  We describe a method to generate scalar-tensor theories with Weyl symmetry,
starting from arbitrary purely metric higher derivative gravity theories. The
method consists in the definition of a conformally-invariant metric
$\hat{g}_{\mu \nu}$, that is a rank (0,2)-tensor constructed out of the metric
tensor and the scalar field. This new object has zero conformal weight and is
given by $\phi^{2/\Delta}g_{\mu \nu}$, where ($-\Delta$) is the conformal
dimension of the scalar. As $g_{\mu \nu}$ has conformal dimension of 2, the
resulting tensor is trivially a conformal invariant. Then, the generated
scalar-tensor theory, which we call the Weyl uplift of the original purely
metric theory, is obtained by replacing the metric by $\hat{g}_{\mu \nu}$ in
the action that defines the original theory. This prescription allowed us to
define the Weyl uplift of theories with terms of higher order in the Riemannian
curvature. Furthermore, the prescription for scalar-tensor theories coming from
terms that have explicit covariant derivatives in the Lagrangian is discussed.
The same mechanism can also be used for the derivation of the equations of
motion of the scalar-tensor theory from the original field equations in the
Einstein frame. Applying this method of Weyl uplift allowed us to reproduce the
known result for the conformal scalar coupling to Lovelock gravity and to
derive that of Einsteinian cubic gravity. Finally, we show that the
renormalization of the theory given by the conformal scalar coupling to
Einstein-Anti-de Sitter gravity originates from the Weyl uplift of the original
renormalized theory, which is relevant in the framework of conformal
renormalization.
