Fermionic fields with mass dimension one as supersymmetric extension of the O'Raifeartaigh model

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. The calculations confirm that ELKO spinors are not eigenspinors of the parity operator and satisfy (CPT)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. 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. 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. 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