Recent observations with ALMA have revealed evidence for non-thermal synchrotron emission from the core regions of two nearby Seyfert galaxies. This suggests that the coronae of accretion disks in active galactic nuclei (AGNs) can be conducive to the acceleration of non-thermal electrons, in addition to the hot, thermal electrons responsible for their X-ray emission through thermal Comptonization. Here we investigate the mechanism of such particle acceleration, based on observationally inferred parameters for AGN disk coronae. One possibility to account for the observed non-thermal electrons is diffusive shock acceleration, as long as the gyrofactor $eta_g$ does not exceed $sim10^6$. These non-thermal electrons can generate gamma rays via inverse Compton scattering of disk photons, which can appear in the MeV band, while those with energies above $sim100$ MeV would be attenuated via internal $gammagamma$ pair production. The integrated emission from all AGNs with thermal and non-thermal Comptonization can reproduce the observed cosmic background radiation in X-rays as well as gamma-rays up to $sim 10$ MeV. Furthermore, if protons are accelerated in the same conditions as electrons and $eta_gsim30$, our observationally motivated model is also able to account for the diffuse neutrino flux at energies below 100-300 TeV. The next generation of MeV gamma-ray and neutrino facilities can test these expectations by searching for signals from bright, nearby Seyfert galaxies such as NGC 4151 and IC 4329A.
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