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Correlators, Feynman diagrams, and quantum no-hair in deSitter spacetime

Stefan Hollands

TL;DR

This work develops a comprehensive parametric framework for renormalized position-space Feynman integrals of a massive self-interacting scalar on deSitter space, using a Mellin-Barnes representation tied to forest-structured graph data. By deriving explicit graph polynomials and a generalized $H$-function MB representation, it provides analytic continuation from Euclidean sphere to Lorentzian deSitter configurations and proves that connected vacuum correlators decay exponentially with both timelike and spacelike separations to all orders in perturbation theory, yielding a quantum no-hair analog. The approach also clarifies renormalization via Zimmermann forests in curved spacetime and relates the deSitter correlators to invariant structures $Z_{rs}$, enabling robust control of their analytic properties. The results extend prior one-loop insights to arbitrary loop orders and arbitrary graphs, and they establish a practical pathway to explore similar decay phenomena for other fields and backgrounds.

Abstract

We provide a parametric representation for position-space Feynman integrals of a massive, self-interacting scalar field in deSitter spacetime, for an arbitrary graph. The expression is given as a multiple contour integral over a kernel whose structure is determined by the set of all trees (or forests) within the graph, and it belongs to a class of generalized hypergeometric functions. We argue from this representation that connected deSitter $n$-point vacuum correlation functions have exponential decay for large proper time-separation, and also decay for large spatial separation, to arbitrary orders in perturbation theory. Our results may be viewed as an analog of the so-called cosmic-no-hair theorem in the context of a quantized test scalar field. This work has significant overlap with a paper by Marolf and Morrison, which is being released simultaneously.

Correlators, Feynman diagrams, and quantum no-hair in deSitter spacetime

TL;DR

This work develops a comprehensive parametric framework for renormalized position-space Feynman integrals of a massive self-interacting scalar on deSitter space, using a Mellin-Barnes representation tied to forest-structured graph data. By deriving explicit graph polynomials and a generalized -function MB representation, it provides analytic continuation from Euclidean sphere to Lorentzian deSitter configurations and proves that connected vacuum correlators decay exponentially with both timelike and spacelike separations to all orders in perturbation theory, yielding a quantum no-hair analog. The approach also clarifies renormalization via Zimmermann forests in curved spacetime and relates the deSitter correlators to invariant structures , enabling robust control of their analytic properties. The results extend prior one-loop insights to arbitrary loop orders and arbitrary graphs, and they establish a practical pathway to explore similar decay phenomena for other fields and backgrounds.

Abstract

We provide a parametric representation for position-space Feynman integrals of a massive, self-interacting scalar field in deSitter spacetime, for an arbitrary graph. The expression is given as a multiple contour integral over a kernel whose structure is determined by the set of all trees (or forests) within the graph, and it belongs to a class of generalized hypergeometric functions. We argue from this representation that connected deSitter -point vacuum correlation functions have exponential decay for large proper time-separation, and also decay for large spatial separation, to arbitrary orders in perturbation theory. Our results may be viewed as an analog of the so-called cosmic-no-hair theorem in the context of a quantized test scalar field. This work has significant overlap with a paper by Marolf and Morrison, which is being released simultaneously.

Paper Structure

This paper contains 20 sections, 4 theorems, 155 equations, 2 tables.

Table of Contents

  1. Introduction
  2. DeSitter space basics
  3. QFT in general curved spacetimes and deSitter space
  4. General spacetimes
  5. DeSitter spacetime
  6. The free field on deSitter
  7. Interacting fields on dS-spacetime
  8. Basic setup
  9. Renormalization
  10. Parametric representation
  11. Analytic continuation and Mellin-Barnes represenation
  12. The graph polynomials $\mathcal{U,F}$
  13. Mellin-Barnes representation
  14. Example
  15. Analytic continuation
  16. ...and 5 more sections

Key Result

Theorem 1

The distribution $t_G(X_1, \dots, X_V)$, initially only defined as a distribution non-coinciding configurations of points in $S^D$, i.e. $X_i \neq X_j$ for all $i \neq j$ has an extension to a distribution $t^{\rm R}_G$ defined for all configurations $(X_1, \dots, X_V) \in (S^D)^V$. For any arbitrar

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4