De Sitter Diagrammar and the Resummation of Time

TL;DR

The paper addresses the infrared pathologies of light scalar fields in De Sitter during inflation by developing a diagrammatic, all-orders leading-log resummation within in-in perturbation theory. It shows that leading IR contributions are captured by retarded-tree subdiagrams dressed with quantum noise, yielding a semiclassical evolution that matches Starobinsky's stochastic inflation and leads to a Fokker-Planck description with a finite late-time distribution. This framework links quantum De Sitter correlators to a stochastic, Markovian evolution and offers insight into eternal inflation measures and potential holographic interpretations. The results provide a controlled, improvable approach to resumming infrared logs and set the stage for exploring higher-order (NLL) corrections and broader cosmological implications.

Abstract

Light scalars in inflationary spacetimes suffer from logarithmic infrared divergences at every order in perturbation theory. This corresponds to the scalar field values in different Hubble patches undergoing a random walk of quantum fluctuations, leading to a simple toy "landscape" on superhorizon scales, in which we can explore questions relevant to eternal inflation. However, for a sufficiently long period of inflation, the infrared divergences appear to spoil computability. Some form of renormalization group approach is thus motivated to resum the log divergences of conformal time. Such a resummation may provide insight into De Sitter holography. We present here a novel diagrammatic analysis of these infrared divergences and their resummation. Basic graph theory observations and momentum power counting for the in-in propagators allow a simple and insightful determination of the leading-log contributions. One thus sees diagrammatically how the superhorizon sector consists of a semiclassical theory with quantum noise evolved by a first-order, interacting classical equation of motion. This rigorously leads to the "Stochastic Inflation" ansatz developed by Starobinsky to cure the scalar infrared pathology nonperturbatively. Our approach is a controlled approximation of the underlying quantum field theory and is systematically improvable.

Figures (7)

• Figure 1: Left: QFT Diagrams for $\left\langle {\phi(\eta, \vec{x}) \phi(\eta, \vec{y})} \right\rangle$ in $\phi^3$ in-in perturbation theory, where time is flowing upward in the graph. As explained in Section \ref{['sec:setup']}, propagators are either retarded, $G_R$ (solid lines), or anticommutator, $G_+ = \left\langle {\{ \phi_1,\, \phi_2 \}} \right\rangle$ (dashed lines). Right: As we will show in Section \ref{['sec:llcorr']}, in all leading contributions to $\left\langle {\phi^n} \right\rangle$ correlation function, the $G_R$ form tree shaped subdiagrams that touch one and only one correlation point. Thus, if we cut each $G_+$ propagator and consider each "$\times$" as an insertion of the zeroth-order solution, $\phi^0$, we see that each diagram decomposes into a product of classical perturbation theory diagrams. In Section \ref{['sec:fp']}, we detail how the inclusion of these $\phi^0$ as a quantum distribution leads to the famous Fokker-Planck description of De Sitter light scalar evolution. The use of $\phi^3$ vertices is purely for visual simplicity. The graph theory statements in this paper hold for a generic, non-derivatively coupled scalar potential. Although a $\phi^3$ interaction is unstable, one can nonetheless consider it as a subsector of an ultimately (meta)stable theory.
• Figure 2: Left: Our graphical notation for the various propagators of interest. Right: In a $\phi^3$ theory, we show second-order contributions to $\left\langle {\phi(\eta, \vec{x}) \phi(\eta, \vec{y})} \right\rangle$, where the solid bar at the top of each diagram indicates the correlation time, $\eta$. The top graph fixes a particular topology for this contribution, all of which will come from various contributions of Wightman functions and time orderings. We can decompose the various contributing two-point functions purely in terms of $G_+$ and $G_R$. The middle graph shows a nonvanishing contribution in this basis that contributes at leading-log level by having the minimum number of allowed $G_R$ factors and places them consistently with rules 1) and 2). Lastly, the bottom graph is subleading-log as it contains more than the minimum number of $G_R$ terms.
• Figure 3: Examples diagrams of $\left\langle {\phi^6} \right\rangle$ evaluated at correlation time $\eta$ in $\lambda \phi^4$ theory at $\mathcal{O}(\lambda^7)$. For visual clarity we have separated out the six correlation points, though we generally take them to be spatially coincident, as well. Left: Illustration of the statement that if the subdiagram containing all $V$ retarded propagators is a connected tree that only touches the correlation point once, then it necessarily touches all $V$ vertices. Solid lines are $G_R$ and dashed lines are $G_+$ propagators. Right: An example of the general situation with multiple disconnected subgraphs. By the same argument, each one is a tree that touches one and only one external correlation point and contains a number of propagators equal to the number of vertices in the subgraph.
• Figure 4: Example of a classical perturbation theory Feynman diagram. In this case, it is the fourth-order perturbative correction, $\phi_4(t, \vec{x})$, to the full field solution, $\phi(t, \vec{x})$, in classical $\phi^3$ theory. Solid lines are retarded propagators, $G_R$, which get convolved with each other, and ultimately the free solutions $\phi_0(t_i, \vec{x}_i)$. To make contact with our quantum field theory diagrams, the free field insertions are depicted as dashed lines leading to a "$\times$". If one were doing classical field theory, one would need to use initial/boundary conditions or a specific inhomogeneous source function to give a particular $\phi_0$ for a corrected solution.
• Figure 5: A schematic representation of Eq. \ref{['eq:qtosc']}. The graph labelled $\phi_0$ represents those contributions arising from the direct insertion of the free-field solution, $\phi_0$, which can be generalized to the inhomogeneous source term in Eq. \ref{['eq:inhomeom']}. The middle graph shows those evaluated at $\mathcal{O}(\lambda)$ in classical perturbation theory, and the last graph is a representative of those at $\mathcal{O}(\lambda^2)$. There may be multiple contributions at each order (the $n_i$ in Eq. \ref{['eq:qtosc']}) and at higher orders. The interpretation of $\left\langle {} \right\rangle$ is that it joins up the dangling $\times$ factors of each classical solution pairwise in all possible combinations, evaluating them as the anticommutator propagator, $G_+$.