  We discuss phase diagrams obtained by calculating temperature dependent
correlation functions for superconductivity and density waves for a single
(5,0) carbon nanotube (CN). We use one-loop renormalization group method within
logarithmic accuracy. In this system we must specify scattering channels in
terms of momentum along the circumferential direction as well as the axis
direction, since (5,0) CN has two degenerate bands crossing the Fermi energy
with circumferential momenta. We find singlet-superconducting phase with
$5\hbar$ circumferential angular momentum. Critical exponents for each
correlation functions are also calculated.
