Exponentiating Higgs

TL;DR

This paper introduces a novel formulation of the Higgs mechanism that relaxes the positivity constraint on the Higgs field, addressing key issues in both perturbative and non-perturbative contexts, and explores scalar theories with exponential interactions.

Contribution

It proposes a new approach to the Higgs mechanism using polar decomposition with a Higgs field in R, avoiding the usual positivity constraint, and analyzes scalar theories with exponential interactions.

Findings

Relaxation of the non-negative definiteness condition for the Higgs field.

The polar decomposition involves U in SU(2)/Z_2 and the Higgs field in R.

Scalar theories with exponential interactions are free for D>2.

Abstract

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.