A Compact Story of Positivity in de Sitter

TL;DR

This work investigates how positivity constraints shape loop-corrected de Sitter correlators, focusing on anomalous dimensions arising for principal-series fields coupled to vertex operators of massless compact scalars. It develops and cross-validates two frameworks—the spectral density/Källén-Lehmann approach with Lorentzian/EAdS inversion and the Soft de Sitter Effective Theory (SdSET)—showing that, when UV data and contact terms are properly treated, the anomalous dimensions are positive and the two methods agree. A central result is that the anomalous dimension γ_φ can be computed either from the spectral density of the vertex operator or from SdSET via RG, with explicit demonstrations for compact scalars and vertex operators V_p. The analysis provides new positivity proofs and clarifies how renormalization-group flow in SdSET corresponds to resummation of bubble diagrams in the spectral representation, contributing to a coherent, cross-checked picture of long-distance dS physics relevant for cosmological correlators and potential bootstrap approaches.

Abstract

Recent developments have yielded significant progress towards systematically understanding loop corrections to de Sitter (dS) correlators. In close analogy with physics in Anti-de Sitter (AdS), large logarithms can result from loops that can be interpreted as corrections to the dimensions of operators. In contrast with AdS, these dimensions are not manifestly real. This implies that the theoretical constraints on the associated correlators are less transparent, particularly in the presence of light scalars. In this paper, we revisit these issues by performing and comparing calculations using the spectral representation approach and the Soft de Sitter Effective Theory (SdSET). We review the general arguments that yield positivity constraints on dS correlators from both perspectives. Our particular focus will be on vertex operators for compact scalar fields, since this case introduces novel complications. We will explain how to resolve apparent disagreements between different techniques for calculating the anomalous dimensions for principal series fields coupled to these vertex operators. Along the way, we will offer new proofs of positivity of the anomalous dimensions, and explain why renormalization group flow associated with these anomalous dimensions in SdSET is the same as resumming bubble diagrams in the spectral representation.

Figures (4)

• Figure 1: We illustrate the contours corresponding to the two inversion integrals for the spectral density. The orange contour represents the Lorentzian integral: this picks up the physical discontinuity of the propagator on time-like separated points, which admits $\xi>1$. The blue contour represents Euclidean superhorizon separations, and are thus accessible in Euclidean AdS. On the Euclidean side, the branch cut of the integration kernel is exploited instead.
• Figure 2: We illustrate two possibilities for the integration contour on the $\xi$-plane over which we could evaluate the Lorentzian inversion integral. The blue line indicates the choice where we evaluate the discontinuity first and then send $d \to 3$. The red contour is the one we obtain if we set $d=3$ first. The difference between the two is essentially a residual integral over short-distances, which is denoted by the small circular contour $\mathcal{C}_\epsilon$.
• Figure 3: We show the result of the two choices for evaluating the Lorentzian inversion formula for a vertex operator, ${\rm L}$ and ${\rm L}+\mathcal{C}_\epsilon$. We pick $\epsilon=0.1$ in order to evaluate (\ref{['eq:lor_circ_corr']}), which we find is sufficient for convergence. Note that we have scaled ${\rm L}+\mathcal{C}_\epsilon$ by a factor of $10^{-3}$ in order to enable a visual comparison. We observe that the result of (\ref{['eq:lor_naive']}), denoted in blue, turns around and takes on negative values for some choices of $\nu$, apparently violating unitarity, whereas the inclusion of the short-distance information restores positivity. We also show the analytical result obtained using the EAdS inversion formula in solid orange which agrees with ${\rm L}+\mathcal{C}_\epsilon$.
• Figure 4: We plot the leading order corrected spectral density for a principal series scalar with $\nu_\phi=1$ coupled to a CFT operator with scaling dimension $\delta$, for various choices of coupling $g$ and scaling dimension. The dashed vertical lines indicate $\nu=\pm \nu_\phi$.