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Baryon number nonconservation in the Standard Model
At first, investigations were carried to check if Sakharov’s conditions can be satisfied within the Standard Model. At least the second condition is satisfied in the Standard Model because CP is violated, as can be seen for instance in kaon mixing [138].
Baryon and lepton number are seemingly conserved, but they are only accidental symmetries. This conservation holds actually only at the perturbative level, while nonperturbatively, baryon and lepton number are broken by the electroweak sphalerons [139], in a way which is related to the anomalies of the associated currents.
The Standard scenario: leptogenesis with right-handed neutrinos
The “standard” leptogenesis scenario [163, 165] involves three heavy right-handed neutrinos, and relies on the Lagrangian of eq. (2.1.42). The Standard Model neutrinos have a mass matrix which is given by the seesaw formula of eq. (2.1.48).
The right-handed neutrinos are assumed to be hierarchical, that is M1 is much smaller than M2 and M3, so that N3 and N2 decay first, usually when the temperature of the Universe is around T M3 and T M2 respectively, and when N1 decays around T M1, the two other species have already disappeared.
An interesting feature of right-handed neutrinos is that they are gauge singlets. Contrary to other particles which are maintained close to thermodynamic equilibrium by fast gauge interactions, their density is usually far from its equilibrium value, which allows to satisfy Sakharov’s third condition. Besides, if there is no other interaction that creates them have to be created and brought to their equilibrium density by inverse decays like `H ! N and `cHc ! N. Because of this, and because of the hierarchy between right-handed neutrinos, any asymmetry generated in the decay of N3 and N2 would be erased to create N1. Thus, it is possible to focus on the decay of the lightest right-handed neutrino N1 only, starting from a time at which the two other species have already decayed and the lepton asymmetry is vanishing.
Leptogenesis with a scalar triplet: a general approach
The type II seesaw is one possible alternative to generate neutrino masses. Like the type I seesaw, it allows to implement leptogenesis [199–204]. Indeed, the decay of the scalar triplet violates lepton number by two units, which matches at least the second Sakharov’s condition. In this section, we study a general scenario of leptogenesis with a scalar triplet, before turning to a more specific model in the next one.
Introduction of flavour and spectator processes
Taking into account flavour effects and spectator processes – by spectator processes we mean processes that do not directly affect B − L – requires us to split the problem in several temperature ranges and keep only the relevant processes in each one of them, in order to simplify a problem otherwise tremendously difficult. For instance, we consider spectator processes, which come from Standard Model physics, to be either negligible or very fast, which is generally a good approximation except at transitions between two temperature ranges. When they are very fast, their effect can be encoded in relations between chemical potentials, which can be translated in relations between the asymmetries. Since the asymmetry stored in lepton doublets is affected by both Yukawa-mediated interactions of the type ` +tR ! e +Q3 and electroweaksphalerons, it is more convenient to study the evolution of the B − L density, which is unaffected by every Standard Model process. Then, since the washout terms due to new physics are still functions of the lepton doublet asymmetry `, we need to express the latter as a function of the asymmetry stored in B − L, or more precisely as a function of the asymmetries stored in the charges B/3−L, just as we expressed H as a function of ` and previously.
Pseudo-Goldstone fermion Lagrangian
As was said previously, a supersymmetric framework with broken R-parity can give rise to nonzero neutrino masses. We consider here such a scenario, in which the masses of neutrinos arise from their mixing with the fermionic partners of neutral bosons. As was mentioned in 2.1.1, some anomalies in neutrino experiments may be explained by the existence of a sterile neutrino with a mass around 1 eV. However, as we said in 2.1.2, a fermionic singlet could have an arbitrarily large Majorana mass, so that such a small value should be justified. There are a few possibilities for this.
(i) For instance, in the singular seesaw mechanism [237, 238], there are three righthanded neutrinos but their mass matrix has only rank 2 because of some symmetry, so that there is an additional light mass eigenstate.
(ii) In theories with more than four space-time dimensions, charged fields live on a 4-dimension brane, but since right-handed neutrinos are gauge singlets, they can live in the bulk. They can then be decomposed in an infinite tower of Kaluza- Klein modes. Standard Model neutrinos get a Dirac mass mostly with zero modes, while sterile neutrinos are mostly made of higher modes [239–241], with masses proportional to 1/R 10−3 eV, R being the compactification radius of the extra dimension. In their minimal realization, such models are now excluded because they are in disagreement with the energy spectrum of solar neutrinos measured by Super-Kamiokande [89].
(iii) In the context of superstring theories, the existence of neutral fields (moduli), which are massless at the perturbative level, is expected. The fermionic components of these fields get a small mass and mixing with neutrino from nonperturbative effects in supergravity and play the role of sterile neutrinos [242, 243].
(iv) Here, similarly to what was proposed in refs. [244–246], the lightness of the new state is justified by the fact that it is the supersymmetric partner of a pseudo- Goldstone boson.
