Gravitational Lorentz-violating $e^-+e^+\to\ell^-+\ell^+$ scattering

Technical Summary: Gravitational Lorentz-violating $e^- + e^+ \to \ell^- + \ell^+$ Scattering

Problem Statement
The Standard Model of particle physics successfully describes fundamental forces but excludes gravity. The Standard Model Extension (SME) addresses this by providing a unified framework that includes all possible Lorentz- and CPT-violating terms coupling Standard Model fields with general relativity. While the minimal SME (renormalizable, dimension $d \le 4$) has been extensively studied, the nonminimal SME (nonrenormalizable, $d > 4$) remains less explored, particularly regarding gravitational interactions. This paper investigates the scattering process $e^- + e^+ \to \ell^- + \ell^+$ (where $\ell$ represents a generic lepton) mediated by graviton exchange within the nonminimal SME gravity sector. Specifically, the study focuses on how a fifth-order Lorentz-violating (LV) background field modifies the scattering cross-section, considering both zero-temperature and finite-temperature scenarios.

Methodology
The authors employ a multi-faceted theoretical approach:

1. Gravitoelectromagnetism (GEM) Framework: The analysis utilizes the Weyl GEM formalism, a weak-field approximation of gravity analogous to Maxwell's theory. In this framework, the Weyl tensor is decomposed into gravitoelectric and gravitomagnetic components, allowing for a Lagrangian formulation where the potential tensor $A_{\mu\nu}$ acts as the fundamental field. This facilitates the quantization of the graviton as a spin-2 particle and simplifies the treatment of fermion-graviton interactions.
2. Lorentz-Violating Sector: The study introduces a dimension-5 Lorentz-violating coefficient, $K^{(5)}_{\mu\nu\alpha\iota\phi}$, into the interaction vertex between fermions and gravitons. This term is structurally analogous to CPT-odd terms found in the QED sector of the SME. The authors redefine this coefficient using the Minkowski metric and the Levi-Civita symbol to express the interaction in terms of a background vector $K_\lambda$.
3. Scattering Amplitude Calculation: The transition amplitude for the $e^- + e^+ \to \ell^- + \ell^+$ process is calculated using Feynman diagrams. The total amplitude includes the standard non-LV contribution ($M_0$) and corrections arising from single LV vertices ($M_1$). Diagrams with two LV vertices are neglected as they represent higher-order corrections. The calculation is performed in the center-of-mass frame, utilizing the ultrarelativistic limit and trace properties of Dirac matrices to compute the squared transition amplitude.
4. Thermal Effects (Thermo Field Dynamics): To analyze the process at finite temperature, the authors adopt the Thermo Field Dynamics (TFD) formalism. This involves doubling the Hilbert space ($H_T = H \otimes \tilde{H}$) and applying Bogoliubov transformations to field operators. This method allows for the consistent treatment of thermal averages as expectation values of quantum operators, specifically modifying the graviton propagator to include temperature-dependent terms.

Key Contributions and Results

• Modified Cross-Section at Zero Temperature: The authors derive the differential and total cross-sections for gravitational scattering in the presence of the LV background. The results show that the squared transition amplitude acquires a term proportional to $K_z^2$ (where $K_z$ is the component of the LV vector along the scattering axis). The total cross-section is expressed as:
  $$\sigma = \sigma_{\text{GEM}} \left( 1 + \frac{5}{2} \frac{K_z^2}{\kappa^2} \right)$$
  where $\sigma_{\text{GEM}}$ is the standard gravitational cross-section and $\kappa$ is the coupling constant. The LV contribution introduces an angular dependence, reaching a maximum at $\theta = 0$ and a minimum at $\theta = \pi/2$.
• Finite Temperature Corrections: By applying the TFD formalism, the authors derive the cross-section at finite temperature. The thermal effects introduce a multiplicative factor $D(\beta)$, where $\beta = 1/k_B T$. The resulting cross-section is:
  $$\sigma(\beta) = \sigma_{\text{GEM}}(\beta) \left( 1 + \frac{5}{2} \frac{K_z^2}{\kappa^2} \right) D(\beta)$$
  The analysis indicates that the thermal factor $D(\beta)$ enhances the scattering amplitude as temperature increases. In the limit $T \to 0$ ($\beta \to \infty$), the results correctly reduce to the zero-temperature case.
• Coupling Constant Rescaling: To facilitate a comparison between GEM and QED, the authors note the dimensional difference in coupling constants (inverse energy in GEM vs. dimensionless in QED) and propose a rescaling $\kappa \to \kappa' = \kappa E_c$ using a characteristic energy scale $E_c$.

Significance and Claims
The paper claims to provide a detailed analysis of how Lorentz-violating and thermal corrections influence gravitational interactions within the SME framework. The authors emphasize that these results are particularly relevant for understanding physical processes in high-energy or astrophysical environments where temperature and potential symmetry-breaking effects at the Planck scale may be significant.

The study highlights that:

1. Lorentz violation in the nonminimal gravity sector introduces specific, calculable corrections to scattering cross-sections that depend on the orientation of the background field.
2. Thermal effects, treated via TFD, do not merely add noise but structurally modify the interaction, enhancing the amplitude in high-temperature regimes.
3. The combination of SME symmetry breaking and finite temperature offers a more comprehensive view of spacetime symmetries and thermodynamics interplay, potentially revealing physical effects absent in standard zero-temperature or Lorentz-invariant scenarios.

The authors explicitly state that direct comparison with experimental data is currently impossible due to the lack of available data on graviton-mediated scattering, but the theoretical framework establishes a basis for future investigations into these fundamental interactions.