Correlated parity violation in gravity and electromagnetism from five-dimensional spacetime

Technical Summary: Correlated Parity Violation in Gravity and Electromagnetism from Five-Dimensional Spacetime

Problem Statement
Parity violation (PV) in the gravitational and electromagnetic sectors has been extensively studied, yet these phenomena are conventionally treated as independent, governed by unrelated model parameters. This separation is natural in four-dimensional spacetime, where gravity and electromagnetism are described by independent degrees of freedom ($g_{\mu\nu}$ and $A_\mu$) with no geometric connection at the Lagrangian level. However, the authors posit that this independence may be a "four-dimensional prejudice." In higher-dimensional unification frameworks, such as Kaluza-Klein (KK) theory, gravity and electromagnetism emerge from a single five-dimensional metric. If the fundamental five-dimensional theory violates parity, the resulting PV in both sectors should not be independent but rather correlated. The paper addresses the lack of literature exploring this unified perspective on parity violation.

Methodology
The authors construct the simplest five-dimensional parity-violating action within the KK framework and perform a dimensional reduction to four dimensions. To ensure a robust comparison and address stability issues, they analyze the reduction in two distinct geometric frameworks:

1. Riemannian Geometry: The standard framework where gravity is described by curvature. The authors construct a parity-odd scalar using the five-dimensional Levi-Civita tensor, the Riemann curvature tensor, and the normalized vector field along the compact fifth dimension.
2. Teleparallel Geometry: A framework where gravity is described by torsion rather than curvature (Teleparallel Equivalent of General Relativity, TEGR). Here, the parity-odd scalar is constructed from the torsion 2-form, the Levi-Civita tensor, and the preferred direction vector.

In both cases, the "cylinder condition" is imposed, assuming all fields are independent of the fifth coordinate ($x^5$), reducing the system to four-dimensional effective fields: the metric $g_{\mu\nu}$, the electromagnetic potential $A_\mu$, and the scalar dilaton/radion $\phi$.

Key Contributions and Results

• Riemannian Case (Curvature-Based):
  The simplest five-dimensional parity-violating term, quadratic in curvature, reduces to a complex set of four-dimensional interactions. These include:

  • The gravitational Chern-Simons (CS) term ($\phi \, ^*R R$).
  • The electromagnetic CS term.
  • Mixed gravity-electromagnetism interactions.
  • Quartic electromagnetic couplings.
  • Derivative interactions involving the KK scalar.
    Crucially, the gravitational sector inherits the well-known ghost instability of CS gravity. The kinetic term for one helicity of gravitational waves (GWs) becomes negative at sufficiently high wave numbers, rendering the theory unstable as a fundamental description and limiting it to a low-energy effective theory with a cutoff.

• Teleparallel Case (Torsion-Based):
  In contrast, the analogous construction in teleparallel geometry yields a strikingly simple result. The simplest five-dimensional parity-violating term reduces to only two four-dimensional terms:

  1. The Nieh-Yan term (the gravitational PV sector).
  2. The standard electromagnetic Chern-Simons term ($\phi^3 \, ^*F F$).
     Notably, this reduction generates no mixed gravity-electromagnetism terms, no quartic electromagnetic couplings, and no higher-derivative terms. Both resulting terms are ghost-free. The Nieh-Yan term introduces no ghost modes in scalar, vector, or tensor sectors, and the electromagnetic CS term preserves a positive kinetic coefficient for the physical branch ($\phi > 0$).

• The Correlated Dispersion Relation:
  The most significant quantitative result is the exact, background-independent relation between the helicity-dependent dispersion shifts for electromagnetic waves ($\Delta \omega^2_{EM}$) and gravitational waves ($\Delta \omega^2_{GW}$). The authors derive:
  $$\Delta \omega^2_{EM} = 6 \Delta \omega^2_{GW}$$
  This factor of six is fixed entirely by the tensor contraction structure of the five-dimensional action and does not depend on the specific evolution of the cosmological background or the KK scalar field.

Significance and Claims
The paper claims that the teleparallel KK model offers a ghost-free, unified realization of parity violation where the gravitational and electromagnetic sectors are intrinsically linked. Unlike the Riemannian case, which suffers from instabilities and complex mixing, the teleparallel model provides a clean theoretical framework where the two sectors share a common geometric origin.

The authors emphasize that the derived relation $\Delta \omega^2_{EM} = 6 \Delta \omega^2_{GW}$ offers a falsifiable test of this unification hypothesis. This can be probed through joint observations of:

1. Cosmic Microwave Background (CMB) Birefringence: Measuring the rotation of the polarization plane (isotropic cosmic birefringence) to constrain the electromagnetic sector.
2. Gravitational Wave Velocity Birefringence: Measuring the difference in propagation speeds for left- and right-handed GWs to constrain the gravitational sector.

The paper concludes that while the Riemannian approach is limited by ghost instabilities, the teleparallel approach provides a stable, minimal model where the correlation between electromagnetic and gravitational parity violation is a direct, testable prediction of five-dimensional geometry. The authors note that a detailed quantitative analysis using specific CMB and GW datasets is left for future work.