We study a series of the Wess-Zumino actions obtained by repeatedly integrating conformal anomalies with respect to the conformal-factor field that appear at higher loops. We show that they arise as physical quantities required to make nonlocal loop correction terms diffeomorphism invariant. Specifically, in a conformally flat spacetime $ds^2=e^{2\phi}(-d\eta^2 + d{\bf x}^2)$, we find that effective actions are described in terms of momentum squared expressed as a physical $Q^2 = q^2/e^{2\phi}$ for $q^2$ measured by the flat metric, which recalls the relationship between physical momentum and comoving momentum in cosmology. It is confirmed by calculating the effective action of QED in such a curved spacetime at the 3-loop level using dimensional regularization. The same applies to the case of QCD, in which we show that the effective action can be summarized in the form of the reciprocal of a running coupling constant squared described by the physical momentum. We also see that the same holds for renormalizable quantum conformal gravity and that conformal anomalies are indispensable for formulating the theory.

中文翻译：

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微分同胚不变性需要保形异常

我们研究了一系列 Wess-Zumino 动作，这些动作是通过针对出现在更高循环的保形因子场反复积分保形异常而获得的。我们表明它们是作为使非局部循环校正项微分同胚不变所需的物理量而出现的。具体来说，在共形平坦时空 $ds^2=e^{2\phi}(-d\eta^2 + d{\bf x}^2)$ 中，我们发现有效的动作是用动量平方来描述的表示为物理 $Q^2 = q^2/e^{2\phi}$ 为 $q^2$ 由平面度量测量，这让人想起宇宙学中物理动量和共动动量之间的关系。通过使用维度正则化在 3-loop 级别计算 QED 在这种弯曲时空中的有效作用来证实这一点。这同样适用于 QCD 的情况，其中我们表明，有效作用可以用物理动量描述的运行耦合常数平方的倒数形式来概括。我们还看到，这同样适用于可重整化的量子共形引力，并且共形异常对于制定该理论是必不可少的。