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Observations of long duration gamma-ray bursts (GRBs) with TeV emission during their afterglow have been on the rise. Recently, GRB 221009A, the most energetic GRB ever observed, was detected by the {LHAASO} experiment in the energy band 0.2 - 7 TeV. Here, we interpret its afterglow in the context of a hybrid model in which the TeV spectral component is explained by the proton-synchrotron process while the low energy emission from optical to X-ray is due to synchrotron radiation from electrons. We constrained the model parameters using the observed optical, X-ray and TeV data. By comparing the parameters of this burst and of GRB 190114C, we deduce that the VHE emission at energies $\geq$ 1 TeV in the GRB afterglow requires large explosion kinetic energy, $E \gtrsim 10^{54}$~erg and a reasonable circumburst density, $n\gtrsim 10$~cm$^{-3}$. This results in a small injection fractions of particles accelerated to a power-law, $\sim 10^{-2}$. {A significant fraction of shock energy must be allocated to a near equipartition magnetic field, $\epsilon_B \sim 10^{-1}$, while electrons should only carry a small fraction of this energy, $\epsilon_e \sim 10^{-3}$. Under these conditions required for a proton synchrotron model, namely $\epsilon_B \gg \epsilon_e$, the SSC component is substantially sub-dominant over proton-synchrotron as a source of TeV photons.} These results lead us to suggest that proton-synchrotron process is a strong contender for the radiative mechanisms explaining GRB afterglows in the TeV band., Comment: 15 pages and 4 figures (Accepted to be published in ApJ)

We consider here a proton-synchrotron model to explain the MAGIC observation of GRB 190114C afterglow in the energy band $0.2 - 1$~TeV, while the X-ray spectra are explained by electron-synchrotron emission. Given the uncertainty of the particle acceleration process, we consider several variations of the model, and show that they all match the data very well. We find that the values of the uncertain model parameters are reasonable: {isotropic} explosion energy $\sim 10^{54.5}$~erg, ambient density $\sim 10-100~{\rm cm^{-3}}$, and fraction of electrons/protons accelerated to a high energy power law of a few per-cents. All these values are directly derived from the observed TeV and X-ray fluxes. The model also requires that protons be accelerated to observed energies as high as a few $10^{20}$~eV. Further, assuming that the jet break takes place after $10^6$~s gives the beaming-corrected energy of the burst to be $\approx 10^{53}$~erg, which is one to two orders of magnitude higher than usually inferred, because of the small fraction of electrons accelerated. Our modeling is consistent with both late time data at all bands, from optical to X-rays, and with numerical models of particle acceleration. Our results thus demonstrate the relevance of proton-synchrotron emission to the high energy observations of GRBs during their afterglow phase., Comment: Revised version with the added subsections and figures (accepted for publication in ApJ)