Particle with torsion on 3d null-curves

The search of Lagrangians, describing spinning particles, both massive and massless, has a long story. The conventional approach in this direction consists in the extension of the initial spacetime by auxiliary odd/even coordinates which equip a system with spinning degrees of freedom. There is another, less developed approach, where the spinning particle systems are described by the Lagrangians, which are formulated in the initial space-time, but depend on higher derivatives. The aesthetically attractive point of the latter approach is that spinning degrees of freedom are encoded in the geometry of trajectories. The Poincaré and reparametrization invariance require actions to be of the form