  $\gamma$-ray bursts (GRBs) are short-lived transients releasing a large
amount of energy ($10^{51}-10^{53} $ erg) in the keV-MeV energy range. GRBs are
thought to originate from internal dissipation of the energy carried by
ultra-relativistic jets launched by the remnant of a massive star's death or a
compact binary coalescence. While thousands of GRBs have been observed over the
last thirty years, we still have an incomplete understanding of where and how
the radiation is generated in the jet. Here we show a relation between the
spectral index and the flux found by investigating the X-ray tails of bright
GRB pulses via time-resolved spectral analysis. This relation is incompatible
with the long standing scenario which invokes the delayed arrival of photons
from high-latitude parts of the jet. While the alternative scenarios cannot be
firmly excluded, the adiabatic cooling of the emitting particles is the most
plausible explanation for the discovered relation, suggesting a
proton-synchrotron origin of the GRB emission.
