Singlet superconductivity phase in carbon nanotubes

We discuss phase diagrams obtained by calculating temperature-dependent correlation functions for superconductivity and density waves for a single $\mathrm{}(5,0)\mathrm{}$ 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 $\mathrm{}(5,0)\mathrm{}$ CN has two degenerate bands crossing the Fermi energy with circumferential momenta. We find singlet-superconducting phase with $5ħ$ circumferential angular momentum. Critical exponents for each correlation functions are also calculated.