Cherenkov radiation in a strong magnetic field

According to quantum electrodynamics, in a strong magnetic field that is constant and spatially uniform, the vacuum becomes polarized with a refractive index greater than unity. As a result, ultra-relativistic charged particles travelling in such media can emit Cherenkov radiation with a power spectrum directly proportional to the photon frequency $\omega$. Therefore, by extrapolating $\omega$ beyond the critical synchrotron frequency $\omega_{c}$, the Cherenkov radiation will eventually dominate over its synchrotron counterpart. However, such an extrapolation is not possible. We show that in the framework of effective field theory, the maximal attainable photon frequency $\omega_{\tiny{\mbox{max}}}$ is about four order of magnitude less than $\omega_{c}$. At $\omega=\omega_{\tiny{\mbox{max}}}$, given the $\gamma_{e}$-factor of an electron travelling normal to a constant and spatially uniform magnetic field $\mathbf{B}$, the spectrum of Cherenkov radiation becomes dominant when $\gamma_{e}(|\mathbf{B}|/\mbox{Gauss})\gtrsim 4.32\times 10^{19}$. Nevertheless, detecting the Cherenkov radiation in astrophysical environments remains challenging since its spectral flux density is about three orders of magnitude less than the synchrotron radiation.