Relation between the keV-MeV and TeV emission of GRB 221009A and its implications

Gamma-ray bursts (GRBs) are believed to launch relativistic jets, which generate prompt emission by internal processes, and produce long-lasting afterglows by driving external shocks into surrounding medium. However, how the jet powers the external shock is poorly known. The unprecedented observations of the keV-MeV emission with GECAM and the TeV emission with LHAASO of the brightest-of-all-time GRB 221009A offer a great opportunity to study the prompt-to-afterglow transition and the impact of jet on the early dynamics of external shock. In this letter, we find that the cumulative light curve of keV-MeV emission could well fit the rising stage of the TeV light curve of GRB 221009A, with a time delay, $4.45^{+0.26}_{-0.26}$\,s, of TeV emission. Moreover, both the rapid increase in the initial stage and the excess from about \T+260\,s to 270\,s in the TeV light curve are tracking the light-curve bumps in the prompt keV-MeV emission. The close relation between the keV-MeV and TeV emission reveals the continuous energy-injection into the external shock. Assuming an energy-injection rate exactly following the keV-MeV flux of GRB 221009A, including the very early precursor, we build a continuous energy-injection model where the jet Lorentz factor is derived from the TeV time delay, and the TeV data is well fitted, with the TeV excesses interpreted by inverse Compton (IC) scatterings of the inner-coming prompt emission by the energetic electrons in external shock.