  We develop a scaling relationship between the exciton binding energy and the
external dielectric function in carbon nanotubes. We show that the
electron-electron and electron-hole interaction energies are strongly affected
by screening yet largely counteract each other, resulting in much smaller
changes in the optical transition energy. The model indicates that the relevant
particle interaction energies are reduced by as much as 50 percent upon
screening by water and that the unscreened electron-electron interaction energy
is larger than the unscreened electron-hole interaction energy, in agreement
with explanations of the "ratio problem." We apply the model to measurements of
the changes in the optical transistion energies in single, suspended carbon
nanotubes as the external dielectric environment is altered.
