Low-energy general relativity with torsion: a systematic derivative expansion

TL;DR

This paper systematically constructs the low-energy effective Lagrangian for Einstein--Cartan gravity with torsion, revealing new invariants, gauge symmetries, and the implications of torsion-induced fermion interactions for cosmology.

Contribution

It introduces a comprehensive method to derive the effective Lagrangian including torsion, identifies new invariants, and explores the gauge symmetries and fermion interactions in this framework.

Findings

Torsion in gravity can be associated with additional gauge symmetries.

Integrating out torsion yields a four-fermion interaction with specific structure.

The effects of torsion-induced interactions are likely unobservable in current experiments.

Abstract

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.