Remarks on the Barbero-Immirzi parameter as a field

We revisit a propagating torsion gravity theory obtained by introducing a field coupled to the Holst term in the first-order Einstein-Cartan action. The resulting theory has second-order field equations, no adjustable coupling constants, and one more propagating degree of freedom as compared to general relativity. When no fermions are present the theory is known to be equivalent to that of a single massless scalar field canonically coupled to gravity. We extend this result to the case with fermions and obtain an effective interaction between the scalar field and the fermionic currents. We also describe a version of the theory with a potential for the scalar field and discuss whether it can be interpreted as the inflaton.