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.
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