Higher-dimensional routes to the Standard Model bosons

TL;DR

This paper explores a higher-dimensional geometric model that naturally incorporates Standard Model gauge bosons and a Higgs-like mechanism, deriving mass terms and coupling constants from a modified Kaluza-Klein framework.

Contribution

It introduces a novel higher-dimensional setup with a non-fully right-invariant metric on SU(3), deriving Standard Model features including gauge bosons, Higgs field dynamics, and mass generation from classical Einstein-Hilbert action.

Findings

Derivation of Standard Model gauge bosons from higher-dimensional geometry

Explicit calculation of Higgs-like and gauge boson masses

Relations between coupling constants and metric parameters

Abstract

In the old spirit of Kaluza-Klein, we consider a spacetime of the form $P = M_4 \times K$, where $K$ is the Lie group $\mathrm{SU}(3)$ equipped with a left-invariant metric that is not fully right-invariant. This metric has a ${\rm U}(1) \times \mathrm{SU}(3)$ isometry group, corresponding to the massless gauge bosons, and depends on a parameter $\phi$ with values in a subspace of $\mathfrak{su}(3)$ isomorphic to $\mathbb{C}^2$. It is shown that the classical Einstein-Hilbert Lagrangian density $R_P - 2 \Lambda$ on the higher-dimensional manifold $P$, after integration over $K$, encodes not only the Yang-Mills terms of the Standard Model over $M_4$, as in the usual Kaluza-Klein calculation, but also a kinetic term $|{\mathrm d}^A \phi|^2$ identical to the covariant derivative of the Higgs field. For $\Lambda$ in an appropriate range, it also encodes a potential $V(| \phi|^2)$ having absolute minima with $|\phi_0|^2 \neq 0$, thereby inducing mass terms for the remaining gauge bosons. The classical masses of the resulting Higgs-like and gauge bosons are explicitly calculated as functions of the vacuum value $|\phi_0|^2$ in the simplest version of the model. In more general versions, the classical values of the strong and electroweak gauge coupling constants are given as functions of the parameters of the left-invariant metric on $K$.