## ABSTRACT

Any renormalizable supersymmetric field theory involving just scalar and spinor fields can then be formulated most succinctly in terms of products of chiral superfields. However, the quarks and leptons, and therefore the squarks and sleptons, all participate in gauge theories, quantum chromodynamics and electroweak theory, which of course involve gauge vector bosons. In this chapter, the authors construct the vector superfield; in general it includes four auxiliary scalar fields as well as an auxiliary fermion field. They investigate spontaneous symmetry breaking of a global and local U(1) gauge invariance in the context of a supersymmetric theory. The renormalized coupling constants necessarily depend upon the scale M used in their definition. The parts of the Lagrangian are written down using the techniques developed in this chapter.