Generating extremal neutrino mixing angles with Higgs family symmetries

The existence of maximal and minimal mixing angles in the neutrino mixing matrix motivates the search for extensions to the standard model that may explain these angles. A previous study [C. I. Low and R. R. Volkas, Phys. Rev. D 68,, 033007 (2003)], began a systematic search to find the minimal extension to the standard model that explains these mixing angles. It was found that in the minimal extensions to the standard model which allow neutrino oscillations, discrete unbroken lepton family symmetries only generate neutrino mixing matrices that are ruled out by experiment. This paper continues the search by investigating all models with two or more Higgs doublets and an Abelian family symmetry. It is found that discrete Abelian family symmetries permit, but cannot explain, maximal atmospheric mixing; however, these models can ensure ${\theta }_{13}=0$.