The motivations for these models were initially the following. Before the Super- Kamiokande results, solar neutrinos oscillations could be explained by active-sterile oscillations, which was ruled out since then. Active-sterile oscillations were also thought to be the best explanation for the LSND anomaly. Moreover, light sterile neutrinos were also seen as candidates for hot dark matter, which is now excluded. On the other hand, recent papers on the topic focus on phenomenological aspect: assuming the existence of light sterile neutrinos, they study the experimental constraints on these neutrinos or try to fit the anomalies (see for instance refs. [247, 248, 104]). Here, we study a model that aims at explaining the origin of the sterile neutrino and the active-sterile mixing in the light of recent experimental results.
Chargino and neutralino mass matrices
In the MSSM, because of the electroweak symmetry breaking, the higgsinos mix with the gauginos of the SU(2) × U(1) gauge group. The mass eigenstates arising from the mixing of charged fields ˜H + u , ˜H − d , ˜W ± are referred to as charginos, while those coming from the mixing of neutral fields ˜H 0 u, ˜H 0 d , ˜W 3 and ˜B (or equivalently ˜ Z0 and ˜ ) are called neutralinos. In the present scenario, since R-parity is broken in the lepton sector, charged leptons and neutrinos mix with charginos and neutralinos respectively. In addition to that, the pseudo-Goldstone fermion also mixes with neutralinos. The mass matrices of the fermions receive three different contributions at tree level. Oneloop corrections may also contribute to the final expression of the neutrino mass matrix [235, 250–261]. However, we assume here that they are negligible, as it is the case if R-parity-violating parameters are small enough. For instance, ref. [262] gives the following estimation for the contribution to the neutrino mass matrix from the ye , assuming that all slepton masses are of the same order of magnitude ˜m and that the mixing between left- and right-handed sleptons is roughly given by (m2 LR) ‘ ˜ mye vd.
Extended gauge mediation
From now on, we will focus on gauge mediation and its extensions. As already said, minimal gauge mediation is predictive and naturally forbids flavour violation. However, the measured mass of the Higgs boson turns out to be problematic, because it is a bit too large to fit naturally in this framework. This stems from the fact that, if one wants to restrict fine-tuning, a large mass for the Higgs boson requires large A-terms, and this condition is not satisfied within minimal gauge mediation. Extended gauge mediation, in which direct couplings between messengers and matter superfields are allowed, is a way to overcome this issue.
Table of contents :
1 A review of the Standard Model
1.1 Historical overview
1.2 The framework
1.2.1 The gauge theory
1.2.2 The Higgs mechanism and its consequences
1.3 Unanswered questions
2 The lepton sector
2.1 Neutrino masses
2.1.1 Neutrino oscillations and their interpretation
2.1.2 The theoretical puzzle
2.2 Charged lepton flavour violation
2.2.1 The Glashow-Iliopoulos-Maiani mechanism
2.2.2 New physics and charged lepton flavour violation
3 Leptogenesis
3.1 From baryogenesis to leptogenesis
3.1.1 Baryon number nonconservation in the Standard Model
3.1.2 Neutrino masses and leptogenesis
3.1.3 The Standard scenario: leptogenesis with right-handed neutrinos
3.2 Leptogenesis with a scalar triplet: a general approach
3.2.1 The setup
3.2.2 A simplified model
3.2.3 Introduction of flavour and spectator processes
3.2.4 Numerical approach
3.3 A more predictive scenario
3.3.1 The setup
3.3.2 Boltzmann equations
3.3.3 Numerical approach
4 Sterile neutrinos as pseudo-Goldstone fermions
4.1 Introduction to Supersymmetry
4.1.1 Superspace and superfields
4.1.2 Supersymmetric theories
4.1.3 The MSSM
4.2 Pseudo-Goldstone fermion Lagrangian
4.2.1 The framework
4.2.2 Chargino and neutralino mass matrices
4.3 The neutrino mass matrix
4.3.1 Parametrization
4.3.2 Numerical search for solutions
5 Supersymmetric seesaw and gauge mediation
5.1 Introduction to Supersymmetry breaking
5.1.1 The vacuum of supersymmetric theories
5.1.2 Explicit Models of supersymmetry breaking
5.1.3 Supersymmetry broken in a separate sector
5.1.4 Minimal gauge mediation
5.2 Extended gauge mediation
5.2.1 The electroweak symmetry breaking in the MSSM
5.2.2 Soft terms from wavefunction renormalization
5.2.3 General matter-messenger mixing
5.3 Application: the supersymmetric seesaw
5.3.1 Flavour violation
5.3.2 Type I seesaw
5.3.3 Type II seesaw
5.3.4 Additional comments
5.3.5 Conclusion
Appendix
A Boltzmann equations
A.1 Classical derivation
A.2 Decays and scattering rates
A.3 CP violating terms
A.4 The closed time-path formalism
B Loop integrals in wavefunction renormalization
B.1 Contribution of the superpotential
B.2 Mixed gauge-superpotential contribution
B.3 Holomorphic wavefunction
Synopsis
I Le Modèle Standard
II Le Mécanisme de seesaw
III La leptogenèse avec un triplet scalaire
IV Introduction à la supersymétrie
V Pseudo-fermions de Goldstone et neutrinos stériles
VI Médiation de jauge étendue
