Scalar-torsion theories of gravity III: analogue of scalar-tensor gravity and conformal invariants

We discuss a class of teleparallel scalar-torsion theories of gravity, which is parametrized by five free functions of the scalar field. The theories are formulated covariantly using a flat, but non-vanishing spin connection. We show how the actions of different theories within this class are related via conformal transformations of the tetrad and redefinitions of the scalar field, and derive the corresponding transformation laws for the free function in the action. From these we construct a number of quantities which are invariant under these transformations, and use them to write the action and field equations in different conformal frames. These results generalize a similar formalism for scalar-tensor theories of gravity, where the invariants have been used to express observables independently of the conformal frame.