We demonstrate that curvature-induced spin-orbit (SO) coupling induces a $0-\pi$ transition in the Josephson current through a carbon nanotube quantum dot coupled to superconducting leads. In the non-interacting regime, the transition can be tuned by applying parallel magnetic field near the critical field where orbital states become degenerate. Moreover, the interplay between charging and SO effects in the Coulomb Blockade and cotunneling regimes leads to a rich phase diagram with well-defined (analytical) boundaries in parameter space. Finally, the 0 phase always prevails in the Kondo regime. Our calculations are relevant in view of recent experimental advances in transport through ultra-clean carbon nanotubes.Comment: 4 (main text) + 10 (appendices) pages, 3 figure
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