Can we probe the Lorentz factor of gamma-ray bursts from GeV-TeV spectra integrated over internal shocks?

We revisit the high-energy spectral cutoff originating from the electron-positron pair creation in the prompt phase of gamma-ray bursts (GRBs) with numerical and analytical calculations. We show that the conventional exponential and/or broken power-law cutoff should be drastically modified to a shallower broken power law in practical observations that integrate emissions from different internal shocks. Since the steepening is tiny for observations, this "smearing" effect can generally reduce the previous estimates of the Lorentz factor of the GRB outflows. We apply our formulation to GRB 080916C, recently detected by the Large Area Telescope detector on the Fermi satellite, and find that the minimum Lorentz factor can be ∼600 (or even smaller values), which is below but consistent with the previous result of ∼900. Observing the steepening energy (the so-called "pair-break energy") is crucial to diagnosing the Lorentz factor and/or the emission site in future observations, especially current and future Cherenkov telescopes such as MAGIC, VERITAS, and CTA.