Understanding the interplay between many-body phenomena and non-equilibrium in systems with entangled spin and orbital degrees of freedom is a central objective in nano-electronics. We demonstrate that the combination of Coulomb interaction, spin-orbit coupling and valley mixing results in a particular selection of the inelastic virtual processes contributing to the Kondo resonance in carbon nanotubes at low temperatures. This effect is dictated by conjugation properties of the underlying carbon nanotube spectrum at zero and finite magnetic field. Our measurements on a clean carbon nanotube are complemented by calculations based on a new approach to the non-equilibrium Kondo problem which well reproduces the rich experimental observations in Kondo transport.
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