We use joint observations by the Neil Gehrels Swift X-ray Telescope (XRT) and the Fermi Large Area Telescope (LAT) of gamma-ray burst (GRB) afterglows to investigate the nature of the long-lived high-energy emission observed by Fermi LAT. Joint broadband spectral modeling of XRT and LAT data reveal that LAT nondetections of bright X-ray afterglows are consistent with a cooling break in the inferred electron synchrotron spectrum below the LAT and/or XRT energy ranges. Such a break is sufficient to suppress the high-energy emission so as to be below the LAT detection threshold. By contrast, LAT-detected bursts are best fit by a synchrotron spectrum with a cooling break that lies either between or above the XRT and LAT energy ranges. We speculate that the primary difference between GRBs with LAT afterglow detections and the non-detected population may be in the type of circumstellar environment in which these bursts occur, with late-time LAT detections preferentially selecting GRBs that occur in low wind-like circumburst density profiles. Furthermore, we find no evidence of high-energy emission in the LAT-detected population significantly in excess of the flux expected from the electron synchrotron spectrum fit to the observed X-ray emission. The lack of excess emission at high energies could be due to a shocked external medium in which the energy density in the magnetic field is stronger than or comparable to that of the relativistic electrons behind the shock, precluding the production of a dominant synchrotron self-Compton (SSC) component in the LAT energy range. Alternatively, the peak of the SSC emission could be beyond the 0.1-100 GeV energy range considered for this analysis.
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