We present results from a deep spectral analysis of all the Swift observations of Mrk 421 from April 2006 to July 2006, when it reached its largest X-ray flux recorded until 2006. The peak flux was about 85 milli-Crab in the 2.0-10.0 keV band, with the peak energy (Ep) of the spectral energy distribution (SED) laying often at energies larger than 10 keV. We performed spectral analysis of the Swift observations investigating the trends of the spectral parameters in terms of acceleration and energetic features phenomenologically linked to the SSC model parameters, predicting their effects in the gamma-ray band, in particular the spectral shape expected in the Fermi Gamma-ray Space Telescope-LAT band. We confirm that the X-ray spectrum is well described by a log-parabolic distribution close to Ep, with the peak flux of the SED (Sp) being correlated with Ep, and Ep anti-correlated with the curvature parameter (b). During the most energetic flares the UV-to-soft-X-ray spectral shape requires an electron distribution spectral index s about 2.3. Present analysis shows that the UV-to-X-ray emission from Mrk 421 is likely to be originated by a population of electrons that is actually curved, with a low energy power-law tail. The observed spectral curvature is consistent both with stochastic acceleration or energy dependent acceleration probability mechanisms, whereas the power-law slope form XRT-UVOT data is very close to that inferred from the GRBs X-ray afterglow and in agreement with the universal first-order relativistic shock acceleration models. This scenario hints that the magnetic turbulence may play a twofold role: spatial diffusion relevant to the first order process and momentum diffusion relevant to the second order process.
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