  We demonstrate the key role of phonon occupation in limiting the high-field
ballistic transport in metallic carbon nanotubes. In particular, we provide a
simple analytic formula for the electron transport scattering length, that we
validate by accurate first principles calculations on (6,6) and (11,11)
nanotubes. The comparison of our results with the scattering lengths fitted
from experimental I-V curves indicates the presence of a non-equilibrium
optical phonon heating induced by electron transport. We predict an effective
temperature for optical phonons of thousands Kelvin.
