  We attempt to build systematically the low-energy effective Lagrangian for
the Einstein--Cartan formulation of gravity theory that generally includes the
torsion field. We list all invariant action terms in certain given order; some
of the invariants are new. We show that in the leading order the fermion action
with torsion possesses additional U(1)_L x U(1)_R gauge symmetry, with 4+4
components of the torsion (out of the general 24) playing the role of Abelian
gauge bosons. The bosonic action quadratic in torsion gives masses to those
gauge bosons. Integrating out torsion one obtains a point-like 4-fermion action
of a general form containing vector-vector, axial-vector and axial-axial
interactions. We present a quantum field-theoretic method to average the
4-fermion interaction over the fermion medium, and perform the explicit
averaging for free fermions with given chemical potential and temperature. The
result is different from that following from the "spin fluid" approach used
previously. On the whole, we arrive to rather pessimistic conclusions on the
possibility to observe effects of the torsion-induced 4-fermion interaction,
although under certain circumstances it may have cosmological consequences.

High Energy Physics - Theory;General Relativity and Quantum Cosmology