Synchrotron radiation from massless charge

Classical radiation power from an accelerated massive charge diverges in the zero-mass limit, while some authors suggest that strictly massless charge does 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 ab initio   massless particle of zero spin. Provided E2≫eHE2≫eH, the maximum in the spectrum is shown to be at ħω=E/3ħω=E/3, and the average photon energy is 4E/94E/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. Classical theory is reliable only as providing the low-frequency part of the true quantum radiation spectrum.