This dissertation reviews the Standard Model formalism as well as the Lepton Flavour Violating (LFV) decay processes which cause its extension, known as the physics beyond the SM. Firstly, using the experimental bounds on three body LFV decays, the corresponding bounds on two body LFV decays are reviewed. The dynamical suppression of three body LFV decays due to momentum dependent couplings is also reviewed. Secondly, the role of the LFV decays to explain the LSND excess is discussed in detail, for which the experimental bounds on three body LFV decays, i.e. mu -> 3e are used to constraint the coupling tilde{g}_{Z_{mu e}}, which is needed to calculate the anomalous muon decay mu -> e u_lbar{ u}_l. Then comparing the effective coupling of anomalous muon decay to r>1.6times 10^{-3} [9809524], it is proved that LFV is not the correct hypothesis to explain the LSND excess. Finally, LFV decays at loop order are studied in Seesaw model of neutrino masses [PRL. 86 2502 (2001)] where the smallness of the Seesaw neutrino mass may be naturally realized with m_N (mass of right-handed singlet neutrinos) of order 1 TeV. It is shown that the Higgs mass of a new scalar doublet with lepton number L=-1 needed in the model has to be larger than 50 TeV to get the branching ratio of mu -> 3e to be consistent with the existing bound on mu -> 3e. This defeats the original motivation of the model, namely that there is no physics beyond the TeV energy scale.
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