Radiation from an emitter in the ghost free scalar theory

We study radiation emitted by a time-dependent source of a scalar massless field in the framework of the ghost-free modifications of the theory. We consider a simple model of the emitter: namely,we assume that it is point-like and monochromatic. We focused on the most common versions of the ghost-free theory, where the propagator $\Box^{-1}$ is modified as follows $\exp(-(\Box/\mu^2)^N) \Box^{-1}$, where $\mu$ is the characteristic mass-scale of such $GF_N$-theory. We demonstrated that far from the source, in the wave-zone, the radiation asymptotically converges to its "classical" value for any $N\ge 1$. However, in the near-zone the behavior of the field is quite different from the "classical" case. The difference of field amplitude for the ghost-free field and for the classical one has an oscillatory behavior in this domain. A number of oscillations increases with $N$. The amplitude of these oscillations remain finite for even $N$, while it infinitely grows with frequency for odd $N$. This behavior indicates that even in the classical domain $GF_N$ theories might have pathological behavior.