  We investigate the effect of electron-phonon coupling on low temperature
phases in metallic single-wall carbon nanotubes. We obtain low-temperature
phase diagrams of armchair and zigzag type nanotubes with screened interactions
with a weak-coupling renormalization group approach. In the absence of
electron-phonon coupling, two types of nanotubes have similar phase diagrams. A
$D$-Mott phase or $d$-wave superconductivity appears when the on-site
interaction is dominant, while a charge-density wave or an excitonic insulator
phase emerges when the nearest neighbor interaction becomes comparable to the
on-site interaction. The electron-phonon coupling, treated by a two-cutoff
scaling scheme, leads to different behavior in two types of nanotubes. For
strong electron-phonon interactions, phonon softening is induced and a Peierls
insulator phase appears in armchair nanotubes. We find that this softening of
phonons may occur for any intraband scattering phonon mode. On the other hand,
the effect of electron-phonon coupling is negligible for zigzag nanotubes. The
distinct behavior of armchair and zigzag nanotubes against lattice distortion
is explained by analysis of the renormalization group equations.
