Exponentiating Higgs

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.