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 cancellation of the volume divergences in the theory given by the conformal scalar coupling to Einstein–Anti-de Sitter gravity is achieved by the Weyl uplift of the original theory augmented by counterterms, which is relevant in the framework of conformal renormalization.

我们描述了一种从任意纯度量高导数引力理论开始生成具有韦尔对称性的标量张量理论的方法。该方法包含共形不变度量\(\hat{g}_{\mu \nu }\)的定义，即由度量张量和标量场构造的秩 (0,2) 张量。这个新对象的共形权重为零，由\(\phi ^{2/\Delta }g_{\mu \nu }\)给出，其中\((-\Delta )\)是标量的共形尺寸。由于\(g_{\mu \nu }\) 的共形维数为 2，因此得到的张量通常是一个共形不变量。然后，生成的标量-张量理论，我们称之为原始纯度量理论的Weyl提升，是通过在定义原来的理论。这个规定使我们能够用黎曼曲率中的高阶项来定义理论的韦尔提升。此外，还讨论了来自拉格朗日中具有显式协变导数的项的标量-张量理论的规定。同样的机制也可以用于从爱因斯坦框架中的原始场方程推导标量张量理论的运动方程。应用这种外尔隆升方法使我们能够重现洛夫洛克引力的共形标量耦合的已知结果，并推导出爱因斯坦立方引力的结果。最后，我们证明了爱因斯坦-反德西特引力的共形标量耦合所给出的理论中体积散度的抵消是通过反项增强的原始理论的韦尔提升来实现的，这在共形的框架中是相关的 重整化。