  The quantum corrections to the frequencies of the $\Gamma$ point longitudinal
optical (LO) and transverse optical (TO) phonon modes in carbon nanotubes are
investigated theoretically. The frequency shift and broadening of the TO phonon
mode strongly depend on the curvature effect due to a special electron-phonon
coupling in carbon nanotubes, which is shown by the Fermi energy dependence of
the frequency shift for different nanotube chiralities. It is also shown that
the TO mode near the $\Gamma$ point decouples from electrons due to local gauge
symmetry and that a phonon mixing between LO and TO modes is absent due to
time-reversal symmetry. Some comparison between theory and experiment is
presented.
