The geometry of CP symmetry 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 metric on $M_4$ and massless gauge fields. In this framework, the interactions of 4D gauge fields with spinors are determined by the Kosmann derivatives of the spinors' internal components. For massive gauge fields, however, those derivatives do not commute with the internal Dirac operator. This causes a natural misalignment between the mass eigenspinors and the spinor representation bases. Thus, a complex, infinite-dimensional, CKM-like matrix emerges directly from the Dirac equation on $M_4 \times K$. Examining this general framework, we prove that it nevertheless cannot produce CP violation in 4D for any compact $K$ with constant internal geometry. The same holds for the Weyl equation unless $\dim K = 2 \pmod{4}$. Using the language of spin geometry, we develop detailed descriptions of parity and conjugation symmetries in KK models. We find that the gauge representations are always self-conjugate when $\dim K \neq 1 \pmod{4}$ (hence anomaly-free), discuss fermion generations, and introduce a new Lie derivative of spinors along non-Killing vector fields induced by actions of compact groups.