Torsion, Scalar Field and f(\mathcal{R}) Gravity

The role of torsion and a scalar field $\phi$ in gravitation in the background of a particular class of the Riemann-Cartan geometry is considered here. Some times ago, a Lagrangian density with Lagrange multipliers has been proposed by the author which has been obtained by picking some particular terms from the SO(4,1) Pontryagin density, where the scalar field $\phi$ causes the de Sitter connection to have the proper dimension of a gauge field. Here it has been shown that the divergence of the axial torsion gives the Newton's constant and the scalar field becomes a function of the Ricci scalar $\mathcal{R}$. The starting Lagrangian then reduces to a Lagrangian representing the metric $f(\mathcal{R})$ gravity theory.