The generation of connected high and very high energy $gamma$-rays and radio emission in active galactic nuclei

We consider relativistic electrons in magnetospheric flows close to a supermassive black hole and study the mechanism of quasi-linear diffusion (QLD) to investigate the correlation between $\gamma$-ray and radio emission in active galactic nuclei. Moving in the nonuniform magnetic field the particles experience a force that is responsible for the conservation of the adiabatic invariant. This force, together with the radiation reaction force, tends to decrease the pitch angles. Contrary to this, the QLD attempts to increase the pitch angle, and this maintains the synchrotron emission regime. To examine the balance between the QLD and the aforementioned dissipative factors we investigate the quasi-stationary state by applying the kinetic equation. Considering the magnetospheric plasma close to the supermassive black hole, we examined the efficiency of the QLD for different parameters. By examining the cyclotron instability, we show that despite the efficient dissipative factors, the cyclotron modes excite transverse and longitudinal-transversal waves which leads to the QLD. We find that the QLD provides a connection of emission in the $\gamma$-ray and radio domains. We show that under favourable conditions the radio emission from 22 MHz to 9 GHz on the mpc scale is associated with $\gamma$-ray emission from 900 GeV down to 9 GeV on the same scale.