The IR stability of de Sitter QFT: results at all orders

TL;DR

The paper proves that the Hartle-Hawking vacuum for interacting massive scalars in de Sitter space is perturbatively IR-stable and well-defined to all orders. It employs a Euclidean $S^D$ formulation, Pauli-Villars regularization, and a systematic Mellin-Barnes representation to compute connected correlators and establish their decay at large separations, with the decay governed by the lightest field in trimmed diagrams. This leads to an IR attractor behavior for local operators in the Hartle-Hawking state, valid for all $M^2>0$ (encompassing the complementary and principal series). The results hold regardless of the specific counter-terms, as regulator-dependent contributions decouple in the $M_{ij}^2\to\infty$ limit, reinforcing the robustness of the Hartle-Hawking vacuum as a physical, IR-stable state in de Sitter QFT.

Abstract

We show that the Hartle-Hawking vacuum for theories of interacting massive scalars in de Sitter space is both perturbatively well-defined and stable in the IR. Correlation functions in this state may be computed on the Euclidean section and Wick-rotated to Lorentz-signature. The results are manifestly de Sitter-invariant and contain only the familiar UV singularities. More importantly, the connected parts of all Lorentz-signature correlators decay at large separations of their arguments. Our results apply to all cases in which the free Euclidean vacuum is well defined, including scalars with masses belonging to both the complementary and principal series of $SO(D,1)$. This suggests that interacting QFTs in de Sitter -- including higher spin fields -- are perturbatively IR-stable at least when i) the Euclidean vacuum of the zero-coupling theory exists and ii) corresponding Lorentz-signature zero-coupling correlators decay at large separations. This work has significant overlap with a paper by Stefan Hollands, which is being released simultaneously.

Figures (4)

• Figure 1: The conformal diagram of global de Sitter. The dashed ends are identified. Shown are the points $X$ and the corresponding antipodal point $-X$. Values of the embedding distance $Z := X Y/\ell^2$ in different regions of de Sitter are labeled; in addition, the dashed red lines denote the lightcone with $Z=1$ and the dotted green lines denote the lightcone with $Z=-1$Allen:1985ux.
• Figure 2: On-shell values of $\sigma$ and $-(\sigma+2\alpha)$ in the complex plane for massive scalar fields. The red dashed line denotes the path of $\sigma$ for increasing $M^2$ starting from at $\sigma=0$ for $M^2 = 0$. The green doted line shows the path of $-(\sigma+2\alpha)$ for increasing $M^2$ starting from $-(\sigma+2\alpha) = -2\alpha$ for $M^2 = 0$. Relatively light fields with $0 < M^2\ell^2 < \alpha^2$ correspond to values of $\sigma$ and $-(\sigma+2\alpha)$ on the negative real axis and belong to the complementary series. Heavier fields with $M^2\ell^2 \ge \alpha^2$ correspond to complex values of $\sigma$ and $-(\sigma+2\alpha)$ on the line defined by ${\rm Re}\, \sigma = {\rm Re}\,(-\sigma-2\alpha) = -\alpha$.
• Figure 3: The single-vertex tree Feynman diagram.
• Figure 4: The process of adding a new vertex to an existing diagram.