  We study spin-orbit coupling in metallic carbon nanotubes (CNTs) within the
many-body Tomonaga-Luttinger liquid (TLL) framework. For a well defined
sub-class of metallic CNTs, that contains both achiral zig-zag as well as a
sub-set of chiral tubes, an effective low energy field theory description is
derived. We aim to describe system at finite dopings, but close to the charge
neutrality point (commensurability). A new regime is identified where
spin-orbit coupling leads to an inverted hierarchy of mini-gaps of bosonic
modes. We then add a proximity coupling to a superconducting (SC) substrate and
show that the only order parameter that is supported within the novel,
spin-orbit induced phase is a topologically trivial s-SC.
