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The geometry of CP violation in Kaluza-Klein models

Joao Baptista

TL;DR

This work demonstrates that the massless Dirac equation on a higher-dimensional manifold $M_4\times K$ with a submersion metric can, upon dimensional reduction, yield 4D equations that inherently violate CP symmetry. It develops a rigorous spin-geometry framework based on Riemannian submersions, generalized reflections, conjugations, and compact-group actions to decompose the higher-dimensional Dirac operator into 4D mass terms, gauge couplings, and Pauli-type interactions, all controlled by the internal geometry. CP violation in 4D arises from three KK-origin mechanisms: misalignment between mass eigenspinors and gauge representations, a non-minimal coupling to massive gauge fields, and a non-abelian Pauli term, with additional contributions from non-Killing internal vector fields that mix generations. The results offer a geometric path to intrinsic CP violation in KK-type models, connecting to gauge anomalies and potential fermion generation structures, and suggesting concrete links between higher-dimensional geometry and observable CP-violating phenomena.

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.

The geometry of CP violation in Kaluza-Klein models

TL;DR

This work demonstrates that the massless Dirac equation on a higher-dimensional manifold with a submersion metric can, upon dimensional reduction, yield 4D equations that inherently violate CP symmetry. It develops a rigorous spin-geometry framework based on Riemannian submersions, generalized reflections, conjugations, and compact-group actions to decompose the higher-dimensional Dirac operator into 4D mass terms, gauge couplings, and Pauli-type interactions, all controlled by the internal geometry. CP violation in 4D arises from three KK-origin mechanisms: misalignment between mass eigenspinors and gauge representations, a non-minimal coupling to massive gauge fields, and a non-abelian Pauli term, with additional contributions from non-Killing internal vector fields that mix generations. The results offer a geometric path to intrinsic CP violation in KK-type models, connecting to gauge anomalies and potential fermion generation structures, and suggesting concrete links between higher-dimensional geometry and observable CP-violating phenomena.

Abstract

We investigate the free, massless Dirac equation on a higher-dimensional manifold 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: a misalignment between the mass eigenspinors and the spinors in the representation basis; a new non-minimal term coupling 4D fermions to massive gauge fields; 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.
Paper Structure (21 sections, 23 theorems, 149 equations)

This paper contains 21 sections, 23 theorems, 149 equations.

Table of Contents

  1. Introduction and overview of results
  2. Introduction
  3. Overview of the main results
  4. Spinors on Riemannian submersions
  5. Riemannian submersions
  6. Spinors on $M_4 \times K$
  7. Horizontal and vertical spinors
  8. Decomposing the higher-dimensional Dirac operator
  9. Spinor symmetries on $M\times K$ induced by reflections and parity transformations on $M$
  10. Conjugation symmetries on spinor bundles
  11. Actions of compact groups on $K$ and its spinors
  12. Representation spaces versus $\slashed{D}$-eigenspaces
  13. Dimensionally reduced equations and CP violation
  14. Dimensional reduction of the Dirac equation on $M_4 \times K$
  15. The CP-transformed equation in 4D
  16. ...and 6 more sections

Key Result

Proposition 2.1

Consider a spinor on $P$ of the form $\Psi = \varphi^\mathcal{H}(x) \otimes \psi(x,y)$, as in TensorHDSpinor. The action of the higher-dimensional Dirac operator on $\Psi$ can be locally decomposed as Here $\mathscr{L}_{e_a}$ denotes the derivative KLDerivative of spinors on $K$; $\mathrm{div} (e_a)$ denotes the divergence of the internal vector field $e_a$ with respect to $g_K$; and $|g_K |$ is

Theorems & Definitions (35)

  • Proposition 2.1
  • Corollary 2.2
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • ...and 25 more