Non-minimally coupled Dirac equation with torsion: Poincar\'e gauge theory of gravity with even and odd parity terms

Understand

We take a Dirac field non-minimally coupled to the gravitational field within the framework of the Poincar\'e gauge theory of gravity with torsion and curvature.

• We study the subcase of "weak" gravity, that is, the gravitational Lagrangian depends only linearly on the curvature and quadratically on the torsion.
• We include all pieces in curvature and torsion that are of odd parity.
• The second field equation of gravity is derived by varying the Lorentz connection.