  In order to investigate the Higgs mechanism nonperturbatively, we compute the
Gaussian effective potential (GEP) of the U(1) Higgs model ("scalar
electrodynamics"). We show that the same simple result is obtained in three
different formalisms. A general covariant gauge is used, with Landau gauge
proving to be optimal. The renormalization generalizes the "autonomous"
renormalization for lambda-phi^4 theory and requires a particular relationship
between the bare gauge coupling e_B and the bare scalar self- coupling
lambda_B. When both couplings are small, then lambda is proportional to e^4 and
the scalar/vector mass-squared ratio is of order e^2, as in the classic 1-loop
analysis of Coleman and Weinberg. However, as lambda increases, e reaches a
maximum value and then decreases, and in this "nonperturbative" regime the
Higgs scalar can be much heavier than the vector boson. We compare our results
to the autonomously renormalized 1-loop effective potential, finding many
similarities. The main phenomenological implication is a Higgs mass of about 2
TeV.
