Operator Origin of Anomalous Dimensions in de Sitter Space

TL;DR

This paper investigates the origin of complex scaling dimensions in de Sitter space, demonstrating how to compute one-loop anomalous dimensions of operators and exploring implications for the Soft de Sitter Effective Theory.

Contribution

It introduces a method to calculate one-loop corrections to operator dimensions in de Sitter space using Mellin space and discusses the necessity of non-dynamical operators in the effective theory.

Findings

One-loop anomalous dimensions can be computed using Mellin space techniques.

Matching UV and IR descriptions requires non-dynamical operators in the effective theory.

Complex scaling dimensions relate to operator corrections in de Sitter quantum fields.

Abstract

The late time limit of the power spectrum for heavy (principal series) fields in de Sitter space yields a series of polynomial terms with complex scaling dimensions. Such scaling behavior is expected to result from an associated operator with a complex dimension. In a free theory, these complex dimensions are known to match the constraints imposed by unitarity on the space of states. Yet, perturbative corrections to the scaling behavior of operators are naively inconsistent with unitary evolution of the quantum fields in dS. This paper demonstrates how to compute one-loop corrections to the scaling dimensions that appear in the two point function from the field theory description in terms of local operators. We first show how to evaluate these anomalous dimensions using Mellin space, which has the feature that it naturally accommodates a scaleless regulator. We then explore the consequences for the Soft de Sitter Effective Theory (SdSET) description that emerges in the long wavelength limit. Carefully matching between the UV and SdSET descriptions requires the introduction of novel non-dynamical "operators" in the effective theory. This is not only necessary to reproduce results extracted from the K\"all\'en-Lehmann representation (that use the space of unitary states directly), but it is also required by general arguments that invoke positivity.