Perturbative unitarity bounds from momentum-space entanglement

TL;DR

This paper introduces a new method using momentum-space entanglement growth as a diagnostic for the breakdown of perturbation theory in curved spacetime, applicable in cosmological contexts where traditional unitarity bounds fail.

Contribution

It proposes a novel entanglement-based diagnostic for perturbation theory validity applicable in curved spacetimes, extending beyond flat spacetime unitarity bounds.

Findings

Purity bounds can be weaker than partial wave bounds in flat spacetime.

Purity bounds sometimes exist where partial wave bounds do not.

Derived scale-invariant purity bounds for scalar fields in de Sitter space.

Abstract

Physical theories have a limited regime of validity and hence must be accompanied by a breakdown diagnostic to establish when they cease to be valid as parameters are varied. For perturbative theories, estimates of the first neglected order offer valuable guidance, but one is often interested in sharp bounds beyond which perturbation theory necessarily fails. In particle physics, it is common to employ the bounds on partial waves imposed by unitarity as such a diagnostic. Unfortunately, these bounds don't extend to curved spacetime, where scattering experiments are challenging to define. Here, we propose to use the growth of entanglement in momentum space as a breakdown diagnostic for perturbation theory in general field theories. This diagnostic can be readily used in cosmological spacetimes and does not require any flat spacetime limit. More in detail, we consider the so-called purity of the reduced density operator constructed by tracing out all but one of the Fourier modes in an effective theory and we present a diagrammatic technique to compute it perturbatively. Constraints on the regime of validity of perturbation theory are then derived when the perturbative purity violates its unitarity bounds. In flat spacetime, we compare these purity bounds to those from partial waves. We find general qualitative agreement but with remarkable differences: purity bounds can be sometimes weaker, but other times they exist when no partial wave bounds exist. We then derive purity bounds for scalar field theories in de Sitter spacetime for a variety of interactions that appear in inflationary models. Importantly, our bounds make no reference to time evolution and in de Sitter they depend exclusively on scale-invariant ratios of the physical kinematics.