Non-minimal scalar theories in Palatini formalism

We show that for a non-minimally coupled theory containing derivatives of the scalar field (Sushkov's action) in Palatini formalism the connection turns out to be metrically associated with the "second" metric, which is non-conformally expressed through the physical metric. We obtain that in the first order approximation in a weak scalar field regime the effective Energy-momentum tensors coincide in the 1st and 2nd order formalisms, but in general they can differ and higher derivatives appear in the equations of motion. We also prove that in this theory, as in the non-minimally coupled theory $R\phi^{2}$, as well as for theories $f(R)$, torsion can be reduced to a gauge transformation with respect to which the action is invariant. We also investigated the Einstein-Chern-Simons action and determined that in the absence of nonmetricity it reduces to the Einstein-Hilbert action with the minimally coupled scalar field.