Synchrotron radiation from massless charge

Classical radiation power from an accelerated massive charge diverges in the zero-mass limit, while some general arguments suggest that strictly massless charge does not not radiate at all. On the other hand, the regularized classical radiation reaction force, though looking odd, is non-zero and finite. To clarify this controversy, we consider radiation problem in massless scalar quantum electrodynamics in the external magnetic field. In this framework, synchrotron radiation is found to be non-zero, finite, and essentially quantum. Its spectral distribution is calculated using Schwinger's proper time technique for {\em ab initio} massless particle of zero spin. Provided $E^2\gg eH$, the maximum in the spectrum is shown to be at $\hbar \omega=E/3$, and the average photon energy is $4E/9$. The normalized spectrum is universal, depending neither on $E$ nor on $H$. Quantum nature of radiation makes classical radiation reaction equation meaningless for massless charge. Our results are consistent with the view (supported by the renormalization group argument) that the correct classical limit of massless quantum electrodynamics is free theory.