New ghost-free gravity Lagrangians with propagating torsion

A new class of $R^2$ type of actions without ghosts and tachyons is found in which both the vierbein field ${e_{\mu }}^a$ and the spin connection ${{\omega }_{\mu }}^{\mathrm{ab}}$ are independent and propagating fields. A complete set of spin-projection operators for ${e_{\mu }}^a$ and ${{\omega }_{\mu }}^{\mathrm{ab}}$ is constructed. The particle content of these actions and other $R^2$ theories in the literature is given. The relation of this work to the interesting work of Neville is discussed.