  We determine the level-shifts induced by cotunneling in a Coulomb blockaded
carbon nanotube quantum dot using leading order quasi-degenerate perturbation
theory within a single nanotube quartet. It is demonstrated that otherwise
degenerate and equally tunnel-coupled $K$ and $K'$ states are mixed by
cotunneling and therefore split up in energy except at the
particle/hole-symmetric midpoints of the Coulomb diamonds. In the presence of
an external magnetic field, we show that cotunneling induces a gate-dependent
$g$-factor renormalization, and we outline different scenarios which might be
observed experimentally, depending on the values of both intrinsic $KK'$
splitting and spin-orbit coupling.
