  We consider two related formulations for mass generation in the $U(1)$
Higgs-Kibble model and in the Standard Model (SM). In the first model there are
no scalar self-interactions and, in the case of the SM, the formulation is
related to the normal subgroup of $G=SU(3)\times SU(2)\times U(1)$, generated
by $(e^{2\pi i/3}I,-I,e^{\pi i/3})\in G$, that acts trivially on all the fields
of the SM. The key step of our construction is to relax the non-negative
definiteness condition for the Higgs field due to the polar decomposition. This
solves several stringent problems, that we will shortly review, both in the
perturbative and non-perturbative formulations. We will show that the usual
polar decomposition of the complex scalar doublet $\Phi$ should be done with
$U\in SU(2)/Z_2\simeq SO(3)$, where $Z_2$ is the group generated by $-I$, and
with the Higgs field $\phi\in R$ rather than $\phi\in R_{\geq0}$. As a
byproduct, the investigation shows how Elitzur theorem may be avoided in the
usual formulation of the SM. It follows that the simplest lagrangian density
for the Higgs mechanism has the standard kinetic term in addition to the mass
term, with the right sign, and to a linear term in $\phi$. The other model
concerns the scalar theories with normal ordered exponential interactions. The
remarkable property of these theories is that for $D>2$ the purely scalar
sector corresponds to a free theory.
