Gravitino and scalar $\tau$-lepton decays in supersymmetric models with broken R-parity

Mildly broken R-parity is known to provide a solution to thecosmological gravitino problem in supergravity extensions of theStandard Model. In this work we consider new effects occurring in theR-parity breaking Minimal Supersymmetric Standard Model includingright-handed neutrino superfields. We calculate the most generalvacuum expectation values of neutral scalar fields including left- andright-handed scalar neutrinos. Additionally, we derive thecorresponding mass mixing matrices of the scalar sector. Werecalculate the neutrino mass generation mechanisms due toright-handed neutrinos as well as by cause of R-parity breaking.Furthermore, we obtain a, so far, unknown formula for the neutrinomasses for the case where both mechanisms are effective. We thenconstrain the couplings to bilinear R-parity violating couplings inorder to accommodate R-parity breaking to experimental results. Inorder to constrain the family structure with a U(1)_Q flavor symmetrywe furthermore embed the particle content into an SU(5) Grand UnifiedTheory. In this model we calculate the signal of decaying gravitinodark matter as well as the dominant decay channel of a likely NLSP,the scalar τ-lepton. Comparing the gravitino signal with results ofthe Fermi Large Area Telescope enables us to find a lower bound on thedecay length of scalar τ-leptons in collider experiments