Quantum electrodynamics based upon Euler-Heisenberg or Born-Infeld Lagrangians predict photons to move according to an effective metric which is dependent on the background electromagnetic field. In other words, photon trajectories are curved in presence of electromagnetic fields, meaning that an effective gravity is acting upon. If part of fermion masses is allegedly of electromagnetic origin, the question why their trajectories are not affected by this effective gravity naturally comes to mind.
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In the presence of a background electromagnetic field, electromagnetic fields travel along a deformed light cone which is smaller than the "relativistic light cone". However, charged fermions can still travel faster than electromagnetic waves as long as they are still slower than the "relativistic speed of light". They emit Cherenkov radiation while doing so.