Synchrotron radiation mechanism , when electrons are accelerated in a relativistic shock , is known to have serious problems to explain the observed gamma-ray spectrum below the peak for most Gamma-Ray Bursts ( GRBs ) ; the synchrotron spectrum below the peak is much softer than observed spectra .
Recently , the possibility that electrons responsible for the radiation cool via Inverse Compton , but in the Klein-Nishina regime , has been proposed as a solution to this problem .
We provide an analytical study of this effect and show that it leads to a hardening of the low energy spectrum but not by enough to make it consistent with the observed spectra for most GRBs ( this is assuming that electrons are injected continuously over a time scale comparable to the dynamical time scale , as is expected for internal shocks of GRBs ) .
In particular , we find that it is not possible to obtain a spectrum with \alpha > -0.1 ( f _ { \nu } \propto \nu ^ { \alpha } ) whereas the typical observed value is \alpha \sim 0 .
Moreover , extreme values for a number of parameters are required in order that \alpha \sim - 0.1 : the energy fraction in magnetic field needs to be less than about 10 ^ { -4 } , the thermal Lorentz factor of electrons should be larger than 10 ^ { 6 } , and the radius where gamma-rays are produced should be not too far away from the deceleration radius .
These difficulties suggest that the synchrotron radiation mechanism in internal shocks does not provide a self-consistent solution when \alpha \lower 2.0 pt \hbox { $ \buildrel \scriptstyle > \over { \scriptstyle \sim } $ } -0.2 .