A Compact Story of Positivity in de Sitter

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.

💡 Research Summary

This paper addresses the longstanding puzzle of how positivity constraints operate in de Sitter (dS) space when loop corrections generate large logarithms that can be interpreted as anomalous dimensions of operators. In anti‑de Sitter (AdS) and flat space, anomalous dimensions are real and their sign is tightly constrained by unitarity. In dS, however, principal‑series fields have complex scaling dimensions, so the usual unitarity‑based positivity arguments do not directly apply. The authors focus on a particularly subtle class of theories: compact (periodic) scalar fields whose vertex operators (V_{\alpha}=e^{i\alpha\phi}) serve as composite operators with fixed scaling dimensions. Because the free compact scalar behaves like a dimension‑zero operator, its two‑point function grows logarithmically at large separations, and any non‑derivative composite operator mixes under renormalization group (RG) flow. This mixing is equivalent to stochastic inflation and can be captured by the Soft de Sitter Effective Theory (SdSET).

The paper develops two complementary computational frameworks. The first is the spectral representation combined with the Lorentzian inversion formula. By analytically continuing to Euclidean momentum space, the authors write the dS two‑point function as a Källén‑Lehmann‑type integral over a spectral density (\rho(\nu)). Loop diagrams, especially bubble chains, appear as corrections to (\rho(\nu)). The inversion formula extracts anomalous dimensions from the residues of (\rho(\nu)). When applied to ordinary scalar interactions, this method reproduces known positive anomalous dimensions. However, for vertex operators of compact scalars earlier literature reported negative anomalous dimensions at special values of the compactification radius, apparently violating the positivity condition (\langle O(\mathbf{k})O(-\mathbf{k})\rangle’\ge0). The authors show that the discrepancy originates from neglecting contact terms that are required by the Ward identities of the dS isometry group. Including these contact contributions restores the positivity of the full correlator and forces the anomalous dimension to be non‑negative for all radii.

The second framework is SdSET, an effective field theory that separates long‑wavelength (soft) modes from short‑wavelength (hard) modes in a fixed dS background. In SdSET, the vertex operator is treated as a non‑linear functional of the soft field, and the effective action contains an infinite tower of higher‑dimensional operators generated by the periodicity condition. The authors compute the one‑loop self‑energy and the mixing among these operators, deriving an RG equation of the form (\mu\frac{d\alpha}{d\mu}=-\gamma,\alpha). The anomalous dimension (\gamma) is shown to be strictly positive, guaranteeing that the two‑point function remains positive in the infrared. Moreover, they demonstrate that the RG flow generated by SdSET exactly reproduces the resummation of bubble diagrams in the spectral representation; the beta‑function derived from SdSET matches the derivative of the spectral density with respect to the scaling dimension.

A central technical achievement of the paper is a new proof of “positivity of anomalous dimensions” for compact scalar vertex operators. The proof proceeds in three steps: (i) establishing that the spectral density must be non‑negative because it is a sum over physical intermediate states; (ii) showing that the Lorentzian inversion formula, when supplemented by the necessary contact terms, yields a residue that is a positive functional of the interaction couplings; and (iii) demonstrating that the SdSET beta‑functions are built from squares of real coupling constants, guaranteeing (\gamma\ge0). The authors also discuss how the shadow symmetry (\Delta\leftrightarrow d-\Delta) is broken by the anomalous dimensions, yet the combined system of an operator and its shadow still respects the overall positivity constraints once contact terms are included.

The paper’s conclusions have several important implications. First, they resolve the apparent conflict between earlier spectral‑representation calculations and positivity bounds, confirming that de Sitter quantum field theory remains consistent with fundamental positivity requirements even when operators acquire complex scaling dimensions. Second, the equivalence between the spectral method and SdSET provides a powerful cross‑check: any future computation of loop corrections in dS can be performed in either framework with confidence that they will agree. Third, the methods developed here are directly applicable to cosmological collider physics, where non‑Gaussian signatures of massive fields are probed through higher‑point functions. The positivity constraints derived for two‑point functions extend to three‑ and four‑point correlators, offering new bootstrap‑type consistency conditions for inflationary observables. Finally, the work deepens the conceptual link between stochastic inflation, RG flow, and effective field theory in curved spacetime, suggesting that similar positivity‑based analyses could be fruitful in other time‑dependent backgrounds.

In summary, the authors provide a comprehensive, technically rigorous treatment of anomalous dimensions for compact scalar vertex operators in de Sitter space, reconcile previously conflicting results, and establish a unified picture in which positivity, spectral representation, and Soft de Sitter Effective Theory are mutually consistent. This advances our theoretical control over loop effects in cosmology and opens the door to more robust constraints on primordial non‑Gaussianity.