We develop a Mellin space approach to boundary correlation functions in anti-de Sitter (AdS) and de Sitter (dS) spaces. Using the Mellin-Barnes representation of correlators in Fourier space, we show that the analytic continuation between AdS d +1 and dS d +1 is encoded in a collection of simple relative phases. This allows us to determine the late-time tree-level three-point correlators of spinning fields in dS d +1 from known results for Witten diagrams in AdS d +1 by multiplication with a simple trigonometric factor. At four point level, we show that Conformal symmetry fixes exchange four-point functions both in AdS d +1 and dS d +1 in terms of the dual Conformal Partial Wave (which in Fourier space is a product of boundary three-point correlators) up to a factor which is determined by the boundary conditions. In this work we focus on late-time four-point correlators with external scalars and an exchanged field of integer spin- ℓ . The Mellin-Barnes representation makes manifest the analytic structure of boundary correlation functions, providing an analytic expression for the exchange four-point function which is valid for general d and generic scaling dimensions, in particular massive, light and (partially-)massless fields. It moreover naturally identifies boundary correlation functions for generic fields with multi-variable Meijer-G functions. When d = 3 we reproduce existing explicit results available in the literature for external conformally coupled and massless scalars. From these results, assuming the weak breaking of the de Sitter isometries, we extract the corresponding correction to the inflationary three-point function of general external scalars induced by a general spin- ℓ field at leading order in slow roll. These results provide a step towards a more systematic understanding of de Sitter observables at tree level and beyond using Mellin space methods.

中文翻译：

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在梅林空间中引导通货膨胀相关器

我们开发了一种梅林空间方法来处理反德西特 (AdS) 和德西特 (dS) 空间中的边界相关函数。使用傅立叶空间中相关器的梅林-巴恩斯表示，我们表明 AdS d +1 和 dS d +1 之间的解析连续被编码在一组简单的相对相位中。这使我们能够从 AdS d +1 中 Witten 图的已知结果通过与简单的三角因子相乘来确定 dS d +1 中旋转场的后期树级三点相关器。在四点水平上，我们证明了共形对称固定在 AdS d +1 和 dS d +1 中交换四点函数，根据双共形部分波（在傅立叶空间中是边界三点相关器的乘积）由边界条件决定的因子。在这项工作中，我们专注于具有外部标量和整数自旋 ℓ 的交换场的后期四点相关器。Mellin-Barnes 表示法展示了边界相关函数的解析结构，为交换四点函数提供了一个解析表达式，该表达式适用于一般 d 和一般标度维度，特别是质量、轻和（部分）无质量场。此外，它自然地识别具有多变量 Meijer-G 函数的通用字段的边界相关函数。当 d = 3 时，我们重现了文献中可用的外部共形耦合和无质量标量的现有显式结果。根据这些结果，假设 de Sitter 等距线的弱破坏，我们提取了对一般外部标量的膨胀三点函数的相应修正，该函数由慢滚中处于领先顺序的一般自旋 ℓ 场引起。这些结果为使用梅林空间方法更系统地理解树级别及其他级别的 de Sitter 可观测值迈出了一步。