Lepton Flavor Violation in the Two Higgs Doublet Model type III

We consider the Two Higgs Doublet Model (2HDM) of type III which leads to Flavour Changing Neutral Currents (FCNC) at tree level in the leptonic sector. In the framework of this model we can have, in principle, two situations: the case (a) when both doublets acquire a vacuum expectation value different from zero and the case (b) when only one of them is not zero. In addition, we show that we can make two types of rotations for the flavor mixing matrices which generates four types of lagrangians, with the rotation of type I we recover the case (b) from the case (a) in the limit $\tan \beta \to \infty$, and with the rotation of type II we obtain the case (b) from (a) in the limit $\tan \beta \to 0.$Moreover, two of the four possible lagrangians correspond to the models of types I and II plus Flavor Changing (FC) interactions. The analitical expressions of the partial lepton number violating widths $\Gamma (\mu \to eee)$ and $\Gamma (\mu \to e\gamma)$are derived for the cases (a) and (b) and both types of rotations.$$In all cases these widths go asymptotically to zero in the decoupling limit for all Higgses. We present from our analysis upper bounds for the flavour changing transition $\mu \to e,$ and we show that such bounds are sensitive to the VEV structure and the type of rotation utilized.