Potential Minimization in Left-Right Symmetric Models

We study the Higgs potentials in the Left-Right symmetric model for various choices of Higgs fields. We give emphasis to the cases when the Higgs field $\xi=(2,2,15)$ is included to give the correct relations of quark masses and a singlet field $\eta$ which breaks the left-right parity. As special cases we also include $\xi^\prime=(2,2,15)$ and $\chi=(2,2,6)$ (which are interesting in the context of the three lepton decay mode of the proton) and field $\delta=(3,3,0)$ none of which acquire {\it vev}. We show that the linear couplings of these fields upon minimization put fine tuning conditions on the parameters of the model. We carry out the minimization of these potentials explicitly. In all the cases the relationship between the {\it vev$\;$}s of the left and right handed triplets $v_L$ and $v_R$ are given. The phenomenological consequences of this minimization regarding the neutrino masses are also studied.