Energies of GRB blast waves and prompt efficiencies as implied by modeling of X-ray and GeV afterglows

We consider a sample of ten GRBs with long lasting ($\gtrsim10^2\rm\,sec$) emission detected by Fermi/LAT and for which X-ray data around $1\,$day are also available. We assume that both the X-rays and the GeV emission are produced by electrons accelerated at the external forward shock, and show that the X-ray and the GeV fluxes lead to very different estimates of the initial kinetic energy of the blast wave. The energy estimated from GeV is on average $\sim50$ times larger than the one estimated from X-rays. We model the data (accounting also for optical detections around $1\,$day, if available) to unveil the reason for this discrepancy and find that good modelling within the forward shock model is always possible and leads to two possibilities: either the X-ray emitting electrons (unlike the GeV emitting electrons) are in the slow cooling regime or ii) the X-ray synchrotron flux is strongly suppressed by Compton cooling, whereas, due to the Klein-Nishina suppression, this effect is much smaller at GeV energies. In both cases the X-ray flux is no longer a robust proxy for the blast wave kinetic energy. On average, both cases require weak magnetic fields ($10^{-6}\lesssim \epsilon_B \lesssim 10^{-3}$) and relatively large isotropic kinetic blast wave energies $10^{53}\rm\,erg<E_{0,kin}<10^{55}\rm\,erg$ corresponding to large lower limits on the collimated energies, in the range $10^{52}\rm\,erg<E_{\theta,kin}<5\times10^{52}\rm\,erg$ for an ISM environment with $n\sim 1\mbox{cm}^{-3}$ and $10^{52}\rm\,erg<E_{\theta,kin}<10^{53}\rm\,erg$ for a wind environment with $A_* \sim 1$. These energies are larger than those estimated from the X-ray flux alone, and imply smaller inferred values of the prompt efficiency mechanism, reducing the efficiency requirements on the still uncertain mechanism responsible for prompt emission.