSM will be presented.
In the first section, the analytical results for the amplitudes of CLFV decays and , as well as their branching ratios will be given. Second section gives analytical expressions for the neutrinoless three-body decays pertinent to muon and tau decays. Third section will deal with coherent conversion in nuclei, giving analytical results for transition amplitudes. All analytical results are expressed in terms of one-loop functions and composite form factors defined in the appendices at the end of this thesis.
Finally, last section will present the numerical results for above mentioned processes, accompanied by the brief description of the numerical methods used and corresponding discussion regarding the very results.
These results are presented in Ref .
3.1 The Decays and
At the one-loop level, the effective and couplings are generated by the Feynman graphs shown in Fig 3.1. The general form of the transition amplitudes associated with these effective couplings is given by
where , , is the electromagnetic coupling constant, is the -boson mass, is the weak mixing angle and is the photon momentum. The form factors , , , and receive contributions from heavy neutrinos , heavy sneutrinos and RGE induced soft SUSY-breaking terms. The analytical expressions for these three individual contributions are given in Appendix C. Note that, according to the normalization used, the composite form factors <