We formulate scalar field theories coupled nonconformally to gravity in a manifestly frame-independent fashion. Physical quantities such as the $S$ matrix should be invariant under field redefinitions, and hence can be represented by the geometry of the target space. This elegant geometric formulation, however, is obscured when considering the coupling to gravity because of the redundancy associated with the Weyl transformation. The well-known example is the Higgs inflation, where the target space of the Higgs fields is flat in the Jordan frame but is curved in the Einstein frame. Furthermore, one can even show that any geometry of $\mathrm{O}(N)$ nonlinear $\sigma$ models can be flattened by an appropriate Weyl transformation. In this Letter, we extend the notion of the target space by including the conformal mode of the metric, and show that the extended geometry provides a compact formulation that is manifestly Weyl-transformation or field-redefinition invariant. We identify the cutoff scale with the inverse of square root of the extended target-space curvature and confirm that it coincides with that obtained from two-to-two scattering amplitudes based on our formalism.

中文翻译：

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与重力耦合的标量场的几何和幺正性

我们以明显与框架无关的方式制定了与引力非共形耦合的标量场论。物理量，例如$S$矩阵在场重定义下应该是不变的，因此可以用目标空间的几何形状来表示。然而，当考虑与重力的耦合时，由于与韦尔变换相关的冗余，这种优雅的几何公式​​就变得模糊了。众所周知的例子是希格斯暴胀，其中希格斯场的目标空间在乔丹坐标系中是平坦的，但在爱因斯坦坐标系中是弯曲的。此外，我们甚至可以证明任何几何形状$\mathrm{氧}（氮）$非线性$\sigma$模型可以通过适当的 Weyl 变换进行扁平化。在这封信中，我们通过包含度量的共形模式来扩展目标空间的概念，并表明扩展的几何提供了一个明显是 Weyl 变换或场重定义不变式的紧凑公式。我们用扩展目标空间曲率的平方根的倒数来确定截止尺度，并确认它与基于我们的形式主义从两到两个散射幅度获得的结果一致。