Vacuum Cherenkov radiation in supercritical magnetic fields

In the presence of very intense electromagnetic fields, the refractive index of vacuum is modified such that light velocity is less than $c$ and ultrarelativistic charged particles can be faster than light and can induce Cherenkov radiation. We present the comparison of the Cherenkov radiation produced by the Euler-Heisenberg theory for critical and supercritical magnetic fields. We also make the comparison between the Cherenkov and the synchrotron radiation produced by the charged particles.