2023ApJ...948...82H

An interesting result that was recently reported for Mrk 421 is the detection of a significant excess at hard X-ray energies, which could provide useful information to investigate particle acceleration and emission mechanisms in the relativistic jet. Considering a two-injection scenario, we develop a self-consistent one-zone leptonic model to understand the origin of the hard X-ray excess in Mrk 421 during the period of extremely low X-ray and very high-energy flux in 2013 January. In the model, two populations of mono-energetic ultra-relativistic electrons are injected into the emission region, which is a magnetized plasmoid propagating along the blazar jet. We numerically calculate the emitting electron energy distribution by solving a kinetic equation that incorporates both shock acceleration and stochastic acceleration processes. Moreover, we infer analytic expressions relating the electrons' acceleration, cooling, escape, and injection to the observed spectra and variability. In particular, for the injection luminosity, we derive a new approximate analytical expression for the case of continual injection with a mono-energetic distribution. Based on a comparison between the theoretical predictions and the observed SED, we conclude that the hard X-ray excess that was observed in Mrk 421 may be due to the synchrotron radiation emitted by an additional electron population, which is co-spatial with an electron population producing simultaneous optical/UV, soft X-ray, and γ-ray emissions. Therefore, stochastic acceleration may play a major role in producing the observed X-ray spectrum.