Towards a unified treatment of Yang-Mills and Higgs fields.

Abstract

Starting from a noncommutative algebra $\mathcal{A}$ of the form $\mathcal{C}\ensuremath{\bigotimes}\mathcal{M}$, where $\mathcal{C}$ is the algebra of smooth functions on space-time and $\mathcal{M}$ is the algebra of $n\ifmmode\times\else\texttimes\fi{}n$ Hermitian matrices, we construct an exterior algebra of differential forms over $\mathcal{A}$. We use the one-forms of this algebra to describe Yang-Mills and Higgs fields on a similar footing and construct a Lagrangian from its two-forms. We show how, in the resulting geometrical description, a Higgs potential that leads to spontaneous symmetry breaking arises naturally. We discuss the application of this formalism to the bosonic sectors of the standard electroweak theory and a grand-unified model based on SU(5)\ensuremath{\bigotimes}U(1).