The geometry of CP violation in Kaluza-Klein models

Abstract: We investigate the free, massless Dirac equation $Dψ= 0$ on a higher-dimensional manifold $M_4 \times K$ equipped with a submersion metric. These background metrics generalize the Kaluza ansatz. They encode 4D massive gauge fields and Higgs-like scalars, alongside the usual 4D metric and massless gauge fields. We show that the dimensional reduction of the Dirac equation on these backgrounds naturally violates CP symmetry in four dimensions. This provides a new geometric path to constructing models with intrinsic CP violation. In this framework, massive gauge fields can break CP for three different reasons: $i)$ a misalignment between the mass eigenspinors and the spinors in the representation basis; $ii)$ a new non-minimal term coupling 4D fermions to massive gauge fields; $iii)$ the presence of a non-abelian Pauli term. All this derives from the higher-dimensional Dirac equation. Technically, the paper uses the language of spin geometry and Riemannian submersions. Along the way, it develops detailed geometric descriptions of several constructions. It discusses gauge anomalies, fermion generations, and introduces a new Lie derivative of spinors along non-Killing vector fields induced by actions of compact groups.