Bootstrapping Inflationary Correlators in Mellin Space. (arXiv:1907.01143v1 [hep-th])
<a href="http://arxiv.org/find/hep-th/1/au:+Sleight_C/0/1/0/all/0/1">Charlotte Sleight</a>, <a href="http://arxiv.org/find/hep-th/1/au:+Taronna_M/0/1/0/all/0/1">Massimo Taronna</a>
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-$ell$. 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. 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-$ell$ 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.
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-$ell$. 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. 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-$ell$ 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.
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