The Three Faces of $U(3)$

Abstract

SU(3) × U (1) as a gauge symmetry is known to be phenomenologically problematic unless the U (1) factor lives in a "dark" or "hidden" sector. By contrast, U (3) is a semi-direct product group that is characterized by non-trivial homomorphisms mapping U (1) into the automorphism group of SU (3). Since the semi-direct product includes mutual coupling among all gauge field mass-energy, U (3) at least has a chance of being phenomenologically sensible as a gauge symmetry outside of any dark or hidden sector. For U (3), there are three different non-trivial homomorphisms that induce three separate defining representations. In a toy model of U (3) Yang-Mills (endowed with a suitable inner product) coupled to massive fermions, this renders three distinct covariant derivatives acting on a single matter field. By employing a mod 3 permutation of the vector space carrying the defining representation induced by a "large" gauge transformation, the three covariant derivatives and one matter field can alternatively be expressed as a single covariant derivative acting on three distinct species of matter fields possessing the same U (3) quantum numbers but different renormalized masses. One can interpret this as three generations of matter fields.