# The geometry of CP violation in Kaluza-Klein models

**Authors:** Joao Baptista  
**arXiv:** [2601.08902](https://arxiv.org/abs/2601.08902)  
**Date:** 2026-01-13  
**Summary page:** https://www.summarizepaper.com/en/paper/2601.08902v1/

## Key points

- The paper explores the free, massless Dirac equation $Dψ= 0$ within a higher-dimensional manifold $M_4 \times K$ with a submersion metric.
- The study reveals a natural violation of CP symmetry in four dimensions by reducing the Dirac equation on specific backgrounds.
- Massive gauge fields can induce CP violation through three mechanisms: misalignment between mass eigenspinors and spinors, non-minimal term coupling 4D fermions to massive gauge fields, and presence of a non-abelian Pauli term.
- Phenomena stem from the higher-dimensional Dirac equation and are elucidated using concepts from spin geometry and Riemannian submersions.
- The paper discusses gauge anomalies, fermion generations, and introduces a novel Lie derivative of spinors along non-Killing vector fields induced by actions of compact groups.

## Summary

The paper "The Geometry of CP Violation in Kaluza-Klein Models" by Joao Baptista explores the free, massless Dirac equation $Dψ= 0$ within a higher-dimensional manifold $M_4 \times K$ with a submersion metric. This metric extends the Kaluza ansatz and encompasses 4D massive gauge fields, Higgs-like scalars, the standard 4D metric, and massless gauge fields. By reducing the Dirac equation on these backgrounds, the study reveals a natural violation of CP symmetry in four dimensions. This discovery offers a novel geometric approach to constructing models with intrinsic CP violation. Within this framework, it is demonstrated that massive gauge fields can induce CP violation through three distinct mechanisms: first, via a misalignment between mass eigenspinors and spinors in the representation basis; secondly, through a new non-minimal term that couples 4D fermions to massive gauge fields; and finally, due to the presence of a non-abelian Pauli term. These phenomena stem directly from the higher-dimensional Dirac equation and are elucidated using concepts from spin geometry and Riemannian submersions. The paper also discusses gauge anomalies and fermion generations while introducing a novel Lie derivative of spinors along non-Killing vector fields induced by actions of compact groups. With its comprehensive analysis and theoretical framework rooted in advanced mathematical concepts, this research significantly contributes to our understanding of CP violation within Kaluza-Klein models.