A convenient and general decomposition of the photon self-energy in a magnetized, but otherwise isotropic, medium is given in terms of the minimal set of tensors consistent with the transversality condition. As we show, the self-energy in such a medium is completely parametrized in terms of nine independent form factors, and they reduce to three in the long wavelength limit. We consider in detail an electron gas with a background magnetic field, and, using finite temperature field theory methods, we obtain the one-loop formulas for the form factors, which are exact to all orders in the magnetic field. Explicit results are derived for a variety of physical conditions. In the appropriate limits, we recover the well-known semiclassical results for the photon dispersion relations and the Faraday effect. In more general cases, where the semiclassical treatment or the linear approximation (weak-field limit) are not applicable, our formulas provide a consistent and systematic way for computing the self-energy form factors and, from them, the photon dispersion relations.

High-energy peaked blazars are known to undergo episodes of flaring in GeV–TeV gamma-rays involving different timescales. This flaring mechanism is not well understood, despite long-term simultaneous multiwavelength observations. These gamma-rays en route to Earth undergo attenuation by the extragalactic background light. Using the photohadronic model, where the seed photons follow a power-law spectrum and a template extragalactic background light model, we derive a simple relation between the observed multi-TeV gamma-ray flux and the intrinsic flux with a single parameter. We study 42 flaring epochs of 23 blazars and excellent fits to most of the observed spectra are obtained, strengthening the photohadronic origin of multi-TeV gamma-rays. We can also constrain the power spectrum of the seed photons during the flaring period. Stringent bounds on the blazars of unknown redshifts, whose multi-TeV flaring spectra are known, can be placed using the photohadronic model.

The detection of multi-TeV gamma rays from the afterglow phase of GRB 190829A by the High Energy Stereoscopic System telescope is an addition to the already existing list of two GRBs observed in very high-energy (VHE) gamma rays in recent years. Jets of blazars and GRBs have many similarities and the photohadronic model is very successful in explaining the VHE gamma-ray spectra from the high-energy blazars. Recently, the photohadronic model has been successfully applied to study the sub-TeV gamma rays from the afterglow phases of GRB 180720B and GRB 190114C. We employed this model again to explain the VHE spectra observed for the two consecutive nights from GRB 190829A. We show that the spectra of GRB 190829A can be due to the interactions of high-energy protons with the synchrotron self-Compton photons in the forward shock region of the GRB jet, similar to the low emission state of the VHE flaring events of high-energy blazars. We speculate that, if in the future, it is possible to observe the VHE gamma-ray spectra from nearby GRBs in their afterglow phases, then some of them could only be explained by employing two different spectral indices. If confirmed, such VHE spectra could be interpreted as a result of the interactions of the high-energy protons with the photons, both from the synchrotron background and the synchrotron self-Compton background in the forward shock region.

The detection of GRB 180722B and GRB 190114C in sub-TeV gamma-rays has opened up a new window to study gamma-ray bursts in high-energy gamma-rays. Recently it has been shown that the synchrotron and inverse Compton processes are responsible for the production of these high-energy gamma-rays during the afterglow. Here, for the first time we demonstrate that the photohadronic scenario that is successful in explaining the multi-TeV flaring in high-energy blazars is also applicable for gamma-ray bursts. We show that the sub-TeV spectra of GRB 190114C and GRB 180720B are due to the interaction of high-energy protons with the background photons in the synchrotron self-Compton region and synchrotron region, respectively. The nature of the background photon distributions help us to constrain their bulk Lorentz factors.

The IceCube Collaboration has observed 37 neutrino events in the energy range \(30\,\mathrm{TeV}\le E_{\nu } \le 2\) PeV and the sources of these neutrinos are unknown. Here we show that the positions of 12 high energy blazars and the position of the FR-I galaxy Centaurus A coincide within the error circles of ten IceCube events, the latter being in the error circle of the highest energy event so far observed by IceCube. Two of the above blazars are simultaneously within the error circles of the Telescope Array hotspot and one IceCube event. We found that the blazar H2356-309 is within the error circles of three IceCube events. We propose that photohadronic interactions of the Fermi accelerated high energy protons with the synchrotron/SSC background photons in the nuclear region of these high energy blazars and AGN are probably responsible for some of the observed IceCube events.