The objective of this thesis is to derive a supersymmetric Lagrangian for fermionic fields with mass dimension one and to discuss their coupling to the O'Raifeartaigh model which is the simplest model permitting supersymmetry breaking. In addition it will be shown that eigenspinors of the charge conjugation operator (ELKO) exhibit a different transformation behaviour under discrete symmetries than previously assumed.<p> The calculations confirm that ELKO spinors are not eigenspinors of the parity operator and satisfy (<i>CPT</i>)² = - 1 which identifies them as representation of a nonstandard Wigner class. However, it is found that ELKO spinors transform symmetrically under parity instead of the previously assumed asymmetry. Furthermore, it is demonstrated that ELKO spinors transform asymmetrically under time reversal which is opposite to the previously reported symmetric behaviour. These changes affect the (anti)commutation relations that are satisfied by the operators acting on ELKO spinors. Therefore, ELKO spinors satisfy the same (anti)commutation relations as Dirac spinors, even though they belong to two different representations of the Lorentz group.<p> Afterwards, a supersymmetric model for fermionic fields with mass dimension one based on a general superfield with one spinor index is formulated. It includes the systematic derivation of all associated chiral and anti-chiral superfields up to third order in covariant derivatives. Starting from these fundamental superfields a supersymmetric on-shell Lagrangian that contains a kinetic term for the fermionic fields with mass dimension one is constructed. This on-shell Lagrangian is subsequently used to derive the on-shell supercurrent and to successfully formulate a consistent second quantisation for the component fields. In addition, the Hamiltonian in position space that corresponds to the supersymmetric Lagrangian is calculated. As the Lagrangian is by construction supersymmetric and the second quantisation of the component fields is consistent with their general supertranslations, the Hamiltonian is positive definite. This is confirmed by the results for the Hamiltonian in momentum space and the derivation of the creation and annihilation operators in momentum space. Based on these results, fermionic fields with mass dimension one represent an intriguing candidate for supersymmetric dark matter.<p> As an application the coupling of the fermionic fields with mass dimension one to the O'Raifeartaigh model is discussed. It turns out that the coupled model has two distinct solutions. The first solution representing a local minimum of the superpotential spontaneously breaks supersymmetry in perfect analogy to the O'Raifeartaigh model. The second solution is more intriguing as it corresponds to a global minimum of the superpotential. In this case the coupling to the fermionic sector restores supersymmetry. This is, however, achieved at the cost of breaking Lorentz invariance. Finally, the mass matrices for the multiplets of the coupled model are presented. It turns out that it contains two bosonic triplets and one fermionic doublet which are mass multiplets. In addition it contains a massless fermionic doublet as well as one fermionic triplet which is not a mass multiplet but rather an interaction multiplet that contains component fields of different mass dimension.<p> These results show that the presented model for fermionic fields with mass dimension one is a viable candidate for supersymmetric dark matter that could be accessible to experiments in the near future.

