I present evidence that the strong electron-electron
interactions in gapped carbon nanotubes lead to finite
hierarchies of excitons within a given nanotube subband. I
study these hierarchies by employing a field theoretic
reduction of the gapped carbon nanotube permitting
electron-electron interactions to be treated exactly. I
analyze this reduction by employing a Wilsonian-like
numerical renormalization group. I am so able to determine
the gap ratios of the one-photon excitons as a function of
the effective strength of interactions. I also determine
within the same subband the gaps of the two-photon
excitons, the single particle gaps, as well as a subset of the
dark excitons. The strong electron-electron interactions in
addition lead to strongly renormalized dispersion relations
where the consequences of spin-charge separation can be
readily observed.
