  We explore several models which might be proposed to explain recent possible
detections of high-energy (TeV) gamma rays in association with low-energy
gamma-ray bursts (GRBs). Likely values (and/or upper limits) for the source
energies in low- and high-energy gamma rays and hadrons are deduced for the
burst sources associated with possible TeV gamma-ray detections by the Project
GRAND array. Possible spectra for energetic gammas are deduced for three
models: 1) inverse-Compton scattering of ambient photons from relativistic
electrons; 2) proton-synchrotron emission; and 3) inelastic scattering of
relativistic protons from ambient photons creating high-energy neutral pions,
which decay into high-energy photons. These models rely on some basic
assumptions about the GRB properties, e.g. that: the low- and high-energy gamma
rays are produced at the same location; the time variability of the high-energy
component can be estimated from the FWHM of the highest peak in the low-energy
gamma ray light curve; and the variability-luminosity relation of Fenimore &
Ramirez-Ruiz (2000) gives a reliable estimate of the redshifts of these bursts.
We also explore the impact of each of these assumptions upon our models. We
conclude that the energetic requirements are difficult to satisfy for any of
these models unless, perhaps, either the photon beaming angle is much narrower
for the high-energy component than for the low-energy GRB or the bursts occur
at very low redshifts (z<0.01). Nevertheless, we find that the energetic
requirements are most easily satisfied if TeV gamma rays are produced
predominantly by inverse-Compton scattering with a magnetic field strength well
below equipartition or by proton-synchrotron emission with a magnetic field
strength near equipartition.