The objective of this thesis is to derive a supersymmetric Lagrangian for fermionic fields with mass dimension one and to discuss their coupling to the O'Raifeartaigh model which is the simplest model permitting supersymmetry breaking. In addition it will be shown that eigenspinors of the charge conjugation operator (ELKO) exhibit a different transformation behaviour under discrete symmetries than previously assumed.<p> The calculations confirm that ELKO spinors are not eigenspinors of the parity operator and satisfy (<i>CPT</i>)² = - 1 which identifies them as representation of a nonstandard Wigner class. However, it is found that ELKO spinors transform symmetrically under parity instead of the previously assumed asymmetry. Furthermore, it is demonstrated that ELKO spinors transform asymmetrically under time reversal which is opposite to the previously reported symmetric behaviour. These changes affect the (anti)commutation relations that are satisfied by the operators acting on ELKO spinors. Therefore, ELKO spinors satisfy the same (anti)commutation relations as Dirac spinors, even though they belong to two different representations of the Lorentz group.<p> Afterwards, a supersymmetric model for fermionic fields with mass dimension one based on a general superfield with one spinor index is formulated. It includes the systematic derivation of all associated chiral and anti-chiral superfields up to third order in covariant derivatives. Starting from these fundamental superfields a supersymmetric on-shell Lagrangian that contains a kinetic term for the fermionic fields with mass dimension one is constructed. This on-shell Lagrangian is subsequently used to derive the on-shell supercurrent and to successfully formulate a consistent second quantisation for the component fields. In addition, the Hamiltonian in position space that corresponds to the supersymmetric Lagrangian is calculated. As the Lagrangian is by construction supersymmetric and the second quantisation of the component fields is consistent with their general supertranslations, the Hamiltonian is positive definite. This is confirmed by the results for the Hamiltonian in momentum space and the derivation of the creation and annihilation operators in momentum space. Based on these results, fermionic fields with mass dimension one represent an intriguing candidate for supersymmetric dark matter.<p> As an application the coupling of the fermionic fields with mass dimension one to the O'Raifeartaigh model is discussed. It turns out that the coupled model has two distinct solutions. The first solution representing a local minimum of the superpotential spontaneously breaks supersymmetry in perfect analogy to the O'Raifeartaigh model. The second solution is more intriguing as it corresponds to a global minimum of the superpotential. In this case the coupling to the fermionic sector restores supersymmetry. This is, however, achieved at the cost of breaking Lorentz invariance. Finally, the mass matrices for the multiplets of the coupled model are presented. It turns out that it contains two bosonic triplets and one fermionic doublet which are mass multiplets. In addition it contains a massless fermionic doublet as well as one fermionic triplet which is not a mass multiplet but rather an interaction multiplet that contains component fields of different mass dimension.<p> These results show that the presented model for fermionic fields with mass dimension one is a viable candidate for supersymmetric dark matter that could be accessible to experiments in the near future.

Orientador: Andrei Mikhailov / Resumo: We review some relevant examples for String Theory of non-linear sigma models. These are bosonic strings propagating in curved background, the Wess-Zumino-Witten model and superstrings in flat and AdS superspace. The mathematical tools required for the study of these models (e.g. topological quantization, Cartan geometry, Lie superalgebras and geometry on coset spaces) are also described. Throughout the dissertation we have focused on classical aspects of these models such as the construction of the action and its symmetries where conditions for holomorphic symmetry of the bosonic string case were found. / Mestre

Submitted by DENNIS EDUARDO ZAVALETA FERRO (dennis.ttf@gmail.com) on 2017-09-18T13:53:32Z No. of bitstreams: 1 msc-thesis.pdf: 505892 bytes, checksum: c724040eff49813a08ac27d82fce286b (MD5) / Approved for entry into archive by Monique Sasaki (sayumi_sasaki@hotmail.com) on 2017-09-19T19:20:32Z (GMT) No. of bitstreams: 1 ferro_dez_me_ift.pdf: 505892 bytes, checksum: c724040eff49813a08ac27d82fce286b (MD5) / Made available in DSpace on 2017-09-19T19:20:32Z (GMT). No. of bitstreams: 1 ferro_dez_me_ift.pdf: 505892 bytes, checksum: c724040eff49813a08ac27d82fce286b (MD5) Previous issue date: 2016-08-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / We review some relevant examples for String Theory of non-linear sigma models. These are bosonic strings propagating in curved background, the Wess-Zumino-Witten model and superstrings in flat and AdS superspace. The mathematical tools required for the study of these models (e.g. topological quantization, Cartan geometry, Lie superalgebras and geometry on coset spaces) are also described. Throughout the dissertation we have focused on classical aspects of these models such as the construction of the action and its symmetries where conditions for holomorphic symmetry of the bosonic string case were found. / Nesta dissertação estudamos alguns exemplos de modelos sigma não lineares em Teoria de cordas. Estes são a corda bosónica se propagando em espaços curvos, o modelo Wess-Zumino-Witten e supercordas em superespaço plano e AdS. As ferramentas matemáticas que se precisam para o estudo destes modelos (e.g. quantização topológica, geometria de Cartan, super-álgebras de Lie e geometria em espaços coset) também são descritas. Ao longo desta dissertação focamos os aspectos clássicos destes modelos tais como a construção da ação e suas simetrias onde condições para serem estas holomorficas no caso da corda bosónica foram achadas.