Cosmic rays

Energetic cosmic rays that continuously reach Earth's atmosphere can serve as powerful tests of Lorentz symmetry in electrodynamics. Photons are versatile particles to search for possible violations of Lorentz invariance because unconventional effects can be studied in a wide range of experiments. The connection with cosmic rays appears for some coefficients for Lorentz violation that modify the way photons interact with charged particles, in particular cosmic-ray protons.

A non-dispersive extension of Maxwell electrodynamics that preserves CPT invariance can be written as
$$\mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} -\frac{1}{4}(\color{red}{k_F})_{\mu\nu\lambda\rho}\,F^{\mu\nu} F^{\lambda\rho},$$
where the coefficients $(\color{red}{k_F})_{\mu\nu\lambda\rho}$ control deviations from exact Lorentz invariance. In the case of isotropic Lorentz violation, the relevant components of $(\color{red}{k_F})_{\mu\nu\lambda\rho}$ can be parametrized in terms of the single parameter ${\color{red}\kappa}$, which makes photons in vacuum behave as photons in a macroscopic medium. In other words, the parameter ${\color{red}\kappa}$ induces a vacuum index of refraction $\color{red}n>1$. Above some threshold energy $E_\text{th}$, a proton moving in this vacuum can then emit Cherenkov radiation. Energy-momentum conservation can be used to determine this threshold energy to be $$E_\text{th} \approx \frac{M}{\sqrt{2{\color{red}\kappa}}},$$
where $M$ is the proton mass. A field-theory calculation gives the same result. Above this threshold energy, the highly efficient process $\color{blue}{\pmb{p}} \to \color{blue}{\pmb{p}} +\color{red}{\pmb{\gamma}}$ would make protons lose energy rapidly until they fall below the threshold. For this reason, the observation of a proton of energy $E$ in high-energy cosmic rays can be used to set the constraint $E<E_\text{th}$, and then determine a numerical value for $\color{red}\kappa$. In a more general scenario, coefficients for Lorentz violation controlling direction-dependent effects can also be studied using this technique, which would require the observation of several high-energy cosmic-ray events distributed in the sky.