Using N = 1 superspace techniques in four dimensions we show how to perturbatively compute the superpotential generated for the glueball superfield upon integrating out massive charged fields. The technique applies to arbitrary gauge groups and representations. Moreover we show that for U (N ) gauge theories admitting a large N expansion the computation dramatically simplifies and we prove the validity of the recently proposed recipe for computation of this quantity in terms of planar diagrams of matrix integrals. November, 2002

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We argue that the best way to determine horizontal symmetry is from neutrino mixing, and proceed to show that the only finite group capable of yielding the tri-bimaximal mixing for all Yukawa couplings is S 4 , or any group containing it. The method used is largely group theoretical, but it can be implemented by dynamical schemes in which the Higgs expectation values introduced to break S 4 spontaneously are uniquely determined up to an unknown scale for each.

There is a group-theoretical connection between fermion mixing matrices and minimal horizontal symmetry groups. Applying this connection to the tri-bimaximal neutrino mixing matrix, we show that the minimal horizontal symmetry group for leptons is uniquely S 4 , the permutation group of four objects.