  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.

High Energy Physics - Theory;High Energy Astrophysical Phenomena;High Energy Physics - Phenomenology