Higher-dimensional routes to the Standard Model bosons

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 $K$.