Robust Constraints on the Physics of the MeV Emission Line in GRB 221009A from Optical Depth Arguments

The brightest-of-all-time gamma-ray burst (GRB), GRB 221009A, is the first GRB observed to have emission line (up to 37 MeV) in its prompt emission spectra. It is naturally explained as \pair annihilation line that was Doppler boosted in the relativistic jet of the GRB. In this work, we repeatedly apply the simple optical depth argument to different physical processes necessary to produce an observable \pair annihilation line. This approach results in robust constraints on the physics of the line: We conclude that in GRB 221009A, the \pair pairs were produced at a radius greater than $4.3\times 10^{15}$\,cm from the central engine, and annihilated in a region between $1.4\times 10^{16}$\,cm and $4.3\times 10^{16}$\,cm. From these constraints, we established a self-consistent picture of \pair production, cooling, and annihilation. We also derived a criterion for pair production in the GRB prompt emission: $E_{\rm{iso}} \gtrsim3.3\times 10^{53} E_{\rm{peak},100} (1+z) R^2_{\rm{prod},16}~\text{erg}$. Using this criterion, we find tens of candidate GRBs that could have produced \pair in prompt emissions to annihilate. GRB 221009A is with the highest likelihood according to this criterion. We also predict the presence of a thermal radiation, with a time-evolving black body temperature, sweeping through soft X-ray during the prompt emission phase.